LMFDB - Embedded newform 729.3.b.a.728.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,3,Mod(728,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, 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("729.728");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	729.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:	$19.8638112719$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			728.15
Character	$\chi$	$=$	729.728
Dual form			729.3.b.a.728.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.117696i q^{2} +3.98615 q^{4} -6.19791i q^{5} +7.97344 q^{7} -0.939939i q^{8} -0.729470 q^{10} -11.4435i q^{11} -8.25567 q^{13} -0.938443i q^{14} +15.8340 q^{16} -23.9171i q^{17} -12.2420 q^{19} -24.7058i q^{20} -1.34685 q^{22} +14.7525i q^{23} -13.4141 q^{25} +0.971660i q^{26} +31.7833 q^{28} +17.7931i q^{29} +13.0209 q^{31} -5.62335i q^{32} -2.81495 q^{34} -49.4187i q^{35} -17.0700 q^{37} +1.44084i q^{38} -5.82566 q^{40} +30.7358i q^{41} -39.8300 q^{43} -45.6153i q^{44} +1.73632 q^{46} +58.3790i q^{47} +14.5757 q^{49} +1.57879i q^{50} -32.9083 q^{52} -91.2612i q^{53} -70.9255 q^{55} -7.49454i q^{56} +2.09418 q^{58} +20.6622i q^{59} -34.9591 q^{61} -1.53250i q^{62} +62.6740 q^{64} +51.1679i q^{65} +51.5623 q^{67} -95.3370i q^{68} -5.81639 q^{70} -2.63480i q^{71} +69.0145 q^{73} +2.00907i q^{74} -48.7986 q^{76} -91.2437i q^{77} +151.435 q^{79} -98.1375i q^{80} +3.61748 q^{82} -54.0683i q^{83} -148.236 q^{85} +4.68784i q^{86} -10.7561 q^{88} -163.072i q^{89} -65.8261 q^{91} +58.8058i q^{92} +6.87098 q^{94} +75.8751i q^{95} +57.0121 q^{97} -1.71551i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) $ 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 0.117696i	− 0.0588481i	−0.999567	−	0.0294240i	$-0.990633\pi$
$2$	− 0.117696i	− 0.0588481i	0.999567	−	0.0294240i	$-0.00936731\pi$
$3$	0	0
$3$	0	0
$4$	3.98615	0.996537
$5$	− 6.19791i	− 1.23958i	−0.784767	−	0.619791i	$-0.787217\pi$
$5$	− 6.19791i	− 1.23958i	0.784767	−	0.619791i	$-0.212783\pi$
$6$	0	0
$7$	7.97344	1.13906	0.569531	−	0.821970i	$-0.307125\pi$
$7$	7.97344	1.13906	0.569531	+	0.821970i	$0.307125\pi$
$8$	− 0.939939i	− 0.117492i
$9$	0	0
$10$	−0.729470	−0.0729470
$11$	− 11.4435i	− 1.04031i	−0.854071	−	0.520157i	$-0.825873\pi$
$11$	− 11.4435i	− 1.04031i	0.854071	−	0.520157i	$-0.174127\pi$
$12$	0	0
$13$	−8.25567	−0.635051	−0.317526	−	0.948250i	$-0.602852\pi$
$13$	−8.25567	−0.635051	−0.317526	+	0.948250i	$0.602852\pi$
$14$	− 0.938443i	− 0.0670316i
$15$	0	0
$16$	15.8340	0.989623
$17$	− 23.9171i	− 1.40689i	−0.710751	−	0.703443i	$-0.751645\pi$
$17$	− 23.9171i	− 1.40689i	0.710751	−	0.703443i	$-0.248355\pi$
$18$	0	0
$19$	−12.2420	−0.644318	−0.322159	−	0.946686i	$-0.604409\pi$
$19$	−12.2420	−0.644318	−0.322159	+	0.946686i	$0.604409\pi$
$20$	− 24.7058i	− 1.23529i
$21$	0	0
$22$	−1.34685	−0.0612205
$23$	14.7525i	0.641414i	0.947178	+	0.320707i	$0.103921\pi$
$23$	14.7525i	0.641414i	−0.947178	+	0.320707i	$0.896079\pi$
$24$	0	0
$25$	−13.4141	−0.536565
$26$	0.971660i	0.0373715i
$27$	0	0
$28$	31.7833	1.13512
$29$	17.7931i	0.613557i	0.951781	+	0.306778i	$0.0992510\pi$
$29$	17.7931i	0.613557i	−0.951781	+	0.306778i	$0.900749\pi$
$30$	0	0
$31$	13.0209	0.420028	0.210014	−	0.977698i	$-0.432649\pi$
$31$	13.0209	0.420028	0.210014	+	0.977698i	$0.432649\pi$
$32$	− 5.62335i	− 0.175730i
$33$	0	0
$34$	−2.81495	−0.0827926
$35$	− 49.4187i	− 1.41196i
$36$	0	0
$37$	−17.0700	−0.461351	−0.230676	−	0.973031i	$-0.574094\pi$
$37$	−17.0700	−0.461351	−0.230676	+	0.973031i	$0.574094\pi$
$38$	1.44084i	0.0379168i
$39$	0	0
$40$	−5.82566	−0.145641
$41$	30.7358i	0.749654i	0.927095	+	0.374827i	$0.122298\pi$
$41$	30.7358i	0.749654i	−0.927095	+	0.374827i	$0.877702\pi$
$42$	0	0
$43$	−39.8300	−0.926280	−0.463140	−	0.886285i	$-0.653277\pi$
$43$	−39.8300	−0.926280	−0.463140	+	0.886285i	$0.653277\pi$
$44$	− 45.6153i	− 1.03671i
$45$	0	0
$46$	1.73632	0.0377460
$47$	58.3790i	1.24211i	0.783769	+	0.621053i	$0.213295\pi$
$47$	58.3790i	1.24211i	−0.783769	+	0.621053i	$0.786705\pi$
$48$	0	0
$49$	14.5757	0.297464
$50$	1.57879i	0.0315758i
$51$	0	0
$52$	−32.9083	−0.632852
$53$	− 91.2612i	− 1.72191i	−0.508681	−	0.860955i	$-0.669867\pi$
$53$	− 91.2612i	− 1.72191i	0.508681	−	0.860955i	$-0.330133\pi$
$54$	0	0
$55$	−70.9255	−1.28956
$56$	− 7.49454i	− 0.133831i
$57$	0	0
$58$	2.09418	0.0361066
$59$	20.6622i	0.350207i	0.984550	+	0.175104i	$0.0560261\pi$
$59$	20.6622i	0.350207i	−0.984550	+	0.175104i	$0.943974\pi$
$60$	0	0
$61$	−34.9591	−0.573101	−0.286550	−	0.958065i	$-0.592509\pi$
$61$	−34.9591	−0.573101	−0.286550	+	0.958065i	$0.592509\pi$
$62$	− 1.53250i	− 0.0247178i
$63$	0	0
$64$	62.6740	0.979281
$65$	51.1679i	0.787199i
$66$	0	0
$67$	51.5623	0.769587	0.384793	−	0.923003i	$-0.374273\pi$
$67$	51.5623	0.769587	0.384793	+	0.923003i	$0.374273\pi$
$68$	− 95.3370i	− 1.40201i
$69$	0	0
$70$	−5.81639	−0.0830912
$71$	− 2.63480i	− 0.0371099i	−0.999828	−	0.0185549i	$-0.994093\pi$
$71$	− 2.63480i	− 0.0371099i	0.999828	−	0.0185549i	$-0.00590656\pi$
$72$	0	0
$73$	69.0145	0.945403	0.472702	−	0.881222i	$-0.343279\pi$
$73$	69.0145	0.945403	0.472702	+	0.881222i	$0.343279\pi$
$74$	2.00907i	0.0271496i
$75$	0	0
$76$	−48.7986	−0.642086
