LMFDB - Embedded newform 1143.3.b.a.890.9

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			890.9
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.76

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22737i q^{2} -6.41592 q^{4} +1.45400i q^{5} +2.15873 q^{7} +7.79707i q^{8} +O(q^{10})$	\(q-3.22737i q^{2} -6.41592 q^{4} +1.45400i q^{5} +2.15873 q^{7} +7.79707i q^{8} +4.69259 q^{10} -6.04547i q^{11} +15.2005 q^{13} -6.96703i q^{14} -0.499646 q^{16} +16.9441i q^{17} -5.63400 q^{19} -9.32873i q^{20} -19.5110 q^{22} +12.2914i q^{23} +22.8859 q^{25} -49.0577i q^{26} -13.8503 q^{28} +20.9473i q^{29} +44.7052 q^{31} +32.8008i q^{32} +54.6849 q^{34} +3.13879i q^{35} +50.2331 q^{37} +18.1830i q^{38} -11.3369 q^{40} -45.3146i q^{41} +63.6233 q^{43} +38.7873i q^{44} +39.6689 q^{46} -57.8837i q^{47} -44.3399 q^{49} -73.8613i q^{50} -97.5252 q^{52} -16.7003i q^{53} +8.79009 q^{55} +16.8318i q^{56} +67.6048 q^{58} -111.728i q^{59} -102.642 q^{61} -144.280i q^{62} +103.862 q^{64} +22.1015i q^{65} -69.5730 q^{67} -108.712i q^{68} +10.1300 q^{70} +87.3147i q^{71} -14.5578 q^{73} -162.121i q^{74} +36.1473 q^{76} -13.0506i q^{77} +146.673 q^{79} -0.726483i q^{80} -146.247 q^{82} +59.3483i q^{83} -24.6367 q^{85} -205.336i q^{86} +47.1370 q^{88} -10.9134i q^{89} +32.8138 q^{91} -78.8606i q^{92} -186.812 q^{94} -8.19182i q^{95} +21.3026 q^{97} +143.101i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.22737i	− 1.61369i	−0.590766	−	0.806843i	$-0.701175\pi$
$2$	− 3.22737i	− 1.61369i	0.590766	−	0.806843i	$-0.298825\pi$
$3$	0	0
$3$	0	0
$4$	−6.41592	−1.60398
$5$	1.45400i	0.290799i	0.989373	+	0.145400i	$0.0464468\pi$
$5$	1.45400i	0.290799i	−0.989373	+	0.145400i	$0.953553\pi$
$6$	0	0
$7$	2.15873	0.308390	0.154195	−	0.988040i	$-0.450722\pi$
$7$	2.15873	0.308390	0.154195	+	0.988040i	$0.450722\pi$
$8$	7.79707i	0.974634i
$9$	0	0
$10$	4.69259	0.469259
$11$	− 6.04547i	− 0.549588i	−0.961503	−	0.274794i	$-0.911390\pi$
$11$	− 6.04547i	− 0.549588i	0.961503	−	0.274794i	$-0.0886097\pi$
$12$	0	0
$13$	15.2005	1.16927	0.584635	−	0.811297i	$-0.301238\pi$
$13$	15.2005	1.16927	0.584635	+	0.811297i	$0.301238\pi$
$14$	− 6.96703i	− 0.497645i
$15$	0	0
$16$	−0.499646	−0.0312279
$17$	16.9441i	0.996713i	0.866972	+	0.498356i	$0.166063\pi$
$17$	16.9441i	0.996713i	−0.866972	+	0.498356i	$0.833937\pi$
$18$	0	0
$19$	−5.63400	−0.296527	−0.148263	−	0.988948i	$-0.547368\pi$
$19$	−5.63400	−0.296527	−0.148263	+	0.988948i	$0.547368\pi$
$20$	− 9.32873i	− 0.466436i
$21$	0	0
$22$	−19.5110	−0.886862
$23$	12.2914i	0.534408i	0.963640	+	0.267204i	$0.0860998\pi$
$23$	12.2914i	0.534408i	−0.963640	+	0.267204i	$0.913900\pi$
$24$	0	0
$25$	22.8859	0.915436
$26$	− 49.0577i	− 1.88683i
$27$	0	0
$28$	−13.8503	−0.494652
$29$	20.9473i	0.722321i	0.932504	+	0.361161i	$0.117619\pi$
$29$	20.9473i	0.722321i	−0.932504	+	0.361161i	$0.882381\pi$
$30$	0	0
$31$	44.7052	1.44210	0.721052	−	0.692881i	$-0.243659\pi$
$31$	44.7052	1.44210	0.721052	+	0.692881i	$0.243659\pi$
$32$	32.8008i	1.02503i
$33$	0	0
$34$	54.6849	1.60838
$35$	3.13879i	0.0896797i
$36$	0	0
$37$	50.2331	1.35765	0.678826	−	0.734299i	$-0.262489\pi$
$37$	50.2331	1.35765	0.678826	+	0.734299i	$0.262489\pi$
$38$	18.1830i	0.478501i
$39$	0	0
$40$	−11.3369	−0.283423
$41$	− 45.3146i	− 1.10523i	−0.833435	−	0.552617i	$-0.813629\pi$
$41$	− 45.3146i	− 1.10523i	0.833435	−	0.552617i	$-0.186371\pi$
$42$	0	0
$43$	63.6233	1.47961	0.739806	−	0.672820i	$-0.234917\pi$
$43$	63.6233	1.47961	0.739806	+	0.672820i	$0.234917\pi$
$44$	38.7873i	0.881529i
$45$	0	0
$46$	39.6689	0.862367
$47$	− 57.8837i	− 1.23157i	−0.787915	−	0.615784i	$-0.788839\pi$
$47$	− 57.8837i	− 1.23157i	0.787915	−	0.615784i	$-0.211161\pi$
$48$	0	0
$49$	−44.3399	−0.904895
$50$	− 73.8613i	− 1.47723i
$51$	0	0
$52$	−97.5252	−1.87548
$53$	− 16.7003i	− 0.315099i	−0.987511	−	0.157550i	$-0.949641\pi$
$53$	− 16.7003i	− 0.315099i	0.987511	−	0.157550i	$-0.0503594\pi$
$54$	0	0
$55$	8.79009	0.159820
$56$	16.8318i	0.300568i
$57$	0	0
$58$	67.6048	1.16560
$59$	− 111.728i	− 1.89370i	−0.321682	−	0.946848i	$-0.604248\pi$
$59$	− 111.728i	− 1.89370i	0.321682	−	0.946848i	$-0.395752\pi$
$60$	0	0
$61$	−102.642	−1.68265	−0.841325	−	0.540529i	$-0.818224\pi$
$61$	−102.642	−1.68265	−0.841325	+	0.540529i	$0.818224\pi$
$62$	− 144.280i	− 2.32710i
$63$	0	0
$64$	103.862	1.62284
$65$	22.1015i	0.340023i
$66$	0	0
$67$	−69.5730	−1.03840	−0.519201	−	0.854652i	$-0.673770\pi$
$67$	−69.5730	−1.03840	−0.519201	+	0.854652i	$0.673770\pi$
$68$	− 108.712i	− 1.59871i
$69$	0	0
$70$	10.1300	0.144715
$71$	87.3147i	1.22979i	0.788611	+	0.614893i	$0.210801\pi$
$71$	87.3147i	1.22979i	−0.788611	+	0.614893i	$0.789199\pi$
$72$	0	0
$73$	−14.5578	−0.199422	−0.0997108	−	0.995016i	$-0.531792\pi$
$73$	−14.5578	−0.199422	−0.0997108	+	0.995016i	$0.531792\pi$
$74$	− 162.121i	− 2.19082i
$75$	0	0
$76$	36.1473	0.475623
$77$	− 13.0506i	− 0.169488i
$78$	0	0
$79$	146.673	1.85662	0.928310	−	0.371807i	$-0.121262\pi$
$79$	146.673	1.85662	0.928310	+	0.371807i	$0.121262\pi$
