LMFDB - Embedded newform 396.4.j.b.37.1

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 = [8,0,0,0,27] 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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} + 58x^{6} - 115x^{5} + 2421x^{4} + 3435x^{3} + 83262x^{2} + 124773x + 4791721$ x^8 - x^7 + 58x^6 - 115x^5 + 2421x^4 + 3435x^3 + 83262x^2 + 124773x + 4791721
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			37.1
Root			$-2.37690 + 7.31535i$ of defining polynomial
Character	$\chi$	$=$	396.37
Dual form			396.4.j.b.289.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.60477 + 1.16593i) q^{5} +(3.78898 + 11.6613i) q^{7} +(8.21788 - 35.5453i) q^{11} +(5.26513 + 3.82534i) q^{13} +(62.9784 - 45.7565i) q^{17} +(-28.2631 + 86.9850i) q^{19} +20.2371 q^{23} +(-37.4112 + 115.140i) q^{25} +(23.5838 + 72.5834i) q^{29} +(92.8764 + 67.4787i) q^{31} +(-19.6767 - 14.2960i) q^{35} +(27.6238 + 85.0174i) q^{37} +(-30.4658 + 93.7641i) q^{41} +488.233 q^{43} +(-33.0191 + 101.622i) q^{47} +(155.864 - 113.242i) q^{49} +(447.009 + 324.771i) q^{53} +(28.2556 + 66.6235i) q^{55} +(52.3705 + 161.180i) q^{59} +(235.345 - 170.988i) q^{61} -12.9094 q^{65} +107.351 q^{67} +(-589.515 + 428.308i) q^{71} +(110.836 + 341.119i) q^{73} +(445.641 - 38.8493i) q^{77} +(413.815 + 300.654i) q^{79} +(-515.712 + 374.687i) q^{83} +(-47.7168 + 146.857i) q^{85} +830.847 q^{89} +(-24.6589 + 75.8924i) q^{91} +(-56.0629 - 172.544i) q^{95} +(-642.830 - 467.043i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 27 q^{5} + 13 q^{7} + 45 q^{11} - 89 q^{13} + 24 q^{17} - 28 q^{19} + 348 q^{23} - 9 q^{25} + 249 q^{29} - 217 q^{31} - 630 q^{35} + 585 q^{37} + 177 q^{41} + 684 q^{43} + 93 q^{47} + 1377 q^{49}+ \cdots - 3783 q^{97}+O(q^{100})$ 8 * q + 27 * q^5 + 13 * q^7 + 45 * q^11 - 89 * q^13 + 24 * q^17 - 28 * q^19 + 348 * q^23 - 9 * q^25 + 249 * q^29 - 217 * q^31 - 630 * q^35 + 585 * q^37 + 177 * q^41 + 684 * q^43 + 93 * q^47 + 1377 * q^49 + 693 * q^53 - 472 * q^55 - 1086 * q^59 + 1069 * q^61 + 114 * q^65 + 470 * q^67 - 1479 * q^71 + 947 * q^73 + 3693 * q^77 + 1251 * q^79 - 1185 * q^83 - 2585 * q^85 + 1716 * q^89 + 4191 * q^91 + 141 * q^95 - 3783 * q^97

Character values

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

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

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
$5$	−1.60477	+	1.16593i	−0.143535	+	0.104284i	−0.657235	−	0.753685i	$-0.728274\pi$
$5$	−1.60477	+	1.16593i	−0.143535	+	0.104284i	0.513700	+	0.857970i	$0.328274\pi$
$6$		0			0
$7$	3.78898	+	11.6613i	0.204586	+	0.629650i	0.999730	+	0.0232301i	$0.00739502\pi$
$7$	3.78898	+	11.6613i	0.204586	+	0.629650i	−0.795144	+	0.606420i	$0.792605\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.21788	−	35.5453i	0.225253	−	0.974300i
$11$	8.21788	−	35.5453i	0.225253	−	0.974300i
$12$		0			0
$13$	5.26513	+	3.82534i	0.112330	+	0.0816123i	0.642532	−	0.766259i	$-0.277884\pi$
$13$	5.26513	+	3.82534i	0.112330	+	0.0816123i	−0.530202	+	0.847871i	$0.677884\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	62.9784	−	45.7565i	0.898500	−	0.652798i	−0.0395805	−	0.999216i	$-0.512602\pi$
$17$	62.9784	−	45.7565i	0.898500	−	0.652798i	0.938080	+	0.346418i	$0.112602\pi$
$18$		0			0
$19$	−28.2631	+	86.9850i	−0.341264	+	1.05030i	0.622290	+	0.782787i	$0.286202\pi$
$19$	−28.2631	+	86.9850i	−0.341264	+	1.05030i	−0.963554	+	0.267515i	$0.913798\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.2371			0.183466			0.0917331	−	0.995784i	$-0.470759\pi$
$23$	20.2371			0.183466			0.0917331	+	0.995784i	$0.470759\pi$
$24$		0			0
$25$	−37.4112	+	115.140i	−0.299290	+	0.921120i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	23.5838	+	72.5834i	0.151014	+	0.464773i	0.997735	−	0.0672639i	$-0.0214269\pi$
$29$	23.5838	+	72.5834i	0.151014	+	0.464773i	−0.846721	+	0.532036i	$0.821427\pi$
$30$		0			0
$31$	92.8764	+	67.4787i	0.538100	+	0.390953i	0.823379	−	0.567492i	$-0.192086\pi$
$31$	92.8764	+	67.4787i	0.538100	+	0.390953i	−0.285279	+	0.958445i	$0.592086\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−19.6767	−	14.2960i	−0.0950279	−	0.0690418i
$36$		0			0
$37$	27.6238	+	85.0174i	0.122739	+	0.377751i	0.993482	−	0.113987i	$-0.0363621\pi$
$37$	27.6238	+	85.0174i	0.122739	+	0.377751i	−0.870744	+	0.491737i	$0.836362\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−30.4658	+	93.7641i	−0.116048	+	0.357159i	−0.992164	−	0.124941i	$-0.960126\pi$
$41$	−30.4658	+	93.7641i	−0.116048	+	0.357159i	0.876116	+	0.482100i	$0.160126\pi$
$42$		0			0
$43$	488.233			1.73151			0.865753	−	0.500471i	$-0.166840\pi$
$43$	488.233			1.73151			0.865753	+	0.500471i	$0.166840\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.0191	+	101.622i	−0.102475	+	0.315386i	−0.989130	−	0.147046i	$-0.953023\pi$
$47$	−33.0191	+	101.622i	−0.102475	+	0.315386i	0.886654	+	0.462433i	$0.153023\pi$
$48$		0			0
$49$	155.864	−	113.242i	0.454413	−	0.330150i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	447.009	+	324.771i	1.15852	+	0.841712i	0.989590	−	0.143916i	$-0.0459695\pi$
$53$	447.009	+	324.771i	1.15852	+	0.841712i	0.168928	+	0.985628i	$0.445970\pi$
$54$		0			0
$55$	28.2556	+	66.6235i	0.0692725	+	0.163337i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	52.3705	+	161.180i	0.115560	+	0.355658i	0.992063	−	0.125738i	$-0.0401299\pi$
$59$	52.3705	+	161.180i	0.115560	+	0.355658i	−0.876503	+	0.481396i	$0.840130\pi$
$60$		0			0
$61$	235.345	−	170.988i	0.493982	−	0.358899i	−0.312732	−	0.949841i	$-0.601244\pi$
$61$	235.345	−	170.988i	0.493982	−	0.358899i	0.806713	+	0.590943i	$0.201244\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.9094			−0.0246341
$66$		0			0
$67$	107.351			0.195747			0.0978736	−	0.995199i	$-0.468796\pi$
$67$	107.351			0.195747			0.0978736	+	0.995199i	$0.468796\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−589.515	+	428.308i	−0.985389	+	0.715927i	−0.958906	−	0.283722i	$-0.908431\pi$
