LMFDB - Embedded newform 2880.2.bl.c.431.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(431,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))

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

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

chi := DirichletCharacter("2880.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bl (of order $4$, degree $2$, not 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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			431.6
Character	$\chi$	$=$	2880.431
Dual form			2880.2.bl.c.1871.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} -1.44621 q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} -1.44621 q^{7} +(-1.89671 + 1.89671i) q^{11} +(0.905090 + 0.905090i) q^{13} -6.42494i q^{17} +(2.02287 - 2.02287i) q^{19} +4.25432i q^{23} +1.00000i q^{25} +(2.71899 - 2.71899i) q^{29} +4.19182i q^{31} +(1.02262 + 1.02262i) q^{35} +(0.0486719 - 0.0486719i) q^{37} -8.11147 q^{41} +(3.17721 + 3.17721i) q^{43} -2.47830 q^{47} -4.90848 q^{49} +(1.67784 + 1.67784i) q^{53} +2.68236 q^{55} +(-4.69563 + 4.69563i) q^{59} +(2.69937 + 2.69937i) q^{61} -1.27999i q^{65} +(-8.13574 + 8.13574i) q^{67} +13.8354i q^{71} +5.23815i q^{73} +(2.74305 - 2.74305i) q^{77} +6.59850i q^{79} +(-12.8367 - 12.8367i) q^{83} +(-4.54312 + 4.54312i) q^{85} -17.0498 q^{89} +(-1.30895 - 1.30895i) q^{91} -2.86077 q^{95} +12.8861 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) $ 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$	−1.44621			−0.546616			−0.273308	−	0.961927i	$-0.588118\pi$
$7$	−1.44621			−0.546616			−0.273308	+	0.961927i	$0.588118\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.89671	+	1.89671i	−0.571881	+	0.571881i	−0.932654	−	0.360773i	$-0.882513\pi$
$11$	−1.89671	+	1.89671i	−0.571881	+	0.571881i	0.360773	+	0.932654i	$0.382513\pi$
$12$		0			0
$13$	0.905090	+	0.905090i	0.251027	+	0.251027i	0.821392	−	0.570365i	$-0.193198\pi$
$13$	0.905090	+	0.905090i	0.251027	+	0.251027i	−0.570365	+	0.821392i	$0.693198\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.42494i		−	1.55828i	−0.626851	−	0.779139i	$-0.715657\pi$
$17$		−	6.42494i		−	1.55828i	0.626851	−	0.779139i	$-0.284343\pi$
$18$		0			0
$19$	2.02287	−	2.02287i	0.464078	−	0.464078i	−0.435912	−	0.899990i	$-0.643574\pi$
$19$	2.02287	−	2.02287i	0.464078	−	0.464078i	0.899990	+	0.435912i	$0.143574\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.25432i			0.887087i	0.896253	+	0.443543i	$0.146279\pi$
$23$			4.25432i			0.887087i	−0.896253	+	0.443543i	$0.853721\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.71899	−	2.71899i	0.504903	−	0.504903i	−0.408055	−	0.912958i	$-0.633793\pi$
$29$	2.71899	−	2.71899i	0.504903	−	0.504903i	0.912958	+	0.408055i	$0.133793\pi$
$30$		0			0
$31$			4.19182i			0.752873i	0.926442	+	0.376437i	$0.122851\pi$
$31$			4.19182i			0.752873i	−0.926442	+	0.376437i	$0.877149\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.02262	+	1.02262i	0.172855	+	0.172855i
$36$		0			0
$37$	0.0486719	−	0.0486719i	0.00800162	−	0.00800162i	−0.703095	−	0.711096i	$-0.748199\pi$
$37$	0.0486719	−	0.0486719i	0.00800162	−	0.00800162i	0.711096	+	0.703095i	$0.248199\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.11147			−1.26680			−0.633399	−	0.773825i	$-0.718341\pi$
$41$	−8.11147			−1.26680			−0.633399	+	0.773825i	$0.718341\pi$
$42$		0			0
$43$	3.17721	+	3.17721i	0.484520	+	0.484520i	0.906572	−	0.422052i	$-0.138690\pi$
$43$	3.17721	+	3.17721i	0.484520	+	0.484520i	−0.422052	+	0.906572i	$0.638690\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.47830			−0.361497			−0.180749	−	0.983529i	$-0.557852\pi$
$47$	−2.47830			−0.361497			−0.180749	+	0.983529i	$0.557852\pi$
$48$		0			0
$49$	−4.90848			−0.701211
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.67784	+	1.67784i	0.230468	+	0.230468i	0.812888	−	0.582420i	$-0.197894\pi$
$53$	1.67784	+	1.67784i	0.230468	+	0.230468i	−0.582420	+	0.812888i	$0.697894\pi$
$54$		0			0
$55$	2.68236			0.361689
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.69563	+	4.69563i	−0.611319	+	0.611319i	−0.943290	−	0.331971i	$-0.892286\pi$
$59$	−4.69563	+	4.69563i	−0.611319	+	0.611319i	0.331971	+	0.943290i	$0.392286\pi$
$60$		0			0
$61$	2.69937	+	2.69937i	0.345619	+	0.345619i	0.858475	−	0.512856i	$-0.171412\pi$
$61$	2.69937	+	2.69937i	0.345619	+	0.345619i	−0.512856	+	0.858475i	$0.671412\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.27999i		−	0.158763i
$66$		0			0
$67$	−8.13574	+	8.13574i	−0.993939	+	0.993939i	−0.999982	−	0.00604256i	$-0.998077\pi$
$67$	−8.13574	+	8.13574i	−0.993939	+	0.993939i	0.00604256	+	0.999982i	$0.498077\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.8354i			1.64196i	0.570956	+	0.820981i	$0.306573\pi$
$71$			13.8354i			1.64196i	−0.570956	+	0.820981i	$0.693427\pi$
$72$		0			0
$73$			5.23815i			0.613079i	0.951858	+	0.306540i	$0.0991712\pi$
$73$			5.23815i			0.613079i	−0.951858	+	0.306540i	$0.900829\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.74305	−	2.74305i	0.312599	−	0.312599i
$78$		0			0
$79$			6.59850i			0.742390i	0.928555	+	0.371195i	$0.121052\pi$
$79$			6.59850i			0.742390i	−0.928555	+	0.371195i	$0.878948\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.8367	−	12.8367i	−1.40901	−	1.40901i	−0.765092	−	0.643920i	$-0.777307\pi$
$83$	−12.8367	−	12.8367i	−1.40901	−	1.40901i	−0.643920	−	0.765092i	$-0.722693\pi$
$84$		0			0
$85$	−4.54312	+	4.54312i	−0.492771	+	0.492771i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−17.0498			−1.80728			−0.903640	−	0.428293i	$-0.859115\pi$
