LMFDB - Embedded newform 27.10.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,10,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

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:	yes
Analytic conductor:	$13.9059675764$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.177113.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 118x + 136 $ x^3 - x^2 - 118*x + 136
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$10.7777$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+32.3331 q^{2} +533.432 q^{4} -2071.63 q^{5} -10517.1 q^{7} +692.954 q^{8} -66982.2 q^{10} +31620.5 q^{11} +47622.2 q^{13} -340051. q^{14} -250712. q^{16} -34440.3 q^{17} +763166. q^{19} -1.10507e6 q^{20} +1.02239e6 q^{22} -1.48534e6 q^{23} +2.33851e6 q^{25} +1.53977e6 q^{26} -5.61016e6 q^{28} -4.80506e6 q^{29} -7.82305e6 q^{31} -8.46109e6 q^{32} -1.11356e6 q^{34} +2.17875e7 q^{35} +9.43456e6 q^{37} +2.46755e7 q^{38} -1.43554e6 q^{40} -1.22899e7 q^{41} +3.74423e7 q^{43} +1.68674e7 q^{44} -4.80258e7 q^{46} -1.98234e7 q^{47} +7.02560e7 q^{49} +7.56113e7 q^{50} +2.54032e7 q^{52} -5.78978e7 q^{53} -6.55059e7 q^{55} -7.28788e6 q^{56} -1.55363e8 q^{58} +2.84988e7 q^{59} +4.45141e7 q^{61} -2.52944e8 q^{62} -1.45209e8 q^{64} -9.86553e7 q^{65} -1.81934e8 q^{67} -1.83715e7 q^{68} +7.04459e8 q^{70} -1.53722e8 q^{71} +1.55706e7 q^{73} +3.05049e8 q^{74} +4.07097e8 q^{76} -3.32557e8 q^{77} -3.26170e8 q^{79} +5.19381e8 q^{80} -3.97371e8 q^{82} -5.01181e7 q^{83} +7.13473e7 q^{85} +1.21063e9 q^{86} +2.19116e7 q^{88} +1.59066e8 q^{89} -5.00848e8 q^{91} -7.92329e8 q^{92} -6.40953e8 q^{94} -1.58099e9 q^{95} -1.49373e8 q^{97} +2.27160e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8} - 18981 q^{10} - 16863 q^{11} + 116916 q^{13} - 503463 q^{14} - 239919 q^{16} - 1014048 q^{17} - 15222 q^{19} - 2548407 q^{20} + 305721 q^{22}+ \cdots + 2592182286 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 597 * q^4 - 1983 * q^5 - 3693 * q^7 - 4503 * q^8 - 18981 * q^10 - 16863 * q^11 + 116916 * q^13 - 503463 * q^14 - 239919 * q^16 - 1014048 * q^17 - 15222 * q^19 - 2548407 * q^20 + 305721 * q^22 - 2927118 * q^23 + 2133732 * q^25 - 4765116 * q^26 - 3535725 * q^28 - 5768790 * q^29 - 6575223 * q^31 + 2687697 * q^32 + 17098128 * q^34 + 17340537 * q^35 - 11686026 * q^37 + 50473374 * q^38 - 4130811 * q^40 + 22213518 * q^41 + 45384414 * q^43 + 57206991 * q^44 - 95638590 * q^46 - 12392034 * q^47 + 18933462 * q^49 + 82984044 * q^50 + 182736492 * q^52 - 80579637 * q^53 - 174735333 * q^55 - 21850521 * q^56 - 102758922 * q^58 - 244026660 * q^59 + 369729960 * q^61 - 166297341 * q^62 - 420692127 * q^64 - 492225684 * q^65 - 252614586 * q^67 - 162270144 * q^68 + 935092377 * q^70 - 403193088 * q^71 - 406626717 * q^73 + 641373558 * q^74 - 39499590 * q^76 - 360443199 * q^77 + 265451856 * q^79 + 1168359837 * q^80 - 884408166 * q^82 - 121625871 * q^83 + 316251216 * q^85 + 2368956570 * q^86 + 219257739 * q^88 - 377904006 * q^89 + 245059140 * q^91 + 1178754558 * q^92 + 86583438 * q^94 - 536878770 * q^95 - 438907539 * q^97 + 2592182286 * q^98

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 $

$2$	32.3331	1.42894	0.714468	−	0.699668i	$-0.246669\pi$
$2$	32.3331	1.42894	0.714468	+	0.699668i	$0.246669\pi$
$3$	0	0
$3$	0	0
$4$	533.432	1.04186
$5$	−2071.63	−1.48233	−0.741167	−	0.671320i	$-0.765728\pi$
$5$	−2071.63	−1.48233	−0.741167	+	0.671320i	$0.765728\pi$
$6$	0	0
$7$	−10517.1	−1.65560	−0.827800	−	0.561024i	$-0.810408\pi$
$7$	−10517.1	−1.65560	−0.827800	+	0.561024i	$0.810408\pi$
$8$	692.954	0.0598136
$9$	0	0
$10$	−66982.2	−2.11816
$11$	31620.5	0.651182	0.325591	−	0.945511i	$-0.394437\pi$
$11$	31620.5	0.651182	0.325591	+	0.945511i	$0.394437\pi$
$12$	0	0
$13$	47622.2	0.462449	0.231225	−	0.972900i	$-0.425727\pi$
$13$	47622.2	0.462449	0.231225	+	0.972900i	$0.425727\pi$
$14$	−340051.	−2.36575
$15$	0	0
$16$	−250712.	−0.956389
$17$	−34440.3	−0.100011	−0.0500053	−	0.998749i	$-0.515924\pi$
$17$	−34440.3	−0.100011	−0.0500053	+	0.998749i	$0.515924\pi$
$18$	0	0
$19$	763166.	1.34347	0.671735	−	0.740792i	$-0.265550\pi$
$19$	763166.	1.34347	0.671735	+	0.740792i	$0.265550\pi$
$20$	−1.10507e6	−1.54438
$21$	0	0
$22$	1.02239e6	0.930497
$23$	−1.48534e6	−1.10675	−0.553377	−	0.832931i	$-0.686661\pi$
$23$	−1.48534e6	−1.10675	−0.553377	+	0.832931i	$0.686661\pi$
$24$	0	0
$25$	2.33851e6	1.19732
$26$	1.53977e6	0.660811
$27$	0	0
$28$	−5.61016e6	−1.72490
$29$	−4.80506e6	−1.26156	−0.630780	−	0.775962i	$-0.717265\pi$
$29$	−4.80506e6	−1.26156	−0.630780	+	0.775962i	$0.717265\pi$
$30$	0	0
$31$	−7.82305e6	−1.52142	−0.760708	−	0.649094i	$-0.775148\pi$
$31$	−7.82305e6	−1.52142	−0.760708	+	0.649094i	$0.775148\pi$
$32$	−8.46109e6	−1.42643
$33$	0	0
$34$	−1.11356e6	−0.142909
$35$	2.17875e7	2.45415
$36$	0	0
$37$	9.43456e6	0.827587	0.413794	−	0.910371i	$-0.364204\pi$
$37$	9.43456e6	0.827587	0.413794	+	0.910371i	$0.364204\pi$
$38$	2.46755e7	1.91973
$39$	0	0
$40$	−1.43554e6	−0.0886637
$41$	−1.22899e7	−0.679237	−0.339618	−	0.940563i	$-0.610298\pi$
$41$	−1.22899e7	−0.679237	−0.339618	+	0.940563i	$0.610298\pi$
$42$	0	0
$43$	3.74423e7	1.67014	0.835072	−	0.550140i	$-0.185426\pi$
$43$	3.74423e7	1.67014	0.835072	+	0.550140i	$0.185426\pi$
$44$	1.68674e7	0.678439
$45$	0	0
$46$	−4.80258e7	−1.58148
$47$	−1.98234e7	−0.592568	−0.296284	−	0.955100i	$-0.595747\pi$
$47$	−1.98234e7	−0.592568	−0.296284	+	0.955100i	$0.595747\pi$
$48$	0	0
$49$	7.02560e7	1.74101
$50$	7.56113e7	1.71089
$51$	0	0
$52$	2.54032e7	0.481807
