LMFDB - Embedded newform 1143.3.b.a.890.70

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,3,Mod(890,1143)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1143.890");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1143.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.1444942164$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			890.70
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.89430i q^{2} -4.37696 q^{4} -0.368899i q^{5} -11.4242 q^{7} -1.09103i q^{8} +O(q^{10})$	\(q+2.89430i q^{2} -4.37696 q^{4} -0.368899i q^{5} -11.4242 q^{7} -1.09103i q^{8} +1.06770 q^{10} -12.4660i q^{11} -14.3568 q^{13} -33.0651i q^{14} -14.3501 q^{16} -4.03686i q^{17} +4.50638 q^{19} +1.61466i q^{20} +36.0802 q^{22} +24.7854i q^{23} +24.8639 q^{25} -41.5529i q^{26} +50.0034 q^{28} +40.2521i q^{29} -0.142872 q^{31} -45.8975i q^{32} +11.6839 q^{34} +4.21439i q^{35} +70.5507 q^{37} +13.0428i q^{38} -0.402480 q^{40} -72.7348i q^{41} +32.2116 q^{43} +54.5630i q^{44} -71.7363 q^{46} -29.8941i q^{47} +81.5130 q^{49} +71.9636i q^{50} +62.8392 q^{52} +46.0232i q^{53} -4.59868 q^{55} +12.4642i q^{56} -116.501 q^{58} -67.6830i q^{59} -40.9491 q^{61} -0.413515i q^{62} +75.4407 q^{64} +5.29621i q^{65} -13.7399 q^{67} +17.6692i q^{68} -12.1977 q^{70} +14.7337i q^{71} +131.028 q^{73} +204.195i q^{74} -19.7242 q^{76} +142.414i q^{77} +86.0968 q^{79} +5.29373i q^{80} +210.516 q^{82} -141.334i q^{83} -1.48920 q^{85} +93.2301i q^{86} -13.6007 q^{88} -136.262i q^{89} +164.016 q^{91} -108.485i q^{92} +86.5224 q^{94} -1.66240i q^{95} +22.3537 q^{97} +235.923i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 160 q^{4}+O(q^{10}) $ 84 * q - 160 * q^4	$ 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) $ 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.89430i	1.44715i	0.690246	+	0.723574i	$0.257502\pi$
$2$	2.89430i	1.44715i	−0.690246	+	0.723574i	$0.742498\pi$
$3$	0	0
$3$	0	0
$4$	−4.37696	−1.09424
$5$	− 0.368899i	− 0.0737798i	−0.999319	−	0.0368899i	$-0.988255\pi$
$5$	− 0.368899i	− 0.0737798i	0.999319	−	0.0368899i	$-0.0117451\pi$
$6$	0	0
$7$	−11.4242	−1.63203	−0.816016	−	0.578029i	$-0.803822\pi$
$7$	−11.4242	−1.63203	−0.816016	+	0.578029i	$0.803822\pi$
$8$	− 1.09103i	− 0.136379i
$9$	0	0
$10$	1.06770	0.106770
$11$	− 12.4660i	− 1.13327i	−0.823969	−	0.566635i	$-0.808245\pi$
$11$	− 12.4660i	− 1.13327i	0.823969	−	0.566635i	$-0.191755\pi$
$12$	0	0
$13$	−14.3568	−1.10437	−0.552185	−	0.833722i	$-0.686206\pi$
$13$	−14.3568	−1.10437	−0.552185	+	0.833722i	$0.686206\pi$
$14$	− 33.0651i	− 2.36179i
$15$	0	0
$16$	−14.3501	−0.896879
$17$	− 4.03686i	− 0.237463i	−0.992926	−	0.118731i	$-0.962117\pi$
$17$	− 4.03686i	− 0.237463i	0.992926	−	0.118731i	$-0.0378827\pi$
$18$	0	0
$19$	4.50638	0.237178	0.118589	−	0.992943i	$-0.462163\pi$
$19$	4.50638	0.237178	0.118589	+	0.992943i	$0.462163\pi$
$20$	1.61466i	0.0807328i
$21$	0	0
$22$	36.0802	1.64001
$23$	24.7854i	1.07763i	0.842425	+	0.538813i	$0.181127\pi$
$23$	24.7854i	1.07763i	−0.842425	+	0.538813i	$0.818873\pi$
$24$	0	0
$25$	24.8639	0.994557
$26$	− 41.5529i	− 1.59819i
$27$	0	0
$28$	50.0034	1.78583
$29$	40.2521i	1.38800i	0.719974	+	0.694001i	$0.244154\pi$
$29$	40.2521i	1.38800i	−0.719974	+	0.694001i	$0.755846\pi$
$30$	0	0
$31$	−0.142872	−0.00460879	−0.00230439	−	0.999997i	$-0.500734\pi$
$31$	−0.142872	−0.00460879	−0.00230439	+	0.999997i	$0.500734\pi$
$32$	− 45.8975i	− 1.43430i
$33$	0	0
$34$	11.6839	0.343644
$35$	4.21439i	0.120411i
$36$	0	0
$37$	70.5507	1.90678	0.953388	−	0.301747i	$-0.0975699\pi$
$37$	70.5507	1.90678	0.953388	+	0.301747i	$0.0975699\pi$
$38$	13.0428i	0.343232i
$39$	0	0
$40$	−0.402480	−0.0100620
$41$	− 72.7348i	− 1.77402i	−0.461751	−	0.887009i	$-0.652779\pi$
$41$	− 72.7348i	− 1.77402i	0.461751	−	0.887009i	$-0.347221\pi$
$42$	0	0
$43$	32.2116	0.749108	0.374554	−	0.927205i	$-0.377796\pi$
$43$	32.2116	0.749108	0.374554	+	0.927205i	$0.377796\pi$
$44$	54.5630i	1.24007i
$45$	0	0
$46$	−71.7363	−1.55949
$47$	− 29.8941i	− 0.636044i	−0.948083	−	0.318022i	$-0.896981\pi$
$47$	− 29.8941i	− 0.636044i	0.948083	−	0.318022i	$-0.103019\pi$
$48$	0	0
$49$	81.5130	1.66353
$50$	71.9636i	1.43927i
$51$	0	0
$52$	62.8392	1.20845
$53$	46.0232i	0.868362i	0.900826	+	0.434181i	$0.142962\pi$
$53$	46.0232i	0.868362i	−0.900826	+	0.434181i	$0.857038\pi$
$54$	0	0
$55$	−4.59868	−0.0836124
$56$	12.4642i	0.222575i
$57$	0	0
$58$	−116.501	−2.00865
$59$	− 67.6830i	− 1.14717i	−0.819146	−	0.573585i	$-0.805552\pi$
$59$	− 67.6830i	− 1.14717i	0.819146	−	0.573585i	$-0.194448\pi$
$60$	0	0
$61$	−40.9491	−0.671297	−0.335648	−	0.941987i	$-0.608955\pi$
$61$	−40.9491	−0.671297	−0.335648	+	0.941987i	$0.608955\pi$
$62$	− 0.413515i	− 0.00666960i
$63$	0	0
$64$	75.4407	1.17876
$65$	5.29621i	0.0814802i
$66$	0	0
$67$	−13.7399	−0.205073	−0.102536	−	0.994729i	$-0.532696\pi$
$67$	−13.7399	−0.205073	−0.102536	+	0.994729i	$0.532696\pi$
$68$	17.6692i	0.259841i
$69$	0	0
$70$	−12.1977	−0.174253
$71$	14.7337i	0.207517i	0.994603	+	0.103759i	$0.0330870\pi$
$71$	14.7337i	0.207517i	−0.994603	+	0.103759i	$0.966913\pi$
