LMFDB - Embedded newform 171.10.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,10,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("171.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	171.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:	$88.0711279840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 2124x^{4} - 384x^{3} + 1071312x^{2} + 1260144x - 135644992 $ x^6 - 3x^5 - 2124x^4 - 384x^3 + 1071312x^2 + 1260144*x - 135644992
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-35.7587$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+41.7587 q^{2} +1231.79 q^{4} +1014.12 q^{5} +11492.9 q^{7} +30057.3 q^{8} +42348.3 q^{10} +51884.4 q^{11} +43035.9 q^{13} +479927. q^{14} +624478. q^{16} -364626. q^{17} -130321. q^{19} +1.24918e6 q^{20} +2.16662e6 q^{22} -1.00095e6 q^{23} -924686. q^{25} +1.79712e6 q^{26} +1.41568e7 q^{28} -793096. q^{29} -7.60200e6 q^{31} +1.06880e7 q^{32} -1.52263e7 q^{34} +1.16551e7 q^{35} -7.70164e6 q^{37} -5.44203e6 q^{38} +3.04817e7 q^{40} +2.21833e7 q^{41} -1.50803e7 q^{43} +6.39104e7 q^{44} -4.17983e7 q^{46} -1.68125e7 q^{47} +9.17325e7 q^{49} -3.86136e7 q^{50} +5.30110e7 q^{52} +2.23907e7 q^{53} +5.26170e7 q^{55} +3.45445e8 q^{56} -3.31186e7 q^{58} -5.70512e7 q^{59} -4.31618e7 q^{61} -3.17450e8 q^{62} +1.26585e8 q^{64} +4.36436e7 q^{65} -3.83656e7 q^{67} -4.49141e8 q^{68} +4.86703e8 q^{70} +1.04064e7 q^{71} -4.12491e8 q^{73} -3.21610e8 q^{74} -1.60528e8 q^{76} +5.96301e8 q^{77} +6.20411e8 q^{79} +6.33295e8 q^{80} +9.26344e8 q^{82} +5.15116e8 q^{83} -3.69774e8 q^{85} -6.29735e8 q^{86} +1.55950e9 q^{88} +3.59933e8 q^{89} +4.94606e8 q^{91} -1.23296e9 q^{92} -7.02067e8 q^{94} -1.32161e8 q^{95} -4.39653e8 q^{97} +3.83063e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 33 q^{2} + 1365 q^{4} + 3612 q^{5} + 4085 q^{7} + 23511 q^{8} - 93884 q^{10} + 69312 q^{11} - 191747 q^{13} + 644691 q^{14} + 13905 q^{16} + 288195 q^{17} - 781926 q^{19} + 1551444 q^{20} + 2409710 q^{22}+ \cdots + 4245664590 q^{98}+O(q^{100}) $ 6 * q + 33 * q^2 + 1365 * q^4 + 3612 * q^5 + 4085 * q^7 + 23511 * q^8 - 93884 * q^10 + 69312 * q^11 - 191747 * q^13 + 644691 * q^14 + 13905 * q^16 + 288195 * q^17 - 781926 * q^19 + 1551444 * q^20 + 2409710 * q^22 + 50697 * q^23 + 7126888 * q^25 - 4394073 * q^26 + 19295623 * q^28 + 7178667 * q^29 - 4310716 * q^31 + 5388975 * q^32 + 18049 * q^34 + 4166490 * q^35 - 15687040 * q^37 - 4300593 * q^38 + 33173148 * q^40 + 25134306 * q^41 - 75118674 * q^43 + 94242102 * q^44 - 85549087 * q^46 + 83731938 * q^47 + 21010019 * q^49 - 141211941 * q^50 + 147001043 * q^52 + 138019203 * q^53 - 1518258 * q^55 + 182020629 * q^56 + 14051197 * q^58 + 7809915 * q^59 + 191946566 * q^61 - 564818736 * q^62 + 167338593 * q^64 + 146177172 * q^65 - 16109787 * q^67 - 561722511 * q^68 + 692634996 * q^70 + 264469698 * q^71 - 287572857 * q^73 - 376807098 * q^74 - 177888165 * q^76 + 1619915436 * q^77 - 86534002 * q^79 + 1300596708 * q^80 + 1582686496 * q^82 + 359214570 * q^83 - 648092718 * q^85 + 1061832924 * q^86 + 779241042 * q^88 + 2263866306 * q^89 + 1172864023 * q^91 + 373904061 * q^92 - 864022488 * q^94 - 470719452 * q^95 + 2705893460 * q^97 + 4245664590 * 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 $

$2$	41.7587	1.84549	0.922745	−	0.385412i	$-0.125941\pi$
$2$	41.7587	1.84549	0.922745	+	0.385412i	$0.125941\pi$
$3$	0	0
$3$	0	0
$4$	1231.79	2.40583
$5$	1014.12	0.725645	0.362823	−	0.931858i	$-0.381813\pi$
$5$	1014.12	0.725645	0.362823	+	0.931858i	$0.381813\pi$
$6$	0	0
$7$	11492.9	1.80920	0.904602	−	0.426258i	$-0.140168\pi$
$7$	11492.9	1.80920	0.904602	+	0.426258i	$0.140168\pi$
$8$	30057.3	2.59445
$9$	0	0
$10$	42348.3	1.33917
$11$	51884.4	1.06849	0.534244	−	0.845330i	$-0.320596\pi$
$11$	51884.4	1.06849	0.534244	+	0.845330i	$0.320596\pi$
$12$	0	0
$13$	43035.9	0.417913	0.208956	−	0.977925i	$-0.432993\pi$
$13$	43035.9	0.417913	0.208956	+	0.977925i	$0.432993\pi$
$14$	479927.	3.33887
$15$	0	0
$16$	624478.	2.38219
$17$	−364626.	−1.05883	−0.529416	−	0.848362i	$-0.677589\pi$
$17$	−364626.	−1.05883	−0.529416	+	0.848362i	$0.677589\pi$
$18$	0	0
$19$	−130321.	−0.229416
$19$	−130321.	−0.229416
$20$	1.24918e6	1.74578
$21$	0	0
$22$	2.16662e6	1.97188
$23$	−1.00095e6	−0.745826	−0.372913	−	0.927866i	$-0.621641\pi$
$23$	−1.00095e6	−0.745826	−0.372913	+	0.927866i	$0.621641\pi$
$24$	0	0
$25$	−924686.	−0.473439
$26$	1.79712e6	0.771254
$27$	0	0
$28$	1.41568e7	4.35264
$29$	−793096.	−0.208226	−0.104113	−	0.994565i	$-0.533200\pi$
$29$	−793096.	−0.208226	−0.104113	+	0.994565i	$0.533200\pi$
$30$	0	0
$31$	−7.60200e6	−1.47843	−0.739215	−	0.673470i	$-0.764803\pi$
$31$	−7.60200e6	−1.47843	−0.739215	+	0.673470i	$0.764803\pi$
$32$	1.06880e7	1.80187
$33$	0	0
$34$	−1.52263e7	−1.95407
$35$	1.16551e7	1.31284
$36$	0	0
$37$	−7.70164e6	−0.675578	−0.337789	−	0.941222i	$-0.609679\pi$
$37$	−7.70164e6	−0.675578	−0.337789	+	0.941222i	$0.609679\pi$
$38$	−5.44203e6	−0.423384
$39$	0	0
$40$	3.04817e7	1.88265
$41$	2.21833e7	1.22602	0.613011	−	0.790074i	$-0.289958\pi$
$41$	2.21833e7	1.22602	0.613011	+	0.790074i	$0.289958\pi$
$42$	0	0
$43$	−1.50803e7	−0.672671	−0.336336	−	0.941742i	$-0.609188\pi$
$43$	−1.50803e7	−0.672671	−0.336336	+	0.941742i	$0.609188\pi$
$44$	6.39104e7	2.57060
$45$	0	0
$46$	−4.17983e7	−1.37641
$47$	−1.68125e7	−0.502564	−0.251282	−	0.967914i	$-0.580852\pi$
$47$	−1.68125e7	−0.502564	−0.251282	+	0.967914i	$0.580852\pi$
$48$	0	0
$49$	9.17325e7	2.27322
$50$	−3.86136e7	−0.873727
$51$	0	0
$52$	5.30110e7	1.00543
