LMFDB - Embedded newform 2340.2.bi.d.1061.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(161,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.161"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.bi (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.58498535041007616.52
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 $ x^12 - 12x^9 + 72x^6 - 324*x^3 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1061.4
Root			$1.72927 + 0.0980500i$ of defining polynomial
Character	$\chi$	$=$	2340.1061
Dual form			2340.2.bi.d.161.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{5} +(-1.63122 + 1.63122i) q^{7} +(-3.24154 - 3.24154i) q^{11} +(-3.21545 - 1.63122i) q^{13} +3.72111 q^{17} +(2.00000 + 2.00000i) q^{19} +4.17618 q^{23} -1.00000i q^{25} +5.06886i q^{29} +(-3.26245 - 3.26245i) q^{31} +2.30690i q^{35} +(-5.21545 + 5.21545i) q^{37} +(-7.85533 + 7.85533i) q^{41} +1.35644i q^{43} +(-4.17618 - 4.17618i) q^{47} +1.67822i q^{49} +11.0304i q^{53} -4.58423 q^{55} -4.49023 q^{61} +(-3.42711 + 1.12022i) q^{65} +(-8.43090 - 8.43090i) q^{67} +(-10.5508 + 10.5508i) q^{71} +(-6.84667 + 6.84667i) q^{73} +10.5753 q^{77} -10.3716 q^{79} +(5.13533 - 5.13533i) q^{83} +(2.63122 - 2.63122i) q^{85} +(3.61269 + 3.61269i) q^{89} +(7.90600 - 2.58423i) q^{91} +2.82843 q^{95} +(2.04700 + 2.04700i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) $ 12 * q + 12 * q^13 + 24 * q^19 - 12 * q^37 - 24 * q^55 - 12 * q^73 + 24 * q^79 + 12 * q^85 + 72 * q^91 + 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i
$6$		0			0
$7$	−1.63122	+	1.63122i	−0.616544	+	0.616544i	−0.944643	−	0.328099i	$-0.893592\pi$
$7$	−1.63122	+	1.63122i	−0.616544	+	0.616544i	0.328099	+	0.944643i	$0.393592\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.24154	−	3.24154i	−0.977360	−	0.977360i	0.0223892	−	0.999749i	$-0.492873\pi$
$11$	−3.24154	−	3.24154i	−0.977360	−	0.977360i	−0.999749	+	0.0223892i	$0.992873\pi$
$12$		0			0
$13$	−3.21545	−	1.63122i	−0.891805	−	0.452420i
$13$	−3.21545	−	1.63122i	−0.891805	−	0.452420i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.72111			0.902502			0.451251	−	0.892397i	$-0.350978\pi$
$17$	3.72111			0.902502			0.451251	+	0.892397i	$0.350978\pi$
$18$		0			0
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	0.898272	−	0.439440i	$-0.144823\pi$
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	−0.439440	+	0.898272i	$0.644823\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.17618			0.870793			0.435396	−	0.900239i	$-0.356608\pi$
$23$	4.17618			0.870793			0.435396	+	0.900239i	$0.356608\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.06886i			0.941264i	0.882330	+	0.470632i	$0.155974\pi$
$29$			5.06886i			0.941264i	−0.882330	+	0.470632i	$0.844026\pi$
$30$		0			0
$31$	−3.26245	−	3.26245i	−0.585953	−	0.585953i	0.350580	−	0.936533i	$-0.385984\pi$
$31$	−3.26245	−	3.26245i	−0.585953	−	0.585953i	−0.936533	+	0.350580i	$0.885984\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.30690i			0.389937i
$36$		0			0
$37$	−5.21545	+	5.21545i	−0.857414	+	0.857414i	−0.991033	−	0.133618i	$-0.957340\pi$
$37$	−5.21545	+	5.21545i	−0.857414	+	0.857414i	0.133618	+	0.991033i	$0.457340\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.85533	+	7.85533i	−1.22680	+	1.22680i	−0.261628	+	0.965169i	$0.584259\pi$
$41$	−7.85533	+	7.85533i	−1.22680	+	1.22680i	−0.965169	+	0.261628i	$0.915741\pi$
$42$		0			0
$43$			1.35644i			0.206856i	0.994637	+	0.103428i	$0.0329811\pi$
$43$			1.35644i			0.206856i	−0.994637	+	0.103428i	$0.967019\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.17618	−	4.17618i	−0.609158	−	0.609158i	0.333568	−	0.942726i	$-0.391747\pi$
$47$	−4.17618	−	4.17618i	−0.609158	−	0.609158i	−0.942726	+	0.333568i	$0.891747\pi$
$48$		0			0
$49$			1.67822i			0.239746i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.0304i			1.51514i	0.652752	+	0.757571i	$0.273614\pi$
$53$			11.0304i			1.51514i	−0.652752	+	0.757571i	$0.726386\pi$
$54$		0			0
$55$	−4.58423			−0.618137
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−4.49023			−0.574915			−0.287458	−	0.957793i	$-0.592810\pi$
$61$	−4.49023			−0.574915			−0.287458	+	0.957793i	$0.592810\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.42711	+	1.12022i	−0.425081	+	0.138946i
$66$		0			0
$67$	−8.43090	−	8.43090i	−1.03000	−	1.03000i	−0.999536	−	0.0304621i	$-0.990302\pi$
$67$	−8.43090	−	8.43090i	−1.03000	−	1.03000i	−0.0304621	−	0.999536i	$-0.509698\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.5508	+	10.5508i	−1.25215	+	1.25215i	−0.297400	+	0.954753i	$0.596120\pi$
$71$	−10.5508	+	10.5508i	−1.25215	+	1.25215i	−0.954753	+	0.297400i	$0.903880\pi$
$72$		0			0
$73$	−6.84667	+	6.84667i	−0.801342	+	0.801342i	−0.983305	−	0.181963i	$-0.941755\pi$
$73$	−6.84667	+	6.84667i	−0.801342	+	0.801342i	0.181963	+	0.983305i	$0.441755\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.5753			1.20517
$78$		0			0
$79$	−10.3716			−1.16689			−0.583446	−	0.812152i	$-0.698296\pi$
$79$	−10.3716			−1.16689			−0.583446	+	0.812152i	$0.698296\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.13533	−	5.13533i	0.563675	−	0.563675i	−0.366674	−	0.930349i	$-0.619504\pi$
$83$	5.13533	−	5.13533i	0.563675	−	0.563675i	0.930349	+	0.366674i	$0.119504\pi$
$84$		0			0
$85$	2.63122	−	2.63122i	0.285396	−	0.285396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.61269	+	3.61269i	0.382945	+	0.382945i	0.872162	−	0.489217i	$-0.162717\pi$