$72$	0	0
$73$	131.028	1.79491	0.897454	−	0.441109i	$-0.145415\pi$
$73$	131.028	1.79491	0.897454	+	0.441109i	$0.145415\pi$
$74$	204.195i	2.75939i
$75$	0	0
$76$	−19.7242	−0.259530
$77$	142.414i	1.84953i
$78$	0	0
$79$	86.0968	1.08983	0.544916	−	0.838490i	$-0.316561\pi$
$79$	86.0968	1.08983	0.544916	+	0.838490i	$0.316561\pi$
$80$	5.29373i	0.0661716i
$81$	0	0
$82$	210.516	2.56727
$83$	− 141.334i	− 1.70281i	−0.524505	−	0.851407i	$-0.675750\pi$
$83$	− 141.334i	− 1.70281i	0.524505	−	0.851407i	$-0.324250\pi$
$84$	0	0
$85$	−1.48920	−0.0175199
$86$	93.2301i	1.08407i
$87$	0	0
$88$	−13.6007	−0.154554
$89$	− 136.262i	− 1.53104i	−0.643413	−	0.765519i	$-0.722482\pi$
$89$	− 136.262i	− 1.53104i	0.643413	−	0.765519i	$-0.277518\pi$
$90$	0	0
$91$	164.016	1.80237
$92$	− 108.485i	− 1.17918i
$93$	0	0
$94$	86.5224	0.920451
$95$	− 1.66240i	− 0.0174989i
$96$	0	0
$97$	22.3537	0.230450	0.115225	−	0.993339i	$-0.463241\pi$
$97$	22.3537	0.230450	0.115225	+	0.993339i	$0.463241\pi$
$98$	235.923i	2.40738i
$99$	0	0
$100$	−108.828	−1.08828
$101$	143.434i	1.42014i	0.704131	+	0.710070i	$0.251337\pi$
$101$	143.434i	1.42014i	−0.704131	+	0.710070i	$0.748663\pi$
$102$	0	0
$103$	48.6766	0.472588	0.236294	−	0.971682i	$-0.424067\pi$
$103$	48.6766	0.472588	0.236294	+	0.971682i	$0.424067\pi$
$104$	15.6637i	0.150613i
$105$	0	0
$106$	−133.205	−1.25665
$107$	59.3338i	0.554522i	0.960795	+	0.277261i	$0.0894266\pi$
$107$	59.3338i	0.554522i	−0.960795	+	0.277261i	$0.910573\pi$
$108$	0	0
$109$	−170.051	−1.56010	−0.780050	−	0.625717i	$-0.784806\pi$
$109$	−170.051	−1.56010	−0.780050	+	0.625717i	$0.784806\pi$
$110$	− 13.3100i	− 0.121000i
$111$	0	0
$112$	163.938	1.46374
$113$	− 75.6100i	− 0.669115i	−0.942375	−	0.334558i	$-0.891413\pi$
$113$	− 75.6100i	− 0.669115i	0.942375	−	0.334558i	$-0.108587\pi$
$114$	0	0
$115$	9.14331	0.0795071
$116$	− 176.182i	− 1.51881i
$117$	0	0
$118$	195.895	1.66012
$119$	46.1181i	0.387547i
$120$	0	0
$121$	−34.4002	−0.284299
$122$	− 118.519i	− 0.971466i
$123$	0	0
$124$	0.625347	0.00504312
$125$	− 18.3948i	− 0.147158i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	34.7579i	0.271546i
$129$	0	0
$130$	−15.3288	−0.117914
$131$	155.996i	1.19081i	0.803425	+	0.595406i	$0.203009\pi$
$131$	155.996i	1.19081i	−0.803425	+	0.595406i	$0.796991\pi$
$132$	0	0
$133$	−51.4819	−0.387082
$134$	− 39.7672i	− 0.296770i
$135$	0	0
$136$	−4.40434	−0.0323848
$137$	− 73.9267i	− 0.539611i	−0.962915	−	0.269806i	$-0.913041\pi$
$137$	− 73.9267i	− 0.539611i	0.962915	−	0.269806i	$-0.0869594\pi$
$138$	0	0
$139$	−264.466	−1.90264	−0.951318	−	0.308212i	$-0.900269\pi$
$139$	−264.466	−1.90264	−0.951318	+	0.308212i	$0.900269\pi$
$140$	− 18.4462i	− 0.131759i
$141$	0	0
$142$	−42.6438	−0.300308
$143$	178.971i	1.25155i
$144$	0	0
$145$	14.8489	0.102407
$146$	379.235i	2.59750i
$147$	0	0
$148$	−308.798	−2.08647
$149$	− 81.6878i	− 0.548240i	−0.961695	−	0.274120i	$-0.911613\pi$
$149$	− 81.6878i	− 0.548240i	0.961695	−	0.274120i	$-0.0883866\pi$
$150$	0	0
$151$	−245.937	−1.62872	−0.814360	−	0.580360i	$-0.802912\pi$
$151$	−245.937	−1.62872	−0.814360	+	0.580360i	$0.802912\pi$
$152$	− 4.91660i	− 0.0323460i
$153$	0	0
$154$	−412.188	−2.67655
$155$	0.0527055i	0 0.000340035i
$156$	0	0
$157$	231.778	1.47629	0.738147	−	0.674640i	$-0.235701\pi$
$157$	231.778	1.47629	0.738147	+	0.674640i	$0.235701\pi$
$158$	249.190i	1.57715i
$159$	0	0
$160$	−16.9315	−0.105822
$161$	− 283.154i	− 1.75872i
$162$	0	0
$163$	124.426	0.763347	0.381673	−	0.924297i	$-0.375348\pi$
$163$	124.426	0.763347	0.381673	+	0.924297i	$0.375348\pi$
$164$	318.357i	1.94120i
$165$	0	0
$166$	409.062	2.46423
$167$	− 47.4894i	− 0.284368i	−0.989840	−	0.142184i	$-0.954588\pi$
$167$	− 47.4894i	− 0.284368i	0.989840	−	0.142184i	$-0.0454124\pi$
$168$	0	0
$169$	37.1181	0.219634
$170$	− 4.31017i	− 0.0253540i
$171$	0	0
$172$	−140.989	−0.819704
$173$	− 220.614i	− 1.27522i	−0.770357	−	0.637612i	$-0.779922\pi$
$173$	− 220.614i	− 1.27522i	0.770357	−	0.637612i	$-0.220078\pi$
$174$	0	0
$175$	−284.051	−1.62315
$176$	178.887i	1.01641i
$177$	0	0
$178$	394.384	2.21564
$179$	− 94.0577i	− 0.525462i	−0.964869	−	0.262731i	$-0.915377\pi$
$179$	− 94.0577i	− 0.525462i	0.964869	−	0.262731i	$-0.0846232\pi$
$180$	0	0
$181$	115.413	0.637642	0.318821	−	0.947815i	$-0.396713\pi$
$181$	115.413	0.637642	0.318821	+	0.947815i	$0.396713\pi$
$182$	474.710i	2.60830i
$183$	0	0
$184$	27.0416	0.146965
$185$	− 26.0261i	− 0.140682i
$186$	0	0
$187$	−50.3234	−0.269109
$188$	130.845i	0.695985i
$189$	0	0
$190$	4.81148	0.0253236
$191$	− 74.0418i	− 0.387654i	−0.981036	−	0.193827i	$-0.937910\pi$
$191$	− 74.0418i	− 0.387654i	0.981036	−	0.193827i	$-0.0620900\pi$
$192$	0	0
$193$	−14.0989	−0.0730514	−0.0365257	−	0.999333i	$-0.511629\pi$
$193$	−14.0989	−0.0730514	−0.0365257	+	0.999333i	$0.511629\pi$
$194$	64.6983i	0.333496i
$195$	0	0
$196$	−356.779	−1.82030
$197$	312.804i	1.58784i	0.608024	+	0.793919i	$0.291962\pi$
$197$	312.804i	1.58784i	−0.608024	+	0.793919i	$0.708038\pi$
$198$	0	0
$199$	69.5241	0.349367	0.174684	−	0.984625i	$-0.444110\pi$
