LMFDB - Embedded newform 2535.2.a.bj.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2535,2,Mod(1,2535)] 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("2535.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2535, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2535 = 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2535.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,4,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.2420769124$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13824.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 6 $ x^4 - 6*x^2 + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.17533$ of defining polynomial
Character	$\chi$	$=$	2535.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.17533 q^{2} -1.00000 q^{3} +2.73205 q^{4} -1.00000 q^{5} -2.17533 q^{6} -0.443277 q^{7} +1.59245 q^{8} +1.00000 q^{9} -2.17533 q^{10} +1.59245 q^{11} -2.73205 q^{12} -0.964273 q^{14} +1.00000 q^{15} -2.00000 q^{16} +3.76778 q^{17} +2.17533 q^{18} +1.54715 q^{19} -2.73205 q^{20} +0.443277 q^{21} +3.46410 q^{22} +0.705897 q^{23} -1.59245 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.21106 q^{28} +2.49983 q^{29} +2.17533 q^{30} +6.18490 q^{31} -7.53556 q^{32} -1.59245 q^{33} +8.19615 q^{34} +0.443277 q^{35} +2.73205 q^{36} +10.3507 q^{37} +3.36556 q^{38} -1.59245 q^{40} +6.02082 q^{41} +0.964273 q^{42} +4.72248 q^{43} +4.35066 q^{44} -1.00000 q^{45} +1.53556 q^{46} +5.41712 q^{47} +2.00000 q^{48} -6.80351 q^{49} +2.17533 q^{50} -3.76778 q^{51} +5.51641 q^{53} -2.17533 q^{54} -1.59245 q^{55} -0.705897 q^{56} -1.54715 q^{57} +5.43795 q^{58} +8.72214 q^{59} +2.73205 q^{60} -9.73205 q^{61} +13.4542 q^{62} -0.443277 q^{63} -12.3923 q^{64} -3.46410 q^{66} -13.6960 q^{67} +10.2938 q^{68} -0.705897 q^{69} +0.964273 q^{70} -15.1488 q^{71} +1.59245 q^{72} +10.7939 q^{73} +22.5161 q^{74} -1.00000 q^{75} +4.22689 q^{76} -0.705897 q^{77} +10.7317 q^{79} +2.00000 q^{80} +1.00000 q^{81} +13.0973 q^{82} -9.12801 q^{83} +1.21106 q^{84} -3.76778 q^{85} +10.2729 q^{86} -2.49983 q^{87} +2.53590 q^{88} +4.75653 q^{89} -2.17533 q^{90} +1.92855 q^{92} -6.18490 q^{93} +11.7840 q^{94} -1.54715 q^{95} +7.53556 q^{96} +8.41712 q^{97} -14.7999 q^{98} +1.59245 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{9} - 4 q^{12} - 12 q^{14} + 4 q^{15} - 8 q^{16} + 12 q^{19} - 4 q^{20} + 4 q^{25} - 4 q^{27} + 12 q^{28} - 12 q^{29} + 12 q^{31} + 12 q^{34} + 4 q^{36} + 24 q^{37}+ \cdots - 24 q^{98}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^9 - 4 * q^12 - 12 * q^14 + 4 * q^15 - 8 * q^16 + 12 * q^19 - 4 * q^20 + 4 * q^25 - 4 * q^27 + 12 * q^28 - 12 * q^29 + 12 * q^31 + 12 * q^34 + 4 * q^36 + 24 * q^37 + 12 * q^41 + 12 * q^42 + 16 * q^43 - 4 * q^45 - 24 * q^46 + 24 * q^47 + 8 * q^48 - 4 * q^49 - 12 * q^57 + 12 * q^58 - 12 * q^59 + 4 * q^60 - 32 * q^61 - 8 * q^64 - 12 * q^67 + 12 * q^70 - 12 * q^71 + 24 * q^73 + 24 * q^74 - 4 * q^75 + 24 * q^76 - 8 * q^79 + 8 * q^80 + 4 * q^81 - 12 * q^82 - 12 * q^84 + 12 * q^86 + 12 * q^87 + 24 * q^88 + 12 * q^89 + 24 * q^92 - 12 * q^93 - 12 * q^94 - 12 * q^95 + 36 * q^97 - 24 * 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 $

$2$	2.17533	1.53819	0.769095	−	0.639135i	$-0.220708\pi$
$2$	2.17533	1.53819	0.769095	+	0.639135i	$0.220708\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.73205	1.36603
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.17533	−0.888074
$7$	−0.443277	−0.167543	−0.0837715	−	0.996485i	$-0.526697\pi$
$7$	−0.443277	−0.167543	−0.0837715	+	0.996485i	$0.526697\pi$
$8$	1.59245	0.563016
$9$	1.00000	0.333333
$10$	−2.17533	−0.687899
$11$	1.59245	0.480142	0.240071	−	0.970755i	$-0.422829\pi$
$11$	1.59245	0.480142	0.240071	+	0.970755i	$0.422829\pi$
$12$	−2.73205	−0.788675
$13$	0	0
$13$	0	0
$14$	−0.964273	−0.257713
$15$	1.00000	0.258199
$16$	−2.00000	−0.500000
$17$	3.76778	0.913820	0.456910	−	0.889513i	$-0.348956\pi$
$17$	3.76778	0.913820	0.456910	+	0.889513i	$0.348956\pi$
$18$	2.17533	0.512730
$19$	1.54715	0.354941	0.177470	−	0.984126i	$-0.443209\pi$
$19$	1.54715	0.354941	0.177470	+	0.984126i	$0.443209\pi$
$20$	−2.73205	−0.610905
$21$	0.443277	0.0967310
$22$	3.46410	0.738549
$23$	0.705897	0.147190	0.0735948	−	0.997288i	$-0.476553\pi$
$23$	0.705897	0.147190	0.0735948	+	0.997288i	$0.476553\pi$
$24$	−1.59245	−0.325058
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.21106	−0.228868
$29$	2.49983	0.464207	0.232103	−	0.972691i	$-0.425439\pi$
$29$	2.49983	0.464207	0.232103	+	0.972691i	$0.425439\pi$
$30$	2.17533	0.397159
$31$	6.18490	1.11084	0.555420	−	0.831570i	$-0.312557\pi$
$31$	6.18490	1.11084	0.555420	+	0.831570i	$0.312557\pi$
$32$	−7.53556	−1.33211
$33$	−1.59245	−0.277210
$34$	8.19615	1.40563
$35$	0.443277	0.0749275
$36$	2.73205	0.455342
$37$	10.3507	1.70164	0.850819	−	0.525459i	$-0.176107\pi$
$37$	10.3507	1.70164	0.850819	+	0.525459i	$0.176107\pi$
$38$	3.36556	0.545966
$39$	0	0
$40$	−1.59245	−0.251789
$41$	6.02082	0.940295	0.470147	−	0.882588i	$-0.344201\pi$
$41$	6.02082	0.940295	0.470147	+	0.882588i	$0.344201\pi$
$42$	0.964273	0.148790
$43$	4.72248	0.720171	0.360086	−	0.932919i	$-0.382748\pi$
$43$	4.72248	0.720171	0.360086	+	0.932919i	$0.382748\pi$
$44$	4.35066	0.655886
$45$	−1.00000	−0.149071
$46$	1.53556	0.226405
$47$	5.41712	0.790169	0.395084	−	0.918645i	$-0.370715\pi$
$47$	5.41712	0.790169	0.395084	+	0.918645i	$0.370715\pi$
$48$	2.00000	0.288675
$49$	−6.80351	−0.971929
$50$	2.17533	0.307638
$51$	−3.76778	−0.527594
$52$	0	0
$53$	5.51641	0.757737	0.378869	−	0.925450i	$-0.376313\pi$
$53$	5.51641	0.757737	0.378869	+	0.925450i	$0.376313\pi$
$54$	−2.17533	−0.296025
$55$	−1.59245	−0.214726
$56$	−0.705897	−0.0943294
$57$	−1.54715	−0.204925
$58$	5.43795	0.714037
