LMFDB - Embedded newform 2277.2.a.r.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2277,2,Mod(1,2277)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2277, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("2277.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2277 = 3^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2277.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,6,8,0,-6,-6,0,-12,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1819365402$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} - 2x^{5} + 36x^{4} + 10x^{3} - 35x^{2} - 14x + 1 $ x^8 - 11x^6 - 2x^5 + 36x^4 + 10x^3 - 35x^2 - 14x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.0620205$ of defining polynomial
Character	$\chi$	$=$	2277.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0620205 q^{2} -1.99615 q^{4} +2.44556 q^{5} +3.80059 q^{7} +0.247843 q^{8} -0.151675 q^{10} -1.00000 q^{11} -5.27271 q^{13} -0.235715 q^{14} +3.97694 q^{16} -4.79799 q^{17} -6.79534 q^{19} -4.88172 q^{20} +0.0620205 q^{22} -1.00000 q^{23} +0.980782 q^{25} +0.327016 q^{26} -7.58657 q^{28} -9.73379 q^{29} -0.886679 q^{31} -0.742338 q^{32} +0.297574 q^{34} +9.29460 q^{35} +1.61979 q^{37} +0.421450 q^{38} +0.606117 q^{40} -9.48834 q^{41} -3.98952 q^{43} +1.99615 q^{44} +0.0620205 q^{46} +12.0364 q^{47} +7.44452 q^{49} -0.0608286 q^{50} +10.5251 q^{52} +8.10494 q^{53} -2.44556 q^{55} +0.941952 q^{56} +0.603695 q^{58} +5.97345 q^{59} -10.1481 q^{61} +0.0549923 q^{62} -7.90783 q^{64} -12.8947 q^{65} -2.13847 q^{67} +9.57753 q^{68} -0.576455 q^{70} -10.3760 q^{71} -15.0878 q^{73} -0.100460 q^{74} +13.5645 q^{76} -3.80059 q^{77} +14.3232 q^{79} +9.72585 q^{80} +0.588471 q^{82} +9.19090 q^{83} -11.7338 q^{85} +0.247432 q^{86} -0.247843 q^{88} +9.15738 q^{89} -20.0394 q^{91} +1.99615 q^{92} -0.746506 q^{94} -16.6184 q^{95} +6.71731 q^{97} -0.461713 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 8 q^{11} - 10 q^{13} - 2 q^{14} - 2 q^{16} - 16 q^{19} + 12 q^{20} - 8 q^{23} - 4 q^{25} - 20 q^{28} - 2 q^{29} - 20 q^{31} - 12 q^{32} - 22 q^{34}+ \cdots + 46 q^{98}+O(q^{100}) $ 8 * q + 6 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 8 * q^11 - 10 * q^13 - 2 * q^14 - 2 * q^16 - 16 * q^19 + 12 * q^20 - 8 * q^23 - 4 * q^25 - 20 * q^28 - 2 * q^29 - 20 * q^31 - 12 * q^32 - 22 * q^34 - 18 * q^35 - 6 * q^37 + 18 * q^38 - 30 * q^40 - 6 * q^41 - 26 * q^43 - 6 * q^44 + 12 * q^47 + 4 * q^49 - 30 * q^50 - 30 * q^52 + 14 * q^53 - 8 * q^55 + 10 * q^56 - 36 * q^58 - 12 * q^61 - 8 * q^64 - 14 * q^65 - 50 * q^67 - 6 * q^68 - 6 * q^70 - 32 * q^71 - 24 * q^73 - 14 * q^74 - 22 * q^76 + 6 * q^77 - 4 * q^79 + 54 * q^80 + 14 * q^82 - 10 * q^83 - 18 * q^85 + 54 * q^86 + 6 * q^88 + 16 * q^89 - 26 * q^91 - 6 * q^92 + 22 * q^94 - 38 * q^95 - 22 * q^97 + 46 * 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 $

$2$	−0.0620205	−0.0438551	−0.0219276	−	0.999760i	$-0.506980\pi$
$2$	−0.0620205	−0.0438551	−0.0219276	+	0.999760i	$0.506980\pi$
$3$	0	0
$3$	0	0
$4$	−1.99615	−0.998077
$5$	2.44556	1.09369	0.546845	−	0.837234i	$-0.315829\pi$
$5$	2.44556	1.09369	0.546845	+	0.837234i	$0.315829\pi$
$6$	0	0
$7$	3.80059	1.43649	0.718245	−	0.695790i	$-0.244946\pi$
$7$	3.80059	1.43649	0.718245	+	0.695790i	$0.244946\pi$
$8$	0.247843	0.0876259
$9$	0	0
$10$	−0.151675	−0.0479639
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.27271	−1.46239	−0.731193	−	0.682170i	$-0.761036\pi$
$13$	−5.27271	−1.46239	−0.731193	+	0.682170i	$0.761036\pi$
$14$	−0.235715	−0.0629974
$15$	0	0
$16$	3.97694	0.994234
$17$	−4.79799	−1.16368	−0.581842	−	0.813302i	$-0.697668\pi$
$17$	−4.79799	−1.16368	−0.581842	+	0.813302i	$0.697668\pi$
$18$	0	0
$19$	−6.79534	−1.55896	−0.779479	−	0.626428i	$-0.784516\pi$
$19$	−6.79534	−1.55896	−0.779479	+	0.626428i	$0.784516\pi$
$20$	−4.88172	−1.09159
$21$	0	0
$22$	0.0620205	0.0132228
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.980782	0.196156
$26$	0.327016	0.0641331
$27$	0	0
$28$	−7.58657	−1.43373
$29$	−9.73379	−1.80752	−0.903760	−	0.428039i	$-0.859204\pi$
$29$	−9.73379	−1.80752	−0.903760	+	0.428039i	$0.859204\pi$
$30$	0	0
$31$	−0.886679	−0.159252	−0.0796262	−	0.996825i	$-0.525373\pi$
$31$	−0.886679	−0.159252	−0.0796262	+	0.996825i	$0.525373\pi$
$32$	−0.742338	−0.131228
$33$	0	0
$34$	0.297574	0.0510335
$35$	9.29460	1.57107
$36$	0	0
$37$	1.61979	0.266293	0.133146	−	0.991096i	$-0.457492\pi$
$37$	1.61979	0.266293	0.133146	+	0.991096i	$0.457492\pi$
$38$	0.421450	0.0683683
$39$	0	0
$40$	0.606117	0.0958355
$41$	−9.48834	−1.48183	−0.740915	−	0.671599i	$-0.765608\pi$
$41$	−9.48834	−1.48183	−0.740915	+	0.671599i	$0.765608\pi$
$42$	0	0
$43$	−3.98952	−0.608396	−0.304198	−	0.952609i	$-0.598388\pi$
$43$	−3.98952	−0.608396	−0.304198	+	0.952609i	$0.598388\pi$
$44$	1.99615	0.300931
$45$	0	0
$46$	0.0620205	0.00914442
$47$	12.0364	1.75570	0.877848	−	0.478940i	$-0.158979\pi$
$47$	12.0364	1.75570	0.877848	+	0.478940i	$0.158979\pi$
$48$	0	0
$49$	7.44452	1.06350
$50$	−0.0608286	−0.00860246
$51$	0	0
$52$	10.5251	1.45957
$53$	8.10494	1.11330	0.556650	−	0.830747i	$-0.312087\pi$
$53$	8.10494	1.11330	0.556650	+	0.830747i	$0.312087\pi$
$54$	0	0