$53$	−5.78978e7	−1.00791	−0.503953	−	0.863731i	$-0.668122\pi$
$53$	−5.78978e7	−1.00791	−0.503953	+	0.863731i	$0.668122\pi$
$54$	0	0
$55$	−6.55059e7	−0.965269
$56$	−7.28788e6	−0.0990273
$57$	0	0
$58$	−1.55363e8	−1.80269
$59$	2.84988e7	0.306191	0.153095	−	0.988211i	$-0.451076\pi$
$59$	2.84988e7	0.306191	0.153095	+	0.988211i	$0.451076\pi$
$60$	0	0
$61$	4.45141e7	0.411636	0.205818	−	0.978590i	$-0.434014\pi$
$61$	4.45141e7	0.411636	0.205818	+	0.978590i	$0.434014\pi$
$62$	−2.52944e8	−2.17401
$63$	0	0
$64$	−1.45209e8	−1.08189
$65$	−9.86553e7	−0.685505
$66$	0	0
$67$	−1.81934e8	−1.10301	−0.551503	−	0.834173i	$-0.685946\pi$
$67$	−1.81934e8	−1.10301	−0.551503	+	0.834173i	$0.685946\pi$
$68$	−1.83715e7	−0.104197
$69$	0	0
$70$	7.04459e8	3.50683
$71$	−1.53722e8	−0.717915	−0.358958	−	0.933354i	$-0.616868\pi$
$71$	−1.53722e8	−0.717915	−0.358958	+	0.933354i	$0.616868\pi$
$72$	0	0
$73$	1.55706e7	0.0641729	0.0320864	−	0.999485i	$-0.489785\pi$
$73$	1.55706e7	0.0641729	0.0320864	+	0.999485i	$0.489785\pi$
$74$	3.05049e8	1.18257
$75$	0	0
$76$	4.07097e8	1.39971
$77$	−3.32557e8	−1.07810
$78$	0	0
$79$	−3.26170e8	−0.942154	−0.471077	−	0.882092i	$-0.656135\pi$
$79$	−3.26170e8	−0.942154	−0.471077	+	0.882092i	$0.656135\pi$
$80$	5.19381e8	1.41769
$81$	0	0
$82$	−3.97371e8	−0.970586
$83$	−5.01181e7	−0.115916	−0.0579580	−	0.998319i	$-0.518459\pi$
$83$	−5.01181e7	−0.115916	−0.0579580	+	0.998319i	$0.518459\pi$
$84$	0	0
$85$	7.13473e7	0.148249
$86$	1.21063e9	2.38653
$87$	0	0
$88$	2.19116e7	0.0389495
$89$	1.59066e8	0.268735	0.134367	−	0.990932i	$-0.457100\pi$
$89$	1.59066e8	0.268735	0.134367	+	0.990932i	$0.457100\pi$
$90$	0	0
$91$	−5.00848e8	−0.765631
$92$	−7.92329e8	−1.15308
$93$	0	0
$94$	−6.40953e8	−0.846742
$95$	−1.58099e9	−1.99147
$96$	0	0
$97$	−1.49373e8	−0.171317	−0.0856583	−	0.996325i	$-0.527299\pi$
$97$	−1.49373e8	−0.171317	−0.0856583	+	0.996325i	$0.527299\pi$
$98$	2.27160e9	2.48779
$99$	0	0
$100$	1.24743e9	1.24743
$101$	1.84265e9	1.76196	0.880981	−	0.473152i	$-0.156884\pi$
$101$	1.84265e9	1.76196	0.880981	+	0.473152i	$0.156884\pi$
$102$	0	0
$103$	−3.50430e8	−0.306785	−0.153392	−	0.988165i	$-0.549020\pi$
$103$	−3.50430e8	−0.306785	−0.153392	+	0.988165i	$0.549020\pi$
$104$	3.30000e7	0.0276607
$105$	0	0
$106$	−1.87202e9	−1.44023
$107$	−7.93460e8	−0.585192	−0.292596	−	0.956236i	$-0.594519\pi$
$107$	−7.93460e8	−0.585192	−0.292596	+	0.956236i	$0.594519\pi$
$108$	0	0
$109$	1.09944e9	0.746020	0.373010	−	0.927827i	$-0.378326\pi$
$109$	1.09944e9	0.746020	0.373010	+	0.927827i	$0.378326\pi$
$110$	−2.11801e9	−1.37931
$111$	0	0
$112$	2.63676e9	1.58340
$113$	−2.91370e9	−1.68109	−0.840547	−	0.541739i	$-0.817766\pi$
$113$	−2.91370e9	−1.68109	−0.840547	+	0.541739i	$0.817766\pi$
$114$	0	0
$115$	3.07707e9	1.64058
$116$	−2.56317e9	−1.31437
$117$	0	0
$118$	9.21455e8	0.437527
$119$	3.62212e8	0.165578
$120$	0	0
$121$	−1.35809e9	−0.575962
$122$	1.43928e9	0.588202
$123$	0	0
$124$	−4.17306e9	−1.58510
$125$	−7.98371e8	−0.292489
$126$	0	0
$127$	1.58765e9	0.541551	0.270776	−	0.962643i	$-0.412720\pi$
$127$	1.58765e9	0.541551	0.270776	+	0.962643i	$0.412720\pi$
$128$	−3.62989e8	−0.119522
$129$	0	0
$130$	−3.18984e9	−0.979543
$131$	3.75999e9	1.11549	0.557745	−	0.830012i	$-0.311667\pi$
$131$	3.75999e9	1.11549	0.557745	+	0.830012i	$0.311667\pi$
$132$	0	0
$133$	−8.02630e9	−2.22425
$134$	−5.88251e9	−1.57613
$135$	0	0
$136$	−2.38655e7	−0.00598199
$137$	−9.88347e8	−0.239699	−0.119850	−	0.992792i	$-0.538241\pi$
$137$	−9.88347e8	−0.239699	−0.119850	+	0.992792i	$0.538241\pi$
$138$	0	0
$139$	4.71601e9	1.07154	0.535769	−	0.844364i	$-0.320022\pi$
$139$	4.71601e9	1.07154	0.535769	+	0.844364i	$0.320022\pi$
$140$	1.16222e10	2.55688
$141$	0	0
$142$	−4.97031e9	−1.02585
$143$	1.50584e9	0.301139
$144$	0	0
$145$	9.95428e9	1.87005
$146$	5.03445e8	0.0916990
$147$	0	0
$148$	5.03269e9	0.862229
$149$	−4.36878e9	−0.726143	−0.363071	−	0.931761i	$-0.618272\pi$
$149$	−4.36878e9	−0.726143	−0.363071	+	0.931761i	$0.618272\pi$
$150$	0	0
$151$	6.47508e9	1.01356	0.506779	−	0.862076i	$-0.330836\pi$
$151$	6.47508e9	1.01356	0.506779	+	0.862076i	$0.330836\pi$
$152$	5.28839e8	0.0803577
$153$	0	0
$154$	−1.07526e10	−1.54053
$155$	1.62064e10	2.25525
$156$	0	0
$157$	−8.50860e8	−0.111766	−0.0558830	−	0.998437i	$-0.517797\pi$
$157$	−8.50860e8	−0.111766	−0.0558830	+	0.998437i	$0.517797\pi$
$158$	−1.05461e10	−1.34628
$159$	0	0
$160$	1.75282e10	2.11445
$161$	1.56215e10	1.83234
$162$	0	0
$163$	−8.54283e9	−0.947889	−0.473945	−	0.880555i	$-0.657170\pi$
$163$	−8.54283e9	−0.947889	−0.473945	+	0.880555i	$0.657170\pi$
$164$	−6.55582e9	−0.707669
$165$	0	0
$166$	−1.62048e9	−0.165637
$167$	−4.57212e8	−0.0454876	−0.0227438	−	0.999741i	$-0.507240\pi$
$167$	−4.57212e8	−0.0454876	−0.0227438	+	0.999741i	$0.507240\pi$
$168$	0	0
$169$	−8.33663e9	−0.786141
$170$	2.30688e9	0.211839
$171$	0	0
$172$	1.99729e10	1.74005
$173$	−9.13108e9	−0.775023	−0.387511	−	0.921865i	$-0.626665\pi$
$173$	−9.13108e9	−0.775023	−0.387511	+	0.921865i	$0.626665\pi$
$174$	0	0
$175$	−2.45944e10	−1.98228
$176$	−7.92764e9	−0.622783
$177$	0	0
$178$	5.14312e9	0.384005
$179$	−1.84541e9	−0.134355	−0.0671777	−	0.997741i	$-0.521399\pi$
$179$	−1.84541e9	−0.134355	−0.0671777	+	0.997741i	$0.521399\pi$
$180$	0	0
$181$	−1.02105e10	−0.707123	−0.353562	−	0.935411i	$-0.615030\pi$
$181$	−1.02105e10	−0.707123	−0.353562	+	0.935411i	$0.615030\pi$
$182$	−1.61940e10	−1.09404
$183$	0	0
$184$	−1.02927e9	−0.0661989
$185$	−1.95449e10	−1.22676
$186$	0	0
$187$	−1.08902e9	−0.0651251
$188$	−1.05744e10	−0.617372
$189$	0	0
$190$	−5.11185e10	−2.84569
