LMFDB - Embedded newform 252.8.t.a.17.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(17,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("252.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.t (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.14
Character	$\chi$	$=$	252.17
Dual form			252.8.t.a.89.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(148.283 + 256.835i) q^{5} +(-446.098 - 790.278i) q^{7} +(4160.98 + 2402.34i) q^{11} -15004.6i q^{13} +(-8187.77 + 14181.6i) q^{17} +(-11329.0 + 6540.82i) q^{19} +(-11372.4 + 6565.88i) q^{23} +(-4913.48 + 8510.40i) q^{25} +133505. i q^{29} +(-151075. - 87223.4i) q^{31} +(136822. - 231759. i) q^{35} +(-93136.1 - 161316. i) q^{37} -490565. q^{41} +233538. q^{43} +(-244830. - 424058. i) q^{47} +(-425536. + 705083. i) q^{49} +(188584. + 108879. i) q^{53} +1.42491e6i q^{55} +(779780. - 1.35062e6i) q^{59} +(2.26287e6 - 1.30647e6i) q^{61} +(3.85370e6 - 2.22493e6i) q^{65} +(220911. - 382629. i) q^{67} +3.92314e6i q^{71} +(-3.41550e6 - 1.97194e6i) q^{73} +(42312.4 - 4.36001e6i) q^{77} +(-1.22413e6 - 2.12026e6i) q^{79} -7.62261e6 q^{83} -4.85645e6 q^{85} +(-3.82575e6 - 6.62639e6i) q^{89} +(-1.18578e7 + 6.69353e6i) q^{91} +(-3.35982e6 - 1.93979e6i) q^{95} -5.40637e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 2634 q^{7} - 47862 q^{19} - 360762 q^{25} + 486018 q^{31} - 972270 q^{37} + 298788 q^{43} + 1556886 q^{49} + 4324644 q^{61} - 2969562 q^{67} - 5157378 q^{73} + 7676514 q^{79} - 15214128 q^{85} - 6114678 q^{91}+O(q^{100}) $ 36 * q + 2634 * q^7 - 47862 * q^19 - 360762 * q^25 + 486018 * q^31 - 972270 * q^37 + 298788 * q^43 + 1556886 * q^49 + 4324644 * q^61 - 2969562 * q^67 - 5157378 * q^73 + 7676514 * q^79 - 15214128 * q^85 - 6114678 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	148.283	+	256.835i	0.530515	+	0.918879i	0.999366	+	0.0356018i	$0.0113348\pi$
$5$	148.283	+	256.835i	0.530515	+	0.918879i	−0.468851	+	0.883277i	$0.655332\pi$
$6$		0			0
$7$	−446.098	−	790.278i	−0.491572	−	0.870837i
$7$	−446.098	−	790.278i	−0.491572	−	0.870837i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4160.98	+	2402.34i	0.942587	+	0.544203i	0.890770	−	0.454454i	$-0.150166\pi$
$11$	4160.98	+	2402.34i	0.942587	+	0.544203i	0.0518164	+	0.998657i	$0.483499\pi$
$12$		0			0
$13$		−	15004.6i		−	1.89419i	−0.320958	−	0.947093i	$-0.604005\pi$
$13$		−	15004.6i		−	1.89419i	0.320958	−	0.947093i	$-0.395995\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8187.77	+	14181.6i	−0.404198	+	0.700092i	−0.994228	−	0.107290i	$-0.965783\pi$
$17$	−8187.77	+	14181.6i	−0.404198	+	0.700092i	0.590029	+	0.807382i	$0.299116\pi$
$18$		0			0
$19$	−11329.0	+	6540.82i	−0.378927	+	0.218773i	−0.677351	−	0.735660i	$-0.736872\pi$
$19$	−11329.0	+	6540.82i	−0.378927	+	0.218773i	0.298424	+	0.954433i	$0.403539\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−11372.4	+	6565.88i	−0.194897	+	0.112524i	−0.594273	−	0.804263i	$-0.702560\pi$
$23$	−11372.4	+	6565.88i	−0.194897	+	0.112524i	0.399376	+	0.916787i	$0.369227\pi$
$24$		0			0
$25$	−4913.48	+	8510.40i	−0.0628926	+	0.108933i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			133505.i			1.01650i	0.861211	+	0.508248i	$0.169707\pi$
$29$			133505.i			1.01650i	−0.861211	+	0.508248i	$0.830293\pi$
$30$		0			0
$31$	−151075.	−	87223.4i	−0.910810	−	0.525856i	−0.0301183	−	0.999546i	$-0.509588\pi$
$31$	−151075.	−	87223.4i	−0.910810	−	0.525856i	−0.880692	+	0.473690i	$0.842922\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	136822.	−	231759.i	0.539407	−	0.913688i
$36$		0			0
$37$	−93136.1	−	161316.i	−0.302282	−	0.523567i	0.674371	−	0.738393i	$-0.264415\pi$
$37$	−93136.1	−	161316.i	−0.302282	−	0.523567i	−0.976652	+	0.214826i	$0.931082\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−490565.			−1.11161			−0.555806	−	0.831312i	$-0.687590\pi$
$41$	−490565.			−1.11161			−0.555806	+	0.831312i	$0.687590\pi$
$42$		0			0
$43$	233538.			0.447939			0.223969	−	0.974596i	$-0.428098\pi$
$43$	233538.			0.447939			0.223969	+	0.974596i	$0.428098\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−244830.	−	424058.i	−0.343971	−	0.595776i	0.641195	−	0.767378i	$-0.278439\pi$
$47$	−244830.	−	424058.i	−0.343971	−	0.595776i	−0.985166	+	0.171602i	$0.945106\pi$
$48$		0			0
$49$	−425536.	+	705083.i	−0.516713	+	0.856159i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	188584.	+	108879.i	0.173996	+	0.100457i	0.584469	−	0.811416i	$-0.301303\pi$
$53$	188584.	+	108879.i	0.173996	+	0.100457i	−0.410473	+	0.911873i	$0.634636\pi$
$54$		0			0
$55$			1.42491e6i			1.15483i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	779780.	−	1.35062e6i	0.494299	−	0.856151i	−0.505679	−	0.862722i	$-0.668758\pi$
$59$	779780.	−	1.35062e6i	0.494299	−	0.856151i	0.999978	+	0.00657014i	$0.00209136\pi$
$60$		0			0
$61$	2.26287e6	−	1.30647e6i	1.27645	−	0.736960i	0.300258	−	0.953858i	$-0.402927\pi$
$61$	2.26287e6	−	1.30647e6i	1.27645	−	0.736960i	0.976194	+	0.216898i	$0.0695938\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.85370e6	−	2.22493e6i	1.74053	−	1.00489i
$66$		0			0
$67$	220911.	−	382629.i	0.0897336	−	0.155423i	−0.817665	−	0.575694i	$-0.804732\pi$
$67$	220911.	−	382629.i	0.0897336	−	0.155423i	0.907398	+	0.420271i	$0.138065\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.92314e6i			1.30086i	0.759568	+	0.650428i	$0.225410\pi$
$71$			3.92314e6i			1.30086i	−0.759568	+	0.650428i	$0.774590\pi$
$72$		0			0
$73$	−3.41550e6	−	1.97194e6i	−1.02760	−	0.593285i	−0.111304	−	0.993786i	$-0.535503\pi$
$73$	−3.41550e6	−	1.97194e6i	−1.02760	−	0.593285i	−0.916296	+	0.400501i	$0.868836\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	42312.4	−	4.36001e6i	0.0105621	−	1.08835i
$78$		0			0
$79$	−1.22413e6	−	2.12026e6i	−0.279340	−	0.483831i	0.691881	−	0.722012i	$-0.256782\pi$
$79$	−1.22413e6	−	2.12026e6i	−0.279340	−	0.483831i	−0.971221	+	0.238181i	$0.923449\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.62261e6			−1.46329			−0.731645	−	0.681686i	$-0.761247\pi$