$55$	−2.44556	−0.329760
$56$	0.941952	0.125874
$57$	0	0
$58$	0.603695	0.0792690
$59$	5.97345	0.777677	0.388839	−	0.921306i	$-0.372876\pi$
$59$	5.97345	0.777677	0.388839	+	0.921306i	$0.372876\pi$
$60$	0	0
$61$	−10.1481	−1.29933	−0.649664	−	0.760222i	$-0.725090\pi$
$61$	−10.1481	−1.29933	−0.649664	+	0.760222i	$0.725090\pi$
$62$	0.0549923	0.00698403
$63$	0	0
$64$	−7.90783	−0.988479
$65$	−12.8947	−1.59940
$66$	0	0
$67$	−2.13847	−0.261255	−0.130628	−	0.991431i	$-0.541699\pi$
$67$	−2.13847	−0.261255	−0.130628	+	0.991431i	$0.541699\pi$
$68$	9.57753	1.16145
$69$	0	0
$70$	−0.576455	−0.0688996
$71$	−10.3760	−1.23140	−0.615700	−	0.787980i	$-0.711127\pi$
$71$	−10.3760	−1.23140	−0.615700	+	0.787980i	$0.711127\pi$
$72$	0	0
$73$	−15.0878	−1.76589	−0.882946	−	0.469475i	$-0.844443\pi$
$73$	−15.0878	−1.76589	−0.882946	+	0.469475i	$0.844443\pi$
$74$	−0.100460	−0.0116783
$75$	0	0
$76$	13.5645	1.55596
$77$	−3.80059	−0.433118
$78$	0	0
$79$	14.3232	1.61148	0.805741	−	0.592267i	$-0.201767\pi$
$79$	14.3232	1.61148	0.805741	+	0.592267i	$0.201767\pi$
$80$	9.72585	1.08738
$81$	0	0
$82$	0.588471	0.0649858
$83$	9.19090	1.00883	0.504416	−	0.863461i	$-0.331708\pi$
$83$	9.19090	1.00883	0.504416	+	0.863461i	$0.331708\pi$
$84$	0	0
$85$	−11.7338	−1.27271
$86$	0.247432	0.0266813
$87$	0	0
$88$	−0.247843	−0.0264202
$89$	9.15738	0.970681	0.485340	−	0.874325i	$-0.338696\pi$
$89$	9.15738	0.970681	0.485340	+	0.874325i	$0.338696\pi$
$90$	0	0
$91$	−20.0394	−2.10070
$92$	1.99615	0.208113
$93$	0	0
$94$	−0.746506	−0.0769962
$95$	−16.6184	−1.70502
$96$	0	0
$97$	6.71731	0.682039	0.341020	−	0.940056i	$-0.389228\pi$
$97$	6.71731	0.682039	0.341020	+	0.940056i	$0.389228\pi$
$98$	−0.461713	−0.0466400
$99$	0	0
$100$	−1.95779	−0.195779
$101$	15.6011	1.55236	0.776182	−	0.630509i	$-0.217154\pi$
$101$	15.6011	1.55236	0.776182	+	0.630509i	$0.217154\pi$
$102$	0	0
$103$	−7.08875	−0.698476	−0.349238	−	0.937034i	$-0.613559\pi$
$103$	−7.08875	−0.698476	−0.349238	+	0.937034i	$0.613559\pi$
$104$	−1.30681	−0.128143
$105$	0	0
$106$	−0.502672	−0.0488239
$107$	0.372348	0.0359962	0.0179981	−	0.999838i	$-0.494271\pi$
$107$	0.372348	0.0359962	0.0179981	+	0.999838i	$0.494271\pi$
$108$	0	0
$109$	−6.12688	−0.586848	−0.293424	−	0.955982i	$-0.594795\pi$
$109$	−6.12688	−0.586848	−0.293424	+	0.955982i	$0.594795\pi$
$110$	0.151675	0.0144617
$111$	0	0
$112$	15.1147	1.42821
$113$	−1.58796	−0.149383	−0.0746913	−	0.997207i	$-0.523797\pi$
$113$	−1.58796	−0.149383	−0.0746913	+	0.997207i	$0.523797\pi$
$114$	0	0
$115$	−2.44556	−0.228050
$116$	19.4301	1.80404
$117$	0	0
$118$	−0.370476	−0.0341051
$119$	−18.2352	−1.67162
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.629389	0.0569822
$123$	0	0
$124$	1.76995	0.158946
$125$	−9.82925	−0.879155
$126$	0	0
$127$	−13.0429	−1.15737	−0.578687	−	0.815550i	$-0.696435\pi$
$127$	−13.0429	−1.15737	−0.578687	+	0.815550i	$0.696435\pi$
$128$	1.97512	0.174578
$129$	0	0
$130$	0.799739	0.0701417
$131$	−0.394677	−0.0344830	−0.0172415	−	0.999851i	$-0.505488\pi$
$131$	−0.394677	−0.0344830	−0.0172415	+	0.999851i	$0.505488\pi$
$132$	0	0
$133$	−25.8263	−2.23943
$134$	0.132629	0.0114574
$135$	0	0
$136$	−1.18915	−0.101969
$137$	4.20100	0.358916	0.179458	−	0.983766i	$-0.442566\pi$
$137$	4.20100	0.358916	0.179458	+	0.983766i	$0.442566\pi$
$138$	0	0
$139$	−17.3537	−1.47192	−0.735962	−	0.677022i	$-0.763270\pi$
$139$	−17.3537	−1.47192	−0.735962	+	0.677022i	$0.763270\pi$
$140$	−18.5534	−1.56805
$141$	0	0
$142$	0.643523	0.0540032
$143$	5.27271	0.440926
$144$	0	0
$145$	−23.8046	−1.97687
$146$	0.935752	0.0774434
$147$	0	0
$148$	−3.23336	−0.265780
$149$	7.35621	0.602644	0.301322	−	0.953522i	$-0.402572\pi$
$149$	7.35621	0.602644	0.301322	+	0.953522i	$0.402572\pi$
$150$	0	0
$151$	−13.2583	−1.07894	−0.539472	−	0.842003i	$-0.681376\pi$
$151$	−13.2583	−1.07894	−0.539472	+	0.842003i	$0.681376\pi$
$152$	−1.68418	−0.136605
$153$	0	0
$154$	0.235715	0.0189944
$155$	−2.16843	−0.174173
$156$	0	0
$157$	1.37042	0.109371	0.0546857	−	0.998504i	$-0.482584\pi$
$157$	1.37042	0.109371	0.0546857	+	0.998504i	$0.482584\pi$
$158$	−0.888330	−0.0706718
$159$	0	0
$160$	−1.81544	−0.143523
$161$	−3.80059	−0.299529
$162$	0	0
$163$	−2.15342	−0.168669	−0.0843344	−	0.996438i	$-0.526876\pi$
$163$	−2.15342	−0.168669	−0.0843344	+	0.996438i	$0.526876\pi$
$164$	18.9402	1.47898
$165$	0	0
$166$	−0.570024	−0.0442425
$167$	2.90785	0.225016	0.112508	−	0.993651i	$-0.464112\pi$
$167$	2.90785	0.225016	0.112508	+	0.993651i	$0.464112\pi$
$168$	0	0
$169$	14.8015	1.13857
$170$	0.727736	0.0558148
$171$	0	0
$172$	7.96369	0.607226
$173$	2.73583	0.208001	0.104001	−	0.994577i	$-0.466836\pi$
$173$	2.73583	0.208001	0.104001	+	0.994577i	$0.466836\pi$
$174$	0	0
$175$	3.72756	0.281777
$176$	−3.97694	−0.299773
$177$	0	0
$178$	−0.567945	−0.0425693
$179$	9.92487	0.741819	0.370910	−	0.928669i	$-0.379046\pi$
$179$	9.92487	0.741819	0.370910	+	0.928669i	$0.379046\pi$
$180$	0	0
$181$	−22.7977	−1.69454	−0.847270	−	0.531163i	$-0.821755\pi$
$181$	−22.7977	−1.69454	−0.847270	+	0.531163i	$0.821755\pi$
$182$	1.24286	0.0921266
$183$	0	0
$184$	−0.247843	−0.0182713
$185$	3.96131	0.291241
$186$	0	0
$187$	4.79799	0.350864
