LMFDB - Embedded newform 1125.3.c.a.251.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1125,3,Mod(251,1125)] 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("1125.251"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1125 = 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1125.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,-76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$30.6540297405$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.10
Character	$\chi$	$=$	1125.251
Dual form			1125.3.c.a.251.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.859005i q^{2} +3.26211 q^{4} -6.29943 q^{7} +6.23819i q^{8} +7.37984i q^{11} -3.27406 q^{13} -5.41124i q^{14} +7.68981 q^{16} +6.55221i q^{17} +17.7738 q^{19} -6.33932 q^{22} +0.135035i q^{23} -2.81244i q^{26} -20.5494 q^{28} +21.0607i q^{29} -18.0374 q^{31} +31.5583i q^{32} -5.62838 q^{34} -32.8447 q^{37} +15.2678i q^{38} -16.0868i q^{41} -78.9676 q^{43} +24.0739i q^{44} -0.115996 q^{46} -23.7246i q^{47} -9.31715 q^{49} -10.6804 q^{52} +79.0737i q^{53} -39.2970i q^{56} -18.0913 q^{58} +80.9060i q^{59} -24.3380 q^{61} -15.4942i q^{62} +3.65049 q^{64} +101.002 q^{67} +21.3741i q^{68} +70.3926i q^{71} -83.5120 q^{73} -28.2137i q^{74} +57.9802 q^{76} -46.4888i q^{77} -7.09649 q^{79} +13.8186 q^{82} +132.342i q^{83} -67.8336i q^{86} -46.0368 q^{88} -48.3394i q^{89} +20.6247 q^{91} +0.440499i q^{92} +20.3796 q^{94} +49.6214 q^{97} -8.00348i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 76 q^{4} + 92 q^{16} - 80 q^{19} + 272 q^{31} + 184 q^{34} - 368 q^{46} + 64 q^{49} + 416 q^{61} + 68 q^{64} + 640 q^{76} - 528 q^{79} - 528 q^{91} - 548 q^{94}+O(q^{100}) $ 32 * q - 76 * q^4 + 92 * q^16 - 80 * q^19 + 272 * q^31 + 184 * q^34 - 368 * q^46 + 64 * q^49 + 416 * q^61 + 68 * q^64 + 640 * q^76 - 528 * q^79 - 528 * q^91 - 548 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1125\mathbb{Z}\right)^\times$.

$n$	$127$	$1001$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.859005i	0.429502i	0.976669	+	0.214751i	$0.0688941\pi$
$2$	0.859005i	0.429502i	−0.976669	+	0.214751i	$0.931106\pi$
$3$	0	0
$3$	0	0
$4$	3.26211	0.815528
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.29943	−0.899919	−0.449959	−	0.893049i	$-0.648562\pi$
$7$	−6.29943	−0.899919	−0.449959	+	0.893049i	$0.648562\pi$
$8$	6.23819i	0.779773i
$9$	0	0
$10$	0	0
$11$	7.37984i	0.670895i	0.942059	+	0.335447i	$0.108887\pi$
$11$	7.37984i	0.670895i	−0.942059	+	0.335447i	$0.891113\pi$
$12$	0	0
$13$	−3.27406	−0.251851	−0.125926	−	0.992040i	$-0.540190\pi$
$13$	−3.27406	−0.251851	−0.125926	+	0.992040i	$0.540190\pi$
$14$	− 5.41124i	− 0.386517i
$15$	0	0
$16$	7.68981	0.480613
$17$	6.55221i	0.385424i	0.981255	+	0.192712i	$0.0617284\pi$
$17$	6.55221i	0.385424i	−0.981255	+	0.192712i	$0.938272\pi$
$18$	0	0
$19$	17.7738	0.935465	0.467733	−	0.883870i	$-0.345071\pi$
$19$	17.7738	0.935465	0.467733	+	0.883870i	$0.345071\pi$
$20$	0	0
$21$	0	0
$22$	−6.33932	−0.288151
$23$	0.135035i	0.00587108i	0.999996	+	0.00293554i	$0.000934413\pi$
$23$	0.135035i	0.00587108i	−0.999996	+	0.00293554i	$0.999066\pi$
$24$	0	0
$25$	0	0
$26$	− 2.81244i	− 0.108171i
$27$	0	0
$28$	−20.5494	−0.733909
$29$	21.0607i	0.726232i	0.931744	+	0.363116i	$0.118287\pi$
$29$	21.0607i	0.726232i	−0.931744	+	0.363116i	$0.881713\pi$
$30$	0	0
$31$	−18.0374	−0.581852	−0.290926	−	0.956746i	$-0.593963\pi$
$31$	−18.0374	−0.581852	−0.290926	+	0.956746i	$0.593963\pi$
$32$	31.5583i	0.986198i
$33$	0	0
$34$	−5.62838	−0.165541
$35$	0	0
$36$	0	0
$37$	−32.8447	−0.887694	−0.443847	−	0.896103i	$-0.646387\pi$
$37$	−32.8447	−0.887694	−0.443847	+	0.896103i	$0.646387\pi$
$38$	15.2678i	0.401784i
$39$	0	0
$40$	0	0
$41$	− 16.0868i	− 0.392361i	−0.980568	−	0.196180i	$-0.937146\pi$
$41$	− 16.0868i	− 0.392361i	0.980568	−	0.196180i	$-0.0628538\pi$
$42$	0	0
$43$	−78.9676	−1.83646	−0.918228	−	0.396052i	$-0.870380\pi$
$43$	−78.9676	−1.83646	−0.918228	+	0.396052i	$0.870380\pi$
$44$	24.0739i	0.547133i
$45$	0	0
$46$	−0.115996	−0.00252164
$47$	− 23.7246i	− 0.504779i	−0.967626	−	0.252390i	$-0.918783\pi$
$47$	− 23.7246i	− 0.504779i	0.967626	−	0.252390i	$-0.0812165\pi$
$48$	0	0
$49$	−9.31715	−0.190146
$50$	0	0
$51$	0	0
$52$	−10.6804	−0.205391
$53$	79.0737i	1.49196i	0.665970	+	0.745979i	$0.268018\pi$
$53$	79.0737i	1.49196i	−0.665970	+	0.745979i	$0.731982\pi$
$54$	0	0
$55$	0	0
$56$	− 39.2970i	− 0.701733i
$57$	0	0
$58$	−18.0913	−0.311919
$59$	80.9060i	1.37129i	0.727937	+	0.685644i	$0.240479\pi$
$59$	80.9060i	1.37129i	−0.727937	+	0.685644i	$0.759521\pi$
$60$	0	0
$61$	−24.3380	−0.398983	−0.199492	−	0.979900i	$-0.563929\pi$
$61$	−24.3380	−0.398983	−0.199492	+	0.979900i	$0.563929\pi$
$62$	− 15.4942i	− 0.249907i
$63$	0	0
$64$	3.65049	0.0570388
$65$	0	0
$66$	0	0
$67$	101.002	1.50749	0.753745	−	0.657167i	$-0.228245\pi$
$67$	101.002	1.50749	0.753745	+	0.657167i	$0.228245\pi$
$68$	21.3741i	0.314324i
$69$	0	0
