LMFDB - Embedded newform 162.8.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,8,Mod(1,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-32,0,256,-528] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.6063741284$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.43103376.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 $ x^4 - 2x^3 - 383x^2 + 384*x + 18612
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.05976$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000 q^{2} +64.0000 q^{4} -325.500 q^{5} -1450.45 q^{7} -512.000 q^{8} +2604.00 q^{10} +5555.80 q^{11} +13503.7 q^{13} +11603.6 q^{14} +4096.00 q^{16} -11828.7 q^{17} +44544.0 q^{19} -20832.0 q^{20} -44446.4 q^{22} -26806.6 q^{23} +27825.1 q^{25} -108030. q^{26} -92829.0 q^{28} -140864. q^{29} +165692. q^{31} -32768.0 q^{32} +94629.7 q^{34} +472122. q^{35} +36343.3 q^{37} -356352. q^{38} +166656. q^{40} +478069. q^{41} -604215. q^{43} +355571. q^{44} +214453. q^{46} -503065. q^{47} +1.28027e6 q^{49} -222601. q^{50} +864238. q^{52} -1.99651e6 q^{53} -1.80841e6 q^{55} +742632. q^{56} +1.12691e6 q^{58} -609072. q^{59} +1.90187e6 q^{61} -1.32554e6 q^{62} +262144. q^{64} -4.39546e6 q^{65} +1.77367e6 q^{67} -757037. q^{68} -3.77698e6 q^{70} -1.00843e6 q^{71} +146188. q^{73} -290746. q^{74} +2.85081e6 q^{76} -8.05843e6 q^{77} -7.27378e6 q^{79} -1.33325e6 q^{80} -3.82456e6 q^{82} +5.37693e6 q^{83} +3.85024e6 q^{85} +4.83372e6 q^{86} -2.84457e6 q^{88} +2.94159e6 q^{89} -1.95865e7 q^{91} -1.71562e6 q^{92} +4.02452e6 q^{94} -1.44991e7 q^{95} -7.91467e6 q^{97} -1.02422e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) $ 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * 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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−325.500	−1.16454	−0.582272	−	0.812994i	$-0.697836\pi$
$5$	−325.500	−1.16454	−0.582272	+	0.812994i	$0.697836\pi$
$6$	0	0
$7$	−1450.45	−1.59831	−0.799155	−	0.601126i	$-0.794719\pi$
$7$	−1450.45	−1.59831	−0.799155	+	0.601126i	$0.794719\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	2604.00	0.823457
$11$	5555.80	1.25855	0.629277	−	0.777181i	$-0.283351\pi$
$11$	5555.80	1.25855	0.629277	+	0.777181i	$0.283351\pi$
$12$	0	0
$13$	13503.7	1.70471	0.852357	−	0.522961i	$-0.175173\pi$
$13$	13503.7	1.70471	0.852357	+	0.522961i	$0.175173\pi$
$14$	11603.6	1.13018
$15$	0	0
$16$	4096.00	0.250000
$17$	−11828.7	−0.583937	−0.291969	−	0.956428i	$-0.594310\pi$
$17$	−11828.7	−0.583937	−0.291969	+	0.956428i	$0.594310\pi$
$18$	0	0
$19$	44544.0	1.48988	0.744940	−	0.667131i	$-0.232478\pi$
$19$	44544.0	1.48988	0.744940	+	0.667131i	$0.232478\pi$
$20$	−20832.0	−0.582272
$21$	0	0
$22$	−44446.4	−0.889932
$23$	−26806.6	−0.459404	−0.229702	−	0.973261i	$-0.573775\pi$
$23$	−26806.6	−0.459404	−0.229702	+	0.973261i	$0.573775\pi$
$24$	0	0
$25$	27825.1	0.356162
$26$	−108030.	−1.20541
$27$	0	0
$28$	−92829.0	−0.799155
$29$	−140864.	−1.07253	−0.536263	−	0.844051i	$-0.680164\pi$
$29$	−140864.	−1.07253	−0.536263	+	0.844051i	$0.680164\pi$
$30$	0	0
$31$	165692.	0.998933	0.499467	−	0.866333i	$-0.333529\pi$
$31$	165692.	0.998933	0.499467	+	0.866333i	$0.333529\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	94629.7	0.412906
$35$	472122.	1.86130
$36$	0	0
$37$	36343.3	0.117956	0.0589778	−	0.998259i	$-0.481216\pi$
$37$	36343.3	0.117956	0.0589778	+	0.998259i	$0.481216\pi$
$38$	−356352.	−1.05350
$39$	0	0
$40$	166656.	0.411728
$41$	478069.	1.08330	0.541648	−	0.840605i	$-0.317800\pi$
$41$	478069.	1.08330	0.541648	+	0.840605i	$0.317800\pi$
$42$	0	0
$43$	−604215.	−1.15892	−0.579458	−	0.815002i	$-0.696736\pi$
$43$	−604215.	−1.15892	−0.579458	+	0.815002i	$0.696736\pi$
$44$	355571.	0.629277
$45$	0	0
$46$	214453.	0.324848
$47$	−503065.	−0.706776	−0.353388	−	0.935477i	$-0.614970\pi$
$47$	−503065.	−0.706776	−0.353388	+	0.935477i	$0.614970\pi$
$48$	0	0
$49$	1.28027e6	1.55459
$50$	−222601.	−0.251844
$51$	0	0
$52$	864238.	0.852357
$53$	−1.99651e6	−1.84207	−0.921036	−	0.389477i	$-0.872656\pi$
$53$	−1.99651e6	−1.84207	−0.921036	+	0.389477i	$0.872656\pi$
$54$	0	0
$55$	−1.80841e6	−1.46564
$56$	742632.	0.565088
$57$	0	0
$58$	1.12691e6	0.758390
$59$	−609072.	−0.386088	−0.193044	−	0.981190i	$-0.561836\pi$
$59$	−609072.	−0.386088	−0.193044	+	0.981190i	$0.561836\pi$
$60$	0	0
$61$	1.90187e6	1.07282	0.536408	−	0.843959i	$-0.319781\pi$
$61$	1.90187e6	1.07282	0.536408	+	0.843959i	$0.319781\pi$
$62$	−1.32554e6	−0.706353
$63$	0	0
$64$	262144.	0.125000
$65$	−4.39546e6	−1.98521
$66$	0	0
$67$	1.77367e6	0.720461	0.360231	−	0.932863i	$-0.382698\pi$
$67$	1.77367e6	0.720461	0.360231	+	0.932863i	$0.382698\pi$
$68$	−757037.	−0.291969
$69$	0	0
$70$	−3.77698e6	−1.31614
$71$	−1.00843e6	−0.334380	−0.167190	−	0.985925i	$-0.553469\pi$
$71$	−1.00843e6	−0.334380	−0.167190	+	0.985925i	$0.553469\pi$
$72$	0	0
$73$	146188.	0.0439825	0.0219913	−	0.999758i	$-0.492999\pi$
$73$	146188.	0.0439825	0.0219913	+	0.999758i	$0.492999\pi$
$74$	−290746.	−0.0834072
$75$	0	0
$76$	2.85081e6	0.744940
$77$	−8.05843e6	−2.01156
$78$	0	0
$79$	−7.27378e6	−1.65984	−0.829919	−	0.557884i	$-0.811613\pi$