$59$	8.72214	1.13553	0.567763	−	0.823192i	$-0.307809\pi$
$59$	8.72214	1.13553	0.567763	+	0.823192i	$0.307809\pi$
$60$	2.73205	0.352706
$61$	−9.73205	−1.24606	−0.623031	−	0.782197i	$-0.714099\pi$
$61$	−9.73205	−1.24606	−0.623031	+	0.782197i	$0.714099\pi$
$62$	13.4542	1.70868
$63$	−0.443277	−0.0558476
$64$	−12.3923	−1.54904
$65$	0	0
$66$	−3.46410	−0.426401
$67$	−13.6960	−1.67323	−0.836615	−	0.547791i	$-0.815469\pi$
$67$	−13.6960	−1.67323	−0.836615	+	0.547791i	$0.815469\pi$
$68$	10.2938	1.24830
$69$	−0.705897	−0.0849800
$70$	0.964273	0.115253
$71$	−15.1488	−1.79784	−0.898918	−	0.438117i	$-0.855645\pi$
$71$	−15.1488	−1.79784	−0.898918	+	0.438117i	$0.855645\pi$
$72$	1.59245	0.187672
$73$	10.7939	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$73$	10.7939	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$74$	22.5161	2.61744
$75$	−1.00000	−0.115470
$76$	4.22689	0.484858
$77$	−0.705897	−0.0804444
$78$	0	0
$79$	10.7317	1.20741	0.603706	−	0.797207i	$-0.293690\pi$
$79$	10.7317	1.20741	0.603706	+	0.797207i	$0.293690\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	13.0973	1.44635
$83$	−9.12801	−1.00193	−0.500964	−	0.865468i	$-0.667021\pi$
$83$	−9.12801	−1.00193	−0.500964	+	0.865468i	$0.667021\pi$
$84$	1.21106	0.132137
$85$	−3.76778	−0.408673
$86$	10.2729	1.10776
$87$	−2.49983	−0.268010
$88$	2.53590	0.270328
$89$	4.75653	0.504191	0.252095	−	0.967702i	$-0.418880\pi$
$89$	4.75653	0.504191	0.252095	+	0.967702i	$0.418880\pi$
$90$	−2.17533	−0.229300
$91$	0	0
$92$	1.92855	0.201065
$93$	−6.18490	−0.641344
$94$	11.7840	1.21543
$95$	−1.54715	−0.158734
$96$	7.53556	0.769095
$97$	8.41712	0.854629	0.427315	−	0.904103i	$-0.359460\pi$
$97$	8.41712	0.854629	0.427315	+	0.904103i	$0.359460\pi$
$98$	−14.7999	−1.49501
$99$	1.59245	0.160047
$100$	2.73205	0.273205
$101$	−6.70590	−0.667262	−0.333631	−	0.942704i	$-0.608274\pi$
$101$	−6.70590	−0.667262	−0.333631	+	0.942704i	$0.608274\pi$
$102$	−8.19615	−0.811540
$103$	1.74197	0.171641	0.0858205	−	0.996311i	$-0.472649\pi$
$103$	1.74197	0.171641	0.0858205	+	0.996311i	$0.472649\pi$
$104$	0	0
$105$	−0.443277	−0.0432594
$106$	12.0000	1.16554
$107$	11.9186	1.15222	0.576109	−	0.817373i	$-0.304570\pi$
$107$	11.9186	1.15222	0.576109	+	0.817373i	$0.304570\pi$
$108$	−2.73205	−0.262892
$109$	17.0407	1.63220	0.816102	−	0.577908i	$-0.196131\pi$
$109$	17.0407	1.63220	0.816102	+	0.577908i	$0.196131\pi$
$110$	−3.46410	−0.330289
$111$	−10.3507	−0.982441
$112$	0.886554	0.0837715
$113$	−3.46410	−0.325875	−0.162938	−	0.986636i	$-0.552097\pi$
$113$	−3.46410	−0.325875	−0.162938	+	0.986636i	$0.552097\pi$
$114$	−3.36556	−0.315213
$115$	−0.705897	−0.0658252
$116$	6.82966	0.634118
$117$	0	0
$118$	18.9735	1.74665
$119$	−1.67017	−0.153104
$120$	1.59245	0.145370
$121$	−8.46410	−0.769464
$122$	−21.1704	−1.91668
$123$	−6.02082	−0.542879
$124$	16.8975	1.51744
$125$	−1.00000	−0.0894427
$126$	−0.964273	−0.0859042
$127$	14.2510	1.26457	0.632287	−	0.774734i	$-0.282116\pi$
$127$	14.2510	1.26457	0.632287	+	0.774734i	$0.282116\pi$
$128$	−11.8862	−1.05060
$129$	−4.72248	−0.415791
$130$	0	0
$131$	10.0354	0.876796	0.438398	−	0.898781i	$-0.355546\pi$
$131$	10.0354	0.876796	0.438398	+	0.898781i	$0.355546\pi$
$132$	−4.35066	−0.378676
$133$	−0.685816	−0.0594678
$134$	−29.7932	−2.57374
$135$	1.00000	0.0860663
$136$	6.00000	0.514496
$137$	−9.28419	−0.793202	−0.396601	−	0.917991i	$-0.629810\pi$
$137$	−9.28419	−0.793202	−0.396601	+	0.917991i	$0.629810\pi$
$138$	−1.53556	−0.130715
$139$	−6.51641	−0.552715	−0.276357	−	0.961055i	$-0.589127\pi$
$139$	−6.51641	−0.552715	−0.276357	+	0.961055i	$0.589127\pi$
$140$	1.21106	0.102353
$141$	−5.41712	−0.456204
$142$	−32.9537	−2.76541
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−2.49983	−0.207599
$146$	23.4803	1.94325
$147$	6.80351	0.561144
$148$	28.2785	2.32448
$149$	0.0568941	0.00466094	0.00233047	−	0.999997i	$-0.499258\pi$
$149$	0.0568941	0.00466094	0.00233047	+	0.999997i	$0.499258\pi$
$150$	−2.17533	−0.177615
$151$	−17.3004	−1.40789	−0.703944	−	0.710255i	$-0.748580\pi$
$151$	−17.3004	−1.40789	−0.703944	+	0.710255i	$0.748580\pi$
$152$	2.46376	0.199837
$153$	3.76778	0.304607
$154$	−1.53556	−0.123739
$155$	−6.18490	−0.496783
$156$	0	0
$157$	−1.34898	−0.107660	−0.0538300	−	0.998550i	$-0.517143\pi$
$157$	−1.34898	−0.107660	−0.0538300	+	0.998550i	$0.517143\pi$
$158$	23.3450	1.85723
$159$	−5.51641	−0.437480
$160$	7.53556	0.595738
$161$	−0.312908	−0.0246606
$162$	2.17533	0.170910
$163$	2.97918	0.233347	0.116674	−	0.993170i	$-0.462777\pi$
$163$	2.97918	0.233347	0.116674	+	0.993170i	$0.462777\pi$
$164$	16.4492	1.28447
$165$	1.59245	0.123972
$166$	−19.8564	−1.54116
$167$	23.8141	1.84279	0.921394	−	0.388629i	$-0.127051\pi$
$167$	23.8141	1.84279	0.921394	+	0.388629i	$0.127051\pi$
$168$	0.705897	0.0544611
$169$	0	0
$170$	−8.19615	−0.628616
$171$	1.54715	0.118314
$172$	12.9020	0.983772
$173$	−24.6762	−1.87609	−0.938047	−	0.346509i	$-0.887367\pi$
$173$	−24.6762	−1.87609	−0.938047	+	0.346509i	$0.887367\pi$
$174$	−5.43795	−0.412250
$175$	−0.443277	−0.0335086
$176$	−3.18490	−0.240071
$177$	−8.72214	−0.655596
$178$	10.3470	0.775541
$179$	−2.88488	−0.215626	−0.107813	−	0.994171i	$-0.534385\pi$
$179$	−2.88488	−0.215626	−0.107813	+	0.994171i	$0.534385\pi$
$180$	−2.73205	−0.203635
$181$	7.41086	0.550845	0.275422	−	0.961323i	$-0.411182\pi$
$181$	7.41086	0.550845	0.275422	+	0.961323i	$0.411182\pi$
$182$	0	0
$183$	9.73205	0.719414
$184$	1.12411	0.0828701
$185$	−10.3507	−0.760995
$186$	−13.4542	−0.986509
$187$	6.00000	0.438763
$188$	14.7999	1.07939
$189$	0.443277	0.0322437
$190$	−3.36556	−0.244163
$191$	−16.2577	−1.17637	−0.588183	−	0.808728i	$-0.700156\pi$
$191$	−16.2577	−1.17637	−0.588183	+	0.808728i	$0.700156\pi$
$192$	12.3923	0.894338
