LMFDB - Embedded newform 396.4.j.d.289.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [396,4,Mod(37,396)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("396.37"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$396 = 2^{2} \cdot 3^{2} \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	396.j (of order $5$ , degree $4$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$23.3647563623$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{11} + 70 x^{10} - 84 x^{9} + 2459 x^{8} - 8514 x^{7} + 54995 x^{6} - 432951 x^{5} + \cdots + 40896025$ x^12 - x^11 + 70x^10 - 84x^9 + 2459x^8 - 8514x^7 + 54995x^6 - 432951x^5 + 3635141x^4 - 13377765x^3 + 34447885x^2 - 51287900x + 40896025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			289.1
Root			$1.50918 + 1.09648i$ of defining polynomial
Character	$\chi$	$=$	396.289
Dual form			396.4.j.d.37.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-14.9139 - 10.8356i) q^{5} +(9.02047 - 27.7622i) q^{7} +(-5.54276 - 36.0594i) q^{11} +(-63.1228 + 45.8614i) q^{13} +(23.1093 + 16.7899i) q^{17} +(-13.4717 - 41.4615i) q^{19} +104.563 q^{23} +(66.3872 + 204.319i) q^{25} +(-13.1435 + 40.4515i) q^{29} +(51.0435 - 37.0853i) q^{31} +(-435.349 + 316.300i) q^{35} +(-98.0877 + 301.883i) q^{37} +(21.1199 + 65.0003i) q^{41} -227.116 q^{43} +(3.32926 + 10.2464i) q^{47} +(-411.875 - 299.245i) q^{49} +(-402.111 + 292.151i) q^{53} +(-308.060 + 597.844i) q^{55} +(130.834 - 402.665i) q^{59} +(154.010 + 111.895i) q^{61} +1438.34 q^{65} -386.223 q^{67} +(-559.414 - 406.438i) q^{71} +(98.3036 - 302.547i) q^{73} +(-1051.08 - 171.393i) q^{77} +(-130.828 + 95.0523i) q^{79} +(318.845 + 231.654i) q^{83} +(-162.721 - 500.804i) q^{85} -108.129 q^{89} +(703.814 + 2166.12i) q^{91} +(-248.344 + 764.325i) q^{95} +(549.111 - 398.952i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 4 q^{5} + 6 q^{7} - 39 q^{11} - 10 q^{13} + 56 q^{17} - 141 q^{19} + 388 q^{23} - 203 q^{25} - 772 q^{29} + 882 q^{31} - 412 q^{35} - 192 q^{37} + 180 q^{41} - 2330 q^{43} - 196 q^{47} - 973 q^{49}+ \cdots + 2651 q^{97}+O(q^{100})$ 12 * q + 4 * q^5 + 6 * q^7 - 39 * q^11 - 10 * q^13 + 56 * q^17 - 141 * q^19 + 388 * q^23 - 203 * q^25 - 772 * q^29 + 882 * q^31 - 412 * q^35 - 192 * q^37 + 180 * q^41 - 2330 * q^43 - 196 * q^47 - 973 * q^49 - 2246 * q^53 + 2672 * q^55 + 1099 * q^59 - 470 * q^61 + 3892 * q^65 - 4394 * q^67 + 2084 * q^71 + 2824 * q^73 - 3654 * q^77 + 838 * q^79 - 789 * q^83 - 2632 * q^85 + 954 * q^89 + 2460 * q^91 - 4362 * q^95 + 2651 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/396\mathbb{Z}\right)^\times$ .

$n$	$145$	$199$	$353$
$\chi(n)$	$e\left(\frac{4}{5}\right)$	$1$	$1$

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
$5$	−14.9139	−	10.8356i	−1.33394	−	0.969163i	−0.999644	−	0.0266942i	$-0.991502\pi$
$5$	−14.9139	−	10.8356i	−1.33394	−	0.969163i	−0.334295	−	0.942469i	$-0.608498\pi$
$6$		0			0
$7$	9.02047	−	27.7622i	0.487060	−	1.49902i	−0.341916	−	0.939731i	$-0.611076\pi$
$7$	9.02047	−	27.7622i	0.487060	−	1.49902i	0.828976	−	0.559285i	$-0.188924\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.54276	−	36.0594i	−0.151928	−	0.988392i
$11$	−5.54276	−	36.0594i	−0.151928	−	0.988392i
$12$		0			0
$13$	−63.1228	+	45.8614i	−1.34670	+	0.978435i	−0.347532	+	0.937668i	$0.612980\pi$
$13$	−63.1228	+	45.8614i	−1.34670	+	0.978435i	−0.999169	+	0.0407672i	$0.987020\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	23.1093	+	16.7899i	0.329695	+	0.239538i	0.740301	−	0.672275i	$-0.234683\pi$
$17$	23.1093	+	16.7899i	0.329695	+	0.239538i	−0.410606	+	0.911813i	$0.634683\pi$
$18$		0			0
$19$	−13.4717	−	41.4615i	−0.162664	−	0.500628i	0.836193	−	0.548436i	$-0.184776\pi$
$19$	−13.4717	−	41.4615i	−0.162664	−	0.500628i	−0.998857	+	0.0478081i	$0.984776\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	104.563			0.947952			0.473976	−	0.880538i	$-0.342818\pi$
$23$	104.563			0.947952			0.473976	+	0.880538i	$0.342818\pi$
$24$		0			0
$25$	66.3872	+	204.319i	0.531098	+	1.63455i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−13.1435	+	40.4515i	−0.0841615	+	0.259022i	−0.984278	−	0.176627i	$-0.943481\pi$
$29$	−13.1435	+	40.4515i	−0.0841615	+	0.259022i	0.900116	+	0.435650i	$0.143481\pi$
$30$		0			0
$31$	51.0435	−	37.0853i	0.295732	−	0.214862i	−0.430018	−	0.902820i	$-0.641493\pi$
$31$	51.0435	−	37.0853i	0.295732	−	0.214862i	0.725750	+	0.687958i	$0.241493\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−435.349	+	316.300i	−2.10250	+	1.52755i
$36$		0			0
$37$	−98.0877	+	301.883i	−0.435825	+	1.34133i	0.456414	+	0.889768i	$0.349134\pi$
$37$	−98.0877	+	301.883i	−0.435825	+	1.34133i	−0.892239	+	0.451564i	$0.850866\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	21.1199	+	65.0003i	0.0804481	+	0.247594i	0.983189	−	0.182591i	$-0.0584484\pi$
$41$	21.1199	+	65.0003i	0.0804481	+	0.247594i	−0.902741	+	0.430185i	$0.858448\pi$
$42$		0			0
$43$	−227.116			−0.805463			−0.402731	−	0.915318i	$-0.631939\pi$
$43$	−227.116			−0.805463			−0.402731	+	0.915318i	$0.631939\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.32926	+	10.2464i	0.0103324	+	0.0317999i	0.956090	−	0.293074i	$-0.0946782\pi$
$47$	3.32926	+	10.2464i	0.0103324	+	0.0317999i	−0.945757	+	0.324874i	$0.894678\pi$
$48$		0			0
$49$	−411.875	−	299.245i	−1.20080	−	0.872434i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−402.111	+	292.151i	−1.04215	+	0.757170i	−0.970705	−	0.240274i	$-0.922763\pi$
$53$	−402.111	+	292.151i	−1.04215	+	0.757170i	−0.0714497	+	0.997444i	$0.522763\pi$
$54$		0			0
$55$	−308.060	+	597.844i	−0.755250	+	1.46570i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	130.834	−	402.665i	0.288697	−	0.888517i	−0.696570	−	0.717489i	$-0.745291\pi$
$59$	130.834	−	402.665i	0.288697	−	0.888517i	0.985266	−	0.171028i	$-0.0547088\pi$
$60$		0			0
$61$	154.010	+	111.895i	0.323262	+	0.234864i	0.737566	−	0.675275i	$-0.235975\pi$