$199$	69.5241	0.349367	0.174684	+	0.984625i	$0.444110\pi$
$200$	− 27.1273i	− 0.135636i
$201$	0	0
$202$	−415.141	−2.05515
$203$	− 459.849i	− 2.26526i
$204$	0	0
$205$	−26.8318	−0.130887
$206$	140.885i	0.683906i
$207$	0	0
$208$	206.021	0.990487
$209$	− 56.1764i	− 0.268787i
$210$	0	0
$211$	211.824	1.00390	0.501952	−	0.864896i	$-0.332615\pi$
$211$	211.824	1.00390	0.501952	+	0.864896i	$0.332615\pi$
$212$	− 201.442i	− 0.950196i
$213$	0	0
$214$	−171.730	−0.802476
$215$	− 11.8828i	− 0.0552691i
$216$	0	0
$217$	1.63221	0.00752169
$218$	− 492.178i	− 2.25770i
$219$	0	0
$220$	20.1282	0.0914920
$221$	57.9565i	0.262247i
$222$	0	0
$223$	−264.549	−1.18632	−0.593159	−	0.805085i	$-0.702120\pi$
$223$	−264.549	−1.18632	−0.593159	+	0.805085i	$0.702120\pi$
$224$	524.343i	2.34082i
$225$	0	0
$226$	218.838	0.968309
$227$	− 390.263i	− 1.71922i	−0.510949	−	0.859611i	$-0.670706\pi$
$227$	− 390.263i	− 1.71922i	0.510949	−	0.859611i	$-0.329294\pi$
$228$	0	0
$229$	−156.112	−0.681714	−0.340857	−	0.940115i	$-0.610717\pi$
$229$	−156.112	−0.681714	−0.340857	+	0.940115i	$0.610717\pi$
$230$	26.4635i	0.115059i
$231$	0	0
$232$	43.9162	0.189294
$233$	327.852i	1.40709i	0.710650	+	0.703545i	$0.248401\pi$
$233$	327.852i	1.40709i	−0.710650	+	0.703545i	$0.751599\pi$
$234$	0	0
$235$	−11.0279	−0.0469272
$236$	296.246i	1.25528i
$237$	0	0
$238$	−133.479	−0.560838
$239$	− 218.750i	− 0.915274i	−0.889139	−	0.457637i	$-0.848696\pi$
$239$	− 218.750i	− 0.915274i	0.889139	−	0.457637i	$-0.151304\pi$
$240$	0	0
$241$	−145.126	−0.602182	−0.301091	−	0.953595i	$-0.597351\pi$
$241$	−145.126	−0.602182	−0.301091	+	0.953595i	$0.597351\pi$
$242$	− 99.5643i	− 0.411423i
$243$	0	0
$244$	179.232	0.734559
$245$	− 30.0701i	− 0.122735i
$246$	0	0
$247$	−64.6973	−0.261932
$248$	0.155878i	0 0.000628540i
$249$	0	0
$250$	53.2399	0.212960
$251$	347.366i	1.38393i	0.721933	+	0.691963i	$0.243254\pi$
$251$	347.366i	1.38393i	−0.721933	+	0.691963i	$0.756746\pi$
$252$	0	0
$253$	308.974	1.22124
$254$	32.6171i	0.128414i
$255$	0	0
$256$	201.163	0.785793
$257$	− 121.233i	− 0.471722i	−0.971787	−	0.235861i	$-0.924209\pi$
$257$	− 121.233i	− 0.471722i	0.971787	−	0.235861i	$-0.0757910\pi$
$258$	0	0
$259$	−805.987	−3.11192
$260$	− 23.1813i	− 0.0891589i
$261$	0	0
$262$	−451.500	−1.72328
$263$	− 288.631i	− 1.09746i	−0.836000	−	0.548729i	$-0.815112\pi$
$263$	− 288.631i	− 1.09746i	0.836000	−	0.548729i	$-0.184888\pi$
$264$	0	0
$265$	16.9779	0.0640676
$266$	− 149.004i	− 0.560166i
$267$	0	0
$268$	60.1388	0.224398
$269$	− 67.7121i	− 0.251718i	−0.992048	−	0.125859i	$-0.959831\pi$
$269$	− 67.7121i	− 0.251718i	0.992048	−	0.125859i	$-0.0401687\pi$
$270$	0	0
$271$	396.016	1.46131	0.730657	−	0.682745i	$-0.239214\pi$
$271$	396.016	1.46131	0.730657	+	0.682745i	$0.239214\pi$
$272$	57.9293i	0.212975i
$273$	0	0
$274$	213.966	0.780898
$275$	− 309.953i	− 1.12710i
$276$	0	0
$277$	−215.982	−0.779718	−0.389859	−	0.920875i	$-0.627476\pi$
$277$	−215.982	−0.779718	−0.389859	+	0.920875i	$0.627476\pi$
$278$	− 765.444i	− 2.75340i
$279$	0	0
$280$	4.59802	0.0164215
$281$	170.535i	0.606886i	0.952850	+	0.303443i	$0.0981362\pi$
$281$	170.535i	0.606886i	−0.952850	+	0.303443i	$0.901864\pi$
$282$	0	0
$283$	−0.946273	−0.00334372	−0.00167186	−	0.999999i	$-0.500532\pi$
$283$	−0.946273	−0.00334372	−0.00167186	+	0.999999i	$0.500532\pi$
$284$	− 64.4889i	− 0.227074i
$285$	0	0
$286$	−517.997	−1.81118
$287$	830.939i	2.89526i
$288$	0	0
$289$	272.704	0.943612
$290$	42.9773i	0.148197i
$291$	0	0
$292$	−573.505	−1.96406
$293$	− 308.048i	− 1.05136i	−0.850683	−	0.525679i	$-0.823811\pi$
$293$	− 308.048i	− 1.05136i	0.850683	−	0.525679i	$-0.176189\pi$
$294$	0	0
$295$	−24.9682	−0.0846380
$296$	− 76.9729i	− 0.260044i
$297$	0	0
$298$	236.429	0.793385
$299$	− 355.839i	− 1.19010i
$300$	0	0
$301$	−367.993	−1.22257
$302$	− 711.814i	− 2.35700i
$303$	0	0
$304$	−64.6669	−0.212720
$305$	15.1061i	0.0495281i
$306$	0	0
$307$	−226.372	−0.737369	−0.368685	−	0.929555i	$-0.620192\pi$
$307$	−226.372	−0.737369	−0.368685	+	0.929555i	$0.620192\pi$
$308$	− 623.340i	− 2.02383i
$309$	0	0
$310$	−0.152545	−0.000492082 0
$311$	− 286.254i	− 0.920431i	−0.887807	−	0.460215i	$-0.847772\pi$
$311$	− 286.254i	− 0.920431i	0.887807	−	0.460215i	$-0.152228\pi$
$312$	0	0
$313$	403.280	1.28844	0.644218	−	0.764842i	$-0.277183\pi$
$313$	403.280	1.28844	0.644218	+	0.764842i	$0.277183\pi$
$314$	670.835i	2.13642i
$315$	0	0
$316$	−376.842	−1.19254
$317$	− 252.786i	− 0.797431i	−0.917075	−	0.398715i	$-0.869456\pi$
$317$	− 252.786i	− 0.797431i	0.917075	−	0.398715i	$-0.130544\pi$
$318$	0	0
$319$	501.780	1.57298
$320$	− 27.8300i	− 0.0869688i
$321$	0	0
$322$	819.532	2.54513
$323$	− 18.1917i	− 0.0563209i
$324$	0	0
$325$	−356.967	−1.09836
$326$	360.125i	1.10468i
$327$	0	0
$328$	−79.3558	−0.241938
$329$	341.517i	1.03804i
$330$	0	0
$331$	493.284	1.49028	0.745142	−	0.666906i	$-0.232382\pi$
$331$	493.284	1.49028	0.745142	+	0.666906i	$0.232382\pi$
$332$	618.611i	1.86329i
$333$	0	0
$334$	137.448	0.411522
$335$	5.06862i	0.0151302i
$336$	0	0
$337$	437.426	1.29800	0.649000	−	0.760789i	$-0.275188\pi$