$70$	0	0
$71$	70.3926i	0.991445i	0.868481	+	0.495722i	$0.165097\pi$
$71$	70.3926i	0.991445i	−0.868481	+	0.495722i	$0.834903\pi$
$72$	0	0
$73$	−83.5120	−1.14400	−0.572000	−	0.820253i	$-0.693832\pi$
$73$	−83.5120	−1.14400	−0.572000	+	0.820253i	$0.693832\pi$
$74$	− 28.2137i	− 0.381266i
$75$	0	0
$76$	57.9802	0.762898
$77$	− 46.4888i	− 0.603751i
$78$	0	0
$79$	−7.09649	−0.0898290	−0.0449145	−	0.998991i	$-0.514302\pi$
$79$	−7.09649	−0.0898290	−0.0449145	+	0.998991i	$0.514302\pi$
$80$	0	0
$81$	0	0
$82$	13.8186	0.168520
$83$	132.342i	1.59448i	0.603661	+	0.797241i	$0.293708\pi$
$83$	132.342i	1.59448i	−0.603661	+	0.797241i	$0.706292\pi$
$84$	0	0
$85$	0	0
$86$	− 67.8336i	− 0.788762i
$87$	0	0
$88$	−46.0368	−0.523146
$89$	− 48.3394i	− 0.543139i	−0.962419	−	0.271570i	$-0.912457\pi$
$89$	− 48.3394i	− 0.543139i	0.962419	−	0.271570i	$-0.0875427\pi$
$90$	0	0
$91$	20.6247	0.226646
$92$	0.440499i	0.00478803i
$93$	0	0
$94$	20.3796	0.216804
$95$	0	0
$96$	0	0
$97$	49.6214	0.511561	0.255780	−	0.966735i	$-0.417668\pi$
$97$	49.6214	0.511561	0.255780	+	0.966735i	$0.417668\pi$
$98$	− 8.00348i	− 0.0816681i
$99$	0	0
$100$	0	0
$101$	144.739i	1.43306i	0.697555	+	0.716531i	$0.254271\pi$
$101$	144.739i	1.43306i	−0.697555	+	0.716531i	$0.745729\pi$
$102$	0	0
$103$	−189.430	−1.83913	−0.919565	−	0.392937i	$-0.871459\pi$
$103$	−189.430	−1.83913	−0.919565	+	0.392937i	$0.871459\pi$
$104$	− 20.4242i	− 0.196387i
$105$	0	0
$106$	−67.9247	−0.640799
$107$	0.596520i	0.00557496i	0.999996	+	0.00278748i	$0.000887283\pi$
$107$	0.596520i	0.00557496i	−0.999996	+	0.00278748i	$0.999113\pi$
$108$	0	0
$109$	178.917	1.64144	0.820719	−	0.571332i	$-0.193573\pi$
$109$	178.917	1.64144	0.820719	+	0.571332i	$0.193573\pi$
$110$	0	0
$111$	0	0
$112$	−48.4414	−0.432513
$113$	37.4130i	0.331088i	0.986202	+	0.165544i	$0.0529380\pi$
$113$	37.4130i	0.331088i	−0.986202	+	0.165544i	$0.947062\pi$
$114$	0	0
$115$	0	0
$116$	68.7025i	0.592263i
$117$	0	0
$118$	−69.4986	−0.588972
$119$	− 41.2752i	− 0.346851i
$120$	0	0
$121$	66.5379	0.549900
$122$	− 20.9064i	− 0.171364i
$123$	0	0
$124$	−58.8400	−0.474516
$125$	0	0
$126$	0	0
$127$	80.3484	0.632665	0.316332	−	0.948648i	$-0.397549\pi$
$127$	80.3484	0.632665	0.316332	+	0.948648i	$0.397549\pi$
$128$	129.369i	1.01070i
$129$	0	0
$130$	0	0
$131$	− 181.897i	− 1.38852i	−0.719723	−	0.694262i	$-0.755731\pi$
$131$	− 181.897i	− 1.38852i	0.719723	−	0.694262i	$-0.244269\pi$
$132$	0	0
$133$	−111.965	−0.841843
$134$	86.7611i	0.647471i
$135$	0	0
$136$	−40.8739	−0.300544
$137$	− 42.2928i	− 0.308706i	−0.988016	−	0.154353i	$-0.950671\pi$
$137$	− 42.2928i	− 0.308706i	0.988016	−	0.154353i	$-0.0493293\pi$
$138$	0	0
$139$	−139.711	−1.00512	−0.502559	−	0.864543i	$-0.667608\pi$
$139$	−139.711	−1.00512	−0.502559	+	0.864543i	$0.667608\pi$
$140$	0	0
$141$	0	0
$142$	−60.4675	−0.425828
$143$	− 24.1621i	− 0.168966i
$144$	0	0
$145$	0	0
$146$	− 71.7372i	− 0.491351i
$147$	0	0
$148$	−107.143	−0.723939
$149$	− 203.483i	− 1.36566i	−0.730580	−	0.682828i	$-0.760750\pi$
$149$	− 203.483i	− 1.36566i	0.730580	−	0.682828i	$-0.239250\pi$
$150$	0	0
$151$	84.5171	0.559716	0.279858	−	0.960041i	$-0.409713\pi$
$151$	84.5171	0.559716	0.279858	+	0.960041i	$0.409713\pi$
$152$	110.877i	0.729451i
$153$	0	0
$154$	39.9341	0.259312
$155$	0	0
$156$	0	0
$157$	62.7363	0.399594	0.199797	−	0.979837i	$-0.435972\pi$
$157$	62.7363	0.399594	0.199797	+	0.979837i	$0.435972\pi$
$158$	− 6.09592i	− 0.0385818i
$159$	0	0
$160$	0	0
$161$	− 0.850643i	− 0.00528350i
$162$	0	0
$163$	−0.618262	−0.00379302	−0.00189651	−	0.999998i	$-0.500604\pi$
$163$	−0.618262	−0.00379302	−0.00189651	+	0.999998i	$0.500604\pi$
$164$	− 52.4769i	− 0.319981i
$165$	0	0
$166$	−113.682	−0.684834
$167$	− 102.221i	− 0.612100i	−0.952015	−	0.306050i	$-0.900993\pi$
$167$	− 102.221i	− 0.612100i	0.952015	−	0.306050i	$-0.0990075\pi$
$168$	0	0
$169$	−158.281	−0.936571
$170$	0	0
$171$	0	0
$172$	−257.601	−1.49768
$173$	259.475i	1.49985i	0.661520	+	0.749927i	$0.269911\pi$
$173$	259.475i	1.49985i	−0.661520	+	0.749927i	$0.730089\pi$
$174$	0	0
$175$	0	0
$176$	56.7496i	0.322441i
$177$	0	0
$178$	41.5238	0.233280
$179$	60.9754i	0.340645i	0.985388	+	0.170322i	$0.0544809\pi$
$179$	60.9754i	0.340645i	−0.985388	+	0.170322i	$0.945519\pi$
$180$	0	0
$181$	−144.102	−0.796142	−0.398071	−	0.917355i	$-0.630320\pi$
$181$	−144.102	−0.796142	−0.398071	+	0.917355i	$0.630320\pi$
$182$	17.7168i	0.0973448i
$183$	0	0
$184$	−0.842373	−0.00457812
$185$	0	0
$186$	0	0
$187$	−48.3543	−0.258579
$188$	− 77.3924i	− 0.411662i
$189$	0	0
$190$	0	0
$191$	6.42807i	0.0336548i	0.999858	+	0.0168274i	$0.00535658\pi$
$191$	6.42807i	0.0336548i	−0.999858	+	0.0168274i	$0.994643\pi$
$192$	0	0
$193$	190.099	0.984970	0.492485	−	0.870321i	$-0.336089\pi$
$193$	190.099	0.984970	0.492485	+	0.870321i	$0.336089\pi$
$194$	42.6250i	0.219717i
$195$	0	0
$196$	−30.3936	−0.155069
$197$	296.892i	1.50707i	0.657409	+	0.753534i	$0.271652\pi$