$89$	3.61269	+	3.61269i	0.382945	+	0.382945i	−0.489217	+	0.872162i	$0.662717\pi$
$90$		0			0
$91$	7.90600	−	2.58423i	0.828774	−	0.270900i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.82843			0.290191
$96$		0			0
$97$	2.04700	+	2.04700i	0.207841	+	0.207841i	0.803349	−	0.595508i	$-0.203049\pi$
$97$	2.04700	+	2.04700i	0.207841	+	0.207841i	−0.595508	+	0.803349i	$0.703049\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.16094			0.613036			0.306518	−	0.951865i	$-0.400836\pi$
$101$	6.16094			0.613036			0.306518	+	0.951865i	$0.400836\pi$
$102$		0			0
$103$		−	17.5993i		−	1.73412i	−0.498208	−	0.867058i	$-0.666008\pi$
$103$		−	17.5993i		−	1.73412i	0.498208	−	0.867058i	$-0.333992\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.34775i			0.130292i	0.997876	+	0.0651459i	$0.0207513\pi$
$107$			1.34775i			0.130292i	−0.997876	+	0.0651459i	$0.979249\pi$
$108$		0			0
$109$	−10.7527	−	10.7527i	−1.02992	−	1.02992i	−0.999538	−	0.0303812i	$-0.990328\pi$
$109$	−10.7527	−	10.7527i	−1.02992	−	1.02992i	−0.0303812	−	0.999538i	$-0.509672\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.11192i		−	0.574961i	−0.957787	−	0.287480i	$-0.907182\pi$
$113$		−	6.11192i		−	0.574961i	0.957787	−	0.287480i	$-0.0928176\pi$
$114$		0			0
$115$	2.95300	−	2.95300i	0.275369	−	0.275369i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.06996	+	6.06996i	−0.556433	+	0.556433i
$120$		0			0
$121$			10.0151i			0.910466i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.0632456	−	0.0632456i
$126$		0			0
$127$		−	0.430897i		−	0.0382360i	−0.999817	−	0.0191180i	$-0.993914\pi$
$127$		−	0.430897i		−	0.0382360i	0.999817	−	0.0191180i	$-0.00608581\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.39917i			0.559098i	0.960131	+	0.279549i	$0.0901849\pi$
$131$			6.39917i			0.559098i	−0.960131	+	0.279549i	$0.909815\pi$
$132$		0			0
$133$	−6.52489			−0.565780
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.388599	+	0.388599i	0.0332003	+	0.0332003i	0.723512	−	0.690312i	$-0.242527\pi$
$137$	0.388599	+	0.388599i	0.0332003	+	0.0332003i	−0.690312	+	0.723512i	$0.742527\pi$
$138$		0			0
$139$	−14.3716			−1.21898			−0.609490	−	0.792794i	$-0.708626\pi$
$139$	−14.3716			−1.21898			−0.609490	+	0.792794i	$0.708626\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.13533	+	15.7107i	0.429437	+	1.31379i
$144$		0			0
$145$	3.58423	+	3.58423i	0.297654	+	0.297654i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.3384	−	14.3384i	1.17465	−	1.17465i	0.193560	−	0.981089i	$-0.437997\pi$
$149$	14.3384	−	14.3384i	1.17465	−	1.17465i	0.981089	−	0.193560i	$-0.0620033\pi$
$150$		0			0
$151$	−13.7527	+	13.7527i	−1.11918	+	1.11918i	−0.127315	+	0.991862i	$0.540636\pi$
$151$	−13.7527	+	13.7527i	−1.11918	+	1.11918i	−0.991862	+	0.127315i	$0.959364\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.61380			−0.370589
$156$		0			0
$157$	−9.04979			−0.722252			−0.361126	−	0.932517i	$-0.617608\pi$
$157$	−9.04979			−0.722252			−0.361126	+	0.932517i	$0.617608\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.81228	+	6.81228i	−0.536882	+	0.536882i
$162$		0			0
$163$	14.1561	−	14.1561i	1.10879	−	1.10879i	0.115483	−	0.993309i	$-0.463158\pi$
$163$	14.1561	−	14.1561i	1.10879	−	1.10879i	0.993309	−	0.115483i	$-0.0368416\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.34775	+	1.34775i	0.104292	+	0.104292i	0.757327	−	0.653035i	$-0.226505\pi$
$167$	1.34775	+	1.34775i	0.104292	+	0.104292i	−0.653035	+	0.757327i	$0.726505\pi$
$168$		0			0
$169$	7.67822	+	10.4902i	0.590632	+	0.806941i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.06886			0.385378			0.192689	−	0.981260i	$-0.438279\pi$
$173$	5.06886			0.385378			0.192689	+	0.981260i	$0.438279\pi$
$174$		0			0
$175$	1.63122	+	1.63122i	0.123309	+	0.123309i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.00221			0.149652			0.0748260	−	0.997197i	$-0.476160\pi$
$179$	2.00221			0.149652			0.0748260	+	0.997197i	$0.476160\pi$
$180$		0			0
$181$			26.0649i			1.93739i	0.248255	+	0.968695i	$0.420143\pi$
$181$			26.0649i			1.93739i	−0.248255	+	0.968695i	$0.579857\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			7.37576i			0.542277i
$186$		0			0
$187$	−12.0621	−	12.0621i	−0.882070	−	0.882070i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.0538i			0.727468i	0.931503	+	0.363734i	$0.118498\pi$
$191$			10.0538i			0.727468i	−0.931503	+	0.363734i	$0.881502\pi$
$192$		0			0
$193$	−13.7403	+	13.7403i	−0.989051	+	0.989051i	−0.999941	−	0.0108893i	$-0.996534\pi$
$193$	−13.7403	+	13.7403i	−0.989051	+	0.989051i	0.0108893	+	0.999941i	$0.496534\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.2472	+	11.2472i	−0.801333	+	0.801333i	−0.983304	−	0.181971i	$-0.941752\pi$
$197$	−11.2472	+	11.2472i	−0.801333	+	0.801333i	0.181971	+	0.983304i	$0.441752\pi$
$198$		0			0
$199$		−	1.16845i		−	0.0828293i	−0.999142	−	0.0414146i	$-0.986814\pi$
$199$		−	1.16845i		−	0.0828293i	0.999142	−	0.0414146i	$-0.0131865\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.26844	−	8.26844i	−0.580331	−	0.580331i
$204$		0			0
$205$			11.1091i			0.775894i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	12.9661i		−	0.896887i
$210$		0			0
$211$	14.2182			0.978824			0.489412	−	0.872053i	$-0.337211\pi$
$211$	14.2182			0.978824			0.489412	+	0.872053i	$0.337211\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.959150	+	0.959150i	0.0654135	+	0.0654135i
