LMFDB - Embedded newform 2520.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(1,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.a (trivial)

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:	yes
Analytic conductor:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.00000 q^{7} -5.65685 q^{11} +2.00000 q^{13} +3.65685 q^{17} -5.65685 q^{19} +5.65685 q^{23} +1.00000 q^{25} -3.65685 q^{29} +4.00000 q^{31} -1.00000 q^{35} +11.6569 q^{37} -2.00000 q^{41} +1.65685 q^{43} +2.34315 q^{47} +1.00000 q^{49} +3.65685 q^{53} +5.65685 q^{55} +4.00000 q^{59} +0.343146 q^{61} -2.00000 q^{65} -9.65685 q^{67} +7.31371 q^{71} +6.00000 q^{73} -5.65685 q^{77} +11.3137 q^{79} +4.00000 q^{83} -3.65685 q^{85} +14.0000 q^{89} +2.00000 q^{91} +5.65685 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} + 4 q^{13} - 4 q^{17} + 2 q^{25} + 4 q^{29} + 8 q^{31} - 2 q^{35} + 12 q^{37} - 4 q^{41} - 8 q^{43} + 16 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} + 12 q^{61} - 4 q^{65} - 8 q^{67} - 8 q^{71} + 12 q^{73} + 8 q^{83} + 4 q^{85} + 28 q^{89} + 4 q^{91} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 4 * q^13 - 4 * q^17 + 2 * q^25 + 4 * q^29 + 8 * q^31 - 2 * q^35 + 12 * q^37 - 4 * q^41 - 8 * q^43 + 16 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^59 + 12 * q^61 - 4 * q^65 - 8 * q^67 - 8 * q^71 + 12 * q^73 + 8 * q^83 + 4 * q^85 + 28 * q^89 + 4 * q^91 + 12 * q^97

Display 100 coefficients 10 coefficients

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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.34315	0.341783	0.170891	−	0.985290i	$-0.445335\pi$
$47$	2.34315	0.341783	0.170891	+	0.985290i	$0.445335\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.31371	0.867978	0.433989	−	0.900918i	$-0.357106\pi$
$71$	7.31371	0.867978	0.433989	+	0.900918i	$0.357106\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.65685	−0.644658
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−3.65685	−0.396642
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.3137	−1.72278	−0.861389	−	0.507946i	$-0.830405\pi$
$101$	−17.3137	−1.72278	−0.861389	+	0.507946i	$0.830405\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.34315	0.613215	0.306608	−	0.951836i	$-0.400806\pi$
$107$	6.34315	0.613215	0.306608	+	0.951836i	$0.400806\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.34315	0.408569	0.204284	−	0.978912i	$-0.434513\pi$
$113$	4.34315	0.408569	0.204284	+	0.978912i	$0.434513\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.65685	0.335223
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.3137	−1.71381	−0.856907	−	0.515471i	$-0.827617\pi$
$127$	−19.3137	−1.71381	−0.856907	+	0.515471i	$0.827617\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.34315	0.371060	0.185530	−	0.982639i	$-0.440600\pi$
$137$	4.34315	0.371060	0.185530	+	0.982639i	$0.440600\pi$
$138$	0	0
$139$	−13.6569	−1.15836	−0.579180	−	0.815200i	$-0.696627\pi$
$139$	−13.6569	−1.15836	−0.579180	+	0.815200i	$0.696627\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.3137	−0.946100
$144$	0	0
$145$	3.65685	0.303685
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.9706	1.55413	0.777065	−	0.629421i	$-0.216708\pi$
$149$	18.9706	1.55413	0.777065	+	0.629421i	$0.216708\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.65685	0.445823
$162$	0	0
$163$	−17.6569	−1.38299	−0.691496	−	0.722380i	$-0.743048\pi$