$80$	− 0.726483i	− 0.00908104i
$81$	0	0
$82$	−146.247	−1.78350
$83$	59.3483i	0.715040i	0.933905	+	0.357520i	$0.116378\pi$
$83$	59.3483i	0.715040i	−0.933905	+	0.357520i	$0.883622\pi$
$84$	0	0
$85$	−24.6367	−0.289843
$86$	− 205.336i	− 2.38763i
$87$	0	0
$88$	47.1370	0.535647
$89$	− 10.9134i	− 0.122623i	−0.998119	−	0.0613114i	$-0.980472\pi$
$89$	− 10.9134i	− 0.122623i	0.998119	−	0.0613114i	$-0.0195283\pi$
$90$	0	0
$91$	32.8138	0.360591
$92$	− 78.8606i	− 0.857180i
$93$	0	0
$94$	−186.812	−1.98736
$95$	− 8.19182i	− 0.0862297i
$96$	0	0
$97$	21.3026	0.219614	0.109807	−	0.993953i	$-0.464977\pi$
$97$	21.3026	0.219614	0.109807	+	0.993953i	$0.464977\pi$
$98$	143.101i	1.46022i
$99$	0	0
$100$	−146.834	−1.46834
$101$	− 191.133i	− 1.89240i	−0.323576	−	0.946202i	$-0.604885\pi$
$101$	− 191.133i	− 1.89240i	0.323576	−	0.946202i	$-0.395115\pi$
$102$	0	0
$103$	27.2875	0.264928	0.132464	−	0.991188i	$-0.457711\pi$
$103$	27.2875	0.264928	0.132464	+	0.991188i	$0.457711\pi$
$104$	118.519i	1.13961i
$105$	0	0
$106$	−53.8980	−0.508471
$107$	77.1669i	0.721186i	0.932723	+	0.360593i	$0.117426\pi$
$107$	77.1669i	0.721186i	−0.932723	+	0.360593i	$0.882574\pi$
$108$	0	0
$109$	−1.49702	−0.0137341	−0.00686707	−	0.999976i	$-0.502186\pi$
$109$	−1.49702	−0.0137341	−0.00686707	+	0.999976i	$0.502186\pi$
$110$	− 28.3689i	− 0.257899i
$111$	0	0
$112$	−1.07860	−0.00963037
$113$	− 131.194i	− 1.16101i	−0.814256	−	0.580506i	$-0.802855\pi$
$113$	− 131.194i	− 1.16101i	0.814256	−	0.580506i	$-0.197145\pi$
$114$	0	0
$115$	−17.8716	−0.155406
$116$	− 134.396i	− 1.15859i
$117$	0	0
$118$	−360.588	−3.05583
$119$	36.5778i	0.307376i
$120$	0	0
$121$	84.4523	0.697953
$122$	331.263i	2.71527i
$123$	0	0
$124$	−286.825	−2.31311
$125$	69.6259i	0.557007i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	− 203.997i	− 1.59373i
$129$	0	0
$130$	71.3297	0.548690
$131$	− 143.034i	− 1.09186i	−0.837830	−	0.545931i	$-0.816176\pi$
$131$	− 143.034i	− 1.09186i	0.837830	−	0.545931i	$-0.183824\pi$
$132$	0	0
$133$	−12.1623	−0.0914459
$134$	224.538i	1.67565i
$135$	0	0
$136$	−132.114	−0.971430
$137$	− 4.85061i	− 0.0354059i	−0.999843	−	0.0177030i	$-0.994365\pi$
$137$	− 4.85061i	− 0.0354059i	0.999843	−	0.0177030i	$-0.00563532\pi$
$138$	0	0
$139$	115.522	0.831097	0.415548	−	0.909571i	$-0.363590\pi$
$139$	115.522	0.831097	0.415548	+	0.909571i	$0.363590\pi$
$140$	− 20.1382i	− 0.143844i
$141$	0	0
$142$	281.797	1.98449
$143$	− 91.8942i	− 0.642617i
$144$	0	0
$145$	−30.4573	−0.210051
$146$	46.9833i	0.321804i
$147$	0	0
$148$	−322.292	−2.17765
$149$	− 101.662i	− 0.682295i	−0.940010	−	0.341148i	$-0.889184\pi$
$149$	− 101.662i	− 0.682295i	0.940010	−	0.341148i	$-0.110816\pi$
$150$	0	0
$151$	147.676	0.977990	0.488995	−	0.872287i	$-0.337364\pi$
$151$	147.676	0.977990	0.488995	+	0.872287i	$0.337364\pi$
$152$	− 43.9287i	− 0.289005i
$153$	0	0
$154$	−42.1190	−0.273500
$155$	65.0012i	0.419363i
$156$	0	0
$157$	−31.8716	−0.203004	−0.101502	−	0.994835i	$-0.532365\pi$
$157$	−31.8716	−0.203004	−0.101502	+	0.994835i	$0.532365\pi$
$158$	− 473.368i	− 2.99600i
$159$	0	0
$160$	−47.6923	−0.298077
$161$	26.5338i	0.164806i
$162$	0	0
$163$	−8.77669	−0.0538447	−0.0269224	−	0.999638i	$-0.508571\pi$
$163$	−8.77669	−0.0538447	−0.0269224	+	0.999638i	$0.508571\pi$
$164$	290.735i	1.77277i
$165$	0	0
$166$	191.539	1.15385
$167$	117.208i	0.701842i	0.936405	+	0.350921i	$0.114131\pi$
$167$	117.208i	0.701842i	−0.936405	+	0.350921i	$0.885869\pi$
$168$	0	0
$169$	62.0553	0.367191
$170$	79.5117i	0.467716i
$171$	0	0
$172$	−408.202	−2.37327
$173$	206.099i	1.19132i	0.803235	+	0.595662i	$0.203110\pi$
$173$	206.099i	1.19132i	−0.803235	+	0.595662i	$0.796890\pi$
$174$	0	0
$175$	49.4045	0.282311
$176$	3.02059i	0.0171625i
$177$	0	0
$178$	−35.2217	−0.197875
$179$	114.251i	0.638271i	0.947709	+	0.319136i	$0.103393\pi$
$179$	114.251i	0.638271i	−0.947709	+	0.319136i	$0.896607\pi$
$180$	0	0
$181$	320.232	1.76923	0.884617	−	0.466317i	$-0.154420\pi$
$181$	320.232	1.76923	0.884617	+	0.466317i	$0.154420\pi$
$182$	− 105.902i	− 0.581881i
$183$	0	0
$184$	−95.8368	−0.520852
$185$	73.0388i	0.394804i
$186$	0	0
$187$	102.435	0.547781
$188$	371.377i	1.97541i
$189$	0	0
$190$	−26.4381	−0.139148
$191$	− 104.041i	− 0.544719i	−0.962196	−	0.272360i	$-0.912196\pi$
$191$	− 104.041i	− 0.544719i	0.962196	−	0.272360i	$-0.0878040\pi$
$192$	0	0
$193$	−333.803	−1.72955	−0.864774	−	0.502162i	$-0.832538\pi$
$193$	−333.803	−1.72955	−0.864774	+	0.502162i	$0.832538\pi$
$194$	− 68.7514i	− 0.354389i
$195$	0	0
$196$	284.481	1.45143
$197$	111.309i	0.565020i	0.959264	+	0.282510i	$0.0911670\pi$
$197$	111.309i	0.565020i	−0.959264	+	0.282510i	$0.908833\pi$
$198$	0	0
$199$	179.209	0.900549	0.450274	−	0.892890i	$-0.351326\pi$
$199$	179.209	0.900549	0.450274	+	0.892890i	$0.351326\pi$
$200$	178.443i	0.892215i
$201$	0	0
$202$	−616.856	−3.05374
$203$	45.2196i	0.222757i
$204$	0	0
$205$	65.8873	0.321401
$206$	− 88.0670i	− 0.427510i
$207$	0	0
$208$	−7.59487	−0.0365138
$209$	34.0602i	0.162968i
