LMFDB - Embedded newform 1143.3.b.a.890.74

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			890.74
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.06240i q^{2} -5.37827 q^{4} +5.25232i q^{5} -2.64254 q^{7} -4.22081i q^{8} +O(q^{10})$	\(q+3.06240i q^{2} -5.37827 q^{4} +5.25232i q^{5} -2.64254 q^{7} -4.22081i q^{8} -16.0847 q^{10} +4.48093i q^{11} -6.53221 q^{13} -8.09251i q^{14} -8.58729 q^{16} +18.5009i q^{17} -4.12806 q^{19} -28.2484i q^{20} -13.7224 q^{22} +35.9663i q^{23} -2.58688 q^{25} -20.0042i q^{26} +14.2123 q^{28} -36.9550i q^{29} -31.4793 q^{31} -43.1809i q^{32} -56.6572 q^{34} -13.8795i q^{35} +60.1826 q^{37} -12.6418i q^{38} +22.1690 q^{40} +19.7137i q^{41} -19.2361 q^{43} -24.0996i q^{44} -110.143 q^{46} -61.4575i q^{47} -42.0170 q^{49} -7.92206i q^{50} +35.1320 q^{52} -46.6228i q^{53} -23.5353 q^{55} +11.1537i q^{56} +113.171 q^{58} +24.6150i q^{59} -43.2555 q^{61} -96.4020i q^{62} +97.8879 q^{64} -34.3093i q^{65} -6.75193 q^{67} -99.5030i q^{68} +42.5044 q^{70} +44.4482i q^{71} +8.96829 q^{73} +184.303i q^{74} +22.2018 q^{76} -11.8410i q^{77} -23.4298 q^{79} -45.1032i q^{80} -60.3710 q^{82} -60.5881i q^{83} -97.1728 q^{85} -58.9085i q^{86} +18.9131 q^{88} +43.3046i q^{89} +17.2616 q^{91} -193.436i q^{92} +188.207 q^{94} -21.6819i q^{95} +64.0922 q^{97} -128.673i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.06240i	1.53120i	0.643318	+	0.765599i	$0.277557\pi$
$2$	3.06240i	1.53120i	−0.643318	+	0.765599i	$0.722443\pi$
$3$	0	0
$3$	0	0
$4$	−5.37827	−1.34457
$5$	5.25232i	1.05046i	0.850959	+	0.525232i	$0.176022\pi$
$5$	5.25232i	1.05046i	−0.850959	+	0.525232i	$0.823978\pi$
$6$	0	0
$7$	−2.64254	−0.377506	−0.188753	−	0.982025i	$-0.560445\pi$
$7$	−2.64254	−0.377506	−0.188753	+	0.982025i	$0.560445\pi$
$8$	− 4.22081i	− 0.527601i
$9$	0	0
$10$	−16.0847	−1.60847
$11$	4.48093i	0.407357i	0.979038	+	0.203678i	$0.0652897\pi$
$11$	4.48093i	0.407357i	−0.979038	+	0.203678i	$0.934710\pi$
$12$	0	0
$13$	−6.53221	−0.502478	−0.251239	−	0.967925i	$-0.580838\pi$
$13$	−6.53221	−0.502478	−0.251239	+	0.967925i	$0.580838\pi$
$14$	− 8.09251i	− 0.578036i
$15$	0	0
$16$	−8.58729	−0.536706
$17$	18.5009i	1.08829i	0.838991	+	0.544145i	$0.183146\pi$
$17$	18.5009i	1.08829i	−0.838991	+	0.544145i	$0.816854\pi$
$18$	0	0
$19$	−4.12806	−0.217266	−0.108633	−	0.994082i	$-0.534647\pi$
$19$	−4.12806	−0.217266	−0.108633	+	0.994082i	$0.534647\pi$
$20$	− 28.2484i	− 1.41242i
$21$	0	0
$22$	−13.7224	−0.623744
$23$	35.9663i	1.56375i	0.623434	+	0.781876i	$0.285737\pi$
$23$	35.9663i	1.56375i	−0.623434	+	0.781876i	$0.714263\pi$
$24$	0	0
$25$	−2.58688	−0.103475
$26$	− 20.0042i	− 0.769393i
$27$	0	0
$28$	14.2123	0.507582
$29$	− 36.9550i	− 1.27431i	−0.770735	−	0.637155i	$-0.780111\pi$
$29$	− 36.9550i	− 1.27431i	0.770735	−	0.637155i	$-0.219889\pi$
$30$	0	0
$31$	−31.4793	−1.01546	−0.507730	−	0.861516i	$-0.669515\pi$
$31$	−31.4793	−1.01546	−0.507730	+	0.861516i	$0.669515\pi$
$32$	− 43.1809i	− 1.34940i
$33$	0	0
$34$	−56.6572	−1.66639
$35$	− 13.8795i	− 0.396556i
$36$	0	0
$37$	60.1826	1.62656	0.813278	−	0.581875i	$-0.197681\pi$
$37$	60.1826	1.62656	0.813278	+	0.581875i	$0.197681\pi$
$38$	− 12.6418i	− 0.332678i
$39$	0	0
$40$	22.1690	0.554226
$41$	19.7137i	0.480821i	0.970671	+	0.240410i	$0.0772820\pi$
$41$	19.7137i	0.480821i	−0.970671	+	0.240410i	$0.922718\pi$
$42$	0	0
$43$	−19.2361	−0.447351	−0.223676	−	0.974664i	$-0.571806\pi$
$43$	−19.2361	−0.447351	−0.223676	+	0.974664i	$0.571806\pi$
$44$	− 24.0996i	− 0.547719i
$45$	0	0
$46$	−110.143	−2.39441
$47$	− 61.4575i	− 1.30761i	−0.756664	−	0.653804i	$-0.773172\pi$
$47$	− 61.4575i	− 1.30761i	0.756664	−	0.653804i	$-0.226828\pi$
$48$	0	0
$49$	−42.0170	−0.857489
$50$	− 7.92206i	− 0.158441i
$51$	0	0
$52$	35.1320	0.675615
$53$	− 46.6228i	− 0.879676i	−0.898077	−	0.439838i	$-0.855036\pi$
$53$	− 46.6228i	− 0.879676i	0.898077	−	0.439838i	$-0.144964\pi$
$54$	0	0
$55$	−23.5353	−0.427914
$56$	11.1537i	0.199172i
$57$	0	0
$58$	113.171	1.95122
$59$	24.6150i	0.417204i	0.978001	+	0.208602i	$0.0668913\pi$
$59$	24.6150i	0.417204i	−0.978001	+	0.208602i	$0.933109\pi$
$60$	0	0
$61$	−43.2555	−0.709107	−0.354553	−	0.935036i	$-0.615367\pi$
$61$	−43.2555	−0.709107	−0.354553	+	0.935036i	$0.615367\pi$
$62$	− 96.4020i	− 1.55487i
$63$	0	0
$64$	97.8879	1.52950
$65$	− 34.3093i	− 0.527835i
$66$	0	0
$67$	−6.75193	−0.100775	−0.0503875	−	0.998730i	$-0.516046\pi$
$67$	−6.75193	−0.100775	−0.0503875	+	0.998730i	$0.516046\pi$
$68$	− 99.5030i	− 1.46328i
$69$	0	0
$70$	42.5044	0.607206
$71$	44.4482i	0.626030i	0.949748	+	0.313015i	$0.101339\pi$
$71$	44.4482i	0.626030i	−0.949748	+	0.313015i	$0.898661\pi$
$72$	0	0
$73$	8.96829	0.122853	0.0614266	−	0.998112i	$-0.480435\pi$
$73$	8.96829	0.122853	0.0614266	+	0.998112i	$0.480435\pi$
$74$	184.303i	2.49058i
$75$	0	0
$76$	22.2018	0.292129
$77$	− 11.8410i	− 0.153780i
$78$	0	0
$79$	−23.4298	−0.296579	−0.148290	−	0.988944i	$-0.547377\pi$
$79$	−23.4298	−0.296579	−0.148290	+	0.988944i	$0.547377\pi$