$71$	−589.515	+	428.308i	−0.985389	+	0.715927i	−0.0264824	+	0.999649i	$0.508431\pi$
$72$		0			0
$73$	110.836	+	341.119i	0.177704	+	0.546917i	0.999747	−	0.0225091i	$-0.00716546\pi$
$73$	110.836	+	341.119i	0.177704	+	0.546917i	−0.822043	+	0.569426i	$0.807165\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	445.641	−	38.8493i	0.659552	−	0.0574973i
$78$		0			0
$79$	413.815	+	300.654i	0.589339	+	0.428180i	0.842079	−	0.539354i	$-0.181332\pi$
$79$	413.815	+	300.654i	0.589339	+	0.428180i	−0.252740	+	0.967534i	$0.581332\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−515.712	+	374.687i	−0.682010	+	0.495509i	−0.874024	−	0.485883i	$-0.838498\pi$
$83$	−515.712	+	374.687i	−0.682010	+	0.495509i	0.192014	+	0.981392i	$0.438498\pi$
$84$		0			0
$85$	−47.7168	+	146.857i	−0.0608896	+	0.187399i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	830.847			0.989547			0.494773	−	0.869022i	$-0.335251\pi$
$89$	830.847			0.989547			0.494773	+	0.869022i	$0.335251\pi$
$90$		0			0
$91$	−24.6589	+	75.8924i	−0.0284061	+	0.0874251i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−56.0629	−	172.544i	−0.0605467	−	0.186344i
$96$		0			0
$97$	−642.830	−	467.043i	−0.672882	−	0.488877i	0.198107	−	0.980180i	$-0.436521\pi$
$97$	−642.830	−	467.043i	−0.672882	−	0.488877i	−0.870989	+	0.491303i	$0.836521\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−381.819	−	277.407i	−0.376162	−	0.273298i	0.383599	−	0.923500i	$-0.374684\pi$
$101$	−381.819	−	277.407i	−0.376162	−	0.273298i	−0.759762	+	0.650202i	$0.774684\pi$
$102$		0			0
$103$	571.034	+	1757.46i	0.546268	+	1.68124i	0.717955	+	0.696090i	$0.245078\pi$
$103$	571.034	+	1757.46i	0.546268	+	1.68124i	−0.171686	+	0.985152i	$0.554922\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	287.507	−	884.855i	0.259760	−	0.799460i	−0.733094	−	0.680127i	$-0.761925\pi$
$107$	287.507	−	884.855i	0.259760	−	0.799460i	0.992854	−	0.119332i	$-0.0380754\pi$
$108$		0			0
$109$	433.332			0.380786			0.190393	−	0.981708i	$-0.439024\pi$
$109$	433.332			0.380786			0.190393	+	0.981708i	$0.439024\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	520.776	−	1602.78i	0.433544	−	1.33431i	−0.461026	−	0.887386i	$-0.652519\pi$
$113$	520.776	−	1602.78i	0.433544	−	1.33431i	0.894571	−	0.446926i	$-0.147481\pi$
$114$		0			0
$115$	−32.4759	+	23.5951i	−0.0263338	+	0.0191327i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	772.203	+	561.038i	0.594855	+	0.432187i
$120$		0			0
$121$	−1195.93	−	584.214i	−0.898522	−	0.438928i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−150.830	−	464.208i	−0.107925	−	0.332160i
$126$		0			0
$127$	−100.382	+	72.9320i	−0.0701377	+	0.0509580i	−0.622302	−	0.782778i	$-0.713802\pi$
$127$	−100.382	+	72.9320i	−0.0701377	+	0.0509580i	0.552164	+	0.833736i	$0.313802\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	140.240			0.0935328			0.0467664	−	0.998906i	$-0.485108\pi$
$131$	140.240			0.0935328			0.0467664	+	0.998906i	$0.485108\pi$
$132$		0			0
$133$	−1121.45			−0.731140
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−610.321	+	443.424i	−0.380608	+	0.276528i	−0.761596	−	0.648052i	$-0.775584\pi$
$137$	−610.321	+	443.424i	−0.380608	+	0.276528i	0.380988	+	0.924580i	$0.375584\pi$
$138$		0			0
$139$	−367.333	−	1130.53i	−0.224149	−	0.689861i	−0.998377	−	0.0569534i	$-0.981861\pi$
$139$	−367.333	−	1130.53i	−0.224149	−	0.689861i	0.774227	−	0.632907i	$-0.218139\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	179.241	−	155.714i	0.104817	−	0.0910594i
$144$		0			0
$145$	−122.474	−	88.9827i	−0.0701443	−	0.0509628i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1173.73	+	852.763i	−0.645339	+	0.468866i	−0.861680	−	0.507452i	$-0.830588\pi$
$149$	−1173.73	+	852.763i	−0.645339	+	0.468866i	0.216341	+	0.976318i	$0.430588\pi$
$150$		0			0
$151$	−870.060	+	2677.77i	−0.468904	+	1.44314i	0.385102	+	0.922874i	$0.374166\pi$
$151$	−870.060	+	2677.77i	−0.468904	+	1.44314i	−0.854006	+	0.520263i	$0.825834\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−227.721			−0.118006
$156$		0			0
$157$	742.481	−	2285.12i	0.377429	−	1.16161i	−0.564396	−	0.825504i	$-0.690891\pi$
$157$	742.481	−	2285.12i	0.377429	−	1.16161i	0.941825	−	0.336104i	$-0.109109\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	76.6779	+	235.990i	0.0375346	+	0.115520i
$162$		0			0
$163$	−1694.23	−	1230.93i	−0.814123	−	0.591495i	0.100900	−	0.994897i	$-0.467828\pi$
$163$	−1694.23	−	1230.93i	−0.814123	−	0.591495i	−0.915023	+	0.403402i	$0.867828\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3089.83	−	2244.90i	−1.43173	−	1.04021i	−0.989692	−	0.143214i	$-0.954256\pi$
$167$	−3089.83	−	2244.90i	−1.43173	−	1.04021i	−0.442036	−	0.896997i	$-0.645744\pi$
$168$		0			0
$169$	−665.822	−	2049.19i	−0.303060	−	0.932722i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	611.373	−	1881.61i	0.268681	−	0.826916i	−0.722141	−	0.691746i	$-0.756842\pi$
$173$	611.373	−	1881.61i	0.268681	−	0.826916i	0.990822	−	0.135170i	$-0.0431581\pi$
$174$		0			0
$175$	−1484.43			−0.641214
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−818.393	+	2518.76i	−0.341729	+	1.05174i	0.621582	+	0.783349i	$0.286490\pi$
$179$	−818.393	+	2518.76i	−0.341729	+	1.05174i	−0.963311	+	0.268386i	$0.913510\pi$
$180$		0			0
$181$	3446.85	−	2504.29i	1.41548	−	1.02841i	0.422988	−	0.906135i	$-0.360981\pi$
$181$	3446.85	−	2504.29i	1.41548	−	1.02841i	0.992496	−	0.122274i	$-0.0390188\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−143.455	−	104.226i	−0.0570108	−	0.0414208i
$186$		0			0
$187$	−1108.88	−	2614.60i	−0.433632	−	1.02245i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	407.397	+	1253.84i	0.154336	+	0.474998i	0.998093	−	0.0617276i	$-0.0196610\pi$
$191$	407.397	+	1253.84i	0.154336	+	0.474998i	−0.843757	+	0.536726i	$0.819661\pi$
