LMFDB - Embedded newform 75.4.e.d.68.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(32,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.e (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 36x^{14} + 562x^{12} - 3672x^{10} + 16413x^{8} - 6588x^{6} + 43024x^{4} + 499896x^{2} + 532900 $ x^16 - 36x^14 + 562x^12 - 3672x^10 + 16413x^8 - 6588x^6 + 43024x^4 + 499896*x^2 + 532900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{8}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			68.4
Root			$-0.0852547 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.68
Dual form			75.4.e.d.32.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39558 - 1.39558i) q^{2} +(-1.62635 + 4.93508i) q^{3} -4.10469i q^{4} +(9.15703 - 4.61761i) q^{6} +(3.80245 - 3.80245i) q^{7} +(-16.8931 + 16.8931i) q^{8} +(-21.7100 - 16.0523i) q^{9} +O(q^{10})$	\(q+(-1.39558 - 1.39558i) q^{2} +(-1.62635 + 4.93508i) q^{3} -4.10469i q^{4} +(9.15703 - 4.61761i) q^{6} +(3.80245 - 3.80245i) q^{7} +(-16.8931 + 16.8931i) q^{8} +(-21.7100 - 16.0523i) q^{9} -61.8903i q^{11} +(20.2569 + 6.67567i) q^{12} +(-48.1497 - 48.1497i) q^{13} -10.6133 q^{14} +14.3141 q^{16} +(-47.5960 - 47.5960i) q^{17} +(7.89568 + 52.7005i) q^{18} +93.4187i q^{19} +(12.5813 + 24.9495i) q^{21} +(-86.3732 + 86.3732i) q^{22} +(33.7862 - 33.7862i) q^{23} +(-55.8947 - 110.843i) q^{24} +134.394i q^{26} +(114.528 - 81.0335i) q^{27} +(-15.6079 - 15.6079i) q^{28} +179.192 q^{29} -123.523 q^{31} +(115.168 + 115.168i) q^{32} +(305.434 + 100.655i) q^{33} +132.848i q^{34} +(-65.8898 + 89.1126i) q^{36} +(10.5672 - 10.5672i) q^{37} +(130.374 - 130.374i) q^{38} +(315.931 - 159.314i) q^{39} +61.8903i q^{41} +(17.2609 - 52.3773i) q^{42} +(-133.882 - 133.882i) q^{43} -254.040 q^{44} -94.3031 q^{46} +(-56.9922 - 56.9922i) q^{47} +(-23.2797 + 70.6410i) q^{48} +314.083i q^{49} +(312.298 - 157.482i) q^{51} +(-197.639 + 197.639i) q^{52} +(-234.750 + 234.750i) q^{53} +(-272.922 - 46.7438i) q^{54} +128.470i q^{56} +(-461.029 - 151.932i) q^{57} +(-250.078 - 250.078i) q^{58} -260.519 q^{59} +240.664 q^{61} +(172.387 + 172.387i) q^{62} +(-143.589 + 21.5128i) q^{63} -435.967i q^{64} +(-285.785 - 566.732i) q^{66} +(320.313 - 320.313i) q^{67} +(-195.367 + 195.367i) q^{68} +(111.789 + 221.686i) q^{69} -1084.53i q^{71} +(637.923 - 95.5747i) q^{72} +(208.052 + 208.052i) q^{73} -29.4950 q^{74} +383.455 q^{76} +(-235.335 - 235.335i) q^{77} +(-663.244 - 218.572i) q^{78} -676.816i q^{79} +(213.645 + 696.991i) q^{81} +(86.3732 - 86.3732i) q^{82} +(-71.3863 + 71.3863i) q^{83} +(102.410 - 51.6421i) q^{84} +373.687i q^{86} +(-291.429 + 884.326i) q^{87} +(1045.52 + 1045.52i) q^{88} +228.124 q^{89} -366.173 q^{91} +(-138.682 - 138.682i) q^{92} +(200.893 - 609.598i) q^{93} +159.075i q^{94} +(-755.670 + 381.061i) q^{96} +(89.5346 - 89.5346i) q^{97} +(438.329 - 438.329i) q^{98} +(-993.485 + 1343.64i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 84 q^{6}+O(q^{10}) $ 16 * q - 84 * q^6	$ 16 q - 84 q^{6} - 232 q^{16} + 816 q^{21} - 1208 q^{31} + 252 q^{36} + 1872 q^{46} + 156 q^{51} - 1528 q^{61} - 3420 q^{66} + 1064 q^{76} + 6876 q^{81} - 10008 q^{91} - 8172 q^{96}+O(q^{100}) $ 16 * q - 84 * q^6 - 232 * q^16 + 816 * q^21 - 1208 * q^31 + 252 * q^36 + 1872 * q^46 + 156 * q^51 - 1528 * q^61 - 3420 * q^66 + 1064 * q^76 + 6876 * q^81 - 10008 * q^91 - 8172 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.39558	−	1.39558i	−0.493414	−	0.493414i	0.415966	−	0.909380i	$-0.363443\pi$
$2$	−1.39558	−	1.39558i	−0.493414	−	0.493414i	−0.909380	+	0.415966i	$0.863443\pi$
$3$	−1.62635	+	4.93508i	−0.312992	+	0.949756i
$3$	−1.62635	+	4.93508i	−0.312992	+	0.949756i
$4$		−	4.10469i		−	0.513086i
$5$		0			0
$5$		0			0
$6$	9.15703	−	4.61761i	0.623057	−	0.314188i
$7$	3.80245	−	3.80245i	0.205313	−	0.205313i	−0.596959	−	0.802272i	$-0.703624\pi$
$7$	3.80245	−	3.80245i	0.205313	−	0.205313i	0.802272	+	0.596959i	$0.203624\pi$
$8$	−16.8931	+	16.8931i	−0.746577	+	0.746577i
$9$	−21.7100	−	16.0523i	−0.804073	−	0.594531i
$10$		0			0
$11$		−	61.8903i		−	1.69642i	−0.529659	−	0.848211i	$-0.677680\pi$
$11$		−	61.8903i		−	1.69642i	0.529659	−	0.848211i	$-0.322320\pi$
$12$	20.2569	+	6.67567i	0.487306	+	0.160592i
$13$	−48.1497	−	48.1497i	−1.02726	−	1.02726i	−0.999618	−	0.0276375i	$-0.991202\pi$
$13$	−48.1497	−	48.1497i	−1.02726	−	1.02726i	−0.0276375	−	0.999618i	$-0.508798\pi$
$14$	−10.6133			−0.202608
$15$		0			0
$16$	14.3141			0.223657
$17$	−47.5960	−	47.5960i	−0.679042	−	0.679042i	0.280741	−	0.959784i	$-0.409420\pi$
$17$	−47.5960	−	47.5960i	−0.679042	−	0.679042i	−0.959784	+	0.280741i	$0.909420\pi$
$18$	7.89568	+	52.7005i	0.103391	+	0.690090i
$19$			93.4187i			1.12799i	0.825780	+	0.563993i	$0.190735\pi$
$19$			93.4187i			1.12799i	−0.825780	+	0.563993i	$0.809265\pi$
$20$		0			0
$21$	12.5813	+	24.9495i	0.130736	+	0.259258i
$22$	−86.3732	+	86.3732i	−0.837038	+	0.837038i
$23$	33.7862	−	33.7862i	0.306301	−	0.306301i	−0.537172	−	0.843473i	$-0.680507\pi$
$23$	33.7862	−	33.7862i	0.306301	−	0.306301i	0.843473	+	0.537172i	$0.180507\pi$
$24$	−55.8947	−	110.843i	−0.475394	−	0.942739i
$25$		0			0
$26$			134.394i			1.01372i
$27$	114.528	−	81.0335i	0.816327	−	0.577589i
$28$	−15.6079	−	15.6079i	−0.105343	−	0.105343i
$29$	179.192			1.14742			0.573709	−	0.819059i	$-0.305504\pi$
$29$	179.192			1.14742			0.573709	+	0.819059i	$0.305504\pi$
$30$		0			0
$31$	−123.523			−0.715660			−0.357830	−	0.933787i	$-0.616483\pi$
$31$	−123.523			−0.715660			−0.357830	+	0.933787i	$0.616483\pi$
$32$	115.168	+	115.168i	0.636222	+	0.636222i
$33$	305.434	+	100.655i	1.61119	+	0.530966i
$34$			132.848i			0.670098i
$35$		0			0
$36$	−65.8898	+	89.1126i	−0.305046	+	0.412558i
$37$	10.5672	−	10.5672i	0.0469525	−	0.0469525i	−0.683241	−	0.730193i	$-0.739430\pi$
$37$	10.5672	−	10.5672i	0.0469525	−	0.0469525i	0.730193	+	0.683241i	$0.239430\pi$
$38$	130.374	−	130.374i	0.556564	−	0.556564i
$39$	315.931	−	159.314i	1.29716	−	0.654120i
$40$		0			0
$41$			61.8903i			0.235747i	0.993029	+	0.117874i	$0.0376078\pi$
$41$			61.8903i			0.235747i	−0.993029	+	0.117874i	$0.962392\pi$
$42$	17.2609	−	52.3773i	0.0634147	−	0.192429i
$43$	−133.882	−	133.882i	−0.474809	−	0.474809i	0.428658	−	0.903467i	$-0.358987\pi$
$43$	−133.882	−	133.882i	−0.474809	−	0.474809i	−0.903467	+	0.428658i	$0.858987\pi$
$44$	−254.040			−0.870410
$45$		0			0
$46$	−94.3031			−0.302266
$47$	−56.9922	−	56.9922i	−0.176876	−	0.176876i	0.613117	−	0.789992i	$-0.289916\pi$
$47$	−56.9922	−	56.9922i	−0.176876	−	0.176876i	−0.789992	+	0.613117i	$0.789916\pi$
$48$	−23.2797	+	70.6410i	−0.0700028	+	0.212420i
$49$			314.083i			0.915693i
$50$		0			0
$51$	312.298	−	157.482i	0.857459	−	0.432390i
$52$	−197.639	+	197.639i	−0.527070	+	0.527070i
$53$	−234.750	+	234.750i	−0.608405	+	0.608405i	−0.942529	−	0.334124i	$-0.891559\pi$
$53$	−234.750	+	234.750i	−0.608405	+	0.608405i	0.334124	+	0.942529i	$0.391559\pi$
$54$	−272.922	−	46.7438i	−0.687778	−	0.117797i
$55$		0			0
$56$			128.470i			0.306564i
$57$	−461.029	−	151.932i	−1.07131	−	0.353050i
$58$	−250.078	−	250.078i	−0.566152	−	0.566152i
$59$	−260.519			−0.574860			−0.287430	−	0.957802i	$-0.592801\pi$
$59$	−260.519			−0.574860			−0.287430	+	0.957802i	$0.592801\pi$
$60$		0			0
$61$	240.664			0.505145			0.252573	−	0.967578i	$-0.418723\pi$
$61$	240.664			0.505145			0.252573	+	0.967578i	$0.418723\pi$
$62$	172.387	+	172.387i	0.353117	+	0.353117i
$63$	−143.589	+	21.5128i	−0.287151	+	0.0430215i
$64$		−	435.967i		−	0.851498i
$65$		0			0
$66$	−285.785	−	566.732i	−0.532996	−	1.05697i
$67$	320.313	−	320.313i	0.584066	−	0.584066i	−0.351952	−	0.936018i	$-0.614482\pi$
$67$	320.313	−	320.313i	0.584066	−	0.584066i	0.936018	+	0.351952i	$0.114482\pi$