$77$	− 91.2437i	− 1.18498i
$78$	0	0
$79$	151.435	1.91690	0.958450	−	0.285261i	$-0.0920802\pi$
$79$	151.435	1.91690	0.958450	+	0.285261i	$0.0920802\pi$
$80$	− 98.1375i	− 1.22672i
$81$	0	0
$82$	3.61748	0.0441157
$83$	− 54.0683i	− 0.651425i	−0.945469	−	0.325713i	$-0.894396\pi$
$83$	− 54.0683i	− 0.651425i	0.945469	−	0.325713i	$-0.105604\pi$
$84$	0	0
$85$	−148.236	−1.74395
$86$	4.68784i	0.0545098i
$87$	0	0
$88$	−10.7561	−0.122229
$89$	− 163.072i	− 1.83227i	−0.400868	−	0.916136i	$-0.631292\pi$
$89$	− 163.072i	− 1.83227i	0.400868	−	0.916136i	$-0.368708\pi$
$90$	0	0
$91$	−65.8261	−0.723363
$92$	58.8058i	0.639193i
$93$	0	0
$94$	6.87098	0.0730955
$95$	75.8751i	0.798685i
$96$	0	0
$97$	57.0121	0.587754	0.293877	−	0.955843i	$-0.405054\pi$
$97$	57.0121	0.587754	0.293877	+	0.955843i	$0.405054\pi$
$98$	− 1.71551i	− 0.0175052i
$99$	0	0
$100$	−53.4707	−0.534707
$101$	− 27.7736i	− 0.274986i	−0.990503	−	0.137493i	$-0.956096\pi$
$101$	− 27.7736i	− 0.274986i	0.990503	−	0.137493i	$-0.0439044\pi$
$102$	0	0
$103$	82.0979	0.797067	0.398533	−	0.917154i	$-0.369519\pi$
$103$	82.0979	0.797067	0.398533	+	0.917154i	$0.369519\pi$
$104$	7.75982i	0.0746137i
$105$	0	0
$106$	−10.7411	−0.101331
$107$	35.8639i	0.335177i	0.985857	+	0.167588i	$0.0535980\pi$
$107$	35.8639i	0.335177i	−0.985857	+	0.167588i	$0.946402\pi$
$108$	0	0
$109$	1.41546	0.0129859	0.00649295	−	0.999979i	$-0.497933\pi$
$109$	1.41546	0.0129859	0.00649295	+	0.999979i	$0.497933\pi$
$110$	8.34766i	0.0758878i
$111$	0	0
$112$	126.251	1.12724
$113$	149.032i	1.31887i	0.751762	+	0.659435i	$0.229204\pi$
$113$	149.032i	1.31887i	−0.751762	+	0.659435i	$0.770796\pi$
$114$	0	0
$115$	91.4349	0.795086
$116$	70.9261i	0.611432i
$117$	0	0
$118$	2.43187	0.0206090
$119$	− 190.701i	− 1.60253i
$120$	0	0
$121$	−9.95262	−0.0822531
$122$	4.11455i	0.0337259i
$123$	0	0
$124$	51.9031	0.418573
$125$	− 71.8083i	− 0.574466i
$126$	0	0
$127$	17.1409	0.134968	0.0674838	−	0.997720i	$-0.478503\pi$
$127$	17.1409	0.134968	0.0674838	+	0.997720i	$0.478503\pi$
$128$	− 29.8699i	− 0.233359i
$129$	0	0
$130$	6.02226	0.0463251
$131$	145.928i	1.11396i	0.830527	+	0.556978i	$0.188039\pi$
$131$	145.928i	1.11396i	−0.830527	+	0.556978i	$0.811961\pi$
$132$	0	0
$133$	−97.6111	−0.733918
$134$	− 6.06869i	− 0.0452887i
$135$	0	0
$136$	−22.4806	−0.165298
$137$	− 87.0154i	− 0.635149i	−0.948233	−	0.317574i	$-0.897132\pi$
$137$	− 87.0154i	− 0.635149i	0.948233	−	0.317574i	$-0.102868\pi$
$138$	0	0
$139$	36.8609	0.265187	0.132593	−	0.991171i	$-0.457670\pi$
$139$	36.8609	0.265187	0.132593	+	0.991171i	$0.457670\pi$
$140$	− 196.990i	− 1.40707i
$141$	0	0
$142$	−0.310106	−0.00218384
$143$	94.4733i	0.660653i
$144$	0	0
$145$	110.280	0.760554
$146$	− 8.12273i	− 0.0556352i
$147$	0	0
$148$	−68.0436	−0.459754
$149$	64.5972i	0.433538i	0.976223	+	0.216769i	$0.0695519\pi$
$149$	64.5972i	0.433538i	−0.976223	+	0.216769i	$0.930448\pi$
$150$	0	0
$151$	−154.456	−1.02289	−0.511445	−	0.859316i	$-0.670890\pi$
$151$	−154.456	−1.02289	−0.511445	+	0.859316i	$0.670890\pi$
$152$	11.5068i	0.0757024i
$153$	0	0
$154$	−10.7390	−0.0697339
$155$	− 80.7021i	− 0.520659i
$156$	0	0
$157$	−192.554	−1.22646	−0.613229	−	0.789905i	$-0.710130\pi$
$157$	−192.554	−1.22646	−0.613229	+	0.789905i	$0.710130\pi$
$158$	− 17.8233i	− 0.112806i
$159$	0	0
$160$	−34.8530	−0.217831
$161$	117.628i	0.730611i
$162$	0	0
$163$	171.309	1.05098	0.525488	−	0.850801i	$-0.323883\pi$
$163$	171.309	1.05098	0.525488	+	0.850801i	$0.323883\pi$
$164$	122.517i	0.747057i
$165$	0	0
$166$	−6.36363	−0.0383351
$167$	− 157.383i	− 0.942416i	−0.882022	−	0.471208i	$-0.843818\pi$
$167$	− 157.383i	− 0.942416i	0.882022	−	0.471208i	$-0.156182\pi$
$168$	0	0
$169$	−100.844	−0.596710
$170$	17.4468i	0.102628i
$171$	0	0
$172$	−158.768	−0.923072
$173$	52.5050i	0.303497i	0.988419	+	0.151749i	$0.0484904\pi$
$173$	52.5050i	0.303497i	−0.988419	+	0.151749i	$0.951510\pi$
$174$	0	0
$175$	−106.957	−0.611181
$176$	− 181.195i	− 1.02952i
$177$	0	0
$178$	−19.1930	−0.107826
$179$	66.9393i	0.373963i	0.982363	+	0.186981i	$0.0598704\pi$
$179$	66.9393i	0.373963i	−0.982363	+	0.186981i	$0.940130\pi$
$180$	0	0
$181$	−129.459	−0.715242	−0.357621	−	0.933867i	$-0.616412\pi$
$181$	−129.459	−0.715242	−0.357621	+	0.933867i	$0.616412\pi$
$182$	7.74747i	0.0425685i
$183$	0	0
$184$	13.8665	0.0753613
$185$	105.798i	0.571883i
$186$	0	0
$187$	−273.694	−1.46360
$188$	232.707i	1.23780i
$189$	0	0
$190$	8.93020	0.0470011
$191$	271.347i	1.42066i	0.703867	+	0.710332i	$0.251455\pi$
$191$	271.347i	1.42066i	−0.703867	+	0.710332i	$0.748545\pi$
$192$	0	0
$193$	207.484	1.07505	0.537524	−	0.843249i	$-0.319360\pi$
$193$	207.484	1.07505	0.537524	+	0.843249i	$0.319360\pi$
$194$	− 6.71011i	− 0.0345882i
$195$	0	0
$196$	58.1010	0.296434
$197$	178.829i	0.907763i	0.891062	+	0.453881i	$0.149961\pi$
$197$	178.829i	0.907763i	−0.891062	+	0.453881i	$0.850039\pi$
$198$	0	0
$199$	−25.6080	−0.128684	−0.0643418	−	0.997928i	$-0.520495\pi$
$199$	−25.6080	−0.128684	−0.0643418	+	0.997928i	$0.520495\pi$
$200$	12.6085i	0.0630423i
$201$	0	0
$202$	−3.26884	−0.0161824
$203$	141.873i	0.698879i
$204$	0	0
$205$	190.498	0.929257
$206$	− 9.66260i	− 0.0469058i
$207$	0	0
$208$	−130.720	−0.628461