$83$	−7.62261e6			−1.46329			−0.731645	+	0.681686i	$0.761247\pi$
$84$		0			0
$85$	−4.85645e6			−0.857733
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.82575e6	−	6.62639e6i	−0.575243	−	0.996349i	−0.996015	−	0.0891833i	$-0.971574\pi$
$89$	−3.82575e6	−	6.62639e6i	−0.575243	−	0.996349i	0.420773	−	0.907166i	$-0.361759\pi$
$90$		0			0
$91$	−1.18578e7	+	6.69353e6i	−1.64953	+	0.931130i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.35982e6	−	1.93979e6i	−0.402053	−	0.232125i
$96$		0			0
$97$		−	5.40637e6i		−	0.601457i	−0.953710	−	0.300728i	$-0.902770\pi$
$97$		−	5.40637e6i		−	0.601457i	0.953710	−	0.300728i	$-0.0972298\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.28145e6	−	7.41569e6i	0.413491	−	0.716188i	−0.581777	−	0.813348i	$-0.697643\pi$
$101$	4.28145e6	−	7.41569e6i	0.413491	−	0.716188i	0.995269	+	0.0971600i	$0.0309759\pi$
$102$		0			0
$103$	−1.10821e7	+	6.39828e6i	−0.999294	+	0.576943i	−0.908039	−	0.418885i	$-0.862421\pi$
$103$	−1.10821e7	+	6.39828e6i	−0.999294	+	0.576943i	−0.0912547	+	0.995828i	$0.529088\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.02378e7	−	5.91082e6i	0.807913	−	0.466449i	−0.0383174	−	0.999266i	$-0.512200\pi$
$107$	1.02378e7	−	5.91082e6i	0.807913	−	0.466449i	0.846231	+	0.532817i	$0.178866\pi$
$108$		0			0
$109$	6.79395e6	−	1.17675e7i	0.502492	−	0.870342i	−0.497504	−	0.867462i	$-0.665750\pi$
$109$	6.79395e6	−	1.17675e7i	0.502492	−	0.870342i	0.999996	−	0.00288017i	$-0.000916787\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.63304e7i		−	1.71666i	−0.513101	−	0.858328i	$-0.671503\pi$
$113$		−	2.63304e7i		−	1.71666i	0.513101	−	0.858328i	$-0.328497\pi$
$114$		0			0
$115$	−3.37269e6	−	1.94722e6i	−0.206792	−	0.119391i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.48600e7	+	144211.i	0.808359	+	0.00784484i
$120$		0			0
$121$	1.79892e6	+	3.11582e6i	0.0923131	+	0.159891i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.02549e7			0.927568
$126$		0			0
$127$	−3.17578e7			−1.37574			−0.687872	−	0.725832i	$-0.741455\pi$
$127$	−3.17578e7			−1.37574			−0.687872	+	0.725832i	$0.741455\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.99677e6	−	1.55829e7i	−0.349653	−	0.605616i	0.636535	−	0.771248i	$-0.280367\pi$
$131$	−8.99677e6	−	1.55829e7i	−0.349653	−	0.605616i	−0.986188	+	0.165631i	$0.947034\pi$
$132$		0			0
$133$	1.02229e7	+	6.03524e6i	0.376786	+	0.222440i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.31159e7	−	2.48930e7i	−1.43257	−	0.827094i	−0.435253	−	0.900308i	$-0.643341\pi$
$137$	−4.31159e7	−	2.48930e7i	−1.43257	−	0.827094i	−0.997316	+	0.0732144i	$0.976674\pi$
$138$		0			0
$139$		−	1.98029e7i		−	0.625428i	−0.949847	−	0.312714i	$-0.898762\pi$
$139$		−	1.98029e7i		−	0.625428i	0.949847	−	0.312714i	$-0.101238\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.60462e7	−	6.24339e7i	1.03082	−	1.78544i
$144$		0			0
$145$	−3.42888e7	+	1.97967e7i	−0.934038	+	0.539267i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.15664e7	+	6.67788e6i	−0.286449	+	0.165381i	−0.636339	−	0.771409i	$-0.719552\pi$
$149$	−1.15664e7	+	6.67788e6i	−0.286449	+	0.165381i	0.349890	+	0.936791i	$0.386219\pi$
$150$		0			0
$151$	−7.99215e6	+	1.38428e7i	−0.188905	+	0.327194i	−0.944886	−	0.327401i	$-0.893827\pi$
$151$	−7.99215e6	+	1.38428e7i	−0.188905	+	0.327194i	0.755980	+	0.654595i	$0.227161\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.17352e7i		−	1.11590i
$156$		0			0
$157$	4.88079e7	+	2.81792e7i	1.00656	+	0.581140i	0.910184	−	0.414205i	$-0.135940\pi$
$157$	4.88079e7	+	2.81792e7i	1.00656	+	0.581140i	0.0963797	+	0.995345i	$0.469274\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.02621e7	+	6.05836e6i	0.193796	+	0.114410i
$162$		0			0
$163$	−3.51998e7	−	6.09678e7i	−0.636624	−	1.10267i	−0.986169	−	0.165746i	$-0.946997\pi$
$163$	−3.51998e7	−	6.09678e7i	−0.636624	−	1.10267i	0.349544	−	0.936920i	$-0.386336\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.75806e7			0.956685			0.478342	−	0.878173i	$-0.341238\pi$
$167$	5.75806e7			0.956685			0.478342	+	0.878173i	$0.341238\pi$
$168$		0			0
$169$	−1.62390e8			−2.58794
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.31540e6	−	7.47450e6i	−0.0633665	−	0.109754i	0.832602	−	0.553872i	$-0.186850\pi$
$173$	−4.31540e6	−	7.47450e6i	−0.0633665	−	0.109754i	−0.895968	+	0.444118i	$0.853517\pi$
$174$		0			0
$175$	8.91748e6	+	86541.0i	0.125779	+	0.00122064i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.24589e7	−	4.76077e7i	−1.07461	−	0.620428i	−0.145175	−	0.989406i	$-0.546374\pi$
$179$	−8.24589e7	−	4.76077e7i	−1.07461	−	0.620428i	−0.929438	+	0.368978i	$0.879708\pi$
$180$		0			0
$181$		−	2.15791e7i		−	0.270495i	−0.990812	−	0.135247i	$-0.956817\pi$
$181$		−	2.15791e7i		−	0.270495i	0.990812	−	0.135247i	$-0.0431829\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.76211e7	−	4.78411e7i	0.320730	−	0.555521i
$186$		0			0
$187$	−6.81383e7	+	3.93397e7i	−0.761984	+	0.439932i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.52177e8	+	8.78597e7i	−1.58028	+	0.912374i	−0.585460	+	0.810702i	$0.699086\pi$
$191$	−1.52177e8	+	8.78597e7i	−1.58028	+	0.912374i	−0.994818	+	0.101672i	$0.967581\pi$
$192$		0			0
$193$	4.11839e7	−	7.13325e7i	0.412360	−	0.714228i	−0.582787	−	0.812625i	$-0.698038\pi$
$193$	4.11839e7	−	7.13325e7i	0.412360	−	0.714228i	0.995147	+	0.0983963i	$0.0313713\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.20671e7i			0.764781i	0.924001	+	0.382391i	$0.124899\pi$
$197$			8.20671e7i			0.764781i	−0.924001	+	0.382391i	$0.875101\pi$
$198$		0			0
$199$	4.00491e7	+	2.31223e7i	0.360252	+	0.207992i	0.669191	−	0.743090i	$-0.266641\pi$
$199$	4.00491e7	+	2.31223e7i	0.360252	+	0.207992i	−0.308939	+	0.951082i	$0.599974\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.05506e8	−	5.95566e7i	0.885203	−	0.499682i
$204$		0			0
$205$	−7.27427e7	−	1.25994e8i	−0.589727	−	1.02144i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.28532e7			−0.476228
$210$		0			0
$211$	2.26680e8			1.66121			0.830603	−	0.556864i	$-0.187996\pi$