$89$	−17.0498			−1.80728			−0.903640	+	0.428293i	$0.859115\pi$
$90$		0			0
$91$	−1.30895	−	1.30895i	−0.137215	−	0.137215i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.86077			−0.293509
$96$		0			0
$97$	12.8861			1.30838			0.654191	−	0.756329i	$-0.273009\pi$
$97$	12.8861			1.30838			0.654191	+	0.756329i	$0.273009\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.15930	−	6.15930i	−0.612873	−	0.612873i	0.330821	−	0.943694i	$-0.392674\pi$
$101$	−6.15930	−	6.15930i	−0.612873	−	0.612873i	−0.943694	+	0.330821i	$0.892674\pi$
$102$		0			0
$103$	−8.83779			−0.870813			−0.435407	−	0.900234i	$-0.643395\pi$
$103$	−8.83779			−0.870813			−0.435407	+	0.900234i	$0.643395\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.78200	−	7.78200i	0.752314	−	0.752314i	−0.222596	−	0.974911i	$-0.571453\pi$
$107$	7.78200	−	7.78200i	0.752314	−	0.752314i	0.974911	+	0.222596i	$0.0714532\pi$
$108$		0			0
$109$	11.8439	+	11.8439i	1.13444	+	1.13444i	0.989431	+	0.145006i	$0.0463201\pi$
$109$	11.8439	+	11.8439i	1.13444	+	1.13444i	0.145006	+	0.989431i	$0.453680\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			15.4074i			1.44941i	0.689061	+	0.724703i	$0.258023\pi$
$113$			15.4074i			1.44941i	−0.689061	+	0.724703i	$0.741977\pi$
$114$		0			0
$115$	3.00826	−	3.00826i	0.280521	−	0.280521i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			9.29181i			0.851779i
$120$		0			0
$121$			3.80495i			0.345904i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$			16.2998i			1.44637i	0.690653	+	0.723186i	$0.257323\pi$
$127$			16.2998i			1.44637i	−0.690653	+	0.723186i	$0.742677\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.51424	+	1.51424i	0.132300	+	0.132300i	0.770156	−	0.637856i	$-0.220178\pi$
$131$	1.51424	+	1.51424i	0.132300	+	0.132300i	−0.637856	+	0.770156i	$0.720178\pi$
$132$		0			0
$133$	−2.92549	+	2.92549i	−0.253672	+	0.253672i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.75908			0.150288			0.0751441	−	0.997173i	$-0.476058\pi$
$137$	1.75908			0.150288			0.0751441	+	0.997173i	$0.476058\pi$
$138$		0			0
$139$	3.66959	+	3.66959i	0.311251	+	0.311251i	0.845394	−	0.534143i	$-0.179366\pi$
$139$	3.66959	+	3.66959i	0.311251	+	0.311251i	−0.534143	+	0.845394i	$0.679366\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.43339			−0.287115
$144$		0			0
$145$	−3.84523			−0.319329
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.34044	+	5.34044i	0.437506	+	0.437506i	0.891172	−	0.453666i	$-0.149884\pi$
$149$	5.34044	+	5.34044i	0.437506	+	0.437506i	−0.453666	+	0.891172i	$0.649884\pi$
$150$		0			0
$151$	−8.00878			−0.651745			−0.325873	−	0.945414i	$-0.605658\pi$
$151$	−8.00878			−0.651745			−0.325873	+	0.945414i	$0.605658\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.96407	−	2.96407i	0.238079	−	0.238079i
$156$		0			0
$157$	14.4335	+	14.4335i	1.15192	+	1.15192i	0.986168	+	0.165747i	$0.0530034\pi$
$157$	14.4335	+	14.4335i	1.15192	+	1.15192i	0.165747	+	0.986168i	$0.446997\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.15263i		−	0.484896i
$162$		0			0
$163$	0.399016	−	0.399016i	0.0312534	−	0.0312534i	−0.691307	−	0.722561i	$-0.742965\pi$
$163$	0.399016	−	0.399016i	0.0312534	−	0.0312534i	0.722561	+	0.691307i	$0.242965\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.278914i		−	0.0215830i	−0.999942	−	0.0107915i	$-0.996565\pi$
$167$		−	0.278914i		−	0.0215830i	0.999942	−	0.0107915i	$-0.00343511\pi$
$168$		0			0
$169$		−	11.3616i		−	0.873971i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.28991	−	2.28991i	0.174099	−	0.174099i	−0.614679	−	0.788778i	$-0.710714\pi$
$173$	2.28991	−	2.28991i	0.174099	−	0.174099i	0.788778	+	0.614679i	$0.210714\pi$
$174$		0			0
$175$		−	1.44621i		−	0.109323i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.469151	−	0.469151i	−0.0350660	−	0.0350660i	0.689356	−	0.724422i	$-0.257893\pi$
$179$	−0.469151	−	0.469151i	−0.0350660	−	0.0350660i	−0.724422	+	0.689356i	$0.757893\pi$
$180$		0			0
$181$	−10.2944	+	10.2944i	−0.765176	+	0.765176i	−0.977253	−	0.212077i	$-0.931977\pi$
$181$	−10.2944	+	10.2944i	−0.765176	+	0.765176i	0.212077	+	0.977253i	$0.431977\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.0688325			−0.00506067
$186$		0			0
$187$	12.1863	+	12.1863i	0.891149	+	0.891149i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.60754			0.622820			0.311410	−	0.950276i	$-0.399199\pi$
$191$	8.60754			0.622820			0.311410	+	0.950276i	$0.399199\pi$
$192$		0			0
$193$	4.82410			0.347246			0.173623	−	0.984812i	$-0.444453\pi$
$193$	4.82410			0.347246			0.173623	+	0.984812i	$0.444453\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.502615	+	0.502615i	0.0358098	+	0.0358098i	0.724785	−	0.688975i	$-0.241939\pi$
$197$	0.502615	+	0.502615i	0.0358098	+	0.0358098i	−0.688975	+	0.724785i	$0.741939\pi$
$198$		0			0
$199$	−25.2595			−1.79060			−0.895299	−	0.445466i	$-0.853038\pi$
$199$	−25.2595			−1.79060			−0.895299	+	0.445466i	$0.853038\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.93222	+	3.93222i	−0.275988	+	0.275988i
$204$		0			0
$205$	5.73567	+	5.73567i	0.400597	+	0.400597i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			7.67361i			0.530795i
$210$		0			0
$211$	6.28367	−	6.28367i	0.432586	−	0.432586i	−0.456921	−	0.889507i	$-0.651048\pi$
$211$	6.28367	−	6.28367i	0.432586	−	0.432586i	0.889507	+	0.456921i	$0.151048\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	4.49325i		−	0.306437i
$216$		0			0
