LMFDB - Embedded newform 4680.2.l.g.2809.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (of order $2$, degree $1$, 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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.57815240704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 $ x^8 - 2x^7 + 2x^6 + 89x^4 - 170x^3 + 162x^2 - 72x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.8
Root			$2.20793 - 2.20793i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.g.2809.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.20793 + 0.353624i) q^{5} -1.65573i q^{7} +O(q^{10})$	$q+(2.20793 + 0.353624i) q^{5} -1.65573i q^{7} +2.94848 q^{11} +1.00000i q^{13} -1.46738i q^{17} +0.532621i q^{23} +(4.74990 + 1.56155i) q^{25} +5.70861 q^{29} +(0.585504 - 3.65573i) q^{35} +8.77883i q^{37} -1.23987 q^{41} +1.70861i q^{43} -2.70725i q^{47} +4.25857 q^{49} -8.77883i q^{53} +(6.51003 + 1.04265i) q^{55} -3.83035 q^{59} -0.241231 q^{61} +(-0.353624 + 2.20793i) q^{65} +2.58550i q^{67} +2.55132 q^{71} -0.188347i q^{73} -4.88187i q^{77} +11.0729 q^{79} -7.91566i q^{83} +(0.518900 - 3.23987i) q^{85} +15.9685 q^{89} +1.65573 q^{91} +16.8641i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5}+O(q^{10}) $ 8 * q + 2 * q^5	$ 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100}) $ 8 * q + 2 * q^5 - 2 * q^11 + 16 * q^29 + 8 * q^35 - 14 * q^41 - 18 * q^49 + 10 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 - 30 * q^71 + 2 * q^79 + 24 * q^85 + 18 * q^89 - 14 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$5$	2.20793	+	0.353624i	0.987416	+	0.158145i
$5$	2.20793	+	0.353624i	0.987416	+	0.158145i
$6$		0			0
$7$		−	1.65573i		−	0.625806i	−0.949785	−	0.312903i	$-0.898699\pi$
$7$		−	1.65573i		−	0.625806i	0.949785	−	0.312903i	$-0.101301\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.94848			0.889000			0.444500	−	0.895779i	$-0.353381\pi$
$11$	2.94848			0.889000			0.444500	+	0.895779i	$0.353381\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.46738i		−	0.355892i	−0.984040	−	0.177946i	$-0.943055\pi$
$17$		−	1.46738i		−	0.355892i	0.984040	−	0.177946i	$-0.0569452\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.532621i			0.111059i	0.998457	+	0.0555296i	$0.0176847\pi$
$23$			0.532621i			0.111059i	−0.998457	+	0.0555296i	$0.982315\pi$
$24$		0			0
$25$	4.74990	+	1.56155i	0.949980	+	0.312311i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.70861			1.06006			0.530031	−	0.847978i	$-0.322180\pi$
$29$	5.70861			1.06006			0.530031	+	0.847978i	$0.322180\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.585504	−	3.65573i	0.0989683	−	0.617931i
$36$		0			0
$37$			8.77883i			1.44323i	0.692294	+	0.721616i	$0.256600\pi$
$37$			8.77883i			1.44323i	−0.692294	+	0.721616i	$0.743400\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.23987			−0.193635			−0.0968175	−	0.995302i	$-0.530866\pi$
$41$	−1.23987			−0.193635			−0.0968175	+	0.995302i	$0.530866\pi$
$42$		0			0
$43$			1.70861i			0.260561i	0.991477	+	0.130280i	$0.0415877\pi$
$43$			1.70861i			0.260561i	−0.991477	+	0.130280i	$0.958412\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.70725i		−	0.394893i	−0.980314	−	0.197446i	$-0.936735\pi$
$47$		−	2.70725i		−	0.394893i	0.980314	−	0.197446i	$-0.0632648\pi$
$48$		0			0
$49$	4.25857			0.608367
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.77883i		−	1.20587i	−0.797792	−	0.602933i	$-0.793999\pi$
$53$		−	8.77883i		−	1.20587i	0.797792	−	0.602933i	$-0.206001\pi$
$54$		0			0
$55$	6.51003	+	1.04265i	0.877812	+	0.140591i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.83035			−0.498670			−0.249335	−	0.968417i	$-0.580212\pi$
$59$	−3.83035			−0.498670			−0.249335	+	0.968417i	$0.580212\pi$
$60$		0			0
$61$	−0.241231			−0.0308865			−0.0154432	−	0.999881i	$-0.504916\pi$
$61$	−0.241231			−0.0308865			−0.0154432	+	0.999881i	$0.504916\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.353624	+	2.20793i	−0.0438616	+	0.273860i
$66$		0			0
$67$			2.58550i			0.315870i	0.987450	+	0.157935i	$0.0504836\pi$
$67$			2.58550i			0.315870i	−0.987450	+	0.157935i	$0.949516\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.55132			0.302786			0.151393	−	0.988474i	$-0.451624\pi$
$71$	2.55132			0.302786			0.151393	+	0.988474i	$0.451624\pi$
$72$		0			0
$73$		−	0.188347i		−	0.0220444i	−0.999939	−	0.0110222i	$-0.996491\pi$
$73$		−	0.188347i		−	0.0220444i	0.999939	−	0.0110222i	$-0.00350855\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	4.88187i		−	0.556341i
$78$		0			0
$79$	11.0729			1.24580			0.622902	−	0.782300i	$-0.285954\pi$
$79$	11.0729			1.24580			0.622902	+	0.782300i	$0.285954\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	7.91566i		−	0.868856i	−0.900706	−	0.434428i	$-0.856950\pi$
$83$		−	7.91566i		−	0.868856i	0.900706	−	0.434428i	$-0.143050\pi$
$84$		0			0
