LMFDB - Embedded newform 1143.3.b.a.890.69

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			890.69
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71611i q^{2} -3.37726 q^{4} +6.95811i q^{5} +4.31599 q^{7} +1.69142i q^{8} +O(q^{10})$	\(q+2.71611i q^{2} -3.37726 q^{4} +6.95811i q^{5} +4.31599 q^{7} +1.69142i q^{8} -18.8990 q^{10} -10.6740i q^{11} +12.5086 q^{13} +11.7227i q^{14} -18.1031 q^{16} +32.6178i q^{17} -15.7237 q^{19} -23.4994i q^{20} +28.9918 q^{22} -11.1858i q^{23} -23.4154 q^{25} +33.9747i q^{26} -14.5762 q^{28} +42.0647i q^{29} -12.3795 q^{31} -42.4045i q^{32} -88.5935 q^{34} +30.0312i q^{35} -7.23275 q^{37} -42.7075i q^{38} -11.7691 q^{40} -3.87379i q^{41} -16.1935 q^{43} +36.0489i q^{44} +30.3820 q^{46} +2.07073i q^{47} -30.3722 q^{49} -63.5987i q^{50} -42.2448 q^{52} +3.76342i q^{53} +74.2709 q^{55} +7.30016i q^{56} -114.252 q^{58} +64.3455i q^{59} -52.2457 q^{61} -33.6241i q^{62} +42.7627 q^{64} +87.0362i q^{65} +112.184 q^{67} -110.159i q^{68} -81.5680 q^{70} +56.2639i q^{71} -25.1935 q^{73} -19.6450i q^{74} +53.1032 q^{76} -46.0689i q^{77} +58.5144 q^{79} -125.964i q^{80} +10.5216 q^{82} -90.6579i q^{83} -226.958 q^{85} -43.9834i q^{86} +18.0542 q^{88} -76.9304i q^{89} +53.9869 q^{91} +37.7775i q^{92} -5.62433 q^{94} -109.408i q^{95} -41.6816 q^{97} -82.4944i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.71611i	1.35806i	0.734112	+	0.679028i	$0.237598\pi$
$2$	2.71611i	1.35806i	−0.734112	+	0.679028i	$0.762402\pi$
$3$	0	0
$3$	0	0
$4$	−3.37726	−0.844316
$5$	6.95811i	1.39162i	0.718224	+	0.695811i	$0.244955\pi$
$5$	6.95811i	1.39162i	−0.718224	+	0.695811i	$0.755045\pi$
$6$	0	0
$7$	4.31599	0.616570	0.308285	−	0.951294i	$-0.400245\pi$
$7$	4.31599	0.616570	0.308285	+	0.951294i	$0.400245\pi$
$8$	1.69142i	0.211428i
$9$	0	0
$10$	−18.8990	−1.88990
$11$	− 10.6740i	− 0.970364i	−0.874413	−	0.485182i	$-0.838753\pi$
$11$	− 10.6740i	− 0.970364i	0.874413	−	0.485182i	$-0.161247\pi$
$12$	0	0
$13$	12.5086	0.962199	0.481099	−	0.876666i	$-0.340238\pi$
$13$	12.5086	0.962199	0.481099	+	0.876666i	$0.340238\pi$
$14$	11.7227i	0.837337i
$15$	0	0
$16$	−18.1031	−1.13145
$17$	32.6178i	1.91869i	0.282232	+	0.959346i	$0.408925\pi$
$17$	32.6178i	1.91869i	−0.282232	+	0.959346i	$0.591075\pi$
$18$	0	0
$19$	−15.7237	−0.827566	−0.413783	−	0.910376i	$-0.635793\pi$
$19$	−15.7237	−0.827566	−0.413783	+	0.910376i	$0.635793\pi$
$20$	− 23.4994i	− 1.17497i
$21$	0	0
$22$	28.9918	1.31781
$23$	− 11.1858i	− 0.486341i	−0.969984	−	0.243170i	$-0.921813\pi$
$23$	− 11.1858i	− 0.486341i	0.969984	−	0.243170i	$-0.0781874\pi$
$24$	0	0
$25$	−23.4154	−0.936615
$26$	33.9747i	1.30672i
$27$	0	0
$28$	−14.5762	−0.520580
$29$	42.0647i	1.45051i	0.688482	+	0.725254i	$0.258278\pi$
$29$	42.0647i	1.45051i	−0.688482	+	0.725254i	$0.741722\pi$
$30$	0	0
$31$	−12.3795	−0.399339	−0.199669	−	0.979863i	$-0.563987\pi$
$31$	−12.3795	−0.399339	−0.199669	+	0.979863i	$0.563987\pi$
$32$	− 42.4045i	− 1.32514i
$33$	0	0
$34$	−88.5935	−2.60569
$35$	30.0312i	0.858033i
$36$	0	0
$37$	−7.23275	−0.195480	−0.0977399	−	0.995212i	$-0.531161\pi$
$37$	−7.23275	−0.195480	−0.0977399	+	0.995212i	$0.531161\pi$
$38$	− 42.7075i	− 1.12388i
$39$	0	0
$40$	−11.7691	−0.294228
$41$	− 3.87379i	− 0.0944826i	−0.998884	−	0.0472413i	$-0.984957\pi$
$41$	− 3.87379i	− 0.0944826i	0.998884	−	0.0472413i	$-0.0150430\pi$
$42$	0	0
$43$	−16.1935	−0.376593	−0.188297	−	0.982112i	$-0.560297\pi$
$43$	−16.1935	−0.376593	−0.188297	+	0.982112i	$0.560297\pi$
$44$	36.0489i	0.819293i
$45$	0	0
$46$	30.3820	0.660478
$47$	2.07073i	0.0440580i	0.999757	+	0.0220290i	$0.00701262\pi$
$47$	2.07073i	0.0440580i	−0.999757	+	0.0220290i	$0.992987\pi$
$48$	0	0
$49$	−30.3722	−0.619841
$50$	− 63.5987i	− 1.27197i
$51$	0	0
$52$	−42.2448	−0.812400
$53$	3.76342i	0.0710079i	0.999370	+	0.0355039i	$0.0113036\pi$
$53$	3.76342i	0.0710079i	−0.999370	+	0.0355039i	$0.988696\pi$
$54$	0	0
$55$	74.2709	1.35038
$56$	7.30016i	0.130360i
$57$	0	0
$58$	−114.252	−1.96987
$59$	64.3455i	1.09060i	0.838240	+	0.545301i	$0.183585\pi$
$59$	64.3455i	1.09060i	−0.838240	+	0.545301i	$0.816415\pi$
$60$	0	0
$61$	−52.2457	−0.856486	−0.428243	−	0.903664i	$-0.640867\pi$
$61$	−52.2457	−0.856486	−0.428243	+	0.903664i	$0.640867\pi$
$62$	− 33.6241i	− 0.542325i
$63$	0	0
$64$	42.7627	0.668167
$65$	87.0362i	1.33902i
$66$	0	0
$67$	112.184	1.67438	0.837192	−	0.546909i	$-0.184196\pi$
$67$	112.184	1.67438	0.837192	+	0.546909i	$0.184196\pi$
$68$	− 110.159i	− 1.61998i
$69$	0	0
$70$	−81.5680	−1.16526
$71$	56.2639i	0.792449i	0.918154	+	0.396224i	$0.129680\pi$
$71$	56.2639i	0.792449i	−0.918154	+	0.396224i	$0.870320\pi$
$72$	0	0
$73$	−25.1935	−0.345117	−0.172558	−	0.984999i	$-0.555203\pi$
$73$	−25.1935	−0.345117	−0.172558	+	0.984999i	$0.555203\pi$
$74$	− 19.6450i	− 0.265472i
$75$	0	0
$76$	53.1032	0.698727
$77$	− 46.0689i	− 0.598297i
$78$	0	0
$79$	58.5144	0.740688	0.370344	−	0.928895i	$-0.379240\pi$
$79$	58.5144	0.740688	0.370344	+	0.928895i	$0.379240\pi$
