LMFDB - Embedded newform 2790.2.a.be.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2790,2,Mod(1,2790)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2790.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,-2,0,1,-2,0,2,-1,0,4] 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:	$22.2782621639$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.56155 q^{11} +2.00000 q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -7.68466 q^{19} -1.00000 q^{20} +2.56155 q^{22} -1.43845 q^{23} +1.00000 q^{25} -2.00000 q^{26} +2.56155 q^{28} -7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.12311 q^{34} -2.56155 q^{35} -3.12311 q^{37} +7.68466 q^{38} +1.00000 q^{40} -7.12311 q^{41} +12.8078 q^{43} -2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} -0.438447 q^{49} -1.00000 q^{50} +2.00000 q^{52} -7.43845 q^{53} +2.56155 q^{55} -2.56155 q^{56} +7.12311 q^{58} +13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -15.3693 q^{67} +3.12311 q^{68} +2.56155 q^{70} +7.68466 q^{71} -10.8078 q^{73} +3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} -4.31534 q^{79} -1.00000 q^{80} +7.12311 q^{82} -14.2462 q^{83} -3.12311 q^{85} -12.8078 q^{86} +2.56155 q^{88} +13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +5.12311 q^{94} +7.68466 q^{95} -6.00000 q^{97} +0.438447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29}+ \cdots + 5 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 3 * q^19 - 2 * q^20 + q^22 - 7 * q^23 + 2 * q^25 - 4 * q^26 + q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 2 * q^37 + 3 * q^38 + 2 * q^40 - 6 * q^41 + 5 * q^43 - q^44 + 7 * q^46 - 2 * q^47 - 5 * q^49 - 2 * q^50 + 4 * q^52 - 19 * q^53 + q^55 - q^56 + 6 * q^58 + 18 * q^59 + 12 * q^61 - 2 * q^62 + 2 * q^64 - 4 * q^65 - 6 * q^67 - 2 * q^68 + q^70 + 3 * q^71 - q^73 - 2 * q^74 - 3 * q^76 - 9 * q^77 - 21 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 2 * q^85 - 5 * q^86 + q^88 + 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^94 + 3 * q^95 - 12 * q^97 + 5 * q^98

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.56155	−0.684604
$15$	0	0
$16$	1.00000	0.250000
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	0	0
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.56155	0.546125
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	2.56155	0.484088
$29$	−7.12311	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$29$	−7.12311	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.12311	−0.535608
$35$	−2.56155	−0.432981
$36$	0	0
$37$	−3.12311	−0.513435	−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311	−0.513435	−0.256718	+	0.966486i	$0.582641\pi$
$38$	7.68466	1.24662
$39$	0	0
$40$	1.00000	0.158114
$41$	−7.12311	−1.11244	−0.556221	−	0.831034i	$-0.687749\pi$
$41$	−7.12311	−1.11244	−0.556221	+	0.831034i	$0.687749\pi$
$42$	0	0
$43$	12.8078	1.95317	0.976583	−	0.215142i	$-0.0690213\pi$
$43$	12.8078	1.95317	0.976583	+	0.215142i	$0.0690213\pi$
$44$	−2.56155	−0.386169
$45$	0	0
$46$	1.43845	0.212087
$47$	−5.12311	−0.747282	−0.373641	−	0.927573i	$-0.621891\pi$
$47$	−5.12311	−0.747282	−0.373641	+	0.927573i	$0.621891\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−7.43845	−1.02175	−0.510875	−	0.859655i	$-0.670678\pi$
$53$	−7.43845	−1.02175	−0.510875	+	0.859655i	$0.670678\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	−2.56155	−0.342302
$57$	0	0
$58$	7.12311	0.935310
$59$	13.1231	1.70848	0.854241	−	0.519877i	$-0.174022\pi$
$59$	13.1231	1.70848	0.854241	+	0.519877i	$0.174022\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−15.3693	−1.87766	−0.938830	−	0.344380i	$-0.888089\pi$
$67$	−15.3693	−1.87766	−0.938830	+	0.344380i	$0.888089\pi$
$68$	3.12311	0.378732
$69$	0	0
$70$	2.56155	0.306164
$71$	7.68466	0.912001	0.456001	−	0.889979i	$-0.349281\pi$
$71$	7.68466	0.912001	0.456001	+	0.889979i	$0.349281\pi$
$72$	0	0
$73$	−10.8078	−1.26495	−0.632477	−	0.774580i	$-0.717962\pi$
$73$	−10.8078	−1.26495	−0.632477	+	0.774580i	$0.717962\pi$
$74$	3.12311	0.363054
$75$	0	0
$76$	−7.68466	−0.881491
$77$	−6.56155	−0.747758
$78$	0	0
$79$	−4.31534	−0.485514	−0.242757	−	0.970087i	$-0.578052\pi$
$79$	−4.31534	−0.485514	−0.242757	+	0.970087i	$0.578052\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	7.12311	0.786615
$83$	−14.2462	−1.56372	−0.781862	−	0.623451i	$-0.785730\pi$
$83$	−14.2462	−1.56372	−0.781862	+	0.623451i	$0.785730\pi$
$84$	0	0
$85$	−3.12311	−0.338748
$86$	−12.8078	−1.38110
$87$	0	0
$88$	2.56155	0.273062
$89$	13.6847	1.45057	0.725285	−	0.688448i	$-0.241708\pi$
