LMFDB - Embedded newform 1458.3.b.c.1457.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,3,Mod(1457,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1458.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1458.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:	$39.7276225437$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.16
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +7.63957i q^{5} +5.75438 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +7.63957i q^{5} +5.75438 q^{7} +2.82843i q^{8} +10.8040 q^{10} -8.15026i q^{11} +23.4613 q^{13} -8.13793i q^{14} +4.00000 q^{16} -7.38231i q^{17} +15.6026 q^{19} -15.2791i q^{20} -11.5262 q^{22} +31.0764i q^{23} -33.3630 q^{25} -33.1793i q^{26} -11.5088 q^{28} -28.3184i q^{29} +16.0044 q^{31} -5.65685i q^{32} -10.4402 q^{34} +43.9610i q^{35} -11.9809 q^{37} -22.0654i q^{38} -21.6080 q^{40} -8.26339i q^{41} -45.0827 q^{43} +16.3005i q^{44} +43.9486 q^{46} -67.6105i q^{47} -15.8871 q^{49} +47.1824i q^{50} -46.9226 q^{52} +16.4115i q^{53} +62.2645 q^{55} +16.2759i q^{56} -40.0482 q^{58} +58.2441i q^{59} +71.3542 q^{61} -22.6337i q^{62} -8.00000 q^{64} +179.234i q^{65} +50.1960 q^{67} +14.7646i q^{68} +62.1703 q^{70} -40.9183i q^{71} +99.5384 q^{73} +16.9436i q^{74} -31.2052 q^{76} -46.8997i q^{77} +77.0529 q^{79} +30.5583i q^{80} -11.6862 q^{82} +105.482i q^{83} +56.3976 q^{85} +63.7566i q^{86} +23.0524 q^{88} -0.348951i q^{89} +135.005 q^{91} -62.1527i q^{92} -95.6157 q^{94} +119.197i q^{95} +63.9662 q^{97} +22.4677i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4}+O(q^{10}) $ 36 * q - 72 * q^4	$ 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) $ 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	7.63957i	1.52791i	0.645268	+	0.763957i	$0.276746\pi$
$5$	7.63957i	1.52791i	−0.645268	+	0.763957i	$0.723254\pi$
$6$	0	0
$7$	5.75438	0.822055	0.411027	−	0.911623i	$-0.365170\pi$
$7$	5.75438	0.822055	0.411027	+	0.911623i	$0.365170\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	10.8040	1.08040
$11$	− 8.15026i	− 0.740933i	−0.928846	−	0.370466i	$-0.879198\pi$
$11$	− 8.15026i	− 0.740933i	0.928846	−	0.370466i	$-0.120802\pi$
$12$	0	0
$13$	23.4613	1.80471	0.902357	−	0.430990i	$-0.141836\pi$
$13$	23.4613	1.80471	0.902357	+	0.430990i	$0.141836\pi$
$14$	− 8.13793i	− 0.581281i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 7.38231i	− 0.434253i	−0.976143	−	0.217127i	$-0.930332\pi$
$17$	− 7.38231i	− 0.434253i	0.976143	−	0.217127i	$-0.0696685\pi$
$18$	0	0
$19$	15.6026	0.821189	0.410595	−	0.911818i	$-0.365321\pi$
$19$	15.6026	0.821189	0.410595	+	0.911818i	$0.365321\pi$
$20$	− 15.2791i	− 0.763957i
$21$	0	0
$22$	−11.5262	−0.523919
$23$	31.0764i	1.35115i	0.737293	+	0.675573i	$0.236104\pi$
$23$	31.0764i	1.35115i	−0.737293	+	0.675573i	$0.763896\pi$
$24$	0	0
$25$	−33.3630	−1.33452
$26$	− 33.1793i	− 1.27613i
$27$	0	0
$28$	−11.5088	−0.411027
$29$	− 28.3184i	− 0.976496i	−0.872705	−	0.488248i	$-0.837636\pi$
$29$	− 28.3184i	− 0.976496i	0.872705	−	0.488248i	$-0.162364\pi$
$30$	0	0
$31$	16.0044	0.516272	0.258136	−	0.966109i	$-0.416892\pi$
$31$	16.0044	0.516272	0.258136	+	0.966109i	$0.416892\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−10.4402	−0.307064
$35$	43.9610i	1.25603i
$36$	0	0
$37$	−11.9809	−0.323809	−0.161904	−	0.986806i	$-0.551764\pi$
$37$	−11.9809	−0.323809	−0.161904	+	0.986806i	$0.551764\pi$
$38$	− 22.0654i	− 0.580669i
$39$	0	0
$40$	−21.6080	−0.540199
$41$	− 8.26339i	− 0.201546i	−0.994909	−	0.100773i	$-0.967868\pi$
$41$	− 8.26339i	− 0.201546i	0.994909	−	0.100773i	$-0.0321316\pi$
$42$	0	0
$43$	−45.0827	−1.04844	−0.524218	−	0.851584i	$-0.675642\pi$
$43$	−45.0827	−1.04844	−0.524218	+	0.851584i	$0.675642\pi$
$44$	16.3005i	0.370466i
$45$	0	0
$46$	43.9486	0.955405
$47$	− 67.6105i	− 1.43852i	−0.694740	−	0.719261i	$-0.744481\pi$
$47$	− 67.6105i	− 1.43852i	0.694740	−	0.719261i	$-0.255519\pi$
$48$	0	0
$49$	−15.8871	−0.324226
$50$	47.1824i	0.943648i
$51$	0	0
$52$	−46.9226	−0.902357
$53$	16.4115i	0.309652i	0.987942	+	0.154826i	$0.0494816\pi$
$53$	16.4115i	0.309652i	−0.987942	+	0.154826i	$0.950518\pi$
$54$	0	0
$55$	62.2645	1.13208
$56$	16.2759i	0.290640i
$57$	0	0
$58$	−40.0482	−0.690487
$59$	58.2441i	0.987187i	0.869693	+	0.493594i	$0.164317\pi$
$59$	58.2441i	0.987187i	−0.869693	+	0.493594i	$0.835683\pi$
$60$	0	0
$61$	71.3542	1.16974	0.584870	−	0.811127i	$-0.301145\pi$
$61$	71.3542	1.16974	0.584870	+	0.811127i	$0.301145\pi$
$62$	− 22.6337i	− 0.365059i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	179.234i	2.75745i
$66$	0	0
$67$	50.1960	0.749193	0.374597	−	0.927188i	$-0.377781\pi$
$67$	50.1960	0.749193	0.374597	+	0.927188i	$0.377781\pi$
$68$	14.7646i	0.217127i
$69$	0	0
$70$	62.1703	0.888146
$71$	− 40.9183i	− 0.576315i	−0.957583	−	0.288157i	$-0.906957\pi$
$71$	− 40.9183i	− 0.576315i	0.957583	−	0.288157i	$-0.0930426\pi$
$72$	0	0
$73$	99.5384	1.36354	0.681770	−	0.731567i	$-0.261211\pi$
$73$	99.5384	1.36354	0.681770	+	0.731567i	$0.261211\pi$
$74$	16.9436i	0.228967i
$75$	0	0
$76$	−31.2052	−0.410595
$77$	− 46.8997i	− 0.609088i
$78$	0	0