$188$	−24.0266	−1.75232
$189$	0	0
$190$	1.03068	0.0747736
$191$	−13.9028	−1.00597	−0.502985	−	0.864295i	$-0.667765\pi$
$191$	−13.9028	−1.00597	−0.502985	+	0.864295i	$0.667765\pi$
$192$	0	0
$193$	9.36653	0.674218	0.337109	−	0.941466i	$-0.390551\pi$
$193$	9.36653	0.674218	0.337109	+	0.941466i	$0.390551\pi$
$194$	−0.416611	−0.0299109
$195$	0	0
$196$	−14.8604	−1.06146
$197$	21.2598	1.51470	0.757348	−	0.653012i	$-0.226495\pi$
$197$	21.2598	1.51470	0.757348	+	0.653012i	$0.226495\pi$
$198$	0	0
$199$	6.05772	0.429420	0.214710	−	0.976678i	$-0.431119\pi$
$199$	6.05772	0.429420	0.214710	+	0.976678i	$0.431119\pi$
$200$	0.243080	0.0171884
$201$	0	0
$202$	−0.967585	−0.0680791
$203$	−36.9942	−2.59648
$204$	0	0
$205$	−23.2043	−1.62066
$206$	0.439648	0.0306317
$207$	0	0
$208$	−20.9692	−1.45395
$209$	6.79534	0.470043
$210$	0	0
$211$	7.83187	0.539168	0.269584	−	0.962977i	$-0.413114\pi$
$211$	7.83187	0.539168	0.269584	+	0.962977i	$0.413114\pi$
$212$	−16.1787	−1.11116
$213$	0	0
$214$	−0.0230932	−0.00157862
$215$	−9.75662	−0.665396
$216$	0	0
$217$	−3.36991	−0.228764
$218$	0.379992	0.0257363
$219$	0	0
$220$	4.88172	0.329126
$221$	25.2984	1.70176
$222$	0	0
$223$	−9.78311	−0.655126	−0.327563	−	0.944829i	$-0.606227\pi$
$223$	−9.78311	−0.655126	−0.327563	+	0.944829i	$0.606227\pi$
$224$	−2.82133	−0.188508
$225$	0	0
$226$	0.0984859	0.00655119
$227$	−21.8481	−1.45011	−0.725055	−	0.688691i	$-0.758186\pi$
$227$	−21.8481	−1.45011	−0.725055	+	0.688691i	$0.758186\pi$
$228$	0	0
$229$	8.39529	0.554776	0.277388	−	0.960758i	$-0.410531\pi$
$229$	8.39529	0.554776	0.277388	+	0.960758i	$0.410531\pi$
$230$	0.151675	0.0100012
$231$	0	0
$232$	−2.41246	−0.158386
$233$	8.59974	0.563388	0.281694	−	0.959504i	$-0.409104\pi$
$233$	8.59974	0.563388	0.281694	+	0.959504i	$0.409104\pi$
$234$	0	0
$235$	29.4359	1.92019
$236$	−11.9239	−0.776181
$237$	0	0
$238$	1.13096	0.0733091
$239$	6.04883	0.391266	0.195633	−	0.980677i	$-0.437324\pi$
$239$	6.04883	0.391266	0.195633	+	0.980677i	$0.437324\pi$
$240$	0	0
$241$	−27.5549	−1.77497	−0.887484	−	0.460839i	$-0.847549\pi$
$241$	−27.5549	−1.77497	−0.887484	+	0.460839i	$0.847549\pi$
$242$	−0.0620205	−0.00398683
$243$	0	0
$244$	20.2571	1.29683
$245$	18.2060	1.16314
$246$	0	0
$247$	35.8299	2.27980
$248$	−0.219758	−0.0139546
$249$	0	0
$250$	0.609615	0.0385554
$251$	−26.7049	−1.68560	−0.842799	−	0.538228i	$-0.819094\pi$
$251$	−26.7049	−1.68560	−0.842799	+	0.538228i	$0.819094\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0.808930	0.0507568
$255$	0	0
$256$	15.6932	0.980823
$257$	22.2255	1.38639	0.693195	−	0.720751i	$-0.256203\pi$
$257$	22.2255	1.38639	0.693195	+	0.720751i	$0.256203\pi$
$258$	0	0
$259$	6.15618	0.382527
$260$	25.7399	1.59632
$261$	0	0
$262$	0.0244780	0.00151226
$263$	6.69665	0.412933	0.206466	−	0.978454i	$-0.433804\pi$
$263$	6.69665	0.412933	0.206466	+	0.978454i	$0.433804\pi$
$264$	0	0
$265$	19.8212	1.21760
$266$	1.60176	0.0982103
$267$	0	0
$268$	4.26871	0.260753
$269$	−20.1313	−1.22743	−0.613713	−	0.789529i	$-0.710325\pi$
$269$	−20.1313	−1.22743	−0.613713	+	0.789529i	$0.710325\pi$
$270$	0	0
$271$	31.9169	1.93882	0.969408	−	0.245455i	$-0.0789373\pi$
$271$	31.9169	1.93882	0.969408	+	0.245455i	$0.0789373\pi$
$272$	−19.0813	−1.15697
$273$	0	0
$274$	−0.260548	−0.0157403
$275$	−0.980782	−0.0591434
$276$	0	0
$277$	13.0651	0.785004	0.392502	−	0.919751i	$-0.371610\pi$
$277$	13.0651	0.785004	0.392502	+	0.919751i	$0.371610\pi$
$278$	1.07629	0.0645514
$279$	0	0
$280$	2.30360	0.137667
$281$	−14.4815	−0.863892	−0.431946	−	0.901899i	$-0.642173\pi$
$281$	−14.4815	−0.863892	−0.431946	+	0.901899i	$0.642173\pi$
$282$	0	0
$283$	−12.6799	−0.753743	−0.376872	−	0.926265i	$-0.623000\pi$
$283$	−12.6799	−0.753743	−0.376872	+	0.926265i	$0.623000\pi$
$284$	20.7120	1.22903
$285$	0	0
$286$	−0.327016	−0.0193369
$287$	−36.0613	−2.12863
$288$	0	0
$289$	6.02072	0.354160
$290$	1.47637	0.0866957
$291$	0	0
$292$	30.1175	1.76250
$293$	−6.02608	−0.352047	−0.176024	−	0.984386i	$-0.556324\pi$
$293$	−6.02608	−0.352047	−0.176024	+	0.984386i	$0.556324\pi$
$294$	0	0
$295$	14.6085	0.850537
$296$	0.401455	0.0233341
$297$	0	0
$298$	−0.456236	−0.0264290
$299$	5.27271	0.304929
$300$	0	0
$301$	−15.1625	−0.873954
$302$	0.822286	0.0473172
$303$	0	0
$304$	−27.0246	−1.54997
$305$	−24.8178	−1.42106
$306$	0	0
$307$	16.0732	0.917347	0.458674	−	0.888605i	$-0.348325\pi$
$307$	16.0732	0.917347	0.458674	+	0.888605i	$0.348325\pi$
$308$	7.58657	0.432285
$309$	0	0
$310$	0.134487	0.00763836
$311$	10.6421	0.603456	0.301728	−	0.953394i	$-0.402436\pi$
$311$	10.6421	0.603456	0.301728	+	0.953394i	$0.402436\pi$
$312$	0	0
$313$	23.2183	1.31238	0.656188	−	0.754598i	$-0.272168\pi$
$313$	23.2183	1.31238	0.656188	+	0.754598i	$0.272168\pi$
$314$	−0.0849942	−0.00479650
$315$	0	0
$316$	−28.5913	−1.60838
$317$	−28.9370	−1.62526	−0.812632	−	0.582778i	$-0.801966\pi$
$317$	−28.9370	−1.62526	−0.812632	+	0.582778i	$0.801966\pi$
$318$	0	0
$319$	9.73379	0.544988
$320$	−19.3391	−1.08109
$321$	0	0
$322$	0.235715	0.0131359
$323$	32.6040	1.81413
$324$	0	0
$325$	−5.17138	−0.286857
$326$	0.133556	0.00739699
$327$	0	0
$328$	−2.35162	−0.129847
$329$	45.7456	2.52204