$80$	− 45.1032i	− 0.563790i
$81$	0	0
$82$	−60.3710	−0.736232
$83$	− 60.5881i	− 0.729977i	−0.931012	−	0.364989i	$-0.881073\pi$
$83$	− 60.5881i	− 0.729977i	0.931012	−	0.364989i	$-0.118927\pi$
$84$	0	0
$85$	−97.1728	−1.14321
$86$	− 58.9085i	− 0.684983i
$87$	0	0
$88$	18.9131	0.214922
$89$	43.3046i	0.486569i	0.969955	+	0.243284i	$0.0782249\pi$
$89$	43.3046i	0.486569i	−0.969955	+	0.243284i	$0.921775\pi$
$90$	0	0
$91$	17.2616	0.189688
$92$	− 193.436i	− 2.10257i
$93$	0	0
$94$	188.207	2.00221
$95$	− 21.6819i	− 0.228231i
$96$	0	0
$97$	64.0922	0.660744	0.330372	−	0.943851i	$-0.392826\pi$
$97$	64.0922	0.660744	0.330372	+	0.943851i	$0.392826\pi$
$98$	− 128.673i	− 1.31299i
$99$	0	0
$100$	13.9130	0.139130
$101$	− 84.6780i	− 0.838396i	−0.907895	−	0.419198i	$-0.862311\pi$
$101$	− 84.6780i	− 0.838396i	0.907895	−	0.419198i	$-0.137689\pi$
$102$	0	0
$103$	81.0163	0.786566	0.393283	−	0.919417i	$-0.371339\pi$
$103$	81.0163	0.786566	0.393283	+	0.919417i	$0.371339\pi$
$104$	27.5712i	0.265108i
$105$	0	0
$106$	142.778	1.34696
$107$	166.803i	1.55891i	0.626460	+	0.779454i	$0.284503\pi$
$107$	166.803i	1.55891i	−0.626460	+	0.779454i	$0.715497\pi$
$108$	0	0
$109$	−6.11967	−0.0561437	−0.0280719	−	0.999606i	$-0.508937\pi$
$109$	−6.11967	−0.0561437	−0.0280719	+	0.999606i	$0.508937\pi$
$110$	− 72.0743i	− 0.655221i
$111$	0	0
$112$	22.6923	0.202610
$113$	− 225.544i	− 1.99597i	−0.0634585	−	0.997984i	$-0.520213\pi$
$113$	− 225.544i	− 1.99597i	0.0634585	−	0.997984i	$-0.479787\pi$
$114$	0	0
$115$	−188.907	−1.64267
$116$	198.754i	1.71340i
$117$	0	0
$118$	−75.3810	−0.638822
$119$	− 48.8895i	− 0.410836i
$120$	0	0
$121$	100.921	0.834060
$122$	− 132.465i	− 1.08578i
$123$	0	0
$124$	169.304	1.36536
$125$	117.721i	0.941767i
$126$	0	0
$127$	11.2694	0.0887357
$127$	11.2694	0.0887357
$128$	127.048i	0.992562i
$129$	0	0
$130$	105.069	0.808220
$131$	− 232.707i	− 1.77639i	−0.459469	−	0.888194i	$-0.651960\pi$
$131$	− 232.707i	− 1.77639i	0.459469	−	0.888194i	$-0.348040\pi$
$132$	0	0
$133$	10.9086	0.0820193
$134$	− 20.6771i	− 0.154307i
$135$	0	0
$136$	78.0889	0.574183
$137$	131.074i	0.956747i	0.878157	+	0.478373i	$0.158773\pi$
$137$	131.074i	0.956747i	−0.878157	+	0.478373i	$0.841227\pi$
$138$	0	0
$139$	212.317	1.52746	0.763732	−	0.645534i	$-0.223365\pi$
$139$	212.317	1.52746	0.763732	+	0.645534i	$0.223365\pi$
$140$	74.6476i	0.533197i
$141$	0	0
$142$	−136.118	−0.958577
$143$	− 29.2704i	− 0.204688i
$144$	0	0
$145$	194.100	1.33862
$146$	27.4645i	0.188113i
$147$	0	0
$148$	−323.678	−2.18702
$149$	− 6.04907i	− 0.0405978i	−0.999794	−	0.0202989i	$-0.993538\pi$
$149$	− 6.04907i	− 0.0405978i	0.999794	−	0.0202989i	$-0.00646178\pi$
$150$	0	0
$151$	114.743	0.759889	0.379945	−	0.925009i	$-0.375943\pi$
$151$	114.743	0.759889	0.379945	+	0.925009i	$0.375943\pi$
$152$	17.4238i	0.114630i
$153$	0	0
$154$	36.2619	0.235467
$155$	− 165.339i	− 1.06671i
$156$	0	0
$157$	−239.194	−1.52353	−0.761765	−	0.647854i	$-0.775667\pi$
$157$	−239.194	−1.52353	−0.761765	+	0.647854i	$0.775667\pi$
$158$	− 71.7512i	− 0.454122i
$159$	0	0
$160$	226.800	1.41750
$161$	− 95.0424i	− 0.590325i
$162$	0	0
$163$	−125.077	−0.767346	−0.383673	−	0.923469i	$-0.625341\pi$
$163$	−125.077	−0.767346	−0.383673	+	0.923469i	$0.625341\pi$
$164$	− 106.025i	− 0.646496i
$165$	0	0
$166$	185.545	1.11774
$167$	14.8310i	0.0888085i	0.999014	+	0.0444043i	$0.0141390\pi$
$167$	14.8310i	0.0888085i	−0.999014	+	0.0444043i	$0.985861\pi$
$168$	0	0
$169$	−126.330	−0.747516
$170$	− 297.582i	− 1.75048i
$171$	0	0
$172$	103.457	0.601494
$173$	− 22.0241i	− 0.127307i	−0.997972	−	0.0636535i	$-0.979725\pi$
$173$	− 22.0241i	− 0.127307i	0.997972	−	0.0636535i	$-0.0202752\pi$
$174$	0	0
$175$	6.83595	0.0390625
$176$	− 38.4790i	− 0.218631i
$177$	0	0
$178$	−132.616	−0.745033
$179$	335.906i	1.87657i	0.345862	+	0.938285i	$0.387586\pi$
$179$	335.906i	1.87657i	−0.345862	+	0.938285i	$0.612414\pi$
$180$	0	0
$181$	−271.379	−1.49933	−0.749665	−	0.661818i	$-0.769785\pi$
$181$	−271.379	−1.49933	−0.749665	+	0.661818i	$0.769785\pi$
$182$	52.8620i	0.290450i
$183$	0	0
$184$	151.807	0.825037
$185$	316.098i	1.70864i
$186$	0	0
$187$	−82.9013	−0.443322
$188$	330.535i	1.75817i
$189$	0	0
$190$	66.3986	0.349466
$191$	114.291i	0.598382i	0.954193	+	0.299191i	$0.0967168\pi$
$191$	114.291i	0.598382i	−0.954193	+	0.299191i	$0.903283\pi$
$192$	0	0
$193$	−119.740	−0.620413	−0.310206	−	0.950669i	$-0.600398\pi$
$193$	−119.740	−0.620413	−0.310206	+	0.950669i	$0.600398\pi$
$194$	196.276i	1.01173i
$195$	0	0
$196$	225.979	1.15295
$197$	272.581i	1.38366i	0.722060	+	0.691830i	$0.243195\pi$
$197$	272.581i	1.38366i	−0.722060	+	0.691830i	$0.756805\pi$
$198$	0	0
$199$	86.4883	0.434615	0.217307	−	0.976103i	$-0.430273\pi$
$199$	86.4883	0.434615	0.217307	+	0.976103i	$0.430273\pi$
$200$	10.9187i	0.0545937i
$201$	0	0
$202$	259.318	1.28375
$203$	97.6551i	0.481060i
$204$	0	0
$205$	−103.542	−0.505085
$206$	248.104i	1.20439i
$207$	0	0
$208$	56.0940	0.269683
$209$	− 18.4975i	− 0.0885050i
$210$	0	0