$193$	11.2775	0.811774	0.405887	−	0.913923i	$-0.366963\pi$
$193$	11.2775	0.811774	0.405887	+	0.913923i	$0.366963\pi$
$194$	18.3100	1.31458
$195$	0	0
$196$	−18.5875	−1.32768
$197$	−24.6669	−1.75745	−0.878723	−	0.477333i	$-0.841604\pi$
$197$	−24.6669	−1.75745	−0.878723	+	0.477333i	$0.841604\pi$
$198$	3.46410	0.246183
$199$	−22.0711	−1.56458	−0.782290	−	0.622915i	$-0.785948\pi$
$199$	−22.0711	−1.56458	−0.782290	+	0.622915i	$0.785948\pi$
$200$	1.59245	0.112603
$201$	13.6960	0.966040
$202$	−14.5875	−1.02637
$203$	−1.10812	−0.0777745
$204$	−10.2938	−0.720707
$205$	−6.02082	−0.420513
$206$	3.78935	0.264016
$207$	0.705897	0.0490632
$208$	0	0
$209$	2.46376	0.170422
$210$	−0.964273	−0.0665411
$211$	−21.5161	−1.48123	−0.740614	−	0.671931i	$-0.765465\pi$
$211$	−21.5161	−1.48123	−0.740614	+	0.671931i	$0.765465\pi$
$212$	15.0711	1.03509
$213$	15.1488	1.03798
$214$	25.9269	1.77233
$215$	−4.72248	−0.322070
$216$	−1.59245	−0.108353
$217$	−2.74162	−0.186114
$218$	37.0691	2.51064
$219$	−10.7939	−0.729386
$220$	−4.35066	−0.293321
$221$	0	0
$222$	−22.5161	−1.51118
$223$	21.2365	1.42210	0.711051	−	0.703140i	$-0.248219\pi$
$223$	21.2365	1.42210	0.711051	+	0.703140i	$0.248219\pi$
$224$	3.34034	0.223186
$225$	1.00000	0.0666667
$226$	−7.53556	−0.501258
$227$	−10.4499	−0.693587	−0.346794	−	0.937941i	$-0.612730\pi$
$227$	−10.4499	−0.693587	−0.346794	+	0.937941i	$0.612730\pi$
$228$	−4.22689	−0.279933
$229$	21.4135	1.41505	0.707523	−	0.706690i	$-0.249813\pi$
$229$	21.4135	1.41505	0.707523	+	0.706690i	$0.249813\pi$
$230$	−1.53556	−0.101252
$231$	0.705897	0.0464446
$232$	3.98085	0.261356
$233$	3.24179	0.212377	0.106189	−	0.994346i	$-0.466135\pi$
$233$	3.24179	0.212377	0.106189	+	0.994346i	$0.466135\pi$
$234$	0	0
$235$	−5.41712	−0.353374
$236$	23.8293	1.55116
$237$	−10.7317	−0.697099
$238$	−3.63317	−0.235503
$239$	−18.3129	−1.18456	−0.592282	−	0.805731i	$-0.701773\pi$
$239$	−18.3129	−1.18456	−0.592282	+	0.805731i	$0.701773\pi$
$240$	−2.00000	−0.129099
$241$	−6.63741	−0.427553	−0.213777	−	0.976883i	$-0.568576\pi$
$241$	−6.63741	−0.427553	−0.213777	+	0.976883i	$0.568576\pi$
$242$	−18.4122	−1.18358
$243$	−1.00000	−0.0641500
$244$	−26.5885	−1.70215
$245$	6.80351	0.434660
$246$	−13.0973	−0.835051
$247$	0	0
$248$	9.84915	0.625422
$249$	9.12801	0.578464
$250$	−2.17533	−0.137580
$251$	−19.8816	−1.25492	−0.627459	−	0.778650i	$-0.715905\pi$
$251$	−19.8816	−1.25492	−0.627459	+	0.778650i	$0.715905\pi$
$252$	−1.21106	−0.0762893
$253$	1.12411	0.0706719
$254$	31.0007	1.94515
$255$	3.76778	0.235947
$256$	−1.07180	−0.0669873
$257$	1.00957	0.0629754	0.0314877	−	0.999504i	$-0.489975\pi$
$257$	1.00957	0.0629754	0.0314877	+	0.999504i	$0.489975\pi$
$258$	−10.2729	−0.639565
$259$	−4.58821	−0.285097
$260$	0	0
$261$	2.49983	0.154736
$262$	21.8303	1.34868
$263$	−21.5719	−1.33018	−0.665089	−	0.746764i	$-0.731607\pi$
$263$	−21.5719	−1.33018	−0.665089	+	0.746764i	$0.731607\pi$
$264$	−2.53590	−0.156074
$265$	−5.51641	−0.338870
$266$	−1.49187	−0.0914727
$267$	−4.75653	−0.291095
$268$	−37.4181	−2.28568
$269$	−4.65068	−0.283557	−0.141779	−	0.989898i	$-0.545282\pi$
$269$	−4.65068	−0.283557	−0.141779	+	0.989898i	$0.545282\pi$
$270$	2.17533	0.132386
$271$	−5.31992	−0.323162	−0.161581	−	0.986859i	$-0.551659\pi$
$271$	−5.31992	−0.323162	−0.161581	+	0.986859i	$0.551659\pi$
$272$	−7.53556	−0.456910
$273$	0	0
$274$	−20.1962	−1.22009
$275$	1.59245	0.0960284
$276$	−1.92855	−0.116085
$277$	9.35948	0.562357	0.281178	−	0.959655i	$-0.409275\pi$
$277$	9.35948	0.562357	0.281178	+	0.959655i	$0.409275\pi$
$278$	−14.1753	−0.850180
$279$	6.18490	0.370280
$280$	0.705897	0.0421854
$281$	13.1449	0.784161	0.392080	−	0.919931i	$-0.371756\pi$
$281$	13.1449	0.784161	0.392080	+	0.919931i	$0.371756\pi$
$282$	−11.7840	−0.701728
$283$	−3.64367	−0.216594	−0.108297	−	0.994119i	$-0.534540\pi$
$283$	−3.64367	−0.216594	−0.108297	+	0.994119i	$0.534540\pi$
$284$	−41.3874	−2.45589
$285$	1.54715	0.0916453
$286$	0	0
$287$	−2.66889	−0.157540
$288$	−7.53556	−0.444037
$289$	−2.80385	−0.164932
$290$	−5.43795	−0.319327
$291$	−8.41712	−0.493420
$292$	29.4896	1.72575
$293$	19.6348	1.14708	0.573540	−	0.819178i	$-0.305570\pi$
$293$	19.6348	1.14708	0.573540	+	0.819178i	$0.305570\pi$
$294$	14.7999	0.863145
$295$	−8.72214	−0.507822
$296$	16.4829	0.958049
$297$	−1.59245	−0.0924033
$298$	0.123763	0.00716941
$299$	0	0
$300$	−2.73205	−0.157735
$301$	−2.09337	−0.120660
$302$	−37.6341	−2.16560
$303$	6.70590	0.385244
$304$	−3.09430	−0.177470
$305$	9.73205	0.557256
$306$	8.19615	0.468543
$307$	4.74863	0.271019	0.135509	−	0.990776i	$-0.456733\pi$
$307$	4.74863	0.271019	0.135509	+	0.990776i	$0.456733\pi$
$308$	−1.92855	−0.109889
$309$	−1.74197	−0.0990970
$310$	−13.4542	−0.764146
$311$	5.45512	0.309332	0.154666	−	0.987967i	$-0.450570\pi$
$311$	5.45512	0.309332	0.154666	+	0.987967i	$0.450570\pi$
$312$	0	0
$313$	−7.23855	−0.409147	−0.204573	−	0.978851i	$-0.565581\pi$
$313$	−7.23855	−0.409147	−0.204573	+	0.978851i	$0.565581\pi$
$314$	−2.93447	−0.165602
$315$	0.443277	0.0249758
$316$	29.3196	1.64935
$317$	−21.0142	−1.18028	−0.590138	−	0.807302i	$-0.700927\pi$
$317$	−21.0142	−1.18028	−0.590138	+	0.807302i	$0.700927\pi$
$318$	−12.0000	−0.672927
$319$	3.98085	0.222885
$320$	12.3923	0.692751
$321$	−11.9186	−0.665233
$322$	−0.680677	−0.0379326
$323$	5.82932	0.324352
$324$	2.73205	0.151781
$325$	0	0
$326$	6.48068	0.358932
$327$	−17.0407	−0.942354
$328$	9.58786	0.529401
$329$	−2.40129	−0.132387
$330$	3.46410	0.190693
$331$	15.5136	0.852707	0.426354	−	0.904557i	$-0.359798\pi$
$331$	15.5136	0.852707	0.426354	+	0.904557i	$0.359798\pi$
$332$	−24.9382	−1.36866
$333$	10.3507	0.567212
$334$	51.8034	2.83456
$335$	13.6960	0.748291
$336$	−0.886554	−0.0483655
$337$	−32.8124	−1.78741	−0.893703	−	0.448660i	$-0.851901\pi$