$79$	−7.27378e6	−1.65984	−0.829919	+	0.557884i	$0.811613\pi$
$80$	−1.33325e6	−0.291136
$81$	0	0
$82$	−3.82456e6	−0.766007
$83$	5.37693e6	1.03219	0.516096	−	0.856531i	$-0.327385\pi$
$83$	5.37693e6	1.03219	0.516096	+	0.856531i	$0.327385\pi$
$84$	0	0
$85$	3.85024e6	0.680020
$86$	4.83372e6	0.819478
$87$	0	0
$88$	−2.84457e6	−0.444966
$89$	2.94159e6	0.442301	0.221150	−	0.975240i	$-0.429019\pi$
$89$	2.94159e6	0.442301	0.221150	+	0.975240i	$0.429019\pi$
$90$	0	0
$91$	−1.95865e7	−2.72466
$92$	−1.71562e6	−0.229702
$93$	0	0
$94$	4.02452e6	0.499766
$95$	−1.44991e7	−1.73503
$96$	0	0
$97$	−7.91467e6	−0.880505	−0.440253	−	0.897874i	$-0.645111\pi$
$97$	−7.91467e6	−0.880505	−0.440253	+	0.897874i	$0.645111\pi$
$98$	−1.02422e7	−1.09926
$99$	0	0
$100$	1.78081e6	0.178081
$101$	−1.85158e7	−1.78820	−0.894101	−	0.447865i	$-0.852185\pi$
$101$	−1.85158e7	−1.78820	−0.894101	+	0.447865i	$0.852185\pi$
$102$	0	0
$103$	1.00406e7	0.905381	0.452691	−	0.891668i	$-0.350464\pi$
$103$	1.00406e7	0.905381	0.452691	+	0.891668i	$0.350464\pi$
$104$	−6.91390e6	−0.602707
$105$	0	0
$106$	1.59721e7	1.30254
$107$	−1.26542e7	−0.998601	−0.499300	−	0.866429i	$-0.666410\pi$
$107$	−1.26542e7	−0.998601	−0.499300	+	0.866429i	$0.666410\pi$
$108$	0	0
$109$	−1.26740e6	−0.0937388	−0.0468694	−	0.998901i	$-0.514924\pi$
$109$	−1.26740e6	−0.0937388	−0.0468694	+	0.998901i	$0.514924\pi$
$110$	1.44673e7	1.03636
$111$	0	0
$112$	−5.94106e6	−0.399577
$113$	1.48420e7	0.967651	0.483825	−	0.875165i	$-0.339247\pi$
$113$	1.48420e7	0.967651	0.483825	+	0.875165i	$0.339247\pi$
$114$	0	0
$115$	8.72555e6	0.534996
$116$	−9.01531e6	−0.536263
$117$	0	0
$118$	4.87257e6	0.273005
$119$	1.71570e7	0.933312
$120$	0	0
$121$	1.13797e7	0.583959
$122$	−1.52149e7	−0.758596
$123$	0	0
$124$	1.06043e7	0.499467
$125$	1.63726e7	0.749778
$126$	0	0
$127$	−9.71798e6	−0.420982	−0.210491	−	0.977596i	$-0.567506\pi$
$127$	−9.71798e6	−0.420982	−0.210491	+	0.977596i	$0.567506\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	3.51636e7	1.40376
$131$	1.83062e7	0.711459	0.355729	−	0.934589i	$-0.384233\pi$
$131$	1.83062e7	0.711459	0.355729	+	0.934589i	$0.384233\pi$
$132$	0	0
$133$	−6.46090e7	−2.38129
$134$	−1.41893e7	−0.509443
$135$	0	0
$136$	6.05630e6	0.206453
$137$	−4.84572e7	−1.61004	−0.805019	−	0.593248i	$-0.797845\pi$
$137$	−4.84572e7	−1.61004	−0.805019	+	0.593248i	$0.797845\pi$
$138$	0	0
$139$	−3.79831e7	−1.19960	−0.599802	−	0.800148i	$-0.704754\pi$
$139$	−3.79831e7	−1.19960	−0.599802	+	0.800148i	$0.704754\pi$
$140$	3.02158e7	0.930650
$141$	0	0
$142$	8.06741e6	0.236442
$143$	7.50239e7	2.14547
$144$	0	0
$145$	4.58513e7	1.24900
$146$	−1.16950e6	−0.0311004
$147$	0	0
$148$	2.32597e6	0.0589778
$149$	−3.58082e7	−0.886810	−0.443405	−	0.896321i	$-0.646230\pi$
$149$	−3.58082e7	−0.886810	−0.443405	+	0.896321i	$0.646230\pi$
$150$	0	0
$151$	4.81020e7	1.13696	0.568478	−	0.822699i	$-0.307533\pi$
$151$	4.81020e7	1.13696	0.568478	+	0.822699i	$0.307533\pi$
$152$	−2.28065e7	−0.526752
$153$	0	0
$154$	6.44674e7	1.42239
$155$	−5.39328e7	−1.16330
$156$	0	0
$157$	−7.95290e7	−1.64012	−0.820062	−	0.572274i	$-0.806061\pi$
$157$	−7.95290e7	−1.64012	−0.820062	+	0.572274i	$0.806061\pi$
$158$	5.81903e7	1.17368
$159$	0	0
$160$	1.06660e7	0.205864
$161$	3.88818e7	0.734269
$162$	0	0
$163$	−4.46340e7	−0.807252	−0.403626	−	0.914924i	$-0.632250\pi$
$163$	−4.46340e7	−0.807252	−0.403626	+	0.914924i	$0.632250\pi$
$164$	3.05964e7	0.541648
$165$	0	0
$166$	−4.30154e7	−0.729870
$167$	7.08646e6	0.117739	0.0588697	−	0.998266i	$-0.481250\pi$
$167$	7.08646e6	0.117739	0.0588697	+	0.998266i	$0.481250\pi$
$168$	0	0
$169$	1.19602e8	1.90605
$170$	−3.08019e7	−0.480847
$171$	0	0
$172$	−3.86698e7	−0.579458
$173$	6.88987e7	1.01170	0.505848	−	0.862623i	$-0.331180\pi$
$173$	6.88987e7	1.01170	0.505848	+	0.862623i	$0.331180\pi$
$174$	0	0
$175$	−4.03591e7	−0.569256
$176$	2.27565e7	0.314639
$177$	0	0
$178$	−2.35327e7	−0.312754
$179$	8.88821e7	1.15832	0.579161	−	0.815214i	$-0.303380\pi$
$179$	8.88821e7	1.15832	0.579161	+	0.815214i	$0.303380\pi$
$180$	0	0
$181$	7.42211e7	0.930363	0.465182	−	0.885215i	$-0.345989\pi$
$181$	7.42211e7	0.930363	0.465182	+	0.885215i	$0.345989\pi$
$182$	1.56692e8	1.92663
$183$	0	0
$184$	1.37250e7	0.162424
$185$	−1.18297e7	−0.137364
$186$	0	0
$187$	−6.57179e7	−0.734917
$188$	−3.21962e7	−0.353388
$189$	0	0
$190$	1.15992e8	1.22685
$191$	−2.55625e7	−0.265452	−0.132726	−	0.991153i	$-0.542373\pi$
$191$	−2.55625e7	−0.265452	−0.132726	+	0.991153i	$0.542373\pi$
$192$	0	0
$193$	−9.83708e7	−0.984953	−0.492477	−	0.870326i	$-0.663908\pi$
$193$	−9.83708e7	−0.984953	−0.492477	+	0.870326i	$0.663908\pi$
$194$	6.33174e7	0.622611
$195$	0	0
$196$	8.19375e7	0.777296
$197$	−1.40581e7	−0.131007	−0.0655036	−	0.997852i	$-0.520865\pi$
$197$	−1.40581e7	−0.131007	−0.0655036	+	0.997852i	$0.520865\pi$
$198$	0	0
$199$	2.87427e7	0.258548	0.129274	−	0.991609i	$-0.458735\pi$
$199$	2.87427e7	0.258548	0.129274	+	0.991609i	$0.458735\pi$
$200$	−1.42465e7	−0.125922
$201$	0	0
$202$	1.48126e8	1.26445
$203$	2.04317e8	1.71423
$204$	0	0
$205$	−1.55612e8	−1.26155
$206$	−8.03252e7	−0.640201
