LMFDB - Embedded newform 2880.2.b.a.2591.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2591,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("2880.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.b (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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.7
Root			$0.437016 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	2880.2591
Dual form			2880.2.b.a.2591.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +3.16228i q^{7} +O(q^{10})$	$q-1.00000 q^{5} +3.16228i q^{7} +5.16228i q^{11} -3.05792i q^{13} -7.30056i q^{17} +7.30056 q^{19} +1.00000 q^{25} +4.32456 q^{29} -8.32456i q^{31} -3.16228i q^{35} +11.5432i q^{37} +10.3585i q^{41} +8.48528 q^{43} -1.18472 q^{47} -3.00000 q^{49} -10.3246 q^{53} -5.16228i q^{55} +6.83772i q^{59} +8.48528i q^{61} +3.05792i q^{65} -6.11584 q^{67} -6.11584 q^{71} -2.00000 q^{73} -16.3246 q^{77} +12.3246i q^{79} +10.3246i q^{83} +7.30056i q^{85} -1.87320i q^{89} +9.67000 q^{91} -7.30056 q^{95} +12.3246 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 8 q^{25} - 16 q^{29} - 24 q^{49} - 32 q^{53} - 16 q^{73} - 80 q^{77} + 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^25 - 16 * q^29 - 24 * q^49 - 32 * q^53 - 16 * q^73 - 80 * q^77 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.16228i	1.19523i	0.801784	+	0.597614i	$0.203885\pi$
$7$	3.16228i	1.19523i	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.16228i	1.55649i	0.627964	+	0.778243i	$0.283889\pi$
$11$	5.16228i	1.55649i	−0.627964	+	0.778243i	$0.716111\pi$
$12$	0	0
$13$	− 3.05792i	− 0.848115i	−0.905635	−	0.424058i	$-0.860605\pi$
$13$	− 3.05792i	− 0.848115i	0.905635	−	0.424058i	$-0.139395\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.30056i	− 1.77065i	−0.464976	−	0.885323i	$-0.653937\pi$
$17$	− 7.30056i	− 1.77065i	0.464976	−	0.885323i	$-0.346063\pi$
$18$	0	0
$19$	7.30056	1.67486	0.837432	−	0.546542i	$-0.184056\pi$
$19$	7.30056	1.67486	0.837432	+	0.546542i	$0.184056\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.32456	0.803050	0.401525	−	0.915848i	$-0.368480\pi$
$29$	4.32456	0.803050	0.401525	+	0.915848i	$0.368480\pi$
$30$	0	0
$31$	− 8.32456i	− 1.49513i	−0.664186	−	0.747567i	$-0.731222\pi$
$31$	− 8.32456i	− 1.49513i	0.664186	−	0.747567i	$-0.268778\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.16228i	− 0.534522i
$36$	0	0
$37$	11.5432i	1.89769i	0.315741	+	0.948846i	$0.397747\pi$
$37$	11.5432i	1.89769i	−0.315741	+	0.948846i	$0.602253\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.3585i	1.61772i	0.587999	+	0.808862i	$0.299916\pi$
$41$	10.3585i	1.61772i	−0.587999	+	0.808862i	$0.700084\pi$
$42$	0	0
$43$	8.48528	1.29399	0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528	1.29399	0.646997	+	0.762493i	$0.276025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.18472	−0.172809	−0.0864045	−	0.996260i	$-0.527538\pi$
$47$	−1.18472	−0.172809	−0.0864045	+	0.996260i	$0.527538\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.3246	−1.41819	−0.709093	−	0.705115i	$-0.750896\pi$
$53$	−10.3246	−1.41819	−0.709093	+	0.705115i	$0.750896\pi$
$54$	0	0
$55$	− 5.16228i	− 0.696081i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.83772i	0.890196i	0.895482	+	0.445098i	$0.146831\pi$
$59$	6.83772i	0.890196i	−0.895482	+	0.445098i	$0.853169\pi$
$60$	0	0
$61$	8.48528i	1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$	8.48528i	1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.05792i	0.379289i
$66$	0	0
$67$	−6.11584	−0.747169	−0.373585	−	0.927596i	$-0.621871\pi$
$67$	−6.11584	−0.747169	−0.373585	+	0.927596i	$0.621871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.11584	−0.725817	−0.362909	−	0.931825i	$-0.618216\pi$
$71$	−6.11584	−0.725817	−0.362909	+	0.931825i	$0.618216\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16.3246	−1.86036
$78$	0	0
$79$	12.3246i	1.38662i	0.720639	+	0.693310i	$0.243849\pi$
$79$	12.3246i	1.38662i	−0.720639	+	0.693310i	$0.756151\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3246i	1.13327i	0.823970	+	0.566634i	$0.191754\pi$
$83$	10.3246i	1.13327i	−0.823970	+	0.566634i	$0.808246\pi$
$84$	0	0
$85$	7.30056i	0.791857i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.87320i	− 0.198559i	−0.995060	−	0.0992796i	$-0.968346\pi$