$79$	77.0529	0.975354	0.487677	−	0.873024i	$-0.337844\pi$
$79$	77.0529	0.975354	0.487677	+	0.873024i	$0.337844\pi$
$80$	30.5583i	0.381978i
$81$	0	0
$82$	−11.6862	−0.142515
$83$	105.482i	1.27087i	0.772156	+	0.635433i	$0.219178\pi$
$83$	105.482i	1.27087i	−0.772156	+	0.635433i	$0.780822\pi$
$84$	0	0
$85$	56.3976	0.663502
$86$	63.7566i	0.741356i
$87$	0	0
$88$	23.0524	0.261959
$89$	− 0.348951i	− 0.00392080i	−0.999998	−	0.00196040i	$-0.999376\pi$
$89$	− 0.348951i	− 0.00392080i	0.999998	−	0.00196040i	$-0.000624015\pi$
$90$	0	0
$91$	135.005	1.48357
$92$	− 62.1527i	− 0.675573i
$93$	0	0
$94$	−95.6157	−1.01719
$95$	119.197i	1.25471i
$96$	0	0
$97$	63.9662	0.659445	0.329723	−	0.944078i	$-0.393045\pi$
$97$	63.9662	0.659445	0.329723	+	0.944078i	$0.393045\pi$
$98$	22.4677i	0.229262i
$99$	0	0
$100$	66.7260	0.667260
$101$	166.725i	1.65074i	0.564593	+	0.825370i	$0.309033\pi$
$101$	166.725i	1.65074i	−0.564593	+	0.825370i	$0.690967\pi$
$102$	0	0
$103$	−100.610	−0.976800	−0.488400	−	0.872620i	$-0.662419\pi$
$103$	−100.610	−0.976800	−0.488400	+	0.872620i	$0.662419\pi$
$104$	66.3585i	0.638063i
$105$	0	0
$106$	23.2094	0.218957
$107$	− 6.57806i	− 0.0614772i	−0.999527	−	0.0307386i	$-0.990214\pi$
$107$	− 6.57806i	− 0.0614772i	0.999527	−	0.0307386i	$-0.00978594\pi$
$108$	0	0
$109$	198.876	1.82455	0.912276	−	0.409576i	$-0.134323\pi$
$109$	198.876	1.82455	0.912276	+	0.409576i	$0.134323\pi$
$110$	− 88.0553i	− 0.800502i
$111$	0	0
$112$	23.0175	0.205514
$113$	− 5.24721i	− 0.0464355i	−0.999730	−	0.0232177i	$-0.992609\pi$
$113$	− 5.24721i	− 0.0464355i	0.999730	−	0.0232177i	$-0.00739110\pi$
$114$	0	0
$115$	−237.410	−2.06443
$116$	56.6368i	0.488248i
$117$	0	0
$118$	82.3695	0.698047
$119$	− 42.4806i	− 0.356980i
$120$	0	0
$121$	54.5732	0.451018
$122$	− 100.910i	− 0.827132i
$123$	0	0
$124$	−32.0088	−0.258136
$125$	− 63.8896i	− 0.511117i
$126$	0	0
$127$	44.4382	0.349907	0.174954	−	0.984577i	$-0.444022\pi$
$127$	44.4382	0.349907	0.174954	+	0.984577i	$0.444022\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	253.475	1.94981
$131$	− 40.1802i	− 0.306719i	−0.988170	−	0.153360i	$-0.950991\pi$
$131$	− 40.1802i	− 0.306719i	0.988170	−	0.153360i	$-0.0490093\pi$
$132$	0	0
$133$	89.7834	0.675063
$134$	− 70.9878i	− 0.529760i
$135$	0	0
$136$	20.8803	0.153532
$137$	51.5369i	0.376182i	0.982152	+	0.188091i	$0.0602300\pi$
$137$	51.5369i	0.376182i	−0.982152	+	0.188091i	$0.939770\pi$
$138$	0	0
$139$	−107.649	−0.774455	−0.387228	−	0.921984i	$-0.626567\pi$
$139$	−107.649	−0.774455	−0.387228	+	0.921984i	$0.626567\pi$
$140$	− 87.9220i	− 0.628014i
$141$	0	0
$142$	−57.8673	−0.407516
$143$	− 191.216i	− 1.33717i
$144$	0	0
$145$	216.340	1.49200
$146$	− 140.769i	− 0.964168i
$147$	0	0
$148$	23.9619	0.161904
$149$	141.101i	0.946987i	0.880797	+	0.473493i	$0.157007\pi$
$149$	141.101i	0.946987i	−0.880797	+	0.473493i	$0.842993\pi$
$150$	0	0
$151$	−131.182	−0.868755	−0.434378	−	0.900731i	$-0.643032\pi$
$151$	−131.182	−0.868755	−0.434378	+	0.900731i	$0.643032\pi$
$152$	44.1308i	0.290334i
$153$	0	0
$154$	−66.3263	−0.430690
$155$	122.267i	0.788818i
$156$	0	0
$157$	192.772	1.22785	0.613924	−	0.789365i	$-0.289590\pi$
$157$	192.772	1.22785	0.613924	+	0.789365i	$0.289590\pi$
$158$	− 108.969i	− 0.689679i
$159$	0	0
$160$	43.2159	0.270099
$161$	178.825i	1.11072i
$162$	0	0
$163$	80.4819	0.493754	0.246877	−	0.969047i	$-0.420596\pi$
$163$	80.4819	0.493754	0.246877	+	0.969047i	$0.420596\pi$
$164$	16.5268i	0.100773i
$165$	0	0
$166$	149.174	0.898638
$167$	− 108.270i	− 0.648321i	−0.946002	−	0.324161i	$-0.894918\pi$
$167$	− 108.270i	− 0.648321i	0.946002	−	0.324161i	$-0.105082\pi$
$168$	0	0
$169$	381.431	2.25699
$170$	− 79.7583i	− 0.469167i
$171$	0	0
$172$	90.1655	0.524218
$173$	− 84.6632i	− 0.489383i	−0.969601	−	0.244691i	$-0.921313\pi$
$173$	− 84.6632i	− 0.489383i	0.969601	−	0.244691i	$-0.0786866\pi$
$174$	0	0
$175$	−191.983	−1.09705
$176$	− 32.6011i	− 0.185233i
$177$	0	0
$178$	−0.493491	−0.00277242
$179$	167.767i	0.937248i	0.883398	+	0.468624i	$0.155250\pi$
$179$	167.767i	0.937248i	−0.883398	+	0.468624i	$0.844750\pi$
$180$	0	0
$181$	−289.465	−1.59925	−0.799627	−	0.600497i	$-0.794970\pi$
$181$	−289.465	−1.59925	−0.799627	+	0.600497i	$0.794970\pi$
$182$	− 190.926i	− 1.04905i
$183$	0	0
$184$	−87.8972	−0.477702
$185$	− 91.5291i	− 0.494752i
$186$	0	0
$187$	−60.1678	−0.321753
$188$	135.221i	0.719261i
$189$	0	0
$190$	168.570	0.887211
$191$	− 40.3377i	− 0.211192i	−0.994409	−	0.105596i	$-0.966325\pi$
$191$	− 40.3377i	− 0.211192i	0.994409	−	0.105596i	$-0.0336751\pi$
$192$	0	0
$193$	−242.523	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$193$	−242.523	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$194$	− 90.4619i	− 0.466298i
$195$	0	0
$196$	31.7741	0.162113
$197$	− 5.45517i	− 0.0276912i	−0.999904	−	0.0138456i	$-0.995593\pi$
$197$	− 5.45517i	− 0.0276912i	0.999904	−	0.0138456i	$-0.00440733\pi$
$198$	0	0
$199$	−143.015	−0.718667	−0.359333	−	0.933209i	$-0.616996\pi$
$199$	−143.015	−0.718667	−0.359333	+	0.933209i	$0.616996\pi$
$200$	− 94.3648i	− 0.471824i
$201$	0	0
$202$	235.784	1.16725
$203$	− 162.955i	− 0.802733i
$204$	0	0
$205$	63.1288	0.307945