$61$	154.010	+	111.895i	0.323262	+	0.234864i	−0.414304	+	0.910139i	$0.635975\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1438.34			2.74468
$66$		0			0
$67$	−386.223			−0.704248			−0.352124	−	0.935953i	$-0.614541\pi$
$67$	−386.223			−0.704248			−0.352124	+	0.935953i	$0.614541\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−559.414	−	406.438i	−0.935074	−	0.679371i	0.0121559	−	0.999926i	$-0.496131\pi$
$71$	−559.414	−	406.438i	−0.935074	−	0.679371i	−0.947230	+	0.320555i	$0.896131\pi$
$72$		0			0
$73$	98.3036	−	302.547i	0.157610	−	0.485075i	−0.840806	−	0.541337i	$-0.817918\pi$
$73$	98.3036	−	302.547i	0.157610	−	0.485075i	0.998416	+	0.0562620i	$0.0179182\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1051.08	−	171.393i	−1.55561	−	0.253664i
$78$		0			0
$79$	−130.828	+	95.0523i	−0.186321	+	0.135370i	−0.677035	−	0.735950i	$-0.736736\pi$
$79$	−130.828	+	95.0523i	−0.186321	+	0.135370i	0.490715	+	0.871320i	$0.336736\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	318.845	+	231.654i	0.421660	+	0.306354i	0.778305	−	0.627886i	$-0.216080\pi$
$83$	318.845	+	231.654i	0.421660	+	0.306354i	−0.356645	+	0.934240i	$0.616080\pi$
$84$		0			0
$85$	−162.721	−	500.804i	−0.207642	−	0.639057i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−108.129			−0.128782			−0.0643912	−	0.997925i	$-0.520511\pi$
$89$	−108.129			−0.128782			−0.0643912	+	0.997925i	$0.520511\pi$
$90$		0			0
$91$	703.814	+	2166.12i	0.810766	+	2.49528i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−248.344	+	764.325i	−0.268206	+	0.825454i
$96$		0			0
$97$	549.111	−	398.952i	0.574781	−	0.417603i	−0.262058	−	0.965052i	$-0.584401\pi$
$97$	549.111	−	398.952i	0.574781	−	0.417603i	0.836839	+	0.547449i	$0.184401\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−913.347	+	663.585i	−0.899816	+	0.653755i	−0.938419	−	0.345500i	$-0.887709\pi$
$101$	−913.347	+	663.585i	−0.899816	+	0.653755i	0.0386027	+	0.999255i	$0.487709\pi$
$102$		0			0
$103$	−30.3811	+	93.5035i	−0.0290635	+	0.0894483i	−0.964536	−	0.263951i	$-0.914974\pi$
$103$	−30.3811	+	93.5035i	−0.0290635	+	0.0894483i	0.935473	+	0.353399i	$0.114974\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	168.311	+	518.006i	0.152067	+	0.468015i	0.997852	−	0.0655096i	$-0.0208673\pi$
$107$	168.311	+	518.006i	0.152067	+	0.468015i	−0.845785	+	0.533524i	$0.820867\pi$
$108$		0			0
$109$	367.085			0.322572			0.161286	−	0.986908i	$-0.448436\pi$
$109$	367.085			0.322572			0.161286	+	0.986908i	$0.448436\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	172.032	+	529.461i	0.143216	+	0.440774i	0.996777	−	0.0802186i	$-0.0255618\pi$
$113$	172.032	+	529.461i	0.143216	+	0.440774i	−0.853561	+	0.520993i	$0.825562\pi$
$114$		0			0
$115$	−1559.44	−	1133.00i	−1.26451	−	0.918720i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	674.579	−	490.110i	0.519652	−	0.377549i
$120$		0			0
$121$	−1269.56	+	399.737i	−0.953836	+	0.300328i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	511.745	−	1574.99i	0.366175	−	1.12697i
$126$		0			0
$127$	1050.67	+	763.356i	0.734109	+	0.533362i	0.890861	−	0.454276i	$-0.150102\pi$
$127$	1050.67	+	763.356i	0.734109	+	0.533362i	−0.156751	+	0.987638i	$0.550102\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1110.64			0.740738			0.370369	−	0.928885i	$-0.379231\pi$
$131$	1110.64			0.740738			0.370369	+	0.928885i	$0.379231\pi$
$132$		0			0
$133$	−1272.58			−0.829675
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−960.004	−	697.484i	−0.598676	−	0.434964i	0.246732	−	0.969084i	$-0.420643\pi$
$137$	−960.004	−	697.484i	−0.598676	−	0.434964i	−0.845409	+	0.534120i	$0.820643\pi$
$138$		0			0
$139$	895.798	−	2756.98i	0.546623	−	1.68233i	−0.170476	−	0.985362i	$-0.554531\pi$
$139$	895.798	−	2756.98i	0.546623	−	1.68233i	0.717099	−	0.696971i	$-0.245469\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2003.61	+	2021.97i	1.17168	+	1.18242i
$144$		0			0
$145$	634.335	−	460.871i	0.363301	−	0.263954i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2376.18	−	1726.40i	−1.30647	−	0.949208i	−0.306476	−	0.951878i	$-0.599150\pi$
$149$	−2376.18	−	1726.40i	−1.30647	−	0.949208i	−0.999996	+	0.00267076i	$0.999150\pi$
$150$		0			0
$151$	944.489	+	2906.84i	0.509016	+	1.56659i	0.793913	+	0.608032i	$0.208041\pi$
$151$	944.489	+	2906.84i	0.509016	+	1.56659i	−0.284897	+	0.958558i	$0.591959\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1163.10			−0.602724
$156$		0			0
$157$	−487.141	−	1499.27i	−0.247631	−	0.762130i	−0.995193	−	0.0979375i	$-0.968775\pi$
$157$	−487.141	−	1499.27i	−0.247631	−	0.762130i	0.747561	−	0.664193i	$-0.231225\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	943.208	−	2902.90i	0.461709	−	1.42100i
$162$		0			0
$163$	2290.77	−	1664.34i	1.10078	−	0.799761i	0.119590	−	0.992823i	$-0.461842\pi$
$163$	2290.77	−	1664.34i	1.10078	−	0.799761i	0.981187	+	0.193062i	$0.0618419\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1008.58	−	732.774i	0.467341	−	0.339543i	−0.329063	−	0.944308i	$-0.606733\pi$
$167$	1008.58	−	732.774i	0.467341	−	0.339543i	0.796404	+	0.604765i	$0.206733\pi$
$168$		0			0
$169$	1202.31	−	3700.32i	0.547250	−	1.68426i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	605.621	+	1863.91i	0.266153	+	0.819136i	0.991425	+	0.130674i	$0.0417140\pi$
$173$	605.621	+	1863.91i	0.266153	+	0.819136i	−0.725272	+	0.688462i	$0.758286\pi$
$174$		0			0
$175$	6271.17			2.70889
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−197.434	−	607.639i	−0.0824408	−	0.253727i	0.901337	−	0.433119i	$-0.142587\pi$
$179$	−197.434	−	607.639i	−0.0824408	−	0.253727i	−0.983778	+	0.179392i	$0.942587\pi$
$180$		0			0
$181$	2545.77	+	1849.61i	1.04544	+	0.759559i	0.971341	−	0.237691i	$-0.0763907\pi$
$181$	2545.77	+	1849.61i	1.04544	+	0.759559i	0.0741029	+	0.997251i	$0.476391\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4733.94	−	3439.41i	1.88133	−	1.36687i
$186$		0			0
$187$	477.343	−	926.367i	0.186667	−	0.362260i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−544.897	+	1677.02i	−0.206426	+	0.635313i	0.793226	+	0.608927i	$0.208400\pi$