$53$	2.23907e7	0.389786	0.194893	−	0.980825i	$-0.437564\pi$
$53$	2.23907e7	0.389786	0.194893	+	0.980825i	$0.437564\pi$
$54$	0	0
$55$	5.26170e7	0.775343
$56$	3.45445e8	4.69388
$57$	0	0
$58$	−3.31186e7	−0.384279
$59$	−5.70512e7	−0.612958	−0.306479	−	0.951877i	$-0.599151\pi$
$59$	−5.70512e7	−0.612958	−0.306479	+	0.951877i	$0.599151\pi$
$60$	0	0
$61$	−4.31618e7	−0.399130	−0.199565	−	0.979885i	$-0.563953\pi$
$61$	−4.31618e7	−0.399130	−0.199565	+	0.979885i	$0.563953\pi$
$62$	−3.17450e8	−2.72843
$63$	0	0
$64$	1.26585e8	0.943131
$65$	4.36436e7	0.303256
$66$	0	0
$67$	−3.83656e7	−0.232598	−0.116299	−	0.993214i	$-0.537103\pi$
$67$	−3.83656e7	−0.232598	−0.116299	+	0.993214i	$0.537103\pi$
$68$	−4.49141e8	−2.54737
$69$	0	0
$70$	4.86703e8	2.42283
$71$	1.04064e7	0.0486004	0.0243002	−	0.999705i	$-0.492264\pi$
$71$	1.04064e7	0.0486004	0.0243002	+	0.999705i	$0.492264\pi$
$72$	0	0
$73$	−4.12491e8	−1.70005	−0.850024	−	0.526744i	$-0.823413\pi$
$73$	−4.12491e8	−1.70005	−0.850024	+	0.526744i	$0.823413\pi$
$74$	−3.21610e8	−1.24677
$75$	0	0
$76$	−1.60528e8	−0.551936
$77$	5.96301e8	1.93311
$78$	0	0
$79$	6.20411e8	1.79208	0.896041	−	0.443972i	$-0.146431\pi$
$79$	6.20411e8	1.79208	0.896041	+	0.443972i	$0.146431\pi$
$80$	6.33295e8	1.72863
$81$	0	0
$82$	9.26344e8	2.26261
$83$	5.15116e8	1.19139	0.595694	−	0.803211i	$-0.296877\pi$
$83$	5.15116e8	1.19139	0.595694	+	0.803211i	$0.296877\pi$
$84$	0	0
$85$	−3.69774e8	−0.768337
$86$	−6.29735e8	−1.24141
$87$	0	0
$88$	1.55950e9	2.77213
$89$	3.59933e8	0.608089	0.304045	−	0.952658i	$-0.401663\pi$
$89$	3.59933e8	0.608089	0.304045	+	0.952658i	$0.401663\pi$
$90$	0	0
$91$	4.94606e8	0.756089
$92$	−1.23296e9	−1.79433
$93$	0	0
$94$	−7.02067e8	−0.927477
$95$	−1.32161e8	−0.166474
$96$	0	0
$97$	−4.39653e8	−0.504240	−0.252120	−	0.967696i	$-0.581128\pi$
$97$	−4.39653e8	−0.504240	−0.252120	+	0.967696i	$0.581128\pi$
$98$	3.83063e9	4.19520
$99$	0	0
$100$	−1.13901e9	−1.13901
$101$	1.16061e9	1.10979	0.554893	−	0.831922i	$-0.312759\pi$
$101$	1.16061e9	1.10979	0.554893	+	0.831922i	$0.312759\pi$
$102$	0	0
$103$	4.11491e8	0.360240	0.180120	−	0.983645i	$-0.442351\pi$
$103$	4.11491e8	0.360240	0.180120	+	0.983645i	$0.442351\pi$
$104$	1.29354e9	1.08425
$105$	0	0
$106$	9.35005e8	0.719345
$107$	4.85221e8	0.357860	0.178930	−	0.983862i	$-0.442737\pi$
$107$	4.85221e8	0.357860	0.178930	+	0.983862i	$0.442737\pi$
$108$	0	0
$109$	−1.60826e9	−1.09128	−0.545642	−	0.838019i	$-0.683714\pi$
$109$	−1.60826e9	−1.09128	−0.545642	+	0.838019i	$0.683714\pi$
$110$	2.19721e9	1.43089
$111$	0	0
$112$	7.17704e9	4.30987
$113$	5.46306e8	0.315198	0.157599	−	0.987503i	$-0.449625\pi$
$113$	5.46306e8	0.315198	0.157599	+	0.987503i	$0.449625\pi$
$114$	0	0
$115$	−1.01508e9	−0.541205
$116$	−9.76925e8	−0.500957
$117$	0	0
$118$	−2.38238e9	−1.13121
$119$	−4.19060e9	−1.91564
$120$	0	0
$121$	3.34041e8	0.141666
$122$	−1.80238e9	−0.736591
$123$	0	0
$124$	−9.36404e9	−3.55685
$125$	−2.91845e9	−1.06919
$126$	0	0
$127$	9.85431e8	0.336132	0.168066	−	0.985776i	$-0.446248\pi$
$127$	9.85431e8	0.336132	0.168066	+	0.985776i	$0.446248\pi$
$128$	−1.86252e8	−0.0613277
$129$	0	0
$130$	1.82250e9	0.559657
$131$	1.07735e9	0.319621	0.159811	−	0.987148i	$-0.448912\pi$
$131$	1.07735e9	0.319621	0.159811	+	0.987148i	$0.448912\pi$
$132$	0	0
$133$	−1.49776e9	−0.415060
$134$	−1.60210e9	−0.429257
$135$	0	0
$136$	−1.09597e10	−2.74709
$137$	−1.97459e9	−0.478888	−0.239444	−	0.970910i	$-0.576965\pi$
$137$	−1.97459e9	−0.478888	−0.239444	+	0.970910i	$0.576965\pi$
$138$	0	0
$139$	−2.73066e9	−0.620442	−0.310221	−	0.950664i	$-0.600403\pi$
$139$	−2.73066e9	−0.620442	−0.310221	+	0.950664i	$0.600403\pi$
$140$	1.43566e10	3.15847
$141$	0	0
$142$	4.34559e8	0.0896916
$143$	2.23289e9	0.446535
$144$	0	0
$145$	−8.04295e8	−0.151098
$146$	−1.72251e10	−3.13742
$147$	0	0
$148$	−9.48677e9	−1.62533
$149$	7.83959e9	1.30303	0.651516	−	0.758635i	$-0.274133\pi$
$149$	7.83959e9	1.30303	0.651516	+	0.758635i	$0.274133\pi$
$150$	0	0
$151$	8.02998e8	0.125695	0.0628475	−	0.998023i	$-0.479982\pi$
$151$	8.02998e8	0.125695	0.0628475	+	0.998023i	$0.479982\pi$
$152$	−3.91710e9	−0.595207
$153$	0	0
$154$	2.49007e10	3.56754
$155$	−7.70934e9	−1.07281
$156$	0	0
$157$	−1.18394e10	−1.55518	−0.777590	−	0.628772i	$-0.783558\pi$
$157$	−1.18394e10	−1.55518	−0.777590	+	0.628772i	$0.783558\pi$
$158$	2.59075e10	3.30727
$159$	0	0
$160$	1.08389e10	1.30752
$161$	−1.15038e10	−1.34935
$162$	0	0
$163$	1.21288e10	1.34577	0.672887	−	0.739746i	$-0.265054\pi$
$163$	1.21288e10	1.34577	0.672887	+	0.739746i	$0.265054\pi$
$164$	2.73251e10	2.94960
$165$	0	0
$166$	2.15105e10	2.19869
$167$	−4.61691e9	−0.459333	−0.229666	−	0.973269i	$-0.573764\pi$
$167$	−4.61691e9	−0.459333	−0.229666	+	0.973269i	$0.573764\pi$
$168$	0	0
$169$	−8.75241e9	−0.825349
$170$	−1.54413e10	−1.41796
$171$	0	0
$172$	−1.85757e10	−1.61833
$173$	−7.93226e9	−0.673271	−0.336635	−	0.941635i	$-0.609289\pi$
$173$	−7.93226e9	−0.673271	−0.336635	+	0.941635i	$0.609289\pi$
$174$	0	0
$175$	−1.06273e10	−0.856548
$176$	3.24006e10	2.54534
$177$	0	0
$178$	1.50303e10	1.12222
$179$	2.14340e10	1.56051	0.780253	−	0.625464i	$-0.215090\pi$
$179$	2.14340e10	1.56051	0.780253	+	0.625464i	$0.215090\pi$
$180$	0	0
$181$	2.17685e10	1.50756	0.753780	−	0.657127i	$-0.228228\pi$
$181$	2.17685e10	1.50756	0.753780	+	0.657127i	$0.228228\pi$
$182$	2.06541e10	1.39535
$183$	0	0
$184$	−3.00858e10	−1.93500
$185$	−7.81039e9	−0.490230
$186$	0	0
$187$	−1.89184e10	−1.13135
$188$	−2.07094e10	−1.20908
$189$	0	0
$190$	−5.51887e9	−0.307227
$191$	2.29510e10	1.24782	0.623909	−	0.781497i	$-0.285544\pi$
