LMFDB - Embedded newform 6321.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6321,2,Mod(1,6321)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6321, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6321.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6321 = 3 \cdot 7^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6321.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:	$50.4734391177$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 129)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -2.00000 q^{4} +2.00000 q^{5} +1.00000 q^{9} -5.00000 q^{11} -2.00000 q^{12} -3.00000 q^{13} +2.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{19} -4.00000 q^{20} -1.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} +5.00000 q^{31} -5.00000 q^{33} -2.00000 q^{36} +8.00000 q^{37} -3.00000 q^{39} +7.00000 q^{41} -1.00000 q^{43} +10.0000 q^{44} +2.00000 q^{45} +8.00000 q^{47} +4.00000 q^{48} +3.00000 q^{51} +6.00000 q^{52} +3.00000 q^{53} -10.0000 q^{55} -2.00000 q^{57} -12.0000 q^{59} -4.00000 q^{60} +8.00000 q^{61} -8.00000 q^{64} -6.00000 q^{65} -15.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} -14.0000 q^{71} -12.0000 q^{73} -1.00000 q^{75} +4.00000 q^{76} -16.0000 q^{79} +8.00000 q^{80} +1.00000 q^{81} -15.0000 q^{83} +6.00000 q^{85} -10.0000 q^{89} +2.00000 q^{92} +5.00000 q^{93} -4.00000 q^{95} -11.0000 q^{97} -5.00000 q^{99} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−2.00000	−0.577350
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	4.00000	1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	10.0000	1.50756
$45$	2.00000	0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	0	0
$51$	3.00000	0.420084
$52$	6.00000	0.832050
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−10.0000	−1.34840
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−4.00000	−0.516398
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	−15.0000	−1.83254	−0.916271	−	0.400559i	$-0.868816\pi$
$67$	−15.0000	−1.83254	−0.916271	+	0.400559i	$0.868816\pi$
$68$	−6.00000	−0.727607
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	8.00000	0.894427
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	2.00000	0.208514
$93$	5.00000	0.518476
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−11.0000	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$97$	−11.0000	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	2.00000	0.200000
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−2.00000	−0.192450
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−3.00000	−0.277350
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	7.00000	0.631169
$124$	−10.0000	−0.898027
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	10.0000	0.870388
$133$	0	0
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	15.0000	1.25436
$144$	4.00000	0.333333
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−16.0000	−1.31519
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	10.0000	0.803219
$156$	6.00000	0.480384
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−14.0000	−1.09322
$165$	−10.0000	−0.778499
$166$	0	0
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−2.00000	−0.152944
$172$	2.00000	0.152499
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	0	0
$176$	−20.0000	−1.50756
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	−4.00000	−0.298142
$181$	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
$183$	8.00000	0.591377
$184$	0	0
$185$	16.0000	1.17634
$186$	0	0
$187$	−15.0000	−1.09691
$188$	−16.0000	−1.16692
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−8.00000	−0.577350
$193$	−21.0000	−1.51161	−0.755807	−	0.654795i	$-0.772755\pi$
$193$	−21.0000	−1.51161	−0.755807	+	0.654795i	$0.772755\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	0	0
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−15.0000	−1.05802
$202$	0	0
$203$	0	0
$204$	−6.00000	−0.420084
$205$	14.0000	0.977802
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	−12.0000	−0.832050
$209$	10.0000	0.691714