$216$		0			0
$217$	10.6436			0.722532
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−11.9650	−	6.06996i	−0.804856	−	0.408310i
$222$		0			0
$223$	7.90600	+	7.90600i	0.529425	+	0.529425i	0.920401	−	0.390976i	$-0.127862\pi$
$223$	7.90600	+	7.90600i	0.529425	+	0.529425i	−0.390976	+	0.920401i	$0.627862\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.17599	+	1.17599i	−0.0780531	+	0.0780531i	−0.745056	−	0.667002i	$-0.767577\pi$
$227$	−1.17599	+	1.17599i	−0.0780531	+	0.0780531i	0.667002	+	0.745056i	$0.267577\pi$
$228$		0			0
$229$	4.26245	−	4.26245i	0.281670	−	0.281670i	−0.552105	−	0.833775i	$-0.686175\pi$
$229$	4.26245	−	4.26245i	0.281670	−	0.281670i	0.833775	+	0.552105i	$0.186175\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.2881			−0.673995			−0.336998	−	0.941505i	$-0.609411\pi$
$233$	−10.2881			−0.673995			−0.336998	+	0.941505i	$0.609411\pi$
$234$		0			0
$235$	−5.90600			−0.385265
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.20069	+	4.20069i	−0.271720	+	0.271720i	−0.829792	−	0.558072i	$-0.811541\pi$
$239$	−4.20069	+	4.20069i	−0.271720	+	0.271720i	0.558072	+	0.829792i	$0.311541\pi$
$240$		0			0
$241$	6.41577	−	6.41577i	0.413276	−	0.413276i	−0.469602	−	0.882878i	$-0.655603\pi$
$241$	6.41577	−	6.41577i	0.413276	−	0.413276i	0.882878	+	0.469602i	$0.155603\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.18668	+	1.18668i	0.0758143	+	0.0758143i
$246$		0			0
$247$	−3.16845	−	9.69334i	−0.201604	−	0.616773i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.26403			−0.269143			−0.134572	−	0.990904i	$-0.542966\pi$
$251$	−4.26403			−0.269143			−0.134572	+	0.990904i	$0.542966\pi$
$252$		0			0
$253$	−13.5372	−	13.5372i	−0.851078	−	0.851078i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22.5159			1.40450			0.702251	−	0.711930i	$-0.252179\pi$
$257$	22.5159			1.40450			0.702251	+	0.711930i	$0.252179\pi$
$258$		0			0
$259$		−	17.0151i		−	1.05727i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.39917i		−	0.394589i	−0.980344	−	0.197295i	$-0.936784\pi$
$263$		−	6.39917i		−	0.394589i	0.980344	−	0.197295i	$-0.0632156\pi$
$264$		0			0
$265$	7.79967	+	7.79967i	0.479130	+	0.479130i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	13.6032i		−	0.829399i	−0.909958	−	0.414700i	$-0.863887\pi$
$269$		−	13.6032i		−	0.829399i	0.909958	−	0.414700i	$-0.136113\pi$
$270$		0			0
$271$	0.584225	−	0.584225i	0.0354892	−	0.0354892i	−0.689140	−	0.724629i	$-0.742011\pi$
$271$	0.584225	−	0.584225i	0.0354892	−	0.0354892i	0.724629	+	0.689140i	$0.242011\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.24154	+	3.24154i	−0.195472	+	0.195472i
$276$		0			0
$277$		−	26.2429i		−	1.57678i	−0.615174	−	0.788392i	$-0.710914\pi$
$277$		−	26.2429i		−	1.57678i	0.615174	−	0.788392i	$-0.289086\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	17.7968	+	17.7968i	1.06167	+	1.06167i	0.997969	+	0.0636978i	$0.0202894\pi$
$281$	17.7968	+	17.7968i	1.06167	+	1.06167i	0.0636978	+	0.997969i	$0.479711\pi$
$282$		0			0
$283$			21.9060i			1.30218i	0.759002	+	0.651088i	$0.225687\pi$
$283$			21.9060i			1.30218i	−0.759002	+	0.651088i	$0.774313\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	25.6276i		−	1.51275i
$288$		0			0
$289$	−3.15333			−0.185490
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.5391	−	18.5391i	−1.08307	−	1.08307i	−0.996222	−	0.0868434i	$-0.972322\pi$
$293$	−18.5391	−	18.5391i	−1.08307	−	1.08307i	−0.0868434	−	0.996222i	$-0.527678\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.4283	−	6.81228i	−0.776577	−	0.393964i
$300$		0			0
$301$	−2.21266	−	2.21266i	−0.127536	−	0.127536i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.17507	+	3.17507i	−0.181804	+	0.181804i
$306$		0			0
$307$	2.70568	−	2.70568i	0.154421	−	0.154421i	−0.625668	−	0.780089i	$-0.715174\pi$
$307$	2.70568	−	2.70568i	0.154421	−	0.154421i	0.780089	+	0.625668i	$0.215174\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.69991			0.379917			0.189959	−	0.981792i	$-0.439165\pi$
$311$	6.69991			0.379917			0.189959	+	0.981792i	$0.439165\pi$
$312$		0			0
$313$	0.187991			0.0106259			0.00531295	−	0.999986i	$-0.498309\pi$
$313$	0.187991			0.0106259			0.00531295	+	0.999986i	$0.498309\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.46582	−	9.46582i	0.531653	−	0.531653i	−0.389411	−	0.921064i	$-0.627321\pi$
$317$	9.46582	−	9.46582i	0.531653	−	0.531653i	0.921064	+	0.389411i	$0.127321\pi$
$318$		0			0
$319$	16.4309	−	16.4309i	0.919954	−	0.919954i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.44222	+	7.44222i	0.414096	+	0.414096i
$324$		0			0
$325$	−1.63122	+	3.21545i	−0.0904840	+	0.178361i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	13.6246			0.751146
$330$		0			0
$331$	10.8618	+	10.8618i	0.597018	+	0.597018i	0.939518	−	0.342500i	$-0.111274\pi$
$331$	10.8618	+	10.8618i	0.597018	+	0.597018i	−0.342500	+	0.939518i	$0.611274\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.9231			−0.651428
$336$		0			0
$337$			13.1685i			0.717331i	0.933466	+	0.358666i	$0.116768\pi$
$337$			13.1685i			0.717331i	−0.933466	+	0.358666i	$0.883232\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			21.1507i			1.14537i
$342$		0			0
$343$	−14.1561	−	14.1561i	−0.764358	−	0.764358i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	31.7260i		−	1.70314i	−0.524239	−	0.851571i	$-0.675650\pi$
$347$		−	31.7260i		−	1.70314i	0.524239	−	0.851571i	$-0.324350\pi$