$197$	296.892i	1.50707i	−0.657409	+	0.753534i	$0.728348\pi$
$198$	0	0
$199$	−271.775	−1.36570	−0.682852	−	0.730556i	$-0.739261\pi$
$199$	−271.775	−1.36570	−0.682852	+	0.730556i	$0.739261\pi$
$200$	0	0
$201$	0	0
$202$	−124.332	−0.615504
$203$	− 132.671i	− 0.653550i
$204$	0	0
$205$	0	0
$206$	− 162.722i	− 0.789911i
$207$	0	0
$208$	−25.1769	−0.121043
$209$	131.168i	0.627599i
$210$	0	0
$211$	259.759	1.23108	0.615542	−	0.788104i	$-0.288937\pi$
$211$	259.759	1.23108	0.615542	+	0.788104i	$0.288937\pi$
$212$	257.947i	1.21673i
$213$	0	0
$214$	−0.512414	−0.00239446
$215$	0	0
$216$	0	0
$217$	113.625	0.523619
$218$	153.690i	0.705002i
$219$	0	0
$220$	0	0
$221$	− 21.4524i	− 0.0970695i
$222$	0	0
$223$	−77.2253	−0.346302	−0.173151	−	0.984895i	$-0.555395\pi$
$223$	−77.2253	−0.346302	−0.173151	+	0.984895i	$0.555395\pi$
$224$	− 198.800i	− 0.887498i
$225$	0	0
$226$	−32.1379	−0.142203
$227$	− 196.878i	− 0.867305i	−0.901080	−	0.433653i	$-0.857225\pi$
$227$	− 196.878i	− 0.867305i	0.901080	−	0.433653i	$-0.142775\pi$
$228$	0	0
$229$	110.560	0.482795	0.241398	−	0.970426i	$-0.422394\pi$
$229$	110.560	0.482795	0.241398	+	0.970426i	$0.422394\pi$
$230$	0	0
$231$	0	0
$232$	−131.381	−0.566297
$233$	− 231.012i	− 0.991467i	−0.868475	−	0.495733i	$-0.834899\pi$
$233$	− 231.012i	− 0.991467i	0.868475	−	0.495733i	$-0.165101\pi$
$234$	0	0
$235$	0	0
$236$	263.924i	1.11832i
$237$	0	0
$238$	35.4556	0.148973
$239$	− 29.0274i	− 0.121454i	−0.998154	−	0.0607268i	$-0.980658\pi$
$239$	− 29.0274i	− 0.121454i	0.998154	−	0.0607268i	$-0.0193418\pi$
$240$	0	0
$241$	111.259	0.461656	0.230828	−	0.972995i	$-0.425857\pi$
$241$	111.259	0.461656	0.230828	+	0.972995i	$0.425857\pi$
$242$	57.1564i	0.236183i
$243$	0	0
$244$	−79.3932	−0.325382
$245$	0	0
$246$	0	0
$247$	−58.1927	−0.235598
$248$	− 112.521i	− 0.453712i
$249$	0	0
$250$	0	0
$251$	364.759i	1.45322i	0.687048	+	0.726612i	$0.258906\pi$
$251$	364.759i	1.45322i	−0.687048	+	0.726612i	$0.741094\pi$
$252$	0	0
$253$	−0.996536	−0.00393888
$254$	69.0197i	0.271731i
$255$	0	0
$256$	−96.5267	−0.377058
$257$	− 484.039i	− 1.88342i	−0.336424	−	0.941711i	$-0.609217\pi$
$257$	− 484.039i	− 1.88342i	0.336424	−	0.941711i	$-0.390783\pi$
$258$	0	0
$259$	206.903	0.798852
$260$	0	0
$261$	0	0
$262$	156.250	0.596374
$263$	− 65.9544i	− 0.250777i	−0.992108	−	0.125389i	$-0.959982\pi$
$263$	− 65.9544i	− 0.250777i	0.992108	−	0.125389i	$-0.0400178\pi$
$264$	0	0
$265$	0	0
$266$	− 96.1785i	− 0.361573i
$267$	0	0
$268$	329.479	1.22940
$269$	− 413.537i	− 1.53731i	−0.639661	−	0.768657i	$-0.720925\pi$
$269$	− 413.537i	− 1.53731i	0.639661	−	0.768657i	$-0.279075\pi$
$270$	0	0
$271$	−123.704	−0.456474	−0.228237	−	0.973606i	$-0.573296\pi$
$271$	−123.704	−0.456474	−0.228237	+	0.973606i	$0.573296\pi$
$272$	50.3853i	0.185240i
$273$	0	0
$274$	36.3297	0.132590
$275$	0	0
$276$	0	0
$277$	359.268	1.29699	0.648497	−	0.761217i	$-0.275398\pi$
$277$	359.268	1.29699	0.648497	+	0.761217i	$0.275398\pi$
$278$	− 120.013i	− 0.431701i
$279$	0	0
$280$	0	0
$281$	− 251.042i	− 0.893387i	−0.894687	−	0.446694i	$-0.852601\pi$
$281$	− 251.042i	− 0.893387i	0.894687	−	0.446694i	$-0.147399\pi$
$282$	0	0
$283$	147.207	0.520166	0.260083	−	0.965586i	$-0.416250\pi$
$283$	147.207	0.520166	0.260083	+	0.965586i	$0.416250\pi$
$284$	229.628i	0.808551i
$285$	0	0
$286$	20.7553	0.0725711
$287$	101.338i	0.353093i
$288$	0	0
$289$	246.068	0.851448
$290$	0	0
$291$	0	0
$292$	−272.426	−0.932964
$293$	− 426.397i	− 1.45528i	−0.685958	−	0.727641i	$-0.740617\pi$
$293$	− 426.397i	− 1.45528i	0.685958	−	0.727641i	$-0.259383\pi$
$294$	0	0
$295$	0	0
$296$	− 204.891i	− 0.692200i
$297$	0	0
$298$	174.793	0.586552
$299$	− 0.442113i	− 0.00147864i
$300$	0	0
$301$	497.451	1.65266
$302$	72.6006i	0.240399i
$303$	0	0
$304$	136.677	0.449597
$305$	0	0
$306$	0	0
$307$	231.780	0.754982	0.377491	−	0.926013i	$-0.376787\pi$
$307$	231.780	0.754982	0.377491	+	0.926013i	$0.376787\pi$
$308$	− 151.652i	− 0.492376i
$309$	0	0
$310$	0	0
$311$	277.755i	0.893104i	0.894758	+	0.446552i	$0.147348\pi$
$311$	277.755i	0.893104i	−0.894758	+	0.446552i	$0.852652\pi$
$312$	0	0
$313$	557.662	1.78167	0.890834	−	0.454330i	$-0.150121\pi$
$313$	557.662	1.78167	0.890834	+	0.454330i	$0.150121\pi$
$314$	53.8907i	0.171627i
$315$	0	0
$316$	−23.1495	−0.0732580
$317$	− 443.318i	− 1.39848i	−0.714887	−	0.699240i	$-0.753522\pi$
$317$	− 443.318i	− 1.39848i	0.714887	−	0.699240i	$-0.246478\pi$
$318$	0	0
$319$	−155.425	−0.487225
$320$	0	0
$321$	0	0
$322$	0.730707	0.00226928
$323$	116.458i	0.360551i
$324$	0	0
$325$	0	0
$326$	− 0.531090i	− 0.00162911i
$327$	0	0
$328$	100.352	0.305953
$329$	149.452i	0.454261i
$330$	0	0
$331$	−453.887	−1.37126	−0.685629	−	0.727951i	$-0.740473\pi$
$331$	−453.887	−1.37126	−0.685629	+	0.727951i	$0.740473\pi$
$332$	431.714i	1.30034i
$333$	0	0
$334$	87.8080	0.262898
$335$	0	0
$336$	0	0
$337$	230.806	0.684885	0.342442	−	0.939539i	$-0.388746\pi$
$337$	230.806	0.684885	0.342442	+	0.939539i	$0.388746\pi$
