LMFDB - Embedded newform 4680.2.l.g.2809.7

Newspace parameters

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

Self dual:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.57815240704.2
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			2809.7
Root			$2.20793 + 2.20793i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.g.2809.8

$q$-expansion

$f(q)$	$=$	$q+(2.20793 - 0.353624i) q^{5} +1.65573i q^{7} +O(q^{10})$	$q+(2.20793 - 0.353624i) q^{5} +1.65573i q^{7} +2.94848 q^{11} -1.00000i q^{13} +1.46738i q^{17} -0.532621i q^{23} +(4.74990 - 1.56155i) q^{25} +5.70861 q^{29} +(0.585504 + 3.65573i) q^{35} -8.77883i q^{37} -1.23987 q^{41} -1.70861i q^{43} +2.70725i q^{47} +4.25857 q^{49} +8.77883i q^{53} +(6.51003 - 1.04265i) q^{55} -3.83035 q^{59} -0.241231 q^{61} +(-0.353624 - 2.20793i) q^{65} -2.58550i q^{67} +2.55132 q^{71} +0.188347i q^{73} +4.88187i q^{77} +11.0729 q^{79} +7.91566i q^{83} +(0.518900 + 3.23987i) q^{85} +15.9685 q^{89} +1.65573 q^{91} -16.8641i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.101301\pi$
$7$			1.65573i			0.625806i	−0.949785	+	0.312903i	$0.898699\pi$
$8$		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	$0.0569452\pi$
$17$			1.46738i			0.355892i	−0.984040	+	0.177946i	$0.943055\pi$
$18$		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	$-0.982315\pi$
$23$		−	0.532621i		−	0.111059i	0.998457	−	0.0555296i	$-0.0176847\pi$
$24$		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	$-0.743400\pi$
$37$		−	8.77883i		−	1.44323i	0.692294	−	0.721616i	$-0.256600\pi$
$38$		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	$-0.958412\pi$
$43$		−	1.70861i		−	0.260561i	0.991477	−	0.130280i	$-0.0415877\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.70725i			0.394893i	0.980314	+	0.197446i	$0.0632648\pi$
$47$			2.70725i			0.394893i	−0.980314	+	0.197446i	$0.936735\pi$
$48$		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	$0.206001\pi$
$53$			8.77883i			1.20587i	−0.797792	+	0.602933i	$0.793999\pi$
$54$		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	$-0.949516\pi$
$67$		−	2.58550i		−	0.315870i	0.987450	−	0.157935i	$-0.0504836\pi$
$68$		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	$0.00350855\pi$
$73$			0.188347i			0.0220444i	−0.999939	+	0.0110222i	$0.996491\pi$
$74$		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	$0.143050\pi$
$83$			7.91566i			0.868856i	−0.900706	+	0.434428i	$0.856950\pi$
$84$		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	$-0.672852\pi$
$97$		−	16.8641i		−	1.71229i	0.516733	−	0.856147i	$-0.327148\pi$
$98$		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	$0.157895\pi$
$103$			9.66071i			0.951898i	−0.879473	+	0.475949i	$0.842105\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.9498i			1.15523i	0.816308	+	0.577617i	$0.196017\pi$
$107$			11.9498i			1.15523i	−0.816308	+	0.577617i	$0.803983\pi$
$108$		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	$-0.964341\pi$
$113$		−	2.37669i		−	0.223581i	0.993732	−	0.111790i	$-0.0356585\pi$
$114$		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	$-0.959259\pi$
$127$		−	2.87689i		−	0.255283i	0.991820	−	0.127642i	$-0.0407407\pi$
$128$		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	$0.00486802\pi$
$137$			0.357994i			0.0305855i	−0.999883	+	0.0152927i	$0.995132\pi$
$138$		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	$-0.879949\pi$
$157$		−	9.22887i		−	0.736544i	0.929718	−	0.368272i	$-0.120051\pi$
$158$		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	$0.224784\pi$
$163$			16.5700i			1.29786i	−0.760846	+	0.648932i	$0.775216\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.5390i			1.20244i	0.799083	+	0.601221i	$0.205319\pi$
$167$			15.5390i			1.20244i	−0.799083	+	0.601221i	$0.794681\pi$
$168$		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	$-0.961536\pi$