$348$		0			0
$349$	−1.73755	+	1.73755i	−0.0930091	+	0.0930091i	−0.752080	−	0.659071i	$-0.770950\pi$
$349$	−1.73755	+	1.73755i	−0.0930091	+	0.0930091i	0.659071	+	0.752080i	$0.270950\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.39935	−	9.39935i	0.500277	−	0.500277i	−0.411247	−	0.911524i	$-0.634907\pi$
$353$	9.39935	−	9.39935i	0.500277	−	0.500277i	0.911524	+	0.411247i	$0.134907\pi$
$354$		0			0
$355$			14.9211i			0.791931i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	23.2368	+	23.2368i	1.22639	+	1.22639i	0.965319	+	0.261072i	$0.0840760\pi$
$359$	23.2368	+	23.2368i	1.22639	+	1.22639i	0.261072	+	0.965319i	$0.415924\pi$
$360$		0			0
$361$		−	11.0000i		−	0.578947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			9.68266i			0.506813i
$366$		0			0
$367$	−24.4309			−1.27528			−0.637641	−	0.770333i	$-0.720090\pi$
$367$	−24.4309			−1.27528			−0.637641	+	0.770333i	$0.720090\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−17.9931	−	17.9931i	−0.934153	−	0.934153i
$372$		0			0
$373$	24.3122			1.25884			0.629420	−	0.777065i	$-0.283293\pi$
$373$	24.3122			1.25884			0.629420	+	0.777065i	$0.283293\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.26844	−	16.2987i	0.425846	−	0.839424i
$378$		0			0
$379$	−22.3122	−	22.3122i	−1.14610	−	1.14610i	−0.987312	−	0.158790i	$-0.949241\pi$
$379$	−22.3122	−	22.3122i	−1.14610	−	1.14610i	−0.158790	−	0.987312i	$-0.550759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.4877	+	13.4877i	−0.689188	+	0.689188i	−0.962052	−	0.272864i	$-0.912029\pi$
$383$	−13.4877	+	13.4877i	−0.689188	+	0.689188i	0.272864	+	0.962052i	$0.412029\pi$
$384$		0			0
$385$	7.47790	−	7.47790i	0.381109	−	0.381109i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.8200			−0.700702			−0.350351	−	0.936618i	$-0.613938\pi$
$389$	−13.8200			−0.700702			−0.350351	+	0.936618i	$0.613938\pi$
$390$		0			0
$391$	15.5400			0.785892
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.33380	+	7.33380i	−0.369004	+	0.369004i
$396$		0			0
$397$	12.5719	−	12.5719i	0.630965	−	0.630965i	−0.317345	−	0.948310i	$-0.602791\pi$
$397$	12.5719	−	12.5719i	0.630965	−	0.630965i	0.948310	+	0.317345i	$0.102791\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.89729	−	7.89729i	−0.394372	−	0.394372i	0.481871	−	0.876242i	$-0.339958\pi$
$401$	−7.89729	−	7.89729i	−0.394372	−	0.394372i	−0.876242	+	0.481871i	$0.839958\pi$
$402$		0			0
$403$	5.16845	+	15.8120i	0.257459	+	0.787652i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	33.8121			1.67601
$408$		0			0
$409$	7.37157	+	7.37157i	0.364500	+	0.364500i	0.865467	−	0.500966i	$-0.167022\pi$
$409$	7.37157	+	7.37157i	0.364500	+	0.364500i	−0.500966	+	0.865467i	$0.667022\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	7.26245i		−	0.356499i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	18.0564i		−	0.882113i	−0.897479	−	0.441056i	$-0.854604\pi$
$419$		−	18.0564i		−	0.882113i	0.897479	−	0.441056i	$-0.145396\pi$
$420$		0			0
$421$	24.9363	+	24.9363i	1.21532	+	1.21532i	0.969253	+	0.246065i	$0.0791377\pi$
$421$	24.9363	+	24.9363i	1.21532	+	1.21532i	0.246065	+	0.969253i	$0.420862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	3.72111i		−	0.180500i
$426$		0			0
$427$	7.32457	−	7.32457i	0.354461	−	0.354461i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.8053	+	24.8053i	−1.19483	+	1.19483i	−0.219136	+	0.975694i	$0.570324\pi$
$431$	−24.8053	+	24.8053i	−1.19483	+	1.19483i	−0.975694	+	0.219136i	$0.929676\pi$
$432$		0			0
$433$			34.5798i			1.66180i	0.556422	+	0.830900i	$0.312174\pi$
$433$			34.5798i			1.66180i	−0.556422	+	0.830900i	$0.687826\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.35235	+	8.35235i	0.399547	+	0.399547i
$438$		0			0
$439$			31.8467i			1.51996i	0.649947	+	0.759979i	$0.274791\pi$
$439$			31.8467i			1.51996i	−0.649947	+	0.759979i	$0.725209\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5.35216i			0.254289i	0.991884	+	0.127144i	$0.0405812\pi$
$443$			5.35216i			0.254289i	−0.991884	+	0.127144i	$0.959419\pi$
$444$		0			0
$445$	5.10912			0.242195
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.1496	+	20.1496i	0.950917	+	0.950917i	0.998851	−	0.0479332i	$-0.0152634\pi$
$449$	20.1496	+	20.1496i	0.950917	+	0.950917i	−0.0479332	+	0.998851i	$0.515263\pi$
$450$		0			0
$451$	50.9267			2.39804
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.76307	−	7.41771i	0.176415	−	0.347748i
$456$		0			0
$457$	−15.9530	−	15.9530i	−0.746250	−	0.746250i	0.227523	−	0.973773i	$-0.426937\pi$
$457$	−15.9530	−	15.9530i	−0.746250	−	0.746250i	−0.973773	+	0.227523i	$0.926937\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.77830	−	1.77830i	0.0828238	−	0.0828238i	−0.664481	−	0.747305i	$-0.731347\pi$
$461$	1.77830	−	1.77830i	0.0828238	−	0.0828238i	0.747305	+	0.664481i	$0.231347\pi$
$462$		0			0
$463$	19.0426	−	19.0426i	0.884984	−	0.884984i	−0.109052	−	0.994036i	$-0.534782\pi$
$463$	19.0426	−	19.0426i	0.884984	−	0.884984i	0.994036	+	0.109052i	$0.0347816\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.69568			0.263565			0.131782	−	0.991279i	$-0.457930\pi$
$467$	5.69568			0.263565			0.131782	+	0.991279i	$0.457930\pi$
$468$		0			0
$469$	27.5054			1.27008
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.39696	−	4.39696i	0.202172	−	0.202172i
$474$		0			0
$475$	2.00000	−	2.00000i	0.0917663	−	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.5251	−	24.5251i	−1.12058	−	1.12058i	−0.991654	−	0.128929i	$-0.958846\pi$