$163$	−17.6569	−1.38299	−0.691496	+	0.722380i	$0.743048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.6569	1.05680	0.528400	−	0.848996i	$-0.322792\pi$
$167$	13.6569	1.05680	0.528400	+	0.848996i	$0.322792\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.3137	1.62045	0.810226	−	0.586118i	$-0.199345\pi$
$173$	21.3137	1.62045	0.810226	+	0.586118i	$0.199345\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.6569	−1.02076	−0.510381	−	0.859949i	$-0.670495\pi$
$179$	−13.6569	−1.02076	−0.510381	+	0.859949i	$0.670495\pi$
$180$	0	0
$181$	11.6569	0.866447	0.433224	−	0.901286i	$-0.357376\pi$
$181$	11.6569	0.866447	0.433224	+	0.901286i	$0.357376\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.6569	−0.857029
$186$	0	0
$187$	−20.6863	−1.51273
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.3137	−1.10806	−0.554031	−	0.832496i	$-0.686911\pi$
$191$	−15.3137	−1.10806	−0.554031	+	0.832496i	$0.686911\pi$
$192$	0	0
$193$	13.3137	0.958342	0.479171	−	0.877722i	$-0.340937\pi$
$193$	13.3137	0.958342	0.479171	+	0.877722i	$0.340937\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.6569	0.830516	0.415258	−	0.909704i	$-0.363691\pi$
$197$	11.6569	0.830516	0.415258	+	0.909704i	$0.363691\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.65685	−0.256661
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	26.6274	1.83311	0.916553	−	0.399912i	$-0.130959\pi$
$211$	26.6274	1.83311	0.916553	+	0.399912i	$0.130959\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.65685	−0.112997
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.31371	0.491973
$222$	0	0
$223$	−19.3137	−1.29334	−0.646671	−	0.762769i	$-0.723839\pi$
$223$	−19.3137	−1.29334	−0.646671	+	0.762769i	$0.723839\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.6274	1.76732	0.883662	−	0.468126i	$-0.155071\pi$
$227$	26.6274	1.76732	0.883662	+	0.468126i	$0.155071\pi$
$228$	0	0
$229$	14.9706	0.989283	0.494641	−	0.869097i	$-0.335299\pi$
$229$	14.9706	0.989283	0.494641	+	0.869097i	$0.335299\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.9706	−0.980754	−0.490377	−	0.871510i	$-0.663141\pi$
$233$	−14.9706	−0.980754	−0.490377	+	0.871510i	$0.663141\pi$
$234$	0	0
$235$	−2.34315	−0.152850
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	−1.31371	−0.0846234	−0.0423117	−	0.999104i	$-0.513472\pi$
$241$	−1.31371	−0.0846234	−0.0423117	+	0.999104i	$0.513472\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−11.3137	−0.719874
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.3137	0.966593	0.483296	−	0.875457i	$-0.339439\pi$
$251$	15.3137	0.966593	0.483296	+	0.875457i	$0.339439\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.6569	1.72519	0.862594	−	0.505898i	$-0.168839\pi$
$257$	27.6569	1.72519	0.862594	+	0.505898i	$0.168839\pi$
$258$	0	0
$259$	11.6569	0.724322
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−29.6569	−1.82872	−0.914360	−	0.404902i	$-0.867306\pi$
$263$	−29.6569	−1.82872	−0.914360	+	0.404902i	$0.867306\pi$
$264$	0	0
$265$	−3.65685	−0.224639
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.65685	−0.341121
$276$	0	0
$277$	−15.6569	−0.940729	−0.470365	−	0.882472i	$-0.655878\pi$
$277$	−15.6569	−0.940729	−0.470365	+	0.882472i	$0.655878\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.31371	0.555609	0.277805	−	0.960638i	$-0.410393\pi$
$281$	9.31371	0.555609	0.277805	+	0.960638i	$0.410393\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.6274	−1.20507	−0.602533	−	0.798094i	$-0.705842\pi$