$337$	437.426	1.29800	0.649000	+	0.760789i	$0.275188\pi$
$338$	107.431i	0.317843i
$339$	0	0
$340$	6.51815	0.0191710
$341$	1.78104i	0.00522300i
$342$	0	0
$343$	−371.436	−1.08290
$344$	− 35.1439i	− 0.102162i
$345$	0	0
$346$	638.522	1.84544
$347$	− 585.153i	− 1.68632i	−0.537663	−	0.843160i	$-0.680693\pi$
$347$	− 585.153i	− 1.68632i	0.537663	−	0.843160i	$-0.319307\pi$
$348$	0	0
$349$	526.031	1.50725	0.753626	−	0.657304i	$-0.228303\pi$
$349$	526.031	1.50725	0.753626	+	0.657304i	$0.228303\pi$
$350$	− 822.128i	− 2.34894i
$351$	0	0
$352$	−572.156	−1.62544
$353$	− 50.5781i	− 0.143281i	−0.997431	−	0.0716404i	$-0.977177\pi$
$353$	− 50.5781i	− 0.143281i	0.997431	−	0.0716404i	$-0.0228234\pi$
$354$	0	0
$355$	5.43526	0.0153106
$356$	596.415i	1.67532i
$357$	0	0
$358$	272.231	0.760422
$359$	− 141.833i	− 0.395078i	−0.980295	−	0.197539i	$-0.936705\pi$
$359$	− 141.833i	− 0.395078i	0.980295	−	0.197539i	$-0.0632949\pi$
$360$	0	0
$361$	−340.693	−0.943747
$362$	334.040i	0.922763i
$363$	0	0
$364$	−717.889	−1.97222
$365$	− 48.3362i	− 0.132428i
$366$	0	0
$367$	−158.951	−0.433110	−0.216555	−	0.976270i	$-0.569482\pi$
$367$	−158.951	−0.433110	−0.216555	+	0.976270i	$0.569482\pi$
$368$	− 355.672i	− 0.966501i
$369$	0	0
$370$	75.3272	0.203587
$371$	− 525.779i	− 1.41719i
$372$	0	0
$373$	530.611	1.42255	0.711275	−	0.702914i	$-0.248118\pi$
$373$	530.611	1.42255	0.711275	+	0.702914i	$0.248118\pi$
$374$	− 145.651i	− 0.389441i
$375$	0	0
$376$	−32.6153	−0.0867429
$377$	− 577.891i	− 1.53287i
$378$	0	0
$379$	244.663	0.645548	0.322774	−	0.946476i	$-0.395385\pi$
$379$	244.663	0.645548	0.322774	+	0.946476i	$0.395385\pi$
$380$	7.27626i	0.0191480i
$381$	0	0
$382$	214.299	0.560992
$383$	167.289i	0.436786i	0.975861	+	0.218393i	$0.0700814\pi$
$383$	167.289i	0.436786i	−0.975861	+	0.218393i	$0.929919\pi$
$384$	0	0
$385$	52.5364	0.136458
$386$	− 40.8065i	− 0.105716i
$387$	0	0
$388$	−97.8412	−0.252168
$389$	412.946i	1.06156i	0.847511	+	0.530779i	$0.178100\pi$
$389$	412.946i	1.06156i	−0.847511	+	0.530779i	$0.821900\pi$
$390$	0	0
$391$	100.055	0.255896
$392$	− 88.9331i	− 0.226870i
$393$	0	0
$394$	−905.348	−2.29784
$395$	− 31.7610i	− 0.0804077i
$396$	0	0
$397$	536.080	1.35033	0.675164	−	0.737668i	$-0.264073\pi$
$397$	536.080	1.35033	0.675164	+	0.737668i	$0.264073\pi$
$398$	201.223i	0.505587i
$399$	0	0
$400$	−356.799	−0.891997
$401$	535.852i	1.33629i	0.744032	+	0.668144i	$0.232911\pi$
$401$	535.852i	1.33629i	−0.744032	+	0.668144i	$0.767089\pi$
$402$	0	0
$403$	2.05119	0.00508981
$404$	− 627.805i	− 1.55397i
$405$	0	0
$406$	1330.94	3.27817
$407$	− 879.482i	− 2.16089i
$408$	0	0
$409$	−357.786	−0.874783	−0.437391	−	0.899271i	$-0.644098\pi$
$409$	−357.786	−0.874783	−0.437391	+	0.899271i	$0.644098\pi$
$410$	− 77.6592i	− 0.189413i
$411$	0	0
$412$	−213.056	−0.517125
$413$	773.226i	1.87222i
$414$	0	0
$415$	−52.1378	−0.125633
$416$	658.942i	1.58399i
$417$	0	0
$418$	162.591	0.388974
$419$	− 613.475i	− 1.46414i	−0.681229	−	0.732070i	$-0.738554\pi$
$419$	− 613.475i	− 1.46414i	0.681229	−	0.732070i	$-0.261446\pi$
$420$	0	0
$421$	569.239	1.35211	0.676056	−	0.736850i	$-0.263688\pi$
$421$	569.239	1.35211	0.676056	+	0.736850i	$0.263688\pi$
$422$	613.081i	1.45280i
$423$	0	0
$424$	50.2127	0.118426
$425$	− 100.372i	− 0.236170i
$426$	0	0
$427$	467.812	1.09558
$428$	− 259.702i	− 0.606780i
$429$	0	0
$430$	34.3925	0.0799826
$431$	− 269.812i	− 0.626014i	−0.949751	−	0.313007i	$-0.898664\pi$
$431$	− 269.812i	− 0.626014i	0.949751	−	0.313007i	$-0.101336\pi$
$432$	0	0
$433$	151.221	0.349241	0.174620	−	0.984636i	$-0.444130\pi$
$433$	151.221	0.349241	0.174620	+	0.984636i	$0.444130\pi$
$434$	4.72409i	0.0108850i
$435$	0	0
$436$	744.306	1.70712
$437$	111.692i	0.255589i
$438$	0	0
$439$	272.925	0.621698	0.310849	−	0.950459i	$-0.399387\pi$
$439$	272.925	0.621698	0.310849	+	0.950459i	$0.399387\pi$
$440$	5.01730i	0.0114029i
$441$	0	0
$442$	−167.743	−0.379510
$443$	173.360i	0.391331i	0.980671	+	0.195666i	$0.0626867\pi$
$443$	173.360i	0.391331i	−0.980671	+	0.195666i	$0.937313\pi$
$444$	0	0
$445$	−50.2671	−0.112960
$446$	− 765.684i	− 1.71678i
$447$	0	0
$448$	−861.852	−1.92378
$449$	− 97.2936i	− 0.216690i	−0.994113	−	0.108345i	$-0.965445\pi$
$449$	− 97.2936i	− 0.216690i	0.994113	−	0.108345i	$-0.0345550\pi$
$450$	0	0
$451$	−906.709	−2.01044
$452$	330.942i	0.732172i
$453$	0	0
$454$	1129.54	2.48797
$455$	− 60.5052i	− 0.132978i
$456$	0	0
$457$	30.5417	0.0668310	0.0334155	−	0.999442i	$-0.489362\pi$
$457$	30.5417	0.0668310	0.0334155	+	0.999442i	$0.489362\pi$
$458$	− 451.836i	− 0.986541i
$459$	0	0
$460$	−40.0199	−0.0869998
$461$	− 88.1669i	− 0.191251i	−0.995417	−	0.0956257i	$-0.969515\pi$
$461$	− 88.1669i	− 0.191251i	0.995417	−	0.0956257i	$-0.0304852\pi$
$462$	0	0
$463$	−794.001	−1.71491	−0.857453	−	0.514563i	$-0.827954\pi$
$463$	−794.001	−1.71491	−0.857453	+	0.514563i	$0.827954\pi$
$464$	− 577.620i	− 1.24487i
$465$	0	0
$466$	−948.902	−2.03627
$467$	142.933i	0.306066i	0.988221	+	0.153033i	$0.0489041\pi$
$467$	142.933i	0.306066i	−0.988221	+	0.153033i	$0.951096\pi$
$468$	0	0
$469$	156.967	0.334685