$192$		0			0
$193$	2230.05	−	1620.23i	0.831724	−	0.604283i	−0.0883223	−	0.996092i	$-0.528151\pi$
$193$	2230.05	−	1620.23i	0.831724	−	0.604283i	0.920047	+	0.391809i	$0.128151\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1901.61			−0.687738			−0.343869	−	0.939018i	$-0.611738\pi$
$197$	−1901.61			−0.687738			−0.343869	+	0.939018i	$0.611738\pi$
$198$		0			0
$199$	−3994.66			−1.42299			−0.711493	−	0.702693i	$-0.751981\pi$
$199$	−3994.66			−1.42299			−0.711493	+	0.702693i	$0.751981\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−757.058	+	550.034i	−0.261749	+	0.190172i
$204$		0			0
$205$	−60.4322	−	185.991i	−0.0205891	−	0.0633668i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2859.64	+	1719.45i	0.946438	+	0.569077i
$210$		0			0
$211$	−417.464	−	303.305i	−0.136206	−	0.0989592i	0.517596	−	0.855625i	$-0.326827\pi$
$211$	−417.464	−	303.305i	−0.136206	−	0.0989592i	−0.653802	+	0.756666i	$0.726827\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−783.502	+	569.247i	−0.248532	+	0.180569i
$216$		0			0
$217$	−434.981	+	1338.73i	−0.136076	+	0.418798i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	506.624			0.154204
$222$		0			0
$223$	−271.131	+	834.455i	−0.0814182	+	0.250579i	−0.983477	−	0.181034i	$-0.942056\pi$
$223$	−271.131	+	834.455i	−0.0814182	+	0.250579i	0.902059	+	0.431614i	$0.142056\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1238.72	+	3812.39i	0.362189	+	1.11470i	0.951723	+	0.306959i	$0.0993113\pi$
$227$	1238.72	+	3812.39i	0.362189	+	1.11470i	−0.589534	+	0.807743i	$0.700689\pi$
$228$		0			0
$229$	−1386.93	−	1007.66i	−0.400222	−	0.290779i	0.369409	−	0.929267i	$-0.379560\pi$
$229$	−1386.93	−	1007.66i	−0.400222	−	0.290779i	−0.769632	+	0.638488i	$0.779560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5121.82	−	3721.22i	−1.44009	−	1.04629i	−0.988023	−	0.154305i	$-0.950686\pi$
$233$	−5121.82	−	3721.22i	−1.44009	−	1.04629i	−0.452068	−	0.891983i	$-0.649314\pi$
$234$		0			0
$235$	−65.4970	−	201.579i	−0.0181811	−	0.0559556i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1863.05	−	5733.87i	0.504228	−	1.55185i	−0.297837	−	0.954617i	$-0.596265\pi$
$239$	1863.05	−	5733.87i	0.504228	−	1.55185i	0.802065	−	0.597237i	$-0.203735\pi$
$240$		0			0
$241$	2528.58			0.675852			0.337926	−	0.941173i	$-0.390275\pi$
$241$	2528.58			0.675852			0.337926	+	0.941173i	$0.390275\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−118.093	+	363.454i	−0.0307947	+	0.0947763i
$246$		0			0
$247$	−481.557	+	349.871i	−0.124051	+	0.0901287i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2313.85	+	1681.11i	0.581867	+	0.422751i	0.839397	−	0.543519i	$-0.182908\pi$
$251$	2313.85	+	1681.11i	0.581867	+	0.422751i	−0.257530	+	0.966270i	$0.582908\pi$
$252$		0			0
$253$	166.306	−	719.333i	0.0413263	−	0.178751i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1454.89	−	4477.69i	−0.353127	−	1.08681i	−0.957088	−	0.289798i	$-0.906412\pi$
$257$	−1454.89	−	4477.69i	−0.353127	−	1.08681i	0.603961	−	0.797014i	$-0.293588\pi$
$258$		0			0
$259$	−886.746	+	644.259i	−0.212740	+	0.154565i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1322.27			0.310018			0.155009	−	0.987913i	$-0.450459\pi$
$263$	1322.27			0.310018			0.155009	+	0.987913i	$0.450459\pi$
$264$		0			0
$265$	−1096.01			−0.254065
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4208.37	−	3057.56i	0.953863	−	0.693022i	0.00214547	−	0.999998i	$-0.499317\pi$
$269$	4208.37	−	3057.56i	0.953863	−	0.693022i	0.951717	+	0.306976i	$0.0993171\pi$
$270$		0			0
$271$	1664.83	+	5123.81i	0.373177	+	1.14852i	0.944700	+	0.327935i	$0.106353\pi$
$271$	1664.83	+	5123.81i	0.373177	+	1.14852i	−0.571523	+	0.820586i	$0.693647\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3785.24	+	2276.00i	0.830031	+	0.499083i
$276$		0			0
$277$	2523.21	+	1833.22i	0.547310	+	0.397644i	0.826793	−	0.562507i	$-0.190163\pi$
$277$	2523.21	+	1833.22i	0.547310	+	0.397644i	−0.279483	+	0.960151i	$0.590163\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5524.63	+	4013.88i	−1.17285	+	0.852127i	−0.991348	−	0.131263i	$-0.958097\pi$
$281$	−5524.63	+	4013.88i	−1.17285	+	0.852127i	−0.181505	+	0.983390i	$0.558097\pi$
$282$		0			0
$283$	−2827.14	+	8701.05i	−0.593838	+	1.82765i	−0.0334114	+	0.999442i	$0.510637\pi$
$283$	−2827.14	+	8701.05i	−0.593838	+	1.82765i	−0.560427	+	0.828204i	$0.689363\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1208.84			−0.248627
$288$		0			0
$289$	354.420	−	1090.79i	0.0721391	−	0.222021i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1000.77	−	3080.04i	−0.199541	−	0.614123i	−0.999894	−	0.0145935i	$-0.995355\pi$
$293$	−1000.77	−	3080.04i	−0.199541	−	0.614123i	0.800353	−	0.599529i	$-0.204645\pi$
$294$		0			0
$295$	−271.968	−	197.596i	−0.0536766	−	0.0389983i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	106.551	+	77.4138i	0.0206087	+	0.0149731i
$300$		0			0
$301$	1849.90	+	5693.42i	0.354242	+	1.09024i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−178.314	+	548.794i	−0.0334762	+	0.103029i
$306$		0			0
$307$	−3909.16			−0.726735			−0.363367	−	0.931646i	$-0.618373\pi$
$307$	−3909.16			−0.726735			−0.363367	+	0.931646i	$0.618373\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2705.87	−	8327.83i	0.493364	−	1.51842i	−0.326128	−	0.945326i	$-0.605744\pi$
$311$	2705.87	−	8327.83i	0.493364	−	1.51842i	0.819491	−	0.573091i	$-0.194256\pi$
$312$		0			0
$313$	4483.78	−	3257.66i	0.809707	−	0.588286i	−0.104039	−	0.994573i	$-0.533177\pi$
$313$	4483.78	−	3257.66i	0.809707	−	0.588286i	0.913746	+	0.406287i	$0.133177\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6558.45	+	4765.00i	1.16202	+	0.844255i	0.990032	−	0.140845i	$-0.0449818\pi$
$317$	6558.45	+	4765.00i	1.16202	+	0.844255i	0.171985	+	0.985099i	$0.444982\pi$
$318$		0			0