$85$	0.518900	−	3.23987i	0.0562826	−	0.351413i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	15.9685			1.69266			0.846331	−	0.532657i	$-0.178807\pi$
$89$	15.9685			1.69266			0.846331	+	0.532657i	$0.178807\pi$
$90$		0			0
$91$	1.65573			0.173567
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			16.8641i			1.71229i	0.516733	+	0.856147i	$0.327148\pi$
$97$			16.8641i			1.71229i	−0.516733	+	0.856147i	$0.672852\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.64337			0.462032			0.231016	−	0.972950i	$-0.425795\pi$
$101$	4.64337			0.462032			0.231016	+	0.972950i	$0.425795\pi$
$102$		0			0
$103$		−	9.66071i		−	0.951898i	−0.879473	−	0.475949i	$-0.842105\pi$
$103$		−	9.66071i		−	0.951898i	0.879473	−	0.475949i	$-0.157895\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	11.9498i		−	1.15523i	−0.816308	−	0.577617i	$-0.803983\pi$
$107$		−	11.9498i		−	1.15523i	0.816308	−	0.577617i	$-0.196017\pi$
$108$		0			0
$109$	−11.8970			−1.13952			−0.569761	−	0.821810i	$-0.692964\pi$
$109$	−11.8970			−1.13952			−0.569761	+	0.821810i	$0.692964\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2.37669i			0.223581i	0.993732	+	0.111790i	$0.0356585\pi$
$113$			2.37669i			0.223581i	−0.993732	+	0.111790i	$0.964341\pi$
$114$		0			0
$115$	−0.188347	+	1.17599i	−0.0175635	+	0.109662i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.42958			−0.222719
$120$		0			0
$121$	−2.30647			−0.209679
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.93524	+	5.12748i	0.888635	+	0.458615i
$126$		0			0
$127$			2.87689i			0.255283i	0.991820	+	0.127642i	$0.0407407\pi$
$127$			2.87689i			0.255283i	−0.991820	+	0.127642i	$0.959259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.3968			−1.69470			−0.847351	−	0.531033i	$-0.821804\pi$
$131$	−19.3968			−1.69470			−0.847351	+	0.531033i	$0.821804\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	0.357994i		−	0.0305855i	−0.999883	−	0.0152927i	$-0.995132\pi$
$137$		−	0.357994i		−	0.0305855i	0.999883	−	0.0152927i	$-0.00486802\pi$
$138$		0			0
$139$	4.88187			0.414075			0.207038	−	0.978333i	$-0.433618\pi$
$139$	4.88187			0.414075			0.207038	+	0.978333i	$0.433618\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.94848i			0.246564i
$144$		0			0
$145$	12.6042	+	2.01870i	1.04672	+	0.167644i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	20.6092			1.68837			0.844185	−	0.536052i	$-0.180085\pi$
$149$	20.6092			1.68837			0.844185	+	0.536052i	$0.180085\pi$
$150$		0			0
$151$	−3.89423			−0.316908			−0.158454	−	0.987366i	$-0.550651\pi$
$151$	−3.89423			−0.316908			−0.158454	+	0.987366i	$0.550651\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			9.22887i			0.736544i	0.929718	+	0.368272i	$0.120051\pi$
$157$			9.22887i			0.736544i	−0.929718	+	0.368272i	$0.879949\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.881875			0.0695015
$162$		0			0
$163$		−	16.5700i		−	1.29786i	−0.760846	−	0.648932i	$-0.775216\pi$
$163$		−	16.5700i		−	1.29786i	0.760846	−	0.648932i	$-0.224784\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.5390i		−	1.20244i	−0.799083	−	0.601221i	$-0.794681\pi$
$167$		−	15.5390i		−	1.20244i	0.799083	−	0.601221i	$-0.205319\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			3.17101i			0.241087i	0.992708	+	0.120544i	$0.0384638\pi$
$173$			3.17101i			0.241087i	−0.992708	+	0.120544i	$0.961536\pi$
$174$		0			0
$175$	2.58550	−	7.86454i	0.195446	−	0.594503i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.29411			0.171470			0.0857351	−	0.996318i	$-0.472676\pi$
$179$	2.29411			0.171470			0.0857351	+	0.996318i	$0.472676\pi$
$180$		0			0
$181$	−17.6105			−1.30898			−0.654491	−	0.756070i	$-0.727117\pi$
$181$	−17.6105			−1.30898			−0.654491	+	0.756070i	$0.727117\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.10440	+	19.3830i	−0.228240	+	1.42507i
$186$		0			0
$187$		−	4.32654i		−	0.316388i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.105767			0.00765304			0.00382652	−	0.999993i	$-0.498782\pi$
$191$	0.105767			0.00765304			0.00382652	+	0.999993i	$0.498782\pi$
$192$		0			0
$193$			3.27903i			0.236030i	0.993012	+	0.118015i	$0.0376531\pi$
$193$			3.27903i			0.236030i	−0.993012	+	0.118015i	$0.962347\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.01598i		−	0.428621i	−0.976766	−	0.214310i	$-0.931250\pi$
$197$		−	6.01598i		−	0.428621i	0.976766	−	0.214310i	$-0.0687504\pi$
$198$		0			0
$199$	5.33192			0.377969			0.188985	−	0.981980i	$-0.439480\pi$
$199$	5.33192			0.377969			0.188985	+	0.981980i	$0.439480\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	9.45190i		−	0.663393i
$204$		0			0
$205$	−2.73754	−	0.438447i	−0.191198	−	0.0306225i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	14.5777			1.00357			0.501786	−	0.864992i	$-0.332676\pi$
$211$	14.5777			1.00357			0.501786	+	0.864992i	$0.332676\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.604205	+	3.77249i	−0.0412065	+	0.257282i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.46738			0.0987066