$206$	142.285i	0.690702i
$207$	0	0
$208$	93.8451	0.451178
$209$	− 127.165i	− 0.608446i
$210$	0	0
$211$	−8.63782	−0.0409375	−0.0204688	−	0.999790i	$-0.506516\pi$
$211$	−8.63782	−0.0409375	−0.0204688	+	0.999790i	$0.506516\pi$
$212$	− 32.8231i	− 0.154826i
$213$	0	0
$214$	−9.30278	−0.0434709
$215$	− 344.413i	− 1.60192i
$216$	0	0
$217$	92.0956	0.424404
$218$	− 281.253i	− 1.29015i
$219$	0	0
$220$	−124.529	−0.566041
$221$	− 173.198i	− 0.783703i
$222$	0	0
$223$	−306.810	−1.37583	−0.687915	−	0.725791i	$-0.741474\pi$
$223$	−306.810	−1.37583	−0.687915	+	0.725791i	$0.741474\pi$
$224$	− 32.5517i	− 0.145320i
$225$	0	0
$226$	−7.42068	−0.0328348
$227$	− 11.8982i	− 0.0524148i	−0.999657	−	0.0262074i	$-0.991657\pi$
$227$	− 11.8982i	− 0.0524148i	0.999657	−	0.0262074i	$-0.00834303\pi$
$228$	0	0
$229$	−385.366	−1.68282	−0.841409	−	0.540398i	$-0.818274\pi$
$229$	−385.366	−1.68282	−0.841409	+	0.540398i	$0.818274\pi$
$230$	335.748i	1.45978i
$231$	0	0
$232$	80.0965	0.345243
$233$	9.60654i	0.0412298i	0.999787	+	0.0206149i	$0.00656239\pi$
$233$	9.60654i	0.0412298i	−0.999787	+	0.0206149i	$0.993438\pi$
$234$	0	0
$235$	516.515	2.19794
$236$	− 116.488i	− 0.493594i
$237$	0	0
$238$	−60.0767	−0.252423
$239$	102.957i	0.430782i	0.976528	+	0.215391i	$0.0691025\pi$
$239$	102.957i	0.430782i	−0.976528	+	0.215391i	$0.930897\pi$
$240$	0	0
$241$	106.473	0.441798	0.220899	−	0.975297i	$-0.429101\pi$
$241$	106.473	0.441798	0.220899	+	0.975297i	$0.429101\pi$
$242$	− 77.1782i	− 0.318918i
$243$	0	0
$244$	−142.708	−0.584870
$245$	− 121.370i	− 0.495389i
$246$	0	0
$247$	366.057	1.48201
$248$	45.2673i	0.182530i
$249$	0	0
$250$	−90.3535	−0.361414
$251$	81.6007i	0.325102i	0.986700	+	0.162551i	$0.0519723\pi$
$251$	81.6007i	0.325102i	−0.986700	+	0.162551i	$0.948028\pi$
$252$	0	0
$253$	253.281	1.00111
$254$	− 62.8451i	− 0.247422i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 439.294i	− 1.70931i	−0.519193	−	0.854657i	$-0.673767\pi$
$257$	− 439.294i	− 1.70931i	0.519193	−	0.854657i	$-0.326233\pi$
$258$	0	0
$259$	−68.9429	−0.266189
$260$	− 358.468i	− 1.37872i
$261$	0	0
$262$	−56.8234	−0.216883
$263$	450.911i	1.71449i	0.514908	+	0.857245i	$0.327826\pi$
$263$	450.911i	1.71449i	−0.514908	+	0.857245i	$0.672174\pi$
$264$	0	0
$265$	−125.377	−0.473121
$266$	− 126.973i	− 0.477342i
$267$	0	0
$268$	−100.392	−0.374597
$269$	363.566i	1.35155i	0.737110	+	0.675773i	$0.236190\pi$
$269$	363.566i	1.35155i	−0.737110	+	0.675773i	$0.763810\pi$
$270$	0	0
$271$	−70.6655	−0.260758	−0.130379	−	0.991464i	$-0.541619\pi$
$271$	−70.6655	−0.260758	−0.130379	+	0.991464i	$0.541619\pi$
$272$	− 29.5292i	− 0.108563i
$273$	0	0
$274$	72.8842	0.266001
$275$	271.917i	0.988789i
$276$	0	0
$277$	−233.500	−0.842962	−0.421481	−	0.906837i	$-0.638490\pi$
$277$	−233.500	−0.842962	−0.421481	+	0.906837i	$0.638490\pi$
$278$	152.239i	0.547622i
$279$	0	0
$280$	−124.341	−0.444073
$281$	289.317i	1.02960i	0.857311	+	0.514799i	$0.172134\pi$
$281$	289.317i	1.02960i	−0.857311	+	0.514799i	$0.827866\pi$
$282$	0	0
$283$	153.403	0.542060	0.271030	−	0.962571i	$-0.412636\pi$
$283$	153.403	0.542060	0.271030	+	0.962571i	$0.412636\pi$
$284$	81.8367i	0.288157i
$285$	0	0
$286$	−270.420	−0.945523
$287$	− 47.5508i	− 0.165682i
$288$	0	0
$289$	234.502	0.811424
$290$	− 305.951i	− 1.05500i
$291$	0	0
$292$	−199.077	−0.681770
$293$	− 163.328i	− 0.557434i	−0.960373	−	0.278717i	$-0.910091\pi$
$293$	− 163.328i	− 0.557434i	0.960373	−	0.278717i	$-0.0899091\pi$
$294$	0	0
$295$	−444.959	−1.50834
$296$	− 33.8872i	− 0.114484i
$297$	0	0
$298$	199.547	0.669621
$299$	729.091i	2.43843i
$300$	0	0
$301$	−259.423	−0.861872
$302$	185.519i	0.614303i
$303$	0	0
$304$	62.4104	0.205297
$305$	545.115i	1.78726i
$306$	0	0
$307$	−326.648	−1.06400	−0.532000	−	0.846744i	$-0.678560\pi$
$307$	−326.648	−1.06400	−0.532000	+	0.846744i	$0.678560\pi$
$308$	93.7995i	0.304544i
$309$	0	0
$310$	172.911	0.557779
$311$	− 465.553i	− 1.49695i	−0.663161	−	0.748477i	$-0.730786\pi$
$311$	− 465.553i	− 1.49695i	0.663161	−	0.748477i	$-0.269214\pi$
$312$	0	0
$313$	219.310	0.700669	0.350335	−	0.936625i	$-0.386068\pi$
$313$	219.310	0.700669	0.350335	+	0.936625i	$0.386068\pi$
$314$	− 272.621i	− 0.868220i
$315$	0	0
$316$	−154.106	−0.487677
$317$	− 127.383i	− 0.401838i	−0.979608	−	0.200919i	$-0.935607\pi$
$317$	− 127.383i	− 0.401838i	0.979608	−	0.200919i	$-0.0643928\pi$
$318$	0	0
$319$	−230.802	−0.723518
$320$	− 61.1165i	− 0.190989i
$321$	0	0
$322$	252.897	0.785395
$323$	− 115.183i	− 0.356604i
$324$	0	0
$325$	−782.738	−2.40843
$326$	− 113.819i	− 0.349137i
$327$	0	0
$328$	23.3724	0.0712573
$329$	− 389.057i	− 1.18254i
$330$	0	0
$331$	271.665	0.820739	0.410370	−	0.911919i	$-0.365400\pi$
$331$	271.665	0.820739	0.410370	+	0.911919i	$0.365400\pi$
$332$	− 210.964i	− 0.635433i
$333$	0	0
$334$	−153.116	−0.458432
$335$	383.475i	1.14470i
$336$	0	0
$337$	513.337	1.52325	0.761627	−	0.648015i	$-0.224401\pi$
$337$	513.337	1.52325	0.761627	+	0.648015i	$0.224401\pi$
$338$	− 539.426i	− 1.59593i
$339$	0	0
$340$	−112.795	−0.331751
$341$	− 130.440i	− 0.382523i
$342$	0	0
$343$	−373.385	−1.08859
$344$	− 127.513i	− 0.370678i