$210$	0	0
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	−6.00000	−0.412082
$213$	−14.0000	−0.959264
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.0000	−0.810885
$220$	20.0000	1.34840
$221$	−9.00000	−0.605406
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	4.00000	0.264906
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	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
$235$	16.0000	1.04372
$236$	24.0000	1.56227
$237$	−16.0000	−1.03931
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	8.00000	0.516398
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−16.0000	−1.02430
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	−25.0000	−1.57799	−0.788993	−	0.614402i	$-0.789397\pi$
$251$	−25.0000	−1.57799	−0.788993	+	0.614402i	$0.789397\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	6.00000	0.375735
$256$	16.0000	1.00000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0	0
$260$	12.0000	0.744208
$261$	0	0
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−10.0000	−0.611990
$268$	30.0000	1.83254
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	17.0000	1.03268	0.516338	−	0.856385i	$-0.327295\pi$
$271$	17.0000	1.03268	0.516338	+	0.856385i	$0.327295\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	0	0
$275$	5.00000	0.301511
$276$	2.00000	0.120386
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	−1.00000	−0.0594438	−0.0297219	−	0.999558i	$-0.509462\pi$
$283$	−1.00000	−0.0594438	−0.0297219	+	0.999558i	$0.509462\pi$
$284$	28.0000	1.66149
$285$	−4.00000	−0.236940
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−11.0000	−0.644831
$292$	24.0000	1.40449
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	3.00000	0.173494
$300$	2.00000	0.115470
$301$	0	0
$302$	0	0
$303$	9.00000	0.517036
$304$	−8.00000	−0.458831
$305$	16.0000	0.916157
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	0	0
$311$	5.00000	0.283524	0.141762	−	0.989901i	$-0.454723\pi$
$311$	5.00000	0.283524	0.141762	+	0.989901i	$0.454723\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	0	0
$316$	32.0000	1.80014
$317$	5.00000	0.280828	0.140414	−	0.990093i	$-0.455157\pi$
$317$	5.00000	0.280828	0.140414	+	0.990093i	$0.455157\pi$
$318$	0	0
$319$	0	0
$320$	−16.0000	−0.894427
$321$	0	0
$322$	0	0
$323$	−6.00000	−0.333849
$324$	−2.00000	−0.111111
$325$	3.00000	0.166410
$326$	0	0
$327$	11.0000	0.608301
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	30.0000	1.64646
$333$	8.00000	0.438397
$334$	0	0
$335$	−30.0000	−1.63908
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	−12.0000	−0.650791
$341$	−25.0000	−1.35383
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	32.0000	1.71292	0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000	1.71292	0.856460	+	0.516213i	$0.172659\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	−28.0000	−1.48609
$356$	20.0000	1.06000
$357$	0	0
$358$	0	0
$359$	−5.00000	−0.263890	−0.131945	−	0.991257i	$-0.542122\pi$
$359$	−5.00000	−0.263890	−0.131945	+	0.991257i	$0.542122\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	14.0000	0.734809
$364$	0	0
$365$	−24.0000	−1.25622
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−4.00000	−0.208514
$369$	7.00000	0.364405
$370$	0	0
$371$	0	0
$372$	−10.0000	−0.518476
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	8.00000	0.410391
$381$	−3.00000	−0.153695
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	22.0000	1.11688
$389$	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
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	−32.0000	−1.61009
$396$	10.0000	0.502519
$397$	−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
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	−18.0000	−0.895533
$405$	2.00000	0.0993808
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	10.0000	0.492665
$413$	0	0
$414$	0	0
$415$	−30.0000	−1.47264
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	−34.0000	−1.66101	−0.830504	−	0.557012i	$-0.811948\pi$
$419$	−34.0000	−1.66101	−0.830504	+	0.557012i	$0.811948\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	0	0