$68$	−195.367	+	195.367i	−0.348407	+	0.348407i
$69$	111.789	+	221.686i	0.195041	+	0.386780i
$70$		0			0
$71$		−	1084.53i		−	1.81282i	−0.422400	−	0.906410i	$-0.638812\pi$
$71$		−	1084.53i		−	1.81282i	0.422400	−	0.906410i	$-0.361188\pi$
$72$	637.923	−	95.5747i	1.04417	−	0.156439i
$73$	208.052	+	208.052i	0.333570	+	0.333570i	0.853940	−	0.520371i	$-0.174206\pi$
$73$	208.052	+	208.052i	0.333570	+	0.333570i	−0.520371	+	0.853940i	$0.674206\pi$
$74$	−29.4950			−0.0463341
$75$		0			0
$76$	383.455			0.578753
$77$	−235.335	−	235.335i	−0.348297	−	0.348297i
$78$	−663.244	−	218.572i	−0.962790	−	0.317287i
$79$		−	676.816i		−	0.963895i	−0.876200	−	0.481947i	$-0.839930\pi$
$79$		−	676.816i		−	0.963895i	0.876200	−	0.481947i	$-0.160070\pi$
$80$		0			0
$81$	213.645	+	696.991i	0.293065	+	0.956092i
$82$	86.3732	−	86.3732i	0.116321	−	0.116321i
$83$	−71.3863	+	71.3863i	−0.0944056	+	0.0944056i	−0.752732	−	0.658327i	$-0.771265\pi$
$83$	−71.3863	+	71.3863i	−0.0944056	+	0.0944056i	0.658327	+	0.752732i	$0.271265\pi$
$84$	102.410	−	51.6421i	0.133022	−	0.0670788i
$85$		0			0
$86$			373.687i			0.468555i
$87$	−291.429	+	884.326i	−0.359132	+	1.08977i
$88$	1045.52	+	1045.52i	1.26651	+	1.26651i
$89$	228.124			0.271698			0.135849	−	0.990730i	$-0.456624\pi$
$89$	228.124			0.271698			0.135849	+	0.990730i	$0.456624\pi$
$90$		0			0
$91$	−366.173			−0.421818
$92$	−138.682	−	138.682i	−0.157159	−	0.157159i
$93$	200.893	−	609.598i	0.223996	−	0.679702i
$94$			159.075i			0.174546i
$95$		0			0
$96$	−755.670	+	381.061i	−0.803387	+	0.405123i
$97$	89.5346	−	89.5346i	0.0937202	−	0.0937202i	−0.658692	−	0.752412i	$-0.728890\pi$
$97$	89.5346	−	89.5346i	0.0937202	−	0.0937202i	0.752412	+	0.658692i	$0.228890\pi$
$98$	438.329	−	438.329i	0.451816	−	0.451816i
$99$	−993.485	+	1343.64i	−1.00858	+	1.36405i
$100$		0			0
$101$			1514.86i			1.49242i	0.665710	+	0.746210i	$0.268129\pi$
$101$			1514.86i			1.49242i	−0.665710	+	0.746210i	$0.731871\pi$
$102$	−655.617	−	216.058i	−0.636429	−	0.209735i
$103$	−71.0084	−	71.0084i	−0.0679288	−	0.0679288i	0.672326	−	0.740255i	$-0.265295\pi$
$103$	−71.0084	−	71.0084i	−0.0679288	−	0.0679288i	−0.740255	+	0.672326i	$0.765295\pi$
$104$	1626.80			1.53385
$105$		0			0
$106$	655.228			0.600391
$107$	−28.8036	−	28.8036i	−0.0260238	−	0.0260238i	0.693975	−	0.719999i	$-0.255858\pi$
$107$	−28.8036	−	28.8036i	−0.0260238	−	0.0260238i	−0.719999	+	0.693975i	$0.755858\pi$
$108$	−332.617	−	470.100i	−0.296353	−	0.418846i
$109$		−	600.245i		−	0.527459i	−0.964597	−	0.263730i	$-0.915047\pi$
$109$		−	600.245i		−	0.527459i	0.964597	−	0.263730i	$-0.0849527\pi$
$110$		0			0
$111$	34.9641	+	69.3362i	0.0298977	+	0.0592892i
$112$	54.4285	−	54.4285i	0.0459197	−	0.0459197i
$113$	1149.62	−	1149.62i	0.957058	−	0.957058i	−0.0420576	−	0.999115i	$-0.513391\pi$
$113$	1149.62	−	1149.62i	0.957058	−	0.957058i	0.999115	+	0.0420576i	$0.0133913\pi$
$114$	431.371	+	855.438i	0.354400	+	0.702799i
$115$		0			0
$116$		−	735.527i		−	0.588724i
$117$	272.412	+	1818.24i	0.215252	+	1.43672i
$118$	363.577	+	363.577i	0.283644	+	0.283644i
$119$	−361.963			−0.278832
$120$		0			0
$121$	−2499.41			−1.87785
$122$	−335.867	−	335.867i	−0.249246	−	0.249246i
$123$	−305.434	−	100.655i	−0.223903	−	0.0737870i
$124$			507.025i			0.367195i
$125$		0			0
$126$	230.414	+	170.368i	0.162912	+	0.120457i
$127$	1327.45	−	1327.45i	0.927501	−	0.927501i	−0.0700433	−	0.997544i	$-0.522314\pi$
$127$	1327.45	−	1327.45i	0.927501	−	0.927501i	0.997544	+	0.0700433i	$0.0223138\pi$
$128$	312.918	−	312.918i	0.216081	−	0.216081i
$129$	878.456	−	442.978i	0.599564	−	0.302341i
$130$		0			0
$131$		−	88.4849i		−	0.0590150i	−0.999565	−	0.0295075i	$-0.990606\pi$
$131$		−	88.4849i		−	0.0590150i	0.999565	−	0.0295075i	$-0.00939389\pi$
$132$	413.159	−	1253.71i	0.272431	−	0.826677i
$133$	355.220	+	355.220i	0.231590	+	0.231590i
$134$	−894.048			−0.576373
$135$		0			0
$136$	1608.09			1.01392
$137$	1491.02	+	1491.02i	0.929830	+	0.929830i	0.997695	−	0.0678646i	$-0.0216186\pi$
$137$	1491.02	+	1491.02i	0.929830	+	0.929830i	−0.0678646	+	0.997695i	$0.521619\pi$
$138$	153.370	−	465.393i	0.0946067	−	0.287079i
$139$		−	2123.04i		−	1.29550i	−0.761854	−	0.647748i	$-0.775711\pi$
$139$		−	2123.04i		−	1.29550i	0.761854	−	0.647748i	$-0.224289\pi$
$140$		0			0
$141$	373.950	−	188.571i	0.223350	−	0.112628i
$142$	−1513.56	+	1513.56i	−0.894470	+	0.894470i
$143$	−2980.00	+	2980.00i	−1.74266	+	1.74266i
$144$	−310.758	−	229.774i	−0.179837	−	0.132971i
$145$		0			0
$146$		−	580.707i		−	0.329176i
$147$	−1550.02	−	510.809i	−0.869685	−	0.286604i
$148$	−43.3752	−	43.3752i	−0.0240907	−	0.0240907i
$149$	−1237.81			−0.680571			−0.340286	−	0.940322i	$-0.610524\pi$
$149$	−1237.81			−0.680571			−0.340286	+	0.940322i	$0.610524\pi$
$150$		0			0
$151$	2364.27			1.27418			0.637092	−	0.770788i	$-0.280137\pi$
$151$	2364.27			1.27418			0.637092	+	0.770788i	$0.280137\pi$
$152$	−1578.13	−	1578.13i	−0.842129	−	0.842129i
$153$	269.280	+	1797.33i	0.142287	+	0.949711i
$154$			656.859i			0.343709i
$155$		0			0
$156$	−653.934	−	1296.80i	−0.335619	−	0.665557i
$157$	709.026	−	709.026i	0.360423	−	0.360423i	−0.503546	−	0.863969i	$-0.667971\pi$
$157$	709.026	−	709.026i	0.360423	−	0.360423i	0.863969	+	0.503546i	$0.167971\pi$
$158$	−944.553	+	944.553i	−0.475599	+	0.475599i
$159$	−776.724	−	1540.30i	−0.387410	−	0.768262i
$160$		0			0
$161$		−	256.941i		−	0.125775i
$162$	674.551	−	1270.87i	0.327147	−	0.616352i
$163$	−1909.69	−	1909.69i	−0.917662	−	0.917662i	0.0791974	−	0.996859i	$-0.474764\pi$
$163$	−1909.69	−	1909.69i	−0.917662	−	0.917662i	−0.996859	+	0.0791974i	$0.974764\pi$
$164$	254.040			0.120959
$165$		0			0
$166$	199.251			0.0931621
$167$	1002.75	+	1002.75i	0.464639	+	0.464639i	0.900173	−	0.435533i	$-0.143440\pi$
$167$	1002.75	+	1002.75i	0.464639	+	0.464639i	−0.435533	+	0.900173i	$0.643440\pi$
$168$	−634.011	−	208.938i	−0.291161	−	0.0959519i
$169$			2439.79i			1.11051i
$170$		0			0
$171$	1499.59	−	2028.12i	0.670623	−	0.906982i
$172$	−549.543	+	549.543i	−0.243618	+	0.243618i
$173$	2360.68	−	2360.68i	1.03745	−	1.03745i	0.0381828	−	0.999271i	$-0.487843\pi$
$173$	2360.68	−	2360.68i	1.03745	−	1.03745i	0.999271	−	0.0381828i	$-0.0121569\pi$
$174$	1640.87	−	827.438i	0.714906	−	0.360505i
$175$		0			0
$176$		−	885.902i		−	0.379417i
$177$	423.696	−	1285.68i	0.179926	−	0.545977i
$178$	−318.367	−	318.367i	−0.134059	−	0.134059i
$179$	−2605.87			−1.08811			−0.544056	−	0.839049i	$-0.683112\pi$
$179$	−2605.87			−1.08811			−0.544056	+	0.839049i	$0.683112\pi$
$180$		0			0
$181$	−3682.37			−1.51220			−0.756100	−	0.654456i	$-0.772898\pi$
$181$	−3682.37			−1.51220			−0.756100	+	0.654456i	$0.772898\pi$
$182$	511.026	+	511.026i	0.208131	+	0.208131i
$183$	−391.404	+	1187.70i	−0.158106	+	0.479765i
$184$			1141.51i			0.457354i
$185$		0			0
$186$	−1131.11	+	570.382i	−0.445897	+	0.224852i
$187$	−2945.73	+	2945.73i	−1.15194	+	1.15194i
$188$	−233.935	+	233.935i	−0.0907525	+	0.0907525i
$189$	127.359	−	743.611i	0.0490160	−	0.286189i
$190$		0			0
$191$			842.771i			0.319271i	0.987176	+	0.159635i	$0.0510319\pi$
$191$			842.771i			0.319271i	−0.987176	+	0.159635i	$0.948968\pi$
$192$	2151.53	+	709.036i	0.808716	+	0.266512i
$193$	−2343.62	−	2343.62i	−0.874079	−	0.874079i	0.118835	−	0.992914i	$-0.462084\pi$
$193$	−2343.62	−	2343.62i	−0.874079	−	0.874079i	−0.992914	+	0.118835i	$0.962084\pi$
$194$	−249.906			−0.0924857
$195$		0			0
$196$	1289.21			0.469829
$197$	−309.658	−	309.658i	−0.111991	−	0.111991i	0.648891	−	0.760882i	$-0.275233\pi$
$197$	−309.658	−	309.658i	−0.111991	−	0.111991i	−0.760882	+	0.648891i	$0.775233\pi$
$198$	3261.65	−	488.666i	1.17068	−	0.175394i
$199$			1759.07i			0.626618i	0.949651	+	0.313309i	$0.101438\pi$