$211$	−38.3542	−0.181774	−0.0908868	−	0.995861i	$-0.528970\pi$
$211$	−38.3542	−0.181774	−0.0908868	+	0.995861i	$0.528970\pi$
$212$	250.750i	1.18278i
$213$	0	0
$214$	−510.817	−2.38700
$215$	− 101.034i	− 0.469926i
$216$	0	0
$217$	83.1853	0.383342
$218$	− 18.7408i	− 0.0859672i
$219$	0	0
$220$	126.579	0.575359
$221$	− 120.852i	− 0.546842i
$222$	0	0
$223$	−79.5135	−0.356563	−0.178281	−	0.983980i	$-0.557054\pi$
$223$	−79.5135	−0.356563	−0.178281	+	0.983980i	$0.557054\pi$
$224$	114.107i	0.509408i
$225$	0	0
$226$	690.707	3.05622
$227$	241.685i	1.06469i	0.846528	+	0.532345i	$0.178689\pi$
$227$	241.685i	1.06469i	−0.846528	+	0.532345i	$0.821311\pi$
$228$	0	0
$229$	−341.127	−1.48964	−0.744820	−	0.667266i	$-0.767464\pi$
$229$	−341.127	−1.48964	−0.744820	+	0.667266i	$0.767464\pi$
$230$	− 578.507i	− 2.51525i
$231$	0	0
$232$	−155.980	−0.672328
$233$	− 237.837i	− 1.02076i	−0.859950	−	0.510379i	$-0.829505\pi$
$233$	− 237.837i	− 1.02076i	0.859950	−	0.510379i	$-0.170495\pi$
$234$	0	0
$235$	322.795	1.37359
$236$	− 132.386i	− 0.560959i
$237$	0	0
$238$	149.719	0.629071
$239$	− 65.3419i	− 0.273397i	−0.990613	−	0.136698i	$-0.956351\pi$
$239$	− 65.3419i	− 0.273397i	0.990613	−	0.136698i	$-0.0436491\pi$
$240$	0	0
$241$	−139.424	−0.578521	−0.289261	−	0.957250i	$-0.593409\pi$
$241$	−139.424	−0.578521	−0.289261	+	0.957250i	$0.593409\pi$
$242$	309.061i	1.27711i
$243$	0	0
$244$	232.640	0.953441
$245$	− 220.687i	− 0.900762i
$246$	0	0
$247$	26.9654	0.109172
$248$	132.868i	0.535758i
$249$	0	0
$250$	−360.508	−1.44203
$251$	250.839i	0.999357i	0.866211	+	0.499679i	$0.166549\pi$
$251$	250.839i	0.999357i	−0.866211	+	0.499679i	$0.833451\pi$
$252$	0	0
$253$	−161.162	−0.637005
$254$	34.5115i	0.135872i
$255$	0	0
$256$	2.48077	0.00969049
$257$	− 91.9826i	− 0.357909i	−0.983857	−	0.178954i	$-0.942729\pi$
$257$	− 91.9826i	− 0.357909i	0.983857	−	0.178954i	$-0.0572715\pi$
$258$	0	0
$259$	−159.035	−0.614035
$260$	184.525i	0.709710i
$261$	0	0
$262$	712.640	2.72000
$263$	358.057i	1.36143i	0.732547	+	0.680716i	$0.238331\pi$
$263$	358.057i	1.36143i	−0.732547	+	0.680716i	$0.761669\pi$
$264$	0	0
$265$	244.878	0.924069
$266$	33.4064i	0.125588i
$267$	0	0
$268$	36.3137	0.135499
$269$	116.606i	0.433479i	0.976229	+	0.216739i	$0.0695422\pi$
$269$	116.606i	0.433479i	−0.976229	+	0.216739i	$0.930458\pi$
$270$	0	0
$271$	−48.0667	−0.177368	−0.0886839	−	0.996060i	$-0.528266\pi$
$271$	−48.0667	−0.177368	−0.0886839	+	0.996060i	$0.528266\pi$
$272$	− 158.873i	− 0.584092i
$273$	0	0
$274$	−401.401	−1.46497
$275$	− 11.5916i	− 0.0421514i
$276$	0	0
$277$	81.2173	0.293203	0.146602	−	0.989196i	$-0.453166\pi$
$277$	81.2173	0.293203	0.146602	+	0.989196i	$0.453166\pi$
$278$	650.200i	2.33885i
$279$	0	0
$280$	−58.5826	−0.209224
$281$	115.455i	0.410870i	0.978671	+	0.205435i	$0.0658610\pi$
$281$	115.455i	0.410870i	−0.978671	+	0.205435i	$0.934139\pi$
$282$	0	0
$283$	−174.042	−0.614991	−0.307495	−	0.951550i	$-0.599491\pi$
$283$	−174.042	−0.614991	−0.307495	+	0.951550i	$0.599491\pi$
$284$	− 239.054i	− 0.841740i
$285$	0	0
$286$	89.6374	0.313418
$287$	− 52.0941i	− 0.181513i
$288$	0	0
$289$	−53.2844	−0.184375
$290$	594.410i	2.04969i
$291$	0	0
$292$	−48.2339	−0.165185
$293$	− 330.591i	− 1.12830i	−0.825673	−	0.564149i	$-0.809205\pi$
$293$	− 330.591i	− 1.12830i	0.825673	−	0.564149i	$-0.190795\pi$
$294$	0	0
$295$	−129.286	−0.438258
$296$	− 254.019i	− 0.858173i
$297$	0	0
$298$	18.5246	0.0621632
$299$	− 234.939i	− 0.785751i
$300$	0	0
$301$	50.8322	0.168878
$302$	351.389i	1.16354i
$303$	0	0
$304$	35.4489	0.116608
$305$	− 227.192i	− 0.744891i
$306$	0	0
$307$	−240.533	−0.783494	−0.391747	−	0.920073i	$-0.628129\pi$
$307$	−240.533	−0.783494	−0.391747	+	0.920073i	$0.628129\pi$
$308$	63.6843i	0.206767i
$309$	0	0
$310$	506.334	1.63334
$311$	103.308i	0.332179i	0.986111	+	0.166089i	$0.0531140\pi$
$311$	103.308i	0.332179i	−0.986111	+	0.166089i	$0.946886\pi$
$312$	0	0
$313$	−173.122	−0.553104	−0.276552	−	0.960999i	$-0.589192\pi$
$313$	−173.122	−0.553104	−0.276552	+	0.960999i	$0.589192\pi$
$314$	− 732.507i	− 2.33283i
$315$	0	0
$316$	126.012	0.398771
$317$	− 301.974i	− 0.952600i	−0.879283	−	0.476300i	$-0.841978\pi$
$317$	− 301.974i	− 0.952600i	0.879283	−	0.476300i	$-0.158022\pi$
$318$	0	0
$319$	165.593	0.519099
$320$	514.139i	1.60668i
$321$	0	0
$322$	291.057	0.903905
$323$	− 76.3730i	− 0.236449i
$324$	0	0
$325$	16.8981	0.0519941
$326$	− 383.036i	− 1.17496i
$327$	0	0
$328$	83.2075	0.253682
$329$	162.404i	0.493629i
$330$	0	0
$331$	−437.919	−1.32302	−0.661509	−	0.749938i	$-0.730083\pi$
$331$	−437.919	−1.32302	−0.661509	+	0.749938i	$0.730083\pi$
$332$	325.859i	0.981503i
$333$	0	0
$334$	−45.4185	−0.135983
$335$	− 35.4633i	− 0.105861i
$336$	0	0
$337$	−359.907	−1.06797	−0.533986	−	0.845493i	$-0.679306\pi$
$337$	−359.907	−1.06797	−0.533986	+	0.845493i	$0.679306\pi$
$338$	− 386.873i	− 1.14459i
$339$	0	0
$340$	522.622	1.53712
$341$	− 141.056i	− 0.413655i
$342$	0	0
$343$	240.516	0.701213
$344$	81.1918i	0.236023i
$345$	0	0
$346$	67.4465	0.194932