$89$	− 1.87320i	− 0.198559i	0.995060	−	0.0992796i	$-0.0316538\pi$
$90$	0	0
$91$	9.67000	1.01369
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.30056	−0.749022
$96$	0	0
$97$	12.3246	1.25137	0.625684	−	0.780076i	$-0.284820\pi$
$97$	12.3246	1.25137	0.625684	+	0.780076i	$0.284820\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	7.16228i	0.705720i	0.935676	+	0.352860i	$0.114791\pi$
$103$	7.16228i	0.705720i	−0.935676	+	0.352860i	$0.885209\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.67544i	0.161971i	0.996715	+	0.0809857i	$0.0258068\pi$
$107$	1.67544i	0.161971i	−0.996715	+	0.0809857i	$0.974193\pi$
$108$	0	0
$109$	8.48528i	0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$	8.48528i	0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.18472i	0.111449i	0.998446	+	0.0557245i	$0.0177468\pi$
$113$	1.18472i	0.111449i	−0.998446	+	0.0557245i	$0.982253\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.0864	2.11633
$120$	0	0
$121$	−15.6491	−1.42265
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	− 8.83772i	− 0.784221i	−0.919918	−	0.392111i	$-0.871745\pi$
$127$	− 8.83772i	− 0.784221i	0.919918	−	0.392111i	$-0.128255\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 3.48683i	− 0.304646i	−0.988331	−	0.152323i	$-0.951325\pi$
$131$	− 3.48683i	− 0.304646i	0.988331	−	0.152323i	$-0.0486754\pi$
$132$	0	0
$133$	23.0864i	2.00185i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.7858i	− 1.34868i	−0.738423	−	0.674338i	$-0.764429\pi$
$137$	− 15.7858i	− 1.34868i	0.738423	−	0.674338i	$-0.235571\pi$
$138$	0	0
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.7858	1.32008
$144$	0	0
$145$	−4.32456	−0.359135
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.3246	−1.33736	−0.668680	−	0.743550i	$-0.733140\pi$
$149$	−16.3246	−1.33736	−0.668680	+	0.743550i	$0.733140\pi$
$150$	0	0
$151$	10.0000i	0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$	10.0000i	0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.32456i	0.668644i
$156$	0	0
$157$	5.42736i	0.433150i	0.976266	+	0.216575i	$0.0694887\pi$
$157$	5.42736i	0.433150i	−0.976266	+	0.216575i	$0.930511\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.36944	−0.183353	−0.0916763	−	0.995789i	$-0.529222\pi$
$167$	−2.36944	−0.183353	−0.0916763	+	0.995789i	$0.529222\pi$
$168$	0	0
$169$	3.64911	0.280701
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.9737	1.44254	0.721271	−	0.692653i	$-0.243558\pi$
$173$	18.9737	1.44254	0.721271	+	0.692653i	$0.243558\pi$
$174$	0	0
$175$	3.16228i	0.239046i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.48683i	0.260618i	0.991473	+	0.130309i	$0.0415970\pi$
$179$	3.48683i	0.260618i	−0.991473	+	0.130309i	$0.958403\pi$
$180$	0	0
$181$	− 6.11584i	− 0.454587i	−0.973826	−	0.227294i	$-0.927012\pi$
$181$	− 6.11584i	− 0.454587i	0.973826	−	0.227294i	$-0.0729877\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 11.5432i	− 0.848673i
$186$	0	0
$187$	37.6875	2.75599
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.6011	−1.05650	−0.528250	−	0.849089i	$-0.677152\pi$
$191$	−14.6011	−1.05650	−0.528250	+	0.849089i	$0.677152\pi$
$192$	0	0
$193$	0.324555	0.0233620	0.0116810	−	0.999932i	$-0.496282\pi$
$193$	0.324555	0.0233620	0.0116810	+	0.999932i	$0.496282\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.6491	1.04371	0.521853	−	0.853035i	$-0.325241\pi$
$197$	14.6491	1.04371	0.521853	+	0.853035i	$0.325241\pi$
$198$	0	0
$199$	− 6.64911i	− 0.471343i	−0.971833	−	0.235671i	$-0.924271\pi$
$199$	− 6.64911i	− 0.471343i	0.971833	−	0.235671i	$-0.0757289\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.6754i	0.959828i
$204$	0	0
$205$	− 10.3585i	− 0.723468i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	37.6875i	2.60690i
$210$	0	0
$211$	12.2317	0.842064	0.421032	−	0.907046i	$-0.361668\pi$
$211$	12.2317	0.842064	0.421032	+	0.907046i	$0.361668\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.48528	−0.578691
$216$	0	0
$217$	26.3246	1.78703
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.3246	−1.50171
$222$	0	0
$223$	− 5.48683i	− 0.367426i	−0.982980	−	0.183713i	$-0.941188\pi$