$209$	140.091i	0.670293i
$210$	0	0
$211$	360.286	1.70752	0.853758	−	0.520671i	$-0.174318\pi$
$211$	360.286	1.70752	0.853758	+	0.520671i	$0.174318\pi$
$212$	− 363.781i	− 1.71595i
$213$	0	0
$214$	4.22104	0.0197245
$215$	246.863i	1.14820i
$216$	0	0
$217$	103.821	0.478438
$218$	− 0.166595i	0 0.000764195i
$219$	0	0
$220$	−282.720	−1.28509
$221$	197.451i	0.893445i
$222$	0	0
$223$	8.98641	0.0402978	0.0201489	−	0.999797i	$-0.493586\pi$
$223$	8.98641	0.0402978	0.0201489	+	0.999797i	$0.493586\pi$
$224$	− 44.8374i	− 0.200167i
$225$	0	0
$226$	17.5405	0.0776129
$227$	71.7800i	0.316212i	0.987422	+	0.158106i	$0.0505387\pi$
$227$	71.7800i	0.316212i	−0.987422	+	0.158106i	$0.949461\pi$
$228$	0	0
$229$	53.9366	0.235531	0.117765	−	0.993041i	$-0.462427\pi$
$229$	53.9366	0.235531	0.117765	+	0.993041i	$0.462427\pi$
$230$	− 10.7615i	− 0.0467893i
$231$	0	0
$232$	16.7245	0.0720882
$233$	362.666i	1.55651i	0.627950	+	0.778254i	$0.283894\pi$
$233$	362.666i	1.55651i	−0.627950	+	0.778254i	$0.716106\pi$
$234$	0	0
$235$	361.828	1.53969
$236$	82.3627i	0.348995i
$237$	0	0
$238$	−22.4448	−0.0943059
$239$	251.571i	1.05260i	0.850299	+	0.526299i	$0.176421\pi$
$239$	251.571i	1.05260i	−0.850299	+	0.526299i	$0.823579\pi$
$240$	0	0
$241$	402.902	1.67179	0.835897	−	0.548886i	$-0.184948\pi$
$241$	402.902	1.67179	0.835897	+	0.548886i	$0.184948\pi$
$242$	1.17139i	0.00484043i
$243$	0	0
$244$	−139.352	−0.571116
$245$	− 90.3391i	− 0.368731i
$246$	0	0
$247$	101.066	0.409175
$248$	− 12.2388i	− 0.0493500i
$249$	0	0
$250$	−8.45155	−0.0338062
$251$	380.219i	1.51482i	0.652941	+	0.757409i	$0.273535\pi$
$251$	380.219i	1.51482i	−0.652941	+	0.757409i	$0.726465\pi$
$252$	0	0
$253$	168.820	0.667272
$254$	− 2.01742i	− 0.00794258i
$255$	0	0
$256$	247.180	0.965549
$257$	− 21.6885i	− 0.0843912i	−0.999109	−	0.0421956i	$-0.986565\pi$
$257$	− 21.6885i	− 0.0843912i	0.999109	−	0.0421956i	$-0.0134353\pi$
$258$	0	0
$259$	−136.107	−0.525508
$260$	203.963i	0.784472i
$261$	0	0
$262$	17.1752	0.0655542
$263$	− 67.8776i	− 0.258090i	−0.991639	−	0.129045i	$-0.958809\pi$
$263$	− 67.8776i	− 0.258090i	0.991639	−	0.129045i	$-0.0411911\pi$
$264$	0	0
$265$	−565.629	−2.13445
$266$	11.4885i	0.0431897i
$267$	0	0
$268$	205.535	0.766922
$269$	− 290.581i	− 1.08023i	−0.841592	−	0.540113i	$-0.818381\pi$
$269$	− 290.581i	− 1.08023i	0.841592	−	0.540113i	$-0.181619\pi$
$270$	0	0
$271$	−414.644	−1.53005	−0.765026	−	0.644000i	$-0.777274\pi$
$271$	−414.644	−1.53005	−0.765026	+	0.644000i	$0.777274\pi$
$272$	− 378.702i	− 1.39229i
$273$	0	0
$274$	−10.2414	−0.0373773
$275$	153.504i	0.558196i
$276$	0	0
$277$	175.130	0.632237	0.316118	−	0.948720i	$-0.397620\pi$
$277$	175.130	0.632237	0.316118	+	0.948720i	$0.397620\pi$
$278$	− 4.33839i	− 0.0156057i
$279$	0	0
$280$	−46.4505	−0.165895
$281$	375.263i	1.33545i	0.744406	+	0.667727i	$0.232733\pi$
$281$	375.263i	1.33545i	−0.744406	+	0.667727i	$0.767267\pi$
$282$	0	0
$283$	14.7766	0.0522141	0.0261070	−	0.999659i	$-0.491689\pi$
$283$	14.7766	0.0522141	0.0261070	+	0.999659i	$0.491689\pi$
$284$	− 10.5027i	− 0.0369814i
$285$	0	0
$286$	11.1191	0.0388781
$287$	245.070i	0.853902i
$288$	0	0
$289$	−283.027	−0.979331
$290$	− 12.9796i	− 0.0447571i
$291$	0	0
$292$	275.102	0.942129
$293$	132.946i	0.453740i	0.973925	+	0.226870i	$0.0728493\pi$
$293$	132.946i	0.453740i	−0.973925	+	0.226870i	$0.927151\pi$
$294$	0	0
$295$	128.063	0.434111
$296$	16.0448i	0.0542053i
$297$	0	0
$298$	7.60284	0.0255129
$299$	− 121.792i	− 0.407331i
$300$	0	0
$301$	−317.582	−1.05509
$302$	18.1789i	0.0601951i
$303$	0	0
$304$	−193.840	−0.637631
$305$	216.674i	0.710405i
$306$	0	0
$307$	70.5177	0.229699	0.114850	−	0.993383i	$-0.463361\pi$
$307$	70.5177	0.229699	0.114850	+	0.993383i	$0.463361\pi$
$308$	− 363.711i	− 1.18088i
$309$	0	0
$310$	−9.49833	−0.0306398
$311$	60.9783i	0.196072i	0.995183	+	0.0980359i	$0.0312560\pi$
$311$	60.9783i	0.196072i	−0.995183	+	0.0980359i	$0.968744\pi$
$312$	0	0
$313$	211.333	0.675184	0.337592	−	0.941292i	$-0.390388\pi$
$313$	211.333	0.675184	0.337592	+	0.941292i	$0.390388\pi$
$314$	22.6629i	0.0721747i
$315$	0	0
$316$	603.643	1.91026
$317$	133.790i	0.422050i	0.977481	+	0.211025i	$0.0676801\pi$
$317$	133.790i	0.422050i	−0.977481	+	0.211025i	$0.932320\pi$
$318$	0	0
$319$	203.615	0.638291
$320$	− 388.448i	− 1.21390i
$321$	0	0
$322$	13.8444	0.0429950
$323$	292.794i	0.906482i
$324$	0	0
$325$	110.743	0.340746
$326$	− 20.1624i	− 0.0618479i
$327$	0	0
$328$	28.8898	0.0880785
$329$	465.481i	1.41484i
$330$	0	0
$331$	−88.2013	−0.266469	−0.133235	−	0.991085i	$-0.542536\pi$
$331$	−88.2013	−0.266469	−0.133235	+	0.991085i	$0.542536\pi$
$332$	− 215.524i	− 0.649169i
$333$	0	0
$334$	−18.5234	−0.0554593
$335$	− 319.579i	− 0.953967i
$336$	0	0
$337$	41.1842	0.122208	0.0611042	−	0.998131i	$-0.480538\pi$
$337$	41.1842	0.122208	0.0611042	+	0.998131i	$0.480538\pi$
$338$	11.8689i	0.0351152i
$339$	0	0
$340$	−590.890	−1.73791
$341$	− 149.004i	− 0.436961i
$342$	0	0
$343$	−274.480	−0.800233
$344$	37.4378i	0.108831i
$345$	0	0
$346$	6.17964	0.0178602
$347$	− 342.939i	− 0.988297i	−0.869378	−	0.494148i	$-0.835480\pi$
$347$	− 342.939i	− 0.988297i	0.869378	−	0.494148i	$-0.164520\pi$
$348$	0	0
$349$	−589.249	−1.68839	−0.844196	−	0.536035i	$-0.819922\pi$