$207$	0	0
$208$	5.53112e7	0.426178
$209$	2.47477e8	1.87510
$210$	0	0
$211$	−1.97309e8	−1.44597	−0.722984	−	0.690865i	$-0.757230\pi$
$211$	−1.97309e8	−1.44597	−0.722984	+	0.690865i	$0.757230\pi$
$212$	−1.27777e8	−0.921036
$213$	0	0
$214$	1.01234e8	0.706117
$215$	1.96672e8	1.34961
$216$	0	0
$217$	−2.40329e8	−1.59660
$218$	1.01392e7	0.0662834
$219$	0	0
$220$	−1.15738e8	−0.732821
$221$	−1.59731e8	−0.995446
$222$	0	0
$223$	−7.48286e7	−0.451857	−0.225928	−	0.974144i	$-0.572542\pi$
$223$	−7.48286e7	−0.451857	−0.225928	+	0.974144i	$0.572542\pi$
$224$	4.75285e7	0.282544
$225$	0	0
$226$	−1.18736e8	−0.684232
$227$	−7.23482e7	−0.410523	−0.205261	−	0.978707i	$-0.565804\pi$
$227$	−7.23482e7	−0.410523	−0.205261	+	0.978707i	$0.565804\pi$
$228$	0	0
$229$	−3.04784e8	−1.67713	−0.838567	−	0.544798i	$-0.816606\pi$
$229$	−3.04784e8	−1.67713	−0.838567	+	0.544798i	$0.816606\pi$
$230$	−6.98044e7	−0.378299
$231$	0	0
$232$	7.21224e7	0.379195
$233$	2.36334e7	0.122400	0.0612000	−	0.998126i	$-0.480507\pi$
$233$	2.36334e7	0.122400	0.0612000	+	0.998126i	$0.480507\pi$
$234$	0	0
$235$	1.63748e8	0.823072
$236$	−3.89806e7	−0.193044
$237$	0	0
$238$	−1.37256e8	−0.659951
$239$	−2.31014e8	−1.09458	−0.547288	−	0.836944i	$-0.684340\pi$
$239$	−2.31014e8	−1.09458	−0.547288	+	0.836944i	$0.684340\pi$
$240$	0	0
$241$	−5.93812e7	−0.273268	−0.136634	−	0.990622i	$-0.543628\pi$
$241$	−5.93812e7	−0.273268	−0.136634	+	0.990622i	$0.543628\pi$
$242$	−9.10377e7	−0.412921
$243$	0	0
$244$	1.21719e8	0.536408
$245$	−4.16729e8	−1.81039
$246$	0	0
$247$	6.01509e8	2.53982
$248$	−8.48345e7	−0.353176
$249$	0	0
$250$	−1.30981e8	−0.530173
$251$	−3.16166e8	−1.26199	−0.630997	−	0.775785i	$-0.717354\pi$
$251$	−3.16166e8	−1.26199	−0.630997	+	0.775785i	$0.717354\pi$
$252$	0	0
$253$	−1.48932e8	−0.578185
$254$	7.77439e7	0.297679
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	5.66309e7	0.208108	0.104054	−	0.994572i	$-0.466819\pi$
$257$	5.66309e7	0.208108	0.104054	+	0.994572i	$0.466819\pi$
$258$	0	0
$259$	−5.27143e7	−0.188529
$260$	−2.81309e8	−0.992607
$261$	0	0
$262$	−1.46450e8	−0.503077
$263$	−2.40640e7	−0.0815685	−0.0407842	−	0.999168i	$-0.512986\pi$
$263$	−2.40640e7	−0.0815685	−0.0407842	+	0.999168i	$0.512986\pi$
$264$	0	0
$265$	6.49865e8	2.14517
$266$	5.16872e8	1.68383
$267$	0	0
$268$	1.13515e8	0.360231
$269$	4.42132e8	1.38490	0.692451	−	0.721465i	$-0.256531\pi$
$269$	4.42132e8	1.38490	0.692451	+	0.721465i	$0.256531\pi$
$270$	0	0
$271$	1.81928e8	0.555272	0.277636	−	0.960686i	$-0.410449\pi$
$271$	1.81928e8	0.555272	0.277636	+	0.960686i	$0.410449\pi$
$272$	−4.84504e7	−0.145984
$273$	0	0
$274$	3.87658e8	1.13847
$275$	1.54591e8	0.448249
$276$	0	0
$277$	2.60786e8	0.737234	0.368617	−	0.929581i	$-0.379831\pi$
$277$	2.60786e8	0.737234	0.368617	+	0.929581i	$0.379831\pi$
$278$	3.03864e8	0.848249
$279$	0	0
$280$	−2.41727e8	−0.658069
$281$	3.55769e8	0.956523	0.478261	−	0.878217i	$-0.341267\pi$
$281$	3.55769e8	0.956523	0.478261	+	0.878217i	$0.341267\pi$
$282$	0	0
$283$	−1.50560e8	−0.394874	−0.197437	−	0.980316i	$-0.563262\pi$
$283$	−1.50560e8	−0.394874	−0.197437	+	0.980316i	$0.563262\pi$
$284$	−6.45393e7	−0.167190
$285$	0	0
$286$	−6.00191e8	−1.51708
$287$	−6.93418e8	−1.73144
$288$	0	0
$289$	−2.70420e8	−0.659017
$290$	−3.66810e8	−0.883178
$291$	0	0
$292$	9.35600e6	0.0219913
$293$	−2.18901e8	−0.508407	−0.254204	−	0.967151i	$-0.581813\pi$
$293$	−2.18901e8	−0.508407	−0.254204	+	0.967151i	$0.581813\pi$
$294$	0	0
$295$	1.98253e8	0.449616
$296$	−1.86078e7	−0.0417036
$297$	0	0
$298$	2.86466e8	0.627070
$299$	−3.61989e8	−0.783152
$300$	0	0
$301$	8.76387e8	1.85231
$302$	−3.84816e8	−0.803949
$303$	0	0
$304$	1.82452e8	0.372470
$305$	−6.19057e8	−1.24934
$306$	0	0
$307$	−5.49207e8	−1.08331	−0.541654	−	0.840602i	$-0.682202\pi$
$307$	−5.49207e8	−1.08331	−0.541654	+	0.840602i	$0.682202\pi$
$308$	−5.15739e8	−1.00578
$309$	0	0
$310$	4.31463e8	0.822578
$311$	−5.09200e8	−0.959901	−0.479951	−	0.877295i	$-0.659345\pi$
$311$	−5.09200e8	−0.959901	−0.479951	+	0.877295i	$0.659345\pi$
$312$	0	0
$313$	5.61841e8	1.03564	0.517819	−	0.855490i	$-0.326744\pi$
$313$	5.61841e8	1.03564	0.517819	+	0.855490i	$0.326744\pi$
$314$	6.36232e8	1.15974
$315$	0	0
$316$	−4.65522e8	−0.829919
$317$	−8.95957e8	−1.57972	−0.789859	−	0.613288i	$-0.789846\pi$
$317$	−8.95957e8	−1.57972	−0.789859	+	0.613288i	$0.789846\pi$
$318$	0	0
$319$	−7.82613e8	−1.34983
$320$	−8.53278e7	−0.145568
$321$	0	0
$322$	−3.11054e8	−0.519207
$323$	−5.26898e8	−0.869997
$324$	0	0
$325$	3.75743e8	0.607154
$326$	3.57072e8	0.570813
$327$	0	0
$328$	−2.44772e8	−0.383003
$329$	7.29673e8	1.12965
$330$	0	0
$331$	−7.05161e8	−1.06879	−0.534393	−	0.845236i	$-0.679460\pi$
$331$	−7.05161e8	−1.06879	−0.534393	+	0.845236i	$0.679460\pi$
$332$	3.44123e8	0.516096
$333$	0	0
$334$	−5.66917e7	−0.0832543
$335$	−5.77329e8	−0.839009
$336$	0	0
$337$	9.94415e8	1.41535	0.707674	−	0.706539i	$-0.249745\pi$
$337$	9.94415e8	1.41535	0.707674	+	0.706539i	$0.249745\pi$
$338$	−9.56814e8	−1.34778
$339$	0	0
$340$	2.46416e8	0.340010
$341$	9.20553e8	1.25721
$342$	0	0
$343$	−6.62466e8	−0.886409