$337$	−32.8124	−1.78741	−0.893703	+	0.448660i	$0.851901\pi$
$338$	0	0
$339$	3.46410	0.188144
$340$	−10.2938	−0.558258
$341$	9.84915	0.533361
$342$	3.36556	0.181989
$343$	6.11878	0.330383
$344$	7.52031	0.405468
$345$	0.705897	0.0380042
$346$	−53.6787	−2.88579
$347$	−19.4634	−1.04485	−0.522426	−	0.852685i	$-0.674973\pi$
$347$	−19.4634	−1.04485	−0.522426	+	0.852685i	$0.674973\pi$
$348$	−6.82966	−0.366108
$349$	9.72114	0.520361	0.260180	−	0.965560i	$-0.416218\pi$
$349$	9.72114	0.520361	0.260180	+	0.965560i	$0.416218\pi$
$350$	−0.964273	−0.0515425
$351$	0	0
$352$	−12.0000	−0.639602
$353$	23.0176	1.22510	0.612551	−	0.790431i	$-0.290143\pi$
$353$	23.0176	1.22510	0.612551	+	0.790431i	$0.290143\pi$
$354$	−18.9735	−1.00843
$355$	15.1488	0.804016
$356$	12.9951	0.688737
$357$	1.67017	0.0883947
$358$	−6.27555	−0.331673
$359$	10.4131	0.549584	0.274792	−	0.961504i	$-0.411391\pi$
$359$	10.4131	0.549584	0.274792	+	0.961504i	$0.411391\pi$
$360$	−1.59245	−0.0839295
$361$	−16.6063	−0.874017
$362$	16.1210	0.847303
$363$	8.46410	0.444250
$364$	0	0
$365$	−10.7939	−0.564980
$366$	21.1704	1.10659
$367$	−3.63153	−0.189565	−0.0947823	−	0.995498i	$-0.530215\pi$
$367$	−3.63153	−0.189565	−0.0947823	+	0.995498i	$0.530215\pi$
$368$	−1.41179	−0.0735948
$369$	6.02082	0.313432
$370$	−22.5161	−1.17055
$371$	−2.44530	−0.126954
$372$	−16.8975	−0.876093
$373$	19.8389	1.02722	0.513609	−	0.858024i	$-0.328308\pi$
$373$	19.8389	1.02722	0.513609	+	0.858024i	$0.328308\pi$
$374$	13.0520	0.674901
$375$	1.00000	0.0516398
$376$	8.62650	0.444878
$377$	0	0
$378$	0.964273	0.0495968
$379$	−6.83390	−0.351034	−0.175517	−	0.984476i	$-0.556160\pi$
$379$	−6.83390	−0.351034	−0.175517	+	0.984476i	$0.556160\pi$
$380$	−4.22689	−0.216835
$381$	−14.2510	−0.730102
$382$	−35.3658	−1.80947
$383$	38.8839	1.98687	0.993437	−	0.114379i	$-0.0364879\pi$
$383$	38.8839	1.98687	0.993437	+	0.114379i	$0.0364879\pi$
$384$	11.8862	0.606566
$385$	0.705897	0.0359758
$386$	24.5323	1.24866
$387$	4.72248	0.240057
$388$	22.9960	1.16745
$389$	8.84881	0.448652	0.224326	−	0.974514i	$-0.427982\pi$
$389$	8.84881	0.448652	0.224326	+	0.974514i	$0.427982\pi$
$390$	0	0
$391$	2.65966	0.134505
$392$	−10.8342	−0.547212
$393$	−10.0354	−0.506218
$394$	−53.6586	−2.70328
$395$	−10.7317	−0.539971
$396$	4.35066	0.218629
$397$	−17.0378	−0.855103	−0.427552	−	0.903991i	$-0.640624\pi$
$397$	−17.0378	−0.855103	−0.427552	+	0.903991i	$0.640624\pi$
$398$	−48.0119	−2.40662
$399$	0.685816	0.0343337
$400$	−2.00000	−0.100000
$401$	7.69099	0.384070	0.192035	−	0.981388i	$-0.438491\pi$
$401$	7.69099	0.384070	0.192035	+	0.981388i	$0.438491\pi$
$402$	29.7932	1.48595
$403$	0	0
$404$	−18.3209	−0.911496
$405$	−1.00000	−0.0496904
$406$	−2.41052	−0.119632
$407$	16.4829	0.817027
$408$	−6.00000	−0.297044
$409$	−9.08330	−0.449140	−0.224570	−	0.974458i	$-0.572098\pi$
$409$	−9.08330	−0.449140	−0.224570	+	0.974458i	$0.572098\pi$
$410$	−13.0973	−0.646828
$411$	9.28419	0.457955
$412$	4.75914	0.234466
$413$	−3.86632	−0.190249
$414$	1.53556	0.0754685
$415$	9.12801	0.448076
$416$	0	0
$417$	6.51641	0.319110
$418$	5.35948	0.262141
$419$	−12.9907	−0.634636	−0.317318	−	0.948319i	$-0.602782\pi$
$419$	−12.9907	−0.634636	−0.317318	+	0.948319i	$0.602782\pi$
$420$	−1.21106	−0.0590934
$421$	−19.8859	−0.969178	−0.484589	−	0.874742i	$-0.661031\pi$
$421$	−19.8859	−0.969178	−0.484589	+	0.874742i	$0.661031\pi$
$422$	−46.8045	−2.27841
$423$	5.41712	0.263390
$424$	8.78461	0.426618
$425$	3.76778	0.182764
$426$	32.9537	1.59661
$427$	4.31399	0.208769
$428$	32.5623	1.57396
$429$	0	0
$430$	−10.2729	−0.495405
$431$	−39.1280	−1.88473	−0.942365	−	0.334587i	$-0.891403\pi$
$431$	−39.1280	−1.88473	−0.942365	+	0.334587i	$0.891403\pi$
$432$	2.00000	0.0962250
$433$	38.1399	1.83289	0.916444	−	0.400164i	$-0.131047\pi$
$433$	38.1399	1.83289	0.916444	+	0.400164i	$0.131047\pi$
$434$	−5.96393	−0.286278
$435$	2.49983	0.119858
$436$	46.5561	2.22963
$437$	1.09213	0.0522436
$438$	−23.4803	−1.12193
$439$	6.75086	0.322201	0.161100	−	0.986938i	$-0.448496\pi$
$439$	6.75086	0.322201	0.161100	+	0.986938i	$0.448496\pi$
$440$	−2.53590	−0.120894
$441$	−6.80351	−0.323976
$442$	0	0
$443$	−12.2647	−0.582713	−0.291357	−	0.956614i	$-0.594107\pi$
$443$	−12.2647	−0.582713	−0.291357	+	0.956614i	$0.594107\pi$
$444$	−28.2785	−1.34204
$445$	−4.75653	−0.225481
$446$	46.1964	2.18746
$447$	−0.0568941	−0.00269100
$448$	5.49322	0.259530
$449$	23.4026	1.10444	0.552219	−	0.833699i	$-0.313781\pi$
$449$	23.4026	1.10444	0.552219	+	0.833699i	$0.313781\pi$
$450$	2.17533	0.102546
$451$	9.58786	0.451475
$452$	−9.46410	−0.445154
$453$	17.3004	0.812845
$454$	−22.7321	−1.06687
$455$	0	0
$456$	−2.46376	−0.115376
$457$	29.5630	1.38290	0.691451	−	0.722424i	$-0.256972\pi$
$457$	29.5630	1.38290	0.691451	+	0.722424i	$0.256972\pi$
$458$	46.5814	2.17661
$459$	−3.76778	−0.175865
$460$	−1.92855	−0.0899189
$461$	9.57498	0.445951	0.222976	−	0.974824i	$-0.428423\pi$
$461$	9.57498	0.445951	0.222976	+	0.974824i	$0.428423\pi$
$462$	1.53556	0.0714405
$463$	20.0241	0.930600	0.465300	−	0.885153i	$-0.345946\pi$
$463$	20.0241	0.930600	0.465300	+	0.885153i	$0.345946\pi$
$464$	−4.99966	−0.232103
$465$	6.18490	0.286818
$466$	7.05197	0.326676
$467$	15.7942	0.730868	0.365434	−	0.930837i	$-0.380921\pi$
$467$	15.7942	0.730868	0.365434	+	0.930837i	$0.380921\pi$
$468$	0	0
$469$	6.07111	0.280338
$470$	−11.7840	−0.543556
$471$	1.34898	0.0621576
$472$	13.8896	0.639319
$473$	7.52031	0.345784
$474$	−23.3450	−1.07227
$475$	1.54715	0.0709881
$476$	−4.56299	−0.209144
$477$	5.51641	0.252579
$478$	−39.8366	−1.82208
$479$	−26.7485	−1.22217	−0.611086	−	0.791564i	$-0.709267\pi$
$479$	−26.7485	−1.22217	−0.611086	+	0.791564i	$0.709267\pi$
$480$	−7.53556	−0.343950
$481$	0	0
$482$	−14.4385	−0.657657