$293$	−20.6274	−1.20507	−0.602533	+	0.798094i	$0.705842\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.3137	0.654289
$300$	0	0
$301$	1.65685	0.0954995
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.343146	−0.0196485
$306$	0	0
$307$	18.6274	1.06312	0.531561	−	0.847020i	$-0.321605\pi$
$307$	18.6274	1.06312	0.531561	+	0.847020i	$0.321605\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−29.3137	−1.65691	−0.828454	−	0.560057i	$-0.810779\pi$
$313$	−29.3137	−1.65691	−0.828454	+	0.560057i	$0.810779\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.6569	0.654714	0.327357	−	0.944901i	$-0.393842\pi$
$317$	11.6569	0.654714	0.327357	+	0.944901i	$0.393842\pi$
$318$	0	0
$319$	20.6863	1.15821
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.6863	−1.15102
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.34315	0.129182
$330$	0	0
$331$	−26.6274	−1.46358	−0.731788	−	0.681533i	$-0.761314\pi$
$331$	−26.6274	−1.46358	−0.731788	+	0.681533i	$0.761314\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.65685	0.527610
$336$	0	0
$337$	13.3137	0.725244	0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137	0.725244	0.362622	+	0.931936i	$0.381882\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.65685	−0.0889446	−0.0444723	−	0.999011i	$-0.514161\pi$
$347$	−1.65685	−0.0889446	−0.0444723	+	0.999011i	$0.514161\pi$
$348$	0	0
$349$	−20.3431	−1.08894	−0.544472	−	0.838779i	$-0.683270\pi$
$349$	−20.3431	−1.08894	−0.544472	+	0.838779i	$0.683270\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.02944	−0.0547914	−0.0273957	−	0.999625i	$-0.508721\pi$
$353$	−1.02944	−0.0547914	−0.0273957	+	0.999625i	$0.508721\pi$
$354$	0	0
$355$	−7.31371	−0.388171
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	14.6274	0.763545	0.381772	−	0.924256i	$-0.375314\pi$
$367$	14.6274	0.763545	0.381772	+	0.924256i	$0.375314\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.65685	0.189854
$372$	0	0
$373$	19.6569	1.01779	0.508897	−	0.860828i	$-0.330054\pi$
$373$	19.6569	1.01779	0.508897	+	0.860828i	$0.330054\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.31371	−0.376675
$378$	0	0
$379$	−34.6274	−1.77869	−0.889345	−	0.457236i	$-0.848840\pi$
$379$	−34.6274	−1.77869	−0.889345	+	0.457236i	$0.848840\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.65685	0.289052	0.144526	−	0.989501i	$-0.453834\pi$
$383$	5.65685	0.289052	0.144526	+	0.989501i	$0.453834\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.9706	−1.57027	−0.785135	−	0.619325i	$-0.787406\pi$
$389$	−30.9706	−1.57027	−0.785135	+	0.619325i	$0.787406\pi$
$390$	0	0
$391$	20.6863	1.04615
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.3137	−0.569254
$396$	0	0
$397$	−9.31371	−0.467442	−0.233721	−	0.972304i	$-0.575090\pi$
$397$	−9.31371	−0.467442	−0.233721	+	0.972304i	$0.575090\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.3137	−0.664855	−0.332427	−	0.943129i	$-0.607868\pi$
$401$	−13.3137	−0.664855	−0.332427	+	0.943129i	$0.607868\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−65.9411	−3.26858
$408$	0	0
$409$	21.3137	1.05390	0.526948	−	0.849898i	$-0.323336\pi$
$409$	21.3137	1.05390	0.526948	+	0.849898i	$0.323336\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.65685	0.177383
$426$	0	0
$427$	0.343146	0.0166060
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.6863	−0.803750	−0.401875	−	0.915694i	$-0.631641\pi$
$431$	−16.6863	−0.803750	−0.401875	+	0.915694i	$0.631641\pi$