$210$	0	0
$211$	−211.720	−1.00341	−0.501706	−	0.865038i	$-0.667294\pi$
$211$	−211.720	−1.00341	−0.501706	+	0.865038i	$0.667294\pi$
$212$	107.148i	0.505413i
$213$	0	0
$214$	249.046	1.16377
$215$	92.5081i	0.430270i
$216$	0	0
$217$	96.5066	0.444731
$218$	4.83144i	0.0221626i
$219$	0	0
$220$	−56.3965	−0.256348
$221$	257.559i	1.16543i
$222$	0	0
$223$	252.821	1.13373	0.566864	−	0.823812i	$-0.308157\pi$
$223$	252.821	1.13373	0.566864	+	0.823812i	$0.308157\pi$
$224$	70.8082i	0.316108i
$225$	0	0
$226$	−423.413	−1.87351
$227$	160.992i	0.709217i	0.935015	+	0.354609i	$0.115386\pi$
$227$	160.992i	0.709217i	−0.935015	+	0.354609i	$0.884614\pi$
$228$	0	0
$229$	16.8026	0.0733739	0.0366870	−	0.999327i	$-0.488320\pi$
$229$	16.8026	0.0733739	0.0366870	+	0.999327i	$0.488320\pi$
$230$	57.6784i	0.250776i
$231$	0	0
$232$	−163.328	−0.703999
$233$	56.2312i	0.241336i	0.992693	+	0.120668i	$0.0385036\pi$
$233$	56.2312i	0.241336i	−0.992693	+	0.120668i	$0.961496\pi$
$234$	0	0
$235$	84.1627	0.358139
$236$	716.838i	3.03745i
$237$	0	0
$238$	118.050	0.496009
$239$	− 365.589i	− 1.52966i	−0.644232	−	0.764830i	$-0.722823\pi$
$239$	− 365.589i	− 1.52966i	0.644232	−	0.764830i	$-0.277177\pi$
$240$	0	0
$241$	−314.533	−1.30512	−0.652558	−	0.757739i	$-0.726304\pi$
$241$	−314.533	−1.30512	−0.652558	+	0.757739i	$0.726304\pi$
$242$	− 272.559i	− 1.12628i
$243$	0	0
$244$	658.541	2.69894
$245$	− 64.4700i	− 0.263143i
$246$	0	0
$247$	−85.6397	−0.346719
$248$	348.570i	1.40552i
$249$	0	0
$250$	224.709	0.898835
$251$	− 413.825i	− 1.64871i	−0.566076	−	0.824353i	$-0.691539\pi$
$251$	− 413.825i	− 1.64871i	0.566076	−	0.824353i	$-0.308461\pi$
$252$	0	0
$253$	74.3072	0.293704
$254$	36.3706i	0.143191i
$255$	0	0
$256$	−242.928	−0.948936
$257$	129.894i	0.505426i	0.967541	+	0.252713i	$0.0813228\pi$
$257$	129.894i	0.505426i	−0.967541	+	0.252713i	$0.918677\pi$
$258$	0	0
$259$	108.440	0.418687
$260$	− 141.801i	− 0.545390i
$261$	0	0
$262$	−461.623	−1.76192
$263$	395.319i	1.50311i	0.659668	+	0.751557i	$0.270697\pi$
$263$	395.319i	1.50311i	−0.659668	+	0.751557i	$0.729303\pi$
$264$	0	0
$265$	24.2821	0.0916307
$266$	39.2523i	0.147565i
$267$	0	0
$268$	446.375	1.66558
$269$	424.438i	1.57784i	0.614497	+	0.788919i	$0.289359\pi$
$269$	424.438i	1.57784i	−0.614497	+	0.788919i	$0.710641\pi$
$270$	0	0
$271$	−296.500	−1.09410	−0.547048	−	0.837101i	$-0.684248\pi$
$271$	−296.500	−1.09410	−0.547048	+	0.837101i	$0.684248\pi$
$272$	− 8.46606i	− 0.0311252i
$273$	0	0
$274$	−15.6547	−0.0571340
$275$	− 138.356i	− 0.503113i
$276$	0	0
$277$	−111.921	−0.404045	−0.202023	−	0.979381i	$-0.564751\pi$
$277$	−111.921	−0.404045	−0.202023	+	0.979381i	$0.564751\pi$
$278$	− 372.834i	− 1.34113i
$279$	0	0
$280$	−24.4734	−0.0874049
$281$	− 202.966i	− 0.722298i	−0.932508	−	0.361149i	$-0.882385\pi$
$281$	− 202.966i	− 0.722298i	0.932508	−	0.361149i	$-0.117615\pi$
$282$	0	0
$283$	20.8527	0.0736843	0.0368422	−	0.999321i	$-0.488270\pi$
$283$	20.8527	0.0736843	0.0368422	+	0.999321i	$0.488270\pi$
$284$	− 560.204i	− 1.97255i
$285$	0	0
$286$	−296.577	−1.03698
$287$	− 97.8221i	− 0.340844i
$288$	0	0
$289$	1.89702	0.00656409
$290$	98.2971i	0.338955i
$291$	0	0
$292$	93.4015	0.319868
$293$	− 506.171i	− 1.72755i	−0.503881	−	0.863773i	$-0.668095\pi$
$293$	− 506.171i	− 1.72755i	0.503881	−	0.863773i	$-0.331905\pi$
$294$	0	0
$295$	162.452	0.550685
$296$	391.671i	1.32321i
$297$	0	0
$298$	−328.101	−1.10101
$299$	186.835i	0.624867i
$300$	0	0
$301$	137.346	0.456298
$302$	− 476.607i	− 1.57817i
$303$	0	0
$304$	2.81501	0.00925989
$305$	− 149.241i	− 0.489314i
$306$	0	0
$307$	446.721	1.45512	0.727559	−	0.686045i	$-0.240655\pi$
$307$	446.721	1.45512	0.727559	+	0.686045i	$0.240655\pi$
$308$	83.7313i	0.271855i
$309$	0	0
$310$	209.783	0.676720
$311$	523.253i	1.68249i	0.540657	+	0.841243i	$0.318176\pi$
$311$	523.253i	1.68249i	−0.540657	+	0.841243i	$0.681824\pi$
$312$	0	0
$313$	354.860	1.13374	0.566869	−	0.823808i	$-0.308155\pi$
$313$	354.860	1.13374	0.566869	+	0.823808i	$0.308155\pi$
$314$	102.861i	0.327584i
$315$	0	0
$316$	−941.042	−2.97798
$317$	387.303i	1.22178i	0.791717	+	0.610888i	$0.209187\pi$
$317$	387.303i	1.22178i	−0.791717	+	0.610888i	$0.790813\pi$
$318$	0	0
$319$	126.636	0.396979
$320$	151.015i	0.471921i
$321$	0	0
$322$	85.6344	0.265945
$323$	− 95.4632i	− 0.295552i
$324$	0	0
$325$	347.877	1.07039
$326$	28.3256i	0.0868884i
$327$	0	0
$328$	353.321	1.07720
$329$	− 124.955i	− 0.379804i
$330$	0	0
$331$	−266.449	−0.804981	−0.402490	−	0.915424i	$-0.631855\pi$
$331$	−266.449	−0.804981	−0.402490	+	0.915424i	$0.631855\pi$
$332$	− 380.774i	− 1.14691i
$333$	0	0
$334$	378.272	1.13255
$335$	− 101.159i	− 0.301967i
$336$	0	0
$337$	555.631	1.64876	0.824379	−	0.566039i	$-0.191525\pi$
$337$	555.631	1.64876	0.824379	+	0.566039i	$0.191525\pi$
$338$	− 200.275i	− 0.592531i
$339$	0	0
$340$	158.067	0.464903
$341$	− 270.264i	− 0.792563i
$342$	0	0
$343$	−201.496	−0.587451
$344$	496.076i	1.44208i
$345$	0	0
$346$	665.158	1.92242