$330$	0	0
$331$	3.90143	0.214442	0.107221	−	0.994235i	$-0.465805\pi$
$331$	3.90143	0.214442	0.107221	+	0.994235i	$0.465805\pi$
$332$	−18.3464	−1.00689
$333$	0	0
$334$	−0.180347	−0.00986812
$335$	−5.22976	−0.285732
$336$	0	0
$337$	25.2478	1.37534	0.687668	−	0.726025i	$-0.258635\pi$
$337$	25.2478	1.37534	0.687668	+	0.726025i	$0.258635\pi$
$338$	−0.917995	−0.0499323
$339$	0	0
$340$	23.4225	1.27026
$341$	0.886679	0.0480164
$342$	0	0
$343$	1.68943	0.0912209
$344$	−0.988776	−0.0533112
$345$	0	0
$346$	−0.169677	−0.00912192
$347$	20.1964	1.08420	0.542100	−	0.840314i	$-0.317630\pi$
$347$	20.1964	1.08420	0.542100	+	0.840314i	$0.317630\pi$
$348$	0	0
$349$	19.6729	1.05307	0.526533	−	0.850155i	$-0.323492\pi$
$349$	19.6729	1.05307	0.526533	+	0.850155i	$0.323492\pi$
$350$	−0.231185	−0.0123573
$351$	0	0
$352$	0.742338	0.0395668
$353$	10.0073	0.532633	0.266317	−	0.963886i	$-0.414193\pi$
$353$	10.0073	0.532633	0.266317	+	0.963886i	$0.414193\pi$
$354$	0	0
$355$	−25.3751	−1.34677
$356$	−18.2795	−0.968814
$357$	0	0
$358$	−0.615545	−0.0325326
$359$	11.1068	0.586194	0.293097	−	0.956083i	$-0.405314\pi$
$359$	11.1068	0.586194	0.293097	+	0.956083i	$0.405314\pi$
$360$	0	0
$361$	27.1766	1.43035
$362$	1.41392	0.0743142
$363$	0	0
$364$	40.0018	2.09666
$365$	−36.8981	−1.93134
$366$	0	0
$367$	−6.28530	−0.328090	−0.164045	−	0.986453i	$-0.552454\pi$
$367$	−6.28530	−0.328090	−0.164045	+	0.986453i	$0.552454\pi$
$368$	−3.97694	−0.207312
$369$	0	0
$370$	−0.245683	−0.0127724
$371$	30.8036	1.59924
$372$	0	0
$373$	17.3123	0.896399	0.448199	−	0.893934i	$-0.352065\pi$
$373$	17.3123	0.896399	0.448199	+	0.893934i	$0.352065\pi$
$374$	−0.297574	−0.0153872
$375$	0	0
$376$	2.98315	0.153844
$377$	51.3235	2.64329
$378$	0	0
$379$	5.24331	0.269331	0.134665	−	0.990891i	$-0.457004\pi$
$379$	5.24331	0.269331	0.134665	+	0.990891i	$0.457004\pi$
$380$	33.1729	1.70174
$381$	0	0
$382$	0.862257	0.0441169
$383$	−18.7675	−0.958975	−0.479487	−	0.877549i	$-0.659177\pi$
$383$	−18.7675	−0.958975	−0.479487	+	0.877549i	$0.659177\pi$
$384$	0	0
$385$	−9.29460	−0.473696
$386$	−0.580917	−0.0295679
$387$	0	0
$388$	−13.4088	−0.680727
$389$	−37.6653	−1.90971	−0.954854	−	0.297075i	$-0.903989\pi$
$389$	−37.6653	−1.90971	−0.954854	+	0.297075i	$0.903989\pi$
$390$	0	0
$391$	4.79799	0.242645
$392$	1.84507	0.0931903
$393$	0	0
$394$	−1.31854	−0.0664271
$395$	35.0282	1.76246
$396$	0	0
$397$	−3.83422	−0.192434	−0.0962171	−	0.995360i	$-0.530674\pi$
$397$	−3.83422	−0.192434	−0.0962171	+	0.995360i	$0.530674\pi$
$398$	−0.375703	−0.0188323
$399$	0	0
$400$	3.90051	0.195025
$401$	33.4473	1.67028	0.835140	−	0.550037i	$-0.185386\pi$
$401$	33.4473	1.67028	0.835140	+	0.550037i	$0.185386\pi$
$402$	0	0
$403$	4.67520	0.232888
$404$	−31.1421	−1.54938
$405$	0	0
$406$	2.29440	0.113869
$407$	−1.61979	−0.0802903
$408$	0	0
$409$	−31.9566	−1.58015	−0.790076	−	0.613009i	$-0.789959\pi$
$409$	−31.9566	−1.58015	−0.790076	+	0.613009i	$0.789959\pi$
$410$	1.43914	0.0710743
$411$	0	0
$412$	14.1502	0.697132
$413$	22.7027	1.11713
$414$	0	0
$415$	22.4769	1.10335
$416$	3.91413	0.191906
$417$	0	0
$418$	−0.421450	−0.0206138
$419$	27.7519	1.35577	0.677884	−	0.735168i	$-0.262897\pi$
$419$	27.7519	1.35577	0.677884	+	0.735168i	$0.262897\pi$
$420$	0	0
$421$	−20.0832	−0.978796	−0.489398	−	0.872061i	$-0.662783\pi$
$421$	−20.0832	−0.978796	−0.489398	+	0.872061i	$0.662783\pi$
$422$	−0.485736	−0.0236453
$423$	0	0
$424$	2.00876	0.0975538
$425$	−4.70578	−0.228264
$426$	0	0
$427$	−38.5687	−1.86647
$428$	−0.743263	−0.0359270
$429$	0	0
$430$	0.605110	0.0291810
$431$	32.1617	1.54918	0.774588	−	0.632467i	$-0.217957\pi$
$431$	32.1617	1.54918	0.774588	+	0.632467i	$0.217957\pi$
$432$	0	0
$433$	−8.04837	−0.386780	−0.193390	−	0.981122i	$-0.561948\pi$
$433$	−8.04837	−0.386780	−0.193390	+	0.981122i	$0.561948\pi$
$434$	0.209003	0.0100325
$435$	0	0
$436$	12.2302	0.585720
$437$	6.79534	0.325065
$438$	0	0
$439$	−25.1695	−1.20128	−0.600638	−	0.799521i	$-0.705087\pi$
$439$	−25.1695	−1.20128	−0.600638	+	0.799521i	$0.705087\pi$
$440$	−0.606117	−0.0288955
$441$	0	0
$442$	−1.56902	−0.0746307
$443$	−13.7201	−0.651860	−0.325930	−	0.945394i	$-0.605677\pi$
$443$	−13.7201	−0.651860	−0.325930	+	0.945394i	$0.605677\pi$
$444$	0	0
$445$	22.3950	1.06162
$446$	0.606753	0.0287306
$447$	0	0
$448$	−30.0545	−1.41994
$449$	−0.588616	−0.0277785	−0.0138892	−	0.999904i	$-0.504421\pi$
$449$	−0.588616	−0.0277785	−0.0138892	+	0.999904i	$0.504421\pi$
$450$	0	0
$451$	9.48834	0.446788
$452$	3.16981	0.149095
$453$	0	0
$454$	1.35503	0.0635948
$455$	−49.0077	−2.29752
$456$	0	0
$457$	−18.4682	−0.863908	−0.431954	−	0.901896i	$-0.642176\pi$
$457$	−18.4682	−0.863908	−0.431954	+	0.901896i	$0.642176\pi$
$458$	−0.520680	−0.0243298
$459$	0	0
$460$	4.88172	0.227611
$461$	−4.17501	−0.194449	−0.0972247	−	0.995262i	$-0.530997\pi$
$461$	−4.17501	−0.194449	−0.0972247	+	0.995262i	$0.530997\pi$
$462$	0	0
$463$	−16.3331	−0.759064	−0.379532	−	0.925179i	$-0.623915\pi$
$463$	−16.3331	−0.759064	−0.379532	+	0.925179i	$0.623915\pi$
$464$	−38.7107	−1.79710
$465$	0	0
$466$	−0.533360	−0.0247074
$467$	−11.6963	−0.541238	−0.270619	−	0.962687i	$-0.587228\pi$