$432$	0	0
$433$	−21.3137	−1.02427	−0.512136	−	0.858905i	$-0.671146\pi$
$433$	−21.3137	−1.02427	−0.512136	+	0.858905i	$0.671146\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−32.0000	−1.53077
$438$	0	0
$439$	23.3137	1.11270	0.556351	−	0.830947i	$-0.312201\pi$
$439$	23.3137	1.11270	0.556351	+	0.830947i	$0.312201\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.97056	0.236159	0.118079	−	0.993004i	$-0.462326\pi$
$443$	4.97056	0.236159	0.118079	+	0.993004i	$0.462326\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.31371	0.439541	0.219771	−	0.975552i	$-0.429469\pi$
$449$	9.31371	0.439541	0.219771	+	0.975552i	$0.429469\pi$
$450$	0	0
$451$	11.3137	0.532742
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.9411	1.67395	0.836973	−	0.547245i	$-0.184323\pi$
$461$	35.9411	1.67395	0.836973	+	0.547245i	$0.184323\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.3137	1.44903	0.724513	−	0.689261i	$-0.242065\pi$
$467$	31.3137	1.44903	0.724513	+	0.689261i	$0.242065\pi$
$468$	0	0
$469$	−9.65685	−0.445912
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.37258	−0.430952
$474$	0	0
$475$	−5.65685	−0.259554
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.6274	1.03387	0.516937	−	0.856024i	$-0.327072\pi$
$479$	22.6274	1.03387	0.516937	+	0.856024i	$0.327072\pi$
$480$	0	0
$481$	23.3137	1.06301
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	22.6274	1.02535	0.512673	−	0.858584i	$-0.328655\pi$
$487$	22.6274	1.02535	0.512673	+	0.858584i	$0.328655\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.65685	−0.255290	−0.127645	−	0.991820i	$-0.540742\pi$
$491$	−5.65685	−0.255290	−0.127645	+	0.991820i	$0.540742\pi$
$492$	0	0
$493$	−13.3726	−0.602271
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.31371	0.328065
$498$	0	0
$499$	34.6274	1.55014	0.775068	−	0.631878i	$-0.217716\pi$
$499$	34.6274	1.55014	0.775068	+	0.631878i	$0.217716\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.6569	−0.608929	−0.304465	−	0.952524i	$-0.598478\pi$
$503$	−13.6569	−0.608929	−0.304465	+	0.952524i	$0.598478\pi$
$504$	0	0
$505$	17.3137	0.770450
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−13.2548	−0.582947
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.3137	−1.28426	−0.642128	−	0.766597i	$-0.721948\pi$
$521$	−29.3137	−1.28426	−0.642128	+	0.766597i	$0.721948\pi$
$522$	0	0
$523$	26.6274	1.16434	0.582168	−	0.813069i	$-0.302205\pi$
$523$	26.6274	1.16434	0.582168	+	0.813069i	$0.302205\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.6274	0.637180
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	−6.34315	−0.274238
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.65685	−0.243658
$540$	0	0
$541$	−6.68629	−0.287466	−0.143733	−	0.989616i	$-0.545911\pi$
$541$	−6.68629	−0.287466	−0.143733	+	0.989616i	$0.545911\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−32.2843	−1.38038	−0.690188	−	0.723630i	$-0.742472\pi$
$547$	−32.2843	−1.38038	−0.690188	+	0.723630i	$0.742472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.6863	0.881266
$552$	0	0
$553$	11.3137	0.481108
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.02944	0.213104	0.106552	−	0.994307i	$-0.466019\pi$
$557$	5.02944	0.213104	0.106552	+	0.994307i	$0.466019\pi$
$558$	0	0
$559$	3.31371	0.140155
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−4.34315	−0.182718
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.9411	−1.17135	−0.585676	−	0.810545i	$-0.699171\pi$