$338$	− 135.964i	− 0.402259i
$339$	0	0
$340$	0	0
$341$	− 133.113i	− 0.390361i
$342$	0	0
$343$	367.365	1.07103
$344$	− 492.615i	− 1.43202i
$345$	0	0
$346$	−222.890	−0.644191
$347$	− 170.231i	− 0.490580i	−0.969450	−	0.245290i	$-0.921117\pi$
$347$	− 170.231i	− 0.490580i	0.969450	−	0.245290i	$-0.0788832\pi$
$348$	0	0
$349$	91.1993	0.261316	0.130658	−	0.991427i	$-0.458291\pi$
$349$	91.1993	0.261316	0.130658	+	0.991427i	$0.458291\pi$
$350$	0	0
$351$	0	0
$352$	−232.896	−0.661635
$353$	− 95.1576i	− 0.269568i	−0.990875	−	0.134784i	$-0.956966\pi$
$353$	− 95.1576i	− 0.269568i	0.990875	−	0.134784i	$-0.0430341\pi$
$354$	0	0
$355$	0	0
$356$	− 157.688i	− 0.442945i
$357$	0	0
$358$	−52.3782	−0.146308
$359$	670.526i	1.86776i	0.357585	+	0.933881i	$0.383600\pi$
$359$	670.526i	1.86776i	−0.357585	+	0.933881i	$0.616400\pi$
$360$	0	0
$361$	−45.0908	−0.124905
$362$	− 123.784i	− 0.341945i
$363$	0	0
$364$	67.2802	0.184836
$365$	0	0
$366$	0	0
$367$	505.765	1.37811	0.689053	−	0.724711i	$-0.258027\pi$
$367$	505.765	1.37811	0.689053	+	0.724711i	$0.258027\pi$
$368$	1.03839i	0.00282172i
$369$	0	0
$370$	0	0
$371$	− 498.120i	− 1.34264i
$372$	0	0
$373$	518.596	1.39034	0.695168	−	0.718847i	$-0.255330\pi$
$373$	518.596	1.39034	0.695168	+	0.718847i	$0.255330\pi$
$374$	− 41.5366i	− 0.111060i
$375$	0	0
$376$	147.999	0.393614
$377$	− 68.9542i	− 0.182902i
$378$	0	0
$379$	−173.866	−0.458750	−0.229375	−	0.973338i	$-0.573668\pi$
$379$	−173.866	−0.458750	−0.229375	+	0.973338i	$0.573668\pi$
$380$	0	0
$381$	0	0
$382$	−5.52174	−0.0144548
$383$	− 146.419i	− 0.382295i	−0.981561	−	0.191147i	$-0.938779\pi$
$383$	− 146.419i	− 0.382295i	0.981561	−	0.191147i	$-0.0612208\pi$
$384$	0	0
$385$	0	0
$386$	163.296i	0.423047i
$387$	0	0
$388$	161.870	0.417192
$389$	762.839i	1.96103i	0.196454	+	0.980513i	$0.437057\pi$
$389$	762.839i	1.96103i	−0.196454	+	0.980513i	$0.562943\pi$
$390$	0	0
$391$	−0.884778	−0.00226286
$392$	− 58.1221i	− 0.148271i
$393$	0	0
$394$	−255.032	−0.647289
$395$	0	0
$396$	0	0
$397$	293.403	0.739050	0.369525	−	0.929221i	$-0.379520\pi$
$397$	293.403	0.739050	0.369525	+	0.929221i	$0.379520\pi$
$398$	− 233.456i	− 0.586573i
$399$	0	0
$400$	0	0
$401$	677.605i	1.68979i	0.534934	+	0.844894i	$0.320336\pi$
$401$	677.605i	1.68979i	−0.534934	+	0.844894i	$0.679664\pi$
$402$	0	0
$403$	59.0556	0.146540
$404$	472.156i	1.16870i
$405$	0	0
$406$	113.965	0.280701
$407$	− 242.388i	− 0.595549i
$408$	0	0
$409$	346.229	0.846526	0.423263	−	0.906007i	$-0.360885\pi$
$409$	346.229	0.846526	0.423263	+	0.906007i	$0.360885\pi$
$410$	0	0
$411$	0	0
$412$	−617.943	−1.49986
$413$	− 509.662i	− 1.23405i
$414$	0	0
$415$	0	0
$416$	− 103.324i	− 0.248375i
$417$	0	0
$418$	−112.674	−0.269555
$419$	− 623.253i	− 1.48748i	−0.668471	−	0.743738i	$-0.733051\pi$
$419$	− 623.253i	− 1.48748i	0.668471	−	0.743738i	$-0.266949\pi$
$420$	0	0
$421$	−605.778	−1.43890	−0.719451	−	0.694543i	$-0.755607\pi$
$421$	−605.778	−1.43890	−0.719451	+	0.694543i	$0.755607\pi$
$422$	223.134i	0.528753i
$423$	0	0
$424$	−493.277	−1.16339
$425$	0	0
$426$	0	0
$427$	153.315	0.359053
$428$	1.94592i	0.00454653i
$429$	0	0
$430$	0	0
$431$	− 349.153i	− 0.810100i	−0.914295	−	0.405050i	$-0.867254\pi$
$431$	− 349.153i	− 0.810100i	0.914295	−	0.405050i	$-0.132746\pi$
$432$	0	0
$433$	644.491	1.48843	0.744216	−	0.667939i	$-0.232823\pi$
$433$	644.491	1.48843	0.744216	+	0.667939i	$0.232823\pi$
$434$	97.6047i	0.224896i
$435$	0	0
$436$	583.646	1.33864
$437$	2.40009i	0.00549219i
$438$	0	0
$439$	335.797	0.764913	0.382457	−	0.923973i	$-0.375078\pi$
$439$	335.797	0.764913	0.382457	+	0.923973i	$0.375078\pi$
$440$	0	0
$441$	0	0
$442$	18.4277	0.0416916
$443$	− 159.172i	− 0.359304i	−0.983730	−	0.179652i	$-0.942503\pi$
$443$	− 159.172i	− 0.359304i	0.983730	−	0.179652i	$-0.0574971\pi$
$444$	0	0
$445$	0	0
$446$	− 66.3369i	− 0.148737i
$447$	0	0
$448$	−22.9960	−0.0513303
$449$	− 312.798i	− 0.696654i	−0.937373	−	0.348327i	$-0.886750\pi$
$449$	− 312.798i	− 0.696654i	0.937373	−	0.348327i	$-0.113250\pi$
$450$	0	0
$451$	118.718	0.263233
$452$	122.045i	0.270012i
$453$	0	0
$454$	169.119	0.372510
$455$	0	0
$456$	0	0
$457$	−239.782	−0.524686	−0.262343	−	0.964975i	$-0.584495\pi$
$457$	−239.782	−0.524686	−0.262343	+	0.964975i	$0.584495\pi$
$458$	94.9717i	0.207362i
$459$	0	0
$460$	0	0
$461$	619.397i	1.34359i	0.740735	+	0.671797i	$0.234477\pi$
$461$	619.397i	1.34359i	−0.740735	+	0.671797i	$0.765523\pi$
$462$	0	0
$463$	0.510937	0.00110354	0.000551768	−	1.00000i	$-0.499824\pi$
$463$	0.510937	0.00110354	0.000551768		1.00000i	$0.499824\pi$
$464$	161.953i	0.349037i
$465$	0	0
$466$	198.440	0.425837
$467$	627.526i	1.34374i	0.740669	+	0.671870i	$0.234508\pi$
$467$	627.526i	1.34374i	−0.740669	+	0.671870i	$0.765492\pi$
$468$	0	0
$469$	−636.254	−1.35662
$470$	0	0
$471$	0	0
$472$	−504.707	−1.06929
$473$	− 582.769i	− 1.23207i
$474$	0	0
$475$	0	0
$476$	− 134.644i	− 0.282866i
$477$	0	0