$344$	3.09358e8	0.409739
$345$	0	0
$346$	−5.51190e8	−0.715377
$347$	−1.63922e7	−0.0210612	−0.0105306	−	0.999945i	$-0.503352\pi$
$347$	−1.63922e7	−0.0210612	−0.0105306	+	0.999945i	$0.503352\pi$
$348$	0	0
$349$	−4.26778e8	−0.537419	−0.268709	−	0.963221i	$-0.586597\pi$
$349$	−4.26778e8	−0.537419	−0.268709	+	0.963221i	$0.586597\pi$
$350$	3.22873e8	0.402525
$351$	0	0
$352$	−1.82052e8	−0.222483
$353$	1.60553e9	1.94271	0.971353	−	0.237641i	$-0.0763743\pi$
$353$	1.60553e9	1.94271	0.971353	+	0.237641i	$0.0763743\pi$
$354$	0	0
$355$	3.28243e8	0.389400
$356$	1.88262e8	0.221150
$357$	0	0
$358$	−7.11057e8	−0.819057
$359$	−1.39612e9	−1.59255	−0.796276	−	0.604934i	$-0.793199\pi$
$359$	−1.39612e9	−1.59255	−0.796276	+	0.604934i	$0.793199\pi$
$360$	0	0
$361$	1.09029e9	1.21974
$362$	−5.93769e8	−0.657866
$363$	0	0
$364$	−1.25354e9	−1.36233
$365$	−4.75840e7	−0.0512196
$366$	0	0
$367$	−1.50761e9	−1.59206	−0.796029	−	0.605258i	$-0.793070\pi$
$367$	−1.50761e9	−1.59206	−0.796029	+	0.605258i	$0.793070\pi$
$368$	−1.09800e8	−0.114851
$369$	0	0
$370$	9.46379e7	0.0971313
$371$	2.89585e9	2.94420
$372$	0	0
$373$	−7.73180e8	−0.771435	−0.385718	−	0.922617i	$-0.626046\pi$
$373$	−7.73180e8	−0.771435	−0.385718	+	0.922617i	$0.626046\pi$
$374$	5.25743e8	0.519664
$375$	0	0
$376$	2.57569e8	0.249883
$377$	−1.90219e9	−1.82835
$378$	0	0
$379$	1.65039e9	1.55722	0.778610	−	0.627508i	$-0.215925\pi$
$379$	1.65039e9	1.55722	0.778610	+	0.627508i	$0.215925\pi$
$380$	−9.27940e8	−0.867515
$381$	0	0
$382$	2.04500e8	0.187703
$383$	6.51438e8	0.592485	0.296242	−	0.955113i	$-0.404266\pi$
$383$	6.51438e8	0.592485	0.296242	+	0.955113i	$0.404266\pi$
$384$	0	0
$385$	2.62302e9	2.34255
$386$	7.86966e8	0.696467
$387$	0	0
$388$	−5.06539e8	−0.440253
$389$	8.72769e8	0.751754	0.375877	−	0.926670i	$-0.377342\pi$
$389$	8.72769e8	0.751754	0.375877	+	0.926670i	$0.377342\pi$
$390$	0	0
$391$	3.17088e8	0.268263
$392$	−6.55500e8	−0.549631
$393$	0	0
$394$	1.12465e8	0.0926360
$395$	2.36761e9	1.93295
$396$	0	0
$397$	3.62633e8	0.290871	0.145436	−	0.989368i	$-0.453542\pi$
$397$	3.62633e8	0.290871	0.145436	+	0.989368i	$0.453542\pi$
$398$	−2.29942e8	−0.182821
$399$	0	0
$400$	1.13972e8	0.0890404
$401$	−6.98398e8	−0.540876	−0.270438	−	0.962737i	$-0.587168\pi$
$401$	−6.98398e8	−0.540876	−0.270438	+	0.962737i	$0.587168\pi$
$402$	0	0
$403$	2.23746e9	1.70290
$404$	−1.18501e9	−0.894101
$405$	0	0
$406$	−1.63454e9	−1.21214
$407$	2.01916e8	0.148453
$408$	0	0
$409$	−1.23087e9	−0.889570	−0.444785	−	0.895637i	$-0.646720\pi$
$409$	−1.23087e9	−0.889570	−0.444785	+	0.895637i	$0.646720\pi$
$410$	1.24489e9	0.892048
$411$	0	0
$412$	6.42601e8	0.452691
$413$	8.83430e8	0.617088
$414$	0	0
$415$	−1.75019e9	−1.20203
$416$	−4.42490e8	−0.301354
$417$	0	0
$418$	−1.97982e9	−1.32589
$419$	1.59613e9	1.06003	0.530016	−	0.847988i	$-0.322186\pi$
$419$	1.59613e9	1.06003	0.530016	+	0.847988i	$0.322186\pi$
$420$	0	0
$421$	−7.72913e8	−0.504828	−0.252414	−	0.967619i	$-0.581224\pi$
$421$	−7.72913e8	−0.504828	−0.252414	+	0.967619i	$0.581224\pi$
$422$	1.57847e9	1.02245
$423$	0	0
$424$	1.02222e9	0.651271
$425$	−3.29135e8	−0.207976
$426$	0	0
$427$	−2.75857e9	−1.71469
$428$	−8.09870e8	−0.499300
$429$	0	0
$430$	−1.57338e9	−0.954318
$431$	−2.41483e8	−0.145284	−0.0726418	−	0.997358i	$-0.523143\pi$
$431$	−2.41483e8	−0.145284	−0.0726418	+	0.997358i	$0.523143\pi$
$432$	0	0
$433$	2.85764e9	1.69161	0.845805	−	0.533493i	$-0.179121\pi$
$433$	2.85764e9	1.69161	0.845805	+	0.533493i	$0.179121\pi$
$434$	1.92263e9	1.12897
$435$	0	0
$436$	−8.11133e7	−0.0468694
$437$	−1.19407e9	−0.684457
$438$	0	0
$439$	3.27002e8	0.184469	0.0922347	−	0.995737i	$-0.470599\pi$
$439$	3.27002e8	0.184469	0.0922347	+	0.995737i	$0.470599\pi$
$440$	9.25906e8	0.518182
$441$	0	0
$442$	1.27785e9	0.703886
$443$	−7.97209e8	−0.435672	−0.217836	−	0.975985i	$-0.569900\pi$
$443$	−7.97209e8	−0.435672	−0.217836	+	0.975985i	$0.569900\pi$
$444$	0	0
$445$	−9.57488e8	−0.515078
$446$	5.98629e8	0.319511
$447$	0	0
$448$	−3.80228e8	−0.199789
$449$	1.41817e9	0.739378	0.369689	−	0.929155i	$-0.379464\pi$
$449$	1.41817e9	0.739378	0.369689	+	0.929155i	$0.379464\pi$
$450$	0	0
$451$	2.65606e9	1.36339
$452$	9.49890e8	0.483825
$453$	0	0
$454$	5.78786e8	0.290284
$455$	6.37541e9	3.17298
$456$	0	0
$457$	−9.74906e8	−0.477811	−0.238905	−	0.971043i	$-0.576789\pi$
$457$	−9.74906e8	−0.477811	−0.238905	+	0.971043i	$0.576789\pi$
$458$	2.43827e9	1.18591
$459$	0	0
$460$	5.58435e8	0.267498
$461$	−1.37286e9	−0.652640	−0.326320	−	0.945259i	$-0.605809\pi$
$461$	−1.37286e9	−0.652640	−0.326320	+	0.945259i	$0.605809\pi$
$462$	0	0
$463$	1.78715e9	0.836809	0.418405	−	0.908261i	$-0.362589\pi$
$463$	1.78715e9	0.836809	0.418405	+	0.908261i	$0.362589\pi$
$464$	−5.76980e8	−0.268131
$465$	0	0
$466$	−1.89068e8	−0.0865499
$467$	−2.25768e9	−1.02578	−0.512889	−	0.858455i	$-0.671425\pi$
$467$	−2.25768e9	−1.02578	−0.512889	+	0.858455i	$0.671425\pi$
$468$	0	0
$469$	−2.57262e9	−1.15152
$470$	−1.30998e9	−0.581999
$471$	0	0
$472$	3.11845e8	0.136503
$473$	−3.35690e9	−1.45856
$474$	0	0
$475$	1.23944e9	0.530638
$476$	1.09805e9	0.466656
$477$	0	0
$478$	1.84811e9	0.773983