$569$	−27.9411	−1.17135	−0.585676	+	0.810545i	$0.699171\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.65685	0.235907
$576$	0	0
$577$	33.3137	1.38687	0.693434	−	0.720520i	$-0.256097\pi$
$577$	33.3137	1.38687	0.693434	+	0.720520i	$0.256097\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−20.6863	−0.856739
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−22.6274	−0.932346
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.97056	−0.121986	−0.0609932	−	0.998138i	$-0.519427\pi$
$593$	−2.97056	−0.121986	−0.0609932	+	0.998138i	$0.519427\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.686292	−0.0280411	−0.0140206	−	0.999902i	$-0.504463\pi$
$599$	−0.686292	−0.0280411	−0.0140206	+	0.999902i	$0.504463\pi$
$600$	0	0
$601$	24.6274	1.00457	0.502287	−	0.864701i	$-0.332492\pi$
$601$	24.6274	1.00457	0.502287	+	0.864701i	$0.332492\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−21.0000	−0.853771
$606$	0	0
$607$	−25.9411	−1.05292	−0.526459	−	0.850201i	$-0.676481\pi$
$607$	−25.9411	−1.05292	−0.526459	+	0.850201i	$0.676481\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.68629	0.189587
$612$	0	0
$613$	−26.9706	−1.08933	−0.544665	−	0.838653i	$-0.683343\pi$
$613$	−26.9706	−1.08933	−0.544665	+	0.838653i	$0.683343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.9706	−1.24683	−0.623414	−	0.781892i	$-0.714255\pi$
$617$	−30.9706	−1.24683	−0.623414	+	0.781892i	$0.714255\pi$
$618$	0	0
$619$	−40.9706	−1.64675	−0.823373	−	0.567501i	$-0.807910\pi$
$619$	−40.9706	−1.64675	−0.823373	+	0.567501i	$0.807910\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	42.6274	1.69967
$630$	0	0
$631$	−14.6274	−0.582308	−0.291154	−	0.956676i	$-0.594039\pi$
$631$	−14.6274	−0.582308	−0.291154	+	0.956676i	$0.594039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.3137	0.766441
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.6274	−1.28871	−0.644353	−	0.764728i	$-0.722873\pi$
$641$	−32.6274	−1.28871	−0.644353	+	0.764728i	$0.722873\pi$
$642$	0	0
$643$	7.31371	0.288425	0.144212	−	0.989547i	$-0.453935\pi$
$643$	7.31371	0.288425	0.144212	+	0.989547i	$0.453935\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.6569	−0.536906	−0.268453	−	0.963293i	$-0.586512\pi$
$647$	−13.6569	−0.536906	−0.268453	+	0.963293i	$0.586512\pi$
$648$	0	0
$649$	−22.6274	−0.888204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.9706	0.585843	0.292922	−	0.956136i	$-0.405372\pi$
$653$	14.9706	0.585843	0.292922	+	0.956136i	$0.405372\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.9706	0.661079	0.330540	−	0.943792i	$-0.392769\pi$
$659$	16.9706	0.661079	0.330540	+	0.943792i	$0.392769\pi$
$660$	0	0
$661$	−46.2843	−1.80025	−0.900125	−	0.435632i	$-0.856525\pi$
$661$	−46.2843	−1.80025	−0.900125	+	0.435632i	$0.856525\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.65685	0.219363
$666$	0	0
$667$	−20.6863	−0.800976
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.94113	−0.0749363
$672$	0	0
$673$	−49.3137	−1.90090	−0.950452	−	0.310872i	$-0.899379\pi$
$673$	−49.3137	−1.90090	−0.950452	+	0.310872i	$0.899379\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−41.3137	−1.58781	−0.793907	−	0.608039i	$-0.791957\pi$
$677$	−41.3137	−1.58781	−0.793907	+	0.608039i	$0.791957\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.3431	−0.854937	−0.427468	−	0.904030i	$-0.640594\pi$
$683$	−22.3431	−0.854937	−0.427468	+	0.904030i	$0.640594\pi$