$211$	2.26680e8			1.66121			0.830603	+	0.556864i	$0.187996\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.46299e7	+	5.99807e7i	0.237638	+	0.411602i
$216$		0			0
$217$	−1.53626e6	+	1.58302e8i	−0.0102060	+	1.05166i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.12790e8	+	1.22854e8i	1.32611	+	0.765627i
$222$		0			0
$223$			7.95339e7i			0.480270i	0.970740	+	0.240135i	$0.0771917\pi$
$223$			7.95339e7i			0.480270i	−0.970740	+	0.240135i	$0.922808\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.00255e7	−	1.03967e8i	0.340601	−	0.589938i	−0.643944	−	0.765073i	$-0.722703\pi$
$227$	6.00255e7	−	1.03967e8i	0.340601	−	0.589938i	0.984544	+	0.175135i	$0.0560363\pi$
$228$		0			0
$229$	6.00167e7	−	3.46507e7i	0.330254	−	0.190672i	−0.325700	−	0.945473i	$-0.605600\pi$
$229$	6.00167e7	−	3.46507e7i	0.330254	−	0.190672i	0.655954	+	0.754801i	$0.272267\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.65882e8	−	1.53507e8i	1.37703	−	0.795027i	0.385227	−	0.922822i	$-0.374123\pi$
$233$	2.65882e8	−	1.53507e8i	1.37703	−	0.795027i	0.991801	+	0.127795i	$0.0407899\pi$
$234$		0			0
$235$	7.26085e7	−	1.25762e8i	0.364964	−	0.632136i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			3.28399e8i			1.55600i	0.628264	+	0.778000i	$0.283766\pi$
$239$			3.28399e8i			1.55600i	−0.628264	+	0.778000i	$0.716234\pi$
$240$		0			0
$241$	2.98238e8	+	1.72188e8i	1.37247	+	0.792397i	0.991239	−	0.132083i	$-0.0421665\pi$
$241$	2.98238e8	+	1.72188e8i	1.37247	+	0.792397i	0.381232	+	0.924479i	$0.375500\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.44190e8	−	4.74000e6i	−1.06083	−	0.0205919i
$246$		0			0
$247$	9.81424e7	+	1.69988e8i	0.414398	+	0.717758i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.83927e8			−1.53246			−0.766232	−	0.642564i	$-0.777871\pi$
$251$	−3.83927e8			−1.53246			−0.766232	+	0.642564i	$0.777871\pi$
$252$		0			0
$253$	−6.30940e7			−0.244944
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.07212e8	+	1.85696e8i	0.393982	+	0.682397i	0.992971	−	0.118361i	$-0.0377639\pi$
$257$	1.07212e8	+	1.85696e8i	0.393982	+	0.682397i	−0.598989	+	0.800758i	$0.704431\pi$
$258$		0			0
$259$	−8.59370e7	+	1.45566e8i	−0.307348	+	0.520609i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.73266e8	−	2.15505e8i	−1.26524	−	0.730488i	−0.291158	−	0.956675i	$-0.594041\pi$
$263$	−3.73266e8	−	2.15505e8i	−1.26524	−	0.730488i	−0.974084	+	0.226187i	$0.927374\pi$
$264$		0			0
$265$			6.45799e7i			0.213175i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.52919e8	+	4.38069e8i	−0.792226	+	1.37218i	0.132360	+	0.991202i	$0.457745\pi$
$269$	−2.52919e8	+	4.38069e8i	−0.792226	+	1.37218i	−0.924586	+	0.380974i	$0.875589\pi$
$270$		0			0
$271$	1.45499e8	−	8.40041e7i	0.444087	−	0.256394i	−0.261242	−	0.965273i	$-0.584132\pi$
$271$	1.45499e8	−	8.40041e7i	0.444087	−	0.256394i	0.705330	+	0.708879i	$0.250799\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.08898e7	+	2.36077e7i	−0.118563	+	0.0684526i
$276$		0			0
$277$	2.11263e8	−	3.65917e8i	0.597232	−	1.03444i	−0.395995	−	0.918253i	$-0.629600\pi$
$277$	2.11263e8	−	3.65917e8i	0.597232	−	1.03444i	0.993228	−	0.116184i	$-0.0370663\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6.51528e8i			1.75170i	0.482580	+	0.875852i	$0.339700\pi$
$281$			6.51528e8i			1.75170i	−0.482580	+	0.875852i	$0.660300\pi$
$282$		0			0
$283$	−3.46880e8	−	2.00271e8i	−0.909759	−	0.525250i	−0.0294055	−	0.999568i	$-0.509361\pi$
$283$	−3.46880e8	−	2.00271e8i	−0.909759	−	0.525250i	−0.880354	+	0.474318i	$0.842695\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.18840e8	+	3.87683e8i	0.546438	+	0.968033i
$288$		0			0
$289$	7.10901e7	+	1.23132e8i	0.173247	+	0.300073i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.68434e8			0.855702			0.427851	−	0.903849i	$-0.359271\pi$
$293$	3.68434e8			0.855702			0.427851	+	0.903849i	$0.359271\pi$
$294$		0			0
$295$	4.62514e8			1.04893
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.85184e7	+	1.70639e8i	0.213142	+	0.369172i
$300$		0			0
$301$	−1.04181e8	−	1.84560e8i	−0.220194	−	0.390082i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.71092e8	+	3.87455e8i	1.35435	+	0.781937i
$306$		0			0
$307$		−	2.43861e8i		−	0.481014i	−0.970647	−	0.240507i	$-0.922686\pi$
$307$		−	2.43861e8i		−	0.481014i	0.970647	−	0.240507i	$-0.0773137\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.74225e8	+	3.01767e8i	−0.328435	+	0.568866i	−0.982201	−	0.187830i	$-0.939855\pi$
$311$	−1.74225e8	+	3.01767e8i	−0.328435	+	0.568866i	0.653767	+	0.756696i	$0.273188\pi$
$312$		0			0
$313$	6.50847e8	−	3.75767e8i	1.19970	−	0.692649i	0.239214	−	0.970967i	$-0.423110\pi$
$313$	6.50847e8	−	3.75767e8i	1.19970	−	0.692649i	0.960489	+	0.278318i	$0.0897769\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.33330e8	+	1.92448e8i	−0.587716	+	0.339318i	−0.764194	−	0.644987i	$-0.776863\pi$
$317$	−3.33330e8	+	1.92448e8i	−0.587716	+	0.339318i	0.176478	+	0.984305i	$0.443530\pi$
$318$		0			0
$319$	−3.20726e8	+	5.55514e8i	−0.553180	+	0.958136i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.14219e8i		−	0.353711i
$324$		0			0
$325$	1.27695e8	+	7.37249e7i	0.206340	+	0.119130i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.25906e8	+	3.82656e8i	−0.349737	+	0.592410i
$330$		0			0
$331$	4.66662e8	+	8.08282e8i	0.707301	+	1.22508i	0.965855	+	0.259084i	$0.0834208\pi$
$331$	4.66662e8	+	8.08282e8i	0.707301	+	1.22508i	−0.258554	+	0.965997i	$0.583246\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.31030e8			0.190420
$336$		0			0
$337$	2.67531e8			0.380776			0.190388	−	0.981709i	$-0.439025\pi$
$337$	2.67531e8			0.380776			0.190388	+	0.981709i	$0.439025\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.19081e8	−	7.25870e8i	−0.572345	−	0.991330i
$342$		0			0
$343$	7.47043e8	+	2.17548e7i	0.999576	+	0.0291089i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.03799e8	−	2.33134e8i	−0.518815	−	0.299538i	0.217635	−	0.976030i	$-0.430166\pi$