$345$	0	0
$346$	−119.732	−0.346046
$347$	179.150i	0.516281i	0.966107	+	0.258141i	$0.0831098\pi$
$347$	179.150i	0.516281i	−0.966107	+	0.258141i	$0.916890\pi$
$348$	0	0
$349$	−345.505	−0.989985	−0.494992	−	0.868897i	$-0.664829\pi$
$349$	−345.505	−0.989985	−0.494992	+	0.868897i	$0.664829\pi$
$350$	271.506i	0.775730i
$351$	0	0
$352$	−46.1048	−0.130980
$353$	245.720i	0.696091i	0.937478	+	0.348046i	$0.113155\pi$
$353$	245.720i	0.696091i	−0.937478	+	0.348046i	$0.886845\pi$
$354$	0	0
$355$	312.598	0.880559
$356$	0.697902i	0.00196040i
$357$	0	0
$358$	237.259	0.662735
$359$	− 555.660i	− 1.54780i	−0.633309	−	0.773899i	$-0.718304\pi$
$359$	− 555.660i	− 1.54780i	0.633309	−	0.773899i	$-0.281696\pi$
$360$	0	0
$361$	−117.559	−0.325648
$362$	409.365i	1.13084i
$363$	0	0
$364$	−270.010	−0.741787
$365$	760.430i	2.08337i
$366$	0	0
$367$	398.565	1.08601	0.543004	−	0.839730i	$-0.317287\pi$
$367$	398.565	1.08601	0.543004	+	0.839730i	$0.317287\pi$
$368$	124.305i	0.337787i
$369$	0	0
$370$	−129.442	−0.349842
$371$	94.4383i	0.254551i
$372$	0	0
$373$	409.675	1.09832	0.549162	−	0.835716i	$-0.314947\pi$
$373$	409.675	1.09832	0.549162	+	0.835716i	$0.314947\pi$
$374$	85.0901i	0.227514i
$375$	0	0
$376$	191.231	0.508594
$377$	− 664.385i	− 1.76230i
$378$	0	0
$379$	−61.5139	−0.162306	−0.0811529	−	0.996702i	$-0.525860\pi$
$379$	−61.5139	−0.162306	−0.0811529	+	0.996702i	$0.525860\pi$
$380$	− 238.394i	− 0.627353i
$381$	0	0
$382$	−57.0462	−0.149335
$383$	203.341i	0.530917i	0.964122	+	0.265458i	$0.0855233\pi$
$383$	203.341i	0.530917i	−0.964122	+	0.265458i	$0.914477\pi$
$384$	0	0
$385$	358.294	0.930633
$386$	342.980i	0.888549i
$387$	0	0
$388$	−127.932	−0.329723
$389$	255.004i	0.655538i	0.944758	+	0.327769i	$0.106297\pi$
$389$	255.004i	0.655538i	−0.944758	+	0.327769i	$0.893703\pi$
$390$	0	0
$391$	229.415	0.586740
$392$	− 44.9354i	− 0.114631i
$393$	0	0
$394$	−7.71477	−0.0195806
$395$	588.651i	1.49026i
$396$	0	0
$397$	−27.6278	−0.0695914	−0.0347957	−	0.999394i	$-0.511078\pi$
$397$	−27.6278	−0.0695914	−0.0347957	+	0.999394i	$0.511078\pi$
$398$	202.253i	0.508174i
$399$	0	0
$400$	−133.452	−0.333630
$401$	374.465i	0.933829i	0.884303	+	0.466914i	$0.154634\pi$
$401$	374.465i	0.933829i	−0.884303	+	0.466914i	$0.845366\pi$
$402$	0	0
$403$	375.484	0.931722
$404$	− 333.449i	− 0.825370i
$405$	0	0
$406$	−230.453	−0.567618
$407$	97.6477i	0.239921i
$408$	0	0
$409$	204.938	0.501071	0.250536	−	0.968107i	$-0.419393\pi$
$409$	204.938	0.501071	0.250536	+	0.968107i	$0.419393\pi$
$410$	− 89.2775i	− 0.217750i
$411$	0	0
$412$	201.221	0.488400
$413$	335.159i	0.811522i
$414$	0	0
$415$	−805.835	−1.94177
$416$	− 132.717i	− 0.319031i
$417$	0	0
$418$	−179.839	−0.430237
$419$	− 239.000i	− 0.570405i	−0.958467	−	0.285202i	$-0.907939\pi$
$419$	− 239.000i	− 0.570405i	0.958467	−	0.285202i	$-0.0920608\pi$
$420$	0	0
$421$	199.488	0.473844	0.236922	−	0.971529i	$-0.423861\pi$
$421$	199.488	0.473844	0.236922	+	0.971529i	$0.423861\pi$
$422$	12.2157i	0.0289472i
$423$	0	0
$424$	−46.4188	−0.109478
$425$	246.296i	0.579520i
$426$	0	0
$427$	410.599	0.961591
$428$	13.1561i	0.0307386i
$429$	0	0
$430$	−487.073	−1.13273
$431$	− 235.094i	− 0.545461i	−0.962090	−	0.272731i	$-0.912073\pi$
$431$	− 235.094i	− 0.545461i	0.962090	−	0.272731i	$-0.0879267\pi$
$432$	0	0
$433$	−391.650	−0.904504	−0.452252	−	0.891890i	$-0.649379\pi$
$433$	−391.650	−0.904504	−0.452252	+	0.891890i	$0.649379\pi$
$434$	− 130.243i	− 0.300099i
$435$	0	0
$436$	−397.752	−0.912276
$437$	484.872i	1.10955i
$438$	0	0
$439$	81.2695	0.185124	0.0925621	−	0.995707i	$-0.470494\pi$
$439$	81.2695	0.185124	0.0925621	+	0.995707i	$0.470494\pi$
$440$	176.111i	0.400251i
$441$	0	0
$442$	−244.940	−0.554162
$443$	− 527.706i	− 1.19121i	−0.803277	−	0.595605i	$-0.796912\pi$
$443$	− 527.706i	− 1.19121i	0.803277	−	0.595605i	$-0.203088\pi$
$444$	0	0
$445$	2.66583	0.00599064
$446$	433.895i	0.972859i
$447$	0	0
$448$	−46.0351	−0.102757
$449$	120.226i	0.267765i	0.990997	+	0.133882i	$0.0427445\pi$
$449$	120.226i	0.267765i	−0.990997	+	0.133882i	$0.957256\pi$
$450$	0	0
$451$	−67.3488	−0.149332
$452$	10.4944i	0.0232177i
$453$	0	0
$454$	−16.8265	−0.0370629
$455$	1031.38i	2.26677i
$456$	0	0
$457$	−143.140	−0.313216	−0.156608	−	0.987661i	$-0.550056\pi$
$457$	−143.140	−0.313216	−0.156608	+	0.987661i	$0.550056\pi$
$458$	544.989i	1.18993i
$459$	0	0
$460$	474.820	1.03222
$461$	− 236.139i	− 0.512232i	−0.966646	−	0.256116i	$-0.917557\pi$
$461$	− 236.139i	− 0.512232i	0.966646	−	0.256116i	$-0.0824429\pi$
$462$	0	0
$463$	148.324	0.320353	0.160177	−	0.987088i	$-0.448794\pi$
$463$	148.324	0.320353	0.160177	+	0.987088i	$0.448794\pi$
$464$	− 113.274i	− 0.244124i
$465$	0	0
$466$	13.5857	0.0291539
$467$	− 90.2013i	− 0.193150i	−0.995326	−	0.0965752i	$-0.969211\pi$
$467$	− 90.2013i	− 0.193150i	0.995326	−	0.0965752i	$-0.0307888\pi$
$468$	0	0
$469$	288.847	0.615878
$470$	− 730.463i	− 1.55418i
$471$	0	0
$472$	−164.739	−0.349023
$473$	367.436i	0.776821i
$474$	0	0
$475$	−520.549	−1.09589
$476$	84.9613i	0.178490i
$477$	0	0
$478$	145.603	0.304609
$479$	− 258.118i	− 0.538868i	−0.963019	−	0.269434i	$-0.913163\pi$
