LMFDB - Embedded newform 1035.2.a.p.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1035,2,Mod(1,1035)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, 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("1035.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,10,-6,0,6,0,0,0,4] 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:	$8.26451660920$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.98838128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 $ x^6 - x^5 - 10x^4 - x^3 + 16x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.09026$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.70496 q^{2} +0.906890 q^{4} -1.00000 q^{5} +3.36634 q^{7} -1.86371 q^{8} -1.70496 q^{10} -2.32955 q^{11} +5.56867 q^{13} +5.73947 q^{14} -4.99133 q^{16} +3.47556 q^{17} +7.73947 q^{19} -0.906890 q^{20} -3.97179 q^{22} +1.00000 q^{23} +1.00000 q^{25} +9.49436 q^{26} +3.05290 q^{28} +4.93501 q^{29} +2.09311 q^{31} -4.78260 q^{32} +5.92569 q^{34} -3.36634 q^{35} -8.67448 q^{37} +13.1955 q^{38} +1.86371 q^{40} +3.88211 q^{41} -3.71200 q^{43} -2.11265 q^{44} +1.70496 q^{46} -6.30134 q^{47} +4.33223 q^{49} +1.70496 q^{50} +5.05017 q^{52} -0.310014 q^{53} +2.32955 q^{55} -6.27387 q^{56} +8.41399 q^{58} -5.61225 q^{59} +4.51577 q^{61} +3.56867 q^{62} +1.82851 q^{64} -5.56867 q^{65} -9.35173 q^{67} +3.15195 q^{68} -5.73947 q^{70} -1.45424 q^{71} -1.56867 q^{73} -14.7896 q^{74} +7.01885 q^{76} -7.84206 q^{77} -1.54120 q^{79} +4.99133 q^{80} +6.61884 q^{82} +14.3209 q^{83} -3.47556 q^{85} -6.32882 q^{86} +4.34161 q^{88} -15.9250 q^{89} +18.7460 q^{91} +0.906890 q^{92} -10.7435 q^{94} -7.73947 q^{95} +1.94184 q^{97} +7.38628 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{5} + 6 q^{7} + 4 q^{11} + 12 q^{13} - 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 10 q^{20} + 8 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26} + 24 q^{28} - 6 q^{29} + 8 q^{31} + 20 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^5 + 6 * q^7 + 4 * q^11 + 12 * q^13 - 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 10 * q^20 + 8 * q^22 + 6 * q^23 + 6 * q^25 - 12 * q^26 + 24 * q^28 - 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 6 * q^35 + 22 * q^37 + 36 * q^38 - 18 * q^41 + 8 * q^43 + 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^55 + 4 * q^56 - 16 * q^58 - 6 * q^59 + 14 * q^64 - 12 * q^65 + 10 * q^67 + 28 * q^68 + 4 * q^70 - 22 * q^71 + 12 * q^73 - 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 - 14 * q^80 - 12 * q^82 + 24 * q^83 - 4 * q^85 + 20 * q^86 - 8 * q^88 - 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 8 * q^95 + 12 * q^97 - 4 * 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 $

$2$	1.70496	1.20559	0.602795	−	0.797896i	$-0.294054\pi$
$2$	1.70496	1.20559	0.602795	+	0.797896i	$0.294054\pi$
$3$	0	0
$3$	0	0
$4$	0.906890	0.453445
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.36634	1.27236	0.636178	−	0.771542i	$-0.280514\pi$
$7$	3.36634	1.27236	0.636178	+	0.771542i	$0.280514\pi$
$8$	−1.86371	−0.658921
$9$	0	0
$10$	−1.70496	−0.539156
$11$	−2.32955	−0.702386	−0.351193	−	0.936303i	$-0.614224\pi$
$11$	−2.32955	−0.702386	−0.351193	+	0.936303i	$0.614224\pi$
$12$	0	0
$13$	5.56867	1.54447	0.772236	−	0.635336i	$-0.219139\pi$
$13$	5.56867	1.54447	0.772236	+	0.635336i	$0.219139\pi$
$14$	5.73947	1.53394
$15$	0	0
$16$	−4.99133	−1.24783
$17$	3.47556	0.842947	0.421474	−	0.906841i	$-0.361513\pi$
$17$	3.47556	0.842947	0.421474	+	0.906841i	$0.361513\pi$
$18$	0	0
$19$	7.73947	1.77556	0.887778	−	0.460271i	$-0.152248\pi$
$19$	7.73947	1.77556	0.887778	+	0.460271i	$0.152248\pi$
$20$	−0.906890	−0.202787
$21$	0	0
$22$	−3.97179	−0.846789
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	9.49436	1.86200
$27$	0	0
$28$	3.05290	0.576944
$29$	4.93501	0.916408	0.458204	−	0.888847i	$-0.348493\pi$
$29$	4.93501	0.916408	0.458204	+	0.888847i	$0.348493\pi$
$30$	0	0
$31$	2.09311	0.375934	0.187967	−	0.982175i	$-0.439810\pi$
$31$	2.09311	0.375934	0.187967	+	0.982175i	$0.439810\pi$
$32$	−4.78260	−0.845453
$33$	0	0
$34$	5.92569	1.01625
$35$	−3.36634	−0.569015
$36$	0	0
$37$	−8.67448	−1.42608	−0.713038	−	0.701126i	$-0.752681\pi$
$37$	−8.67448	−1.42608	−0.713038	+	0.701126i	$0.752681\pi$
$38$	13.1955	2.14059
$39$	0	0
$40$	1.86371	0.294678
$41$	3.88211	0.606283	0.303142	−	0.952945i	$-0.401964\pi$
$41$	3.88211	0.606283	0.303142	+	0.952945i	$0.401964\pi$
$42$	0	0
$43$	−3.71200	−0.566075	−0.283037	−	0.959109i	$-0.591342\pi$
$43$	−3.71200	−0.566075	−0.283037	+	0.959109i	$0.591342\pi$
$44$	−2.11265	−0.318494
$45$	0	0
$46$	1.70496	0.251383
$47$	−6.30134	−0.919146	−0.459573	−	0.888140i	$-0.651997\pi$
$47$	−6.30134	−0.919146	−0.459573	+	0.888140i	$0.651997\pi$
$48$	0	0
$49$	4.33223	0.618890
$50$	1.70496	0.241118
$51$	0	0
$52$	5.05017	0.700333
$53$	−0.310014	−0.0425837	−0.0212918	−	0.999773i	$-0.506778\pi$
$53$	−0.310014	−0.0425837	−0.0212918	+	0.999773i	$0.506778\pi$
$54$	0	0
$55$	2.32955	0.314117
$56$	−6.27387	−0.838382
$57$	0	0
$58$	8.41399	1.10481
$59$	−5.61225	−0.730653	−0.365327	−	0.930879i	$-0.619043\pi$
