LMFDB - Embedded newform 2646.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2646,2,Mod(1,2646)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2646.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2646 = 2 \cdot 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2646.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,2,0,0,1,0,2,-2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$21.1284163748$
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	$\chi$	$=$	2646.1

$q$ -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^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +2.00000 q^{10} -2.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} +4.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} +3.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} +7.00000 q^{29} +11.0000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} +1.00000 q^{43} -2.00000 q^{44} +3.00000 q^{46} +8.00000 q^{47} -1.00000 q^{50} +1.00000 q^{52} +1.00000 q^{53} -4.00000 q^{55} +7.00000 q^{58} -7.00000 q^{59} +14.0000 q^{61} +11.0000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -11.0000 q^{67} -3.00000 q^{68} -11.0000 q^{71} +8.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +4.00000 q^{79} +2.00000 q^{80} +6.00000 q^{82} -6.00000 q^{85} +1.00000 q^{86} -2.00000 q^{88} +1.00000 q^{89} +3.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} -16.0000 q^{97} +O(q^{100})

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	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
$8$	1.00000	0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−2.00000	−0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	0	0
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	11.0000	1.97566	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	11.0000	1.97566	0.987829	+	0.155543i	$0.0497126\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	7.00000	0.919145
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	11.0000	1.39700
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	0	0
$71$	−11.0000	−1.30546	−0.652730	−	0.757591i	$-0.726376\pi$
$71$	−11.0000	−1.30546	−0.652730	+	0.757591i	$0.726376\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	6.00000	0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	1.00000	0.107833
$87$	0	0
$88$	−2.00000	−0.213201
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	0	0
$94$	8.00000	0.825137
$95$	8.00000	0.820783
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	1.00000	0.0971286
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	7.00000	0.649934
$117$	0	0
$118$	−7.00000	−0.644402
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	14.0000	1.26750
$123$	0	0
$124$	11.0000	0.987829
$125$	−12.0000	−1.07331
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	0	0
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	0	0
$142$	−11.0000	−0.923099
$143$	−2.00000	−0.167248
$144$	0	0
$145$	14.0000	1.16264
$146$	8.00000	0.662085
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	22.0000	1.76708
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	2.00000	0.158114
$161$	0	0
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−6.00000	−0.460179
$171$	0	0
$172$	1.00000	0.0762493
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	1.00000	0.0749532
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	3.00000	0.221163
$185$	−12.0000	−0.882258
$186$	0	0
$187$	6.00000	0.438763
$188$	8.00000	0.583460
$189$	0	0
$190$	8.00000	0.580381
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−16.0000	−1.12576
$203$	0	0
$204$	0	0
$205$	12.0000	0.838116
$206$	13.0000	0.905753
$207$	0	0
$208$	1.00000	0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	1.00000	0.0686803
$213$	0	0
$214$	−16.0000	−1.09374
$215$	2.00000	0.136399
$216$	0	0
$217$	0	0
$218$	16.0000	1.08366
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−3.00000	−0.201802
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	−10.0000	−0.665190
$227$	13.0000	0.862840	0.431420	−	0.902151i	$-0.358013\pi$
$227$	13.0000	0.862840	0.431420	+	0.902151i	$0.358013\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	7.00000	0.459573
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	−7.00000	−0.455661
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	11.0000	0.698501
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−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
$260$	2.00000	0.124035
$261$	0	0
$262$	15.0000	0.926703
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	0	0
$268$	−11.0000	−0.671932
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−8.00000	−0.483298
$275$	2.00000	0.120605
$276$	0	0
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−11.0000	−0.652730
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	14.0000	0.822108
$291$	0	0