$223$	− 5.48683i	− 0.367426i	0.982980	−	0.183713i	$-0.0588116\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.64911i	0.574062i	0.957921	+	0.287031i	$0.0926682\pi$
$227$	8.64911i	0.574062i	−0.957921	+	0.287031i	$0.907332\pi$
$228$	0	0
$229$	2.36944i	0.156577i	0.996931	+	0.0782884i	$0.0249455\pi$
$229$	2.36944i	0.156577i	−0.996931	+	0.0782884i	$0.975055\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.9017i	1.43483i	0.696647	+	0.717414i	$0.254674\pi$
$233$	21.9017i	1.43483i	−0.696647	+	0.717414i	$0.745326\pi$
$234$	0	0
$235$	1.18472	0.0772825
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.36944	0.153266	0.0766331	−	0.997059i	$-0.475583\pi$
$239$	2.36944	0.153266	0.0766331	+	0.997059i	$0.475583\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	− 22.3246i	− 1.42048i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.4868i	0.977520i	0.872418	+	0.488760i	$0.162551\pi$
$251$	15.4868i	0.977520i	−0.872418	+	0.488760i	$0.837449\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.1553i	− 1.13250i	−0.824235	−	0.566248i	$-0.808395\pi$
$257$	− 18.1553i	− 1.13250i	0.824235	−	0.566248i	$-0.191605\pi$
$258$	0	0
$259$	−36.5028	−2.26817
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.6011	−0.900344	−0.450172	−	0.892942i	$-0.648637\pi$
$263$	−14.6011	−0.900344	−0.450172	+	0.892942i	$0.648637\pi$
$264$	0	0
$265$	10.3246	0.634232
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.9737	−0.791018	−0.395509	−	0.918462i	$-0.629432\pi$
$269$	−12.9737	−0.791018	−0.395509	+	0.918462i	$0.629432\pi$
$270$	0	0
$271$	6.64911i	0.403905i	0.979395	+	0.201952i	$0.0647286\pi$
$271$	6.64911i	0.403905i	−0.979395	+	0.201952i	$0.935271\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.16228i	0.311297i
$276$	0	0
$277$	− 26.1443i	− 1.57086i	−0.618950	−	0.785430i	$-0.712442\pi$
$277$	− 26.1443i	− 1.57086i	0.618950	−	0.785430i	$-0.287558\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 10.3585i	− 0.617935i	−0.951073	−	0.308968i	$-0.900016\pi$
$281$	− 10.3585i	− 0.617935i	0.951073	−	0.308968i	$-0.0999835\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.7564	−1.93355
$288$	0	0
$289$	−36.2982	−2.13519
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	− 6.83772i	− 0.398108i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	26.8328i	1.54662i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.48528i	− 0.485866i
$306$	0	0
$307$	−2.36944	−0.135231	−0.0676154	−	0.997711i	$-0.521539\pi$
$307$	−2.36944	−0.135231	−0.0676154	+	0.997711i	$0.521539\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.2023	1.65591	0.827954	−	0.560796i	$-0.189505\pi$
$311$	29.2023	1.65591	0.827954	+	0.560796i	$0.189505\pi$
$312$	0	0
$313$	−20.9737	−1.18550	−0.592751	−	0.805386i	$-0.701958\pi$
$313$	−20.9737	−1.18550	−0.592751	+	0.805386i	$0.701958\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	22.3246i	1.24994i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 53.2982i	− 2.96559i
$324$	0	0
$325$	− 3.05792i	− 0.169623i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.74641i	− 0.206546i
$330$	0	0
$331$	−4.93113	−0.271039	−0.135520	−	0.990775i	$-0.543270\pi$
$331$	−4.93113	−0.271039	−0.135520	+	0.990775i	$0.543270\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.11584	0.334144
$336$	0	0
$337$	5.35089	0.291482	0.145741	−	0.989323i	$-0.453443\pi$
$337$	5.35089	0.291482	0.145741	+	0.989323i	$0.453443\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	42.9737	2.32715
$342$	0	0
$343$	12.6491i	0.682988i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	20.7170i	1.10895i	0.832199	+	0.554477i	$0.187082\pi$
$349$	20.7170i	1.10895i	−0.832199	+	0.554477i	$0.812918\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 19.5323i	− 1.03960i	−0.854289	−	0.519798i	$-0.826007\pi$
$353$	− 19.5323i	− 1.03960i	0.854289	−	0.519798i	$-0.173993\pi$
$354$	0	0
$355$	6.11584	0.324595
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.8547	0.572890	0.286445	−	0.958097i	$-0.407526\pi$
$359$	10.8547	0.572890	0.286445	+	0.958097i	$0.407526\pi$
$360$	0	0
$361$	34.2982	1.80517
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	− 25.4868i	− 1.33040i	−0.746664	−	0.665201i	$-0.768346\pi$
$367$	− 25.4868i	− 1.33040i	0.746664	−	0.665201i	$-0.231654\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 32.6491i	− 1.69506i
