LMFDB - Embedded newform 2100.2.a.b.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:	yes
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)

$f(q)$

$=$

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

$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +4.00000 q^{13} +6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -1.00000 q^{27} +3.00000 q^{29} -10.0000 q^{31} +3.00000 q^{33} +7.00000 q^{37} -4.00000 q^{39} +1.00000 q^{43} +12.0000 q^{47} +1.00000 q^{49} -6.00000 q^{51} -6.00000 q^{53} +4.00000 q^{57} +12.0000 q^{59} -4.00000 q^{61} -1.00000 q^{63} +7.00000 q^{67} +3.00000 q^{69} +9.00000 q^{71} -2.00000 q^{73} +3.00000 q^{77} +17.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -3.00000 q^{87} -4.00000 q^{91} +10.0000 q^{93} -2.00000 q^{97} -3.00000 q^{99} +O(q^{100})$

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
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	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
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	10.0000	1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	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
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−18.0000	−1.31629
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	25.0000	1.79954	0.899770	−	0.436365i	$-0.143734\pi$
$193$	25.0000	1.79954	0.899770	+	0.436365i	$0.143734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	−3.00000	−0.210559
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	−9.00000	−0.616670
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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
$231$	−3.00000	−0.197386
$232$	0	0
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−17.0000	−1.10427
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	−26.0000	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$283$	−26.0000	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.00000	0.387101
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	7.00000	0.383598
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	−15.0000	−0.814688
$340$	0	0
$341$	30.0000	1.62459
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	33.0000	1.74167	0.870837	−	0.491572i	$-0.163578\pi$
$359$	33.0000	1.74167	0.870837	+	0.491572i	$0.163578\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	37.0000	1.91579	0.957894	−	0.287123i	$-0.0926989\pi$
$373$	37.0000	1.91579	0.957894	+	0.287123i	$0.0926989\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−31.0000	−1.59236	−0.796182	−	0.605058i	$-0.793150\pi$
$379$	−31.0000	−1.59236	−0.796182	+	0.605058i	$0.793150\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	−40.0000	−1.99254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.0000	−1.04093
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.0000	−0.685583
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000	0.193574
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0000	0.574038
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	−39.0000	−1.84052	−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000	−1.84052	−0.920262	+	0.391303i	$0.872024\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.0000	0.610793
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.0000	−1.63723	−0.818615	−	0.574342i	$-0.805258\pi$
$457$	−35.0000	−1.63723	−0.818615	+	0.574342i	$0.805258\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	−3.00000	−0.137940
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	28.0000	1.27669
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.0000	−0.498458	−0.249229	−	0.968445i	$-0.580177\pi$
$487$	−11.0000	−0.498458	−0.249229	+	0.968445i	$0.580177\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−36.0000	−1.58328
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−60.0000	−2.61364
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−17.0000	−0.722914
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	−42.0000	−1.77009	−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000	−1.77009	−0.885044	+	0.465506i	$0.845872\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	0	0
$579$	−25.0000	−1.03896
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	42.0000	1.67465
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	9.00000	0.356034
$640$	0	0
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	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
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−24.0000	−0.914327
$690$	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
$693$	3.00000	0.113961
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9.00000	−0.340411
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−28.0000	−1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	17.0000	0.637550
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	16.0000	0.595046
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.00000	0.221918
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.0000	−0.773545
$738$	0	0
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	0	0
$759$	−9.00000	−0.326679
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	7.00000	0.253417
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.00000	0.251124
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−27.0000	−0.966136
$782$	0	0
$783$	−3.00000	−0.107211
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.00000	−0.0712923	−0.0356462	−	0.999364i	$-0.511349\pi$
$787$	−2.00000	−0.0712923	−0.0356462	+	0.999364i	$0.511349\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	−15.0000	−0.533339
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	45.0000	1.58212	0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000	1.58212	0.791058	+	0.611741i	$0.209531\pi$
$810$	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
$813$	16.0000	0.561144
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	−5.00000	−0.174289	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−5.00000	−0.174289	−0.0871445	+	0.996196i	$0.527774\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000	0.345651
$838$	0	0
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−15.0000	−0.516627
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	26.0000	0.892318
$850$	0	0
$851$	−21.0000	−0.719871
$852$	0	0
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.0000	1.73606	0.868030	−	0.496512i	$-0.165386\pi$
$863$	51.0000	1.73606	0.868030	+	0.496512i	$0.165386\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−51.0000	−1.73006
$870$	0	0
$871$	28.0000	0.948744
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−48.0000	−1.60626
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	−30.0000	−1.00056
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	1.00000	0.0332779
$904$	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
$909$	6.00000	0.199007
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00000	0.198137
$918$	0	0
$919$	−19.0000	−0.626752	−0.313376	−	0.949629i	$-0.601460\pi$
$919$	−19.0000	−0.626752	−0.313376	+	0.949629i	$0.601460\pi$
$920$	0	0
$921$	26.0000	0.856729
$922$	0	0
$923$	36.0000	1.18495
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	6.00000	0.196431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	0	0
$953$	−57.0000	−1.84641	−0.923206	−	0.384307i	$-0.874441\pi$
$953$	−57.0000	−1.84641	−0.923206	+	0.384307i	$0.874441\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.00000	0.290929
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	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
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−7.00000	−0.223493
$982$	0	0
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	0	0
$993$	−17.0000	−0.539479
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	0	0
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.a.b.1.1	✓	1
3.2	odd	2		6300.2.a.m.1.1		1
4.3	odd	2		8400.2.a.cs.1.1		1
5.2	odd	4		2100.2.k.b.1849.2		2
5.3	odd	4		2100.2.k.b.1849.1		2
5.4	even	2		2100.2.a.o.1.1	yes	1
15.2	even	4		6300.2.k.n.6049.1		2
15.8	even	4		6300.2.k.n.6049.2		2
15.14	odd	2		6300.2.a.bb.1.1		1
20.19	odd	2		8400.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.a.b.1.1	✓	1	1.1	even	1	trivial
2100.2.a.o.1.1	yes	1	5.4	even	2
2100.2.k.b.1849.1		2	5.3	odd	4
2100.2.k.b.1849.2		2	5.2	odd	4
6300.2.a.m.1.1		1	3.2	odd	2
6300.2.a.bb.1.1		1	15.14	odd	2
6300.2.k.n.6049.1		2	15.2	even	4
6300.2.k.n.6049.2		2	15.8	even	4
8400.2.a.i.1.1		1	20.19	odd	2
8400.2.a.cs.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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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