$479$	−24.5251	−	24.5251i	−1.12058	−	1.12058i	−0.128929	−	0.991654i	$-0.541154\pi$
$480$		0			0
$481$	25.2776	−	8.26245i	1.15256	−	0.376735i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.89489			0.131450
$486$		0			0
$487$	−11.4186	−	11.4186i	−0.517424	−	0.517424i	0.399367	−	0.916791i	$-0.369230\pi$
$487$	−11.4186	−	11.4186i	−0.517424	−	0.517424i	−0.916791	+	0.399367i	$0.869230\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−32.7240			−1.47681			−0.738407	−	0.674356i	$-0.764422\pi$
$491$	−32.7240			−1.47681			−0.738407	+	0.674356i	$0.764422\pi$
$492$		0			0
$493$			18.8618i			0.849493i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	34.4215i		−	1.54402i
$498$		0			0
$499$	20.0649	+	20.0649i	0.898229	+	0.898229i	0.995279	−	0.0970507i	$-0.0309409\pi$
$499$	20.0649	+	20.0649i	0.898229	+	0.898229i	−0.0970507	+	0.995279i	$0.530941\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.91896i		−	0.397677i	−0.980032	−	0.198838i	$-0.936283\pi$
$503$		−	8.91896i		−	0.397677i	0.980032	−	0.198838i	$-0.0637169\pi$
$504$		0			0
$505$	4.35644	−	4.35644i	0.193859	−	0.193859i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.4927	−	14.4927i	0.642379	−	0.642379i	−0.308761	−	0.951140i	$-0.599914\pi$
$509$	14.4927	−	14.4927i	0.642379	−	0.642379i	0.951140	+	0.308761i	$0.0999143\pi$
$510$		0			0
$511$		−	22.3369i		−	0.988126i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.4446	−	12.4446i	−0.548375	−	0.548375i
$516$		0			0
$517$			27.0745i			1.19073i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	15.6893i		−	0.687360i	−0.939087	−	0.343680i	$-0.888326\pi$
$521$		−	15.6893i		−	0.687360i	0.939087	−	0.343680i	$-0.111674\pi$
$522$		0			0
$523$	−9.04421			−0.395476			−0.197738	−	0.980255i	$-0.563359\pi$
$523$	−9.04421			−0.395476			−0.197738	+	0.980255i	$0.563359\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.1399	−	12.1399i	−0.528824	−	0.528824i
$528$		0			0
$529$	−5.55956			−0.241720
$530$		0			0
$531$		0			0
$532$		0			0
$533$	38.0722	−	12.4446i	1.64909	−	0.539036i
$534$		0			0
$535$	0.953002	+	0.953002i	0.0412019	+	0.0412019i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.44002	−	5.44002i	0.234318	−	0.234318i
$540$		0			0
$541$	−12.1338	+	12.1338i	−0.521672	+	0.521672i	−0.918076	−	0.396404i	$-0.870258\pi$
$541$	−12.1338	+	12.1338i	−0.521672	+	0.521672i	0.396404	+	0.918076i	$0.370258\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.2066			−0.651378
$546$		0			0
$547$	44.3671			1.89700			0.948501	−	0.316774i	$-0.102600\pi$
$547$	44.3671			1.89700			0.948501	+	0.316774i	$0.102600\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.1377	+	10.1377i	−0.431881	+	0.431881i
$552$		0			0
$553$	16.9183	−	16.9183i	0.719441	−	0.719441i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.26844	+	8.26844i	0.350345	+	0.350345i	0.860238	−	0.509893i	$-0.170315\pi$
$557$	8.26844	+	8.26844i	0.350345	+	0.350345i	−0.509893	+	0.860238i	$0.670315\pi$
$558$		0			0
$559$	2.21266	−	4.36157i	0.0935856	−	0.184475i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.5367			−0.570504			−0.285252	−	0.958453i	$-0.592077\pi$
$563$	−13.5367			−0.570504			−0.285252	+	0.958453i	$0.592077\pi$
$564$		0			0
$565$	−4.32178	−	4.32178i	−0.181819	−	0.181819i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	41.1389			1.72463			0.862316	−	0.506371i	$-0.169013\pi$
$569$	41.1389			1.72463			0.862316	+	0.506371i	$0.169013\pi$
$570$		0			0
$571$		−	5.57910i		−	0.233478i	−0.993163	−	0.116739i	$-0.962756\pi$
$571$		−	5.57910i		−	0.233478i	0.993163	−	0.116739i	$-0.0372441\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.17618i		−	0.174159i
$576$		0			0
$577$	3.30944	+	3.30944i	0.137774	+	0.137774i	0.772630	−	0.634856i	$-0.218941\pi$
$577$	3.30944	+	3.30944i	0.137774	+	0.137774i	−0.634856	+	0.772630i	$0.718941\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			16.7537i			0.695062i
$582$		0			0
$583$	35.7555	−	35.7555i	1.48084	−	1.48084i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.1917	−	23.1917i	0.957225	−	0.957225i	−0.0418972	−	0.999122i	$-0.513340\pi$
$587$	23.1917	−	23.1917i	0.957225	−	0.957225i	0.999122	+	0.0418972i	$0.0133402\pi$
$588$		0			0
$589$		−	13.0498i		−	0.537707i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.06665	−	3.06665i	−0.125932	−	0.125932i	0.641332	−	0.767264i	$-0.278382\pi$
$593$	−3.06665	−	3.06665i	−0.125932	−	0.125932i	−0.767264	+	0.641332i	$0.778382\pi$
$594$		0			0
$595$			8.58423i			0.351919i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.658403i		−	0.0269016i	−0.999910	−	0.0134508i	$-0.995718\pi$
$599$		−	0.658403i		−	0.0269016i	0.999910	−	0.0134508i	$-0.00428165\pi$
$600$		0			0
$601$	−23.9653			−0.977566			−0.488783	−	0.872405i	$-0.662559\pi$
$601$	−23.9653			−0.977566			−0.488783	+	0.872405i	$0.662559\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.08176	+	7.08176i	0.287915	+	0.287915i
$606$		0			0
$607$	−2.33690			−0.0948519			−0.0474260	−	0.998875i	$-0.515102\pi$
$607$	−2.33690			−0.0948519			−0.0474260	+	0.998875i	$0.515102\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.61600	+	20.2406i	0.267655	+	0.818845i
$612$		0			0
$613$	20.6961	+	20.6961i	0.835909	+	0.835909i	0.988318	−	0.152409i	$-0.0487030\pi$
