LMFDB - Embedded newform 378.4.k.b.269.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} - 86 x^{14} + 5225 x^{12} - 158916 x^{10} + 3517046 x^{8} - 29955345 x^{6} + 190411550 x^{4} + \cdots + 1500625 $ x^16 - 86x^14 + 5225x^12 - 158916x^10 + 3517046x^8 - 29955345x^6 + 190411550x^4 - 17021375*x^2 + 1500625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.6
Root			$-5.68309 + 3.28113i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.b.215.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-1.09136 - 1.89028i) q^{5} +(-9.39922 + 15.9579i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-1.09136 - 1.89028i) q^{5} +(-9.39922 + 15.9579i) q^{7} -8.00000i q^{8} +(-3.78057 - 2.18271i) q^{10} +(17.3389 + 10.0106i) q^{11} -46.7045i q^{13} +(-0.322020 + 37.0391i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(56.6046 - 98.0420i) q^{17} +(76.4791 - 44.1553i) q^{19} -8.73084 q^{20} +40.0425 q^{22} +(12.5199 - 7.22837i) q^{23} +(60.1179 - 104.127i) q^{25} +(-46.7045 - 80.8945i) q^{26} +(36.4814 + 64.4757i) q^{28} -16.2305i q^{29} +(-77.9934 - 45.0295i) q^{31} +(-27.7128 - 16.0000i) q^{32} -226.418i q^{34} +(40.4228 + 0.351438i) q^{35} +(-83.7622 - 145.080i) q^{37} +(88.3105 - 152.958i) q^{38} +(-15.1223 + 8.73084i) q^{40} +92.1826 q^{41} -159.007 q^{43} +(69.3556 - 40.0425i) q^{44} +(14.4567 - 25.0398i) q^{46} +(182.310 + 315.771i) q^{47} +(-166.309 - 299.984i) q^{49} -240.472i q^{50} +(-161.789 - 93.4090i) q^{52} +(-210.389 - 121.468i) q^{53} -43.7006i q^{55} +(127.663 + 75.1937i) q^{56} +(-16.2305 - 28.1121i) q^{58} +(-257.947 + 446.777i) q^{59} +(543.582 - 313.837i) q^{61} -180.118 q^{62} -64.0000 q^{64} +(-88.2847 + 50.9712i) q^{65} +(430.357 - 745.400i) q^{67} +(-226.418 - 392.168i) q^{68} +(70.3659 - 39.8141i) q^{70} +534.461i q^{71} +(451.186 + 260.492i) q^{73} +(-290.161 - 167.524i) q^{74} -353.242i q^{76} +(-322.721 + 182.601i) q^{77} +(-525.897 - 910.881i) q^{79} +(-17.4617 + 30.2445i) q^{80} +(159.665 - 92.1826i) q^{82} +1472.54 q^{83} -247.103 q^{85} +(-275.408 + 159.007i) q^{86} +(80.0850 - 138.711i) q^{88} +(233.582 + 404.576i) q^{89} +(745.306 + 438.986i) q^{91} -57.8270i q^{92} +(631.542 + 364.621i) q^{94} +(-166.932 - 96.3782i) q^{95} -411.573i q^{97} +(-588.040 - 353.278i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 52 q^{7}+O(q^{10}) $ 16 * q + 32 * q^4 - 52 * q^7	$ 16 q + 32 q^{4} - 52 q^{7} - 24 q^{10} - 128 q^{16} - 186 q^{19} - 216 q^{22} - 44 q^{25} - 200 q^{28} + 408 q^{31} - 704 q^{37} - 96 q^{40} - 2036 q^{43} + 144 q^{46} - 20 q^{49} + 480 q^{52} - 672 q^{58} - 1242 q^{61} - 1024 q^{64} + 596 q^{67} - 48 q^{70} + 852 q^{73} - 2914 q^{79} + 1344 q^{82} - 10980 q^{85} - 432 q^{88} + 4134 q^{91} - 492 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 - 52 * q^7 - 24 * q^10 - 128 * q^16 - 186 * q^19 - 216 * q^22 - 44 * q^25 - 200 * q^28 + 408 * q^31 - 704 * q^37 - 96 * q^40 - 2036 * q^43 + 144 * q^46 - 20 * q^49 + 480 * q^52 - 672 * q^58 - 1242 * q^61 - 1024 * q^64 + 596 * q^67 - 48 * q^70 + 852 * q^73 - 2914 * q^79 + 1344 * q^82 - 10980 * q^85 - 432 * q^88 + 4134 * q^91 - 492 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$2$	1.73205	−	1.00000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−1.09136	−	1.89028i	−0.0976138	−	0.169072i	0.813083	−	0.582148i	$-0.197788\pi$
$5$	−1.09136	−	1.89028i	−0.0976138	−	0.169072i	−0.910696	+	0.413076i	$0.864454\pi$
$6$		0			0
$7$	−9.39922	+	15.9579i	−0.507510	+	0.861646i
$7$	−9.39922	+	15.9579i	−0.507510	+	0.861646i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$	−3.78057	−	2.18271i	−0.119552	−	0.0690234i
$11$	17.3389	+	10.0106i	0.475262	+	0.274392i	0.718440	−	0.695589i	$-0.244857\pi$
$11$	17.3389	+	10.0106i	0.475262	+	0.274392i	−0.243178	+	0.969982i	$0.578190\pi$
$12$		0			0
$13$		−	46.7045i		−	0.996423i	−0.867056	−	0.498211i	$-0.833990\pi$
$13$		−	46.7045i		−	0.996423i	0.867056	−	0.498211i	$-0.166010\pi$
$14$	−0.322020	+	37.0391i	−0.00614739	+	0.707080i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	56.6046	−	98.0420i	0.807566	−	1.39875i	−0.106979	−	0.994261i	$-0.534118\pi$
$17$	56.6046	−	98.0420i	0.807566	−	1.39875i	0.914545	−	0.404484i	$-0.132549\pi$
$18$		0			0
$19$	76.4791	−	44.1553i	0.923448	−	0.533153i	0.0387148	−	0.999250i	$-0.487674\pi$
$19$	76.4791	−	44.1553i	0.923448	−	0.533153i	0.884734	+	0.466097i	$0.154340\pi$
$20$	−8.73084			−0.0976138
$21$		0			0
$22$	40.0425			0.388049
$23$	12.5199	−	7.22837i	0.113503	−	0.0655313i	−0.442174	−	0.896930i	$-0.645792\pi$
$23$	12.5199	−	7.22837i	0.113503	−	0.0655313i	0.555677	+	0.831398i	$0.312459\pi$
$24$		0			0
$25$	60.1179	−	104.127i	0.480943	−	0.833018i
$26$	−46.7045	−	80.8945i	−0.352289	−	0.610182i
$27$		0			0
$28$	36.4814	+	64.4757i	0.246226	+	0.435170i
$29$		−	16.2305i		−	0.103929i	−0.998649	−	0.0519644i	$-0.983452\pi$
$29$		−	16.2305i		−	0.103929i	0.998649	−	0.0519644i	$-0.0165482\pi$
$30$		0			0
$31$	−77.9934	−	45.0295i	−0.451872	−	0.260888i	0.256749	−	0.966478i	$-0.417349\pi$
$31$	−77.9934	−	45.0295i	−0.451872	−	0.260888i	−0.708620	+	0.705590i	$0.750682\pi$
$32$	−27.7128	−	16.0000i	−0.153093	−	0.0883883i
$33$		0			0
$34$		−	226.418i		−	1.14207i
$35$	40.4228	+	0.351438i	0.195220	+	0.00169725i
$36$		0			0
$37$	−83.7622	−	145.080i	−0.372173	−	0.644623i	0.617726	−	0.786393i	$-0.288054\pi$
$37$	−83.7622	−	145.080i	−0.372173	−	0.644623i	−0.989900	+	0.141770i	$0.954721\pi$
$38$	88.3105	−	152.958i	0.376996	−	0.652977i
$39$		0			0
$40$	−15.1223	+	8.73084i	−0.0597760	+	0.0345117i
$41$	92.1826			0.351134			0.175567	−	0.984467i	$-0.443824\pi$
$41$	92.1826			0.351134			0.175567	+	0.984467i	$0.443824\pi$
$42$		0			0
$43$	−159.007			−0.563915			−0.281958	−	0.959427i	$-0.590984\pi$
$43$	−159.007			−0.563915			−0.281958	+	0.959427i	$0.590984\pi$
$44$	69.3556	−	40.0425i	0.237631	−	0.137196i
$45$		0			0
$46$	14.4567	−	25.0398i	0.0463376	−	0.0802591i
$47$	182.310	+	315.771i	0.565802	+	0.979998i	0.996975	+	0.0777287i	$0.0247668\pi$
$47$	182.310	+	315.771i	0.565802	+	0.979998i	−0.431172	+	0.902270i	$0.641900\pi$
$48$		0			0
$49$	−166.309	−	299.984i	−0.484867	−	0.874588i
$50$		−	240.472i		−	0.680156i
$51$		0			0
$52$	−161.789	−	93.4090i	−0.431464	−	0.249106i
$53$	−210.389	−	121.468i	−0.545268	−	0.314810i	0.201944	−	0.979397i	$-0.435274\pi$
$53$	−210.389	−	121.468i	−0.545268	−	0.314810i	−0.747211	+	0.664587i	$0.768608\pi$
$54$		0			0
$55$		−	43.7006i		−	0.107138i
$56$	127.663	+	75.1937i	0.304638	+	0.179432i
$57$		0			0
$58$	−16.2305	−	28.1121i	−0.0367444	−	0.0636432i
$59$	−257.947	+	446.777i	−0.569183	+	0.985854i	0.427464	+	0.904032i	$0.359407\pi$
$59$	−257.947	+	446.777i	−0.569183	+	0.985854i	−0.996647	+	0.0818216i	$0.973926\pi$
$60$		0			0
$61$	543.582	−	313.837i	1.14096	−	0.658733i	0.194292	−	0.980944i	$-0.437759\pi$
$61$	543.582	−	313.837i	1.14096	−	0.658733i	0.946668	+	0.322210i	$0.104426\pi$
$62$	−180.118			−0.368952
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−88.2847	+	50.9712i	−0.168467	+	0.0972646i
$66$		0			0
$67$	430.357	−	745.400i	0.784723	−	1.35918i	−0.144442	−	0.989513i	$-0.546139\pi$
$67$	430.357	−	745.400i	0.784723	−	1.35918i	0.929164	−	0.369667i	$-0.120528\pi$
$68$	−226.418	−	392.168i	−0.403783	−	0.699373i
$69$		0			0
$70$	70.3659	−	39.8141i	0.120148	−	0.0679814i