$347$	−4.03799e8	−	2.33134e8i	−0.518815	−	0.299538i	−0.736450	+	0.676492i	$0.763499\pi$
$348$		0			0
$349$		−	5.70673e8i		−	0.718618i	−0.933219	−	0.359309i	$-0.883012\pi$
$349$		−	5.70673e8i		−	0.718618i	0.933219	−	0.359309i	$-0.116988\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.80978e8	+	3.13463e8i	−0.218985	+	0.379293i	−0.954498	−	0.298218i	$-0.903608\pi$
$353$	−1.80978e8	+	3.13463e8i	−0.218985	+	0.379293i	0.735513	+	0.677510i	$0.236941\pi$
$354$		0			0
$355$	−1.00760e9	+	5.81737e8i	−1.19533	+	0.690124i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.63579e8	+	2.67647e8i	−0.528802	+	0.305304i	−0.740528	−	0.672025i	$-0.765425\pi$
$359$	−4.63579e8	+	2.67647e8i	−0.528802	+	0.305304i	0.211727	+	0.977329i	$0.432091\pi$
$360$		0			0
$361$	−3.61371e8	+	6.25913e8i	−0.404276	+	0.700227i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1.16962e9i		−	1.25899i
$366$		0			0
$367$	−5.84564e8	−	3.37498e8i	−0.617306	−	0.356402i	0.158513	−	0.987357i	$-0.449330\pi$
$367$	−5.84564e8	−	3.37498e8i	−0.617306	−	0.356402i	−0.775820	+	0.630955i	$0.782663\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.91768e6	−	1.97605e8i	0.00194970	−	0.200904i
$372$		0			0
$373$	−1.31502e7	−	2.27767e7i	−0.0131205	−	0.0227253i	0.859391	−	0.511320i	$-0.170843\pi$
$373$	−1.31502e7	−	2.27767e7i	−0.0131205	−	0.0227253i	−0.872511	+	0.488594i	$0.837510\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.00320e9			1.92543
$378$		0			0
$379$	8.80005e7			0.0830325			0.0415162	−	0.999138i	$-0.486781\pi$
$379$	8.80005e7			0.0830325			0.0415162	+	0.999138i	$0.486781\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.74302e8	−	3.01899e8i	−0.158528	−	0.274579i	0.775810	−	0.630966i	$-0.217341\pi$
$383$	−1.74302e8	−	3.01899e8i	−0.158528	−	0.274579i	−0.934338	+	0.356388i	$0.884008\pi$
$384$		0			0
$385$	1.12608e9	−	6.35651e8i	1.00567	−	0.567683i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.83947e8	+	2.21672e8i	0.330710	+	0.190936i	0.656156	−	0.754625i	$-0.272181\pi$
$389$	3.83947e8	+	2.21672e8i	0.330710	+	0.190936i	−0.325446	+	0.945561i	$0.605515\pi$
$390$		0			0
$391$		−	2.15040e8i		−	0.181928i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.63037e8	−	6.28798e8i	0.296388	−	0.513359i
$396$		0			0
$397$	8.43601e8	−	4.87053e8i	0.676660	−	0.390670i	−0.121936	−	0.992538i	$-0.538910\pi$
$397$	8.43601e8	−	4.87053e8i	0.676660	−	0.390670i	0.798595	+	0.601868i	$0.205577\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.53739e8	−	2.61967e8i	0.351400	−	0.202881i	−0.313902	−	0.949455i	$-0.601636\pi$
$401$	4.53739e8	−	2.61967e8i	0.351400	−	0.202881i	0.665302	+	0.746575i	$0.268303\pi$
$402$		0			0
$403$	−1.30875e9	+	2.26683e9i	−0.996070	+	1.72524i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.94979e8i		−	0.658010i
$408$		0			0
$409$	−6.22353e8	−	3.59316e8i	−0.449785	−	0.259684i	0.257954	−	0.966157i	$-0.416952\pi$
$409$	−6.22353e8	−	3.59316e8i	−0.449785	−	0.259684i	−0.707740	+	0.706473i	$0.750285\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.41522e9	−	1.37342e7i	−0.988552	−	0.00959355i
$414$		0			0
$415$	−1.13031e9	−	1.95775e9i	−0.776297	−	1.34459i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.52574e8			−0.499804			−0.249902	−	0.968271i	$-0.580398\pi$
$419$	−7.52574e8			−0.499804			−0.249902	+	0.968271i	$0.580398\pi$
$420$		0			0
$421$	−2.00300e8			−0.130826			−0.0654131	−	0.997858i	$-0.520837\pi$
$421$	−2.00300e8			−0.130826			−0.0654131	+	0.997858i	$0.520837\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.04610e7	−	1.39362e8i	−0.0508422	−	0.0880612i
$426$		0			0
$427$	−2.04193e9	−	1.20548e9i	−1.26924	−	0.749312i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.17582e8	−	6.78858e7i	−0.0707407	−	0.0408422i	0.464212	−	0.885724i	$-0.346337\pi$
$431$	−1.17582e8	−	6.78858e7i	−0.0707407	−	0.0408422i	−0.534953	+	0.844882i	$0.679671\pi$
$432$		0			0
$433$			1.37105e9i			0.811606i	0.913961	+	0.405803i	$0.133008\pi$
$433$			1.37105e9i			0.811606i	−0.913961	+	0.405803i	$0.866992\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.58925e7	−	1.48770e8i	0.0492346	−	0.0852768i
$438$		0			0
$439$	−2.44096e9	+	1.40929e9i	−1.37700	+	0.795012i	−0.991798	−	0.127819i	$-0.959202\pi$
$439$	−2.44096e9	+	1.40929e9i	−1.37700	+	0.795012i	−0.385204	+	0.922831i	$0.625869\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.02780e9	−	1.17075e9i	1.10819	−	0.639812i	0.169828	−	0.985474i	$-0.445679\pi$
$443$	2.02780e9	−	1.17075e9i	1.10819	−	0.639812i	0.938359	+	0.345662i	$0.112346\pi$
$444$		0			0
$445$	1.13459e9	−	1.96517e9i	0.610350	−	1.05716i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.28435e9i			0.669608i	0.942288	+	0.334804i	$0.108670\pi$
$449$			1.28435e9i			0.669608i	−0.942288	+	0.334804i	$0.891330\pi$
$450$		0			0
$451$	−2.04123e9	−	1.17851e9i	−1.04779	−	0.604942i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.47745e9	−	2.05295e9i	−1.73069	−	1.02174i
$456$		0			0
$457$	−9.63551e8	−	1.66892e9i	−0.472246	−	0.817953i	0.527250	−	0.849710i	$-0.323223\pi$
$457$	−9.63551e8	−	1.66892e9i	−0.472246	−	0.817953i	−0.999496	+	0.0317569i	$0.989890\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.06415e9			0.981270			0.490635	−	0.871365i	$-0.336765\pi$
$461$	2.06415e9			0.981270			0.490635	+	0.871365i	$0.336765\pi$
$462$		0			0
$463$	2.12484e9			0.994932			0.497466	−	0.867483i	$-0.334264\pi$
$463$	2.12484e9			0.994932			0.497466	+	0.867483i	$0.334264\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.60319e9	−	2.77681e9i	−0.728412	−	1.26165i	−0.957554	−	0.288254i	$-0.906925\pi$
$467$	−1.60319e9	−	2.77681e9i	−0.728412	−	1.26165i	0.229142	−	0.973393i	$-0.426408\pi$
$468$		0			0
$469$	−4.00931e8	−	3.89090e6i	−0.179459	−	0.00174159i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.71749e8	+	5.61039e8i	0.422221	+	0.243770i
$474$		0			0
$475$		−	1.28553e8i		−	0.0550369i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.72905e9	−	2.99481e9i	0.718842	−	1.24507i	−0.242616	−	0.970122i	$-0.578006\pi$