$478$	24.9347	0.0521646
$479$	56.3083i	0.117554i	0.998271	+	0.0587770i	$0.0187201\pi$
$479$	56.3083i	0.117554i	−0.998271	+	0.0587770i	$0.981280\pi$
$480$	0	0
$481$	107.536	0.223567
$482$	95.5720i	0.198282i
$483$	0	0
$484$	217.054	0.448459
$485$	0	0
$486$	0	0
$487$	−364.899	−0.749279	−0.374640	−	0.927171i	$-0.622234\pi$
$487$	−364.899	−0.749279	−0.374640	+	0.927171i	$0.622234\pi$
$488$	− 151.825i	− 0.311117i
$489$	0	0
$490$	0	0
$491$	256.588i	0.522582i	0.965260	+	0.261291i	$0.0841482\pi$
$491$	256.588i	0.522582i	−0.965260	+	0.261291i	$0.915852\pi$
$492$	0	0
$493$	−137.994	−0.279908
$494$	− 49.9878i	− 0.101190i
$495$	0	0
$496$	−138.704	−0.279646
$497$	− 443.433i	− 0.892220i
$498$	0	0
$499$	91.3613	0.183089	0.0915444	−	0.995801i	$-0.470820\pi$
$499$	91.3613	0.183089	0.0915444	+	0.995801i	$0.470820\pi$
$500$	0	0
$501$	0	0
$502$	−313.330	−0.624163
$503$	322.707i	0.641565i	0.947153	+	0.320783i	$0.103946\pi$
$503$	322.707i	0.641565i	−0.947153	+	0.320783i	$0.896054\pi$
$504$	0	0
$505$	0	0
$506$	− 0.856030i	− 0.00169176i
$507$	0	0
$508$	262.105	0.515956
$509$	427.435i	0.839755i	0.907581	+	0.419877i	$0.137927\pi$
$509$	427.435i	0.839755i	−0.907581	+	0.419877i	$0.862073\pi$
$510$	0	0
$511$	526.078	1.02951
$512$	434.560i	0.848749i
$513$	0	0
$514$	415.792	0.808934
$515$	0	0
$516$	0	0
$517$	175.084	0.338654
$518$	177.730i	0.343109i
$519$	0	0
$520$	0	0
$521$	− 674.311i	− 1.29426i	−0.762379	−	0.647131i	$-0.775969\pi$
$521$	− 674.311i	− 1.29426i	0.762379	−	0.647131i	$-0.224031\pi$
$522$	0	0
$523$	−137.540	−0.262984	−0.131492	−	0.991317i	$-0.541977\pi$
$523$	−137.540	−0.262984	−0.131492	+	0.991317i	$0.541977\pi$
$524$	− 593.367i	− 1.13238i
$525$	0	0
$526$	56.6552	0.107709
$527$	− 118.185i	− 0.224260i
$528$	0	0
$529$	528.982	0.999966
$530$	0	0
$531$	0	0
$532$	−365.242	−0.686546
$533$	52.6692i	0.0988165i
$534$	0	0
$535$	0	0
$536$	630.068i	1.17550i
$537$	0	0
$538$	355.231	0.660280
$539$	− 68.7591i	− 0.127568i
$540$	0	0
$541$	−729.719	−1.34883	−0.674417	−	0.738351i	$-0.735605\pi$
$541$	−729.719	−1.34883	−0.674417	+	0.738351i	$0.735605\pi$
$542$	− 106.263i	− 0.196057i
$543$	0	0
$544$	−206.777	−0.380105
$545$	0	0
$546$	0	0
$547$	539.685	0.986626	0.493313	−	0.869852i	$-0.335786\pi$
$547$	539.685	0.986626	0.493313	+	0.869852i	$0.335786\pi$
$548$	− 137.964i	− 0.251758i
$549$	0	0
$550$	0	0
$551$	374.330i	0.679365i
$552$	0	0
$553$	44.7039	0.0808388
$554$	308.612i	0.557062i
$555$	0	0
$556$	−455.754	−0.819702
$557$	1015.38i	1.82295i	0.411354	+	0.911476i	$0.365056\pi$
$557$	1015.38i	1.82295i	−0.411354	+	0.911476i	$0.634944\pi$
$558$	0	0
$559$	258.545	0.462513
$560$	0	0
$561$	0	0
$562$	215.646	0.383712
$563$	865.244i	1.53685i	0.639942	+	0.768423i	$0.278958\pi$
$563$	865.244i	1.53685i	−0.639942	+	0.768423i	$0.721042\pi$
$564$	0	0
$565$	0	0
$566$	126.451i	0.223412i
$567$	0	0
$568$	−439.122	−0.773102
$569$	− 853.749i	− 1.50044i	−0.661189	−	0.750219i	$-0.729948\pi$
$569$	− 853.749i	− 1.50044i	0.661189	−	0.750219i	$-0.270052\pi$
$570$	0	0
$571$	−233.600	−0.409107	−0.204553	−	0.978855i	$-0.565574\pi$
$571$	−233.600	−0.409107	−0.204553	+	0.978855i	$0.565574\pi$
$572$	− 78.8194i	− 0.137796i
$573$	0	0
$574$	−87.0496	−0.151654
$575$	0	0
$576$	0	0
$577$	104.108	0.180431	0.0902153	−	0.995922i	$-0.471244\pi$
$577$	104.108	0.180431	0.0902153	+	0.995922i	$0.471244\pi$
$578$	211.374i	0.365699i
$579$	0	0
$580$	0	0
$581$	− 833.679i	− 1.43490i
$582$	0	0
$583$	−583.552	−1.00095
$584$	− 520.964i	− 0.892061i
$585$	0	0
$586$	366.277	0.625047
$587$	− 1089.94i	− 1.85679i	−0.371589	−	0.928397i	$-0.621187\pi$
$587$	− 1089.94i	− 1.85679i	0.371589	−	0.928397i	$-0.378813\pi$
$588$	0	0
$589$	−320.594	−0.544302
$590$	0	0
$591$	0	0
$592$	−252.569	−0.426637
$593$	− 138.078i	− 0.232847i	−0.993200	−	0.116423i	$-0.962857\pi$
$593$	− 138.078i	− 0.232847i	0.993200	−	0.116423i	$-0.0371430\pi$
$594$	0	0
$595$	0	0
$596$	− 663.783i	− 1.11373i
$597$	0	0
$598$	0.379777	0.000635079 0
$599$	− 242.905i	− 0.405517i	−0.979229	−	0.202758i	$-0.935009\pi$
$599$	− 242.905i	− 0.405517i	0.979229	−	0.202758i	$-0.0649906\pi$
$600$	0	0
$601$	504.058	0.838699	0.419349	−	0.907825i	$-0.362258\pi$
$601$	504.058	0.838699	0.419349	+	0.907825i	$0.362258\pi$
$602$	427.313i	0.709822i
$603$	0	0
$604$	275.704	0.456464
$605$	0	0
$606$	0	0
$607$	376.002	0.619443	0.309721	−	0.950827i	$-0.399764\pi$
$607$	376.002	0.619443	0.309721	+	0.950827i	$0.399764\pi$
$608$	560.913i	0.922554i
$609$	0	0
$610$	0	0
$611$	77.6760i	0.127129i
$612$	0	0
$613$	−708.555	−1.15588	−0.577940	−	0.816079i	$-0.696143\pi$
$613$	−708.555	−1.15588	−0.577940	+	0.816079i	$0.696143\pi$
$614$	199.100i	0.324267i
$615$	0	0
$616$	290.006	0.470789
$617$	− 208.972i	− 0.338690i	−0.985557	−	0.169345i	$-0.945835\pi$
$617$	− 208.972i	− 0.338690i	0.985557	−	0.169345i	$-0.0541652\pi$
$618$	0	0
$619$	−672.298	−1.08610	−0.543052	−	0.839699i	$-0.682731\pi$
$619$	−672.298	−1.08610	−0.543052	+	0.839699i	$0.682731\pi$