$71$			534.461i			0.893364i	0.894693	+	0.446682i	$0.147394\pi$
$71$			534.461i			0.893364i	−0.894693	+	0.446682i	$0.852606\pi$
$72$		0			0
$73$	451.186	+	260.492i	0.723387	+	0.417648i	0.815998	−	0.578055i	$-0.196188\pi$
$73$	451.186	+	260.492i	0.723387	+	0.417648i	−0.0926108	+	0.995702i	$0.529521\pi$
$74$	−290.161	−	167.524i	−0.455817	−	0.263166i
$75$		0			0
$76$		−	353.242i		−	0.533153i
$77$	−322.721	+	182.601i	−0.477629	+	0.270250i
$78$		0			0
$79$	−525.897	−	910.881i	−0.748963	−	1.29724i	−0.948320	−	0.317315i	$-0.897219\pi$
$79$	−525.897	−	910.881i	−0.748963	−	1.29724i	0.199358	−	0.979927i	$-0.436114\pi$
$80$	−17.4617	+	30.2445i	−0.0244034	+	0.0422680i
$81$		0			0
$82$	159.665	−	92.1826i	0.215025	−	0.124145i
$83$	1472.54			1.94738			0.973689	−	0.227880i	$-0.0731793\pi$
$83$	1472.54			1.94738			0.973689	+	0.227880i	$0.0731793\pi$
$84$		0			0
$85$	−247.103			−0.315318
$86$	−275.408	+	159.007i	−0.345326	+	0.199374i
$87$		0			0
$88$	80.0850	−	138.711i	0.0970124	−	0.168030i
$89$	233.582	+	404.576i	0.278199	+	0.481854i	0.970937	−	0.239335i	$-0.0769294\pi$
$89$	233.582	+	404.576i	0.278199	+	0.481854i	−0.692739	+	0.721189i	$0.743596\pi$
$90$		0			0
$91$	745.306	+	438.986i	0.858563	+	0.505695i
$92$		−	57.8270i		−	0.0655313i
$93$		0			0
$94$	631.542	+	364.621i	0.692963	+	0.400083i
$95$	−166.932	−	96.3782i	−0.180283	−	0.104086i
$96$		0			0
$97$		−	411.573i		−	0.430813i	−0.976524	−	0.215407i	$-0.930892\pi$
$97$		−	411.573i		−	0.430813i	0.976524	−	0.215407i	$-0.0691077\pi$
$98$	−588.040	−	353.278i	−0.606133	−	0.364147i
$99$		0			0
$100$	−240.472	−	416.509i	−0.240472	−	0.416509i
$101$	−707.575	+	1225.56i	−0.697092	+	1.20740i	0.272378	+	0.962190i	$0.412190\pi$
$101$	−707.575	+	1225.56i	−0.697092	+	1.20740i	−0.969470	+	0.245209i	$0.921144\pi$
$102$		0			0
$103$	−797.914	+	460.676i	−0.763309	+	0.440696i	−0.830482	−	0.557045i	$-0.811935\pi$
$103$	−797.914	+	460.676i	−0.763309	+	0.440696i	0.0671738	+	0.997741i	$0.478602\pi$
$104$	−373.636			−0.352289
$105$		0			0
$106$	−485.873			−0.445209
$107$	−485.609	+	280.367i	−0.438744	+	0.253309i	−0.703065	−	0.711126i	$-0.748186\pi$
$107$	−485.609	+	280.367i	−0.438744	+	0.253309i	0.264321	+	0.964435i	$0.414852\pi$
$108$		0			0
$109$	−506.720	+	877.664i	−0.445275	+	0.771238i	−0.998071	−	0.0620778i	$-0.980227\pi$
$109$	−506.720	+	877.664i	−0.445275	+	0.771238i	0.552797	+	0.833316i	$0.313561\pi$
$110$	−43.7006	−	75.6916i	−0.0378790	−	0.0656083i
$111$		0			0
$112$	296.313	+	2.57616i	0.249991	+	0.00217343i
$113$		−	995.485i		−	0.828738i	−0.910109	−	0.414369i	$-0.864002\pi$
$113$		−	995.485i		−	0.828738i	0.910109	−	0.414369i	$-0.135998\pi$
$114$		0			0
$115$	−27.3273	−	15.7774i	−0.0221590	−	0.0127935i
$116$	−56.2243	−	32.4611i	−0.0450025	−	0.0259822i
$117$		0			0
$118$			1031.79i			0.804946i
$119$	1032.51	+	1824.81i	0.795375	+	1.40571i
$120$		0			0
$121$	−465.075	−	805.533i	−0.349418	−	0.605209i
$122$	627.674	−	1087.16i	0.465795	−	0.806780i
$123$		0			0
$124$	−311.973	+	180.118i	−0.225936	+	0.130444i
$125$	−535.279			−0.383014
$126$		0			0
$127$	319.850			0.223481			0.111740	−	0.993737i	$-0.464357\pi$
$127$	319.850			0.223481			0.111740	+	0.993737i	$0.464357\pi$
$128$	−110.851	+	64.0000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−101.942	+	176.569i	−0.0687765	+	0.119124i
$131$	−861.833	−	1492.74i	−0.574799	−	0.995582i	−0.996063	−	0.0886438i	$-0.971747\pi$
$131$	−861.833	−	1492.74i	−0.574799	−	0.995582i	0.421264	−	0.906938i	$-0.361587\pi$
$132$		0			0
$133$	−14.2189	+	1635.47i	−0.00927017	+	1.06627i
$134$		−	1721.43i		−	1.10977i
$135$		0			0
$136$	−784.336	−	452.836i	−0.494531	−	0.285518i
$137$	2670.20	+	1541.64i	1.66518	+	0.961395i	0.970179	+	0.242390i	$0.0779315\pi$
$137$	2670.20	+	1541.64i	1.66518	+	0.961395i	0.695006	+	0.719004i	$0.255402\pi$
$138$		0			0
$139$			1647.06i			1.00505i	0.864562	+	0.502526i	$0.167596\pi$
$139$			1647.06i			1.00505i	−0.864562	+	0.502526i	$0.832404\pi$
$140$	82.0631	−	139.326i	0.0495400	−	0.0841085i
$141$		0			0
$142$	534.461	+	925.713i	0.315852	+	0.547071i
$143$	467.541	−	809.805i	0.273411	−	0.473561i
$144$		0			0
$145$	−30.6803	+	17.7133i	−0.0175715	+	0.0101449i
$146$	1041.97			0.590643
$147$		0			0
$148$	−670.097			−0.372173
$149$	−1374.06	+	793.314i	−0.755486	+	0.436180i	−0.827673	−	0.561211i	$-0.810336\pi$
$149$	−1374.06	+	793.314i	−0.755486	+	0.436180i	0.0721867	+	0.997391i	$0.477002\pi$
$150$		0			0
$151$	−806.312	+	1396.57i	−0.434548	+	0.752659i	−0.997259	−	0.0739948i	$-0.976425\pi$
$151$	−806.312	+	1396.57i	−0.434548	+	0.752659i	0.562711	+	0.826654i	$0.309759\pi$
$152$	−353.242	−	611.833i	−0.188498	−	0.326488i
$153$		0			0
$154$	−376.368	+	638.994i	−0.196939	+	0.334361i
$155$			196.573i			0.101865i
$156$		0			0
$157$	−241.423	−	139.386i	−0.122724	−	0.0708547i	0.437381	−	0.899276i	$-0.355906\pi$
$157$	−241.423	−	139.386i	−0.122724	−	0.0708547i	−0.560105	+	0.828421i	$0.689239\pi$
$158$	−1821.76	−	1051.79i	−0.917288	−	0.529597i
$159$		0			0
$160$			69.8467i			0.0345117i
$161$	−2.32768	+	267.732i	−0.00113942	+	0.131058i
$162$		0			0
$163$	758.314	+	1313.44i	0.364391	+	0.631144i	0.988678	−	0.150051i	$-0.0479438\pi$
$163$	758.314	+	1313.44i	0.364391	+	0.631144i	−0.624287	+	0.781195i	$0.714610\pi$
$164$	184.365	−	319.330i	0.0877835	−	0.152046i
$165$		0			0
$166$	2550.52	−	1472.54i	1.19252	−	0.688502i
$167$	1530.40			0.709136			0.354568	−	0.935030i	$-0.384628\pi$
$167$	1530.40			0.709136			0.354568	+	0.935030i	$0.384628\pi$
$168$		0			0
$169$	15.6909			0.00714197
$170$	−427.995	+	247.103i	−0.193092	+	0.111482i
$171$		0			0
$172$	−318.014	+	550.817i	−0.140979	+	0.244183i
$173$	660.239	+	1143.57i	0.290156	+	0.502565i	0.973847	−	0.227207i	$-0.0729594\pi$
$173$	660.239	+	1143.57i	0.290156	+	0.502565i	−0.683690	+	0.729772i	$0.739626\pi$
$174$		0			0
$175$	1096.59	+	1938.07i	0.473683	+	0.837168i
$176$		−	320.340i		−	0.137196i
$177$		0			0
$178$	809.153	+	467.164i	0.340722	+	0.196716i
$179$	−73.3579	−	42.3532i	−0.0306314	−	0.0176851i	0.484606	−	0.874732i	$-0.338963\pi$
$179$	−73.3579	−	42.3532i	−0.0306314	−	0.0176851i	−0.515238	+	0.857047i	$0.672296\pi$
$180$		0			0
$181$			594.940i			0.244318i	0.992511	+	0.122159i	$0.0389817\pi$
$181$			594.940i			0.244318i	−0.992511	+	0.122159i	$0.961018\pi$
$182$	1729.89	+	15.0398i	0.704551	+	0.00612540i
$183$		0			0
$184$	−57.8270	−	100.159i	−0.0231688	−	0.0401295i
$185$	−182.829	+	316.668i	−0.0726585	+	0.125848i
$186$		0			0
$187$	1962.92	−	1133.29i	0.767610	−	0.443180i
$188$	1458.48			0.565802
$189$		0			0
$190$	−385.513			−0.147200
$191$	2265.44	−	1307.95i	0.858228	−	0.495498i	−0.00519060	−	0.999987i	$-0.501652\pi$
$191$	2265.44	−	1307.95i	0.858228	−	0.495498i	0.863418	+	0.504488i	$0.168319\pi$
$192$		0			0
$193$	−2071.09	+	3587.23i	−0.772436	+	1.33790i	0.163789	+	0.986495i	$0.447628\pi$
$193$	−2071.09	+	3587.23i	−0.772436	+	1.33790i	−0.936224	+	0.351403i	$0.885705\pi$
$194$	−411.573	−	712.865i	−0.152315	−	0.263818i
$195$		0			0
$196$	−1371.79	−	23.8547i	−0.499924	−	0.00869339i
$197$		−	3880.46i		−	1.40341i	−0.712469	−	0.701704i	$-0.752423\pi$
$197$		−	3880.46i		−	1.40341i	0.712469	−	0.701704i	$-0.247577\pi$
$198$		0			0
$199$	−161.294	−	93.1233i	−0.0574565	−	0.0331725i	0.470997	−	0.882135i	$-0.343894\pi$
