LMFDB - Embedded newform 378.4.k.b.269.3

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			269.3
Root			$5.68309 - 3.28113i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.b.215.3

$q$-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	0.910696	−	0.413076i	$-0.135546\pi$
$5$	1.09136	+	1.89028i	0.0976138	+	0.169072i	−0.813083	+	0.582148i	$0.802212\pi$
$6$		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	0.243178	−	0.969982i	$-0.421810\pi$
$11$	−17.3389	−	10.0106i	−0.475262	−	0.274392i	−0.718440	+	0.695589i	$0.755143\pi$
$12$		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	$0.465882\pi$
$17$	−56.6046	+	98.0420i	−0.807566	+	1.39875i	−0.914545	+	0.404484i	$0.867451\pi$
$18$		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	−0.555677	−	0.831398i	$-0.687541\pi$
$23$	−12.5199	+	7.22837i	−0.113503	+	0.0655313i	0.442174	+	0.896930i	$0.354208\pi$
$24$		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	$0.0165482\pi$
$29$			16.2305i			0.103929i	−0.998649	+	0.0519644i	$0.983452\pi$
$30$		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	$-0.556176\pi$
$41$	−92.1826			−0.351134			−0.175567	+	0.984467i	$0.556176\pi$
$42$		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	$-0.975233\pi$
$47$	−182.310	−	315.771i	−0.565802	−	0.979998i	0.431172	−	0.902270i	$-0.358100\pi$
$48$		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	0.747211	−	0.664587i	$-0.231392\pi$
$53$	210.389	+	121.468i	0.545268	+	0.314810i	−0.201944	+	0.979397i	$0.564726\pi$
$54$		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	$-0.640593\pi$
$59$	257.947	−	446.777i	0.569183	−	0.985854i	0.996647	−	0.0818216i	$-0.0260738\pi$
$60$		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	$-0.852606\pi$
$71$		−	534.461i		−	0.893364i	0.894693	−	0.446682i	$-0.147394\pi$
$72$		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	$-0.926821\pi$
$83$	−1472.54			−1.94738			−0.973689	+	0.227880i	$0.926821\pi$
$84$		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	0.692739	−	0.721189i	$-0.256404\pi$
$89$	−233.582	−	404.576i	−0.278199	−	0.481854i	−0.970937	+	0.239335i	$0.923071\pi$
$90$		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	$-0.587810\pi$
$101$	707.575	−	1225.56i	0.697092	−	1.20740i	0.969470	−	0.245209i	$-0.0788565\pi$
$102$		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	−0.264321	−	0.964435i	$-0.585148\pi$
$107$	485.609	−	280.367i	0.438744	−	0.253309i	0.703065	+	0.711126i	$0.251814\pi$
$108$		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	$0.135998\pi$
$113$			995.485i			0.828738i	−0.910109	+	0.414369i	$0.864002\pi$
$114$		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	$0.0282533\pi$
$131$	861.833	+	1492.74i	0.574799	+	0.995582i	−0.421264	+	0.906938i	$0.638413\pi$
$132$		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	$-0.922069\pi$
$137$	−2670.20	−	1541.64i	−1.66518	−	0.961395i	−0.695006	−	0.719004i	$-0.744598\pi$
$138$		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	−0.0721867	−	0.997391i	$-0.522998\pi$
$149$	1374.06	−	793.314i	0.755486	−	0.436180i	0.827673	+	0.561211i	$0.189664\pi$
$150$		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	$-0.615372\pi$
$167$	−1530.40			−0.709136			−0.354568	+	0.935030i	$0.615372\pi$
$168$		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	0.683690	−	0.729772i	$-0.260374\pi$
$173$	−660.239	−	1143.57i	−0.290156	−	0.502565i	−0.973847	+	0.227207i	$0.927041\pi$
$174$		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	0.515238	−	0.857047i	$-0.327704\pi$
$179$	73.3579	+	42.3532i	0.0306314	+	0.0176851i	−0.484606	+	0.874732i	$0.661037\pi$
$180$		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	−0.863418	−	0.504488i	$-0.831681\pi$