$191$	−544.897	+	1677.02i	−0.206426	+	0.635313i	−0.999652	+	0.0263861i	$0.991600\pi$
$192$		0			0
$193$	−3375.40	−	2452.37i	−1.25889	−	0.914639i	−0.260190	−	0.965558i	$-0.583785\pi$
$193$	−3375.40	−	2452.37i	−1.25889	−	0.914639i	−0.998703	+	0.0509185i	$0.983785\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2892.24			−1.04601			−0.523005	−	0.852330i	$-0.675189\pi$
$197$	−2892.24			−1.04601			−0.523005	+	0.852330i	$0.675189\pi$
$198$		0			0
$199$	−4281.26			−1.52508			−0.762539	−	0.646943i	$-0.776047\pi$
$199$	−4281.26			−1.52508			−0.762539	+	0.646943i	$0.776047\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1004.46	+	729.783i	0.347287	+	0.252319i
$204$		0			0
$205$	389.336	−	1198.25i	0.132646	−	0.408242i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1420.41	+	715.591i	−0.470103	+	0.236835i
$210$		0			0
$211$	−2222.99	+	1615.09i	−0.725293	+	0.526956i	−0.888071	−	0.459707i	$-0.847955\pi$
$211$	−2222.99	+	1615.09i	−0.725293	+	0.526956i	0.162778	+	0.986663i	$0.447955\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3387.18	+	2460.93i	1.07444	+	0.780624i
$216$		0			0
$217$	−569.131	−	1751.61i	−0.178042	−	0.547957i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2228.73			−0.678373
$222$		0			0
$223$	−268.699	−	826.970i	−0.0806880	−	0.248332i	0.902572	−	0.430538i	$-0.141676\pi$
$223$	−268.699	−	826.970i	−0.0806880	−	0.248332i	−0.983260	+	0.182206i	$0.941676\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	882.731	−	2716.77i	0.258101	−	0.794353i	−0.735102	−	0.677957i	$-0.762866\pi$
$227$	882.731	−	2716.77i	0.258101	−	0.794353i	0.993203	−	0.116397i	$-0.0371344\pi$
$228$		0			0
$229$	−4269.92	+	3102.28i	−1.23216	+	0.895215i	−0.997050	−	0.0767540i	$-0.975544\pi$
$229$	−4269.92	+	3102.28i	−1.23216	+	0.895215i	−0.235108	+	0.971969i	$0.575544\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	740.566	−	538.052i	0.208223	−	0.151283i	−0.478786	−	0.877932i	$-0.658923\pi$
$233$	740.566	−	538.052i	0.208223	−	0.151283i	0.687010	+	0.726648i	$0.258923\pi$
$234$		0			0
$235$	61.3736	−	188.888i	0.0170365	−	0.0524329i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1955.53	−	6018.51i	−0.529259	−	1.62889i	−0.755737	−	0.654876i	$-0.772721\pi$
$239$	−1955.53	−	6018.51i	−0.529259	−	1.62889i	0.226477	−	0.974016i	$-0.427279\pi$
$240$		0			0
$241$	−540.257			−0.144403			−0.0722013	−	0.997390i	$-0.523002\pi$
$241$	−540.257			−0.144403			−0.0722013	+	0.997390i	$0.523002\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2900.17	+	8925.81i	0.756266	+	2.32755i
$246$		0			0
$247$	2751.85	+	1999.34i	0.708891	+	0.515039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4314.83	+	3134.91i	−1.08506	+	0.788341i	−0.978558	−	0.205972i	$-0.933965\pi$
$251$	−4314.83	+	3134.91i	−1.08506	+	0.788341i	−0.106500	+	0.994313i	$0.533965\pi$
$252$		0			0
$253$	−579.568	−	3770.48i	−0.144020	−	0.936948i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−425.852	+	1310.64i	−0.103362	+	0.318114i	−0.989342	−	0.145608i	$-0.953486\pi$
$257$	−425.852	+	1310.64i	−0.103362	+	0.318114i	0.885981	+	0.463722i	$0.153486\pi$
$258$		0			0
$259$	7496.12	+	5446.25i	1.79840	+	1.30662i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7490.14			−1.75613			−0.878064	−	0.478543i	$-0.841165\pi$
$263$	−7490.14			−1.75613			−0.878064	+	0.478543i	$0.841165\pi$
$264$		0			0
$265$	9162.66			2.12399
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5898.73	+	4285.68i	1.33700	+	0.971384i	0.999549	+	0.0300390i	$0.00956315\pi$
$269$	5898.73	+	4285.68i	1.33700	+	0.971384i	0.337446	+	0.941345i	$0.390437\pi$
$270$		0			0
$271$	−241.614	+	743.610i	−0.0541586	+	0.166683i	−0.974477	−	0.224487i	$-0.927929\pi$
$271$	−241.614	+	743.610i	−0.0541586	+	0.166683i	0.920319	+	0.391170i	$0.127929\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6999.64	−	3526.37i	1.53489	−	0.773266i
$276$		0			0
$277$	465.219	−	338.001i	0.100911	−	0.0733160i	−0.536186	−	0.844100i	$-0.680135\pi$
$277$	465.219	−	338.001i	0.100911	−	0.0733160i	0.637096	+	0.770784i	$0.280135\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4532.66	−	3293.17i	−0.962262	−	0.699125i	−0.00858736	−	0.999963i	$-0.502733\pi$
$281$	−4532.66	−	3293.17i	−0.962262	−	0.699125i	−0.953675	+	0.300839i	$0.902733\pi$
$282$		0			0
$283$	−277.501	−	854.061i	−0.0582888	−	0.179395i	0.917673	−	0.397337i	$-0.130066\pi$
$283$	−277.501	−	854.061i	−0.0582888	−	0.179395i	−0.975962	+	0.217942i	$0.930066\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1995.06			0.410330
$288$		0			0
$289$	−1266.06	−	3896.54i	−0.257696	−	0.793108i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2484.65	−	7646.98i	0.495410	−	1.52471i	−0.320907	−	0.947111i	$-0.603988\pi$
$293$	2484.65	−	7646.98i	0.495410	−	1.52471i	0.816317	−	0.577604i	$-0.196012\pi$
$294$		0			0
$295$	−6314.34	+	4587.64i	−1.24622	+	0.905433i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6600.31	+	4795.41i	−1.27661	+	0.927510i
$300$		0			0
$301$	−2048.69	+	6305.23i	−0.392308	+	1.20740i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1084.44	−	3337.58i	−0.203591	−	0.626587i
$306$		0			0
$307$	−5491.51			−1.02090			−0.510451	−	0.859907i	$-0.670522\pi$
$307$	−5491.51			−1.02090			−0.510451	+	0.859907i	$0.670522\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−987.327	−	3038.68i	−0.180020	−	0.554044i	0.819807	−	0.572640i	$-0.194081\pi$
$311$	−987.327	−	3038.68i	−0.180020	−	0.554044i	−0.999827	+	0.0185954i	$0.994081\pi$
$312$		0			0
$313$	2522.04	+	1832.37i	0.455445	+	0.330900i	0.791742	−	0.610856i	$-0.209175\pi$
$313$	2522.04	+	1832.37i	0.455445	+	0.330900i	−0.336297	+	0.941756i	$0.609175\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	194.378	−	141.224i	0.0344396	−	0.0250219i	−0.570432	−	0.821345i	$-0.693224\pi$
$317$	194.378	−	141.224i	0.0344396	−	0.0250219i	0.604872	+	0.796323i	$0.293224\pi$
$318$		0			0