$222$		0			0
$223$		−	19.8518i		−	1.32937i	−0.747122	−	0.664687i	$-0.768565\pi$
$223$		−	19.8518i		−	1.32937i	0.747122	−	0.664687i	$-0.231435\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.85042i			0.454678i	0.973816	+	0.227339i	$0.0730026\pi$
$227$			6.85042i			0.454678i	−0.973816	+	0.227339i	$0.926997\pi$
$228$		0			0
$229$	−18.0374			−1.19195			−0.595973	−	0.803005i	$-0.703233\pi$
$229$	−18.0374			−1.19195			−0.595973	+	0.803005i	$0.703233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.4049i			0.943694i	0.881680	+	0.471847i	$0.156413\pi$
$233$			14.4049i			0.943694i	−0.881680	+	0.471847i	$0.843587\pi$
$234$		0			0
$235$	0.957347	−	5.97741i	0.0624505	−	0.389923i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.7350			1.40592			0.702961	−	0.711229i	$-0.251861\pi$
$239$	21.7350			1.40592			0.702961	+	0.711229i	$0.251861\pi$
$240$		0			0
$241$	17.5604			1.13116			0.565582	−	0.824692i	$-0.308652\pi$
$241$	17.5604			1.13116			0.565582	+	0.824692i	$0.308652\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.40262	+	1.50593i	0.600711	+	0.0962105i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.7082			1.43333			0.716665	−	0.697418i	$-0.245668\pi$
$251$	22.7082			1.43333			0.716665	+	0.697418i	$0.245668\pi$
$252$		0			0
$253$			1.57042i			0.0987316i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.58278i		−	0.410623i	−0.978697	−	0.205311i	$-0.934179\pi$
$257$		−	6.58278i		−	0.410623i	0.978697	−	0.205311i	$-0.0658207\pi$
$258$		0			0
$259$	14.5353			0.903182
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.24349i			0.261665i	0.991405	+	0.130832i	$0.0417649\pi$
$263$			4.24349i			0.261665i	−0.991405	+	0.130832i	$0.958235\pi$
$264$		0			0
$265$	3.10440	−	19.3830i	0.190702	−	1.19069i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.6434			0.770880			0.385440	−	0.922733i	$-0.374050\pi$
$269$	12.6434			0.770880			0.385440	+	0.922733i	$0.374050\pi$
$270$		0			0
$271$	−23.7665			−1.44371			−0.721855	−	0.692044i	$-0.756710\pi$
$271$	−23.7665			−1.44371			−0.721855	+	0.692044i	$0.756710\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.0050	+	4.60421i	0.844532	+	0.277644i
$276$		0			0
$277$			17.8765i			1.07409i	0.843552	+	0.537047i	$0.180460\pi$
$277$			17.8765i			1.07409i	−0.843552	+	0.537047i	$0.819540\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.39540			−0.500827			−0.250414	−	0.968139i	$-0.580567\pi$
$281$	−8.39540			−0.500827			−0.250414	+	0.968139i	$0.580567\pi$
$282$		0			0
$283$			13.9344i			0.828312i	0.910206	+	0.414156i	$0.135923\pi$
$283$			13.9344i			0.828312i	−0.910206	+	0.414156i	$0.864077\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.05288i			0.121178i
$288$		0			0
$289$	14.8468			0.873341
$290$		0			0
$291$		0			0
$292$		0			0
$293$			11.4085i			0.666491i	0.942840	+	0.333245i	$0.108144\pi$
$293$			11.4085i			0.666491i	−0.942840	+	0.333245i	$0.891856\pi$
$294$		0			0
$295$	−8.45715	−	1.35450i	−0.492394	−	0.0788623i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.532621			−0.0308023
$300$		0			0
$301$	2.82899			0.163060
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.532621	−	0.0853050i	−0.0304978	−	0.00488455i
$306$		0			0
$307$		−	2.50790i		−	0.143134i	−0.997436	−	0.0715668i	$-0.977200\pi$
$307$		−	2.50790i		−	0.143134i	0.997436	−	0.0715668i	$-0.0227999\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.96220			0.394790			0.197395	−	0.980324i	$-0.436752\pi$
$311$	6.96220			0.394790			0.197395	+	0.980324i	$0.436752\pi$
$312$		0			0
$313$			16.0880i			0.909349i	0.890658	+	0.454675i	$0.150244\pi$
$313$			16.0880i			0.909349i	−0.890658	+	0.454675i	$0.849756\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.25263i		−	0.0703545i	−0.999381	−	0.0351772i	$-0.988800\pi$
$317$		−	1.25263i		−	0.0703545i	0.999381	−	0.0351772i	$-0.0111996\pi$
$318$		0			0
$319$	16.8317			0.942395
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.56155	+	4.74990i	−0.0866194	+	0.263477i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.48246			−0.247126
$330$		0			0
$331$	−8.24349			−0.453103			−0.226552	−	0.973999i	$-0.572745\pi$
$331$	−8.24349			−0.453103			−0.226552	+	0.973999i	$0.572745\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.914296	+	5.70861i	−0.0499533	+	0.311895i
$336$		0			0
$337$		−	22.1227i		−	1.20510i	−0.798081	−	0.602550i	$-0.794151\pi$
$337$		−	22.1227i		−	1.20510i	0.798081	−	0.602550i	$-0.205849\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	18.6411i		−	1.00653i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.4395i		−	0.989886i	−0.868925	−	0.494943i	$-0.835189\pi$
$347$		−	18.4395i		−	0.989886i	0.868925	−	0.494943i	$-0.164811\pi$
$348$		0			0
$349$	−19.6634			−1.05256			−0.526280	−	0.850312i	$-0.676414\pi$