$479$	− 258.118i	− 0.538868i	0.963019	−	0.269434i	$-0.0868366\pi$
$480$	0	0
$481$	−281.088	−0.584382
$482$	− 150.576i	− 0.312398i
$483$	0	0
$484$	−109.146	−0.225509
$485$	488.674i	1.00758i
$486$	0	0
$487$	−103.107	−0.211718	−0.105859	−	0.994381i	$-0.533759\pi$
$487$	−103.107	−0.211718	−0.105859	+	0.994381i	$0.533759\pi$
$488$	201.820i	0.413566i
$489$	0	0
$490$	−171.643	−0.350293
$491$	− 764.752i	− 1.55754i	−0.627310	−	0.778770i	$-0.715844\pi$
$491$	− 764.752i	− 1.55754i	0.627310	−	0.778770i	$-0.284156\pi$
$492$	0	0
$493$	−209.055	−0.424047
$494$	− 517.683i	− 1.04794i
$495$	0	0
$496$	64.0177	0.129068
$497$	− 235.460i	− 0.473762i
$498$	0	0
$499$	−718.977	−1.44084	−0.720418	−	0.693540i	$-0.756050\pi$
$499$	−718.977	−1.44084	−0.720418	+	0.693540i	$0.756050\pi$
$500$	127.779i	0.255558i
$501$	0	0
$502$	115.401	0.229882
$503$	− 585.028i	− 1.16308i	−0.813519	−	0.581538i	$-0.802451\pi$
$503$	− 585.028i	− 1.16308i	0.813519	−	0.581538i	$-0.197549\pi$
$504$	0	0
$505$	−1273.70	−2.52219
$506$	− 358.193i	− 0.707891i
$507$	0	0
$508$	−88.8764	−0.174954
$509$	− 883.436i	− 1.73563i	−0.496886	−	0.867816i	$-0.665523\pi$
$509$	− 883.436i	− 1.73563i	0.496886	−	0.867816i	$-0.334477\pi$
$510$	0	0
$511$	572.782	1.12090
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−621.255	−1.20867
$515$	− 768.620i	− 1.49247i
$516$	0	0
$517$	−551.044	−1.06585
$518$	97.5000i	0.188224i
$519$	0	0
$520$	−506.950	−0.974904
$521$	26.0398i	0.0499804i	0.999688	+	0.0249902i	$0.00795545\pi$
$521$	26.0398i	0.0499804i	−0.999688	+	0.0249902i	$0.992045\pi$
$522$	0	0
$523$	578.821	1.10673	0.553366	−	0.832938i	$-0.313343\pi$
$523$	578.821	1.10673	0.553366	+	0.832938i	$0.313343\pi$
$524$	80.3605i	0.153360i
$525$	0	0
$526$	637.684	1.21233
$527$	− 118.150i	− 0.224193i
$528$	0	0
$529$	−436.741	−0.825597
$530$	177.310i	0.334547i
$531$	0	0
$532$	−179.567	−0.337531
$533$	− 193.870i	− 0.363733i
$534$	0	0
$535$	50.2535	0.0939318
$536$	141.976i	0.264880i
$537$	0	0
$538$	514.160	0.955688
$539$	129.484i	0.240229i
$540$	0	0
$541$	−84.9565	−0.157036	−0.0785180	−	0.996913i	$-0.525019\pi$
$541$	−84.9565	−0.157036	−0.0785180	+	0.996913i	$0.525019\pi$
$542$	99.9360i	0.184384i
$543$	0	0
$544$	−41.7606	−0.0767659
$545$	1519.33i	2.78776i
$546$	0	0
$547$	−864.185	−1.57986	−0.789931	−	0.613195i	$-0.789884\pi$
$547$	−864.185	−1.57986	−0.789931	+	0.613195i	$0.789884\pi$
$548$	− 103.074i	− 0.188091i
$549$	0	0
$550$	384.549	0.699180
$551$	− 441.840i	− 0.801888i
$552$	0	0
$553$	443.392	0.801794
$554$	330.219i	0.596064i
$555$	0	0
$556$	215.299	0.387228
$557$	261.397i	0.469294i	0.972081	+	0.234647i	$0.0753934\pi$
$557$	261.397i	0.469294i	−0.972081	+	0.234647i	$0.924607\pi$
$558$	0	0
$559$	−1057.70	−1.89213
$560$	175.844i	0.314007i
$561$	0	0
$562$	409.156	0.728036
$563$	− 787.033i	− 1.39793i	−0.715158	−	0.698963i	$-0.753645\pi$
$563$	− 787.033i	− 1.39793i	0.715158	−	0.698963i	$-0.246355\pi$
$564$	0	0
$565$	40.0864	0.0709494
$566$	− 216.944i	− 0.383294i
$567$	0	0
$568$	115.735	0.203758
$569$	264.418i	0.464707i	0.972631	+	0.232354i	$0.0746427\pi$
$569$	264.418i	0.464707i	−0.972631	+	0.232354i	$0.925357\pi$
$570$	0	0
$571$	−63.6987	−0.111556	−0.0557782	−	0.998443i	$-0.517764\pi$
$571$	−63.6987	−0.111556	−0.0557782	+	0.998443i	$0.517764\pi$
$572$	382.431i	0.668586i
$573$	0	0
$574$	−67.2469	−0.117155
$575$	− 1036.80i	− 1.80313i
$576$	0	0
$577$	−970.722	−1.68236	−0.841180	−	0.540755i	$-0.818139\pi$
$577$	−970.722	−1.68236	−0.841180	+	0.540755i	$0.818139\pi$
$578$	− 331.635i	− 0.573763i
$579$	0	0
$580$	−432.680	−0.746001
$581$	606.983i	1.04472i
$582$	0	0
$583$	133.758	0.229431
$584$	281.537i	0.482084i
$585$	0	0
$586$	−230.981	−0.394165
$587$	101.662i	0.173190i	0.996244	+	0.0865948i	$0.0275985\pi$
$587$	101.662i	0.173190i	−0.996244	+	0.0865948i	$0.972401\pi$
$588$	0	0
$589$	249.711	0.423957
$590$	629.268i	1.06656i
$591$	0	0
$592$	−47.9237	−0.0809522
$593$	926.755i	1.56282i	0.624016	+	0.781412i	$0.285500\pi$
$593$	926.755i	1.56282i	−0.624016	+	0.781412i	$0.714500\pi$
$594$	0	0
$595$	324.534	0.545435
$596$	− 282.202i	− 0.473493i
$597$	0	0
$598$	1031.09	1.72423
$599$	− 562.278i	− 0.938695i	−0.883014	−	0.469347i	$-0.844489\pi$
$599$	− 562.278i	− 0.938695i	0.883014	−	0.469347i	$-0.155511\pi$
$600$	0	0
$601$	380.292	0.632765	0.316382	−	0.948632i	$-0.397532\pi$
$601$	380.292	0.632765	0.316382	+	0.948632i	$0.397532\pi$
$602$	366.880i	0.609436i
$603$	0	0
$604$	262.364	0.434378
$605$	416.916i	0.689117i
$606$	0	0
$607$	−1124.48	−1.85251	−0.926257	−	0.376893i	$-0.876992\pi$
$607$	−1124.48	−1.85251	−0.926257	+	0.376893i	$0.876992\pi$
$608$	− 88.2616i	− 0.145167i
$609$	0	0
$610$	770.909	1.26379
$611$	− 1586.23i	− 2.59612i
$612$	0	0
$613$	626.817	1.02254	0.511270	−	0.859420i	$-0.329175\pi$
$613$	626.817	1.02254	0.511270	+	0.859420i	$0.329175\pi$
$614$	461.950i	0.752362i
$615$	0	0
$616$	132.653	0.215345
$617$	− 176.294i	− 0.285727i	−0.989742	−	0.142864i	$-0.954369\pi$
$617$	− 176.294i	− 0.285727i	0.989742	−	0.142864i	$-0.0456311\pi$
$618$	0	0
$619$	929.905	1.50227	0.751135	−	0.660149i	$-0.229507\pi$
$619$	929.905	1.50227	0.751135	+	0.660149i	$0.229507\pi$