$483$	0.312908	0.0142378
$484$	−23.1244	−1.05111
$485$	−8.41712	−0.382202
$486$	−2.17533	−0.0986749
$487$	−28.3238	−1.28347	−0.641737	−	0.766925i	$-0.721786\pi$
$487$	−28.3238	−1.28347	−0.641737	+	0.766925i	$0.721786\pi$
$488$	−15.4978	−0.701553
$489$	−2.97918	−0.134723
$490$	14.7999	0.668589
$491$	9.50017	0.428737	0.214368	−	0.976753i	$-0.431231\pi$
$491$	9.50017	0.428737	0.214368	+	0.976753i	$0.431231\pi$
$492$	−16.4492	−0.741587
$493$	9.41880	0.424201
$494$	0	0
$495$	−1.59245	−0.0715753
$496$	−12.3698	−0.555420
$497$	6.71513	0.301215
$498$	19.8564	0.889787
$499$	23.1767	1.03753	0.518765	−	0.854917i	$-0.326392\pi$
$499$	23.1767	1.03753	0.518765	+	0.854917i	$0.326392\pi$
$500$	−2.73205	−0.122181
$501$	−23.8141	−1.06393
$502$	−43.2491	−1.93030
$503$	−40.3533	−1.79927	−0.899633	−	0.436647i	$-0.856166\pi$
$503$	−40.3533	−1.79927	−0.899633	+	0.436647i	$0.856166\pi$
$504$	−0.705897	−0.0314431
$505$	6.70590	0.298408
$506$	2.44530	0.108707
$507$	0	0
$508$	38.9345	1.72744
$509$	−10.2938	−0.456263	−0.228131	−	0.973630i	$-0.573262\pi$
$509$	−10.2938	−0.456263	−0.228131	+	0.973630i	$0.573262\pi$
$510$	8.19615	0.362932
$511$	−4.78470	−0.211663
$512$	21.4409	0.947564
$513$	−1.54715	−0.0683083
$514$	2.19615	0.0968681
$515$	−1.74197	−0.0767602
$516$	−12.9020	−0.567981
$517$	8.62650	0.379393
$518$	−9.98085	−0.438534
$519$	24.6762	1.08316
$520$	0	0
$521$	43.7218	1.91549	0.957743	−	0.287624i	$-0.0928655\pi$
$521$	43.7218	1.91549	0.957743	+	0.287624i	$0.0928655\pi$
$522$	5.43795	0.238012
$523$	7.19556	0.314640	0.157320	−	0.987548i	$-0.449715\pi$
$523$	7.19556	0.314640	0.157320	+	0.987548i	$0.449715\pi$
$524$	27.4172	1.19773
$525$	0.443277	0.0193462
$526$	−46.9259	−2.04607
$527$	23.3033	1.01511
$528$	3.18490	0.138605
$529$	−22.5017	−0.978335
$530$	−12.0000	−0.521247
$531$	8.72214	0.378508
$532$	−1.87368	−0.0812345
$533$	0	0
$534$	−10.3470	−0.447759
$535$	−11.9186	−0.515287
$536$	−21.8102	−0.942056
$537$	2.88488	0.124492
$538$	−10.1168	−0.436164
$539$	−10.8342	−0.466664
$540$	2.73205	0.117569
$541$	−24.7159	−1.06262	−0.531310	−	0.847177i	$-0.678300\pi$
$541$	−24.7159	−1.06262	−0.531310	+	0.847177i	$0.678300\pi$
$542$	−11.5726	−0.497084
$543$	−7.41086	−0.318030
$544$	−28.3923	−1.21731
$545$	−17.0407	−0.729944
$546$	0	0
$547$	−6.97187	−0.298096	−0.149048	−	0.988830i	$-0.547621\pi$
$547$	−6.97187	−0.298096	−0.149048	+	0.988830i	$0.547621\pi$
$548$	−25.3649	−1.08353
$549$	−9.73205	−0.415354
$550$	3.46410	0.147710
$551$	3.86761	0.164766
$552$	−1.12411	−0.0478451
$553$	−4.75712	−0.202293
$554$	20.3599	0.865011
$555$	10.3507	0.439361
$556$	−17.8032	−0.755022
$557$	1.67575	0.0710038	0.0355019	−	0.999370i	$-0.488697\pi$
$557$	1.67575	0.0710038	0.0355019	+	0.999370i	$0.488697\pi$
$558$	13.4542	0.569561
$559$	0	0
$560$	−0.886554	−0.0374637
$561$	−6.00000	−0.253320
$562$	28.5945	1.20619
$563$	2.05231	0.0864945	0.0432472	−	0.999064i	$-0.486230\pi$
$563$	2.05231	0.0864945	0.0432472	+	0.999064i	$0.486230\pi$
$564$	−14.7999	−0.623186
$565$	3.46410	0.145736
$566$	−7.92618	−0.333162
$567$	−0.443277	−0.0186159
$568$	−24.1238	−1.01221
$569$	−41.2191	−1.72799	−0.863996	−	0.503498i	$-0.832046\pi$
$569$	−41.2191	−1.72799	−0.863996	+	0.503498i	$0.832046\pi$
$570$	3.36556	0.140968
$571$	30.7115	1.28524	0.642619	−	0.766186i	$-0.277848\pi$
$571$	30.7115	1.28524	0.642619	+	0.766186i	$0.277848\pi$
$572$	0	0
$573$	16.2577	0.679175
$574$	−5.80572	−0.242326
$575$	0.705897	0.0294379
$576$	−12.3923	−0.516346
$577$	44.8752	1.86818	0.934090	−	0.357038i	$-0.116213\pi$
$577$	44.8752	1.86818	0.934090	+	0.357038i	$0.116213\pi$
$578$	−6.09929	−0.253697
$579$	−11.2775	−0.468678
$580$	−6.82966	−0.283586
$581$	4.04623	0.167866
$582$	−18.3100	−0.758974
$583$	8.78461	0.363821
$584$	17.1888	0.711278
$585$	0	0
$586$	42.7122	1.76443
$587$	−23.8717	−0.985291	−0.492645	−	0.870230i	$-0.663970\pi$
$587$	−23.8717	−0.985291	−0.492645	+	0.870230i	$0.663970\pi$
$588$	18.5875	0.766537
$589$	9.56897	0.394282
$590$	−18.9735	−0.781127
$591$	24.6669	1.01466
$592$	−20.7013	−0.850819
$593$	−26.8263	−1.10162	−0.550811	−	0.834630i	$-0.685682\pi$
$593$	−26.8263	−1.10162	−0.550811	+	0.834630i	$0.685682\pi$
$594$	−3.46410	−0.142134
$595$	1.67017	0.0684703
$596$	0.155437	0.00636697
$597$	22.0711	0.903311
$598$	0	0
$599$	−26.4919	−1.08243	−0.541215	−	0.840885i	$-0.682035\pi$
$599$	−26.4919	−1.08243	−0.541215	+	0.840885i	$0.682035\pi$
$600$	−1.59245	−0.0650115
$601$	23.2290	0.947530	0.473765	−	0.880651i	$-0.342895\pi$
$601$	23.2290	0.947530	0.473765	+	0.880651i	$0.342895\pi$
$602$	−4.55376	−0.185597
$603$	−13.6960	−0.557743
$604$	−47.2656	−1.92321
$605$	8.46410	0.344115
$606$	14.5875	0.592578
$607$	10.5812	0.429478	0.214739	−	0.976672i	$-0.431110\pi$
$607$	10.5812	0.429478	0.214739	+	0.976672i	$0.431110\pi$
$608$	−11.6586	−0.472820
$609$	1.10812	0.0449031
$610$	21.1704	0.857164
$611$	0	0
$612$	10.2938	0.416101
$613$	6.31735	0.255155	0.127578	−	0.991829i	$-0.459280\pi$
$613$	6.31735	0.255155	0.127578	+	0.991829i	$0.459280\pi$
$614$	10.3298	0.416878
$615$	6.02082	0.242783
$616$	−1.12411	−0.0452915
$617$	−32.7277	−1.31757	−0.658784	−	0.752332i	$-0.728929\pi$
$617$	−32.7277	−1.31757	−0.658784	+	0.752332i	$0.728929\pi$
$618$	−3.78935	−0.152430
$619$	21.0143	0.844635	0.422317	−	0.906448i	$-0.361217\pi$
$619$	21.0143	0.844635	0.422317	+	0.906448i	$0.361217\pi$
$620$	−16.8975	−0.678618
$621$	−0.705897	−0.0283267
$622$	11.8667	0.475810
$623$	−2.10846	−0.0844736
$624$	0	0
$625$	1.00000	0.0400000
$626$	−15.7462	−0.629345
$627$	−2.46376	−0.0983931
$628$	−3.68547	−0.147066
$629$	38.9990	1.55499
$630$	0.964273	0.0384175
$631$	15.5657	0.619661	0.309830	−	0.950792i	$-0.399728\pi$
$631$	15.5657	0.619661	0.309830	+	0.950792i	$0.399728\pi$
$632$	17.0897	0.679792