$217$		−	6.06225i		−	0.411532i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.81515	−	5.81515i	0.391169	−	0.391169i
$222$		0			0
$223$		−	3.63497i		−	0.243415i	−0.992566	−	0.121708i	$-0.961163\pi$
$223$		−	3.63497i		−	0.243415i	0.992566	−	0.121708i	$-0.0388370\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.1488	−	19.1488i	−1.27095	−	1.27095i	−0.945590	−	0.325360i	$-0.894514\pi$
$227$	−19.1488	−	19.1488i	−1.27095	−	1.27095i	−0.325360	−	0.945590i	$-0.605486\pi$
$228$		0			0
$229$	1.43515	−	1.43515i	0.0948377	−	0.0948377i	−0.658096	−	0.752934i	$-0.728638\pi$
$229$	1.43515	−	1.43515i	0.0948377	−	0.0948377i	0.752934	+	0.658096i	$0.228638\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.9161			−1.17373			−0.586863	−	0.809686i	$-0.699637\pi$
$233$	−17.9161			−1.17373			−0.586863	+	0.809686i	$0.699637\pi$
$234$		0			0
$235$	1.75242	+	1.75242i	0.114315	+	0.114315i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−19.7782			−1.27934			−0.639672	−	0.768648i	$-0.720930\pi$
$239$	−19.7782			−1.27934			−0.639672	+	0.768648i	$0.720930\pi$
$240$		0			0
$241$	−12.5869			−0.810795			−0.405397	−	0.914141i	$-0.632867\pi$
$241$	−12.5869			−0.810795			−0.405397	+	0.914141i	$0.632867\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.47082	+	3.47082i	0.221742	+	0.221742i
$246$		0			0
$247$	3.66176			0.232992
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.70399	−	7.70399i	0.486271	−	0.486271i	−0.420856	−	0.907127i	$-0.638270\pi$
$251$	7.70399	−	7.70399i	0.486271	−	0.486271i	0.907127	+	0.420856i	$0.138270\pi$
$252$		0			0
$253$	−8.06923	−	8.06923i	−0.507308	−	0.507308i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.7818i			1.17158i	0.810463	+	0.585789i	$0.199215\pi$
$257$			18.7818i			1.17158i	−0.810463	+	0.585789i	$0.800785\pi$
$258$		0			0
$259$	−0.0703898	+	0.0703898i	−0.00437381	+	0.00437381i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1.41951i			0.0875307i	0.999042	+	0.0437654i	$0.0139354\pi$
$263$			1.41951i			0.0875307i	−0.999042	+	0.0437654i	$0.986065\pi$
$264$		0			0
$265$		−	2.37282i		−	0.145761i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.8497	+	15.8497i	−0.966376	+	0.966376i	−0.999453	−	0.0330772i	$-0.989469\pi$
$269$	−15.8497	+	15.8497i	−0.966376	+	0.966376i	0.0330772	+	0.999453i	$0.489469\pi$
$270$		0			0
$271$		−	22.8614i		−	1.38873i	−0.719623	−	0.694365i	$-0.755685\pi$
$271$		−	22.8614i		−	1.38873i	0.719623	−	0.694365i	$-0.244315\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.89671	−	1.89671i	−0.114376	−	0.114376i
$276$		0			0
$277$	3.31509	−	3.31509i	0.199185	−	0.199185i	−0.600466	−	0.799650i	$-0.705018\pi$
$277$	3.31509	−	3.31509i	0.199185	−	0.199185i	0.799650	+	0.600466i	$0.205018\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−29.7563			−1.77511			−0.887557	−	0.460697i	$-0.847599\pi$
$281$	−29.7563			−1.77511			−0.887557	+	0.460697i	$0.847599\pi$
$282$		0			0
$283$	7.55871	+	7.55871i	0.449319	+	0.449319i	0.895128	−	0.445809i	$-0.147084\pi$
$283$	7.55871	+	7.55871i	0.449319	+	0.449319i	−0.445809	+	0.895128i	$0.647084\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.7309			0.692452
$288$		0			0
$289$	−24.2799			−1.42823
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.1816	+	13.1816i	0.770079	+	0.770079i	0.978120	−	0.208041i	$-0.0667089\pi$
$293$	13.1816	+	13.1816i	0.770079	+	0.770079i	−0.208041	+	0.978120i	$0.566709\pi$
$294$		0			0
$295$	6.64062			0.386632
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.85054	+	3.85054i	−0.222682	+	0.222682i
$300$		0			0
$301$	−4.59491	−	4.59491i	−0.264846	−	0.264846i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.81749i		−	0.218589i
$306$		0			0
$307$	14.1255	−	14.1255i	0.806185	−	0.806185i	−0.177869	−	0.984054i	$-0.556920\pi$
$307$	14.1255	−	14.1255i	0.806185	−	0.806185i	0.984054	+	0.177869i	$0.0569203\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.9415i			1.18748i	0.804656	+	0.593741i	$0.202350\pi$
$311$			20.9415i			1.18748i	−0.804656	+	0.593741i	$0.797650\pi$
$312$		0			0
$313$		−	5.02307i		−	0.283920i	−0.989872	−	0.141960i	$-0.954659\pi$
$313$		−	5.02307i		−	0.283920i	0.989872	−	0.141960i	$-0.0453405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	23.9908	−	23.9908i	1.34746	−	1.34746i	0.459040	−	0.888416i	$-0.348193\pi$
$317$	23.9908	−	23.9908i	1.34746	−	1.34746i	0.888416	−	0.459040i	$-0.151807\pi$
$318$		0			0
$319$			10.3143i			0.577489i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.9968	−	12.9968i	−0.723162	−	0.723162i
$324$		0			0
$325$	−0.905090	+	0.905090i	−0.0502054	+	0.0502054i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.58414			0.197600
$330$		0			0
$331$	23.2892	+	23.2892i	1.28009	+	1.28009i	0.940614	+	0.339479i	$0.110251\pi$
$331$	23.2892	+	23.2892i	1.28009	+	1.28009i	0.339479	+	0.940614i	$0.389749\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.5057			0.628622
$336$		0			0
$337$	13.4580			0.733106			0.366553	−	0.930397i	$-0.380538\pi$
$337$	13.4580			0.733106			0.366553	+	0.930397i	$0.380538\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.95069	−	7.95069i	−0.430554	−	0.430554i
$342$		0			0
$343$	17.2222			0.929909
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.9661	−	16.9661i	0.910787	−	0.910787i	−0.0855473	−	0.996334i	$-0.527264\pi$
$347$	16.9661	−	16.9661i	0.910787	−	0.910787i	0.996334	+	0.0855473i	$0.0272639\pi$
$348$		0			0
$349$	−11.9455	−	11.9455i	−0.639426	−	0.639426i	0.310988	−	0.950414i	$-0.399340\pi$