$191$	−1.00230e10	−0.544938	−0.272469	−	0.962164i	$-0.587840\pi$
$191$	−1.00230e10	−0.544938	−0.272469	+	0.962164i	$0.587840\pi$
$192$	0	0
$193$	5.87244e9	0.304657	0.152328	−	0.988330i	$-0.451323\pi$
$193$	5.87244e9	0.304657	0.152328	+	0.988330i	$0.451323\pi$
$194$	−4.82970e9	−0.244801
$195$	0	0
$196$	3.74768e10	1.81389
$197$	2.73628e10	1.29438	0.647191	−	0.762328i	$-0.275944\pi$
$197$	2.73628e10	1.29438	0.647191	+	0.762328i	$0.275944\pi$
$198$	0	0
$199$	2.21639e9	0.100186	0.0500930	−	0.998745i	$-0.484048\pi$
$199$	2.21639e9	0.100186	0.0500930	+	0.998745i	$0.484048\pi$
$200$	1.62048e9	0.0716158
$201$	0	0
$202$	5.95786e10	2.51773
$203$	5.05353e10	2.08864
$204$	0	0
$205$	2.54601e10	1.00686
$206$	−1.13305e10	−0.438376
$207$	0	0
$208$	−1.19394e10	−0.442281
$209$	2.41317e10	0.874843
$210$	0	0
$211$	−1.35207e10	−0.469599	−0.234799	−	0.972044i	$-0.575443\pi$
$211$	−1.35207e10	−0.469599	−0.234799	+	0.972044i	$0.575443\pi$
$212$	−3.08845e10	−1.05010
$213$	0	0
$214$	−2.56551e10	−0.836202
$215$	−7.75664e10	−2.47571
$216$	0	0
$217$	8.22758e10	2.51886
$218$	3.55482e10	1.06601
$219$	0	0
$220$	−3.49429e10	−1.00567
$221$	−1.64012e9	−0.0462499
$222$	0	0
$223$	−2.97283e9	−0.0805004	−0.0402502	−	0.999190i	$-0.512815\pi$
$223$	−2.97283e9	−0.0805004	−0.0402502	+	0.999190i	$0.512815\pi$
$224$	8.89862e10	2.36160
$225$	0	0
$226$	−9.42090e10	−2.40218
$227$	3.34599e10	0.836388	0.418194	−	0.908358i	$-0.362663\pi$
$227$	3.34599e10	0.836388	0.418194	+	0.908358i	$0.362663\pi$
$228$	0	0
$229$	−7.60785e10	−1.82811	−0.914055	−	0.405590i	$-0.867066\pi$
$229$	−7.60785e10	−1.82811	−0.914055	+	0.405590i	$0.867066\pi$
$230$	9.94914e10	2.34429
$231$	0	0
$232$	−3.32969e9	−0.0754584
$233$	−6.49552e10	−1.44382	−0.721909	−	0.691988i	$-0.756735\pi$
$233$	−6.49552e10	−1.44382	−0.721909	+	0.691988i	$0.756735\pi$
$234$	0	0
$235$	4.10667e10	0.878384
$236$	1.52022e10	0.319008
$237$	0	0
$238$	1.17115e10	0.236600
$239$	3.98471e10	0.789963	0.394981	−	0.918689i	$-0.370751\pi$
$239$	3.98471e10	0.789963	0.394981	+	0.918689i	$0.370751\pi$
$240$	0	0
$241$	1.18614e10	0.226496	0.113248	−	0.993567i	$-0.463875\pi$
$241$	1.18614e10	0.226496	0.113248	+	0.993567i	$0.463875\pi$
$242$	−4.39113e10	−0.823014
$243$	0	0
$244$	2.37452e10	0.428867
$245$	−1.45544e11	−2.58076
$246$	0	0
$247$	3.63436e10	0.621286
$248$	−5.42101e9	−0.0910014
$249$	0	0
$250$	−2.58138e10	−0.417948
$251$	6.51966e10	1.03680	0.518398	−	0.855139i	$-0.326529\pi$
$251$	6.51966e10	1.03680	0.518398	+	0.855139i	$0.326529\pi$
$252$	0	0
$253$	−4.69673e10	−0.720698
$254$	5.13339e10	0.773842
$255$	0	0
$256$	6.26105e10	0.911102
$257$	−3.80164e10	−0.543591	−0.271796	−	0.962355i	$-0.587617\pi$
$257$	−3.80164e10	−0.543591	−0.271796	+	0.962355i	$0.587617\pi$
$258$	0	0
$259$	−9.92243e10	−1.37015
$260$	−5.26259e10	−0.714199
$261$	0	0
$262$	1.21572e11	1.59397
$263$	−3.53924e10	−0.456152	−0.228076	−	0.973643i	$-0.573243\pi$
$263$	−3.53924e10	−0.456152	−0.228076	+	0.973643i	$0.573243\pi$
$264$	0	0
$265$	1.19943e11	1.49406
$266$	−2.59515e11	−3.17831
$267$	0	0
$268$	−9.70495e10	−1.14918
$269$	6.44475e9	0.0750448	0.0375224	−	0.999296i	$-0.488053\pi$
$269$	6.44475e9	0.0750448	0.0375224	+	0.999296i	$0.488053\pi$
$270$	0	0
$271$	1.68603e11	1.89890	0.949452	−	0.313912i	$-0.101640\pi$
$271$	1.68603e11	1.89890	0.949452	+	0.313912i	$0.101640\pi$
$272$	8.63458e9	0.0956491
$273$	0	0
$274$	−3.19563e10	−0.342515
$275$	7.39449e10	0.779671
$276$	0	0
$277$	−4.87122e10	−0.497140	−0.248570	−	0.968614i	$-0.579961\pi$
$277$	−4.87122e10	−0.497140	−0.248570	+	0.968614i	$0.579961\pi$
$278$	1.52483e11	1.53116
$279$	0	0
$280$	1.50978e10	0.146792
$281$	−1.51361e11	−1.44822	−0.724110	−	0.689684i	$-0.757749\pi$
$281$	−1.51361e11	−1.44822	−0.724110	+	0.689684i	$0.757749\pi$
$282$	0	0
$283$	−7.14282e10	−0.661959	−0.330979	−	0.943638i	$-0.607379\pi$
$283$	−7.14282e10	−0.661959	−0.330979	+	0.943638i	$0.607379\pi$
$284$	−8.20001e10	−0.747966
$285$	0	0
$286$	4.86885e10	0.430308
$287$	1.29254e11	1.12454
$288$	0	0
$289$	−1.17402e11	−0.989998
$290$	3.21853e11	2.67219
$291$	0	0
$292$	8.30584e9	0.0668591
$293$	−2.22271e11	−1.76189	−0.880945	−	0.473218i	$-0.843092\pi$
$293$	−2.22271e11	−1.76189	−0.880945	+	0.473218i	$0.843092\pi$
$294$	0	0
$295$	−5.90388e10	−0.453878
$296$	6.53772e9	0.0495009
$297$	0	0
$298$	−1.41256e11	−1.03761
$299$	−7.07352e10	−0.511818
$300$	0	0
$301$	−3.93784e11	−2.76509
$302$	2.09360e11	1.44831
$303$	0	0
$304$	−1.91335e11	−1.28488
$305$	−9.22166e10	−0.610183
$306$	0	0
$307$	2.24657e11	1.44343	0.721717	−	0.692188i	$-0.243353\pi$
$307$	2.24657e11	1.44343	0.721717	+	0.692188i	$0.243353\pi$
$308$	−1.77396e11	−1.12322
$309$	0	0
$310$	5.24005e11	3.22261
$311$	2.99889e11	1.81777	0.908886	−	0.417046i	$-0.136934\pi$
$311$	2.99889e11	1.81777	0.908886	+	0.417046i	$0.136934\pi$
$312$	0	0
$313$	1.50502e11	0.886325	0.443162	−	0.896441i	$-0.353857\pi$
$313$	1.50502e11	0.886325	0.443162	+	0.896441i	$0.353857\pi$
$314$	−2.75110e10	−0.159706
$315$	0	0
$316$	−1.73989e11	−0.981591
$317$	2.49896e11	1.38993	0.694966	−	0.719043i	$-0.255420\pi$
$317$	2.49896e11	1.38993	0.694966	+	0.719043i	$0.255420\pi$
$318$	0	0
$319$	−1.51939e11	−0.821504
$320$	3.00819e11	1.60373
$321$	0	0
$322$	5.05092e11	2.61830
$323$	−2.62836e10	−0.134361
$324$	0	0
$325$	1.11365e11	0.553698
$326$	−2.76216e11	−1.35447
$327$	0	0
$328$	−8.51634e9	−0.0406276
$329$	2.08485e11	0.981055
$330$	0	0
$331$	5.32449e10	0.243810	0.121905	−	0.992542i	$-0.461100\pi$
$331$	5.32449e10	0.243810	0.121905	+	0.992542i	$0.461100\pi$