$319$	1531.51	+	249.733i	0.268802	+	0.0438318i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	384.813	−	1184.33i	0.0662897	−	0.204019i
$324$		0			0
$325$	−13560.9	−	9852.56i	−2.31453	−	1.68161i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	314.494			0.0527010
$330$		0			0
$331$	−9405.79			−1.56190			−0.780950	−	0.624594i	$-0.785265\pi$
$331$	−9405.79			−1.56190			−0.780950	+	0.624594i	$0.785265\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5760.08	+	4184.95i	0.939424	+	0.682531i
$336$		0			0
$337$	−1449.42	+	4460.85i	−0.234288	+	0.721063i	0.762928	+	0.646484i	$0.223761\pi$
$337$	−1449.42	+	4460.85i	−0.234288	+	0.721063i	−0.997215	+	0.0745790i	$0.976239\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1620.19	−	1635.04i	−0.257298	−	0.259656i
$342$		0			0
$343$	−3922.74	+	2850.04i	−0.617516	+	0.448652i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4334.77	−	3149.39i	−0.670612	−	0.487228i	0.199618	−	0.979874i	$-0.436030\pi$
$347$	−4334.77	−	3149.39i	−0.670612	−	0.487228i	−0.870230	+	0.492645i	$0.836030\pi$
$348$		0			0
$349$	−305.037	−	938.808i	−0.0467859	−	0.143992i	0.924935	−	0.380126i	$-0.124120\pi$
$349$	−305.037	−	938.808i	−0.0467859	−	0.143992i	−0.971720	+	0.236134i	$0.924120\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	130.359			0.0196552			0.00982762	−	0.999952i	$-0.496872\pi$
$353$	130.359			0.0196552			0.00982762	+	0.999952i	$0.496872\pi$
$354$		0			0
$355$	3939.05	+	12123.1i	0.588910	+	1.81248i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2852.55	+	8779.23i	−0.419364	+	1.29067i	0.488925	+	0.872326i	$0.337389\pi$
$359$	−2852.55	+	8779.23i	−0.419364	+	1.29067i	−0.908289	+	0.418343i	$0.862611\pi$
$360$		0			0
$361$	4011.48	−	2914.51i	0.584849	−	0.424917i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4744.36	+	3446.98i	−0.680360	+	0.494310i
$366$		0			0
$367$	2625.81	−	8081.41i	0.373477	−	1.14944i	−0.571023	−	0.820934i	$-0.693453\pi$
$367$	2625.81	−	8081.41i	0.373477	−	1.14944i	0.944500	−	0.328510i	$-0.106547\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4483.50	+	13798.8i	0.627418	+	1.93099i
$372$		0			0
$373$	−12810.7			−1.77832			−0.889160	−	0.457597i	$-0.848710\pi$
$373$	−12810.7			−1.77832			−0.889160	+	0.457597i	$0.848710\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1025.51	−	3156.19i	−0.140096	−	0.431172i
$378$		0			0
$379$	5475.79	+	3978.39i	0.742143	+	0.539199i	0.893382	−	0.449299i	$-0.148326\pi$
$379$	5475.79	+	3978.39i	0.742143	+	0.539199i	−0.151238	+	0.988497i	$0.548326\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9385.29	+	6818.81i	−1.25213	+	0.909726i	−0.998344	−	0.0575338i	$-0.981676\pi$
$383$	−9385.29	+	6818.81i	−1.25213	+	0.909726i	−0.253787	+	0.967260i	$0.581676\pi$
$384$		0			0
$385$	13818.6	+	13945.2i	1.82925	+	1.84601i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−983.756	+	3027.69i	−0.128222	+	0.394627i	−0.994474	−	0.104980i	$-0.966522\pi$
$389$	−983.756	+	3027.69i	−0.128222	+	0.394627i	0.866252	+	0.499607i	$0.166522\pi$
$390$		0			0
$391$	2416.37	+	1755.60i	0.312535	+	0.227070i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2981.10			0.379736
$396$		0			0
$397$	4412.72			0.557854			0.278927	−	0.960312i	$-0.410021\pi$
$397$	4412.72			0.557854			0.278927	+	0.960312i	$0.410021\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3050.01	+	2215.96i	0.379826	+	0.275959i	0.761274	−	0.648431i	$-0.224574\pi$
$401$	3050.01	+	2215.96i	0.379826	+	0.275959i	−0.381448	+	0.924390i	$0.624574\pi$
$402$		0			0
$403$	−1521.23	+	4681.85i	−0.188034	+	0.578709i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	11429.4	+	1863.72i	1.39197	+	0.226980i
$408$		0			0
$409$	−5154.89	+	3745.25i	−0.623210	+	0.452789i	−0.854041	−	0.520205i	$-0.825856\pi$
$409$	−5154.89	+	3745.25i	−0.623210	+	0.452789i	0.230831	+	0.972994i	$0.425856\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9998.66	−	7264.45i	−1.19129	−	0.865522i
$414$		0			0
$415$	−2245.11	−	6909.74i	−0.265562	−	0.817315i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2107.98			−0.245779			−0.122890	−	0.992420i	$-0.539216\pi$
$419$	−2107.98			−0.245779			−0.122890	+	0.992420i	$0.539216\pi$
$420$		0			0
$421$	−2548.58	−	7843.73i	−0.295036	−	0.908029i	−0.983209	−	0.182482i	$-0.941587\pi$
$421$	−2548.58	−	7843.73i	−0.295036	−	0.908029i	0.688173	−	0.725547i	$-0.258413\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1896.32	+	5836.29i	−0.216436	+	0.666121i
$426$		0			0
$427$	4495.69	−	3266.31i	0.509512	−	0.370182i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10459.3	+	7599.12i	−1.16892	+	0.849273i	−0.990880	−	0.134749i	$-0.956977\pi$
$431$	−10459.3	+	7599.12i	−1.16892	+	0.849273i	−0.178045	+	0.984022i	$0.556977\pi$
$432$		0			0
$433$	−2413.49	+	7427.96i	−0.267864	+	0.824400i	0.723156	+	0.690685i	$0.242691\pi$
$433$	−2413.49	+	7427.96i	−0.267864	+	0.824400i	−0.991020	+	0.133715i	$0.957309\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1408.64	−	4335.34i	−0.154197	−	0.474571i
$438$		0			0
$439$	−17491.4			−1.90163			−0.950817	−	0.309752i	$-0.899754\pi$
$439$	−17491.4			−1.90163			−0.950817	+	0.309752i	$0.899754\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−794.117	−	2444.04i	−0.0851685	−	0.262122i	0.899399	−	0.437130i	$-0.144005\pi$
$443$	−794.117	−	2444.04i	−0.0851685	−	0.262122i	−0.984567	+	0.175008i	$0.944005\pi$
$444$		0			0
$445$	1612.62	+	1171.64i	0.171788	+	0.124811i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4334.56	−	3149.24i	0.455591	−	0.331007i	−0.336208	−	0.941788i	$-0.609145\pi$
$449$	4334.56	−	3149.24i	0.455591	−	0.331007i	0.791799	+	0.610781i	$0.209145\pi$
$450$		0			0
$451$	2226.81	−	1121.85i	0.232497	−	0.117131i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12974.5	−	39931.4i	1.33682	−	4.11432i
$456$		0			0
$457$	−3912.81	−	2842.82i	−0.400511	−	0.290988i	0.369238	−	0.929335i	$-0.379619\pi$