$372$	0	0
$373$	− 5.42736i	− 0.281018i	−0.990079	−	0.140509i	$-0.955126\pi$
$373$	− 5.42736i	− 0.281018i	0.990079	−	0.140509i	$-0.0448739\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13.2242i	− 0.681079i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.7564	1.67377	0.836887	−	0.547376i	$-0.184373\pi$
$383$	32.7564	1.67377	0.836887	+	0.547376i	$0.184373\pi$
$384$	0	0
$385$	16.3246	0.831976
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 12.3246i	− 0.620116i
$396$	0	0
$397$	0.688486i	0.0345541i	0.999851	+	0.0172771i	$0.00549973\pi$
$397$	0.688486i	0.0345541i	−0.999851	+	0.0172771i	$0.994500\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7.98905i	− 0.398954i	−0.979903	−	0.199477i	$-0.936076\pi$
$401$	− 7.98905i	− 0.398954i	0.979903	−	0.199477i	$-0.0639243\pi$
$402$	0	0
$403$	−25.4558	−1.26805
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−59.5892	−2.95373
$408$	0	0
$409$	−5.35089	−0.264584	−0.132292	−	0.991211i	$-0.542234\pi$
$409$	−5.35089	−0.264584	−0.132292	+	0.991211i	$0.542234\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−21.6228	−1.06399
$414$	0	0
$415$	− 10.3246i	− 0.506812i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 15.4868i	− 0.756581i	−0.925687	−	0.378291i	$-0.876512\pi$
$419$	− 15.4868i	− 0.756581i	0.925687	−	0.378291i	$-0.123488\pi$
$420$	0	0
$421$	− 14.6011i	− 0.711615i	−0.934559	−	0.355808i	$-0.884206\pi$
$421$	− 14.6011i	− 0.711615i	0.934559	−	0.355808i	$-0.115794\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7.30056i	− 0.354129i
$426$	0	0
$427$	−26.8328	−1.29853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.9487	1.58708	0.793541	−	0.608517i	$-0.208235\pi$
$431$	32.9487	1.58708	0.793541	+	0.608517i	$0.208235\pi$
$432$	0	0
$433$	15.2982	0.735186	0.367593	−	0.929987i	$-0.380182\pi$
$433$	15.2982	0.735186	0.367593	+	0.929987i	$0.380182\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	34.6491i	1.65371i	0.562414	+	0.826856i	$0.309873\pi$
$439$	34.6491i	1.65371i	−0.562414	+	0.826856i	$0.690127\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.64911i	0.410932i	0.978664	+	0.205466i	$0.0658709\pi$
$443$	8.64911i	0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$444$	0	0
$445$	1.87320i	0.0887984i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.4449i	1.57836i	0.614161	+	0.789181i	$0.289495\pi$
$449$	33.4449i	1.57836i	−0.614161	+	0.789181i	$0.710505\pi$
$450$	0	0
$451$	−53.4734	−2.51796
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.67000	−0.453337
$456$	0	0
$457$	−29.6228	−1.38570	−0.692848	−	0.721084i	$-0.743644\pi$
$457$	−29.6228	−1.38570	−0.692848	+	0.721084i	$0.743644\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.67544	0.357481	0.178741	−	0.983896i	$-0.442798\pi$
$461$	7.67544	0.357481	0.178741	+	0.983896i	$0.442798\pi$
$462$	0	0
$463$	28.8377i	1.34020i	0.742270	+	0.670101i	$0.233749\pi$
$463$	28.8377i	1.34020i	−0.742270	+	0.670101i	$0.766251\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 25.6754i	− 1.18812i	−0.804422	−	0.594059i	$-0.797525\pi$
$467$	− 25.6754i	− 1.18812i	0.804422	−	0.594059i	$-0.202475\pi$
$468$	0	0
$469$	− 19.3400i	− 0.893038i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	43.8034i	2.01408i
$474$	0	0
$475$	7.30056	0.334973
$476$	0	0
$477$	0	0
$478$	0	0
$479$	40.0570	1.83025	0.915125	−	0.403171i	$-0.132092\pi$
$479$	40.0570	1.83025	0.915125	+	0.403171i	$0.132092\pi$
$480$	0	0
$481$	35.2982	1.60946
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.3246	−0.559629
$486$	0	0
$487$	− 8.83772i	− 0.400475i	−0.979747	−	0.200238i	$-0.935829\pi$
$487$	− 8.83772i	− 0.400475i	0.979747	−	0.200238i	$-0.0641714\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.4868i	1.24046i	0.784419	+	0.620232i	$0.212961\pi$
$491$	27.4868i	1.24046i	−0.784419	+	0.620232i	$0.787039\pi$
$492$	0	0
$493$	− 31.5717i	− 1.42192i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 19.3400i	− 0.867518i
$498$	0	0
$499$	−12.0394	−0.538959	−0.269480	−	0.963006i	$-0.586852\pi$
$499$	−12.0394	−0.538959	−0.269480	+	0.963006i	$0.586852\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.18472	−0.0528240	−0.0264120	−	0.999651i	$-0.508408\pi$