$349$	−19.6634			−1.05256			−0.526280	+	0.850312i	$0.676414\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	1.18971i		−	0.0633219i	−0.999499	−	0.0316609i	$-0.989920\pi$
$353$		−	1.18971i		−	0.0633219i	0.999499	−	0.0316609i	$-0.0100797\pi$
$354$		0			0
$355$	5.63314	+	0.902208i	0.298976	+	0.0478842i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.98090			0.262882			0.131441	−	0.991324i	$-0.458040\pi$
$359$	4.98090			0.262882			0.131441	+	0.991324i	$0.458040\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.0666042	−	0.415858i	0.00348622	−	0.0217670i
$366$		0			0
$367$			10.7160i			0.559370i	0.960092	+	0.279685i	$0.0902300\pi$
$367$			10.7160i			0.559370i	−0.960092	+	0.279685i	$0.909770\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.5353			−0.754638
$372$		0			0
$373$		−	22.8691i		−	1.18412i	−0.805895	−	0.592059i	$-0.798315\pi$
$373$		−	22.8691i		−	1.18412i	0.805895	−	0.592059i	$-0.201685\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			5.70861i			0.294008i
$378$		0			0
$379$	−26.5325			−1.36289			−0.681443	−	0.731871i	$-0.738647\pi$
$379$	−26.5325			−1.36289			−0.681443	+	0.731871i	$0.738647\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			38.3325i			1.95870i	0.202176	+	0.979349i	$0.435199\pi$
$383$			38.3325i			1.95870i	−0.202176	+	0.979349i	$0.564801\pi$
$384$		0			0
$385$	1.72635	−	10.7788i	0.0879828	−	0.549340i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.87650			−0.399354			−0.199677	−	0.979862i	$-0.563989\pi$
$389$	−7.87650			−0.399354			−0.199677	+	0.979862i	$0.563989\pi$
$390$		0			0
$391$	0.781557			0.0395250
$392$		0			0
$393$		0			0
$394$		0			0
$395$	24.4483	+	3.91566i	1.23013	+	0.197018i
$396$		0			0
$397$		−	0.673065i		−	0.0337802i	−0.999857	−	0.0168901i	$-0.994623\pi$
$397$		−	0.673065i		−	0.0337802i	0.999857	−	0.0168901i	$-0.00537654\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.3981			1.31826			0.659130	−	0.752029i	$-0.270925\pi$
$401$	26.3981			1.31826			0.659130	+	0.752029i	$0.270925\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			25.8842i			1.28303i
$408$		0			0
$409$	33.4646			1.65472			0.827359	−	0.561674i	$-0.189843\pi$
$409$	33.4646			1.65472			0.827359	+	0.561674i	$0.189843\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			6.34202i			0.312070i
$414$		0			0
$415$	2.79917	−	17.4772i	0.137406	−	0.857923i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−32.3068			−1.57829			−0.789145	−	0.614207i	$-0.789476\pi$
$419$	−32.3068			−1.57829			−0.789145	+	0.614207i	$0.789476\pi$
$420$		0			0
$421$	2.33929			0.114010			0.0570051	−	0.998374i	$-0.481845\pi$
$421$	2.33929			0.114010			0.0570051	+	0.998374i	$0.481845\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.29139	−	6.96990i	0.111149	−	0.338090i
$426$		0			0
$427$			0.399413i			0.0193289i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.5138			0.988117			0.494059	−	0.869429i	$-0.335513\pi$
$431$	20.5138			0.988117			0.494059	+	0.869429i	$0.335513\pi$
$432$		0			0
$433$			24.8514i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$433$			24.8514i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	8.18337			0.390571			0.195285	−	0.980746i	$-0.437437\pi$
$439$	8.18337			0.390571			0.195285	+	0.980746i	$0.437437\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			26.1961i			1.24461i	0.782774	+	0.622306i	$0.213804\pi$
$443$			26.1961i			1.24461i	−0.782774	+	0.622306i	$0.786196\pi$
$444$		0			0
$445$	35.2574	+	5.64686i	1.67136	+	0.267687i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.8034			−0.557036			−0.278518	−	0.960431i	$-0.589843\pi$
$449$	−11.8034			−0.557036			−0.278518	+	0.960431i	$0.589843\pi$
$450$		0			0
$451$	−3.65573			−0.172141
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.65573	+	0.585504i	0.171383	+	0.0274489i
$456$		0			0
$457$			1.86454i			0.0872193i	0.999049	+	0.0436097i	$0.0138858\pi$
$457$			1.86454i			0.0872193i	−0.999049	+	0.0436097i	$0.986114\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.42822			−0.345967			−0.172983	−	0.984925i	$-0.555341\pi$
$461$	−7.42822			−0.345967			−0.172983	+	0.984925i	$0.555341\pi$
$462$		0			0
$463$		−	29.0829i		−	1.35160i	−0.737086	−	0.675799i	$-0.763799\pi$
$463$		−	29.0829i		−	1.35160i	0.737086	−	0.675799i	$-0.236201\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	10.5727i		−	0.489248i	−0.969618	−	0.244624i	$-0.921335\pi$
$467$		−	10.5727i		−	0.489248i	0.969618	−	0.244624i	$-0.0786646\pi$
$468$		0			0
$469$	4.28089			0.197673
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.03780i			0.231638i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.6166			−0.896304			−0.448152	−	0.893957i	$-0.647918\pi$
$479$	−19.6166			−0.896304			−0.448152	+	0.893957i	$0.647918\pi$
$480$		0			0
$481$	−8.77883			−0.400280
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.96356	+	37.2348i	−0.270791	+	1.69075i
$486$		0			0
$487$			0.235782i			0.0106843i	0.999986	+	0.00534215i	$0.00170047\pi$