$428$	0	0
$429$	15.0000	0.724207
$430$	0	0
$431$	31.0000	1.49322	0.746609	−	0.665263i	$-0.231681\pi$
$431$	31.0000	1.49322	0.746609	+	0.665263i	$0.231681\pi$
$432$	4.00000	0.192450
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	−22.0000	−1.05361
$437$	2.00000	0.0956730
$438$	0	0
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−16.0000	−0.759326
$445$	−20.0000	−0.948091
$446$	0	0
$447$	−16.0000	−0.756774
$448$	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
$451$	−35.0000	−1.64809
$452$	8.00000	0.376288
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	4.00000	0.186501
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	0	0
$467$	14.0000	0.647843	0.323921	−	0.946084i	$-0.394999\pi$
$467$	14.0000	0.647843	0.323921	+	0.946084i	$0.394999\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	5.00000	0.229900
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	−13.0000	−0.593985	−0.296993	−	0.954880i	$-0.595984\pi$
$479$	−13.0000	−0.593985	−0.296993	+	0.954880i	$0.595984\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	0	0
$484$	−28.0000	−1.27273
$485$	−22.0000	−0.998969
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	−14.0000	−0.631169
$493$	0	0
$494$	0	0
$495$	−10.0000	−0.449467
$496$	20.0000	0.898027
$497$	0	0
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	24.0000	1.07331
$501$	1.00000	0.0446767
$502$	0	0
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	−4.00000	−0.177646
$508$	6.00000	0.266207
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	−10.0000	−0.440653
$516$	2.00000	0.0880451
$517$	−40.0000	−1.75920
$518$	0	0
$519$	10.0000	0.438951
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	32.0000	1.39793
$525$	0	0
$526$	0	0
$527$	15.0000	0.653410
$528$	−20.0000	−0.870388
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−21.0000	−0.909611
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−26.0000	−1.12198
$538$	0	0
$539$	0	0
$540$	−4.00000	−0.172133
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	22.0000	0.942376
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	−36.0000	−1.53784
$549$	8.00000	0.341432
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	16.0000	0.679162
$556$	−10.0000	−0.424094
$557$	−43.0000	−1.82197	−0.910984	−	0.412441i	$-0.864676\pi$
$557$	−43.0000	−1.82197	−0.910984	+	0.412441i	$0.864676\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	−1.00000	−0.0421450	−0.0210725	−	0.999778i	$-0.506708\pi$
$563$	−1.00000	−0.0421450	−0.0210725	+	0.999778i	$0.506708\pi$
$564$	−16.0000	−0.673722
$565$	−8.00000	−0.336563
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−30.0000	−1.25436
$573$	−8.00000	−0.334205
$574$	0	0
$575$	1.00000	0.0417029
$576$	−8.00000	−0.333333
$577$	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
$579$	−21.0000	−0.872730
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−15.0000	−0.621237
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	−14.0000	−0.575883
$592$	32.0000	1.31519
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	32.0000	1.31077
$597$	14.0000	0.572982
$598$	0	0
$599$	11.0000	0.449448	0.224724	−	0.974422i	$-0.427852\pi$
$599$	11.0000	0.449448	0.224724	+	0.974422i	$0.427852\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	0	0
$603$	−15.0000	−0.610847
$604$	4.00000	0.162758
$605$	28.0000	1.13836
$606$	0	0
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	−6.00000	−0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	14.0000	0.564534
$616$	0	0
$617$	−25.0000	−1.00646	−0.503231	−	0.864152i	$-0.667856\pi$
$617$	−25.0000	−1.00646	−0.503231	+	0.864152i	$0.667856\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−20.0000	−0.803219
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0	0
$624$	−12.0000	−0.480384
$625$	−19.0000	−0.760000
$626$	0	0
$627$	10.0000	0.399362
$628$	−12.0000	−0.478852
$629$	24.0000	0.956943
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	0	0
$633$	−6.00000	−0.238479
$634$	0	0
$635$	−6.00000	−0.238103