$613$	20.6961	+	20.6961i	0.835909	+	0.835909i	−0.152409	+	0.988318i	$0.548703\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.7623	−	23.7623i	0.956633	−	0.956633i	−0.0424650	−	0.999098i	$-0.513521\pi$
$617$	23.7623	−	23.7623i	0.956633	−	0.956633i	0.999098	+	0.0424650i	$0.0135211\pi$
$618$		0			0
$619$	−0.0593323	+	0.0593323i	−0.00238477	+	0.00238477i	−0.708298	−	0.705913i	$-0.750537\pi$
$619$	−0.0593323	+	0.0593323i	−0.00238477	+	0.00238477i	0.705913	+	0.708298i	$0.250537\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.7862			−0.472205
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−19.4073	+	19.4073i	−0.773818	+	0.773818i
$630$		0			0
$631$	6.09400	−	6.09400i	0.242598	−	0.242598i	−0.575326	−	0.817924i	$-0.695125\pi$
$631$	6.09400	−	6.09400i	0.242598	−	0.242598i	0.817924	+	0.575326i	$0.195125\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.304691	−	0.304691i	−0.0120913	−	0.0120913i
$636$		0			0
$637$	2.73755	−	5.39623i	0.108466	−	0.213807i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−32.6187			−1.28836			−0.644181	−	0.764873i	$-0.722802\pi$
$641$	−32.6187			−1.28836			−0.644181	+	0.764873i	$0.722802\pi$
$642$		0			0
$643$	−31.9435	−	31.9435i	−1.25973	−	1.25973i	−0.951223	−	0.308504i	$-0.900172\pi$
$643$	−31.9435	−	31.9435i	−1.25973	−	1.25973i	−0.308504	−	0.951223i	$-0.599828\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	26.6358			1.04716			0.523580	−	0.851976i	$-0.324596\pi$
$647$	26.6358			1.04716			0.523580	+	0.851976i	$0.324596\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	0.154317i		−	0.00603889i	−0.999995	−	0.00301945i	$-0.999039\pi$
$653$		−	0.154317i		−	0.00603889i	0.999995	−	0.00301945i	$-0.000961121\pi$
$654$		0			0
$655$	4.52489	+	4.52489i	0.176802	+	0.176802i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	45.9070i		−	1.78828i	−0.447785	−	0.894141i	$-0.647787\pi$
$659$		−	45.9070i		−	1.78828i	0.447785	−	0.894141i	$-0.352213\pi$
$660$		0			0
$661$	−31.0302	+	31.0302i	−1.20694	+	1.20694i	−0.234923	+	0.972014i	$0.575484\pi$
$661$	−31.0302	+	31.0302i	−1.20694	+	1.20694i	−0.972014	+	0.234923i	$0.924516\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.61380	+	4.61380i	−0.178915	+	0.178915i
$666$		0			0
$667$			21.1685i			0.819646i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.5552	+	14.5552i	0.561899	+	0.561899i
$672$		0			0
$673$			17.3811i			0.669993i	0.942219	+	0.334996i	$0.108735\pi$
$673$			17.3811i			0.669993i	−0.942219	+	0.334996i	$0.891265\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	37.0607i		−	1.42436i	−0.701997	−	0.712180i	$-0.747708\pi$
$677$		−	37.0607i		−	1.42436i	0.701997	−	0.712180i	$-0.252292\pi$
$678$		0			0
$679$	−6.67822			−0.256287
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.09668	−	8.09668i	−0.309811	−	0.309811i	0.535025	−	0.844836i	$-0.320302\pi$
$683$	−8.09668	−	8.09668i	−0.309811	−	0.309811i	−0.844836	+	0.535025i	$0.820302\pi$
$684$		0			0
$685$	0.549562			0.0209977
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17.9931	−	35.4677i	0.685481	−	1.35121i
$690$		0			0
$691$	−21.4158	−	21.4158i	−0.814694	−	0.814694i	0.170639	−	0.985334i	$-0.445417\pi$
$691$	−21.4158	−	21.4158i	−0.814694	−	0.814694i	−0.985334	+	0.170639i	$0.945417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.1622	+	10.1622i	−0.385475	+	0.385475i
$696$		0			0
$697$	−29.2306	+	29.2306i	−1.10719	+	1.10719i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.1775			−1.44194			−0.720972	−	0.692964i	$-0.756305\pi$
$701$	−38.1775			−1.44194			−0.720972	+	0.692964i	$0.756305\pi$
$702$		0			0
$703$	−20.8618			−0.786817
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.0499	+	10.0499i	−0.377964	+	0.377964i
$708$		0			0
$709$	−6.13379	+	6.13379i	−0.230359	+	0.230359i	−0.812843	−	0.582483i	$-0.802081\pi$
$709$	−6.13379	+	6.13379i	−0.230359	+	0.230359i	0.582483	+	0.812843i	$0.302081\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.6246	−	13.6246i	−0.510243	−	0.510243i
$714$		0			0
$715$	14.7403	+	7.47790i	0.551258	+	0.279657i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.78758			0.141253			0.0706264	−	0.997503i	$-0.477500\pi$
$719$	3.78758			0.141253			0.0706264	+	0.997503i	$0.477500\pi$
$720$		0			0
$721$	28.7085	+	28.7085i	1.06916	+	1.06916i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.06886			0.188253
$726$		0			0
$727$		−	38.0302i		−	1.41046i	−0.708977	−	0.705232i	$-0.750843\pi$
$727$		−	38.0302i		−	1.41046i	0.708977	−	0.705232i	$-0.249157\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.04747i			0.186688i
$732$		0			0
$733$	−26.8148	−	26.8148i	−0.990427	−	0.990427i	0.00952748	−	0.999955i	$-0.496967\pi$
$733$	−26.8148	−	26.8148i	−0.990427	−	0.990427i	−0.999955	+	0.00952748i	$0.996967\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			54.6581i			2.01336i
$738$		0			0
$739$	7.78734	−	7.78734i	0.286462	−	0.286462i	−0.549218	−	0.835679i	$-0.685074\pi$
$739$	7.78734	−	7.78734i	0.286462	−	0.286462i	0.835679	+	0.549218i	$0.185074\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.4165	+	21.4165i	−0.785697	+	0.785697i	−0.980786	−	0.195089i	$-0.937500\pi$
$743$	−21.4165	+	21.4165i	−0.785697	+	0.785697i	0.195089	+	0.980786i	$0.437500\pi$
$744$		0			0
$745$		−	20.2776i		−	0.742913i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.19848	−	2.19848i	−0.0803307	−	0.0803307i
$750$		0			0
$751$			10.8662i			0.396514i	0.980150	+	0.198257i	$0.0635280\pi$
$751$			10.8662i			0.396514i	−0.980150	+	0.198257i	$0.936472\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			19.4492i			0.707830i