$191$	2.29510e10	1.24782	0.623909	+	0.781497i	$0.285544\pi$
$192$	0	0
$193$	−3.70544e10	−1.92234	−0.961172	−	0.275948i	$-0.911008\pi$
$193$	−3.70544e10	−1.92234	−0.961172	+	0.275948i	$0.911008\pi$
$194$	−1.83593e10	−0.930569
$195$	0	0
$196$	1.12995e11	5.46898
$197$	−7.47377e9	−0.353543	−0.176771	−	0.984252i	$-0.556565\pi$
$197$	−7.47377e9	−0.353543	−0.176771	+	0.984252i	$0.556565\pi$
$198$	0	0
$199$	3.18116e10	1.43796	0.718980	−	0.695031i	$-0.244609\pi$
$199$	3.18116e10	1.43796	0.718980	+	0.695031i	$0.244609\pi$
$200$	−2.77935e10	−1.22831
$201$	0	0
$202$	4.84654e10	2.04810
$203$	−9.11495e9	−0.376723
$204$	0	0
$205$	2.24965e10	0.889657
$206$	1.71833e10	0.664820
$207$	0	0
$208$	2.68750e10	0.995549
$209$	−6.76162e9	−0.245128
$210$	0	0
$211$	−2.37411e10	−0.824574	−0.412287	−	0.911054i	$-0.635270\pi$
$211$	−2.37411e10	−0.824574	−0.412287	+	0.911054i	$0.635270\pi$
$212$	2.75805e10	0.937758
$213$	0	0
$214$	2.02622e10	0.660426
$215$	−1.52933e10	−0.488121
$216$	0	0
$217$	−8.73689e10	−2.67478
$218$	−6.71589e10	−2.01395
$219$	0	0
$220$	6.48128e10	1.86534
$221$	−1.56920e10	−0.442500
$222$	0	0
$223$	2.52999e10	0.685089	0.342545	−	0.939502i	$-0.388711\pi$
$223$	2.52999e10	0.685089	0.342545	+	0.939502i	$0.388711\pi$
$224$	1.22836e11	3.25994
$225$	0	0
$226$	2.28130e10	0.581694
$227$	6.77167e9	0.169270	0.0846349	−	0.996412i	$-0.473028\pi$
$227$	6.77167e9	0.169270	0.0846349	+	0.996412i	$0.473028\pi$
$228$	0	0
$229$	−7.64883e9	−0.183796	−0.0918979	−	0.995768i	$-0.529293\pi$
$229$	−7.64883e9	−0.183796	−0.0918979	+	0.995768i	$0.529293\pi$
$230$	−4.23885e10	−0.998788
$231$	0	0
$232$	−2.38383e10	−0.540231
$233$	4.69852e10	1.04438	0.522191	−	0.852828i	$-0.325115\pi$
$233$	4.69852e10	1.04438	0.522191	+	0.852828i	$0.325115\pi$
$234$	0	0
$235$	−1.70499e10	−0.364683
$236$	−7.02748e10	−1.47467
$237$	0	0
$238$	−1.74994e11	−3.53530
$239$	7.82314e9	0.155092	0.0775462	−	0.996989i	$-0.475291\pi$
$239$	7.82314e9	0.155092	0.0775462	+	0.996989i	$0.475291\pi$
$240$	0	0
$241$	−3.32182e9	−0.0634308	−0.0317154	−	0.999497i	$-0.510097\pi$
$241$	−3.32182e9	−0.0634308	−0.0317154	+	0.999497i	$0.510097\pi$
$242$	1.39491e10	0.261443
$243$	0	0
$244$	−5.31660e10	−0.960241
$245$	9.30277e10	1.64955
$246$	0	0
$247$	−5.60848e9	−0.0958758
$248$	−2.28496e11	−3.83571
$249$	0	0
$250$	−1.21870e11	−1.97319
$251$	−1.71033e10	−0.271987	−0.135994	−	0.990710i	$-0.543423\pi$
$251$	−1.71033e10	−0.271987	−0.135994	+	0.990710i	$0.543423\pi$
$252$	0	0
$253$	−5.19337e10	−0.796905
$254$	4.11503e10	0.620327
$255$	0	0
$256$	−7.25891e10	−1.05631
$257$	−9.57932e10	−1.36973	−0.684866	−	0.728669i	$-0.740139\pi$
$257$	−9.57932e10	−1.36973	−0.684866	+	0.728669i	$0.740139\pi$
$258$	0	0
$259$	−8.85140e10	−1.22226
$260$	5.37595e10	0.729584
$261$	0	0
$262$	4.49886e10	0.589857
$263$	−6.34297e9	−0.0817507	−0.0408754	−	0.999164i	$-0.513015\pi$
$263$	−6.34297e9	−0.0817507	−0.0408754	+	0.999164i	$0.513015\pi$
$264$	0	0
$265$	2.27068e10	0.282846
$266$	−6.25446e10	−0.765988
$267$	0	0
$268$	−4.72582e10	−0.559591
$269$	3.08503e10	0.359231	0.179615	−	0.983737i	$-0.442515\pi$
$269$	3.08503e10	0.359231	0.179615	+	0.983737i	$0.442515\pi$
$270$	0	0
$271$	1.45802e11	1.64210	0.821052	−	0.570853i	$-0.193387\pi$
$271$	1.45802e11	1.64210	0.821052	+	0.570853i	$0.193387\pi$
$272$	−2.27701e11	−2.52234
$273$	0	0
$274$	−8.24562e10	−0.883783
$275$	−4.79768e10	−0.505864
$276$	0	0
$277$	3.79755e10	0.387565	0.193783	−	0.981044i	$-0.437924\pi$
$277$	3.79755e10	0.387565	0.193783	+	0.981044i	$0.437924\pi$
$278$	−1.14029e11	−1.14502
$279$	0	0
$280$	3.50322e11	3.40609
$281$	1.21279e11	1.16039	0.580197	−	0.814476i	$-0.302975\pi$
$281$	1.21279e11	1.16039	0.580197	+	0.814476i	$0.302975\pi$
$282$	0	0
$283$	−6.88549e10	−0.638110	−0.319055	−	0.947736i	$-0.603366\pi$
$283$	−6.88549e10	−0.638110	−0.319055	+	0.947736i	$0.603366\pi$
$284$	1.28185e10	0.116924
$285$	0	0
$286$	9.32425e10	0.824075
$287$	2.54950e11	2.21812
$288$	0	0
$289$	1.43642e10	0.121127
$290$	−3.35863e10	−0.278850
$291$	0	0
$292$	−5.08100e11	−4.09003
$293$	−2.17421e11	−1.72344	−0.861721	−	0.507383i	$-0.830613\pi$
$293$	−2.17421e11	−1.72344	−0.861721	+	0.507383i	$0.830613\pi$
$294$	0	0
$295$	−5.78567e10	−0.444790
$296$	−2.31490e11	−1.75275
$297$	0	0
$298$	3.27371e11	2.40473
$299$	−4.30768e10	−0.311690
$300$	0	0
$301$	−1.73316e11	−1.21700
$302$	3.35321e10	0.231969
$303$	0	0
$304$	−8.13826e10	−0.546513
$305$	−4.37712e10	−0.289627
$306$	0	0
$307$	−9.71479e10	−0.624181	−0.312091	−	0.950052i	$-0.601029\pi$
$307$	−9.71479e10	−0.624181	−0.312091	+	0.950052i	$0.601029\pi$
$308$	7.34514e11	4.65074
$309$	0	0
$310$	−3.21932e11	−1.97987
$311$	−1.26366e11	−0.765965	−0.382983	−	0.923756i	$-0.625103\pi$
$311$	−1.26366e11	−0.765965	−0.382983	+	0.923756i	$0.625103\pi$
$312$	0	0
$313$	−1.26191e11	−0.743156	−0.371578	−	0.928402i	$-0.621183\pi$
$313$	−1.26191e11	−0.743156	−0.371578	+	0.928402i	$0.621183\pi$
$314$	−4.94397e11	−2.87007
$315$	0	0
$316$	7.64214e11	4.31145
$317$	−2.10513e11	−1.17088	−0.585439	−	0.810716i	$-0.699078\pi$
$317$	−2.10513e11	−1.17088	−0.585439	+	0.810716i	$0.699078\pi$
$318$	0	0
$319$	−4.11493e10	−0.222487
$320$	1.28372e11	0.684378
$321$	0	0
$322$	−4.80383e11	−2.49021
$323$	4.75184e10	0.242913
$324$	0	0
$325$	−3.97947e10	−0.197856
$326$	5.06480e11	2.48361
$327$	0	0
$328$	6.66769e11	3.18085
$329$	−1.93224e11	−0.909240
$330$	0	0
$331$	−2.62984e11	−1.20421	−0.602106	−	0.798416i	$-0.705671\pi$
$331$	−2.62984e11	−1.20421	−0.602106	+	0.798416i	$0.705671\pi$