$80$	− 125.964i	− 1.57455i
$81$	0	0
$82$	10.5216	0.128313
$83$	− 90.6579i	− 1.09226i	−0.837699	−	0.546132i	$-0.816100\pi$
$83$	− 90.6579i	− 1.09226i	0.837699	−	0.546132i	$-0.183900\pi$
$84$	0	0
$85$	−226.958	−2.67010
$86$	− 43.9834i	− 0.511435i
$87$	0	0
$88$	18.0542	0.205162
$89$	− 76.9304i	− 0.864387i	−0.901781	−	0.432193i	$-0.857740\pi$
$89$	− 76.9304i	− 0.864387i	0.901781	−	0.432193i	$-0.142260\pi$
$90$	0	0
$91$	53.9869	0.593263
$92$	37.7775i	0.410625i
$93$	0	0
$94$	−5.62433	−0.0598333
$95$	− 109.408i	− 1.15166i
$96$	0	0
$97$	−41.6816	−0.429707	−0.214854	−	0.976646i	$-0.568927\pi$
$97$	−41.6816	−0.429707	−0.214854	+	0.976646i	$0.568927\pi$
$98$	− 82.4944i	− 0.841779i
$99$	0	0
$100$	79.0798	0.790798
$101$	− 18.4549i	− 0.182722i	−0.995818	−	0.0913608i	$-0.970878\pi$
$101$	− 18.4549i	− 0.182722i	0.995818	−	0.0913608i	$-0.0291216\pi$
$102$	0	0
$103$	4.37878	0.0425124	0.0212562	−	0.999774i	$-0.493233\pi$
$103$	4.37878	0.0425124	0.0212562	+	0.999774i	$0.493233\pi$
$104$	21.1573i	0.203436i
$105$	0	0
$106$	−10.2219	−0.0964326
$107$	− 120.961i	− 1.13048i	−0.824927	−	0.565240i	$-0.808784\pi$
$107$	− 120.961i	− 1.13048i	0.824927	−	0.565240i	$-0.191216\pi$
$108$	0	0
$109$	22.7798	0.208989	0.104494	−	0.994525i	$-0.466678\pi$
$109$	22.7798	0.208989	0.104494	+	0.994525i	$0.466678\pi$
$110$	201.728i	1.83389i
$111$	0	0
$112$	−78.1330	−0.697616
$113$	22.4867i	0.198997i	0.995038	+	0.0994985i	$0.0317239\pi$
$113$	22.4867i	0.198997i	−0.995038	+	0.0994985i	$0.968276\pi$
$114$	0	0
$115$	77.8323	0.676803
$116$	− 142.064i	− 1.22469i
$117$	0	0
$118$	−174.770	−1.48110
$119$	140.778i	1.18301i
$120$	0	0
$121$	7.06569	0.0583942
$122$	− 141.905i	− 1.16316i
$123$	0	0
$124$	41.8088	0.337168
$125$	11.0261i	0.0882086i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	− 53.4696i	− 0.417731i
$129$	0	0
$130$	−236.400	−1.81846
$131$	− 91.7796i	− 0.700608i	−0.936636	−	0.350304i	$-0.886078\pi$
$131$	− 91.7796i	− 0.700608i	0.936636	−	0.350304i	$-0.113922\pi$
$132$	0	0
$133$	−67.8636	−0.510252
$134$	304.703i	2.27391i
$135$	0	0
$136$	−55.1704	−0.405665
$137$	− 89.3843i	− 0.652440i	−0.945294	−	0.326220i	$-0.894225\pi$
$137$	− 89.3843i	− 0.652440i	0.945294	−	0.326220i	$-0.105775\pi$
$138$	0	0
$139$	−149.686	−1.07687	−0.538437	−	0.842666i	$-0.680985\pi$
$139$	−149.686	−1.07687	−0.538437	+	0.842666i	$0.680985\pi$
$140$	− 101.423i	− 0.724451i
$141$	0	0
$142$	−152.819	−1.07619
$143$	− 133.517i	− 0.933683i
$144$	0	0
$145$	−292.691	−2.01856
$146$	− 68.4284i	− 0.468688i
$147$	0	0
$148$	24.4269	0.165047
$149$	124.068i	0.832674i	0.909210	+	0.416337i	$0.136686\pi$
$149$	124.068i	0.832674i	−0.909210	+	0.416337i	$0.863314\pi$
$150$	0	0
$151$	216.165	1.43156	0.715779	−	0.698327i	$-0.246072\pi$
$151$	216.165	1.43156	0.715779	+	0.698327i	$0.246072\pi$
$152$	− 26.5955i	− 0.174970i
$153$	0	0
$154$	125.128	0.812521
$155$	− 86.1380i	− 0.555729i
$156$	0	0
$157$	−229.144	−1.45952	−0.729758	−	0.683706i	$-0.760367\pi$
$157$	−229.144	−1.45952	−0.729758	+	0.683706i	$0.760367\pi$
$158$	158.932i	1.00590i
$159$	0	0
$160$	295.055	1.84410
$161$	− 48.2780i	− 0.299863i
$162$	0	0
$163$	−123.348	−0.756736	−0.378368	−	0.925655i	$-0.623515\pi$
$163$	−123.348	−0.756736	−0.378368	+	0.925655i	$0.623515\pi$
$164$	13.0828i	0.0797732i
$165$	0	0
$166$	246.237	1.48336
$167$	− 63.7232i	− 0.381576i	−0.981631	−	0.190788i	$-0.938896\pi$
$167$	− 63.7232i	− 0.381576i	0.981631	−	0.190788i	$-0.0611043\pi$
$168$	0	0
$169$	−12.5353	−0.0741735
$170$	− 616.444i	− 3.62614i
$171$	0	0
$172$	54.6898	0.317964
$173$	− 35.2926i	− 0.204004i	−0.994784	−	0.102002i	$-0.967475\pi$
$173$	− 35.2926i	− 0.204004i	0.994784	−	0.102002i	$-0.0325248\pi$
$174$	0	0
$175$	−101.060	−0.577489
$176$	193.233i	1.09791i
$177$	0	0
$178$	208.952	1.17389
$179$	− 241.836i	− 1.35104i	−0.737342	−	0.675519i	$-0.763920\pi$
$179$	− 241.836i	− 1.35104i	0.737342	−	0.675519i	$-0.236080\pi$
$180$	0	0
$181$	288.741	1.59526	0.797628	−	0.603150i	$-0.206088\pi$
$181$	288.741	1.59526	0.797628	+	0.603150i	$0.206088\pi$
$182$	146.635i	0.805684i
$183$	0	0
$184$	18.9200	0.102826
$185$	− 50.3263i	− 0.272034i
$186$	0	0
$187$	348.162	1.86183
$188$	− 6.99339i	− 0.0371989i
$189$	0	0
$190$	297.163	1.56402
$191$	91.5192i	0.479158i	0.970877	+	0.239579i	$0.0770095\pi$
$191$	91.5192i	0.479158i	−0.970877	+	0.239579i	$0.922991\pi$
$192$	0	0
$193$	226.506	1.17361	0.586803	−	0.809730i	$-0.300386\pi$
$193$	226.506	1.17361	0.586803	+	0.809730i	$0.300386\pi$
$194$	− 113.212i	− 0.583566i
$195$	0	0
$196$	102.575	0.523342
$197$	272.020i	1.38081i	0.723423	+	0.690405i	$0.242568\pi$
$197$	272.020i	1.38081i	−0.723423	+	0.690405i	$0.757432\pi$
$198$	0	0
$199$	−152.048	−0.764059	−0.382029	−	0.924150i	$-0.624775\pi$
$199$	−152.048	−0.764059	−0.382029	+	0.924150i	$0.624775\pi$
$200$	− 39.6053i	− 0.198026i
$201$	0	0
$202$	50.1255	0.248146
$203$	181.551i	0.894340i
$204$	0	0
$205$	26.9543	0.131484
$206$	11.8933i	0.0577343i
$207$	0	0
$208$	−226.445	−1.08868
$209$	167.835i	0.803040i