$89$	13.6847	1.45057	0.725285	+	0.688448i	$0.241708\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	−1.43845	−0.149968
$93$	0	0
$94$	5.12311	0.528408
$95$	7.68466	0.788429
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0.438447	0.0442899
$99$	0	0
$100$	1.00000	0.100000
$101$	0.561553	0.0558766	0.0279383	−	0.999610i	$-0.491106\pi$
$101$	0.561553	0.0558766	0.0279383	+	0.999610i	$0.491106\pi$
$102$	0	0
$103$	−1.75379	−0.172806	−0.0864030	−	0.996260i	$-0.527537\pi$
$103$	−1.75379	−0.172806	−0.0864030	+	0.996260i	$0.527537\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	7.43845	0.722486
$107$	2.56155	0.247635	0.123817	−	0.992305i	$-0.460486\pi$
$107$	2.56155	0.247635	0.123817	+	0.992305i	$0.460486\pi$
$108$	0	0
$109$	5.36932	0.514287	0.257144	−	0.966373i	$-0.417219\pi$
$109$	5.36932	0.514287	0.257144	+	0.966373i	$0.417219\pi$
$110$	−2.56155	−0.244234
$111$	0	0
$112$	2.56155	0.242044
$113$	1.68466	0.158479	0.0792397	−	0.996856i	$-0.474751\pi$
$113$	1.68466	0.158479	0.0792397	+	0.996856i	$0.474751\pi$
$114$	0	0
$115$	1.43845	0.134136
$116$	−7.12311	−0.661364
$117$	0	0
$118$	−13.1231	−1.20808
$119$	8.00000	0.733359
$120$	0	0
$121$	−4.43845	−0.403495
$122$	−6.00000	−0.543214
$123$	0	0
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	15.3693	1.34282	0.671412	−	0.741085i	$-0.265688\pi$
$131$	15.3693	1.34282	0.671412	+	0.741085i	$0.265688\pi$
$132$	0	0
$133$	−19.6847	−1.70688
$134$	15.3693	1.32771
$135$	0	0
$136$	−3.12311	−0.267804
$137$	−20.2462	−1.72975	−0.864875	−	0.501987i	$-0.832603\pi$
$137$	−20.2462	−1.72975	−0.864875	+	0.501987i	$0.832603\pi$
$138$	0	0
$139$	−17.1231	−1.45236	−0.726181	−	0.687503i	$-0.758707\pi$
$139$	−17.1231	−1.45236	−0.726181	+	0.687503i	$0.758707\pi$
$140$	−2.56155	−0.216491
$141$	0	0
$142$	−7.68466	−0.644882
$143$	−5.12311	−0.428416
$144$	0	0
$145$	7.12311	0.591542
$146$	10.8078	0.894457
$147$	0	0
$148$	−3.12311	−0.256718
$149$	−17.0540	−1.39712	−0.698558	−	0.715553i	$-0.746175\pi$
$149$	−17.0540	−1.39712	−0.698558	+	0.715553i	$0.746175\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	7.68466	0.623308
$153$	0	0
$154$	6.56155	0.528745
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	17.0540	1.36106	0.680528	−	0.732722i	$-0.261751\pi$
$157$	17.0540	1.36106	0.680528	+	0.732722i	$0.261751\pi$
$158$	4.31534	0.343310
$159$	0	0
$160$	1.00000	0.0790569
$161$	−3.68466	−0.290392
$162$	0	0
$163$	−15.3693	−1.20382	−0.601909	−	0.798565i	$-0.705593\pi$
$163$	−15.3693	−1.20382	−0.601909	+	0.798565i	$0.705593\pi$
$164$	−7.12311	−0.556221
$165$	0	0
$166$	14.2462	1.10572
$167$	6.56155	0.507748	0.253874	−	0.967237i	$-0.418295\pi$
$167$	6.56155	0.507748	0.253874	+	0.967237i	$0.418295\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	3.12311	0.239531
$171$	0	0
$172$	12.8078	0.976583
$173$	−8.24621	−0.626948	−0.313474	−	0.949597i	$-0.601493\pi$
$173$	−8.24621	−0.626948	−0.313474	+	0.949597i	$0.601493\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	−2.56155	−0.193084
$177$	0	0
$178$	−13.6847	−1.02571
$179$	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
$181$	−13.0540	−0.970294	−0.485147	−	0.874433i	$-0.661234\pi$
$181$	−13.0540	−0.970294	−0.485147	+	0.874433i	$0.661234\pi$
$182$	−5.12311	−0.379750
$183$	0	0
$184$	1.43845	0.106044
$185$	3.12311	0.229615
$186$	0	0
$187$	−8.00000	−0.585018
$188$	−5.12311	−0.373641
$189$	0	0
$190$	−7.68466	−0.557504
$191$	14.2462	1.03082	0.515410	−	0.856944i	$-0.327640\pi$
$191$	14.2462	1.03082	0.515410	+	0.856944i	$0.327640\pi$
$192$	0	0
$193$	14.4924	1.04319	0.521594	−	0.853194i	$-0.325338\pi$
$193$	14.4924	1.04319	0.521594	+	0.853194i	$0.325338\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−0.438447	−0.0313177
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−16.8078	−1.19147	−0.595735	−	0.803181i	$-0.703139\pi$
$199$	−16.8078	−1.19147	−0.595735	+	0.803181i	$0.703139\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−0.561553	−0.0395107
$203$	−18.2462	−1.28063
$204$	0	0
$205$	7.12311	0.497499
$206$	1.75379	0.122192
$207$	0	0
$208$	2.00000	0.138675
$209$	19.6847	1.36162
$210$	0	0
$211$	−23.6847	−1.63052	−0.815260	−	0.579096i	$-0.803406\pi$
$211$	−23.6847	−1.63052	−0.815260	+	0.579096i	$0.803406\pi$
$212$	−7.43845	−0.510875
$213$	0	0
$214$	−2.56155	−0.175104
$215$	−12.8078	−0.873482
$216$	0	0
$217$	2.56155	0.173890
$218$	−5.36932	−0.363656
$219$	0	0
$220$	2.56155	0.172700
$221$	6.24621	0.420166
$222$	0	0
$223$	−21.1231	−1.41451	−0.707254	−	0.706960i	$-0.750066\pi$
$223$	−21.1231	−1.41451	−0.707254	+	0.706960i	$0.750066\pi$
$224$	−2.56155	−0.171151
$225$	0	0