$199$			1759.07i			0.626618i	−0.949651	+	0.313309i	$0.898562\pi$
$200$		0			0
$201$	1059.83	+	2101.71i	0.371913	+	0.737528i
$202$	2114.12	−	2114.12i	0.736381	−	0.736381i
$203$	681.368	−	681.368i	0.235580	−	0.235580i
$204$	−646.414	−	1281.88i	−0.221853	−	0.439950i
$205$		0			0
$206$			198.196i			0.0670340i
$207$	−1275.85	+	191.149i	−0.428393	+	0.0641826i
$208$	−689.218	−	689.218i	−0.229753	−	0.229753i
$209$	5781.72			1.91354
$210$		0			0
$211$	−1344.63			−0.438710			−0.219355	−	0.975645i	$-0.570395\pi$
$211$	−1344.63			−0.438710			−0.219355	+	0.975645i	$0.570395\pi$
$212$	963.577	+	963.577i	0.312164	+	0.312164i
$213$	5352.24	+	1763.83i	1.72174	+	0.567397i
$214$			80.3957i			0.0256810i
$215$		0			0
$216$	−565.819	+	3303.64i	−0.178236	+	1.04067i
$217$	−469.692	+	469.692i	−0.146934	+	0.146934i
$218$	−837.693	+	837.693i	−0.260256	+	0.260256i
$219$	−1365.12	+	688.385i	−0.421214	+	0.212405i
$220$		0			0
$221$			4583.46i			1.39510i
$222$	47.9692	−	145.560i	0.0145022	−	0.0440060i
$223$	3146.44	+	3146.44i	0.944850	+	0.944850i	0.998557	−	0.0537070i	$-0.0171037\pi$
$223$	3146.44	+	3146.44i	0.944850	+	0.944850i	−0.0537070	+	0.998557i	$0.517104\pi$
$224$	875.844			0.261249
$225$		0			0
$226$	−3208.79			−0.944451
$227$	−4427.84	−	4427.84i	−1.29465	−	1.29465i	−0.931876	−	0.362776i	$-0.881829\pi$
$227$	−4427.84	−	4427.84i	−1.29465	−	1.29465i	−0.362776	−	0.931876i	$-0.618171\pi$
$228$	−623.632	+	1892.38i	−0.181145	+	0.549675i
$229$			2059.29i			0.594242i	0.954840	+	0.297121i	$0.0960265\pi$
$229$			2059.29i			0.594242i	−0.954840	+	0.297121i	$0.903973\pi$
$230$		0			0
$231$	1544.13	−	778.658i	0.439812	−	0.221783i
$232$	−3027.11	+	3027.11i	−0.856636	+	0.856636i
$233$	2302.49	−	2302.49i	0.647387	−	0.647387i	−0.304973	−	0.952361i	$-0.598648\pi$
$233$	2302.49	−	2302.49i	0.647387	−	0.647387i	0.952361	+	0.304973i	$0.0986476\pi$
$234$	2157.34	−	2917.69i	0.602691	−	0.815108i
$235$		0			0
$236$			1069.35i			0.294953i
$237$	3340.14	+	1100.74i	0.915465	+	0.301691i
$238$	505.149	+	505.149i	0.137580	+	0.137580i
$239$	2459.08			0.665542			0.332771	−	0.943008i	$-0.392016\pi$
$239$	2459.08			0.665542			0.332771	+	0.943008i	$0.392016\pi$
$240$		0			0
$241$	4461.08			1.19238			0.596189	−	0.802844i	$-0.296681\pi$
$241$	4461.08			1.19238			0.596189	+	0.802844i	$0.296681\pi$
$242$	3488.14	+	3488.14i	0.926555	+	0.926555i
$243$	−3787.17	−	79.2012i	−0.999781	−	0.0209085i
$244$		−	987.850i		−	0.259183i
$245$		0			0
$246$	285.785	+	566.732i	0.0740691	+	0.146884i
$247$	4498.08	−	4498.08i	1.15873	−	1.15873i
$248$	2086.70	−	2086.70i	0.534296	−	0.534296i
$249$	−236.198	−	468.396i	−0.0601141	−	0.119210i
$250$		0			0
$251$		−	5602.53i		−	1.40888i	−0.709765	−	0.704439i	$-0.751199\pi$
$251$		−	5602.53i		−	1.40888i	0.709765	−	0.704439i	$-0.248801\pi$
$252$	88.3033	+	589.389i	0.0220737	+	0.147333i
$253$	−2091.04	−	2091.04i	−0.519615	−	0.519615i
$254$	−3705.15			−0.915283
$255$		0			0
$256$	−4361.15			−1.06473
$257$	−980.647	−	980.647i	−0.238020	−	0.238020i	0.578010	−	0.816030i	$-0.303829\pi$
$257$	−980.647	−	980.647i	−0.238020	−	0.238020i	−0.816030	+	0.578010i	$0.803829\pi$
$258$	−1844.17	−	607.746i	−0.445012	−	0.146654i
$259$		−	80.3628i		−	0.0192799i
$260$		0			0
$261$	−3890.25	−	2876.45i	−0.922607	−	0.682176i
$262$	−123.488	+	123.488i	−0.0291188	+	0.0291188i
$263$	−1277.16	+	1277.16i	−0.299440	+	0.299440i	−0.840795	−	0.541354i	$-0.817912\pi$
$263$	−1277.16	+	1277.16i	−0.299440	+	0.299440i	0.541354	+	0.840795i	$0.317912\pi$
$264$	−6860.11	+	3459.34i	−1.59928	+	0.806468i
$265$		0			0
$266$		−	991.479i		−	0.228539i
$267$	−371.010	+	1125.81i	−0.0850391	+	0.258047i
$268$	−1314.78	−	1314.78i	−0.299676	−	0.299676i
$269$	1742.31			0.394909			0.197454	−	0.980312i	$-0.436733\pi$
$269$	1742.31			0.394909			0.197454	+	0.980312i	$0.436733\pi$
$270$		0			0
$271$	2262.36			0.507116			0.253558	−	0.967320i	$-0.418399\pi$
$271$	2262.36			0.507116			0.253558	+	0.967320i	$0.418399\pi$
$272$	−681.292	−	681.292i	−0.151873	−	0.151873i
$273$	595.527	−	1807.09i	0.132025	−	0.400624i
$274$		−	4161.70i		−	0.917582i
$275$		0			0
$276$	909.951	−	458.860i	0.198452	−	0.100073i
$277$	−3801.88	+	3801.88i	−0.824668	+	0.824668i	−0.986773	−	0.162106i	$-0.948171\pi$
$277$	−3801.88	+	3801.88i	−0.824668	+	0.824668i	0.162106	+	0.986773i	$0.448171\pi$
$278$	−2962.89	+	2962.89i	−0.639216	+	0.639216i
$279$	2681.69	+	1982.84i	0.575443	+	0.425482i
$280$		0			0
$281$		−	3737.11i		−	0.793372i	−0.917954	−	0.396686i	$-0.870160\pi$
$281$		−	3737.11i		−	0.793372i	0.917954	−	0.396686i	$-0.129840\pi$
$282$	−785.046	−	258.712i	−0.165776	−	0.0546314i
$283$	549.478	+	549.478i	0.115417	+	0.115417i	0.762457	−	0.647039i	$-0.223993\pi$
$283$	549.478	+	549.478i	0.115417	+	0.115417i	−0.647039	+	0.762457i	$0.723993\pi$
$284$	−4451.66			−0.930132
$285$		0			0
$286$	8317.69			1.71970
$287$	235.335	+	235.335i	0.0484020	+	0.0484020i
$288$	−651.579	−	4349.03i	−0.133315	−	0.889822i
$289$		−	382.245i		−	0.0778028i
$290$		0			0
$291$	296.245	+	587.475i	0.0596777	+	0.118345i
$292$	853.986	−	853.986i	0.171150	−	0.171150i
$293$	−5554.60	+	5554.60i	−1.10752	+	1.10752i	−0.114043	+	0.993476i	$0.536380\pi$
$293$	−5554.60	+	5554.60i	−1.10752	+	1.10752i	−0.993476	+	0.114043i	$0.963620\pi$
$294$	1450.31	+	2876.07i	0.287700	+	0.570529i
$295$		0			0
$296$			357.027i			0.0701074i
$297$	−5015.19	−	7088.15i	−0.979835	−	1.38484i
$298$	1727.46	+	1727.46i	0.335803	+	0.335803i
$299$	−3253.59			−0.629298
$300$		0			0
$301$	−1018.16			−0.194969
$302$	−3299.54	−	3299.54i	−0.628700	−	0.628700i
$303$	−7475.96	−	2463.70i	−1.41744	−	0.467115i
$304$			1337.20i			0.252282i
$305$		0			0
$306$	2132.53	−	2884.13i	0.398394	−	0.538807i
$307$	5210.62	−	5210.62i	0.968684	−	0.968684i	−0.0308404	−	0.999524i	$-0.509818\pi$
$307$	5210.62	−	5210.62i	0.968684	−	0.968684i	0.999524	+	0.0308404i	$0.00981837\pi$
$308$	−965.976	+	965.976i	−0.178706	+	0.178706i
$309$	465.916	−	234.947i	0.0857769	−	0.0432546i
$310$		0			0
$311$		−	2717.37i		−	0.495460i	−0.968829	−	0.247730i	$-0.920315\pi$
$311$		−	2717.37i		−	0.495460i	0.968829	−	0.247730i	$-0.0796846\pi$
$312$	−2645.74	+	8028.37i	−0.480083	+	1.45678i
$313$	6206.89	+	6206.89i	1.12088	+	1.12088i	0.991610	+	0.129266i	$0.0412621\pi$
$313$	6206.89	+	6206.89i	1.12088	+	1.12088i	0.129266	+	0.991610i	$0.458738\pi$
$314$	−1979.01			−0.355675
$315$		0			0
$316$	−2778.12			−0.494561
$317$	3909.79	+	3909.79i	0.692731	+	0.692731i	0.962832	−	0.270101i	$-0.0870571\pi$
$317$	3909.79	+	3909.79i	0.692731	+	0.692731i	−0.270101	+	0.962832i	$0.587057\pi$
$318$	−1065.63	+	3233.60i	−0.187917	+	0.570224i
$319$		−	11090.2i		−	1.94650i
$320$		0			0
$321$	188.993	−	95.3031i	0.0328615	−	0.0165710i
$322$	−358.583	+	358.583i	−0.0620591	+	0.0620591i
$323$	4446.36	−	4446.36i	0.765950	−	0.765950i
$324$	2860.93	−	876.944i	0.490557	−	0.150368i
$325$		0			0
$326$			5330.28i			0.905574i
$327$	2962.26	+	976.210i	0.500958	+	0.165090i
$328$	−1045.52	−	1045.52i	−0.176004	−	0.176004i
$329$	−433.420			−0.0726298
$330$		0			0
$331$	3332.16			0.553330			0.276665	−	0.960966i	$-0.410771\pi$
$331$	3332.16			0.553330			0.276665	+	0.960966i	$0.410771\pi$
$332$	293.018	+	293.018i	0.0484382	+	0.0484382i
$333$	−399.043	+	59.7854i	−0.0656680	+	0.00983850i
$334$		−	2798.83i		−	0.458519i
$335$		0			0
$336$	180.089	+	357.129i	0.0292400	+	0.0579850i
$337$	2214.55	−	2214.55i	0.357965	−	0.357965i	−0.505097	−	0.863063i	$-0.668543\pi$
$337$	2214.55	−	2214.55i	0.357965	−	0.357965i	0.863063	+	0.505097i	$0.168543\pi$
$338$	3404.93	−	3404.93i	0.547940	−	0.547940i
$339$	3803.79	+	7543.17i	0.609420	+	1.20852i
$340$		0			0
$341$			7644.91i			1.21406i