$199$	−161.294	−	93.1233i	−0.0574565	−	0.0331725i	−0.528453	+	0.848963i	$0.677228\pi$
$200$	−833.018	−	480.943i	−0.294516	−	0.170039i
$201$		0			0
$202$			2830.30i			0.985837i
$203$	259.006	+	152.554i	0.0895499	+	0.0527450i
$204$		0			0
$205$	−100.604	−	174.251i	−0.0342755	−	0.0593670i
$206$	−921.352	+	1595.83i	−0.311619	+	0.539741i
$207$		0			0
$208$	−647.156	+	373.636i	−0.215732	+	0.124553i
$209$	1768.09			0.585173
$210$		0			0
$211$	−5159.23			−1.68330			−0.841649	−	0.540026i	$-0.818415\pi$
$211$	−5159.23			−1.68330			−0.841649	+	0.540026i	$0.818415\pi$
$212$	−841.557	+	485.873i	−0.272634	+	0.157405i
$213$		0			0
$214$	−560.733	+	971.218i	−0.179116	+	0.310239i
$215$	173.533	+	300.569i	0.0550459	+	0.0953423i
$216$		0			0
$217$	1451.65	−	821.369i	0.454123	−	0.256950i
$218$			2026.88i			0.629714i
$219$		0			0
$220$	−151.383	−	87.4012i	−0.0463921	−	0.0267845i
$221$	−4579.00	−	2643.69i	−1.39374	−	0.804677i
$222$		0			0
$223$			2149.72i			0.645541i	0.946477	+	0.322771i	$0.104614\pi$
$223$			2149.72i			0.645541i	−0.946477	+	0.322771i	$0.895386\pi$
$224$	515.805	−	291.851i	0.153856	−	0.0870541i
$225$		0			0
$226$	−995.485	−	1724.23i	−0.293003	−	0.507496i
$227$	−637.706	+	1104.54i	−0.186458	+	0.322955i	−0.944067	−	0.329754i	$-0.893034\pi$
$227$	−637.706	+	1104.54i	−0.186458	+	0.322955i	0.757609	+	0.652709i	$0.226368\pi$
$228$		0			0
$229$	2665.53	−	1538.95i	0.769185	−	0.444089i	−0.0633987	−	0.997988i	$-0.520194\pi$
$229$	2665.53	−	1538.95i	0.769185	−	0.444089i	0.832584	+	0.553899i	$0.186861\pi$
$230$	−63.1098			−0.0180928
$231$		0			0
$232$	−129.844			−0.0367444
$233$	−5676.43	+	3277.29i	−1.59603	+	0.921469i	−0.603789	+	0.797144i	$0.706343\pi$
$233$	−5676.43	+	3277.29i	−1.59603	+	0.921469i	−0.992242	+	0.124324i	$0.960324\pi$
$234$		0			0
$235$	397.931	−	689.237i	0.110460	−	0.191323i
$236$	1031.79	+	1787.11i	0.284592	+	0.492927i
$237$		0			0
$238$	3613.16	+	2128.15i	0.984060	+	0.579612i
$239$			108.203i			0.0292849i	0.999893	+	0.0146425i	$0.00466100\pi$
$239$			108.203i			0.0292849i	−0.999893	+	0.0146425i	$0.995339\pi$
$240$		0			0
$241$	2749.77	+	1587.58i	0.734972	+	0.424336i	0.820238	−	0.572022i	$-0.193841\pi$
$241$	2749.77	+	1587.58i	0.734972	+	0.424336i	−0.0852665	+	0.996358i	$0.527174\pi$
$242$	−1611.07	−	930.150i	−0.427947	−	0.247076i
$243$		0			0
$244$		−	2510.70i		−	0.658733i
$245$	−385.551	+	641.761i	−0.100539	+	0.167349i
$246$		0			0
$247$	−2062.25	−	3571.92i	−0.531246	−	0.920145i
$248$	−360.236	+	623.947i	−0.0922379	+	0.159761i
$249$		0			0
$250$	−927.130	+	535.279i	−0.234547	+	0.135416i
$251$	2760.22			0.694119			0.347059	−	0.937843i	$-0.387180\pi$
$251$	2760.22			0.694119			0.347059	+	0.937843i	$0.387180\pi$
$252$		0			0
$253$	289.442			0.0719251
$254$	553.996	−	319.850i	0.136853	−	0.0790124i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−750.559	−	1300.01i	−0.182174	−	0.315534i	0.760447	−	0.649400i	$-0.224980\pi$
$257$	−750.559	−	1300.01i	−0.182174	−	0.315534i	−0.942620	+	0.333866i	$0.891647\pi$
$258$		0			0
$259$	3102.48	+	26.9731i	0.744319	+	0.00647114i
$260$			407.770i			0.0972646i
$261$		0			0
$262$	−2985.48	−	1723.67i	−0.703983	−	0.406445i
$263$	2160.23	+	1247.21i	0.506485	+	0.292419i	0.731388	−	0.681962i	$-0.238873\pi$
$263$	2160.23	+	1247.21i	0.506485	+	0.292419i	−0.224903	+	0.974381i	$0.572206\pi$
$264$		0			0
$265$			530.260i			0.122919i
$266$	1610.84	+	2846.94i	0.371305	+	0.656229i
$267$		0			0
$268$	−1721.43	−	2981.60i	−0.392361	−	0.679590i
$269$	−577.799	+	1000.78i	−0.130963	+	0.226834i	−0.924048	−	0.382276i	$-0.875140\pi$
$269$	−577.799	+	1000.78i	−0.130963	+	0.226834i	0.793085	+	0.609111i	$0.208474\pi$
$270$		0			0
$271$	4110.34	−	2373.10i	0.921347	−	0.531940i	0.0372827	−	0.999305i	$-0.488130\pi$
$271$	4110.34	−	2373.10i	0.921347	−	0.531940i	0.884065	+	0.467365i	$0.154796\pi$
$272$	−1811.35			−0.403783
$273$		0			0
$274$	6166.55			1.35962
$275$	2084.76	−	1203.64i	0.457148	−	0.263934i
$276$		0			0
$277$	−3778.51	+	6544.58i	−0.819599	+	1.41959i	0.0863794	+	0.996262i	$0.472470\pi$
$277$	−3778.51	+	6544.58i	−0.819599	+	1.41959i	−0.905978	+	0.423324i	$0.860863\pi$
$278$	1647.06	+	2852.80i	0.355339	+	0.615466i
$279$		0			0
$280$	2.81150	−	323.383i	0.000600070	−	0.0690208i
$281$		−	3703.99i		−	0.786339i	−0.919466	−	0.393170i	$-0.871379\pi$
$281$		−	3703.99i		−	0.786339i	0.919466	−	0.393170i	$-0.128621\pi$
$282$		0			0
$283$	−6097.78	−	3520.56i	−1.28083	−	0.739489i	−0.303832	−	0.952726i	$-0.598266\pi$
$283$	−6097.78	−	3520.56i	−1.28083	−	0.739489i	−0.977001	+	0.213236i	$0.931600\pi$
$284$	1851.43	+	1068.92i	0.386838	+	0.223341i
$285$		0			0
$286$		−	1870.16i		−	0.386661i
$287$	−866.444	+	1471.04i	−0.178204	+	0.302553i
$288$		0			0
$289$	−3951.65	−	6844.46i	−0.804326	−	1.39313i
$290$	−35.4266	+	61.3607i	−0.00717352	+	0.0124249i
$291$		0			0
$292$	1804.74	−	1041.97i	0.361694	−	0.208824i
$293$	−2797.65			−0.557816			−0.278908	−	0.960318i	$-0.589973\pi$
$293$	−2797.65			−0.557816			−0.278908	+	0.960318i	$0.589973\pi$
$294$		0			0
$295$	1126.05			0.222240
$296$	−1160.64	+	670.097i	−0.227909	+	0.131583i
$297$		0			0
$298$	−1586.63	+	2748.12i	−0.308426	+	0.534209i
$299$	−337.597	−	584.736i	−0.0652968	−	0.113097i
$300$		0			0
$301$	1494.54	−	2537.42i	0.286193	−	0.485895i
$302$			3225.25i			0.614544i
$303$		0			0
$304$	−1223.67	−	706.484i	−0.230862	−	0.133288i
$305$	−1186.48	−	685.016i	−0.222747	−	0.128603i
$306$		0			0
$307$		−	188.225i		−	0.0349922i	−0.999847	−	0.0174961i	$-0.994431\pi$
$307$		−	188.225i		−	0.0349922i	0.999847	−	0.0174961i	$-0.00556946\pi$
$308$	−12.8945	+	1483.14i	−0.00238549	+	0.274382i
$309$		0			0
$310$	196.573	+	340.474i	0.0360148	+	0.0623794i
$311$	−1417.00	+	2454.31i	−0.258362	+	0.447495i	−0.965803	−	0.259276i	$-0.916516\pi$
$311$	−1417.00	+	2454.31i	−0.258362	+	0.447495i	0.707442	+	0.706772i	$0.249849\pi$
$312$		0			0
$313$	6867.98	−	3965.23i	1.24026	−	0.716064i	0.271113	−	0.962548i	$-0.412608\pi$
$313$	6867.98	−	3965.23i	1.24026	−	0.716064i	0.969147	+	0.246483i	$0.0792751\pi$
$314$	−557.543			−0.100204
$315$		0			0
$316$	−4207.18			−0.748963
$317$	−3382.54	+	1952.91i	−0.599314	+	0.346014i	−0.768772	−	0.639523i	$-0.779132\pi$
$317$	−3382.54	+	1952.91i	−0.599314	+	0.346014i	0.169458	+	0.985537i	$0.445798\pi$
$318$		0			0
$319$	162.478	−	281.420i	0.0285173	−	0.0493934i
$320$	69.8467	+	120.978i	0.0122017	+	0.0211340i
$321$		0			0
$322$	263.701	+	466.054i	0.0456381	+	0.0806589i
$323$		−	9997.55i		−	1.72223i
$324$		0			0
$325$	−4863.21	−	2807.78i	−0.830038	−	0.479223i
$326$	2626.88	+	1516.63i	0.446286	+	0.257664i
$327$		0			0
$328$		−	737.461i		−	0.124145i
$329$	−6752.62	−	58.7076i	−1.13156	−	0.00983785i
$330$		0			0
$331$	−4844.47	−	8390.87i	−0.804460	−	1.39337i	−0.916655	−	0.399680i	$-0.869121\pi$
$331$	−4844.47	−	8390.87i	−0.804460	−	1.39337i	0.112195	−	0.993686i	$-0.464212\pi$
$332$	2945.08	−	5101.03i	0.486845	−	0.843240i
$333$		0			0
$334$	2650.73	−	1530.40i	0.434256	−	0.250718i
$335$	−1878.69			−0.306399
$336$		0			0
$337$	−1491.53			−0.241094			−0.120547	−	0.992708i	$-0.538465\pi$
$337$	−1491.53			−0.241094			−0.120547	+	0.992708i	$0.538465\pi$
$338$	27.1774	−	15.6909i	0.00437354	−	0.00252507i
$339$		0			0
$340$	−494.205	+	855.989i	−0.0788296	+	0.136537i