$620$	− 244.534i	− 0.394409i
$621$	0	0
$622$	−658.391	−1.05851
$623$	− 2.00800i	− 0.00322311i
$624$	0	0
$625$	−345.986	−0.553577
$626$	− 310.151i	− 0.495448i
$627$	0	0
$628$	−385.544	−0.613924
$629$	88.4469i	0.140615i
$630$	0	0
$631$	−654.106	−1.03662	−0.518309	−	0.855193i	$-0.673438\pi$
$631$	−654.106	−1.03662	−0.518309	+	0.855193i	$0.673438\pi$
$632$	217.939i	0.344840i
$633$	0	0
$634$	−180.146	−0.284142
$635$	339.489i	0.534628i
$636$	0	0
$637$	−372.731	−0.585134
$638$	326.404i	0.511605i
$639$	0	0
$640$	−86.4318	−0.135050
$641$	736.054i	1.14829i	0.818754	+	0.574145i	$0.194665\pi$
$641$	736.054i	1.14829i	−0.818754	+	0.574145i	$0.805335\pi$
$642$	0	0
$643$	−712.261	−1.10771	−0.553857	−	0.832612i	$-0.686845\pi$
$643$	−712.261	−1.10771	−0.553857	+	0.832612i	$0.686845\pi$
$644$	− 357.651i	− 0.555358i
$645$	0	0
$646$	−162.894	−0.252157
$647$	291.136i	0.449978i	0.974361	+	0.224989i	$0.0722346\pi$
$647$	291.136i	0.449978i	−0.974361	+	0.224989i	$0.927765\pi$
$648$	0	0
$649$	474.704	0.731440
$650$	1106.96i	1.70301i
$651$	0	0
$652$	−160.964	−0.246877
$653$	− 1162.93i	− 1.78090i	−0.455081	−	0.890450i	$-0.650390\pi$
$653$	− 1162.93i	− 1.78090i	0.455081	−	0.890450i	$-0.349610\pi$
$654$	0	0
$655$	306.960	0.468641
$656$	− 33.0536i	− 0.0503866i
$657$	0	0
$658$	−550.210	−0.836185
$659$	− 529.292i	− 0.803174i	−0.915821	−	0.401587i	$-0.868459\pi$
$659$	− 529.292i	− 0.803174i	0.915821	−	0.401587i	$-0.131541\pi$
$660$	0	0
$661$	3.85043	0.00582516	0.00291258	−	0.999996i	$-0.499073\pi$
$661$	3.85043	0.00582516	0.00291258	+	0.999996i	$0.499073\pi$
$662$	− 384.192i	− 0.580350i
$663$	0	0
$664$	−298.348	−0.449319
$665$	685.906i	1.03144i
$666$	0	0
$667$	880.033	1.31939
$668$	216.539i	0.324161i
$669$	0	0
$670$	542.316	0.809427
$671$	− 581.555i	− 0.866700i
$672$	0	0
$673$	−414.813	−0.616364	−0.308182	−	0.951327i	$-0.599721\pi$
$673$	−414.813	−0.616364	−0.308182	+	0.951327i	$0.599721\pi$
$674$	− 725.968i	− 1.07710i
$675$	0	0
$676$	−762.863	−1.12850
$677$	− 1187.49i	− 1.75405i	−0.480444	−	0.877025i	$-0.659524\pi$
$677$	− 1187.49i	− 1.75405i	0.480444	−	0.877025i	$-0.340476\pi$
$678$	0	0
$679$	368.086	0.542100
$680$	159.517i	0.234583i
$681$	0	0
$682$	−184.470	−0.270484
$683$	513.834i	0.752320i	0.926555	+	0.376160i	$0.122756\pi$
$683$	513.834i	0.752320i	−0.926555	+	0.376160i	$0.877244\pi$
$684$	0	0
$685$	−393.720	−0.574774
$686$	528.046i	0.769747i
$687$	0	0
$688$	−180.331	−0.262109
$689$	385.036i	0.558833i
$690$	0	0
$691$	529.360	0.766079	0.383039	−	0.923732i	$-0.374877\pi$
$691$	529.360	0.766079	0.383039	+	0.923732i	$0.374877\pi$
$692$	169.326i	0.244691i
$693$	0	0
$694$	253.356	0.365066
$695$	− 822.394i	− 1.18330i
$696$	0	0
$697$	−61.0029	−0.0875221
$698$	488.617i	0.700025i
$699$	0	0
$700$	383.967	0.548524
$701$	13.0718i	0.0186474i	0.999957	+	0.00932369i	$0.00296787\pi$
$701$	13.0718i	0.0186474i	−0.999957	+	0.00932369i	$0.997032\pi$
$702$	0	0
$703$	−186.934	−0.265908
$704$	65.2021i	0.0926166i
$705$	0	0
$706$	347.501	0.492211
$707$	959.398i	1.35700i
$708$	0	0
$709$	−762.984	−1.07614	−0.538071	−	0.842900i	$-0.680847\pi$
$709$	−762.984	−1.07614	−0.538071	+	0.842900i	$0.680847\pi$
$710$	− 442.081i	− 0.622649i
$711$	0	0
$712$	0.986982	0.00138621
$713$	497.359i	0.697559i
$714$	0	0
$715$	1460.80	2.04308
$716$	− 335.535i	− 0.468624i
$717$	0	0
$718$	−785.821	−1.09446
$719$	903.516i	1.25663i	0.777959	+	0.628315i	$0.216255\pi$
$719$	903.516i	1.25663i	−0.777959	+	0.628315i	$0.783745\pi$
$720$	0	0
$721$	−578.951	−0.802983
$722$	166.253i	0.230268i
$723$	0	0
$724$	578.930	0.799627
$725$	944.786i	1.30315i
$726$	0	0
$727$	−964.599	−1.32682	−0.663411	−	0.748256i	$-0.730892\pi$
$727$	−964.599	−1.32682	−0.663411	+	0.748256i	$0.730892\pi$
$728$	381.852i	0.524523i
$729$	0	0
$730$	1075.41	1.47317
$731$	332.815i	0.455287i
$732$	0	0
$733$	549.731	0.749974	0.374987	−	0.927030i	$-0.377647\pi$
$733$	549.731	0.749974	0.374987	+	0.927030i	$0.377647\pi$
$734$	− 563.656i	− 0.767924i
$735$	0	0
$736$	175.794	0.238851
$737$	− 409.110i	− 0.555102i
$738$	0	0
$739$	522.615	0.707192	0.353596	−	0.935398i	$-0.384959\pi$
$739$	522.615	0.707192	0.353596	+	0.935398i	$0.384959\pi$
$740$	183.058i	0.247376i
$741$	0	0
$742$	133.556	0.179995
$743$	498.311i	0.670674i	0.942098	+	0.335337i	$0.108850\pi$
$743$	498.311i	0.670674i	−0.942098	+	0.335337i	$0.891150\pi$
$744$	0	0
$745$	−1077.95	−1.44691
$746$	− 579.367i	− 0.776632i
$747$	0	0
$748$	120.336	0.160876
$749$	− 37.8527i	− 0.0505376i
$750$	0	0
$751$	401.886	0.535135	0.267567	−	0.963539i	$-0.413780\pi$
$751$	401.886	0.535135	0.267567	+	0.963539i	$0.413780\pi$
$752$	− 270.442i	− 0.359631i
$753$	0	0
$754$	−939.583	−1.24613
$755$	− 1002.17i	− 1.32738i
$756$	0	0
$757$	564.958	0.746312	0.373156	−	0.927769i	$-0.378276\pi$
$757$	564.958	0.746312	0.373156	+	0.927769i	$0.378276\pi$
$758$	86.9938i	0.114768i
$759$	0	0
$760$	−337.140	−0.443606
$761$	− 359.307i	− 0.472151i	−0.971735	−	0.236076i	$-0.924139\pi$
$761$	− 359.307i	− 0.472151i	0.971735	−	0.236076i	$-0.0758613\pi$
$762$	0	0
$763$	1144.41	1.49988
$764$	80.6754i	0.105596i
$765$	0	0
