LMFDB - Embedded newform 6720.2.a.t.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3360)
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)

$f(q)$

$=$

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

$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} -1.00000 q^{15} -6.00000 q^{17} +4.00000 q^{19} +1.00000 q^{21} +8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -8.00000 q^{31} -1.00000 q^{35} +2.00000 q^{37} +2.00000 q^{39} -6.00000 q^{41} +1.00000 q^{45} +1.00000 q^{49} +6.00000 q^{51} -6.00000 q^{53} -4.00000 q^{57} +4.00000 q^{59} +6.00000 q^{61} -1.00000 q^{63} -2.00000 q^{65} +8.00000 q^{67} -8.00000 q^{69} -14.0000 q^{73} -1.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} +4.00000 q^{83} -6.00000 q^{85} -2.00000 q^{87} +2.00000 q^{89} +2.00000 q^{91} +8.00000 q^{93} +4.00000 q^{95} -14.0000 q^{97} +O(q^{100})$

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	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
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	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
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	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
$49$	1.00000	0.142857
$50$	0	0
$51$	6.00000	0.840168
$52$	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
$55$	0	0
$56$	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
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	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
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$85$	−6.00000	−0.650791
$86$	0	0
$87$	−2.00000	−0.214423
$88$	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
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	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
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000	0.541002
$124$	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
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.00000	0.166091
$146$	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
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	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
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	14.0000	0.946032
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	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
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	14.0000	0.760376
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.00000	−0.317554
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	4.00000	0.195881
$418$	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
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	32.0000	1.53077
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	−8.00000	−0.340195
$554$	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
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−10.0000	−0.411345
$592$	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
$595$	6.00000	0.245976
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	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
$627$	0	0
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	0	0
$646$	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
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	22.0000	0.832116
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0000	0.376089
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	−64.0000	−2.39682
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−26.0000	−0.966950
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	0	0
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	−16.0000	−0.569615
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	26.0000	0.915243
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	0	0
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	−42.0000	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	28.0000	0.960958
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−14.0000	−0.473828
$874$	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
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−16.0000	−0.533630
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.00000	−0.198354
$916$	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
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−32.0000	−1.04763
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	30.0000	0.972817
$952$	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
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	50.0000	1.59964	0.799821	−	0.600239i	$-0.204928\pi$
$977$	50.0000	1.59964	0.799821	+	0.600239i	$0.204928\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	−24.0000	−0.761617
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	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.t.1.1		1
4.3	odd	2		6720.2.a.cl.1.1		1
8.3	odd	2		3360.2.a.e.1.1	✓	1
8.5	even	2		3360.2.a.o.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.e.1.1	✓	1	8.3	odd	2
3360.2.a.o.1.1	yes	1	8.5	even	2
6720.2.a.t.1.1		1	1.1	even	1	trivial
6720.2.a.cl.1.1		1	4.3	odd	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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