$347$	− 51.5775i	− 0.148638i	−0.997235	−	0.0743192i	$-0.976322\pi$
$347$	− 51.5775i	− 0.148638i	0.997235	−	0.0743192i	$-0.0236784\pi$
$348$	0	0
$349$	593.799	1.70143	0.850716	−	0.525626i	$-0.176169\pi$
$349$	593.799	1.70143	0.850716	+	0.525626i	$0.176169\pi$
$350$	− 159.447i	− 0.455562i
$351$	0	0
$352$	198.296	0.563342
$353$	− 461.155i	− 1.30639i	−0.757190	−	0.653195i	$-0.773428\pi$
$353$	− 461.155i	− 1.30639i	0.757190	−	0.653195i	$-0.226572\pi$
$354$	0	0
$355$	−126.955	−0.357621
$356$	70.0197i	0.196685i
$357$	0	0
$358$	368.729	1.02997
$359$	− 231.114i	− 0.643771i	−0.946779	−	0.321885i	$-0.895683\pi$
$359$	− 231.114i	− 0.643771i	0.946779	−	0.321885i	$-0.104317\pi$
$360$	0	0
$361$	−329.258	−0.912072
$362$	− 1033.51i	− 2.85499i
$363$	0	0
$364$	−210.531	−0.578381
$365$	− 21.1670i	− 0.0579916i
$366$	0	0
$367$	−26.4050	−0.0719482	−0.0359741	−	0.999353i	$-0.511453\pi$
$367$	−26.4050	−0.0719482	−0.0359741	+	0.999353i	$0.511453\pi$
$368$	− 6.14134i	− 0.0166884i
$369$	0	0
$370$	235.723	0.637090
$371$	− 36.0514i	− 0.0971736i
$372$	0	0
$373$	−529.266	−1.41894	−0.709472	−	0.704734i	$-0.751066\pi$
$373$	−529.266	−1.41894	−0.709472	+	0.704734i	$0.751066\pi$
$374$	− 330.596i	− 0.883947i
$375$	0	0
$376$	451.323	1.20033
$377$	318.410i	0.844588i
$378$	0	0
$379$	−361.693	−0.954336	−0.477168	−	0.878812i	$-0.658337\pi$
$379$	−361.693	−0.954336	−0.477168	+	0.878812i	$0.658337\pi$
$380$	52.5581i	0.138311i
$381$	0	0
$382$	−335.780	−0.879005
$383$	− 93.1447i	− 0.243198i	−0.992579	−	0.121599i	$-0.961198\pi$
$383$	− 93.1447i	− 0.243198i	0.992579	−	0.121599i	$-0.0388021\pi$
$384$	0	0
$385$	18.9755	0.0492869
$386$	1077.30i	2.79095i
$387$	0	0
$388$	−136.676	−0.352257
$389$	205.427i	0.528091i	0.964510	+	0.264045i	$0.0850569\pi$
$389$	205.427i	0.528091i	−0.964510	+	0.264045i	$0.914943\pi$
$390$	0	0
$391$	−208.267	−0.532651
$392$	− 345.721i	− 0.881942i
$393$	0	0
$394$	359.235	0.911764
$395$	213.262i	0.539904i
$396$	0	0
$397$	178.575	0.449812	0.224906	−	0.974380i	$-0.427793\pi$
$397$	178.575	0.449812	0.224906	+	0.974380i	$0.427793\pi$
$398$	− 578.375i	− 1.45320i
$399$	0	0
$400$	−11.4348	−0.0285871
$401$	− 80.5359i	− 0.200838i	−0.994945	−	0.100419i	$-0.967982\pi$
$401$	− 80.5359i	− 0.200838i	0.994945	−	0.100419i	$-0.0320183\pi$
$402$	0	0
$403$	679.542	1.68621
$404$	1226.29i	3.03538i
$405$	0	0
$406$	145.941	0.359459
$407$	− 303.683i	− 0.746149i
$408$	0	0
$409$	−50.7083	−0.123981	−0.0619906	−	0.998077i	$-0.519745\pi$
$409$	−50.7083	−0.123981	−0.0619906	+	0.998077i	$0.519745\pi$
$410$	− 212.643i	− 0.518641i
$411$	0	0
$412$	−175.075	−0.424939
$413$	− 241.191i	− 0.583997i
$414$	0	0
$415$	−86.2923	−0.207933
$416$	498.589i	1.19853i
$417$	0	0
$418$	109.925	0.262978
$419$	421.505i	1.00598i	0.864292	+	0.502990i	$0.167767\pi$
$419$	421.505i	1.00598i	−0.864292	+	0.502990i	$0.832233\pi$
$420$	0	0
$421$	126.531	0.300548	0.150274	−	0.988644i	$-0.451984\pi$
$421$	126.531	0.300548	0.150274	+	0.988644i	$0.451984\pi$
$422$	683.298i	1.61919i
$423$	0	0
$424$	130.213	0.307107
$425$	387.781i	0.912426i
$426$	0	0
$427$	−221.576	−0.518913
$428$	− 495.097i	− 1.15677i
$429$	0	0
$430$	298.558	0.694321
$431$	69.7232i	0.161771i	0.996723	+	0.0808854i	$0.0257748\pi$
$431$	69.7232i	0.161771i	−0.996723	+	0.0808854i	$0.974225\pi$
$432$	0	0
$433$	433.061	1.00014	0.500070	−	0.865985i	$-0.333308\pi$
$433$	433.061	1.00014	0.500070	+	0.865985i	$0.333308\pi$
$434$	− 311.462i	− 0.717656i
$435$	0	0
$436$	9.60477	0.0220293
$437$	− 69.2497i	− 0.158466i
$438$	0	0
$439$	730.260	1.66346	0.831731	−	0.555179i	$-0.187350\pi$
$439$	730.260	1.66346	0.831731	+	0.555179i	$0.187350\pi$
$440$	68.5370i	0.155766i
$441$	0	0
$442$	831.238	1.88063
$443$	113.431i	0.256053i	0.991771	+	0.128026i	$0.0408642\pi$
$443$	113.431i	0.256053i	−0.991771	+	0.128026i	$0.959136\pi$
$444$	0	0
$445$	15.8681	0.0356586
$446$	− 815.948i	− 1.82948i
$447$	0	0
$448$	224.210	0.500468
$449$	372.379i	0.829352i	0.909969	+	0.414676i	$0.136105\pi$
$449$	372.379i	0.829352i	−0.909969	+	0.414676i	$0.863895\pi$
$450$	0	0
$451$	−273.948	−0.607424
$452$	841.733i	1.86224i
$453$	0	0
$454$	519.582	1.14445
$455$	47.7112i	0.104860i
$456$	0	0
$457$	−525.639	−1.15020	−0.575098	−	0.818085i	$-0.695036\pi$
$457$	−525.639	−1.15020	−0.575098	+	0.818085i	$0.695036\pi$
$458$	− 54.2283i	− 0.118402i
$459$	0	0
$460$	114.663	0.249267
$461$	86.0937i	0.186754i	0.995631	+	0.0933771i	$0.0297662\pi$
$461$	86.0937i	0.186754i	−0.995631	+	0.0933771i	$0.970234\pi$
$462$	0	0
$463$	395.423	0.854046	0.427023	−	0.904241i	$-0.359562\pi$
$463$	395.423	0.854046	0.427023	+	0.904241i	$0.359562\pi$
$464$	− 10.4662i	− 0.0225566i
$465$	0	0
$466$	181.479	0.389440
$467$	398.158i	0.852587i	0.904585	+	0.426294i	$0.140181\pi$
$467$	398.158i	0.852587i	−0.904585	+	0.426294i	$0.859819\pi$
$468$	0	0
$469$	−150.189	−0.320233
$470$	− 271.624i	− 0.577924i
$471$	0	0
$472$	871.151	1.84566
$473$	− 384.633i	− 0.813177i
$474$	0	0
$475$	−128.939	−0.271451
$476$	− 234.680i	− 0.493026i
$477$	0	0
$478$	−1179.89	−2.46839