$349$	−11.9455	−	11.9455i	−0.639426	−	0.639426i	−0.950414	+	0.310988i	$0.899340\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			7.28622i			0.387806i	0.981021	+	0.193903i	$0.0621148\pi$
$353$			7.28622i			0.387806i	−0.981021	+	0.193903i	$0.937885\pi$
$354$		0			0
$355$	9.78311	−	9.78311i	0.519234	−	0.519234i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.9889i		−	0.843861i	−0.906628	−	0.421931i	$-0.861353\pi$
$359$		−	15.9889i		−	0.843861i	0.906628	−	0.421931i	$-0.138647\pi$
$360$		0			0
$361$			10.8160i			0.569263i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.70393	−	3.70393i	0.193873	−	0.193873i
$366$		0			0
$367$			23.1759i			1.20977i	0.796311	+	0.604887i	$0.206782\pi$
$367$			23.1759i			1.20977i	−0.796311	+	0.604887i	$0.793218\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.42650	−	2.42650i	−0.125978	−	0.125978i
$372$		0			0
$373$	−21.1943	+	21.1943i	−1.09740	+	1.09740i	−0.102685	+	0.994714i	$0.532744\pi$
$373$	−21.1943	+	21.1943i	−1.09740	+	1.09740i	−0.994714	+	0.102685i	$0.967256\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.92185			0.253488
$378$		0			0
$379$	−17.0350	−	17.0350i	−0.875031	−	0.875031i	0.117984	−	0.993015i	$-0.462357\pi$
$379$	−17.0350	−	17.0350i	−0.875031	−	0.875031i	−0.993015	+	0.117984i	$0.962357\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.53775			0.334064			0.167032	−	0.985952i	$-0.446582\pi$
$383$	6.53775			0.334064			0.167032	+	0.985952i	$0.446582\pi$
$384$		0			0
$385$	−3.87925			−0.197705
$386$		0			0
$387$		0			0
$388$		0			0
$389$	17.4230	+	17.4230i	0.883379	+	0.883379i	0.993876	−	0.110497i	$-0.0352444\pi$
$389$	17.4230	+	17.4230i	0.883379	+	0.883379i	−0.110497	+	0.993876i	$0.535244\pi$
$390$		0			0
$391$	27.3338			1.38233
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.66585	−	4.66585i	0.234764	−	0.234764i
$396$		0			0
$397$	4.56336	+	4.56336i	0.229028	+	0.229028i	0.812287	−	0.583258i	$-0.198222\pi$
$397$	4.56336	+	4.56336i	0.229028	+	0.229028i	−0.583258	+	0.812287i	$0.698222\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.11274i			0.0555674i	0.999614	+	0.0277837i	$0.00884496\pi$
$401$			1.11274i			0.0555674i	−0.999614	+	0.0277837i	$0.991155\pi$
$402$		0			0
$403$	−3.79398	+	3.79398i	−0.188991	+	0.188991i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.184634i			0.00915195i
$408$		0			0
$409$		−	29.7892i		−	1.47298i	−0.676447	−	0.736491i	$-0.736481\pi$
$409$		−	29.7892i		−	1.47298i	0.676447	−	0.736491i	$-0.263519\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.79087	−	6.79087i	0.334157	−	0.334157i
$414$		0			0
$415$			18.1539i			0.891138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.142401	−	0.142401i	−0.00695673	−	0.00695673i	0.703620	−	0.710577i	$-0.251566\pi$
$419$	−0.142401	−	0.142401i	−0.00695673	−	0.00695673i	−0.710577	+	0.703620i	$0.751566\pi$
$420$		0			0
$421$	−11.2150	+	11.2150i	−0.546588	+	0.546588i	−0.925452	−	0.378864i	$-0.876315\pi$
$421$	−11.2150	+	11.2150i	−0.546588	+	0.546588i	0.378864	+	0.925452i	$0.376315\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.42494			0.311656
$426$		0			0
$427$	−3.90386	−	3.90386i	−0.188921	−	0.188921i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.3679			1.22193			0.610965	−	0.791658i	$-0.290782\pi$
$431$	25.3679			1.22193			0.610965	+	0.791658i	$0.290782\pi$
$432$		0			0
$433$	−13.7112			−0.658917			−0.329459	−	0.944170i	$-0.606866\pi$
$433$	−13.7112			−0.658917			−0.329459	+	0.944170i	$0.606866\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.60593	+	8.60593i	0.411677	+	0.411677i
$438$		0			0
$439$	−29.7327			−1.41906			−0.709532	−	0.704673i	$-0.751094\pi$
$439$	−29.7327			−1.41906			−0.709532	+	0.704673i	$0.751094\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.83238	−	2.83238i	0.134571	−	0.134571i	−0.636613	−	0.771184i	$-0.719665\pi$
$443$	2.83238	−	2.83238i	0.134571	−	0.134571i	0.771184	+	0.636613i	$0.219665\pi$
$444$		0			0
$445$	12.0561	+	12.0561i	0.571512	+	0.571512i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	0.146417i		−	0.00690984i	−0.999994	−	0.00345492i	$-0.998900\pi$
$449$		−	0.146417i		−	0.00690984i	0.999994	−	0.00345492i	$-0.00109974\pi$
$450$		0			0
$451$	15.3851	−	15.3851i	0.724458	−	0.724458i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1.85113i			0.0867825i
$456$		0			0
$457$		−	30.2095i		−	1.41314i	−0.707642	−	0.706571i	$-0.750241\pi$
$457$		−	30.2095i		−	1.41314i	0.707642	−	0.706571i	$-0.249759\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.94894	−	3.94894i	0.183920	−	0.183920i	−0.609141	−	0.793062i	$-0.708486\pi$
$461$	3.94894	−	3.94894i	0.183920	−	0.183920i	0.793062	+	0.609141i	$0.208486\pi$
$462$		0			0
$463$		−	38.5905i		−	1.79345i	−0.442586	−	0.896726i	$-0.645939\pi$
$463$		−	38.5905i		−	1.79345i	0.442586	−	0.896726i	$-0.354061\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.93298	−	7.93298i	−0.367094	−	0.367094i	0.499322	−	0.866416i	$-0.333582\pi$
$467$	−7.93298	−	7.93298i	−0.367094	−	0.367094i	−0.866416	+	0.499322i	$0.833582\pi$
$468$		0			0
$469$	11.7660	−	11.7660i	0.543303	−	0.543303i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.0525			−0.554176
$474$		0			0
$475$	2.02287	+	2.02287i	0.0928156	+	0.0928156i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−29.4981			−1.34780			−0.673901	−	0.738821i	$-0.735383\pi$
$479$	−29.4981			−1.34780			−0.673901	+	0.738821i	$0.735383\pi$
$480$		0			0
$481$	0.0881050			0.00401724
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.11183	−	9.11183i	−0.413747	−	0.413747i