$319$	2773.81	−	241.810i	0.486844	−	0.0424413i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2200.16	+	6771.39i	0.379010	+	1.16647i
$324$		0			0
$325$	−637.425	+	463.116i	−0.108794	+	0.0790433i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1310.16			−0.219548
$330$		0			0
$331$	2603.54			0.432336			0.216168	−	0.976356i	$-0.430644\pi$
$331$	2603.54			0.432336			0.216168	+	0.976356i	$0.430644\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−172.274	+	125.165i	−0.0280966	+	0.0204134i
$336$		0			0
$337$	−238.684	−	734.594i	−0.0385814	−	0.118741i	0.929911	−	0.367785i	$-0.119884\pi$
$337$	−238.684	−	734.594i	−0.0385814	−	0.118741i	−0.968492	+	0.249044i	$0.919884\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3161.80	−	2746.79i	0.502114	−	0.436208i
$342$		0			0
$343$	5313.56	+	3860.53i	0.836458	+	0.607722i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3408.56	+	2476.46i	−0.527323	+	0.383123i	−0.819355	−	0.573286i	$-0.805668\pi$
$347$	−3408.56	+	2476.46i	−0.527323	+	0.383123i	0.292032	+	0.956408i	$0.405668\pi$
$348$		0			0
$349$	1483.19	−	4564.79i	0.227488	−	0.700137i	−0.770541	−	0.637390i	$-0.780014\pi$
$349$	1483.19	−	4564.79i	0.227488	−	0.700137i	0.998029	−	0.0627467i	$-0.0199860\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	448.615			0.0676412			0.0338206	−	0.999428i	$-0.489233\pi$
$353$	448.615			0.0676412			0.0338206	+	0.999428i	$0.489233\pi$
$354$		0			0
$355$	446.658	−	1374.67i	0.0667779	−	0.205521i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3435.35	+	10572.9i	0.505045	+	1.55437i	0.800695	+	0.599072i	$0.204464\pi$
$359$	3435.35	+	10572.9i	0.505045	+	1.55437i	−0.295651	+	0.955296i	$0.595536\pi$
$360$		0			0
$361$	−1218.54	−	885.320i	−0.177655	−	0.129074i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−575.589	−	418.190i	−0.0825417	−	0.0599700i
$366$		0			0
$367$	3149.15	+	9692.08i	0.447913	+	1.37854i	0.879257	+	0.476348i	$0.158040\pi$
$367$	3149.15	+	9692.08i	0.447913	+	1.37854i	−0.431344	+	0.902188i	$0.641960\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2093.54	+	6443.25i	−0.292968	+	0.901663i
$372$		0			0
$373$	11174.0			1.55112			0.775560	−	0.631274i	$-0.217468\pi$
$373$	11174.0			1.55112			0.775560	+	0.631274i	$0.217468\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−153.485	+	472.377i	−0.0209678	+	0.0645323i
$378$		0			0
$379$	7717.65	−	5607.20i	1.04599	−	0.759954i	0.0745413	−	0.997218i	$-0.476251\pi$
$379$	7717.65	−	5607.20i	1.04599	−	0.759954i	0.971445	+	0.237264i	$0.0762507\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9441.77	−	6859.85i	−1.25967	−	0.915201i	−0.260925	−	0.965359i	$-0.584027\pi$
$383$	−9441.77	−	6859.85i	−1.25967	−	0.915201i	−0.998741	+	0.0501584i	$0.984027\pi$
$384$		0			0
$385$	−669.856	+	581.932i	−0.0886728	+	0.0770338i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3268.87	+	10060.6i	0.426063	+	1.31129i	0.901973	+	0.431793i	$0.142119\pi$
$389$	3268.87	+	10060.6i	0.426063	+	1.31129i	−0.475910	+	0.879494i	$0.657881\pi$
$390$		0			0
$391$	1274.50	−	925.977i	0.164844	−	0.119766i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1014.62			−0.129243
$396$		0			0
$397$	−337.577			−0.0426763			−0.0213381	−	0.999772i	$-0.506793\pi$
$397$	−337.577			−0.0426763			−0.0213381	+	0.999772i	$0.506793\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4762.00	−	3459.79i	0.593024	−	0.430857i	−0.250372	−	0.968150i	$-0.580553\pi$
$401$	4762.00	−	3459.79i	0.593024	−	0.430857i	0.843396	+	0.537292i	$0.180553\pi$
$402$		0			0
$403$	230.878	+	710.569i	0.0285381	+	0.0878311i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3248.98	−	283.233i	0.395690	−	0.0344947i
$408$		0			0
$409$	−10318.7	−	7496.99i	−1.24750	−	0.906363i	−0.249427	−	0.968394i	$-0.580242\pi$
$409$	−10318.7	−	7496.99i	−1.24750	−	0.906363i	−0.998074	+	0.0620309i	$0.980242\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1681.13	+	1221.41i	−0.200298	+	0.145525i
$414$		0			0
$415$	390.740	−	1202.57i	0.0462185	−	0.142246i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4686.15			−0.546380			−0.273190	−	0.961960i	$-0.588079\pi$
$419$	−4686.15			−0.546380			−0.273190	+	0.961960i	$0.588079\pi$
$420$		0			0
$421$	−479.525	+	1475.83i	−0.0555121	+	0.170849i	−0.974968	−	0.222344i	$-0.928629\pi$
$421$	−479.525	+	1475.83i	−0.0555121	+	0.170849i	0.919456	+	0.393192i	$0.128629\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2912.30	+	8963.13i	0.332393	+	1.02300i
$426$		0			0
$427$	2885.66	+	2096.56i	0.327042	+	0.237610i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9433.01	−	6853.48i	−1.05423	−	0.765941i	−0.0812156	−	0.996697i	$-0.525880\pi$
$431$	−9433.01	−	6853.48i	−1.05423	−	0.765941i	−0.973012	+	0.230756i	$0.925880\pi$
$432$		0			0
$433$	785.662	+	2418.02i	0.0871975	+	0.268366i	0.985142	−	0.171743i	$-0.0549398\pi$
$433$	785.662	+	2418.02i	0.0871975	+	0.268366i	−0.897944	+	0.440109i	$0.854940\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−571.963	+	1760.32i	−0.0626103	+	0.192695i
$438$		0			0
$439$	−5101.99			−0.554681			−0.277340	−	0.960772i	$-0.589453\pi$
$439$	−5101.99			−0.554681			−0.277340	+	0.960772i	$0.589453\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−846.872	+	2606.40i	−0.0908264	+	0.279535i	−0.986144	−	0.165894i	$-0.946949\pi$
$443$	−846.872	+	2606.40i	−0.0908264	+	0.279535i	0.895317	+	0.445429i	$0.146949\pi$
$444$		0			0
$445$	−1333.32	+	968.714i	−0.142035	+	0.103194i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8818.11	+	6406.73i	0.926843	+	0.673391i	0.945218	−	0.326440i	$-0.105849\pi$
$449$	8818.11	+	6406.73i	0.926843	+	0.673391i	−0.0183750	+	0.999831i	$0.505849\pi$
$450$		0			0
$451$	3082.51	+	1853.46i	0.321840	+	0.193517i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−48.9136	−	150.541i	−0.00503979	−	0.0155109i
$456$		0			0
$457$	−11106.2	+	8069.11i	−1.13682	+	0.825946i	−0.986673	−	0.162718i	$-0.947974\pi$