$332$	6.34512e11	2.86628
$333$	0	0
$334$	−1.92796e11	−0.847694
$335$	−3.89074e10	−0.168784
$336$	0	0
$337$	8.10072e10	0.342128	0.171064	−	0.985260i	$-0.445279\pi$
$337$	8.10072e10	0.342128	0.171064	+	0.985260i	$0.445279\pi$
$338$	−3.65489e11	−1.52317
$339$	0	0
$340$	−4.55483e11	−1.84849
$341$	−3.94425e11	−1.57968
$342$	0	0
$343$	5.90491e11	2.30351
$344$	−4.53274e11	−1.74521
$345$	0	0
$346$	−3.31241e11	−1.24251
$347$	2.78162e11	1.02995	0.514974	−	0.857206i	$-0.327801\pi$
$347$	2.78162e11	1.02995	0.514974	+	0.857206i	$0.327801\pi$
$348$	0	0
$349$	−1.23111e11	−0.444204	−0.222102	−	0.975023i	$-0.571292\pi$
$349$	−1.23111e11	−0.444204	−0.222102	+	0.975023i	$0.571292\pi$
$350$	−4.43782e11	−1.58075
$351$	0	0
$352$	5.54541e11	1.92527
$353$	4.83818e11	1.65842	0.829212	−	0.558934i	$-0.188790\pi$
$353$	4.83818e11	1.65842	0.829212	+	0.558934i	$0.188790\pi$
$354$	0	0
$355$	1.05534e10	0.0352667
$356$	4.43361e11	1.46296
$357$	0	0
$358$	8.95057e11	2.87990
$359$	1.29651e10	0.0411957	0.0205979	−	0.999788i	$-0.493443\pi$
$359$	1.29651e10	0.0411957	0.0205979	+	0.999788i	$0.493443\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	9.09023e11	2.78219
$363$	0	0
$364$	6.09249e11	1.81902
$365$	−4.18315e11	−1.23363
$366$	0	0
$367$	−4.07401e11	−1.17226	−0.586131	−	0.810217i	$-0.699349\pi$
$367$	−4.07401e11	−1.17226	−0.586131	+	0.810217i	$0.699349\pi$
$368$	−6.25071e11	−1.77670
$369$	0	0
$370$	−3.26151e11	−0.904714
$371$	2.57333e11	0.705201
$372$	0	0
$373$	2.68806e10	0.0719034	0.0359517	−	0.999354i	$-0.488554\pi$
$373$	2.68806e10	0.0719034	0.0359517	+	0.999354i	$0.488554\pi$
$374$	−7.90007e11	−2.08789
$375$	0	0
$376$	−5.05337e11	−1.30388
$377$	−3.41316e10	−0.0870203
$378$	0	0
$379$	6.33155e10	0.157628	0.0788140	−	0.996889i	$-0.474887\pi$
$379$	6.33155e10	0.157628	0.0788140	+	0.996889i	$0.474887\pi$
$380$	−1.62794e11	−0.400509
$381$	0	0
$382$	9.58402e11	2.30283
$383$	−3.25509e11	−0.772979	−0.386490	−	0.922294i	$-0.626313\pi$
$383$	−3.25509e11	−0.772979	−0.386490	+	0.922294i	$0.626313\pi$
$384$	0	0
$385$	6.04720e11	1.40275
$386$	−1.54734e12	−3.54767
$387$	0	0
$388$	−5.41558e11	−1.21312
$389$	1.55902e11	0.345206	0.172603	−	0.984991i	$-0.444782\pi$
$389$	1.55902e11	0.345206	0.172603	+	0.984991i	$0.444782\pi$
$390$	0	0
$391$	3.64972e11	0.789705
$392$	2.75723e12	5.89774
$393$	0	0
$394$	−3.12095e11	−0.652459
$395$	6.29171e11	1.30042
$396$	0	0
$397$	8.10900e11	1.63836	0.819181	−	0.573535i	$-0.194428\pi$
$397$	8.10900e11	1.63836	0.819181	+	0.573535i	$0.194428\pi$
$398$	1.32841e12	2.65374
$399$	0	0
$400$	−5.77446e11	−1.12782
$401$	8.40675e11	1.62360	0.811799	−	0.583937i	$-0.198489\pi$
$401$	8.40675e11	1.62360	0.811799	+	0.583937i	$0.198489\pi$
$402$	0	0
$403$	−3.27159e11	−0.617855
$404$	1.42962e12	2.66996
$405$	0	0
$406$	−3.80628e11	−0.695239
$407$	−3.99595e11	−0.721847
$408$	0	0
$409$	4.85989e11	0.858760	0.429380	−	0.903124i	$-0.358732\pi$
$409$	4.85989e11	0.858760	0.429380	+	0.903124i	$0.358732\pi$
$410$	9.39424e11	1.64185
$411$	0	0
$412$	5.06868e11	0.866678
$413$	−6.55682e11	−1.10897
$414$	0	0
$415$	5.22389e11	0.864525
$416$	4.59969e11	0.753023
$417$	0	0
$418$	−2.82356e11	−0.452381
$419$	−1.12599e12	−1.78473	−0.892366	−	0.451312i	$-0.850956\pi$
$419$	−1.12599e12	−1.78473	−0.892366	+	0.451312i	$0.850956\pi$
$420$	0	0
$421$	5.18013e11	0.803658	0.401829	−	0.915715i	$-0.368375\pi$
$421$	5.18013e11	0.803658	0.401829	+	0.915715i	$0.368375\pi$
$422$	−9.91396e11	−1.52174
$423$	0	0
$424$	6.73003e11	1.01128
$425$	3.37164e11	0.501293
$426$	0	0
$427$	−4.96053e11	−0.722108
$428$	5.97688e11	0.860950
$429$	0	0
$430$	−6.38626e11	−0.900822
$431$	9.63258e11	1.34461	0.672303	−	0.740276i	$-0.265305\pi$
$431$	9.63258e11	1.34461	0.672303	+	0.740276i	$0.265305\pi$
$432$	0	0
$433$	−4.49398e11	−0.614378	−0.307189	−	0.951648i	$-0.599388\pi$
$433$	−4.49398e11	−0.614378	−0.307189	+	0.951648i	$0.599388\pi$
$434$	−3.64841e12	−4.93628
$435$	0	0
$436$	−1.98103e12	−2.62544
$437$	1.30445e11	0.171104
$438$	0	0
$439$	−2.47686e11	−0.318281	−0.159141	−	0.987256i	$-0.550872\pi$
$439$	−2.47686e11	−0.318281	−0.159141	+	0.987256i	$0.550872\pi$
$440$	1.58152e12	2.01159
$441$	0	0
$442$	−6.55277e11	−0.816629
$443$	1.45436e12	1.79414	0.897068	−	0.441893i	$-0.145693\pi$
$443$	1.45436e12	1.79414	0.897068	+	0.441893i	$0.145693\pi$
$444$	0	0
$445$	3.65016e11	0.441257
$446$	1.05649e12	1.26433
$447$	0	0
$448$	1.45482e12	1.70632
$449$	1.07615e12	1.24958	0.624790	−	0.780793i	$-0.285185\pi$
$449$	1.07615e12	1.24958	0.624790	+	0.780793i	$0.285185\pi$
$450$	0	0
$451$	1.15097e12	1.30999
$452$	6.72932e11	0.758313
$453$	0	0
$454$	2.82776e11	0.312386
$455$	5.01590e11	0.548652
$456$	0	0
$457$	5.14155e11	0.551406	0.275703	−	0.961243i	$-0.411089\pi$
$457$	5.14155e11	0.551406	0.275703	+	0.961243i	$0.411089\pi$
$458$	−3.19405e11	−0.339193
$459$	0	0
$460$	−1.25037e12	−1.30205
$461$	−4.22940e11	−0.436139	−0.218070	−	0.975933i	$-0.569976\pi$
$461$	−4.22940e11	−0.436139	−0.218070	+	0.975933i	$0.569976\pi$
$462$	0	0
$463$	−7.36798e11	−0.745134	−0.372567	−	0.928005i	$-0.621522\pi$
$463$	−7.36798e11	−0.745134	−0.372567	+	0.928005i	$0.621522\pi$
$464$	−4.95271e11	−0.496035
$465$	0	0
$466$	1.96204e12	1.92740
$467$	−5.12604e11	−0.498719	−0.249360	−	0.968411i	$-0.580220\pi$
$467$	−5.12604e11	−0.498719	−0.249360	+	0.968411i	$0.580220\pi$
$468$	0	0
$469$	−4.40931e11	−0.420817
$470$	−7.11980e11	−0.673019
$471$	0	0
$472$	−1.71480e12	−1.59029
$473$	−7.82434e11	−0.718741
$474$	0	0
$475$	1.20506e11	0.108614
$476$	−5.16192e12	−4.60872
$477$	0	0