$486$		0			0
$487$	−16.7651			−0.759700			−0.379850	−	0.925048i	$-0.624024\pi$
$487$	−16.7651			−0.759700			−0.379850	+	0.925048i	$0.624024\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.8049	+	14.8049i	−0.668137	+	0.668137i	−0.957285	−	0.289147i	$-0.906628\pi$
$491$	−14.8049	+	14.8049i	−0.668137	+	0.668137i	0.289147	+	0.957285i	$0.406628\pi$
$492$		0			0
$493$	−17.4693	−	17.4693i	−0.786779	−	0.786779i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	20.0089i		−	0.897522i
$498$		0			0
$499$	25.1047	−	25.1047i	1.12384	−	1.12384i	0.132682	−	0.991159i	$-0.457641\pi$
$499$	25.1047	−	25.1047i	1.12384	−	1.12384i	0.991159	−	0.132682i	$-0.0423589\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			31.8786i			1.42140i	0.703497	+	0.710698i	$0.251621\pi$
$503$			31.8786i			1.42140i	−0.703497	+	0.710698i	$0.748379\pi$
$504$		0			0
$505$			8.71056i			0.387615i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.2195	+	30.2195i	−1.33946	+	1.33946i	−0.442871	+	0.896585i	$0.646040\pi$
$509$	−30.2195	+	30.2195i	−1.33946	+	1.33946i	−0.896585	+	0.442871i	$0.853960\pi$
$510$		0			0
$511$		−	7.57546i		−	0.335119i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.24926	+	6.24926i	0.275375	+	0.275375i
$516$		0			0
$517$	4.70063	−	4.70063i	0.206733	−	0.206733i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.1934			0.709447			0.354723	−	0.934971i	$-0.384575\pi$
$521$	16.1934			0.709447			0.354723	+	0.934971i	$0.384575\pi$
$522$		0			0
$523$	−17.1547	−	17.1547i	−0.750123	−	0.750123i	0.224379	−	0.974502i	$-0.427965\pi$
$523$	−17.1547	−	17.1547i	−0.750123	−	0.750123i	−0.974502	+	0.224379i	$0.927965\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	26.9322			1.17319
$528$		0			0
$529$	4.90078			0.213077
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.34161	−	7.34161i	−0.318000	−	0.318000i
$534$		0			0
$535$	−11.0054			−0.475805
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.30998	−	9.30998i	0.401009	−	0.401009i
$540$		0			0
$541$	0.546983	+	0.546983i	0.0235166	+	0.0235166i	0.718767	−	0.695251i	$-0.244707\pi$
$541$	0.546983	+	0.546983i	0.0235166	+	0.0235166i	−0.695251	+	0.718767i	$0.744707\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	16.7498i		−	0.717481i
$546$		0			0
$547$	−23.9635	+	23.9635i	−1.02460	+	1.02460i	−0.0249138	+	0.999690i	$0.507931\pi$
$547$	−23.9635	+	23.9635i	−1.02460	+	1.02460i	−0.999690	+	0.0249138i	$0.992069\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.0003i		−	0.468629i
$552$		0			0
$553$		−	9.54282i		−	0.405802i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.9083	−	26.9083i	1.14014	−	1.14014i	0.151717	−	0.988424i	$-0.451520\pi$
$557$	26.9083	−	26.9083i	1.14014	−	1.14014i	0.988424	−	0.151717i	$-0.0484801\pi$
$558$		0			0
$559$			5.75132i			0.243255i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	24.4491	+	24.4491i	1.03040	+	1.03040i	0.999523	+	0.0308820i	$0.00983159\pi$
$563$	24.4491	+	24.4491i	1.03040	+	1.03040i	0.0308820	+	0.999523i	$0.490168\pi$
$564$		0			0
$565$	10.8947	−	10.8947i	0.458342	−	0.458342i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	22.1607			0.929024			0.464512	−	0.885567i	$-0.346230\pi$
$569$	22.1607			0.929024			0.464512	+	0.885567i	$0.346230\pi$
$570$		0			0
$571$	−31.5395	−	31.5395i	−1.31989	−	1.31989i	−0.913861	−	0.406028i	$-0.866914\pi$
$571$	−31.5395	−	31.5395i	−1.31989	−	1.31989i	−0.406028	−	0.913861i	$-0.633086\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.25432			−0.177417
$576$		0			0
$577$	6.13571			0.255433			0.127717	−	0.991811i	$-0.459235\pi$
$577$	6.13571			0.255433			0.127717	+	0.991811i	$0.459235\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	18.5646	+	18.5646i	0.770189	+	0.770189i
$582$		0			0
$583$	−6.36475			−0.263601
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.30473	−	8.30473i	0.342773	−	0.342773i	−0.514636	−	0.857409i	$-0.672073\pi$
$587$	8.30473	−	8.30473i	0.342773	−	0.342773i	0.857409	+	0.514636i	$0.172073\pi$
$588$		0			0
$589$	8.47951	+	8.47951i	0.349392	+	0.349392i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.47693i			0.183845i	0.995766	+	0.0919227i	$0.0293013\pi$
$593$			4.47693i			0.183845i	−0.995766	+	0.0919227i	$0.970699\pi$
$594$		0			0
$595$	6.57030	−	6.57030i	0.269356	−	0.269356i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			46.3235i			1.89273i	0.323104	+	0.946363i	$0.395274\pi$
$599$			46.3235i			1.89273i	−0.323104	+	0.946363i	$0.604726\pi$
$600$		0			0
$601$		−	13.5155i		−	0.551309i	−0.961257	−	0.275654i	$-0.911105\pi$
$601$		−	13.5155i		−	0.551309i	0.961257	−	0.275654i	$-0.0888946\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.69050	−	2.69050i	0.109384	−	0.109384i
$606$		0			0
$607$		−	30.2228i		−	1.22671i	−0.789809	−	0.613353i	$-0.789820\pi$
$607$		−	30.2228i		−	1.22671i	0.789809	−	0.613353i	$-0.210180\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.24308	−	2.24308i	−0.0907454	−	0.0907454i
$612$		0			0
$613$	31.5921	−	31.5921i	1.27599	−	1.27599i	0.333100	−	0.942892i	$-0.391905\pi$
$613$	31.5921	−	31.5921i	1.27599	−	1.27599i	0.942892	−	0.333100i	$-0.108095\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.7884			0.716133			0.358067	−	0.933696i	$-0.383436\pi$
$617$	17.7884			0.716133			0.358067	+	0.933696i	$0.383436\pi$
$618$		0			0
$619$	−13.6322	−	13.6322i	−0.547926	−	0.547926i	0.377914	−	0.925841i	$-0.376641\pi$
$619$	−13.6322	−	13.6322i	−0.547926	−	0.547926i	−0.925841	+	0.377914i	$0.876641\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.6577			0.987888