$341$	−901.547	−	1561.52i	−0.143172	−	0.247980i
$342$		0			0
$343$	6350.29	+	165.662i	0.999660	+	0.0260785i
$344$			1272.06i			0.199374i
$345$		0			0
$346$	2287.13	+	1320.48i	0.355367	+	0.205171i
$347$	6579.57	+	3798.72i	1.01790	+	0.587682i	0.913494	−	0.406853i	$-0.133374\pi$
$347$	6579.57	+	3798.72i	1.01790	+	0.587682i	0.104402	+	0.994535i	$0.466707\pi$
$348$		0			0
$349$		−	3696.49i		−	0.566958i	−0.958978	−	0.283479i	$-0.908511\pi$
$349$		−	3696.49i		−	0.566958i	0.958978	−	0.283479i	$-0.0914887\pi$
$350$	3837.42	+	2260.24i	0.586054	+	0.345186i
$351$		0			0
$352$	−320.340	−	554.845i	−0.0485062	−	0.0840152i
$353$	−227.406	+	393.878i	−0.0342878	+	0.0593881i	−0.882660	−	0.470012i	$-0.844250\pi$
$353$	−227.406	+	393.878i	−0.0342878	+	0.0593881i	0.848372	+	0.529400i	$0.177583\pi$
$354$		0			0
$355$	1010.28	−	583.287i	0.151043	−	0.0872046i
$356$	1868.66			0.278199
$357$		0			0
$358$	−169.413			−0.0250104
$359$	5454.70	−	3149.27i	0.801917	−	0.462987i	−0.0422241	−	0.999108i	$-0.513444\pi$
$359$	5454.70	−	3149.27i	0.801917	−	0.462987i	0.844141	+	0.536121i	$0.180111\pi$
$360$		0			0
$361$	469.873	−	813.844i	0.0685046	−	0.118653i
$362$	594.940	+	1030.47i	0.0863794	+	0.149613i
$363$		0			0
$364$	3011.30	−	1703.84i	0.433613	−	0.245345i
$365$		−	1137.16i		−	0.163073i
$366$		0			0
$367$	4756.21	+	2746.00i	0.676491	+	0.390572i	0.798532	−	0.601953i	$-0.205610\pi$
$367$	4756.21	+	2746.00i	0.676491	+	0.390572i	−0.122041	+	0.992525i	$0.538944\pi$
$368$	−200.318	−	115.654i	−0.0283759	−	0.0163828i
$369$		0			0
$370$			731.314i			0.102755i
$371$	3915.87	−	2215.66i	0.547984	−	0.310058i
$372$		0			0
$373$	−1630.53	−	2824.17i	−0.226343	−	0.392037i	0.730379	−	0.683042i	$-0.239344\pi$
$373$	−1630.53	−	2824.17i	−0.226343	−	0.392037i	−0.956721	+	0.291005i	$0.906010\pi$
$374$	2266.59	−	3925.84i	0.313376	−	0.542782i
$375$		0			0
$376$	2526.17	−	1458.48i	0.346482	−	0.200041i
$377$	−758.040			−0.103557
$378$		0			0
$379$	2634.41			0.357046			0.178523	−	0.983936i	$-0.442868\pi$
$379$	2634.41			0.357046			0.178523	+	0.983936i	$0.442868\pi$
$380$	−667.727	+	385.513i	−0.0901413	+	0.0520431i
$381$		0			0
$382$	2615.91	−	4530.88i	0.350370	−	0.606859i
$383$	−3901.67	−	6757.89i	−0.520538	−	0.901598i	−0.999715	−	0.0238797i	$-0.992398\pi$
$383$	−3901.67	−	6757.89i	−0.520538	−	0.901598i	0.479177	−	0.877718i	$-0.340935\pi$
$384$		0			0
$385$	697.370	+	410.751i	0.0923150	+	0.0543736i
$386$			8284.35i			1.09239i
$387$		0			0
$388$	−1425.73	−	823.145i	−0.186548	−	0.107703i
$389$	−11120.4	−	6420.37i	−1.44943	−	0.836827i	−0.450980	−	0.892534i	$-0.648925\pi$
$389$	−11120.4	−	6420.37i	−1.44943	−	0.836827i	−0.998447	+	0.0557073i	$0.982259\pi$
$390$		0			0
$391$		−	1636.63i		−	0.211683i
$392$	−2399.87	+	1330.48i	−0.309214	+	0.171426i
$393$		0			0
$394$	−3880.46	−	6721.15i	−0.496180	−	0.859408i
$395$	−1147.88	+	1988.19i	−0.146218	+	0.253257i
$396$		0			0
$397$	13283.1	−	7669.02i	1.67925	−	0.969514i	0.717105	−	0.696965i	$-0.245467\pi$
$397$	13283.1	−	7669.02i	1.67925	−	0.969514i	0.962142	−	0.272549i	$-0.0878667\pi$
$398$	−372.493			−0.0469130
$399$		0			0
$400$	−1923.77			−0.240472
$401$	9661.06	−	5577.81i	1.20312	−	0.694620i	0.241871	−	0.970309i	$-0.422239\pi$
$401$	9661.06	−	5577.81i	1.20312	−	0.694620i	0.961247	+	0.275688i	$0.0889057\pi$
$402$		0			0
$403$	−2103.08	+	3642.64i	−0.259955	+	0.450255i
$404$	2830.30	+	4902.22i	0.348546	+	0.603699i
$405$		0			0
$406$	601.165	+	5.22656i	0.0734860	+	0.000638891i
$407$		−	3354.05i		−	0.408486i
$408$		0			0
$409$	−10451.1	−	6033.96i	−1.26351	−	0.729487i	−0.289756	−	0.957100i	$-0.593574\pi$
$409$	−10451.1	−	6033.96i	−1.26351	−	0.729487i	−0.973751	+	0.227614i	$0.926908\pi$
$410$	−348.502	−	201.208i	−0.0419788	−	0.0242365i
$411$		0			0
$412$			3685.41i			0.440696i
$413$	−4705.12	−	8315.64i	−0.560591	−	0.990765i
$414$		0			0
$415$	−1607.07	−	2783.52i	−0.190091	−	0.329247i
$416$	−747.272	+	1294.31i	−0.0880721	+	0.152545i
$417$		0			0
$418$	3062.42	−	1768.09i	0.358344	−	0.206890i
$419$	16571.6			1.93216			0.966080	−	0.258243i	$-0.0831435\pi$
$419$	16571.6			1.93216			0.966080	+	0.258243i	$0.0831435\pi$
$420$		0			0
$421$	−7310.98			−0.846355			−0.423178	−	0.906047i	$-0.639085\pi$
$421$	−7310.98			−0.846355			−0.423178	+	0.906047i	$0.639085\pi$
$422$	−8936.04	+	5159.23i	−1.03080	+	0.595135i
$423$		0			0
$424$	−971.746	+	1683.11i	−0.111302	+	0.192781i
$425$	−6805.89	−	11788.2i	−0.776787	−	1.34543i
$426$		0			0
$427$	−101.062	+	11624.3i	−0.0114537	+	1.31742i
$428$			2242.93i			0.253309i
$429$		0			0
$430$	601.137	+	347.067i	0.0674172	+	0.0389233i
$431$	11669.5	+	6737.41i	1.30418	+	0.752969i	0.981118	−	0.193409i	$-0.0619546\pi$
$431$	11669.5	+	6737.41i	1.30418	+	0.752969i	0.323062	+	0.946378i	$0.395288\pi$
$432$		0			0
$433$			15794.5i			1.75297i	0.481428	+	0.876485i	$0.340118\pi$
$433$			15794.5i			1.75297i	−0.481428	+	0.876485i	$0.659882\pi$
$434$	1692.97	−	2874.31i	0.187247	−	0.317906i
$435$		0			0
$436$	2026.88	+	3510.66i	0.222637	+	0.385619i
$437$	638.341	−	1105.64i	0.0698764	−	0.121029i
$438$		0			0
$439$	10120.7	−	5843.18i	1.10030	−	0.635261i	0.164004	−	0.986460i	$-0.447559\pi$
$439$	10120.7	−	5843.18i	1.10030	−	0.635261i	0.936301	+	0.351199i	$0.114226\pi$
$440$	−349.605			−0.0378790
$441$		0			0
$442$	−10574.7			−1.13799
$443$	7209.31	−	4162.30i	0.773193	−	0.446403i	−0.0608195	−	0.998149i	$-0.519371\pi$
$443$	7209.31	−	4162.30i	0.773193	−	0.446403i	0.834012	+	0.551746i	$0.186038\pi$
$444$		0			0
$445$	509.842	−	883.073i	0.0543120	−	0.0940712i
$446$	2149.72	+	3723.42i	0.228233	+	0.395312i
$447$		0			0
$448$	601.550	−	1021.31i	0.0634388	−	0.107706i
$449$			3856.07i			0.405299i	0.979251	+	0.202649i	$0.0649552\pi$
$449$			3856.07i			0.405299i	−0.979251	+	0.202649i	$0.935045\pi$
$450$		0			0
$451$	1598.35	+	922.805i	0.166881	+	0.0963486i
$452$	−3448.46	−	1990.97i	−0.358854	−	0.207184i
$453$		0			0
$454$			2550.82i			0.263692i
$455$	16.4137	−	1887.93i	0.00169118	−	0.194522i
$456$		0			0
$457$	−525.994	−	911.049i	−0.0538402	−	0.0932540i	0.837849	−	0.545902i	$-0.183813\pi$
$457$	−525.994	−	911.049i	−0.0538402	−	0.0932540i	−0.891689	+	0.452648i	$0.850479\pi$
$458$	3077.89	−	5331.07i	0.314019	−	0.543896i
$459$		0			0
$460$	−109.309	+	63.1098i	−0.0110795	+	0.00639676i
$461$	−15470.1			−1.56294			−0.781470	−	0.623943i	$-0.785530\pi$
$461$	−15470.1			−1.56294			−0.781470	+	0.623943i	$0.785530\pi$
$462$		0			0
$463$	−7935.31			−0.796512			−0.398256	−	0.917274i	$-0.630385\pi$
$463$	−7935.31			−0.796512			−0.398256	+	0.917274i	$0.630385\pi$
$464$	−224.897	+	129.844i	−0.0225013	+	0.0129911i
$465$		0			0
$466$	−6554.57	+	11352.9i	−0.651577	+	1.12856i
$467$	8047.97	+	13939.5i	0.797464	+	1.38125i	0.921263	+	0.388941i	$0.127159\pi$
$467$	8047.97	+	13939.5i	0.797464	+	1.38125i	−0.123799	+	0.992307i	$0.539508\pi$
$468$		0			0
$469$	7850.00	+	13873.8i	0.772877	+	1.36595i
$470$		−	1591.72i		−	0.156214i
$471$		0			0
$472$	3574.21	+	2063.57i	0.348552	+	0.201237i
$473$	−2757.01	−	1591.76i	−0.268007	−	0.154734i
$474$		0			0
$475$		−	10618.1i		−	1.02567i
$476$	8386.33	+	72.9112i	0.807535	+	0.00702075i
$477$		0			0
$478$	108.203	+	187.414i	0.0103538	+	0.0179333i
$479$	−3080.96	+	5336.39i	−0.293889	+	0.509031i	−0.974726	−	0.223404i	$-0.928283\pi$