$467$	−11.6963	−0.541238	−0.270619	+	0.962687i	$0.587228\pi$
$468$	0	0
$469$	−8.12745	−0.375291
$470$	−1.82563	−0.0842100
$471$	0	0
$472$	1.48048	0.0681446
$473$	3.98952	0.183438
$474$	0	0
$475$	−6.66475	−0.305800
$476$	36.4003	1.66840
$477$	0	0
$478$	−0.375151	−0.0171590
$479$	−36.3441	−1.66060	−0.830302	−	0.557314i	$-0.811832\pi$
$479$	−36.3441	−1.66060	−0.830302	+	0.557314i	$0.811832\pi$
$480$	0	0
$481$	−8.54071	−0.389423
$482$	1.70897	0.0778414
$483$	0	0
$484$	−1.99615	−0.0907342
$485$	16.4276	0.745939
$486$	0	0
$487$	−2.42176	−0.109740	−0.0548702	−	0.998493i	$-0.517475\pi$
$487$	−2.42176	−0.109740	−0.0548702	+	0.998493i	$0.517475\pi$
$488$	−2.51513	−0.113855
$489$	0	0
$490$	−1.12915	−0.0510097
$491$	−36.5163	−1.64796	−0.823979	−	0.566621i	$-0.808250\pi$
$491$	−36.5163	−1.64796	−0.823979	+	0.566621i	$0.808250\pi$
$492$	0	0
$493$	46.7027	2.10338
$494$	−2.22219	−0.0999808
$495$	0	0
$496$	−3.52627	−0.158334
$497$	−39.4348	−1.76889
$498$	0	0
$499$	−34.7189	−1.55423	−0.777117	−	0.629357i	$-0.783318\pi$
$499$	−34.7189	−1.55423	−0.777117	+	0.629357i	$0.783318\pi$
$500$	19.6207	0.877464
$501$	0	0
$502$	1.65625	0.0739221
$503$	16.1502	0.720101	0.360050	−	0.932933i	$-0.382760\pi$
$503$	16.1502	0.720101	0.360050	+	0.932933i	$0.382760\pi$
$504$	0	0
$505$	38.1534	1.69780
$506$	−0.0620205	−0.00275715
$507$	0	0
$508$	26.0357	1.15515
$509$	−5.78369	−0.256357	−0.128179	−	0.991751i	$-0.540913\pi$
$509$	−5.78369	−0.256357	−0.128179	+	0.991751i	$0.540913\pi$
$510$	0	0
$511$	−57.3426	−2.53669
$512$	−4.92355	−0.217592
$513$	0	0
$514$	−1.37844	−0.0608002
$515$	−17.3360	−0.763915
$516$	0	0
$517$	−12.0364	−0.529362
$518$	−0.381810	−0.0167757
$519$	0	0
$520$	−3.19588	−0.140149
$521$	−11.5404	−0.505594	−0.252797	−	0.967519i	$-0.581351\pi$
$521$	−11.5404	−0.505594	−0.252797	+	0.967519i	$0.581351\pi$
$522$	0	0
$523$	−28.4732	−1.24505	−0.622523	−	0.782602i	$-0.713892\pi$
$523$	−28.4732	−1.24505	−0.622523	+	0.782602i	$0.713892\pi$
$524$	0.787835	0.0344167
$525$	0	0
$526$	−0.415329	−0.0181092
$527$	4.25428	0.185319
$528$	0	0
$529$	1.00000	0.0434783
$530$	−1.22932	−0.0533981
$531$	0	0
$532$	51.5533	2.23512
$533$	50.0292	2.16701
$534$	0	0
$535$	0.910600	0.0393687
$536$	−0.530005	−0.0228927
$537$	0	0
$538$	1.24855	0.0538289
$539$	−7.44452	−0.320658
$540$	0	0
$541$	−18.9960	−0.816701	−0.408350	−	0.912825i	$-0.633896\pi$
$541$	−18.9960	−0.816701	−0.408350	+	0.912825i	$0.633896\pi$
$542$	−1.97950	−0.0850270
$543$	0	0
$544$	3.56173	0.152708
$545$	−14.9837	−0.641830
$546$	0	0
$547$	16.0028	0.684229	0.342114	−	0.939658i	$-0.388857\pi$
$547$	16.0028	0.684229	0.342114	+	0.939658i	$0.388857\pi$
$548$	−8.38584	−0.358225
$549$	0	0
$550$	0.0608286	0.00259374
$551$	66.1444	2.81785
$552$	0	0
$553$	54.4366	2.31488
$554$	−0.810302	−0.0344264
$555$	0	0
$556$	34.6407	1.46909
$557$	14.0545	0.595509	0.297755	−	0.954642i	$-0.403762\pi$
$557$	14.0545	0.595509	0.297755	+	0.954642i	$0.403762\pi$
$558$	0	0
$559$	21.0356	0.889710
$560$	36.9640	1.56201
$561$	0	0
$562$	0.898148	0.0378861
$563$	11.8821	0.500771	0.250386	−	0.968146i	$-0.419443\pi$
$563$	11.8821	0.500771	0.250386	+	0.968146i	$0.419443\pi$
$564$	0	0
$565$	−3.88345	−0.163378
$566$	0.786415	0.0330555
$567$	0	0
$568$	−2.57162	−0.107903
$569$	−26.0071	−1.09027	−0.545137	−	0.838347i	$-0.683522\pi$
$569$	−26.0071	−1.09027	−0.545137	+	0.838347i	$0.683522\pi$
$570$	0	0
$571$	5.42386	0.226981	0.113491	−	0.993539i	$-0.463797\pi$
$571$	5.42386	0.226981	0.113491	+	0.993539i	$0.463797\pi$
$572$	−10.5251	−0.440078
$573$	0	0
$574$	2.23654	0.0933514
$575$	−0.980782	−0.0409014
$576$	0	0
$577$	29.8686	1.24345	0.621723	−	0.783237i	$-0.286433\pi$
$577$	29.8686	1.24345	0.621723	+	0.783237i	$0.286433\pi$
$578$	−0.373408	−0.0155317
$579$	0	0
$580$	47.5177	1.97306
$581$	34.9309	1.44918
$582$	0	0
$583$	−8.10494	−0.335672
$584$	−3.73941	−0.154738
$585$	0	0
$586$	0.373740	0.0154391
$587$	8.36955	0.345448	0.172724	−	0.984970i	$-0.444743\pi$
$587$	8.36955	0.345448	0.172724	+	0.984970i	$0.444743\pi$
$588$	0	0
$589$	6.02529	0.248268
$590$	−0.906024	−0.0373004
$591$	0	0
$592$	6.44182	0.264757
$593$	7.21144	0.296139	0.148069	−	0.988977i	$-0.452694\pi$
$593$	7.21144	0.296139	0.148069	+	0.988977i	$0.452694\pi$
$594$	0	0
$595$	−44.5954	−1.82823
$596$	−14.6841	−0.601485
$597$	0	0
$598$	−0.327016	−0.0133727
$599$	14.3381	0.585838	0.292919	−	0.956137i	$-0.405373\pi$
$599$	14.3381	0.585838	0.292919	+	0.956137i	$0.405373\pi$
$600$	0	0
$601$	10.2123	0.416568	0.208284	−	0.978068i	$-0.433212\pi$
$601$	10.2123	0.416568	0.208284	+	0.978068i	$0.433212\pi$
$602$	0.940388	0.0383274
$603$	0	0
$604$	26.4656	1.07687
$605$	2.44556	0.0994263
$606$	0	0
$607$	20.1377	0.817366	0.408683	−	0.912676i	$-0.365988\pi$
$607$	20.1377	0.817366	0.408683	+	0.912676i	$0.365988\pi$
$608$	5.04444	0.204579
$609$	0	0
$610$	1.53921	0.0623208
$611$	−63.4647	−2.56751
$612$	0	0
$613$	−8.78165	−0.354687	−0.177344	−	0.984149i	$-0.556750\pi$
$613$	−8.78165	−0.354687	−0.177344	+	0.984149i	$0.556750\pi$
$614$	−0.996869	−0.0402304
$615$	0	0
$616$	−0.941952	−0.0379523
$617$	21.2291	0.854653	0.427326	−	0.904097i	$-0.359456\pi$
$617$	21.2291	0.854653	0.427326	+	0.904097i	$0.359456\pi$