$342$	−4923.21	+	737.605i	−0.778412	+	0.116623i
$343$	2498.52	+	2498.52i	0.393317	+	0.393317i
$344$	4523.36			0.708963
$345$		0			0
$346$	−6589.07			−1.02379
$347$	5447.38	+	5447.38i	0.842740	+	0.842740i	0.989214	−	0.146475i	$-0.0467927\pi$
$347$	5447.38	+	5447.38i	0.842740	+	0.842740i	−0.146475	+	0.989214i	$0.546793\pi$
$348$	3629.88	+	1196.23i	0.559144	+	0.184266i
$349$			2153.29i			0.330267i	0.986271	+	0.165134i	$0.0528055\pi$
$349$			2153.29i			0.330267i	−0.986271	+	0.165134i	$0.947194\pi$
$350$		0			0
$351$	−9416.21	−	1612.73i	−1.43191	−	0.245245i
$352$	7127.81	−	7127.81i	1.07930	−	1.07930i
$353$	723.013	−	723.013i	0.109014	−	0.109014i	−0.650496	−	0.759510i	$-0.725439\pi$
$353$	723.013	−	723.013i	0.109014	−	0.109014i	0.759510	+	0.650496i	$0.225439\pi$
$354$	−2385.58	+	1202.98i	−0.358171	+	0.180614i
$355$		0			0
$356$		−	936.378i		−	0.139404i
$357$	588.679	−	1786.31i	0.0872722	−	0.264823i
$358$	3636.72	+	3636.72i	0.536890	+	0.536890i
$359$	2816.10			0.414006			0.207003	−	0.978340i	$-0.433629\pi$
$359$	2816.10			0.414006			0.207003	+	0.978340i	$0.433629\pi$
$360$		0			0
$361$	−1868.06			−0.272352
$362$	5139.06	+	5139.06i	0.746140	+	0.746140i
$363$	4064.93	−	12334.8i	0.587750	−	1.78350i
$364$			1503.03i			0.216429i
$365$		0			0
$366$	2203.77	−	1111.29i	0.314734	−	0.158711i
$367$	−7080.66	+	7080.66i	−1.00710	+	1.00710i	−0.00712974	+	0.999975i	$0.502269\pi$
$367$	−7080.66	+	7080.66i	−1.00710	+	1.00710i	−0.999975	+	0.00712974i	$0.997731\pi$
$368$	483.618	−	483.618i	0.0685064	−	0.0685064i
$369$	993.485	−	1343.64i	0.140159	−	0.189558i
$370$		0			0
$371$			1785.25i			0.249827i
$372$	−2502.21	−	824.601i	−0.348746	−	0.114929i
$373$	−3207.86	−	3207.86i	−0.445299	−	0.445299i	0.448489	−	0.893788i	$-0.351962\pi$
$373$	−3207.86	−	3207.86i	−0.445299	−	0.445299i	−0.893788	+	0.448489i	$0.851962\pi$
$374$	8222.04			1.13677
$375$		0			0
$376$	1925.55			0.264103
$377$	−8628.04	−	8628.04i	−1.17869	−	1.17869i
$378$	−1215.51	+	860.032i	−0.165395	+	0.117024i
$379$			2987.72i			0.404930i	0.979289	+	0.202465i	$0.0648953\pi$
$379$			2987.72i			0.404930i	−0.979289	+	0.202465i	$0.935105\pi$
$380$		0			0
$381$	4392.18	+	8710.00i	0.590599	+	1.17120i
$382$	1176.16	−	1176.16i	0.157533	−	0.157533i
$383$	993.565	−	993.565i	0.132556	−	0.132556i	−0.637716	−	0.770272i	$-0.720121\pi$
$383$	993.565	−	993.565i	0.132556	−	0.132556i	0.770272	+	0.637716i	$0.220121\pi$
$384$	1035.36	+	2053.19i	0.137593	+	0.272856i
$385$		0			0
$386$			6541.44i			0.862566i
$387$	757.452	+	5055.69i	0.0994921	+	0.664070i
$388$	−367.511	−	367.511i	−0.0480865	−	0.0480865i
$389$	−12754.3			−1.66239			−0.831197	−	0.555979i	$-0.812344\pi$
$389$	−12754.3			−1.66239			−0.831197	+	0.555979i	$0.812344\pi$
$390$		0			0
$391$	−3216.18			−0.415982
$392$	−5305.84	−	5305.84i	−0.683636	−	0.683636i
$393$	436.680	+	143.908i	0.0560498	+	0.0184712i
$394$			864.309i			0.110516i
$395$		0			0
$396$	5515.21	+	4077.94i	0.699873	+	0.517486i
$397$	−8826.34	+	8826.34i	−1.11582	+	1.11582i	−0.123474	+	0.992348i	$0.539404\pi$
$397$	−8826.34	+	8826.34i	−1.11582	+	1.11582i	−0.992348	+	0.123474i	$0.960596\pi$
$398$	2454.93	−	2454.93i	0.309182	−	0.309182i
$399$	−2330.75	+	1175.33i	−0.292440	+	0.147468i
$400$		0			0
$401$		−	4022.87i		−	0.500979i	−0.968119	−	0.250490i	$-0.919408\pi$
$401$		−	4022.87i		−	0.500979i	0.968119	−	0.250490i	$-0.0805916\pi$
$402$	1454.04	−	4412.19i	0.180400	−	0.547413i
$403$	5947.62	+	5947.62i	0.735166	+	0.735166i
$404$	6218.04			0.765740
$405$		0			0
$406$	−1901.81			−0.232476
$407$	−654.010	−	654.010i	−0.0796513	−	0.0796513i
$408$	−2615.32	+	7936.04i	−0.317347	+	0.962972i
$409$			12309.0i			1.48812i	0.668110	+	0.744062i	$0.267103\pi$
$409$			12309.0i			1.48812i	−0.668110	+	0.744062i	$0.732897\pi$
$410$		0			0
$411$	−9783.24	+	4933.38i	−1.17414	+	0.592082i
$412$	−291.467	+	291.467i	−0.0348533	+	0.0348533i
$413$	−990.612	+	990.612i	−0.118026	+	0.118026i
$414$	2047.32	+	1513.79i	0.243044	+	0.179707i
$415$		0			0
$416$		−	11090.7i		−	1.30712i
$417$	10477.4	+	3452.81i	1.23041	+	0.405480i
$418$	−8068.88	−	8068.88i	−0.944167	−	0.944167i
$419$	−3075.95			−0.358639			−0.179320	−	0.983791i	$-0.557390\pi$
$419$	−3075.95			−0.358639			−0.179320	+	0.983791i	$0.557390\pi$
$420$		0			0
$421$	12055.0			1.39554			0.697770	−	0.716322i	$-0.254176\pi$
$421$	12055.0			1.39554			0.697770	+	0.716322i	$0.254176\pi$
$422$	1876.54	+	1876.54i	0.216466	+	0.216466i
$423$	322.440	+	2152.16i	0.0370628	+	0.247379i
$424$		−	7931.33i		−	0.908442i
$425$		0			0
$426$	−5007.94	−	9931.08i	−0.569566	−	1.12949i
$427$	915.113	−	915.113i	0.103713	−	0.103713i
$428$	−118.230	+	118.230i	−0.0133524	+	0.0133524i
$429$	−9860.00	−	19553.1i	−1.10966	−	2.20054i
$430$		0			0
$431$			6265.92i			0.700275i	0.936698	+	0.350138i	$0.113865\pi$
$431$			6265.92i			0.700275i	−0.936698	+	0.350138i	$0.886135\pi$
$432$	1639.35	−	1159.92i	0.182577	−	0.129182i
$433$	−8190.21	−	8190.21i	−0.908998	−	0.908998i	0.0871933	−	0.996191i	$-0.472210\pi$
$433$	−8190.21	−	8190.21i	−0.908998	−	0.908998i	−0.996191	+	0.0871933i	$0.972210\pi$
$434$	1310.99			0.144999
$435$		0			0
$436$	−2463.82			−0.270632
$437$	3156.27	+	3156.27i	0.345503	+	0.345503i
$438$	2865.83	+	944.434i	0.312637	+	0.103029i
$439$		−	8494.79i		−	0.923541i	−0.887000	−	0.461770i	$-0.847214\pi$
$439$		−	8494.79i		−	0.923541i	0.887000	−	0.461770i	$-0.152786\pi$
$440$		0			0
$441$	5041.76	−	6818.72i	0.544408	−	0.736284i
$442$	6396.61	−	6396.61i	0.688362	−	0.688362i
$443$	−9078.70	+	9078.70i	−0.973684	+	0.973684i	−0.999663	−	0.0259785i	$-0.991730\pi$
$443$	−9078.70	+	9078.70i	−0.973684	+	0.973684i	0.0259785	+	0.999663i	$0.491730\pi$
$444$	284.603	−	143.517i	0.0304204	−	0.0153401i
$445$		0			0
$446$		−	8782.26i		−	0.932404i
$447$	2013.11	−	6108.67i	0.213013	−	0.646376i
$448$	−1657.74	−	1657.74i	−0.174824	−	0.174824i
$449$	4883.04			0.513241			0.256620	−	0.966512i	$-0.417391\pi$
$449$	4883.04			0.513241			0.256620	+	0.966512i	$0.417391\pi$
$450$		0			0
$451$	3830.41			0.399927
$452$	−4718.84	−	4718.84i	−0.491053	−	0.491053i
$453$	−3845.14	+	11667.9i	−0.398809	+	1.21016i
$454$			12358.9i			1.27760i
$455$		0			0
$456$	10354.8	−	5221.61i	1.06340	−	0.536237i
$457$	−5081.87	+	5081.87i	−0.520175	+	0.520175i	−0.917624	−	0.397449i	$-0.869896\pi$
$457$	−5081.87	+	5081.87i	−0.520175	+	0.520175i	0.397449	+	0.917624i	$0.369896\pi$
$458$	2873.91	−	2873.91i	0.293207	−	0.293207i
$459$	−9307.92	−	1594.18i	−0.946529	−	0.162113i
$460$		0			0
$461$		−	17258.2i		−	1.74359i	−0.489871	−	0.871795i	$-0.662956\pi$
$461$		−	17258.2i		−	1.74359i	0.489871	−	0.871795i	$-0.337044\pi$
$462$	−3241.65	−	1068.28i	−0.326440	−	0.107578i
$463$	2519.17	+	2519.17i	0.252864	+	0.252864i	0.822144	−	0.569280i	$-0.192778\pi$
$463$	2519.17	+	2519.17i	0.252864	+	0.252864i	−0.569280	+	0.822144i	$0.692778\pi$
$464$	2564.96			0.256628
$465$		0			0
$466$	−6426.65			−0.638860
$467$	4169.76	+	4169.76i	0.413177	+	0.413177i	0.882844	−	0.469667i	$-0.155626\pi$
$467$	4169.76	+	4169.76i	0.413177	+	0.413177i	−0.469667	+	0.882844i	$0.655626\pi$
$468$	7463.32	−	1118.17i	0.737162	−	0.110443i
$469$		−	2435.95i		−	0.239833i
$470$		0			0
$471$	2345.97	+	4652.22i	0.229505	+	0.455123i
$472$	4400.99	−	4400.99i	0.429178	−	0.429178i
$473$	−8285.99	+	8285.99i	−0.805476	+	0.805476i
$474$	−3125.27	−	6197.62i	−0.302844	−	0.600561i
$475$		0			0
$476$			1485.74i			0.143065i
$477$	8864.72	−	1328.13i	0.850917	−	0.127486i
$478$	−3431.85	−	3431.85i	−0.328387	−	0.328387i
$479$	4040.09			0.385378			0.192689	−	0.981260i	$-0.438279\pi$
$479$	4040.09			0.385378			0.192689	+	0.981260i	$0.438279\pi$
$480$		0			0
$481$	−1017.62			−0.0964645
$482$	−6225.81	−	6225.81i	−0.588336	−	0.588336i