$479$	1.72905e9	−	2.99481e9i	0.718842	−	1.24507i	0.961459	−	0.274949i	$-0.0886611\pi$
$480$		0			0
$481$	−2.42049e9	+	1.39747e9i	−0.991734	+	0.572578i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.38854e9	−	8.01675e8i	0.552666	−	0.319082i
$486$		0			0
$487$	−4.45520e8	+	7.71663e8i	−0.174790	+	0.302744i	−0.940088	−	0.340931i	$-0.889258\pi$
$487$	−4.45520e8	+	7.71663e8i	−0.174790	+	0.302744i	0.765299	+	0.643675i	$0.222591\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	1.37391e9i		−	0.523807i	−0.965094	−	0.261904i	$-0.915650\pi$
$491$		−	1.37391e9i		−	0.523807i	0.965094	−	0.261904i	$-0.0843503\pi$
$492$		0			0
$493$	−1.89333e9	−	1.09311e9i	−0.711641	−	0.410866i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.10037e9	−	1.75011e9i	1.13283	−	0.639465i
$498$		0			0
$499$	2.75111e9	+	4.76506e9i	0.991188	+	1.71679i	0.610310	+	0.792163i	$0.291045\pi$
$499$	2.75111e9	+	4.76506e9i	0.991188	+	1.71679i	0.380878	+	0.924625i	$0.375622\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.14208e9			−1.45121			−0.725606	−	0.688110i	$-0.758441\pi$
$503$	−4.14208e9			−1.45121			−0.725606	+	0.688110i	$0.758441\pi$
$504$		0			0
$505$	2.53948e9			0.877454
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.37541e8	−	7.57844e8i	−0.147064	−	0.254723i	0.783077	−	0.621925i	$-0.213649\pi$
$509$	−4.37541e8	−	7.57844e8i	−0.147064	−	0.254723i	−0.930141	+	0.367202i	$0.880316\pi$
$510$		0			0
$511$	−3.47317e7	+	3.57887e9i	−0.0115147	+	1.18651i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.28660e9	−	1.89752e9i	−1.06028	−	0.612154i
$516$		0			0
$517$		−	2.35266e9i		−	0.748761i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.26369e9	−	3.92083e9i	0.701270	−	1.21463i	−0.266751	−	0.963765i	$-0.585950\pi$
$521$	2.26369e9	−	3.92083e9i	0.701270	−	1.21463i	0.968021	−	0.250869i	$-0.0807165\pi$
$522$		0			0
$523$	−1.68534e8	+	9.73032e7i	−0.0515148	+	0.0297421i	−0.525536	−	0.850771i	$-0.676135\pi$
$523$	−1.68534e8	+	9.73032e7i	−0.0515148	+	0.0297421i	0.474021	+	0.880513i	$0.342802\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.47394e9	−	1.42833e9i	0.736296	−	0.425101i
$528$		0			0
$529$	−1.61619e9	+	2.79933e9i	−0.474677	+	0.822164i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			7.36074e9i			2.10560i
$534$		0			0
$535$	3.03620e9	+	1.75295e9i	0.857221	+	0.494916i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.46450e9	+	1.91156e9i	−0.952971	+	0.525807i
$540$		0			0
$541$	−3.07229e9	−	5.32136e9i	−0.834203	−	1.44488i	−0.894678	−	0.446712i	$-0.852595\pi$
$541$	−3.07229e9	−	5.32136e9i	−0.834203	−	1.44488i	0.0604749	−	0.998170i	$-0.480738\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.02972e9			1.06632
$546$		0			0
$547$	−2.19151e9			−0.572517			−0.286258	−	0.958152i	$-0.592412\pi$
$547$	−2.19151e9			−0.572517			−0.286258	+	0.958152i	$0.592412\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.73235e8	−	1.51249e9i	−0.222382	−	0.385178i
$552$		0			0
$553$	−1.12951e9	+	1.91325e9i	−0.284022	+	0.481097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.65472e9	+	9.55351e8i	0.405724	+	0.234245i	0.688951	−	0.724808i	$-0.258072\pi$
$557$	1.65472e9	+	9.55351e8i	0.405724	+	0.234245i	−0.283227	+	0.959053i	$0.591405\pi$
$558$		0			0
$559$		−	3.50415e9i		−	0.848480i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.88736e9	+	5.00105e9i	−0.681901	+	1.18109i	0.292499	+	0.956266i	$0.405513\pi$
$563$	−2.88736e9	+	5.00105e9i	−0.681901	+	1.18109i	−0.974400	+	0.224821i	$0.927820\pi$
$564$		0			0
$565$	6.76256e9	−	3.90437e9i	1.57740	−	0.910712i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.31588e9	+	7.59725e8i	−0.299450	+	0.172887i	−0.642196	−	0.766541i	$-0.721976\pi$
$569$	−1.31588e9	+	7.59725e8i	−0.299450	+	0.172887i	0.342746	+	0.939428i	$0.388643\pi$
$570$		0			0
$571$	3.72410e9	−	6.45034e9i	0.837135	−	1.44996i	−0.0551452	−	0.998478i	$-0.517562\pi$
$571$	3.72410e9	−	6.45034e9i	0.837135	−	1.44996i	0.892280	−	0.451482i	$-0.149104\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1.29045e8i		−	0.0283077i
$576$		0			0
$577$	−5.29913e9	−	3.05945e9i	−1.14839	−	0.663023i	−0.199895	−	0.979817i	$-0.564060\pi$
$577$	−5.29913e9	−	3.05945e9i	−1.14839	−	0.663023i	−0.948494	+	0.316795i	$0.897393\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.40043e9	+	6.02398e9i	0.719313	+	1.27429i
$582$		0			0
$583$	5.23130e8	+	9.06088e8i	0.109338	+	0.189378i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.69324e9			1.56992			0.784958	−	0.619549i	$-0.212685\pi$
$587$	7.69324e9			1.56992			0.784958	+	0.619549i	$0.212685\pi$
$588$		0			0
$589$	2.28205e9			0.460174
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.19701e8	+	2.07328e8i	0.0235725	+	0.0408288i	0.877571	−	0.479447i	$-0.159163\pi$
$593$	1.19701e8	+	2.07328e8i	0.0235725	+	0.0408288i	−0.853998	+	0.520275i	$0.825829\pi$
$594$		0			0
$595$	2.16645e9	+	3.83794e9i	0.421638	+	0.746946i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.80378e9	−	3.92816e9i	−1.29347	−	0.746785i	−0.314202	−	0.949356i	$-0.601737\pi$
$599$	−6.80378e9	−	3.92816e9i	−1.29347	−	0.746785i	−0.979268	+	0.202571i	$0.935070\pi$
$600$		0			0
$601$		−	3.92725e9i		−	0.737951i	−0.929439	−	0.368975i	$-0.879709\pi$
$601$		−	3.92725e9i		−	0.737951i	0.929439	−	0.368975i	$-0.120291\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.33501e8	+	9.24050e8i	−0.0979470	+	0.169649i
$606$		0			0
$607$	−8.90080e9	+	5.13888e9i	−1.61536	+	0.932627i	−0.627257	+	0.778812i	$0.715823\pi$
$607$	−8.90080e9	+	5.13888e9i	−1.61536	+	0.932627i	−0.988100	+	0.153815i	$0.950844\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.36283e9	+	3.67358e9i	−1.12851	+	0.651546i
$612$		0			0
$613$	−2.34777e9	+	4.06645e9i	−0.411665	+	0.713024i	−0.995072	−	0.0991557i	$-0.968386\pi$
$613$	−2.34777e9	+	4.06645e9i	−0.411665	+	0.713024i	0.583407	+	0.812180i	$0.301719\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.05312e10i			1.80501i	0.430683	+	0.902503i	$0.358273\pi$
$617$			1.05312e10i			1.80501i	−0.430683	+	0.902503i	$0.641727\pi$
$618$		0			0