$226$	−1.68466	−0.112062
$227$	−7.68466	−0.510049	−0.255024	−	0.966935i	$-0.582083\pi$
$227$	−7.68466	−0.510049	−0.255024	+	0.966935i	$0.582083\pi$
$228$	0	0
$229$	−16.5616	−1.09442	−0.547209	−	0.836996i	$-0.684310\pi$
$229$	−16.5616	−1.09442	−0.547209	+	0.836996i	$0.684310\pi$
$230$	−1.43845	−0.0948484
$231$	0	0
$232$	7.12311	0.467655
$233$	17.0540	1.11724	0.558622	−	0.829423i	$-0.311330\pi$
$233$	17.0540	1.11724	0.558622	+	0.829423i	$0.311330\pi$
$234$	0	0
$235$	5.12311	0.334195
$236$	13.1231	0.854241
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	4.43845	0.285314
$243$	0	0
$244$	6.00000	0.384111
$245$	0.438447	0.0280114
$246$	0	0
$247$	−15.3693	−0.977926
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	−16.4924	−1.04099	−0.520496	−	0.853864i	$-0.674253\pi$
$251$	−16.4924	−1.04099	−0.520496	+	0.853864i	$0.674253\pi$
$252$	0	0
$253$	3.68466	0.231652
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.68466	0.105086	0.0525431	−	0.998619i	$-0.483267\pi$
$257$	1.68466	0.105086	0.0525431	+	0.998619i	$0.483267\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	−2.00000	−0.124035
$261$	0	0
$262$	−15.3693	−0.949520
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	0	0
$265$	7.43845	0.456940
$266$	19.6847	1.20694
$267$	0	0
$268$	−15.3693	−0.938830
$269$	0.246211	0.0150118	0.00750588	−	0.999972i	$-0.497611\pi$
$269$	0.246211	0.0150118	0.00750588	+	0.999972i	$0.497611\pi$
$270$	0	0
$271$	27.0540	1.64341	0.821706	−	0.569912i	$-0.193023\pi$
$271$	27.0540	1.64341	0.821706	+	0.569912i	$0.193023\pi$
$272$	3.12311	0.189366
$273$	0	0
$274$	20.2462	1.22312
$275$	−2.56155	−0.154467
$276$	0	0
$277$	−5.36932	−0.322611	−0.161305	−	0.986905i	$-0.551570\pi$
$277$	−5.36932	−0.322611	−0.161305	+	0.986905i	$0.551570\pi$
$278$	17.1231	1.02698
$279$	0	0
$280$	2.56155	0.153082
$281$	−4.87689	−0.290931	−0.145466	−	0.989363i	$-0.546468\pi$
$281$	−4.87689	−0.290931	−0.145466	+	0.989363i	$0.546468\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	7.68466	0.456001
$285$	0	0
$286$	5.12311	0.302936
$287$	−18.2462	−1.07704
$288$	0	0
$289$	−7.24621	−0.426248
$290$	−7.12311	−0.418283
$291$	0	0
$292$	−10.8078	−0.632477
$293$	−26.4924	−1.54770	−0.773852	−	0.633367i	$-0.781673\pi$
$293$	−26.4924	−1.54770	−0.773852	+	0.633367i	$0.781673\pi$
$294$	0	0
$295$	−13.1231	−0.764057
$296$	3.12311	0.181527
$297$	0	0
$298$	17.0540	0.987910
$299$	−2.87689	−0.166375
$300$	0	0
$301$	32.8078	1.89101
$302$	0	0
$303$	0	0
$304$	−7.68466	−0.440745
$305$	−6.00000	−0.343559
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−6.56155	−0.373879
$309$	0	0
$310$	1.00000	0.0567962
$311$	1.75379	0.0994482	0.0497241	−	0.998763i	$-0.484166\pi$
$311$	1.75379	0.0994482	0.0497241	+	0.998763i	$0.484166\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−17.0540	−0.962412
$315$	0	0
$316$	−4.31534	−0.242757
$317$	−16.8769	−0.947901	−0.473950	−	0.880552i	$-0.657172\pi$
$317$	−16.8769	−0.947901	−0.473950	+	0.880552i	$0.657172\pi$
$318$	0	0
$319$	18.2462	1.02159
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	3.68466	0.205338
$323$	−24.0000	−1.33540
$324$	0	0
$325$	2.00000	0.110940
$326$	15.3693	0.851228
$327$	0	0
$328$	7.12311	0.393308
$329$	−13.1231	−0.723500
$330$	0	0
$331$	−6.24621	−0.343323	−0.171661	−	0.985156i	$-0.554914\pi$
$331$	−6.24621	−0.343323	−0.171661	+	0.985156i	$0.554914\pi$
$332$	−14.2462	−0.781862
$333$	0	0
$334$	−6.56155	−0.359032
$335$	15.3693	0.839715
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−3.12311	−0.169374
$341$	−2.56155	−0.138716
$342$	0	0
$343$	−19.0540	−1.02882
$344$	−12.8078	−0.690548
$345$	0	0
$346$	8.24621	0.443319
$347$	−14.2462	−0.764777	−0.382388	−	0.924002i	$-0.624898\pi$
$347$	−14.2462	−0.764777	−0.382388	+	0.924002i	$0.624898\pi$
$348$	0	0
$349$	5.36932	0.287413	0.143706	−	0.989620i	$-0.454098\pi$
$349$	5.36932	0.287413	0.143706	+	0.989620i	$0.454098\pi$
$350$	−2.56155	−0.136921
$351$	0	0
$352$	2.56155	0.136531
$353$	0.246211	0.0131045	0.00655225	−	0.999979i	$-0.497914\pi$
$353$	0.246211	0.0131045	0.00655225	+	0.999979i	$0.497914\pi$
$354$	0	0
$355$	−7.68466	−0.407859
$356$	13.6847	0.725285
$357$	0	0
$358$	−12.0000	−0.634220
$359$	31.6847	1.67225	0.836126	−	0.548537i	$-0.184815\pi$
$359$	31.6847	1.67225	0.836126	+	0.548537i	$0.184815\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	13.0540	0.686102
$363$	0	0
$364$	5.12311	0.268524
$365$	10.8078	0.565704
$366$	0	0
$367$	15.3693	0.802272	0.401136	−	0.916019i	$-0.368616\pi$
$367$	15.3693	0.802272	0.401136	+	0.916019i	$0.368616\pi$
$368$	−1.43845	−0.0749842
$369$	0	0