$457$	−3912.81	−	2842.82i	−0.400511	−	0.290988i	−0.769749	+	0.638347i	$0.779619\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18173.0			1.83601			0.918007	−	0.396564i	$-0.129797\pi$
$461$	18173.0			1.83601			0.918007	+	0.396564i	$0.129797\pi$
$462$		0			0
$463$	13451.4			1.35019			0.675095	−	0.737731i	$-0.264103\pi$
$463$	13451.4			1.35019			0.675095	+	0.737731i	$0.264103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13010.5	+	9452.69i	1.28920	+	0.936656i	0.999789	−	0.0205495i	$-0.00654158\pi$
$467$	13010.5	+	9452.69i	1.28920	+	0.936656i	0.289408	+	0.957206i	$0.406542\pi$
$468$		0			0
$469$	−3483.91	+	10722.4i	−0.343011	+	1.05568i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1258.85	+	8189.66i	0.122372	+	0.796112i
$474$		0			0
$475$	7577.02	−	5505.03i	0.731911	−	0.531764i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2651.17	+	1926.19i	0.252892	+	0.183737i	0.707008	−	0.707206i	$-0.250045\pi$
$479$	2651.17	+	1926.19i	0.252892	+	0.183737i	−0.454116	+	0.890943i	$0.650045\pi$
$480$		0			0
$481$	−7653.20	−	23554.1i	−0.725480	−	2.23280i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12512.3			−1.17145
$486$		0			0
$487$	498.075	+	1532.92i	0.0463448	+	0.142635i	0.971551	−	0.236830i	$-0.0761083\pi$
$487$	498.075	+	1532.92i	0.0463448	+	0.142635i	−0.925206	+	0.379464i	$0.876108\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3326.22	−	10237.0i	0.305723	−	0.940919i	−0.673683	−	0.739020i	$-0.735289\pi$
$491$	3326.22	−	10237.0i	0.305723	−	0.940919i	0.979406	−	0.201899i	$-0.0647112\pi$
$492$		0			0
$493$	−982.911	+	714.126i	−0.0897932	+	0.0652386i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16329.8	+	11864.3i	−1.47382	+	1.07080i
$498$		0			0
$499$	4149.15	−	12769.8i	0.372228	−	1.14560i	−0.573103	−	0.819484i	$-0.694260\pi$
$499$	4149.15	−	12769.8i	0.372228	−	1.14560i	0.945330	−	0.326115i	$-0.105740\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5840.89	−	17976.4i	−0.517758	−	1.59350i	−0.778207	−	0.628008i	$-0.783870\pi$
$503$	−5840.89	−	17976.4i	−0.517758	−	1.59350i	0.260448	−	0.965488i	$-0.416130\pi$
$504$		0			0
$505$	20811.9			1.83389
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3253.27	−	10012.5i	−0.283298	−	0.871902i	−0.986904	−	0.161311i	$-0.948428\pi$
$509$	−3253.27	−	10012.5i	−0.283298	−	0.871902i	0.703606	−	0.710591i	$-0.251572\pi$
$510$		0			0
$511$	−7512.62	−	5458.24i	−0.650369	−	0.472521i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1466.26	−	1065.30i	0.125459	−	0.0911513i
$516$		0			0
$517$	351.026	−	176.845i	0.0298610	−	0.0150437i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	573.594	−	1765.34i	0.0482334	−	0.148447i	−0.924039	−	0.382298i	$-0.875133\pi$
$521$	573.594	−	1765.34i	0.0482334	−	0.148447i	0.972272	+	0.233851i	$0.0751328\pi$
$522$		0			0
$523$	1578.10	+	1146.56i	0.131941	+	0.0958611i	0.651799	−	0.758392i	$-0.274015\pi$
$523$	1578.10	+	1146.56i	0.131941	+	0.0958611i	−0.519857	+	0.854253i	$0.674015\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1802.24			0.148969
$528$		0			0
$529$	−1233.57			−0.101386
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4314.15	−	3134.41i	−0.350594	−	0.254721i
$534$		0			0
$535$	3102.73	−	9549.23i	0.250734	−	0.771681i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8507.66	+	16510.6i	−0.679872	+	1.31941i
$540$		0			0
$541$	−70.7645	+	51.4135i	−0.00562367	+	0.00408584i	−0.590594	−	0.806969i	$-0.701106\pi$
$541$	−70.7645	+	51.4135i	−0.00562367	+	0.00408584i	0.584970	+	0.811055i	$0.301106\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5474.66	−	3977.57i	−0.430291	−	0.312625i
$546$		0			0
$547$	4171.58	+	12838.8i	0.326077	+	1.00356i	0.970952	+	0.239274i	$0.0769093\pi$
$547$	4171.58	+	12838.8i	0.326077	+	1.00356i	−0.644875	+	0.764288i	$0.723091\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1854.24			0.143364
$552$		0			0
$553$	1458.72	+	4489.49i	0.112172	+	0.345231i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2141.73	+	6591.58i	−0.162923	+	0.501426i	−0.998877	−	0.0473735i	$-0.984915\pi$
$557$	−2141.73	+	6591.58i	−0.162923	+	0.501426i	0.835954	+	0.548799i	$0.184915\pi$
$558$		0			0
$559$	14336.2	−	10415.9i	1.08472	−	0.788093i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12283.7	−	8924.65i	0.919534	−	0.668080i	−0.0238741	−	0.999715i	$-0.507600\pi$
$563$	12283.7	−	8924.65i	0.919534	−	0.668080i	0.943408	+	0.331635i	$0.107600\pi$
$564$		0			0
$565$	3171.34	−	9760.38i	0.236140	−	0.726765i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2671.84	+	8223.09i	0.196853	+	0.605852i	0.999950	+	0.0100035i	$0.00318427\pi$
$569$	2671.84	+	8223.09i	0.196853	+	0.605852i	−0.803097	+	0.595849i	$0.796816\pi$
$570$		0			0
$571$	306.105			0.0224345			0.0112172	−	0.999937i	$-0.496429\pi$
$571$	306.105			0.0224345			0.0112172	+	0.999937i	$0.496429\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6941.65	+	21364.2i	0.503455	+	1.54948i
$576$		0			0
$577$	12018.4	+	8731.90i	0.867130	+	0.630007i	0.929815	−	0.368026i	$-0.119966\pi$
$577$	12018.4	+	8731.90i	0.867130	+	0.630007i	−0.0626853	+	0.998033i	$0.519966\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9307.36	−	6762.19i	0.664603	−	0.482862i
$582$		0			0
$583$	12763.6	+	12880.5i	0.906712	+	0.915022i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−379.264	+	1167.26i	−0.0266677	+	0.0820746i	−0.963505	−	0.267692i	$-0.913739\pi$
$587$	−379.264	+	1167.26i	−0.0266677	+	0.0820746i	0.936837	+	0.349766i	$0.113739\pi$
$588$		0			0
$589$	−2225.25	−	1616.74i	−0.155671	−	0.113101i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14255.5			0.987189			0.493594	−	0.869692i	$-0.335683\pi$
$593$	14255.5			0.987189			0.493594	+	0.869692i	$0.335683\pi$
$594$		0			0
$595$	−15371.2			−1.05909
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12643.3	−	9185.92i	−0.862426	−	0.626589i	0.0661182	−	0.997812i	$-0.478939\pi$
$599$	−12643.3	−	9185.92i	−0.862426	−	0.626589i	−0.928544	+	0.371223i	$0.878939\pi$
$600$		0			0
$601$	7972.25	−	24536.1i	0.541090	−	1.66530i	−0.189020	−	0.981973i	$-0.560531\pi$