$470$	− 31.9180i	− 0.0679107i
$471$	0	0
$472$	−73.8442	−0.156450
$473$	− 401.549i	− 0.848941i
$474$	0	0
$475$	112.046	0.235887
$476$	− 201.857i	− 0.424069i
$477$	0	0
$478$	633.129	1.32454
$479$	555.490i	1.15969i	0.814728	+	0.579843i	$0.196886\pi$
$479$	555.490i	1.15969i	−0.814728	+	0.579843i	$0.803114\pi$
$480$	0	0
$481$	−1012.88	−2.10579
$482$	− 420.037i	− 0.871447i
$483$	0	0
$484$	150.568	0.311091
$485$	− 8.24626i	− 0.0170026i
$486$	0	0
$487$	456.287	0.936934	0.468467	−	0.883481i	$-0.344807\pi$
$487$	456.287	0.936934	0.468467	+	0.883481i	$0.344807\pi$
$488$	44.6767i	0.0915506i
$489$	0	0
$490$	87.0317	0.177616
$491$	428.089i	0.871872i	0.899978	+	0.435936i	$0.143583\pi$
$491$	428.089i	0.871872i	−0.899978	+	0.435936i	$0.856417\pi$
$492$	0	0
$493$	162.492	0.329598
$494$	− 187.253i	− 0.379055i
$495$	0	0
$496$	2.05023	0.00413353
$497$	− 168.321i	− 0.338675i
$498$	0	0
$499$	−216.198	−0.433263	−0.216632	−	0.976253i	$-0.569507\pi$
$499$	−216.198	−0.433263	−0.216632	+	0.976253i	$0.569507\pi$
$500$	80.5131i	0.161026i
$501$	0	0
$502$	−1005.38	−2.00275
$503$	− 484.500i	− 0.963220i	−0.876386	−	0.481610i	$-0.840052\pi$
$503$	− 484.500i	− 0.963220i	0.876386	−	0.481610i	$-0.159948\pi$
$504$	0	0
$505$	52.9127	0.104778
$506$	894.262i	1.76732i
$507$	0	0
$508$	−49.3258	−0.0970981
$509$	− 618.428i	− 1.21499i	−0.794325	−	0.607493i	$-0.792175\pi$
$509$	− 618.428i	− 1.21499i	0.794325	−	0.607493i	$-0.207825\pi$
$510$	0	0
$511$	−1496.90	−2.92935
$512$	721.258i	1.40871i
$513$	0	0
$514$	350.883	0.682652
$515$	− 17.9568i	− 0.0348675i
$516$	0	0
$517$	−372.658	−0.720809
$518$	− 2332.77i	− 4.50341i
$519$	0	0
$520$	5.77833	0.0111122
$521$	− 15.6085i	− 0.0299587i	−0.999888	−	0.0149793i	$-0.995232\pi$
$521$	− 15.6085i	− 0.0299587i	0.999888	−	0.0149793i	$-0.00476825\pi$
$522$	0	0
$523$	−422.072	−0.807022	−0.403511	−	0.914975i	$-0.632210\pi$
$523$	−422.072	−0.807022	−0.403511	+	0.914975i	$0.632210\pi$
$524$	− 682.789i	− 1.30303i
$525$	0	0
$526$	835.385	1.58818
$527$	0.576756i	0.00109441i
$528$	0	0
$529$	−85.3161	−0.161278
$530$	49.1391i	0.0927153i
$531$	0	0
$532$	225.334	0.423561
$533$	1044.24i	1.95917i
$534$	0	0
$535$	21.8882	0.0409125
$536$	14.9906i	0.0279675i
$537$	0	0
$538$	195.979	0.364273
$539$	− 1016.14i	− 1.88523i
$540$	0	0
$541$	−797.374	−1.47389	−0.736944	−	0.675953i	$-0.763732\pi$
$541$	−797.374	−1.47389	−0.736944	+	0.675953i	$0.763732\pi$
$542$	1146.19i	2.11474i
$543$	0	0
$544$	−185.282	−0.340592
$545$	62.7316i	0.115104i
$546$	0	0
$547$	607.308	1.11025	0.555126	−	0.831766i	$-0.312670\pi$
$547$	607.308	1.11025	0.555126	+	0.831766i	$0.312670\pi$
$548$	323.574i	0.590464i
$549$	0	0
$550$	897.095	1.63108
$551$	181.391i	0.329204i
$552$	0	0
$553$	−983.589	−1.77864
$554$	− 625.116i	− 1.12837i
$555$	0	0
$556$	1157.56	2.08194
$557$	− 763.344i	− 1.37046i	−0.728329	−	0.685228i	$-0.759703\pi$
$557$	− 763.344i	− 1.37046i	0.728329	−	0.685228i	$-0.240297\pi$
$558$	0	0
$559$	−462.457	−0.827293
$560$	− 60.4767i	− 0.107994i
$561$	0	0
$562$	−493.579	−0.878254
$563$	− 911.688i	− 1.61934i	−0.586886	−	0.809670i	$-0.699646\pi$
$563$	− 911.688i	− 1.61934i	0.586886	−	0.809670i	$-0.300354\pi$
$564$	0	0
$565$	−27.8925	−0.0493672
$566$	− 2.73880i	− 0.00483886i
$567$	0	0
$568$	16.0749	0.0283009
$569$	− 680.350i	− 1.19569i	−0.801610	−	0.597847i	$-0.796023\pi$
$569$	− 680.350i	− 1.19569i	0.801610	−	0.597847i	$-0.203977\pi$
$570$	0	0
$571$	74.3734	0.130251	0.0651256	−	0.997877i	$-0.479255\pi$
$571$	74.3734	0.130251	0.0651256	+	0.997877i	$0.479255\pi$
$572$	− 783.351i	− 1.36949i
$573$	0	0
$574$	−2404.98	−4.18987
$575$	616.262i	1.07176i
$576$	0	0
$577$	234.061	0.405652	0.202826	−	0.979215i	$-0.434987\pi$
$577$	234.061	0.405652	0.202826	+	0.979215i	$0.434987\pi$
$578$	789.286i	1.36555i
$579$	0	0
$580$	−64.9932	−0.112057
$581$	1614.63i	2.77905i
$582$	0	0
$583$	573.723	0.984088
$584$	− 142.956i	− 0.244787i
$585$	0	0
$586$	891.583	1.52147
$587$	547.257i	0.932295i	0.884707	+	0.466147i	$0.154358\pi$
$587$	547.257i	0.932295i	−0.884707	+	0.466147i	$0.845642\pi$
$588$	0	0
$589$	−0.643838	−0.00109310
$590$	− 72.2654i	− 0.122484i
$591$	0	0
$592$	−1012.41	−1.71015
$593$	453.958i	0.765528i	0.923846	+	0.382764i	$0.125028\pi$
$593$	453.958i	0.765528i	−0.923846	+	0.382764i	$0.874972\pi$
$594$	0	0
$595$	17.0129	0.0285931
$596$	357.544i	0.599906i
$597$	0	0
$598$	1029.90	1.72225
$599$	− 852.190i	− 1.42269i	−0.702844	−	0.711344i	$-0.748087\pi$
$599$	− 852.190i	− 1.42269i	0.702844	−	0.711344i	$-0.251913\pi$
$600$	0	0
$601$	70.0421	0.116543	0.0582713	−	0.998301i	$-0.481441\pi$
$601$	70.0421	0.116543	0.0582713	+	0.998301i	$0.481441\pi$
$602$	− 1065.08i	− 1.76924i
$603$	0	0
$604$	1076.45	1.78221
$605$	12.6902i	0.0209755i
$606$	0	0
$607$	569.835	0.938773	0.469387	−	0.882993i	$-0.344475\pi$
$607$	569.835	0.938773	0.469387	+	0.882993i	$0.344475\pi$
$608$	− 206.832i	− 0.340184i
$609$	0	0
$610$	−43.7215	−0.0716746
$611$	429.184i	0.702428i
$612$	0	0
$613$	477.707	0.779294	0.389647	−	0.920964i	$-0.372597\pi$
$613$	477.707	0.779294	0.389647	+	0.920964i	$0.372597\pi$