$210$	0	0
$211$	210.301	0.996685	0.498342	−	0.866980i	$-0.333942\pi$
$211$	210.301	0.996685	0.498342	+	0.866980i	$0.333942\pi$
$212$	− 12.7100i	− 0.0599531i
$213$	0	0
$214$	328.544	1.53525
$215$	− 112.676i	− 0.524076i
$216$	0	0
$217$	−53.4298	−0.246220
$218$	61.8725i	0.283819i
$219$	0	0
$220$	−250.832	−1.14015
$221$	408.002i	1.84616i
$222$	0	0
$223$	198.451	0.889913	0.444956	−	0.895552i	$-0.353219\pi$
$223$	198.451	0.889913	0.444956	+	0.895552i	$0.353219\pi$
$224$	− 183.017i	− 0.817042i
$225$	0	0
$226$	−61.0763	−0.270249
$227$	161.325i	0.710685i	0.934736	+	0.355342i	$0.115636\pi$
$227$	161.325i	0.710685i	−0.934736	+	0.355342i	$0.884364\pi$
$228$	0	0
$229$	394.501	1.72271	0.861356	−	0.508002i	$-0.169616\pi$
$229$	394.501	1.72271	0.861356	+	0.508002i	$0.169616\pi$
$230$	211.401i	0.919136i
$231$	0	0
$232$	−71.1492	−0.306678
$233$	385.938i	1.65639i	0.560442	+	0.828194i	$0.310631\pi$
$233$	385.938i	1.65639i	−0.560442	+	0.828194i	$0.689369\pi$
$234$	0	0
$235$	−14.4084	−0.0613122
$236$	− 217.312i	− 0.920812i
$237$	0	0
$238$	−382.369	−1.60659
$239$	393.051i	1.64456i	0.569081	+	0.822281i	$0.307299\pi$
$239$	393.051i	1.64456i	−0.569081	+	0.822281i	$0.692701\pi$
$240$	0	0
$241$	80.5872	0.334387	0.167193	−	0.985924i	$-0.446530\pi$
$241$	80.5872	0.334387	0.167193	+	0.985924i	$0.446530\pi$
$242$	19.1912i	0.0793025i
$243$	0	0
$244$	176.447	0.723145
$245$	− 211.333i	− 0.862585i
$246$	0	0
$247$	−196.682	−0.796283
$248$	− 20.9390i	− 0.0844314i
$249$	0	0
$250$	−29.9481	−0.119792
$251$	156.298i	0.622699i	0.950295	+	0.311350i	$0.100781\pi$
$251$	156.298i	0.622699i	−0.950295	+	0.311350i	$0.899219\pi$
$252$	0	0
$253$	−119.398	−0.471927
$254$	− 30.6090i	− 0.120508i
$255$	0	0
$256$	316.280	1.23547
$257$	− 334.933i	− 1.30324i	−0.758546	−	0.651620i	$-0.774090\pi$
$257$	− 334.933i	− 1.30324i	0.758546	−	0.651620i	$-0.225910\pi$
$258$	0	0
$259$	−31.2165	−0.120527
$260$	− 293.944i	− 1.13055i
$261$	0	0
$262$	249.284	0.951465
$263$	403.738i	1.53513i	0.640973	+	0.767563i	$0.278531\pi$
$263$	403.738i	1.53513i	−0.640973	+	0.767563i	$0.721469\pi$
$264$	0	0
$265$	−26.1863	−0.0988162
$266$	− 184.325i	− 0.692951i
$267$	0	0
$268$	−378.874	−1.41371
$269$	96.1414i	0.357403i	0.983903	+	0.178701i	$0.0571896\pi$
$269$	96.1414i	0.357403i	−0.983903	+	0.178701i	$0.942810\pi$
$270$	0	0
$271$	189.580	0.699556	0.349778	−	0.936833i	$-0.386257\pi$
$271$	189.580	0.699556	0.349778	+	0.936833i	$0.386257\pi$
$272$	− 590.484i	− 2.17090i
$273$	0	0
$274$	242.778	0.886050
$275$	249.936i	0.908857i
$276$	0	0
$277$	171.642	0.619646	0.309823	−	0.950794i	$-0.399730\pi$
$277$	171.642	0.619646	0.309823	+	0.950794i	$0.399730\pi$
$278$	− 406.563i	− 1.46246i
$279$	0	0
$280$	−50.7954	−0.181412
$281$	45.9693i	0.163592i	0.996649	+	0.0817959i	$0.0260656\pi$
$281$	45.9693i	0.163592i	−0.996649	+	0.0817959i	$0.973934\pi$
$282$	0	0
$283$	−130.208	−0.460098	−0.230049	−	0.973179i	$-0.573889\pi$
$283$	−130.208	−0.460098	−0.230049	+	0.973179i	$0.573889\pi$
$284$	− 190.018i	− 0.669077i
$285$	0	0
$286$	362.646	1.26799
$287$	− 16.7192i	− 0.0582552i
$288$	0	0
$289$	−774.919	−2.68138
$290$	− 794.982i	− 2.74132i
$291$	0	0
$292$	85.0851	0.291387
$293$	500.515i	1.70824i	0.520075	+	0.854121i	$0.325904\pi$
$293$	500.515i	1.70824i	−0.520075	+	0.854121i	$0.674096\pi$
$294$	0	0
$295$	−447.723	−1.51771
$296$	− 12.2336i	− 0.0413299i
$297$	0	0
$298$	−336.984	−1.13082
$299$	− 139.919i	− 0.467956i
$300$	0	0
$301$	−69.8911	−0.232196
$302$	587.129i	1.94414i
$303$	0	0
$304$	284.649	0.936347
$305$	− 363.531i	− 1.19191i
$306$	0	0
$307$	65.9196	0.214722	0.107361	−	0.994220i	$-0.465760\pi$
$307$	65.9196	0.214722	0.107361	+	0.994220i	$0.465760\pi$
$308$	155.587i	0.505152i
$309$	0	0
$310$	233.961	0.754711
$311$	− 309.588i	− 0.995459i	−0.867332	−	0.497730i	$-0.834167\pi$
$311$	− 309.588i	− 0.995459i	0.867332	−	0.497730i	$-0.165833\pi$
$312$	0	0
$313$	147.814	0.472248	0.236124	−	0.971723i	$-0.424123\pi$
$313$	147.814	0.472248	0.236124	+	0.971723i	$0.424123\pi$
$314$	− 622.381i	− 1.98210i
$315$	0	0
$316$	−197.618	−0.625375
$317$	− 148.170i	− 0.467413i	−0.972307	−	0.233706i	$-0.924915\pi$
$317$	− 148.170i	− 0.467413i	0.972307	−	0.233706i	$-0.0750854\pi$
$318$	0	0
$319$	448.999	1.40752
$320$	297.548i	0.929837i
$321$	0	0
$322$	131.128	0.407231
$323$	− 512.874i	− 1.58784i
$324$	0	0
$325$	−292.893	−0.901209
$326$	− 335.027i	− 1.02769i
$327$	0	0
$328$	6.55221	0.0199763
$329$	8.93724i	0.0271649i
$330$	0	0
$331$	−286.761	−0.866348	−0.433174	−	0.901310i	$-0.642606\pi$
$331$	−286.761	−0.866348	−0.433174	+	0.901310i	$0.642606\pi$
$332$	306.176i	0.922216i
$333$	0	0
$334$	173.079	0.518201
$335$	780.587i	2.33011i
$336$	0	0
$337$	−480.605	−1.42613	−0.713063	−	0.701100i	$-0.752693\pi$
$337$	−480.605	−1.42613	−0.713063	+	0.701100i	$0.752693\pi$
$338$	− 34.0473i	− 0.100732i
$339$	0	0
$340$	766.498	2.25440
$341$	132.139i	0.387504i
$342$	0	0
$343$	−342.570	−0.998746
$344$	− 27.3901i	− 0.0796223i
$345$	0	0
$346$	95.8588	0.277049