$483$	1268.02	+	417.876i	0.119456	+	0.0393665i
$484$			10259.3i			0.963496i
$485$		0			0
$486$	5174.78	+	5395.84i	0.482989	+	0.503622i
$487$	−5543.45	+	5543.45i	−0.515806	+	0.515806i	−0.916300	−	0.400493i	$-0.868839\pi$
$487$	−5543.45	+	5543.45i	−0.515806	+	0.515806i	0.400493	+	0.916300i	$0.368839\pi$
$488$	−4065.57	+	4065.57i	−0.377130	+	0.377130i
$489$	12530.3	−	6318.65i	1.15877	−	0.584334i
$490$		0			0
$491$			9978.85i			0.917187i	0.888646	+	0.458594i	$0.151647\pi$
$491$			9978.85i			0.917187i	−0.888646	+	0.458594i	$0.848353\pi$
$492$	−413.159	+	1253.71i	−0.0378591	+	0.114881i
$493$	−8528.82	−	8528.82i	−0.779145	−	0.779145i
$494$	−12554.9			−1.14347
$495$		0			0
$496$	−1768.12			−0.160063
$497$	−4123.87	−	4123.87i	−0.372195	−	0.372195i
$498$	−324.053	+	983.321i	−0.0291589	+	0.0884812i
$499$			1383.97i			0.124159i	0.998071	+	0.0620793i	$0.0197732\pi$
$499$			1383.97i			0.124159i	−0.998071	+	0.0620793i	$0.980227\pi$
$500$		0			0
$501$	−6579.44	+	3317.81i	−0.586722	+	0.295866i
$502$	−7818.80	+	7818.80i	−0.695160	+	0.695160i
$503$	11241.0	−	11241.0i	0.996447	−	0.996447i	−0.00354658	−	0.999994i	$-0.501129\pi$
$503$	11241.0	−	11241.0i	0.996447	−	0.996447i	0.999994	+	0.00354658i	$0.00112892\pi$
$504$	2062.25	−	2789.09i	0.182262	−	0.246500i
$505$		0			0
$506$			5836.45i			0.512771i
$507$	−12040.5	−	3967.95i	−1.05471	−	0.347580i
$508$	−5448.79	−	5448.79i	−0.475887	−	0.475887i
$509$	−20619.5			−1.79557			−0.897785	−	0.440434i	$-0.854825\pi$
$509$	−20619.5			−1.79557			−0.897785	+	0.440434i	$0.854825\pi$
$510$		0			0
$511$	1582.21			0.136972
$512$	3583.00	+	3583.00i	0.309273	+	0.309273i
$513$	7570.05	+	10699.0i	0.651513	+	0.920806i
$514$			2737.15i			0.234884i
$515$		0			0
$516$	−1818.29	−	3605.79i	−0.155127	−	0.307628i
$517$	−3527.27	+	3527.27i	−0.300056	+	0.300056i
$518$	−112.153	+	112.153i	−0.00951298	+	0.00951298i
$519$	7810.85	+	15489.5i	0.660613	+	1.31004i
$520$		0			0
$521$		−	3009.42i		−	0.253062i	−0.991963	−	0.126531i	$-0.959616\pi$
$521$		−	3009.42i		−	0.253062i	0.991963	−	0.126531i	$-0.0403843\pi$
$522$	1414.84	+	9443.50i	0.118632	+	0.791822i
$523$	3916.11	+	3916.11i	0.327418	+	0.327418i	0.851604	−	0.524186i	$-0.175630\pi$
$523$	3916.11	+	3916.11i	0.327418	+	0.327418i	−0.524186	+	0.851604i	$0.675630\pi$
$524$	−363.203			−0.0302797
$525$		0			0
$526$	3564.76			0.295496
$527$	5879.22	+	5879.22i	0.485964	+	0.485964i
$528$	4371.99	+	1440.79i	0.360353	+	0.118754i
$529$			9883.98i			0.812360i
$530$		0			0
$531$	5655.87	+	4181.95i	0.462229	+	0.341772i
$532$	1458.07	−	1458.07i	0.118826	−	0.118826i
$533$	2980.00	−	2980.00i	0.242173	−	0.242173i
$534$	2088.94	−	1053.39i	0.169283	−	0.0853643i
$535$		0			0
$536$			10822.2i			0.872101i
$537$	4238.07	−	12860.2i	0.340570	−	1.03344i
$538$	−2431.54	−	2431.54i	−0.194853	−	0.194853i
$539$	19438.7			1.55340
$540$		0			0
$541$	−13166.8			−1.04637			−0.523184	−	0.852220i	$-0.675256\pi$
$541$	−13166.8			−1.04637			−0.523184	+	0.852220i	$0.675256\pi$
$542$	−3157.31	−	3157.31i	−0.250218	−	0.250218i
$543$	5988.83	−	18172.8i	0.473306	−	1.43622i
$544$		−	10963.1i		−	0.864043i
$545$		0			0
$546$	−3353.06	+	1690.84i	−0.262816	+	0.132530i
$547$	8856.30	−	8856.30i	0.692263	−	0.692263i	−0.270466	−	0.962729i	$-0.587178\pi$
$547$	8856.30	−	8856.30i	0.692263	−	0.692263i	0.962729	+	0.270466i	$0.0871779\pi$
$548$	6120.18	−	6120.18i	0.477083	−	0.477083i
$549$	−5224.81	−	3863.22i	−0.406174	−	0.300325i
$550$		0			0
$551$			16739.9i			1.29427i
$552$	−5633.44	−	1856.50i	−0.434375	−	0.143148i
$553$	−2573.56	−	2573.56i	−0.197900	−	0.197900i
$554$	10611.7			0.813805
$555$		0			0
$556$	−8714.42			−0.664701
$557$	14446.6	+	14446.6i	1.09896	+	1.09896i	0.994532	+	0.104431i	$0.0333020\pi$
$557$	14446.6	+	14446.6i	1.09896	+	1.09896i	0.104431	+	0.994532i	$0.466698\pi$
$558$	−975.302	−	6509.75i	−0.0739925	−	0.493870i
$559$			12892.7i			0.975500i
$560$		0			0
$561$	−9746.61	−	19328.2i	−0.733516	−	1.45461i
$562$	−5215.46	+	5215.46i	−0.391461	+	0.391461i
$563$	8100.71	−	8100.71i	0.606402	−	0.606402i	−0.335602	−	0.942004i	$-0.608940\pi$
$563$	8100.71	−	8100.71i	0.606402	−	0.606402i	0.942004	+	0.335602i	$0.108940\pi$
$564$	−774.027	−	1534.95i	−0.0577879	−	0.114597i
$565$		0			0
$566$		−	1533.69i		−	0.113897i
$567$	3462.65	+	1837.90i	0.256468	+	0.136128i
$568$	18321.1	+	18321.1i	1.35341	+	1.35341i
$569$	11158.8			0.822149			0.411074	−	0.911602i	$-0.365154\pi$
$569$	11158.8			0.822149			0.411074	+	0.911602i	$0.365154\pi$
$570$		0			0
$571$	−7508.76			−0.550318			−0.275159	−	0.961399i	$-0.588731\pi$
$571$	−7508.76			−0.550318			−0.275159	+	0.961399i	$0.588731\pi$
$572$	12232.0	+	12232.0i	0.894133	+	0.894133i
$573$	−4159.14	−	1370.64i	−0.303229	−	0.0999291i
$574$		−	656.859i		−	0.0477644i
$575$		0			0
$576$	−6998.29	+	9464.83i	−0.506242	+	0.684666i
$577$	5375.41	−	5375.41i	0.387836	−	0.387836i	−0.486079	−	0.873915i	$-0.661573\pi$
$577$	5375.41	−	5375.41i	0.387836	−	0.387836i	0.873915	+	0.486079i	$0.161573\pi$
$578$	−533.456	+	533.456i	−0.0383890	+	0.0383890i
$579$	15377.5	−	7754.39i	1.10374	−	0.556583i
$580$		0			0
$581$			542.886i			0.0387654i
$582$	406.435	−	1233.31i	0.0289472	−	0.0878388i
$583$	14528.8	+	14528.8i	1.03211	+	1.03211i
$584$	−7029.28			−0.498071
$585$		0			0
$586$	15503.8			1.09293
$587$	−13341.5	−	13341.5i	−0.938094	−	0.938094i	0.0600982	−	0.998192i	$-0.480859\pi$
$587$	−13341.5	−	13341.5i	−0.938094	−	0.938094i	−0.998192	+	0.0600982i	$0.980859\pi$
$588$	−2096.71	+	6362.36i	−0.147053	+	0.446223i
$589$		−	11539.4i		−	0.807254i
$590$		0			0
$591$	2031.80	−	1024.57i	0.141416	−	0.0713119i
$592$	151.260	−	151.260i	0.0105013	−	0.0105013i
$593$	−1787.90	+	1787.90i	−0.123812	+	0.123812i	−0.766298	−	0.642486i	$-0.777903\pi$
$593$	−1787.90	+	1787.90i	−0.123812	+	0.123812i	0.642486	+	0.766298i	$0.277903\pi$
$594$	−2892.99	+	16891.2i	−0.199833	+	1.16676i
$595$		0			0
$596$			5080.81i			0.349191i
$597$	−8681.13	−	2860.86i	−0.595134	−	0.196126i
$598$	4540.67	+	4540.67i	0.310504	+	0.310504i
$599$	−12112.1			−0.826190			−0.413095	−	0.910688i	$-0.635552\pi$
$599$	−12112.1			−0.826190			−0.413095	+	0.910688i	$0.635552\pi$
$600$		0			0
$601$	16891.6			1.14646			0.573230	−	0.819394i	$-0.305690\pi$
$601$	16891.6			1.14646			0.573230	+	0.819394i	$0.305690\pi$
$602$	1420.93	+	1420.93i	0.0962003	+	0.0962003i
$603$	−12095.8	+	1812.21i	−0.816877	+	0.122386i
$604$		−	9704.60i		−	0.653766i
$605$		0			0
$606$	6995.04	+	13871.6i	0.468901	+	0.929863i
$607$	910.639	−	910.639i	0.0608924	−	0.0608924i	−0.676005	−	0.736897i	$-0.736290\pi$
$607$	910.639	−	910.639i	0.0608924	−	0.0608924i	0.736897	+	0.676005i	$0.236290\pi$
$608$	−10758.9	+	10758.9i	−0.717649	+	0.717649i
$609$	2254.46	+	4470.75i	0.150009	+	0.297478i
$610$		0			0
$611$			5488.31i			0.363393i
$612$	7377.49	−	1105.31i	0.487283	−	0.0730057i
$613$	−12990.7	−	12990.7i	−0.855934	−	0.855934i	0.134922	−	0.990856i	$-0.456921\pi$
$613$	−12990.7	−	12990.7i	−0.855934	−	0.855934i	−0.990856	+	0.134922i	$0.956921\pi$
$614$	−14543.7			−0.955924
$615$		0			0
$616$	7951.08			0.520062
$617$	−15676.0	−	15676.0i	−1.02284	−	1.02284i	−0.999733	−	0.0231063i	$-0.992644\pi$
$617$	−15676.0	−	15676.0i	−1.02284	−	1.02284i	−0.0231063	−	0.999733i	$-0.507356\pi$
$618$	−978.114	−	322.337i	−0.0636659	−	0.0209811i
$619$		−	22193.5i		−	1.44109i	−0.693411	−	0.720543i	$-0.743893\pi$
$619$		−	22193.5i		−	1.44109i	0.693411	−	0.720543i	$-0.256107\pi$
$620$		0			0
$621$	1131.64	−	6607.27i	0.0731257	−	0.426958i
$622$	−3792.33	+	3792.33i	−0.244467	+	0.244467i
$623$	867.430	−	867.430i	0.0557831	−	0.0557831i
$624$	4522.25	−	2280.43i	0.290120	−	0.146299i
$625$		0			0
$626$		−	17324.5i		−	1.10611i