$59$	−5.61225	−0.730653	−0.365327	+	0.930879i	$0.619043\pi$
$60$	0	0
$61$	4.51577	0.578185	0.289093	−	0.957301i	$-0.406646\pi$
$61$	4.51577	0.578185	0.289093	+	0.957301i	$0.406646\pi$
$62$	3.56867	0.453221
$63$	0	0
$64$	1.82851	0.228564
$65$	−5.56867	−0.690708
$66$	0	0
$67$	−9.35173	−1.14250	−0.571248	−	0.820778i	$-0.693540\pi$
$67$	−9.35173	−1.14250	−0.571248	+	0.820778i	$0.693540\pi$
$68$	3.15195	0.382230
$69$	0	0
$70$	−5.73947	−0.685998
$71$	−1.45424	−0.172587	−0.0862933	−	0.996270i	$-0.527502\pi$
$71$	−1.45424	−0.172587	−0.0862933	+	0.996270i	$0.527502\pi$
$72$	0	0
$73$	−1.56867	−0.183599	−0.0917994	−	0.995778i	$-0.529262\pi$
$73$	−1.56867	−0.183599	−0.0917994	+	0.995778i	$0.529262\pi$
$74$	−14.7896	−1.71926
$75$	0	0
$76$	7.01885	0.805118
$77$	−7.84206	−0.893685
$78$	0	0
$79$	−1.54120	−0.173398	−0.0866992	−	0.996235i	$-0.527632\pi$
$79$	−1.54120	−0.173398	−0.0866992	+	0.996235i	$0.527632\pi$
$80$	4.99133	0.558048
$81$	0	0
$82$	6.61884	0.730929
$83$	14.3209	1.57192	0.785961	−	0.618277i	$-0.212169\pi$
$83$	14.3209	1.57192	0.785961	+	0.618277i	$0.212169\pi$
$84$	0	0
$85$	−3.47556	−0.376977
$86$	−6.32882	−0.682454
$87$	0	0
$88$	4.34161	0.462817
$89$	−15.9250	−1.68804	−0.844021	−	0.536310i	$-0.819818\pi$
$89$	−15.9250	−1.68804	−0.844021	+	0.536310i	$0.819818\pi$
$90$	0	0
$91$	18.7460	1.96512
$92$	0.906890	0.0945498
$93$	0	0
$94$	−10.7435	−1.10811
$95$	−7.73947	−0.794053
$96$	0	0
$97$	1.94184	0.197164	0.0985822	−	0.995129i	$-0.468569\pi$
$97$	1.94184	0.197164	0.0985822	+	0.995129i	$0.468569\pi$
$98$	7.38628	0.746127
$99$	0	0
$100$	0.906890	0.0906890
$101$	0.776994	0.0773138	0.0386569	−	0.999253i	$-0.487692\pi$
$101$	0.776994	0.0773138	0.0386569	+	0.999253i	$0.487692\pi$
$102$	0	0
$103$	9.87274	0.972790	0.486395	−	0.873739i	$-0.338312\pi$
$103$	9.87274	0.972790	0.486395	+	0.873739i	$0.338312\pi$
$104$	−10.3784	−1.01768
$105$	0	0
$106$	−0.528561	−0.0513384
$107$	10.9691	1.06042	0.530212	−	0.847865i	$-0.322112\pi$
$107$	10.9691	1.06042	0.530212	+	0.847865i	$0.322112\pi$
$108$	0	0
$109$	13.2552	1.26962	0.634808	−	0.772670i	$-0.281079\pi$
$109$	13.2552	1.26962	0.634808	+	0.772670i	$0.281079\pi$
$110$	3.97179	0.378696
$111$	0	0
$112$	−16.8025	−1.58769
$113$	−18.9263	−1.78044	−0.890218	−	0.455534i	$-0.849448\pi$
$113$	−18.9263	−1.78044	−0.890218	+	0.455534i	$0.849448\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	4.47551	0.415541
$117$	0	0
$118$	−9.56867	−0.880867
$119$	11.6999	1.07253
$120$	0	0
$121$	−5.57319	−0.506654
$122$	7.69921	0.697054
$123$	0	0
$124$	1.89822	0.170465
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.3074	−1.53578	−0.767892	−	0.640579i	$-0.778694\pi$
$127$	−17.3074	−1.53578	−0.767892	+	0.640579i	$0.778694\pi$
$128$	12.6827	1.12101
$129$	0	0
$130$	−9.49436	−0.832711
$131$	−19.7273	−1.72359	−0.861793	−	0.507261i	$-0.830658\pi$
$131$	−19.7273	−1.72359	−0.861793	+	0.507261i	$0.830658\pi$
$132$	0	0
$133$	26.0537	2.25914
$134$	−15.9443	−1.37738
$135$	0	0
$136$	−6.47743	−0.555435
$137$	15.0033	1.28182	0.640911	−	0.767615i	$-0.278557\pi$
$137$	15.0033	1.28182	0.640911	+	0.767615i	$0.278557\pi$
$138$	0	0
$139$	15.1287	1.28320	0.641598	−	0.767041i	$-0.278272\pi$
$139$	15.1287	1.28320	0.641598	+	0.767041i	$0.278272\pi$
$140$	−3.05290	−0.258017
$141$	0	0
$142$	−2.47942	−0.208069
$143$	−12.9725	−1.08482
$144$	0	0
$145$	−4.93501	−0.409830
$146$	−2.67452	−0.221345
$147$	0	0
$148$	−7.86680	−0.646647
$149$	−20.6147	−1.68883	−0.844413	−	0.535693i	$-0.820050\pi$
$149$	−20.6147	−1.68883	−0.844413	+	0.535693i	$0.820050\pi$
$150$	0	0
$151$	−22.3696	−1.82041	−0.910207	−	0.414153i	$-0.864078\pi$
$151$	−22.3696	−1.82041	−0.910207	+	0.414153i	$0.864078\pi$
$152$	−14.4241	−1.16995
$153$	0	0
$154$	−13.3704	−1.07742
$155$	−2.09311	−0.168123
$156$	0	0
$157$	−7.18092	−0.573100	−0.286550	−	0.958065i	$-0.592508\pi$
$157$	−7.18092	−0.573100	−0.286550	+	0.958065i	$0.592508\pi$
$158$	−2.62768	−0.209047
$159$	0	0
$160$	4.78260	0.378098
$161$	3.36634	0.265305
$162$	0	0
$163$	−11.3236	−0.886930	−0.443465	−	0.896292i	$-0.646251\pi$
$163$	−11.3236	−0.886930	−0.443465	+	0.896292i	$0.646251\pi$
$164$	3.52065	0.274916
$165$	0	0
$166$	24.4165	1.89509
$167$	−21.9987	−1.70231	−0.851157	−	0.524912i	$-0.824098\pi$
$167$	−21.9987	−1.70231	−0.851157	+	0.524912i	$0.824098\pi$
$168$	0	0
$169$	18.0101	1.38539
$170$	−5.92569	−0.454480
$171$	0	0
$172$	−3.36638	−0.256684
$173$	6.32882	0.481171	0.240585	−	0.970628i	$-0.422661\pi$
$173$	6.32882	0.481171	0.240585	+	0.970628i	$0.422661\pi$
$174$	0	0
$175$	3.36634	0.254471
$176$	11.6276	0.876460
$177$	0	0
$178$	−27.1514	−2.03509
$179$	4.71726	0.352584	0.176292	−	0.984338i	$-0.443590\pi$
$179$	4.71726	0.352584	0.176292	+	0.984338i	$0.443590\pi$
$180$	0	0
$181$	−12.1434	−0.902612	−0.451306	−	0.892369i	$-0.649042\pi$
$181$	−12.1434	−0.902612	−0.451306	+	0.892369i	$0.649042\pi$
$182$	31.9612	2.36912
$183$	0	0
$184$	−1.86371	−0.137394
$185$	8.67448	0.637760
$186$	0	0
$187$	−8.09649	−0.592074
$188$	−5.71463	−0.416782
$189$	0	0
$190$	−13.1955	−0.957302
$191$	4.30407	0.311432	0.155716	−	0.987802i	$-0.450232\pi$