$618$	0	0
$619$	−16.6270	−0.668295	−0.334147	−	0.942521i	$-0.608448\pi$
$619$	−16.6270	−0.668295	−0.334147	+	0.942521i	$0.608448\pi$
$620$	4.32852	0.173838
$621$	0	0
$622$	−0.660026	−0.0264646
$623$	34.8035	1.39437
$624$	0	0
$625$	−28.9420	−1.15768
$626$	−1.44001	−0.0575544
$627$	0	0
$628$	−2.73557	−0.109161
$629$	−7.77176	−0.309880
$630$	0	0
$631$	−2.58482	−0.102900	−0.0514499	−	0.998676i	$-0.516384\pi$
$631$	−2.58482	−0.102900	−0.0514499	+	0.998676i	$0.516384\pi$
$632$	3.54990	0.141208
$633$	0	0
$634$	1.79469	0.0712761
$635$	−31.8974	−1.26581
$636$	0	0
$637$	−39.2528	−1.55525
$638$	−0.603695	−0.0239005
$639$	0	0
$640$	4.83029	0.190934
$641$	−2.45708	−0.0970488	−0.0485244	−	0.998822i	$-0.515452\pi$
$641$	−2.45708	−0.0970488	−0.0485244	+	0.998822i	$0.515452\pi$
$642$	0	0
$643$	−18.8261	−0.742428	−0.371214	−	0.928547i	$-0.621058\pi$
$643$	−18.8261	−0.742428	−0.371214	+	0.928547i	$0.621058\pi$
$644$	7.58657	0.298953
$645$	0	0
$646$	−2.02212	−0.0795591
$647$	12.9598	0.509501	0.254751	−	0.967007i	$-0.418007\pi$
$647$	12.9598	0.509501	0.254751	+	0.967007i	$0.418007\pi$
$648$	0	0
$649$	−5.97345	−0.234478
$650$	0.320732	0.0125801
$651$	0	0
$652$	4.29856	0.168344
$653$	−36.8991	−1.44397	−0.721987	−	0.691907i	$-0.756771\pi$
$653$	−36.8991	−1.44397	−0.721987	+	0.691907i	$0.756771\pi$
$654$	0	0
$655$	−0.965207	−0.0377137
$656$	−37.7345	−1.47328
$657$	0	0
$658$	−2.83717	−0.110604
$659$	34.1824	1.33156	0.665778	−	0.746150i	$-0.268100\pi$
$659$	34.1824	1.33156	0.665778	+	0.746150i	$0.268100\pi$
$660$	0	0
$661$	21.2402	0.826150	0.413075	−	0.910697i	$-0.364455\pi$
$661$	21.2402	0.826150	0.413075	+	0.910697i	$0.364455\pi$
$662$	−0.241968	−0.00940437
$663$	0	0
$664$	2.27790	0.0883998
$665$	−63.1599	−2.44924
$666$	0	0
$667$	9.73379	0.376894
$668$	−5.80452	−0.224584
$669$	0	0
$670$	0.324352	0.0125308
$671$	10.1481	0.391762
$672$	0	0
$673$	21.0215	0.810320	0.405160	−	0.914246i	$-0.367216\pi$
$673$	21.0215	0.810320	0.405160	+	0.914246i	$0.367216\pi$
$674$	−1.56588	−0.0603155
$675$	0	0
$676$	−29.5460	−1.13638
$677$	33.2633	1.27841	0.639207	−	0.769035i	$-0.279263\pi$
$677$	33.2633	1.27841	0.639207	+	0.769035i	$0.279263\pi$
$678$	0	0
$679$	25.5298	0.979742
$680$	−2.90814	−0.111522
$681$	0	0
$682$	−0.0549923	−0.00210576
$683$	−24.6295	−0.942423	−0.471211	−	0.882020i	$-0.656183\pi$
$683$	−24.6295	−0.942423	−0.471211	+	0.882020i	$0.656183\pi$
$684$	0	0
$685$	10.2738	0.392542
$686$	−0.104780	−0.00400050
$687$	0	0
$688$	−15.8661	−0.604888
$689$	−42.7350	−1.62807
$690$	0	0
$691$	−25.3441	−0.964134	−0.482067	−	0.876134i	$-0.660114\pi$
$691$	−25.3441	−0.964134	−0.482067	+	0.876134i	$0.660114\pi$
$692$	−5.46114	−0.207601
$693$	0	0
$694$	−1.25259	−0.0475477
$695$	−42.4397	−1.60983
$696$	0	0
$697$	45.5250	1.72438
$698$	−1.22012	−0.0461823
$699$	0	0
$700$	−7.44077	−0.281235
$701$	−42.8313	−1.61772	−0.808858	−	0.588004i	$-0.799914\pi$
$701$	−42.8313	−1.61772	−0.808858	+	0.588004i	$0.799914\pi$
$702$	0	0
$703$	−11.0071	−0.415139
$704$	7.90783	0.298038
$705$	0	0
$706$	−0.620656	−0.0233587
$707$	59.2933	2.22995
$708$	0	0
$709$	−15.7544	−0.591668	−0.295834	−	0.955239i	$-0.595597\pi$
$709$	−15.7544	−0.591668	−0.295834	+	0.955239i	$0.595597\pi$
$710$	1.57378	0.0590628
$711$	0	0
$712$	2.26960	0.0850567
$713$	0.886679	0.0332064
$714$	0	0
$715$	12.8947	0.482236
$716$	−19.8116	−0.740393
$717$	0	0
$718$	−0.688849	−0.0257076
$719$	−21.6047	−0.805719	−0.402860	−	0.915262i	$-0.631984\pi$
$719$	−21.6047	−0.805719	−0.402860	+	0.915262i	$0.631984\pi$
$720$	0	0
$721$	−26.9415	−1.00335
$722$	−1.68551	−0.0627281
$723$	0	0
$724$	45.5077	1.69128
$725$	−9.54673	−0.354557
$726$	0	0
$727$	−1.85572	−0.0688249	−0.0344124	−	0.999408i	$-0.510956\pi$
$727$	−1.85572	−0.0688249	−0.0344124	+	0.999408i	$0.510956\pi$
$728$	−4.96664	−0.184076
$729$	0	0
$730$	2.28844	0.0846990
$731$	19.1417	0.707980
$732$	0	0
$733$	−6.87471	−0.253923	−0.126962	−	0.991908i	$-0.540522\pi$
$733$	−6.87471	−0.253923	−0.126962	+	0.991908i	$0.540522\pi$
$734$	0.389817	0.0143884
$735$	0	0
$736$	0.742338	0.0273630
$737$	2.13847	0.0787715
$738$	0	0
$739$	16.4364	0.604621	0.302311	−	0.953209i	$-0.402242\pi$
$739$	16.4364	0.604621	0.302311	+	0.953209i	$0.402242\pi$
$740$	−7.90739	−0.290681
$741$	0	0
$742$	−1.91045	−0.0701350
$743$	−11.7330	−0.430442	−0.215221	−	0.976565i	$-0.569047\pi$
$743$	−11.7330	−0.430442	−0.215221	+	0.976565i	$0.569047\pi$
$744$	0	0
$745$	17.9901	0.659105
$746$	−1.07372	−0.0393117
$747$	0	0
$748$	−9.57753	−0.350189
$749$	1.41514	0.0517082
$750$	0	0
$751$	25.4835	0.929904	0.464952	−	0.885336i	$-0.346071\pi$
$751$	25.4835	0.929904	0.464952	+	0.885336i	$0.346071\pi$
$752$	47.8682	1.74557
$753$	0	0
$754$	−3.18311	−0.115922
$755$	−32.4240	−1.18003
$756$	0	0
$757$	−28.7418	−1.04464	−0.522319	−	0.852750i	$-0.674933\pi$
$757$	−28.7418	−1.04464	−0.522319	+	0.852750i	$0.674933\pi$
$758$	−0.325193	−0.0118115
$759$	0	0
$760$	−4.11877	−0.149403
$761$	41.2289	1.49455	0.747273	−	0.664517i	$-0.231363\pi$
$761$	41.2289	1.49455	0.747273	+	0.664517i	$0.231363\pi$
$762$	0	0
$763$	−23.2858	−0.843001
$764$	27.7521	1.00404
$765$	0	0
$766$	1.16397	0.0420559