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.312714	−	0.312714i	−0.0124687	−	0.0124687i
$630$		0			0
$631$	8.17320			0.325370			0.162685	−	0.986678i	$-0.447985\pi$
$631$	8.17320			0.325370			0.162685	+	0.986678i	$0.447985\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.5257	−	11.5257i	0.457383	−	0.457383i
$636$		0			0
$637$	−4.44261	−	4.44261i	−0.176023	−	0.176023i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			20.7621i			0.820052i	0.912074	+	0.410026i	$0.134480\pi$
$641$			20.7621i			0.820052i	−0.912074	+	0.410026i	$0.865520\pi$
$642$		0			0
$643$	−27.2576	+	27.2576i	−1.07493	+	1.07493i	−0.0779790	+	0.996955i	$0.524847\pi$
$643$	−27.2576	+	27.2576i	−1.07493	+	1.07493i	−0.996955	+	0.0779790i	$0.975153\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	46.6454i		−	1.83382i	−0.399093	−	0.916911i	$-0.630675\pi$
$647$		−	46.6454i		−	1.83382i	0.399093	−	0.916911i	$-0.369325\pi$
$648$		0			0
$649$		−	17.8125i		−	0.699204i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16.7411	−	16.7411i	0.655129	−	0.655129i	−0.299095	−	0.954223i	$-0.596685\pi$
$653$	16.7411	−	16.7411i	0.655129	−	0.655129i	0.954223	+	0.299095i	$0.0966847\pi$
$654$		0			0
$655$		−	2.14147i		−	0.0836740i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.8079	+	19.8079i	0.771606	+	0.771606i	0.978387	−	0.206781i	$-0.0662989\pi$
$659$	19.8079	+	19.8079i	0.771606	+	0.771606i	−0.206781	+	0.978387i	$0.566299\pi$
$660$		0			0
$661$	−4.83176	+	4.83176i	−0.187934	+	0.187934i	−0.794802	−	0.606868i	$-0.792425\pi$
$661$	−4.83176	+	4.83176i	−0.187934	+	0.187934i	0.606868	+	0.794802i	$0.292425\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.13727			0.160436
$666$		0			0
$667$	11.5674	+	11.5674i	0.447893	+	0.447893i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.2399			−0.395306
$672$		0			0
$673$	40.1613			1.54810			0.774052	−	0.633122i	$-0.218227\pi$
$673$	40.1613			1.54810			0.774052	+	0.633122i	$0.218227\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.78450	+	7.78450i	0.299183	+	0.299183i	0.840694	−	0.541511i	$-0.182148\pi$
$677$	7.78450	+	7.78450i	0.299183	+	0.299183i	−0.541511	+	0.840694i	$0.682148\pi$
$678$		0			0
$679$	−18.6360			−0.715183
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.2413	+	16.2413i	−0.621454	+	0.621454i	−0.945903	−	0.324449i	$-0.894821\pi$
$683$	−16.2413	+	16.2413i	−0.621454	+	0.621454i	0.324449	+	0.945903i	$0.394821\pi$
$684$		0			0
$685$	−1.24386	−	1.24386i	−0.0475253	−	0.0475253i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			3.03718i			0.115708i
$690$		0			0
$691$	−19.2435	+	19.2435i	−0.732057	+	0.732057i	−0.971027	−	0.238970i	$-0.923190\pi$
$691$	−19.2435	+	19.2435i	−0.732057	+	0.732057i	0.238970	+	0.971027i	$0.423190\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	5.18958i		−	0.196852i
$696$		0			0
$697$			52.1157i			1.97402i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.0853	−	27.0853i	1.02300	−	1.02300i	0.0232695	−	0.999729i	$-0.492592\pi$
$701$	27.0853	−	27.0853i	1.02300	−	1.02300i	0.999729	−	0.0232695i	$-0.00740757\pi$
$702$		0			0
$703$		−	0.196914i		−	0.00742675i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.90763	+	8.90763i	0.335006	+	0.335006i
$708$		0			0
$709$	11.6774	−	11.6774i	0.438555	−	0.438555i	−0.452971	−	0.891525i	$-0.649636\pi$
$709$	11.6774	−	11.6774i	0.438555	−	0.438555i	0.891525	+	0.452971i	$0.149636\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17.8333			−0.667864
$714$		0			0
$715$	2.42778	+	2.42778i	0.0907937	+	0.0907937i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5.14095			−0.191725			−0.0958625	−	0.995395i	$-0.530561\pi$
$719$	−5.14095			−0.191725			−0.0958625	+	0.995395i	$0.530561\pi$
$720$		0			0
$721$	12.7813			0.476000
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.71899	+	2.71899i	0.100981	+	0.100981i
$726$		0			0
$727$	34.7084			1.28726			0.643631	−	0.765336i	$-0.277427\pi$
$727$	34.7084			1.28726			0.643631	+	0.765336i	$0.277427\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	20.4134	−	20.4134i	0.755017	−	0.755017i
$732$		0			0
$733$	−18.8406	−	18.8406i	−0.695893	−	0.695893i	0.267629	−	0.963522i	$-0.413760\pi$
$733$	−18.8406	−	18.8406i	−0.695893	−	0.695893i	−0.963522	+	0.267629i	$0.913760\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	30.8624i		−	1.13683i
$738$		0			0
$739$	16.5358	−	16.5358i	0.608278	−	0.608278i	−0.334218	−	0.942496i	$-0.608472\pi$
$739$	16.5358	−	16.5358i	0.608278	−	0.608278i	0.942496	+	0.334218i	$0.108472\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			15.3994i			0.564950i	0.959275	+	0.282475i	$0.0911554\pi$
$743$			15.3994i			0.564950i	−0.959275	+	0.282475i	$0.908845\pi$
$744$		0			0
$745$		−	7.55252i		−	0.276703i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.2544	+	11.2544i	−0.411227	+	0.411227i
$750$		0			0
$751$		−	43.2461i		−	1.57807i	−0.614348	−	0.789035i	$-0.710581\pi$
$751$		−	43.2461i		−	1.57807i	0.614348	−	0.789035i	$-0.289419\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.66306	+	5.66306i	0.206100	+	0.206100i
$756$		0			0
$757$	24.3062	−	24.3062i	0.883426	−	0.883426i	−0.110456	−	0.993881i	$-0.535231\pi$
$757$	24.3062	−	24.3062i	0.883426	−	0.883426i	0.993881	+	0.110456i	$0.0352310\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.36638			0.267031			0.133516	−	0.991047i	$-0.457373\pi$
$761$	7.36638			0.267031			0.133516	+	0.991047i	$0.457373\pi$
$762$		0			0