$347$	220.886i	0.636560i	0.947997	+	0.318280i	$0.103105\pi$
$347$	220.886i	0.636560i	−0.947997	+	0.318280i	$0.896895\pi$
$348$	0	0
$349$	−354.027	−1.01440	−0.507201	−	0.861828i	$-0.669320\pi$
$349$	−354.027	−1.01440	−0.507201	+	0.861828i	$0.669320\pi$
$350$	20.9344i	0.0598125i
$351$	0	0
$352$	193.491	0.549689
$353$	− 620.179i	− 1.75688i	−0.477851	−	0.878441i	$-0.658584\pi$
$353$	− 620.179i	− 1.75688i	0.477851	−	0.878441i	$-0.341416\pi$
$354$	0	0
$355$	−233.456	−0.657623
$356$	− 232.904i	− 0.654225i
$357$	0	0
$358$	−1028.68	−2.87340
$359$	589.579i	1.64228i	0.570726	+	0.821141i	$0.306662\pi$
$359$	589.579i	1.64228i	−0.570726	+	0.821141i	$0.693338\pi$
$360$	0	0
$361$	−343.959	−0.952795
$362$	− 831.069i	− 2.29577i
$363$	0	0
$364$	−92.8377	−0.255049
$365$	47.1043i	0.129053i
$366$	0	0
$367$	−371.572	−1.01246	−0.506228	−	0.862399i	$-0.668961\pi$
$367$	−371.572	−1.01246	−0.506228	+	0.862399i	$0.668961\pi$
$368$	− 308.853i	− 0.839275i
$369$	0	0
$370$	−968.019	−2.61627
$371$	123.203i	0.332083i
$372$	0	0
$373$	−57.7415	−0.154803	−0.0774015	−	0.997000i	$-0.524662\pi$
$373$	−57.7415	−0.154803	−0.0774015	+	0.997000i	$0.524662\pi$
$374$	− 253.877i	− 0.678814i
$375$	0	0
$376$	−259.400	−0.689895
$377$	241.398i	0.640313i
$378$	0	0
$379$	−368.555	−0.972441	−0.486220	−	0.873836i	$-0.661625\pi$
$379$	−368.555	−0.972441	−0.486220	+	0.873836i	$0.661625\pi$
$380$	116.611i	0.306871i
$381$	0	0
$382$	−350.004	−0.916242
$383$	165.786i	0.432861i	0.976298	+	0.216431i	$0.0694415\pi$
$383$	165.786i	0.432861i	−0.976298	+	0.216431i	$0.930558\pi$
$384$	0	0
$385$	62.1929	0.161540
$386$	− 366.690i	− 0.949975i
$387$	0	0
$388$	−344.705	−0.888415
$389$	24.7882i	0.0637228i	0.999492	+	0.0318614i	$0.0101435\pi$
$389$	24.7882i	0.0637228i	−0.999492	+	0.0318614i	$0.989856\pi$
$390$	0	0
$391$	−665.410	−1.70182
$392$	177.346i	0.452412i
$393$	0	0
$394$	−834.751	−2.11866
$395$	− 123.061i	− 0.311546i
$396$	0	0
$397$	397.699	1.00176	0.500881	−	0.865516i	$-0.333010\pi$
$397$	397.699	1.00176	0.500881	+	0.865516i	$0.333010\pi$
$398$	264.861i	0.665481i
$399$	0	0
$400$	22.2143	0.0555358
$401$	336.736i	0.839742i	0.907584	+	0.419871i	$0.137925\pi$
$401$	336.736i	0.839742i	−0.907584	+	0.419871i	$0.862075\pi$
$402$	0	0
$403$	205.629	0.510247
$404$	455.421i	1.12728i
$405$	0	0
$406$	−299.059	−0.736598
$407$	269.674i	0.662589i
$408$	0	0
$409$	334.028	0.816695	0.408347	−	0.912827i	$-0.366105\pi$
$409$	334.028	0.816695	0.408347	+	0.912827i	$0.366105\pi$
$410$	− 317.088i	− 0.773385i
$411$	0	0
$412$	−435.728	−1.05759
$413$	− 65.0462i	− 0.157497i
$414$	0	0
$415$	318.228	0.766815
$416$	282.067i	0.678046i
$417$	0	0
$418$	56.6468	0.135519
$419$	− 293.225i	− 0.699821i	−0.936783	−	0.349910i	$-0.886212\pi$
$419$	− 293.225i	− 0.699821i	0.936783	−	0.349910i	$-0.113788\pi$
$420$	0	0
$421$	−154.980	−0.368123	−0.184061	−	0.982915i	$-0.558924\pi$
$421$	−154.980	−0.368123	−0.184061	+	0.982915i	$0.558924\pi$
$422$	− 117.456i	− 0.278331i
$423$	0	0
$424$	−196.786	−0.464118
$425$	− 47.8597i	− 0.112611i
$426$	0	0
$427$	114.304	0.267692
$428$	− 897.112i	− 2.09606i
$429$	0	0
$430$	309.407	0.719550
$431$	− 273.927i	− 0.635562i	−0.948164	−	0.317781i	$-0.897062\pi$
$431$	− 273.927i	− 0.635562i	0.948164	−	0.317781i	$-0.102938\pi$
$432$	0	0
$433$	−505.428	−1.16727	−0.583635	−	0.812016i	$-0.698370\pi$
$433$	−505.428	−1.16727	−0.583635	+	0.812016i	$0.698370\pi$
$434$	254.746i	0.586973i
$435$	0	0
$436$	32.9132	0.0754890
$437$	− 148.471i	− 0.339751i
$438$	0	0
$439$	−60.9146	−0.138758	−0.0693789	−	0.997590i	$-0.522102\pi$
$439$	−60.9146	−0.138758	−0.0693789	+	0.997590i	$0.522102\pi$
$440$	99.3378i	0.225768i
$441$	0	0
$442$	370.097	0.837323
$443$	24.2180i	0.0546682i	0.999626	+	0.0273341i	$0.00870180\pi$
$443$	24.2180i	0.0546682i	−0.999626	+	0.0273341i	$0.991298\pi$
$444$	0	0
$445$	−227.450	−0.511123
$446$	− 243.502i	− 0.545968i
$447$	0	0
$448$	−258.673	−0.577395
$449$	29.2631i	0.0651740i	0.999469	+	0.0325870i	$0.0103746\pi$
$449$	29.2631i	0.0651740i	−0.999469	+	0.0325870i	$0.989625\pi$
$450$	0	0
$451$	−88.3354	−0.195866
$452$	1213.04i	2.68371i
$453$	0	0
$454$	−740.134	−1.63025
$455$	90.6637i	0.199261i
$456$	0	0
$457$	284.619	0.622799	0.311399	−	0.950279i	$-0.399202\pi$
$457$	284.619	0.622799	0.311399	+	0.950279i	$0.399202\pi$
$458$	− 1044.67i	− 2.28093i
$459$	0	0
$460$	1015.99	2.20867
$461$	246.275i	0.534218i	0.963666	+	0.267109i	$0.0860684\pi$
$461$	246.275i	0.534218i	−0.963666	+	0.267109i	$0.913932\pi$
$462$	0	0
$463$	47.9854	0.103640	0.0518201	−	0.998656i	$-0.483498\pi$
$463$	47.9854	0.103640	0.0518201	+	0.998656i	$0.483498\pi$
$464$	317.344i	0.683930i
$465$	0	0
$466$	728.350	1.56298
$467$	376.578i	0.806377i	0.915117	+	0.403189i	$0.132098\pi$
$467$	376.578i	0.806377i	−0.915117	+	0.403189i	$0.867902\pi$
$468$	0	0
$469$	17.8422	0.0380432
$470$	988.526i	2.10325i
$471$	0	0
$472$	103.895	0.220117
$473$	− 86.1955i	− 0.182232i
$474$	0	0
$475$	10.6788	0.0224817
$476$	262.941i	0.552396i
$477$	0	0
$478$	200.103	0.418625