$479$	−1.74267e9	−0.724503	−0.362252	−	0.932080i	$-0.617992\pi$
$479$	−1.74267e9	−0.724503	−0.362252	+	0.932080i	$0.617992\pi$
$480$	0	0
$481$	4.90769e8	0.201080
$482$	4.75049e8	0.193230
$483$	0	0
$484$	7.28301e8	0.291979
$485$	2.57623e9	1.02539
$486$	0	0
$487$	−2.68849e9	−1.05477	−0.527384	−	0.849627i	$-0.676827\pi$
$487$	−2.68849e9	−1.05477	−0.527384	+	0.849627i	$0.676827\pi$
$488$	−9.73756e8	−0.379298
$489$	0	0
$490$	3.33383e9	1.28014
$491$	2.34881e9	0.895495	0.447747	−	0.894160i	$-0.352226\pi$
$491$	2.34881e9	0.895495	0.447747	+	0.894160i	$0.352226\pi$
$492$	0	0
$493$	1.66624e9	0.626287
$494$	−4.81207e9	−1.79592
$495$	0	0
$496$	6.78676e8	0.249733
$497$	1.46268e9	0.534442
$498$	0	0
$499$	−1.66742e9	−0.600751	−0.300375	−	0.953821i	$-0.597112\pi$
$499$	−1.66742e9	−0.600751	−0.300375	+	0.953821i	$0.597112\pi$
$500$	1.04785e9	0.374889
$501$	0	0
$502$	2.52933e9	0.892365
$503$	−3.41605e9	−1.19684	−0.598419	−	0.801183i	$-0.704204\pi$
$503$	−3.41605e9	−1.19684	−0.598419	+	0.801183i	$0.704204\pi$
$504$	0	0
$505$	6.02688e9	2.08244
$506$	1.19146e9	0.408838
$507$	0	0
$508$	−6.21951e8	−0.210491
$509$	−5.04639e9	−1.69617	−0.848083	−	0.529864i	$-0.822243\pi$
$509$	−5.04639e9	−1.69617	−0.848083	+	0.529864i	$0.822243\pi$
$510$	0	0
$511$	−2.12038e8	−0.0702977
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	−4.53047e8	−0.147154
$515$	−3.26823e9	−1.05436
$516$	0	0
$517$	−2.79493e9	−0.889516
$518$	4.21714e8	0.133310
$519$	0	0
$520$	2.25047e9	0.701879
$521$	−2.07610e9	−0.643156	−0.321578	−	0.946883i	$-0.604213\pi$
$521$	−2.07610e9	−0.643156	−0.321578	+	0.946883i	$0.604213\pi$
$522$	0	0
$523$	−2.82869e9	−0.864627	−0.432314	−	0.901723i	$-0.642303\pi$
$523$	−2.82869e9	−0.864627	−0.432314	+	0.901723i	$0.642303\pi$
$524$	1.17160e9	0.355729
$525$	0	0
$526$	1.92512e8	0.0576776
$527$	−1.95993e9	−0.583314
$528$	0	0
$529$	−2.68623e9	−0.788948
$530$	−5.19892e9	−1.51687
$531$	0	0
$532$	−4.13498e9	−1.19064
$533$	6.45571e9	1.84671
$534$	0	0
$535$	4.11894e9	1.16291
$536$	−9.08118e8	−0.254722
$537$	0	0
$538$	−3.53706e9	−0.979274
$539$	7.11294e9	1.95654
$540$	0	0
$541$	−3.38888e9	−0.920166	−0.460083	−	0.887876i	$-0.652180\pi$
$541$	−3.38888e9	−0.920166	−0.460083	+	0.887876i	$0.652180\pi$
$542$	−1.45542e9	−0.392637
$543$	0	0
$544$	3.87603e8	0.103226
$545$	4.12537e8	0.109163
$546$	0	0
$547$	−1.66155e9	−0.434067	−0.217034	−	0.976164i	$-0.569638\pi$
$547$	−1.66155e9	−0.434067	−0.217034	+	0.976164i	$0.569638\pi$
$548$	−3.10126e9	−0.805019
$549$	0	0
$550$	−1.23673e9	−0.316960
$551$	−6.27465e9	−1.59793
$552$	0	0
$553$	1.05503e10	2.65293
$554$	−2.08629e9	−0.521303
$555$	0	0
$556$	−2.43092e9	−0.599802
$557$	6.21412e9	1.52365	0.761827	−	0.647780i	$-0.224303\pi$
$557$	6.21412e9	1.52365	0.761827	+	0.647780i	$0.224303\pi$
$558$	0	0
$559$	−8.15915e9	−1.97562
$560$	1.93381e9	0.465325
$561$	0	0
$562$	−2.84615e9	−0.676364
$563$	6.09480e8	0.143939	0.0719697	−	0.997407i	$-0.477071\pi$
$563$	6.09480e8	0.143939	0.0719697	+	0.997407i	$0.477071\pi$
$564$	0	0
$565$	−4.83108e9	−1.12687
$566$	1.20448e9	0.279218
$567$	0	0
$568$	5.16314e8	0.118221
$569$	−4.55115e9	−1.03569	−0.517844	−	0.855475i	$-0.673265\pi$
$569$	−4.55115e9	−1.03569	−0.517844	+	0.855475i	$0.673265\pi$
$570$	0	0
$571$	−3.33002e9	−0.748549	−0.374275	−	0.927318i	$-0.622108\pi$
$571$	−3.33002e9	−0.748549	−0.374275	+	0.927318i	$0.622108\pi$
$572$	4.80153e9	1.07274
$573$	0	0
$574$	5.54734e9	1.22432
$575$	−7.45898e8	−0.163622
$576$	0	0
$577$	−1.26141e9	−0.273363	−0.136682	−	0.990615i	$-0.543644\pi$
$577$	−1.26141e9	−0.273363	−0.136682	+	0.990615i	$0.543644\pi$
$578$	2.16336e9	0.465996
$579$	0	0
$580$	2.93448e9	0.624501
$581$	−7.79898e9	−1.64976
$582$	0	0
$583$	−1.10922e10	−2.31835
$584$	−7.48480e7	−0.0155502
$585$	0	0
$586$	1.75121e9	0.359498
$587$	4.31767e9	0.881082	0.440541	−	0.897732i	$-0.354787\pi$
$587$	4.31767e9	0.881082	0.440541	+	0.897732i	$0.354787\pi$
$588$	0	0
$589$	7.38060e9	1.48829
$590$	−1.58602e9	−0.317927
$591$	0	0
$592$	1.48862e8	0.0294889
$593$	−2.64830e9	−0.521525	−0.260762	−	0.965403i	$-0.583974\pi$
$593$	−2.64830e9	−0.521525	−0.260762	+	0.965403i	$0.583974\pi$
$594$	0	0
$595$	−5.58460e9	−1.08688
$596$	−2.29173e9	−0.443405
$597$	0	0
$598$	2.89591e9	0.553772
$599$	−7.12463e9	−1.35447	−0.677233	−	0.735768i	$-0.736821\pi$
$599$	−7.12463e9	−1.35447	−0.677233	+	0.735768i	$0.736821\pi$
$600$	0	0
$601$	2.09427e8	0.0393525	0.0196762	−	0.999806i	$-0.493736\pi$
$601$	2.09427e8	0.0393525	0.0196762	+	0.999806i	$0.493736\pi$
$602$	−7.01109e9	−1.30978
$603$	0	0
$604$	3.07853e9	0.568478
$605$	−3.70409e9	−0.680046
$606$	0	0
$607$	−7.79489e9	−1.41465	−0.707326	−	0.706888i	$-0.750099\pi$
$607$	−7.79489e9	−1.41465	−0.707326	+	0.706888i	$0.750099\pi$
$608$	−1.45962e9	−0.263376
$609$	0	0
$610$	4.95246e9	0.883418
$611$	−6.79325e9	−1.20485
$612$	0	0
$613$	−9.11476e8	−0.159821	−0.0799105	−	0.996802i	$-0.525463\pi$
$613$	−9.11476e8	−0.159821	−0.0799105	+	0.996802i	$0.525463\pi$
$614$	4.39366e9	0.766014
$615$	0	0
$616$	4.12591e9	0.711193
$617$	−2.35163e9	−0.403060	−0.201530	−	0.979482i	$-0.564591\pi$