$487$			0.235782i			0.0106843i	−0.999986	+	0.00534215i	$0.998300\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	23.6183			1.06588			0.532938	−	0.846154i	$-0.321088\pi$
$491$	23.6183			1.06588			0.532938	+	0.846154i	$0.321088\pi$
$492$		0			0
$493$		−	8.37669i		−	0.377267i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	4.22429i		−	0.189485i
$498$		0			0
$499$	18.2737			0.818041			0.409021	−	0.912525i	$-0.365870\pi$
$499$	18.2737			0.818041			0.409021	+	0.912525i	$0.365870\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	27.8722i		−	1.24276i	−0.783509	−	0.621381i	$-0.786572\pi$
$503$		−	27.8722i		−	1.24276i	0.783509	−	0.621381i	$-0.213428\pi$
$504$		0			0
$505$	10.2522	+	1.64201i	0.456218	+	0.0730683i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.2878			0.766267			0.383134	−	0.923693i	$-0.374845\pi$
$509$	17.2878			0.766267			0.383134	+	0.923693i	$0.374845\pi$
$510$		0			0
$511$	−0.311852			−0.0137955
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.41626	−	21.3302i	0.150538	−	0.939919i
$516$		0			0
$517$		−	7.98226i		−	0.351060i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.9474			−1.18059			−0.590294	−	0.807188i	$-0.700988\pi$
$521$	−26.9474			−1.18059			−0.590294	+	0.807188i	$0.700988\pi$
$522$		0			0
$523$		−	0.150946i		−	0.00660040i	−0.999995	−	0.00330020i	$-0.998950\pi$
$523$		−	0.150946i		−	0.00660040i	0.999995	−	0.00330020i	$-0.00105049\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	22.7163			0.987666
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.23987i		−	0.0537047i
$534$		0			0
$535$	4.22575	−	26.3844i	0.182695	−	1.14070i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.5563			0.540838
$540$		0			0
$541$	−1.97256			−0.0848069			−0.0424035	−	0.999101i	$-0.513501\pi$
$541$	−1.97256			−0.0848069			−0.0424035	+	0.999101i	$0.513501\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−26.2676	−	4.20705i	−1.12518	−	0.180210i
$546$		0			0
$547$		−	21.6128i		−	0.924097i	−0.886855	−	0.462048i	$-0.847115\pi$
$547$		−	21.6128i		−	0.924097i	0.886855	−	0.462048i	$-0.152885\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	18.3338i		−	0.779631i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	36.1619i		−	1.53223i	−0.642705	−	0.766114i	$-0.722188\pi$
$557$		−	36.1619i		−	1.53223i	0.642705	−	0.766114i	$-0.277812\pi$
$558$		0			0
$559$	−1.70861			−0.0722665
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	9.05248i		−	0.381517i	−0.981637	−	0.190758i	$-0.938905\pi$
$563$		−	9.05248i		−	0.381517i	0.981637	−	0.190758i	$-0.0610947\pi$
$564$		0			0
$565$	−0.840456	+	5.24757i	−0.0353583	+	0.220767i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.7255			0.743094			0.371547	−	0.928414i	$-0.378828\pi$
$569$	17.7255			0.743094			0.371547	+	0.928414i	$0.378828\pi$
$570$		0			0
$571$	−17.4095			−0.728566			−0.364283	−	0.931288i	$-0.618686\pi$
$571$	−17.4095			−0.728566			−0.364283	+	0.931288i	$0.618686\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.831716	+	2.52990i	−0.0346849	+	0.105504i
$576$		0			0
$577$			27.0729i			1.12706i	0.826095	+	0.563531i	$0.190557\pi$
$577$			27.0729i			1.12706i	−0.826095	+	0.563531i	$0.809443\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13.1062			−0.543735
$582$		0			0
$583$		−	25.8842i		−	1.07201i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			21.5417i			0.889121i	0.895749	+	0.444560i	$0.146640\pi$
$587$			21.5417i			0.889121i	−0.895749	+	0.444560i	$0.853360\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	0.542087i		−	0.0222608i	−0.999938	−	0.0111304i	$-0.996457\pi$
$593$		−	0.542087i		−	0.0222608i	0.999938	−	0.0111304i	$-0.00354300\pi$
$594$		0			0
$595$	−5.36434	−	0.859157i	−0.219916	−	0.0352220i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.72595			−0.193097			−0.0965485	−	0.995328i	$-0.530780\pi$
$599$	−4.72595			−0.193097			−0.0965485	+	0.995328i	$0.530780\pi$
$600$		0			0
$601$	−2.47203			−0.100836			−0.0504182	−	0.998728i	$-0.516055\pi$
$601$	−2.47203			−0.100836			−0.0504182	+	0.998728i	$0.516055\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.09253	−	0.815624i	−0.207041	−	0.0331598i
$606$		0			0
$607$			42.8966i			1.74112i	0.492064	+	0.870559i	$0.336242\pi$
$607$			42.8966i			1.74112i	−0.492064	+	0.870559i	$0.663758\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.70725			0.109524
$612$		0			0
$613$			39.5175i			1.59610i	0.602594	+	0.798048i	$0.294134\pi$
$613$			39.5175i			1.59610i	−0.602594	+	0.798048i	$0.705866\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.2951i		−	1.30015i	−0.759870	−	0.650075i	$-0.774737\pi$
$617$		−	32.2951i		−	1.30015i	0.759870	−	0.650075i	$-0.225263\pi$
$618$		0			0
$619$	17.4793			0.702554			0.351277	−	0.936272i	$-0.385748\pi$
$619$	17.4793			0.702554			0.351277	+	0.936272i	$0.385748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	26.4395i		−	1.05928i