$292$	8.00000	0.468165
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−5.00000	−0.289642
$299$	3.00000	0.173494
$300$	0	0
$301$	0	0
$302$	4.00000	0.230174
$303$	0	0
$304$	4.00000	0.229416
$305$	28.0000	1.60328
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	0	0
$310$	22.0000	1.24952
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	4.00000	0.225018
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	−14.0000	−0.783850
$320$	2.00000	0.111803
$321$	0	0
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	9.00000	0.498464
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−22.0000	−1.20199
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−22.0000	−1.19137
$342$	0	0
$343$	0	0
$344$	1.00000	0.0539164
$345$	0	0
$346$	−8.00000	−0.430083
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	27.0000	1.44528	0.722638	−	0.691226i	$-0.242929\pi$
$349$	27.0000	1.44528	0.722638	+	0.691226i	$0.242929\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−33.0000	−1.75641	−0.878206	−	0.478282i	$-0.841260\pi$
$353$	−33.0000	−1.75641	−0.878206	+	0.478282i	$0.841260\pi$
$354$	0	0
$355$	−22.0000	−1.16764
$356$	1.00000	0.0529999
$357$	0	0
$358$	6.00000	0.317110
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−7.00000	−0.367912
$363$	0	0
$364$	0	0
$365$	16.0000	0.837478
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	−12.0000	−0.623850
$371$	0	0
$372$	0	0
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	8.00000	0.412568
$377$	7.00000	0.360518
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	24.0000	1.22795
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	−19.0000	−0.967075
$387$	0	0
$388$	−16.0000	−0.812277
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	0	0
$394$	−18.0000	−0.906827
$395$	8.00000	0.402524
$396$	0	0
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	5.00000	0.250627
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	11.0000	0.547949
$404$	−16.0000	−0.796030
$405$	0	0
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	13.0000	0.640464
$413$	0	0
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	−8.00000	−0.391293
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	0	0
$421$	24.0000	1.16969	0.584844	−	0.811146i	$-0.301156\pi$
$421$	24.0000	1.16969	0.584844	+	0.811146i	$0.301156\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	1.00000	0.0485643
$425$	3.00000	0.145521
$426$	0	0
$427$	0	0
$428$	−16.0000	−0.773389
$429$	0	0
$430$	2.00000	0.0964486
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	0	0
$436$	16.0000	0.766261
$437$	12.0000	0.574038
$438$	0	0
$439$	−3.00000	−0.143182	−0.0715911	−	0.997434i	$-0.522808\pi$
$439$	−3.00000	−0.143182	−0.0715911	+	0.997434i	$0.522808\pi$
$440$	−4.00000	−0.190693
$441$	0	0
$442$	−3.00000	−0.142695
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	8.00000	0.378811
$447$	0	0
$448$	0	0
$449$	−42.0000	−1.98210	−0.991051	−	0.133482i	$-0.957384\pi$
$449$	−42.0000	−1.98210	−0.991051	+	0.133482i	$0.957384\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−10.0000	−0.470360
$453$	0	0
$454$	13.0000	0.610120
$455$	0	0
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	6.00000	0.279751
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	7.00000	0.324967
$465$	0	0
$466$	−8.00000	−0.370593
$467$	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
$469$	0	0
$470$	16.0000	0.738025
$471$	0	0
$472$	−7.00000	−0.322201
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	8.00000	0.364390
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−32.0000	−1.45305
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	14.0000	0.633750
$489$	0	0
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	−21.0000	−0.945792
$494$	4.00000	0.179969
$495$	0	0
$496$	11.0000	0.493915
$497$	0	0
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−28.0000	−1.24970
$503$	22.0000	0.980932	0.490466	−	0.871460i	$-0.336827\pi$
$503$	22.0000	0.980932	0.490466	+	0.871460i	$0.336827\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	−6.00000	−0.266733
$507$	0	0
$508$	14.0000	0.621150
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.0000	−0.617514
$515$	26.0000	1.14570
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	0	0
$520$	2.00000	0.0877058
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	−23.0000	−1.00285
$527$	−33.0000	−1.43750
$528$	0	0
$529$	−14.0000	−0.608696
$530$	2.00000	0.0868744
$531$	0	0
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−32.0000	−1.38348
$536$	−11.0000	−0.475128
$537$	0	0
$538$	−10.0000	−0.431131
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	7.00000	0.300676
$543$	0	0
$544$	−3.00000	−0.128624
$545$	32.0000	1.37073
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	−8.00000	−0.341743
$549$	0	0
$550$	2.00000	0.0852803
$551$	28.0000	1.19284
$552$	0	0