$347$	421.056i	1.21342i	0.794924	+	0.606709i	$0.207511\pi$
$347$	421.056i	1.21342i	−0.794924	+	0.606709i	$0.792489\pi$
$348$	0	0
$349$	−192.726	−0.552224	−0.276112	−	0.961125i	$-0.589046\pi$
$349$	−192.726	−0.552224	−0.276112	+	0.961125i	$0.589046\pi$
$350$	− 274.492i	− 0.784262i
$351$	0	0
$352$	−452.625	−1.28587
$353$	687.481i	1.94754i	0.227541	+	0.973769i	$0.426931\pi$
$353$	687.481i	1.94754i	−0.227541	+	0.973769i	$0.573069\pi$
$354$	0	0
$355$	−391.490	−1.10279
$356$	259.814i	0.729815i
$357$	0	0
$358$	656.853	1.83479
$359$	620.707i	1.72899i	0.502643	+	0.864494i	$0.332361\pi$
$359$	620.707i	1.72899i	−0.502643	+	0.864494i	$0.667639\pi$
$360$	0	0
$361$	−113.764	−0.315135
$362$	784.254i	2.16645i
$363$	0	0
$364$	−182.328	−0.500901
$365$	− 175.299i	− 0.480272i
$366$	0	0
$367$	189.875	0.517371	0.258686	−	0.965962i	$-0.416711\pi$
$367$	189.875	0.517371	0.258686	+	0.965962i	$0.416711\pi$
$368$	202.499i	0.550268i
$369$	0	0
$370$	136.692	0.369438
$371$	16.2429i	0.0437813i
$372$	0	0
$373$	439.303	1.17776	0.588878	−	0.808222i	$-0.299570\pi$
$373$	439.303	1.17776	0.588878	+	0.808222i	$0.299570\pi$
$374$	945.647i	2.52847i
$375$	0	0
$376$	−3.50247	−0.00931509
$377$	526.170i	1.39568i
$378$	0	0
$379$	−514.039	−1.35630	−0.678152	−	0.734922i	$-0.737219\pi$
$379$	−514.039	−1.35630	−0.678152	+	0.734922i	$0.737219\pi$
$380$	369.498i	0.972364i
$381$	0	0
$382$	−248.576	−0.650724
$383$	− 400.504i	− 1.04570i	−0.852424	−	0.522851i	$-0.824868\pi$
$383$	− 400.504i	− 1.04570i	0.852424	−	0.522851i	$-0.175132\pi$
$384$	0	0
$385$	320.553	0.832604
$386$	615.215i	1.59382i
$387$	0	0
$388$	140.770	0.362808
$389$	714.324i	1.83631i	0.396222	+	0.918155i	$0.370321\pi$
$389$	714.324i	1.83631i	−0.396222	+	0.918155i	$0.629679\pi$
$390$	0	0
$391$	364.857	0.933138
$392$	− 51.3723i	− 0.131052i
$393$	0	0
$394$	−738.836	−1.87522
$395$	407.150i	1.03076i
$396$	0	0
$397$	45.5606	0.114762	0.0573811	−	0.998352i	$-0.481725\pi$
$397$	45.5606	0.114762	0.0573811	+	0.998352i	$0.481725\pi$
$398$	− 412.979i	− 1.03763i
$399$	0	0
$400$	423.892	1.05973
$401$	− 378.312i	− 0.943423i	−0.881753	−	0.471711i	$-0.843636\pi$
$401$	− 378.312i	− 0.943423i	0.881753	−	0.471711i	$-0.156364\pi$
$402$	0	0
$403$	−154.850	−0.384243
$404$	62.3270i	0.154275i
$405$	0	0
$406$	−493.113	−1.21456
$407$	77.2024i	0.189686i
$408$	0	0
$409$	−226.210	−0.553080	−0.276540	−	0.961002i	$-0.589188\pi$
$409$	−226.210	−0.553080	−0.276540	+	0.961002i	$0.589188\pi$
$410$	73.2108i	0.178563i
$411$	0	0
$412$	−14.7883	−0.0358939
$413$	277.715i	0.672432i
$414$	0	0
$415$	630.808	1.52002
$416$	− 530.420i	− 1.27505i
$417$	0	0
$418$	−455.860	−1.09057
$419$	− 36.9462i	− 0.0881771i	−0.999028	−	0.0440885i	$-0.985962\pi$
$419$	− 36.9462i	− 0.0881771i	0.999028	−	0.0440885i	$-0.0140384\pi$
$420$	0	0
$421$	162.731	0.386535	0.193268	−	0.981146i	$-0.438091\pi$
$421$	162.731	0.386535	0.193268	+	0.981146i	$0.438091\pi$
$422$	571.200i	1.35355i
$423$	0	0
$424$	−6.36553	−0.0150130
$425$	− 763.757i	− 1.79708i
$426$	0	0
$427$	−225.492	−0.528084
$428$	408.518i	0.954482i
$429$	0	0
$430$	306.042	0.711725
$431$	− 680.810i	− 1.57960i	−0.613361	−	0.789802i	$-0.710183\pi$
$431$	− 680.810i	− 1.57960i	0.613361	−	0.789802i	$-0.289817\pi$
$432$	0	0
$433$	−43.8082	−0.101174	−0.0505869	−	0.998720i	$-0.516109\pi$
$433$	−43.8082	−0.101174	−0.0505869	+	0.998720i	$0.516109\pi$
$434$	− 145.121i	− 0.334381i
$435$	0	0
$436$	−76.9334	−0.176453
$437$	175.883i	0.402479i
$438$	0	0
$439$	−208.320	−0.474532	−0.237266	−	0.971445i	$-0.576251\pi$
$439$	−208.320	−0.474532	−0.237266	+	0.971445i	$0.576251\pi$
$440$	125.624i	0.285508i
$441$	0	0
$442$	−1108.18	−2.50719
$443$	63.6117i	0.143593i	0.997419	+	0.0717965i	$0.0228732\pi$
$443$	63.6117i	0.143593i	−0.997419	+	0.0717965i	$0.977127\pi$
$444$	0	0
$445$	535.291	1.20290
$446$	539.014i	1.20855i
$447$	0	0
$448$	184.563	0.411972
$449$	803.599i	1.78975i	0.446314	+	0.894876i	$0.352736\pi$
$449$	803.599i	1.78975i	−0.446314	+	0.894876i	$0.647264\pi$
$450$	0	0
$451$	−41.3488	−0.0916825
$452$	− 75.9434i	− 0.168016i
$453$	0	0
$454$	−438.178	−0.965150
$455$	375.647i	0.825598i
$456$	0	0
$457$	576.381	1.26123	0.630613	−	0.776097i	$-0.282803\pi$
$457$	576.381	1.26123	0.630613	+	0.776097i	$0.282803\pi$
$458$	1071.51i	2.33954i
$459$	0	0
$460$	−262.860	−0.571435
$461$	309.110i	0.670521i	0.942126	+	0.335260i	$0.108824\pi$
$461$	309.110i	0.670521i	−0.942126	+	0.335260i	$0.891176\pi$
$462$	0	0
$463$	699.328	1.51043	0.755214	−	0.655479i	$-0.227533\pi$
$463$	699.328	1.51043	0.755214	+	0.655479i	$0.227533\pi$
$464$	− 761.504i	− 1.64117i
$465$	0	0
$466$	−1048.25	−2.24947
$467$	− 785.231i	− 1.68144i	−0.541472	−	0.840719i	$-0.682133\pi$
$467$	− 785.231i	− 1.68144i	0.541472	−	0.840719i	$-0.317867\pi$
$468$	0	0
$469$	484.184	1.03237
$470$	− 39.1347i	− 0.0832653i
$471$	0	0
$472$	−108.835	−0.230584
$473$	172.850i	0.365433i
$474$	0	0
$475$	368.177	0.775110
$476$	− 475.444i	− 0.998833i
$477$	0	0
$478$	−1067.57	−2.23341