$478$	3.26684e11	0.286221
$479$	9.76695e11	0.847714	0.423857	−	0.905729i	$-0.360676\pi$
$479$	9.76695e11	0.847714	0.423857	+	0.905729i	$0.360676\pi$
$480$	0	0
$481$	−3.31447e11	−0.282333
$482$	−1.38715e11	−0.117061
$483$	0	0
$484$	4.11467e11	0.340825
$485$	−4.45861e11	−0.365899
$486$	0	0
$487$	−1.55134e11	−0.124976	−0.0624881	−	0.998046i	$-0.519904\pi$
$487$	−1.55134e11	−0.124976	−0.0624881	+	0.998046i	$0.519904\pi$
$488$	−1.29733e12	−1.03552
$489$	0	0
$490$	3.88471e12	3.04422
$491$	−9.27255e11	−0.720000	−0.360000	−	0.932952i	$-0.617223\pi$
$491$	−9.27255e11	−0.720000	−0.360000	+	0.932952i	$0.617223\pi$
$492$	0	0
$493$	2.89183e11	0.220477
$494$	−2.34203e11	−0.176938
$495$	0	0
$496$	−4.74728e12	−3.52190
$497$	1.19600e11	0.0879280
$498$	0	0
$499$	−2.24714e12	−1.62247	−0.811236	−	0.584719i	$-0.801205\pi$
$499$	−2.24714e12	−1.62247	−0.811236	+	0.584719i	$0.801205\pi$
$500$	−3.59490e12	−2.57230
$501$	0	0
$502$	−7.14212e11	−0.501950
$503$	−1.48116e12	−1.03168	−0.515840	−	0.856685i	$-0.672520\pi$
$503$	−1.48116e12	−1.03168	−0.515840	+	0.856685i	$0.672520\pi$
$504$	0	0
$505$	1.17700e12	0.805311
$506$	−2.16868e12	−1.47068
$507$	0	0
$508$	1.21384e12	0.808676
$509$	2.73246e12	1.80436	0.902180	−	0.431360i	$-0.141966\pi$
$509$	2.73246e12	1.80436	0.902180	+	0.431360i	$0.141966\pi$
$510$	0	0
$511$	−4.74070e12	−3.07573
$512$	−2.93586e12	−1.88808
$513$	0	0
$514$	−4.00020e12	−2.52783
$515$	4.17301e11	0.261407
$516$	0	0
$517$	−8.72305e11	−0.536984
$518$	−3.69623e12	−2.25566
$519$	0	0
$520$	1.31181e12	0.786783
$521$	3.34389e11	0.198830	0.0994150	−	0.995046i	$-0.468303\pi$
$521$	3.34389e11	0.198830	0.0994150	+	0.995046i	$0.468303\pi$
$522$	0	0
$523$	2.73251e12	1.59700	0.798498	−	0.601997i	$-0.205628\pi$
$523$	2.73251e12	1.59700	0.798498	+	0.601997i	$0.205628\pi$
$524$	1.32706e12	0.768954
$525$	0	0
$526$	−2.64874e11	−0.150870
$527$	2.77189e12	1.56541
$528$	0	0
$529$	−7.99251e11	−0.443744
$530$	9.48207e11	0.521989
$531$	0	0
$532$	−1.84492e12	−0.998564
$533$	9.54677e11	0.512371
$534$	0	0
$535$	4.92072e11	0.259679
$536$	−1.15317e12	−0.603463
$537$	0	0
$538$	1.28827e12	0.662957
$539$	4.75948e12	2.42890
$540$	0	0
$541$	5.64522e11	0.283330	0.141665	−	0.989915i	$-0.454754\pi$
$541$	5.64522e11	0.283330	0.141665	+	0.989915i	$0.454754\pi$
$542$	6.08849e12	3.03049
$543$	0	0
$544$	−3.89713e12	−1.90787
$545$	−1.63097e12	−0.791884
$546$	0	0
$547$	1.15089e12	0.549654	0.274827	−	0.961494i	$-0.411379\pi$
$547$	1.15089e12	0.549654	0.274827	+	0.961494i	$0.411379\pi$
$548$	−2.43227e12	−1.15212
$549$	0	0
$550$	−2.00344e12	−0.933567
$551$	1.03357e11	0.0477703
$552$	0	0
$553$	7.13031e12	3.24224
$554$	1.58581e12	0.715248
$555$	0	0
$556$	−3.36359e12	−1.49268
$557$	2.90286e11	0.127784	0.0638921	−	0.997957i	$-0.479649\pi$
$557$	2.90286e11	0.127784	0.0638921	+	0.997957i	$0.479649\pi$
$558$	0	0
$559$	−6.48996e11	−0.281118
$560$	7.27838e12	3.12744
$561$	0	0
$562$	5.06443e12	2.14150
$563$	7.83279e11	0.328571	0.164285	−	0.986413i	$-0.447468\pi$
$563$	7.83279e11	0.328571	0.164285	+	0.986413i	$0.447468\pi$
$564$	0	0
$565$	5.54020e11	0.228722
$566$	−2.87529e12	−1.17763
$567$	0	0
$568$	3.12790e11	0.126091
$569$	3.73407e12	1.49340	0.746702	−	0.665159i	$-0.231636\pi$
$569$	3.73407e12	1.49340	0.746702	+	0.665159i	$0.231636\pi$
$570$	0	0
$571$	3.23290e12	1.27271	0.636355	−	0.771396i	$-0.280441\pi$
$571$	3.23290e12	1.27271	0.636355	+	0.771396i	$0.280441\pi$
$572$	2.75044e12	1.07429
$573$	0	0
$574$	1.06464e13	4.09352
$575$	9.25565e11	0.353103
$576$	0	0
$577$	3.14886e12	1.18267	0.591333	−	0.806427i	$-0.298602\pi$
$577$	3.14886e12	1.18267	0.591333	+	0.806427i	$0.298602\pi$
$578$	5.99831e11	0.223539
$579$	0	0
$580$	−9.90719e11	−0.363517
$581$	5.92016e12	2.15546
$582$	0	0
$583$	1.16173e12	0.416481
$584$	−1.23983e13	−4.41068
$585$	0	0
$586$	−9.07920e12	−3.18059
$587$	−4.38854e12	−1.52563	−0.762815	−	0.646617i	$-0.776183\pi$
$587$	−4.38854e12	−1.52563	−0.762815	+	0.646617i	$0.776183\pi$
$588$	0	0
$589$	9.90701e11	0.339175
$590$	−2.41602e12	−0.820855
$591$	0	0
$592$	−4.80950e12	−1.60936
$593$	−3.83643e12	−1.27404	−0.637018	−	0.770849i	$-0.719832\pi$
$593$	−3.83643e12	−1.27404	−0.637018	+	0.770849i	$0.719832\pi$
$594$	0	0
$595$	−4.24977e12	−1.39008
$596$	9.65670e12	3.13488
$597$	0	0
$598$	−1.79883e12	−0.575221
$599$	−3.29183e12	−1.04476	−0.522380	−	0.852713i	$-0.674956\pi$
$599$	−3.29183e12	−1.04476	−0.522380	+	0.852713i	$0.674956\pi$
$600$	0	0
$601$	4.43624e12	1.38701	0.693506	−	0.720451i	$-0.256065\pi$
$601$	4.43624e12	1.38701	0.693506	+	0.720451i	$0.256065\pi$
$602$	−7.23746e12	−2.24596
$603$	0	0
$604$	9.89122e11	0.302401
$605$	3.38758e11	0.102799
$606$	0	0
$607$	1.67409e12	0.500529	0.250264	−	0.968178i	$-0.419482\pi$
$607$	1.67409e12	0.500529	0.250264	+	0.968178i	$0.419482\pi$
$608$	−1.39287e12	−0.413376
$609$	0	0
$610$	−1.82783e12	−0.534504
$611$	−7.23540e11	−0.210028
$612$	0	0
$613$	1.31692e12	0.376694	0.188347	−	0.982103i	$-0.439687\pi$
$613$	1.31692e12	0.376694	0.188347	+	0.982103i	$0.439687\pi$
$614$	−4.05677e12	−1.15192
$615$	0	0
$616$	1.79232e13	5.01536
$617$	5.34801e12	1.48562	0.742812	−	0.669500i	$-0.233492\pi$
$617$	5.34801e12	1.48562	0.742812	+	0.669500i	$0.233492\pi$
$618$	0	0
$619$	1.83252e12	0.501697	0.250849	−	0.968026i	$-0.419290\pi$
$619$	1.83252e12	0.501697	0.250849	+	0.968026i	$0.419290\pi$
$620$	−9.49626e12	−2.58101
$621$	0	0
$622$	−5.27688e12	−1.41358
$623$	4.13667e12	1.10016
$624$	0	0
$625$	−1.15363e12	−0.302416
$626$	−5.26958e12	−1.37149
$627$	0	0
$628$	−1.45836e13	−3.74150