$619$	7.12145e9	+	4.11157e9i	1.20684	+	0.696772i	0.962069	−	0.272808i	$-0.0879523\pi$
$619$	7.12145e9	+	4.11157e9i	1.20684	+	0.696772i	0.244776	+	0.969580i	$0.421286\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.53003e9	+	5.97942e9i	−0.584884	+	0.990720i
$624$		0			0
$625$	3.38734e9	+	5.86704e9i	0.554982	+	0.961256i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.05031e9			0.488727
$630$		0			0
$631$	−9.93041e9			−1.57349			−0.786746	−	0.617277i	$-0.788236\pi$
$631$	−9.93041e9			−1.57349			−0.786746	+	0.617277i	$0.788236\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.70916e9	−	8.15650e9i	−0.729853	−	1.26414i
$636$		0			0
$637$	1.05795e10	+	6.38499e9i	1.62172	+	0.978751i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.59992e9	−	2.07842e9i	−0.539871	−	0.311695i	0.205156	−	0.978729i	$-0.434230\pi$
$641$	−3.59992e9	−	2.07842e9i	−0.539871	−	0.311695i	−0.745027	+	0.667035i	$0.767563\pi$
$642$		0			0
$643$		−	3.36943e9i		−	0.499826i	−0.968268	−	0.249913i	$-0.919598\pi$
$643$		−	3.36943e9i		−	0.499826i	0.968268	−	0.249913i	$-0.0804020\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.84561e9	−	3.19669e9i	0.267901	−	0.464018i	−0.700419	−	0.713732i	$-0.747003\pi$
$647$	1.84561e9	−	3.19669e9i	0.267901	−	0.464018i	0.968320	+	0.249714i	$0.0803366\pi$
$648$		0			0
$649$	6.48930e9	−	3.74660e9i	0.931840	−	0.537998i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.77199e9	+	4.48716e9i	−1.09229	+	0.630631i	−0.934184	−	0.356792i	$-0.883871\pi$
$653$	−7.77199e9	+	4.48716e9i	−1.09229	+	0.630631i	−0.158101	+	0.987423i	$0.550537\pi$
$654$		0			0
$655$	2.66814e9	−	4.62136e9i	0.370992	−	0.642577i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			8.41379e9i			1.14523i	0.819825	+	0.572615i	$0.194071\pi$
$659$			8.41379e9i			1.14523i	−0.819825	+	0.572615i	$0.805929\pi$
$660$		0			0
$661$	1.10340e10	+	6.37050e9i	1.48604	+	0.857963i	0.999874	−	0.0159054i	$-0.00506306\pi$
$661$	1.10340e10	+	6.37050e9i	1.48604	+	0.857963i	0.486162	+	0.873869i	$0.338396\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.41655e7	+	3.52053e9i	−0.00450518	+	0.464229i
$666$		0			0
$667$	−8.76581e8	−	1.51828e9i	−0.114380	−	0.198113i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.25543e10			1.60422
$672$		0			0
$673$	1.25069e10			1.58161			0.790803	−	0.612070i	$-0.209663\pi$
$673$	1.25069e10			1.58161			0.790803	+	0.612070i	$0.209663\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.32371e9	+	4.02479e9i	0.287821	+	0.498520i	0.973289	−	0.229582i	$-0.0737360\pi$
$677$	2.32371e9	+	4.02479e9i	0.287821	+	0.498520i	−0.685469	+	0.728102i	$0.740403\pi$
$678$		0			0
$679$	−4.27253e9	+	2.41177e9i	−0.523771	+	0.295660i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.23635e9	+	4.17791e9i	0.869054	+	0.501749i	0.867034	−	0.498249i	$-0.166024\pi$
$683$	7.23635e9	+	4.17791e9i	0.869054	+	0.501749i	0.00202047	+	0.999998i	$0.499357\pi$
$684$		0			0
$685$		−	1.47649e10i		−	1.75514i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.63369e9	−	2.82963e9i	0.190284	−	0.329581i
$690$		0			0
$691$	−1.03606e10	+	5.98169e9i	−1.19457	+	0.689685i	−0.959340	−	0.282254i	$-0.908918\pi$
$691$	−1.03606e10	+	5.98169e9i	−1.19457	+	0.689685i	−0.235230	+	0.971940i	$0.575584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.08607e9	−	2.93644e9i	0.574693	−	0.331799i
$696$		0			0
$697$	4.01664e9	−	6.95702e9i	0.449312	−	0.778231i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.07994e10i			1.18410i	0.805903	+	0.592048i	$0.201680\pi$
$701$			1.07994e10i			1.18410i	−0.805903	+	0.592048i	$0.798320\pi$
$702$		0			0
$703$	2.11028e9	+	1.21837e9i	0.229085	+	0.132262i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.77041e9	−	7.54091e7i	−0.826944	−	0.00802520i
$708$		0			0
$709$	−2.62754e9	−	4.55103e9i	−0.276878	−	0.479566i	0.693730	−	0.720236i	$-0.255966\pi$
$709$	−2.62754e9	−	4.55103e9i	−0.276878	−	0.479566i	−0.970607	+	0.240670i	$0.922633\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.29079e9			0.236686
$714$		0			0
$715$	2.13802e10			2.18747
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.32117e9	+	7.48448e9i	0.433561	+	0.750949i	0.997177	−	0.0750878i	$-0.0239237\pi$
$719$	4.32117e9	+	7.48448e9i	0.433561	+	0.750949i	−0.563616	+	0.826037i	$0.690590\pi$
$720$		0			0
$721$	1.00001e10	+	5.90371e9i	0.993648	+	0.586613i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.13618e9	−	6.55977e8i	−0.110730	−	0.0639301i
$726$		0			0
$727$			1.14277e10i			1.10304i	0.834163	+	0.551518i	$0.185951\pi$
$727$			1.14277e10i			1.10304i	−0.834163	+	0.551518i	$0.814049\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.91216e9	+	3.31196e9i	−0.181056	+	0.313598i
$732$		0			0
$733$	−1.48014e10	+	8.54561e9i	−1.38816	+	0.801455i	−0.993108	−	0.117203i	$-0.962607\pi$
$733$	−1.48014e10	+	8.54561e9i	−1.38816	+	0.801455i	−0.395053	+	0.918658i	$0.629274\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.83841e9	−	1.06141e9i	0.169163	−	0.0976666i
$738$		0			0
$739$	4.82373e8	−	8.35495e8i	0.0439671	−	0.0761533i	−0.843204	−	0.537593i	$-0.819334\pi$
$739$	4.82373e8	−	8.35495e8i	0.0439671	−	0.0761533i	0.887171	+	0.461440i	$0.152667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.20799e10i			1.08045i	0.841522	+	0.540223i	$0.181660\pi$
$743$			1.20799e10i			1.08045i	−0.841522	+	0.540223i	$0.818340\pi$
$744$		0			0
$745$	−3.43022e9	−	1.98044e9i	−0.303931	−	0.175475i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.23827e9	−	5.45393e9i	−0.803349	−	0.474267i
$750$		0			0
$751$	5.45370e8	+	9.44609e8i	0.0469842	+	0.0813790i	0.888561	−	0.458758i	$-0.151706\pi$
$751$	5.45370e8	+	9.44609e8i	0.0469842	+	0.0813790i	−0.841577	+	0.540137i	$0.818372\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.74042e9			−0.400869
$756$		0			0
$757$	6.24346e9			0.523106			0.261553	−	0.965189i	$-0.415765\pi$
$757$	6.24346e9			0.523106			0.261553	+	0.965189i	$0.415765\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.59752e8	−	1.14272e9i	−0.0542668	−	0.0939929i	0.837616	−	0.546260i	$-0.183949\pi$