$191$	−2265.44	+	1307.95i	−0.858228	+	0.495498i	0.00519060	+	0.999987i	$0.498348\pi$
$192$		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	$0.247577\pi$
$197$			3880.46i			1.40341i	−0.712469	+	0.701704i	$0.752423\pi$
$198$		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	−0.757609	−	0.652709i	$-0.773632\pi$
$227$	637.706	−	1104.54i	0.186458	−	0.322955i	0.944067	+	0.329754i	$0.106966\pi$
$228$		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	$-0.293657\pi$
$233$	5676.43	−	3277.29i	1.59603	−	0.921469i	0.992242	−	0.124324i	$-0.0396764\pi$
$234$		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	$-0.995339\pi$
$239$		−	108.203i		−	0.0292849i	0.999893	−	0.0146425i	$-0.00466100\pi$
$240$		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	$-0.612820\pi$
$251$	−2760.22			−0.694119			−0.347059	+	0.937843i	$0.612820\pi$
$252$		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	0.942620	−	0.333866i	$-0.108353\pi$
$257$	750.559	+	1300.01i	0.182174	+	0.315534i	−0.760447	+	0.649400i	$0.775020\pi$
$258$		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	0.224903	−	0.974381i	$-0.427794\pi$
$263$	−2160.23	−	1247.21i	−0.506485	−	0.292419i	−0.731388	+	0.681962i	$0.761127\pi$
$264$		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	−0.793085	−	0.609111i	$-0.791526\pi$
$269$	577.799	−	1000.78i	0.130963	−	0.226834i	0.924048	+	0.382276i	$0.124860\pi$
$270$		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	$0.128621\pi$
$281$			3703.99i			0.786339i	−0.919466	+	0.393170i	$0.871379\pi$
$282$		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	$-0.410027\pi$
$293$	2797.65			0.557816			0.278908	+	0.960318i	$0.410027\pi$
$294$		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	−0.707442	−	0.706772i	$-0.750151\pi$
$311$	1417.00	−	2454.31i	0.258362	−	0.447495i	0.965803	+	0.259276i	$0.0834841\pi$
$312$		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	−0.169458	−	0.985537i	$-0.554202\pi$
$317$	3382.54	−	1952.91i	0.599314	−	0.346014i	0.768772	+	0.639523i	$0.220868\pi$
$318$		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	−0.104402	−	0.994535i	$-0.533293\pi$
$347$	−6579.57	−	3798.72i	−1.01790	−	0.587682i	−0.913494	+	0.406853i	$0.866626\pi$
$348$		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	−0.848372	−	0.529400i	$-0.822417\pi$
$353$	227.406	−	393.878i	0.0342878	−	0.0593881i	0.882660	+	0.470012i	$0.155750\pi$
$354$		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	−0.844141	−	0.536121i	$-0.819889\pi$
$359$	−5454.70	+	3149.27i	−0.801917	+	0.462987i	0.0422241	+	0.999108i	$0.486556\pi$
$360$		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	$0.00760186\pi$
$383$	3901.67	+	6757.89i	0.520538	+	0.901598i	−0.479177	+	0.877718i	$0.659065\pi$
$384$		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	0.998447	−	0.0557073i	$-0.0177414\pi$
$389$	11120.4	+	6420.37i	1.44943	+	0.836827i	0.450980	+	0.892534i	$0.351075\pi$
$390$		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	−0.961247	−	0.275688i	$-0.911094\pi$
$401$	−9661.06	+	5577.81i	−1.20312	+	0.694620i	−0.241871	+	0.970309i	$0.577761\pi$
$402$		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	$-0.916857\pi$
$419$	−16571.6			−1.93216			−0.966080	+	0.258243i	$0.916857\pi$
$420$		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	−0.323062	−	0.946378i	$-0.604712\pi$
$431$	−11669.5	−	6737.41i	−1.30418	−	0.752969i	−0.981118	+	0.193409i	$0.938045\pi$
$432$		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	−0.834012	−	0.551746i	$-0.813962\pi$
$443$	−7209.31	+	4162.30i	−0.773193	+	0.446403i	0.0608195	+	0.998149i	$0.480629\pi$
$444$		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	$-0.935045\pi$