$479$	298.231i	0.622611i	0.950310	+	0.311305i	$0.100766\pi$
$479$	298.231i	0.622611i	−0.950310	+	0.311305i	$0.899234\pi$
$480$	0	0
$481$	763.568	1.58746
$482$	1015.11i	2.10605i
$483$	0	0
$484$	−541.839	−1.11950
$485$	30.9739i	0.0638637i
$486$	0	0
$487$	4.07802	0.00837376	0.00418688	−	0.999991i	$-0.498667\pi$
$487$	4.07802	0.00837376	0.00418688	+	0.999991i	$0.498667\pi$
$488$	− 800.304i	− 1.63997i
$489$	0	0
$490$	−208.069	−0.424630
$491$	− 515.872i	− 1.05065i	−0.850900	−	0.525327i	$-0.823943\pi$
$491$	− 515.872i	− 1.05065i	0.850900	−	0.525327i	$-0.176057\pi$
$492$	0	0
$493$	−354.934	−0.719947
$494$	276.391i	0.559496i
$495$	0	0
$496$	−22.3368	−0.0450338
$497$	188.489i	0.379254i
$498$	0	0
$499$	−26.3742	−0.0528541	−0.0264270	−	0.999651i	$-0.508413\pi$
$499$	−26.3742	−0.0528541	−0.0264270	+	0.999651i	$0.508413\pi$
$500$	− 446.714i	− 0.893429i
$501$	0	0
$502$	−1335.57	−2.66049
$503$	11.6037i	0.0230690i	0.999933	+	0.0115345i	$0.00367163\pi$
$503$	11.6037i	0.0230690i	−0.999933	+	0.0115345i	$0.996328\pi$
$504$	0	0
$505$	277.907	0.550310
$506$	− 239.817i	− 0.473946i
$507$	0	0
$508$	72.3038	0.142330
$509$	623.478i	1.22491i	0.790506	+	0.612454i	$0.209818\pi$
$509$	623.478i	1.22491i	−0.790506	+	0.612454i	$0.790182\pi$
$510$	0	0
$511$	−31.4263	−0.0614997
$512$	− 31.9719i	− 0.0624451i
$513$	0	0
$514$	419.218	0.815598
$515$	39.6760i	0.0770408i
$516$	0	0
$517$	−349.934	−0.676855
$518$	− 349.975i	− 0.675628i
$519$	0	0
$520$	−172.327	−0.331398
$521$	126.020i	0.241881i	0.992660	+	0.120941i	$0.0385911\pi$
$521$	126.020i	0.241881i	−0.992660	+	0.120941i	$0.961409\pi$
$522$	0	0
$523$	272.380	0.520802	0.260401	−	0.965500i	$-0.416145\pi$
$523$	272.380	0.520802	0.260401	+	0.965500i	$0.416145\pi$
$524$	917.694i	1.75132i
$525$	0	0
$526$	1275.84	2.42555
$527$	757.490i	1.43736i
$528$	0	0
$529$	377.922	0.714408
$530$	− 78.3674i	− 0.147863i
$531$	0	0
$532$	78.0324	0.146677
$533$	− 688.805i	− 1.29232i
$534$	0	0
$535$	−112.200	−0.209721
$536$	− 542.465i	− 1.01206i
$537$	0	0
$538$	1369.82	2.54613
$539$	268.055i	0.497320i
$540$	0	0
$541$	−893.640	−1.65183	−0.825915	−	0.563795i	$-0.809341\pi$
$541$	−893.640	−1.65183	−0.825915	+	0.563795i	$0.809341\pi$
$542$	956.915i	1.76553i
$543$	0	0
$544$	−555.781	−1.02166
$545$	− 2.17666i	− 0.00399388i
$546$	0	0
$547$	271.399	0.496158	0.248079	−	0.968740i	$-0.420201\pi$
$547$	271.399	0.496158	0.248079	+	0.968740i	$0.420201\pi$
$548$	31.1211i	0.0567904i
$549$	0	0
$550$	−446.526	−0.811866
$551$	− 118.017i	− 0.214187i
$552$	0	0
$553$	316.628	0.572564
$554$	361.209i	0.652002i
$555$	0	0
$556$	−741.183	−1.33306
$557$	356.461i	0.639967i	0.947423	+	0.319983i	$0.103677\pi$
$557$	356.461i	0.639967i	−0.947423	+	0.319983i	$0.896323\pi$
$558$	0	0
$559$	967.106	1.73007
$560$	− 1.56828i	− 0.00280051i
$561$	0	0
$562$	−655.045	−1.16556
$563$	255.522i	0.453858i	0.973911	+	0.226929i	$0.0728686\pi$
$563$	255.522i	0.453858i	−0.973911	+	0.226929i	$0.927131\pi$
$564$	0	0
$565$	190.756	0.337622
$566$	− 67.2993i	− 0.118903i
$567$	0	0
$568$	−680.799	−1.19859
$569$	182.702i	0.321093i	0.987028	+	0.160547i	$0.0513257\pi$
$569$	182.702i	0.321093i	−0.987028	+	0.160547i	$0.948674\pi$
$570$	0	0
$571$	159.637	0.279574	0.139787	−	0.990182i	$-0.455358\pi$
$571$	159.637	0.279574	0.139787	+	0.990182i	$0.455358\pi$
$572$	589.586i	1.03074i
$573$	0	0
$574$	−315.708	−0.550014
$575$	281.299i	0.489216i
$576$	0	0
$577$	−920.566	−1.59543	−0.797717	−	0.603031i	$-0.793959\pi$
$577$	−920.566	−1.59543	−0.797717	+	0.603031i	$0.793959\pi$
$578$	− 6.12239i	− 0.0105924i
$579$	0	0
$580$	195.412	0.336917
$581$	128.117i	0.220511i
$582$	0	0
$583$	−100.961	−0.173175
$584$	− 113.508i	− 0.194363i
$585$	0	0
$586$	−1633.60	−2.78772
$587$	790.574i	1.34680i	0.739276	+	0.673402i	$0.235168\pi$
$587$	790.574i	1.34680i	−0.739276	+	0.673402i	$0.764832\pi$
$588$	0	0
$589$	−251.869	−0.427622
$590$	− 524.293i	− 0.888633i
$591$	0	0
$592$	−25.0988	−0.0423966
$593$	− 42.5943i	− 0.0718285i	−0.999355	−	0.0359142i	$-0.988566\pi$
$593$	− 42.5943i	− 0.0718285i	0.999355	−	0.0359142i	$-0.0114343\pi$
$594$	0	0
$595$	−53.1840	−0.0893849
$596$	652.255i	1.09439i
$597$	0	0
$598$	602.987	1.00834
$599$	532.138i	0.888378i	0.895933	+	0.444189i	$0.146508\pi$
$599$	532.138i	0.888378i	−0.895933	+	0.444189i	$0.853492\pi$
$600$	0	0
$601$	−20.6373	−0.0343382	−0.0171691	−	0.999853i	$-0.505465\pi$
$601$	−20.6373	−0.0343382	−0.0171691	+	0.999853i	$0.505465\pi$
$602$	− 443.265i	− 0.736321i
$603$	0	0
$604$	−947.480	−1.56868
$605$	122.793i	0.202964i
$606$	0	0
$607$	611.889	1.00805	0.504027	−	0.863688i	$-0.331851\pi$
$607$	611.889	1.00805	0.504027	+	0.863688i	$0.331851\pi$
$608$	− 184.800i	− 0.303947i
$609$	0	0
$610$	−481.655	−0.789598
$611$	− 879.861i	− 1.44003i
$612$	0	0
$613$	939.218	1.53217	0.766083	−	0.642742i	$-0.222203\pi$
$613$	939.218	1.53217	0.766083	+	0.642742i	$0.222203\pi$
$614$	− 1441.74i	− 2.34810i
$615$	0	0
$616$	101.756	0.165188
$617$	− 1182.83i	− 1.91707i	−0.284978	−	0.958534i	$-0.591986\pi$