$761$	−6.59752e8	−	1.14272e9i	−0.0542668	−	0.0939929i	−0.891883	+	0.452267i	$0.850615\pi$
$762$		0			0
$763$	−1.23303e10	−	1.19662e8i	−1.00494	−	0.00975256i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.02655e10	−	1.17003e10i	−1.62171	−	0.936295i
$768$		0			0
$769$		−	6.37720e9i		−	0.505694i	−0.967506	−	0.252847i	$-0.918633\pi$
$769$		−	6.37720e9i		−	0.505694i	0.967506	−	0.252847i	$-0.0813670\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.07507e10	−	1.86208e10i	0.837164	−	1.45001i	−0.0550928	−	0.998481i	$-0.517545\pi$
$773$	1.07507e10	−	1.86208e10i	0.837164	−	1.45001i	0.892257	−	0.451529i	$-0.149121\pi$
$774$		0			0
$775$	1.48461e9	−	8.57141e8i	0.114566	−	0.0661449i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.55763e9	−	3.20870e9i	0.421220	−	0.243191i
$780$		0			0
$781$	−9.42472e9	+	1.63241e10i	−0.707929	+	1.22617i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.67141e10i			1.23321i
$786$		0			0
$787$	−1.14934e10	−	6.63569e9i	−0.840495	−	0.485260i	0.0169373	−	0.999857i	$-0.494608\pi$
$787$	−1.14934e10	−	6.63569e9i	−0.840495	−	0.485260i	−0.857433	+	0.514596i	$0.827942\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.08084e10	+	1.17460e10i	−1.49493	+	0.843861i
$792$		0			0
$793$	−1.96030e10	−	3.39534e10i	−1.39594	−	2.41784i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.43558e10			1.00444			0.502218	−	0.864741i	$-0.332517\pi$
$797$	1.43558e10			1.00444			0.502218	+	0.864741i	$0.332517\pi$
$798$		0			0
$799$	8.01846e9			0.556131
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.47455e9	−	1.64104e10i	−0.645735	−	1.11845i
$804$		0			0
$805$	−3.42964e7	+	3.53402e9i	−0.00231720	+	0.238772i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.18343e10	+	6.83256e9i	0.785822	+	0.453695i	0.838490	−	0.544917i	$-0.183439\pi$
$809$	1.18343e10	+	6.83256e9i	0.785822	+	0.453695i	−0.0526675	+	0.998612i	$0.516772\pi$
$810$		0			0
$811$		−	9.51125e9i		−	0.626130i	−0.949732	−	0.313065i	$-0.898644\pi$
$811$		−	9.51125e9i		−	0.626130i	0.949732	−	0.313065i	$-0.101356\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.04391e10	−	1.80810e10i	0.675478	−	1.16996i
$816$		0			0
$817$	−2.64576e9	+	1.52753e9i	−0.169736	+	0.0979971i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.74213e9	−	3.89257e9i	0.425203	−	0.245491i	−0.272098	−	0.962270i	$-0.587717\pi$
$821$	6.74213e9	−	3.89257e9i	0.425203	−	0.245491i	0.697301	+	0.716779i	$0.254384\pi$
$822$		0			0
$823$	−3.36811e9	+	5.83374e9i	−0.210614	+	0.364794i	−0.951907	−	0.306388i	$-0.900880\pi$
$823$	−3.36811e9	+	5.83374e9i	−0.210614	+	0.364794i	0.741293	+	0.671182i	$0.234213\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.20211e10i			1.35385i	0.736054	+	0.676923i	$0.236687\pi$
$827$			2.20211e10i			1.35385i	−0.736054	+	0.676923i	$0.763313\pi$
$828$		0			0
$829$	−1.93019e10	−	1.11439e10i	−1.17668	−	0.679357i	−0.221436	−	0.975175i	$-0.571074\pi$
$829$	−1.93019e10	−	1.11439e10i	−1.17668	−	0.679357i	−0.955244	+	0.295818i	$0.904408\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.51505e9	−	1.18079e10i	−0.390535	−	0.707805i
$834$		0			0
$835$	8.53826e9	+	1.47887e10i	0.507536	+	0.879078i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.95746e9			0.406709			0.203354	−	0.979105i	$-0.434816\pi$
$839$	6.95746e9			0.406709			0.203354	+	0.979105i	$0.434816\pi$
$840$		0			0
$841$	−5.73827e8			−0.0332656
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.40797e10	−	4.17072e10i	−1.37294	−	2.37801i
$846$		0			0
$847$	1.65987e9	−	2.81161e9i	0.0938604	−	0.158988i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.11837e9	+	1.22304e9i	0.117828	+	0.0680280i
$852$		0			0
$853$			7.56137e9i			0.417137i	0.978008	+	0.208569i	$0.0668804\pi$
$853$			7.56137e9i			0.417137i	−0.978008	+	0.208569i	$0.933120\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.21090e9	+	1.07576e10i	−0.337071	+	0.583824i	−0.983880	−	0.178828i	$-0.942770\pi$
$857$	−6.21090e9	+	1.07576e10i	−0.337071	+	0.583824i	0.646809	+	0.762652i	$0.276103\pi$
$858$		0			0
$859$	−1.36107e10	+	7.85815e9i	−0.732664	+	0.423004i	−0.819396	−	0.573228i	$-0.805691\pi$
$859$	−1.36107e10	+	7.85815e9i	−0.732664	+	0.423004i	0.0867322	+	0.996232i	$0.472358\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.60910e10	−	1.50637e10i	1.38183	−	0.797798i	0.389451	−	0.921047i	$-0.372665\pi$
$863$	2.60910e10	−	1.50637e10i	1.38183	−	0.797798i	0.992376	+	0.123249i	$0.0393313\pi$
$864$		0			0
$865$	1.27981e9	−	2.21669e9i	0.0672338	−	0.116452i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.17631e10i		−	0.608070i
$870$		0			0
$871$	−5.74119e9	−	3.31468e9i	−0.294401	−	0.169972i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.03570e9	−	1.60070e10i	−0.455967	−	0.807761i
$876$		0			0
$877$	1.44954e10	+	2.51068e10i	0.725659	+	1.25688i	0.958702	+	0.284412i	$0.0917985\pi$
$877$	1.44954e10	+	2.51068e10i	0.725659	+	1.25688i	−0.233043	+	0.972467i	$0.574868\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.49222e10			1.22792			0.613960	−	0.789337i	$-0.289576\pi$
$881$	2.49222e10			1.22792			0.613960	+	0.789337i	$0.289576\pi$
$882$		0			0
$883$	2.17555e10			1.06343			0.531713	−	0.846925i	$-0.321549\pi$
$883$	2.17555e10			1.06343			0.531713	+	0.846925i	$0.321549\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.21111e9	−	9.02590e9i	−0.250725	−	0.434268i	0.713001	−	0.701163i	$-0.247336\pi$
$887$	−5.21111e9	−	9.02590e9i	−0.250725	−	0.434268i	−0.963726	+	0.266895i	$0.914002\pi$
$888$		0			0
$889$	1.41671e10	+	2.50975e10i	0.676278	+	1.19805i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.54738e9	+	3.20278e9i	0.260680	+	0.150504i
$894$		0			0
$895$		−	2.82377e10i		−	1.31659i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.16448e10	−	2.01694e10i	0.534531	−	0.925835i
$900$		0			0
$901$	−3.08817e9	+	1.78295e9i	−0.140658	+	0.0812089i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.54227e9	−	3.19983e9i	0.248552	−	0.143502i
$906$		0			0