$684$	0	0
$685$	−4.34315	−0.165943
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.31371	0.278630
$690$	0	0
$691$	−8.97056	−0.341256	−0.170628	−	0.985335i	$-0.554580\pi$
$691$	−8.97056	−0.341256	−0.170628	+	0.985335i	$0.554580\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.6569	0.518034
$696$	0	0
$697$	−7.31371	−0.277026
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.65685	−0.138117	−0.0690587	−	0.997613i	$-0.522000\pi$
$701$	−3.65685	−0.138117	−0.0690587	+	0.997613i	$0.522000\pi$
$702$	0	0
$703$	−65.9411	−2.48702
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.3137	−0.651149
$708$	0	0
$709$	17.3137	0.650230	0.325115	−	0.945674i	$-0.394597\pi$
$709$	17.3137	0.650230	0.325115	+	0.945674i	$0.394597\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.6274	0.847403
$714$	0	0
$715$	11.3137	0.423109
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−35.3137	−1.31698	−0.658490	−	0.752590i	$-0.728804\pi$
$719$	−35.3137	−1.31698	−0.658490	+	0.752590i	$0.728804\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.65685	−0.135812
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.05887	0.224096
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.6274	2.01223
$738$	0	0
$739$	23.3137	0.857609	0.428804	−	0.903397i	$-0.358935\pi$
$739$	23.3137	0.857609	0.428804	+	0.903397i	$0.358935\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.6569	−0.501021	−0.250511	−	0.968114i	$-0.580599\pi$
$743$	−13.6569	−0.501021	−0.250511	+	0.968114i	$0.580599\pi$
$744$	0	0
$745$	−18.9706	−0.695028
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.34315	0.231774
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	48.9117	1.77773	0.888863	−	0.458174i	$-0.151496\pi$
$757$	48.9117	1.77773	0.888863	+	0.458174i	$0.151496\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.9411	1.15786	0.578932	−	0.815376i	$-0.303469\pi$
$761$	31.9411	1.15786	0.578932	+	0.815376i	$0.303469\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	27.9411	1.00758	0.503791	−	0.863825i	$-0.331938\pi$
$769$	27.9411	1.00758	0.503791	+	0.863825i	$0.331938\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.3137	0.405356
$780$	0	0
$781$	−41.3726	−1.48043
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	16.6863	0.594802	0.297401	−	0.954753i	$-0.403880\pi$
$787$	16.6863	0.594802	0.297401	+	0.954753i	$0.403880\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.34315	0.154424
$792$	0	0
$793$	0.686292	0.0243709
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.3137	−0.613283	−0.306642	−	0.951825i	$-0.599205\pi$
$797$	−17.3137	−0.613283	−0.306642	+	0.951825i	$0.599205\pi$
$798$	0	0
$799$	8.56854	0.303133
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−33.9411	−1.19776
$804$	0	0
$805$	−5.65685	−0.199378
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.62742	0.162691	0.0813457	−	0.996686i	$-0.474078\pi$
$809$	4.62742	0.162691	0.0813457	+	0.996686i	$0.474078\pi$
$810$	0	0
$811$	−13.6569	−0.479557	−0.239779	−	0.970828i	$-0.577075\pi$
$811$	−13.6569	−0.479557	−0.239779	+	0.970828i	$0.577075\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.6569	0.618493
$816$	0	0
$817$	−9.37258	−0.327905
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.2843	1.33613	0.668065	−	0.744103i	$-0.267123\pi$
$821$	38.2843	1.33613	0.668065	+	0.744103i	$0.267123\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.3431	−1.05513	−0.527567	−	0.849513i	$-0.676896\pi$