$370$	−3.12311	−0.162363
$371$	−19.0540	−0.989233
$372$	0	0
$373$	−5.68466	−0.294340	−0.147170	−	0.989111i	$-0.547017\pi$
$373$	−5.68466	−0.294340	−0.147170	+	0.989111i	$0.547017\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	5.12311	0.264204
$377$	−14.2462	−0.733717
$378$	0	0
$379$	−7.05398	−0.362338	−0.181169	−	0.983452i	$-0.557988\pi$
$379$	−7.05398	−0.362338	−0.181169	+	0.983452i	$0.557988\pi$
$380$	7.68466	0.394215
$381$	0	0
$382$	−14.2462	−0.728900
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	0	0
$385$	6.56155	0.334408
$386$	−14.4924	−0.737645
$387$	0	0
$388$	−6.00000	−0.304604
$389$	−7.75379	−0.393133	−0.196566	−	0.980491i	$-0.562979\pi$
$389$	−7.75379	−0.393133	−0.196566	+	0.980491i	$0.562979\pi$
$390$	0	0
$391$	−4.49242	−0.227192
$392$	0.438447	0.0221449
$393$	0	0
$394$	6.00000	0.302276
$395$	4.31534	0.217128
$396$	0	0
$397$	9.05398	0.454406	0.227203	−	0.973847i	$-0.427042\pi$
$397$	9.05398	0.454406	0.227203	+	0.973847i	$0.427042\pi$
$398$	16.8078	0.842497
$399$	0	0
$400$	1.00000	0.0500000
$401$	13.6847	0.683379	0.341690	−	0.939813i	$-0.389001\pi$
$401$	13.6847	0.683379	0.341690	+	0.939813i	$0.389001\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0.561553	0.0279383
$405$	0	0
$406$	18.2462	0.905544
$407$	8.00000	0.396545
$408$	0	0
$409$	−37.3693	−1.84779	−0.923897	−	0.382642i	$-0.875014\pi$
$409$	−37.3693	−1.84779	−0.923897	+	0.382642i	$0.875014\pi$
$410$	−7.12311	−0.351785
$411$	0	0
$412$	−1.75379	−0.0864030
$413$	33.6155	1.65411
$414$	0	0
$415$	14.2462	0.699319
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−19.6847	−0.962808
$419$	−5.75379	−0.281091	−0.140545	−	0.990074i	$-0.544886\pi$
$419$	−5.75379	−0.281091	−0.140545	+	0.990074i	$0.544886\pi$
$420$	0	0
$421$	−14.4924	−0.706317	−0.353159	−	0.935563i	$-0.614892\pi$
$421$	−14.4924	−0.706317	−0.353159	+	0.935563i	$0.614892\pi$
$422$	23.6847	1.15295
$423$	0	0
$424$	7.43845	0.361243
$425$	3.12311	0.151493
$426$	0	0
$427$	15.3693	0.743773
$428$	2.56155	0.123817
$429$	0	0
$430$	12.8078	0.617645
$431$	30.2462	1.45691	0.728454	−	0.685094i	$-0.240239\pi$
$431$	30.2462	1.45691	0.728454	+	0.685094i	$0.240239\pi$
$432$	0	0
$433$	35.3002	1.69642	0.848209	−	0.529661i	$-0.177681\pi$
$433$	35.3002	1.69642	0.848209	+	0.529661i	$0.177681\pi$
$434$	−2.56155	−0.122958
$435$	0	0
$436$	5.36932	0.257144
$437$	11.0540	0.528783
$438$	0	0
$439$	9.61553	0.458924	0.229462	−	0.973318i	$-0.426303\pi$
$439$	9.61553	0.458924	0.229462	+	0.973318i	$0.426303\pi$
$440$	−2.56155	−0.122117
$441$	0	0
$442$	−6.24621	−0.297102
$443$	−23.0540	−1.09533	−0.547664	−	0.836699i	$-0.684483\pi$
$443$	−23.0540	−1.09533	−0.547664	+	0.836699i	$0.684483\pi$
$444$	0	0
$445$	−13.6847	−0.648715
$446$	21.1231	1.00021
$447$	0	0
$448$	2.56155	0.121022
$449$	−10.4924	−0.495168	−0.247584	−	0.968866i	$-0.579637\pi$
$449$	−10.4924	−0.495168	−0.247584	+	0.968866i	$0.579637\pi$
$450$	0	0
$451$	18.2462	0.859181
$452$	1.68466	0.0792397
$453$	0	0
$454$	7.68466	0.360659
$455$	−5.12311	−0.240175
$456$	0	0
$457$	−24.7386	−1.15722	−0.578612	−	0.815603i	$-0.696406\pi$
$457$	−24.7386	−1.15722	−0.578612	+	0.815603i	$0.696406\pi$
$458$	16.5616	0.773871
$459$	0	0
$460$	1.43845	0.0670679
$461$	−9.36932	−0.436373	−0.218186	−	0.975907i	$-0.570014\pi$
$461$	−9.36932	−0.436373	−0.218186	+	0.975907i	$0.570014\pi$
$462$	0	0
$463$	30.7386	1.42855	0.714273	−	0.699867i	$-0.246758\pi$
$463$	30.7386	1.42855	0.714273	+	0.699867i	$0.246758\pi$
$464$	−7.12311	−0.330682
$465$	0	0
$466$	−17.0540	−0.790010
$467$	9.75379	0.451352	0.225676	−	0.974202i	$-0.427541\pi$
$467$	9.75379	0.451352	0.225676	+	0.974202i	$0.427541\pi$
$468$	0	0
$469$	−39.3693	−1.81791
$470$	−5.12311	−0.236311
$471$	0	0
$472$	−13.1231	−0.604040
$473$	−32.8078	−1.50850
$474$	0	0
$475$	−7.68466	−0.352596
$476$	8.00000	0.366679
$477$	0	0
$478$	24.0000	1.09773
$479$	−10.5616	−0.482570	−0.241285	−	0.970454i	$-0.577569\pi$
$479$	−10.5616	−0.482570	−0.241285	+	0.970454i	$0.577569\pi$
$480$	0	0
$481$	−6.24621	−0.284803
$482$	−12.2462	−0.557800
$483$	0	0
$484$	−4.43845	−0.201748
$485$	6.00000	0.272446
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	−0.438447	−0.0198070
$491$	25.9309	1.17024	0.585122	−	0.810945i	$-0.301047\pi$
$491$	25.9309	1.17024	0.585122	+	0.810945i	$0.301047\pi$
$492$	0	0
$493$	−22.2462	−1.00192
$494$	15.3693	0.691498
$495$	0	0
$496$	1.00000	0.0449013
$497$	19.6847	0.882978
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	16.4924	0.736093
$503$	−26.2462	−1.17026	−0.585130	−	0.810939i	$-0.698957\pi$
$503$	−26.2462	−1.17026	−0.585130	+	0.810939i	$0.698957\pi$