$767$	−31.4963	−1.13726
$768$	0	0
$769$	−3.24451	−0.117000	−0.0585000	−	0.998287i	$-0.518632\pi$
$769$	−3.24451	−0.117000	−0.0585000	+	0.998287i	$0.518632\pi$
$770$	0.576455	0.0207740
$771$	0	0
$772$	−18.6970	−0.672921
$773$	−24.0624	−0.865465	−0.432732	−	0.901522i	$-0.642450\pi$
$773$	−24.0624	−0.865465	−0.432732	+	0.901522i	$0.642450\pi$
$774$	0	0
$775$	−0.869639	−0.0312384
$776$	1.66484	0.0597643
$777$	0	0
$778$	2.33602	0.0837505
$779$	64.4765	2.31011
$780$	0	0
$781$	10.3760	0.371281
$782$	−0.297574	−0.0106412
$783$	0	0
$784$	29.6064	1.05737
$785$	3.35145	0.119618
$786$	0	0
$787$	−48.3519	−1.72356	−0.861779	−	0.507284i	$-0.830650\pi$
$787$	−48.3519	−1.72356	−0.861779	+	0.507284i	$0.830650\pi$
$788$	−42.4377	−1.51178
$789$	0	0
$790$	−2.17247	−0.0772930
$791$	−6.03518	−0.214586
$792$	0	0
$793$	53.5078	1.90012
$794$	0.237800	0.00843922
$795$	0	0
$796$	−12.0921	−0.428594
$797$	30.3183	1.07393	0.536965	−	0.843605i	$-0.319571\pi$
$797$	30.3183	1.07393	0.536965	+	0.843605i	$0.319571\pi$
$798$	0	0
$799$	−57.7508	−2.04307
$800$	−0.728072	−0.0257412
$801$	0	0
$802$	−2.07442	−0.0732503
$803$	15.0878	0.532436
$804$	0	0
$805$	−9.29460	−0.327591
$806$	−0.289958	−0.0102134
$807$	0	0
$808$	3.86662	0.136027
$809$	48.7336	1.71338	0.856691	−	0.515830i	$-0.172516\pi$
$809$	48.7336	1.71338	0.856691	+	0.515830i	$0.172516\pi$
$810$	0	0
$811$	15.7407	0.552732	0.276366	−	0.961052i	$-0.410870\pi$
$811$	15.7407	0.552732	0.276366	+	0.961052i	$0.410870\pi$
$812$	73.8461	2.59149
$813$	0	0
$814$	0.100460	0.00352114
$815$	−5.26633	−0.184471
$816$	0	0
$817$	27.1101	0.948463
$818$	1.98196	0.0692977
$819$	0	0
$820$	46.3194	1.61754
$821$	−20.4763	−0.714627	−0.357313	−	0.933985i	$-0.616307\pi$
$821$	−20.4763	−0.714627	−0.357313	+	0.933985i	$0.616307\pi$
$822$	0	0
$823$	27.3973	0.955010	0.477505	−	0.878629i	$-0.341541\pi$
$823$	27.3973	0.955010	0.477505	+	0.878629i	$0.341541\pi$
$824$	−1.75690	−0.0612045
$825$	0	0
$826$	−1.40803	−0.0489916
$827$	−29.4934	−1.02559	−0.512793	−	0.858512i	$-0.671389\pi$
$827$	−29.4934	−1.02559	−0.512793	+	0.858512i	$0.671389\pi$
$828$	0	0
$829$	−29.9652	−1.04073	−0.520366	−	0.853943i	$-0.674205\pi$
$829$	−29.9652	−1.04073	−0.520366	+	0.853943i	$0.674205\pi$
$830$	−1.39403	−0.0483875
$831$	0	0
$832$	41.6957	1.44554
$833$	−35.7187	−1.23758
$834$	0	0
$835$	7.11134	0.246098
$836$	−13.5645	−0.469139
$837$	0	0
$838$	−1.72119	−0.0594574
$839$	−22.4124	−0.773761	−0.386881	−	0.922130i	$-0.626447\pi$
$839$	−22.4124	−0.773761	−0.386881	+	0.922130i	$0.626447\pi$
$840$	0	0
$841$	65.7468	2.26713
$842$	1.24557	0.0429252
$843$	0	0
$844$	−15.6336	−0.538131
$845$	36.1979	1.24525
$846$	0	0
$847$	3.80059	0.130590
$848$	32.2328	1.10688
$849$	0	0
$850$	0.291855	0.0100105
$851$	−1.61979	−0.0555259
$852$	0	0
$853$	41.7909	1.43089	0.715447	−	0.698667i	$-0.246223\pi$
$853$	41.7909	1.43089	0.715447	+	0.698667i	$0.246223\pi$
$854$	2.39205	0.0818543
$855$	0	0
$856$	0.0922839	0.00315420
$857$	−18.6358	−0.636588	−0.318294	−	0.947992i	$-0.603110\pi$
$857$	−18.6358	−0.636588	−0.318294	+	0.947992i	$0.603110\pi$
$858$	0	0
$859$	30.1013	1.02704	0.513521	−	0.858077i	$-0.328341\pi$
$859$	30.1013	1.02704	0.513521	+	0.858077i	$0.328341\pi$
$860$	19.4757	0.664116
$861$	0	0
$862$	−1.99469	−0.0679393
$863$	−1.09826	−0.0373853	−0.0186927	−	0.999825i	$-0.505950\pi$
$863$	−1.09826	−0.0373853	−0.0186927	+	0.999825i	$0.505950\pi$
$864$	0	0
$865$	6.69065	0.227489
$866$	0.499164	0.0169623
$867$	0	0
$868$	6.72686	0.228324
$869$	−14.3232	−0.485880
$870$	0	0
$871$	11.2755	0.382056
$872$	−1.51851	−0.0514231
$873$	0	0
$874$	−0.421450	−0.0142558
$875$	−37.3570	−1.26290
$876$	0	0
$877$	13.3223	0.449861	0.224930	−	0.974375i	$-0.427785\pi$
$877$	13.3223	0.449861	0.224930	+	0.974375i	$0.427785\pi$
$878$	1.56103	0.0526821
$879$	0	0
$880$	−9.72585	−0.327858
$881$	53.6169	1.80640	0.903200	−	0.429220i	$-0.141211\pi$
$881$	53.6169	1.80640	0.903200	+	0.429220i	$0.141211\pi$
$882$	0	0
$883$	25.3026	0.851502	0.425751	−	0.904840i	$-0.360010\pi$
$883$	25.3026	0.851502	0.425751	+	0.904840i	$0.360010\pi$
$884$	−50.4995	−1.69848
$885$	0	0
$886$	0.850925	0.0285874
$887$	−50.6072	−1.69923	−0.849613	−	0.527407i	$-0.823164\pi$
$887$	−50.6072	−1.69923	−0.849613	+	0.527407i	$0.823164\pi$
$888$	0	0
$889$	−49.5710	−1.66256
$890$	−1.38895	−0.0465576
$891$	0	0
$892$	19.5286	0.653866
$893$	−81.7917	−2.73706
$894$	0	0
$895$	24.2719	0.811320
$896$	7.50665	0.250779
$897$	0	0
$898$	0.0365062	0.00121823
$899$	8.63076	0.287852
$900$	0	0
$901$	−38.8874	−1.29553
$902$	−0.588471	−0.0195940
$903$	0	0
$904$	−0.393565	−0.0130898
$905$	−55.7532	−1.85330
$906$	0	0
$907$	−25.8419	−0.858066	−0.429033	−	0.903289i	$-0.641146\pi$
$907$	−25.8419	−0.858066	−0.429033	+	0.903289i	$0.641146\pi$
$908$	43.6122	1.44732
$909$	0	0
$910$	3.03948	0.100758
$911$	−0.789957	−0.0261725	−0.0130862	−	0.999914i	$-0.504166\pi$
$911$	−0.789957	−0.0261725	−0.0130862	+	0.999914i	$0.504166\pi$
$912$	0	0
$913$	−9.19090	−0.304174
$914$	1.14541	0.0378868
$915$	0	0
$916$	−16.7583	−0.553709
$917$	−1.50001	−0.0495345
$918$	0	0
$919$	−13.0849	−0.431631	−0.215815	−	0.976434i	$-0.569241\pi$