$617$	−2.35163e9	−0.403060	−0.201530	+	0.979482i	$0.564591\pi$
$618$	0	0
$619$	−1.75254e9	−0.296996	−0.148498	−	0.988913i	$-0.547444\pi$
$619$	−1.75254e9	−0.296996	−0.148498	+	0.988913i	$0.547444\pi$
$620$	−3.45170e9	−0.581651
$621$	0	0
$622$	4.07360e9	0.678753
$623$	−4.26665e9	−0.706933
$624$	0	0
$625$	−7.50312e9	−1.22931
$626$	−4.49473e9	−0.732307
$627$	0	0
$628$	−5.08986e9	−0.820062
$629$	−4.29894e8	−0.0688786
$630$	0	0
$631$	1.95065e9	0.309084	0.154542	−	0.987986i	$-0.450610\pi$
$631$	1.95065e9	0.309084	0.154542	+	0.987986i	$0.450610\pi$
$632$	3.72418e9	0.586841
$633$	0	0
$634$	7.16766e9	1.11703
$635$	3.16320e9	0.490251
$636$	0	0
$637$	1.72884e10	2.65013
$638$	6.26090e9	0.954475
$639$	0	0
$640$	6.82623e8	0.102932
$641$	1.36527e9	0.204746	0.102373	−	0.994746i	$-0.467356\pi$
$641$	1.36527e9	0.204746	0.102373	+	0.994746i	$0.467356\pi$
$642$	0	0
$643$	−7.71048e9	−1.14378	−0.571891	−	0.820330i	$-0.693790\pi$
$643$	−7.71048e9	−1.14378	−0.571891	+	0.820330i	$0.693790\pi$
$644$	2.48843e9	0.367135
$645$	0	0
$646$	4.21518e9	0.615180
$647$	−1.07543e10	−1.56106	−0.780529	−	0.625120i	$-0.785050\pi$
$647$	−1.07543e10	−1.56106	−0.780529	+	0.625120i	$0.785050\pi$
$648$	0	0
$649$	−3.38388e9	−0.485913
$650$	−3.00594e9	−0.429322
$651$	0	0
$652$	−2.85657e9	−0.403626
$653$	1.28284e10	1.80292	0.901458	−	0.432867i	$-0.142498\pi$
$653$	1.28284e10	1.80292	0.901458	+	0.432867i	$0.142498\pi$
$654$	0	0
$655$	−5.95868e9	−0.828524
$656$	1.95817e9	0.270824
$657$	0	0
$658$	−5.83738e9	−0.798781
$659$	−8.09108e9	−1.10130	−0.550652	−	0.834735i	$-0.685621\pi$
$659$	−8.09108e9	−1.10130	−0.550652	+	0.834735i	$0.685621\pi$
$660$	0	0
$661$	1.08520e10	1.46151	0.730756	−	0.682638i	$-0.239167\pi$
$661$	1.08520e10	1.46151	0.730756	+	0.682638i	$0.239167\pi$
$662$	5.64129e9	0.755745
$663$	0	0
$664$	−2.75299e9	−0.364935
$665$	2.10302e10	2.77312
$666$	0	0
$667$	3.77609e9	0.492722
$668$	4.53534e8	0.0588697
$669$	0	0
$670$	4.61863e9	0.593269
$671$	1.05664e10	1.35020
$672$	0	0
$673$	8.46680e9	1.07070	0.535348	−	0.844631i	$-0.320180\pi$
$673$	8.46680e9	1.07070	0.535348	+	0.844631i	$0.320180\pi$
$674$	−7.95532e9	−1.00080
$675$	0	0
$676$	7.65451e9	0.953024
$677$	−7.39370e9	−0.915802	−0.457901	−	0.889003i	$-0.651399\pi$
$677$	−7.39370e9	−0.915802	−0.457901	+	0.889003i	$0.651399\pi$
$678$	0	0
$679$	1.14799e10	1.40732
$680$	−1.97132e9	−0.240423
$681$	0	0
$682$	−7.36442e9	−0.888983
$683$	−4.64936e9	−0.558368	−0.279184	−	0.960238i	$-0.590064\pi$
$683$	−4.64936e9	−0.558368	−0.279184	+	0.960238i	$0.590064\pi$
$684$	0	0
$685$	1.57728e10	1.87496
$686$	5.29973e9	0.626786
$687$	0	0
$688$	−2.47487e9	−0.289729
$689$	−2.69603e10	−3.14021
$690$	0	0
$691$	−1.16767e10	−1.34632	−0.673158	−	0.739499i	$-0.735063\pi$
$691$	−1.16767e10	−1.34632	−0.673158	+	0.739499i	$0.735063\pi$
$692$	4.40952e9	0.505848
$693$	0	0
$694$	1.31137e8	0.0148925
$695$	1.23635e10	1.39699
$696$	0	0
$697$	−5.65494e9	−0.632577
$698$	3.41422e9	0.380012
$699$	0	0
$700$	−2.58298e9	−0.284628
$701$	1.48274e10	1.62575	0.812873	−	0.582441i	$-0.197902\pi$
$701$	1.48274e10	1.62575	0.812873	+	0.582441i	$0.197902\pi$
$702$	0	0
$703$	1.61888e9	0.175740
$704$	1.45642e9	0.157319
$705$	0	0
$706$	−1.28442e10	−1.37370
$707$	2.68563e10	2.85810
$708$	0	0
$709$	6.16649e9	0.649795	0.324898	−	0.945749i	$-0.394670\pi$
$709$	6.16649e9	0.649795	0.324898	+	0.945749i	$0.394670\pi$
$710$	−2.62594e9	−0.275347
$711$	0	0
$712$	−1.50610e9	−0.156377
$713$	−4.44165e9	−0.458914
$714$	0	0
$715$	−2.44203e10	−2.49850
$716$	5.68846e9	0.579161
$717$	0	0
$718$	1.11690e10	1.12610
$719$	1.31343e10	1.31782	0.658912	−	0.752220i	$-0.271017\pi$
$719$	1.31343e10	1.31782	0.658912	+	0.752220i	$0.271017\pi$
$720$	0	0
$721$	−1.45635e10	−1.44708
$722$	−8.72236e9	−0.862489
$723$	0	0
$724$	4.75015e9	0.465182
$725$	−3.91956e9	−0.381992
$726$	0	0
$727$	4.45556e9	0.430063	0.215031	−	0.976607i	$-0.431015\pi$
$727$	4.45556e9	0.430063	0.215031	+	0.976607i	$0.431015\pi$
$728$	1.00283e10	0.963313
$729$	0	0
$730$	3.80672e8	0.0362177
$731$	7.14709e9	0.676735
$732$	0	0
$733$	2.99736e9	0.281109	0.140555	−	0.990073i	$-0.455111\pi$
$733$	2.99736e9	0.281109	0.140555	+	0.990073i	$0.455111\pi$
$734$	1.20609e10	1.12576
$735$	0	0
$736$	8.78399e8	0.0812119
$737$	9.85414e9	0.906740
$738$	0	0
$739$	8.82559e9	0.804429	0.402215	−	0.915545i	$-0.368241\pi$
$739$	8.82559e9	0.804429	0.402215	+	0.915545i	$0.368241\pi$
$740$	−7.57103e8	−0.0686822
$741$	0	0
$742$	−2.31668e10	−2.08186
$743$	−2.00697e10	−1.79507	−0.897534	−	0.440946i	$-0.854643\pi$
$743$	−2.00697e10	−1.79507	−0.897534	+	0.440946i	$0.854643\pi$
$744$	0	0
$745$	1.16556e10	1.03273
$746$	6.18544e9	0.545487
$747$	0	0
$748$	−4.20595e9	−0.367458
$749$	1.83544e10	1.59607
$750$	0	0
$751$	−3.94246e9	−0.339647	−0.169824	−	0.985474i	$-0.554320\pi$
$751$	−3.94246e9	−0.339647	−0.169824	+	0.985474i	$0.554320\pi$
$752$	−2.06056e9	−0.176694
$753$	0	0
$754$	1.52175e10	1.29284
$755$	−1.56572e10	−1.32403
$756$	0	0
$757$	1.54038e10	1.29060	0.645301	−	0.763928i	$-0.276732\pi$
$757$	1.54038e10	1.29060	0.645301	+	0.763928i	$0.276732\pi$
$758$	−1.32031e10	−1.10112
$759$	0	0