$504$	0	0
$505$	−0.561553	−0.0249888
$506$	−3.68466	−0.163803
$507$	0	0
$508$	0	0
$509$	−19.6155	−0.869443	−0.434721	−	0.900565i	$-0.643153\pi$
$509$	−19.6155	−0.869443	−0.434721	+	0.900565i	$0.643153\pi$
$510$	0	0
$511$	−27.6847	−1.22470
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−1.68466	−0.0743071
$515$	1.75379	0.0772812
$516$	0	0
$517$	13.1231	0.577154
$518$	8.00000	0.351500
$519$	0	0
$520$	2.00000	0.0877058
$521$	32.2462	1.41273	0.706366	−	0.707847i	$-0.250333\pi$
$521$	32.2462	1.41273	0.706366	+	0.707847i	$0.250333\pi$
$522$	0	0
$523$	−22.4233	−0.980502	−0.490251	−	0.871581i	$-0.663095\pi$
$523$	−22.4233	−0.980502	−0.490251	+	0.871581i	$0.663095\pi$
$524$	15.3693	0.671412
$525$	0	0
$526$	−20.4924	−0.893512
$527$	3.12311	0.136045
$528$	0	0
$529$	−20.9309	−0.910038
$530$	−7.43845	−0.323105
$531$	0	0
$532$	−19.6847	−0.853438
$533$	−14.2462	−0.617072
$534$	0	0
$535$	−2.56155	−0.110746
$536$	15.3693	0.663853
$537$	0	0
$538$	−0.246211	−0.0106149
$539$	1.12311	0.0483756
$540$	0	0
$541$	19.7538	0.849282	0.424641	−	0.905362i	$-0.360400\pi$
$541$	19.7538	0.849282	0.424641	+	0.905362i	$0.360400\pi$
$542$	−27.0540	−1.16207
$543$	0	0
$544$	−3.12311	−0.133902
$545$	−5.36932	−0.229996
$546$	0	0
$547$	26.8769	1.14917	0.574587	−	0.818444i	$-0.305163\pi$
$547$	26.8769	1.14917	0.574587	+	0.818444i	$0.305163\pi$
$548$	−20.2462	−0.864875
$549$	0	0
$550$	2.56155	0.109225
$551$	54.7386	2.33194
$552$	0	0
$553$	−11.0540	−0.470063
$554$	5.36932	0.228120
$555$	0	0
$556$	−17.1231	−0.726181
$557$	26.8078	1.13588	0.567941	−	0.823069i	$-0.307740\pi$
$557$	26.8078	1.13588	0.567941	+	0.823069i	$0.307740\pi$
$558$	0	0
$559$	25.6155	1.08342
$560$	−2.56155	−0.108245
$561$	0	0
$562$	4.87689	0.205719
$563$	−16.4924	−0.695073	−0.347536	−	0.937667i	$-0.612982\pi$
$563$	−16.4924	−0.695073	−0.347536	+	0.937667i	$0.612982\pi$
$564$	0	0
$565$	−1.68466	−0.0708741
$566$	0	0
$567$	0	0
$568$	−7.68466	−0.322441
$569$	−30.8078	−1.29153	−0.645764	−	0.763537i	$-0.723461\pi$
$569$	−30.8078	−1.29153	−0.645764	+	0.763537i	$0.723461\pi$
$570$	0	0
$571$	41.1231	1.72095	0.860474	−	0.509494i	$-0.170167\pi$
$571$	41.1231	1.72095	0.860474	+	0.509494i	$0.170167\pi$
$572$	−5.12311	−0.214208
$573$	0	0
$574$	18.2462	0.761582
$575$	−1.43845	−0.0599874
$576$	0	0
$577$	15.1231	0.629583	0.314792	−	0.949161i	$-0.398065\pi$
$577$	15.1231	0.629583	0.314792	+	0.949161i	$0.398065\pi$
$578$	7.24621	0.301403
$579$	0	0
$580$	7.12311	0.295771
$581$	−36.4924	−1.51396
$582$	0	0
$583$	19.0540	0.789135
$584$	10.8078	0.447228
$585$	0	0
$586$	26.4924	1.09439
$587$	0.492423	0.0203245	0.0101622	−	0.999948i	$-0.496765\pi$
$587$	0.492423	0.0203245	0.0101622	+	0.999948i	$0.496765\pi$
$588$	0	0
$589$	−7.68466	−0.316641
$590$	13.1231	0.540270
$591$	0	0
$592$	−3.12311	−0.128359
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	−17.0540	−0.698558
$597$	0	0
$598$	2.87689	0.117645
$599$	−38.4233	−1.56993	−0.784967	−	0.619538i	$-0.787320\pi$
$599$	−38.4233	−1.56993	−0.784967	+	0.619538i	$0.787320\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−32.8078	−1.33714
$603$	0	0
$604$	0	0
$605$	4.43845	0.180449
$606$	0	0
$607$	7.05398	0.286312	0.143156	−	0.989700i	$-0.454275\pi$
$607$	7.05398	0.286312	0.143156	+	0.989700i	$0.454275\pi$
$608$	7.68466	0.311654
$609$	0	0
$610$	6.00000	0.242933
$611$	−10.2462	−0.414517
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	0	0
$616$	6.56155	0.264372
$617$	15.4384	0.621528	0.310764	−	0.950487i	$-0.399415\pi$
$617$	15.4384	0.621528	0.310764	+	0.950487i	$0.399415\pi$
$618$	0	0
$619$	11.3693	0.456971	0.228486	−	0.973547i	$-0.426623\pi$
$619$	11.3693	0.456971	0.228486	+	0.973547i	$0.426623\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	−1.75379	−0.0703205
$623$	35.0540	1.40441
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	0	0
$628$	17.0540	0.680528
$629$	−9.75379	−0.388909
$630$	0	0
$631$	29.3002	1.16642	0.583211	−	0.812321i	$-0.301796\pi$
$631$	29.3002	1.16642	0.583211	+	0.812321i	$0.301796\pi$
$632$	4.31534	0.171655
$633$	0	0
$634$	16.8769	0.670267
$635$	0	0
$636$	0	0
$637$	−0.876894	−0.0347438
$638$	−18.2462	−0.722374
$639$	0	0
$640$	1.00000	0.0395285
$641$	5.50758	0.217536	0.108768	−	0.994067i	$-0.465309\pi$
$641$	5.50758	0.217536	0.108768	+	0.994067i	$0.465309\pi$
$642$	0	0
$643$	−7.05398	−0.278182	−0.139091	−	0.990280i	$-0.544418\pi$
$643$	−7.05398	−0.278182	−0.139091	+	0.990280i	$0.544418\pi$
$644$	−3.68466	−0.145196
$645$	0	0
$646$	24.0000	0.944267
$647$	45.9309	1.80573	0.902864	−	0.429925i	$-0.141460\pi$
