LMFDB - Embedded newform 6720.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6720,2,Mod(1,6720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6720, 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("6720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6720.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:	$53.6594701583$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{15} +2.00000 q^{17} -4.00000 q^{19} +1.00000 q^{21} +8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +8.00000 q^{31} -4.00000 q^{33} +1.00000 q^{35} +2.00000 q^{37} -2.00000 q^{39} +2.00000 q^{41} -12.0000 q^{43} -1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} -4.00000 q^{55} +4.00000 q^{57} +4.00000 q^{59} +2.00000 q^{61} -1.00000 q^{63} -2.00000 q^{65} +12.0000 q^{67} -8.00000 q^{69} -8.00000 q^{71} -14.0000 q^{73} -1.00000 q^{75} -4.00000 q^{77} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{85} +6.00000 q^{87} +2.00000 q^{89} -2.00000 q^{91} -8.00000 q^{93} +4.00000 q^{95} +10.0000 q^{97} +4.00000 q^{99} +O(q^{100})$

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
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	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
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	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
$103$	−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
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−2.00000	−0.180334
$124$	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
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	14.0000	0.946032
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	16.0000	0.925304
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	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
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	20.0000	1.11629
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	14.0000	0.760376
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	−12.0000	−0.609994
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−32.0000	−1.53077
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−48.0000	−2.20704
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	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
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	9.00000	0.399704
$508$	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
$511$	14.0000	0.619324
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	32.0000	1.40736
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	20.0000	0.864675
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−48.0000	−1.85857
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	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
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−10.0000	−0.378235
$700$	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
$703$	−8.00000	−0.301726
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	64.0000	2.39682
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	20.0000	0.730784
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−56.0000	−1.97620
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	20.0000	0.668526
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−12.0000	−0.399335
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	−34.0000	−1.10955
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	24.0000	0.775810
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−20.0000	−0.644491
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−96.0000	−3.05262
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.a.k.1.1		1
4.3	odd	2		6720.2.a.bp.1.1		1
8.3	odd	2		210.2.a.c.1.1	✓	1
8.5	even	2		1680.2.a.q.1.1		1
24.5	odd	2		5040.2.a.i.1.1		1
24.11	even	2		630.2.a.b.1.1		1
40.3	even	4		1050.2.g.d.799.1		2
40.19	odd	2		1050.2.a.h.1.1		1
40.27	even	4		1050.2.g.d.799.2		2
40.29	even	2		8400.2.a.p.1.1		1
56.3	even	6		1470.2.i.b.961.1		2
56.11	odd	6		1470.2.i.f.961.1		2
56.19	even	6		1470.2.i.b.361.1		2
56.27	even	2		1470.2.a.q.1.1		1
56.51	odd	6		1470.2.i.f.361.1		2
120.59	even	2		3150.2.a.w.1.1		1
120.83	odd	4		3150.2.g.e.2899.2		2
120.107	odd	4		3150.2.g.e.2899.1		2
168.83	odd	2		4410.2.a.l.1.1		1
280.139	even	2		7350.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.c.1.1	✓	1	8.3	odd	2
630.2.a.b.1.1		1	24.11	even	2
1050.2.a.h.1.1		1	40.19	odd	2
1050.2.g.d.799.1		2	40.3	even	4
1050.2.g.d.799.2		2	40.27	even	4
1470.2.a.q.1.1		1	56.27	even	2
1470.2.i.b.361.1		2	56.19	even	6
1470.2.i.b.961.1		2	56.3	even	6
1470.2.i.f.361.1		2	56.51	odd	6
1470.2.i.f.961.1		2	56.11	odd	6
1680.2.a.q.1.1		1	8.5	even	2
3150.2.a.w.1.1		1	120.59	even	2
3150.2.g.e.2899.1		2	120.107	odd	4
3150.2.g.e.2899.2		2	120.83	odd	4
4410.2.a.l.1.1		1	168.83	odd	2
5040.2.a.i.1.1		1	24.5	odd	2
6720.2.a.k.1.1		1	1.1	even	1	trivial
6720.2.a.bp.1.1		1	4.3	odd	2
7350.2.a.p.1.1		1	280.139	even	2
8400.2.a.p.1.1		1	40.29	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:	\( N \)	\(=\)	\( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6720.a (trivial)

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