$614$	− 655.189i	− 1.06708i
$615$	0	0
$616$	155.378	0.252237
$617$	1014.71i	1.64459i	0.569061	+	0.822295i	$0.307307\pi$
$617$	1014.71i	1.64459i	−0.569061	+	0.822295i	$0.692693\pi$
$618$	0	0
$619$	−401.464	−0.648569	−0.324284	−	0.945960i	$-0.605123\pi$
$619$	−401.464	−0.648569	−0.324284	+	0.945960i	$0.605123\pi$
$620$	− 0.230690i	0 0.000372080i
$621$	0	0
$622$	828.504	1.33200
$623$	1556.69i	2.49871i
$624$	0	0
$625$	614.812	0.983699
$626$	1167.21i	1.86456i
$627$	0	0
$628$	−1014.48	−1.61542
$629$	− 284.804i	− 0.452788i
$630$	0	0
$631$	−807.950	−1.28043	−0.640214	−	0.768197i	$-0.721154\pi$
$631$	−807.950	−1.28043	−0.640214	+	0.768197i	$0.721154\pi$
$632$	− 93.9342i	− 0.148630i
$633$	0	0
$634$	731.637	1.15400
$635$	− 4.15728i	− 0.00654690i
$636$	0	0
$637$	−1170.27	−1.83715
$638$	1452.30i	2.27634i
$639$	0	0
$640$	12.8222	0.0200346
$641$	− 225.405i	− 0.351646i	−0.984422	−	0.175823i	$-0.943741\pi$
$641$	− 225.405i	− 0.351646i	0.984422	−	0.175823i	$-0.0562587\pi$
$642$	0	0
$643$	−282.242	−0.438946	−0.219473	−	0.975619i	$-0.570434\pi$
$643$	−282.242	−0.438946	−0.219473	+	0.975619i	$0.570434\pi$
$644$	1239.35i	1.92446i
$645$	0	0
$646$	52.6521	0.0815047
$647$	927.159i	1.43301i	0.697581	+	0.716506i	$0.254260\pi$
$647$	927.159i	1.43301i	−0.697581	+	0.716506i	$0.745740\pi$
$648$	0	0
$649$	−843.734	−1.30005
$650$	− 1033.17i	− 1.58949i
$651$	0	0
$652$	−544.605	−0.835284
$653$	− 91.8009i	− 0.140583i	−0.997526	−	0.0702916i	$-0.977607\pi$
$653$	− 91.8009i	− 0.140583i	0.997526	−	0.0702916i	$-0.0223930\pi$
$654$	0	0
$655$	57.5469	0.0878578
$656$	1043.75i	1.59108i
$657$	0	0
$658$	−988.451	−1.50221
$659$	− 201.283i	− 0.305437i	−0.988270	−	0.152718i	$-0.951197\pi$
$659$	− 201.283i	− 0.305437i	0.988270	−	0.152718i	$-0.0488027\pi$
$660$	0	0
$661$	−1000.56	−1.51371	−0.756856	−	0.653582i	$-0.773266\pi$
$661$	−1000.56	−1.51371	−0.756856	+	0.653582i	$0.773266\pi$
$662$	1427.71i	2.15666i
$663$	0	0
$664$	−154.199	−0.232228
$665$	18.9916i	0.0285589i
$666$	0	0
$667$	−997.663	−1.49575
$668$	207.859i	0.311166i
$669$	0	0
$670$	−14.6701	−0.0218957
$671$	510.470i	0.760760i
$672$	0	0
$673$	−787.143	−1.16960	−0.584801	−	0.811177i	$-0.698828\pi$
$673$	−787.143	−1.16960	−0.584801	+	0.811177i	$0.698828\pi$
$674$	1266.04i	1.87840i
$675$	0	0
$676$	−162.464	−0.240332
$677$	846.014i	1.24965i	0.780764	+	0.624825i	$0.214830\pi$
$677$	846.014i	1.24965i	−0.780764	+	0.624825i	$0.785170\pi$
$678$	0	0
$679$	−255.374	−0.376103
$680$	1.62476i	0.00238935i
$681$	0	0
$682$	−5.15486	−0.00755845
$683$	− 24.7020i	− 0.0361669i	−0.999836	−	0.0180834i	$-0.994244\pi$
$683$	− 24.7020i	− 0.0361669i	0.999836	−	0.0180834i	$-0.00575645\pi$
$684$	0	0
$685$	−27.2715	−0.0398124
$686$	− 1075.05i	− 1.56712i
$687$	0	0
$688$	−462.239	−0.671860
$689$	− 660.746i	− 0.958993i
$690$	0	0
$691$	1031.11	1.49219	0.746097	−	0.665838i	$-0.231926\pi$
$691$	1031.11	1.49219	0.746097	+	0.665838i	$0.231926\pi$
$692$	965.618i	1.39540i
$693$	0	0
$694$	1693.61	2.44036
$695$	97.5614i	0.140376i
$696$	0	0
$697$	−293.620	−0.421263
$698$	1522.49i	2.18122i
$699$	0	0
$700$	1243.28	1.77611
$701$	131.146i	0.187085i	0.995615	+	0.0935424i	$0.0298191\pi$
$701$	131.146i	0.187085i	−0.995615	+	0.0935424i	$0.970181\pi$
$702$	0	0
$703$	317.928	0.452245
$704$	− 940.441i	− 1.33585i
$705$	0	0
$706$	146.388	0.207349
$707$	− 1638.62i	− 2.31772i
$708$	0	0
$709$	488.183	0.688552	0.344276	−	0.938869i	$-0.388124\pi$
$709$	488.183	0.688552	0.344276	+	0.938869i	$0.388124\pi$
$710$	15.7313i	0.0221567i
$711$	0	0
$712$	−148.666	−0.208801
$713$	− 3.54115i	− 0.00496655i
$714$	0	0
$715$	66.0224	0.0923390
$716$	411.687i	0.574982i
$717$	0	0
$718$	410.507	0.571737
$719$	14.8164i	0.0206069i	0.999947	+	0.0103035i	$0.00327975\pi$
$719$	14.8164i	0.0206069i	−0.999947	+	0.0103035i	$0.996720\pi$
$720$	0	0
$721$	−556.093	−0.771280
$722$	− 986.066i	− 1.36574i
$723$	0	0
$724$	−505.159	−0.697733
$725$	1000.82i	1.38045i
$726$	0	0
$727$	288.303	0.396565	0.198282	−	0.980145i	$-0.436464\pi$
$727$	288.303	0.396565	0.198282	+	0.980145i	$0.436464\pi$
$728$	− 178.946i	− 0.245805i
$729$	0	0
$730$	139.899	0.191643
$731$	− 130.034i	− 0.177885i
$732$	0	0
$733$	1127.93	1.53879	0.769393	−	0.638776i	$-0.220559\pi$
$733$	1127.93	1.53879	0.769393	+	0.638776i	$0.220559\pi$
$734$	− 460.053i	− 0.626775i
$735$	0	0
$736$	1137.59	1.54564
$737$	171.281i	0.232402i
$738$	0	0
$739$	289.034	0.391115	0.195557	−	0.980692i	$-0.437348\pi$
$739$	289.034	0.391115	0.195557	+	0.980692i	$0.437348\pi$
$740$	113.915i	0.153939i
$741$	0	0
$742$	1521.76	2.05089
$743$	1057.13i	1.42279i	0.702793	+	0.711395i	$0.251936\pi$
$743$	1057.13i	1.42279i	−0.702793	+	0.711395i	$0.748064\pi$
$744$	0	0
$745$	−30.1345	−0.0404491
$746$	1535.75i	2.05864i
$747$	0	0
$748$	220.263	0.294470
$749$	− 677.843i	− 0.904998i
$750$	0	0
$751$	528.864	0.704214	0.352107	−	0.935960i	$-0.385465\pi$
$751$	528.864	0.704214	0.352107	+	0.935960i	$0.385465\pi$
$752$	428.982i	0.570455i
$753$	0	0
$754$	1672.59	2.21829
$755$	90.7258i	0.120167i
$756$	0	0
$757$	1351.92	1.78589	0.892943	−	0.450170i	$-0.148637\pi$