$601$	7972.25	−	24536.1i	0.541090	−	1.66530i	0.730110	−	0.683330i	$-0.239469\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23265.4	+	7794.73i	1.56343	+	0.523803i
$606$		0			0
$607$	−17691.2	+	12853.4i	−1.18297	+	0.859479i	−0.992504	−	0.122215i	$-0.961000\pi$
$607$	−17691.2	+	12853.4i	−1.18297	+	0.859479i	−0.190468	+	0.981693i	$0.561000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−680.067	−	494.098i	−0.0450288	−	0.0327153i
$612$		0			0
$613$	7044.02	+	21679.3i	0.464120	+	1.42841i	0.860086	+	0.510148i	$0.170410\pi$
$613$	7044.02	+	21679.3i	0.464120	+	1.42841i	−0.395967	+	0.918265i	$0.629590\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16843.3			1.09901			0.549504	−	0.835491i	$-0.314817\pi$
$617$	16843.3			1.09901			0.549504	+	0.835491i	$0.314817\pi$
$618$		0			0
$619$	−5621.76	−	17302.0i	−0.365037	−	1.12347i	−0.949958	−	0.312378i	$-0.898875\pi$
$619$	−5621.76	−	17302.0i	−0.365037	−	1.12347i	0.584921	−	0.811090i	$-0.301125\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−975.374	+	3001.89i	−0.0627247	+	0.193047i
$624$		0			0
$625$	−2972.52	+	2159.66i	−0.190241	+	0.138218i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7335.31	+	5329.41i	−0.464989	+	0.337834i
$630$		0			0
$631$	4988.01	−	15351.5i	0.314690	−	0.968516i	−0.661192	−	0.750217i	$-0.729949\pi$
$631$	4988.01	−	15351.5i	0.314690	−	0.968516i	0.975882	−	0.218299i	$-0.0700510\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7398.17	−	22769.2i	−0.462342	−	1.42294i
$636$		0			0
$637$	39722.5			2.47074
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6301.60	+	19394.3i	0.388297	+	1.19506i	0.934060	+	0.357116i	$0.116240\pi$
$641$	6301.60	+	19394.3i	0.388297	+	1.19506i	−0.545763	+	0.837939i	$0.683760\pi$
$642$		0			0
$643$	−9208.86	−	6690.63i	−0.564793	−	0.410346i	0.268417	−	0.963303i	$-0.413500\pi$
$643$	−9208.86	−	6690.63i	−0.564793	−	0.410346i	−0.833210	+	0.552957i	$0.813500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10915.7	−	7930.73i	0.663278	−	0.481900i	−0.204490	−	0.978869i	$-0.565554\pi$
$647$	10915.7	−	7930.73i	0.663278	−	0.481900i	0.867768	+	0.496969i	$0.165554\pi$
$648$		0			0
$649$	−15245.0	−	2485.91i	−0.922064	−	0.150355i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2631.59	−	8099.21i	0.157706	−	0.485370i	−0.840719	−	0.541472i	$-0.817867\pi$
$653$	2631.59	−	8099.21i	0.157706	−	0.485370i	0.998425	+	0.0561021i	$0.0178672\pi$
$654$		0			0
$655$	−16563.9	−	12034.4i	−0.988099	−	0.717896i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−29637.1			−1.75189			−0.875946	−	0.482409i	$-0.839762\pi$
$659$	−29637.1			−1.75189			−0.875946	+	0.482409i	$0.839762\pi$
$660$		0			0
$661$	7113.07			0.418557			0.209279	−	0.977856i	$-0.432888\pi$
$661$	7113.07			0.418557			0.209279	+	0.977856i	$0.432888\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18979.1	+	13789.1i	1.10674	+	0.804090i
$666$		0			0
$667$	−1374.32	+	4229.73i	−0.0797811	+	0.245541i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3181.22	−	6173.72i	0.183025	−	0.355192i
$672$		0			0
$673$	−9566.47	+	6950.45i	−0.547935	+	0.398098i	−0.827024	−	0.562167i	$-0.809968\pi$
$673$	−9566.47	+	6950.45i	−0.547935	+	0.398098i	0.279088	+	0.960265i	$0.409968\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19761.7	−	14357.7i	−1.12187	−	0.815083i	−0.137375	−	0.990519i	$-0.543867\pi$
$677$	−19761.7	−	14357.7i	−1.12187	−	0.815083i	−0.984491	+	0.175436i	$0.943867\pi$
$678$		0			0
$679$	−6122.54	−	18843.2i	−0.346041	−	1.06500i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21680.4			−1.21461			−0.607304	−	0.794469i	$-0.707749\pi$
$683$	−21680.4			−1.21461			−0.607304	+	0.794469i	$0.707749\pi$
$684$		0			0
$685$	6759.75	+	20804.4i	0.377047	+	1.16043i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11983.9	−	36882.7i	0.662629	−	2.03936i
$690$		0			0
$691$	−5707.55	+	4146.78i	−0.314219	+	0.228294i	−0.733705	−	0.679468i	$-0.762211\pi$
$691$	−5707.55	+	4146.78i	−0.314219	+	0.228294i	0.419485	+	0.907762i	$0.362211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−43233.3	+	31410.8i	−2.35962	+	1.71436i
$696$		0			0
$697$	−603.282	+	1856.71i	−0.0327847	+	0.100901i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9684.94	−	29807.2i	−0.521819	−	1.60599i	−0.770522	−	0.637414i	$-0.780004\pi$
$701$	−9684.94	−	29807.2i	−0.521819	−	1.60599i	0.248703	−	0.968580i	$-0.419996\pi$
$702$		0			0
$703$	13837.9			0.742400
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10183.7	+	31342.3i	0.541724	+	1.66726i
$708$		0			0
$709$	−3823.80	−	2778.15i	−0.202547	−	0.147159i	0.481888	−	0.876233i	$-0.339951\pi$
$709$	−3823.80	−	2778.15i	−0.202547	−	0.147159i	−0.684435	+	0.729074i	$0.739951\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5337.27	−	3877.75i	0.280340	−	0.203679i
$714$		0			0
$715$	−7972.37	−	51865.6i	−0.416993	−	2.71282i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8402.15	−	25859.2i	0.435810	−	1.34129i	−0.456444	−	0.889752i	$-0.650877\pi$
$719$	8402.15	−	25859.2i	0.435810	−	1.34129i	0.892254	−	0.451533i	$-0.149123\pi$
$720$		0			0
$721$	2321.81	+	1686.89i	0.119929	+	0.0871333i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9137.55			−0.468083
$726$		0			0
$727$	−14387.6			−0.733986			−0.366993	−	0.930224i	$-0.619613\pi$
$727$	−14387.6			−0.733986			−0.366993	+	0.930224i	$0.619613\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5248.49	−	3813.25i	−0.265557	−	0.192939i
$732$		0			0
$733$	9206.38	−	28334.3i	0.463909	−	1.42777i	−0.396440	−	0.918061i	$-0.629754\pi$
$733$	9206.38	−	28334.3i	0.463909	−	1.42777i	0.860349	−	0.509705i	$-0.170246\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2140.74	+	13927.0i	0.106995	+	0.696073i
$738$		0			0
$739$	−18153.6	+	13189.4i	−0.903641	+	0.656534i	−0.939399	−	0.342827i	$-0.888616\pi$
$739$	−18153.6	+	13189.4i	−0.903641	+	0.656534i	0.0357574	+	0.999361i	$0.488616\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5660.31	+	4112.45i	0.279484	+	0.203057i	0.718692	−	0.695328i	$-0.244741\pi$