$479$	−3080.96	+	5336.39i	−0.293889	+	0.509031i	0.680837	+	0.732435i	$0.261616\pi$
$480$		0			0
$481$	−6775.90	+	3912.07i	−0.642317	+	0.370842i
$482$	6350.32			0.600102
$483$		0			0
$484$	−3720.60			−0.349418
$485$	−777.989	+	449.172i	−0.0728385	+	0.0420533i
$486$		0			0
$487$	−10431.3	+	18067.6i	−0.970614	+	1.68115i	−0.276904	+	0.960898i	$0.589308\pi$
$487$	−10431.3	+	18067.6i	−0.970614	+	1.68115i	−0.693710	+	0.720255i	$0.744025\pi$
$488$	−2510.70	−	4348.66i	−0.232897	−	0.403390i
$489$		0			0
$490$	−26.0339	+	1497.11i	−0.00240019	+	0.138026i
$491$			17286.0i			1.58881i	0.607388	+	0.794406i	$0.292217\pi$
$491$			17286.0i			1.58881i	−0.607388	+	0.794406i	$0.707783\pi$
$492$		0			0
$493$	−1591.27	−	918.723i	−0.145370	−	0.0839294i
$494$	−7143.84	−	4124.50i	−0.650641	−	0.375648i
$495$		0			0
$496$			1440.94i			0.130444i
$497$	−8528.87	−	5023.51i	−0.769763	−	0.453391i
$498$		0			0
$499$	−1939.89	−	3359.99i	−0.174031	−	0.301430i	0.765795	−	0.643085i	$-0.222346\pi$
$499$	−1939.89	−	3359.99i	−0.174031	−	0.301430i	−0.939825	+	0.341655i	$0.889013\pi$
$500$	−1070.56	+	1854.26i	−0.0957536	+	0.165850i
$501$		0			0
$502$	4780.85	−	2760.22i	0.425059	−	0.245408i
$503$	−9538.83			−0.845557			−0.422779	−	0.906233i	$-0.638945\pi$
$503$	−9538.83			−0.845557			−0.422779	+	0.906233i	$0.638945\pi$
$504$		0			0
$505$	3088.86			0.272183
$506$	501.328	−	289.442i	0.0440450	−	0.0254294i
$507$		0			0
$508$	639.699	−	1107.99i	0.0558702	−	0.0967700i
$509$	2012.69	+	3486.08i	0.175267	+	0.303572i	0.940254	−	0.340475i	$-0.110588\pi$
$509$	2012.69	+	3486.08i	0.175267	+	0.303572i	−0.764987	+	0.644046i	$0.777254\pi$
$510$		0			0
$511$	−8397.70	+	4751.55i	−0.726991	+	0.411343i
$512$			512.000i			0.0441942i
$513$		0			0
$514$	−2600.01	−	1501.12i	−0.223116	−	0.128816i
$515$	1741.62	+	1005.52i	0.149019	+	0.0860361i
$516$		0			0
$517$			7300.16i			0.621007i
$518$	5400.62	−	3055.76i	0.458088	−	0.259194i
$519$		0			0
$520$	407.770	+	706.278i	0.0343882	+	0.0595622i
$521$	−6184.05	+	10711.1i	−0.520015	+	0.900693i	0.479714	+	0.877425i	$0.340740\pi$
$521$	−6184.05	+	10711.1i	−0.520015	+	0.900693i	−0.999729	+	0.0232682i	$0.992593\pi$
$522$		0			0
$523$	−4466.84	+	2578.93i	−0.373464	+	0.215619i	−0.674971	−	0.737845i	$-0.735844\pi$
$523$	−4466.84	+	2578.93i	−0.373464	+	0.215619i	0.301507	+	0.953464i	$0.402510\pi$
$524$	−6894.67			−0.574799
$525$		0			0
$526$	4988.84			0.413543
$527$	−8829.56	+	5097.75i	−0.729832	+	0.421369i
$528$		0			0
$529$	−5979.00	+	10355.9i	−0.491411	+	0.851149i
$530$	530.260	+	918.438i	0.0434585	+	0.0752724i
$531$		0			0
$532$	5637.00	+	3320.20i	0.459389	+	0.270581i
$533$		−	4305.34i		−	0.349878i
$534$		0			0
$535$	1059.94	+	611.959i	0.0856549	+	0.0494529i
$536$	−5963.20	−	3442.85i	−0.480543	−	0.277441i
$537$		0			0
$538$			2311.19i			0.185209i
$539$	119.400	−	6866.25i	0.00954160	−	0.548702i
$540$		0			0
$541$	−292.567	−	506.742i	−0.0232504	−	0.0402708i	0.854166	−	0.520000i	$-0.174068\pi$
$541$	−292.567	−	506.742i	−0.0232504	−	0.0402708i	−0.877416	+	0.479729i	$0.840735\pi$
$542$	4746.21	−	8220.67i	0.376139	−	0.651491i
$543$		0			0
$544$	−3137.34	+	1811.35i	−0.247266	+	0.142759i
$545$	2212.05			0.173860
$546$		0			0
$547$	9758.76			0.762805			0.381403	−	0.924409i	$-0.375441\pi$
$547$	9758.76			0.762805			0.381403	+	0.924409i	$0.375441\pi$
$548$	10680.8	−	6166.55i	0.832592	−	0.480697i
$549$		0			0
$550$	2407.27	−	4169.51i	0.186630	−	0.323252i
$551$	−716.664	−	1241.30i	−0.0554100	−	0.0959730i
$552$		0			0
$553$	19478.8	+	169.349i	1.49787	+	0.0130225i
$554$			15114.1i			1.15909i
$555$		0			0
$556$	5705.60	+	3294.13i	0.435200	+	0.251263i
$557$	21014.7	+	12132.8i	1.59860	+	0.922953i	0.991757	+	0.128131i	$0.0408979\pi$
$557$	21014.7	+	12132.8i	1.59860	+	0.922953i	0.606844	+	0.794821i	$0.292435\pi$
$558$		0			0
$559$			7426.35i			0.561898i
$560$	−318.513	−	562.927i	−0.0240351	−	0.0424786i
$561$		0			0
$562$	−3703.99	−	6415.49i	−0.278013	−	0.481533i
$563$	5704.41	−	9880.33i	0.427020	−	0.739620i	−0.569587	−	0.821931i	$-0.692897\pi$
$563$	5704.41	−	9880.33i	0.427020	−	0.739620i	0.996607	+	0.0823109i	$0.0262301\pi$
$564$		0			0
$565$	−1881.75	+	1086.43i	−0.140116	+	0.0808962i
$566$	−14082.2			−1.04580
$567$		0			0
$568$	4275.69			0.315852
$569$	−11281.1	+	6513.14i	−0.831157	+	0.479868i	−0.854249	−	0.519865i	$-0.825982\pi$
$569$	−11281.1	+	6513.14i	−0.831157	+	0.479868i	0.0230919	+	0.999733i	$0.492649\pi$
$570$		0			0
$571$	−4872.42	+	8439.28i	−0.357101	+	0.618516i	−0.987475	−	0.157775i	$-0.949568\pi$
$571$	−4872.42	+	8439.28i	−0.357101	+	0.618516i	0.630375	+	0.776291i	$0.282901\pi$
$572$	−1870.16	−	3239.22i	−0.136705	−	0.236781i
$573$		0			0
$574$	−29.6846	+	3414.36i	−0.00215856	+	0.248280i
$575$		−	1738.22i		−	0.126067i
$576$		0			0
$577$	20008.1	+	11551.7i	1.44358	+	0.833452i	0.998087	−	0.0618323i	$-0.0196944\pi$
$577$	20008.1	+	11551.7i	1.44358	+	0.833452i	0.445495	+	0.895284i	$0.353028\pi$
$578$	−13688.9	−	7903.30i	−0.985094	−	0.568744i
$579$		0			0
$580$			141.706i			0.0101449i
$581$	−13840.7	+	23498.7i	−0.988314	+	1.67795i
$582$		0			0
$583$	−2431.95	−	4212.25i	−0.172763	−	0.299235i
$584$	2083.94	−	3609.48i	0.147661	−	0.255756i
$585$		0			0
$586$	−4845.67	+	2797.65i	−0.341591	+	0.197218i
$587$	−1016.23			−0.0714551			−0.0357276	−	0.999362i	$-0.511375\pi$
$587$	−1016.23			−0.0714551			−0.0357276	+	0.999362i	$0.511375\pi$
$588$		0			0
$589$	−7953.15			−0.556374
$590$	1950.37	−	1126.05i	0.136094	−	0.0785739i
$591$		0			0
$592$	−1340.19	+	2321.29i	−0.0930433	+	0.161156i
$593$	−3277.56	−	5676.90i	−0.226970	−	0.393124i	0.729939	−	0.683513i	$-0.239549\pi$
$593$	−3277.56	−	5676.90i	−0.226970	−	0.393124i	−0.956909	+	0.290389i	$0.906215\pi$
$594$		0			0
$595$	2322.57	−	3943.24i	0.160027	−	0.271693i
$596$			6346.51i			0.436180i
$597$		0			0
$598$	−1169.47	−	675.195i	−0.0799720	−	0.0461718i
$599$	24856.2	+	14350.7i	1.69548	+	0.978888i	0.949940	+	0.312432i	$0.101144\pi$
$599$	24856.2	+	14350.7i	1.69548	+	0.978888i	0.745544	+	0.666456i	$0.232190\pi$
$600$		0			0
$601$			5078.15i			0.344662i	0.985039	+	0.172331i	$0.0551299\pi$
$601$			5078.15i			0.344662i	−0.985039	+	0.172331i	$0.944870\pi$
$602$	51.2035	−	5889.48i	0.00346661	−	0.398733i
$603$		0			0
$604$	3225.25	+	5586.29i	0.217274	+	0.376330i
$605$	−1015.12	+	1758.25i	−0.0682160	+	0.118153i
$606$		0			0
$607$	6689.48	−	3862.17i	0.447311	−	0.258255i	−0.259383	−	0.965775i	$-0.583519\pi$
$607$	6689.48	−	3862.17i	0.447311	−	0.258255i	0.706694	+	0.707520i	$0.250186\pi$
$608$	−2825.94			−0.188498
$609$		0			0
$610$	−2740.06			−0.181872
$611$	14747.9	−	8514.71i	0.976492	−	0.563778i
$612$		0			0
$613$	11102.4	−	19229.9i	0.731518	−	1.26703i	−0.224716	−	0.974424i	$-0.572146\pi$
$613$	11102.4	−	19229.9i	0.731518	−	1.26703i	0.956234	−	0.292602i	$-0.0945211\pi$
$614$	−188.225	−	326.016i	−0.0123716	−	0.0214282i
$615$		0			0
$616$	1460.80	+	2581.77i	0.0955479	+	0.168867i
$617$			24170.2i			1.57708i	0.614985	+	0.788539i	$0.289162\pi$
$617$			24170.2i			1.57708i	−0.614985	+	0.788539i	$0.710838\pi$
$618$		0			0
$619$	15905.4	+	9182.98i	1.03278	+	0.596277i	0.917780	−	0.397089i	$-0.129980\pi$
$619$	15905.4	+	9182.98i	1.03278	+	0.596277i	0.115001	+	0.993365i	$0.463313\pi$
$620$	680.948	+	393.145i	0.0441089	+	0.0254663i