$627$	−9403.11	+	28533.2i	−0.598922	+	1.81740i
$628$	−2910.33	−	2910.33i	−0.184928	−	0.184928i
$629$	−1005.92			−0.0637655
$630$		0			0
$631$	−3000.41			−0.189294			−0.0946470	−	0.995511i	$-0.530172\pi$
$631$	−3000.41			−0.189294			−0.0946470	+	0.995511i	$0.530172\pi$
$632$	11433.5	+	11433.5i	0.719622	+	0.719622i
$633$	2186.83	−	6635.83i	0.137313	−	0.416667i
$634$		−	10912.9i		−	0.683606i
$635$		0			0
$636$	−6322.44	+	3188.21i	−0.394184	+	0.198775i
$637$	15123.0	−	15123.0i	0.940651	−	0.940651i
$638$	−15477.4	+	15477.4i	−0.960432	+	0.960432i
$639$	−17409.3	+	23545.1i	−1.07778	+	1.45764i
$640$		0			0
$641$		−	26111.9i		−	1.60898i	−0.593963	−	0.804492i	$-0.702438\pi$
$641$		−	26111.9i		−	1.60898i	0.593963	−	0.804492i	$-0.297562\pi$
$642$	−396.759	−	130.752i	−0.0243907	−	0.00803794i
$643$	15717.3	+	15717.3i	0.963964	+	0.963964i	0.999373	−	0.0354091i	$-0.0112734\pi$
$643$	15717.3	+	15717.3i	0.963964	+	0.963964i	−0.0354091	+	0.999373i	$0.511273\pi$
$644$	−1054.66			−0.0645334
$645$		0			0
$646$	−12410.5			−0.755861
$647$	−16612.6	−	16612.6i	−1.00944	−	1.00944i	−0.999955	−	0.00948360i	$-0.996981\pi$
$647$	−16612.6	−	16612.6i	−1.00944	−	1.00944i	−0.00948360	−	0.999955i	$-0.503019\pi$
$648$	−15383.5	−	8165.23i	−0.932593	−	0.495001i
$649$			16123.6i			0.975205i
$650$		0			0
$651$	−1554.08	−	3081.85i	−0.0935625	−	0.185541i
$652$	−7838.70	+	7838.70i	−0.470839	+	0.470839i
$653$	592.526	−	592.526i	0.0355090	−	0.0355090i	−0.689129	−	0.724638i	$-0.742007\pi$
$653$	592.526	−	592.526i	0.0355090	−	0.0355090i	0.724638	+	0.689129i	$0.242007\pi$
$654$	−2771.70	−	5496.46i	−0.165722	−	0.328637i
$655$		0			0
$656$			885.902i			0.0527266i
$657$	−1177.08	−	7856.51i	−0.0698966	−	0.466532i
$658$	604.874	+	604.874i	0.0358365	+	0.0358365i
$659$	16623.0			0.982608			0.491304	−	0.870988i	$-0.336520\pi$
$659$	16623.0			0.982608			0.491304	+	0.870988i	$0.336520\pi$
$660$		0			0
$661$	10421.2			0.613217			0.306609	−	0.951836i	$-0.400806\pi$
$661$	10421.2			0.613217			0.306609	+	0.951836i	$0.400806\pi$
$662$	−4650.32	−	4650.32i	−0.273021	−	0.273021i
$663$	−22619.7	−	7454.33i	−1.32500	−	0.436655i
$664$		−	2411.87i		−	0.140962i
$665$		0			0
$666$	640.335	+	473.463i	0.0372559	+	0.0275470i
$667$	6054.22	−	6054.22i	0.351455	−	0.351455i
$668$	4115.95	−	4115.95i	0.238400	−	0.238400i
$669$	−20645.2	+	10410.7i	−1.19311	+	0.601647i
$670$		0			0
$671$		−	14894.8i		−	0.856940i
$672$	−1424.43	+	4322.36i	−0.0817688	+	0.248123i
$673$	−11678.9	−	11678.9i	−0.668930	−	0.668930i	0.288539	−	0.957468i	$-0.406831\pi$
$673$	−11678.9	−	11678.9i	−0.668930	−	0.668930i	−0.957468	+	0.288539i	$0.906831\pi$
$674$	−6181.19			−0.353250
$675$		0			0
$676$	10014.6			0.569786
$677$	15362.2	+	15362.2i	0.872111	+	0.872111i	0.992702	−	0.120591i	$-0.0384791\pi$
$677$	15362.2	+	15362.2i	0.872111	+	0.872111i	−0.120591	+	0.992702i	$0.538479\pi$
$678$	5218.63	−	15835.6i	0.295605	−	0.896998i
$679$		−	680.901i		−	0.0384839i
$680$		0			0
$681$	29053.0	−	14650.5i	1.63482	−	0.824389i
$682$	10669.1	−	10669.1i	0.599035	−	0.599035i
$683$	−494.598	+	494.598i	−0.0277090	+	0.0277090i	−0.720826	−	0.693117i	$-0.756237\pi$
$683$	−494.598	+	494.598i	−0.0277090	+	0.0277090i	0.693117	+	0.720826i	$0.256237\pi$
$684$	−8324.78	−	6155.35i	−0.465360	−	0.344087i
$685$		0			0
$686$		−	6973.80i		−	0.388136i
$687$	−10162.7	−	3349.12i	−0.564385	−	0.185993i
$688$	−1916.39	−	1916.39i	−0.106194	−	0.106194i
$689$	22606.3			1.24997
$690$		0			0
$691$	−16751.9			−0.922247			−0.461124	−	0.887336i	$-0.652554\pi$
$691$	−16751.9			−0.922247			−0.461124	+	0.887336i	$0.652554\pi$
$692$	−9689.87	−	9689.87i	−0.532303	−	0.532303i
$693$	1331.43	+	8886.78i	0.0729827	+	0.487130i
$694$		−	15204.6i		−	0.831639i
$695$		0			0
$696$	−10015.9	−	19862.2i	−0.545475	−	1.08171i
$697$	2945.73	−	2945.73i	0.160083	−	0.160083i
$698$	3005.10	−	3005.10i	0.162958	−	0.162958i
$699$	7618.31	+	15107.6i	0.412233	+	0.817487i
$700$		0			0
$701$		−	3170.42i		−	0.170820i	−0.996346	−	0.0854102i	$-0.972780\pi$
$701$		−	3170.42i		−	0.170820i	0.996346	−	0.0854102i	$-0.0272201\pi$
$702$	10890.4	+	15391.8i	0.585516	+	0.827531i
$703$	987.179	+	987.179i	0.0529618	+	0.0529618i
$704$	−26982.2			−1.44450
$705$		0			0
$706$	−2018.05			−0.107578
$707$	5760.19	+	5760.19i	0.306413	+	0.306413i
$708$	−5277.33	−	1739.14i	−0.280133	−	0.0923177i
$709$		−	17069.4i		−	0.904166i	−0.891976	−	0.452083i	$-0.850681\pi$
$709$		−	17069.4i		−	0.904166i	0.891976	−	0.452083i	$-0.149319\pi$
$710$		0			0
$711$	−10864.5	+	14693.6i	−0.573066	+	0.775041i
$712$	−3853.73	+	3853.73i	−0.202843	+	0.202843i
$713$	−4173.39	+	4173.39i	−0.219207	+	0.219207i
$714$	−3314.50	+	1671.40i	−0.173728	+	0.0876059i
$715$		0			0
$716$			10696.3i			0.558295i
$717$	−3999.32	+	12135.7i	−0.208309	+	0.632102i
$718$	−3930.11	−	3930.11i	−0.204276	−	0.204276i
$719$	1123.41			0.0582698			0.0291349	−	0.999575i	$-0.490725\pi$
$719$	1123.41			0.0582698			0.0291349	+	0.999575i	$0.490725\pi$
$720$		0			0
$721$	−540.011			−0.0278933
$722$	2607.04	+	2607.04i	0.134382	+	0.134382i
$723$	−7255.28	+	22015.8i	−0.373205	+	1.13247i
$724$			15115.0i			0.775888i
$725$		0			0
$726$	−22887.2	+	11541.3i	−1.17001	+	0.589997i
$727$	8990.29	−	8990.29i	0.458640	−	0.458640i	−0.439569	−	0.898209i	$-0.644869\pi$
$727$	8990.29	−	8990.29i	0.458640	−	0.458640i	0.898209	+	0.439569i	$0.144869\pi$
$728$	6185.81	−	6185.81i	0.314920	−	0.314920i
$729$	6550.13	−	18561.1i	0.332781	−	0.943004i
$730$		0			0
$731$			12744.5i			0.644831i
$732$	4875.12	+	1606.59i	0.246161	+	0.0811221i
$733$	20609.7	+	20609.7i	1.03852	+	1.03852i	0.999228	+	0.0392966i	$0.0125117\pi$
$733$	20609.7	+	20609.7i	1.03852	+	1.03852i	0.0392966	+	0.999228i	$0.487488\pi$
$734$	19763.3			0.993838
$735$		0			0
$736$	7782.22			0.389750
$737$	−19824.3	−	19824.3i	−0.990823	−	0.990823i
$738$	−3261.65	+	488.666i	−0.162687	+	0.0243741i
$739$		−	14256.7i		−	0.709664i	−0.934930	−	0.354832i	$-0.884538\pi$
$739$		−	14256.7i		−	0.709664i	0.934930	−	0.354832i	$-0.115462\pi$
$740$		0			0
$741$	14882.9	+	29513.9i	0.737838	+	1.46318i
$742$	2491.47	−	2491.47i	0.123268	−	0.123268i
$743$	14523.7	−	14523.7i	0.717123	−	0.717123i	−0.250892	−	0.968015i	$-0.580724\pi$
$743$	14523.7	−	14523.7i	0.717123	−	0.717123i	0.968015	+	0.250892i	$0.0807240\pi$
$744$	6904.30	+	13691.7i	0.340220	+	0.674680i
$745$		0			0
$746$			8953.67i			0.439433i
$747$	2695.71	−	403.876i	0.132036	−	0.0197819i
$748$	12091.3	+	12091.3i	0.591045	+	0.591045i
$749$	−219.048			−0.0106860
$750$		0			0
$751$	32325.6			1.57068			0.785339	−	0.619066i	$-0.212489\pi$
$751$	32325.6			1.57068			0.785339	+	0.619066i	$0.212489\pi$
$752$	−815.789	−	815.789i	−0.0395595	−	0.0395595i
$753$	27648.9	+	9111.68i	1.33809	+	0.440967i
$754$			24082.3i			1.16316i
$755$		0			0
$756$	−3052.29	−	522.770i	−0.146840	−	0.0251494i
$757$	−26700.7	+	26700.7i	−1.28197	+	1.28197i	−0.342430	+	0.939544i	$0.611250\pi$
$757$	−26700.7	+	26700.7i	−1.28197	+	1.28197i	−0.939544	+	0.342430i	$0.888750\pi$
$758$	4169.61	−	4169.61i	0.199798	−	0.199798i
$759$	13720.2	−	6918.68i	0.656143	−	0.330872i
$760$		0			0
$761$			29296.8i			1.39554i	0.716319	+	0.697772i	$0.245825\pi$
$761$			29296.8i			1.39554i	−0.716319	+	0.697772i	$0.754175\pi$
$762$	6025.88	−	18285.2i	0.286476	−	0.869295i
$763$	−2282.40	−	2282.40i	−0.108294	−	0.108294i
$764$	3459.31			0.163813
$765$		0			0
$766$	−2773.21			−0.130809
$767$	12543.9	+	12543.9i	0.590528	+	0.590528i
$768$	7092.76	−	21522.6i	0.333252	−	1.01124i
$769$			3977.36i			0.186511i	0.995642	+	0.0932556i	$0.0297274\pi$
$769$			3977.36i			0.186511i	−0.995642	+	0.0932556i	$0.970273\pi$
$770$		0			0
$771$	6434.44	−	3244.69i	0.300559	−	0.151562i