$457$	−11106.2	+	8069.11i	−1.13682	+	0.825946i	−0.150144	+	0.988664i	$0.547974\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3104.40			−0.313636			−0.156818	−	0.987628i	$-0.550124\pi$
$461$	−3104.40			−0.313636			−0.156818	+	0.987628i	$0.550124\pi$
$462$		0			0
$463$	3771.28			0.378545			0.189272	−	0.981925i	$-0.439387\pi$
$463$	3771.28			0.378545			0.189272	+	0.981925i	$0.439387\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9914.73	−	7203.47i	0.982439	−	0.713784i	0.0241865	−	0.999707i	$-0.492300\pi$
$467$	9914.73	−	7203.47i	0.982439	−	0.713784i	0.958252	+	0.285924i	$0.0923005\pi$
$468$		0			0
$469$	406.753	+	1251.86i	0.0400471	+	0.123252i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4012.24	−	17354.4i	0.390027	−	1.68701i
$474$		0			0
$475$	−8958.09	−	6508.43i	−0.865316	−	0.628689i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−843.488	+	612.830i	−0.0804592	+	0.0584570i	−0.627288	−	0.778788i	$-0.715835\pi$
$479$	−843.488	+	612.830i	−0.0804592	+	0.0584570i	0.546828	+	0.837245i	$0.315835\pi$
$480$		0			0
$481$	−179.778	+	553.298i	−0.0170419	+	0.0524496i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1576.14			0.147564
$486$		0			0
$487$	536.464	−	1651.07i	0.0499169	−	0.153628i	−0.922991	−	0.384822i	$-0.874263\pi$
$487$	536.464	−	1651.07i	0.0499169	−	0.153628i	0.972908	+	0.231193i	$0.0742630\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3489.83	−	10740.6i	−0.320761	−	0.987202i	−0.973318	−	0.229462i	$-0.926303\pi$
$491$	−3489.83	−	10740.6i	−0.320761	−	0.987202i	0.652556	−	0.757740i	$-0.273697\pi$
$492$		0			0
$493$	4806.43	+	3492.07i	0.439089	+	0.319016i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7228.28	−	5251.66i	−0.652380	−	0.473982i
$498$		0			0
$499$	−4140.83	−	12744.2i	−0.371481	−	1.14330i	−0.945822	−	0.324686i	$-0.894741\pi$
$499$	−4140.83	−	12744.2i	−0.371481	−	1.14330i	0.574341	−	0.818616i	$-0.305259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2410.37	+	7418.36i	−0.213664	+	0.657591i	0.785582	+	0.618758i	$0.212364\pi$
$503$	−2410.37	+	7418.36i	−0.213664	+	0.657591i	−0.999246	+	0.0388326i	$0.987636\pi$
$504$		0			0
$505$	936.171			0.0824932
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6733.56	−	20723.8i	0.586365	−	1.80465i	−0.00735129	−	0.999973i	$-0.502340\pi$
$509$	6733.56	−	20723.8i	0.586365	−	1.80465i	0.593717	−	0.804674i	$-0.297660\pi$
$510$		0			0
$511$	−3557.93	+	2584.99i	−0.308011	+	0.223783i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2965.46	−	2154.54i	−0.253736	−	0.184350i
$516$		0			0
$517$	3340.85	+	2008.80i	0.284198	+	0.170883i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	258.126	+	794.431i	0.0217058	+	0.0668036i	0.961323	−	0.275425i	$-0.0888184\pi$
$521$	258.126	+	794.431i	0.0217058	+	0.0668036i	−0.939617	+	0.342228i	$0.888818\pi$
$522$		0			0
$523$	−664.886	+	483.068i	−0.0555898	+	0.0403883i	−0.615233	−	0.788345i	$-0.710938\pi$
$523$	−664.886	+	483.068i	−0.0555898	+	0.0403883i	0.559643	+	0.828734i	$0.310938\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8936.79			0.738696
$528$		0			0
$529$	−11757.5			−0.966340
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−519.087	+	377.138i	−0.0421841	+	0.0306486i
$534$		0			0
$535$	570.300	+	1755.20i	0.0460864	+	0.141839i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2744.33	−	6470.82i	−0.219308	−	0.517102i
$540$		0			0
$541$	−16604.9	−	12064.2i	−1.31960	−	0.958744i	−0.999937	−	0.0112114i	$-0.996431\pi$
$541$	−16604.9	−	12064.2i	−1.31960	−	0.958744i	−0.319660	−	0.947532i	$-0.603569\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−695.399	+	505.237i	−0.0546562	+	0.0397101i
$546$		0			0
$547$	1843.40	−	5673.41i	0.144092	−	0.443469i	−0.852801	−	0.522236i	$-0.825098\pi$
$547$	1843.40	−	5673.41i	0.144092	−	0.443469i	0.996893	+	0.0787666i	$0.0250982\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6980.22			−0.539687
$552$		0			0
$553$	−1938.08	+	5964.79i	−0.149033	+	0.458677i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	451.401	+	1389.27i	0.0343383	+	0.105683i	0.966757	−	0.255698i	$-0.0823054\pi$
$557$	451.401	+	1389.27i	0.0343383	+	0.105683i	−0.932418	+	0.361381i	$0.882305\pi$
$558$		0			0
$559$	2570.61	+	1867.66i	0.194499	+	0.141312i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20000.9	−	14531.5i	−1.49723	−	1.08780i	−0.971469	−	0.237167i	$-0.923781\pi$
$563$	−20000.9	−	14531.5i	−1.49723	−	1.08780i	−0.525759	−	0.850633i	$-0.676219\pi$
$564$		0			0
$565$	1033.02	+	3179.29i	0.0769191	+	0.236733i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7905.61	+	24331.0i	−0.582461	+	1.79263i	0.0267756	+	0.999641i	$0.491476\pi$
$569$	−7905.61	+	24331.0i	−0.582461	+	1.79263i	−0.609236	+	0.792989i	$0.708524\pi$
$570$		0			0
$571$	−8840.84			−0.647947			−0.323973	−	0.946066i	$-0.605019\pi$
$571$	−8840.84			−0.647947			−0.323973	+	0.946066i	$0.605019\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−757.094	+	2330.10i	−0.0549096	+	0.168994i
$576$		0			0
$577$	−14479.3	+	10519.8i	−1.04468	+	0.759003i	−0.971193	−	0.238293i	$-0.923412\pi$
$577$	−14479.3	+	10519.8i	−1.04468	+	0.759003i	−0.0734847	+	0.997296i	$0.523412\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6323.36	−	4594.19i	−0.451527	−	0.328053i
$582$		0			0
$583$	15217.6	−	13220.1i	1.08104	−	0.939146i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8445.52	+	25992.6i	0.593840	+	1.82765i	0.560419	+	0.828209i	$0.310640\pi$
$587$	8445.52	+	25992.6i	0.593840	+	1.82765i	0.0334206	+	0.999441i	$0.489360\pi$
$588$		0			0
$589$	−8494.61	+	6171.70i	−0.594252	+	0.431749i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	389.353			0.0269626			0.0134813	−	0.999909i	$-0.495709\pi$
$593$	389.353			0.0269626			0.0134813	+	0.999909i	$0.495709\pi$
$594$		0			0
$595$	−1893.34			−0.130453