$553$	0	0
$554$	16.0000	0.679775
$555$	0	0
$556$	−6.00000	−0.254457
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	30.0000	1.26547
$563$	1.00000	0.0421450	0.0210725	−	0.999778i	$-0.493292\pi$
$563$	1.00000	0.0421450	0.0210725	+	0.999778i	$0.493292\pi$
$564$	0	0
$565$	−20.0000	−0.841406
$566$	−16.0000	−0.672530
$567$	0	0
$568$	−11.0000	−0.461550
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	14.0000	0.581318
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	8.00000	0.331042
$585$	0	0
$586$	−6.00000	−0.247858
$587$	29.0000	1.19696	0.598479	−	0.801138i	$-0.295772\pi$
$587$	29.0000	1.19696	0.598479	+	0.801138i	$0.295772\pi$
$588$	0	0
$589$	44.0000	1.81299
$590$	−14.0000	−0.576371
$591$	0	0
$592$	−6.00000	−0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	−5.00000	−0.204808
$597$	0	0
$598$	3.00000	0.122679
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	4.00000	0.162758
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−33.0000	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$607$	−33.0000	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	28.0000	1.13369
$611$	8.00000	0.323645
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	22.0000	0.883541
$621$	0	0
$622$	−8.00000	−0.320771
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−24.0000	−0.959233
$627$	0	0
$628$	7.00000	0.279330
$629$	18.0000	0.717707
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	30.0000	1.19145
$635$	28.0000	1.11115
$636$	0	0
$637$	0	0
$638$	−14.0000	−0.554265
$639$	0	0
$640$	2.00000	0.0790569
$641$	−28.0000	−1.10593	−0.552967	−	0.833203i	$-0.686504\pi$
$641$	−28.0000	−1.10593	−0.552967	+	0.833203i	$0.686504\pi$
$642$	0	0
$643$	42.0000	1.65632	0.828159	−	0.560493i	$-0.189388\pi$
$643$	42.0000	1.65632	0.828159	+	0.560493i	$0.189388\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	14.0000	0.549548
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	9.00000	0.352467
$653$	25.0000	0.978326	0.489163	−	0.872192i	$-0.337302\pi$
$653$	25.0000	0.978326	0.489163	+	0.872192i	$0.337302\pi$
$654$	0	0
$655$	30.0000	1.17220
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	11.0000	0.427527
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.0000	0.813123
$668$	−14.0000	−0.541676
$669$	0	0
$670$	−22.0000	−0.849934
$671$	−28.0000	−1.08093
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	0	0
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−22.0000	−0.842424
$683$	14.0000	0.535695	0.267848	−	0.963461i	$-0.413688\pi$
$683$	14.0000	0.535695	0.267848	+	0.963461i	$0.413688\pi$
$684$	0	0
$685$	−16.0000	−0.611329
$686$	0	0
$687$	0	0
$688$	1.00000	0.0381246
$689$	1.00000	0.0380970
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	−8.00000	−0.304114
$693$	0	0
$694$	−6.00000	−0.227757
$695$	−12.0000	−0.455186
$696$	0	0
$697$	−18.0000	−0.681799
$698$	27.0000	1.02197
$699$	0	0
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−33.0000	−1.24197
$707$	0	0
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−22.0000	−0.825645
$711$	0	0
$712$	1.00000	0.0374766
$713$	33.0000	1.23586
$714$	0	0
$715$	−4.00000	−0.149592
$716$	6.00000	0.224231
$717$	0	0
$718$	−19.0000	−0.709074
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−7.00000	−0.260153
$725$	−7.00000	−0.259973
$726$	0	0
$727$	−17.0000	−0.630495	−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000	−0.630495	−0.315248	+	0.949009i	$0.602088\pi$
$728$	0	0
$729$	0	0
$730$	16.0000	0.592187
$731$	−3.00000	−0.110959
$732$	0	0
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	11.0000	0.406017
$735$	0	0
$736$	3.00000	0.110581
$737$	22.0000	0.810380
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	0	0
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−20.0000	−0.732252
$747$	0	0
$748$	6.00000	0.219382
$749$	0	0
$750$	0	0
$751$	−30.0000	−1.09472	−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	7.00000	0.254925
$755$	8.00000	0.291150
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	8.00000	0.290191
$761$	−1.00000	−0.0362500	−0.0181250	−	0.999836i	$-0.505770\pi$
$761$	−1.00000	−0.0362500	−0.0181250	+	0.999836i	$0.505770\pi$
$762$	0	0
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	18.0000	0.650366
$767$	−7.00000	−0.252755
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	0	0
$772$	−19.0000	−0.683825
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	−11.0000	−0.395132
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−6.00000	−0.215110
$779$	24.0000	0.859889
$780$	0	0
$781$	22.0000	0.787222
$782$	−9.00000	−0.321839
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	14.0000	0.497155
$794$	−38.0000	−1.34857
$795$	0	0
$796$	5.00000	0.177220
$797$	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
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−18.0000	−0.635602