$760$	7.42352e9	0.613426
$761$	1.92170e10	1.58066	0.790330	−	0.612682i	$-0.209909\pi$
$761$	1.92170e10	1.58066	0.790330	+	0.612682i	$0.209909\pi$
$762$	0	0
$763$	1.83830e9	0.149824
$764$	−1.63600e9	−0.132726
$765$	0	0
$766$	−5.21150e9	−0.418950
$767$	−8.22473e9	−0.658170
$768$	0	0
$769$	−1.20525e8	−0.00955726	−0.00477863	−	0.999989i	$-0.501521\pi$
$769$	−1.20525e8	−0.00955726	−0.00477863	+	0.999989i	$0.501521\pi$
$770$	−2.09841e10	−1.65643
$771$	0	0
$772$	−6.29573e9	−0.492477
$773$	−2.06347e10	−1.60683	−0.803415	−	0.595420i	$-0.796986\pi$
$773$	−2.06347e10	−1.60683	−0.803415	+	0.595420i	$0.796986\pi$
$774$	0	0
$775$	4.61041e9	0.355782
$776$	4.05231e9	0.311306
$777$	0	0
$778$	−6.98215e9	−0.531570
$779$	2.12951e10	1.61398
$780$	0	0
$781$	−5.60261e9	−0.420835
$782$	−2.53670e9	−0.189691
$783$	0	0
$784$	5.24400e9	0.388648
$785$	2.58867e10	1.91000
$786$	0	0
$787$	1.69071e9	0.123639	0.0618197	−	0.998087i	$-0.480310\pi$
$787$	1.69071e9	0.123639	0.0618197	+	0.998087i	$0.480310\pi$
$788$	−8.99719e8	−0.0655036
$789$	0	0
$790$	−1.89409e10	−1.36680
$791$	−2.15277e10	−1.54660
$792$	0	0
$793$	2.56823e10	1.82885
$794$	−2.90106e9	−0.205677
$795$	0	0
$796$	1.83953e9	0.129274
$797$	1.35975e9	0.0951384	0.0475692	−	0.998868i	$-0.484853\pi$
$797$	1.35975e9	0.0951384	0.0475692	+	0.998868i	$0.484853\pi$
$798$	0	0
$799$	5.95061e9	0.412713
$800$	−9.11774e8	−0.0629611
$801$	0	0
$802$	5.58718e9	0.382457
$803$	8.12188e8	0.0553544
$804$	0	0
$805$	−1.26560e10	−0.855089
$806$	−1.78997e10	−1.20413
$807$	0	0
$808$	9.48007e9	0.632225
$809$	1.53965e10	1.02235	0.511177	−	0.859475i	$-0.329210\pi$
$809$	1.53965e10	1.02235	0.511177	+	0.859475i	$0.329210\pi$
$810$	0	0
$811$	−9.67476e9	−0.636894	−0.318447	−	0.947941i	$-0.603161\pi$
$811$	−9.67476e9	−0.636894	−0.318447	+	0.947941i	$0.603161\pi$
$812$	1.30763e10	0.857113
$813$	0	0
$814$	−1.61533e9	−0.104972
$815$	1.45283e10	0.940080
$816$	0	0
$817$	−2.69142e10	−1.72665
$818$	9.84695e9	0.629021
$819$	0	0
$820$	−9.95914e9	−0.630773
$821$	2.46343e10	1.55360	0.776799	−	0.629749i	$-0.216842\pi$
$821$	2.46343e10	1.55360	0.776799	+	0.629749i	$0.216842\pi$
$822$	0	0
$823$	−2.14849e9	−0.134349	−0.0671743	−	0.997741i	$-0.521398\pi$
$823$	−2.14849e9	−0.134349	−0.0671743	+	0.997741i	$0.521398\pi$
$824$	−5.14081e9	−0.320101
$825$	0	0
$826$	−7.06744e9	−0.436347
$827$	1.31873e10	0.810751	0.405375	−	0.914150i	$-0.367141\pi$
$827$	1.31873e10	0.810751	0.405375	+	0.914150i	$0.367141\pi$
$828$	0	0
$829$	6.41872e8	0.0391298	0.0195649	−	0.999809i	$-0.493772\pi$
$829$	6.41872e8	0.0391298	0.0195649	+	0.999809i	$0.493772\pi$
$830$	1.40015e10	0.849966
$831$	0	0
$832$	3.53992e9	0.213089
$833$	−1.51440e10	−0.907784
$834$	0	0
$835$	−2.30664e9	−0.137113
$836$	1.58385e10	0.937548
$837$	0	0
$838$	−1.27690e10	−0.749556
$839$	−1.01344e10	−0.592425	−0.296212	−	0.955122i	$-0.595724\pi$
$839$	−1.01344e10	−0.592425	−0.296212	+	0.955122i	$0.595724\pi$
$840$	0	0
$841$	2.59283e9	0.150310
$842$	6.18330e9	0.356967
$843$	0	0
$844$	−1.26278e10	−0.722984
$845$	−3.89303e10	−2.21968
$846$	0	0
$847$	−1.65057e10	−0.933347
$848$	−8.17772e9	−0.460518
$849$	0	0
$850$	2.63308e9	0.147061
$851$	−9.74241e8	−0.0541892
$852$	0	0
$853$	1.35033e10	0.744933	0.372467	−	0.928046i	$-0.378512\pi$
$853$	1.35033e10	0.744933	0.372467	+	0.928046i	$0.378512\pi$
$854$	2.20686e10	1.21247
$855$	0	0
$856$	6.47896e9	0.353059
$857$	5.51538e9	0.299324	0.149662	−	0.988737i	$-0.452181\pi$
$857$	5.51538e9	0.299324	0.149662	+	0.988737i	$0.452181\pi$
$858$	0	0
$859$	1.21351e10	0.653233	0.326617	−	0.945157i	$-0.394091\pi$
$859$	1.21351e10	0.653233	0.326617	+	0.945157i	$0.394091\pi$
$860$	1.25870e10	0.674805
$861$	0	0
$862$	1.93187e9	0.102731
$863$	−2.09525e9	−0.110968	−0.0554839	−	0.998460i	$-0.517670\pi$
$863$	−2.09525e9	−0.110968	−0.0554839	+	0.998460i	$0.517670\pi$
$864$	0	0
$865$	−2.24265e10	−1.17816
$866$	−2.28611e10	−1.19615
$867$	0	0
$868$	−1.53811e10	−0.798302
$869$	−4.04117e10	−2.08900
$870$	0	0
$871$	2.39511e10	1.22818
$872$	6.48907e8	0.0331417
$873$	0	0
$874$	9.55259e9	0.483984
$875$	−2.37477e10	−1.19838
$876$	0	0
$877$	3.53585e10	1.77009	0.885045	−	0.465505i	$-0.154127\pi$
$877$	3.53585e10	1.77009	0.885045	+	0.465505i	$0.154127\pi$
$878$	−2.61602e9	−0.130440
$879$	0	0
$880$	−7.40725e9	−0.366410
$881$	7.96497e8	0.0392436	0.0196218	−	0.999807i	$-0.493754\pi$
$881$	7.96497e8	0.0392436	0.0196218	+	0.999807i	$0.493754\pi$
$882$	0	0
$883$	−4.24897e9	−0.207693	−0.103846	−	0.994593i	$-0.533115\pi$
$883$	−4.24897e9	−0.207693	−0.103846	+	0.994593i	$0.533115\pi$
$884$	−1.02228e10	−0.497723
$885$	0	0
$886$	6.37768e9	0.308066
$887$	1.76980e10	0.851512	0.425756	−	0.904838i	$-0.360008\pi$
$887$	1.76980e10	0.851512	0.425756	+	0.904838i	$0.360008\pi$
$888$	0	0
$889$	1.40955e10	0.672859
$890$	7.65991e9	0.364215
$891$	0	0
$892$	−4.78903e9	−0.225928
$893$	−2.24085e10	−1.05301
$894$	0	0
$895$	−2.89311e10	−1.34892
$896$	3.04182e9	0.141272
$897$	0	0
$898$	−1.13454e10	−0.522819
$899$	−2.33401e10	−1.07138
$900$	0	0
$901$	2.36162e10	1.07565
$902$	−2.12485e10	−0.964061
$903$	0	0
$904$	−7.59912e9	−0.342116
