LMFDB - Embedded newform 8880.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.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:	$70.9071569949$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} -6.00000 q^{19} +1.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} +9.00000 q^{29} -3.00000 q^{31} -5.00000 q^{33} +1.00000 q^{35} -1.00000 q^{37} -2.00000 q^{39} +9.00000 q^{41} -1.00000 q^{43} -1.00000 q^{45} -6.00000 q^{49} -1.00000 q^{51} +9.00000 q^{53} -5.00000 q^{55} +6.00000 q^{57} +10.0000 q^{59} -5.00000 q^{61} -1.00000 q^{63} -2.00000 q^{65} -16.0000 q^{67} +12.0000 q^{73} -1.00000 q^{75} -5.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} -2.00000 q^{83} -1.00000 q^{85} -9.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} +3.00000 q^{93} +6.00000 q^{95} +17.0000 q^{97} +5.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	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
$15$	1.00000	0.258199
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	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
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	−13.0000	−1.22294	−0.611469	−	0.791269i	$-0.709421\pi$
$113$	−13.0000	−1.22294	−0.611469	+	0.791269i	$0.709421\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.0000	0.836242
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	−7.00000	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$173$	−7.00000	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	5.00000	0.365636
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−9.00000	−0.631676
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	−12.0000	−0.810885
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−17.0000	−0.996558
$292$	0	0
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	30.0000	1.71219	0.856095	−	0.516818i	$-0.172884\pi$
$307$	30.0000	1.71219	0.856095	+	0.516818i	$0.172884\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	0	0
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−7.00000	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$317$	−7.00000	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$318$	0	0
$319$	45.0000	2.51952
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	5.00000	0.276501
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	13.0000	0.706063
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	1.00000	0.0532246	0.0266123	−	0.999646i	$-0.491528\pi$
$353$	1.00000	0.0532246	0.0266123	+	0.999646i	$0.491528\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−12.0000	−0.628109
$366$	0	0
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	0	0
$369$	9.00000	0.468521
$370$	0	0
$371$	−9.00000	−0.467257
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	18.0000	0.927047
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	5.00000	0.254824
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	10.0000	0.504433
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−5.00000	−0.247841
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	−10.0000	−0.492068
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	−19.0000	−0.930434
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	5.00000	0.241967
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	36.0000	1.73005	0.865025	−	0.501729i	$-0.167303\pi$
$433$	36.0000	1.73005	0.865025	+	0.501729i	$0.167303\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	0	0
$438$	0	0
$439$	27.0000	1.28864	0.644320	−	0.764756i	$-0.277141\pi$
$439$	27.0000	1.28864	0.644320	+	0.764756i	$0.277141\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	45.0000	2.11897
$452$	0	0
$453$	18.0000	0.845714
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	35.0000	1.63011	0.815056	−	0.579382i	$-0.196706\pi$
$461$	35.0000	1.63011	0.815056	+	0.579382i	$0.196706\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	−7.00000	−0.322543
$472$	0	0
$473$	−5.00000	−0.229900
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	9.00000	0.412082
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.0000	−0.771930
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	−5.00000	−0.224733
$496$	0	0
$497$	0	0
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.00000	0.307266
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−3.00000	−0.130682
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	18.0000	0.779667
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	−30.0000	−1.29219
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	0	0
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−54.0000	−2.30048
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	−13.0000	−0.547885	−0.273942	−	0.961746i	$-0.588328\pi$
$563$	−13.0000	−0.547885	−0.273942	+	0.961746i	$0.588328\pi$
$564$	0	0
$565$	13.0000	0.546914
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	−35.0000	−1.46470	−0.732352	−	0.680926i	$-0.761578\pi$
$571$	−35.0000	−1.46470	−0.732352	+	0.680926i	$0.761578\pi$
$572$	0	0
$573$	11.0000	0.459532
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	45.0000	1.86371