$621$		0			0
$622$			5667.98i			0.365378i
$623$	−8651.68	−	75.2181i	−0.556376	−	0.00483716i
$624$		0			0
$625$	−6930.56	−	12004.1i	−0.443556	−	0.768261i
$626$	7930.46	−	13736.0i	0.506334	−	0.876996i
$627$		0			0
$628$	−965.693	+	557.543i	−0.0613620	+	0.0354274i
$629$	−18965.3			−1.20222
$630$		0			0
$631$	−24645.1			−1.55484			−0.777421	−	0.628981i	$-0.783472\pi$
$631$	−24645.1			−1.55484			−0.777421	+	0.628981i	$0.783472\pi$
$632$	−7287.04	+	4207.18i	−0.458644	+	0.264798i
$633$		0			0
$634$	−3905.83	+	6765.09i	−0.244669	+	0.423779i
$635$	−349.070	−	604.606i	−0.0218148	−	0.0377844i
$636$		0			0
$637$	−14010.6	+	7767.39i	−0.871459	+	0.483132i
$638$		−	649.912i		−	0.0403295i
$639$		0			0
$640$	241.956	+	139.693i	0.0149440	+	0.00862792i
$641$	10258.8	+	5922.93i	0.632136	+	0.364964i	0.781579	−	0.623807i	$-0.214415\pi$
$641$	10258.8	+	5922.93i	0.632136	+	0.364964i	−0.149443	+	0.988770i	$0.547748\pi$
$642$		0			0
$643$			22884.8i			1.40356i	0.712393	+	0.701781i	$0.247611\pi$
$643$			22884.8i			1.40356i	−0.712393	+	0.701781i	$0.752389\pi$
$644$	922.797	+	543.528i	0.0564647	+	0.0332578i
$645$		0			0
$646$	−9997.55	−	17316.3i	−0.608899	−	1.05464i
$647$	−2387.51	+	4135.29i	−0.145074	+	0.251275i	−0.929401	−	0.369073i	$-0.879675\pi$
$647$	−2387.51	+	4135.29i	−0.145074	+	0.251275i	0.784327	+	0.620348i	$0.213009\pi$
$648$		0			0
$649$	−8945.03	+	5164.41i	−0.541022	+	0.312359i
$650$	−11231.1			−0.677723
$651$		0			0
$652$	6066.52			0.364391
$653$	21258.8	−	12273.8i	1.27400	−	0.735543i	0.298261	−	0.954484i	$-0.403594\pi$
$653$	21258.8	−	12273.8i	1.27400	−	0.735543i	0.975738	+	0.218941i	$0.0702602\pi$
$654$		0			0
$655$	−1881.13	+	3258.22i	−0.112217	+	0.194365i
$656$	−737.461	−	1277.32i	−0.0438918	−	0.0760228i
$657$		0			0
$658$	−11754.6	+	6650.93i	−0.696415	+	0.394043i
$659$		−	30064.1i		−	1.77713i	−0.458749	−	0.888566i	$-0.651702\pi$
$659$		−	30064.1i		−	1.77713i	0.458749	−	0.888566i	$-0.348298\pi$
$660$		0			0
$661$	−16277.7	−	9397.91i	−0.957832	−	0.553005i	−0.0623272	−	0.998056i	$-0.519852\pi$
$661$	−16277.7	−	9397.91i	−0.957832	−	0.553005i	−0.895505	+	0.445051i	$0.853186\pi$
$662$	−16781.7	−	9688.95i	−0.985258	−	0.568839i
$663$		0			0
$664$		−	11780.3i		−	0.688502i
$665$	3107.02	−	1758.00i	0.181181	−	0.102515i
$666$		0			0
$667$	−117.320	−	203.205i	−0.00681059	−	0.0117963i
$668$	3060.80	−	5301.46i	0.177284	−	0.307065i
$669$		0			0
$670$	−3253.98	+	1878.69i	−0.187630	+	0.108328i
$671$	12566.8			0.723006
$672$		0			0
$673$	−17940.2			−1.02756			−0.513778	−	0.857923i	$-0.671754\pi$
$673$	−17940.2			−1.02756			−0.513778	+	0.857923i	$0.671754\pi$
$674$	−2583.40	+	1491.53i	−0.147639	+	0.0852397i
$675$		0			0
$676$	31.3818	−	54.3549i	0.00178549	−	0.00309256i
$677$	9829.19	+	17024.7i	0.558001	+	0.966485i	0.997663	+	0.0683224i	$0.0217646\pi$
$677$	9829.19	+	17024.7i	0.558001	+	0.966485i	−0.439663	+	0.898163i	$0.644902\pi$
$678$		0			0
$679$	6567.84	+	3868.46i	0.371208	+	0.218642i
$680$			1976.82i			0.111482i
$681$		0			0
$682$	−3123.05	−	1803.09i	−0.175349	−	0.101238i
$683$	812.211	+	468.930i	0.0455028	+	0.0262710i	0.522579	−	0.852591i	$-0.324970\pi$
$683$	812.211	+	468.930i	0.0455028	+	0.0262710i	−0.477076	+	0.878862i	$0.658303\pi$
$684$		0			0
$685$		−	6729.90i		−	0.375382i
$686$	11164.7	−	6063.35i	0.621384	−	0.337463i
$687$		0			0
$688$	1272.06	+	2203.27i	0.0704894	+	0.122091i
$689$	−5673.11	+	9826.12i	−0.313684	+	0.543317i
$690$		0			0
$691$	18626.5	−	10754.0i	1.02545	−	0.592044i	0.109772	−	0.993957i	$-0.464988\pi$
$691$	18626.5	−	10754.0i	1.02545	−	0.592044i	0.915678	+	0.401913i	$0.131654\pi$
$692$	5281.91			0.290156
$693$		0			0
$694$	15194.9			0.831108
$695$	3113.42	−	1797.53i	0.169926	−	0.0981069i
$696$		0			0
$697$	5217.95	−	9037.76i	0.283564	−	0.491147i
$698$	−3696.49	−	6402.50i	−0.200450	−	0.347190i
$699$		0			0
$700$	8906.85	+	77.4366i	0.480925	+	0.00418118i
$701$		−	13721.8i		−	0.739321i	−0.929167	−	0.369661i	$-0.879474\pi$
$701$		−	13721.8i		−	0.739321i	0.929167	−	0.369661i	$-0.120526\pi$
$702$		0			0
$703$	−12812.1	−	7397.08i	−0.687366	−	0.396851i
$704$	−1109.69	−	640.680i	−0.0594077	−	0.0342991i
$705$		0			0
$706$			909.622i			0.0484902i
$707$	−12906.6	−	22810.7i	−0.686569	−	1.21341i
$708$		0			0
$709$	11739.8	+	20334.0i	0.621860	+	1.07709i	0.989139	+	0.146982i	$0.0469559\pi$
$709$	11739.8	+	20334.0i	0.621860	+	1.07709i	−0.367280	+	0.930111i	$0.619711\pi$
$710$	1166.57	−	2020.56i	0.0616630	−	0.106803i
$711$		0			0
$712$	3236.61	−	1868.66i	0.170361	−	0.0983580i
$713$	−1301.96			−0.0683853
$714$		0			0
$715$	−2041.01			−0.106755
$716$	−293.431	+	169.413i	−0.0153157	+	0.00884253i
$717$		0			0
$718$	6298.55	−	10909.4i	0.327381	−	0.567041i
$719$	13332.6	+	23092.8i	0.691549	+	1.19780i	0.971330	+	0.237734i	$0.0764046\pi$
$719$	13332.6	+	23092.8i	0.691549	+	1.19780i	−0.279782	+	0.960064i	$0.590262\pi$
$720$		0			0
$721$	148.347	−	17063.0i	0.00766258	−	0.881360i
$722$		−	1879.49i		−	0.0968801i
$723$		0			0
$724$	2060.93	+	1189.88i	0.105793	+	0.0610794i
$725$	−1690.04	−	975.746i	−0.0865746	−	0.0499839i
$726$		0			0
$727$		−	20815.1i		−	1.06188i	−0.847409	−	0.530941i	$-0.821838\pi$
$727$		−	20815.1i		−	1.06188i	0.847409	−	0.530941i	$-0.178162\pi$
$728$	3511.89	−	5962.45i	0.178790	−	0.303548i
$729$		0			0
$730$	−1137.16	−	1969.62i	−0.0576549	−	0.0998613i
$731$	−9000.53	+	15589.4i	−0.455399	+	0.788774i
$732$		0			0
$733$	28981.9	−	16732.7i	1.46040	−	0.843161i	0.461368	−	0.887209i	$-0.347359\pi$
$733$	28981.9	−	16732.7i	1.46040	−	0.843161i	0.999029	+	0.0440478i	$0.0140254\pi$
$734$	10984.0			0.552353
$735$		0			0
$736$	−462.616			−0.0231688
$737$	14923.8	−	8616.28i	0.745897	−	0.430644i
$738$		0			0
$739$	4324.52	−	7490.29i	0.215264	−	0.372848i	−0.738090	−	0.674702i	$-0.764272\pi$
$739$	4324.52	−	7490.29i	0.215264	−	0.372848i	0.953354	+	0.301854i	$0.0976054\pi$
$740$	731.314	+	1266.67i	0.0363293	+	0.0629241i
$741$		0			0
$742$	4566.83	−	7753.52i	0.225948	−	0.383613i
$743$			16239.5i			0.801842i	0.916113	+	0.400921i	$0.131310\pi$
$743$			16239.5i			0.801842i	−0.916113	+	0.400921i	$0.868690\pi$
$744$		0			0
$745$	2999.18	+	1731.58i	0.147492	+	0.0851544i
$746$	−5648.34	−	3261.07i	−0.277212	−	0.160049i
$747$		0			0
$748$		−	9066.35i		−	0.443180i
$749$	90.2836	−	10384.5i	0.00440439	−	0.506599i
$750$		0			0
$751$	189.964	+	329.028i	0.00923022	+	0.0159872i	0.870604	−	0.491985i	$-0.163729\pi$
$751$	189.964	+	329.028i	0.00923022	+	0.0159872i	−0.861373	+	0.507972i	$0.830395\pi$
$752$	2916.97	−	5052.33i	0.141451	−	0.245000i
$753$		0			0
$754$	−1312.96	+	758.040i	−0.0634155	+	0.0366130i
$755$	3519.89			0.169672
$756$		0			0
$757$	4089.09			0.196328			0.0981642	−	0.995170i	$-0.468703\pi$
$757$	4089.09			0.196328			0.0981642	+	0.995170i	$0.468703\pi$
$758$	4562.92	−	2634.41i	0.218645	−	0.126235i
$759$		0			0
$760$	−771.025	+	1335.45i	−0.0368000	+	0.0637395i
$761$	−14791.1	−	25619.0i	−0.704569	−	1.22035i	−0.966847	−	0.255357i	$-0.917807\pi$
$761$	−14791.1	−	25619.0i	−0.704569	−	1.22035i	0.262277	−	0.964993i	$-0.415526\pi$
$762$		0			0
$763$	−9242.91	−	16335.5i	−0.438553	−	0.775080i
$764$		−	10463.6i		−	0.495498i
$765$		0			0
$766$	−13515.8	−	7803.34i	−0.637526	−	0.368076i
$767$	20866.5	+	12047.3i	0.982327	+	0.567147i
$768$		0			0