$905$	−2.41590e10	−1.08345
$906$	0	0
$907$	−1.15096e9	−0.0512193	−0.0256096	−	0.999672i	$-0.508153\pi$
$907$	−1.15096e9	−0.0512193	−0.0256096	+	0.999672i	$0.508153\pi$
$908$	−4.63029e9	−0.205261
$909$	0	0
$910$	−5.10032e10	−2.24364
$911$	1.14666e10	0.502480	0.251240	−	0.967925i	$-0.419162\pi$
$911$	1.14666e10	0.502480	0.251240	+	0.967925i	$0.419162\pi$
$912$	0	0
$913$	2.98731e10	1.29907
$914$	7.79925e9	0.337863
$915$	0	0
$916$	−1.95062e10	−0.838567
$917$	−2.65524e10	−1.13713
$918$	0	0
$919$	−3.59560e10	−1.52815	−0.764076	−	0.645126i	$-0.776805\pi$
$919$	−3.59560e10	−1.52815	−0.764076	+	0.645126i	$0.776805\pi$
$920$	−4.46748e9	−0.189150
$921$	0	0
$922$	1.09829e10	0.461486
$923$	−1.36175e10	−0.570022
$924$	0	0
$925$	1.01126e9	0.0420112
$926$	−1.42972e10	−0.591714
$927$	0	0
$928$	4.61584e9	0.189597
$929$	−5.73116e9	−0.234524	−0.117262	−	0.993101i	$-0.537412\pi$
$929$	−5.73116e9	−0.234524	−0.117262	+	0.993101i	$0.537412\pi$
$930$	0	0
$931$	5.70285e10	2.31616
$932$	1.51254e9	0.0612000
$933$	0	0
$934$	1.80614e10	0.725334
$935$	2.13912e10	0.855842
$936$	0	0
$937$	−1.02489e10	−0.406995	−0.203497	−	0.979076i	$-0.565231\pi$
$937$	−1.02489e10	−0.406995	−0.203497	+	0.979076i	$0.565231\pi$
$938$	2.05810e10	0.814247
$939$	0	0
$940$	1.04799e10	0.411536
$941$	−1.13948e10	−0.445801	−0.222901	−	0.974841i	$-0.571553\pi$
$941$	−1.13948e10	−0.445801	−0.222901	+	0.974841i	$0.571553\pi$
$942$	0	0
$943$	−1.28154e10	−0.497671
$944$	−2.49476e9	−0.0965220
$945$	0	0
$946$	2.68552e10	1.03136
$947$	−2.40000e10	−0.918303	−0.459151	−	0.888358i	$-0.651846\pi$
$947$	−2.40000e10	−0.918303	−0.459151	+	0.888358i	$0.651846\pi$
$948$	0	0
$949$	1.97407e9	0.0749777
$950$	−9.91554e9	−0.375218
$951$	0	0
$952$	−8.78438e9	−0.329976
$953$	1.56339e10	0.585117	0.292558	−	0.956248i	$-0.405493\pi$
$953$	1.56339e10	0.585117	0.292558	+	0.956248i	$0.405493\pi$
$954$	0	0
$955$	8.32058e9	0.309130
$956$	−1.47849e10	−0.547288
$957$	0	0
$958$	1.39413e10	0.512301
$959$	7.02849e10	2.57334
$960$	0	0
$961$	−5.86612e7	−0.00213216
$962$	−3.92616e9	−0.142185
$963$	0	0
$964$	−3.80039e9	−0.136634
$965$	3.20197e10	1.14702
$966$	0	0
$967$	4.21929e10	1.50054	0.750268	−	0.661134i	$-0.229925\pi$
$967$	4.21929e10	1.50054	0.750268	+	0.661134i	$0.229925\pi$
$968$	−5.82641e9	−0.206461
$969$	0	0
$970$	−2.06098e10	−0.725058
$971$	−8.93754e9	−0.313293	−0.156646	−	0.987655i	$-0.550068\pi$
$971$	−8.93754e9	−0.313293	−0.156646	+	0.987655i	$0.550068\pi$
$972$	0	0
$973$	5.50927e10	1.91734
$974$	2.15079e10	0.745834
$975$	0	0
$976$	7.79004e9	0.268204
$977$	2.12577e10	0.729266	0.364633	−	0.931151i	$-0.381194\pi$
$977$	2.12577e10	0.729266	0.364633	+	0.931151i	$0.381194\pi$
$978$	0	0
$979$	1.63429e10	0.556659
$980$	−2.66706e10	−0.905195
$981$	0	0
$982$	−1.87905e10	−0.633210
$983$	1.03351e10	0.347038	0.173519	−	0.984831i	$-0.444486\pi$
$983$	1.03351e10	0.347038	0.173519	+	0.984831i	$0.444486\pi$
$984$	0	0
$985$	4.57591e9	0.152564
$986$	−1.33299e10	−0.442852
$987$	0	0
$988$	3.84966e10	1.26991
$989$	1.61970e10	0.532411
$990$	0	0
$991$	5.00170e10	1.63253	0.816263	−	0.577681i	$-0.196042\pi$
$991$	5.00170e10	1.63253	0.816263	+	0.577681i	$0.196042\pi$
$992$	−5.42941e9	−0.176588
$993$	0	0
$994$	−1.17014e10	−0.377908
$995$	−9.35575e9	−0.301091
$996$	0	0
$997$	−3.38087e10	−1.08043	−0.540214	−	0.841528i	$-0.681657\pi$
$997$	−3.38087e10	−1.08043	−0.540214	+	0.841528i	$0.681657\pi$
$998$	1.33394e10	0.424795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.a.g.1.2	✓	4
3.2	odd	2		162.8.a.j.1.3	yes	4
9.2	odd	6		162.8.c.q.109.2		8
9.4	even	3		162.8.c.r.55.3		8
9.5	odd	6		162.8.c.q.55.2		8
9.7	even	3		162.8.c.r.109.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.g.1.2	✓	4	1.1	even	1	trivial
162.8.a.j.1.3	yes	4	3.2	odd	2
162.8.c.q.55.2		8	9.5	odd	6
162.8.c.q.109.2		8	9.2	odd	6
162.8.c.r.55.3		8	9.4	even	3
162.8.c.r.109.3		8	9.7	even	3

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} -325.500 q^{5} -1450.45 q^{7} -512.000 q^{8} +2604.00 q^{10} +5555.80 q^{11} +13503.7 q^{13} +11603.6 q^{14} +4096.00 q^{16} -11828.7 q^{17} +44544.0 q^{19} -20832.0 q^{20} -44446.4 q^{22} -26806.6 q^{23} +27825.1 q^{25} -108030. q^{26} -92829.0 q^{28} -140864. q^{29} +165692. q^{31} -32768.0 q^{32} +94629.7 q^{34} +472122. q^{35} +36343.3 q^{37} -356352. q^{38} +166656. q^{40} +478069. q^{41} -604215. q^{43} +355571. q^{44} +214453. q^{46} -503065. q^{47} +1.28027e6 q^{49} -222601. q^{50} +864238. q^{52} -1.99651e6 q^{53} -1.80841e6 q^{55} +742632. q^{56} +1.12691e6 q^{58} -609072. q^{59} +1.90187e6 q^{61} -1.32554e6 q^{62} +262144. q^{64} -4.39546e6 q^{65} +1.77367e6 q^{67} -757037. q^{68} -3.77698e6 q^{70} -1.00843e6 q^{71} +146188. q^{73} -290746. q^{74} +2.85081e6 q^{76} -8.05843e6 q^{77} -7.27378e6 q^{79} -1.33325e6 q^{80} -3.82456e6 q^{82} +5.37693e6 q^{83} +3.85024e6 q^{85} +4.83372e6 q^{86} -2.84457e6 q^{88} +2.94159e6 q^{89} -1.95865e7 q^{91} -1.71562e6 q^{92} +4.02452e6 q^{94} -1.44991e7 q^{95} -7.91467e6 q^{97} -1.02422e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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