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	−14.0000	−0.553831
$640$	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
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	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
$649$	60.0000	2.35521
$650$	0	0
$651$	0	0
$652$	−32.0000	−1.25322
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	−32.0000	−1.25034
$656$	28.0000	1.09322
$657$	−12.0000	−0.468165
$658$	0	0
$659$	1.00000	0.0389545	0.0194772	−	0.999810i	$-0.493800\pi$
$659$	1.00000	0.0389545	0.0194772	+	0.999810i	$0.493800\pi$
$660$	20.0000	0.778499
$661$	1.00000	0.0388955	0.0194477	−	0.999811i	$-0.493809\pi$
$661$	1.00000	0.0388955	0.0194477	+	0.999811i	$0.493809\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−2.00000	−0.0773823
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	8.00000	0.307692
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	4.00000	0.152944
$685$	36.0000	1.37549
$686$	0	0
$687$	−5.00000	−0.190762
$688$	−4.00000	−0.152499
$689$	−9.00000	−0.342873
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	−20.0000	−0.760286
$693$	0	0
$694$	0	0
$695$	10.0000	0.379322
$696$	0	0
$697$	21.0000	0.795432
$698$	0	0
$699$	−22.0000	−0.832116
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	40.0000	1.50756
$705$	16.0000	0.602595
$706$	0	0
$707$	0	0
$708$	24.0000	0.901975
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	−5.00000	−0.187251
$714$	0	0
$715$	30.0000	1.12194
$716$	52.0000	1.94333
$717$	16.0000	0.597531
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	8.00000	0.298142
$721$	0	0
$722$	0	0
$723$	4.00000	0.148762
$724$	−20.0000	−0.743294
$725$	0	0
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.00000	−0.110959
$732$	−16.0000	−0.591377
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	75.0000	2.76266
$738$	0	0
$739$	−54.0000	−1.98642	−0.993211	−	0.116326i	$-0.962888\pi$
$739$	−54.0000	−1.98642	−0.993211	+	0.116326i	$0.962888\pi$
$740$	−32.0000	−1.17634
$741$	6.00000	0.220416
$742$	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
$745$	−32.0000	−1.17239
$746$	0	0
$747$	−15.0000	−0.548821
$748$	30.0000	1.09691
$749$	0	0
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	32.0000	1.16692
$753$	−25.0000	−0.911051
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	0	0
$759$	5.00000	0.181489
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	0	0
$763$	0	0
$764$	16.0000	0.578860
$765$	6.00000	0.216930
$766$	0	0
$767$	36.0000	1.29988
$768$	16.0000	0.577350
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	42.0000	1.51161
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.0000	−0.501602
$780$	12.0000	0.429669
$781$	70.0000	2.50480
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	28.0000	0.997459
$789$	18.0000	0.640817
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	6.00000	0.212798
$796$	−28.0000	−0.992434
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	60.0000	2.11735
$804$	30.0000	1.05802
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	0	0
$813$	17.0000	0.596216
$814$	0	0
$815$	32.0000	1.12091
$816$	12.0000	0.420084
$817$	2.00000	0.0699711
$818$	0	0
$819$	0	0
$820$	−28.0000	−0.977802
$821$	53.0000	1.84971	0.924856	−	0.380317i	$-0.124185\pi$
$821$	53.0000	1.84971	0.924856	+	0.380317i	$0.124185\pi$
$822$	0	0
$823$	−9.00000	−0.313720	−0.156860	−	0.987621i	$-0.550137\pi$
$823$	−9.00000	−0.313720	−0.156860	+	0.987621i	$0.550137\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	2.00000	0.0695048
$829$	−12.0000	−0.416777	−0.208389	−	0.978046i	$-0.566822\pi$
$829$	−12.0000	−0.416777	−0.208389	+	0.978046i	$0.566822\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	24.0000	0.832050
$833$	0	0
$834$	0	0
$835$	2.00000	0.0692129
$836$	−20.0000	−0.691714
$837$	5.00000	0.172825
$838$	0	0
$839$	38.0000	1.31191	0.655953	−	0.754802i	$-0.272267\pi$
$839$	38.0000	1.31191	0.655953	+	0.754802i	$0.272267\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	3.00000	0.103325
$844$	12.0000	0.413057
$845$	−8.00000	−0.275208
$846$	0	0