$332$	−2.67346e10	−0.120768
$333$	0	0
$334$	−1.47831e10	−0.0649989
$335$	3.76900e11	1.63502
$336$	0	0
$337$	3.54762e10	0.149831	0.0749156	−	0.997190i	$-0.476131\pi$
$337$	3.54762e10	0.149831	0.0749156	+	0.997190i	$0.476131\pi$
$338$	−2.69549e11	−1.12334
$339$	0	0
$340$	3.80589e10	0.154455
$341$	−2.47369e11	−0.990719
$342$	0	0
$343$	−3.14487e11	−1.22681
$344$	2.59458e10	0.0998973
$345$	0	0
$346$	−2.95236e11	−1.10746
$347$	−3.92158e11	−1.45204	−0.726019	−	0.687675i	$-0.758632\pi$
$347$	−3.92158e11	−1.45204	−0.726019	+	0.687675i	$0.758632\pi$
$348$	0	0
$349$	−8.30145e10	−0.299530	−0.149765	−	0.988722i	$-0.547852\pi$
$349$	−8.30145e10	−0.299530	−0.149765	+	0.988722i	$0.547852\pi$
$350$	−7.95213e11	−2.83255
$351$	0	0
$352$	−2.67544e11	−0.928867
$353$	−1.85625e11	−0.636281	−0.318141	−	0.948044i	$-0.603058\pi$
$353$	−1.85625e11	−0.636281	−0.318141	+	0.948044i	$0.603058\pi$
$354$	0	0
$355$	3.18454e11	1.06419
$356$	8.48511e10	0.279984
$357$	0	0
$358$	−5.96680e10	−0.191985
$359$	3.92754e11	1.24794	0.623972	−	0.781447i	$-0.285518\pi$
$359$	3.92754e11	1.24794	0.623972	+	0.781447i	$0.285518\pi$
$360$	0	0
$361$	2.59734e11	0.804909
$362$	−3.30139e11	−1.01043
$363$	0	0
$364$	−2.67168e11	−0.797679
$365$	−3.22564e10	−0.0951257
$366$	0	0
$367$	4.97459e10	0.143140	0.0715698	−	0.997436i	$-0.477199\pi$
$367$	4.97459e10	0.143140	0.0715698	+	0.997436i	$0.477199\pi$
$368$	3.72393e11	1.05849
$369$	0	0
$370$	−6.31947e11	−1.75296
$371$	6.08917e11	1.66869
$372$	0	0
$373$	4.03156e11	1.07841	0.539204	−	0.842175i	$-0.318725\pi$
$373$	4.03156e11	1.07841	0.539204	+	0.842175i	$0.318725\pi$
$374$	−3.52114e10	−0.0930596
$375$	0	0
$376$	−1.37367e10	−0.0354436
$377$	−2.28827e11	−0.583407
$378$	0	0
$379$	−3.41222e11	−0.849494	−0.424747	−	0.905312i	$-0.639637\pi$
$379$	−3.41222e11	−0.849494	−0.424747	+	0.905312i	$0.639637\pi$
$380$	−8.43352e11	−2.07483
$381$	0	0
$382$	−3.24075e11	−0.778682
$383$	−3.16998e11	−0.752770	−0.376385	−	0.926463i	$-0.622833\pi$
$383$	−3.16998e11	−0.752770	−0.376385	+	0.926463i	$0.622833\pi$
$384$	0	0
$385$	6.88933e11	1.59810
$386$	1.89874e11	0.435335
$387$	0	0
$388$	−7.96803e10	−0.178488
$389$	1.11909e11	0.247795	0.123898	−	0.992295i	$-0.460461\pi$
$389$	1.11909e11	0.247795	0.123898	+	0.992295i	$0.460461\pi$
$390$	0	0
$391$	5.11556e10	0.110687
$392$	4.86842e10	0.104136
$393$	0	0
$394$	8.84725e11	1.84959
$395$	6.75702e11	1.39659
$396$	0	0
$397$	3.48788e11	0.704700	0.352350	−	0.935868i	$-0.385383\pi$
$397$	3.48788e11	0.704700	0.352350	+	0.935868i	$0.385383\pi$
$398$	7.16627e10	0.143159
$399$	0	0
$400$	−5.86291e11	−1.14510
$401$	−2.54977e11	−0.492437	−0.246218	−	0.969214i	$-0.579188\pi$
$401$	−2.54977e11	−0.492437	−0.246218	+	0.969214i	$0.579188\pi$
$402$	0	0
$403$	−3.72550e11	−0.703578
$404$	9.82928e11	1.83572
$405$	0	0
$406$	1.63397e12	2.98453
$407$	2.98326e11	0.538910
$408$	0	0
$409$	2.68334e10	0.0474156	0.0237078	−	0.999719i	$-0.492453\pi$
$409$	2.68334e10	0.0474156	0.0237078	+	0.999719i	$0.492453\pi$
$410$	8.23204e11	1.43873
$411$	0	0
$412$	−1.86930e11	−0.319626
$413$	−2.99725e11	−0.506930
$414$	0	0
$415$	1.03826e11	0.171826
$416$	−4.02935e11	−0.659653
$417$	0	0
$418$	7.80254e11	1.25009
$419$	8.67220e9	0.0137457	0.00687284	−	0.999976i	$-0.497812\pi$
$419$	8.67220e9	0.0137457	0.00687284	+	0.999976i	$0.497812\pi$
$420$	0	0
$421$	−9.68105e11	−1.50194	−0.750971	−	0.660335i	$-0.770414\pi$
$421$	−9.68105e11	−1.50194	−0.750971	+	0.660335i	$0.770414\pi$
$422$	−4.37166e11	−0.671027
$423$	0	0
$424$	−4.01205e10	−0.0602865
$425$	−8.05389e10	−0.119744
$426$	0	0
$427$	−4.68160e11	−0.681504
$428$	−4.23257e11	−0.609687
$429$	0	0
$430$	−2.50796e12	−3.53764
$431$	1.38264e12	1.93001	0.965006	−	0.262227i	$-0.0844570\pi$
$431$	1.38264e12	1.93001	0.965006	+	0.262227i	$0.0844570\pi$
$432$	0	0
$433$	−1.26247e12	−1.72594	−0.862971	−	0.505254i	$-0.831399\pi$
$433$	−1.26247e12	−1.72594	−0.862971	+	0.505254i	$0.831399\pi$
$434$	2.66024e12	3.59929
$435$	0	0
$436$	5.86474e11	0.777247
$437$	−1.13356e12	−1.48689
$438$	0	0
$439$	4.36242e11	0.560579	0.280290	−	0.959915i	$-0.409570\pi$
$439$	4.36242e11	0.560579	0.280290	+	0.959915i	$0.409570\pi$
$440$	−4.53926e10	−0.0577362
$441$	0	0
$442$	−5.30302e10	−0.0660881
$443$	−5.07995e11	−0.626675	−0.313337	−	0.949642i	$-0.601447\pi$
$443$	−5.07995e11	−0.626675	−0.313337	+	0.949642i	$0.601447\pi$
$444$	0	0
$445$	−3.29526e11	−0.398355
$446$	−9.61208e10	−0.115030
$447$	0	0
$448$	1.52718e12	1.79118
$449$	−1.47077e12	−1.70780	−0.853899	−	0.520439i	$-0.825768\pi$
$449$	−1.47077e12	−1.70780	−0.853899	+	0.520439i	$0.825768\pi$
$450$	0	0
$451$	−3.88613e11	−0.442307
$452$	−1.55426e12	−1.75146
$453$	0	0
$454$	1.08186e12	1.19515
$455$	1.03757e12	1.13492
$456$	0	0
$457$	−1.11991e12	−1.20105	−0.600526	−	0.799605i	$-0.705042\pi$
$457$	−1.11991e12	−1.20105	−0.600526	+	0.799605i	$0.705042\pi$
$458$	−2.45986e12	−2.61225
$459$	0	0
$460$	1.64141e12	1.70925
$461$	−1.72669e12	−1.78057	−0.890286	−	0.455402i	$-0.849496\pi$
$461$	−1.72669e12	−1.78057	−0.890286	+	0.455402i	$0.849496\pi$
$462$	0	0
$463$	2.74673e11	0.277780	0.138890	−	0.990308i	$-0.455647\pi$
$463$	2.74673e11	0.277780	0.138890	+	0.990308i	$0.455647\pi$
$464$	1.20468e12	1.20654
$465$	0	0
$466$	−2.10021e12	−2.06312
$467$	−1.14495e12	−1.11394	−0.556971	−	0.830532i	$-0.688036\pi$
$467$	−1.14495e12	−1.11394	−0.556971	+	0.830532i	$0.688036\pi$
$468$	0	0
$469$	1.91342e12	1.82614
$470$	1.32782e12	1.25516
$471$	0	0
$472$	1.97484e10	0.0183144
$473$	1.18394e12	1.08757
$474$	0	0
$475$	1.78467e12	1.60856