$349$	−589.249	−1.68839	−0.844196	+	0.536035i	$0.819922\pi$
$350$	12.5884i	0.0359668i
$351$	0	0
$352$	−64.3505	−0.182814
$353$	− 126.001i	− 0.356943i	−0.983945	−	0.178472i	$-0.942885\pi$
$353$	− 126.001i	− 0.356943i	0.983945	−	0.178472i	$-0.0571153\pi$
$354$	0	0
$355$	−16.3303	−0.0460008
$356$	− 650.030i	− 1.82593i
$357$	0	0
$358$	7.87850	0.0220070
$359$	− 344.484i	− 0.959565i	−0.877388	−	0.479782i	$-0.840716\pi$
$359$	− 344.484i	− 0.959565i	0.877388	−	0.479782i	$-0.159284\pi$
$360$	0	0
$361$	−211.133	−0.584855
$362$	15.2368i	0.0420906i
$363$	0	0
$364$	−262.392	−0.720858
$365$	− 427.746i	− 1.17191i
$366$	0	0
$367$	337.725	0.920230	0.460115	−	0.887859i	$-0.347808\pi$
$367$	337.725	0.920230	0.460115	+	0.887859i	$0.347808\pi$
$368$	233.591i	0.634758i
$369$	0	0
$370$	12.4521	0.0336542
$371$	− 727.666i	− 1.96136i
$372$	0	0
$373$	99.1007	0.265685	0.132843	−	0.991137i	$-0.457589\pi$
$373$	99.1007	0.265685	0.132843	+	0.991137i	$0.457589\pi$
$374$	32.2127i	0.0861303i
$375$	0	0
$376$	54.8726	0.145938
$377$	− 146.894i	− 0.389640i
$378$	0	0
$379$	599.859	1.58274	0.791370	−	0.611337i	$-0.209368\pi$
$379$	599.859	1.58274	0.791370	+	0.611337i	$0.209368\pi$
$380$	302.449i	0.795919i
$381$	0	0
$382$	31.9365	0.0836033
$383$	− 113.665i	− 0.296776i	−0.988929	−	0.148388i	$-0.952592\pi$
$383$	− 113.665i	− 0.296776i	0.988929	−	0.148388i	$-0.0474084\pi$
$384$	0	0
$385$	−565.520	−1.46888
$386$	− 24.4201i	− 0.0632644i
$387$	0	0
$388$	227.259	0.585718
$389$	362.878i	0.932850i	0.884561	+	0.466425i	$0.154458\pi$
$389$	362.878i	0.932850i	−0.884561	+	0.466425i	$0.845542\pi$
$390$	0	0
$391$	352.837	0.902397
$392$	− 13.7003i	− 0.0349497i
$393$	0	0
$394$	21.0475	0.0534201
$395$	− 938.581i	− 2.37616i
$396$	0	0
$397$	−478.416	−1.20508	−0.602539	−	0.798089i	$-0.705844\pi$
$397$	−478.416	−1.20508	−0.602539	+	0.798089i	$0.705844\pi$
$398$	3.01397i	0.00757278i
$399$	0	0
$400$	−212.399	−0.530997
$401$	638.015i	1.59106i	0.605914	+	0.795530i	$0.292807\pi$
$401$	638.015i	1.59106i	−0.605914	+	0.795530i	$0.707193\pi$
$402$	0	0
$403$	−107.496	−0.266739
$404$	− 110.709i	− 0.274033i
$405$	0	0
$406$	16.6978	0.0411277
$407$	195.340i	0.479950i
$408$	0	0
$409$	612.478	1.49750	0.748750	−	0.662852i	$-0.230654\pi$
$409$	612.478	1.49750	0.748750	+	0.662852i	$0.230654\pi$
$410$	− 22.4208i	− 0.0546850i
$411$	0	0
$412$	327.254	0.794306
$413$	164.749i	0.398908i
$414$	0	0
$415$	−335.111	−0.807495
$416$	46.4245i	0.111597i
$417$	0	0
$418$	16.4882	0.0394454
$419$	524.631i	1.25210i	0.779782	+	0.626051i	$0.215330\pi$
$419$	524.631i	1.25210i	−0.779782	+	0.626051i	$0.784670\pi$
$420$	0	0
$421$	−368.263	−0.874733	−0.437366	−	0.899283i	$-0.644089\pi$
$421$	−368.263	−0.874733	−0.437366	+	0.899283i	$0.644089\pi$
$422$	− 42.4042i	− 0.100484i
$423$	0	0
$424$	−85.7799	−0.202311
$425$	320.827i	0.754886i
$426$	0	0
$427$	−278.745	−0.652797
$428$	142.959i	0.334016i
$429$	0	0
$430$	29.0548	0.0675693
$431$	− 178.021i	− 0.413043i	−0.978442	−	0.206521i	$-0.933786\pi$
$431$	− 178.021i	− 0.413043i	0.978442	−	0.206521i	$-0.0662143\pi$
$432$	0	0
$433$	−710.746	−1.64144	−0.820722	−	0.571327i	$-0.806429\pi$
$433$	−710.746	−1.64144	−0.820722	+	0.571327i	$0.806429\pi$
$434$	− 12.2193i	− 0.0281551i
$435$	0	0
$436$	5.64225	0.0129409
$437$	− 180.601i	− 0.413275i
$438$	0	0
$439$	636.795	1.45056	0.725279	−	0.688455i	$-0.241711\pi$
$439$	636.795	1.45056	0.725279	+	0.688455i	$0.241711\pi$
$440$	66.6656i	0.151513i
$441$	0	0
$442$	23.2393	0.0525775
$443$	− 322.865i	− 0.728815i	−0.931240	−	0.364407i	$-0.881272\pi$
$443$	− 322.865i	− 0.728815i	0.931240	−	0.364407i	$-0.118728\pi$
$444$	0	0
$445$	−1010.71	−2.27125
$446$	− 1.05767i	− 0.00237145i
$447$	0	0
$448$	499.727	1.11546
$449$	− 25.8984i	− 0.0576803i	−0.999584	−	0.0288401i	$-0.990819\pi$
$449$	− 25.8984i	− 0.0576803i	0.999584	−	0.0288401i	$-0.00918138\pi$
$450$	0	0
$451$	351.724	0.779875
$452$	594.065i	1.31430i
$453$	0	0
$454$	8.44823	0.0186084
$455$	407.984i	0.896669i
$456$	0	0
$457$	−395.461	−0.865341	−0.432671	−	0.901552i	$-0.642429\pi$
$457$	−395.461	−0.865341	−0.432671	+	0.901552i	$0.642429\pi$
$458$	− 6.34813i	− 0.0138605i
$459$	0	0
$460$	364.473	0.792332
$461$	896.893i	1.94554i	0.231776	+	0.972769i	$0.425546\pi$
$461$	896.893i	1.94554i	−0.231776	+	0.972769i	$0.574454\pi$
$462$	0	0
$463$	−297.598	−0.642760	−0.321380	−	0.946950i	$-0.604147\pi$
$463$	−297.598	−0.642760	−0.321380	+	0.946950i	$0.604147\pi$
$464$	281.736i	0.607189i
$465$	0	0
$466$	42.6844	0.0915974
$467$	− 790.347i	− 1.69239i	−0.532872	−	0.846196i	$-0.678887\pi$
$467$	− 790.347i	− 1.69239i	0.532872	−	0.846196i	$-0.321113\pi$
$468$	0	0
$469$	411.129	0.876608
$470$	− 42.5857i	− 0.0906079i
$471$	0	0
$472$	19.4212	0.0411467
$473$	455.793i	0.963621i
$474$	0	0
$475$	164.216	0.345718
$476$	− 760.164i	− 1.59698i
$477$	0	0
$478$	29.6089	0.0619434
$479$	− 1.08180i	− 0.00225845i	−0.999999	−	0.00112922i	$-0.999641\pi$
$479$	− 1.08180i	− 0.00225845i	0.999999	−	0.00112922i	$-0.000359443\pi$
$480$	0	0
$481$	140.924	0.292982
$482$	− 47.4201i	− 0.0983819i
$483$	0	0
$484$	−39.6726	−0.0819682
$485$	− 353.356i	− 0.728569i
$486$	0	0
$487$	647.606	1.32979	0.664893	−	0.746939i	$-0.268477\pi$