$620$	0	0
$621$	0	0
$622$	−238.593	−0.383590
$623$	304.511i	0.488781i
$624$	0	0
$625$	0	0
$626$	479.034i	0.765230i
$627$	0	0
$628$	204.653	0.325880
$629$	− 215.205i	− 0.342139i
$630$	0	0
$631$	−733.124	−1.16185	−0.580923	−	0.813959i	$-0.697308\pi$
$631$	−733.124	−1.16185	−0.580923	+	0.813959i	$0.697308\pi$
$632$	− 44.2692i	− 0.0700463i
$633$	0	0
$634$	380.812	0.600650
$635$	0	0
$636$	0	0
$637$	30.5049	0.0478884
$638$	− 133.511i	− 0.209265i
$639$	0	0
$640$	0	0
$641$	− 547.868i	− 0.854708i	−0.904084	−	0.427354i	$-0.859446\pi$
$641$	− 547.868i	− 0.854708i	0.904084	−	0.427354i	$-0.140554\pi$
$642$	0	0
$643$	−497.012	−0.772958	−0.386479	−	0.922298i	$-0.626309\pi$
$643$	−497.012	−0.772958	−0.386479	+	0.922298i	$0.626309\pi$
$644$	− 2.77489i	− 0.00430884i
$645$	0	0
$646$	−100.038	−0.154858
$647$	95.8316i	0.148117i	0.997254	+	0.0740584i	$0.0235951\pi$
$647$	95.8316i	0.148117i	−0.997254	+	0.0740584i	$0.976405\pi$
$648$	0	0
$649$	−597.074	−0.919990
$650$	0	0
$651$	0	0
$652$	−2.01684	−0.00309331
$653$	− 56.9896i	− 0.0872735i	−0.999047	−	0.0436367i	$-0.986106\pi$
$653$	− 56.9896i	− 0.0872735i	0.999047	−	0.0436367i	$-0.0138944\pi$
$654$	0	0
$655$	0	0
$656$	− 123.704i	− 0.188574i
$657$	0	0
$658$	−128.380	−0.195106
$659$	680.817i	1.03311i	0.856255	+	0.516553i	$0.172785\pi$
$659$	680.817i	1.03311i	−0.856255	+	0.516553i	$0.827215\pi$
$660$	0	0
$661$	−86.6374	−0.131070	−0.0655351	−	0.997850i	$-0.520875\pi$
$661$	−86.6374	−0.131070	−0.0655351	+	0.997850i	$0.520875\pi$
$662$	− 389.891i	− 0.588959i
$663$	0	0
$664$	−825.574	−1.24333
$665$	0	0
$666$	0	0
$667$	−2.84394	−0.00426377
$668$	− 333.455i	− 0.499184i
$669$	0	0
$670$	0	0
$671$	− 179.610i	− 0.267676i
$672$	0	0
$673$	−453.328	−0.673593	−0.336796	−	0.941578i	$-0.609343\pi$
$673$	−453.328	−0.673593	−0.336796	+	0.941578i	$0.609343\pi$
$674$	198.264i	0.294160i
$675$	0	0
$676$	−516.329	−0.763800
$677$	− 751.020i	− 1.10934i	−0.832072	−	0.554668i	$-0.812845\pi$
$677$	− 751.020i	− 1.10934i	0.832072	−	0.554668i	$-0.187155\pi$
$678$	0	0
$679$	−312.587	−0.460363
$680$	0	0
$681$	0	0
$682$	114.345	0.167661
$683$	538.016i	0.787725i	0.919169	+	0.393863i	$0.128861\pi$
$683$	538.016i	0.787725i	−0.919169	+	0.393863i	$0.871139\pi$
$684$	0	0
$685$	0	0
$686$	315.568i	0.460012i
$687$	0	0
$688$	−607.246	−0.882625
$689$	− 258.892i	− 0.375751i
$690$	0	0
$691$	−240.776	−0.348446	−0.174223	−	0.984706i	$-0.555741\pi$
$691$	−240.776	−0.348446	−0.174223	+	0.984706i	$0.555741\pi$
$692$	846.435i	1.22317i
$693$	0	0
$694$	146.230	0.210705
$695$	0	0
$696$	0	0
$697$	105.404	0.151225
$698$	78.3407i	0.112236i
$699$	0	0
$700$	0	0
$701$	779.611i	1.11214i	0.831135	+	0.556070i	$0.187692\pi$
$701$	779.611i	1.11214i	−0.831135	+	0.556070i	$0.812308\pi$
$702$	0	0
$703$	−583.776	−0.830406
$704$	26.9400i	0.0382671i
$705$	0	0
$706$	81.7408	0.115780
$707$	− 911.776i	− 1.28964i
$708$	0	0
$709$	−124.053	−0.174970	−0.0874848	−	0.996166i	$-0.527883\pi$
$709$	−124.053	−0.174970	−0.0874848	+	0.996166i	$0.527883\pi$
$710$	0	0
$711$	0	0
$712$	301.550	0.423526
$713$	− 2.43568i	− 0.00341610i
$714$	0	0
$715$	0	0
$716$	198.909i	0.277805i
$717$	0	0
$718$	−575.985	−0.802208
$719$	− 734.460i	− 1.02150i	−0.859729	−	0.510751i	$-0.829367\pi$
$719$	− 734.460i	− 1.02150i	0.859729	−	0.510751i	$-0.170633\pi$
$720$	0	0
$721$	1193.30	1.65507
$722$	− 38.7332i	− 0.0536471i
$723$	0	0
$724$	−470.076	−0.649276
$725$	0	0
$726$	0	0
$727$	390.967	0.537781	0.268891	−	0.963171i	$-0.413343\pi$
$727$	390.967	0.537781	0.268891	+	0.963171i	$0.413343\pi$
$728$	128.661i	0.176732i
$729$	0	0
$730$	0	0
$731$	− 517.413i	− 0.707815i
$732$	0	0
$733$	175.190	0.239004	0.119502	−	0.992834i	$-0.461870\pi$
$733$	175.190	0.239004	0.119502	+	0.992834i	$0.461870\pi$
$734$	434.455i	0.591900i
$735$	0	0
$736$	−4.26148	−0.00579005
$737$	745.378i	1.01137i
$738$	0	0
$739$	745.922	1.00937	0.504683	−	0.863305i	$-0.331609\pi$
$739$	745.922	1.00937	0.504683	+	0.863305i	$0.331609\pi$
$740$	0	0
$741$	0	0
$742$	427.887	0.576667
$743$	1249.70i	1.68196i	0.541065	+	0.840981i	$0.318021\pi$
$743$	1249.70i	1.68196i	−0.541065	+	0.840981i	$0.681979\pi$
$744$	0	0
$745$	0	0
$746$	445.476i	0.597153i
$747$	0	0
$748$	−157.737	−0.210879
$749$	− 3.75774i	− 0.00501701i
$750$	0	0
$751$	1220.63	1.62534	0.812672	−	0.582721i	$-0.198012\pi$
$751$	1220.63	1.62534	0.812672	+	0.582721i	$0.198012\pi$
$752$	− 182.438i	− 0.242604i
$753$	0	0
$754$	59.2320	0.0785570
$755$	0	0
$756$	0	0
$757$	672.349	0.888175	0.444088	−	0.895983i	$-0.353528\pi$
$757$	672.349	0.888175	0.444088	+	0.895983i	$0.353528\pi$
$758$	− 149.352i	− 0.197034i
$759$	0	0
$760$	0	0
$761$	− 565.183i	− 0.742684i	−0.928496	−	0.371342i	$-0.878898\pi$
$761$	− 565.183i	− 0.742684i	0.928496	−	0.371342i	$-0.121102\pi$
$762$	0	0
$763$	−1127.07	−1.47716
$764$	20.9691i	0.0274464i
$765$	0	0
$766$	125.774	0.164196
$767$	− 264.891i	− 0.345360i
$768$	0	0