$847$	0	0
$848$	12.0000	0.412082
$849$	−1.00000	−0.0343199
$850$	0	0
$851$	−8.00000	−0.274236
$852$	28.0000	0.959264
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	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
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	0	0
$863$	2.00000	0.0680808	0.0340404	−	0.999420i	$-0.489163\pi$
$863$	2.00000	0.0680808	0.0340404	+	0.999420i	$0.489163\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	80.0000	2.71381
$870$	0	0
$871$	45.0000	1.52477
$872$	0	0
$873$	−11.0000	−0.372294
$874$	0	0
$875$	0	0
$876$	24.0000	0.810885
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	−40.0000	−1.34840
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	15.0000	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$883$	15.0000	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$884$	18.0000	0.605406
$885$	−24.0000	−0.806751
$886$	0	0
$887$	58.0000	1.94745	0.973725	−	0.227728i	$-0.0731298\pi$
$887$	58.0000	1.94745	0.973725	+	0.227728i	$0.0731298\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.00000	−0.167506
$892$	28.0000	0.937509
$893$	−16.0000	−0.535420
$894$	0	0
$895$	−52.0000	−1.73817
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	0	0
$900$	2.00000	0.0666667
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−21.0000	−0.697294	−0.348647	−	0.937254i	$-0.613359\pi$
$907$	−21.0000	−0.697294	−0.348647	+	0.937254i	$0.613359\pi$
$908$	−16.0000	−0.530979
$909$	9.00000	0.298511
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−8.00000	−0.264906
$913$	75.0000	2.48214
$914$	0	0
$915$	16.0000	0.528944
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	42.0000	1.38245
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−5.00000	−0.164222
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	0	0
$932$	44.0000	1.44127
$933$	5.00000	0.163693
$934$	0	0
$935$	−30.0000	−0.981105
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	−32.0000	−1.04372
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	−7.00000	−0.227951
$944$	−48.0000	−1.56227
$945$	0	0
$946$	0	0
$947$	−13.0000	−0.422443	−0.211222	−	0.977438i	$-0.567744\pi$
$947$	−13.0000	−0.422443	−0.211222	+	0.977438i	$0.567744\pi$
$948$	32.0000	1.03931
$949$	36.0000	1.16861
$950$	0	0
$951$	5.00000	0.162136
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	−32.0000	−1.03495
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−16.0000	−0.516398
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	−8.00000	−0.257663
$965$	−42.0000	−1.35203
$966$	0	0
$967$	53.0000	1.70437	0.852183	−	0.523245i	$-0.175279\pi$
$967$	53.0000	1.70437	0.852183	+	0.523245i	$0.175279\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	0	0
$975$	3.00000	0.0960769
$976$	32.0000	1.02430
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	0	0
$979$	50.0000	1.59801
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	0	0
$985$	−28.0000	−0.892154
$986$	0	0
$987$	0	0
$988$	−12.0000	−0.381771
$989$	1.00000	0.0317982
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	28.0000	0.887660
$996$	30.0000	0.950586
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6321.2.a.e.1.1		1
7.6	odd	2		129.2.a.a.1.1	✓	1
21.20	even	2		387.2.a.c.1.1		1
28.27	even	2		2064.2.a.k.1.1		1
35.34	odd	2		3225.2.a.g.1.1		1
56.13	odd	2		8256.2.a.bm.1.1		1
56.27	even	2		8256.2.a.s.1.1		1
84.83	odd	2		6192.2.a.v.1.1		1
105.104	even	2		9675.2.a.m.1.1		1
301.300	even	2		5547.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.a.a.1.1	✓	1	7.6	odd	2
387.2.a.c.1.1		1	21.20	even	2
2064.2.a.k.1.1		1	28.27	even	2
3225.2.a.g.1.1		1	35.34	odd	2
5547.2.a.c.1.1		1	301.300	even	2
6192.2.a.v.1.1		1	84.83	odd	2
6321.2.a.e.1.1		1	1.1	even	1	trivial
8256.2.a.s.1.1		1	56.27	even	2
8256.2.a.bm.1.1		1	56.13	odd	2
9675.2.a.m.1.1		1	105.104	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:	\( N \)	\(=\)	\( 6321 = 3 \cdot 7^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6321.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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