$503$	−1.18472	−0.0528240	−0.0264120	+	0.999651i	$0.508408\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.2982	0.500785	0.250392	−	0.968144i	$-0.419440\pi$
$509$	11.2982	0.500785	0.250392	+	0.968144i	$0.419440\pi$
$510$	0	0
$511$	− 6.32456i	− 0.279782i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 7.16228i	− 0.315608i
$516$	0	0
$517$	− 6.11584i	− 0.268975i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7279i	0.557620i	0.960346	+	0.278810i	$0.0899400\pi$
$521$	12.7279i	0.557620i	−0.960346	+	0.278810i	$0.910060\pi$
$522$	0	0
$523$	14.6011	0.638463	0.319231	−	0.947677i	$-0.396575\pi$
$523$	14.6011	0.638463	0.319231	+	0.947677i	$0.396575\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−60.7739	−2.64735
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.6754	1.37202
$534$	0	0
$535$	− 1.67544i	− 0.0724358i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 15.4868i	− 0.667065i
$540$	0	0
$541$	− 35.3181i	− 1.51844i	−0.650832	−	0.759222i	$-0.725580\pi$
$541$	− 35.3181i	− 1.51844i	0.650832	−	0.759222i	$-0.274420\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.48528i	− 0.363470i
$546$	0	0
$547$	−27.8253	−1.18972	−0.594862	−	0.803828i	$-0.702793\pi$
$547$	−27.8253	−1.18972	−0.594862	+	0.803828i	$0.702793\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	31.5717	1.34500
$552$	0	0
$553$	−38.9737	−1.65733
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.3246	1.45438	0.727189	−	0.686437i	$-0.240826\pi$
$557$	34.3246	1.45438	0.727189	+	0.686437i	$0.240826\pi$
$558$	0	0
$559$	− 25.9473i	− 1.09746i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.9473i	1.59929i	0.600473	+	0.799645i	$0.294979\pi$
$563$	37.9473i	1.59929i	−0.600473	+	0.799645i	$0.705021\pi$
$564$	0	0
$565$	− 1.18472i	− 0.0498415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1838i	1.60075i	0.599502	+	0.800373i	$0.295365\pi$
$569$	38.1838i	1.60075i	−0.599502	+	0.800373i	$0.704635\pi$
$570$	0	0
$571$	19.5323	0.817399	0.408700	−	0.912669i	$-0.365982\pi$
$571$	19.5323	0.817399	0.408700	+	0.912669i	$0.365982\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.6754	0.486055	0.243028	−	0.970019i	$-0.421859\pi$
$577$	11.6754	0.486055	0.243028	+	0.970019i	$0.421859\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−32.6491	−1.35451
$582$	0	0
$583$	− 53.2982i	− 2.20739i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 42.9737i	− 1.77371i	−0.462045	−	0.886857i	$-0.652884\pi$
$587$	− 42.9737i	− 1.77371i	0.462045	−	0.886857i	$-0.347116\pi$
$588$	0	0
$589$	− 60.7739i	− 2.50415i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.55415i	0.145952i	0.997334	+	0.0729758i	$0.0232496\pi$
$593$	3.55415i	0.145952i	−0.997334	+	0.0729758i	$0.976750\pi$
$594$	0	0
$595$	−23.0864	−0.946450
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.74641	−0.153074	−0.0765370	−	0.997067i	$-0.524386\pi$
$599$	−3.74641	−0.153074	−0.0765370	+	0.997067i	$0.524386\pi$
$600$	0	0
$601$	−36.6491	−1.49495	−0.747474	−	0.664291i	$-0.768734\pi$
$601$	−36.6491	−1.49495	−0.747474	+	0.664291i	$0.768734\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.6491	0.636227
$606$	0	0
$607$	− 20.8377i	− 0.845777i	−0.906182	−	0.422889i	$-0.861016\pi$
$607$	− 20.8377i	− 0.845777i	0.906182	−	0.422889i	$-0.138984\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.62278i	0.146562i
$612$	0	0
$613$	− 17.6590i	− 0.713242i	−0.934249	−	0.356621i	$-0.883929\pi$
$613$	− 17.6590i	− 0.713242i	0.934249	−	0.356621i	$-0.116071\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 24.2711i	− 0.977119i	−0.872531	−	0.488559i	$-0.837523\pi$
$617$	− 24.2711i	− 0.977119i	0.872531	−	0.488559i	$-0.162477\pi$
$618$	0	0
$619$	41.4339	1.66537	0.832685	−	0.553746i	$-0.186802\pi$
$619$	41.4339	1.66537	0.832685	+	0.553746i	$0.186802\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.92359	0.237324
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	84.2719	3.36014
$630$	0	0
$631$	28.9737i	1.15342i	0.816948	+	0.576712i	$0.195664\pi$
$631$	28.9737i	1.15342i	−0.816948	+	0.576712i	$0.804336\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.83772i	0.350714i
$636$	0	0
$637$	9.17377i	0.363478i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10.3585i	− 0.409136i	−0.978852	−	0.204568i	$-0.934421\pi$