$769$	−884.684	−1.15043	−0.575217	−	0.818001i	$-0.695083\pi$
$769$	−884.684	−1.15043	−0.575217	+	0.818001i	$0.695083\pi$
$770$	0	0
$771$	0	0
$772$	620.125	0.803270
$773$	963.181i	1.24603i	0.782210	+	0.623015i	$0.214092\pi$
$773$	963.181i	1.24603i	−0.782210	+	0.623015i	$0.785908\pi$
$774$	0	0
$775$	0	0
$776$	309.547i	0.398901i
$777$	0	0
$778$	−655.282	−0.842265
$779$	− 285.924i	− 0.367040i
$780$	0	0
$781$	−519.486	−0.665155
$782$	− 0.760028i	0 0.000971903i
$783$	0	0
$784$	−71.6471	−0.0913866
$785$	0	0
$786$	0	0
$787$	557.992	0.709011	0.354505	−	0.935054i	$-0.384649\pi$
$787$	557.992	0.709011	0.354505	+	0.935054i	$0.384649\pi$
$788$	968.495i	1.22906i
$789$	0	0
$790$	0	0
$791$	− 235.680i	− 0.297952i
$792$	0	0
$793$	79.6841	0.100484
$794$	252.035i	0.317424i
$795$	0	0
$796$	−886.561	−1.11377
$797$	− 588.584i	− 0.738500i	−0.929330	−	0.369250i	$-0.879615\pi$
$797$	− 588.584i	− 0.738500i	0.929330	−	0.369250i	$-0.120385\pi$
$798$	0	0
$799$	155.449	0.194554
$800$	0	0
$801$	0	0
$802$	−582.066	−0.725768
$803$	− 616.306i	− 0.767504i
$804$	0	0
$805$	0	0
$806$	50.7290i	0.0629392i
$807$	0	0
$808$	−902.911	−1.11746
$809$	1256.74i	1.55344i	0.629844	+	0.776722i	$0.283119\pi$
$809$	1256.74i	1.55344i	−0.629844	+	0.776722i	$0.716881\pi$
$810$	0	0
$811$	−534.786	−0.659416	−0.329708	−	0.944083i	$-0.606950\pi$
$811$	−534.786	−0.659416	−0.329708	+	0.944083i	$0.606950\pi$
$812$	− 432.787i	− 0.532988i
$813$	0	0
$814$	208.213	0.255790
$815$	0	0
$816$	0	0
$817$	−1403.56	−1.71794
$818$	297.412i	0.363585i
$819$	0	0
$820$	0	0
$821$	1134.73i	1.38213i	0.722795	+	0.691063i	$0.242857\pi$
$821$	1134.73i	1.38213i	−0.722795	+	0.691063i	$0.757143\pi$
$822$	0	0
$823$	−1327.75	−1.61330	−0.806651	−	0.591028i	$-0.798722\pi$
$823$	−1327.75	−1.61330	−0.806651	+	0.591028i	$0.798722\pi$
$824$	− 1181.70i	− 1.43411i
$825$	0	0
$826$	437.802	0.530027
$827$	1063.68i	1.28619i	0.765785	+	0.643097i	$0.222351\pi$
$827$	1063.68i	1.28619i	−0.765785	+	0.643097i	$0.777649\pi$
$828$	0	0
$829$	−14.9731	−0.0180616	−0.00903081	−	0.999959i	$-0.502875\pi$
$829$	−14.9731	−0.0180616	−0.00903081	+	0.999959i	$0.502875\pi$
$830$	0	0
$831$	0	0
$832$	−11.9519	−0.0143653
$833$	− 61.0480i	− 0.0732869i
$834$	0	0
$835$	0	0
$836$	427.885i	0.511824i
$837$	0	0
$838$	535.377	0.638875
$839$	− 513.250i	− 0.611740i	−0.952073	−	0.305870i	$-0.901053\pi$
$839$	− 513.250i	− 0.611740i	0.952073	−	0.305870i	$-0.0989473\pi$
$840$	0	0
$841$	397.445	0.472587
$842$	− 520.366i	− 0.618012i
$843$	0	0
$844$	847.362	1.00398
$845$	0	0
$846$	0	0
$847$	−419.151	−0.494866
$848$	608.062i	0.717054i
$849$	0	0
$850$	0	0
$851$	− 4.43518i	− 0.00521172i
$852$	0	0
$853$	1536.78	1.80162	0.900811	−	0.434212i	$-0.142973\pi$
$853$	1536.78	1.80162	0.900811	+	0.434212i	$0.142973\pi$
$854$	131.699i	0.154214i
$855$	0	0
$856$	−3.72121	−0.00434720
$857$	− 355.426i	− 0.414733i	−0.978263	−	0.207366i	$-0.933511\pi$
$857$	− 355.426i	− 0.414733i	0.978263	−	0.207366i	$-0.0664892\pi$
$858$	0	0
$859$	1348.95	1.57038	0.785188	−	0.619258i	$-0.212566\pi$
$859$	1348.95	1.57038	0.785188	+	0.619258i	$0.212566\pi$
$860$	0	0
$861$	0	0
$862$	299.924	0.347940
$863$	1284.29i	1.48817i	0.668088	+	0.744083i	$0.267113\pi$
$863$	1284.29i	1.48817i	−0.668088	+	0.744083i	$0.732887\pi$
$864$	0	0
$865$	0	0
$866$	553.621i	0.639285i
$867$	0	0
$868$	370.659	0.427026
$869$	− 52.3710i	− 0.0602658i
$870$	0	0
$871$	−330.686	−0.379663
$872$	1116.12i	1.27995i
$873$	0	0
$874$	−2.06169	−0.00235891
$875$	0	0
$876$	0	0
$877$	−460.304	−0.524862	−0.262431	−	0.964951i	$-0.584524\pi$
$877$	−460.304	−0.524862	−0.262431	+	0.964951i	$0.584524\pi$
$878$	288.451i	0.328532i
$879$	0	0
$880$	0	0
$881$	− 926.501i	− 1.05165i	−0.850594	−	0.525823i	$-0.823757\pi$
$881$	− 926.501i	− 1.05165i	0.850594	−	0.525823i	$-0.176243\pi$
$882$	0	0
$883$	−1444.20	−1.63556	−0.817782	−	0.575528i	$-0.804797\pi$
$883$	−1444.20	−1.63556	−0.817782	+	0.575528i	$0.804797\pi$
$884$	− 69.9800i	− 0.0791629i
$885$	0	0
$886$	136.729	0.154322
$887$	806.959i	0.909762i	0.890552	+	0.454881i	$0.150318\pi$
$887$	806.959i	0.909762i	−0.890552	+	0.454881i	$0.849682\pi$
$888$	0	0
$889$	−506.150	−0.569347
$890$	0	0
$891$	0	0
$892$	−251.917	−0.282419
$893$	− 421.678i	− 0.472204i
$894$	0	0
$895$	0	0
$896$	− 814.952i	− 0.909545i
$897$	0	0
$898$	268.695	0.299215
$899$	− 379.881i	− 0.422559i
$900$	0	0
$901$	−518.108	−0.575037
$902$	101.979i	0.113059i
$903$	0	0
$904$	−233.389	−0.258174
$905$	0	0
$906$	0	0
$907$	−819.986	−0.904064	−0.452032	−	0.892002i	$-0.649301\pi$
$907$	−819.986	−0.904064	−0.452032	+	0.892002i	$0.649301\pi$
$908$	− 642.239i	− 0.707311i
$909$	0	0
$910$	0	0
$911$	− 1037.96i	− 1.13936i	−0.821865	−	0.569682i	$-0.807066\pi$
$911$	− 1037.96i	− 1.13936i	0.821865	−	0.569682i	$-0.192934\pi$
$912$	0	0
$913$	−976.663	−1.06973
$914$	− 205.974i	− 0.225354i
$915$	0	0
$916$	360.659	0.393733
$917$	1145.84i	1.24956i
$918$	0	0
$919$	996.753	1.08461	0.542303	−	0.840183i	$-0.317552\pi$