$647$	45.9309	1.80573	0.902864	+	0.429925i	$0.141460\pi$
$648$	0	0
$649$	−33.6155	−1.31952
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−15.3693	−0.601909
$653$	−40.2462	−1.57496	−0.787478	−	0.616343i	$-0.788614\pi$
$653$	−40.2462	−1.57496	−0.787478	+	0.616343i	$0.788614\pi$
$654$	0	0
$655$	−15.3693	−0.600529
$656$	−7.12311	−0.278111
$657$	0	0
$658$	13.1231	0.511592
$659$	30.7386	1.19741	0.598704	−	0.800971i	$-0.295683\pi$
$659$	30.7386	1.19741	0.598704	+	0.800971i	$0.295683\pi$
$660$	0	0
$661$	26.4924	1.03044	0.515218	−	0.857059i	$-0.327711\pi$
$661$	26.4924	1.03044	0.515218	+	0.857059i	$0.327711\pi$
$662$	6.24621	0.242766
$663$	0	0
$664$	14.2462	0.552860
$665$	19.6847	0.763338
$666$	0	0
$667$	10.2462	0.396735
$668$	6.56155	0.253874
$669$	0	0
$670$	−15.3693	−0.593769
$671$	−15.3693	−0.593326
$672$	0	0
$673$	11.7538	0.453075	0.226538	−	0.974002i	$-0.427259\pi$
$673$	11.7538	0.453075	0.226538	+	0.974002i	$0.427259\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−32.4233	−1.24613	−0.623064	−	0.782171i	$-0.714112\pi$
$677$	−32.4233	−1.24613	−0.623064	+	0.782171i	$0.714112\pi$
$678$	0	0
$679$	−15.3693	−0.589820
$680$	3.12311	0.119766
$681$	0	0
$682$	2.56155	0.0980869
$683$	−31.6847	−1.21238	−0.606190	−	0.795320i	$-0.707303\pi$
$683$	−31.6847	−1.21238	−0.606190	+	0.795320i	$0.707303\pi$
$684$	0	0
$685$	20.2462	0.773568
$686$	19.0540	0.727484
$687$	0	0
$688$	12.8078	0.488291
$689$	−14.8769	−0.566765
$690$	0	0
$691$	15.0540	0.572680	0.286340	−	0.958128i	$-0.407561\pi$
$691$	15.0540	0.572680	0.286340	+	0.958128i	$0.407561\pi$
$692$	−8.24621	−0.313474
$693$	0	0
$694$	14.2462	0.540779
$695$	17.1231	0.649516
$696$	0	0
$697$	−22.2462	−0.842635
$698$	−5.36932	−0.203232
$699$	0	0
$700$	2.56155	0.0968176
$701$	−5.19224	−0.196108	−0.0980540	−	0.995181i	$-0.531262\pi$
$701$	−5.19224	−0.196108	−0.0980540	+	0.995181i	$0.531262\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	−2.56155	−0.0965422
$705$	0	0
$706$	−0.246211	−0.00926628
$707$	1.43845	0.0540984
$708$	0	0
$709$	17.0540	0.640475	0.320238	−	0.947337i	$-0.396237\pi$
$709$	17.0540	0.640475	0.320238	+	0.947337i	$0.396237\pi$
$710$	7.68466	0.288400
$711$	0	0
$712$	−13.6847	−0.512854
$713$	−1.43845	−0.0538703
$714$	0	0
$715$	5.12311	0.191593
$716$	12.0000	0.448461
$717$	0	0
$718$	−31.6847	−1.18246
$719$	34.8769	1.30069	0.650344	−	0.759640i	$-0.274625\pi$
$719$	34.8769	1.30069	0.650344	+	0.759640i	$0.274625\pi$
$720$	0	0
$721$	−4.49242	−0.167307
$722$	−40.0540	−1.49065
$723$	0	0
$724$	−13.0540	−0.485147
$725$	−7.12311	−0.264546
$726$	0	0
$727$	23.0540	0.855025	0.427512	−	0.904010i	$-0.359390\pi$
$727$	23.0540	0.855025	0.427512	+	0.904010i	$0.359390\pi$
$728$	−5.12311	−0.189875
$729$	0	0
$730$	−10.8078	−0.400013
$731$	40.0000	1.47945
$732$	0	0
$733$	−6.49242	−0.239803	−0.119902	−	0.992786i	$-0.538258\pi$
$733$	−6.49242	−0.239803	−0.119902	+	0.992786i	$0.538258\pi$
$734$	−15.3693	−0.567292
$735$	0	0
$736$	1.43845	0.0530219
$737$	39.3693	1.45019
$738$	0	0
$739$	52.9848	1.94908	0.974540	−	0.224216i	$-0.0719821\pi$
$739$	52.9848	1.94908	0.974540	+	0.224216i	$0.0719821\pi$
$740$	3.12311	0.114808
$741$	0	0
$742$	19.0540	0.699493
$743$	50.4233	1.84985	0.924926	−	0.380148i	$-0.124127\pi$
$743$	50.4233	1.84985	0.924926	+	0.380148i	$0.124127\pi$
$744$	0	0
$745$	17.0540	0.624809
$746$	5.68466	0.208130
$747$	0	0
$748$	−8.00000	−0.292509
$749$	6.56155	0.239754
$750$	0	0
$751$	45.1231	1.64657	0.823283	−	0.567631i	$-0.192140\pi$
$751$	45.1231	1.64657	0.823283	+	0.567631i	$0.192140\pi$
$752$	−5.12311	−0.186820
$753$	0	0
$754$	14.2462	0.518816
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	7.05398	0.256212
$759$	0	0
$760$	−7.68466	−0.278752
$761$	20.4233	0.740344	0.370172	−	0.928963i	$-0.379299\pi$
$761$	20.4233	0.740344	0.370172	+	0.928963i	$0.379299\pi$
$762$	0	0
$763$	13.7538	0.497921
$764$	14.2462	0.515410
$765$	0	0
$766$	−10.2462	−0.370211
$767$	26.2462	0.947696
$768$	0	0
$769$	−20.5616	−0.741469	−0.370734	−	0.928739i	$-0.620894\pi$
$769$	−20.5616	−0.741469	−0.370734	+	0.928739i	$0.620894\pi$
$770$	−6.56155	−0.236462
$771$	0	0
$772$	14.4924	0.521594
$773$	11.4384	0.411412	0.205706	−	0.978614i	$-0.434051\pi$
$773$	11.4384	0.411412	0.205706	+	0.978614i	$0.434051\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	6.00000	0.215387
$777$	0	0
$778$	7.75379	0.277987
$779$	54.7386	1.96122
$780$	0	0
$781$	−19.6847	−0.704372
$782$	4.49242	0.160649
$783$	0	0
$784$	−0.438447	−0.0156588
$785$	−17.0540	−0.608682
$786$	0	0
$787$	−17.9309	−0.639166	−0.319583	−	0.947558i	$-0.603543\pi$