$743$	5660.31	+	4112.45i	0.279484	+	0.203057i	−0.439208	+	0.898385i	$0.644741\pi$
$744$		0			0
$745$	16731.6	+	51494.6i	0.822817	+	2.53237i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15899.2			0.775627
$750$		0			0
$751$	8967.71	+	27599.8i	0.435734	+	1.34105i	0.892333	+	0.451379i	$0.149068\pi$
$751$	8967.71	+	27599.8i	0.435734	+	1.34105i	−0.456599	+	0.889673i	$0.650932\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17411.2	−	53586.3i	0.839285	−	2.58305i
$756$		0			0
$757$	−12090.7	+	8784.44i	−0.580509	+	0.421765i	−0.838908	−	0.544274i	$-0.816805\pi$
$757$	−12090.7	+	8784.44i	−0.580509	+	0.421765i	0.258398	+	0.966038i	$0.416805\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12571.8	+	9133.92i	−0.598852	+	0.435091i	−0.845471	−	0.534021i	$-0.820680\pi$
$761$	−12571.8	+	9133.92i	−0.598852	+	0.435091i	0.246619	+	0.969112i	$0.420680\pi$
$762$		0			0
$763$	3311.28	−	10191.1i	0.157112	−	0.483541i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10208.2	+	31417.5i	0.480568	+	1.47904i
$768$		0			0
$769$	1512.44			0.0709233			0.0354616	−	0.999371i	$-0.488710\pi$
$769$	1512.44			0.0709233			0.0354616	+	0.999371i	$0.488710\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12574.3	+	38699.8i	0.585081	+	1.80069i	0.598945	+	0.800790i	$0.295587\pi$
$773$	12574.3	+	38699.8i	0.585081	+	1.80069i	−0.0138643	+	0.999904i	$0.504413\pi$
$774$		0			0
$775$	10965.9	+	7967.16i	0.508265	+	0.369276i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2410.49	−	1751.32i	0.110866	−	0.0805491i
$780$		0			0
$781$	−11555.2	+	22424.9i	−0.529421	+	1.02743i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8980.24	+	27638.3i	−0.408304	+	1.25663i
$786$		0			0
$787$	27079.9	+	19674.7i	1.22655	+	0.891140i	0.996627	−	0.0820688i	$-0.0261527\pi$
$787$	27079.9	+	19674.7i	1.22655	+	0.891140i	0.229923	+	0.973209i	$0.426153\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16250.8			0.730482
$792$		0			0
$793$	−14853.2			−0.665136
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15215.6	+	11054.8i	0.676240	+	0.491317i	0.872108	−	0.489313i	$-0.162752\pi$
$797$	15215.6	+	11054.8i	0.676240	+	0.491317i	−0.195868	+	0.980630i	$0.562752\pi$
$798$		0			0
$799$	−95.0991	+	292.685i	−0.00421072	+	0.0129593i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11454.5	−	1867.82i	−0.503390	−	0.0820845i
$804$		0			0
$805$	−45521.4	+	33073.2i	−1.99307	+	1.44805i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24141.4	+	17539.8i	1.04916	+	0.762256i	0.972052	−	0.234767i	$-0.0754326\pi$
$809$	24141.4	+	17539.8i	1.04916	+	0.762256i	0.0771041	+	0.997023i	$0.475433\pi$
$810$		0			0
$811$	−6559.41	−	20187.8i	−0.284010	−	0.874092i	−0.986694	−	0.162590i	$-0.948015\pi$
$811$	−6559.41	−	20187.8i	−0.284010	−	0.874092i	0.702684	−	0.711502i	$-0.251985\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−52198.3			−2.24347
$816$		0			0
$817$	3059.63	+	9416.58i	0.131020	+	0.403237i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1212.55	+	3731.84i	−0.0515447	+	0.158638i	−0.973515	−	0.228621i	$-0.926578\pi$
$821$	−1212.55	+	3731.84i	−0.0515447	+	0.158638i	0.921971	+	0.387260i	$0.126578\pi$
$822$		0			0
$823$	−16307.8	+	11848.3i	−0.690708	+	0.501829i	−0.876893	−	0.480686i	$-0.840388\pi$
$823$	−16307.8	+	11848.3i	−0.690708	+	0.501829i	0.186185	+	0.982515i	$0.440388\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31618.0	−	22971.9i	1.32946	−	0.965912i	0.329703	−	0.944085i	$-0.393051\pi$
$827$	31618.0	−	22971.9i	1.32946	−	0.965912i	0.999762	−	0.0218277i	$-0.00694852\pi$
$828$		0			0
$829$	−5841.62	+	17978.7i	−0.244738	+	0.753226i	0.750941	+	0.660369i	$0.229600\pi$
$829$	−5841.62	+	17978.7i	−0.244738	+	0.753226i	−0.995679	+	0.0928575i	$0.970400\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4493.85	−	13830.7i	−0.186918	−	0.575275i
$834$		0			0
$835$	−22981.8			−0.952477
$836$		0			0
$837$		0			0
$838$		0			0
$839$	417.741	+	1285.67i	0.0171895	+	0.0529039i	0.959283	−	0.282446i	$-0.0911456\pi$
$839$	417.741	+	1285.67i	0.0171895	+	0.0529039i	−0.942094	+	0.335349i	$0.891146\pi$
$840$		0			0
$841$	18267.5	+	13272.1i	0.749008	+	0.544186i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−58026.2	+	42158.5i	−2.36232	+	1.71633i
$846$		0			0
$847$	−354.435	+	38851.4i	−0.0143784	+	1.57609i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−10256.4	+	31565.8i	−0.413141	+	1.27152i
$852$		0			0
$853$	−27556.6	−	20021.0i	−1.10612	−	0.803643i	−0.124071	−	0.992273i	$-0.539595\pi$
$853$	−27556.6	−	20021.0i	−1.10612	−	0.803643i	−0.982048	+	0.188630i	$0.939595\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−40085.5			−1.59778			−0.798888	−	0.601480i	$-0.794578\pi$
$857$	−40085.5			−1.59778			−0.798888	+	0.601480i	$0.794578\pi$
$858$		0			0
$859$	−20827.3			−0.827261			−0.413631	−	0.910445i	$-0.635739\pi$
$859$	−20827.3			−0.827261			−0.413631	+	0.910445i	$0.635739\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	32530.0	+	23634.4i	1.28312	+	0.932242i	0.999643	−	0.0267360i	$-0.00851135\pi$
$863$	32530.0	+	23634.4i	1.28312	+	0.932242i	0.283479	+	0.958978i	$0.408511\pi$
$864$		0			0
$865$	11164.4	−	34360.4i	0.438844	−	1.35062i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4152.67	+	4190.73i	0.162106	+	0.163591i
$870$		0			0
$871$	24379.5	−	17712.7i	0.948412	−	0.689061i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−39108.9	−	28414.3i	−1.51100	−	1.09780i
$876$		0			0
$877$	2172.95	+	6687.64i	0.0836661	+	0.257498i	0.984135	−	0.177423i	$-0.0567762\pi$
$877$	2172.95	+	6687.64i	0.0836661	+	0.257498i	−0.900469	+	0.434921i	$0.856776\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30396.7			−1.16242			−0.581209	−	0.813754i	$-0.697420\pi$
$881$	−30396.7			−1.16242			−0.581209	+	0.813754i	$0.697420\pi$
$882$		0			0
$883$	−582.341	−	1792.26i	−0.0221940	−	0.0683063i	0.939346	−	0.342971i	$-0.111433\pi$