$191$	4.30407	0.311432	0.155716	+	0.987802i	$0.450232\pi$
$192$	0	0
$193$	1.81378	0.130559	0.0652794	−	0.997867i	$-0.479206\pi$
$193$	1.81378	0.130559	0.0652794	+	0.997867i	$0.479206\pi$
$194$	3.31077	0.237699
$195$	0	0
$196$	3.92886	0.280633
$197$	18.5150	1.31914	0.659571	−	0.751642i	$-0.270738\pi$
$197$	18.5150	1.31914	0.659571	+	0.751642i	$0.270738\pi$
$198$	0	0
$199$	−1.93098	−0.136883	−0.0684416	−	0.997655i	$-0.521803\pi$
$199$	−1.93098	−0.136883	−0.0684416	+	0.997655i	$0.521803\pi$
$200$	−1.86371	−0.131784
$201$	0	0
$202$	1.32474	0.0932087
$203$	16.6129	1.16600
$204$	0	0
$205$	−3.88211	−0.271138
$206$	16.8326	1.17279
$207$	0	0
$208$	−27.7951	−1.92724
$209$	−18.0295	−1.24713
$210$	0	0
$211$	−9.63249	−0.663128	−0.331564	−	0.943433i	$-0.607576\pi$
$211$	−9.63249	−0.663128	−0.331564	+	0.943433i	$0.607576\pi$
$212$	−0.281148	−0.0193094
$213$	0	0
$214$	18.7019	1.27844
$215$	3.71200	0.253156
$216$	0	0
$217$	7.04611	0.478321
$218$	22.5996	1.53064
$219$	0	0
$220$	2.11265	0.142435
$221$	19.3542	1.30191
$222$	0	0
$223$	16.9632	1.13594	0.567971	−	0.823048i	$-0.307728\pi$
$223$	16.9632	1.13594	0.567971	+	0.823048i	$0.307728\pi$
$224$	−16.0999	−1.07572
$225$	0	0
$226$	−32.2686	−2.14647
$227$	12.7785	0.848138	0.424069	−	0.905630i	$-0.360601\pi$
$227$	12.7785	0.848138	0.424069	+	0.905630i	$0.360601\pi$
$228$	0	0
$229$	−5.75737	−0.380457	−0.190229	−	0.981740i	$-0.560923\pi$
$229$	−5.75737	−0.380457	−0.190229	+	0.981740i	$0.560923\pi$
$230$	−1.70496	−0.112422
$231$	0	0
$232$	−9.19742	−0.603840
$233$	−26.6304	−1.74461	−0.872307	−	0.488959i	$-0.837377\pi$
$233$	−26.6304	−1.74461	−0.872307	+	0.488959i	$0.837377\pi$
$234$	0	0
$235$	6.30134	0.411054
$236$	−5.08970	−0.331311
$237$	0	0
$238$	19.9479	1.29303
$239$	11.0068	0.711968	0.355984	−	0.934492i	$-0.384146\pi$
$239$	11.0068	0.711968	0.355984	+	0.934492i	$0.384146\pi$
$240$	0	0
$241$	6.49029	0.418076	0.209038	−	0.977907i	$-0.432967\pi$
$241$	6.49029	0.418076	0.209038	+	0.977907i	$0.432967\pi$
$242$	−9.50207	−0.610816
$243$	0	0
$244$	4.09531	0.262175
$245$	−4.33223	−0.276776
$246$	0	0
$247$	43.0986	2.74230
$248$	−3.90095	−0.247710
$249$	0	0
$250$	−1.70496	−0.107831
$251$	4.47288	0.282326	0.141163	−	0.989986i	$-0.454916\pi$
$251$	4.47288	0.282326	0.141163	+	0.989986i	$0.454916\pi$
$252$	0	0
$253$	−2.32955	−0.146458
$254$	−29.5084	−1.85152
$255$	0	0
$256$	17.9666	1.12291
$257$	−4.68657	−0.292340	−0.146170	−	0.989259i	$-0.546695\pi$
$257$	−4.68657	−0.292340	−0.146170	+	0.989259i	$0.546695\pi$
$258$	0	0
$259$	−29.2012	−1.81448
$260$	−5.05017	−0.313198
$261$	0	0
$262$	−33.6343	−2.07794
$263$	13.8747	0.855553	0.427776	−	0.903885i	$-0.359297\pi$
$263$	13.8747	0.855553	0.427776	+	0.903885i	$0.359297\pi$
$264$	0	0
$265$	0.310014	0.0190440
$266$	44.4205	2.72359
$267$	0	0
$268$	−8.48099	−0.518059
$269$	11.7548	0.716706	0.358353	−	0.933586i	$-0.383338\pi$
$269$	11.7548	0.716706	0.358353	+	0.933586i	$0.383338\pi$
$270$	0	0
$271$	−7.49950	−0.455562	−0.227781	−	0.973712i	$-0.573147\pi$
$271$	−7.49950	−0.455562	−0.227781	+	0.973712i	$0.573147\pi$
$272$	−17.3477	−1.05186
$273$	0	0
$274$	25.5801	1.54535
$275$	−2.32955	−0.140477
$276$	0	0
$277$	−12.3296	−0.740815	−0.370408	−	0.928869i	$-0.620782\pi$
$277$	−12.3296	−0.740815	−0.370408	+	0.928869i	$0.620782\pi$
$278$	25.7938	1.54701
$279$	0	0
$280$	6.27387	0.374936
$281$	33.3077	1.98697	0.993485	−	0.113965i	$-0.0363551\pi$
$281$	33.3077	1.98697	0.993485	+	0.113965i	$0.0363551\pi$
$282$	0	0
$283$	12.6644	0.752821	0.376411	−	0.926453i	$-0.377158\pi$
$283$	12.6644	0.752821	0.376411	+	0.926453i	$0.377158\pi$
$284$	−1.31884	−0.0782585
$285$	0	0
$286$	−22.1176	−1.30784
$287$	13.0685	0.771408
$288$	0	0
$289$	−4.92049	−0.289440
$290$	−8.41399	−0.494087
$291$	0	0
$292$	−1.42261	−0.0832520
$293$	14.7133	0.859563	0.429781	−	0.902933i	$-0.358591\pi$
$293$	14.7133	0.859563	0.429781	+	0.902933i	$0.358591\pi$
$294$	0	0
$295$	5.61225	0.326758
$296$	16.1667	0.939671
$297$	0	0
$298$	−35.1473	−2.03603
$299$	5.56867	0.322044
$300$	0	0
$301$	−12.4958	−0.720249
$302$	−38.1393	−2.19467
$303$	0	0
$304$	−38.6303	−2.21560
$305$	−4.51577	−0.258572
$306$	0	0
$307$	−1.37571	−0.0785157	−0.0392578	−	0.999229i	$-0.512499\pi$
$307$	−1.37571	−0.0785157	−0.0392578	+	0.999229i	$0.512499\pi$
$308$	−7.11188	−0.405237
$309$	0	0
$310$	−3.56867	−0.202687
$311$	−25.6524	−1.45461	−0.727306	−	0.686313i	$-0.759228\pi$
$311$	−25.6524	−1.45461	−0.727306	+	0.686313i	$0.759228\pi$
$312$	0	0
$313$	21.0483	1.18972	0.594860	−	0.803829i	$-0.297207\pi$
$313$	21.0483	1.18972	0.594860	+	0.803829i	$0.297207\pi$
$314$	−12.2432	−0.690923
$315$	0	0
$316$	−1.39770	−0.0786266
$317$	−12.2096	−0.685761	−0.342881	−	0.939379i	$-0.611403\pi$
$317$	−12.2096	−0.685761	−0.342881	+	0.939379i	$0.611403\pi$
$318$	0	0
$319$	−11.4964	−0.643672
$320$	−1.82851	−0.102217
$321$	0	0
$322$	5.73947	0.319848
$323$	26.8990	1.49670
$324$	0	0
$325$	5.56867	0.308894
$326$	−19.3062	−1.06927
$327$	0	0
$328$	−7.23512	−0.399493
$329$	−21.2125	−1.16948
$330$	0	0
$331$	−13.8386	−0.760638	−0.380319	−	0.924855i	$-0.624186\pi$