$757$	1351.92	1.78589	0.892943	+	0.450170i	$0.148637\pi$
$758$	708.126i	0.934203i
$759$	0	0
$760$	−1.81373	−0.00238648
$761$	615.698i	0.809064i	0.914524	+	0.404532i	$0.132566\pi$
$761$	615.698i	0.809064i	−0.914524	+	0.404532i	$0.867434\pi$
$762$	0	0
$763$	1942.70	2.54613
$764$	324.078i	0.424186i
$765$	0	0
$766$	−484.184	−0.632094
$767$	971.712i	1.26690i
$768$	0	0
$769$	−1291.47	−1.67942	−0.839711	−	0.543034i	$-0.817275\pi$
$769$	−1291.47	−1.67942	−0.839711	+	0.543034i	$0.817275\pi$
$770$	152.056i	0.197475i
$771$	0	0
$772$	61.7104	0.0799357
$773$	354.437i	0.458521i	0.973365	+	0.229261i	$0.0736308\pi$
$773$	354.437i	0.458521i	−0.973365	+	0.229261i	$0.926369\pi$
$774$	0	0
$775$	−3.55237	−0.00458370
$776$	− 24.3885i	− 0.0314285i
$777$	0	0
$778$	−1195.19	−1.53623
$779$	− 327.771i	− 0.420758i
$780$	0	0
$781$	183.670	0.235173
$782$	289.590i	0.370319i
$783$	0	0
$784$	−1169.72	−1.49199
$785$	− 85.5027i	− 0.108921i
$786$	0	0
$787$	−1040.99	−1.32273	−0.661363	−	0.750066i	$-0.730022\pi$
$787$	−1040.99	−1.32273	−0.661363	+	0.750066i	$0.730022\pi$
$788$	− 1369.13i	− 1.73747i
$789$	0	0
$790$	91.9259	0.116362
$791$	863.786i	1.09202i
$792$	0	0
$793$	587.898	0.741360
$794$	1551.58i	1.95413i
$795$	0	0
$796$	−304.304	−0.382292
$797$	423.595i	0.531486i	0.964044	+	0.265743i	$0.0856173\pi$
$797$	423.595i	0.531486i	−0.964044	+	0.265743i	$0.914383\pi$
$798$	0	0
$799$	−120.678	−0.151037
$800$	− 1141.19i	− 1.42649i
$801$	0	0
$802$	−1550.91	−1.93381
$803$	− 1633.39i	− 2.03411i
$804$	0	0
$805$	−104.455	−0.129758
$806$	5.93676i	0.00736571i
$807$	0	0
$808$	156.491	0.193677
$809$	− 643.837i	− 0.795843i	−0.917419	−	0.397922i	$-0.869732\pi$
$809$	− 643.837i	− 0.795843i	0.917419	−	0.397922i	$-0.130268\pi$
$810$	0	0
$811$	42.7687	0.0527357	0.0263679	−	0.999652i	$-0.491606\pi$
$811$	42.7687	0.0527357	0.0263679	+	0.999652i	$0.491606\pi$
$812$	2012.74i	2.47874i
$813$	0	0
$814$	2545.48	3.12713
$815$	− 45.9005i	− 0.0563196i
$816$	0	0
$817$	145.158	0.177672
$818$	− 1035.54i	− 1.26594i
$819$	0	0
$820$	117.442	0.143221
$821$	155.646i	0.189581i	0.995497	+	0.0947906i	$0.0302182\pi$
$821$	155.646i	0.189581i	−0.995497	+	0.0947906i	$0.969782\pi$
$822$	0	0
$823$	−835.537	−1.01523	−0.507617	−	0.861583i	$-0.669473\pi$
$823$	−835.537	−1.01523	−0.507617	+	0.861583i	$0.669473\pi$
$824$	− 53.1076i	− 0.0644510i
$825$	0	0
$826$	−2237.95	−2.70938
$827$	319.635i	0.386499i	0.981150	+	0.193250i	$0.0619027\pi$
$827$	319.635i	0.386499i	−0.981150	+	0.193250i	$0.938097\pi$
$828$	0	0
$829$	−745.509	−0.899288	−0.449644	−	0.893208i	$-0.648449\pi$
$829$	−745.509	−0.899288	−0.449644	+	0.893208i	$0.648449\pi$
$830$	− 150.902i	− 0.181810i
$831$	0	0
$832$	−1083.09	−1.30179
$833$	− 329.057i	− 0.395026i
$834$	0	0
$835$	−17.5188	−0.0209806
$836$	245.882i	0.294117i
$837$	0	0
$838$	1775.58	2.11883
$839$	− 1295.12i	− 1.54365i	−0.635838	−	0.771823i	$-0.719345\pi$
$839$	− 1295.12i	− 1.54365i	0.635838	−	0.771823i	$-0.280655\pi$
$840$	0	0
$841$	−779.228	−0.926549
$842$	1647.55i	1.95671i
$843$	0	0
$844$	−927.143	−1.09851
$845$	− 13.6928i	− 0.0162045i
$846$	0	0
$847$	392.995	0.463985
$848$	− 660.436i	− 0.778816i
$849$	0	0
$850$	290.507	0.341773
$851$	1748.63i	2.05479i
$852$	0	0
$853$	30.6295	0.0359080	0.0179540	−	0.999839i	$-0.494285\pi$
$853$	30.6295	0.0359080	0.0179540	+	0.999839i	$0.494285\pi$
$854$	1353.99i	1.58546i
$855$	0	0
$856$	64.7350	0.0756250
$857$	446.511i	0.521016i	0.965472	+	0.260508i	$0.0838901\pi$
$857$	446.511i	0.521016i	−0.965472	+	0.260508i	$0.916110\pi$
$858$	0	0
$859$	−1409.65	−1.64103	−0.820516	−	0.571624i	$-0.806314\pi$
$859$	−1409.65	−1.64103	−0.820516	+	0.571624i	$0.806314\pi$
$860$	52.0107i	0.0604776i
$861$	0	0
$862$	780.916	0.905935
$863$	− 1037.85i	− 1.20260i	−0.799022	−	0.601301i	$-0.794649\pi$
$863$	− 1037.85i	− 1.20260i	0.799022	−	0.601301i	$-0.205351\pi$
$864$	0	0
$865$	−81.3842	−0.0940858
$866$	437.679i	0.505403i
$867$	0	0
$868$	−7.14410	−0.00823053
$869$	− 1073.28i	− 1.23507i
$870$	0	0
$871$	197.261	0.226476
$872$	185.531i	0.212764i
$873$	0	0
$874$	−323.271	−0.369876
$875$	210.146i	0.240167i
$876$	0	0
$877$	74.7317	0.0852129	0.0426065	−	0.999092i	$-0.486434\pi$
$877$	74.7317	0.0852129	0.0426065	+	0.999092i	$0.486434\pi$
$878$	789.927i	0.899689i
$879$	0	0
$880$	65.9914	0.0749902
$881$	− 278.915i	− 0.316589i	−0.987392	−	0.158294i	$-0.949400\pi$
$881$	− 278.915i	− 0.316589i	0.987392	−	0.158294i	$-0.0505995\pi$
$882$	0	0
$883$	172.659	0.195537	0.0977686	−	0.995209i	$-0.468829\pi$
$883$	172.659	0.195537	0.0977686	+	0.995209i	$0.468829\pi$
$884$	− 253.673i	− 0.286961i
$885$	0	0
$886$	−501.755	−0.566314
$887$	− 221.858i	− 0.250122i	−0.992149	−	0.125061i	$-0.960087\pi$
$887$	− 221.858i	− 0.250122i	0.992149	−	0.125061i	$-0.0399127\pi$
$888$	0	0
$889$	−128.745	−0.144819
$890$	− 145.488i	− 0.163470i
$891$	0	0
$892$	1157.92	1.29812
$893$	− 134.714i	− 0.150856i
$894$	0	0
$895$	−34.6978	−0.0387685
$896$	− 397.082i	− 0.443172i
$897$	0	0
$898$	281.597	0.313582
$899$	− 5.75091i	− 0.00639700i
$900$	0	0
$901$	185.789	0.206203
$902$	− 2624.28i	− 2.90941i