$641$	− 10.3585i	− 0.409136i	0.978852	−	0.204568i	$-0.0655789\pi$
$642$	0	0
$643$	−31.5717	−1.24507	−0.622533	−	0.782594i	$-0.713896\pi$
$643$	−31.5717	−1.24507	−0.622533	+	0.782594i	$0.713896\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.0470	0.434301	0.217151	−	0.976138i	$-0.430324\pi$
$647$	11.0470	0.434301	0.217151	+	0.976138i	$0.430324\pi$
$648$	0	0
$649$	−35.2982	−1.38558
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.2982	0.911730	0.455865	−	0.890049i	$-0.349330\pi$
$653$	23.2982	0.911730	0.455865	+	0.890049i	$0.349330\pi$
$654$	0	0
$655$	3.48683i	0.136242i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 27.4868i	− 1.07073i	−0.844619	−	0.535367i	$-0.820173\pi$
$659$	− 27.4868i	− 1.07073i	0.844619	−	0.535367i	$-0.179827\pi$
$660$	0	0
$661$	− 27.8253i	− 1.08228i	−0.840933	−	0.541139i	$-0.817993\pi$
$661$	− 27.8253i	− 1.08228i	0.840933	−	0.541139i	$-0.182007\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 23.0864i	− 0.895252i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−43.8034	−1.69101
$672$	0	0
$673$	1.35089	0.0520730	0.0260365	−	0.999661i	$-0.491711\pi$
$673$	1.35089	0.0520730	0.0260365	+	0.999661i	$0.491711\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.9473	−1.68903	−0.844517	−	0.535529i	$-0.820112\pi$
$677$	−43.9473	−1.68903	−0.844517	+	0.535529i	$0.820112\pi$
$678$	0	0
$679$	38.9737i	1.49567i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 13.6754i	− 0.523276i	−0.965166	−	0.261638i	$-0.915737\pi$
$683$	− 13.6754i	− 0.523276i	0.965166	−	0.261638i	$-0.0842627\pi$
$684$	0	0
$685$	15.7858i	0.603146i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	31.5717i	1.20279i
$690$	0	0
$691$	−9.67000	−0.367864	−0.183932	−	0.982939i	$-0.558883\pi$
$691$	−9.67000	−0.367864	−0.183932	+	0.982939i	$0.558883\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.9706	−0.643730
$696$	0	0
$697$	75.6228	2.86442
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	84.2719i	3.17837i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.9737i	− 0.713578i
$708$	0	0
$709$	14.6011i	0.548357i	0.961679	+	0.274178i	$0.0884059\pi$
$709$	14.6011i	0.548357i	−0.961679	+	0.274178i	$0.911594\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−15.7858	−0.590357
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.4558	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$719$	25.4558	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$720$	0	0
$721$	−22.6491	−0.843497
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.32456	0.160610
$726$	0	0
$727$	− 27.8114i	− 1.03147i	−0.856749	−	0.515734i	$-0.827519\pi$
$727$	− 27.8114i	− 1.03147i	0.856749	−	0.515734i	$-0.172481\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 61.9473i	− 2.29120i
$732$	0	0
$733$	− 9.17377i	− 0.338841i	−0.985544	−	0.169420i	$-0.945810\pi$
$733$	− 9.17377i	− 0.338841i	0.985544	−	0.169420i	$-0.0541896\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 31.5717i	− 1.16296i
$738$	0	0
$739$	2.56169	0.0942333	0.0471166	−	0.998889i	$-0.484997\pi$
$739$	2.56169	0.0942333	0.0471166	+	0.998889i	$0.484997\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.0470	0.405274	0.202637	−	0.979254i	$-0.435049\pi$
$743$	11.0470	0.405274	0.202637	+	0.979254i	$0.435049\pi$
$744$	0	0
$745$	16.3246	0.598085
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.29822	−0.193593
$750$	0	0
$751$	8.97367i	0.327454i	0.986506	+	0.163727i	$0.0523516\pi$
$751$	8.97367i	0.327454i	−0.986506	+	0.163727i	$0.947648\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 10.0000i	− 0.363937i
$756$	0	0
$757$	− 1.68095i	− 0.0610952i	−0.999533	−	0.0305476i	$-0.990275\pi$
$757$	− 1.68095i	− 0.0610952i	0.999533	−	0.0305476i	$-0.00972512\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 12.7279i	− 0.461387i	−0.973026	−	0.230693i	$-0.925901\pi$
$761$	− 12.7279i	− 0.461387i	0.973026	−	0.230693i	$-0.0740994\pi$
$762$	0	0
$763$	−26.8328	−0.971413
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.9092	0.754988
$768$	0	0
$769$	4.64911	0.167651	0.0838256	−	0.996480i	$-0.473286\pi$
$769$	4.64911	0.167651	0.0838256	+	0.996480i	$0.473286\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.2982	1.26959	0.634794	−	0.772681i	$-0.281085\pi$