$633$	21.5161	0.855187
$634$	−45.7128	−1.81549
$635$	−14.2510	−0.565535
$636$	−15.0711	−0.597608
$637$	0	0
$638$	8.65966	0.342839
$639$	−15.1488	−0.599279
$640$	11.8862	0.469844
$641$	13.7910	0.544713	0.272356	−	0.962196i	$-0.412197\pi$
$641$	13.7910	0.544713	0.272356	+	0.962196i	$0.412197\pi$
$642$	−25.9269	−1.02325
$643$	−27.6698	−1.09119	−0.545596	−	0.838049i	$-0.683697\pi$
$643$	−27.6698	−1.09119	−0.545596	+	0.838049i	$0.683697\pi$
$644$	−0.854880	−0.0336870
$645$	4.72248	0.185947
$646$	12.6807	0.498915
$647$	−35.8293	−1.40860	−0.704298	−	0.709905i	$-0.748738\pi$
$647$	−35.8293	−1.40860	−0.704298	+	0.709905i	$0.748738\pi$
$648$	1.59245	0.0625574
$649$	13.8896	0.545213
$650$	0	0
$651$	2.74162	0.107453
$652$	8.13926	0.318758
$653$	−45.2305	−1.77001	−0.885003	−	0.465585i	$-0.845844\pi$
$653$	−45.2305	−1.77001	−0.885003	+	0.465585i	$0.845844\pi$
$654$	−37.0691	−1.44952
$655$	−10.0354	−0.392115
$656$	−12.0416	−0.470147
$657$	10.7939	0.421111
$658$	−5.22358	−0.203636
$659$	−2.37316	−0.0924451	−0.0462226	−	0.998931i	$-0.514718\pi$
$659$	−2.37316	−0.0924451	−0.0462226	+	0.998931i	$0.514718\pi$
$660$	4.35066	0.169349
$661$	−13.1763	−0.512500	−0.256250	−	0.966611i	$-0.582487\pi$
$661$	−13.1763	−0.512500	−0.256250	+	0.966611i	$0.582487\pi$
$662$	33.7473	1.31162
$663$	0	0
$664$	−14.5359	−0.564102
$665$	0.685816	0.0265948
$666$	22.5161	0.872480
$667$	1.76462	0.0683264
$668$	65.0613	2.51730
$669$	−21.2365	−0.821052
$670$	29.7932	1.15101
$671$	−15.4978	−0.598286
$672$	−3.34034	−0.128856
$673$	12.7869	0.492900	0.246450	−	0.969156i	$-0.420736\pi$
$673$	12.7869	0.492900	0.246450	+	0.969156i	$0.420736\pi$
$674$	−71.3777	−2.74937
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	16.0794	0.617981	0.308991	−	0.951065i	$-0.400009\pi$
$677$	16.0794	0.617981	0.308991	+	0.951065i	$0.400009\pi$
$678$	7.53556	0.289401
$679$	−3.73112	−0.143187
$680$	−6.00000	−0.230089
$681$	10.4499	0.400443
$682$	21.4251	0.820410
$683$	−6.31162	−0.241507	−0.120754	−	0.992682i	$-0.538531\pi$
$683$	−6.31162	−0.241507	−0.120754	+	0.992682i	$0.538531\pi$
$684$	4.22689	0.161619
$685$	9.28419	0.354731
$686$	13.3103	0.508191
$687$	−21.4135	−0.816977
$688$	−9.44496	−0.360086
$689$	0	0
$690$	1.53556	0.0584576
$691$	−36.7143	−1.39668	−0.698339	−	0.715767i	$-0.746077\pi$
$691$	−36.7143	−1.39668	−0.698339	+	0.715767i	$0.746077\pi$
$692$	−67.4165	−2.56279
$693$	−0.705897	−0.0268148
$694$	−42.3393	−1.60718
$695$	6.51641	0.247182
$696$	−3.98085	−0.150894
$697$	22.6851	0.859261
$698$	21.1467	0.800413
$699$	−3.24179	−0.122616
$700$	−1.21106	−0.0457736
$701$	−52.0509	−1.96594	−0.982968	−	0.183774i	$-0.941168\pi$
$701$	−52.0509	−1.96594	−0.982968	+	0.183774i	$0.941168\pi$
$702$	0	0
$703$	16.0140	0.603980
$704$	−19.7341	−0.743758
$705$	5.41712	0.204021
$706$	50.0708	1.88444
$707$	2.97257	0.111795
$708$	−23.8293	−0.895561
$709$	0.936098	0.0351559	0.0175779	−	0.999845i	$-0.494404\pi$
$709$	0.936098	0.0351559	0.0175779	+	0.999845i	$0.494404\pi$
$710$	32.9537	1.23673
$711$	10.7317	0.402471
$712$	7.57453	0.283868
$713$	4.36590	0.163504
$714$	3.63317	0.135968
$715$	0	0
$716$	−7.88163	−0.294550
$717$	18.3129	0.683908
$718$	22.6520	0.845364
$719$	0.997435	0.0371980	0.0185990	−	0.999827i	$-0.494079\pi$
$719$	0.997435	0.0371980	0.0185990	+	0.999827i	$0.494079\pi$
$720$	2.00000	0.0745356
$721$	−0.772173	−0.0287572
$722$	−36.1242	−1.34440
$723$	6.63741	0.246848
$724$	20.2468	0.752468
$725$	2.49983	0.0928413
$726$	18.4122	0.683341
$727$	15.4879	0.574416	0.287208	−	0.957868i	$-0.407273\pi$
$727$	15.4879	0.574416	0.287208	+	0.957868i	$0.407273\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−23.4803	−0.869046
$731$	17.7932	0.658107
$732$	26.5885	0.982738
$733$	−3.34533	−0.123562	−0.0617812	−	0.998090i	$-0.519678\pi$
$733$	−3.34533	−0.123562	−0.0617812	+	0.998090i	$0.519678\pi$
$734$	−7.89978	−0.291586
$735$	−6.80351	−0.250951
$736$	−5.31932	−0.196073
$737$	−21.8102	−0.803388
$738$	13.0973	0.482117
$739$	13.0936	0.481656	0.240828	−	0.970568i	$-0.422581\pi$
$739$	13.0936	0.481656	0.240828	+	0.970568i	$0.422581\pi$
$740$	−28.2785	−1.03954
$741$	0	0
$742$	−5.31932	−0.195279
$743$	5.07373	0.186137	0.0930685	−	0.995660i	$-0.470332\pi$
$743$	5.07373	0.186137	0.0930685	+	0.995660i	$0.470332\pi$
$744$	−9.84915	−0.361087
$745$	−0.0568941	−0.00208444
$746$	43.1561	1.58006
$747$	−9.12801	−0.333976
$748$	16.3923	0.599362
$749$	−5.28325	−0.193046
$750$	2.17533	0.0794317
$751$	19.0338	0.694552	0.347276	−	0.937763i	$-0.387107\pi$
$751$	19.0338	0.694552	0.347276	+	0.937763i	$0.387107\pi$
$752$	−10.8342	−0.395084
$753$	19.8816	0.724527
$754$	0	0
$755$	17.3004	0.629627
$756$	1.21106	0.0440456
$757$	44.8089	1.62861	0.814303	−	0.580439i	$-0.197119\pi$
$757$	44.8089	1.62861	0.814303	+	0.580439i	$0.197119\pi$
$758$	−14.8660	−0.539957
$759$	−1.12411	−0.0408024
$760$	−2.46376	−0.0893700
$761$	37.9623	1.37613	0.688065	−	0.725649i	$-0.258460\pi$
$761$	37.9623	1.37613	0.688065	+	0.725649i	$0.258460\pi$
$762$	−31.0007	−1.12304
$763$	−7.55376	−0.273464
$764$	−44.4168	−1.60694
$765$	−3.76778	−0.136224
$766$	84.5852	3.05619
$767$	0	0
$768$	1.07180	0.0386751
$769$	24.8650	0.896654	0.448327	−	0.893870i	$-0.352020\pi$
$769$	24.8650	0.896654	0.448327	+	0.893870i	$0.352020\pi$
$770$	1.53556	0.0553376
$771$	−1.00957	−0.0363589
$772$	30.8108	1.10890
$773$	29.3026	1.05394	0.526970	−	0.849884i	$-0.323328\pi$
$773$	29.3026	1.05394	0.526970	+	0.849884i	$0.323328\pi$
$774$	10.2729	0.369253
$775$	6.18490	0.222168
$776$	13.4039	0.481170
$777$	4.58821	0.164601
$778$	19.2491	0.690112
$779$	9.31512	0.333749
$780$	0	0
$781$	−24.1238	−0.863216
$782$	5.78564	0.206894
$783$	−2.49983	−0.0893366
$784$	13.6070	0.485965
$785$	1.34898	0.0481471
$786$	−21.8303	−0.778659