$331$	−13.8386	−0.760638	−0.380319	+	0.924855i	$0.624186\pi$
$332$	12.9875	0.712780
$333$	0	0
$334$	−37.5070	−2.05229
$335$	9.35173	0.510939
$336$	0	0
$337$	−6.20030	−0.337752	−0.168876	−	0.985637i	$-0.554014\pi$
$337$	−6.20030	−0.337752	−0.168876	+	0.985637i	$0.554014\pi$
$338$	30.7065	1.67021
$339$	0	0
$340$	−3.15195	−0.170939
$341$	−4.87601	−0.264051
$342$	0	0
$343$	−8.98062	−0.484908
$344$	6.91809	0.372998
$345$	0	0
$346$	10.7904	0.580094
$347$	−0.203558	−0.0109276	−0.00546379	−	0.999985i	$-0.501739\pi$
$347$	−0.203558	−0.0109276	−0.00546379	+	0.999985i	$0.501739\pi$
$348$	0	0
$349$	−36.0301	−1.92865	−0.964325	−	0.264722i	$-0.914720\pi$
$349$	−36.0301	−1.92865	−0.964325	+	0.264722i	$0.914720\pi$
$350$	5.73947	0.306788
$351$	0	0
$352$	11.1413	0.593834
$353$	14.0529	0.747959	0.373979	−	0.927437i	$-0.377993\pi$
$353$	14.0529	0.747959	0.373979	+	0.927437i	$0.377993\pi$
$354$	0	0
$355$	1.45424	0.0771831
$356$	−14.4422	−0.765434
$357$	0	0
$358$	8.04274	0.425072
$359$	14.5840	0.769714	0.384857	−	0.922976i	$-0.374251\pi$
$359$	14.5840	0.769714	0.384857	+	0.922976i	$0.374251\pi$
$360$	0	0
$361$	40.8994	2.15260
$362$	−20.7040	−1.08818
$363$	0	0
$364$	17.0006	0.891073
$365$	1.56867	0.0821079
$366$	0	0
$367$	−29.6047	−1.54535	−0.772676	−	0.634800i	$-0.781082\pi$
$367$	−29.6047	−1.54535	−0.772676	+	0.634800i	$0.781082\pi$
$368$	−4.99133	−0.260191
$369$	0	0
$370$	14.7896	0.768877
$371$	−1.04361	−0.0541816
$372$	0	0
$373$	31.8682	1.65007	0.825036	−	0.565080i	$-0.191155\pi$
$373$	31.8682	1.65007	0.825036	+	0.565080i	$0.191155\pi$
$374$	−13.8042	−0.713798
$375$	0	0
$376$	11.7439	0.605644
$377$	27.4814	1.41537
$378$	0	0
$379$	26.3732	1.35470	0.677349	−	0.735662i	$-0.263129\pi$
$379$	26.3732	1.35470	0.677349	+	0.735662i	$0.263129\pi$
$380$	−7.01885	−0.360059
$381$	0	0
$382$	7.33827	0.375459
$383$	7.12860	0.364254	0.182127	−	0.983275i	$-0.441702\pi$
$383$	7.12860	0.364254	0.182127	+	0.983275i	$0.441702\pi$
$384$	0	0
$385$	7.84206	0.399668
$386$	3.09242	0.157400
$387$	0	0
$388$	1.76104	0.0894032
$389$	2.49183	0.126341	0.0631704	−	0.998003i	$-0.479879\pi$
$389$	2.49183	0.126341	0.0631704	+	0.998003i	$0.479879\pi$
$390$	0	0
$391$	3.47556	0.175767
$392$	−8.07401	−0.407799
$393$	0	0
$394$	31.5674	1.59034
$395$	1.54120	0.0775461
$396$	0	0
$397$	26.8571	1.34792	0.673960	−	0.738768i	$-0.264592\pi$
$397$	26.8571	1.34792	0.673960	+	0.738768i	$0.264592\pi$
$398$	−3.29224	−0.165025
$399$	0	0
$400$	−4.99133	−0.249567
$401$	−4.98334	−0.248856	−0.124428	−	0.992229i	$-0.539710\pi$
$401$	−4.98334	−0.248856	−0.124428	+	0.992229i	$0.539710\pi$
$402$	0	0
$403$	11.6558	0.580619
$404$	0.704649	0.0350576
$405$	0	0
$406$	28.3243	1.40571
$407$	20.2076	1.00166
$408$	0	0
$409$	19.6429	0.971279	0.485639	−	0.874159i	$-0.338587\pi$
$409$	19.6429	0.971279	0.485639	+	0.874159i	$0.338587\pi$
$410$	−6.61884	−0.326881
$411$	0	0
$412$	8.95349	0.441107
$413$	−18.8927	−0.929651
$414$	0	0
$415$	−14.3209	−0.702984
$416$	−26.6327	−1.30578
$417$	0	0
$418$	−30.7396	−1.50352
$419$	−22.2454	−1.08676	−0.543379	−	0.839487i	$-0.682855\pi$
$419$	−22.2454	−1.08676	−0.543379	+	0.839487i	$0.682855\pi$
$420$	0	0
$421$	33.5231	1.63382	0.816909	−	0.576767i	$-0.195686\pi$
$421$	33.5231	1.63382	0.816909	+	0.576767i	$0.195686\pi$
$422$	−16.4230	−0.799460
$423$	0	0
$424$	0.577775	0.0280593
$425$	3.47556	0.168589
$426$	0	0
$427$	15.2016	0.735657
$428$	9.94778	0.480844
$429$	0	0
$430$	6.32882	0.305203
$431$	−28.1248	−1.35472	−0.677361	−	0.735651i	$-0.736877\pi$
$431$	−28.1248	−1.35472	−0.677361	+	0.735651i	$0.736877\pi$
$432$	0	0
$433$	0.328856	0.0158038	0.00790190	−	0.999969i	$-0.497485\pi$
$433$	0.328856	0.0158038	0.00790190	+	0.999969i	$0.497485\pi$
$434$	12.0133	0.576659
$435$	0	0
$436$	12.0210	0.575702
$437$	7.73947	0.370229
$438$	0	0
$439$	−5.91409	−0.282264	−0.141132	−	0.989991i	$-0.545074\pi$
$439$	−5.91409	−0.282264	−0.141132	+	0.989991i	$0.545074\pi$
$440$	−4.34161	−0.206978
$441$	0	0
$442$	32.9982	1.56957
$443$	4.83291	0.229619	0.114809	−	0.993388i	$-0.463374\pi$
$443$	4.83291	0.229619	0.114809	+	0.993388i	$0.463374\pi$
$444$	0	0
$445$	15.9250	0.754915
$446$	28.9217	1.36948
$447$	0	0
$448$	6.15539	0.290815
$449$	4.10457	0.193707	0.0968533	−	0.995299i	$-0.469122\pi$
$449$	4.10457	0.193707	0.0968533	+	0.995299i	$0.469122\pi$
$450$	0	0
$451$	−9.04357	−0.425845
$452$	−17.1641	−0.807330
$453$	0	0
$454$	21.7868	1.02251
$455$	−18.7460	−0.878827
$456$	0	0
$457$	22.8312	1.06800	0.534000	−	0.845484i	$-0.320688\pi$
$457$	22.8312	1.06800	0.534000	+	0.845484i	$0.320688\pi$
$458$	−9.81608	−0.458675
$459$	0	0
$460$	−0.906890	−0.0422840
$461$	−41.1132	−1.91483	−0.957416	−	0.288713i	$-0.906773\pi$
$461$	−41.1132	−1.91483	−0.957416	+	0.288713i	$0.906773\pi$
$462$	0	0
$463$	−25.9987	−1.20826	−0.604132	−	0.796884i	$-0.706480\pi$
$463$	−25.9987	−1.20826	−0.604132	+	0.796884i	$0.706480\pi$
$464$	−24.6323	−1.14352
$465$	0	0
$466$	−45.4037	−2.10329
$467$	−39.6897	−1.83662	−0.918311	−	0.395860i	$-0.870446\pi$
$467$	−39.6897	−1.83662	−0.918311	+	0.395860i	$0.870446\pi$
$468$	0	0
$469$	−31.4811	−1.45366
$470$	10.7435	0.495563