$756$		0			0
$757$	−32.3122			−1.17441			−0.587204	−	0.809439i	$-0.699771\pi$
$757$	−32.3122			−1.17441			−0.587204	+	0.809439i	$0.699771\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.7537	−	16.7537i	−0.607322	−	0.607322i	0.334923	−	0.942245i	$-0.391290\pi$
$761$	−16.7537	−	16.7537i	−0.607322	−	0.607322i	−0.942245	+	0.334923i	$0.891290\pi$
$762$		0			0
$763$	35.0800			1.26998
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−23.6436	−	23.6436i	−0.852608	−	0.852608i	0.137845	−	0.990454i	$-0.455982\pi$
$769$	−23.6436	−	23.6436i	−0.852608	−	0.852608i	−0.990454	+	0.137845i	$0.955982\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.7454	−	12.7454i	0.458419	−	0.458419i	−0.439717	−	0.898136i	$-0.644921\pi$
$773$	12.7454	−	12.7454i	0.458419	−	0.458419i	0.898136	+	0.439717i	$0.144921\pi$
$774$		0			0
$775$	−3.26245	+	3.26245i	−0.117191	+	0.117191i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−31.4213			−1.12579
$780$		0			0
$781$	68.4018			2.44761
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.39917	+	6.39917i	−0.228396	+	0.228396i
$786$		0			0
$787$	4.11866	−	4.11866i	0.146815	−	0.146815i	−0.629879	−	0.776693i	$-0.716895\pi$
$787$	4.11866	−	4.11866i	0.146815	−	0.146815i	0.776693	+	0.629879i	$0.216895\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.96990	+	9.96990i	0.354489	+	0.354489i
$792$		0			0
$793$	14.4381	+	7.32457i	0.512712	+	0.260103i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−38.2367			−1.35441			−0.677207	−	0.735792i	$-0.736810\pi$
$797$	−38.2367			−1.35441			−0.677207	+	0.735792i	$0.736810\pi$
$798$		0			0
$799$	−15.5400	−	15.5400i	−0.549766	−	0.549766i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	44.3875			1.56640
$804$		0			0
$805$			9.63401i			0.339554i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			35.4471i			1.24625i	0.782120	+	0.623127i	$0.214138\pi$
$809$			35.4471i			1.24625i	−0.782120	+	0.623127i	$0.785862\pi$
$810$		0			0
$811$	12.9558	+	12.9558i	0.454939	+	0.454939i	0.896990	−	0.442051i	$-0.145749\pi$
$811$	12.9558	+	12.9558i	0.454939	+	0.454939i	−0.442051	+	0.896990i	$0.645749\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	20.0198i		−	0.701262i
$816$		0			0
$817$	−2.71288	+	2.71288i	−0.0949118	+	0.0949118i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.1156	+	15.1156i	−0.527538	+	0.527538i	−0.919838	−	0.392299i	$-0.871680\pi$
$821$	−15.1156	+	15.1156i	−0.527538	+	0.527538i	0.392299	+	0.919838i	$0.371680\pi$
$822$		0			0
$823$			3.75709i			0.130964i	0.997854	+	0.0654820i	$0.0208585\pi$
$823$			3.75709i			0.130964i	−0.997854	+	0.0654820i	$0.979141\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.3525	+	11.3525i	0.394766	+	0.394766i	0.876382	−	0.481616i	$-0.159950\pi$
$827$	11.3525	+	11.3525i	0.394766	+	0.394766i	−0.481616	+	0.876382i	$0.659950\pi$
$828$		0			0
$829$			8.83155i			0.306732i	0.988169	+	0.153366i	$0.0490114\pi$
$829$			8.83155i			0.306732i	−0.988169	+	0.153366i	$0.950989\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			6.24485i			0.216371i
$834$		0			0
$835$	1.90600			0.0659600
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−29.2577	−	29.2577i	−1.01009	−	1.01009i	−0.999949	−	0.0101408i	$-0.996772\pi$
$839$	−29.2577	−	29.2577i	−1.01009	−	1.01009i	−0.0101408	−	0.999949i	$-0.503228\pi$
$840$		0			0
$841$	3.30666			0.114023
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.8470	+	1.98839i	0.441951	+	0.0684027i
$846$		0			0
$847$	−16.3369	−	16.3369i	−0.561343	−	0.561343i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21.7806	+	21.7806i	−0.746630	+	0.746630i
$852$		0			0
$853$	−7.30944	+	7.30944i	−0.250271	+	0.250271i	−0.821082	−	0.570811i	$-0.806629\pi$
$853$	−7.30944	+	7.30944i	−0.250271	+	0.250271i	0.570811	+	0.821082i	$0.306629\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.28743			0.112297			0.0561483	−	0.998422i	$-0.482118\pi$
$857$	3.28743			0.112297			0.0561483	+	0.998422i	$0.482118\pi$
$858$		0			0
$859$	−3.77735			−0.128881			−0.0644407	−	0.997922i	$-0.520526\pi$
$859$	−3.77735			−0.128881			−0.0644407	+	0.997922i	$0.520526\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.0196	−	17.0196i	0.579353	−	0.579353i	−0.355372	−	0.934725i	$-0.615646\pi$
$863$	17.0196	−	17.0196i	0.579353	−	0.579353i	0.934725	+	0.355372i	$0.115646\pi$
$864$		0			0
$865$	3.58423	−	3.58423i	0.121867	−	0.121867i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33.6198	+	33.6198i	1.14047	+	1.14047i
$870$		0			0
$871$	13.3564	+	40.8618i	0.452566	+	1.38455i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.30690			0.0779874
$876$		0			0
$877$	8.96534	+	8.96534i	0.302738	+	0.302738i	0.842084	−	0.539346i	$-0.181329\pi$
$877$	8.96534	+	8.96534i	0.302738	+	0.302738i	−0.539346	+	0.842084i	$0.681329\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.2607			0.884745			0.442372	−	0.896831i	$-0.354137\pi$
$881$	26.2607			0.884745			0.442372	+	0.896831i	$0.354137\pi$
$882$		0			0
$883$		−	43.9363i		−	1.47857i	−0.673391	−	0.739286i	$-0.735163\pi$
$883$		−	43.9363i		−	1.47857i	0.673391	−	0.739286i	$-0.264837\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			30.3743i			1.01987i	0.860213	+	0.509935i	$0.170331\pi$
$887$			30.3743i			1.01987i	−0.860213	+	0.509935i	$0.829669\pi$
$888$		0			0