$487$	647.606	1.32979	0.664893	+	0.746939i	$0.268477\pi$
$488$	32.8594i	0.0673349i
$489$	0	0
$490$	−10.6326	−0.0216991
$491$	− 312.314i	− 0.636078i	−0.948078	−	0.318039i	$-0.896976\pi$
$491$	− 312.314i	− 0.636078i	0.948078	−	0.318039i	$-0.103024\pi$
$492$	0	0
$493$	425.560	0.863205
$494$	− 11.8951i	− 0.0240791i
$495$	0	0
$496$	206.172	0.415669
$497$	− 21.0084i	− 0.0422705i
$498$	0	0
$499$	45.8927	0.0919694	0.0459847	−	0.998942i	$-0.485357\pi$
$499$	45.8927	0.0919694	0.0459847	+	0.998942i	$0.485357\pi$
$500$	− 286.238i	− 0.572477i
$501$	0	0
$502$	44.7503	0.0891441
$503$	− 570.225i	− 1.13365i	−0.823839	−	0.566824i	$-0.808172\pi$
$503$	− 570.225i	− 1.13365i	0.823839	−	0.566824i	$-0.191828\pi$
$504$	0	0
$505$	−172.138	−0.340867
$506$	− 19.8694i	− 0.0392677i
$507$	0	0
$508$	68.3261	0.134500
$509$	− 520.078i	− 1.02176i	−0.859651	−	0.510882i	$-0.829319\pi$
$509$	− 520.078i	− 1.02176i	0.859651	−	0.510882i	$-0.170681\pi$
$510$	0	0
$511$	550.283	1.07687
$512$	− 148.572i	− 0.290179i
$513$	0	0
$514$	−2.55266	−0.00496626
$515$	− 508.835i	− 0.988030i
$516$	0	0
$517$	668.057	1.29218
$518$	16.0192i	0.0309251i
$519$	0	0
$520$	48.0947	0.0924898
$521$	− 422.296i	− 0.810549i	−0.914195	−	0.405275i	$-0.867176\pi$
$521$	− 422.296i	− 0.810549i	0.914195	−	0.405275i	$-0.132824\pi$
$522$	0	0
$523$	−1001.06	−1.91407	−0.957035	−	0.289972i	$-0.906354\pi$
$523$	−1001.06	−1.91407	−0.957035	+	0.289972i	$0.906354\pi$
$524$	581.692i	1.11010i
$525$	0	0
$526$	−7.98893	−0.0151881
$527$	− 311.421i	− 0.590931i
$528$	0	0
$529$	311.363	0.588588
$530$	66.5723i	0.125608i
$531$	0	0
$532$	−389.092	−0.731377
$533$	− 253.745i	− 0.476068i
$534$	0	0
$535$	222.281	0.415479
$536$	− 48.4654i	− 0.0904206i
$537$	0	0
$538$	−34.2003	−0.0635692
$539$	− 166.797i	− 0.309456i
$540$	0	0
$541$	−192.818	−0.356410	−0.178205	−	0.983993i	$-0.557029\pi$
$541$	−192.818	−0.356410	−0.178205	+	0.983993i	$0.557029\pi$
$542$	48.8020i	0.0900406i
$543$	0	0
$544$	−134.494	−0.247232
$545$	− 8.77292i	− 0.0160971i
$546$	0	0
$547$	−755.590	−1.38134	−0.690668	−	0.723172i	$-0.742683\pi$
$547$	−755.590	−1.38134	−0.690668	+	0.723172i	$0.742683\pi$
$548$	− 346.856i	− 0.632949i
$549$	0	0
$550$	18.0668	0.0328487
$551$	− 217.824i	− 0.395325i
$552$	0	0
$553$	1207.46	2.18347
$554$	− 20.6121i	− 0.0372059i
$555$	0	0
$556$	146.933	0.264268
$557$	855.844i	1.53652i	0.640135	+	0.768262i	$0.278878\pi$
$557$	855.844i	1.53652i	−0.640135	+	0.768262i	$0.721122\pi$
$558$	0	0
$559$	328.823	0.588235
$560$	− 782.494i	− 1.39731i
$561$	0	0
$562$	44.1670	0.0785889
$563$	− 155.756i	− 0.276653i	−0.990387	−	0.138327i	$-0.955828\pi$
$563$	− 155.756i	− 0.276653i	0.990387	−	0.138327i	$-0.0441723\pi$
$564$	0	0
$565$	923.689	1.63485
$566$	− 1.73915i	− 0.00307270i
$567$	0	0
$568$	−2.47655	−0.00436013
$569$	− 636.016i	− 1.11778i	−0.829242	−	0.558889i	$-0.811228\pi$
$569$	− 636.016i	− 1.11778i	0.829242	−	0.558889i	$-0.188772\pi$
$570$	0	0
$571$	−1004.18	−1.75863	−0.879317	−	0.476236i	$-0.842001\pi$
$571$	−1004.18	−1.75863	−0.879317	+	0.476236i	$0.842001\pi$
$572$	376.585i	0.658365i
$573$	0	0
$574$	28.8438	0.0502505
$575$	− 197.892i	− 0.344160i
$576$	0	0
$577$	829.817	1.43816	0.719079	−	0.694929i	$-0.244564\pi$
$577$	829.817	1.43816	0.719079	+	0.694929i	$0.244564\pi$
$578$	33.3111i	0.0576317i
$579$	0	0
$580$	439.594	0.757920
$581$	− 431.110i	− 0.742014i
$582$	0	0
$583$	−1044.34	−1.79133
$584$	− 64.8693i	− 0.111078i
$585$	0	0
$586$	15.6472	0.0267017
$587$	293.552i	0.500089i	0.968234	+	0.250044i	$0.0804452\pi$
$587$	293.552i	0.500089i	−0.968234	+	0.250044i	$0.919555\pi$
$588$	0	0
$589$	−159.402	−0.270631
$590$	− 15.0725i	− 0.0255466i
$591$	0	0
$592$	−270.286	−0.456564
$593$	− 720.027i	− 1.21421i	−0.794621	−	0.607106i	$-0.792330\pi$
$593$	− 720.027i	− 1.21421i	0.794621	−	0.607106i	$-0.207670\pi$
$594$	0	0
$595$	−1181.95	−1.98647
$596$	257.494i	0.432037i
$597$	0	0
$598$	−14.3344	−0.0239706
$599$	463.470i	0.773740i	0.922134	+	0.386870i	$0.126444\pi$
$599$	463.470i	0.773740i	−0.922134	+	0.386870i	$0.873556\pi$
$600$	0	0
$601$	−545.070	−0.906939	−0.453469	−	0.891272i	$-0.649814\pi$
$601$	−545.070	−0.906939	−0.453469	+	0.891272i	$0.649814\pi$
$602$	37.3782i	0.0620900i
$603$	0	0
$604$	−615.686	−1.01935
$605$	61.6855i	0.101959i
$606$	0	0
$607$	−1019.29	−1.67922	−0.839609	−	0.543191i	$-0.817216\pi$
$607$	−1019.29	−1.67922	−0.839609	+	0.543191i	$0.817216\pi$
$608$	68.8413i	0.113226i
$609$	0	0
$610$	25.5016	0.0418060
$611$	− 481.957i	− 0.788801i
$612$	0	0
$613$	993.926	1.62141	0.810706	−	0.585453i	$-0.199083\pi$
$613$	993.926	1.62141	0.810706	+	0.585453i	$0.199083\pi$
$614$	− 8.29966i	− 0.0135174i
$615$	0	0
$616$	−85.7635	−0.139226
$617$	− 944.535i	− 1.53085i	−0.643524	−	0.765426i	$-0.722528\pi$
$617$	− 944.535i	− 1.53085i	0.643524	−	0.765426i	$-0.277472\pi$
$618$	0	0
$619$	−469.213	−0.758018	−0.379009	−	0.925393i	$-0.623735\pi$
$619$	−469.213	−0.758018	−0.379009	+	0.925393i	$0.623735\pi$
$620$	− 321.691i	− 0.518856i
$621$	0	0
$622$	7.17691	0.0115384
$623$	− 1300.25i	− 2.08707i
$624$	0	0
$625$	−780.414	−1.24866
$626$	− 24.8730i	− 0.0397333i
$627$	0	0
$628$	−767.549	−1.22221
$629$	408.265i	0.649069i
$630$	0	0