$471$	0	0
$472$	10.4596	0.481442
$473$	8.64730	0.397603
$474$	0	0
$475$	7.73947	0.355111
$476$	10.6105	0.486333
$477$	0	0
$478$	18.7661	0.858341
$479$	17.5510	0.801925	0.400962	−	0.916095i	$-0.368676\pi$
$479$	17.5510	0.801925	0.400962	+	0.916095i	$0.368676\pi$
$480$	0	0
$481$	−48.3053	−2.20253
$482$	11.0657	0.504028
$483$	0	0
$484$	−5.05427	−0.229740
$485$	−1.94184	−0.0881746
$486$	0	0
$487$	−36.1405	−1.63768	−0.818842	−	0.574019i	$-0.805383\pi$
$487$	−36.1405	−1.63768	−0.818842	+	0.574019i	$0.805383\pi$
$488$	−8.41608	−0.380978
$489$	0	0
$490$	−7.38628	−0.333678
$491$	11.0375	0.498116	0.249058	−	0.968489i	$-0.419879\pi$
$491$	11.0375	0.498116	0.249058	+	0.968489i	$0.419879\pi$
$492$	0	0
$493$	17.1519	0.772483
$494$	73.4813	3.30608
$495$	0	0
$496$	−10.4474	−0.469102
$497$	−4.89546	−0.219592
$498$	0	0
$499$	−18.2229	−0.815770	−0.407885	−	0.913033i	$-0.633734\pi$
$499$	−18.2229	−0.815770	−0.407885	+	0.913033i	$0.633734\pi$
$500$	−0.906890	−0.0405574
$501$	0	0
$502$	7.62609	0.340369
$503$	−30.3839	−1.35475	−0.677375	−	0.735638i	$-0.736882\pi$
$503$	−30.3839	−1.35475	−0.677375	+	0.735638i	$0.736882\pi$
$504$	0	0
$505$	−0.776994	−0.0345758
$506$	−3.97179	−0.176568
$507$	0	0
$508$	−15.6959	−0.696394
$509$	11.4948	0.509500	0.254750	−	0.967007i	$-0.418007\pi$
$509$	11.4948	0.509500	0.254750	+	0.967007i	$0.418007\pi$
$510$	0	0
$511$	−5.28067	−0.233603
$512$	5.26678	0.232761
$513$	0	0
$514$	−7.99042	−0.352442
$515$	−9.87274	−0.435045
$516$	0	0
$517$	14.6793	0.645595
$518$	−49.7869	−2.18751
$519$	0	0
$520$	10.3784	0.455122
$521$	−38.4421	−1.68418	−0.842090	−	0.539337i	$-0.818675\pi$
$521$	−38.4421	−1.68418	−0.842090	+	0.539337i	$0.818675\pi$
$522$	0	0
$523$	−0.951200	−0.0415931	−0.0207965	−	0.999784i	$-0.506620\pi$
$523$	−0.951200	−0.0415931	−0.0207965	+	0.999784i	$0.506620\pi$
$524$	−17.8905	−0.781551
$525$	0	0
$526$	23.6559	1.03145
$527$	7.27473	0.316892
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.528561	0.0229592
$531$	0	0
$532$	23.6278	1.02440
$533$	21.6182	0.936387
$534$	0	0
$535$	−10.9691	−0.474236
$536$	17.4289	0.752814
$537$	0	0
$538$	20.0416	0.864053
$539$	−10.0921	−0.434700
$540$	0	0
$541$	−2.81851	−0.121177	−0.0605885	−	0.998163i	$-0.519298\pi$
$541$	−2.81851	−0.121177	−0.0605885	+	0.998163i	$0.519298\pi$
$542$	−12.7864	−0.549221
$543$	0	0
$544$	−16.6222	−0.712672
$545$	−13.2552	−0.567790
$546$	0	0
$547$	36.5853	1.56427	0.782137	−	0.623107i	$-0.214130\pi$
$547$	36.5853	1.56427	0.782137	+	0.623107i	$0.214130\pi$
$548$	13.6064	0.581236
$549$	0	0
$550$	−3.97179	−0.169358
$551$	38.1943	1.62713
$552$	0	0
$553$	−5.18819	−0.220624
$554$	−21.0215	−0.893119
$555$	0	0
$556$	13.7200	0.581859
$557$	7.67104	0.325032	0.162516	−	0.986706i	$-0.448039\pi$
$557$	7.67104	0.325032	0.162516	+	0.986706i	$0.448039\pi$
$558$	0	0
$559$	−20.6709	−0.874286
$560$	16.8025	0.710035
$561$	0	0
$562$	56.7883	2.39547
$563$	4.50675	0.189937	0.0949684	−	0.995480i	$-0.469725\pi$
$563$	4.50675	0.189937	0.0949684	+	0.995480i	$0.469725\pi$
$564$	0	0
$565$	18.9263	0.796235
$566$	21.5923	0.907593
$567$	0	0
$568$	2.71028	0.113721
$569$	−11.7146	−0.491103	−0.245551	−	0.969384i	$-0.578969\pi$
$569$	−11.7146	−0.491103	−0.245551	+	0.969384i	$0.578969\pi$
$570$	0	0
$571$	−9.71776	−0.406676	−0.203338	−	0.979109i	$-0.565179\pi$
$571$	−9.71776	−0.406676	−0.203338	+	0.979109i	$0.565179\pi$
$572$	−11.7646	−0.491904
$573$	0	0
$574$	22.2813	0.930002
$575$	1.00000	0.0417029
$576$	0	0
$577$	7.50066	0.312256	0.156128	−	0.987737i	$-0.450099\pi$
$577$	7.50066	0.312256	0.156128	+	0.987737i	$0.450099\pi$
$578$	−8.38923	−0.348946
$579$	0	0
$580$	−4.47551	−0.185835
$581$	48.2089	2.00004
$582$	0	0
$583$	0.722193	0.0299102
$584$	2.92354	0.120977
$585$	0	0
$586$	25.0857	1.03628
$587$	−6.49012	−0.267876	−0.133938	−	0.990990i	$-0.542762\pi$
$587$	−6.49012	−0.267876	−0.133938	+	0.990990i	$0.542762\pi$
$588$	0	0
$589$	16.1996	0.667491
$590$	9.56867	0.393936
$591$	0	0
$592$	43.2972	1.77950
$593$	6.73743	0.276673	0.138337	−	0.990385i	$-0.455824\pi$
$593$	6.73743	0.276673	0.138337	+	0.990385i	$0.455824\pi$
$594$	0	0
$595$	−11.6999	−0.479649
$596$	−18.6953	−0.765790
$597$	0	0
$598$	9.49436	0.388253
$599$	−11.1192	−0.454318	−0.227159	−	0.973858i	$-0.572944\pi$
$599$	−11.1192	−0.454318	−0.227159	+	0.973858i	$0.572944\pi$
$600$	0	0
$601$	−20.8033	−0.848585	−0.424292	−	0.905525i	$-0.639477\pi$
$601$	−20.8033	−0.848585	−0.424292	+	0.905525i	$0.639477\pi$
$602$	−21.3049	−0.868324
$603$	0	0
$604$	−20.2868	−0.825458
$605$	5.57319	0.226582
$606$	0	0
$607$	−9.26841	−0.376193	−0.188097	−	0.982151i	$-0.560232\pi$
$607$	−9.26841	−0.376193	−0.188097	+	0.982151i	$0.560232\pi$
$608$	−37.0148	−1.50115
$609$	0	0
$610$	−7.69921	−0.311732
$611$	−35.0901	−1.41959
$612$	0	0
$613$	28.2753	1.14203	0.571014	−	0.820941i	$-0.306550\pi$
$613$	28.2753	1.14203	0.571014	+	0.820941i	$0.306550\pi$
$614$	−2.34552	−0.0946576
$615$	0	0
$616$	14.6153	0.588868
$617$	46.0046	1.85208	0.926038	−	0.377431i	$-0.123193\pi$
$617$	46.0046	1.85208	0.926038	+	0.377431i	$0.123193\pi$
$618$	0	0
$619$	−7.60289	−0.305586	−0.152793	−	0.988258i	$-0.548827\pi$