$624$		0			0
$625$	20.1231	+	14.8344i	0.804924	+	0.593378i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.8819			0.513634
$630$		0			0
$631$	−0.376695			−0.0149960			−0.00749799	−	0.999972i	$-0.502387\pi$
$631$	−0.376695			−0.0149960			−0.00749799	+	0.999972i	$0.502387\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.01734	+	6.35198i	−0.0403718	+	0.252071i
$636$		0			0
$637$			4.25857i			0.168731i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.3921			−1.20042			−0.600208	−	0.799844i	$-0.704916\pi$
$641$	−30.3921			−1.20042			−0.600208	+	0.799844i	$0.704916\pi$
$642$		0			0
$643$			17.3918i			0.685865i	0.939360	+	0.342932i	$0.111420\pi$
$643$			17.3918i			0.685865i	−0.939360	+	0.342932i	$0.888580\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	34.1331i		−	1.34191i	−0.741497	−	0.670956i	$-0.765884\pi$
$647$		−	34.1331i		−	1.34191i	0.741497	−	0.670956i	$-0.234116\pi$
$648$		0			0
$649$	−11.2937			−0.443317
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	20.6284i		−	0.807250i	−0.914925	−	0.403625i	$-0.867750\pi$
$653$		−	20.6284i		−	0.807250i	0.914925	−	0.403625i	$-0.132250\pi$
$654$		0			0
$655$	−42.8267	−	6.85916i	−1.67338	−	0.268009i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.3940			−0.677575			−0.338788	−	0.940863i	$-0.610017\pi$
$659$	−17.3940			−0.677575			−0.338788	+	0.940863i	$0.610017\pi$
$660$		0			0
$661$	−9.04053			−0.351636			−0.175818	−	0.984423i	$-0.556257\pi$
$661$	−9.04053			−0.351636			−0.175818	+	0.984423i	$0.556257\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			3.04053i			0.117730i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.711265			−0.0274581
$672$		0			0
$673$			12.7634i			0.491991i	0.969271	+	0.245996i	$0.0791148\pi$
$673$			12.7634i			0.491991i	−0.969271	+	0.245996i	$0.920885\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	15.9251i		−	0.612052i	−0.952023	−	0.306026i	$-0.901001\pi$
$677$		−	15.9251i		−	0.612052i	0.952023	−	0.306026i	$-0.0989995\pi$
$678$		0			0
$679$	27.9224			1.07156
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	9.04927i		−	0.346261i	−0.984899	−	0.173130i	$-0.944612\pi$
$683$		−	9.04927i		−	0.346261i	0.984899	−	0.173130i	$-0.0553882\pi$
$684$		0			0
$685$	0.126595	−	0.790425i	0.00483696	−	0.0302006i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.77883			0.334447
$690$		0			0
$691$	−41.4848			−1.57816			−0.789078	−	0.614293i	$-0.789441\pi$
$691$	−41.4848			−1.57816			−0.789078	+	0.614293i	$0.789441\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.7788	+	1.72635i	0.408864	+	0.0654841i
$696$		0			0
$697$			1.81936i			0.0689131i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.08298			−0.0409036			−0.0204518	−	0.999791i	$-0.506510\pi$
$701$	−1.08298			−0.0409036			−0.0204518	+	0.999791i	$0.506510\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.68815i		−	0.289143i
$708$		0			0
$709$	29.9371			1.12431			0.562155	−	0.827032i	$-0.309972\pi$
$709$	29.9371			1.12431			0.562155	+	0.827032i	$0.309972\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1.04265	+	6.51003i	−0.0389930	+	0.243461i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.4793			−1.09939			−0.549697	−	0.835364i	$-0.685257\pi$
$719$	−29.4793			−1.09939			−0.549697	+	0.835364i	$0.685257\pi$
$720$		0			0
$721$	−15.9955			−0.595703
$722$		0			0
$723$		0			0
$724$		0			0
$725$	27.1153	+	8.91430i	1.00704	+	0.331069i
$726$		0			0
$727$		−	8.43456i		−	0.312820i	−0.987692	−	0.156410i	$-0.950008\pi$
$727$		−	8.43456i		−	0.312820i	0.987692	−	0.156410i	$-0.0499922\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.50718			0.0927314
$732$		0			0
$733$			11.5449i			0.426421i	0.977006	+	0.213210i	$0.0683920\pi$
$733$			11.5449i			0.426421i	−0.977006	+	0.213210i	$0.931608\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7.62331i			0.280808i
$738$		0			0
$739$	−14.2891			−0.525632			−0.262816	−	0.964846i	$-0.584651\pi$
$739$	−14.2891			−0.525632			−0.262816	+	0.964846i	$0.584651\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.60693i			0.0956390i	0.998856	+	0.0478195i	$0.0152272\pi$
$743$			2.60693i			0.0956390i	−0.998856	+	0.0478195i	$0.984773\pi$
$744$		0			0
$745$	45.5036	+	7.28790i	1.66712	+	0.267008i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.7857			−0.722953
$750$		0			0
$751$	−46.1277			−1.68322			−0.841612	−	0.540083i	$-0.818393\pi$
$751$	−46.1277			−1.68322			−0.841612	+	0.540083i	$0.818393\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.59819	−	1.37709i	−0.312920	−	0.0501176i
$756$		0			0
$757$			20.6229i			0.749552i	0.927115	+	0.374776i	$0.122280\pi$
$757$			20.6229i			0.749552i	−0.927115	+	0.374776i	$0.877720\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−52.6872			−1.90991			−0.954954	−	0.296752i	$-0.904096\pi$
$761$	−52.6872			−1.90991			−0.954954	+	0.296752i	$0.904096\pi$