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	−14.0000	−0.575883
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	0	0
$615$	9.00000	0.362915
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	30.0000	1.19808
$628$	0	0
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	50.0000	1.96267
$650$	0	0
$651$	−3.00000	−0.117579
$652$	0	0
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−5.00000	−0.193311
$670$	0	0
$671$	−25.0000	−0.965114
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	−7.00000	−0.268241
$682$	0	0
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18.0000	0.685745
$690$	0	0
$691$	1.00000	0.0380418	0.0190209	−	0.999819i	$-0.493945\pi$
$691$	1.00000	0.0380418	0.0190209	+	0.999819i	$0.493945\pi$
$692$	0	0
$693$	−5.00000	−0.189934
$694$	0	0
$695$	−19.0000	−0.720711
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−10.0000	−0.373979
$716$	0	0
$717$	−21.0000	−0.784259
$718$	0	0
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	18.0000	0.669427
$724$	0	0
$725$	9.00000	0.334252
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	−47.0000	−1.73598	−0.867992	−	0.496578i	$-0.834590\pi$
$733$	−47.0000	−1.73598	−0.867992	+	0.496578i	$0.834590\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−80.0000	−2.94684
$738$	0	0
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	−1.00000	−0.0366864	−0.0183432	−	0.999832i	$-0.505839\pi$
$743$	−1.00000	−0.0366864	−0.0183432	+	0.999832i	$0.505839\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	18.0000	0.655087
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	0	0
$763$	5.00000	0.181012
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	−3.00000	−0.107763
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−54.0000	−1.93475
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	−7.00000	−0.249841
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	−15.0000	−0.534014
$790$	0	0
$791$	13.0000	0.462227
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	9.00000	0.319197
$796$	0	0
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	60.0000	2.11735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	1.00000	0.0350285
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	−5.00000	−0.174078
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	−17.0000	−0.590434	−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000	−0.590434	−0.295217	+	0.955430i	$0.595392\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	20.0000	0.692129
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	2.00000	0.0688837
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	7.00000	0.238007
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	60.0000	2.03536
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	17.0000	0.575363
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	47.0000	1.58708	0.793539	−	0.608520i	$-0.208236\pi$
$877$	47.0000	1.58708	0.793539	+	0.608520i	$0.208236\pi$
$878$	0	0
$879$	−19.0000	−0.640854
$880$	0	0
$881$	41.0000	1.38133	0.690663	−	0.723177i	$-0.257319\pi$
$881$	41.0000	1.38133	0.690663	+	0.723177i	$0.257319\pi$
$882$	0	0
$883$	9.00000	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$884$	0	0
$885$	10.0000	0.336146
$886$	0	0
$887$	35.0000	1.17518	0.587592	−	0.809157i	$-0.300076\pi$
$887$	35.0000	1.17518	0.587592	+	0.809157i	$0.300076\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.0000	−0.900500
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	0	0
$915$	−5.00000	−0.165295
$916$	0	0
$917$	10.0000	0.330229
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	−25.0000	−0.818463
$934$	0	0
$935$	−5.00000	−0.163517
$936$	0	0
$937$	56.0000	1.82944	0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000	1.82944	0.914720	+	0.404088i	$0.132411\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−1.00000	−0.0324956	−0.0162478	−	0.999868i	$-0.505172\pi$
$947$	−1.00000	−0.0324956	−0.0162478	+	0.999868i	$0.505172\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	7.00000	0.226991
$952$	0	0
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	0	0
$955$	11.0000	0.355952
$956$	0	0
$957$	−45.0000	−1.45464
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	−29.0000	−0.930654	−0.465327	−	0.885139i	$-0.654063\pi$
$971$	−29.0000	−0.930654	−0.465327	+	0.885139i	$0.654063\pi$
$972$	0	0
$973$	−19.0000	−0.609112
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−5.00000	−0.159638
$982$	0	0
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	31.0000	0.984747	0.492374	−	0.870384i	$-0.336129\pi$
$991$	31.0000	0.984747	0.492374	+	0.870384i	$0.336129\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.d.1.1		1
4.3	odd	2		1110.2.a.f.1.1	✓	1
12.11	even	2		3330.2.a.y.1.1		1
20.19	odd	2		5550.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.f.1.1	✓	1	4.3	odd	2
3330.2.a.y.1.1		1	12.11	even	2
5550.2.a.y.1.1		1	20.19	odd	2
8880.2.a.d.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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