$449$		−	3856.07i		−	0.405299i	0.979251	−	0.202649i	$-0.0649552\pi$
$450$		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	$-0.214470\pi$
$461$	15470.1			1.56294			0.781470	+	0.623943i	$0.214470\pi$
$462$		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	$-0.872841\pi$
$467$	−8047.97	−	13939.5i	−0.797464	−	1.38125i	0.123799	−	0.992307i	$-0.460492\pi$
$468$		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	−0.680837	−	0.732435i	$-0.738384\pi$
$479$	3080.96	−	5336.39i	0.293889	−	0.509031i	0.974726	+	0.223404i	$0.0717170\pi$
$480$		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	$-0.707783\pi$
$491$		−	17286.0i		−	1.58881i	0.607388	−	0.794406i	$-0.292217\pi$
$492$		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	$-0.361055\pi$
$503$	9538.83			0.845557			0.422779	+	0.906233i	$0.361055\pi$
$504$		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	0.764987	−	0.644046i	$-0.222746\pi$
$509$	−2012.69	−	3486.08i	−0.175267	−	0.303572i	−0.940254	+	0.340475i	$0.889412\pi$
$510$		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	$-0.659260\pi$
$521$	6184.05	−	10711.1i	0.520015	−	0.900693i	0.999729	−	0.0232682i	$-0.00740716\pi$
$522$		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	$-0.959102\pi$
$557$	−21014.7	−	12132.8i	−1.59860	−	0.922953i	−0.606844	−	0.794821i	$-0.707565\pi$
$558$		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	−0.996607	−	0.0823109i	$-0.973770\pi$
$563$	−5704.41	+	9880.33i	−0.427020	+	0.739620i	0.569587	+	0.821931i	$0.307103\pi$
$564$		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	−0.0230919	−	0.999733i	$-0.507351\pi$
$569$	11281.1	−	6513.14i	0.831157	−	0.479868i	0.854249	+	0.519865i	$0.174018\pi$
$570$		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	$-0.488625\pi$
$587$	1016.23			0.0714551			0.0357276	+	0.999362i	$0.488625\pi$
$588$		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	0.956909	−	0.290389i	$-0.0937847\pi$
$593$	3277.56	+	5676.90i	0.226970	+	0.393124i	−0.729939	+	0.683513i	$0.760451\pi$
$594$		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	$-0.898856\pi$
$599$	−24856.2	−	14350.7i	−1.69548	−	0.978888i	−0.745544	−	0.666456i	$-0.767810\pi$
$600$		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	$-0.710838\pi$
$617$		−	24170.2i		−	1.57708i	0.614985	−	0.788539i	$-0.289162\pi$
$618$		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	0.149443	−	0.988770i	$-0.452252\pi$
$641$	−10258.8	−	5922.93i	−0.632136	−	0.364964i	−0.781579	+	0.623807i	$0.785585\pi$
$642$		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	−0.784327	−	0.620348i	$-0.786991\pi$
$647$	2387.51	−	4135.29i	0.145074	−	0.251275i	0.929401	+	0.369073i	$0.120325\pi$
$648$		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	−0.975738	−	0.218941i	$-0.929740\pi$
$653$	−21258.8	+	12273.8i	−1.27400	+	0.735543i	−0.298261	+	0.954484i	$0.596406\pi$
$654$		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	$0.348298\pi$
$659$			30064.1i			1.77713i	−0.458749	+	0.888566i	$0.651702\pi$
$660$		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	$-0.978235\pi$
$677$	−9829.19	−	17024.7i	−0.558001	−	0.966485i	0.439663	−	0.898163i	$-0.355098\pi$
$678$		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	0.477076	−	0.878862i	$-0.341697\pi$
$683$	−812.211	−	468.930i	−0.0455028	−	0.0262710i	−0.522579	+	0.852591i	$0.675030\pi$
$684$		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	$0.120526\pi$
$701$			13721.8i			0.739321i	−0.929167	+	0.369661i	$0.879474\pi$
$702$		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	$-0.923595\pi$
$719$	−13332.6	−	23092.8i	−0.691549	−	1.19780i	0.279782	−	0.960064i	$-0.409738\pi$
$720$		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	$-0.868690\pi$