$479$	930.115i	1.94179i	0.239514	+	0.970893i	$0.423012\pi$
$479$	930.115i	1.94179i	−0.239514	+	0.970893i	$0.576988\pi$
$480$	0	0
$481$	−393.126	−0.817309
$482$	− 426.970i	− 0.885830i
$483$	0	0
$484$	−542.782	−1.12145
$485$	336.633i	0.694088i
$486$	0	0
$487$	109.016	0.223853	0.111926	−	0.993717i	$-0.464298\pi$
$487$	109.016	0.223853	0.111926	+	0.993717i	$0.464298\pi$
$488$	182.573i	0.374125i
$489$	0	0
$490$	675.830	1.37925
$491$	16.1679i	0.0329286i	0.999864	+	0.0164643i	$0.00524098\pi$
$491$	16.1679i	0.0329286i	−0.999864	+	0.0164643i	$0.994759\pi$
$492$	0	0
$493$	683.702	1.38682
$494$	82.5787i	0.167163i
$495$	0	0
$496$	270.322	0.545004
$497$	− 117.456i	− 0.236330i
$498$	0	0
$499$	−496.964	−0.995919	−0.497960	−	0.867200i	$-0.665917\pi$
$499$	−496.964	−0.995919	−0.497960	+	0.867200i	$0.665917\pi$
$500$	− 633.135i	− 1.26627i
$501$	0	0
$502$	−768.167	−1.53021
$503$	− 72.7868i	− 0.144705i	−0.997379	−	0.0723527i	$-0.976949\pi$
$503$	− 72.7868i	− 0.144705i	0.997379	−	0.0723527i	$-0.0230507\pi$
$504$	0	0
$505$	444.756	0.880706
$506$	− 493.543i	− 0.975381i
$507$	0	0
$508$	−60.6100	−0.119311
$509$	− 804.529i	− 1.58061i	−0.612716	−	0.790303i	$-0.709923\pi$
$509$	− 804.529i	− 1.58061i	0.612716	−	0.790303i	$-0.290077\pi$
$510$	0	0
$511$	−23.6991	−0.0463778
$512$	515.789i	1.00740i
$513$	0	0
$514$	281.687	0.548029
$515$	425.524i	0.826260i
$516$	0	0
$517$	275.387	0.532663
$518$	− 487.028i	− 0.940209i
$519$	0	0
$520$	−144.813	−0.278486
$521$	367.880i	0.706103i	0.935604	+	0.353052i	$0.114856\pi$
$521$	367.880i	0.706103i	−0.935604	+	0.353052i	$0.885144\pi$
$522$	0	0
$523$	−252.753	−0.483276	−0.241638	−	0.970366i	$-0.577685\pi$
$523$	−252.753	−0.483276	−0.241638	+	0.970366i	$0.577685\pi$
$524$	1251.56i	2.38847i
$525$	0	0
$526$	−1096.51	−2.08462
$527$	− 582.396i	− 1.10512i
$528$	0	0
$529$	−764.574	−1.44532
$530$	749.914i	1.41493i
$531$	0	0
$532$	−58.6692	−0.110281
$533$	− 128.774i	− 0.241602i
$534$	0	0
$535$	−876.104	−1.63758
$536$	28.4986i	0.0531690i
$537$	0	0
$538$	−357.093	−0.663742
$539$	− 188.275i	− 0.349304i
$540$	0	0
$541$	709.058	1.31064	0.655321	−	0.755350i	$-0.272533\pi$
$541$	709.058	1.31064	0.655321	+	0.755350i	$0.272533\pi$
$542$	− 147.199i	− 0.271585i
$543$	0	0
$544$	798.887	1.46854
$545$	− 32.1425i	− 0.0589770i
$546$	0	0
$547$	−84.1095	−0.153765	−0.0768826	−	0.997040i	$-0.524497\pi$
$547$	−84.1095	−0.153765	−0.0768826	+	0.997040i	$0.524497\pi$
$548$	− 704.953i	− 1.28641i
$549$	0	0
$550$	35.4982	0.0645421
$551$	152.553i	0.276865i
$552$	0	0
$553$	61.9141	0.111960
$554$	248.720i	0.448952i
$555$	0	0
$556$	−1141.90	−2.05378
$557$	− 17.5000i	− 0.0314183i	−0.999877	−	0.0157091i	$-0.994999\pi$
$557$	− 17.5000i	− 0.0314183i	0.999877	−	0.0157091i	$-0.00500058\pi$
$558$	0	0
$559$	125.654	0.224784
$560$	119.187i	0.212834i
$561$	0	0
$562$	−353.568	−0.629124
$563$	− 123.655i	− 0.219635i	−0.993952	−	0.109818i	$-0.964973\pi$
$563$	− 123.655i	− 0.219635i	0.993952	−	0.109818i	$-0.0350266\pi$
$564$	0	0
$565$	1184.63	2.09669
$566$	− 532.987i	− 0.941673i
$567$	0	0
$568$	187.607	0.330294
$569$	587.125i	1.03185i	0.856632	+	0.515927i	$0.172553\pi$
$569$	587.125i	1.03185i	−0.856632	+	0.515927i	$0.827447\pi$
$570$	0	0
$571$	531.233	0.930356	0.465178	−	0.885217i	$-0.345990\pi$
$571$	531.233	0.930356	0.465178	+	0.885217i	$0.345990\pi$
$572$	157.424i	0.275217i
$573$	0	0
$574$	159.533	0.277932
$575$	− 93.0406i	− 0.161810i
$576$	0	0
$577$	89.0333	0.154304	0.0771519	−	0.997019i	$-0.475417\pi$
$577$	89.0333	0.154304	0.0771519	+	0.997019i	$0.475417\pi$
$578$	− 163.178i	− 0.282315i
$579$	0	0
$580$	−1043.92	−1.79986
$581$	160.107i	0.275571i
$582$	0	0
$583$	208.914	0.358342
$584$	− 37.8534i	− 0.0648175i
$585$	0	0
$586$	1012.40	1.72765
$587$	165.262i	0.281536i	0.990043	+	0.140768i	$0.0449572\pi$
$587$	165.262i	0.281536i	−0.990043	+	0.140768i	$0.955043\pi$
$588$	0	0
$589$	129.948	0.220625
$590$	− 395.925i	− 0.671060i
$591$	0	0
$592$	−516.806	−0.872983
$593$	1069.09i	1.80285i	0.432935	+	0.901425i	$0.357478\pi$
$593$	1069.09i	1.80285i	−0.432935	+	0.901425i	$0.642522\pi$
$594$	0	0
$595$	256.783	0.431568
$596$	32.5335i	0.0545865i
$597$	0	0
$598$	719.478	1.20314
$599$	− 541.081i	− 0.903308i	−0.892193	−	0.451654i	$-0.850834\pi$
$599$	− 541.081i	− 0.903308i	0.892193	−	0.451654i	$-0.149166\pi$
$600$	0	0
$601$	964.958	1.60559	0.802793	−	0.596257i	$-0.203346\pi$
$601$	964.958	1.60559	0.802793	+	0.596257i	$0.203346\pi$
$602$	155.668i	0.258585i
$603$	0	0
$604$	−617.120	−1.02172
$605$	530.071i	0.876151i
$606$	0	0
$607$	−270.756	−0.446057	−0.223028	−	0.974812i	$-0.571594\pi$
$607$	−270.756	−0.446057	−0.223028	+	0.974812i	$0.571594\pi$
$608$	178.254i	0.293180i
$609$	0	0
$610$	695.751	1.14058
$611$	401.454i	0.657044i
$612$	0	0
$613$	374.707	0.611267	0.305634	−	0.952149i	$-0.401132\pi$
$613$	374.707	0.611267	0.305634	+	0.952149i	$0.401132\pi$
$614$	− 736.606i	− 1.19968i
$615$	0	0
$616$	−49.9787	−0.0811343
$617$	458.820i	0.743630i	0.928307	+	0.371815i	$0.121264\pi$
