LMFDB - Embedded newform 2100.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.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:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} +2.00000 q^{17} -2.00000 q^{19} -1.00000 q^{21} -6.00000 q^{23} -1.00000 q^{27} +6.00000 q^{29} +6.00000 q^{31} -4.00000 q^{33} -4.00000 q^{37} -2.00000 q^{39} -4.00000 q^{43} +4.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -2.00000 q^{53} +2.00000 q^{57} +4.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -12.0000 q^{67} +6.00000 q^{69} -8.00000 q^{71} +14.0000 q^{73} +4.00000 q^{77} +16.0000 q^{79} +1.00000 q^{81} +16.0000 q^{83} -6.00000 q^{87} +16.0000 q^{89} +2.00000 q^{91} -6.00000 q^{93} -14.0000 q^{97} +4.00000 q^{99} +O(q^{100})$

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
$5$	0	0
$5$	0	0
$6$	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
$15$	0	0
$16$	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
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	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	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	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	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	0	0
$96$	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
$99$	4.00000	0.402015
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	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
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	0	0
$136$	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
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	0	0
$176$	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
$181$	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
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	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
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−16.0000	−0.979184
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	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	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.0000	−0.995402
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	0	0
$356$	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
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0000	0.574038
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.00000	−0.283790
$448$	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
$453$	16.0000	0.751746
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.0000	−0.409273
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−30.0000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−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
$655$	0	0
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	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
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36.0000	−1.39393
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−8.00000	−0.308837
$672$	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
$675$	0	0
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−36.0000	−1.34821
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−26.0000	−0.966950
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	34.0000	1.24734	0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000	1.24734	0.623670	+	0.781688i	$0.285641\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.0000	0.585409
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	18.0000	0.651644
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	0	0
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	−30.0000	−1.06803
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	16.0000	0.565332
$802$	0	0
$803$	56.0000	1.97620
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	22.0000	0.771574
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.00000	−0.207390
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	64.0000	2.17105
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	0	0
$879$	22.0000	0.742042
$880$	0	0
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	36.0000	1.20067
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	64.0000	2.11809
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	20.0000	0.654771
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	26.0000	0.843108
$952$	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
$955$	0	0
$956$	0	0
$957$	−24.0000	−0.775810
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	18.0000	0.580042
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	64.0000	2.04545
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.00000	−0.127321
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	0	0
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.a.i.1.1		1
3.2	odd	2		6300.2.a.r.1.1		1
4.3	odd	2		8400.2.a.bm.1.1		1
5.2	odd	4		420.2.k.b.169.2	yes	2
5.3	odd	4		420.2.k.b.169.1	✓	2
5.4	even	2		2100.2.a.n.1.1		1
15.2	even	4		1260.2.k.a.1009.2		2
15.8	even	4		1260.2.k.a.1009.1		2
15.14	odd	2		6300.2.a.b.1.1		1
20.3	even	4		1680.2.t.g.1009.2		2
20.7	even	4		1680.2.t.g.1009.1		2
20.19	odd	2		8400.2.a.o.1.1		1
35.2	odd	12		2940.2.bb.a.949.1		4
35.3	even	12		2940.2.bb.f.1549.2		4
35.12	even	12		2940.2.bb.f.949.2		4
35.13	even	4		2940.2.k.b.589.2		2
35.17	even	12		2940.2.bb.f.1549.1		4
35.18	odd	12		2940.2.bb.a.1549.1		4
35.23	odd	12		2940.2.bb.a.949.2		4
35.27	even	4		2940.2.k.b.589.1		2
35.32	odd	12		2940.2.bb.a.1549.2		4
35.33	even	12		2940.2.bb.f.949.1		4
60.23	odd	4		5040.2.t.d.1009.1		2
60.47	odd	4		5040.2.t.d.1009.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.k.b.169.1	✓	2	5.3	odd	4
420.2.k.b.169.2	yes	2	5.2	odd	4
1260.2.k.a.1009.1		2	15.8	even	4
1260.2.k.a.1009.2		2	15.2	even	4
1680.2.t.g.1009.1		2	20.7	even	4
1680.2.t.g.1009.2		2	20.3	even	4
2100.2.a.i.1.1		1	1.1	even	1	trivial
2100.2.a.n.1.1		1	5.4	even	2
2940.2.k.b.589.1		2	35.27	even	4
2940.2.k.b.589.2		2	35.13	even	4
2940.2.bb.a.949.1		4	35.2	odd	12
2940.2.bb.a.949.2		4	35.23	odd	12
2940.2.bb.a.1549.1		4	35.18	odd	12
2940.2.bb.a.1549.2		4	35.32	odd	12
2940.2.bb.f.949.1		4	35.33	even	12
2940.2.bb.f.949.2		4	35.12	even	12
2940.2.bb.f.1549.1		4	35.17	even	12
2940.2.bb.f.1549.2		4	35.3	even	12
5040.2.t.d.1009.1		2	60.23	odd	4
5040.2.t.d.1009.2		2	60.47	odd	4
6300.2.a.b.1.1		1	15.14	odd	2
6300.2.a.r.1.1		1	3.2	odd	2
8400.2.a.o.1.1		1	20.19	odd	2
8400.2.a.bm.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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.a (trivial)

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L-functions

Modular forms

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Fields

Representations

Groups

Database

Properties

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