$629$	2.80822e12	0.715324
$630$	0	0
$631$	6.87154e11	0.172553	0.0862764	−	0.996271i	$-0.472503\pi$
$631$	6.87154e11	0.172553	0.0862764	+	0.996271i	$0.472503\pi$
$632$	1.86479e13	4.64946
$633$	0	0
$634$	−8.79073e12	−2.16084
$635$	9.99345e11	0.243912
$636$	0	0
$637$	3.94779e12	0.950006
$638$	−1.71834e12	−0.410597
$639$	0	0
$640$	−1.88882e11	−0.0445022
$641$	−2.50011e12	−0.584922	−0.292461	−	0.956277i	$-0.594474\pi$
$641$	−2.50011e12	−0.584922	−0.292461	+	0.956277i	$0.594474\pi$
$642$	0	0
$643$	−1.48345e12	−0.342235	−0.171117	−	0.985251i	$-0.554738\pi$
$643$	−1.48345e12	−0.342235	−0.171117	+	0.985251i	$0.554738\pi$
$644$	−1.41702e13	−3.24631
$645$	0	0
$646$	1.98431e12	0.448293
$647$	5.16393e12	1.15854	0.579270	−	0.815135i	$-0.303338\pi$
$647$	5.16393e12	1.15854	0.579270	+	0.815135i	$0.303338\pi$
$648$	0	0
$649$	−2.96007e12	−0.654938
$650$	−1.66177e12	−0.365142
$651$	0	0
$652$	1.49400e13	3.23770
$653$	−8.25854e12	−1.77744	−0.888718	−	0.458455i	$-0.848403\pi$
$653$	−8.25854e12	−1.77744	−0.888718	+	0.458455i	$0.848403\pi$
$654$	0	0
$655$	1.09256e12	0.231931
$656$	1.38530e13	2.92062
$657$	0	0
$658$	−8.06876e12	−1.67799
$659$	−5.19776e12	−1.07357	−0.536787	−	0.843718i	$-0.680362\pi$
$659$	−5.19776e12	−1.07357	−0.536787	+	0.843718i	$0.680362\pi$
$660$	0	0
$661$	−2.39001e12	−0.486961	−0.243480	−	0.969906i	$-0.578289\pi$
$661$	−2.39001e12	−0.486961	−0.243480	+	0.969906i	$0.578289\pi$
$662$	−1.09818e13	−2.22236
$663$	0	0
$664$	1.54830e13	3.09099
$665$	−1.51891e12	−0.301186
$666$	0	0
$667$	7.93850e11	0.155300
$668$	−5.68705e12	−1.10508
$669$	0	0
$670$	−1.62472e12	−0.311488
$671$	−2.23942e12	−0.426466
$672$	0	0
$673$	1.30739e12	0.245662	0.122831	−	0.992428i	$-0.460803\pi$
$673$	1.30739e12	0.245662	0.122831	+	0.992428i	$0.460803\pi$
$674$	3.38275e12	0.631394
$675$	0	0
$676$	−1.07811e13	−1.98565
$677$	4.51590e12	0.826219	0.413110	−	0.910681i	$-0.364443\pi$
$677$	4.51590e12	0.826219	0.413110	+	0.910681i	$0.364443\pi$
$678$	0	0
$679$	−5.05287e12	−0.912272
$680$	−1.11144e13	−1.99341
$681$	0	0
$682$	−1.64707e13	−2.91529
$683$	−6.82404e12	−1.19991	−0.599955	−	0.800034i	$-0.704815\pi$
$683$	−6.82404e12	−1.19991	−0.599955	+	0.800034i	$0.704815\pi$
$684$	0	0
$685$	−2.00247e12	−0.347503
$686$	2.46581e13	4.25110
$687$	0	0
$688$	−9.41733e12	−1.60243
$689$	9.63603e11	0.162896
$690$	0	0
$691$	5.63718e12	0.940613	0.470306	−	0.882503i	$-0.344143\pi$
$691$	5.63718e12	0.940613	0.470306	+	0.882503i	$0.344143\pi$
$692$	−9.77085e12	−1.61978
$693$	0	0
$694$	1.16157e13	1.90076
$695$	−2.76922e12	−0.450221
$696$	0	0
$697$	−8.08860e12	−1.29815
$698$	−5.14095e12	−0.819774
$699$	0	0
$700$	−1.30906e13	−2.06071
$701$	−7.92649e12	−1.23979	−0.619897	−	0.784683i	$-0.712826\pi$
$701$	−7.92649e12	−1.23979	−0.619897	+	0.784683i	$0.712826\pi$
$702$	0	0
$703$	1.00369e12	0.154988
$704$	6.56778e12	1.00772
$705$	0	0
$706$	2.02036e13	3.06060
$707$	1.33387e13	2.00783
$708$	0	0
$709$	−9.03605e12	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$709$	−9.03605e12	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$710$	4.40695e11	0.0650842
$711$	0	0
$712$	1.08186e13	1.57765
$713$	7.60923e12	1.10265
$714$	0	0
$715$	2.26442e12	0.324026
$716$	2.64022e13	3.75432
$717$	0	0
$718$	5.41407e11	0.0760263
$719$	−4.32764e12	−0.603909	−0.301954	−	0.953322i	$-0.597639\pi$
$719$	−4.32764e12	−0.603909	−0.301954	+	0.953322i	$0.597639\pi$
$720$	0	0
$721$	4.72921e12	0.651748
$722$	7.09211e11	0.0971310
$723$	0	0
$724$	2.68141e13	3.62694
$725$	7.33365e11	0.0985823
$726$	0	0
$727$	−1.03025e13	−1.36785	−0.683926	−	0.729551i	$-0.739729\pi$
$727$	−1.03025e13	−1.36785	−0.683926	+	0.729551i	$0.739729\pi$
$728$	1.48665e13	1.96163
$729$	0	0
$730$	−1.74683e13	−2.27665
$731$	5.49868e12	0.712247
$732$	0	0
$733$	−1.24191e13	−1.58900	−0.794499	−	0.607265i	$-0.792267\pi$
$733$	−1.24191e13	−1.58900	−0.794499	+	0.607265i	$0.792267\pi$
$734$	−1.70125e13	−2.16340
$735$	0	0
$736$	−1.06982e13	−1.34388
$737$	−1.99058e12	−0.248528
$738$	0	0
$739$	9.29716e12	1.14670	0.573351	−	0.819310i	$-0.305643\pi$
$739$	9.29716e12	1.14670	0.573351	+	0.819310i	$0.305643\pi$
$740$	−9.62073e12	−1.17941
$741$	0	0
$742$	1.07459e13	1.30144
$743$	5.92293e12	0.712996	0.356498	−	0.934296i	$-0.383971\pi$
$743$	5.92293e12	0.712996	0.356498	+	0.934296i	$0.383971\pi$
$744$	0	0
$745$	7.95029e12	0.945539
$746$	1.12250e12	0.132697
$747$	0	0
$748$	−2.33034e13	−2.72184
$749$	5.57658e12	0.647441
$750$	0	0
$751$	8.05210e12	0.923696	0.461848	−	0.886959i	$-0.347186\pi$
$751$	8.05210e12	0.923696	0.461848	+	0.886959i	$0.347186\pi$
$752$	−1.04990e13	−1.19720
$753$	0	0
$754$	−1.42529e12	−0.160595
$755$	8.14336e11	0.0912100
$756$	0	0
$757$	1.02771e13	1.13746	0.568731	−	0.822523i	$-0.307434\pi$
$757$	1.02771e13	1.13746	0.568731	+	0.822523i	$0.307434\pi$
$758$	2.64397e12	0.290901
$759$	0	0
$760$	−3.97240e12	−0.431909
$761$	−4.78270e11	−0.0516943	−0.0258471	−	0.999666i	$-0.508228\pi$
$761$	−4.78270e11	−0.0516943	−0.0258471	+	0.999666i	$0.508228\pi$
$762$	0	0
$763$	−1.84835e13	−1.97435
$764$	2.82707e13	3.00204
$765$	0	0
$766$	−1.35928e13	−1.42653
$767$	−2.45525e12	−0.256163
$768$	0	0
$769$	−1.06587e11	−0.0109909	−0.00549546	−	0.999985i	$-0.501749\pi$
$769$	−1.06587e11	−0.0109909	−0.00549546	+	0.999985i	$0.501749\pi$
$770$	2.52523e13	2.58877
$771$	0	0
$772$	−4.56430e13	−4.62484
$773$	−1.57116e13	−1.58276	−0.791378	−	0.611327i	$-0.790636\pi$
$773$	−1.57116e13	−1.58276	−0.791378	+	0.611327i	$0.790636\pi$
$774$	0	0
$775$	7.02947e12	0.699946
$776$	−1.32148e13	−1.30822