$479$	404.198i	0.843836i	0.906634	+	0.421918i	$0.138643\pi$
$479$	404.198i	0.843836i	−0.906634	+	0.421918i	$0.861357\pi$
$480$	0	0
$481$	−90.4715	−0.188090
$482$	218.884i	0.454116i
$483$	0	0
$484$	−23.8627	−0.0493031
$485$	− 290.025i	− 0.597990i
$486$	0	0
$487$	−301.492	−0.619081	−0.309541	−	0.950886i	$-0.600175\pi$
$487$	−301.492	−0.619081	−0.309541	+	0.950886i	$0.600175\pi$
$488$	− 88.3695i	− 0.181085i
$489$	0	0
$490$	574.005	1.17144
$491$	825.810i	1.68189i	0.541118	+	0.840947i	$0.318001\pi$
$491$	825.810i	1.68189i	−0.541118	+	0.840947i	$0.681999\pi$
$492$	0	0
$493$	−1372.06	−2.78308
$494$	− 534.210i	− 1.08140i
$495$	0	0
$496$	224.108	0.451831
$497$	242.834i	0.488600i
$498$	0	0
$499$	814.889	1.63304	0.816522	−	0.577315i	$-0.195899\pi$
$499$	814.889	1.63304	0.816522	+	0.577315i	$0.195899\pi$
$500$	− 37.2380i	− 0.0744760i
$501$	0	0
$502$	−424.522	−0.845661
$503$	− 727.680i	− 1.44668i	−0.690493	−	0.723340i	$-0.742606\pi$
$503$	− 727.680i	− 1.44668i	0.690493	−	0.723340i	$-0.257394\pi$
$504$	0	0
$505$	128.411	0.254280
$506$	− 324.297i	− 0.640904i
$507$	0	0
$508$	38.0598	0.0749209
$509$	81.3689i	0.159860i	0.996800	+	0.0799302i	$0.0254697\pi$
$509$	81.3689i	0.159860i	−0.996800	+	0.0799302i	$0.974530\pi$
$510$	0	0
$511$	−108.735	−0.212789
$512$	645.174i	1.26011i
$513$	0	0
$514$	909.714	1.76987
$515$	30.4681i	0.0591613i
$516$	0	0
$517$	22.1029	0.0427523
$518$	− 84.7875i	− 0.163682i
$519$	0	0
$520$	−147.215	−0.283106
$521$	− 980.158i	− 1.88130i	−0.339376	−	0.940651i	$-0.610216\pi$
$521$	− 980.158i	− 1.88130i	0.339376	−	0.940651i	$-0.389784\pi$
$522$	0	0
$523$	7.12508	0.0136235	0.00681174	−	0.999977i	$-0.497832\pi$
$523$	7.12508	0.0136235	0.00681174	+	0.999977i	$0.497832\pi$
$524$	309.964i	0.591534i
$525$	0	0
$526$	−1096.60	−2.08479
$527$	− 403.792i	− 0.766209i
$528$	0	0
$529$	403.877	0.763473
$530$	− 71.1249i	− 0.134198i
$531$	0	0
$532$	229.193	0.430814
$533$	− 48.4556i	− 0.0909111i
$534$	0	0
$535$	841.663	1.57320
$536$	189.750i	0.354011i
$537$	0	0
$538$	−261.131	−0.485373
$539$	324.193i	0.601472i
$540$	0	0
$541$	880.813	1.62812	0.814060	−	0.580780i	$-0.197252\pi$
$541$	880.813	1.62812	0.814060	+	0.580780i	$0.197252\pi$
$542$	514.919i	0.950036i
$543$	0	0
$544$	1383.14	2.54254
$545$	158.504i	0.290834i
$546$	0	0
$547$	1050.90	1.92121	0.960607	−	0.277910i	$-0.0896417\pi$
$547$	1050.90	1.92121	0.960607	+	0.277910i	$0.0896417\pi$
$548$	301.874i	0.550865i
$549$	0	0
$550$	−678.853	−1.23428
$551$	− 661.415i	− 1.20039i
$552$	0	0
$553$	252.548	0.456686
$554$	466.199i	0.841514i
$555$	0	0
$556$	505.527	0.909222
$557$	− 1054.84i	− 1.89379i	−0.321544	−	0.946895i	$-0.604202\pi$
$557$	− 1054.84i	− 1.89379i	0.321544	−	0.946895i	$-0.395798\pi$
$558$	0	0
$559$	−202.558	−0.362358
$560$	− 543.658i	− 0.970819i
$561$	0	0
$562$	−124.858	−0.222167
$563$	125.889i	0.223603i	0.993731	+	0.111802i	$0.0356621\pi$
$563$	125.889i	0.223603i	−0.993731	+	0.111802i	$0.964338\pi$
$564$	0	0
$565$	−156.465	−0.276929
$566$	− 353.659i	− 0.624839i
$567$	0	0
$568$	−95.1660	−0.167546
$569$	68.3275i	0.120083i	0.998196	+	0.0600417i	$0.0191234\pi$
$569$	68.3275i	0.120083i	−0.998196	+	0.0600417i	$0.980877\pi$
$570$	0	0
$571$	−517.216	−0.905807	−0.452904	−	0.891559i	$-0.649612\pi$
$571$	−517.216	−0.905807	−0.452904	+	0.891559i	$0.649612\pi$
$572$	450.921i	0.788323i
$573$	0	0
$574$	45.4113	0.0791138
$575$	261.920i	0.455514i
$576$	0	0
$577$	724.772	1.25610	0.628052	−	0.778171i	$-0.283852\pi$
$577$	724.772	1.25610	0.628052	+	0.778171i	$0.283852\pi$
$578$	− 2104.77i	− 3.64147i
$579$	0	0
$580$	988.495	1.70430
$581$	− 391.279i	− 0.673458i
$582$	0	0
$583$	40.1707	0.0689035
$584$	− 42.6129i	− 0.0729673i
$585$	0	0
$586$	−1359.45	−2.31989
$587$	− 96.7186i	− 0.164768i	−0.996601	−	0.0823838i	$-0.973747\pi$
$587$	− 96.7186i	− 0.164768i	0.996601	−	0.0823838i	$-0.0262533\pi$
$588$	0	0
$589$	194.652	0.330479
$590$	− 1216.07i	− 2.06113i
$591$	0	0
$592$	130.936	0.221175
$593$	− 644.219i	− 1.08637i	−0.839612	−	0.543186i	$-0.817218\pi$
$593$	− 644.219i	− 1.08637i	0.839612	−	0.543186i	$-0.182782\pi$
$594$	0	0
$595$	−979.550	−1.64630
$596$	− 419.012i	− 0.703040i
$597$	0	0
$598$	380.036	0.635511
$599$	756.993i	1.26376i	0.775066	+	0.631881i	$0.217717\pi$
$599$	756.993i	1.26376i	−0.775066	+	0.631881i	$0.782283\pi$
$600$	0	0
$601$	−898.672	−1.49529	−0.747647	−	0.664096i	$-0.768817\pi$
$601$	−898.672	−1.49529	−0.747647	+	0.664096i	$0.768817\pi$
$602$	− 189.832i	− 0.315335i
$603$	0	0
$604$	−730.047	−1.20869
$605$	49.1639i	0.0812627i
$606$	0	0
$607$	1046.26	1.72366	0.861832	−	0.507193i	$-0.169317\pi$
$607$	1046.26	1.72366	0.861832	+	0.507193i	$0.169317\pi$
$608$	666.757i	1.09664i
$609$	0	0
$610$	987.392	1.61867
$611$	25.9019i	0.0423926i
$612$	0	0
$613$	−328.003	−0.535078	−0.267539	−	0.963547i	$-0.586210\pi$
$613$	−328.003	−0.535078	−0.267539	+	0.963547i	$0.586210\pi$
$614$	179.045i	0.291604i
$615$	0	0
$616$	77.9220	0.126497
$617$	638.293i	1.03451i	0.855831	+	0.517255i	$0.173046\pi$