$631$	1020.56	1.61738	0.808689	−	0.588237i	$-0.200178\pi$
$631$	1020.56	1.61738	0.808689	+	0.588237i	$0.200178\pi$
$632$	− 142.340i	− 0.225221i
$633$	0	0
$634$	15.7465	0.0248368
$635$	− 106.238i	− 0.167304i
$636$	0	0
$637$	−120.332	−0.188905
$638$	− 23.9647i	− 0.0375622i
$639$	0	0
$640$	−185.131	−0.289267
$641$	1125.61i	1.75603i	0.478637	+	0.878013i	$0.341131\pi$
$641$	1125.61i	1.75603i	−0.478637	+	0.878013i	$0.658869\pi$
$642$	0	0
$643$	−213.518	−0.332065	−0.166033	−	0.986120i	$-0.553096\pi$
$643$	−213.518	−0.332065	−0.166033	+	0.986120i	$0.553096\pi$
$644$	468.884i	0.728081i
$645$	0	0
$646$	34.4607	0.0533447
$647$	− 267.943i	− 0.414132i	−0.978327	−	0.207066i	$-0.933609\pi$
$647$	− 267.943i	− 0.414132i	0.978327	−	0.207066i	$-0.0663914\pi$
$648$	0	0
$649$	236.447	0.364326
$650$	− 13.0340i	− 0.0200523i
$651$	0	0
$652$	682.864	1.04734
$653$	− 53.1648i	− 0.0814162i	−0.999171	−	0.0407081i	$-0.987039\pi$
$653$	− 53.1648i	− 0.0814162i	0.999171	−	0.0407081i	$-0.0129614\pi$
$654$	0	0
$655$	904.451	1.38084
$656$	486.669i	0.741874i
$657$	0	0
$658$	54.7853	0.0832604
$659$	− 501.397i	− 0.760845i	−0.924813	−	0.380423i	$-0.875779\pi$
$659$	− 501.397i	− 0.760845i	0.924813	−	0.380423i	$-0.124221\pi$
$660$	0	0
$661$	580.522	0.878249	0.439124	−	0.898426i	$-0.355289\pi$
$661$	580.522	0.878249	0.439124	+	0.898426i	$0.355289\pi$
$662$	10.3810i	0.0156812i
$663$	0	0
$664$	−50.8209	−0.0765375
$665$	604.985i	0.909752i
$666$	0	0
$667$	−262.494	−0.393544
$668$	− 627.354i	− 0.939152i
$669$	0	0
$670$	−37.6132	−0.0561391
$671$	400.053i	0.596204i
$672$	0	0
$673$	−119.109	−0.176982	−0.0884912	−	0.996077i	$-0.528205\pi$
$673$	−119.109	−0.176982	−0.0884912	+	0.996077i	$0.528205\pi$
$674$	− 4.84723i	− 0.00719173i
$675$	0	0
$676$	−401.979	−0.594643
$677$	467.691i	0.690828i	0.938450	+	0.345414i	$0.112262\pi$
$677$	467.691i	0.690828i	−0.938450	+	0.345414i	$0.887738\pi$
$678$	0	0
$679$	454.583	0.669489
$680$	139.333i	0.204901i
$681$	0	0
$682$	−17.5371	−0.0257143
$683$	− 313.488i	− 0.458987i	−0.973310	−	0.229494i	$-0.926293\pi$
$683$	− 313.488i	− 0.458987i	0.973310	−	0.229494i	$-0.0737070\pi$
$684$	0	0
$685$	−539.314	−0.787319
$686$	32.3052i	0.0470921i
$687$	0	0
$688$	−630.667	−0.916667
$689$	753.422i	1.09350i
$690$	0	0
$691$	−672.604	−0.973378	−0.486689	−	0.873575i	$-0.661795\pi$
$691$	−672.604	−0.973378	−0.486689	+	0.873575i	$0.661795\pi$
$692$	209.293i	0.302446i
$693$	0	0
$694$	−40.3626	−0.0581594
$695$	− 228.461i	− 0.328721i
$696$	0	0
$697$	735.110	1.05468
$698$	69.3523i	0.0993586i
$699$	0	0
$700$	−426.345	−0.609065
$701$	906.580i	1.29327i	0.762801	+	0.646633i	$0.223824\pi$
$701$	906.580i	1.29327i	−0.762801	+	0.646633i	$0.776176\pi$
$702$	0	0
$703$	208.972	0.297257
$704$	− 717.207i	− 1.01876i
$705$	0	0
$706$	−14.8298	−0.0210054
$707$	− 221.451i	− 0.313226i
$708$	0	0
$709$	−29.0415	−0.0409612	−0.0204806	−	0.999790i	$-0.506520\pi$
$709$	−29.0415	−0.0409612	−0.0204806	+	0.999790i	$0.506520\pi$
$710$	1.92201i	0.00270706i
$711$	0	0
$712$	−153.278	−0.215278
$713$	192.091i	0.269412i
$714$	0	0
$715$	585.538	0.818934
$716$	266.830i	0.372668i
$717$	0	0
$718$	−40.5444	−0.0564685
$719$	966.847i	1.34471i	0.740229	+	0.672355i	$0.234717\pi$
$719$	966.847i	1.34471i	−0.740229	+	0.672355i	$0.765283\pi$
$720$	0	0
$721$	654.602	0.907909
$722$	24.8495i	0.0344176i
$723$	0	0
$724$	−516.042	−0.712765
$725$	− 238.679i	− 0.329213i
$726$	0	0
$727$	877.362	1.20682	0.603412	−	0.797429i	$-0.293807\pi$
$727$	877.362	1.20682	0.603412	+	0.797429i	$0.293807\pi$
$728$	61.8725i	0.0849896i
$729$	0	0
$730$	−50.3440	−0.0689644
$731$	952.618i	1.30317i
$732$	0	0
$733$	36.1953	0.0493796	0.0246898	−	0.999695i	$-0.492140\pi$
$733$	36.1953	0.0493796	0.0246898	+	0.999695i	$0.492140\pi$
$734$	− 39.7489i	− 0.0541538i
$735$	0	0
$736$	82.9586	0.112716
$737$	− 590.051i	− 0.800612i
$738$	0	0
$739$	−475.629	−0.643612	−0.321806	−	0.946806i	$-0.604290\pi$
$739$	−475.629	−0.643612	−0.321806	+	0.946806i	$0.604290\pi$
$740$	421.728i	0.569903i
$741$	0	0
$742$	−85.6434	−0.115422
$743$	50.8826i	0.0684827i	0.999414	+	0.0342413i	$0.0109015\pi$
$743$	50.8826i	0.0684827i	−0.999414	+	0.0342413i	$0.989099\pi$
$744$	0	0
$745$	400.368	0.537407
$746$	− 11.6638i	− 0.0156351i
$747$	0	0
$748$	−1090.98	−1.45854
$749$	285.959i	0.381787i
$750$	0	0
$751$	47.1961	0.0628444	0.0314222	−	0.999506i	$-0.489996\pi$
$751$	47.1961	0.0628444	0.0314222	+	0.999506i	$0.489996\pi$
$752$	924.370i	1.22922i
$753$	0	0
$754$	−17.2889	−0.0229296
$755$	957.307i	1.26796i
$756$	0	0
$757$	973.584	1.28611	0.643054	−	0.765821i	$-0.277667\pi$
$757$	973.584	1.28611	0.643054	+	0.765821i	$0.277667\pi$
$758$	− 70.6010i	− 0.0931412i
$759$	0	0
$760$	71.3179	0.0938393
$761$	16.3062i	0.0214274i	0.999943	+	0.0107137i	$0.00341034\pi$
$761$	16.3062i	0.0214274i	−0.999943	+	0.0107137i	$0.996590\pi$
$762$	0	0
$763$	11.2861	0.0147918
$764$	1081.63i	1.41574i
$765$	0	0
$766$	−13.3779	−0.0174647
$767$	− 170.581i	− 0.222400i
$768$	0	0
$769$	142.031	0.184695	0.0923477	−	0.995727i	$-0.470563\pi$
$769$	142.031	0.184695	0.0923477	+	0.995727i	$0.470563\pi$
$770$	66.5596i	0.0864410i
$771$	0	0
$772$	827.062	1.07132
$773$	− 1144.42i	− 1.48050i	−0.672334	−	0.740248i	$-0.734708\pi$