$173$		−	3.17101i		−	0.241087i	0.992708	−	0.120544i	$-0.0384638\pi$
$174$		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	$-0.962347\pi$
$193$		−	3.27903i		−	0.236030i	0.993012	−	0.118015i	$-0.0376531\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.01598i			0.428621i	0.976766	+	0.214310i	$0.0687504\pi$
$197$			6.01598i			0.428621i	−0.976766	+	0.214310i	$0.931250\pi$
$198$		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	$0.231435\pi$
$223$			19.8518i			1.32937i	−0.747122	+	0.664687i	$0.768565\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	6.85042i		−	0.454678i	−0.973816	−	0.227339i	$-0.926997\pi$
$227$		−	6.85042i		−	0.454678i	0.973816	−	0.227339i	$-0.0730026\pi$
$228$		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	$-0.843587\pi$
$233$		−	14.4049i		−	0.943694i	0.881680	−	0.471847i	$-0.156413\pi$
$234$		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	$0.0658207\pi$
$257$			6.58278i			0.410623i	−0.978697	+	0.205311i	$0.934179\pi$
$258$		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	$-0.958235\pi$
$263$		−	4.24349i		−	0.261665i	0.991405	−	0.130832i	$-0.0417649\pi$
$264$		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	$-0.819540\pi$
$277$		−	17.8765i		−	1.07409i	0.843552	−	0.537047i	$-0.180460\pi$
$278$		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	$-0.864077\pi$
$283$		−	13.9344i		−	0.828312i	0.910206	−	0.414156i	$-0.135923\pi$
$284$		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	$-0.891856\pi$
$293$		−	11.4085i		−	0.666491i	0.942840	−	0.333245i	$-0.108144\pi$
$294$		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	$0.0227999\pi$
$307$			2.50790i			0.143134i	−0.997436	+	0.0715668i	$0.977200\pi$
$308$		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	$-0.849756\pi$
$313$		−	16.0880i		−	0.909349i	0.890658	−	0.454675i	$-0.150244\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.25263i			0.0703545i	0.999381	+	0.0351772i	$0.0111996\pi$
$317$			1.25263i			0.0703545i	−0.999381	+	0.0351772i	$0.988800\pi$
$318$		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	$0.205849\pi$
$337$			22.1227i			1.20510i	−0.798081	+	0.602550i	$0.794151\pi$
$338$		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	$0.164811\pi$
$347$			18.4395i			0.989886i	−0.868925	+	0.494943i	$0.835189\pi$
$348$		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	$0.0100797\pi$
$353$			1.18971i			0.0633219i	−0.999499	+	0.0316609i	$0.989920\pi$
$354$		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	$-0.909770\pi$
$367$		−	10.7160i		−	0.559370i	0.960092	−	0.279685i	$-0.0902300\pi$
$368$		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	$0.201685\pi$
$373$			22.8691i			1.18412i	−0.805895	+	0.592059i	$0.798315\pi$
$374$		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	$-0.564801\pi$
$383$		−	38.3325i		−	1.95870i	0.202176	−	0.979349i	$-0.435199\pi$
$384$		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	$0.00537654\pi$
$397$			0.673065i			0.0337802i	−0.999857	+	0.0168901i	$0.994623\pi$
$398$		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	$-0.796303\pi$
$433$		−	24.8514i		−	1.19428i	0.802137	−	0.597141i	$-0.203697\pi$
$434$		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	$-0.786196\pi$
$443$		−	26.1961i		−	1.24461i	0.782774	−	0.622306i	$-0.213804\pi$
$444$		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	$-0.986114\pi$
$457$		−	1.86454i		−	0.0872193i	0.999049	−	0.0436097i	$-0.0138858\pi$
$458$		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	$0.236201\pi$
$463$			29.0829i			1.35160i	−0.737086	+	0.675799i	$0.763799\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.5727i			0.489248i	0.969618	+	0.244624i	$0.0786646\pi$
$467$			10.5727i			0.489248i	−0.969618	+	0.244624i	$0.921335\pi$
$468$		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	$-0.998300\pi$