$619$	−7.60289	−0.305586	−0.152793	+	0.988258i	$0.548827\pi$
$620$	−1.89822	−0.0762344
$621$	0	0
$622$	−43.7363	−1.75367
$623$	−53.6088	−2.14779
$624$	0	0
$625$	1.00000	0.0400000
$626$	35.8865	1.43431
$627$	0	0
$628$	−6.51231	−0.259869
$629$	−30.1487	−1.20211
$630$	0	0
$631$	−44.9382	−1.78896	−0.894481	−	0.447106i	$-0.852455\pi$
$631$	−44.9382	−1.78896	−0.894481	+	0.447106i	$0.852455\pi$
$632$	2.87235	0.114256
$633$	0	0
$634$	−20.8169	−0.826746
$635$	17.3074	0.686823
$636$	0	0
$637$	24.1247	0.955857
$638$	−19.6008	−0.776004
$639$	0	0
$640$	−12.6827	−0.501330
$641$	−5.52160	−0.218090	−0.109045	−	0.994037i	$-0.534779\pi$
$641$	−5.52160	−0.218090	−0.109045	+	0.994037i	$0.534779\pi$
$642$	0	0
$643$	−31.7764	−1.25314	−0.626570	−	0.779365i	$-0.715542\pi$
$643$	−31.7764	−1.25314	−0.626570	+	0.779365i	$0.715542\pi$
$644$	3.05290	0.120301
$645$	0	0
$646$	45.8617	1.80441
$647$	49.3890	1.94168	0.970842	−	0.239721i	$-0.0770561\pi$
$647$	49.3890	1.94168	0.970842	+	0.239721i	$0.0770561\pi$
$648$	0	0
$649$	13.0740	0.513201
$650$	9.49436	0.372399
$651$	0	0
$652$	−10.2692	−0.402174
$653$	41.3269	1.61725	0.808623	−	0.588328i	$-0.200213\pi$
$653$	41.3269	1.61725	0.808623	+	0.588328i	$0.200213\pi$
$654$	0	0
$655$	19.7273	0.770811
$656$	−19.3769	−0.756540
$657$	0	0
$658$	−36.1664	−1.40991
$659$	9.71854	0.378581	0.189290	−	0.981921i	$-0.439381\pi$
$659$	9.71854	0.378581	0.189290	+	0.981921i	$0.439381\pi$
$660$	0	0
$661$	24.0766	0.936470	0.468235	−	0.883604i	$-0.344890\pi$
$661$	24.0766	0.936470	0.468235	+	0.883604i	$0.344890\pi$
$662$	−23.5943	−0.917017
$663$	0	0
$664$	−26.6900	−1.03577
$665$	−26.0537	−1.01032
$666$	0	0
$667$	4.93501	0.191084
$668$	−19.9504	−0.771906
$669$	0	0
$670$	15.9443	0.615983
$671$	−10.5197	−0.406109
$672$	0	0
$673$	−29.6276	−1.14206	−0.571030	−	0.820929i	$-0.693456\pi$
$673$	−29.6276	−1.14206	−0.571030	+	0.820929i	$0.693456\pi$
$674$	−10.5713	−0.407190
$675$	0	0
$676$	16.3332	0.628199
$677$	17.9990	0.691760	0.345880	−	0.938279i	$-0.387580\pi$
$677$	17.9990	0.691760	0.345880	+	0.938279i	$0.387580\pi$
$678$	0	0
$679$	6.53690	0.250863
$680$	6.47743	0.248398
$681$	0	0
$682$	−8.31340	−0.318336
$683$	48.7141	1.86399	0.931996	−	0.362468i	$-0.118066\pi$
$683$	48.7141	1.86399	0.931996	+	0.362468i	$0.118066\pi$
$684$	0	0
$685$	−15.0033	−0.573248
$686$	−15.3116	−0.584600
$687$	0	0
$688$	18.5278	0.706367
$689$	−1.72636	−0.0657692
$690$	0	0
$691$	−15.4134	−0.586352	−0.293176	−	0.956059i	$-0.594712\pi$
$691$	−15.4134	−0.586352	−0.293176	+	0.956059i	$0.594712\pi$
$692$	5.73954	0.218185
$693$	0	0
$694$	−0.347059	−0.0131742
$695$	−15.1287	−0.573863
$696$	0	0
$697$	13.4925	0.511065
$698$	−61.4300	−2.32516
$699$	0	0
$700$	3.05290	0.115389
$701$	6.21048	0.234566	0.117283	−	0.993099i	$-0.462581\pi$
$701$	6.21048	0.234566	0.117283	+	0.993099i	$0.462581\pi$
$702$	0	0
$703$	−67.1359	−2.53208
$704$	−4.25961	−0.160540
$705$	0	0
$706$	23.9596	0.901731
$707$	2.61563	0.0983707
$708$	0	0
$709$	51.9143	1.94968	0.974842	−	0.222899i	$-0.0715520\pi$
$709$	51.9143	1.94968	0.974842	+	0.222899i	$0.0715520\pi$
$710$	2.47942	0.0930511
$711$	0	0
$712$	29.6795	1.11229
$713$	2.09311	0.0783876
$714$	0	0
$715$	12.9725	0.485144
$716$	4.27804	0.159878
$717$	0	0
$718$	24.8651	0.927958
$719$	37.5760	1.40135	0.700674	−	0.713482i	$-0.252883\pi$
$719$	37.5760	1.40135	0.700674	+	0.713482i	$0.252883\pi$
$720$	0	0
$721$	33.2350	1.23774
$722$	69.7319	2.59515
$723$	0	0
$724$	−11.0127	−0.409285
$725$	4.93501	0.183282
$726$	0	0
$727$	−7.31279	−0.271216	−0.135608	−	0.990763i	$-0.543299\pi$
$727$	−7.31279	−0.271216	−0.135608	+	0.990763i	$0.543299\pi$
$728$	−34.9371	−1.29486
$729$	0	0
$730$	2.67452	0.0989884
$731$	−12.9013	−0.477171
$732$	0	0
$733$	−47.7195	−1.76256	−0.881281	−	0.472593i	$-0.843318\pi$
$733$	−47.7195	−1.76256	−0.881281	+	0.472593i	$0.843318\pi$
$734$	−50.4748	−1.86306
$735$	0	0
$736$	−4.78260	−0.176289
$737$	21.7853	0.802473
$738$	0	0
$739$	−27.4272	−1.00893	−0.504463	−	0.863434i	$-0.668309\pi$
$739$	−27.4272	−1.00893	−0.504463	+	0.863434i	$0.668309\pi$
$740$	7.86680	0.289189
$741$	0	0
$742$	−1.77932	−0.0653207
$743$	22.9968	0.843669	0.421835	−	0.906673i	$-0.361386\pi$
$743$	22.9968	0.843669	0.421835	+	0.906673i	$0.361386\pi$
$744$	0	0
$745$	20.6147	0.755266
$746$	54.3340	1.98931
$747$	0	0
$748$	−7.34263	−0.268473
$749$	36.9257	1.34924
$750$	0	0
$751$	−40.3144	−1.47109	−0.735547	−	0.677474i	$-0.763075\pi$
$751$	−40.3144	−1.47109	−0.735547	+	0.677474i	$0.763075\pi$
$752$	31.4521	1.14694
$753$	0	0
$754$	46.8547	1.70635
$755$	22.3696	0.814114
$756$	0	0
$757$	18.7386	0.681065	0.340533	−	0.940233i	$-0.389393\pi$
$757$	18.7386	0.681065	0.340533	+	0.940233i	$0.389393\pi$
$758$	44.9652	1.63321
$759$	0	0
$760$	14.4241	0.523218
$761$	−49.3367	−1.78845	−0.894227	−	0.447613i	$-0.852274\pi$
$761$	−49.3367	−1.78845	−0.894227	+	0.447613i	$0.852274\pi$
$762$	0	0
$763$	44.6214	1.61540
$764$	3.90332	0.141217
$765$	0	0
$766$	12.1540	0.439141
$767$	−31.2528	−1.12847
$768$	0	0
$769$	−11.2643	−0.406202	−0.203101	−	0.979158i	$-0.565102\pi$