$773$	− 1144.42i	− 1.48050i	0.672334	−	0.740248i	$-0.265292\pi$
$774$	0	0
$775$	−174.663	−0.225372
$776$	− 53.5879i	− 0.0690566i
$777$	0	0
$778$	42.7094	0.0548964
$779$	− 376.269i	− 0.483015i
$780$	0	0
$781$	−30.1512	−0.0386059
$782$	− 41.5276i	− 0.0531043i
$783$	0	0
$784$	230.792	0.294377
$785$	1193.43i	1.52030i
$786$	0	0
$787$	−259.179	−0.329325	−0.164663	−	0.986350i	$-0.552654\pi$
$787$	−259.179	−0.329325	−0.164663	+	0.986350i	$0.552654\pi$
$788$	712.840i	0.904619i
$789$	0	0
$790$	−110.467	−0.139832
$791$	1188.30i	1.50228i
$792$	0	0
$793$	288.611	0.363948
$794$	56.3077i	0.0709165i
$795$	0	0
$796$	−102.077	−0.128238
$797$	− 1061.42i	− 1.33177i	−0.746055	−	0.665885i	$-0.768054\pi$
$797$	− 1061.42i	− 1.33177i	0.746055	−	0.665885i	$-0.231946\pi$
$798$	0	0
$799$	1396.25	1.74750
$800$	75.4323i	0.0942904i
$801$	0	0
$802$	75.0919	0.0936308
$803$	− 789.764i	− 0.983516i
$804$	0	0
$805$	729.050	0.905653
$806$	12.6518i	0.0156971i
$807$	0	0
$808$	−26.1054	−0.0323087
$809$	1471.80i	1.81928i	0.415398	+	0.909640i	$0.363642\pi$
$809$	1471.80i	1.81928i	−0.415398	+	0.909640i	$0.636358\pi$
$810$	0	0
$811$	128.574	0.158537	0.0792685	−	0.996853i	$-0.474742\pi$
$811$	128.574	0.158537	0.0792685	+	0.996853i	$0.474742\pi$
$812$	565.525i	0.696459i
$813$	0	0
$814$	22.9907	0.0282441
$815$	− 1061.76i	− 1.30277i
$816$	0	0
$817$	487.601	0.596818
$818$	− 72.0863i	− 0.0881250i
$819$	0	0
$820$	759.352	0.926039
$821$	1487.34i	1.81162i	0.423689	+	0.905808i	$0.360735\pi$
$821$	1487.34i	1.81162i	−0.423689	+	0.905808i	$0.639265\pi$
$822$	0	0
$823$	−1199.72	−1.45774	−0.728869	−	0.684653i	$-0.759954\pi$
$823$	−1199.72	−1.45774	−0.728869	+	0.684653i	$0.759954\pi$
$824$	− 77.1670i	− 0.0936492i
$825$	0	0
$826$	19.3903	0.0234750
$827$	703.596i	0.850781i	0.905010	+	0.425391i	$0.139863\pi$
$827$	703.596i	0.850781i	−0.905010	+	0.425391i	$0.860137\pi$
$828$	0	0
$829$	−225.598	−0.272133	−0.136066	−	0.990700i	$-0.543446\pi$
$829$	−225.598	−0.272133	−0.136066	+	0.990700i	$0.543446\pi$
$830$	39.4412i	0.0475195i
$831$	0	0
$832$	−517.416	−0.621894
$833$	− 348.609i	− 0.418498i
$834$	0	0
$835$	−975.449	−1.16820
$836$	558.424i	0.667971i
$837$	0	0
$838$	61.7470	0.0736838
$839$	714.317i	0.851391i	0.904867	+	0.425695i	$0.139970\pi$
$839$	714.317i	0.851391i	−0.904867	+	0.425695i	$0.860030\pi$
$840$	0	0
$841$	524.404	0.623548
$842$	43.3431i	0.0514763i
$843$	0	0
$844$	1436.15	1.70160
$845$	625.022i	0.739671i
$846$	0	0
$847$	−79.3566	−0.0936914
$848$	− 1445.03i	− 1.70404i
$849$	0	0
$850$	37.7600	0.0444236
$851$	− 251.826i	− 0.295917i
$852$	0	0
$853$	260.598	0.305507	0.152754	−	0.988264i	$-0.451186\pi$
$853$	260.598	0.305507	0.152754	+	0.988264i	$0.451186\pi$
$854$	32.8071i	0.0384159i
$855$	0	0
$856$	33.7099	0.0393807
$857$	1035.16i	1.20789i	0.797026	+	0.603944i	$0.206405\pi$
$857$	1035.16i	1.20789i	−0.797026	+	0.603944i	$0.793595\pi$
$858$	0	0
$859$	−558.645	−0.650344	−0.325172	−	0.945655i	$-0.605422\pi$
$859$	−558.645	−0.650344	−0.325172	+	0.945655i	$0.605422\pi$
$860$	984.032i	1.14422i
$861$	0	0
$862$	−20.9524	−0.0243068
$863$	− 251.585i	− 0.291523i	−0.989320	−	0.145762i	$-0.953437\pi$
$863$	− 251.585i	− 0.291523i	0.989320	−	0.145762i	$-0.0465633\pi$
$864$	0	0
$865$	325.422	0.376210
$866$	83.6520i	0.0965958i
$867$	0	0
$868$	413.846	0.476781
$869$	− 1732.94i	− 1.99418i
$870$	0	0
$871$	−425.681	−0.488727
$872$	− 1.33045i	− 0.00152574i
$873$	0	0
$874$	−21.2560	−0.0243204
$875$	− 572.559i	− 0.654353i
$876$	0	0
$877$	−487.231	−0.555565	−0.277783	−	0.960644i	$-0.589599\pi$
$877$	−487.231	−0.555565	−0.277783	+	0.960644i	$0.589599\pi$
$878$	− 74.9483i	− 0.0853625i
$879$	0	0
$880$	−1123.03	−1.27617
$881$	61.9179i	0.0702813i	0.999382	+	0.0351407i	$0.0111879\pi$
$881$	61.9179i	0.0702813i	−0.999382	+	0.0351407i	$0.988812\pi$
$882$	0	0
$883$	−101.853	−0.115349	−0.0576744	−	0.998335i	$-0.518369\pi$
$883$	−101.853	−0.115349	−0.0576744	+	0.998335i	$0.518369\pi$
$884$	787.071i	0.890351i
$885$	0	0
$886$	−38.0000	−0.0428893
$887$	− 1257.63i	− 1.41785i	−0.705284	−	0.708924i	$-0.749181\pi$
$887$	− 1257.63i	− 1.41785i	0.705284	−	0.708924i	$-0.250819\pi$
$888$	0	0
$889$	136.672	0.153737
$890$	118.956i	0.133659i
$891$	0	0
$892$	35.8212	0.0401582
$893$	− 714.677i	− 0.800311i
$894$	0	0
$895$	414.884	0.463557
$896$	− 238.166i	− 0.265810i
$897$	0	0
$898$	−3.04815	−0.00339437
$899$	231.682i	0.257711i
$900$	0	0
$901$	−2182.70	−2.42253
$902$	− 41.3965i	− 0.0458941i
$903$	0	0
$904$	140.081	0.154957
$905$	802.374i	0.886601i
$906$	0	0
$907$	−834.289	−0.919833	−0.459917	−	0.887962i	$-0.652121\pi$
$907$	−834.289	−0.919833	−0.459917	+	0.887962i	$0.652121\pi$
$908$	286.126i	0.315117i
$909$	0	0
$910$	48.0182	0.0527672
$911$	− 715.751i	− 0.785676i	−0.919608	−	0.392838i	$-0.871493\pi$
$911$	− 715.751i	− 0.785676i	0.919608	−	0.392838i	$-0.128507\pi$
$912$	0	0
$913$	−618.728	−0.677687
$914$	46.5442i	0.0509237i
$915$	0	0
$916$	214.999	0.234715
$917$	1163.55i	1.26887i
$918$	0	0
$919$	−714.401	−0.777367	−0.388684	−	0.921371i	$-0.627070\pi$
$919$	−714.401	−0.777367	−0.388684	+	0.921371i	$0.627070\pi$
$920$	− 85.9432i	− 0.0934165i
$921$	0	0
$922$	105.561	0.114491
$923$	21.7520i	0.0235667i
$924$	0	0