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20429.9	−	14843.2i	1.39356	−	1.01248i	0.398096	−	0.917344i	$-0.369671\pi$
$599$	20429.9	−	14843.2i	1.39356	−	1.01248i	0.995464	−	0.0951368i	$-0.0303288\pi$
$600$		0			0
$601$	−2181.63	−	6714.37i	−0.148071	−	0.455715i	0.849322	−	0.527875i	$-0.177011\pi$
$601$	−2181.63	−	6714.37i	−0.148071	−	0.455715i	−0.997393	+	0.0721596i	$0.977011\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2600.35	−	456.850i	0.174743	−	0.0307002i
$606$		0			0
$607$	−12283.3	−	8924.34i	−0.821357	−	0.596751i	0.0957435	−	0.995406i	$-0.469477\pi$
$607$	−12283.3	−	8924.34i	−0.821357	−	0.596751i	−0.917101	+	0.398655i	$0.869477\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−562.591	+	408.746i	−0.0372504	+	0.0270640i
$612$		0			0
$613$	4796.66	−	14762.6i	0.316044	−	0.972685i	−0.659278	−	0.751899i	$-0.729138\pi$
$613$	4796.66	−	14762.6i	0.316044	−	0.972685i	0.975322	−	0.220785i	$-0.0708620\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27430.4			−1.78980			−0.894899	−	0.446269i	$-0.852752\pi$
$617$	−27430.4			−1.78980			−0.894899	+	0.446269i	$0.852752\pi$
$618$		0			0
$619$	6668.83	−	20524.5i	0.433025	−	1.33272i	−0.462071	−	0.886843i	$-0.652894\pi$
$619$	6668.83	−	20524.5i	0.433025	−	1.33272i	0.895097	−	0.445872i	$-0.147106\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3148.07	+	9688.75i	0.202447	+	0.623068i
$624$		0			0
$625$	−11459.7	−	8325.96i	−0.733421	−	0.532862i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5629.80	+	4090.29i	0.356875	+	0.259285i
$630$		0			0
$631$	−2320.52	−	7141.83i	−0.146400	−	0.450573i	0.850788	−	0.525509i	$-0.176125\pi$
$631$	−2320.52	−	7141.83i	−0.146400	−	0.450573i	−0.997188	+	0.0749353i	$0.976125\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	76.0566	−	234.078i	0.00475310	−	0.0146285i
$636$		0			0
$637$	1253.83			0.0779883
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2424.82	−	7462.83i	0.149414	−	0.459850i	−0.848138	−	0.529776i	$-0.822276\pi$
$641$	2424.82	−	7462.83i	0.149414	−	0.459850i	0.997552	+	0.0699253i	$0.0222761\pi$
$642$		0			0
$643$	706.659	−	513.418i	0.0433404	−	0.0314887i	−0.565904	−	0.824471i	$-0.691473\pi$
$643$	706.659	−	513.418i	0.0433404	−	0.0314887i	0.609245	+	0.792982i	$0.291473\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20663.0	−	15012.6i	−1.25556	−	0.912217i	−0.257028	−	0.966404i	$-0.582743\pi$
$647$	−20663.0	−	15012.6i	−1.25556	−	0.912217i	−0.998531	+	0.0541866i	$0.982743\pi$
$648$		0			0
$649$	6159.56	−	536.967i	0.372548	−	0.0324774i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3190.62	+	9819.73i	0.191208	+	0.588477i	1.00000		0.000382475i	$0.000121746\pi$
$653$	3190.62	+	9819.73i	0.191208	+	0.588477i	−0.808792	+	0.588095i	$0.799878\pi$
$654$		0			0
$655$	−225.053	+	163.510i	−0.0134252	+	0.00975401i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	28905.7			1.70866			0.854329	−	0.519733i	$-0.173969\pi$
$659$	28905.7			1.70866			0.854329	+	0.519733i	$0.173969\pi$
$660$		0			0
$661$	28041.5			1.65006			0.825028	−	0.565092i	$-0.191159\pi$
$661$	28041.5			1.65006			0.825028	+	0.565092i	$0.191159\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1799.66	−	1307.53i	0.104944	−	0.0762465i
$666$		0			0
$667$	477.267	+	1468.88i	0.0277059	+	0.0852701i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4143.79	−	9770.57i	−0.238404	−	0.562129i
$672$		0			0
$673$	−7569.87	−	5499.83i	−0.433576	−	0.315012i	0.349501	−	0.936936i	$-0.386351\pi$
$673$	−7569.87	−	5499.83i	−0.433576	−	0.315012i	−0.783077	+	0.621924i	$0.786351\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18651.9	+	13551.4i	−1.05886	+	0.769309i	−0.973878	−	0.227073i	$-0.927084\pi$
$677$	−18651.9	+	13551.4i	−1.05886	+	0.769309i	−0.0849856	+	0.996382i	$0.527084\pi$
$678$		0			0
$679$	3010.66	−	9265.84i	0.170160	−	0.523697i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18443.9			−1.03329			−0.516645	−	0.856200i	$-0.672819\pi$
$683$	−18443.9			−1.03329			−0.516645	+	0.856200i	$0.672819\pi$
$684$		0			0
$685$	462.422	−	1423.19i	0.0257931	−	0.0793829i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1111.20	+	3419.93i	0.0614418	+	0.189098i
$690$		0			0
$691$	20208.4	+	14682.3i	1.11254	+	0.808306i	0.983061	−	0.183276i	$-0.0586703\pi$
$691$	20208.4	+	14682.3i	1.11254	+	0.808306i	0.129476	+	0.991582i	$0.458670\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1907.61	+	1385.96i	0.104115	+	0.0756440i
$696$		0			0
$697$	2371.63	+	7299.12i	0.128884	+	0.396663i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7753.13	+	23861.7i	−0.417734	+	1.28565i	0.492048	+	0.870568i	$0.336249\pi$
$701$	−7753.13	+	23861.7i	−0.417734	+	1.28565i	−0.909782	+	0.415086i	$0.863751\pi$
$702$		0			0
$703$	−8175.97			−0.438638
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1788.22	−	5503.59i	0.0951246	−	0.292763i
$708$		0			0
$709$	12414.6	−	9019.76i	0.657604	−	0.477777i	−0.208249	−	0.978076i	$-0.566776\pi$
$709$	12414.6	−	9019.76i	0.657604	−	0.477777i	0.865853	+	0.500298i	$0.166776\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1879.55	+	1365.57i	0.0987232	+	0.0717266i
$714$		0			0
$715$	−106.088	+	458.869i	−0.00554891	+	0.0240010i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9216.75	+	28366.3i	0.478063	+	1.47133i	0.841783	+	0.539817i	$0.181506\pi$
$719$	9216.75	+	28366.3i	0.478063	+	1.47133i	−0.363720	+	0.931508i	$0.618494\pi$
$720$		0			0
$721$	−18330.6	+	13318.0i	−0.946835	+	0.687916i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9239.55			−0.473308
$726$		0			0
$727$	−19398.7			−0.989626			−0.494813	−	0.869000i	$-0.664763\pi$
$727$	−19398.7			−0.989626			−0.494813	+	0.869000i	$0.664763\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	30748.1	−	22339.8i	1.55576	−	1.13032i
$732$		0			0
$733$	10180.5	+	31332.4i	0.512996	+	1.57884i	0.786900	+	0.617081i	$0.211685\pi$