$617$	638.293i	1.03451i	−0.855831	+	0.517255i	$0.826954\pi$
$618$	0	0
$619$	713.100	1.15202	0.576010	−	0.817443i	$-0.304609\pi$
$619$	713.100	1.15202	0.576010	+	0.817443i	$0.304609\pi$
$620$	290.911i	0.469211i
$621$	0	0
$622$	840.875	1.35189
$623$	− 332.031i	− 0.532955i
$624$	0	0
$625$	−662.105	−1.05937
$626$	401.478i	0.641339i
$627$	0	0
$628$	773.879	1.23229
$629$	− 235.916i	− 0.375066i
$630$	0	0
$631$	376.709	0.597003	0.298502	−	0.954409i	$-0.403513\pi$
$631$	376.709	0.597003	0.298502	+	0.954409i	$0.403513\pi$
$632$	98.9725i	0.156602i
$633$	0	0
$634$	402.446	0.634772
$635$	− 78.4140i	− 0.123487i
$636$	0	0
$637$	−379.914	−0.596411
$638$	1219.53i	1.91149i
$639$	0	0
$640$	372.048	0.581324
$641$	− 7.88021i	− 0.0122936i	−0.999981	−	0.00614681i	$-0.998043\pi$
$641$	− 7.88021i	− 0.0122936i	0.999981	−	0.00614681i	$-0.00195660\pi$
$642$	0	0
$643$	115.955	0.180335	0.0901674	−	0.995927i	$-0.471260\pi$
$643$	115.955	0.180335	0.0901674	+	0.995927i	$0.471260\pi$
$644$	163.047i	0.253179i
$645$	0	0
$646$	1393.02	2.15638
$647$	− 509.586i	− 0.787613i	−0.919193	−	0.393806i	$-0.871158\pi$
$647$	− 509.586i	− 0.787613i	0.919193	−	0.393806i	$-0.128842\pi$
$648$	0	0
$649$	686.824	1.05828
$650$	− 795.530i	− 1.22389i
$651$	0	0
$652$	416.579	0.638924
$653$	430.316i	0.658983i	0.944159	+	0.329491i	$0.106877\pi$
$653$	430.316i	0.658983i	−0.944159	+	0.329491i	$0.893123\pi$
$654$	0	0
$655$	638.613	0.974982
$656$	70.1277i	0.106902i
$657$	0	0
$658$	−24.2745	−0.0368914
$659$	497.970i	0.755646i	0.925878	+	0.377823i	$0.123327\pi$
$659$	497.970i	0.755646i	−0.925878	+	0.377823i	$0.876673\pi$
$660$	0	0
$661$	−55.0164	−0.0832321	−0.0416161	−	0.999134i	$-0.513251\pi$
$661$	−55.0164	−0.0832321	−0.0416161	+	0.999134i	$0.513251\pi$
$662$	− 778.875i	− 1.17655i
$663$	0	0
$664$	153.341	0.230935
$665$	− 472.202i	− 0.710079i
$666$	0	0
$667$	470.529	0.705441
$668$	215.210i	0.322171i
$669$	0	0
$670$	−2120.16	−3.16442
$671$	557.670i	0.831103i
$672$	0	0
$673$	819.232	1.21728	0.608642	−	0.793445i	$-0.291715\pi$
$673$	819.232	1.21728	0.608642	+	0.793445i	$0.291715\pi$
$674$	− 1305.38i	− 1.93676i
$675$	0	0
$676$	42.3351	0.0626258
$677$	876.071i	1.29405i	0.762469	+	0.647025i	$0.223987\pi$
$677$	876.071i	1.29405i	−0.762469	+	0.647025i	$0.776013\pi$
$678$	0	0
$679$	−179.897	−0.264945
$680$	− 383.882i	− 0.564533i
$681$	0	0
$682$	−358.904	−0.526252
$683$	− 1239.06i	− 1.81414i	−0.420978	−	0.907071i	$-0.638313\pi$
$683$	− 1239.06i	− 1.81414i	0.420978	−	0.907071i	$-0.361687\pi$
$684$	0	0
$685$	621.946	0.907950
$686$	− 930.458i	− 1.35635i
$687$	0	0
$688$	293.154	0.426095
$689$	47.0750i	0.0683237i
$690$	0	0
$691$	−737.495	−1.06729	−0.533643	−	0.845710i	$-0.679178\pi$
$691$	−737.495	−1.06729	−0.533643	+	0.845710i	$0.679178\pi$
$692$	119.193i	0.172244i
$693$	0	0
$694$	−1143.63	−1.64789
$695$	− 1041.53i	− 1.49860i
$696$	0	0
$697$	126.354	0.181283
$698$	− 523.466i	− 0.749951i
$699$	0	0
$700$	341.308	0.487583
$701$	− 86.5794i	− 0.123508i	−0.998091	−	0.0617542i	$-0.980330\pi$
$701$	− 86.5794i	− 0.123508i	0.998091	−	0.0617542i	$-0.0196695\pi$
$702$	0	0
$703$	113.726	0.161772
$704$	− 456.449i	− 0.648365i
$705$	0	0
$706$	−1867.27	−2.64486
$707$	− 79.6511i	− 0.112661i
$708$	0	0
$709$	−879.191	−1.24004	−0.620022	−	0.784585i	$-0.712876\pi$
$709$	−879.191	−1.24004	−0.620022	+	0.784585i	$0.712876\pi$
$710$	− 1063.33i	− 1.49765i
$711$	0	0
$712$	130.122	0.182755
$713$	138.475i	0.194215i
$714$	0	0
$715$	929.024	1.29933
$716$	816.743i	1.14070i
$717$	0	0
$718$	−1685.91	−2.34806
$719$	− 553.678i	− 0.770066i	−0.922903	−	0.385033i	$-0.874190\pi$
$719$	− 553.678i	− 0.770066i	0.922903	−	0.385033i	$-0.125810\pi$
$720$	0	0
$721$	18.8988	0.0262119
$722$	− 308.995i	− 0.427971i
$723$	0	0
$724$	−975.155	−1.34690
$725$	− 984.961i	− 1.35857i
$726$	0	0
$727$	8.52118	0.0117210	0.00586051	−	0.999983i	$-0.498135\pi$
$727$	8.52118	0.0117210	0.00586051	+	0.999983i	$0.498135\pi$
$728$	91.3147i	0.125432i
$729$	0	0
$730$	476.133	0.652237
$731$	− 528.196i	− 0.722567i
$732$	0	0
$733$	−1240.93	−1.69294	−0.846472	−	0.532433i	$-0.821278\pi$
$733$	−1240.93	−1.69294	−0.846472	+	0.532433i	$0.821278\pi$
$734$	515.722i	0.702619i
$735$	0	0
$736$	−474.329	−0.644469
$737$	− 1197.45i	− 1.62476i
$738$	0	0
$739$	−236.646	−0.320225	−0.160113	−	0.987099i	$-0.551186\pi$
$739$	−236.646	−0.320225	−0.160113	+	0.987099i	$0.551186\pi$
$740$	169.965i	0.229683i
$741$	0	0
$742$	−44.1175	−0.0594575
$743$	962.986i	1.29608i	0.761607	+	0.648039i	$0.224411\pi$
$743$	962.986i	1.29608i	−0.761607	+	0.648039i	$0.775589\pi$
$744$	0	0
$745$	−863.282	−1.15877
$746$	1193.20i	1.59946i
$747$	0	0
$748$	−1175.84	−1.57197
$749$	− 522.068i	− 0.697020i
$750$	0	0
$751$	192.651	0.256526	0.128263	−	0.991740i	$-0.459060\pi$
$751$	192.651	0.256526	0.128263	+	0.991740i	$0.459060\pi$
$752$	− 37.4867i	− 0.0498493i
$753$	0	0
$754$	−1429.14	−1.89541
$755$	1504.10i	1.99219i
$756$	0	0
$757$	−473.144	−0.625026	−0.312513	−	0.949914i	$-0.601171\pi$