$476$	1.93215e11	0.172508
$477$	0	0
$478$	1.28838e12	1.12881
$479$	−9.32652e11	−0.809487	−0.404744	−	0.914430i	$-0.632639\pi$
$479$	−9.32652e11	−0.809487	−0.404744	+	0.914430i	$0.632639\pi$
$480$	0	0
$481$	4.49294e11	0.382717
$482$	3.83517e11	0.323648
$483$	0	0
$484$	−7.24448e11	−0.600071
$485$	3.09445e11	0.253949
$486$	0	0
$487$	−1.02912e12	−0.829059	−0.414530	−	0.910036i	$-0.636054\pi$
$487$	−1.02912e12	−0.829059	−0.414530	+	0.910036i	$0.636054\pi$
$488$	3.08462e10	0.0246214
$489$	0	0
$490$	−4.70590e12	−3.68774
$491$	−1.72889e12	−1.34246	−0.671228	−	0.741251i	$-0.734233\pi$
$491$	−1.72889e12	−1.34246	−0.671228	+	0.741251i	$0.734233\pi$
$492$	0	0
$493$	1.65487e11	0.126169
$494$	1.17510e12	0.887779
$495$	0	0
$496$	1.96133e12	1.45507
$497$	1.61671e12	1.18858
$498$	0	0
$499$	2.39976e12	1.73267	0.866336	−	0.499462i	$-0.166469\pi$
$499$	2.39976e12	1.73267	0.866336	+	0.499462i	$0.166469\pi$
$500$	−4.25876e11	−0.304732
$501$	0	0
$502$	2.10801e12	1.48152
$503$	−1.98201e12	−1.38054	−0.690271	−	0.723551i	$-0.742509\pi$
$503$	−1.98201e12	−1.38054	−0.690271	+	0.723551i	$0.742509\pi$
$504$	0	0
$505$	−3.81728e12	−2.61182
$506$	−1.51860e12	−1.02983
$507$	0	0
$508$	8.46905e11	0.564220
$509$	2.25559e12	1.48946	0.744731	−	0.667364i	$-0.232578\pi$
$509$	2.25559e12	1.48946	0.744731	+	0.667364i	$0.232578\pi$
$510$	0	0
$511$	−1.63757e11	−0.106245
$512$	2.21024e12	1.42143
$513$	0	0
$514$	−1.22919e12	−0.776757
$515$	7.25959e11	0.454757
$516$	0	0
$517$	−6.26827e11	−0.385869
$518$	−3.20823e12	−1.95786
$519$	0	0
$520$	−6.83636e10	−0.0410025
$521$	7.71315e11	0.458630	0.229315	−	0.973352i	$-0.426351\pi$
$521$	7.71315e11	0.458630	0.229315	+	0.973352i	$0.426351\pi$
$522$	0	0
$523$	−1.54315e12	−0.901886	−0.450943	−	0.892553i	$-0.648912\pi$
$523$	−1.54315e12	−0.901886	−0.450943	+	0.892553i	$0.648912\pi$
$524$	2.00570e12	1.16218
$525$	0	0
$526$	−1.14435e12	−0.651812
$527$	2.69428e11	0.152158
$528$	0	0
$529$	4.05089e11	0.224905
$530$	3.87812e12	2.13491
$531$	0	0
$532$	−4.28148e12	−2.31735
$533$	−5.85272e11	−0.314113
$534$	0	0
$535$	1.64375e12	0.867450
$536$	−1.26072e11	−0.0659747
$537$	0	0
$538$	2.08379e11	0.107234
$539$	2.22153e12	1.13371
$540$	0	0
$541$	3.58683e12	1.80021	0.900105	−	0.435674i	$-0.143490\pi$
$541$	3.58683e12	1.80021	0.900105	+	0.435674i	$0.143490\pi$
$542$	5.45146e12	2.71341
$543$	0	0
$544$	2.91402e11	0.142658
$545$	−2.27762e12	−1.10585
$546$	0	0
$547$	−2.61010e12	−1.24656	−0.623282	−	0.781997i	$-0.714201\pi$
$547$	−2.61010e12	−1.24656	−0.623282	+	0.781997i	$0.714201\pi$
$548$	−5.27215e11	−0.249733
$549$	0	0
$550$	2.39087e12	1.11410
$551$	−3.66706e12	−1.69487
$552$	0	0
$553$	3.43036e12	1.55983
$554$	−1.57502e12	−0.710382
$555$	0	0
$556$	2.51567e12	1.11639
$557$	1.44707e12	0.637002	0.318501	−	0.947923i	$-0.396821\pi$
$557$	1.44707e12	0.637002	0.318501	+	0.947923i	$0.396821\pi$
$558$	0	0
$559$	1.78308e12	0.772357
$560$	−5.46238e12	−2.34712
$561$	0	0
$562$	−4.89397e12	−2.06942
$563$	3.02936e12	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$563$	3.02936e12	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$564$	0	0
$565$	6.03610e12	2.49194
$566$	−2.30950e12	−0.945897
$567$	0	0
$568$	−1.06522e11	−0.0429411
$569$	8.23594e11	0.329388	0.164694	−	0.986345i	$-0.447336\pi$
$569$	8.23594e11	0.329388	0.164694	+	0.986345i	$0.447336\pi$
$570$	0	0
$571$	−7.54932e11	−0.297198	−0.148599	−	0.988898i	$-0.547476\pi$
$571$	−7.54932e11	−0.297198	−0.148599	+	0.988898i	$0.547476\pi$
$572$	8.03262e11	0.313744
$573$	0	0
$574$	4.17920e12	1.60690
$575$	−3.47349e12	−1.32514
$576$	0	0
$577$	−4.22193e12	−1.58569	−0.792847	−	0.609420i	$-0.791402\pi$
$577$	−4.22193e12	−1.58569	−0.792847	+	0.609420i	$0.791402\pi$
$578$	−3.79597e12	−1.41464
$579$	0	0
$580$	5.30993e12	1.94833
$581$	5.27098e11	0.191910
$582$	0	0
$583$	−1.83076e12	−0.656330
$584$	1.07897e10	0.00383841
$585$	0	0
$586$	−7.18673e12	−2.51763
$587$	2.52727e12	0.878578	0.439289	−	0.898346i	$-0.355230\pi$
$587$	2.52727e12	0.878578	0.439289	+	0.898346i	$0.355230\pi$
$588$	0	0
$589$	−5.97028e12	−2.04398
$590$	−1.90891e12	−0.648562
$591$	0	0
$592$	−2.36535e12	−0.791495
$593$	2.59659e12	0.862299	0.431150	−	0.902280i	$-0.358108\pi$
$593$	2.59659e12	0.862299	0.431150	+	0.902280i	$0.358108\pi$
$594$	0	0
$595$	−7.50368e11	−0.245441
$596$	−2.33045e12	−0.756538
$597$	0	0
$598$	−2.28709e12	−0.731355
$599$	3.56005e12	1.12989	0.564944	−	0.825129i	$-0.308898\pi$
$599$	3.56005e12	1.12989	0.564944	+	0.825129i	$0.308898\pi$
$600$	0	0
$601$	1.93346e12	0.604506	0.302253	−	0.953228i	$-0.402261\pi$
$601$	1.93346e12	0.604506	0.302253	+	0.953228i	$0.402261\pi$
$602$	−1.27323e13	−3.95114
$603$	0	0
$604$	3.45401e12	1.05598
$605$	2.81345e12	0.853769
$606$	0	0
$607$	1.32290e12	0.395528	0.197764	−	0.980250i	$-0.436632\pi$
$607$	1.32290e12	0.395528	0.197764	+	0.980250i	$0.436632\pi$
$608$	−6.45721e12	−1.91637
$609$	0	0
$610$	−2.98165e12	−0.871912
$611$	−9.44034e11	−0.274033
$612$	0	0
$613$	−9.74984e11	−0.278885	−0.139443	−	0.990230i	$-0.544531\pi$
$613$	−9.74984e11	−0.278885	−0.139443	+	0.990230i	$0.544531\pi$
$614$	7.26386e12	2.06257
$615$	0	0
$616$	−2.30447e11	−0.0644848
$617$	2.87228e12	0.797890	0.398945	−	0.916975i	$-0.369376\pi$
$617$	2.87228e12	0.797890	0.398945	+	0.916975i	$0.369376\pi$
$618$	0	0
$619$	−4.16143e12	−1.13929	−0.569645	−	0.821891i	$-0.692919\pi$
$619$	−4.16143e12	−1.13929	−0.569645	+	0.821891i	$0.692919\pi$
$620$	8.64502e12	2.34965
$621$	0	0
$622$	9.69636e12	2.59748
$623$	−1.67292e12	−0.444917
$624$	0	0
$625$	−2.91347e12	−0.763750