$743$		−	16239.5i		−	0.801842i	0.916113	−	0.400921i	$-0.131310\pi$
$744$		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	$0.0821931\pi$
$761$	14791.1	+	25619.0i	0.704569	+	1.22035i	−0.262277	+	0.964993i	$0.584474\pi$
$762$		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	$-0.553807\pi$
$773$	16539.2	−	28646.7i	0.769564	−	1.33292i	0.937800	−	0.347176i	$-0.112860\pi$
$774$		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	$-0.622750\pi$
$797$	−16926.6			−0.752286			−0.376143	+	0.926562i	$0.622750\pi$
$798$		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	0.804579	−	0.593845i	$-0.202391\pi$
$809$	15936.6	+	9200.99i	0.692584	+	0.399863i	−0.111996	+	0.993709i	$0.535724\pi$
$810$		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	−0.840675	−	0.541541i	$-0.817841\pi$
$821$	−18631.7	+	10757.0i	−0.792024	+	0.457275i	0.0486508	+	0.998816i	$0.484508\pi$
$822$		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	$-0.997830\pi$
$827$		−	324.218i		−	0.0136326i	0.999977	−	0.00681631i	$-0.00216972\pi$
$828$		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	$-0.435114\pi$
$839$	9839.29			0.404875			0.202437	+	0.979295i	$0.435114\pi$
$840$		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	−0.628482	−	0.777824i	$-0.716324\pi$
$857$	9016.09	−	15616.3i	0.359374	−	0.622454i	0.987856	+	0.155370i	$0.0496569\pi$
$858$		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	0.330784	−	0.943706i	$-0.392687\pi$
$863$	33298.9	−	19225.1i	1.31345	−	0.758321i	0.982666	+	0.185386i	$0.0593535\pi$
$864$		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	$-0.375716\pi$
$881$	19905.3			0.761209			0.380605	+	0.924738i	$0.375716\pi$
$882$		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	$0.0826527\pi$
$887$	18639.7	+	32285.0i	0.705593	+	1.22212i	−0.260884	+	0.965370i	$0.584014\pi$
$888$		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	$-0.722221\pi$
$911$		−	42127.2i		−	1.53209i	0.642786	−	0.766046i	$-0.277779\pi$
$912$		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	0.648953	−	0.760829i	$-0.275207\pi$
$929$	−9469.27	−	16401.3i	−0.334421	−	0.579233i	−0.983373	+	0.181595i	$0.941874\pi$
$930$		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	−0.840773	−	0.541387i	$-0.817899\pi$
$941$	1399.08	−	2423.28i	0.0484684	−	0.0839497i	0.889242	+	0.457437i	$0.151233\pi$
$942$		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	−0.472010	−	0.881593i	$-0.656472\pi$
$947$	1616.42	−	933.242i	0.0554664	−	0.0320235i	0.527477	+	0.849569i	$0.323138\pi$
$948$		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	$0.448608\pi$
$953$			58074.3i			1.97399i	−0.160752	+	0.986995i	$0.551392\pi$
$954$		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	$0.108787\pi$
$971$	23035.8	+	39899.1i	0.761331	+	1.31866i	−0.180833	+	0.983514i	$0.557879\pi$
$972$		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	0.703162	−	0.711029i	$-0.251771\pi$
$977$	13405.4	+	7739.62i	0.438974	+	0.253442i	−0.264188	+	0.964471i	$0.585104\pi$
$978$		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	−0.871912	−	0.489662i	$-0.837120\pi$
$983$	−366.633	+	635.027i	−0.0118960	+	0.0206045i	0.860016	+	0.510267i	$0.170453\pi$
$984$		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.269.3	yes	16
3.2	odd	2	inner	378.4.k.b.269.6	yes	16
7.5	odd	6	inner	378.4.k.b.215.6	yes	16
21.5	even	6	inner	378.4.k.b.215.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.b.215.3	✓	16	21.5	even	6	inner
378.4.k.b.215.6	yes	16	7.5	odd	6	inner
378.4.k.b.269.3	yes	16	1.1	even	1	trivial
378.4.k.b.269.6	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
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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