$827$	−30.3431	−1.05513	−0.527567	+	0.849513i	$0.676896\pi$
$828$	0	0
$829$	−23.6569	−0.821637	−0.410818	−	0.911717i	$-0.634757\pi$
$829$	−23.6569	−0.821637	−0.410818	+	0.911717i	$0.634757\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.65685	0.126702
$834$	0	0
$835$	−13.6569	−0.472615
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.3137	0.942974	0.471487	−	0.881873i	$-0.343717\pi$
$839$	27.3137	0.942974	0.471487	+	0.881873i	$0.343717\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	21.0000	0.721569
$848$	0	0
$849$	0	0
$850$	0	0
$851$	65.9411	2.26043
$852$	0	0
$853$	27.9411	0.956686	0.478343	−	0.878173i	$-0.341238\pi$
$853$	27.9411	0.956686	0.478343	+	0.878173i	$0.341238\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.3431	−1.51473	−0.757367	−	0.652990i	$-0.773514\pi$
$857$	−44.3431	−1.51473	−0.757367	+	0.652990i	$0.773514\pi$
$858$	0	0
$859$	48.9706	1.67085	0.835427	−	0.549601i	$-0.185220\pi$
$859$	48.9706	1.67085	0.835427	+	0.549601i	$0.185220\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.3431	0.624408	0.312204	−	0.950015i	$-0.398933\pi$
$863$	18.3431	0.624408	0.312204	+	0.950015i	$0.398933\pi$
$864$	0	0
$865$	−21.3137	−0.724688
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−19.3137	−0.654420
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	46.9706	1.58608	0.793042	−	0.609167i	$-0.208496\pi$
$877$	46.9706	1.58608	0.793042	+	0.609167i	$0.208496\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.627417	−0.0211382	−0.0105691	−	0.999944i	$-0.503364\pi$
$881$	−0.627417	−0.0211382	−0.0105691	+	0.999944i	$0.503364\pi$
$882$	0	0
$883$	33.6569	1.13264	0.566322	−	0.824184i	$-0.308366\pi$
$883$	33.6569	1.13264	0.566322	+	0.824184i	$0.308366\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.2843	1.48692	0.743460	−	0.668780i	$-0.233183\pi$
$887$	44.2843	1.48692	0.743460	+	0.668780i	$0.233183\pi$
$888$	0	0
$889$	−19.3137	−0.647761
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.2548	−0.443556
$894$	0	0
$895$	13.6569	0.456498
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.6274	−0.487852
$900$	0	0
$901$	13.3726	0.445505
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.6569	−0.387487
$906$	0	0
$907$	−28.9706	−0.961952	−0.480976	−	0.876734i	$-0.659718\pi$
$907$	−28.9706	−0.961952	−0.480976	+	0.876734i	$0.659718\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.9411	0.991994	0.495997	−	0.868324i	$-0.334803\pi$
$911$	29.9411	0.991994	0.495997	+	0.868324i	$0.334803\pi$
$912$	0	0
$913$	−22.6274	−0.748858
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	−41.9411	−1.38351	−0.691755	−	0.722132i	$-0.743162\pi$
$919$	−41.9411	−1.38351	−0.691755	+	0.722132i	$0.743162\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.6274	0.481467
$924$	0	0
$925$	11.6569	0.383275
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.6863	0.350606	0.175303	−	0.984515i	$-0.443910\pi$
$929$	10.6863	0.350606	0.175303	+	0.984515i	$0.443910\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.6863	0.676514
$936$	0	0
$937$	26.6863	0.871803	0.435902	−	0.899994i	$-0.356430\pi$
$937$	26.6863	0.871803	0.435902	+	0.899994i	$0.356430\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−11.3137	−0.368425
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.2843	−0.529168	−0.264584	−	0.964363i	$-0.585235\pi$
$947$	−16.2843	−0.529168	−0.264584	+	0.964363i	$0.585235\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.65685	−0.118457	−0.0592286	−	0.998244i	$-0.518864\pi$