$903$	0	0
$904$	−82.4928	−0.0912531
$905$	− 42.5758i	− 0.0470451i
$906$	0	0
$907$	680.121	0.749858	0.374929	−	0.927054i	$-0.377667\pi$
$907$	680.121	0.749858	0.374929	+	0.927054i	$0.377667\pi$
$908$	1708.17i	1.88124i
$909$	0	0
$910$	175.120	0.192440
$911$	− 1046.68i	− 1.14894i	−0.818527	−	0.574469i	$-0.805209\pi$
$911$	− 1046.68i	− 1.14894i	0.818527	−	0.574469i	$-0.194791\pi$
$912$	0	0
$913$	−1761.86	−1.92975
$914$	88.3969i	0.0967143i
$915$	0	0
$916$	683.298	0.745958
$917$	− 1782.14i	− 1.94344i
$918$	0	0
$919$	−244.736	−0.266306	−0.133153	−	0.991095i	$-0.542510\pi$
$919$	−244.736	−0.266306	−0.133153	+	0.991095i	$0.542510\pi$
$920$	− 9.97562i	− 0.0108431i
$921$	0	0
$922$	255.181	0.276769
$923$	− 211.529i	− 0.229176i
$924$	0	0
$925$	1754.17	1.89640
$926$	− 2298.08i	− 2.48172i
$927$	0	0
$928$	1847.47	1.99081
$929$	− 120.321i	− 0.129516i	−0.997901	−	0.0647582i	$-0.979372\pi$
$929$	− 120.321i	− 0.129516i	0.997901	−	0.0647582i	$-0.0206276\pi$
$930$	0	0
$931$	367.329	0.394553
$932$	− 1435.00i	− 1.53969i
$933$	0	0
$934$	−413.690	−0.442923
$935$	18.5642i	0.0198548i
$936$	0	0
$937$	412.610	0.440353	0.220176	−	0.975460i	$-0.429337\pi$
$937$	412.610	0.440353	0.220176	+	0.975460i	$0.429337\pi$
$938$	454.310i	0.484339i
$939$	0	0
$940$	48.2687	0.0513496
$941$	− 288.520i	− 0.306610i	−0.988179	−	0.153305i	$-0.951008\pi$
$941$	− 288.520i	− 0.306610i	0.988179	−	0.153305i	$-0.0489917\pi$
$942$	0	0
$943$	1802.76	1.91173
$944$	971.256i	1.02887i
$945$	0	0
$946$	1162.20	1.22854
$947$	− 301.186i	− 0.318042i	−0.987275	−	0.159021i	$-0.949166\pi$
$947$	− 301.186i	− 0.318042i	0.987275	−	0.159021i	$-0.0508338\pi$
$948$	0	0
$949$	−1881.15	−1.98224
$950$	324.295i	0.341364i
$951$	0	0
$952$	50.3162	0.0528531
$953$	791.241i	0.830263i	0.909761	+	0.415132i	$0.136265\pi$
$953$	791.241i	0.830263i	−0.909761	+	0.415132i	$0.863735\pi$
$954$	0	0
$955$	−27.3140	−0.0286010
$956$	957.462i	1.00153i
$957$	0	0
$958$	−1607.75	−1.67824
$959$	844.556i	0.880663i
$960$	0	0
$961$	−960.980	−0.999979
$962$	− 2931.59i	− 3.04739i
$963$	0	0
$964$	635.210	0.658931
$965$	5.20108i	0.00538972i
$966$	0	0
$967$	110.779	0.114560	0.0572798	−	0.998358i	$-0.481757\pi$
$967$	110.779	0.114560	0.0572798	+	0.998358i	$0.481757\pi$
$968$	37.5316i	0.0387723i
$969$	0	0
$970$	23.8671	0.0246053
$971$	1479.69i	1.52388i	0.647648	+	0.761940i	$0.275753\pi$
$971$	1479.69i	1.52388i	−0.647648	+	0.761940i	$0.724247\pi$
$972$	0	0
$973$	3021.32	3.10516
$974$	1320.63i	1.35588i
$975$	0	0
$976$	587.622	0.602072
$977$	565.276i	0.578584i	0.957241	+	0.289292i	$0.0934198\pi$
$977$	565.276i	0.578584i	−0.957241	+	0.289292i	$0.906580\pi$
$978$	0	0
$979$	−1698.64	−1.73508
$980$	131.615i	0.134301i
$981$	0	0
$982$	−1239.02	−1.26173
$983$	− 1189.08i	− 1.20965i	−0.796360	−	0.604823i	$-0.793244\pi$
$983$	− 1189.08i	− 1.20965i	0.796360	−	0.604823i	$-0.206756\pi$
$984$	0	0
$985$	115.393	0.117150
$986$	470.300i	0.476978i
$987$	0	0
$988$	283.177	0.286617
$989$	798.379i	0.807258i
$990$	0	0
$991$	−1580.49	−1.59484	−0.797421	−	0.603424i	$-0.793803\pi$
$991$	−1580.49	−1.59484	−0.797421	+	0.603424i	$0.793803\pi$
$992$	6.55748i	0.00661037i
$993$	0	0
$994$	487.172	0.490113
$995$	− 25.6474i	− 0.0257763i
$996$	0	0
$997$	−1820.34	−1.82582	−0.912908	−	0.408166i	$-0.866168\pi$
$997$	−1820.34	−1.82582	−0.912908	+	0.408166i	$0.866168\pi$
$998$	− 625.742i	− 0.626996i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.70	yes	84
3.2	odd	2	inner	1143.3.b.a.890.15	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.15	✓	84	3.2	odd	2	inner
1143.3.b.a.890.70	yes	84	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.89430i q^{2} -4.37696 q^{4} -0.368899i q^{5} -11.4242 q^{7} -1.09103i q^{8} +O(q^{10})\)	\(q+2.89430i q^{2} -4.37696 q^{4} -0.368899i q^{5} -11.4242 q^{7} -1.09103i q^{8} +1.06770 q^{10} -12.4660i q^{11} -14.3568 q^{13} -33.0651i q^{14} -14.3501 q^{16} -4.03686i q^{17} +4.50638 q^{19} +1.61466i q^{20} +36.0802 q^{22} +24.7854i q^{23} +24.8639 q^{25} -41.5529i q^{26} +50.0034 q^{28} +40.2521i q^{29} -0.142872 q^{31} -45.8975i q^{32} +11.6839 q^{34} +4.21439i q^{35} +70.5507 q^{37} +13.0428i q^{38} -0.402480 q^{40} -72.7348i q^{41} +32.2116 q^{43} +54.5630i q^{44} -71.7363 q^{46} -29.8941i q^{47} +81.5130 q^{49} +71.9636i q^{50} +62.8392 q^{52} +46.0232i q^{53} -4.59868 q^{55} +12.4642i q^{56} -116.501 q^{58} -67.6830i q^{59} -40.9491 q^{61} -0.413515i q^{62} +75.4407 q^{64} +5.29621i q^{65} -13.7399 q^{67} +17.6692i q^{68} -12.1977 q^{70} +14.7337i q^{71} +131.028 q^{73} +204.195i q^{74} -19.7242 q^{76} +142.414i q^{77} +86.0968 q^{79} +5.29373i q^{80} +210.516 q^{82} -141.334i q^{83} -1.48920 q^{85} +93.2301i q^{86} -13.6007 q^{88} -136.262i q^{89} +164.016 q^{91} -108.485i q^{92} +86.5224 q^{94} -1.66240i q^{95} +22.3537 q^{97} +235.923i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 84 * q - 160 * q^4	\( 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) \) 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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