$769$	−11.2643	−0.406202	−0.203101	+	0.979158i	$0.565102\pi$
$770$	13.3704	0.481836
$771$	0	0
$772$	1.64490	0.0592012
$773$	41.4171	1.48967	0.744834	−	0.667249i	$-0.232528\pi$
$773$	41.4171	1.48967	0.744834	+	0.667249i	$0.232528\pi$
$774$	0	0
$775$	2.09311	0.0751867
$776$	−3.61903	−0.129916
$777$	0	0
$778$	4.24847	0.152315
$779$	30.0455	1.07649
$780$	0	0
$781$	3.38773	0.121222
$782$	5.92569	0.211902
$783$	0	0
$784$	−21.6236	−0.772271
$785$	7.18092	0.256298
$786$	0	0
$787$	−14.0350	−0.500294	−0.250147	−	0.968208i	$-0.580479\pi$
$787$	−14.0350	−0.500294	−0.250147	+	0.968208i	$0.580479\pi$
$788$	16.7911	0.598158
$789$	0	0
$790$	2.62768	0.0934887
$791$	−63.7123	−2.26535
$792$	0	0
$793$	25.1468	0.892990
$794$	45.7903	1.62504
$795$	0	0
$796$	−1.75118	−0.0620691
$797$	21.4491	0.759765	0.379882	−	0.925035i	$-0.375964\pi$
$797$	21.4491	0.759765	0.379882	+	0.925035i	$0.375964\pi$
$798$	0	0
$799$	−21.9007	−0.774791
$800$	−4.78260	−0.169091
$801$	0	0
$802$	−8.49640	−0.300018
$803$	3.65430	0.128957
$804$	0	0
$805$	−3.36634	−0.118648
$806$	19.8727	0.699987
$807$	0	0
$808$	−1.44809	−0.0509437
$809$	8.83096	0.310480	0.155240	−	0.987877i	$-0.450385\pi$
$809$	8.83096	0.310480	0.155240	+	0.987877i	$0.450385\pi$
$810$	0	0
$811$	8.39739	0.294872	0.147436	−	0.989072i	$-0.452898\pi$
$811$	8.39739	0.294872	0.147436	+	0.989072i	$0.452898\pi$
$812$	15.0661	0.528716
$813$	0	0
$814$	34.4532	1.20759
$815$	11.3236	0.396647
$816$	0	0
$817$	−28.7289	−1.00510
$818$	33.4904	1.17096
$819$	0	0
$820$	−3.52065	−0.122946
$821$	−47.7871	−1.66778	−0.833891	−	0.551930i	$-0.813892\pi$
$821$	−47.7871	−1.66778	−0.833891	+	0.551930i	$0.813892\pi$
$822$	0	0
$823$	21.2580	0.741008	0.370504	−	0.928831i	$-0.379185\pi$
$823$	21.2580	0.741008	0.370504	+	0.928831i	$0.379185\pi$
$824$	−18.3999	−0.640991
$825$	0	0
$826$	−32.2114	−1.12078
$827$	−23.2919	−0.809939	−0.404970	−	0.914330i	$-0.632718\pi$
$827$	−23.2919	−0.809939	−0.404970	+	0.914330i	$0.632718\pi$
$828$	0	0
$829$	17.1286	0.594902	0.297451	−	0.954737i	$-0.403864\pi$
$829$	17.1286	0.594902	0.297451	+	0.954737i	$0.403864\pi$
$830$	−24.4165	−0.847510
$831$	0	0
$832$	10.1824	0.353010
$833$	15.0569	0.521691
$834$	0	0
$835$	21.9987	0.761298
$836$	−16.3508	−0.565503
$837$	0	0
$838$	−37.9275	−1.31018
$839$	41.9077	1.44681	0.723407	−	0.690422i	$-0.242575\pi$
$839$	41.9077	1.44681	0.723407	+	0.690422i	$0.242575\pi$
$840$	0	0
$841$	−4.64571	−0.160197
$842$	57.1556	1.96971
$843$	0	0
$844$	−8.73561	−0.300692
$845$	−18.0101	−0.619566
$846$	0	0
$847$	−18.7612	−0.644644
$848$	1.54738	0.0531373
$849$	0	0
$850$	5.92569	0.203250
$851$	−8.67448	−0.297357
$852$	0	0
$853$	28.4367	0.973654	0.486827	−	0.873498i	$-0.338154\pi$
$853$	28.4367	0.973654	0.486827	+	0.873498i	$0.338154\pi$
$854$	25.9181	0.886901
$855$	0	0
$856$	−20.4432	−0.698736
$857$	40.7539	1.39213	0.696064	−	0.717980i	$-0.254933\pi$
$857$	40.7539	1.39213	0.696064	+	0.717980i	$0.254933\pi$
$858$	0	0
$859$	−34.5086	−1.17742	−0.588710	−	0.808344i	$-0.700364\pi$
$859$	−34.5086	−1.17742	−0.588710	+	0.808344i	$0.700364\pi$
$860$	3.36638	0.114793
$861$	0	0
$862$	−47.9516	−1.63324
$863$	−43.3918	−1.47707	−0.738537	−	0.674213i	$-0.764483\pi$
$863$	−43.3918	−1.47707	−0.738537	+	0.674213i	$0.764483\pi$
$864$	0	0
$865$	−6.32882	−0.215186
$866$	0.560686	0.0190529
$867$	0	0
$868$	6.39005	0.216892
$869$	3.59030	0.121793
$870$	0	0
$871$	−52.0767	−1.76455
$872$	−24.7038	−0.836577
$873$	0	0
$874$	13.1955	0.446344
$875$	−3.36634	−0.113803
$876$	0	0
$877$	3.82061	0.129013	0.0645063	−	0.997917i	$-0.479453\pi$
$877$	3.82061	0.129013	0.0645063	+	0.997917i	$0.479453\pi$
$878$	−10.0833	−0.340294
$879$	0	0
$880$	−11.6276	−0.391965
$881$	41.0582	1.38329	0.691643	−	0.722240i	$-0.256887\pi$
$881$	41.0582	1.38329	0.691643	+	0.722240i	$0.256887\pi$
$882$	0	0
$883$	−21.6971	−0.730166	−0.365083	−	0.930975i	$-0.618959\pi$
$883$	−21.6971	−0.730166	−0.365083	+	0.930975i	$0.618959\pi$
$884$	17.5522	0.590343
$885$	0	0
$886$	8.23993	0.276826
$887$	−50.8277	−1.70663	−0.853314	−	0.521397i	$-0.825411\pi$
$887$	−50.8277	−1.70663	−0.853314	+	0.521397i	$0.825411\pi$
$888$	0	0
$889$	−58.2626	−1.95406
$890$	27.1514	0.910118
$891$	0	0
$892$	15.3838	0.515088
$893$	−48.7691	−1.63200
$894$	0	0
$895$	−4.71726	−0.157681
$896$	42.6944	1.42632
$897$	0	0
$898$	6.99813	0.233531
$899$	10.3295	0.344508
$900$	0	0
$901$	−1.07747	−0.0358958
$902$	−15.4189	−0.513394
$903$	0	0
$904$	35.2731	1.17317
$905$	12.1434	0.403660
$906$	0	0
$907$	−43.7967	−1.45424	−0.727122	−	0.686508i	$-0.759143\pi$
$907$	−43.7967	−1.45424	−0.727122	+	0.686508i	$0.759143\pi$
$908$	11.5887	0.384584
$909$	0	0
$910$	−31.9612	−1.05950
$911$	4.63898	0.153696	0.0768481	−	0.997043i	$-0.475514\pi$
$911$	4.63898	0.153696	0.0768481	+	0.997043i	$0.475514\pi$
$912$	0	0
$913$	−33.3612	−1.10410
$914$	38.9264	1.28757
$915$	0	0
$916$	−5.22130	−0.172517
$917$	−66.4089	−2.19301
$918$	0	0
$919$	−7.27190	−0.239878	−0.119939	−	0.992781i	$-0.538270\pi$
$919$	−7.27190	−0.239878	−0.119939	+	0.992781i	$0.538270\pi$
$920$	1.86371	0.0614447
$921$	0	0
$922$	−70.0963	−2.30850
$923$	−8.09818	−0.266555