$787$	10.6702	0.380350	0.190175	−	0.981750i	$-0.439094\pi$
$787$	10.6702	0.380350	0.190175	+	0.981750i	$0.439094\pi$
$788$	−67.3913	−2.40071
$789$	21.5719	0.767979
$790$	−23.3450	−0.830577
$791$	1.53556	0.0545981
$792$	2.53590	0.0901092
$793$	0	0
$794$	−37.0628	−1.31531
$795$	5.51641	0.195647
$796$	−60.2994	−2.13726
$797$	18.8800	0.668764	0.334382	−	0.942438i	$-0.391472\pi$
$797$	18.8800	0.668764	0.334382	+	0.942438i	$0.391472\pi$
$798$	1.49187	0.0528118
$799$	20.4105	0.722072
$800$	−7.53556	−0.266422
$801$	4.75653	0.168064
$802$	16.7304	0.590772
$803$	17.1888	0.606580
$804$	37.4181	1.31964
$805$	0.312908	0.0110285
$806$	0	0
$807$	4.65068	0.163712
$808$	−10.6788	−0.375679
$809$	43.6203	1.53361	0.766805	−	0.641880i	$-0.221845\pi$
$809$	43.6203	1.53361	0.766805	+	0.641880i	$0.221845\pi$
$810$	−2.17533	−0.0764332
$811$	3.31656	0.116460	0.0582301	−	0.998303i	$-0.481454\pi$
$811$	3.31656	0.116460	0.0582301	+	0.998303i	$0.481454\pi$
$812$	−3.02743	−0.106242
$813$	5.31992	0.186578
$814$	35.8557	1.25674
$815$	−2.97918	−0.104356
$816$	7.53556	0.263797
$817$	7.30638	0.255618
$818$	−19.7592	−0.690863
$819$	0	0
$820$	−16.4492	−0.574431
$821$	7.20740	0.251540	0.125770	−	0.992059i	$-0.459860\pi$
$821$	7.20740	0.251540	0.125770	+	0.992059i	$0.459860\pi$
$822$	20.1962	0.704422
$823$	9.80291	0.341708	0.170854	−	0.985296i	$-0.445347\pi$
$823$	9.80291	0.341708	0.170854	+	0.985296i	$0.445347\pi$
$824$	2.77399	0.0966367
$825$	−1.59245	−0.0554420
$826$	−8.41052	−0.292639
$827$	−3.92322	−0.136424	−0.0682118	−	0.997671i	$-0.521729\pi$
$827$	−3.92322	−0.136424	−0.0682118	+	0.997671i	$0.521729\pi$
$828$	1.92855	0.0670216
$829$	10.4242	0.362047	0.181024	−	0.983479i	$-0.442059\pi$
$829$	10.4242	0.362047	0.181024	+	0.983479i	$0.442059\pi$
$830$	19.8564	0.689226
$831$	−9.35948	−0.324677
$832$	0	0
$833$	−25.6341	−0.888169
$834$	14.1753	0.490851
$835$	−23.8141	−0.824120
$836$	6.73112	0.232801
$837$	−6.18490	−0.213781
$838$	−28.2590	−0.976190
$839$	−10.2521	−0.353942	−0.176971	−	0.984216i	$-0.556630\pi$
$839$	−10.2521	−0.353942	−0.176971	+	0.984216i	$0.556630\pi$
$840$	−0.705897	−0.0243557
$841$	−22.7509	−0.784512
$842$	−43.2583	−1.49078
$843$	−13.1449	−0.452735
$844$	−58.7830	−2.02339
$845$	0	0
$846$	11.7840	0.405143
$847$	3.75194	0.128918
$848$	−11.0328	−0.378869
$849$	3.64367	0.125051
$850$	8.19615	0.281126
$851$	7.30649	0.250463
$852$	41.3874	1.41791
$853$	−26.3671	−0.902792	−0.451396	−	0.892324i	$-0.649074\pi$
$853$	−26.3671	−0.902792	−0.451396	+	0.892324i	$0.649074\pi$
$854$	9.38435	0.321126
$855$	−1.54715	−0.0529114
$856$	18.9798	0.648717
$857$	1.04855	0.0358178	0.0179089	−	0.999840i	$-0.494299\pi$
$857$	1.04855	0.0358178	0.0179089	+	0.999840i	$0.494299\pi$
$858$	0	0
$859$	−19.6076	−0.669003	−0.334501	−	0.942395i	$-0.608568\pi$
$859$	−19.6076	−0.669003	−0.334501	+	0.942395i	$0.608568\pi$
$860$	−12.9020	−0.439956
$861$	2.66889	0.0909556
$862$	−85.1162	−2.89907
$863$	−25.6925	−0.874582	−0.437291	−	0.899320i	$-0.644062\pi$
$863$	−25.6925	−0.874582	−0.437291	+	0.899320i	$0.644062\pi$
$864$	7.53556	0.256365
$865$	24.6762	0.839014
$866$	82.9668	2.81933
$867$	2.80385	0.0952237
$868$	−7.49026	−0.254236
$869$	17.0897	0.579729
$870$	5.43795	0.184364
$871$	0	0
$872$	27.1365	0.918958
$873$	8.41712	0.284876
$874$	2.37574	0.0803605
$875$	0.443277	0.0149855
$876$	−29.4896	−0.996360
$877$	36.4141	1.22962	0.614809	−	0.788676i	$-0.289233\pi$
$877$	36.4141	1.22962	0.614809	+	0.788676i	$0.289233\pi$
$878$	14.6853	0.495606
$879$	−19.6348	−0.662267
$880$	3.18490	0.107363
$881$	16.0383	0.540344	0.270172	−	0.962812i	$-0.412919\pi$
$881$	16.0383	0.540344	0.270172	+	0.962812i	$0.412919\pi$
$882$	−14.7999	−0.498337
$883$	27.4548	0.923927	0.461963	−	0.886899i	$-0.347145\pi$
$883$	27.4548	0.923927	0.461963	+	0.886899i	$0.347145\pi$
$884$	0	0
$885$	8.72214	0.293191
$886$	−26.6797	−0.896323
$887$	−31.9642	−1.07325	−0.536626	−	0.843820i	$-0.680301\pi$
$887$	−31.9642	−1.07325	−0.536626	+	0.843820i	$0.680301\pi$
$888$	−16.4829	−0.553130
$889$	−6.31715	−0.211870
$890$	−10.3470	−0.346832
$891$	1.59245	0.0533491
$892$	58.0193	1.94263
$893$	8.38110	0.280463
$894$	−0.123763	−0.00413926
$895$	2.88488	0.0964308
$896$	5.26888	0.176021
$897$	0	0
$898$	50.9084	1.69883
$899$	15.4612	0.515660
$900$	2.73205	0.0910684
$901$	20.7846	0.692436
$902$	20.8567	0.694454
$903$	2.09337	0.0696628
$904$	−5.51641	−0.183473
$905$	−7.41086	−0.246345
$906$	37.6341	1.25031
$907$	31.6797	1.05191	0.525953	−	0.850514i	$-0.323709\pi$
$907$	31.6797	1.05191	0.525953	+	0.850514i	$0.323709\pi$
$908$	−28.5498	−0.947458
$909$	−6.70590	−0.222421
$910$	0	0
$911$	−25.9227	−0.858858	−0.429429	−	0.903101i	$-0.641285\pi$
$911$	−25.9227	−0.858858	−0.429429	+	0.903101i	$0.641285\pi$
$912$	3.09430	0.102463
$913$	−14.5359	−0.481068
$914$	64.3093	2.12716
$915$	−9.73205	−0.321732
$916$	58.5029	1.93299
$917$	−4.44845	−0.146901
$918$	−8.19615	−0.270513
$919$	−12.0194	−0.396483	−0.198242	−	0.980153i	$-0.563523\pi$
$919$	−12.0194	−0.396483	−0.198242	+	0.980153i	$0.563523\pi$
$920$	−1.12411	−0.0370607
$921$	−4.74863	−0.156473
$922$	20.8287	0.685958
$923$	0	0
$924$	1.92855	0.0634445
$925$	10.3507	0.340327
$926$	43.5591	1.43144
$927$	1.74197	0.0572137
$928$	−18.8376	−0.618375
$929$	21.3576	0.700721	0.350360	−	0.936615i	$-0.386059\pi$
$929$	21.3576	0.700721	0.350360	+	0.936615i	$0.386059\pi$
$930$	13.4542	0.441180
$931$	−10.5260	−0.344977
$932$	8.85675	0.290112
$933$	−5.45512	−0.178593
$934$	34.3575	1.12421
$935$	−6.00000	−0.196221
$936$	0	0
$937$	15.8029	0.516259	0.258129	−	0.966110i	$-0.416894\pi$
$937$	15.8029	0.516259	0.258129	+	0.966110i	$0.416894\pi$
$938$	13.2067	0.431213
$939$	7.23855	0.236221
$940$	−14.7999	−0.482718
$941$	6.50590	0.212086	0.106043	−	0.994362i	$-0.466182\pi$