$487$		−	0.235782i		−	0.0106843i	0.999986	−	0.00534215i	$-0.00170047\pi$
$488$		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	$0.213428\pi$
$503$			27.8722i			1.24276i	−0.783509	+	0.621381i	$0.786572\pi$
$504$		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	$0.00105049\pi$
$523$			0.150946i			0.00660040i	−0.999995	+	0.00330020i	$0.998950\pi$
$524$		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	$0.152885\pi$
$547$			21.6128i			0.924097i	−0.886855	+	0.462048i	$0.847115\pi$
$548$		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	$0.277812\pi$
$557$			36.1619i			1.53223i	−0.642705	+	0.766114i	$0.722188\pi$
$558$		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	$0.0610947\pi$
$563$			9.05248i			0.381517i	−0.981637	+	0.190758i	$0.938905\pi$
$564$		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	$-0.809443\pi$
$577$		−	27.0729i		−	1.12706i	0.826095	−	0.563531i	$-0.190557\pi$
$578$		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	$-0.853360\pi$
$587$		−	21.5417i		−	0.889121i	0.895749	−	0.444560i	$-0.146640\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			0.542087i			0.0222608i	0.999938	+	0.0111304i	$0.00354300\pi$
$593$			0.542087i			0.0222608i	−0.999938	+	0.0111304i	$0.996457\pi$
$594$		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	$-0.663758\pi$
$607$		−	42.8966i		−	1.74112i	0.492064	−	0.870559i	$-0.336242\pi$
$608$		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	$-0.705866\pi$
$613$		−	39.5175i		−	1.59610i	0.602594	−	0.798048i	$-0.294134\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			32.2951i			1.30015i	0.759870	+	0.650075i	$0.225263\pi$
$617$			32.2951i			1.30015i	−0.759870	+	0.650075i	$0.774737\pi$
$618$		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	$-0.888580\pi$
$643$		−	17.3918i		−	0.685865i	0.939360	−	0.342932i	$-0.111420\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			34.1331i			1.34191i	0.741497	+	0.670956i	$0.234116\pi$
$647$			34.1331i			1.34191i	−0.741497	+	0.670956i	$0.765884\pi$
$648$		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	$0.132250\pi$
$653$			20.6284i			0.807250i	−0.914925	+	0.403625i	$0.867750\pi$
$654$		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	$-0.920885\pi$
$673$		−	12.7634i		−	0.491991i	0.969271	−	0.245996i	$-0.0791148\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			15.9251i			0.612052i	0.952023	+	0.306026i	$0.0989995\pi$
$677$			15.9251i			0.612052i	−0.952023	+	0.306026i	$0.901001\pi$
$678$		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	$0.0553882\pi$
$683$			9.04927i			0.346261i	−0.984899	+	0.173130i	$0.944612\pi$
$684$		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	$0.0499922\pi$
$727$			8.43456i			0.312820i	−0.987692	+	0.156410i	$0.950008\pi$
$728$		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	$-0.931608\pi$
$733$		−	11.5449i		−	0.426421i	0.977006	−	0.213210i	$-0.0683920\pi$
$734$		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	$-0.984773\pi$
$743$		−	2.60693i		−	0.0956390i	0.998856	−	0.0478195i	$-0.0152272\pi$
$744$		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	$-0.877720\pi$
$757$		−	20.6229i		−	0.749552i	0.927115	−	0.374776i	$-0.122280\pi$
$758$		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	$-0.735589\pi$
$773$		−	41.0584i		−	1.47677i	0.674380	−	0.738385i	$-0.264411\pi$
$774$		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	$-0.842088\pi$
$787$		−	26.7067i		−	0.951990i	0.879448	−	0.475995i	$-0.157912\pi$
$788$		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	$-0.909549\pi$
$797$		−	15.8293i		−	0.560703i	0.959897	−	0.280352i	$-0.0904511\pi$
$798$		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

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

Related objects

Downloads

Learn more