$907$	−1.23802e10	+	2.14431e10i	−0.550937	+	0.954251i	0.447270	+	0.894399i	$0.352396\pi$
$907$	−1.23802e10	+	2.14431e10i	−0.550937	+	0.954251i	−0.998207	+	0.0598523i	$0.980937\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1.88957e10i		−	0.828034i	−0.910269	−	0.414017i	$-0.864125\pi$
$911$		−	1.88957e10i		−	0.828034i	0.910269	−	0.414017i	$-0.135875\pi$
$912$		0			0
$913$	−3.17175e10	−	1.83121e10i	−1.37928	−	0.796326i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.30135e9	+	1.40614e10i	−0.355513	+	0.602195i
$918$		0			0
$919$	−8.39687e9	−	1.45438e10i	−0.356873	−	0.618122i	0.630564	−	0.776137i	$-0.282824\pi$
$919$	−8.39687e9	−	1.45438e10i	−0.356873	−	0.618122i	−0.987437	+	0.158016i	$0.949490\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.88651e10			2.46406
$924$		0			0
$925$	1.83049e9			0.0760451
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.27996e10	−	3.94901e10i	−0.932980	−	1.61597i	−0.778198	−	0.628019i	$-0.783866\pi$
$929$	−2.27996e10	−	3.94901e10i	−0.932980	−	1.61597i	−0.154782	−	0.987949i	$-0.549468\pi$
$930$		0			0
$931$	2.09082e8	−	1.07713e10i	0.00849168	−	0.437464i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.02076e10	−	1.16668e10i	−0.808488	−	0.466781i
$936$		0			0
$937$		−	3.94197e10i		−	1.56540i	−0.622401	−	0.782699i	$-0.713843\pi$
$937$		−	3.94197e10i		−	1.56540i	0.622401	−	0.782699i	$-0.286157\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.42415e10	+	2.46670e10i	−0.557175	+	0.965056i	0.440556	+	0.897725i	$0.354781\pi$
$941$	−1.42415e10	+	2.46670e10i	−0.557175	+	0.965056i	−0.997731	+	0.0673303i	$0.978552\pi$
$942$		0			0
$943$	5.57892e9	−	3.22099e9i	0.216650	−	0.125083i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.82461e9	+	5.67224e9i	−0.375916	+	0.217035i	−0.676040	−	0.736865i	$-0.736305\pi$
$947$	−9.82461e9	+	5.67224e9i	−0.375916	+	0.217035i	0.300124	+	0.953900i	$0.402972\pi$
$948$		0			0
$949$	−2.95882e10	+	5.12482e10i	−1.12379	+	1.94647i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.80193e10i		−	0.674394i	−0.941434	−	0.337197i	$-0.890521\pi$
$953$		−	1.80193e10i		−	0.674394i	0.941434	−	0.337197i	$-0.109479\pi$
$954$		0			0
$955$	−4.51308e10	−	2.60563e10i	−1.67672	−	0.968056i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.38439e8	+	4.51783e10i	−0.0160526	+	1.65411i
$960$		0			0
$961$	1.45954e9	+	2.52800e9i	0.0530498	+	0.0918850i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.44275e10			0.875053
$966$		0			0
$967$	1.31545e10			0.467822			0.233911	−	0.972258i	$-0.424848\pi$
$967$	1.31545e10			0.467822			0.233911	+	0.972258i	$0.424848\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.50229e10	−	2.60204e10i	−0.526607	−	0.912110i	−0.999519	−	0.0310006i	$-0.990131\pi$
$971$	−1.50229e10	−	2.60204e10i	−0.526607	−	0.912110i	0.472912	−	0.881109i	$-0.343203\pi$
$972$		0			0
$973$	−1.56498e10	+	8.83404e9i	−0.544645	+	0.307443i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.03800e10	−	1.75399e10i	−1.04221	−	0.601722i	−0.121754	−	0.992560i	$-0.538852\pi$
$977$	−3.03800e10	−	1.75399e10i	−1.04221	−	0.601722i	−0.920459	+	0.390838i	$0.872185\pi$
$978$		0			0
$979$		−	3.67630e10i		−	1.25219i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.59535e9	−	6.22732e9i	0.120727	−	0.209105i	−0.799328	−	0.600895i	$-0.794811\pi$
$983$	3.59535e9	−	6.22732e9i	0.120727	−	0.209105i	0.920054	+	0.391790i	$0.128144\pi$
$984$		0			0
$985$	−2.10777e10	+	1.21692e10i	−0.702742	+	0.405728i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.65590e9	+	1.53339e9i	−0.0873022	+	0.0504039i
$990$		0			0
$991$	6.35818e9	−	1.10127e10i	0.207527	−	0.359448i	−0.743408	−	0.668839i	$-0.766792\pi$
$991$	6.35818e9	−	1.10127e10i	0.207527	−	0.359448i	0.950935	+	0.309391i	$0.100125\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			1.37146e10i			0.441371i
$996$		0			0
$997$	1.11315e10	+	6.42679e9i	0.355731	+	0.205381i	0.667207	−	0.744873i	$-0.267490\pi$
$997$	1.11315e10	+	6.42679e9i	0.355731	+	0.205381i	−0.311475	+	0.950254i	$0.600823\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.t.a.17.14	yes	36
3.2	odd	2	inner	252.8.t.a.17.5	✓	36
7.5	odd	6	inner	252.8.t.a.89.5	yes	36
21.5	even	6	inner	252.8.t.a.89.14	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.t.a.17.5	✓	36	3.2	odd	2	inner
252.8.t.a.17.14	yes	36	1.1	even	1	trivial
252.8.t.a.89.5	yes	36	7.5	odd	6	inner
252.8.t.a.89.14	yes	36	21.5	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(148.283 + 256.835i) q^{5} +(-446.098 - 790.278i) q^{7} +(4160.98 + 2402.34i) q^{11} -15004.6i q^{13} +(-8187.77 + 14181.6i) q^{17} +(-11329.0 + 6540.82i) q^{19} +(-11372.4 + 6565.88i) q^{23} +(-4913.48 + 8510.40i) q^{25} +133505. i q^{29} +(-151075. - 87223.4i) q^{31} +(136822. - 231759. i) q^{35} +(-93136.1 - 161316. i) q^{37} -490565. q^{41} +233538. q^{43} +(-244830. - 424058. i) q^{47} +(-425536. + 705083. i) q^{49} +(188584. + 108879. i) q^{53} +1.42491e6i q^{55} +(779780. - 1.35062e6i) q^{59} +(2.26287e6 - 1.30647e6i) q^{61} +(3.85370e6 - 2.22493e6i) q^{65} +(220911. - 382629. i) q^{67} +3.92314e6i q^{71} +(-3.41550e6 - 1.97194e6i) q^{73} +(42312.4 - 4.36001e6i) q^{77} +(-1.22413e6 - 2.12026e6i) q^{79} -7.62261e6 q^{83} -4.85645e6 q^{85} +(-3.82575e6 - 6.62639e6i) q^{89} +(-1.18578e7 + 6.69353e6i) q^{91} +(-3.35982e6 - 1.93979e6i) q^{95} -5.40637e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2634 q^{7} - 47862 q^{19} - 360762 q^{25} + 486018 q^{31} - 972270 q^{37} + 298788 q^{43} + 1556886 q^{49} + 4324644 q^{61} - 2969562 q^{67} - 5157378 q^{73} + 7676514 q^{79} - 15214128 q^{85} - 6114678 q^{91}+O(q^{100}) \) 36 * q + 2634 * q^7 - 47862 * q^19 - 360762 * q^25 + 486018 * q^31 - 972270 * q^37 + 298788 * q^43 + 1556886 * q^49 + 4324644 * q^61 - 2969562 * q^67 - 5157378 * q^73 + 7676514 * q^79 - 15214128 * q^85 - 6114678 * q^91

Introduction

L-functions

Modular forms

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Database

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