$919$	−13.0849	−0.431631	−0.215815	+	0.976434i	$0.569241\pi$
$920$	−0.606117	−0.0199831
$921$	0	0
$922$	0.258936	0.00852760
$923$	54.7095	1.80078
$924$	0	0
$925$	1.58867	0.0522350
$926$	1.01299	0.0332888
$927$	0	0
$928$	7.22577	0.237197
$929$	−1.21726	−0.0399371	−0.0199686	−	0.999801i	$-0.506357\pi$
$929$	−1.21726	−0.0399371	−0.0199686	+	0.999801i	$0.506357\pi$
$930$	0	0
$931$	−50.5880	−1.65796
$932$	−17.1664	−0.562304
$933$	0	0
$934$	0.725407	0.0237361
$935$	11.7338	0.383736
$936$	0	0
$937$	15.2407	0.497892	0.248946	−	0.968517i	$-0.419916\pi$
$937$	15.2407	0.497892	0.248946	+	0.968517i	$0.419916\pi$
$938$	0.504068	0.0164584
$939$	0	0
$940$	−58.7586	−1.91649
$941$	19.2595	0.627843	0.313921	−	0.949449i	$-0.398357\pi$
$941$	19.2595	0.627843	0.313921	+	0.949449i	$0.398357\pi$
$942$	0	0
$943$	9.48834	0.308983
$944$	23.7560	0.773193
$945$	0	0
$946$	−0.247432	−0.00804470
$947$	36.4068	1.18306	0.591531	−	0.806282i	$-0.298524\pi$
$947$	36.4068	1.18306	0.591531	+	0.806282i	$0.298524\pi$
$948$	0	0
$949$	79.5535	2.58242
$950$	0.413351	0.0134109
$951$	0	0
$952$	−4.51948	−0.146477
$953$	5.05538	0.163760	0.0818799	−	0.996642i	$-0.473908\pi$
$953$	5.05538	0.163760	0.0818799	+	0.996642i	$0.473908\pi$
$954$	0	0
$955$	−34.0001	−1.10022
$956$	−12.0744	−0.390513
$957$	0	0
$958$	2.25408	0.0728259
$959$	15.9663	0.515579
$960$	0	0
$961$	−30.2138	−0.974639
$962$	0.529699	0.0170782
$963$	0	0
$964$	55.0038	1.77155
$965$	22.9065	0.737385
$966$	0	0
$967$	−56.4879	−1.81653	−0.908264	−	0.418397i	$-0.862592\pi$
$967$	−56.4879	−1.81653	−0.908264	+	0.418397i	$0.862592\pi$
$968$	0.247843	0.00796599
$969$	0	0
$970$	−1.01885	−0.0327132
$971$	16.0663	0.515593	0.257796	−	0.966199i	$-0.417004\pi$
$971$	16.0663	0.515593	0.257796	+	0.966199i	$0.417004\pi$
$972$	0	0
$973$	−65.9545	−2.11440
$974$	0.150199	0.00481268
$975$	0	0
$976$	−40.3582	−1.29184
$977$	49.2278	1.57494	0.787468	−	0.616356i	$-0.211392\pi$
$977$	49.2278	1.57494	0.787468	+	0.616356i	$0.211392\pi$
$978$	0	0
$979$	−9.15738	−0.292671
$980$	−36.3421	−1.16090
$981$	0	0
$982$	2.26476	0.0722714
$983$	−0.915251	−0.0291920	−0.0145960	−	0.999893i	$-0.504646\pi$
$983$	−0.915251	−0.0291920	−0.0145960	+	0.999893i	$0.504646\pi$
$984$	0	0
$985$	51.9921	1.65661
$986$	−2.89652	−0.0922441
$987$	0	0
$988$	−71.5219	−2.27541
$989$	3.98952	0.126859
$990$	0	0
$991$	−53.1117	−1.68715	−0.843575	−	0.537012i	$-0.819553\pi$
$991$	−53.1117	−1.68715	−0.843575	+	0.537012i	$0.819553\pi$
$992$	0.658216	0.0208984
$993$	0	0
$994$	2.44577	0.0775751
$995$	14.8145	0.469652
$996$	0	0
$997$	23.9662	0.759016	0.379508	−	0.925189i	$-0.376093\pi$
$997$	23.9662	0.759016	0.379508	+	0.925189i	$0.376093\pi$
$998$	2.15329	0.0681611
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2277.2.a.r.1.4	yes	8
3.2	odd	2		2277.2.a.q.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2277.2.a.q.1.5	✓	8	3.2	odd	2
2277.2.a.r.1.4	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2277 = 3^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2277.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0620205 q^{2} -1.99615 q^{4} +2.44556 q^{5} +3.80059 q^{7} +0.247843 q^{8} -0.151675 q^{10} -1.00000 q^{11} -5.27271 q^{13} -0.235715 q^{14} +3.97694 q^{16} -4.79799 q^{17} -6.79534 q^{19} -4.88172 q^{20} +0.0620205 q^{22} -1.00000 q^{23} +0.980782 q^{25} +0.327016 q^{26} -7.58657 q^{28} -9.73379 q^{29} -0.886679 q^{31} -0.742338 q^{32} +0.297574 q^{34} +9.29460 q^{35} +1.61979 q^{37} +0.421450 q^{38} +0.606117 q^{40} -9.48834 q^{41} -3.98952 q^{43} +1.99615 q^{44} +0.0620205 q^{46} +12.0364 q^{47} +7.44452 q^{49} -0.0608286 q^{50} +10.5251 q^{52} +8.10494 q^{53} -2.44556 q^{55} +0.941952 q^{56} +0.603695 q^{58} +5.97345 q^{59} -10.1481 q^{61} +0.0549923 q^{62} -7.90783 q^{64} -12.8947 q^{65} -2.13847 q^{67} +9.57753 q^{68} -0.576455 q^{70} -10.3760 q^{71} -15.0878 q^{73} -0.100460 q^{74} +13.5645 q^{76} -3.80059 q^{77} +14.3232 q^{79} +9.72585 q^{80} +0.588471 q^{82} +9.19090 q^{83} -11.7338 q^{85} +0.247432 q^{86} -0.247843 q^{88} +9.15738 q^{89} -20.0394 q^{91} +1.99615 q^{92} -0.746506 q^{94} -16.6184 q^{95} +6.71731 q^{97} -0.461713 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 8 q^{11} - 10 q^{13} - 2 q^{14} - 2 q^{16} - 16 q^{19} + 12 q^{20} - 8 q^{23} - 4 q^{25} - 20 q^{28} - 2 q^{29} - 20 q^{31} - 12 q^{32} - 22 q^{34}+ \cdots + 46 q^{98}+O(q^{100}) \) 8 * q + 6 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 8 * q^11 - 10 * q^13 - 2 * q^14 - 2 * q^16 - 16 * q^19 + 12 * q^20 - 8 * q^23 - 4 * q^25 - 20 * q^28 - 2 * q^29 - 20 * q^31 - 12 * q^32 - 22 * q^34 - 18 * q^35 - 6 * q^37 + 18 * q^38 - 30 * q^40 - 6 * q^41 - 26 * q^43 - 6 * q^44 + 12 * q^47 + 4 * q^49 - 30 * q^50 - 30 * q^52 + 14 * q^53 - 8 * q^55 + 10 * q^56 - 36 * q^58 - 12 * q^61 - 8 * q^64 - 14 * q^65 - 50 * q^67 - 6 * q^68 - 6 * q^70 - 32 * q^71 - 24 * q^73 - 14 * q^74 - 22 * q^76 + 6 * q^77 - 4 * q^79 + 54 * q^80 + 14 * q^82 - 10 * q^83 - 18 * q^85 + 54 * q^86 + 6 * q^88 + 16 * q^89 - 26 * q^91 - 6 * q^92 + 22 * q^94 - 38 * q^95 - 22 * q^97 + 46 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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