$769$		−	14802.4i		−	0.694133i	−0.937841	−	0.347066i	$-0.887178\pi$
$769$		−	14802.4i		−	0.694133i	0.937841	−	0.347066i	$-0.112822\pi$
$770$	1618.63	+	14.0725i	0.0757551	+	0.000658618i
$771$		0			0
$772$	8284.35	+	14348.9i	0.386218	+	0.668949i
$773$	−16539.2	+	28646.7i	−0.769564	+	1.33292i	0.168236	+	0.985747i	$0.446193\pi$
$773$	−16539.2	+	28646.7i	−0.769564	+	1.33292i	−0.937800	+	0.347176i	$0.887140\pi$
$774$		0			0
$775$	−9377.59	+	5414.16i	−0.434649	+	0.250945i
$776$	−3292.58			−0.152315
$777$		0			0
$778$	−25681.5			−1.18345
$779$	7050.05	−	4070.35i	0.324254	−	0.187208i
$780$		0			0
$781$	−5350.29	+	9266.97i	−0.245132	+	0.424581i
$782$	−1636.63	−	2834.73i	−0.0748413	−	0.129629i
$783$		0			0
$784$	−2826.22	+	4704.32i	−0.128745	+	0.214300i
$785$			608.478i			0.0276656i
$786$		0			0
$787$	6224.59	+	3593.77i	0.281935	+	0.162775i	0.634299	−	0.773088i	$-0.281289\pi$
$787$	6224.59	+	3593.77i	0.281935	+	0.162775i	−0.352364	+	0.935863i	$0.614622\pi$
$788$	−13442.3	−	7760.92i	−0.607693	−	0.350852i
$789$		0			0
$790$			4591.53i			0.206784i
$791$	15885.9	+	9356.78i	0.714078	+	0.420593i
$792$		0			0
$793$	−14657.6	−	25387.7i	−0.656377	−	1.13688i
$794$	15338.0	−	26566.3i	0.685550	−	1.18741i
$795$		0			0
$796$	−645.177	+	372.493i	−0.0287283	+	0.0165863i
$797$	16926.6			0.752286			0.376143	−	0.926562i	$-0.377250\pi$
$797$	16926.6			0.752286			0.376143	+	0.926562i	$0.377250\pi$
$798$		0			0
$799$	41278.4			1.82769
$800$	−3332.07	+	1923.77i	−0.147258	+	0.0850195i
$801$		0			0
$802$	11155.6	−	19322.1i	0.491171	−	0.850733i
$803$	5215.38	+	9033.30i	0.229199	+	0.396984i
$804$		0			0
$805$	508.630	−	287.791i	0.0222694	−	0.0126004i
$806$			8412.32i			0.367632i
$807$		0			0
$808$	9804.44	+	5660.60i	0.426880	+	0.246459i
$809$	−15936.6	−	9200.99i	−0.692584	−	0.399863i	0.111996	−	0.993709i	$-0.464276\pi$
$809$	−15936.6	−	9200.99i	−0.692584	−	0.399863i	−0.804579	+	0.593845i	$0.797609\pi$
$810$		0			0
$811$		−	36097.0i		−	1.56293i	−0.623947	−	0.781466i	$-0.714472\pi$
$811$		−	36097.0i		−	1.56293i	0.623947	−	0.781466i	$-0.285528\pi$
$812$	1046.48	−	592.113i	0.0452267	−	0.0255900i
$813$		0			0
$814$	−3354.05	−	5809.38i	−0.144422	−	0.250146i
$815$	1655.18	−	2866.86i	0.0711392	−	0.123217i
$816$		0			0
$817$	−12160.7	+	7021.00i	−0.520747	+	0.300653i
$818$	−24135.8			−1.03165
$819$		0			0
$820$	−804.832			−0.0342755
$821$	18631.7	−	10757.0i	0.792024	−	0.457275i	−0.0486508	−	0.998816i	$-0.515492\pi$
$821$	18631.7	−	10757.0i	0.792024	−	0.457275i	0.840675	+	0.541541i	$0.182159\pi$
$822$		0			0
$823$	12833.0	−	22227.5i	0.543537	−	0.941435i	−0.455160	−	0.890410i	$-0.650418\pi$
$823$	12833.0	−	22227.5i	0.543537	−	0.941435i	0.998697	−	0.0510248i	$-0.0162488\pi$
$824$	3685.41	+	6383.31i	0.155810	+	0.269870i
$825$		0			0
$826$	−16465.2	−	9697.99i	−0.693579	−	0.408518i
$827$			324.218i			0.0136326i	0.999977	+	0.00681631i	$0.00216972\pi$
$827$			324.218i			0.0136326i	−0.999977	+	0.00681631i	$0.997830\pi$
$828$		0			0
$829$	−15561.7	−	8984.56i	−0.651967	−	0.376414i	0.137242	−	0.990538i	$-0.456176\pi$
$829$	−15561.7	−	8984.56i	−0.651967	−	0.376414i	−0.789210	+	0.614124i	$0.789509\pi$
$830$	−5567.04	−	3214.13i	−0.232813	−	0.134415i
$831$		0			0
$832$			2989.09i			0.124553i
$833$	−38824.9	−	675.141i	−1.61489	−	0.0280819i
$834$		0			0
$835$	−1670.21	−	2892.89i	−0.0692215	−	0.119895i
$836$	3536.17	−	6124.83i	0.146293	−	0.253387i
$837$		0			0
$838$	28702.8	−	16571.6i	1.18320	−	0.683122i
$839$	−9839.29			−0.404875			−0.202437	−	0.979295i	$-0.564886\pi$
$839$	−9839.29			−0.404875			−0.202437	+	0.979295i	$0.564886\pi$
$840$		0			0
$841$	24125.6			0.989199
$842$	−12663.0	+	7310.98i	−0.518285	+	0.299232i
$843$		0			0
$844$	−10318.5	+	17872.1i	−0.420824	+	0.728889i
$845$	−17.1243	−	29.6602i	−0.000697154	−	0.00120751i
$846$		0			0
$847$	17226.0	+	149.763i	0.698809	+	0.00607548i
$848$			3886.98i			0.157405i
$849$		0			0
$850$	−23576.3	−	13611.8i	−0.951365	−	0.549271i
$851$	−2097.39	−	1210.93i	−0.0844860	−	0.0487780i
$852$		0			0
$853$		−	8049.23i		−	0.323096i	−0.986865	−	0.161548i	$-0.948351\pi$
$853$		−	8049.23i		−	0.323096i	0.986865	−	0.161548i	$-0.0516486\pi$
$854$	11449.2	+	20234.9i	0.458763	+	0.810799i
$855$		0			0
$856$	2242.93	+	3884.87i	0.0895582	+	0.155119i
$857$	−9016.09	+	15616.3i	−0.359374	+	0.622454i	−0.987856	−	0.155370i	$-0.950343\pi$
$857$	−9016.09	+	15616.3i	−0.359374	+	0.622454i	0.628482	+	0.777824i	$0.283676\pi$
$858$		0			0
$859$	−25271.0	+	14590.2i	−1.00377	+	0.579524i	−0.909360	−	0.416009i	$-0.863428\pi$
$859$	−25271.0	+	14590.2i	−1.00377	+	0.579524i	−0.0944054	+	0.995534i	$0.530095\pi$
$860$	1388.27			0.0550459
$861$		0			0
$862$	26949.6			1.06486
$863$	−33298.9	+	19225.1i	−1.31345	+	0.758321i	−0.982666	−	0.185386i	$-0.940647\pi$
$863$	−33298.9	+	19225.1i	−1.31345	+	0.758321i	−0.330784	+	0.943706i	$0.607313\pi$
$864$		0			0
$865$	1441.11	−	2496.08i	0.0566465	−	0.0981146i
$866$	15794.5	+	27356.9i	0.619769	+	1.07347i
$867$		0			0
$868$	58.0016	−	6671.41i	0.00226809	−	0.260878i
$869$		−	21058.2i		−	0.822039i
$870$		0			0
$871$	−34813.5	−	20099.6i	−1.35432	−	0.781916i
$872$	7021.31	+	4053.76i	0.272674	+	0.157428i
$873$		0			0
$874$		−	2553.36i		−	0.0988202i
$875$	5031.20	−	8541.93i	0.194384	−	0.330023i
$876$		0			0
$877$	20826.3	+	36072.1i	0.801884	+	1.38890i	0.918374	+	0.395713i	$0.129502\pi$
$877$	20826.3	+	36072.1i	0.801884	+	1.38890i	−0.116490	+	0.993192i	$0.537164\pi$
$878$	11686.4	−	20241.4i	0.449198	−	0.778033i
$879$		0			0
$880$	−605.533	+	349.605i	−0.0231960	+	0.0133922i
$881$	−19905.3			−0.761209			−0.380605	−	0.924738i	$-0.624284\pi$
$881$	−19905.3			−0.761209			−0.380605	+	0.924738i	$0.624284\pi$
$882$		0			0
$883$	5203.86			0.198328			0.0991642	−	0.995071i	$-0.468383\pi$
$883$	5203.86			0.198328			0.0991642	+	0.995071i	$0.468383\pi$
$884$	−18316.0	+	10574.7i	−0.696871	+	0.402338i
$885$		0			0
$886$	8324.59	−	14418.6i	0.315655	−	0.546730i
$887$	−18639.7	−	32285.0i	−0.705593	−	1.22212i	−0.966477	−	0.256753i	$-0.917347\pi$
$887$	−18639.7	−	32285.0i	−0.705593	−	1.22212i	0.260884	−	0.965370i	$-0.415986\pi$
$888$		0			0
$889$	−3006.34	+	5104.13i	−0.113419	+	0.192561i
$890$		−	2039.37i		−	0.0768088i
$891$		0			0
$892$	7446.84	+	4299.43i	0.279527	+	0.161385i
$893$	27885.9	+	16099.9i	1.04498	+	0.603319i
$894$		0			0
$895$			184.890i			0.00690522i
$896$	20.6093	−	2370.50i	0.000768423	−	0.0883850i
$897$		0			0
$898$	3856.07	+	6678.91i	0.143295	+	0.248194i
$899$	−730.853	+	1265.88i	−0.0271138	+	0.0469625i
$900$		0			0
$901$	−23818.0	+	13751.3i	−0.880679	+	0.508460i
$902$	3691.22			0.136257
$903$		0			0
$904$	−7963.88			−0.293003
$905$	1124.60	−	649.290i	0.0413073	−	0.0238488i
$906$		0			0
$907$	6611.15	−	11450.9i	0.242028	−	0.419205i	−0.719264	−	0.694737i	$-0.755521\pi$
$907$	6611.15	−	11450.9i	0.242028	−	0.419205i	0.961292	+	0.275532i	$0.0888540\pi$
$908$	2550.82	+	4418.15i	0.0932291	+	0.161478i
$909$		0			0
$910$	−1859.50	−	3286.40i	−0.0677382	−	0.119718i
$911$			42127.2i			1.53209i	0.642786	+	0.766046i	$0.277779\pi$
$911$			42127.2i			1.53209i	−0.642786	+	0.766046i	$0.722221\pi$
$912$		0			0
$913$	25532.3	+	14741.1i	0.925514	+	0.534346i
$914$	−1822.10	−	1051.99i	−0.0659405	−	0.0380708i
$915$		0			0
$916$		−	12311.6i		−	0.444089i