$757$	−473.144	−0.625026	−0.312513	+	0.949914i	$0.601171\pi$
$758$	− 1396.19i	− 1.84194i
$759$	0	0
$760$	185.055	0.243493
$761$	306.374i	0.402593i	0.979530	+	0.201297i	$0.0645155\pi$
$761$	306.374i	0.402593i	−0.979530	+	0.201297i	$0.935484\pi$
$762$	0	0
$763$	98.3174	0.128856
$764$	− 309.085i	− 0.404561i
$765$	0	0
$766$	1087.81	1.42012
$767$	804.871i	1.04938i
$768$	0	0
$769$	1329.71	1.72914	0.864572	−	0.502508i	$-0.167589\pi$
$769$	1329.71	1.72914	0.864572	+	0.502508i	$0.167589\pi$
$770$	870.657i	1.13072i
$771$	0	0
$772$	−764.970	−0.990894
$773$	− 1473.99i	− 1.90685i	−0.301634	−	0.953424i	$-0.597532\pi$
$773$	− 1473.99i	− 1.90685i	0.301634	−	0.953424i	$-0.402468\pi$
$774$	0	0
$775$	289.871	0.374027
$776$	− 70.5012i	− 0.0908520i
$777$	0	0
$778$	−1940.18	−2.49381
$779$	60.9105i	0.0781906i
$780$	0	0
$781$	600.561	0.768964
$782$	990.992i	1.26725i
$783$	0	0
$784$	549.833	0.701317
$785$	− 1594.41i	− 2.03110i
$786$	0	0
$787$	542.649	0.689516	0.344758	−	0.938692i	$-0.387961\pi$
$787$	542.649	0.689516	0.344758	+	0.938692i	$0.387961\pi$
$788$	− 918.682i	− 1.16584i
$789$	0	0
$790$	−1105.86	−1.39983
$791$	97.0523i	0.122696i
$792$	0	0
$793$	−653.519	−0.824110
$794$	123.748i	0.155853i
$795$	0	0
$796$	513.505	0.645107
$797$	− 1010.34i	− 1.26768i	−0.773462	−	0.633842i	$-0.781477\pi$
$797$	− 1010.34i	− 1.26768i	0.773462	−	0.633842i	$-0.218523\pi$
$798$	0	0
$799$	−67.5425	−0.0845338
$800$	992.916i	1.24115i
$801$	0	0
$802$	1027.54	1.28122
$803$	268.916i	0.334889i
$804$	0	0
$805$	335.924	0.417296
$806$	− 420.590i	− 0.521824i
$807$	0	0
$808$	31.2150	0.0386324
$809$	476.543i	0.589051i	0.955644	+	0.294526i	$0.0951616\pi$
$809$	476.543i	0.589051i	−0.955644	+	0.294526i	$0.904838\pi$
$810$	0	0
$811$	−1459.91	−1.80014	−0.900069	−	0.435748i	$-0.856484\pi$
$811$	−1459.91	−1.80014	−0.900069	+	0.435748i	$0.856484\pi$
$812$	− 613.145i	− 0.755105i
$813$	0	0
$814$	−209.690	−0.257605
$815$	− 858.269i	− 1.05309i
$816$	0	0
$817$	254.623	0.311656
$818$	− 614.411i	− 0.751113i
$819$	0	0
$820$	−91.0316	−0.111014
$821$	− 990.138i	− 1.20602i	−0.797735	−	0.603008i	$-0.793969\pi$
$821$	− 990.138i	− 1.20602i	0.797735	−	0.603008i	$-0.206031\pi$
$822$	0	0
$823$	−108.153	−0.131413	−0.0657065	−	0.997839i	$-0.520930\pi$
$823$	−108.153	−0.131413	−0.0657065	+	0.997839i	$0.520930\pi$
$824$	7.40637i	0.00898831i
$825$	0	0
$826$	−754.304	−0.913201
$827$	755.525i	0.913573i	0.889576	+	0.456787i	$0.151000\pi$
$827$	755.525i	0.913573i	−0.889576	+	0.456787i	$0.849000\pi$
$828$	0	0
$829$	603.366	0.727824	0.363912	−	0.931433i	$-0.381441\pi$
$829$	603.366	0.727824	0.363912	+	0.931433i	$0.381441\pi$
$830$	1713.35i	2.06427i
$831$	0	0
$832$	534.901	0.642910
$833$	− 990.674i	− 1.18928i
$834$	0	0
$835$	443.393	0.531010
$836$	− 566.824i	− 0.678019i
$837$	0	0
$838$	100.350	0.119749
$839$	81.2325i	0.0968206i	0.998828	+	0.0484103i	$0.0154155\pi$
$839$	81.2325i	0.0968206i	−0.998828	+	0.0484103i	$0.984585\pi$
$840$	0	0
$841$	−928.440	−1.10397
$842$	441.996i	0.524936i
$843$	0	0
$844$	−710.240	−0.841517
$845$	− 87.2222i	− 0.103222i
$846$	0	0
$847$	30.4955	0.0360041
$848$	− 68.1297i	− 0.0803416i
$849$	0	0
$850$	2074.45	2.44053
$851$	80.9044i	0.0950698i
$852$	0	0
$853$	532.064	0.623756	0.311878	−	0.950122i	$-0.399042\pi$
$853$	532.064	0.623756	0.311878	+	0.950122i	$0.399042\pi$
$854$	− 612.461i	− 0.717167i
$855$	0	0
$856$	204.597	0.239015
$857$	98.9583i	0.115471i	0.998332	+	0.0577353i	$0.0183879\pi$
$857$	98.9583i	0.115471i	−0.998332	+	0.0577353i	$0.981612\pi$
$858$	0	0
$859$	−124.582	−0.145032	−0.0725158	−	0.997367i	$-0.523103\pi$
$859$	−124.582	−0.145032	−0.0725158	+	0.997367i	$0.523103\pi$
$860$	380.538i	0.442486i
$861$	0	0
$862$	1849.16	2.14519
$863$	1163.67i	1.34840i	0.738547	+	0.674202i	$0.235512\pi$
$863$	1163.67i	1.34840i	−0.738547	+	0.674202i	$0.764488\pi$
$864$	0	0
$865$	245.570	0.283896
$866$	− 118.988i	− 0.137400i
$867$	0	0
$868$	180.447	0.207888
$869$	− 624.583i	− 0.718737i
$870$	0	0
$871$	1403.26	1.61109
$872$	38.5303i	0.0441861i
$873$	0	0
$874$	−477.719	−0.546589
$875$	47.5885i	0.0543868i
$876$	0	0
$877$	93.2868	0.106370	0.0531852	−	0.998585i	$-0.483063\pi$
$877$	93.2868	0.106370	0.0531852	+	0.998585i	$0.483063\pi$
$878$	− 565.819i	− 0.644441i
$879$	0	0
$880$	−1344.54	−1.52788
$881$	− 674.774i	− 0.765918i	−0.923765	−	0.382959i	$-0.874905\pi$
$881$	− 674.774i	− 0.765918i	0.923765	−	0.382959i	$-0.125095\pi$
$882$	0	0
$883$	−595.635	−0.674558	−0.337279	−	0.941405i	$-0.609507\pi$
$883$	−595.635	−0.674558	−0.337279	+	0.941405i	$0.609507\pi$
$884$	− 1377.93i	− 1.55875i
$885$	0	0
$886$	−172.776	−0.195007
$887$	− 432.077i	− 0.487121i	−0.969886	−	0.243561i	$-0.921684\pi$
$887$	− 432.077i	− 0.487121i	0.969886	−	0.243561i	$-0.0783155\pi$
$888$	0	0
$889$	−48.6387	−0.0547118
$890$	1453.91i	1.63361i
$891$	0	0
$892$	−670.220	−0.751368
$893$	− 32.5596i	− 0.0364609i
$894$	0	0
$895$	1682.72	1.88014
$896$	− 230.774i	− 0.257561i
$897$	0	0
$898$	−2182.66	−2.43058
$899$	− 520.740i	− 0.579244i