$773$	35.2982	1.26959	0.634794	+	0.772681i	$0.281085\pi$
$774$	0	0
$775$	− 8.32456i	− 0.299027i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	75.6228i	2.70947i
$780$	0	0
$781$	− 31.5717i	− 1.12972i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 5.42736i	− 0.193711i
$786$	0	0
$787$	−46.1728	−1.64588	−0.822942	−	0.568126i	$-0.807669\pi$
$787$	−46.1728	−1.64588	−0.822942	+	0.568126i	$0.807669\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.74641	−0.133207
$792$	0	0
$793$	25.9473	0.921417
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.6754	−0.909471	−0.454735	−	0.890627i	$-0.650266\pi$
$797$	−25.6754	−0.909471	−0.454735	+	0.890627i	$0.650266\pi$
$798$	0	0
$799$	8.64911i	0.305984i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 10.3246i	− 0.364346i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1.87320i	− 0.0658583i	−0.999458	−	0.0329292i	$-0.989516\pi$
$809$	− 1.87320i	− 0.0658583i	0.999458	−	0.0329292i	$-0.0104836\pi$
$810$	0	0
$811$	29.2023	1.02543	0.512715	−	0.858559i	$-0.328640\pi$
$811$	29.2023	1.02543	0.512715	+	0.858559i	$0.328640\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	61.9473	2.16726
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.6228	−0.754640	−0.377320	−	0.926083i	$-0.623154\pi$
$821$	−21.6228	−0.754640	−0.377320	+	0.926083i	$0.623154\pi$
$822$	0	0
$823$	− 46.7851i	− 1.63082i	−0.578881	−	0.815412i	$-0.696510\pi$
$823$	− 46.7851i	− 1.63082i	0.578881	−	0.815412i	$-0.303490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.9473i	0.902277i	0.892454	+	0.451139i	$0.148982\pi$
$827$	25.9473i	0.902277i	−0.892454	+	0.451139i	$0.851018\pi$
$828$	0	0
$829$	− 23.0864i	− 0.801824i	−0.916117	−	0.400912i	$-0.868693\pi$
$829$	− 23.0864i	− 0.801824i	0.916117	−	0.400912i	$-0.131307\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	21.9017i	0.758849i
$834$	0	0
$835$	2.36944	0.0819977
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.0864	0.797031	0.398516	−	0.917162i	$-0.369525\pi$
$839$	23.0864	0.797031	0.398516	+	0.917162i	$0.369525\pi$
$840$	0	0
$841$	−10.2982	−0.355111
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.64911	−0.125533
$846$	0	0
$847$	− 49.4868i	− 1.70039i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 22.3979i	− 0.766890i	−0.923564	−	0.383445i	$-0.874738\pi$
$853$	− 22.3979i	− 0.766890i	0.923564	−	0.383445i	$-0.125262\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 26.6406i	− 0.910024i	−0.890485	−	0.455012i	$-0.849635\pi$
$857$	− 26.6406i	− 0.910024i	0.890485	−	0.455012i	$-0.150365\pi$
$858$	0	0
$859$	−53.6656	−1.83105	−0.915524	−	0.402264i	$-0.868224\pi$
$859$	−53.6656	−1.83105	−0.915524	+	0.402264i	$0.868224\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.3400	0.658341	0.329171	−	0.944270i	$-0.393231\pi$
$863$	19.3400	0.658341	0.329171	+	0.944270i	$0.393231\pi$
$864$	0	0
$865$	−18.9737	−0.645124
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−63.6228	−2.15825
$870$	0	0
$871$	18.7018i	0.633686i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.16228i	− 0.106904i
$876$	0	0
$877$	− 17.6590i	− 0.596304i	−0.954518	−	0.298152i	$-0.903630\pi$
$877$	− 17.6590i	− 0.596304i	0.954518	−	0.298152i	$-0.0963702\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.87320i	0.0631098i	0.999502	+	0.0315549i	$0.0100459\pi$
$881$	1.87320i	0.0631098i	−0.999502	+	0.0315549i	$0.989954\pi$
$882$	0	0
$883$	−16.9706	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$883$	−16.9706	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.36944	−0.0795579	−0.0397789	−	0.999209i	$-0.512665\pi$
$887$	−2.36944	−0.0795579	−0.0397789	+	0.999209i	$0.512665\pi$
$888$	0	0
$889$	27.9473	0.937323
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.64911	−0.289431
$894$	0	0
$895$	− 3.48683i	− 0.116552i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 36.0000i	− 1.20067i
$900$	0	0
$901$	75.3751i	2.51111i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.11584i	0.203298i
$906$	0	0
$907$	−20.7170	−0.687896	−0.343948	−	0.938989i	$-0.611764\pi$
$907$	−20.7170	−0.687896	−0.343948	+	0.938989i	$0.611764\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.5423	−1.60828	−0.804138	−	0.594442i	$-0.797373\pi$