$626$	4.86620e12	1.26650
$627$	0	0
$628$	−4.53876e11	−0.116444
$629$	−3.24929e11	−0.0827675
$630$	0	0
$631$	−3.22906e12	−0.810855	−0.405428	−	0.914127i	$-0.632877\pi$
$631$	−3.22906e12	−0.810855	−0.405428	+	0.914127i	$0.632877\pi$
$632$	−2.26021e11	−0.0563536
$633$	0	0
$634$	8.07993e12	1.98612
$635$	−3.28903e12	−0.802760
$636$	0	0
$637$	3.34574e12	0.805129
$638$	−4.91265e12	−1.17388
$639$	0	0
$640$	7.51978e11	0.177172
$641$	−4.91306e12	−1.14945	−0.574727	−	0.818345i	$-0.694892\pi$
$641$	−4.91306e12	−1.14945	−0.574727	+	0.818345i	$0.694892\pi$
$642$	0	0
$643$	1.18575e12	0.273553	0.136777	−	0.990602i	$-0.456326\pi$
$643$	1.18575e12	0.273553	0.136777	+	0.990602i	$0.456326\pi$
$644$	8.33301e12	1.90904
$645$	0	0
$646$	−8.49832e11	−0.191994
$647$	−6.72898e12	−1.50966	−0.754832	−	0.655919i	$-0.772281\pi$
$647$	−6.72898e12	−1.50966	−0.754832	+	0.655919i	$0.772281\pi$
$648$	0	0
$649$	9.01147e11	0.199386
$650$	3.60078e12	0.791199
$651$	0	0
$652$	−4.55702e12	−0.987567
$653$	−7.24926e11	−0.156021	−0.0780107	−	0.996953i	$-0.524857\pi$
$653$	−7.24926e11	−0.156021	−0.0780107	+	0.996953i	$0.524857\pi$
$654$	0	0
$655$	−7.78929e12	−1.65353
$656$	3.08122e12	0.649615
$657$	0	0
$658$	6.74098e12	1.40187
$659$	6.02523e12	1.24448	0.622242	−	0.782825i	$-0.286222\pi$
$659$	6.02523e12	1.24448	0.622242	+	0.782825i	$0.286222\pi$
$660$	0	0
$661$	1.74974e12	0.356506	0.178253	−	0.983985i	$-0.442956\pi$
$661$	1.74974e12	0.356506	0.178253	+	0.983985i	$0.442956\pi$
$662$	1.72158e12	0.348390
$663$	0	0
$664$	−3.47296e10	−0.00693335
$665$	1.66275e13	3.29708
$666$	0	0
$667$	7.13716e12	1.39624
$668$	−2.43891e11	−0.0473917
$669$	0	0
$670$	1.21864e13	2.33635
$671$	1.40756e12	0.268050
$672$	0	0
$673$	−3.59466e12	−0.675445	−0.337722	−	0.941246i	$-0.609656\pi$
$673$	−3.59466e12	−0.675445	−0.337722	+	0.941246i	$0.609656\pi$
$674$	1.14706e12	0.214099
$675$	0	0
$676$	−4.44702e12	−0.819048
$677$	2.33912e12	0.427960	0.213980	−	0.976838i	$-0.431357\pi$
$677$	2.33912e12	0.427960	0.213980	+	0.976838i	$0.431357\pi$
$678$	0	0
$679$	1.57097e12	0.283632
$680$	4.94404e10	0.00886732
$681$	0	0
$682$	−7.99821e12	−1.41567
$683$	−4.03165e11	−0.0708907	−0.0354453	−	0.999372i	$-0.511285\pi$
$683$	−4.03165e11	−0.0708907	−0.0354453	+	0.999372i	$0.511285\pi$
$684$	0	0
$685$	2.04748e12	0.355315
$686$	−1.01683e13	−1.75304
$687$	0	0
$688$	−9.38721e12	−1.59731
$689$	−2.75722e12	−0.466106
$690$	0	0
$691$	7.79472e12	1.30062	0.650308	−	0.759670i	$-0.274640\pi$
$691$	7.79472e12	1.30062	0.650308	+	0.759670i	$0.274640\pi$
$692$	−4.87081e12	−0.807465
$693$	0	0
$694$	−1.26797e13	−2.07487
$695$	−9.76980e12	−1.58838
$696$	0	0
$697$	4.23268e11	0.0679309
$698$	−2.68412e12	−0.428009
$699$	0	0
$700$	−1.31194e13	−2.06525
$701$	2.40116e12	0.375569	0.187784	−	0.982210i	$-0.439869\pi$
$701$	2.40116e12	0.375569	0.187784	+	0.982210i	$0.439869\pi$
$702$	0	0
$703$	7.20013e12	1.11184
$704$	−4.59159e12	−0.704508
$705$	0	0
$706$	−6.00182e12	−0.909206
$707$	−1.93793e13	−2.91710
$708$	0	0
$709$	−5.36135e12	−0.796830	−0.398415	−	0.917205i	$-0.630440\pi$
$709$	−5.36135e12	−0.796830	−0.398415	+	0.917205i	$0.630440\pi$
$710$	1.02966e13	1.52066
$711$	0	0
$712$	1.10226e11	0.0160740
$713$	1.16199e13	1.68383
$714$	0	0
$715$	−3.11953e12	−0.446388
$716$	−9.84402e11	−0.139979
$717$	0	0
$718$	1.26990e13	1.78323
$719$	−5.40289e12	−0.753956	−0.376978	−	0.926222i	$-0.623037\pi$
$719$	−5.40289e12	−0.753956	−0.376978	+	0.926222i	$0.623037\pi$
$720$	0	0
$721$	3.68551e12	0.507912
$722$	8.39803e12	1.15016
$723$	0	0
$724$	−5.44662e12	−0.736722
$725$	−1.12367e13	−1.51049
$726$	0	0
$727$	3.15572e12	0.418980	0.209490	−	0.977811i	$-0.432820\pi$
$727$	3.15572e12	0.418980	0.209490	+	0.977811i	$0.432820\pi$
$728$	−3.47065e11	−0.0457951
$729$	0	0
$730$	−1.04295e12	−0.135929
$731$	−1.28952e12	−0.167032
$732$	0	0
$733$	−1.20641e13	−1.54357	−0.771784	−	0.635885i	$-0.780635\pi$
$733$	−1.20641e13	−1.54357	−0.771784	+	0.635885i	$0.780635\pi$
$734$	1.60844e12	0.204537
$735$	0	0
$736$	1.25676e13	1.57871
$737$	−5.75286e12	−0.718257
$738$	0	0
$739$	8.39060e12	1.03489	0.517444	−	0.855717i	$-0.326884\pi$
$739$	8.39060e12	1.03489	0.517444	+	0.855717i	$0.326884\pi$
$740$	−1.04259e13	−1.27811
$741$	0	0
$742$	1.96882e13	2.38445
$743$	−7.40995e11	−0.0892001	−0.0446001	−	0.999005i	$-0.514201\pi$
$743$	−7.40995e11	−0.0892001	−0.0446001	+	0.999005i	$0.514201\pi$
$744$	0	0
$745$	9.05048e12	1.07639
$746$	1.30353e13	1.54098
$747$	0	0
$748$	−5.80918e11	−0.0678512
$749$	8.34491e12	0.968843
$750$	0	0
$751$	1.41125e13	1.61891	0.809457	−	0.587180i	$-0.199762\pi$
$751$	1.41125e13	1.61891	0.809457	+	0.587180i	$0.199762\pi$
$752$	4.96996e12	0.566726
$753$	0	0
$754$	−7.39871e12	−0.833652
$755$	−1.34139e13	−1.50243
$756$	0	0
$757$	5.66623e12	0.627138	0.313569	−	0.949565i	$-0.398475\pi$
$757$	5.66623e12	0.627138	0.313569	+	0.949565i	$0.398475\pi$
$758$	−1.10328e13	−1.21387
$759$	0	0
$760$	−1.09556e12	−0.119117
$761$	1.34131e13	1.44977	0.724883	−	0.688872i	$-0.241894\pi$
$761$	1.34131e13	1.44977	0.724883	+	0.688872i	$0.241894\pi$
$762$	0	0
$763$	−1.15629e13	−1.23511
$764$	−5.34658e12	−0.567749
$765$	0	0
$766$	−1.02495e13	−1.07566
$767$	1.35717e12	0.141598
$768$	0	0
$769$	−6.64729e12	−0.685450	−0.342725	−	0.939436i	$-0.611350\pi$
$769$	−6.64729e12	−0.685450	−0.342725	+	0.939436i	$0.611350\pi$
$770$	2.22754e13	2.28358
$771$	0	0
$772$	3.13255e12	0.317409
$773$	−1.79549e13	−1.80874	−0.904369	−	0.426752i	$-0.859658\pi$
$773$	−1.79549e13	−1.80874	−0.904369	+	0.426752i	$0.859658\pi$
$774$	0	0