$941$	6.50590	0.212086	0.106043	+	0.994362i	$0.466182\pi$
$942$	2.93447	0.0956101
$943$	4.25008	0.138402
$944$	−17.4443	−0.567763
$945$	−0.443277	−0.0144198
$946$	16.3591	0.531882
$947$	−27.8836	−0.906095	−0.453048	−	0.891486i	$-0.649663\pi$
$947$	−27.8836	−0.906095	−0.453048	+	0.891486i	$0.649663\pi$
$948$	−29.3196	−0.952255
$949$	0	0
$950$	3.36556	0.109193
$951$	21.0142	0.681433
$952$	−2.65966	−0.0862001
$953$	51.8861	1.68075	0.840377	−	0.542002i	$-0.182333\pi$
$953$	51.8861	1.68075	0.840377	+	0.542002i	$0.182333\pi$
$954$	12.0000	0.388514
$955$	16.2577	0.526087
$956$	−50.0318	−1.61814
$957$	−3.98085	−0.128683
$958$	−58.1868	−1.87993
$959$	4.11547	0.132895
$960$	−12.3923	−0.399960
$961$	7.25300	0.233968
$962$	0	0
$963$	11.9186	0.384072
$964$	−18.1337	−0.584048
$965$	−11.2775	−0.363036
$966$	0.680677	0.0219004
$967$	21.4251	0.688986	0.344493	−	0.938789i	$-0.388051\pi$
$967$	21.4251	0.688986	0.344493	+	0.938789i	$0.388051\pi$
$968$	−13.4787	−0.433221
$969$	−5.82932	−0.187265
$970$	−18.3100	−0.587899
$971$	−6.67273	−0.214138	−0.107069	−	0.994252i	$-0.534147\pi$
$971$	−6.67273	−0.214138	−0.107069	+	0.994252i	$0.534147\pi$
$972$	−2.73205	−0.0876306
$973$	2.88857	0.0926034
$974$	−61.6136	−1.97423
$975$	0	0
$976$	19.4641	0.623031
$977$	9.82542	0.314343	0.157171	−	0.987571i	$-0.449762\pi$
$977$	9.82542	0.314343	0.157171	+	0.987571i	$0.449762\pi$
$978$	−6.48068	−0.207229
$979$	7.57453	0.242083
$980$	18.5875	0.593757
$981$	17.0407	0.544068
$982$	20.6660	0.659478
$983$	8.83034	0.281644	0.140822	−	0.990035i	$-0.455025\pi$
$983$	8.83034	0.281644	0.140822	+	0.990035i	$0.455025\pi$
$984$	−9.58786	−0.305650
$985$	24.6669	0.785953
$986$	20.4890	0.652502
$987$	2.40129	0.0764338
$988$	0	0
$989$	3.33358	0.106002
$990$	−3.46410	−0.110096
$991$	12.3332	0.391779	0.195889	−	0.980626i	$-0.437241\pi$
$991$	12.3332	0.391779	0.195889	+	0.980626i	$0.437241\pi$
$992$	−46.6067	−1.47976
$993$	−15.5136	−0.492311
$994$	14.6076	0.463325
$995$	22.0711	0.699701
$996$	24.9382	0.790196
$997$	−13.0503	−0.413308	−0.206654	−	0.978414i	$-0.566257\pi$
$997$	−13.0503	−0.413308	−0.206654	+	0.978414i	$0.566257\pi$
$998$	50.4168	1.59592
$999$	−10.3507	−0.327480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2535.2.a.bj.1.4		4
3.2	odd	2		7605.2.a.ci.1.1		4
13.2	odd	12		195.2.bb.b.121.4	✓	8
13.7	odd	12		195.2.bb.b.166.4	yes	8
13.12	even	2		2535.2.a.bk.1.1		4
39.2	even	12		585.2.bu.d.316.1		8
39.20	even	12		585.2.bu.d.361.1		8
39.38	odd	2		7605.2.a.ch.1.4		4
65.2	even	12		975.2.w.h.199.1		8
65.7	even	12		975.2.w.i.49.4		8
65.28	even	12		975.2.w.i.199.4		8
65.33	even	12		975.2.w.h.49.1		8
65.54	odd	12		975.2.bc.j.901.1		8
65.59	odd	12		975.2.bc.j.751.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.b.121.4	✓	8	13.2	odd	12
195.2.bb.b.166.4	yes	8	13.7	odd	12
585.2.bu.d.316.1		8	39.2	even	12
585.2.bu.d.361.1		8	39.20	even	12
975.2.w.h.49.1		8	65.33	even	12
975.2.w.h.199.1		8	65.2	even	12
975.2.w.i.49.4		8	65.7	even	12
975.2.w.i.199.4		8	65.28	even	12
975.2.bc.j.751.1		8	65.59	odd	12
975.2.bc.j.901.1		8	65.54	odd	12
2535.2.a.bj.1.4		4	1.1	even	1	trivial
2535.2.a.bk.1.1		4	13.12	even	2
7605.2.a.ch.1.4		4	39.38	odd	2
7605.2.a.ci.1.1		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2535.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17533 q^{2} -1.00000 q^{3} +2.73205 q^{4} -1.00000 q^{5} -2.17533 q^{6} -0.443277 q^{7} +1.59245 q^{8} +1.00000 q^{9} -2.17533 q^{10} +1.59245 q^{11} -2.73205 q^{12} -0.964273 q^{14} +1.00000 q^{15} -2.00000 q^{16} +3.76778 q^{17} +2.17533 q^{18} +1.54715 q^{19} -2.73205 q^{20} +0.443277 q^{21} +3.46410 q^{22} +0.705897 q^{23} -1.59245 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.21106 q^{28} +2.49983 q^{29} +2.17533 q^{30} +6.18490 q^{31} -7.53556 q^{32} -1.59245 q^{33} +8.19615 q^{34} +0.443277 q^{35} +2.73205 q^{36} +10.3507 q^{37} +3.36556 q^{38} -1.59245 q^{40} +6.02082 q^{41} +0.964273 q^{42} +4.72248 q^{43} +4.35066 q^{44} -1.00000 q^{45} +1.53556 q^{46} +5.41712 q^{47} +2.00000 q^{48} -6.80351 q^{49} +2.17533 q^{50} -3.76778 q^{51} +5.51641 q^{53} -2.17533 q^{54} -1.59245 q^{55} -0.705897 q^{56} -1.54715 q^{57} +5.43795 q^{58} +8.72214 q^{59} +2.73205 q^{60} -9.73205 q^{61} +13.4542 q^{62} -0.443277 q^{63} -12.3923 q^{64} -3.46410 q^{66} -13.6960 q^{67} +10.2938 q^{68} -0.705897 q^{69} +0.964273 q^{70} -15.1488 q^{71} +1.59245 q^{72} +10.7939 q^{73} +22.5161 q^{74} -1.00000 q^{75} +4.22689 q^{76} -0.705897 q^{77} +10.7317 q^{79} +2.00000 q^{80} +1.00000 q^{81} +13.0973 q^{82} -9.12801 q^{83} +1.21106 q^{84} -3.76778 q^{85} +10.2729 q^{86} -2.49983 q^{87} +2.53590 q^{88} +4.75653 q^{89} -2.17533 q^{90} +1.92855 q^{92} -6.18490 q^{93} +11.7840 q^{94} -1.54715 q^{95} +7.53556 q^{96} +8.41712 q^{97} -14.7999 q^{98} +1.59245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{9} - 4 q^{12} - 12 q^{14} + 4 q^{15} - 8 q^{16} + 12 q^{19} - 4 q^{20} + 4 q^{25} - 4 q^{27} + 12 q^{28} - 12 q^{29} + 12 q^{31} + 12 q^{34} + 4 q^{36} + 24 q^{37}+ \cdots - 24 q^{98}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^9 - 4 * q^12 - 12 * q^14 + 4 * q^15 - 8 * q^16 + 12 * q^19 - 4 * q^20 + 4 * q^25 - 4 * q^27 + 12 * q^28 - 12 * q^29 + 12 * q^31 + 12 * q^34 + 4 * q^36 + 24 * q^37 + 12 * q^41 + 12 * q^42 + 16 * q^43 - 4 * q^45 - 24 * q^46 + 24 * q^47 + 8 * q^48 - 4 * q^49 - 12 * q^57 + 12 * q^58 - 12 * q^59 + 4 * q^60 - 32 * q^61 - 8 * q^64 - 12 * q^67 + 12 * q^70 - 12 * q^71 + 24 * q^73 + 24 * q^74 - 4 * q^75 + 24 * q^76 - 8 * q^79 + 8 * q^80 + 4 * q^81 - 12 * q^82 - 12 * q^84 + 12 * q^86 + 12 * q^87 + 24 * q^88 + 12 * q^89 + 24 * q^92 - 12 * q^93 - 12 * q^94 - 12 * q^95 + 36 * q^97 - 24 * q^98

Introduction

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