$763$	−17.1287	−	17.1287i	−0.620101	−	0.620101i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.49994			−0.306915
$768$		0			0
$769$	−50.8707			−1.83445			−0.917223	−	0.398375i	$-0.869574\pi$
$769$	−50.8707			−1.83445			−0.917223	+	0.398375i	$0.869574\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.4213	−	26.4213i	−0.950309	−	0.950309i	0.0485139	−	0.998823i	$-0.484551\pi$
$773$	−26.4213	−	26.4213i	−0.950309	−	0.950309i	−0.998823	+	0.0485139i	$0.984551\pi$
$774$		0			0
$775$	−4.19182			−0.150575
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.4084	+	16.4084i	−0.587893	+	0.587893i
$780$		0			0
$781$	−26.2418	−	26.2418i	−0.939007	−	0.939007i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	20.4120i		−	0.728535i
$786$		0			0
$787$	12.5886	−	12.5886i	0.448734	−	0.448734i	−0.446200	−	0.894933i	$-0.647223\pi$
$787$	12.5886	−	12.5886i	0.448734	−	0.448734i	0.894933	+	0.446200i	$0.147223\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	22.2823i		−	0.792268i
$792$		0			0
$793$			4.88635i			0.173519i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.1728	+	18.1728i	−0.643712	+	0.643712i	−0.951466	−	0.307754i	$-0.900423\pi$
$797$	−18.1728	+	18.1728i	−0.643712	+	0.643712i	0.307754	+	0.951466i	$0.400423\pi$
$798$		0			0
$799$			15.9229i			0.563313i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.93528	−	9.93528i	−0.350608	−	0.350608i
$804$		0			0
$805$	−4.35057	+	4.35057i	−0.153337	+	0.153337i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.9840			−0.421334			−0.210667	−	0.977558i	$-0.567564\pi$
$809$	−11.9840			−0.421334			−0.210667	+	0.977558i	$0.567564\pi$
$810$		0			0
$811$	−7.69393	−	7.69393i	−0.270170	−	0.270170i	0.558998	−	0.829169i	$-0.311186\pi$
$811$	−7.69393	−	7.69393i	−0.270170	−	0.270170i	−0.829169	+	0.558998i	$0.811186\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.564294			−0.0197664
$816$		0			0
$817$	12.8542			0.449710
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.18537	−	5.18537i	−0.180971	−	0.180971i	0.610808	−	0.791779i	$-0.290845\pi$
$821$	−5.18537	−	5.18537i	−0.180971	−	0.180971i	−0.791779	+	0.610808i	$0.790845\pi$
$822$		0			0
$823$	−8.52021			−0.296996			−0.148498	−	0.988913i	$-0.547444\pi$
$823$	−8.52021			−0.296996			−0.148498	+	0.988913i	$0.547444\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.8664	−	15.8664i	0.551730	−	0.551730i	−0.375210	−	0.926940i	$-0.622429\pi$
$827$	15.8664	−	15.8664i	0.551730	−	0.551730i	0.926940	+	0.375210i	$0.122429\pi$
$828$		0			0
$829$	24.3341	+	24.3341i	0.845157	+	0.845157i	0.989524	−	0.144367i	$-0.0461147\pi$
$829$	24.3341	+	24.3341i	0.845157	+	0.845157i	−0.144367	+	0.989524i	$0.546115\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			31.5367i			1.09268i
$834$		0			0
$835$	−0.197222	+	0.197222i	−0.00682515	+	0.00682515i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	0.188170i		−	0.00649635i	−0.999995	−	0.00324818i	$-0.998966\pi$
$839$		−	0.188170i		−	0.00649635i	0.999995	−	0.00324818i	$-0.00103393\pi$
$840$		0			0
$841$			14.2142i			0.490146i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.03388	+	8.03388i	−0.276374	+	0.276374i
$846$		0			0
$847$		−	5.50275i		−	0.189077i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.207066	+	0.207066i	0.00709813	+	0.00709813i
$852$		0			0
$853$	17.1321	−	17.1321i	0.586592	−	0.586592i	−0.350115	−	0.936707i	$-0.613857\pi$
$853$	17.1321	−	17.1321i	0.586592	−	0.586592i	0.936707	+	0.350115i	$0.113857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	29.5262			1.00860			0.504298	−	0.863529i	$-0.331751\pi$
$857$	29.5262			1.00860			0.504298	+	0.863529i	$0.331751\pi$
$858$		0			0
$859$	−18.3139	−	18.3139i	−0.624861	−	0.624861i	0.321909	−	0.946771i	$-0.395675\pi$
$859$	−18.3139	−	18.3139i	−0.624861	−	0.624861i	−0.946771	+	0.321909i	$0.895675\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.282260			−0.00960823			−0.00480412	−	0.999988i	$-0.501529\pi$
$863$	−0.282260			−0.00960823			−0.00480412	+	0.999988i	$0.501529\pi$
$864$		0			0
$865$	−3.23843			−0.110110
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.5155	−	12.5155i	−0.424559	−	0.424559i
$870$		0			0
$871$	−14.7272			−0.499011
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.02262	+	1.02262i	−0.0345710	+	0.0345710i
$876$		0			0
$877$	5.46063	+	5.46063i	0.184392	+	0.184392i	0.793267	−	0.608874i	$-0.208379\pi$
$877$	5.46063	+	5.46063i	0.184392	+	0.184392i	−0.608874	+	0.793267i	$0.708379\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.59959i			0.188655i	0.995541	+	0.0943275i	$0.0300701\pi$
$881$			5.59959i			0.188655i	−0.995541	+	0.0943275i	$0.969930\pi$
$882$		0			0
$883$	−25.6352	+	25.6352i	−0.862694	+	0.862694i	−0.991650	−	0.128956i	$-0.958837\pi$
$883$	−25.6352	+	25.6352i	−0.862694	+	0.862694i	0.128956	+	0.991650i	$0.458837\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	30.5589i		−	1.02607i	−0.858369	−	0.513033i	$-0.828522\pi$
$887$		−	30.5589i		−	1.02607i	0.858369	−	0.513033i	$-0.171478\pi$
$888$		0			0
$889$		−	23.5729i		−	0.790610i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.01327	+	5.01327i	−0.167763	+	0.167763i
$894$		0			0
$895$			0.663480i			0.0221777i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.3975	+	11.3975i	0.380128	+	0.380128i
$900$		0			0
$901$	10.7800	−	10.7800i	0.359134	−	0.359134i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	14.5585			0.483940
$906$		0			0
$907$	24.4856	+	24.4856i	0.813030	+	0.813030i	0.985087	−	0.172057i	$-0.0550413\pi$