$787$	−17.9309	−0.639166	−0.319583	+	0.947558i	$0.603543\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−4.31534	−0.153533
$791$	4.31534	0.153436
$792$	0	0
$793$	12.0000	0.426132
$794$	−9.05398	−0.321314
$795$	0	0
$796$	−16.8078	−0.595735
$797$	26.9848	0.955852	0.477926	−	0.878400i	$-0.341389\pi$
$797$	26.9848	0.955852	0.477926	+	0.878400i	$0.341389\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−13.6847	−0.483222
$803$	27.6847	0.976970
$804$	0	0
$805$	3.68466	0.129867
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	−0.561553	−0.0197554
$809$	37.6847	1.32492	0.662461	−	0.749096i	$-0.269512\pi$
$809$	37.6847	1.32492	0.662461	+	0.749096i	$0.269512\pi$
$810$	0	0
$811$	−24.6695	−0.866263	−0.433132	−	0.901331i	$-0.642592\pi$
$811$	−24.6695	−0.866263	−0.433132	+	0.901331i	$0.642592\pi$
$812$	−18.2462	−0.640316
$813$	0	0
$814$	−8.00000	−0.280400
$815$	15.3693	0.538364
$816$	0	0
$817$	−98.4233	−3.44340
$818$	37.3693	1.30659
$819$	0	0
$820$	7.12311	0.248750
$821$	−19.6155	−0.684587	−0.342293	−	0.939593i	$-0.611204\pi$
$821$	−19.6155	−0.684587	−0.342293	+	0.939593i	$0.611204\pi$
$822$	0	0
$823$	−25.6155	−0.892901	−0.446451	−	0.894808i	$-0.647312\pi$
$823$	−25.6155	−0.892901	−0.446451	+	0.894808i	$0.647312\pi$
$824$	1.75379	0.0610961
$825$	0	0
$826$	−33.6155	−1.16963
$827$	−1.75379	−0.0609852	−0.0304926	−	0.999535i	$-0.509708\pi$
$827$	−1.75379	−0.0609852	−0.0304926	+	0.999535i	$0.509708\pi$
$828$	0	0
$829$	24.4233	0.848256	0.424128	−	0.905602i	$-0.360581\pi$
$829$	24.4233	0.848256	0.424128	+	0.905602i	$0.360581\pi$
$830$	−14.2462	−0.494493
$831$	0	0
$832$	2.00000	0.0693375
$833$	−1.36932	−0.0474440
$834$	0	0
$835$	−6.56155	−0.227072
$836$	19.6847	0.680808
$837$	0	0
$838$	5.75379	0.198761
$839$	−6.06913	−0.209530	−0.104765	−	0.994497i	$-0.533409\pi$
$839$	−6.06913	−0.209530	−0.104765	+	0.994497i	$0.533409\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	14.4924	0.499442
$843$	0	0
$844$	−23.6847	−0.815260
$845$	9.00000	0.309609
$846$	0	0
$847$	−11.3693	−0.390654
$848$	−7.43845	−0.255437
$849$	0	0
$850$	−3.12311	−0.107122
$851$	4.49242	0.153998
$852$	0	0
$853$	47.7926	1.63639	0.818194	−	0.574942i	$-0.194976\pi$
$853$	47.7926	1.63639	0.818194	+	0.574942i	$0.194976\pi$
$854$	−15.3693	−0.525927
$855$	0	0
$856$	−2.56155	−0.0875521
$857$	−29.2311	−0.998514	−0.499257	−	0.866454i	$-0.666394\pi$
$857$	−29.2311	−0.998514	−0.499257	+	0.866454i	$0.666394\pi$
$858$	0	0
$859$	26.7386	0.912310	0.456155	−	0.889900i	$-0.349226\pi$
$859$	26.7386	0.912310	0.456155	+	0.889900i	$0.349226\pi$
$860$	−12.8078	−0.436741
$861$	0	0
$862$	−30.2462	−1.03019
$863$	−39.5464	−1.34618	−0.673088	−	0.739563i	$-0.735032\pi$
$863$	−39.5464	−1.34618	−0.673088	+	0.739563i	$0.735032\pi$
$864$	0	0
$865$	8.24621	0.280380
$866$	−35.3002	−1.19955
$867$	0	0
$868$	2.56155	0.0869448
$869$	11.0540	0.374980
$870$	0	0
$871$	−30.7386	−1.04154
$872$	−5.36932	−0.181828
$873$	0	0
$874$	−11.0540	−0.373906
$875$	−2.56155	−0.0865963
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−9.61553	−0.324508
$879$	0	0
$880$	2.56155	0.0863499
$881$	−9.50758	−0.320318	−0.160159	−	0.987091i	$-0.551201\pi$
$881$	−9.50758	−0.320318	−0.160159	+	0.987091i	$0.551201\pi$
$882$	0	0
$883$	30.4233	1.02383	0.511913	−	0.859038i	$-0.328937\pi$
$883$	30.4233	1.02383	0.511913	+	0.859038i	$0.328937\pi$
$884$	6.24621	0.210083
$885$	0	0
$886$	23.0540	0.774513
$887$	32.6307	1.09563	0.547816	−	0.836599i	$-0.315460\pi$
$887$	32.6307	1.09563	0.547816	+	0.836599i	$0.315460\pi$
$888$	0	0
$889$	0	0
$890$	13.6847	0.458711
$891$	0	0
$892$	−21.1231	−0.707254
$893$	39.3693	1.31744
$894$	0	0
$895$	−12.0000	−0.401116
$896$	−2.56155	−0.0855755
$897$	0	0
$898$	10.4924	0.350137
$899$	−7.12311	−0.237569
$900$	0	0
$901$	−23.2311	−0.773939
$902$	−18.2462	−0.607532
$903$	0	0
$904$	−1.68466	−0.0560309
$905$	13.0540	0.433929
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−7.68466	−0.255024
$909$	0	0
$910$	5.12311	0.169829
$911$	11.5076	0.381263	0.190632	−	0.981662i	$-0.438946\pi$
$911$	11.5076	0.381263	0.190632	+	0.981662i	$0.438946\pi$
$912$	0	0
$913$	36.4924	1.20772
$914$	24.7386	0.818281
$915$	0	0
$916$	−16.5616	−0.547209
$917$	39.3693	1.30009
$918$	0	0
$919$	−1.61553	−0.0532914	−0.0266457	−	0.999645i	$-0.508483\pi$
$919$	−1.61553	−0.0532914	−0.0266457	+	0.999645i	$0.508483\pi$
$920$	−1.43845	−0.0474242
$921$	0	0
$922$	9.36932	0.308562
$923$	15.3693	0.505887
$924$	0	0
$925$	−3.12311	−0.102687
$926$	−30.7386	−1.01013
$927$	0	0
$928$	7.12311	0.233827
$929$	−7.43845	−0.244048	−0.122024	−	0.992527i	$-0.538938\pi$