$953$	−3.65685	−0.118457	−0.0592286	+	0.998244i	$0.518864\pi$
$954$	0	0
$955$	15.3137	0.495540
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.34315	0.140247
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.3137	−0.428583
$966$	0	0
$967$	41.9411	1.34874	0.674368	−	0.738396i	$-0.264416\pi$
$967$	41.9411	1.34874	0.674368	+	0.738396i	$0.264416\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	58.6274	1.88144	0.940722	−	0.339180i	$-0.110149\pi$
$971$	58.6274	1.88144	0.940722	+	0.339180i	$0.110149\pi$
$972$	0	0
$973$	−13.6569	−0.437819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.6569	1.52468	0.762339	−	0.647178i	$-0.224051\pi$
$977$	47.6569	1.52468	0.762339	+	0.647178i	$0.224051\pi$
$978$	0	0
$979$	−79.1960	−2.53111
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.6569	0.690746	0.345373	−	0.938465i	$-0.387752\pi$
$983$	21.6569	0.690746	0.345373	+	0.938465i	$0.387752\pi$
$984$	0	0
$985$	−11.6569	−0.371418
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.37258	0.298031
$990$	0	0
$991$	11.3137	0.359392	0.179696	−	0.983722i	$-0.442489\pi$
$991$	11.3137	0.359392	0.179696	+	0.983722i	$0.442489\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	−17.3137	−0.548331	−0.274165	−	0.961683i	$-0.588402\pi$
$997$	−17.3137	−0.548331	−0.274165	+	0.961683i	$0.588402\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.a.v.1.1		2
3.2	odd	2		840.2.a.k.1.2	✓	2
4.3	odd	2		5040.2.a.br.1.2		2
12.11	even	2		1680.2.a.u.1.1		2
15.2	even	4		4200.2.t.u.1849.2		4
15.8	even	4		4200.2.t.u.1849.4		4
15.14	odd	2		4200.2.a.bg.1.2		2
21.20	even	2		5880.2.a.bl.1.2		2
24.5	odd	2		6720.2.a.co.1.1		2
24.11	even	2		6720.2.a.cu.1.2		2
60.59	even	2		8400.2.a.de.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.a.k.1.2	✓	2	3.2	odd	2
1680.2.a.u.1.1		2	12.11	even	2
2520.2.a.v.1.1		2	1.1	even	1	trivial
4200.2.a.bg.1.2		2	15.14	odd	2
4200.2.t.u.1849.2		4	15.2	even	4
4200.2.t.u.1849.4		4	15.8	even	4
5040.2.a.br.1.2		2	4.3	odd	2
5880.2.a.bl.1.2		2	21.20	even	2
6720.2.a.co.1.1		2	24.5	odd	2
6720.2.a.cu.1.2		2	24.11	even	2
8400.2.a.de.1.1		2	60.59	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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.00000 q^{7} -5.65685 q^{11} +2.00000 q^{13} +3.65685 q^{17} -5.65685 q^{19} +5.65685 q^{23} +1.00000 q^{25} -3.65685 q^{29} +4.00000 q^{31} -1.00000 q^{35} +11.6569 q^{37} -2.00000 q^{41} +1.65685 q^{43} +2.34315 q^{47} +1.00000 q^{49} +3.65685 q^{53} +5.65685 q^{55} +4.00000 q^{59} +0.343146 q^{61} -2.00000 q^{65} -9.65685 q^{67} +7.31371 q^{71} +6.00000 q^{73} -5.65685 q^{77} +11.3137 q^{79} +4.00000 q^{83} -3.65685 q^{85} +14.0000 q^{89} +2.00000 q^{91} +5.65685 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{13} - 4 q^{17} + 2 q^{25} + 4 q^{29} + 8 q^{31} - 2 q^{35} + 12 q^{37} - 4 q^{41} - 8 q^{43} + 16 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} + 12 q^{61} - 4 q^{65} - 8 q^{67} - 8 q^{71} + 12 q^{73} + 8 q^{83} + 4 q^{85} + 28 q^{89} + 4 q^{91} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 4 * q^13 - 4 * q^17 + 2 * q^25 + 4 * q^29 + 8 * q^31 - 2 * q^35 + 12 * q^37 - 4 * q^41 - 8 * q^43 + 16 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^59 + 12 * q^61 - 4 * q^65 - 8 * q^67 - 8 * q^71 + 12 * q^73 + 8 * q^83 + 4 * q^85 + 28 * q^89 + 4 * q^91 + 12 * q^97

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