$772$	−9619.82	+	9619.82i	−0.448478	+	0.448478i
$773$	−21519.4	+	21519.4i	−1.00129	+	1.00129i	−0.00129452	+	0.999999i	$0.500412\pi$
$773$	−21519.4	+	21519.4i	−1.00129	+	1.00129i	−0.999999	+	0.00129452i	$0.999588\pi$
$774$	5998.55	−	8112.73i	0.278570	−	0.376752i
$775$		0			0
$776$			3025.04i			0.139939i
$777$	396.597	+	130.698i	0.0183112	+	0.00603446i
$778$	17799.8	+	17799.8i	0.820248	+	0.820248i
$779$	−5781.72			−0.265920
$780$		0			0
$781$	−67122.0			−3.07531
$782$	4488.45	+	4488.45i	0.205251	+	0.205251i
$783$	20522.4	−	14520.6i	0.936668	−	0.662736i
$784$			4495.80i			0.204801i
$785$		0			0
$786$	−408.588	−	810.259i	−0.0185418	−	0.0367697i
$787$	11206.4	−	11206.4i	0.507578	−	0.507578i	−0.406204	−	0.913782i	$-0.633148\pi$
$787$	11206.4	−	11206.4i	0.507578	−	0.507578i	0.913782	+	0.406204i	$0.133148\pi$
$788$	−1271.05	+	1271.05i	−0.0574610	+	0.0574610i
$789$	−4225.76	−	8379.97i	−0.190673	−	0.378118i
$790$		0			0
$791$		−	8742.77i		−	0.392993i
$792$	−5915.15	−	39481.3i	−0.265386	−	1.77135i
$793$	−11587.9	−	11587.9i	−0.518913	−	0.518913i
$794$	24635.8			1.10112
$795$		0			0
$796$	7220.42			0.321509
$797$	16231.3	+	16231.3i	0.721382	+	0.721382i	0.968887	−	0.247504i	$-0.0796104\pi$
$797$	16231.3	+	16231.3i	0.721382	+	0.721382i	−0.247504	+	0.968887i	$0.579610\pi$
$798$	4893.03	+	1612.49i	0.217057	+	0.0715309i
$799$			5425.20i			0.240212i
$800$		0			0
$801$	−4952.57	−	3661.93i	−0.218465	−	0.161533i
$802$	−5614.26	+	5614.26i	−0.247190	+	0.247190i
$803$	12876.4	−	12876.4i	0.565875	−	0.565875i
$804$	8626.86	−	4350.26i	0.378415	−	0.190823i
$805$		0			0
$806$		−	16600.8i		−	0.725482i
$807$	−2833.61	+	8598.43i	−0.123603	+	0.375067i
$808$	−25590.8	−	25590.8i	−1.11421	−	1.11421i
$809$	−10404.2			−0.452155			−0.226078	−	0.974109i	$-0.572590\pi$
$809$	−10404.2			−0.452155			−0.226078	+	0.974109i	$0.572590\pi$
$810$		0			0
$811$	6213.04			0.269013			0.134506	−	0.990913i	$-0.457055\pi$
$811$	6213.04			0.269013			0.134506	+	0.990913i	$0.457055\pi$
$812$	−2796.80	−	2796.80i	−0.120873	−	0.120873i
$813$	−3679.39	+	11164.9i	−0.158723	+	0.481637i
$814$			1825.45i			0.0786021i
$815$		0			0
$816$	4470.25	−	2254.21i	0.191777	−	0.0967071i
$817$	12507.1	−	12507.1i	0.535578	−	0.535578i
$818$	17178.3	−	17178.3i	0.734261	−	0.734261i
$819$	7949.61	+	5877.94i	0.339172	+	0.250784i
$820$		0			0
$821$			46193.4i			1.96365i	0.189776	+	0.981827i	$0.439224\pi$
$821$			46193.4i			1.96365i	−0.189776	+	0.981827i	$0.560776\pi$
$822$	20538.3	+	6768.39i	0.871479	+	0.287195i
$823$	−24122.0	−	24122.0i	−1.02168	−	1.02168i	−0.999760	−	0.0219179i	$-0.993023\pi$
$823$	−24122.0	−	24122.0i	−1.02168	−	1.02168i	−0.0219179	−	0.999760i	$-0.506977\pi$
$824$	2399.10			0.101428
$825$		0			0
$826$	2764.97			0.116472
$827$	25996.5	+	25996.5i	1.09309	+	1.09309i	0.995197	+	0.0978962i	$0.0312113\pi$
$827$	25996.5	+	25996.5i	1.09309	+	1.09309i	0.0978962	+	0.995197i	$0.468789\pi$
$828$	784.609	+	5236.95i	0.0329312	+	0.219803i
$829$			45991.6i			1.92684i	0.267988	+	0.963422i	$0.413641\pi$
$829$			45991.6i			1.92684i	−0.267988	+	0.963422i	$0.586359\pi$
$830$		0			0
$831$	−12579.4	−	24945.8i	−0.525119	−	1.04135i
$832$	−20991.7	+	20991.7i	−0.874706	+	0.874706i
$833$	14949.1	−	14949.1i	0.621795	−	0.621795i
$834$	−9803.37	−	19440.8i	−0.407030	−	0.807168i
$835$		0			0
$836$		−	23732.1i		−	0.981810i
$837$	−14146.8	+	10009.5i	−0.584213	+	0.413358i
$838$	4292.74	+	4292.74i	0.176957	+	0.176957i
$839$	−29393.2			−1.20949			−0.604747	−	0.796418i	$-0.706726\pi$
$839$	−29393.2			−1.20949			−0.604747	+	0.796418i	$0.706726\pi$
$840$		0			0
$841$	7720.75			0.316567
$842$	−16823.7	−	16823.7i	−0.688579	−	0.688579i
$843$	18442.9	+	6077.86i	0.753510	+	0.248319i
$844$			5519.26i			0.225096i
$845$		0			0
$846$	2553.52	−	3453.51i	0.103773	−	0.140348i
$847$	−9503.89	+	9503.89i	−0.385546	+	0.385546i
$848$	−3360.23	+	3360.23i	−0.136074	+	0.136074i
$849$	−3605.36	+	1818.07i	−0.145743	+	0.0734936i
$850$		0			0
$851$		−	714.055i		−	0.0287632i
$852$	7239.97	−	21969.3i	0.291123	−	0.883398i
$853$	−12249.5	−	12249.5i	−0.491693	−	0.491693i	0.417146	−	0.908839i	$-0.363030\pi$
$853$	−12249.5	−	12249.5i	−0.491693	−	0.491693i	−0.908839	+	0.417146i	$0.863030\pi$
$854$	−2554.23			−0.102347
$855$		0			0
$856$	973.164			0.0388576
$857$	−18196.1	−	18196.1i	−0.725282	−	0.725282i	0.244394	−	0.969676i	$-0.421411\pi$
$857$	−18196.1	−	18196.1i	−0.725282	−	0.725282i	−0.969676	+	0.244394i	$0.921411\pi$
$858$	−13527.5	+	41048.4i	−0.538253	+	1.63330i
$859$		−	7995.99i		−	0.317602i	−0.987311	−	0.158801i	$-0.949237\pi$
$859$		−	7995.99i		−	0.317602i	0.987311	−	0.158801i	$-0.0507628\pi$
$860$		0			0
$861$	−1544.13	+	778.658i	−0.0611195	+	0.0308207i
$862$	8744.62	−	8744.62i	0.345525	−	0.345525i
$863$	−23615.8	+	23615.8i	−0.931509	+	0.931509i	−0.997800	−	0.0662917i	$-0.978883\pi$
$863$	−23615.8	+	23615.8i	−0.931509	+	0.931509i	0.0662917	+	0.997800i	$0.478883\pi$
$864$	22522.5	+	3857.46i	0.886840	+	0.151890i
$865$		0			0
$866$			22860.3i			0.897024i
$867$	1886.41	+	621.665i	0.0738937	+	0.0243516i
$868$	1927.94	+	1927.94i	0.0753899	+	0.0753899i
$869$	−41888.3			−1.63517
$870$		0			0
$871$	−30845.9			−1.19997
$872$	10140.0	+	10140.0i	0.393789	+	0.393789i
$873$	−3381.03	+	506.552i	−0.131077	+	0.0196383i
$874$		−	8809.68i		−	0.340952i
$875$		0			0
$876$	2825.61	+	5603.37i	0.108982	+	0.216119i
$877$	−27239.2	+	27239.2i	−1.04880	+	1.04880i	−0.0500573	+	0.998746i	$0.515940\pi$
$877$	−27239.2	+	27239.2i	−1.04880	+	1.04880i	−0.998746	+	0.0500573i	$0.984060\pi$
$878$	−11855.2	+	11855.2i	−0.455688	+	0.455688i
$879$	−18378.6	−	36446.1i	−0.705228	−	1.39852i
$880$		0			0
$881$		−	43322.3i		−	1.65671i	−0.560202	−	0.828356i	$-0.689276\pi$
$881$		−	43322.3i		−	1.65671i	0.560202	−	0.828356i	$-0.310724\pi$
$882$	−16552.3	+	2479.90i	−0.631911	+	0.0946740i
$883$	−27549.3	−	27549.3i	−1.04995	−	1.04995i	−0.998685	−	0.0512690i	$-0.983673\pi$
$883$	−27549.3	−	27549.3i	−1.04995	−	1.04995i	−0.0512690	−	0.998685i	$-0.516327\pi$
$884$	18813.7			0.715806
$885$		0			0
$886$	25340.2			0.960858
$887$	−5088.76	−	5088.76i	−0.192631	−	0.192631i	0.604201	−	0.796832i	$-0.293492\pi$
$887$	−5088.76	−	5088.76i	−0.192631	−	0.192631i	−0.796832	+	0.604201i	$0.793492\pi$
$888$	−1761.96	−	580.652i	−0.0665849	−	0.0219430i
$889$		−	10095.2i		−	0.380856i
$890$		0			0
$891$	43137.0	−	13222.5i	1.62194	−	0.497162i
$892$	12915.2	−	12915.2i	0.484789	−	0.484789i
$893$	5324.14	−	5324.14i	0.199513	−	0.199513i
$894$	−11334.6	+	5715.70i	−0.424034	+	0.213827i
$895$		0			0
$896$		−	2379.71i		−	0.0887284i
$897$	5291.49	−	16056.7i	0.196965	−	0.597680i
$898$	−6814.70	−	6814.70i	−0.253240	−	0.253240i
$899$	−22134.4			−0.821161
$900$		0			0
$901$	22346.4			0.826265
$902$	−5345.67	−	5345.67i	−0.197330	−	0.197330i
$903$	1655.88	−	5024.69i	0.0610236	−	0.185173i
$904$			38841.5i			1.42903i
$905$		0			0
$906$	21649.7	−	10917.3i	0.793889	−	0.400334i
$907$	38018.9	−	38018.9i	1.39184	−	1.39184i	0.570627	−	0.821210i	$-0.306700\pi$
$907$	38018.9	−	38018.9i	1.39184	−	1.39184i	0.821210	−	0.570627i	$-0.193300\pi$
$908$	−18174.9	+	18174.9i	−0.664268	+	0.664268i
$909$	24317.1	−	32887.6i	0.887291	−	1.20001i
$910$		0			0
$911$		−	31712.0i		−	1.15331i	−0.816987	−	0.576656i	$-0.804357\pi$
$911$		−	31712.0i		−	1.15331i	0.816987	−	0.576656i	$-0.195643\pi$
$912$	−6599.19	−	2174.76i	−0.239606	−	0.0789622i
$913$	4418.12	+	4418.12i	0.160152	+	0.160152i
$914$	14184.4			0.513323
$915$		0			0
$916$	8452.72			0.304897
$917$	−336.459	−	336.459i	−0.0121165	−	0.0121165i
$918$	10765.2	+	15214.8i	0.387041	+	0.547019i
$919$		−	3339.87i		−	0.119883i	−0.998202	−	0.0599413i	$-0.980909\pi$
$919$		−	3339.87i		−	0.119883i	0.998202	−	0.0599413i	$-0.0190913\pi$