$929$	−7.43845	−0.244048	−0.122024	+	0.992527i	$0.538938\pi$
$930$	0	0
$931$	3.36932	0.110425
$932$	17.0540	0.558622
$933$	0	0
$934$	−9.75379	−0.319154
$935$	8.00000	0.261628
$936$	0	0
$937$	41.3693	1.35148	0.675738	−	0.737142i	$-0.263825\pi$
$937$	41.3693	1.35148	0.675738	+	0.737142i	$0.263825\pi$
$938$	39.3693	1.28545
$939$	0	0
$940$	5.12311	0.167097
$941$	−2.63068	−0.0857578	−0.0428789	−	0.999080i	$-0.513653\pi$
$941$	−2.63068	−0.0857578	−0.0428789	+	0.999080i	$0.513653\pi$
$942$	0	0
$943$	10.2462	0.333663
$944$	13.1231	0.427121
$945$	0	0
$946$	32.8078	1.06667
$947$	9.12311	0.296461	0.148231	−	0.988953i	$-0.452642\pi$
$947$	9.12311	0.296461	0.148231	+	0.988953i	$0.452642\pi$
$948$	0	0
$949$	−21.6155	−0.701670
$950$	7.68466	0.249323
$951$	0	0
$952$	−8.00000	−0.259281
$953$	−16.1080	−0.521788	−0.260894	−	0.965367i	$-0.584017\pi$
$953$	−16.1080	−0.521788	−0.260894	+	0.965367i	$0.584017\pi$
$954$	0	0
$955$	−14.2462	−0.460997
$956$	−24.0000	−0.776215
$957$	0	0
$958$	10.5616	0.341228
$959$	−51.8617	−1.67470
$960$	0	0
$961$	1.00000	0.0322581
$962$	6.24621	0.201386
$963$	0	0
$964$	12.2462	0.394424
$965$	−14.4924	−0.466528
$966$	0	0
$967$	−42.2462	−1.35855	−0.679273	−	0.733885i	$-0.737705\pi$
$967$	−42.2462	−1.35855	−0.679273	+	0.733885i	$0.737705\pi$
$968$	4.43845	0.142657
$969$	0	0
$970$	−6.00000	−0.192648
$971$	37.1231	1.19134	0.595669	−	0.803230i	$-0.296887\pi$
$971$	37.1231	1.19134	0.595669	+	0.803230i	$0.296887\pi$
$972$	0	0
$973$	−43.8617	−1.40614
$974$	−24.0000	−0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	−42.9848	−1.37521	−0.687604	−	0.726086i	$-0.741337\pi$
$977$	−42.9848	−1.37521	−0.687604	+	0.726086i	$0.741337\pi$
$978$	0	0
$979$	−35.0540	−1.12033
$980$	0.438447	0.0140057
$981$	0	0
$982$	−25.9309	−0.827487
$983$	−40.9848	−1.30721	−0.653607	−	0.756834i	$-0.726745\pi$
$983$	−40.9848	−1.30721	−0.653607	+	0.756834i	$0.726745\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	22.2462	0.708464
$987$	0	0
$988$	−15.3693	−0.488963
$989$	−18.4233	−0.585827
$990$	0	0
$991$	35.6847	1.13356	0.566780	−	0.823869i	$-0.308189\pi$
$991$	35.6847	1.13356	0.566780	+	0.823869i	$0.308189\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−19.6847	−0.624359
$995$	16.8078	0.532842
$996$	0	0
$997$	−29.5076	−0.934514	−0.467257	−	0.884121i	$-0.654758\pi$
$997$	−29.5076	−0.934514	−0.467257	+	0.884121i	$0.654758\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.be.1.2		2
3.2	odd	2		930.2.a.p.1.2	✓	2
12.11	even	2		7440.2.a.bl.1.1		2
15.2	even	4		4650.2.d.bd.3349.4		4
15.8	even	4		4650.2.d.bd.3349.1		4
15.14	odd	2		4650.2.a.ce.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.2	✓	2	3.2	odd	2
2790.2.a.be.1.2		2	1.1	even	1	trivial
4650.2.a.ce.1.1		2	15.14	odd	2
4650.2.d.bd.3349.1		4	15.8	even	4
4650.2.d.bd.3349.4		4	15.2	even	4
7440.2.a.bl.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.56155 q^{11} +2.00000 q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.12311 q^{17} -7.68466 q^{19} -1.00000 q^{20} +2.56155 q^{22} -1.43845 q^{23} +1.00000 q^{25} -2.00000 q^{26} +2.56155 q^{28} -7.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.12311 q^{34} -2.56155 q^{35} -3.12311 q^{37} +7.68466 q^{38} +1.00000 q^{40} -7.12311 q^{41} +12.8078 q^{43} -2.56155 q^{44} +1.43845 q^{46} -5.12311 q^{47} -0.438447 q^{49} -1.00000 q^{50} +2.00000 q^{52} -7.43845 q^{53} +2.56155 q^{55} -2.56155 q^{56} +7.12311 q^{58} +13.1231 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -15.3693 q^{67} +3.12311 q^{68} +2.56155 q^{70} +7.68466 q^{71} -10.8078 q^{73} +3.12311 q^{74} -7.68466 q^{76} -6.56155 q^{77} -4.31534 q^{79} -1.00000 q^{80} +7.12311 q^{82} -14.2462 q^{83} -3.12311 q^{85} -12.8078 q^{86} +2.56155 q^{88} +13.6847 q^{89} +5.12311 q^{91} -1.43845 q^{92} +5.12311 q^{94} +7.68466 q^{95} -6.00000 q^{97} +0.438447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29}+ \cdots + 5 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 3 * q^19 - 2 * q^20 + q^22 - 7 * q^23 + 2 * q^25 - 4 * q^26 + q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 2 * q^37 + 3 * q^38 + 2 * q^40 - 6 * q^41 + 5 * q^43 - q^44 + 7 * q^46 - 2 * q^47 - 5 * q^49 - 2 * q^50 + 4 * q^52 - 19 * q^53 + q^55 - q^56 + 6 * q^58 + 18 * q^59 + 12 * q^61 - 2 * q^62 + 2 * q^64 - 4 * q^65 - 6 * q^67 - 2 * q^68 + q^70 + 3 * q^71 - q^73 - 2 * q^74 - 3 * q^76 - 9 * q^77 - 21 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 2 * q^85 - 5 * q^86 + q^88 + 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^94 + 3 * q^95 - 12 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more