$803$	−16.0000	−0.564628
$804$	0	0
$805$	0	0
$806$	11.0000	0.387458
$807$	0	0
$808$	−16.0000	−0.562878
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	0	0
$813$	0	0
$814$	12.0000	0.420600
$815$	18.0000	0.630512
$816$	0	0
$817$	4.00000	0.139942
$818$	−18.0000	−0.629355
$819$	0	0
$820$	12.0000	0.419058
$821$	−35.0000	−1.22151	−0.610754	−	0.791820i	$-0.709134\pi$
$821$	−35.0000	−1.22151	−0.610754	+	0.791820i	$0.709134\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	−28.0000	−0.968980
$836$	−8.00000	−0.276686
$837$	0	0
$838$	−19.0000	−0.656344
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	24.0000	0.827095
$843$	0	0
$844$	5.00000	0.172107
$845$	−24.0000	−0.825625
$846$	0	0
$847$	0	0
$848$	1.00000	0.0343401
$849$	0	0
$850$	3.00000	0.102899
$851$	−18.0000	−0.617032
$852$	0	0
$853$	−17.0000	−0.582069	−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000	−0.582069	−0.291034	+	0.956713i	$0.593999\pi$
$854$	0	0
$855$	0	0
$856$	−16.0000	−0.546869
$857$	−9.00000	−0.307434	−0.153717	−	0.988115i	$-0.549124\pi$
$857$	−9.00000	−0.307434	−0.153717	+	0.988115i	$0.549124\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	−4.00000	−0.136241
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	0	0
$865$	−16.0000	−0.544016
$866$	28.0000	0.951479
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−11.0000	−0.372721
$872$	16.0000	0.541828
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	−36.0000	−1.21563	−0.607817	−	0.794077i	$-0.707955\pi$
$877$	−36.0000	−1.21563	−0.607817	+	0.794077i	$0.707955\pi$
$878$	−3.00000	−0.101245
$879$	0	0
$880$	−4.00000	−0.134840
$881$	47.0000	1.58347	0.791735	−	0.610865i	$-0.209178\pi$
$881$	47.0000	1.58347	0.791735	+	0.610865i	$0.209178\pi$
$882$	0	0
$883$	−5.00000	−0.168263	−0.0841317	−	0.996455i	$-0.526812\pi$
$883$	−5.00000	−0.168263	−0.0841317	+	0.996455i	$0.526812\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	18.0000	0.604722
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	2.00000	0.0670402
$891$	0	0
$892$	8.00000	0.267860
$893$	32.0000	1.07084
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	−42.0000	−1.40156
$899$	77.0000	2.56809
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	−12.0000	−0.399556
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−14.0000	−0.465376
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	13.0000	0.431420
$909$	0	0
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	0	0
$914$	23.0000	0.760772
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	−12.0000	−0.395199
$923$	−11.0000	−0.362069
$924$	0	0
$925$	6.00000	0.197279
$926$	20.0000	0.657241
$927$	0	0
$928$	7.00000	0.229786
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	0	0
$932$	−8.00000	−0.262049
$933$	0	0
$934$	28.0000	0.916188
$935$	12.0000	0.392442
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	0	0
$940$	16.0000	0.521862
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	−7.00000	−0.227831
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	48.0000	1.55324
$956$	0	0
$957$	0	0
$958$	30.0000	0.969256
$959$	0	0
$960$	0	0
$961$	90.0000	2.90323
$962$	−6.00000	−0.193448
$963$	0	0
$964$	8.00000	0.257663
$965$	−38.0000	−1.22326
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	−32.0000	−1.02746
$971$	23.0000	0.738105	0.369053	−	0.929409i	$-0.379682\pi$
$971$	23.0000	0.738105	0.369053	+	0.929409i	$0.379682\pi$
$972$	0	0
$973$	0	0
$974$	−20.0000	−0.640841
$975$	0	0
$976$	14.0000	0.448129
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	0	0
$982$	−30.0000	−0.957338
$983$	62.0000	1.97749	0.988746	−	0.149601i	$-0.0477989\pi$
$983$	62.0000	1.97749	0.988746	+	0.149601i	$0.0477989\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	−21.0000	−0.668776
$987$	0	0
$988$	4.00000	0.127257
$989$	3.00000	0.0953945
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	11.0000	0.349250
$993$	0	0
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	37.0000	1.17180	0.585901	−	0.810383i	$-0.300741\pi$
$997$	37.0000	1.17180	0.585901	+	0.810383i	$0.300741\pi$
$998$	−36.0000	−1.13956
$999$	0	0

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a_n

instead) (See

a_n

instead) Display

a_n

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n

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a_p

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.a.ba.1.1	yes	1
3.2	odd	2		2646.2.a.d.1.1	✓	1
7.6	odd	2		2646.2.a.s.1.1	yes	1
21.20	even	2		2646.2.a.l.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.a.d.1.1	✓	1	3.2	odd	2
2646.2.a.l.1.1	yes	1	21.20	even	2
2646.2.a.s.1.1	yes	1	7.6	odd	2
2646.2.a.ba.1.1	yes	1	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2646.2.a.ba.1.1

Level	$2646$
Weight	$2$
Character	2646.1
Self dual	yes
Analytic conductor	$21.128$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$