$900$	0	0
$901$	−122.754	−0.136242
$902$	− 112.308i	− 0.124510i
$903$	0	0
$904$	−38.0345	−0.0420735
$905$	2009.10i	2.21999i
$906$	0	0
$907$	1369.79	1.51025	0.755123	−	0.655583i	$-0.227577\pi$
$907$	1369.79	1.51025	0.755123	+	0.655583i	$0.227577\pi$
$908$	− 544.839i	− 0.600043i
$909$	0	0
$910$	−1020.30	−1.12121
$911$	759.290i	0.833469i	0.909028	+	0.416734i	$0.136825\pi$
$911$	759.290i	0.833469i	−0.909028	+	0.416734i	$0.863175\pi$
$912$	0	0
$913$	−967.683	−1.05989
$914$	1565.51i	1.71282i
$915$	0	0
$916$	−1332.33	−1.45451
$917$	− 396.120i	− 0.431974i
$918$	0	0
$919$	560.626	0.610039	0.305019	−	0.952346i	$-0.401337\pi$
$919$	560.626	0.610039	0.305019	+	0.952346i	$0.401337\pi$
$920$	131.647i	0.143095i
$921$	0	0
$922$	−839.578	−0.910605
$923$	703.781i	0.762493i
$924$	0	0
$925$	169.358	0.183089
$926$	1899.45i	2.05125i
$927$	0	0
$928$	1783.73	1.92213
$929$	− 91.4794i	− 0.0984708i	−0.998787	−	0.0492354i	$-0.984322\pi$
$929$	− 91.4794i	− 0.0984708i	0.998787	−	0.0492354i	$-0.0156785\pi$
$930$	0	0
$931$	477.565	0.512959
$932$	− 1303.42i	− 1.39851i
$933$	0	0
$934$	2132.78	2.28349
$935$	2422.55i	2.59097i
$936$	0	0
$937$	−858.127	−0.915824	−0.457912	−	0.888998i	$-0.651402\pi$
$937$	−858.127	−0.915824	−0.457912	+	0.888998i	$0.651402\pi$
$938$	1315.10i	1.40202i
$939$	0	0
$940$	48.6608	0.0517668
$941$	− 475.385i	− 0.505191i	−0.967572	−	0.252596i	$-0.918716\pi$
$941$	− 475.385i	− 0.505191i	0.967572	−	0.252596i	$-0.0812842\pi$
$942$	0	0
$943$	−43.3315	−0.0459507
$944$	− 1164.86i	− 1.23396i
$945$	0	0
$946$	−469.479	−0.496278
$947$	955.194i	1.00865i	0.863513	+	0.504326i	$0.168259\pi$
$947$	955.194i	1.00865i	−0.863513	+	0.504326i	$0.831741\pi$
$948$	0	0
$949$	−315.135	−0.332071
$950$	1000.01i	1.05264i
$951$	0	0
$952$	−238.115	−0.250121
$953$	− 1128.75i	− 1.18441i	−0.805786	−	0.592206i	$-0.798257\pi$
$953$	− 1128.75i	− 1.18441i	0.805786	−	0.592206i	$-0.201743\pi$
$954$	0	0
$955$	−636.801	−0.666808
$956$	− 1327.44i	− 1.38853i
$957$	0	0
$958$	−1097.85	−1.14598
$959$	− 385.782i	− 0.402275i
$960$	0	0
$961$	−807.748	−0.840528
$962$	− 245.731i	− 0.255437i
$963$	0	0
$964$	−272.164	−0.282328
$965$	1576.05i	1.63322i
$966$	0	0
$967$	−873.591	−0.903404	−0.451702	−	0.892169i	$-0.649183\pi$
$967$	−873.591	−0.903404	−0.451702	+	0.892169i	$0.649183\pi$
$968$	11.9511i	0.0123462i
$969$	0	0
$970$	787.741	0.812104
$971$	− 1506.38i	− 1.55137i	−0.631120	−	0.775685i	$-0.717404\pi$
$971$	− 1506.38i	− 1.55137i	0.631120	−	0.775685i	$-0.282596\pi$
$972$	0	0
$973$	−646.041	−0.663969
$974$	− 818.887i	− 0.840747i
$975$	0	0
$976$	945.811	0.969068
$977$	648.296i	0.663558i	0.943357	+	0.331779i	$0.107649\pi$
$977$	648.296i	0.663558i	−0.943357	+	0.331779i	$0.892351\pi$
$978$	0	0
$979$	−821.155	−0.838769
$980$	713.729i	0.728294i
$981$	0	0
$982$	−2242.99	−2.28411
$983$	1659.56i	1.68826i	0.536140	+	0.844129i	$0.319882\pi$
$983$	1659.56i	1.68826i	−0.536140	+	0.844129i	$0.680118\pi$
$984$	0	0
$985$	−1892.74	−1.92157
$986$	− 3726.66i	− 3.77958i
$987$	0	0
$988$	664.246	0.672314
$989$	181.138i	0.183153i
$990$	0	0
$991$	106.575	0.107542	0.0537712	−	0.998553i	$-0.482876\pi$
$991$	106.575	0.107542	0.0537712	+	0.998553i	$0.482876\pi$
$992$	524.946i	0.529180i
$993$	0	0
$994$	−659.565	−0.663546
$995$	− 1057.97i	− 1.06328i
$996$	0	0
$997$	888.924	0.891598	0.445799	−	0.895133i	$-0.352920\pi$
$997$	888.924	0.891598	0.445799	+	0.895133i	$0.352920\pi$
$998$	2213.33i	2.21776i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.69	yes	84
3.2	odd	2	inner	1143.3.b.a.890.16	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.16	✓	84	3.2	odd	2	inner
1143.3.b.a.890.69	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)

\(f(q)\)	\(=\)	\(q+2.71611i q^{2} -3.37726 q^{4} +6.95811i q^{5} +4.31599 q^{7} +1.69142i q^{8} +O(q^{10})\)	\(q+2.71611i q^{2} -3.37726 q^{4} +6.95811i q^{5} +4.31599 q^{7} +1.69142i q^{8} -18.8990 q^{10} -10.6740i q^{11} +12.5086 q^{13} +11.7227i q^{14} -18.1031 q^{16} +32.6178i q^{17} -15.7237 q^{19} -23.4994i q^{20} +28.9918 q^{22} -11.1858i q^{23} -23.4154 q^{25} +33.9747i q^{26} -14.5762 q^{28} +42.0647i q^{29} -12.3795 q^{31} -42.4045i q^{32} -88.5935 q^{34} +30.0312i q^{35} -7.23275 q^{37} -42.7075i q^{38} -11.7691 q^{40} -3.87379i q^{41} -16.1935 q^{43} +36.0489i q^{44} +30.3820 q^{46} +2.07073i q^{47} -30.3722 q^{49} -63.5987i q^{50} -42.2448 q^{52} +3.76342i q^{53} +74.2709 q^{55} +7.30016i q^{56} -114.252 q^{58} +64.3455i q^{59} -52.2457 q^{61} -33.6241i q^{62} +42.7627 q^{64} +87.0362i q^{65} +112.184 q^{67} -110.159i q^{68} -81.5680 q^{70} +56.2639i q^{71} -25.1935 q^{73} -19.6450i q^{74} +53.1032 q^{76} -46.0689i q^{77} +58.5144 q^{79} -125.964i q^{80} +10.5216 q^{82} -90.6579i q^{83} -226.958 q^{85} -43.9834i q^{86} +18.0542 q^{88} -76.9304i q^{89} +53.9869 q^{91} +37.7775i q^{92} -5.62433 q^{94} -109.408i q^{95} -41.6816 q^{97} -82.4944i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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