$777$	0	0
$778$	6.51026e12	0.637074
$779$	−2.89095e12	−0.281269
$780$	0	0
$781$	5.39932e11	0.0519290
$782$	1.52408e13	1.45739
$783$	0	0
$784$	5.72849e13	5.41524
$785$	−1.20065e13	−1.12851
$786$	0	0
$787$	−1.23117e13	−1.14401	−0.572006	−	0.820250i	$-0.693834\pi$
$787$	−1.23117e13	−1.14401	−0.572006	+	0.820250i	$0.693834\pi$
$788$	−9.20609e12	−0.850564
$789$	0	0
$790$	2.62734e13	2.39990
$791$	6.27863e12	0.570257
$792$	0	0
$793$	−1.85750e12	−0.166802
$794$	3.38621e13	3.02358
$795$	0	0
$796$	3.91851e13	3.45949
$797$	1.25015e13	1.09749	0.548746	−	0.835989i	$-0.315105\pi$
$797$	1.25015e13	1.09749	0.548746	+	0.835989i	$0.315105\pi$
$798$	0	0
$799$	6.13027e12	0.532131
$800$	−9.88306e12	−0.853074
$801$	0	0
$802$	3.51055e13	2.99633
$803$	−2.14018e13	−1.81648
$804$	0	0
$805$	−1.16662e13	−0.979149
$806$	−1.36617e13	−1.14024
$807$	0	0
$808$	3.48847e13	2.87928
$809$	−2.18197e12	−0.179094	−0.0895470	−	0.995983i	$-0.528542\pi$
$809$	−2.18197e12	−0.179094	−0.0895470	+	0.995983i	$0.528542\pi$
$810$	0	0
$811$	−6.14819e12	−0.499061	−0.249530	−	0.968367i	$-0.580276\pi$
$811$	−6.14819e12	−0.499061	−0.249530	+	0.968367i	$0.580276\pi$
$812$	−1.12277e13	−0.906332
$813$	0	0
$814$	−1.66866e13	−1.33216
$815$	1.23000e13	0.976554
$816$	0	0
$817$	1.96528e12	0.154321
$818$	2.02943e13	1.58483
$819$	0	0
$820$	2.77109e13	2.14037
$821$	−1.50871e13	−1.15894	−0.579471	−	0.814993i	$-0.696741\pi$
$821$	−1.50871e13	−1.15894	−0.579471	+	0.814993i	$0.696741\pi$
$822$	0	0
$823$	−2.28414e13	−1.73550	−0.867748	−	0.497005i	$-0.834433\pi$
$823$	−2.28414e13	−1.73550	−0.867748	+	0.497005i	$0.834433\pi$
$824$	1.23683e13	0.934625
$825$	0	0
$826$	−2.73804e13	−2.04658
$827$	2.56123e13	1.90403	0.952015	−	0.306051i	$-0.0990080\pi$
$827$	2.56123e13	1.90403	0.952015	+	0.306051i	$0.0990080\pi$
$828$	0	0
$829$	1.76802e13	1.30015	0.650074	−	0.759871i	$-0.274738\pi$
$829$	1.76802e13	1.30015	0.650074	+	0.759871i	$0.274738\pi$
$830$	2.18143e13	1.59547
$831$	0	0
$832$	5.44769e12	0.394146
$833$	−3.34480e13	−2.40696
$834$	0	0
$835$	−4.68210e12	−0.333313
$836$	−8.32887e12	−0.589736
$837$	0	0
$838$	−4.70200e13	−3.29370
$839$	−8.38339e12	−0.584105	−0.292052	−	0.956402i	$-0.594338\pi$
$839$	−8.38339e12	−0.584105	−0.292052	+	0.956402i	$0.594338\pi$
$840$	0	0
$841$	−1.38781e13	−0.956642
$842$	2.16315e13	1.48314
$843$	0	0
$844$	−2.92439e13	−1.98379
$845$	−8.87599e12	−0.598910
$846$	0	0
$847$	3.83909e12	0.256303
$848$	1.39825e13	0.928545
$849$	0	0
$850$	1.40795e13	0.925131
$851$	7.70896e12	0.503863
$852$	0	0
$853$	−1.52642e13	−0.987197	−0.493598	−	0.869690i	$-0.664319\pi$
$853$	−1.52642e13	−0.987197	−0.493598	+	0.869690i	$0.664319\pi$
$854$	−2.07145e13	−1.33264
$855$	0	0
$856$	1.45844e13	0.928448
$857$	4.72637e12	0.299305	0.149653	−	0.988739i	$-0.452184\pi$
$857$	4.72637e12	0.299305	0.149653	+	0.988739i	$0.452184\pi$
$858$	0	0
$859$	−1.43858e13	−0.901497	−0.450749	−	0.892651i	$-0.648843\pi$
$859$	−1.43858e13	−0.901497	−0.450749	+	0.892651i	$0.648843\pi$
$860$	−1.88380e13	−1.17434
$861$	0	0
$862$	4.02244e13	2.48146
$863$	−3.24081e13	−1.98887	−0.994433	−	0.105375i	$-0.966396\pi$
$863$	−3.24081e13	−1.98887	−0.994433	+	0.105375i	$0.966396\pi$
$864$	0	0
$865$	−8.04427e12	−0.488556
$866$	−1.87663e13	−1.13383
$867$	0	0
$868$	−1.07620e14	−6.43507
$869$	3.21897e13	1.91482
$870$	0	0
$871$	−1.65110e12	−0.0972057
$872$	−4.83400e13	−2.83128
$873$	0	0
$874$	5.44720e12	0.315771
$875$	−3.35413e13	−1.93439
$876$	0	0
$877$	−3.63648e11	−0.0207579	−0.0103789	−	0.999946i	$-0.503304\pi$
$877$	−3.63648e11	−0.0207579	−0.0103789	+	0.999946i	$0.503304\pi$
$878$	−1.03430e13	−0.587385
$879$	0	0
$880$	3.28581e13	1.84702
$881$	−1.02213e13	−0.571632	−0.285816	−	0.958285i	$-0.592265\pi$
$881$	−1.02213e13	−0.571632	−0.285816	+	0.958285i	$0.592265\pi$
$882$	0	0
$883$	−3.02685e13	−1.67559	−0.837795	−	0.545985i	$-0.816156\pi$
$883$	−3.02685e13	−1.67559	−0.837795	+	0.545985i	$0.816156\pi$
$884$	−1.93292e13	−1.06458
$885$	0	0
$886$	6.07321e13	3.31106
$887$	1.17495e12	0.0637328	0.0318664	−	0.999492i	$-0.489855\pi$
$887$	1.17495e12	0.0637328	0.0318664	+	0.999492i	$0.489855\pi$
$888$	0	0
$889$	1.13254e13	0.608130
$890$	1.52426e13	0.814335
$891$	0	0
$892$	3.11641e13	1.64821
$893$	2.19102e12	0.115296
$894$	0	0
$895$	2.17367e13	1.13237
$896$	−2.14057e12	−0.110954
$897$	0	0
$898$	4.49386e13	2.30609
$899$	6.02912e12	0.307847
$900$	0	0
$901$	−8.16422e12	−0.412718
$902$	4.80628e13	2.41757
$903$	0	0
$904$	1.64205e13	0.817764
$905$	2.20759e13	1.09395
$906$	0	0
$907$	2.49201e13	1.22269	0.611347	−	0.791363i	$-0.290628\pi$
$907$	2.49201e13	1.22269	0.611347	+	0.791363i	$0.290628\pi$
$908$	8.34124e12	0.407234
$909$	0	0
$910$	2.09457e13	1.01253
$911$	3.47770e13	1.67286	0.836428	−	0.548076i	$-0.184640\pi$
$911$	3.47770e13	1.67286	0.836428	+	0.548076i	$0.184640\pi$
$912$	0	0
$913$	2.67265e13	1.27298
$914$	2.14704e13	1.01761
$915$	0	0
$916$	−9.42172e12	−0.442182
$917$	1.23818e13	0.578259
$918$	0	0
$919$	1.09456e13	0.506198	0.253099	−	0.967440i	$-0.418550\pi$
$919$	1.09456e13	0.506198	0.253099	+	0.967440i	$0.418550\pi$
$920$	−3.05107e13	−1.40413
$921$	0	0
$922$	−1.76614e13	−0.804890
$923$	4.47851e11	0.0203107
$924$	0	0
$925$	7.12160e12	0.319845
$926$	−3.07677e13	−1.37514
$927$	0	0
$928$	−8.47663e12	−0.375195
$929$	−1.92304e13	−0.847065	−0.423533	−	0.905881i	$-0.639210\pi$
$929$	−1.92304e13	−0.847065	−0.423533	+	0.905881i	$0.639210\pi$
$930$	0	0
$931$	−1.19547e13	−0.521512
$932$	5.78757e13	2.51261
$933$	0	0
$934$	−2.14057e13	−0.920381
$935$	−1.91855e13	−0.820959