$924$	0	0
$925$	−8.67448	−0.285215
$926$	−44.3268	−1.45667
$927$	0	0
$928$	−23.6022	−0.774780
$929$	−1.48464	−0.0487096	−0.0243548	−	0.999703i	$-0.507753\pi$
$929$	−1.48464	−0.0487096	−0.0243548	+	0.999703i	$0.507753\pi$
$930$	0	0
$931$	33.5292	1.09887
$932$	−24.1508	−0.791086
$933$	0	0
$934$	−67.6694	−2.21421
$935$	8.09649	0.264784
$936$	0	0
$937$	2.63415	0.0860540	0.0430270	−	0.999074i	$-0.486300\pi$
$937$	2.63415	0.0860540	0.0430270	+	0.999074i	$0.486300\pi$
$938$	−53.6740	−1.75252
$939$	0	0
$940$	5.71463	0.186391
$941$	−46.0509	−1.50122	−0.750608	−	0.660748i	$-0.770239\pi$
$941$	−46.0509	−1.50122	−0.750608	+	0.660748i	$0.770239\pi$
$942$	0	0
$943$	3.88211	0.126419
$944$	28.0126	0.911733
$945$	0	0
$946$	14.7433	0.479346
$947$	−46.2264	−1.50216	−0.751078	−	0.660213i	$-0.770466\pi$
$947$	−46.2264	−1.50216	−0.751078	+	0.660213i	$0.770466\pi$
$948$	0	0
$949$	−8.73540	−0.283563
$950$	13.1955	0.428118
$951$	0	0
$952$	−21.8052	−0.706711
$953$	13.0234	0.421869	0.210935	−	0.977500i	$-0.432349\pi$
$953$	13.0234	0.421869	0.210935	+	0.977500i	$0.432349\pi$
$954$	0	0
$955$	−4.30407	−0.139276
$956$	9.98192	0.322838
$957$	0	0
$958$	29.9237	0.966792
$959$	50.5063	1.63093
$960$	0	0
$961$	−26.6189	−0.858674
$962$	−82.3586	−2.65535
$963$	0	0
$964$	5.88598	0.189575
$965$	−1.81378	−0.0583877
$966$	0	0
$967$	3.69987	0.118980	0.0594898	−	0.998229i	$-0.481053\pi$
$967$	3.69987	0.118980	0.0594898	+	0.998229i	$0.481053\pi$
$968$	10.3868	0.333845
$969$	0	0
$970$	−3.31077	−0.106302
$971$	−21.6156	−0.693677	−0.346838	−	0.937925i	$-0.612745\pi$
$971$	−21.6156	−0.693677	−0.346838	+	0.937925i	$0.612745\pi$
$972$	0	0
$973$	50.9282	1.63268
$974$	−61.6182	−1.97437
$975$	0	0
$976$	−22.5397	−0.721478
$977$	26.5647	0.849881	0.424940	−	0.905221i	$-0.360295\pi$
$977$	26.5647	0.849881	0.424940	+	0.905221i	$0.360295\pi$
$978$	0	0
$979$	37.0980	1.18566
$980$	−3.92886	−0.125503
$981$	0	0
$982$	18.8185	0.600523
$983$	−15.3307	−0.488973	−0.244486	−	0.969653i	$-0.578619\pi$
$983$	−15.3307	−0.488973	−0.244486	+	0.969653i	$0.578619\pi$
$984$	0	0
$985$	−18.5150	−0.589938
$986$	29.2433	0.931297
$987$	0	0
$988$	39.0857	1.24348
$989$	−3.71200	−0.118035
$990$	0	0
$991$	7.53563	0.239377	0.119689	−	0.992811i	$-0.461810\pi$
$991$	7.53563	0.239377	0.119689	+	0.992811i	$0.461810\pi$
$992$	−10.0105	−0.317834
$993$	0	0
$994$	−8.34657	−0.264737
$995$	1.93098	0.0612161
$996$	0	0
$997$	35.6936	1.13043	0.565215	−	0.824944i	$-0.308793\pi$
$997$	35.6936	1.13043	0.565215	+	0.824944i	$0.308793\pi$
$998$	−31.0694	−0.983483
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.a.p.1.5	✓	6
3.2	odd	2		1035.2.a.q.1.2	yes	6
5.4	even	2		5175.2.a.bz.1.2		6
15.14	odd	2		5175.2.a.by.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.2.a.p.1.5	✓	6	1.1	even	1	trivial
1035.2.a.q.1.2	yes	6	3.2	odd	2
5175.2.a.by.1.5		6	15.14	odd	2
5175.2.a.bz.1.2		6	5.4	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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70496 q^{2} +0.906890 q^{4} -1.00000 q^{5} +3.36634 q^{7} -1.86371 q^{8} -1.70496 q^{10} -2.32955 q^{11} +5.56867 q^{13} +5.73947 q^{14} -4.99133 q^{16} +3.47556 q^{17} +7.73947 q^{19} -0.906890 q^{20} -3.97179 q^{22} +1.00000 q^{23} +1.00000 q^{25} +9.49436 q^{26} +3.05290 q^{28} +4.93501 q^{29} +2.09311 q^{31} -4.78260 q^{32} +5.92569 q^{34} -3.36634 q^{35} -8.67448 q^{37} +13.1955 q^{38} +1.86371 q^{40} +3.88211 q^{41} -3.71200 q^{43} -2.11265 q^{44} +1.70496 q^{46} -6.30134 q^{47} +4.33223 q^{49} +1.70496 q^{50} +5.05017 q^{52} -0.310014 q^{53} +2.32955 q^{55} -6.27387 q^{56} +8.41399 q^{58} -5.61225 q^{59} +4.51577 q^{61} +3.56867 q^{62} +1.82851 q^{64} -5.56867 q^{65} -9.35173 q^{67} +3.15195 q^{68} -5.73947 q^{70} -1.45424 q^{71} -1.56867 q^{73} -14.7896 q^{74} +7.01885 q^{76} -7.84206 q^{77} -1.54120 q^{79} +4.99133 q^{80} +6.61884 q^{82} +14.3209 q^{83} -3.47556 q^{85} -6.32882 q^{86} +4.34161 q^{88} -15.9250 q^{89} +18.7460 q^{91} +0.906890 q^{92} -10.7435 q^{94} -7.73947 q^{95} +1.94184 q^{97} +7.38628 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{5} + 6 q^{7} + 4 q^{11} + 12 q^{13} - 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 10 q^{20} + 8 q^{22} + 6 q^{23} + 6 q^{25} - 12 q^{26} + 24 q^{28} - 6 q^{29} + 8 q^{31} + 20 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^5 + 6 * q^7 + 4 * q^11 + 12 * q^13 - 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 10 * q^20 + 8 * q^22 + 6 * q^23 + 6 * q^25 - 12 * q^26 + 24 * q^28 - 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 6 * q^35 + 22 * q^37 + 36 * q^38 - 18 * q^41 + 8 * q^43 + 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^55 + 4 * q^56 - 16 * q^58 - 6 * q^59 + 14 * q^64 - 12 * q^65 + 10 * q^67 + 28 * q^68 + 4 * q^70 - 22 * q^71 + 12 * q^73 - 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 - 14 * q^80 - 12 * q^82 + 24 * q^83 - 4 * q^85 + 20 * q^86 - 8 * q^88 - 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 8 * q^95 + 12 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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