$617$	458.820i	0.743630i	−0.928307	+	0.371815i	$0.878736\pi$
$618$	0	0
$619$	−470.296	−0.759767	−0.379884	−	0.925034i	$-0.624036\pi$
$619$	−470.296	−0.759767	−0.379884	+	0.925034i	$0.624036\pi$
$620$	889.239i	1.43426i
$621$	0	0
$622$	−316.369	−0.508631
$623$	− 114.434i	− 0.183683i
$624$	0	0
$625$	−682.980	−1.09277
$626$	− 530.167i	− 0.846912i
$627$	0	0
$628$	1286.45	2.04849
$629$	1113.43i	1.77017i
$630$	0	0
$631$	949.552	1.50484	0.752418	−	0.658685i	$-0.228887\pi$
$631$	949.552	1.50484	0.752418	+	0.658685i	$0.228887\pi$
$632$	98.8925i	0.156476i
$633$	0	0
$634$	924.765	1.45862
$635$	59.1907i	0.0932136i
$636$	0	0
$637$	274.464	0.430869
$638$	507.110i	0.794844i
$639$	0	0
$640$	−667.296	−1.04265
$641$	− 163.550i	− 0.255149i	−0.991829	−	0.127574i	$-0.959281\pi$
$641$	− 163.550i	− 0.255149i	0.991829	−	0.127574i	$-0.0407192\pi$
$642$	0	0
$643$	547.664	0.851732	0.425866	−	0.904786i	$-0.359970\pi$
$643$	547.664	0.851732	0.425866	+	0.904786i	$0.359970\pi$
$644$	511.164i	0.793732i
$645$	0	0
$646$	233.884	0.362050
$647$	− 151.339i	− 0.233909i	−0.993137	−	0.116954i	$-0.962687\pi$
$647$	− 151.339i	− 0.233909i	0.993137	−	0.116954i	$-0.0373131\pi$
$648$	0	0
$649$	−110.298	−0.169951
$650$	51.7486i	0.0796132i
$651$	0	0
$652$	672.700	1.03175
$653$	− 935.358i	− 1.43240i	−0.697894	−	0.716201i	$-0.745879\pi$
$653$	− 935.358i	− 1.43240i	0.697894	−	0.716201i	$-0.254121\pi$
$654$	0	0
$655$	1222.25	1.86603
$656$	− 169.287i	− 0.258059i
$657$	0	0
$658$	−497.346	−0.755844
$659$	577.471i	0.876283i	0.898906	+	0.438142i	$0.144363\pi$
$659$	577.471i	0.876283i	−0.898906	+	0.438142i	$0.855637\pi$
$660$	0	0
$661$	518.169	0.783916	0.391958	−	0.919983i	$-0.371798\pi$
$661$	518.169	0.783916	0.391958	+	0.919983i	$0.371798\pi$
$662$	− 1341.08i	− 2.02580i
$663$	0	0
$664$	−255.731	−0.385137
$665$	57.2953i	0.0861584i
$666$	0	0
$667$	1329.14	1.99271
$668$	− 79.7652i	− 0.119409i
$669$	0	0
$670$	108.603	0.162094
$671$	− 193.825i	− 0.288859i
$672$	0	0
$673$	−212.932	−0.316393	−0.158196	−	0.987408i	$-0.550568\pi$
$673$	−212.932	−0.316393	−0.158196	+	0.987408i	$0.550568\pi$
$674$	− 1102.18i	− 1.63528i
$675$	0	0
$676$	679.438	1.00509
$677$	293.836i	0.434026i	0.976169	+	0.217013i	$0.0696315\pi$
$677$	293.836i	0.434026i	−0.976169	+	0.217013i	$0.930369\pi$
$678$	0	0
$679$	−169.366	−0.249435
$680$	410.148i	0.603159i
$681$	0	0
$682$	431.970	0.633388
$683$	800.471i	1.17199i	0.810314	+	0.585996i	$0.199297\pi$
$683$	800.471i	1.17199i	−0.810314	+	0.585996i	$0.800703\pi$
$684$	0	0
$685$	−688.444	−1.00503
$686$	736.555i	1.07370i
$687$	0	0
$688$	165.186	0.240096
$689$	304.550i	0.442018i
$690$	0	0
$691$	1089.35	1.57649	0.788244	−	0.615363i	$-0.210991\pi$
$691$	1089.35	1.57649	0.788244	+	0.615363i	$0.210991\pi$
$692$	118.452i	0.171173i
$693$	0	0
$694$	−676.441	−0.974699
$695$	1115.16i	1.60455i
$696$	0	0
$697$	−364.721	−0.523273
$698$	− 1084.17i	− 1.55325i
$699$	0	0
$700$	−36.7656	−0.0525222
$701$	− 1214.68i	− 1.73278i	−0.499370	−	0.866389i	$-0.666435\pi$
$701$	− 1214.68i	− 1.73278i	0.499370	−	0.866389i	$-0.333565\pi$
$702$	0	0
$703$	−248.437	−0.353396
$704$	438.629i	0.623052i
$705$	0	0
$706$	1899.23	2.69013
$707$	223.765i	0.316500i
$708$	0	0
$709$	571.377	0.805891	0.402945	−	0.915224i	$-0.367986\pi$
$709$	571.377	0.805891	0.402945	+	0.915224i	$0.367986\pi$
$710$	− 714.935i	− 1.00695i
$711$	0	0
$712$	182.781	0.256714
$713$	− 1132.19i	− 1.58793i
$714$	0	0
$715$	153.737	0.215017
$716$	− 1806.59i	− 2.52318i
$717$	0	0
$718$	−1805.52	−2.51466
$719$	618.040i	0.859583i	0.902928	+	0.429791i	$0.141413\pi$
$719$	618.040i	0.859583i	−0.902928	+	0.429791i	$0.858587\pi$
$720$	0	0
$721$	−214.089	−0.296933
$722$	− 1053.34i	− 1.45892i
$723$	0	0
$724$	1459.55	2.01595
$725$	95.5983i	0.131860i
$726$	0	0
$727$	−533.575	−0.733940	−0.366970	−	0.930233i	$-0.619605\pi$
$727$	−533.575	−0.733940	−0.366970	+	0.930233i	$0.619605\pi$
$728$	− 72.8580i	− 0.100080i
$729$	0	0
$730$	−144.252	−0.197606
$731$	− 355.886i	− 0.486848i
$732$	0	0
$733$	−569.444	−0.776867	−0.388434	−	0.921477i	$-0.626984\pi$
$733$	−569.444	−0.776867	−0.388434	+	0.921477i	$0.626984\pi$
$734$	− 1137.90i	− 1.55027i
$735$	0	0
$736$	1553.06	2.11013
$737$	− 30.2549i	− 0.0410514i
$738$	0	0
$739$	52.8399	0.0715019	0.0357510	−	0.999361i	$-0.488618\pi$
$739$	52.8399	0.0715019	0.0357510	+	0.999361i	$0.488618\pi$
$740$	− 1700.06i	− 2.29738i
$741$	0	0
$742$	−377.296	−0.508485
$743$	980.221i	1.31927i	0.751584	+	0.659637i	$0.229290\pi$
$743$	980.221i	1.31927i	−0.751584	+	0.659637i	$0.770710\pi$
$744$	0	0
$745$	31.7717	0.0426465
$746$	− 176.827i	− 0.237034i
$747$	0	0
$748$	445.866	0.596077
$749$	− 440.784i	− 0.588497i
$750$	0	0
$751$	932.837	1.24213	0.621063	−	0.783760i	$-0.286701\pi$
$751$	932.837	1.24213	0.621063	+	0.783760i	$0.286701\pi$
$752$	527.754i	0.701801i
$753$	0	0
$754$	−739.256	−0.980446
$755$	602.669i	0.798236i
$756$	0	0
$757$	−701.478	−0.926655	−0.463328	−	0.886187i	$-0.653345\pi$
$757$	−701.478	−0.926655	−0.463328	+	0.886187i	$0.653345\pi$