$883$	−582.341	−	1792.26i	−0.0221940	−	0.0683063i	−0.961540	+	0.274665i	$0.911433\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−442.860	+	1362.98i	−0.0167641	+	0.0515947i	−0.959089	−	0.283106i	$-0.908635\pi$
$887$	−442.860	+	1362.98i	−0.0167641	+	0.0515947i	0.942325	+	0.334701i	$0.108635\pi$
$888$		0			0
$889$	30670.0	−	22283.0i	1.15707	−	0.840662i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	379.981	−	276.073i	0.0142392	−	0.0103454i
$894$		0			0
$895$	−3639.61	+	11201.6i	−0.135931	+	0.418354i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	829.265	+	2552.22i	0.0307648	+	0.0946843i
$900$		0			0
$901$	−14197.7			−0.524964
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17925.7	−	55169.7i	−0.658421	−	2.02641i
$906$		0			0
$907$	22216.3	+	16141.1i	0.813319	+	0.590911i	0.914791	−	0.403927i	$-0.132355\pi$
$907$	22216.3	+	16141.1i	0.813319	+	0.590911i	−0.101472	+	0.994838i	$0.532355\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	37947.6	−	27570.5i	1.38009	−	1.00269i	0.383215	−	0.923659i	$-0.374817\pi$
$911$	37947.6	−	27570.5i	1.38009	−	1.00269i	0.996872	−	0.0790330i	$-0.0251833\pi$
$912$		0			0
$913$	6586.03	−	12781.4i	0.238736	−	0.463309i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10018.5	−	30833.6i	0.360784	−	1.11038i
$918$		0			0
$919$	−27166.7	−	19737.7i	−0.975131	−	0.708474i	−0.0185159	−	0.999829i	$-0.505894\pi$
$919$	−27166.7	−	19737.7i	−0.975131	−	0.708474i	−0.956615	+	0.291354i	$0.905894\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	53951.6			1.92399
$924$		0			0
$925$	−68192.1			−2.42394
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15803.6	+	11482.0i	0.558127	+	0.405503i	0.830773	−	0.556612i	$-0.187899\pi$
$929$	15803.6	+	11482.0i	0.558127	+	0.405503i	−0.272646	+	0.962114i	$0.587899\pi$
$930$		0			0
$931$	−6858.50	+	21108.3i	−0.241438	+	0.743068i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17156.8	+	8643.46i	−0.600092	+	0.302322i
$936$		0			0
$937$	42555.5	−	30918.4i	1.48370	−	1.07797i	0.507361	−	0.861734i	$-0.330621\pi$
$937$	42555.5	−	30918.4i	1.48370	−	1.07797i	0.976341	−	0.216239i	$-0.0693789\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28258.3	−	20530.8i	−0.978951	−	0.711250i	−0.0214772	−	0.999769i	$-0.506837\pi$
$941$	−28258.3	−	20530.8i	−0.978951	−	0.711250i	−0.957474	+	0.288520i	$0.906837\pi$
$942$		0			0
$943$	2208.36	+	6796.63i	0.0762610	+	0.234707i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4206.50			−0.144343			−0.0721715	−	0.997392i	$-0.522993\pi$
$947$	−4206.50			−0.144343			−0.0721715	+	0.997392i	$0.522993\pi$
$948$		0			0
$949$	7670.04	+	23606.0i	0.262361	+	0.807463i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	226.324	−	696.553i	0.00769291	−	0.0236763i	−0.947136	−	0.320831i	$-0.896038\pi$
$953$	226.324	−	696.553i	0.00769291	−	0.0236763i	0.954829	+	0.297155i	$0.0960378\pi$
$954$		0			0
$955$	26298.0	−	19106.6i	0.891082	−	0.647409i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−28023.3	+	20360.1i	−0.943609	+	0.685572i
$960$		0			0
$961$	−7975.80	+	24547.0i	−0.267725	+	0.823973i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	23767.4	+	73148.7i	0.792851	+	2.44014i
$966$		0			0
$967$	−6877.86			−0.228725			−0.114363	−	0.993439i	$-0.536483\pi$
$967$	−6877.86			−0.228725			−0.114363	+	0.993439i	$0.536483\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11642.4	−	35831.6i	−0.384781	−	1.18423i	−0.936639	−	0.350295i	$-0.886081\pi$
$971$	−11642.4	−	35831.6i	−0.384781	−	1.18423i	0.551859	−	0.833938i	$-0.313919\pi$
$972$		0			0
$973$	−68459.3	−	49738.6i	−2.25560	−	1.63879i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25728.6	−	18692.9i	0.842508	−	0.612118i	−0.0805620	−	0.996750i	$-0.525672\pi$
$977$	25728.6	−	18692.9i	0.842508	−	0.612118i	0.923070	+	0.384632i	$0.125672\pi$
$978$		0			0
$979$	599.333	+	3899.06i	0.0195656	+	0.127288i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10041.4	−	30904.2i	0.325809	−	1.00274i	−0.645265	−	0.763959i	$-0.723253\pi$
$983$	10041.4	−	30904.2i	0.325809	−	1.00274i	0.971074	−	0.238779i	$-0.0767470\pi$
$984$		0			0
$985$	43134.6	+	31339.1i	1.39531	+	1.01375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−23748.0			−0.763540
$990$		0			0
$991$	3024.29			0.0969423			0.0484712	−	0.998825i	$-0.484565\pi$
$991$	3024.29			0.0969423			0.0484712	+	0.998825i	$0.484565\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	63850.2	+	46389.9i	2.03436	+	1.47805i
$996$		0			0
$997$	−3247.69	+	9995.35i	−0.103165	+	0.317509i	−0.989295	−	0.145927i	$-0.953383\pi$
$997$	−3247.69	+	9995.35i	−0.103165	+	0.317509i	0.886131	+	0.463436i	$0.153383\pi$
$998$		0			0
$999$		0			0

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

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a_n

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n

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	396.4.j.d.289.1		12
3.2	odd	2		44.4.e.a.25.2	✓	12
11.4	even	5	inner	396.4.j.d.37.1		12
12.11	even	2		176.4.m.d.113.2		12
33.2	even	10		484.4.a.h.1.3		6
33.20	odd	10		484.4.a.i.1.3		6
33.26	odd	10		44.4.e.a.37.2	yes	12
132.35	odd	10		1936.4.a.bs.1.4		6
132.59	even	10		176.4.m.d.81.2		12
132.119	even	10		1936.4.a.br.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.4.e.a.25.2	✓	12	3.2	odd	2
44.4.e.a.37.2	yes	12	33.26	odd	10
176.4.m.d.81.2		12	132.59	even	10
176.4.m.d.113.2		12	12.11	even	2
396.4.j.d.37.1		12	11.4	even	5	inner
396.4.j.d.289.1		12	1.1	even	1	trivial
484.4.a.h.1.3		6	33.2	even	10
484.4.a.i.1.3		6	33.20	odd	10
1936.4.a.br.1.4		6	132.119	even	10
1936.4.a.bs.1.4		6	132.35	odd	10

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

Newform invariants

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$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	396.4.j.d.289.1

Level	$396$
Weight	$4$
Character	396.289
Analytic conductor	$23.365$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$