$920$		0			0
$921$	17240.5	+	34189.1i	0.616823	+	1.22320i
$922$	−24085.3	+	24085.3i	−0.860311	+	0.860311i
$923$	−52219.8	+	52219.8i	−1.86223	+	1.86223i
$924$	−3196.15	−	6338.18i	−0.113794	−	0.225661i
$925$		0			0
$926$		−	7031.44i		−	0.249533i
$927$	401.738	+	2681.44i	0.0142339	+	0.0950054i
$928$	20637.3	+	20637.3i	0.730012	+	0.730012i
$929$	27741.7			0.979736			0.489868	−	0.871797i	$-0.337045\pi$
$929$	27741.7			0.979736			0.489868	+	0.871797i	$0.337045\pi$
$930$		0			0
$931$	−29341.2			−1.03289
$932$	−9451.01	−	9451.01i	−0.332165	−	0.332165i
$933$	13410.4	+	4419.41i	0.470566	+	0.155075i
$934$		−	11638.5i		−	0.407734i
$935$		0			0
$936$	−35317.7	−	26113.9i	−1.23333	−	0.911923i
$937$	−1426.31	+	1426.31i	−0.0497284	+	0.0497284i	−0.731534	−	0.681805i	$-0.761195\pi$
$937$	−1426.31	+	1426.31i	−0.0497284	+	0.0497284i	0.681805	+	0.731534i	$0.261195\pi$
$938$	−3399.57	+	3399.57i	−0.118337	+	0.118337i
$939$	−40726.1	+	20536.9i	−1.41538	+	0.713734i
$940$		0			0
$941$			16728.8i			0.579535i	0.957097	+	0.289768i	$0.0935780\pi$
$941$			16728.8i			0.579535i	−0.957097	+	0.289768i	$0.906422\pi$
$942$	3218.57	−	9766.57i	0.111323	−	0.337805i
$943$	2091.04	+	2091.04i	0.0722096	+	0.0722096i
$944$	−3729.09			−0.128572
$945$		0			0
$946$	23127.6			0.794866
$947$	173.759	+	173.759i	0.00596243	+	0.00596243i	0.710082	−	0.704119i	$-0.248658\pi$
$947$	173.759	+	173.759i	0.00596243	+	0.00596243i	−0.704119	+	0.710082i	$0.748658\pi$
$948$	4518.19	−	13710.2i	0.154793	−	0.469712i
$949$		−	20035.2i		−	0.685323i
$950$		0			0
$951$	−25653.8	+	12936.4i	−0.874744	+	0.441106i
$952$	6114.68	−	6114.68i	0.208170	−	0.208170i
$953$	20457.0	−	20457.0i	0.695350	−	0.695350i	−0.268054	−	0.963404i	$-0.586380\pi$
$953$	20457.0	−	20457.0i	0.695350	−	0.695350i	0.963404	+	0.268054i	$0.0863805\pi$
$954$	−14225.0	−	10517.9i	−0.482758	−	0.356951i
$955$		0			0
$956$		−	10093.7i		−	0.341480i
$957$	54731.2	+	18036.6i	1.84870	+	0.609239i
$958$	−5638.28	−	5638.28i	−0.190151	−	0.190151i
$959$	11339.1			0.381812
$960$		0			0
$961$	−14533.0			−0.487831
$962$	1420.17	+	1420.17i	0.0475969	+	0.0475969i
$963$	162.960	+	1087.69i	0.00545306	+	0.0363970i
$964$		−	18311.3i		−	0.611793i
$965$		0			0
$966$	−1186.45	−	2352.81i	−0.0395170	−	0.0783650i
$967$	−22650.2	+	22650.2i	−0.753237	+	0.753237i	−0.975082	−	0.221845i	$-0.928792\pi$
$967$	−22650.2	+	22650.2i	−0.753237	+	0.753237i	0.221845	+	0.975082i	$0.428792\pi$
$968$	42222.9	−	42222.9i	1.40196	−	1.40196i
$969$	14711.8	+	29174.5i	0.487730	+	0.967202i
$970$		0			0
$971$		−	41716.5i		−	1.37873i	−0.724415	−	0.689365i	$-0.757890\pi$
$971$		−	41716.5i		−	1.37873i	0.724415	−	0.689365i	$-0.242110\pi$
$972$	−325.096	+	15545.1i	−0.0107278	+	0.512974i
$973$	−8072.76	−	8072.76i	−0.265982	−	0.265982i
$974$	15472.7			0.509012
$975$		0			0
$976$	3444.88			0.112979
$977$	29670.6	+	29670.6i	0.971595	+	0.971595i	0.999608	−	0.0280129i	$-0.00891795\pi$
$977$	29670.6	+	29670.6i	0.971595	+	0.971595i	−0.0280129	+	0.999608i	$0.508918\pi$
$978$	−26305.3	−	8668.91i	−0.860074	−	0.283437i
$979$		−	14118.7i		−	0.460914i
$980$		0			0
$981$	−9635.34	+	13031.3i	−0.313591	+	0.424116i
$982$	13926.3	−	13926.3i	0.452553	−	0.452553i
$983$	14537.3	−	14537.3i	0.471686	−	0.471686i	−0.430774	−	0.902460i	$-0.641759\pi$
$983$	14537.3	−	14537.3i	0.471686	−	0.471686i	0.902460	+	0.430774i	$0.141759\pi$
$984$	6860.11	−	3459.34i	0.222248	−	0.112073i
$985$		0			0
$986$			23805.4i			0.768882i
$987$	704.893	−	2138.96i	0.0227325	−	0.0689806i
$988$	−18463.2	−	18463.2i	−0.594528	−	0.594528i
$989$	−9046.73			−0.290869
$990$		0			0
$991$	2344.55			0.0751536			0.0375768	−	0.999294i	$-0.488036\pi$
$991$	2344.55			0.0751536			0.0375768	+	0.999294i	$0.488036\pi$
$992$	−14226.0	−	14226.0i	−0.455319	−	0.455319i
$993$	−5419.27	+	16444.5i	−0.173188	+	0.525529i
$994$			11510.4i			0.367292i
$995$		0			0
$996$	−1922.62	+	969.517i	−0.0611652	+	0.0308437i
$997$	24252.2	−	24252.2i	0.770385	−	0.770385i	−0.207789	−	0.978174i	$-0.566627\pi$
$997$	24252.2	−	24252.2i	0.770385	−	0.770385i	0.978174	+	0.207789i	$0.0666268\pi$
$998$	1931.45	−	1931.45i	0.0612615	−	0.0612615i
$999$	353.940	−	2066.54i	0.0112094	−	0.0654479i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.e.d.68.4	yes	16
3.2	odd	2	inner	75.4.e.d.68.6	yes	16
5.2	odd	4	inner	75.4.e.d.32.6	yes	16
5.3	odd	4	inner	75.4.e.d.32.3	✓	16
5.4	even	2	inner	75.4.e.d.68.5	yes	16
15.2	even	4	inner	75.4.e.d.32.4	yes	16
15.8	even	4	inner	75.4.e.d.32.5	yes	16
15.14	odd	2	inner	75.4.e.d.68.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.e.d.32.3	✓	16	5.3	odd	4	inner
75.4.e.d.32.4	yes	16	15.2	even	4	inner
75.4.e.d.32.5	yes	16	15.8	even	4	inner
75.4.e.d.32.6	yes	16	5.2	odd	4	inner
75.4.e.d.68.3	yes	16	15.14	odd	2	inner
75.4.e.d.68.4	yes	16	1.1	even	1	trivial
75.4.e.d.68.5	yes	16	5.4	even	2	inner
75.4.e.d.68.6	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.39558 - 1.39558i) q^{2} +(-1.62635 + 4.93508i) q^{3} -4.10469i q^{4} +(9.15703 - 4.61761i) q^{6} +(3.80245 - 3.80245i) q^{7} +(-16.8931 + 16.8931i) q^{8} +(-21.7100 - 16.0523i) q^{9} +O(q^{10})\)	\(q+(-1.39558 - 1.39558i) q^{2} +(-1.62635 + 4.93508i) q^{3} -4.10469i q^{4} +(9.15703 - 4.61761i) q^{6} +(3.80245 - 3.80245i) q^{7} +(-16.8931 + 16.8931i) q^{8} +(-21.7100 - 16.0523i) q^{9} -61.8903i q^{11} +(20.2569 + 6.67567i) q^{12} +(-48.1497 - 48.1497i) q^{13} -10.6133 q^{14} +14.3141 q^{16} +(-47.5960 - 47.5960i) q^{17} +(7.89568 + 52.7005i) q^{18} +93.4187i q^{19} +(12.5813 + 24.9495i) q^{21} +(-86.3732 + 86.3732i) q^{22} +(33.7862 - 33.7862i) q^{23} +(-55.8947 - 110.843i) q^{24} +134.394i q^{26} +(114.528 - 81.0335i) q^{27} +(-15.6079 - 15.6079i) q^{28} +179.192 q^{29} -123.523 q^{31} +(115.168 + 115.168i) q^{32} +(305.434 + 100.655i) q^{33} +132.848i q^{34} +(-65.8898 + 89.1126i) q^{36} +(10.5672 - 10.5672i) q^{37} +(130.374 - 130.374i) q^{38} +(315.931 - 159.314i) q^{39} +61.8903i q^{41} +(17.2609 - 52.3773i) q^{42} +(-133.882 - 133.882i) q^{43} -254.040 q^{44} -94.3031 q^{46} +(-56.9922 - 56.9922i) q^{47} +(-23.2797 + 70.6410i) q^{48} +314.083i q^{49} +(312.298 - 157.482i) q^{51} +(-197.639 + 197.639i) q^{52} +(-234.750 + 234.750i) q^{53} +(-272.922 - 46.7438i) q^{54} +128.470i q^{56} +(-461.029 - 151.932i) q^{57} +(-250.078 - 250.078i) q^{58} -260.519 q^{59} +240.664 q^{61} +(172.387 + 172.387i) q^{62} +(-143.589 + 21.5128i) q^{63} -435.967i q^{64} +(-285.785 - 566.732i) q^{66} +(320.313 - 320.313i) q^{67} +(-195.367 + 195.367i) q^{68} +(111.789 + 221.686i) q^{69} -1084.53i q^{71} +(637.923 - 95.5747i) q^{72} +(208.052 + 208.052i) q^{73} -29.4950 q^{74} +383.455 q^{76} +(-235.335 - 235.335i) q^{77} +(-663.244 - 218.572i) q^{78} -676.816i q^{79} +(213.645 + 696.991i) q^{81} +(86.3732 - 86.3732i) q^{82} +(-71.3863 + 71.3863i) q^{83} +(102.410 - 51.6421i) q^{84} +373.687i q^{86} +(-291.429 + 884.326i) q^{87} +(1045.52 + 1045.52i) q^{88} +228.124 q^{89} -366.173 q^{91} +(-138.682 - 138.682i) q^{92} +(200.893 - 609.598i) q^{93} +159.075i q^{94} +(-755.670 + 381.061i) q^{96} +(89.5346 - 89.5346i) q^{97} +(438.329 - 438.329i) q^{98} +(-993.485 + 1343.64i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 84 q^{6}+O(q^{10}) \) 16 * q - 84 * q^6	\( 16 q - 84 q^{6} - 232 q^{16} + 816 q^{21} - 1208 q^{31} + 252 q^{36} + 1872 q^{46} + 156 q^{51} - 1528 q^{61} - 3420 q^{66} + 1064 q^{76} + 6876 q^{81} - 10008 q^{91} - 8172 q^{96}+O(q^{100}) \) 16 * q - 84 * q^6 - 232 * q^16 + 816 * q^21 - 1208 * q^31 + 252 * q^36 + 1872 * q^46 + 156 * q^51 - 1528 * q^61 - 3420 * q^66 + 1064 * q^76 + 6876 * q^81 - 10008 * q^91 - 8172 * q^96

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