$617$	− 1182.83i	− 1.91707i	0.284978	−	0.958534i	$-0.408014\pi$
$618$	0	0
$619$	−535.727	−0.865472	−0.432736	−	0.901521i	$-0.642452\pi$
$619$	−535.727	−0.865472	−0.432736	+	0.901521i	$0.642452\pi$
$620$	− 417.043i	− 0.672650i
$621$	0	0
$622$	1688.73	2.71500
$623$	− 23.5592i	− 0.0378157i
$624$	0	0
$625$	470.911	0.753458
$626$	− 1145.26i	− 1.82950i
$627$	0	0
$628$	204.485	0.325614
$629$	851.155i	1.35319i
$630$	0	0
$631$	−517.306	−0.819819	−0.409910	−	0.912126i	$-0.634440\pi$
$631$	−517.306	−0.819819	−0.409910	+	0.912126i	$0.634440\pi$
$632$	1143.62i	1.80952i
$633$	0	0
$634$	1249.97	1.97156
$635$	− 16.3857i	− 0.0258043i
$636$	0	0
$637$	−673.988	−1.05807
$638$	− 408.703i	− 0.640600i
$639$	0	0
$640$	296.611	0.463455
$641$	289.081i	0.450984i	0.974245	+	0.225492i	$0.0723989\pi$
$641$	289.081i	0.450984i	−0.974245	+	0.225492i	$0.927601\pi$
$642$	0	0
$643$	−311.244	−0.484050	−0.242025	−	0.970270i	$-0.577812\pi$
$643$	−311.244	−0.484050	−0.242025	+	0.970270i	$0.577812\pi$
$644$	− 170.239i	− 0.264346i
$645$	0	0
$646$	−308.095	−0.476928
$647$	764.050i	1.18091i	0.807070	+	0.590456i	$0.201052\pi$
$647$	764.050i	1.18091i	−0.807070	+	0.590456i	$0.798948\pi$
$648$	0	0
$649$	−675.449	−1.04075
$650$	− 1122.73i	− 1.72727i
$651$	0	0
$652$	56.3105	0.0863659
$653$	442.765i	0.678047i	0.940778	+	0.339023i	$0.110097\pi$
$653$	442.765i	0.678047i	−0.940778	+	0.339023i	$0.889903\pi$
$654$	0	0
$655$	207.971	0.317513
$656$	22.6413i	0.0345141i
$657$	0	0
$658$	−403.277	−0.612883
$659$	− 67.8358i	− 0.102937i	−0.998675	−	0.0514687i	$-0.983610\pi$
$659$	− 67.8358i	− 0.102937i	0.998675	−	0.0514687i	$-0.0163903\pi$
$660$	0	0
$661$	776.372	1.17454	0.587271	−	0.809391i	$-0.300202\pi$
$661$	776.372	1.17454	0.587271	+	0.809391i	$0.300202\pi$
$662$	859.929i	1.29899i
$663$	0	0
$664$	−462.743	−0.696902
$665$	− 17.6840i	− 0.0265924i
$666$	0	0
$667$	−257.472	−0.386014
$668$	− 751.995i	− 1.12574i
$669$	0	0
$670$	−326.477	−0.487279
$671$	620.517i	0.924765i
$672$	0	0
$673$	−1207.59	−1.79434	−0.897172	−	0.441681i	$-0.854382\pi$
$673$	−1207.59	−1.79434	−0.897172	+	0.441681i	$0.854382\pi$
$674$	− 1793.23i	− 2.66058i
$675$	0	0
$676$	−398.142	−0.588967
$677$	− 920.516i	− 1.35970i	−0.733352	−	0.679849i	$-0.762045\pi$
$677$	− 920.516i	− 1.35970i	0.733352	−	0.679849i	$-0.237955\pi$
$678$	0	0
$679$	45.9866	0.0677270
$680$	− 192.094i	− 0.282491i
$681$	0	0
$682$	−872.242	−1.27895
$683$	− 382.840i	− 0.560528i	−0.959923	−	0.280264i	$-0.909578\pi$
$683$	− 382.840i	− 0.560528i	0.959923	−	0.280264i	$-0.0904220\pi$
$684$	0	0
$685$	7.05277	0.0102960
$686$	650.302i	0.947961i
$687$	0	0
$688$	−31.7891	−0.0462051
$689$	− 253.852i	− 0.368436i
$690$	0	0
$691$	−678.861	−0.982433	−0.491217	−	0.871037i	$-0.663448\pi$
$691$	−678.861	−0.982433	−0.491217	+	0.871037i	$0.663448\pi$
$692$	− 1322.31i	− 1.91086i
$693$	0	0
$694$	−166.460	−0.239856
$695$	167.969i	0.241682i
$696$	0	0
$697$	767.816	1.10160
$698$	− 1916.41i	− 2.74557i
$699$	0	0
$700$	−316.975	−0.452822
$701$	684.318i	0.976202i	0.872787	+	0.488101i	$0.162310\pi$
$701$	684.318i	0.976202i	−0.872787	+	0.488101i	$0.837690\pi$
$702$	0	0
$703$	−283.014	−0.402580
$704$	− 627.894i	− 0.891894i
$705$	0	0
$706$	−1488.32	−2.10810
$707$	− 412.605i	− 0.583599i
$708$	0	0
$709$	−1302.62	−1.83727	−0.918633	−	0.395112i	$-0.870706\pi$
$709$	−1302.62	−1.83727	−0.918633	+	0.395112i	$0.870706\pi$
$710$	409.732i	0.577087i
$711$	0	0
$712$	85.0928	0.119512
$713$	549.489i	0.770672i
$714$	0	0
$715$	133.614	0.186873
$716$	− 733.023i	− 1.02377i
$717$	0	0
$718$	−745.890	−1.03884
$719$	− 733.683i	− 1.02042i	−0.860050	−	0.510210i	$-0.829568\pi$
$719$	− 733.683i	− 1.02042i	0.860050	−	0.510210i	$-0.170432\pi$
$720$	0	0
$721$	58.9065	0.0817011
$722$	1062.64i	1.47180i
$723$	0	0
$724$	−2054.58	−2.83782
$725$	479.398i	0.661239i
$726$	0	0
$727$	−509.080	−0.700247	−0.350123	−	0.936704i	$-0.613860\pi$
$727$	−509.080	−0.700247	−0.350123	+	0.936704i	$0.613860\pi$
$728$	255.852i	0.351445i
$729$	0	0
$730$	−68.3136	−0.0935803
$731$	1078.04i	1.47475i
$732$	0	0
$733$	−1281.50	−1.74830	−0.874149	−	0.485658i	$-0.838580\pi$
$733$	−1281.50	−1.74830	−0.874149	+	0.485658i	$0.838580\pi$
$734$	85.2186i	0.116102i
$735$	0	0
$736$	−403.168	−0.547782
$737$	420.601i	0.570694i
$738$	0	0
$739$	−651.069	−0.881013	−0.440507	−	0.897749i	$-0.645201\pi$
$739$	−651.069	−0.881013	−0.440507	+	0.897749i	$0.645201\pi$
$740$	− 468.611i	− 0.633258i
$741$	0	0
$742$	−116.351	−0.156808
$743$	1290.81i	1.73730i	0.495429	+	0.868649i	$0.335011\pi$
$743$	1290.81i	1.73730i	−0.495429	+	0.868649i	$0.664989\pi$
$744$	0	0
$745$	147.816	0.198411
$746$	1708.14i	2.28973i
$747$	0	0
$748$	−657.216	−0.878631
$749$	166.583i	0.222407i
$750$	0	0
$751$	−1270.00	−1.69107	−0.845536	−	0.533918i	$-0.820719\pi$
$751$	−1270.00	−1.69107	−0.845536	+	0.533918i	$0.820719\pi$
$752$	28.9213i	0.0384592i
$753$	0	0
$754$	1027.63	1.36290
$755$	214.721i	0.284399i
$756$	0	0
$757$	−835.003	−1.10304	−0.551521	−	0.834161i	$-0.685952\pi$