$889$	0.702890	+	0.702890i	0.0235742	+	0.0235742i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	16.7047i		−	0.559002i
$894$		0			0
$895$	1.41577	−	1.41577i	0.0473241	−	0.0473241i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.5369	−	16.5369i	0.551536	−	0.551536i
$900$		0			0
$901$			41.0454i			1.36742i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.4307	+	18.4307i	0.612656	+	0.612656i
$906$		0			0
$907$		−	27.5109i		−	0.913485i	−0.889599	−	0.456743i	$-0.849016\pi$
$907$		−	27.5109i		−	0.913485i	0.889599	−	0.456743i	$-0.150984\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	39.1153i		−	1.29595i	−0.761663	−	0.647974i	$-0.775617\pi$
$911$		−	39.1153i		−	1.29595i	0.761663	−	0.647974i	$-0.224383\pi$
$912$		0			0
$913$	−33.2927			−1.10183
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.4385	−	10.4385i	−0.344709	−	0.344709i
$918$		0			0
$919$	−24.0952			−0.794826			−0.397413	−	0.917640i	$-0.630092\pi$
$919$	−24.0952			−0.794826			−0.397413	+	0.917640i	$0.630092\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	51.1364	−	16.7149i	1.68318	−	0.550177i
$924$		0			0
$925$	5.21545	+	5.21545i	0.171483	+	0.171483i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.7725	+	20.7725i	−0.681522	+	0.681522i	−0.960343	−	0.278821i	$-0.910056\pi$
$929$	−20.7725	+	20.7725i	−0.681522	+	0.681522i	0.278821	+	0.960343i	$0.410056\pi$
$930$		0			0
$931$	−3.35644	+	3.35644i	−0.110003	+	0.110003i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.0584			−0.557870
$936$		0			0
$937$	−4.88646			−0.159634			−0.0798169	−	0.996810i	$-0.525434\pi$
$937$	−4.88646			−0.159634			−0.0798169	+	0.996810i	$0.525434\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.713857	+	0.713857i	−0.0232711	+	0.0232711i	−0.718647	−	0.695376i	$-0.755238\pi$
$941$	−0.713857	+	0.713857i	−0.0232711	+	0.0232711i	0.695376	+	0.718647i	$0.255238\pi$
$942$		0			0
$943$	−32.8053	+	32.8053i	−1.06829	+	1.06829i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.7643	−	22.7643i	−0.739740	−	0.739740i	0.232788	−	0.972528i	$-0.425215\pi$
$947$	−22.7643	−	22.7643i	−0.739740	−	0.739740i	−0.972528	+	0.232788i	$0.925215\pi$
$948$		0			0
$949$	33.1836	−	10.8467i	1.07718	−	0.352098i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.6093			−0.505635			−0.252818	−	0.967514i	$-0.581357\pi$
$953$	−15.6093			−0.505635			−0.252818	+	0.967514i	$0.581357\pi$
$954$		0			0
$955$	7.10912	+	7.10912i	0.230046	+	0.230046i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.26778			−0.0409389
$960$		0			0
$961$		−	9.71288i		−	0.313319i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			19.4318i			0.625531i
$966$		0			0
$967$	−12.9558	−	12.9558i	−0.416630	−	0.416630i	0.467410	−	0.884040i	$-0.345187\pi$
$967$	−12.9558	−	12.9558i	−0.416630	−	0.416630i	−0.884040	+	0.467410i	$0.845187\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			41.3422i			1.32673i	0.748294	+	0.663367i	$0.230873\pi$
$971$			41.3422i			1.32673i	−0.748294	+	0.663367i	$0.769127\pi$
$972$		0			0
$973$	23.4432	−	23.4432i	0.751556	−	0.751556i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.50648	−	5.50648i	0.176168	−	0.176168i	−0.613515	−	0.789683i	$-0.710245\pi$
$977$	5.50648	−	5.50648i	0.176168	−	0.176168i	0.789683	+	0.613515i	$0.210245\pi$
$978$		0			0
$979$		−	23.4214i		−	0.748550i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.8031	+	22.8031i	0.727307	+	0.727307i	0.970082	−	0.242776i	$-0.0780579\pi$
$983$	22.8031	+	22.8031i	0.727307	+	0.727307i	−0.242776	+	0.970082i	$0.578058\pi$
$984$		0			0
$985$			15.9060i			0.506807i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			5.66474i			0.180128i
$990$		0			0
$991$	−49.8276			−1.58283			−0.791413	−	0.611282i	$-0.790654\pi$
$991$	−49.8276			−1.58283			−0.791413	+	0.611282i	$0.790654\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.826220	−	0.826220i	−0.0261929	−	0.0261929i
$996$		0			0
$997$	16.2820			0.515656			0.257828	−	0.966191i	$-0.416993\pi$
$997$	16.2820			0.515656			0.257828	+	0.966191i	$0.416993\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bi.d.1061.4	yes	12
3.2	odd	2	inner	2340.2.bi.d.1061.1	yes	12
13.5	odd	4	inner	2340.2.bi.d.161.1	✓	12
39.5	even	4	inner	2340.2.bi.d.161.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.bi.d.161.1	✓	12	13.5	odd	4	inner
2340.2.bi.d.161.4	yes	12	39.5	even	4	inner
2340.2.bi.d.1061.1	yes	12	3.2	odd	2	inner
2340.2.bi.d.1061.4	yes	12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{5} +(-1.63122 + 1.63122i) q^{7} +(-3.24154 - 3.24154i) q^{11} +(-3.21545 - 1.63122i) q^{13} +3.72111 q^{17} +(2.00000 + 2.00000i) q^{19} +4.17618 q^{23} -1.00000i q^{25} +5.06886i q^{29} +(-3.26245 - 3.26245i) q^{31} +2.30690i q^{35} +(-5.21545 + 5.21545i) q^{37} +(-7.85533 + 7.85533i) q^{41} +1.35644i q^{43} +(-4.17618 - 4.17618i) q^{47} +1.67822i q^{49} +11.0304i q^{53} -4.58423 q^{55} -4.49023 q^{61} +(-3.42711 + 1.12022i) q^{65} +(-8.43090 - 8.43090i) q^{67} +(-10.5508 + 10.5508i) q^{71} +(-6.84667 + 6.84667i) q^{73} +10.5753 q^{77} -10.3716 q^{79} +(5.13533 - 5.13533i) q^{83} +(2.63122 - 2.63122i) q^{85} +(3.61269 + 3.61269i) q^{89} +(7.90600 - 2.58423i) q^{91} +2.82843 q^{95} +(2.04700 + 2.04700i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) \) 12 * q + 12 * q^13 + 24 * q^19 - 12 * q^37 - 24 * q^55 - 12 * q^73 + 24 * q^79 + 12 * q^85 + 72 * q^91 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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