$907$	24.4856	+	24.4856i	0.813030	+	0.813030i	−0.172057	+	0.985087i	$0.555041\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.42981			0.0473716			0.0236858	−	0.999719i	$-0.492460\pi$
$911$	1.42981			0.0473716			0.0236858	+	0.999719i	$0.492460\pi$
$912$		0			0
$913$	48.6952			1.61158
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.18992	−	2.18992i	−0.0723174	−	0.0723174i
$918$		0			0
$919$	10.2127			0.336886			0.168443	−	0.985711i	$-0.446126\pi$
$919$	10.2127			0.336886			0.168443	+	0.985711i	$0.446126\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12.5223	+	12.5223i	−0.412176	+	0.412176i
$924$		0			0
$925$	0.0486719	+	0.0486719i	0.00160032	+	0.00160032i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	32.3475i		−	1.06129i	−0.847595	−	0.530644i	$-0.821950\pi$
$929$		−	32.3475i		−	1.06129i	0.847595	−	0.530644i	$-0.178050\pi$
$930$		0			0
$931$	−9.92921	+	9.92921i	−0.325417	+	0.325417i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	17.2340i		−	0.563612i
$936$		0			0
$937$			29.4307i			0.961458i	0.876869	+	0.480729i	$0.159628\pi$
$937$			29.4307i			0.961458i	−0.876869	+	0.480729i	$0.840372\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.8046	+	22.8046i	−0.743409	+	0.743409i	−0.973232	−	0.229823i	$-0.926185\pi$
$941$	−22.8046	+	22.8046i	−0.743409	+	0.743409i	0.229823	+	0.973232i	$0.426185\pi$
$942$		0			0
$943$		−	34.5088i		−	1.12376i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.7144	+	17.7144i	0.575642	+	0.575642i	0.933700	−	0.358058i	$-0.116561\pi$
$947$	17.7144	+	17.7144i	0.575642	+	0.575642i	−0.358058	+	0.933700i	$0.616561\pi$
$948$		0			0
$949$	−4.74100	+	4.74100i	−0.153899	+	0.153899i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17.3904			0.563331			0.281666	−	0.959513i	$-0.409113\pi$
$953$	17.3904			0.563331			0.281666	+	0.959513i	$0.409113\pi$
$954$		0			0
$955$	−6.08645	−	6.08645i	−0.196953	−	0.196953i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.54399			−0.0821499
$960$		0			0
$961$	13.4286			0.433182
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.41115	−	3.41115i	−0.109809	−	0.109809i
$966$		0			0
$967$	−1.28716			−0.0413924			−0.0206962	−	0.999786i	$-0.506588\pi$
$967$	−1.28716			−0.0413924			−0.0206962	+	0.999786i	$0.506588\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.2856	+	25.2856i	−0.811452	+	0.811452i	−0.984852	−	0.173400i	$-0.944525\pi$
$971$	−25.2856	+	25.2856i	−0.811452	+	0.811452i	0.173400	+	0.984852i	$0.444525\pi$
$972$		0			0
$973$	−5.30700	−	5.30700i	−0.170135	−	0.170135i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	26.9554i		−	0.862380i	−0.902261	−	0.431190i	$-0.858094\pi$
$977$		−	26.9554i		−	0.862380i	0.902261	−	0.431190i	$-0.141906\pi$
$978$		0			0
$979$	32.3387	−	32.3387i	1.03355	−	1.03355i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			11.8859i			0.379100i	0.981871	+	0.189550i	$0.0607029\pi$
$983$			11.8859i			0.379100i	−0.981871	+	0.189550i	$0.939297\pi$
$984$		0			0
$985$		−	0.710804i		−	0.0226481i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.5169	+	13.5169i	−0.429811	+	0.429811i
$990$		0			0
$991$			34.9586i			1.11050i	0.831685	+	0.555248i	$0.187377\pi$
$991$			34.9586i			1.11050i	−0.831685	+	0.555248i	$0.812623\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.8612	+	17.8612i	0.566237	+	0.566237i
$996$		0			0
$997$	−29.0306	+	29.0306i	−0.919408	+	0.919408i	−0.996986	−	0.0775785i	$-0.975281\pi$
$997$	−29.0306	+	29.0306i	−0.919408	+	0.919408i	0.0775785	+	0.996986i	$0.475281\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bl.c.431.6		32
3.2	odd	2	inner	2880.2.bl.c.431.12		32
4.3	odd	2		720.2.bl.c.611.8	yes	32
12.11	even	2		720.2.bl.c.611.9	yes	32
16.5	even	4		720.2.bl.c.251.9	yes	32
16.11	odd	4	inner	2880.2.bl.c.1871.12		32
48.5	odd	4		720.2.bl.c.251.8	✓	32
48.11	even	4	inner	2880.2.bl.c.1871.6		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bl.c.251.8	✓	32	48.5	odd	4
720.2.bl.c.251.9	yes	32	16.5	even	4
720.2.bl.c.611.8	yes	32	4.3	odd	2
720.2.bl.c.611.9	yes	32	12.11	even	2
2880.2.bl.c.431.6		32	1.1	even	1	trivial
2880.2.bl.c.431.12		32	3.2	odd	2	inner
2880.2.bl.c.1871.6		32	48.11	even	4	inner
2880.2.bl.c.1871.12		32	16.11	odd	4	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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bl (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} -1.44621 q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} -1.44621 q^{7} +(-1.89671 + 1.89671i) q^{11} +(0.905090 + 0.905090i) q^{13} -6.42494i q^{17} +(2.02287 - 2.02287i) q^{19} +4.25432i q^{23} +1.00000i q^{25} +(2.71899 - 2.71899i) q^{29} +4.19182i q^{31} +(1.02262 + 1.02262i) q^{35} +(0.0486719 - 0.0486719i) q^{37} -8.11147 q^{41} +(3.17721 + 3.17721i) q^{43} -2.47830 q^{47} -4.90848 q^{49} +(1.67784 + 1.67784i) q^{53} +2.68236 q^{55} +(-4.69563 + 4.69563i) q^{59} +(2.69937 + 2.69937i) q^{61} -1.27999i q^{65} +(-8.13574 + 8.13574i) q^{67} +13.8354i q^{71} +5.23815i q^{73} +(2.74305 - 2.74305i) q^{77} +6.59850i q^{79} +(-12.8367 - 12.8367i) q^{83} +(-4.54312 + 4.54312i) q^{85} -17.0498 q^{89} +(-1.30895 - 1.30895i) q^{91} -2.86077 q^{95} +12.8861 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{19} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 16 q^{67} + 16 q^{85} + 16 q^{91}+O(q^{100}) \) 32 * q + 8 * q^19 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 16 * q^67 + 16 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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