$925$	228.979	0.247545
$926$	35.0261i	0.0378252i
$927$	0	0
$928$	100.057	0.107820
$929$	− 323.383i	− 0.348098i	−0.984737	−	0.174049i	$-0.944315\pi$
$929$	− 323.383i	− 0.348098i	0.984737	−	0.174049i	$-0.0556852\pi$
$930$	0	0
$931$	−178.437	−0.191661
$932$	1445.64i	1.55112i
$933$	0	0
$934$	−93.0207	−0.0995939
$935$	1696.33i	1.81426i
$936$	0	0
$937$	−1060.12	−1.13140	−0.565701	−	0.824610i	$-0.691394\pi$
$937$	−1060.12	−1.13140	−0.565701	+	0.824610i	$0.691394\pi$
$938$	− 48.3883i	− 0.0515867i
$939$	0	0
$940$	1442.30	1.53436
$941$	1300.38i	1.38192i	0.722895	+	0.690958i	$0.242811\pi$
$941$	1300.38i	1.38192i	−0.722895	+	0.690958i	$0.757189\pi$
$942$	0	0
$943$	−453.431	−0.480838
$944$	327.165i	0.346573i
$945$	0	0
$946$	53.6451	0.0567073
$947$	− 1643.50i	− 1.73549i	−0.497014	−	0.867743i	$-0.665570\pi$
$947$	− 1643.50i	− 1.73549i	0.497014	−	0.867743i	$-0.334430\pi$
$948$	0	0
$949$	−569.760	−0.600380
$950$	− 19.3276i	− 0.0203448i
$951$	0	0
$952$	−179.248	−0.188285
$953$	466.801i	0.489823i	0.969545	+	0.244912i	$0.0787589\pi$
$953$	466.801i	0.489823i	−0.969545	+	0.244912i	$0.921241\pi$
$954$	0	0
$955$	1681.78	1.76103
$956$	1002.80i	1.04895i
$957$	0	0
$958$	−0.127323	−0.000132905 0
$959$	− 693.812i	− 0.723474i
$960$	0	0
$961$	−791.457	−0.823577
$962$	− 16.5862i	− 0.0172414i
$963$	0	0
$964$	1606.03	1.66600
$965$	− 1285.97i	− 1.33261i
$966$	0	0
$967$	−1146.23	−1.18534	−0.592671	−	0.805445i	$-0.701927\pi$
$967$	−1146.23	−1.18534	−0.592671	+	0.805445i	$0.701927\pi$
$968$	9.35485i	0.00966411i
$969$	0	0
$970$	−41.5887	−0.0428749
$971$	− 1439.65i	− 1.48264i	−0.671150	−	0.741322i	$-0.734199\pi$
$971$	− 1439.65i	− 1.48264i	0.671150	−	0.741322i	$-0.265801\pi$
$972$	0	0
$973$	293.908	0.302064
$974$	− 76.2207i	− 0.0782553i
$975$	0	0
$976$	−553.542	−0.567153
$977$	1377.09i	1.40951i	0.709453	+	0.704753i	$0.248942\pi$
$977$	1377.09i	1.40951i	−0.709453	+	0.704753i	$0.751058\pi$
$978$	0	0
$979$	−1866.11	−1.90614
$980$	− 360.105i	− 0.367454i
$981$	0	0
$982$	−36.7582	−0.0374320
$983$	− 1232.99i	− 1.25431i	−0.778893	−	0.627157i	$-0.784219\pi$
$983$	− 1232.99i	− 1.25431i	0.778893	−	0.627157i	$-0.215781\pi$
$984$	0	0
$985$	1108.37	1.12525
$986$	− 50.0867i	− 0.0507979i
$987$	0	0
$988$	402.865	0.407758
$989$	− 587.593i	− 0.594129i
$990$	0	0
$991$	−51.5755	−0.0520439	−0.0260220	−	0.999661i	$-0.508284\pi$
$991$	−51.5755	−0.0520439	−0.0260220	+	0.999661i	$0.508284\pi$
$992$	− 73.2208i	− 0.0738113i
$993$	0	0
$994$	−2.47261	−0.00248754
$995$	158.716i	0.159514i
$996$	0	0
$997$	−825.561	−0.828045	−0.414023	−	0.910267i	$-0.635877\pi$
$997$	−825.561	−0.828045	−0.414023	+	0.910267i	$0.635877\pi$
$998$	− 5.40139i	− 0.00541222i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.3.b.a.728.15		30
3.2	odd	2	inner	729.3.b.a.728.16		30
27.2	odd	18		243.3.f.b.53.3		30
27.4	even	9		243.3.f.a.107.3		30
27.5	odd	18		27.3.f.a.2.3	✓	30
27.7	even	9		243.3.f.d.134.3		30
27.11	odd	18		81.3.f.a.71.3		30
27.13	even	9		243.3.f.b.188.3		30
27.14	odd	18		243.3.f.c.188.3		30
27.16	even	9		27.3.f.a.14.3	yes	30
27.20	odd	18		243.3.f.a.134.3		30
27.22	even	9		81.3.f.a.8.3		30
27.23	odd	18		243.3.f.d.107.3		30
27.25	even	9		243.3.f.c.53.3		30
108.43	odd	18		432.3.bc.a.257.1		30
108.59	even	18		432.3.bc.a.353.1		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.3	✓	30	27.5	odd	18
27.3.f.a.14.3	yes	30	27.16	even	9
81.3.f.a.8.3		30	27.22	even	9
81.3.f.a.71.3		30	27.11	odd	18
243.3.f.a.107.3		30	27.4	even	9
243.3.f.a.134.3		30	27.20	odd	18
243.3.f.b.53.3		30	27.2	odd	18
243.3.f.b.188.3		30	27.13	even	9
243.3.f.c.53.3		30	27.25	even	9
243.3.f.c.188.3		30	27.14	odd	18
243.3.f.d.107.3		30	27.23	odd	18
243.3.f.d.134.3		30	27.7	even	9
432.3.bc.a.257.1		30	108.43	odd	18
432.3.bc.a.353.1		30	108.59	even	18
729.3.b.a.728.15		30	1.1	even	1	trivial
729.3.b.a.728.16		30	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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	729.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.117696i q^{2} +3.98615 q^{4} -6.19791i q^{5} +7.97344 q^{7} -0.939939i q^{8} -0.729470 q^{10} -11.4435i q^{11} -8.25567 q^{13} -0.938443i q^{14} +15.8340 q^{16} -23.9171i q^{17} -12.2420 q^{19} -24.7058i q^{20} -1.34685 q^{22} +14.7525i q^{23} -13.4141 q^{25} +0.971660i q^{26} +31.7833 q^{28} +17.7931i q^{29} +13.0209 q^{31} -5.62335i q^{32} -2.81495 q^{34} -49.4187i q^{35} -17.0700 q^{37} +1.44084i q^{38} -5.82566 q^{40} +30.7358i q^{41} -39.8300 q^{43} -45.6153i q^{44} +1.73632 q^{46} +58.3790i q^{47} +14.5757 q^{49} +1.57879i q^{50} -32.9083 q^{52} -91.2612i q^{53} -70.9255 q^{55} -7.49454i q^{56} +2.09418 q^{58} +20.6622i q^{59} -34.9591 q^{61} -1.53250i q^{62} +62.6740 q^{64} +51.1679i q^{65} +51.5623 q^{67} -95.3370i q^{68} -5.81639 q^{70} -2.63480i q^{71} +69.0145 q^{73} +2.00907i q^{74} -48.7986 q^{76} -91.2437i q^{77} +151.435 q^{79} -98.1375i q^{80} +3.61748 q^{82} -54.0683i q^{83} -148.236 q^{85} +4.68784i q^{86} -10.7561 q^{88} -163.072i q^{89} -65.8261 q^{91} +58.8058i q^{92} +6.87098 q^{94} +75.8751i q^{95} +57.0121 q^{97} -1.71551i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) \) 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more