$766$	287.568	0.375415
$767$	1366.48i	1.78159i
$768$	0	0
$769$	−600.467	−0.780841	−0.390420	−	0.920637i	$-0.627670\pi$
$769$	−600.467	−0.780841	−0.390420	+	0.920637i	$0.627670\pi$
$770$	− 506.704i	− 0.658057i
$771$	0	0
$772$	485.047	0.628299
$773$	229.599i	0.297023i	0.988911	+	0.148512i	$0.0474482\pi$
$773$	229.599i	0.297023i	−0.988911	+	0.148512i	$0.952552\pi$
$774$	0	0
$775$	−533.955	−0.688974
$776$	180.924i	0.233149i
$777$	0	0
$778$	360.630	0.463535
$779$	− 128.930i	− 0.165508i
$780$	0	0
$781$	−333.495	−0.427010
$782$	− 324.442i	− 0.414888i
$783$	0	0
$784$	−63.5482	−0.0810564
$785$	1472.70i	1.87605i
$786$	0	0
$787$	−1049.16	−1.33311	−0.666554	−	0.745456i	$-0.732232\pi$
$787$	−1049.16	−1.33311	−0.666554	+	0.745456i	$0.732232\pi$
$788$	10.9103i	0.0138456i
$789$	0	0
$790$	832.478	1.05377
$791$	− 30.1945i	− 0.0381725i
$792$	0	0
$793$	1674.06	2.11105
$794$	39.0716i	0.0492086i
$795$	0	0
$796$	286.029	0.359333
$797$	− 1490.08i	− 1.86961i	−0.355162	−	0.934805i	$-0.615574\pi$
$797$	− 1490.08i	− 1.86961i	0.355162	−	0.934805i	$-0.384426\pi$
$798$	0	0
$799$	−499.122	−0.624683
$800$	188.730i	0.235912i
$801$	0	0
$802$	529.574	0.660317
$803$	− 811.264i	− 1.01029i
$804$	0	0
$805$	−1366.15	−1.69708
$806$	− 531.015i	− 0.658827i
$807$	0	0
$808$	−471.569	−0.583625
$809$	1326.92i	1.64020i	0.572222	+	0.820099i	$0.306082\pi$
$809$	1326.92i	1.64020i	−0.572222	+	0.820099i	$0.693918\pi$
$810$	0	0
$811$	412.746	0.508935	0.254467	−	0.967081i	$-0.418100\pi$
$811$	412.746	0.508935	0.254467	+	0.967081i	$0.418100\pi$
$812$	325.910i	0.401367i
$813$	0	0
$814$	138.095	0.169650
$815$	614.847i	0.754413i
$816$	0	0
$817$	−703.408	−0.860965
$818$	− 289.826i	− 0.354311i
$819$	0	0
$820$	−126.258	−0.153973
$821$	− 270.948i	− 0.330022i	−0.986292	−	0.165011i	$-0.947234\pi$
$821$	− 270.948i	− 0.330022i	0.986292	−	0.165011i	$-0.0527660\pi$
$822$	0	0
$823$	718.060	0.872491	0.436246	−	0.899828i	$-0.356308\pi$
$823$	718.060	0.872491	0.436246	+	0.899828i	$0.356308\pi$
$824$	− 284.569i	− 0.345351i
$825$	0	0
$826$	473.986	0.573833
$827$	− 1115.81i	− 1.34923i	−0.738171	−	0.674614i	$-0.764310\pi$
$827$	− 1115.81i	− 1.34923i	0.738171	−	0.674614i	$-0.235690\pi$
$828$	0	0
$829$	1123.75	1.35555	0.677774	−	0.735270i	$-0.262945\pi$
$829$	1123.75	1.35555	0.677774	+	0.735270i	$0.262945\pi$
$830$	1139.62i	1.37304i
$831$	0	0
$832$	−187.690	−0.225589
$833$	117.283i	0.140796i
$834$	0	0
$835$	827.133	0.990579
$836$	254.331i	0.304223i
$837$	0	0
$838$	−337.996	−0.403337
$839$	− 253.356i	− 0.301973i	−0.988536	−	0.150987i	$-0.951755\pi$
$839$	− 253.356i	− 0.301973i	0.988536	−	0.150987i	$-0.0482451\pi$
$840$	0	0
$841$	39.0691	0.0464556
$842$	− 282.119i	− 0.335058i
$843$	0	0
$844$	17.2756	0.0204688
$845$	2913.97i	3.44849i
$846$	0	0
$847$	314.035	0.370762
$848$	65.6462i	0.0774129i
$849$	0	0
$850$	348.315	0.409782
$851$	− 372.324i	− 0.437513i
$852$	0	0
$853$	797.504	0.934940	0.467470	−	0.884009i	$-0.345166\pi$
$853$	797.504	0.934940	0.467470	+	0.884009i	$0.345166\pi$
$854$	− 580.675i	− 0.679948i
$855$	0	0
$856$	18.6056	0.0217355
$857$	− 1117.98i	− 1.30453i	−0.757990	−	0.652266i	$-0.773818\pi$
$857$	− 1117.98i	− 1.30453i	0.757990	−	0.652266i	$-0.226182\pi$
$858$	0	0
$859$	−1024.40	−1.19255	−0.596276	−	0.802779i	$-0.703354\pi$
$859$	−1024.40	−1.19255	−0.596276	+	0.802779i	$0.703354\pi$
$860$	688.825i	0.800960i
$861$	0	0
$862$	−332.473	−0.385699
$863$	− 1691.86i	− 1.96044i	−0.197921	−	0.980218i	$-0.563419\pi$
$863$	− 1691.86i	− 1.96044i	0.197921	−	0.980218i	$-0.436581\pi$
$864$	0	0
$865$	646.790	0.747734
$866$	553.877i	0.639581i
$867$	0	0
$868$	−184.191	−0.212202
$869$	− 628.002i	− 0.722672i
$870$	0	0
$871$	1177.66	1.35208
$872$	562.507i	0.645076i
$873$	0	0
$874$	685.713	0.784568
$875$	− 367.645i	− 0.420166i
$876$	0	0
$877$	210.381	0.239887	0.119944	−	0.992781i	$-0.461729\pi$
$877$	210.381	0.239887	0.119944	+	0.992781i	$0.461729\pi$
$878$	− 114.932i	− 0.130903i
$879$	0	0
$880$	249.058	0.283020
$881$	756.115i	0.858247i	0.903246	+	0.429123i	$0.141177\pi$
$881$	756.115i	0.858247i	−0.903246	+	0.429123i	$0.858823\pi$
$882$	0	0
$883$	1283.92	1.45404	0.727021	−	0.686615i	$-0.240904\pi$
$883$	1283.92	1.45404	0.727021	+	0.686615i	$0.240904\pi$
$884$	346.397i	0.391852i
$885$	0	0
$886$	−746.289	−0.842313
$887$	− 454.977i	− 0.512939i	−0.966552	−	0.256469i	$-0.917441\pi$
$887$	− 454.977i	− 0.512939i	0.966552	−	0.256469i	$-0.0825593\pi$
$888$	0	0
$889$	255.715	0.287643
$890$	− 3.77006i	− 0.00423602i
$891$	0	0
$892$	613.620	0.687915
$893$	− 1054.90i	− 1.18130i
$894$	0	0
$895$	−1281.67	−1.43203
$896$	65.1034i	0.0726601i
$897$	0	0
$898$	170.026	0.189338
$899$	− 453.219i	− 0.504137i
$900$	0	0
$901$	121.155	0.134467
$902$	95.2456i	0.105594i
$903$	0	0
$904$	14.8414	0.0164174
$905$	− 2211.39i	− 2.44352i
$906$	0	0
$907$	1565.38	1.72589	0.862946	−	0.505296i	$-0.168617\pi$
$907$	1565.38	1.72589	0.862946	+	0.505296i	$0.168617\pi$
$908$	23.7963i	0.0262074i
$909$	0	0
$910$	1458.59	1.60285
$911$	962.212i	1.05622i	0.849177	+	0.528108i	$0.177098\pi$
$911$	962.212i	1.05622i	−0.849177	+	0.528108i	$0.822902\pi$
$912$	0	0
$913$	859.705	0.941626