$917$	31921.5	+	277.527i	1.14956	+	0.00999429i
$918$		0			0
$919$	3857.77	+	6681.86i	0.138472	+	0.239841i	0.926919	−	0.375263i	$-0.122447\pi$
$919$	3857.77	+	6681.86i	0.138472	+	0.239841i	−0.788446	+	0.615104i	$0.789114\pi$
$920$	−126.220	+	218.619i	−0.00452319	+	0.00783439i
$921$		0			0
$922$	−26795.0	+	15470.1i	−0.957101	+	0.552582i
$923$	24961.7			0.890168
$924$		0			0
$925$	−20142.4			−0.715977
$926$	−13744.4	+	7935.31i	−0.487762	+	0.281610i
$927$		0			0
$928$	−259.689	+	449.794i	−0.00918610	+	0.0159108i
$929$	9469.27	+	16401.3i	0.334421	+	0.579233i	0.983373	−	0.181595i	$-0.0581261\pi$
$929$	9469.27	+	16401.3i	0.334421	+	0.579233i	−0.648953	+	0.760829i	$0.724793\pi$
$930$		0			0
$931$	−25965.1	−	15599.1i	−0.914039	−	0.549128i
$932$			26218.3i			0.921469i
$933$		0			0
$934$	27879.0	+	16095.9i	0.976690	+	0.563892i
$935$	−4284.49	−	2473.65i	−0.149859	−	0.0865210i
$936$		0			0
$937$			8289.37i			0.289009i	0.989504	+	0.144505i	$0.0461589\pi$
$937$			8289.37i			0.289009i	−0.989504	+	0.144505i	$0.953841\pi$
$938$	27470.4	+	16180.1i	0.956225	+	0.563217i
$939$		0			0
$940$	−1591.72	−	2756.95i	−0.0552301	−	0.0956614i
$941$	−1399.08	+	2423.28i	−0.0484684	+	0.0839497i	−0.889242	−	0.457437i	$-0.848767\pi$
$941$	−1399.08	+	2423.28i	−0.0484684	+	0.0839497i	0.840773	+	0.541387i	$0.182101\pi$
$942$		0			0
$943$	1154.12	−	666.330i	0.0398550	−	0.0230103i
$944$	8254.29			0.284592
$945$		0			0
$946$	−6367.04			−0.218827
$947$	−1616.42	+	933.242i	−0.0554664	+	0.0320235i	−0.527477	−	0.849569i	$-0.676862\pi$
$947$	−1616.42	+	933.242i	−0.0554664	+	0.0320235i	0.472010	+	0.881593i	$0.343528\pi$
$948$		0			0
$949$	12166.1	−	21072.4i	0.416154	−	0.720799i
$950$	−10618.1	−	18391.1i	−0.362627	−	0.628089i
$951$		0			0
$952$	14598.5	−	8260.05i	0.496995	−	0.281208i
$953$		−	58074.3i		−	1.97399i	−0.160752	−	0.986995i	$-0.551392\pi$
$953$		−	58074.3i		−	1.97399i	0.160752	−	0.986995i	$-0.448608\pi$
$954$		0			0
$955$	−4944.80	−	2854.88i	−0.167550	−	0.0967349i
$956$	374.827	+	216.407i	0.0126807	+	0.00732123i
$957$		0			0
$958$			12323.9i			0.415622i
$959$	−49699.1	+	28120.5i	−1.67348	+	0.946882i
$960$		0			0
$961$	−10840.2	−	18775.8i	−0.363875	−	0.630249i
$962$	−7824.14	+	13551.8i	−0.262225	+	0.454187i
$963$		0			0
$964$	10999.1	−	6350.32i	0.367486	−	0.212168i
$965$	9041.17			0.301602
$966$		0			0
$967$	26569.0			0.883560			0.441780	−	0.897123i	$-0.354347\pi$
$967$	26569.0			0.883560			0.441780	+	0.897123i	$0.354347\pi$
$968$	−6444.27	+	3720.60i	−0.213974	+	0.123538i
$969$		0			0
$970$	−898.344	+	1555.98i	−0.0297362	+	0.0515046i
$971$	−23035.8	−	39899.1i	−0.761331	−	1.31866i	−0.942164	−	0.335151i	$-0.891213\pi$
$971$	−23035.8	−	39899.1i	−0.761331	−	1.31866i	0.180833	−	0.983514i	$-0.442121\pi$
$972$		0			0
$973$	−26283.7	−	15481.1i	−0.865998	−	0.510074i
$974$			41725.3i			1.37266i
$975$		0			0
$976$	−8697.31	−	5021.40i	−0.285240	−	0.164683i
$977$	−13405.4	−	7739.62i	−0.438974	−	0.253442i	0.264188	−	0.964471i	$-0.414896\pi$
$977$	−13405.4	−	7739.62i	−0.438974	−	0.253442i	−0.703162	+	0.711029i	$0.748229\pi$
$978$		0			0
$979$			9353.22i			0.305342i
$980$	1452.02	+	2619.11i	0.0473297	+	0.0853718i
$981$		0			0
$982$	17286.0	+	29940.2i	0.561730	+	0.972944i
$983$	366.633	−	635.027i	0.0118960	−	0.0206045i	−0.860016	−	0.510267i	$-0.829547\pi$
$983$	366.633	−	635.027i	0.0118960	−	0.0206045i	0.871912	+	0.489662i	$0.162880\pi$
$984$		0			0
$985$	−7335.17	+	4234.96i	−0.237277	+	0.136992i
$986$	−3674.89			−0.118694
$987$		0			0
$988$	−16498.0			−0.531246
$989$	−1990.75	+	1149.36i	−0.0640064	+	0.0369541i
$990$		0			0
$991$	−6306.80	+	10923.7i	−0.202161	+	0.350154i	−0.949225	−	0.314599i	$-0.898130\pi$
$991$	−6306.80	+	10923.7i	−0.202161	+	0.350154i	0.747063	+	0.664753i	$0.231463\pi$
$992$	1440.94	+	2495.79i	0.0461190	+	0.0798804i
$993$		0			0
$994$	−19796.0	−	172.107i	−0.631680	−	0.00549185i
$995$			406.522i			0.0129524i
$996$		0			0
$997$	31623.9	+	18258.1i	1.00455	+	0.579978i	0.909592	−	0.415503i	$-0.136394\pi$
$997$	31623.9	+	18258.1i	1.00455	+	0.579978i	0.0949601	+	0.995481i	$0.469728\pi$
$998$	−6719.97	−	3879.78i	−0.213143	−	0.123058i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.b.269.6	yes	16
3.2	odd	2	inner	378.4.k.b.269.3	yes	16
7.5	odd	6	inner	378.4.k.b.215.3	✓	16
21.5	even	6	inner	378.4.k.b.215.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.b.215.3	✓	16	7.5	odd	6	inner
378.4.k.b.215.6	yes	16	21.5	even	6	inner
378.4.k.b.269.3	yes	16	3.2	odd	2	inner
378.4.k.b.269.6	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-1.09136 - 1.89028i) q^{5} +(-9.39922 + 15.9579i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-1.09136 - 1.89028i) q^{5} +(-9.39922 + 15.9579i) q^{7} -8.00000i q^{8} +(-3.78057 - 2.18271i) q^{10} +(17.3389 + 10.0106i) q^{11} -46.7045i q^{13} +(-0.322020 + 37.0391i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(56.6046 - 98.0420i) q^{17} +(76.4791 - 44.1553i) q^{19} -8.73084 q^{20} +40.0425 q^{22} +(12.5199 - 7.22837i) q^{23} +(60.1179 - 104.127i) q^{25} +(-46.7045 - 80.8945i) q^{26} +(36.4814 + 64.4757i) q^{28} -16.2305i q^{29} +(-77.9934 - 45.0295i) q^{31} +(-27.7128 - 16.0000i) q^{32} -226.418i q^{34} +(40.4228 + 0.351438i) q^{35} +(-83.7622 - 145.080i) q^{37} +(88.3105 - 152.958i) q^{38} +(-15.1223 + 8.73084i) q^{40} +92.1826 q^{41} -159.007 q^{43} +(69.3556 - 40.0425i) q^{44} +(14.4567 - 25.0398i) q^{46} +(182.310 + 315.771i) q^{47} +(-166.309 - 299.984i) q^{49} -240.472i q^{50} +(-161.789 - 93.4090i) q^{52} +(-210.389 - 121.468i) q^{53} -43.7006i q^{55} +(127.663 + 75.1937i) q^{56} +(-16.2305 - 28.1121i) q^{58} +(-257.947 + 446.777i) q^{59} +(543.582 - 313.837i) q^{61} -180.118 q^{62} -64.0000 q^{64} +(-88.2847 + 50.9712i) q^{65} +(430.357 - 745.400i) q^{67} +(-226.418 - 392.168i) q^{68} +(70.3659 - 39.8141i) q^{70} +534.461i q^{71} +(451.186 + 260.492i) q^{73} +(-290.161 - 167.524i) q^{74} -353.242i q^{76} +(-322.721 + 182.601i) q^{77} +(-525.897 - 910.881i) q^{79} +(-17.4617 + 30.2445i) q^{80} +(159.665 - 92.1826i) q^{82} +1472.54 q^{83} -247.103 q^{85} +(-275.408 + 159.007i) q^{86} +(80.0850 - 138.711i) q^{88} +(233.582 + 404.576i) q^{89} +(745.306 + 438.986i) q^{91} -57.8270i q^{92} +(631.542 + 364.621i) q^{94} +(-166.932 - 96.3782i) q^{95} -411.573i q^{97} +(-588.040 - 353.278i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 52 q^{7}+O(q^{10}) \) 16 * q + 32 * q^4 - 52 * q^7	\( 16 q + 32 q^{4} - 52 q^{7} - 24 q^{10} - 128 q^{16} - 186 q^{19} - 216 q^{22} - 44 q^{25} - 200 q^{28} + 408 q^{31} - 704 q^{37} - 96 q^{40} - 2036 q^{43} + 144 q^{46} - 20 q^{49} + 480 q^{52} - 672 q^{58} - 1242 q^{61} - 1024 q^{64} + 596 q^{67} - 48 q^{70} + 852 q^{73} - 2914 q^{79} + 1344 q^{82} - 10980 q^{85} - 432 q^{88} + 4134 q^{91} - 492 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 - 52 * q^7 - 24 * q^10 - 128 * q^16 - 186 * q^19 - 216 * q^22 - 44 * q^25 - 200 * q^28 + 408 * q^31 - 704 * q^37 - 96 * q^40 - 2036 * q^43 + 144 * q^46 - 20 * q^49 + 480 * q^52 - 672 * q^58 - 1242 * q^61 - 1024 * q^64 + 596 * q^67 - 48 * q^70 + 852 * q^73 - 2914 * q^79 + 1344 * q^82 - 10980 * q^85 - 432 * q^88 + 4134 * q^91 - 492 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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