$757$	−835.003	−1.10304	−0.551521	+	0.834161i	$0.685952\pi$
$758$	1167.32i	1.54000i
$759$	0	0
$760$	63.8722	0.0840424
$761$	− 907.753i	− 1.19284i	−0.802672	−	0.596421i	$-0.796589\pi$
$761$	− 907.753i	− 1.19284i	0.802672	−	0.596421i	$-0.203411\pi$
$762$	0	0
$763$	−3.23167	−0.00423548
$764$	667.521i	0.873719i
$765$	0	0
$766$	−300.613	−0.392445
$767$	− 1698.32i	− 2.21424i
$768$	0	0
$769$	−396.941	−0.516178	−0.258089	−	0.966121i	$-0.583093\pi$
$769$	−396.941	−0.516178	−0.258089	+	0.966121i	$0.583093\pi$
$770$	− 61.2408i	− 0.0795335i
$771$	0	0
$772$	2141.65	2.77416
$773$	− 1381.35i	− 1.78700i	−0.449059	−	0.893502i	$-0.648241\pi$
$773$	− 1381.35i	− 1.78700i	0.449059	−	0.893502i	$-0.351759\pi$
$774$	0	0
$775$	1023.12	1.32015
$776$	166.098i	0.214044i
$777$	0	0
$778$	662.990	0.852172
$779$	255.303i	0.327731i
$780$	0	0
$781$	527.859	0.675875
$782$	672.154i	0.859532i
$783$	0	0
$784$	22.1542	0.0282580
$785$	− 46.3412i	− 0.0590333i
$786$	0	0
$787$	−113.599	−0.144344	−0.0721722	−	0.997392i	$-0.522993\pi$
$787$	−113.599	−0.144344	−0.0721722	+	0.997392i	$0.522993\pi$
$788$	− 714.149i	− 0.906281i
$789$	0	0
$790$	688.276	0.871235
$791$	− 283.214i	− 0.358045i
$792$	0	0
$793$	−1560.20	−1.96747
$794$	− 576.329i	− 0.725855i
$795$	0	0
$796$	−1149.79	−1.44446
$797$	− 81.8545i	− 0.102703i	−0.998681	−	0.0513516i	$-0.983647\pi$
$797$	− 81.8545i	− 0.102703i	0.998681	−	0.0513516i	$-0.0163529\pi$
$798$	0	0
$799$	980.788	1.22752
$800$	750.676i	0.938345i
$801$	0	0
$802$	−259.919	−0.324089
$803$	88.0086i	0.109600i
$804$	0	0
$805$	−38.5801	−0.0479256
$806$	− 2193.13i	− 2.72101i
$807$	0	0
$808$	1490.28	1.84440
$809$	330.667i	0.408735i	0.978894	+	0.204368i	$0.0655138\pi$
$809$	330.667i	0.408735i	−0.978894	+	0.204368i	$0.934486\pi$
$810$	0	0
$811$	−1106.38	−1.36422	−0.682110	−	0.731249i	$-0.738938\pi$
$811$	−1106.38	−1.36422	−0.682110	+	0.731249i	$0.738938\pi$
$812$	− 290.126i	− 0.357298i
$813$	0	0
$814$	−980.097	−1.20405
$815$	− 12.7613i	− 0.0156580i
$816$	0	0
$817$	−358.454	−0.438744
$818$	163.655i	0.200067i
$819$	0	0
$820$	−422.728	−0.515521
$821$	− 1212.58i	− 1.47696i	−0.674276	−	0.738479i	$-0.735544\pi$
$821$	− 1212.58i	− 1.47696i	0.674276	−	0.738479i	$-0.264456\pi$
$822$	0	0
$823$	384.766	0.467517	0.233759	−	0.972295i	$-0.424897\pi$
$823$	384.766	0.467517	0.233759	+	0.972295i	$0.424897\pi$
$824$	212.763i	0.258207i
$825$	0	0
$826$	−778.412	−0.942388
$827$	836.135i	1.01105i	0.862813	+	0.505523i	$0.168700\pi$
$827$	836.135i	1.01105i	−0.862813	+	0.505523i	$0.831300\pi$
$828$	0	0
$829$	−1008.06	−1.21599	−0.607995	−	0.793941i	$-0.708026\pi$
$829$	−1008.06	−1.21599	−0.607995	+	0.793941i	$0.708026\pi$
$830$	278.497i	0.335539i
$831$	0	0
$832$	1578.75	1.89754
$833$	− 751.300i	− 0.901921i
$834$	0	0
$835$	−170.419	−0.204095
$836$	− 218.528i	− 0.261397i
$837$	0	0
$838$	1360.35	1.62333
$839$	788.289i	0.939558i	0.882784	+	0.469779i	$0.155666\pi$
$839$	788.289i	0.939558i	−0.882784	+	0.469779i	$0.844334\pi$
$840$	0	0
$841$	402.210	0.478252
$842$	− 408.361i	− 0.484990i
$843$	0	0
$844$	1358.38	1.60945
$845$	90.2281i	0.106779i
$846$	0	0
$847$	182.310	0.215242
$848$	8.34422i	0.00983988i
$849$	0	0
$850$	1251.51	1.47237
$851$	617.434i	0.725540i
$852$	0	0
$853$	−355.835	−0.417157	−0.208579	−	0.978006i	$-0.566884\pi$
$853$	−355.835	−0.417157	−0.208579	+	0.978006i	$0.566884\pi$
$854$	715.107i	0.837362i
$855$	0	0
$856$	−601.676	−0.702893
$857$	160.752i	0.187576i	0.995592	+	0.0937878i	$0.0298975\pi$
$857$	160.752i	0.187576i	−0.995592	+	0.0937878i	$0.970102\pi$
$858$	0	0
$859$	378.660	0.440815	0.220408	−	0.975408i	$-0.429261\pi$
$859$	378.660	0.440815	0.220408	+	0.975408i	$0.429261\pi$
$860$	− 593.525i	− 0.690145i
$861$	0	0
$862$	225.023	0.261047
$863$	− 116.099i	− 0.134529i	−0.997735	−	0.0672646i	$-0.978573\pi$
$863$	− 116.099i	− 0.134529i	0.997735	−	0.0672646i	$-0.0214272\pi$
$864$	0	0
$865$	−299.667	−0.346436
$866$	− 1397.65i	− 1.61391i
$867$	0	0
$868$	−619.179	−0.713339
$869$	− 886.707i	− 1.02038i
$870$	0	0
$871$	−1057.54	−1.21417
$872$	− 11.6724i	− 0.0133858i
$873$	0	0
$874$	−223.495	−0.255715
$875$	150.304i	0.171776i
$876$	0	0
$877$	−21.0296	−0.0239790	−0.0119895	−	0.999928i	$-0.503816\pi$
$877$	−21.0296	−0.0239790	−0.0119895	+	0.999928i	$0.503816\pi$
$878$	− 2356.82i	− 2.68430i
$879$	0	0
$880$	−4.39193	−0.00499083
$881$	− 176.808i	− 0.200690i	−0.994953	−	0.100345i	$-0.968005\pi$
$881$	− 176.808i	− 0.200690i	0.994953	−	0.100345i	$-0.0319947\pi$
$882$	0	0
$883$	1552.93	1.75870	0.879349	−	0.476177i	$-0.157978\pi$
$883$	1552.93	1.75870	0.879349	+	0.476177i	$0.157978\pi$
$884$	− 1652.48i	− 1.86932i
$885$	0	0
$886$	366.085	0.413189
$887$	1436.68i	1.61971i	0.586631	+	0.809854i	$0.300454\pi$
$887$	1436.68i	1.61971i	−0.586631	+	0.809854i	$0.699546\pi$
$888$	0	0
$889$	−24.3277	−0.0273652
$890$	− 51.2122i	− 0.0575418i
$891$	0	0
$892$	−1622.08	−1.81848
$893$	326.117i	0.365193i
$894$	0	0
$895$	−166.120	−0.185609
$896$	− 440.376i	− 0.491491i
$897$	0	0
$898$	1201.80	1.33831
$899$	936.454i	1.04166i