$919$	996.753	1.08461	0.542303	+	0.840183i	$0.317552\pi$
$920$	0	0
$921$	0	0
$922$	−532.065	−0.577077
$923$	− 230.470i	− 0.249696i
$924$	0	0
$925$	0	0
$926$	0.438897i	0 0.000473971i
$927$	0	0
$928$	−664.642	−0.716209
$929$	588.005i	0.632944i	0.948602	+	0.316472i	$0.102498\pi$
$929$	588.005i	0.632944i	−0.948602	+	0.316472i	$0.897502\pi$
$930$	0	0
$931$	−165.601	−0.177875
$932$	− 753.586i	− 0.808569i
$933$	0	0
$934$	−539.048	−0.577139
$935$	0	0
$936$	0	0
$937$	724.502	0.773215	0.386607	−	0.922244i	$-0.373647\pi$
$937$	724.502	0.773215	0.386607	+	0.922244i	$0.373647\pi$
$938$	− 546.545i	− 0.582671i
$939$	0	0
$940$	0	0
$941$	− 528.469i	− 0.561604i	−0.959766	−	0.280802i	$-0.909400\pi$
$941$	− 528.469i	− 0.561604i	0.959766	−	0.280802i	$-0.0906004\pi$
$942$	0	0
$943$	2.17228	0.00230358
$944$	622.152i	0.659059i
$945$	0	0
$946$	500.601	0.529177
$947$	− 878.450i	− 0.927614i	−0.885936	−	0.463807i	$-0.846483\pi$
$947$	− 878.450i	− 0.927614i	0.885936	−	0.463807i	$-0.153517\pi$
$948$	0	0
$949$	273.424	0.288118
$950$	0	0
$951$	0	0
$952$	257.483	0.270465
$953$	422.690i	0.443536i	0.975099	+	0.221768i	$0.0711828\pi$
$953$	422.690i	0.443536i	−0.975099	+	0.221768i	$0.928817\pi$
$954$	0	0
$955$	0	0
$956$	− 94.6906i	− 0.0990487i
$957$	0	0
$958$	−48.3691	−0.0504897
$959$	266.420i	0.277811i
$960$	0	0
$961$	−635.652	−0.661449
$962$	92.3735i	0.0960224i
$963$	0	0
$964$	362.939	0.376493
$965$	0	0
$966$	0	0
$967$	−1394.75	−1.44234	−0.721172	−	0.692756i	$-0.756396\pi$
$967$	−1394.75	−1.44234	−0.721172	+	0.692756i	$0.756396\pi$
$968$	415.076i	0.428798i
$969$	0	0
$970$	0	0
$971$	620.475i	0.639006i	0.947585	+	0.319503i	$0.103516\pi$
$971$	620.475i	0.639006i	−0.947585	+	0.319503i	$0.896484\pi$
$972$	0	0
$973$	880.103	0.904525
$974$	− 313.450i	− 0.321817i
$975$	0	0
$976$	−187.154	−0.191757
$977$	− 24.9244i	− 0.0255112i	−0.999919	−	0.0127556i	$-0.995940\pi$
$977$	− 24.9244i	− 0.0255112i	0.999919	−	0.0127556i	$-0.00406034\pi$
$978$	0	0
$979$	356.737	0.364389
$980$	0	0
$981$	0	0
$982$	−220.410	−0.224450
$983$	38.9442i	0.0396177i	0.999804	+	0.0198089i	$0.00630577\pi$
$983$	38.9442i	0.0396177i	−0.999804	+	0.0198089i	$0.993694\pi$
$984$	0	0
$985$	0	0
$986$	− 118.538i	− 0.120221i
$987$	0	0
$988$	−189.831	−0.192137
$989$	− 10.6634i	− 0.0107820i
$990$	0	0
$991$	−80.4596	−0.0811903	−0.0405951	−	0.999176i	$-0.512925\pi$
$991$	−80.4596	−0.0811903	−0.0405951	+	0.999176i	$0.512925\pi$
$992$	− 569.230i	− 0.573821i
$993$	0	0
$994$	380.911	0.383210
$995$	0	0
$996$	0	0
$997$	655.418	0.657390	0.328695	−	0.944436i	$-0.393391\pi$
$997$	655.418	0.657390	0.328695	+	0.944436i	$0.393391\pi$
$998$	78.4798i	0.0786371i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1125.3.c.a.251.10	yes	32
3.2	odd	2	inner	1125.3.c.a.251.9	✓	32
5.2	odd	4		1125.3.d.b.1124.5		16
5.3	odd	4		1125.3.d.a.1124.12		16
5.4	even	2	inner	1125.3.c.a.251.23	yes	32
15.2	even	4		1125.3.d.a.1124.11		16
15.8	even	4		1125.3.d.b.1124.6		16
15.14	odd	2	inner	1125.3.c.a.251.24	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1125.3.c.a.251.9	✓	32	3.2	odd	2	inner
1125.3.c.a.251.10	yes	32	1.1	even	1	trivial
1125.3.c.a.251.23	yes	32	5.4	even	2	inner
1125.3.c.a.251.24	yes	32	15.14	odd	2	inner
1125.3.d.a.1124.11		16	15.2	even	4
1125.3.d.a.1124.12		16	5.3	odd	4
1125.3.d.b.1124.5		16	5.2	odd	4
1125.3.d.b.1124.6		16	15.8	even	4

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:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1125.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.859005i q^{2} +3.26211 q^{4} -6.29943 q^{7} +6.23819i q^{8} +7.37984i q^{11} -3.27406 q^{13} -5.41124i q^{14} +7.68981 q^{16} +6.55221i q^{17} +17.7738 q^{19} -6.33932 q^{22} +0.135035i q^{23} -2.81244i q^{26} -20.5494 q^{28} +21.0607i q^{29} -18.0374 q^{31} +31.5583i q^{32} -5.62838 q^{34} -32.8447 q^{37} +15.2678i q^{38} -16.0868i q^{41} -78.9676 q^{43} +24.0739i q^{44} -0.115996 q^{46} -23.7246i q^{47} -9.31715 q^{49} -10.6804 q^{52} +79.0737i q^{53} -39.2970i q^{56} -18.0913 q^{58} +80.9060i q^{59} -24.3380 q^{61} -15.4942i q^{62} +3.65049 q^{64} +101.002 q^{67} +21.3741i q^{68} +70.3926i q^{71} -83.5120 q^{73} -28.2137i q^{74} +57.9802 q^{76} -46.4888i q^{77} -7.09649 q^{79} +13.8186 q^{82} +132.342i q^{83} -67.8336i q^{86} -46.0368 q^{88} -48.3394i q^{89} +20.6247 q^{91} +0.440499i q^{92} +20.3796 q^{94} +49.6214 q^{97} -8.00348i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 76 q^{4} + 92 q^{16} - 80 q^{19} + 272 q^{31} + 184 q^{34} - 368 q^{46} + 64 q^{49} + 416 q^{61} + 68 q^{64} + 640 q^{76} - 528 q^{79} - 528 q^{91} - 548 q^{94}+O(q^{100}) \) 32 * q - 76 * q^4 + 92 * q^16 - 80 * q^19 + 272 * q^31 + 184 * q^34 - 368 * q^46 + 64 * q^49 + 416 * q^61 + 68 * q^64 + 640 * q^76 - 528 * q^79 - 528 * q^91 - 548 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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