$914$	202.430i	0.221477i
$915$	0	0
$916$	770.731	0.841409
$917$	− 231.213i	− 0.252140i
$918$	0	0
$919$	−1645.52	−1.79055	−0.895277	−	0.445509i	$-0.853023\pi$
$919$	−1645.52	−1.79055	−0.895277	+	0.445509i	$0.853023\pi$
$920$	− 671.497i	− 0.729888i
$921$	0	0
$922$	−333.951	−0.362203
$923$	− 959.996i	− 1.04008i
$924$	0	0
$925$	399.720	0.432129
$926$	− 209.761i	− 0.226524i
$927$	0	0
$928$	−160.193	−0.172622
$929$	− 1362.56i	− 1.46670i	−0.679853	−	0.733348i	$-0.737957\pi$
$929$	− 1362.56i	− 1.46670i	0.679853	−	0.733348i	$-0.262043\pi$
$930$	0	0
$931$	−247.879	−0.266251
$932$	− 19.2131i	− 0.0206149i
$933$	0	0
$934$	−127.564	−0.136578
$935$	− 459.656i	− 0.491610i
$936$	0	0
$937$	350.586	0.374157	0.187079	−	0.982345i	$-0.440098\pi$
$937$	350.586	0.374157	0.187079	+	0.982345i	$0.440098\pi$
$938$	− 408.491i	− 0.435492i
$939$	0	0
$940$	−1033.03	−1.09897
$941$	303.734i	0.322778i	0.986891	+	0.161389i	$0.0515974\pi$
$941$	303.734i	0.322778i	−0.986891	+	0.161389i	$0.948403\pi$
$942$	0	0
$943$	256.796	0.272318
$944$	232.976i	0.246797i
$945$	0	0
$946$	519.633	0.549295
$947$	− 1424.55i	− 1.50428i	−0.659005	−	0.752139i	$-0.729022\pi$
$947$	− 1424.55i	− 1.50428i	0.659005	−	0.752139i	$-0.270978\pi$
$948$	0	0
$949$	2335.30	2.46080
$950$	736.168i	0.774914i
$951$	0	0
$952$	120.153	0.126212
$953$	− 636.885i	− 0.668294i	−0.942521	−	0.334147i	$-0.891552\pi$
$953$	− 636.885i	− 0.668294i	0.942521	−	0.334147i	$-0.108448\pi$
$954$	0	0
$955$	308.163	0.322683
$956$	− 205.914i	− 0.215391i
$957$	0	0
$958$	−365.034	−0.381037
$959$	296.563i	0.309242i
$960$	0	0
$961$	−704.859	−0.733464
$962$	397.518i	0.413221i
$963$	0	0
$964$	−212.946	−0.220899
$965$	− 1852.77i	− 1.91997i
$966$	0	0
$967$	−618.697	−0.639811	−0.319905	−	0.947450i	$-0.603651\pi$
$967$	−618.697	−0.639811	−0.319905	+	0.947450i	$0.603651\pi$
$968$	154.356i	0.159459i
$969$	0	0
$970$	691.090	0.712463
$971$	− 1356.90i	− 1.39743i	−0.715402	−	0.698713i	$-0.753756\pi$
$971$	− 1356.90i	− 1.39743i	0.715402	−	0.698713i	$-0.246244\pi$
$972$	0	0
$973$	−619.455	−0.636645
$974$	145.815i	0.149707i
$975$	0	0
$976$	285.417	0.292435
$977$	− 182.369i	− 0.186663i	−0.995635	−	0.0933314i	$-0.970248\pi$
$977$	− 182.369i	− 0.186663i	0.995635	−	0.0933314i	$-0.0297516\pi$
$978$	0	0
$979$	−2.84404	−0.00290505
$980$	242.740i	0.247694i
$981$	0	0
$982$	−1081.52	−1.10135
$983$	− 1270.39i	− 1.29236i	−0.763187	−	0.646178i	$-0.776366\pi$
$983$	− 1270.39i	− 1.29236i	0.763187	−	0.646178i	$-0.223634\pi$
$984$	0	0
$985$	41.6751	0.0423098
$986$	295.648i	0.299846i
$987$	0	0
$988$	−732.114	−0.741006
$989$	− 1401.01i	− 1.41659i
$990$	0	0
$991$	549.168	0.554155	0.277078	−	0.960848i	$-0.410634\pi$
$991$	549.168	0.554155	0.277078	+	0.960848i	$0.410634\pi$
$992$	− 90.5347i	− 0.0912648i
$993$	0	0
$994$	−332.990	−0.335000
$995$	− 1092.57i	− 1.09806i
$996$	0	0
$997$	−1378.50	−1.38265	−0.691325	−	0.722544i	$-0.742973\pi$
$997$	−1378.50	−1.38265	−0.691325	+	0.722544i	$0.742973\pi$
$998$	1016.79i	1.01883i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.16		36
3.2	odd	2	inner	1458.3.b.c.1457.21		36
27.2	odd	18		162.3.f.a.17.4		36
27.13	even	9		162.3.f.a.143.4		36
27.14	odd	18		54.3.f.a.47.3	yes	36
27.25	even	9		54.3.f.a.23.3	✓	36
108.79	odd	18		432.3.bc.c.401.3		36
108.95	even	18		432.3.bc.c.209.3		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.23.3	✓	36	27.25	even	9
54.3.f.a.47.3	yes	36	27.14	odd	18
162.3.f.a.17.4		36	27.2	odd	18
162.3.f.a.143.4		36	27.13	even	9
432.3.bc.c.209.3		36	108.95	even	18
432.3.bc.c.401.3		36	108.79	odd	18
1458.3.b.c.1457.16		36	1.1	even	1	trivial
1458.3.b.c.1457.21		36	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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1458.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +7.63957i q^{5} +5.75438 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +7.63957i q^{5} +5.75438 q^{7} +2.82843i q^{8} +10.8040 q^{10} -8.15026i q^{11} +23.4613 q^{13} -8.13793i q^{14} +4.00000 q^{16} -7.38231i q^{17} +15.6026 q^{19} -15.2791i q^{20} -11.5262 q^{22} +31.0764i q^{23} -33.3630 q^{25} -33.1793i q^{26} -11.5088 q^{28} -28.3184i q^{29} +16.0044 q^{31} -5.65685i q^{32} -10.4402 q^{34} +43.9610i q^{35} -11.9809 q^{37} -22.0654i q^{38} -21.6080 q^{40} -8.26339i q^{41} -45.0827 q^{43} +16.3005i q^{44} +43.9486 q^{46} -67.6105i q^{47} -15.8871 q^{49} +47.1824i q^{50} -46.9226 q^{52} +16.4115i q^{53} +62.2645 q^{55} +16.2759i q^{56} -40.0482 q^{58} +58.2441i q^{59} +71.3542 q^{61} -22.6337i q^{62} -8.00000 q^{64} +179.234i q^{65} +50.1960 q^{67} +14.7646i q^{68} +62.1703 q^{70} -40.9183i q^{71} +99.5384 q^{73} +16.9436i q^{74} -31.2052 q^{76} -46.8997i q^{77} +77.0529 q^{79} +30.5583i q^{80} -11.6862 q^{82} +105.482i q^{83} +56.3976 q^{85} +63.7566i q^{86} +23.0524 q^{88} -0.348951i q^{89} +135.005 q^{91} -62.1527i q^{92} -95.6157 q^{94} +119.197i q^{95} +63.9662 q^{97} +22.4677i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4}+O(q^{10}) \) 36 * q - 72 * q^4	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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