$900$	0	0
$901$	282.971	0.314064
$902$	884.132i	0.980191i
$903$	0	0
$904$	1022.93	1.13156
$905$	465.616i	0.514492i
$906$	0	0
$907$	−1361.80	−1.50144	−0.750718	−	0.660623i	$-0.770292\pi$
$907$	−1361.80	−1.50144	−0.750718	+	0.660623i	$0.770292\pi$
$908$	− 1032.91i	− 1.13757i
$909$	0	0
$910$	153.982	0.169211
$911$	− 1434.07i	− 1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$911$	− 1434.07i	− 1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$912$	0	0
$913$	358.789	0.392978
$914$	1696.43i	1.85605i
$915$	0	0
$916$	−107.804	−0.117690
$917$	− 308.772i	− 0.336720i
$918$	0	0
$919$	752.268	0.818573	0.409286	−	0.912406i	$-0.365778\pi$
$919$	752.268	0.818573	0.409286	+	0.912406i	$0.365778\pi$
$920$	− 139.346i	− 0.151463i
$921$	0	0
$922$	277.856	0.301363
$923$	1327.23i	1.43795i
$924$	0	0
$925$	1149.63	1.24284
$926$	− 1276.18i	− 1.37816i
$927$	0	0
$928$	−687.089	−0.740398
$929$	− 74.4733i	− 0.0801650i	−0.999196	−	0.0400825i	$-0.987238\pi$
$929$	− 74.4733i	− 0.0801650i	0.999196	−	0.0400825i	$-0.0127621\pi$
$930$	0	0
$931$	249.811	0.268326
$932$	− 360.775i	− 0.387098i
$933$	0	0
$934$	1285.00	1.37581
$935$	148.940i	0.159294i
$936$	0	0
$937$	363.649	0.388099	0.194050	−	0.980992i	$-0.437838\pi$
$937$	363.649	0.388099	0.194050	+	0.980992i	$0.437838\pi$
$938$	484.717i	0.516756i
$939$	0	0
$940$	−539.981	−0.574448
$941$	− 201.166i	− 0.213779i	−0.994271	−	0.106890i	$-0.965911\pi$
$941$	− 201.166i	− 0.213779i	0.994271	−	0.106890i	$-0.0340892\pi$
$942$	0	0
$943$	556.979	0.590646
$944$	55.8244i	0.0591361i
$945$	0	0
$946$	−1241.35	−1.31221
$947$	− 1001.85i	− 1.05792i	−0.848647	−	0.528959i	$-0.822582\pi$
$947$	− 1001.85i	− 1.05792i	0.848647	−	0.528959i	$-0.177418\pi$
$948$	0	0
$949$	−221.285	−0.233177
$950$	416.135i	0.438037i
$951$	0	0
$952$	−285.200	−0.299580
$953$	− 1431.96i	− 1.50258i	−0.659971	−	0.751291i	$-0.729432\pi$
$953$	− 1431.96i	− 1.50258i	0.659971	−	0.751291i	$-0.270568\pi$
$954$	0	0
$955$	151.276	0.158404
$956$	2345.59i	2.45355i
$957$	0	0
$958$	962.500	1.00470
$959$	− 10.4712i	− 0.0109188i
$960$	0	0
$961$	1037.56	1.07966
$962$	− 2464.32i	− 2.56166i
$963$	0	0
$964$	2018.02	2.09338
$965$	− 485.348i	− 0.502951i
$966$	0	0
$967$	812.242	0.839961	0.419980	−	0.907533i	$-0.362037\pi$
$967$	812.242	0.839961	0.419980	+	0.907533i	$0.362037\pi$
$968$	658.480i	0.680248i
$969$	0	0
$970$	99.9643	0.103056
$971$	1589.54i	1.63701i	0.574496	+	0.818507i	$0.305198\pi$
$971$	1589.54i	1.63701i	−0.574496	+	0.818507i	$0.694802\pi$
$972$	0	0
$973$	249.382	0.256302
$974$	− 13.1613i	− 0.0135126i
$975$	0	0
$976$	51.2845	0.0525456
$977$	− 1147.24i	− 1.17425i	−0.809495	−	0.587126i	$-0.800259\pi$
$977$	− 1147.24i	− 1.17425i	0.809495	−	0.587126i	$-0.199741\pi$
$978$	0	0
$979$	−65.9768	−0.0673920
$980$	413.635i	0.422076i
$981$	0	0
$982$	−1664.91	−1.69543
$983$	− 1293.14i	− 1.31551i	−0.753234	−	0.657753i	$-0.771507\pi$
$983$	− 1293.14i	− 1.31551i	0.753234	−	0.657753i	$-0.228493\pi$
$984$	0	0
$985$	−161.843	−0.164307
$986$	1145.50i	1.16177i
$987$	0	0
$988$	549.458	0.556131
$989$	782.019i	0.790717i
$990$	0	0
$991$	−605.355	−0.610852	−0.305426	−	0.952216i	$-0.598799\pi$
$991$	−605.355	−0.610852	−0.305426	+	0.952216i	$0.598799\pi$
$992$	1466.37i	1.47819i
$993$	0	0
$994$	608.324	0.611996
$995$	260.570i	0.261879i
$996$	0	0
$997$	927.609	0.930400	0.465200	−	0.885206i	$-0.345982\pi$
$997$	927.609	0.930400	0.465200	+	0.885206i	$0.345982\pi$
$998$	85.1193i	0.0852898i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.9	✓	84
3.2	odd	2	inner	1143.3.b.a.890.76	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.9	✓	84	1.1	even	1	trivial
1143.3.b.a.890.76	yes	84	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-3.22737i q^{2} -6.41592 q^{4} +1.45400i q^{5} +2.15873 q^{7} +7.79707i q^{8} +O(q^{10})\)	\(q-3.22737i q^{2} -6.41592 q^{4} +1.45400i q^{5} +2.15873 q^{7} +7.79707i q^{8} +4.69259 q^{10} -6.04547i q^{11} +15.2005 q^{13} -6.96703i q^{14} -0.499646 q^{16} +16.9441i q^{17} -5.63400 q^{19} -9.32873i q^{20} -19.5110 q^{22} +12.2914i q^{23} +22.8859 q^{25} -49.0577i q^{26} -13.8503 q^{28} +20.9473i q^{29} +44.7052 q^{31} +32.8008i q^{32} +54.6849 q^{34} +3.13879i q^{35} +50.2331 q^{37} +18.1830i q^{38} -11.3369 q^{40} -45.3146i q^{41} +63.6233 q^{43} +38.7873i q^{44} +39.6689 q^{46} -57.8837i q^{47} -44.3399 q^{49} -73.8613i q^{50} -97.5252 q^{52} -16.7003i q^{53} +8.79009 q^{55} +16.8318i q^{56} +67.6048 q^{58} -111.728i q^{59} -102.642 q^{61} -144.280i q^{62} +103.862 q^{64} +22.1015i q^{65} -69.5730 q^{67} -108.712i q^{68} +10.1300 q^{70} +87.3147i q^{71} -14.5578 q^{73} -162.121i q^{74} +36.1473 q^{76} -13.0506i q^{77} +146.673 q^{79} -0.726483i q^{80} -146.247 q^{82} +59.3483i q^{83} -24.6367 q^{85} -205.336i q^{86} +47.1370 q^{88} -10.9134i q^{89} +32.8138 q^{91} -78.8606i q^{92} -186.812 q^{94} -8.19182i q^{95} +21.3026 q^{97} +143.101i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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