$936$	0	0
$937$	2.66920e13	1.13123	0.565617	−	0.824668i	$-0.308638\pi$
$937$	2.66920e13	1.13123	0.565617	+	0.824668i	$0.308638\pi$
$938$	−1.84127e13	−0.776613
$939$	0	0
$940$	−2.10018e13	−0.877366
$941$	8.85521e12	0.368168	0.184084	−	0.982911i	$-0.441068\pi$
$941$	8.85521e12	0.368168	0.184084	+	0.982911i	$0.441068\pi$
$942$	0	0
$943$	−2.22044e13	−0.914399
$944$	−3.56272e13	−1.46018
$945$	0	0
$946$	−3.26734e13	−1.32643
$947$	2.96409e13	1.19761	0.598807	−	0.800893i	$-0.295641\pi$
$947$	2.96409e13	1.19761	0.598807	+	0.800893i	$0.295641\pi$
$948$	0	0
$949$	−1.77519e13	−0.710472
$950$	5.03217e12	0.200447
$951$	0	0
$952$	−1.25958e14	−4.97004
$953$	−3.63447e13	−1.42733	−0.713663	−	0.700489i	$-0.752965\pi$
$953$	−3.63447e13	−1.42733	−0.713663	+	0.700489i	$0.752965\pi$
$954$	0	0
$955$	2.32750e13	0.905473
$956$	9.63643e12	0.373126
$957$	0	0
$958$	4.07855e13	1.56445
$959$	−2.26937e13	−0.866406
$960$	0	0
$961$	3.13509e13	1.18575
$962$	−1.38408e13	−0.521042
$963$	0	0
$964$	−4.09177e12	−0.152604
$965$	−3.75776e13	−1.39494
$966$	0	0
$967$	1.76693e13	0.649829	0.324915	−	0.945743i	$-0.394664\pi$
$967$	1.76693e13	0.649829	0.324915	+	0.945743i	$0.394664\pi$
$968$	1.00404e13	0.367545
$969$	0	0
$970$	−1.86185e13	−0.675263
$971$	7.14363e12	0.257889	0.128944	−	0.991652i	$-0.458841\pi$
$971$	7.14363e12	0.257889	0.128944	+	0.991652i	$0.458841\pi$
$972$	0	0
$973$	−3.13831e13	−1.12251
$974$	−6.47819e12	−0.230642
$975$	0	0
$976$	−2.69536e13	−0.950806
$977$	−2.58824e13	−0.908821	−0.454411	−	0.890792i	$-0.650150\pi$
$977$	−2.58824e13	−0.908821	−0.454411	+	0.890792i	$0.650150\pi$
$978$	0	0
$979$	1.86749e13	0.649736
$980$	1.14590e14	3.96854
$981$	0	0
$982$	−3.87209e13	−1.32875
$983$	3.21189e13	1.09716	0.548579	−	0.836099i	$-0.315169\pi$
$983$	3.21189e13	1.09716	0.548579	+	0.836099i	$0.315169\pi$
$984$	0	0
$985$	−7.57930e12	−0.256547
$986$	1.20759e13	0.406887
$987$	0	0
$988$	−6.90845e12	−0.230661
$989$	1.50947e13	0.501696
$990$	0	0
$991$	−3.52619e13	−1.16138	−0.580690	−	0.814125i	$-0.697217\pi$
$991$	−3.52619e13	−1.16138	−0.580690	+	0.814125i	$0.697217\pi$
$992$	−8.12504e13	−2.66393
$993$	0	0
$994$	4.99433e12	0.162270
$995$	3.22608e13	1.04345
$996$	0	0
$997$	2.19749e13	0.704365	0.352183	−	0.935931i	$-0.385440\pi$
$997$	2.19749e13	0.704365	0.352183	+	0.935931i	$0.385440\pi$
$998$	−9.38374e13	−2.99425
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.10.a.c.1.6		6
3.2	odd	2		19.10.a.a.1.1	✓	6
12.11	even	2		304.10.a.f.1.6		6
57.56	even	2		361.10.a.b.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.a.a.1.1	✓	6	3.2	odd	2
171.10.a.c.1.6		6	1.1	even	1	trivial
304.10.a.f.1.6		6	12.11	even	2
361.10.a.b.1.6		6	57.56	even	2

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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+41.7587 q^{2} +1231.79 q^{4} +1014.12 q^{5} +11492.9 q^{7} +30057.3 q^{8} +42348.3 q^{10} +51884.4 q^{11} +43035.9 q^{13} +479927. q^{14} +624478. q^{16} -364626. q^{17} -130321. q^{19} +1.24918e6 q^{20} +2.16662e6 q^{22} -1.00095e6 q^{23} -924686. q^{25} +1.79712e6 q^{26} +1.41568e7 q^{28} -793096. q^{29} -7.60200e6 q^{31} +1.06880e7 q^{32} -1.52263e7 q^{34} +1.16551e7 q^{35} -7.70164e6 q^{37} -5.44203e6 q^{38} +3.04817e7 q^{40} +2.21833e7 q^{41} -1.50803e7 q^{43} +6.39104e7 q^{44} -4.17983e7 q^{46} -1.68125e7 q^{47} +9.17325e7 q^{49} -3.86136e7 q^{50} +5.30110e7 q^{52} +2.23907e7 q^{53} +5.26170e7 q^{55} +3.45445e8 q^{56} -3.31186e7 q^{58} -5.70512e7 q^{59} -4.31618e7 q^{61} -3.17450e8 q^{62} +1.26585e8 q^{64} +4.36436e7 q^{65} -3.83656e7 q^{67} -4.49141e8 q^{68} +4.86703e8 q^{70} +1.04064e7 q^{71} -4.12491e8 q^{73} -3.21610e8 q^{74} -1.60528e8 q^{76} +5.96301e8 q^{77} +6.20411e8 q^{79} +6.33295e8 q^{80} +9.26344e8 q^{82} +5.15116e8 q^{83} -3.69774e8 q^{85} -6.29735e8 q^{86} +1.55950e9 q^{88} +3.59933e8 q^{89} +4.94606e8 q^{91} -1.23296e9 q^{92} -7.02067e8 q^{94} -1.32161e8 q^{95} -4.39653e8 q^{97} +3.83063e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 33 q^{2} + 1365 q^{4} + 3612 q^{5} + 4085 q^{7} + 23511 q^{8} - 93884 q^{10} + 69312 q^{11} - 191747 q^{13} + 644691 q^{14} + 13905 q^{16} + 288195 q^{17} - 781926 q^{19} + 1551444 q^{20} + 2409710 q^{22}+ \cdots + 4245664590 q^{98}+O(q^{100}) \) 6 * q + 33 * q^2 + 1365 * q^4 + 3612 * q^5 + 4085 * q^7 + 23511 * q^8 - 93884 * q^10 + 69312 * q^11 - 191747 * q^13 + 644691 * q^14 + 13905 * q^16 + 288195 * q^17 - 781926 * q^19 + 1551444 * q^20 + 2409710 * q^22 + 50697 * q^23 + 7126888 * q^25 - 4394073 * q^26 + 19295623 * q^28 + 7178667 * q^29 - 4310716 * q^31 + 5388975 * q^32 + 18049 * q^34 + 4166490 * q^35 - 15687040 * q^37 - 4300593 * q^38 + 33173148 * q^40 + 25134306 * q^41 - 75118674 * q^43 + 94242102 * q^44 - 85549087 * q^46 + 83731938 * q^47 + 21010019 * q^49 - 141211941 * q^50 + 147001043 * q^52 + 138019203 * q^53 - 1518258 * q^55 + 182020629 * q^56 + 14051197 * q^58 + 7809915 * q^59 + 191946566 * q^61 - 564818736 * q^62 + 167338593 * q^64 + 146177172 * q^65 - 16109787 * q^67 - 561722511 * q^68 + 692634996 * q^70 + 264469698 * q^71 - 287572857 * q^73 - 376807098 * q^74 - 177888165 * q^76 + 1619915436 * q^77 - 86534002 * q^79 + 1300596708 * q^80 + 1582686496 * q^82 + 359214570 * q^83 - 648092718 * q^85 + 1061832924 * q^86 + 779241042 * q^88 + 2263866306 * q^89 + 1172864023 * q^91 + 373904061 * q^92 - 864022488 * q^94 - 470719452 * q^95 + 2705893460 * q^97 + 4245664590 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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