$775$	−1.82943e13	−1.82162
$776$	−1.03509e11	−0.0102471
$777$	0	0
$778$	3.61838e12	0.354083
$779$	−9.37924e12	−0.912534
$780$	0	0
$781$	−4.86077e12	−0.467493
$782$	1.65402e12	0.158165
$783$	0	0
$784$	−1.76140e13	−1.66508
$785$	1.76266e12	0.165675
$786$	0	0
$787$	9.36005e12	0.869745	0.434873	−	0.900492i	$-0.356793\pi$
$787$	9.36005e12	0.869745	0.434873	+	0.900492i	$0.356793\pi$
$788$	1.45962e13	1.34856
$789$	0	0
$790$	2.18476e13	1.99563
$791$	3.06437e13	2.78322
$792$	0	0
$793$	2.11986e12	0.190361
$794$	1.12774e13	1.00697
$795$	0	0
$796$	1.18229e12	0.104380
$797$	1.60113e13	1.40561	0.702803	−	0.711384i	$-0.251931\pi$
$797$	1.60113e13	1.40561	0.702803	+	0.711384i	$0.251931\pi$
$798$	0	0
$799$	6.82724e11	0.0592631
$800$	−1.97863e13	−1.70789
$801$	0	0
$802$	−8.24419e12	−0.703661
$803$	4.92350e11	0.0417882
$804$	0	0
$805$	−3.23619e13	−2.71614
$806$	−1.20457e13	−1.00537
$807$	0	0
$808$	1.27687e12	0.105389
$809$	−1.53912e13	−1.26329	−0.631645	−	0.775258i	$-0.717620\pi$
$809$	−1.53912e13	−1.26329	−0.631645	+	0.775258i	$0.717620\pi$
$810$	0	0
$811$	−1.81531e13	−1.47352	−0.736760	−	0.676154i	$-0.763645\pi$
$811$	−1.81531e13	−1.47352	−0.736760	+	0.676154i	$0.763645\pi$
$812$	2.69571e13	2.17606
$813$	0	0
$814$	9.64581e12	0.770068
$815$	1.76975e13	1.40509
$816$	0	0
$817$	2.85747e13	2.24379
$818$	8.67609e11	0.0677539
$819$	0	0
$820$	1.35812e13	1.04900
$821$	−2.20209e13	−1.69157	−0.845786	−	0.533523i	$-0.820868\pi$
$821$	−2.20209e13	−1.69157	−0.845786	+	0.533523i	$0.820868\pi$
$822$	0	0
$823$	−1.72414e13	−1.31000	−0.655002	−	0.755627i	$-0.727332\pi$
$823$	−1.72414e13	−1.31000	−0.655002	+	0.755627i	$0.727332\pi$
$824$	−2.42832e11	−0.0183499
$825$	0	0
$826$	−9.69105e12	−0.724370
$827$	3.54008e12	0.263171	0.131585	−	0.991305i	$-0.457993\pi$
$827$	3.54008e12	0.263171	0.131585	+	0.991305i	$0.457993\pi$
$828$	0	0
$829$	1.85206e13	1.36194	0.680971	−	0.732310i	$-0.261558\pi$
$829$	1.85206e13	1.36194	0.680971	+	0.732310i	$0.261558\pi$
$830$	3.35702e12	0.245529
$831$	0	0
$832$	−6.91517e12	−0.500320
$833$	−2.41964e12	−0.174120
$834$	0	0
$835$	9.47171e11	0.0674279
$836$	1.28726e13	0.911462
$837$	0	0
$838$	2.80399e11	0.0196417
$839$	−1.10025e13	−0.766591	−0.383295	−	0.923626i	$-0.625211\pi$
$839$	−1.10025e13	−0.766591	−0.383295	+	0.923626i	$0.625211\pi$
$840$	0	0
$841$	8.58144e12	0.591532
$842$	−3.13019e13	−2.14618
$843$	0	0
$844$	−7.21235e12	−0.489256
$845$	1.72704e13	1.16532
$846$	0	0
$847$	1.42832e13	0.953563
$848$	1.45156e13	0.963951
$849$	0	0
$850$	−2.60407e12	−0.171107
$851$	−1.40135e13	−0.915936
$852$	0	0
$853$	−1.48235e13	−0.958693	−0.479347	−	0.877626i	$-0.659126\pi$
$853$	−1.48235e13	−0.958693	−0.479347	+	0.877626i	$0.659126\pi$
$854$	−1.51371e13	−0.973826
$855$	0	0
$856$	−5.49832e11	−0.0350024
$857$	7.68399e12	0.486601	0.243301	−	0.969951i	$-0.421770\pi$
$857$	7.68399e12	0.486601	0.243301	+	0.969951i	$0.421770\pi$
$858$	0	0
$859$	−1.90428e12	−0.119333	−0.0596667	−	0.998218i	$-0.519004\pi$
$859$	−1.90428e12	−0.119333	−0.0596667	+	0.998218i	$0.519004\pi$
$860$	−4.13764e13	−2.57934
$861$	0	0
$862$	4.47050e13	2.75786
$863$	−3.09626e12	−0.190015	−0.0950077	−	0.995477i	$-0.530288\pi$
$863$	−3.09626e12	−0.190015	−0.0950077	+	0.995477i	$0.530288\pi$
$864$	0	0
$865$	1.89162e13	1.14884
$866$	−4.08197e13	−2.46626
$867$	0	0
$868$	4.38885e13	2.62429
$869$	−1.03137e13	−0.613513
$870$	0	0
$871$	−8.66411e12	−0.510084
$872$	7.61858e11	0.0446221
$873$	0	0
$874$	−3.66516e13	−2.12467
$875$	8.39656e12	0.484245
$876$	0	0
$877$	4.14925e12	0.236849	0.118425	−	0.992963i	$-0.462216\pi$
$877$	4.14925e12	0.236849	0.118425	+	0.992963i	$0.462216\pi$
$878$	1.41051e13	0.801032
$879$	0	0
$880$	1.64231e13	0.923173
$881$	1.36244e13	0.761948	0.380974	−	0.924586i	$-0.375589\pi$
$881$	1.36244e13	0.761948	0.380974	+	0.924586i	$0.375589\pi$
$882$	0	0
$883$	−1.83775e13	−1.01734	−0.508668	−	0.860963i	$-0.669862\pi$
$883$	−1.83775e13	−1.01734	−0.508668	+	0.860963i	$0.669862\pi$
$884$	−8.74892e11	−0.0481858
$885$	0	0
$886$	−1.64251e13	−0.895479
$887$	2.97823e13	1.61548	0.807741	−	0.589538i	$-0.200690\pi$
$887$	2.97823e13	1.61548	0.807741	+	0.589538i	$0.200690\pi$
$888$	0	0
$889$	−1.66975e13	−0.896592
$890$	−1.06546e13	−0.569223
$891$	0	0
$892$	−1.58580e12	−0.0838700
$893$	−1.51286e13	−0.796097
$894$	0	0
$895$	3.82301e12	0.199160
$896$	3.81760e12	0.197881
$897$	0	0
$898$	−4.75546e13	−2.44033
$899$	3.75902e13	1.91936
$900$	0	0
$901$	1.99401e12	0.100801
$902$	−1.25651e13	−0.632028
$903$	0	0
$904$	−2.01906e12	−0.100552
$905$	2.11524e13	1.04819
$906$	0	0
$907$	−3.49542e13	−1.71501	−0.857504	−	0.514477i	$-0.827986\pi$
$907$	−3.49542e13	−1.71501	−0.857504	+	0.514477i	$0.827986\pi$
$908$	1.78486e13	0.871398
$909$	0	0
$910$	3.35479e13	1.62173
$911$	−2.02274e13	−0.972989	−0.486494	−	0.873684i	$-0.661725\pi$
$911$	−2.02274e13	−0.972989	−0.486494	+	0.873684i	$0.661725\pi$
$912$	0	0
$913$	−1.58476e12	−0.0754824
$914$	−3.62104e13	−1.71623
$915$	0	0
$916$	−4.05827e13	−1.90463
$917$	−3.95442e13	−1.84681
$918$	0	0
$919$	4.43929e12	0.205302	0.102651	−	0.994717i	$-0.467268\pi$
$919$	4.43929e12	0.205302	0.102651	+	0.994717i	$0.467268\pi$
$920$	2.13227e12	0.0981290
$921$	0	0
$922$	−5.58292e13	−2.54432
$923$	−7.32057e12	−0.331999
$924$	0	0
$925$	2.20628e13	0.990884
$926$	8.88104e12	0.396930
$927$	0	0
$928$	4.06560e13	1.79953
$929$	1.16451e13	0.512948	0.256474	−	0.966551i	$-0.417439\pi$
$929$	1.16451e13	0.512948	0.256474	+	0.966551i	$0.417439\pi$
$930$	0	0
$931$	5.36170e13	2.33899
$932$	−3.46492e13	−1.50425