$762$		0			0
$763$			19.6981i			0.713119i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.83035i		−	0.138306i
$768$		0			0
$769$	−31.2458			−1.12675			−0.563376	−	0.826200i	$-0.690498\pi$
$769$	−31.2458			−1.12675			−0.563376	+	0.826200i	$0.690498\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			41.0584i			1.47677i	0.674380	+	0.738385i	$0.264411\pi$
$773$			41.0584i			1.47677i	−0.674380	+	0.738385i	$0.735589\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	7.52252			0.269177
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.26355	+	20.3767i	−0.116481	+	0.727275i
$786$		0			0
$787$			26.7067i			0.951990i	0.879448	+	0.475995i	$0.157912\pi$
$787$			26.7067i			0.951990i	−0.879448	+	0.475995i	$0.842088\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.93516			0.139918
$792$		0			0
$793$		−	0.241231i		−	0.00856636i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			15.8293i			0.560703i	0.959897	+	0.280352i	$0.0904511\pi$
$797$			15.8293i			0.560703i	−0.959897	+	0.280352i	$0.909549\pi$
$798$		0			0
$799$	−3.97256			−0.140539
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	0.555339i		−	0.0195975i
$804$		0			0
$805$	1.94712	+	0.311852i	0.0686268	+	0.0109913i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20.3894			−0.716852			−0.358426	−	0.933558i	$-0.616687\pi$
$809$	−20.3894			−0.716852			−0.358426	+	0.933558i	$0.616687\pi$
$810$		0			0
$811$	−35.7445			−1.25516			−0.627579	−	0.778553i	$-0.715954\pi$
$811$	−35.7445			−1.25516			−0.627579	+	0.778553i	$0.715954\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.85956	−	36.5854i	0.205251	−	1.28153i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−32.4660			−1.13307			−0.566536	−	0.824037i	$-0.691717\pi$
$821$	−32.4660			−1.13307			−0.566536	+	0.824037i	$0.691717\pi$
$822$		0			0
$823$			36.0775i			1.25758i	0.777574	+	0.628792i	$0.216450\pi$
$823$			36.0775i			1.25758i	−0.777574	+	0.628792i	$0.783550\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			10.7247i			0.372935i	0.982461	+	0.186468i	$0.0597040\pi$
$827$			10.7247i			0.372935i	−0.982461	+	0.186468i	$0.940296\pi$
$828$		0			0
$829$	−44.6206			−1.54974			−0.774868	−	0.632123i	$-0.782184\pi$
$829$	−44.6206			−1.54974			−0.774868	+	0.632123i	$0.782184\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	6.24894i		−	0.216513i
$834$		0			0
$835$	5.49495	−	34.3089i	0.190161	−	1.18731i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−34.8786			−1.20414			−0.602071	−	0.798442i	$-0.705658\pi$
$839$	−34.8786			−1.20414			−0.602071	+	0.798442i	$0.705658\pi$
$840$		0			0
$841$	3.58823			0.123732
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.20793	−	0.353624i	−0.0759551	−	0.0121650i
$846$		0			0
$847$			3.81889i			0.131219i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.67579			−0.160284
$852$		0			0
$853$			20.5673i			0.704211i	0.935960	+	0.352105i	$0.114534\pi$
$853$			20.5673i			0.704211i	−0.935960	+	0.352105i	$0.885466\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	36.7060i		−	1.25385i	−0.779079	−	0.626926i	$-0.784313\pi$
$857$		−	36.7060i		−	1.25385i	0.779079	−	0.626926i	$-0.215687\pi$
$858$		0			0
$859$	13.2543			0.452231			0.226116	−	0.974100i	$-0.427397\pi$
$859$	13.2543			0.452231			0.226116	+	0.974100i	$0.427397\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	21.0966i		−	0.718138i	−0.933311	−	0.359069i	$-0.883094\pi$
$863$		−	21.0966i		−	0.718138i	0.933311	−	0.359069i	$-0.116906\pi$
$864$		0			0
$865$	−1.12134	+	7.00136i	−0.0381269	+	0.238053i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.6483			1.10752
$870$		0			0
$871$	−2.58550			−0.0876065
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.48970	−	16.4500i	0.287004	−	0.556113i
$876$		0			0
$877$		−	5.54498i		−	0.187241i	−0.995608	−	0.0936203i	$-0.970156\pi$
$877$		−	5.54498i		−	0.187241i	0.995608	−	0.0936203i	$-0.0298440\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.0378			0.506636			0.253318	−	0.967383i	$-0.418478\pi$
$881$	15.0378			0.506636			0.253318	+	0.967383i	$0.418478\pi$
$882$		0			0
$883$			6.55806i			0.220696i	0.993893	+	0.110348i	$0.0351966\pi$
$883$			6.55806i			0.220696i	−0.993893	+	0.110348i	$0.964803\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	20.3045i		−	0.681760i	−0.940107	−	0.340880i	$-0.889275\pi$
$887$		−	20.3045i		−	0.681760i	0.940107	−	0.340880i	$-0.110725\pi$
$888$		0			0
$889$	4.76335			0.159758
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	5.06524	+	0.811254i	0.169312	+	0.0271172i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−12.8819			−0.429157
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−38.8828	−	6.22751i	−1.29251	−	0.207009i
$906$		0			0
$907$			32.8791i			1.09173i	0.837873	+	0.545866i	$0.183799\pi$
$907$			32.8791i			1.09173i	−0.837873	+	0.545866i	$0.816201\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.27638			−0.274209			−0.137104	−	0.990557i	$-0.543780\pi$