$733$	10180.5	+	31332.4i	0.512996	+	1.57884i	−0.273904	+	0.961757i	$0.588315\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	882.201	−	3815.84i	0.0440927	−	0.190717i
$738$		0			0
$739$	5707.17	+	4146.50i	0.284089	+	0.206403i	0.720699	−	0.693248i	$-0.243821\pi$
$739$	5707.17	+	4146.50i	0.284089	+	0.206403i	−0.436610	+	0.899651i	$0.643821\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4025.05	−	2924.37i	0.198741	−	0.144394i	−0.483964	−	0.875088i	$-0.660803\pi$
$743$	4025.05	−	2924.37i	0.198741	−	0.144394i	0.682705	+	0.730694i	$0.260803\pi$
$744$		0			0
$745$	889.298	−	2736.98i	0.0437334	−	0.134598i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11407.9			0.556523
$750$		0			0
$751$	10127.4	−	31168.8i	0.492081	−	1.51447i	−0.329376	−	0.944199i	$-0.606838\pi$
$751$	10127.4	−	31168.8i	0.492081	−	1.51447i	0.821457	−	0.570270i	$-0.193162\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1725.86	−	5311.64i	−0.0831925	−	0.256040i
$756$		0			0
$757$	25632.9	+	18623.4i	1.23070	+	0.894158i	0.996943	−	0.0781357i	$-0.0248968\pi$
$757$	25632.9	+	18623.4i	1.23070	+	0.894158i	0.233761	+	0.972294i	$0.424897\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16845.6	−	12239.0i	−0.802432	−	0.583001i	0.109195	−	0.994020i	$-0.465173\pi$
$761$	−16845.6	−	12239.0i	−0.802432	−	0.583001i	−0.911627	+	0.411019i	$0.865173\pi$
$762$		0			0
$763$	1641.89	+	5053.21i	0.0779035	+	0.239762i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−340.831	+	1048.97i	−0.0160452	+	0.0493821i
$768$		0			0
$769$	−15356.6			−0.720123			−0.360062	−	0.932929i	$-0.617244\pi$
$769$	−15356.6			−0.720123			−0.360062	+	0.932929i	$0.617244\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7407.09	−	22796.7i	0.344650	−	1.06072i	−0.617121	−	0.786868i	$-0.711701\pi$
$773$	7407.09	−	22796.7i	0.344650	−	1.06072i	0.961771	−	0.273855i	$-0.0882990\pi$
$774$		0			0
$775$	−11244.1	+	8169.33i	−0.521162	+	0.378646i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7295.01	−	5300.14i	−0.335521	−	0.243770i
$780$		0			0
$781$	10379.8	+	24474.3i	0.475566	+	1.12133i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1472.79	+	4532.78i	0.0669632	+	0.206092i
$786$		0			0
$787$	−20832.6	+	15135.7i	−0.943584	+	0.685554i	−0.949281	−	0.314430i	$-0.898187\pi$
$787$	−20832.6	+	15135.7i	−0.943584	+	0.685554i	0.00569705	+	0.999984i	$0.498187\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20663.7			0.928847
$792$		0			0
$793$	1893.21			0.0847793
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18104.8	−	13153.9i	0.804651	−	0.584613i	−0.107624	−	0.994192i	$-0.534324\pi$
$797$	18104.8	−	13153.9i	0.804651	−	0.584613i	0.912275	+	0.409579i	$0.134324\pi$
$798$		0			0
$799$	2570.39	+	7910.86i	0.113810	+	0.350270i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13036.0	−	1136.43i	0.572890	−	0.0499424i
$804$		0			0
$805$	−398.200	−	289.309i	−0.0174344	−	0.0126668i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−327.625	+	238.034i	−0.0142382	+	0.0103446i	−0.594882	−	0.803813i	$-0.702801\pi$
$809$	−327.625	+	238.034i	−0.0142382	+	0.0103446i	0.580643	+	0.814158i	$0.302801\pi$
$810$		0			0
$811$	−1964.56	+	6046.29i	−0.0850616	+	0.261793i	−0.984536	−	0.175180i	$-0.943949\pi$
$811$	−1964.56	+	6046.29i	−0.0850616	+	0.261793i	0.899475	+	0.436973i	$0.143949\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4154.03			0.178539
$816$		0			0
$817$	−13799.0	+	42468.9i	−0.590900	+	1.81860i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13029.9	+	40102.0i	0.553895	+	1.70471i	0.698844	+	0.715274i	$0.253698\pi$
$821$	13029.9	+	40102.0i	0.553895	+	1.70471i	−0.144949	+	0.989439i	$0.546302\pi$
$822$		0			0
$823$	12019.8	+	8732.87i	0.509092	+	0.369877i	0.812479	−	0.582991i	$-0.198118\pi$
$823$	12019.8	+	8732.87i	0.509092	+	0.369877i	−0.303387	+	0.952867i	$0.598118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−296.623	−	215.509i	−0.0124723	−	0.00906167i	0.581532	−	0.813524i	$-0.302454\pi$
$827$	−296.623	−	215.509i	−0.0124723	−	0.00906167i	−0.594004	+	0.804462i	$0.702454\pi$
$828$		0			0
$829$	9464.11	+	29127.5i	0.396504	+	1.22032i	0.927784	+	0.373118i	$0.121711\pi$
$829$	9464.11	+	29127.5i	0.396504	+	1.22032i	−0.531279	+	0.847197i	$0.678289\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4634.50	−	14263.5i	0.192768	−	0.593280i
$834$		0			0
$835$	7575.88			0.313981
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6047.10	+	18611.1i	−0.248831	+	0.765822i	0.746152	+	0.665776i	$0.231899\pi$
$839$	−6047.10	+	18611.1i	−0.248831	+	0.765822i	−0.994983	+	0.100047i	$0.968101\pi$
$840$		0			0
$841$	15019.0	−	10911.9i	0.615809	−	0.447411i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3457.71	+	2512.18i	0.140768	+	0.102274i
$846$		0			0
$847$	2281.32	−	16159.7i	0.0925466	−	0.655553i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	559.025	+	1720.50i	0.0225184	+	0.0693045i
$852$		0			0
$853$	6707.36	−	4873.18i	0.269233	−	0.195609i	−0.444975	−	0.895543i	$-0.646787\pi$
$853$	6707.36	−	4873.18i	0.269233	−	0.195609i	0.714207	+	0.699934i	$0.246787\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−36841.7			−1.46848			−0.734240	−	0.678890i	$-0.762461\pi$
$857$	−36841.7			−1.46848			−0.734240	+	0.678890i	$0.762461\pi$
$858$		0			0
$859$	24003.0			0.953400			0.476700	−	0.879066i	$-0.341833\pi$
$859$	24003.0			0.953400			0.476700	+	0.879066i	$0.341833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15761.4	−	11451.3i	0.621697	−	0.451689i	−0.231817	−	0.972759i	$-0.574467\pi$
$863$	15761.4	−	11451.3i	0.621697	−	0.451689i	0.853514	+	0.521070i	$0.174467\pi$
$864$		0			0
$865$	1212.72	+	3732.38i	0.0476692	+	0.146711i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14087.5	−	12238.4i	0.549927	−	0.477745i
$870$		0			0
$871$	565.220	+	410.656i	0.0219882	+	0.0159754i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4841.76	−	3517.75i	0.187065	−	0.135910i