$758$	− 1128.66i	− 1.48900i
$759$	0	0
$760$	−91.5151	−0.120415
$761$	− 248.730i	− 0.326846i	−0.986556	−	0.163423i	$-0.947747\pi$
$761$	− 248.730i	− 0.326846i	0.986556	−	0.163423i	$-0.0522535\pi$
$762$	0	0
$763$	16.1715	0.0211946
$764$	− 614.688i	− 0.804565i
$765$	0	0
$766$	−507.702	−0.662796
$767$	− 160.791i	− 0.209636i
$768$	0	0
$769$	−645.918	−0.839945	−0.419972	−	0.907537i	$-0.637960\pi$
$769$	−645.918	−0.839945	−0.419972	+	0.907537i	$0.637960\pi$
$770$	190.459i	0.247350i
$771$	0	0
$772$	643.992	0.834187
$773$	233.335i	0.301857i	0.988545	+	0.150928i	$0.0482263\pi$
$773$	233.335i	0.301857i	−0.988545	+	0.150928i	$0.951774\pi$
$774$	0	0
$775$	81.4332	0.105075
$776$	− 270.521i	− 0.348609i
$777$	0	0
$778$	−75.9112	−0.0975722
$779$	− 81.3792i	− 0.104466i
$780$	0	0
$781$	−199.169	−0.255018
$782$	− 2037.75i	− 2.60582i
$783$	0	0
$784$	360.812	0.460220
$785$	− 1256.32i	− 1.60041i
$786$	0	0
$787$	1346.42	1.71082	0.855411	−	0.517950i	$-0.173305\pi$
$787$	1346.42	1.71082	0.855411	+	0.517950i	$0.173305\pi$
$788$	− 1466.01i	− 1.86043i
$789$	0	0
$790$	376.861	0.477039
$791$	596.010i	0.753490i
$792$	0	0
$793$	282.554	0.356310
$794$	1217.91i	1.53389i
$795$	0	0
$796$	−465.157	−0.584369
$797$	605.301i	0.759474i	0.925095	+	0.379737i	$0.123985\pi$
$797$	605.301i	0.759474i	−0.925095	+	0.379737i	$0.876015\pi$
$798$	0	0
$799$	1137.02	1.42306
$800$	111.704i	0.139630i
$801$	0	0
$802$	−1031.22	−1.28581
$803$	40.1862i	0.0500451i
$804$	0	0
$805$	499.193	0.620116
$806$	629.719i	0.781289i
$807$	0	0
$808$	−357.410	−0.442339
$809$	1066.82i	1.31869i	0.751841	+	0.659344i	$0.229166\pi$
$809$	1066.82i	1.31869i	−0.751841	+	0.659344i	$0.770834\pi$
$810$	0	0
$811$	−528.604	−0.651793	−0.325897	−	0.945405i	$-0.605666\pi$
$811$	−528.604	−0.651793	−0.325897	+	0.945405i	$0.605666\pi$
$812$	− 525.216i	− 0.646817i
$813$	0	0
$814$	−825.848	−1.01456
$815$	− 656.946i	− 0.806069i
$816$	0	0
$817$	79.4078	0.0971943
$818$	1022.93i	1.25052i
$819$	0	0
$820$	556.879	0.679121
$821$	− 881.520i	− 1.07371i	−0.843673	−	0.536857i	$-0.819611\pi$
$821$	− 881.520i	− 1.07371i	0.843673	−	0.536857i	$-0.180389\pi$
$822$	0	0
$823$	−141.552	−0.171995	−0.0859977	−	0.996295i	$-0.527408\pi$
$823$	−141.552	−0.171995	−0.0859977	+	0.996295i	$0.527408\pi$
$824$	− 341.954i	− 0.414993i
$825$	0	0
$826$	199.197	0.241159
$827$	− 1063.38i	− 1.28583i	−0.765938	−	0.642914i	$-0.777725\pi$
$827$	− 1063.38i	− 1.28583i	0.765938	−	0.642914i	$-0.222275\pi$
$828$	0	0
$829$	1374.71	1.65828	0.829139	−	0.559043i	$-0.188831\pi$
$829$	1374.71	1.65828	0.829139	+	0.559043i	$0.188831\pi$
$830$	974.541i	1.17415i
$831$	0	0
$832$	−639.425	−0.768539
$833$	− 777.353i	− 0.933197i
$834$	0	0
$835$	−77.8973	−0.0932902
$836$	99.4847i	0.119001i
$837$	0	0
$838$	897.971	1.07156
$839$	65.4450i	0.0780036i	0.999239	+	0.0390018i	$0.0124178\pi$
$839$	65.4450i	0.0780036i	−0.999239	+	0.0390018i	$0.987582\pi$
$840$	0	0
$841$	−524.673	−0.623868
$842$	− 474.609i	− 0.563669i
$843$	0	0
$844$	206.279	0.244407
$845$	− 663.527i	− 0.785239i
$846$	0	0
$847$	−266.689	−0.314863
$848$	400.364i	0.472127i
$849$	0	0
$850$	146.566	0.172430
$851$	2164.55i	2.54353i
$852$	0	0
$853$	250.955	0.294203	0.147101	−	0.989121i	$-0.453006\pi$
$853$	250.955	0.294203	0.147101	+	0.989121i	$0.453006\pi$
$854$	350.045i	0.409889i
$855$	0	0
$856$	704.044	0.822481
$857$	506.000i	0.590432i	0.955431	+	0.295216i	$0.0953916\pi$
$857$	506.000i	0.590432i	−0.955431	+	0.295216i	$0.904608\pi$
$858$	0	0
$859$	−1358.56	−1.58156	−0.790780	−	0.612101i	$-0.790325\pi$
$859$	−1358.56	−1.58156	−0.790780	+	0.612101i	$0.790325\pi$
$860$	543.389i	0.631848i
$861$	0	0
$862$	838.874	0.973172
$863$	932.140i	1.08012i	0.841628	+	0.540058i	$0.181598\pi$
$863$	932.140i	1.08012i	−0.841628	+	0.540058i	$0.818402\pi$
$864$	0	0
$865$	115.678	0.133731
$866$	− 1547.82i	− 1.78732i
$867$	0	0
$868$	−447.393	−0.515430
$869$	− 104.987i	− 0.120814i
$870$	0	0
$871$	44.1050	0.0506372
$872$	25.8299i	0.0296215i
$873$	0	0
$874$	454.677	0.520226
$875$	− 311.082i	− 0.355523i
$876$	0	0
$877$	714.424	0.814623	0.407311	−	0.913289i	$-0.366466\pi$
$877$	714.424	0.814623	0.407311	+	0.913289i	$0.366466\pi$
$878$	− 186.545i	− 0.212466i
$879$	0	0
$880$	202.104	0.229664
$881$	1279.83i	1.45270i	0.687325	+	0.726350i	$0.258785\pi$
$881$	1279.83i	1.45270i	−0.687325	+	0.726350i	$0.741215\pi$
$882$	0	0
$883$	−13.5483	−0.0153435	−0.00767175	−	0.999971i	$-0.502442\pi$
$883$	−13.5483	−0.0153435	−0.00767175	+	0.999971i	$0.502442\pi$
$884$	649.975i	0.735265i
$885$	0	0
$886$	−74.1652	−0.0837079
$887$	294.505i	0.332024i	0.986124	+	0.166012i	$0.0530890\pi$
$887$	294.505i	0.332024i	−0.986124	+	0.166012i	$0.946911\pi$
$888$	0	0
$889$	−29.7799	−0.0334982
$890$	− 696.542i	− 0.782631i
$891$	0	0
$892$	427.645	0.479422
$893$	253.701i	0.284099i
$894$	0	0
$895$	−1764.29	−1.97127
$896$	− 335.729i	− 0.374698i
$897$	0	0
$898$	−89.6153	−0.0997943
$899$	1163.32i	1.29401i
$900$	0	0
$901$	862.566	0.957343
$902$	− 270.518i	− 0.299909i