$911$	−8.27638			−0.274209			−0.137104	+	0.990557i	$0.543780\pi$
$912$		0			0
$913$		−	23.3392i		−	0.772413i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			32.1157i			1.06055i
$918$		0			0
$919$	−53.4997			−1.76479			−0.882397	−	0.470506i	$-0.844071\pi$
$919$	−53.4997			−1.76479			−0.882397	+	0.470506i	$0.844071\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.55132i			0.0839778i
$924$		0			0
$925$	−13.7086	+	41.6986i	−0.450736	+	1.37104i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.9384			−0.851011			−0.425505	−	0.904956i	$-0.639904\pi$
$929$	−25.9384			−0.851011			−0.425505	+	0.904956i	$0.639904\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.52997	−	9.55268i	0.0500353	−	0.312406i
$936$		0			0
$937$		−	14.1537i		−	0.462380i	−0.972909	−	0.231190i	$-0.925738\pi$
$937$		−	14.1537i		−	0.462380i	0.972909	−	0.231190i	$-0.0742619\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.9458			−0.878406			−0.439203	−	0.898388i	$-0.644739\pi$
$941$	−26.9458			−0.878406			−0.439203	+	0.898388i	$0.644739\pi$
$942$		0			0
$943$		−	0.660380i		−	0.0215049i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			48.6570i			1.58114i	0.612371	+	0.790570i	$0.290216\pi$
$947$			48.6570i			1.58114i	−0.612371	+	0.790570i	$0.709784\pi$
$948$		0			0
$949$	0.188347			0.00611402
$950$		0			0
$951$		0			0
$952$		0			0
$953$			41.7911i			1.35375i	0.736099	+	0.676874i	$0.236666\pi$
$953$			41.7911i			1.35375i	−0.736099	+	0.676874i	$0.763334\pi$
$954$		0			0
$955$	0.233526	+	0.0374018i	0.00755673	+	0.00121029i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.592740			−0.0191406
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.15954	+	7.23987i	−0.0373270	+	0.233060i
$966$		0			0
$967$			0.524916i			0.0168802i	0.999964	+	0.00844008i	$0.00268659\pi$
$967$			0.524916i			0.0168802i	−0.999964	+	0.00844008i	$0.997313\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.4036			−1.93844			−0.969222	−	0.246189i	$-0.920822\pi$
$971$	−60.4036			−1.93844			−0.969222	+	0.246189i	$0.920822\pi$
$972$		0			0
$973$		−	8.08305i		−	0.259131i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			41.6412i			1.33222i	0.745853	+	0.666110i	$0.232042\pi$
$977$			41.6412i			1.33222i	−0.745853	+	0.666110i	$0.767958\pi$
$978$		0			0
$979$	47.0829			1.50478
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	58.6498i		−	1.87064i	−0.353806	−	0.935319i	$-0.615113\pi$
$983$		−	58.6498i		−	1.87064i	0.353806	−	0.935319i	$-0.384887\pi$
$984$		0			0
$985$	2.12739	−	13.2828i	0.0677844	−	0.423227i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.910041			−0.0289376
$990$		0			0
$991$	−59.2987			−1.88369			−0.941843	−	0.336054i	$-0.890907\pi$
$991$	−59.2987			−1.88369			−0.941843	+	0.336054i	$0.890907\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.7725	+	1.88549i	0.373213	+	0.0597741i
$996$		0			0
$997$		−	36.9170i		−	1.16917i	−0.811331	−	0.584587i	$-0.801257\pi$
$997$		−	36.9170i		−	1.16917i	0.811331	−	0.584587i	$-0.198743\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.g.2809.8		8
3.2	odd	2		1560.2.l.d.1249.1	✓	8
5.4	even	2	inner	4680.2.l.g.2809.7		8
12.11	even	2		3120.2.l.n.1249.5		8
15.2	even	4		7800.2.a.bt.1.4		4
15.8	even	4		7800.2.a.by.1.1		4
15.14	odd	2		1560.2.l.d.1249.5	yes	8
60.59	even	2		3120.2.l.n.1249.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.1	✓	8	3.2	odd	2
1560.2.l.d.1249.5	yes	8	15.14	odd	2
3120.2.l.n.1249.1		8	60.59	even	2
3120.2.l.n.1249.5		8	12.11	even	2
4680.2.l.g.2809.7		8	5.4	even	2	inner
4680.2.l.g.2809.8		8	1.1	even	1	trivial
7800.2.a.bt.1.4		4	15.2	even	4
7800.2.a.by.1.1		4	15.8	even	4

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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.20793 + 0.353624i) q^{5} -1.65573i q^{7} +O(q^{10})\)	\(q+(2.20793 + 0.353624i) q^{5} -1.65573i q^{7} +2.94848 q^{11} +1.00000i q^{13} -1.46738i q^{17} +0.532621i q^{23} +(4.74990 + 1.56155i) q^{25} +5.70861 q^{29} +(0.585504 - 3.65573i) q^{35} +8.77883i q^{37} -1.23987 q^{41} +1.70861i q^{43} -2.70725i q^{47} +4.25857 q^{49} -8.77883i q^{53} +(6.51003 + 1.04265i) q^{55} -3.83035 q^{59} -0.241231 q^{61} +(-0.353624 + 2.20793i) q^{65} +2.58550i q^{67} +2.55132 q^{71} -0.188347i q^{73} -4.88187i q^{77} +11.0729 q^{79} -7.91566i q^{83} +(0.518900 - 3.23987i) q^{85} +15.9685 q^{89} +1.65573 q^{91} +16.8641i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5}+O(q^{10}) \) 8 * q + 2 * q^5	\( 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100}) \) 8 * q + 2 * q^5 - 2 * q^11 + 16 * q^29 + 8 * q^35 - 14 * q^41 - 18 * q^49 + 10 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 - 30 * q^71 + 2 * q^79 + 24 * q^85 + 18 * q^89 - 14 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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