$933$	0	0
$934$	−3.70200e13	−1.59175
$935$	2.25604e12	0.0965372
$936$	0	0
$937$	−6.68375e12	−0.283265	−0.141632	−	0.989919i	$-0.545235\pi$
$937$	−6.68375e12	−0.283265	−0.141632	+	0.989919i	$0.545235\pi$
$938$	6.18670e13	2.60943
$939$	0	0
$940$	2.19063e13	0.915152
$941$	2.77843e13	1.15517	0.577586	−	0.816330i	$-0.303995\pi$
$941$	2.77843e13	1.15517	0.577586	+	0.816330i	$0.303995\pi$
$942$	0	0
$943$	1.82547e13	0.751748
$944$	−7.14498e12	−0.292838
$945$	0	0
$946$	3.82806e13	1.55406
$947$	2.02254e13	0.817189	0.408595	−	0.912716i	$-0.366019\pi$
$947$	2.02254e13	0.817189	0.408595	+	0.912716i	$0.366019\pi$
$948$	0	0
$949$	7.41504e11	0.0296767
$950$	5.77040e13	2.29853
$951$	0	0
$952$	2.50996e11	0.00990379
$953$	1.95165e13	0.766450	0.383225	−	0.923655i	$-0.374813\pi$
$953$	1.95165e13	0.766450	0.383225	+	0.923655i	$0.374813\pi$
$954$	0	0
$955$	2.07639e13	0.807781
$956$	2.12557e13	0.823030
$957$	0	0
$958$	−3.01556e13	−1.15671
$959$	1.03946e13	0.396846
$960$	0	0
$961$	3.47604e13	1.31471
$962$	1.45271e13	0.546878
$963$	0	0
$964$	6.32727e12	0.235977
$965$	−1.21655e13	−0.451603
$966$	0	0
$967$	4.20323e13	1.54584	0.772919	−	0.634504i	$-0.218796\pi$
$967$	4.20323e13	1.54584	0.772919	+	0.634504i	$0.218796\pi$
$968$	−9.41094e11	−0.0344504
$969$	0	0
$970$	1.00053e13	0.362876
$971$	−8.15237e12	−0.294305	−0.147152	−	0.989114i	$-0.547011\pi$
$971$	−8.15237e12	−0.294305	−0.147152	+	0.989114i	$0.547011\pi$
$972$	0	0
$973$	−4.95988e13	−1.77404
$974$	−3.32747e13	−1.18467
$975$	0	0
$976$	−1.11602e13	−0.393684
$977$	1.48575e13	0.521699	0.260849	−	0.965379i	$-0.415997\pi$
$977$	1.48575e13	0.521699	0.260849	+	0.965379i	$0.415997\pi$
$978$	0	0
$979$	5.02977e12	0.174995
$980$	−7.76379e13	−2.68879
$981$	0	0
$982$	−5.59004e13	−1.91828
$983$	8.38233e12	0.286335	0.143167	−	0.989698i	$-0.454271\pi$
$983$	8.38233e12	0.286335	0.143167	+	0.989698i	$0.454271\pi$
$984$	0	0
$985$	−5.66855e13	−1.91871
$986$	5.35073e12	0.180288
$987$	0	0
$988$	1.93868e13	0.647293
$989$	−5.56146e13	−1.84844
$990$	0	0
$991$	3.54621e13	1.16797	0.583986	−	0.811763i	$-0.301492\pi$
$991$	3.54621e13	1.16797	0.583986	+	0.811763i	$0.301492\pi$
$992$	6.61915e13	2.17020
$993$	0	0
$994$	5.22733e13	1.69840
$995$	−4.59152e12	−0.148509
$996$	0	0
$997$	4.17955e13	1.33968	0.669841	−	0.742505i	$-0.266362\pi$
$997$	4.17955e13	1.33968	0.669841	+	0.742505i	$0.266362\pi$
$998$	7.75919e13	2.47588
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.a.c.1.3	yes	3
3.2	odd	2		27.10.a.b.1.1	✓	3
9.2	odd	6		81.10.c.h.28.3		6
9.4	even	3		81.10.c.g.55.1		6
9.5	odd	6		81.10.c.h.55.3		6
9.7	even	3		81.10.c.g.28.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.b.1.1	✓	3	3.2	odd	2
27.10.a.c.1.3	yes	3	1.1	even	1	trivial
81.10.c.g.28.1		6	9.7	even	3
81.10.c.g.55.1		6	9.4	even	3
81.10.c.h.28.3		6	9.2	odd	6
81.10.c.h.55.3		6	9.5	odd	6

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+32.3331 q^{2} +533.432 q^{4} -2071.63 q^{5} -10517.1 q^{7} +692.954 q^{8} -66982.2 q^{10} +31620.5 q^{11} +47622.2 q^{13} -340051. q^{14} -250712. q^{16} -34440.3 q^{17} +763166. q^{19} -1.10507e6 q^{20} +1.02239e6 q^{22} -1.48534e6 q^{23} +2.33851e6 q^{25} +1.53977e6 q^{26} -5.61016e6 q^{28} -4.80506e6 q^{29} -7.82305e6 q^{31} -8.46109e6 q^{32} -1.11356e6 q^{34} +2.17875e7 q^{35} +9.43456e6 q^{37} +2.46755e7 q^{38} -1.43554e6 q^{40} -1.22899e7 q^{41} +3.74423e7 q^{43} +1.68674e7 q^{44} -4.80258e7 q^{46} -1.98234e7 q^{47} +7.02560e7 q^{49} +7.56113e7 q^{50} +2.54032e7 q^{52} -5.78978e7 q^{53} -6.55059e7 q^{55} -7.28788e6 q^{56} -1.55363e8 q^{58} +2.84988e7 q^{59} +4.45141e7 q^{61} -2.52944e8 q^{62} -1.45209e8 q^{64} -9.86553e7 q^{65} -1.81934e8 q^{67} -1.83715e7 q^{68} +7.04459e8 q^{70} -1.53722e8 q^{71} +1.55706e7 q^{73} +3.05049e8 q^{74} +4.07097e8 q^{76} -3.32557e8 q^{77} -3.26170e8 q^{79} +5.19381e8 q^{80} -3.97371e8 q^{82} -5.01181e7 q^{83} +7.13473e7 q^{85} +1.21063e9 q^{86} +2.19116e7 q^{88} +1.59066e8 q^{89} -5.00848e8 q^{91} -7.92329e8 q^{92} -6.40953e8 q^{94} -1.58099e9 q^{95} -1.49373e8 q^{97} +2.27160e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8} - 18981 q^{10} - 16863 q^{11} + 116916 q^{13} - 503463 q^{14} - 239919 q^{16} - 1014048 q^{17} - 15222 q^{19} - 2548407 q^{20} + 305721 q^{22}+ \cdots + 2592182286 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 597 * q^4 - 1983 * q^5 - 3693 * q^7 - 4503 * q^8 - 18981 * q^10 - 16863 * q^11 + 116916 * q^13 - 503463 * q^14 - 239919 * q^16 - 1014048 * q^17 - 15222 * q^19 - 2548407 * q^20 + 305721 * q^22 - 2927118 * q^23 + 2133732 * q^25 - 4765116 * q^26 - 3535725 * q^28 - 5768790 * q^29 - 6575223 * q^31 + 2687697 * q^32 + 17098128 * q^34 + 17340537 * q^35 - 11686026 * q^37 + 50473374 * q^38 - 4130811 * q^40 + 22213518 * q^41 + 45384414 * q^43 + 57206991 * q^44 - 95638590 * q^46 - 12392034 * q^47 + 18933462 * q^49 + 82984044 * q^50 + 182736492 * q^52 - 80579637 * q^53 - 174735333 * q^55 - 21850521 * q^56 - 102758922 * q^58 - 244026660 * q^59 + 369729960 * q^61 - 166297341 * q^62 - 420692127 * q^64 - 492225684 * q^65 - 252614586 * q^67 - 162270144 * q^68 + 935092377 * q^70 - 403193088 * q^71 - 406626717 * q^73 + 641373558 * q^74 - 39499590 * q^76 - 360443199 * q^77 + 265451856 * q^79 + 1168359837 * q^80 - 884408166 * q^82 - 121625871 * q^83 + 316251216 * q^85 + 2368956570 * q^86 + 219257739 * q^88 - 377904006 * q^89 + 245059140 * q^91 + 1178754558 * q^92 + 86583438 * q^94 - 536878770 * q^95 - 438907539 * q^97 + 2592182286 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more