$903$	0	0
$904$	−951.980	−1.05308
$905$	− 1425.37i	− 1.57499i
$906$	0	0
$907$	−1050.45	−1.15816	−0.579078	−	0.815272i	$-0.696587\pi$
$907$	−1050.45	−1.15816	−0.579078	+	0.815272i	$0.696587\pi$
$908$	− 1299.85i	− 1.43155i
$909$	0	0
$910$	−277.648	−0.305108
$911$	− 1613.61i	− 1.77125i	−0.464404	−	0.885623i	$-0.653731\pi$
$911$	− 1613.61i	− 1.77125i	0.464404	−	0.885623i	$-0.346269\pi$
$912$	0	0
$913$	271.491	0.297361
$914$	871.616i	0.953628i
$915$	0	0
$916$	1834.67	2.00292
$917$	614.937i	0.670597i
$918$	0	0
$919$	−442.772	−0.481797	−0.240899	−	0.970550i	$-0.577442\pi$
$919$	−442.772	−0.481797	−0.240899	+	0.970550i	$0.577442\pi$
$920$	797.338i	0.866672i
$921$	0	0
$922$	−754.191	−0.817994
$923$	− 290.345i	− 0.314566i
$924$	0	0
$925$	−155.685	−0.168309
$926$	146.950i	0.158694i
$927$	0	0
$928$	−1595.75	−1.71956
$929$	− 1413.29i	− 1.52131i	−0.649159	−	0.760653i	$-0.724879\pi$
$929$	− 1413.29i	− 1.52131i	0.649159	−	0.760653i	$-0.275121\pi$
$930$	0	0
$931$	173.449	0.186304
$932$	1279.15i	1.37248i
$933$	0	0
$934$	−1153.23	−1.23472
$935$	− 435.424i	− 0.465694i
$936$	0	0
$937$	310.345	0.331212	0.165606	−	0.986192i	$-0.447042\pi$
$937$	310.345	0.331212	0.165606	+	0.986192i	$0.447042\pi$
$938$	54.6400i	0.0582516i
$939$	0	0
$940$	−1736.08	−1.84689
$941$	1123.68i	1.19413i	0.802191	+	0.597067i	$0.203667\pi$
$941$	1123.68i	1.19413i	−0.802191	+	0.597067i	$0.796333\pi$
$942$	0	0
$943$	−709.027	−0.751885
$944$	− 211.377i	− 0.223916i
$945$	0	0
$946$	263.965	0.279033
$947$	− 817.493i	− 0.863245i	−0.902054	−	0.431622i	$-0.857941\pi$
$947$	− 817.493i	− 0.863245i	0.902054	−	0.431622i	$-0.142059\pi$
$948$	0	0
$949$	−58.5828	−0.0617311
$950$	32.7028i	0.0344240i
$951$	0	0
$952$	−206.353	−0.216757
$953$	− 637.458i	− 0.668896i	−0.942414	−	0.334448i	$-0.891450\pi$
$953$	− 637.458i	− 0.668896i	0.942414	−	0.334448i	$-0.108550\pi$
$954$	0	0
$955$	−600.293	−0.628579
$956$	351.426i	0.367601i
$957$	0	0
$958$	−2848.38	−2.97326
$959$	− 346.369i	− 0.361177i
$960$	0	0
$961$	29.9452	0.0311604
$962$	− 1203.91i	− 1.25146i
$963$	0	0
$964$	749.858	0.777861
$965$	− 628.911i	− 0.651722i
$966$	0	0
$967$	−796.666	−0.823853	−0.411927	−	0.911217i	$-0.635144\pi$
$967$	−796.666	−0.823853	−0.411927	+	0.911217i	$0.635144\pi$
$968$	− 425.969i	− 0.440051i
$969$	0	0
$970$	−1030.90	−1.06279
$971$	− 459.354i	− 0.473073i	−0.971623	−	0.236536i	$-0.923988\pi$
$971$	− 459.354i	− 0.473073i	0.971623	−	0.236536i	$-0.0760122\pi$
$972$	0	0
$973$	−561.057	−0.576626
$974$	333.851i	0.342763i
$975$	0	0
$976$	371.448	0.380582
$977$	1545.88i	1.58227i	0.611639	+	0.791137i	$0.290511\pi$
$977$	1545.88i	1.58227i	−0.611639	+	0.791137i	$0.709489\pi$
$978$	0	0
$979$	−194.045	−0.198207
$980$	1186.91i	1.21114i
$981$	0	0
$982$	−49.5126	−0.0504201
$983$	842.380i	0.856948i	0.903554	+	0.428474i	$0.140949\pi$
$983$	842.380i	0.856948i	−0.903554	+	0.428474i	$0.859051\pi$
$984$	0	0
$985$	−1431.68	−1.45349
$986$	2093.77i	2.12350i
$987$	0	0
$988$	−145.027	−0.146789
$989$	− 691.851i	− 0.699546i
$990$	0	0
$991$	−965.398	−0.974166	−0.487083	−	0.873356i	$-0.661939\pi$
$991$	−965.398	−0.974166	−0.487083	+	0.873356i	$0.661939\pi$
$992$	1359.30i	1.37027i
$993$	0	0
$994$	359.697	0.361868
$995$	454.264i	0.456547i
$996$	0	0
$997$	1834.22	1.83974	0.919871	−	0.392221i	$-0.128293\pi$
$997$	1834.22	1.83974	0.919871	+	0.392221i	$0.128293\pi$
$998$	− 1521.90i	− 1.52495i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.74	yes	84
3.2	odd	2	inner	1143.3.b.a.890.11	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.11	✓	84	3.2	odd	2	inner
1143.3.b.a.890.74	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)

\(f(q)\)	\(=\)	\(q+3.06240i q^{2} -5.37827 q^{4} +5.25232i q^{5} -2.64254 q^{7} -4.22081i q^{8} +O(q^{10})\)	\(q+3.06240i q^{2} -5.37827 q^{4} +5.25232i q^{5} -2.64254 q^{7} -4.22081i q^{8} -16.0847 q^{10} +4.48093i q^{11} -6.53221 q^{13} -8.09251i q^{14} -8.58729 q^{16} +18.5009i q^{17} -4.12806 q^{19} -28.2484i q^{20} -13.7224 q^{22} +35.9663i q^{23} -2.58688 q^{25} -20.0042i q^{26} +14.2123 q^{28} -36.9550i q^{29} -31.4793 q^{31} -43.1809i q^{32} -56.6572 q^{34} -13.8795i q^{35} +60.1826 q^{37} -12.6418i q^{38} +22.1690 q^{40} +19.7137i q^{41} -19.2361 q^{43} -24.0996i q^{44} -110.143 q^{46} -61.4575i q^{47} -42.0170 q^{49} -7.92206i q^{50} +35.1320 q^{52} -46.6228i q^{53} -23.5353 q^{55} +11.1537i q^{56} +113.171 q^{58} +24.6150i q^{59} -43.2555 q^{61} -96.4020i q^{62} +97.8879 q^{64} -34.3093i q^{65} -6.75193 q^{67} -99.5030i q^{68} +42.5044 q^{70} +44.4482i q^{71} +8.96829 q^{73} +184.303i q^{74} +22.2018 q^{76} -11.8410i q^{77} -23.4298 q^{79} -45.1032i q^{80} -60.3710 q^{82} -60.5881i q^{83} -97.1728 q^{85} -58.9085i q^{86} +18.9131 q^{88} +43.3046i q^{89} +17.2616 q^{91} -193.436i q^{92} +188.207 q^{94} -21.6819i q^{95} +64.0922 q^{97} -128.673i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 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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