$911$	−48.5423	−1.60828	−0.804138	+	0.594442i	$0.797373\pi$
$912$	0	0
$913$	−53.2982	−1.76391
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.0263	0.364122
$918$	0	0
$919$	3.02633i	0.0998295i	0.998753	+	0.0499148i	$0.0158950\pi$
$919$	3.02633i	0.0998295i	−0.998753	+	0.0499148i	$0.984105\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18.7018i	0.615577i
$924$	0	0
$925$	11.5432i	0.379538i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 7.98905i	− 0.262112i	−0.991375	−	0.131056i	$-0.958163\pi$
$929$	− 7.98905i	− 0.262112i	0.991375	−	0.131056i	$-0.0418368\pi$
$930$	0	0
$931$	−21.9017	−0.717799
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−37.6875	−1.23251
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.9473	0.650264	0.325132	−	0.945669i	$-0.394591\pi$
$941$	19.9473	0.650264	0.325132	+	0.945669i	$0.394591\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 46.5964i	− 1.51418i	−0.653310	−	0.757090i	$-0.726620\pi$
$947$	− 46.5964i	− 1.51418i	0.653310	−	0.757090i	$-0.273380\pi$
$948$	0	0
$949$	6.11584i	0.198529i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 25.6481i	− 0.830823i	−0.909634	−	0.415412i	$-0.863638\pi$
$953$	− 25.6481i	− 0.830823i	0.909634	−	0.415412i	$-0.136362\pi$
$954$	0	0
$955$	14.6011	0.472481
$956$	0	0
$957$	0	0
$958$	0	0
$959$	49.9192	1.61198
$960$	0	0
$961$	−38.2982	−1.23543
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.324555	−0.0104478
$966$	0	0
$967$	2.13594i	0.0686873i	0.999410	+	0.0343437i	$0.0109341\pi$
$967$	2.13594i	0.0686873i	−0.999410	+	0.0343437i	$0.989066\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 12.1359i	− 0.389461i	−0.980857	−	0.194730i	$-0.937617\pi$
$971$	− 12.1359i	− 0.389461i	0.980857	−	0.194730i	$-0.0623832\pi$
$972$	0	0
$973$	53.6656i	1.72044i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.2711i	− 0.776502i	−0.921554	−	0.388251i	$-0.873079\pi$
$977$	− 24.2711i	− 0.776502i	0.921554	−	0.388251i	$-0.126921\pi$
$978$	0	0
$979$	9.67000	0.309055
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.55415	−0.113360	−0.0566800	−	0.998392i	$-0.518051\pi$
$983$	−3.55415	−0.113360	−0.0566800	+	0.998392i	$0.518051\pi$
$984$	0	0
$985$	−14.6491	−0.466759
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 51.2982i	− 1.62954i	−0.579783	−	0.814771i	$-0.696863\pi$
$991$	− 51.2982i	− 1.62954i	0.579783	−	0.814771i	$-0.303137\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.64911i	0.210791i
$996$	0	0
$997$	34.6296i	1.09673i	0.836239	+	0.548365i	$0.184750\pi$
$997$	34.6296i	1.09673i	−0.836239	+	0.548365i	$0.815250\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.b.a.2591.7	yes	8
3.2	odd	2		2880.2.b.d.2591.5	yes	8
4.3	odd	2	inner	2880.2.b.a.2591.1	✓	8
8.3	odd	2		2880.2.b.d.2591.4	yes	8
8.5	even	2		2880.2.b.d.2591.6	yes	8
12.11	even	2		2880.2.b.d.2591.3	yes	8
24.5	odd	2	inner	2880.2.b.a.2591.8	yes	8
24.11	even	2	inner	2880.2.b.a.2591.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.b.a.2591.1	✓	8	4.3	odd	2	inner
2880.2.b.a.2591.2	yes	8	24.11	even	2	inner
2880.2.b.a.2591.7	yes	8	1.1	even	1	trivial
2880.2.b.a.2591.8	yes	8	24.5	odd	2	inner
2880.2.b.d.2591.3	yes	8	12.11	even	2
2880.2.b.d.2591.4	yes	8	8.3	odd	2
2880.2.b.d.2591.5	yes	8	3.2	odd	2
2880.2.b.d.2591.6	yes	8	8.5	even	2

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.16228i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +3.16228i q^{7} +5.16228i q^{11} -3.05792i q^{13} -7.30056i q^{17} +7.30056 q^{19} +1.00000 q^{25} +4.32456 q^{29} -8.32456i q^{31} -3.16228i q^{35} +11.5432i q^{37} +10.3585i q^{41} +8.48528 q^{43} -1.18472 q^{47} -3.00000 q^{49} -10.3246 q^{53} -5.16228i q^{55} +6.83772i q^{59} +8.48528i q^{61} +3.05792i q^{65} -6.11584 q^{67} -6.11584 q^{71} -2.00000 q^{73} -16.3246 q^{77} +12.3246i q^{79} +10.3246i q^{83} +7.30056i q^{85} -1.87320i q^{89} +9.67000 q^{91} -7.30056 q^{95} +12.3246 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 8 q^{25} - 16 q^{29} - 24 q^{49} - 32 q^{53} - 16 q^{73} - 80 q^{77} + 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^25 - 16 * q^29 - 24 * q^49 - 32 * q^53 - 16 * q^73 - 80 * q^77 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more