$876$		0			0
$877$	761.474	−	2343.58i	0.0293194	−	0.0902360i	−0.935326	−	0.353787i	$-0.884894\pi$
$877$	761.474	−	2343.58i	0.0293194	−	0.0902360i	0.964645	+	0.263551i	$0.0848938\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12953.1			−0.495349			−0.247675	−	0.968843i	$-0.579666\pi$
$881$	−12953.1			−0.495349			−0.247675	+	0.968843i	$0.579666\pi$
$882$		0			0
$883$	3138.31	−	9658.74i	0.119607	−	0.368111i	−0.873273	−	0.487231i	$-0.838007\pi$
$883$	3138.31	−	9658.74i	0.119607	−	0.368111i	0.992880	+	0.119119i	$0.0380071\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3710.82	+	11420.7i	0.140470	+	0.432323i	0.996401	−	0.0847679i	$-0.0270149\pi$
$887$	3710.82	+	11420.7i	0.140470	+	0.432323i	−0.855930	+	0.517091i	$0.827015\pi$
$888$		0			0
$889$	−1230.83	−	894.248i	−0.0464349	−	0.0337369i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7906.41	−	5744.34i	−0.296280	−	0.215260i
$894$		0			0
$895$	−1623.37	−	4996.22i	−0.0606294	−	0.186598i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2707.46	+	8332.69i	−0.100444	+	0.309133i
$900$		0			0
$901$	43012.3			1.59040
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2611.58	+	8037.61i	−0.0959247	+	0.295226i
$906$		0			0
$907$	−5268.47	+	3827.77i	−0.192874	+	0.140131i	−0.680031	−	0.733183i	$-0.738034\pi$
$907$	−5268.47	+	3827.77i	−0.192874	+	0.140131i	0.487157	+	0.873314i	$0.338034\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13675.2	+	9935.64i	0.497345	+	0.361342i	0.808002	−	0.589180i	$-0.200549\pi$
$911$	13675.2	+	9935.64i	0.497345	+	0.361342i	−0.310657	+	0.950522i	$0.600549\pi$
$912$		0			0
$913$	9080.29	+	21410.3i	0.329150	+	0.776097i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	531.366	+	1635.38i	0.0191355	+	0.0588930i
$918$		0			0
$919$	27195.5	−	19758.7i	0.976166	−	0.709226i	0.0193174	−	0.999813i	$-0.493851\pi$
$919$	27195.5	−	19758.7i	0.976166	−	0.709226i	0.956848	+	0.290587i	$0.0938507\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4742.30			−0.169117
$924$		0			0
$925$	−10822.3			−0.384688
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14105.6	+	10248.3i	−0.498160	+	0.361934i	−0.808314	−	0.588752i	$-0.799619\pi$
$929$	−14105.6	+	10248.3i	−0.498160	+	0.361934i	0.310154	+	0.950686i	$0.399619\pi$
$930$		0			0
$931$	5445.12	+	16758.4i	0.191683	+	0.589939i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4827.95	+	2902.96i	0.168867	+	0.101537i
$936$		0			0
$937$	−32056.6	−	23290.5i	−1.11766	−	0.812024i	−0.133804	−	0.991008i	$-0.542719\pi$
$937$	−32056.6	−	23290.5i	−1.11766	−	0.812024i	−0.983852	+	0.178983i	$0.942719\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1854.88	+	1347.65i	−0.0642586	+	0.0466866i	−0.619451	−	0.785035i	$-0.712645\pi$
$941$	−1854.88	+	1347.65i	−0.0642586	+	0.0466866i	0.555192	+	0.831722i	$0.312645\pi$
$942$		0			0
$943$	−616.539	+	1897.51i	−0.0212909	+	0.0655265i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22750.9			−0.780680			−0.390340	−	0.920671i	$-0.627643\pi$
$947$	−22750.9			−0.780680			−0.390340	+	0.920671i	$0.627643\pi$
$948$		0			0
$949$	−721.329	+	2220.02i	−0.0246737	+	0.0759378i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12914.9	+	39747.8i	0.438985	+	1.35106i	0.888947	+	0.458010i	$0.151438\pi$
$953$	12914.9	+	39747.8i	0.438985	+	1.35106i	−0.449961	+	0.893048i	$0.648562\pi$
$954$		0			0
$955$	−2115.67	−	1537.13i	−0.0716875	−	0.0520840i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7483.39	−	5437.00i	−0.251983	−	0.183076i
$960$		0			0
$961$	−5133.26	−	15798.6i	−0.172309	−	0.530313i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1689.65	+	5200.19i	−0.0563644	+	0.173472i
$966$		0			0
$967$	−17075.1			−0.567836			−0.283918	−	0.958849i	$-0.591634\pi$
$967$	−17075.1			−0.567836			−0.283918	+	0.958849i	$0.591634\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5934.01	−	18263.0i	0.196119	−	0.603592i	−0.803843	−	0.594842i	$-0.797215\pi$
$971$	5934.01	−	18263.0i	0.196119	−	0.603592i	0.999962	−	0.00875008i	$-0.00278527\pi$
$972$		0			0
$973$	11791.7	−	8567.15i	0.388513	−	0.282271i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27764.0	+	20171.8i	0.909161	+	0.660544i	0.940803	−	0.338955i	$-0.110074\pi$
$977$	27764.0	+	20171.8i	0.909161	+	0.660544i	−0.0316416	+	0.999499i	$0.510074\pi$
$978$		0			0
$979$	6827.81	−	29532.7i	0.222899	−	0.964116i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1435.17	−	4416.99i	−0.0465664	−	0.143317i	0.925070	−	0.379797i	$-0.124006\pi$
$983$	−1435.17	−	4416.99i	−0.0465664	−	0.143317i	−0.971636	+	0.236480i	$0.924006\pi$
$984$		0			0
$985$	3051.66	−	2217.16i	0.0987146	−	0.0717203i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9880.40			0.317673
$990$		0			0
$991$	−14255.1			−0.456941			−0.228471	−	0.973551i	$-0.573372\pi$
$991$	−14255.1			−0.456941			−0.228471	+	0.973551i	$0.573372\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6410.52	−	4657.52i	0.204248	−	0.148395i
$996$		0			0
$997$	−4467.91	−	13750.8i	−0.141926	−	0.436803i	0.854677	−	0.519160i	$-0.173755\pi$
$997$	−4467.91	−	13750.8i	−0.141926	−	0.436803i	−0.996603	+	0.0823571i	$0.973755\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	396.4.j.b.37.1		8
3.2	odd	2		132.4.i.b.37.2	yes	8
11.3	even	5	inner	396.4.j.b.289.1		8
33.5	odd	10		1452.4.a.o.1.2		4
33.14	odd	10		132.4.i.b.25.2	✓	8
33.17	even	10		1452.4.a.p.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.4.i.b.25.2	✓	8	33.14	odd	10
132.4.i.b.37.2	yes	8	3.2	odd	2
396.4.j.b.37.1		8	1.1	even	1	trivial
396.4.j.b.289.1		8	11.3	even	5	inner
1452.4.a.o.1.2		4	33.5	odd	10
1452.4.a.p.1.2		4	33.17	even	10

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -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.b.37.1

Level	$396$
Weight	$4$
Character	396.37
Analytic conductor	$23.365$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$