LMFDB - Embedded newform 4.34.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,34,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 34, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 34);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 34 $
Character orbit:	$[\chi]$	$=$	4.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:	$27.5931315524$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 65185566x - 173679864984 $ x^3 - x^2 - 65185566*x - 173679864984
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3139.05$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34728e8 q^{3} -4.39893e11 q^{5} -5.84388e13 q^{7} +1.25925e16 q^{9} +O(q^{10})$	$q+1.34728e8 q^{3} -4.39893e11 q^{5} -5.84388e13 q^{7} +1.25925e16 q^{9} +2.38684e17 q^{11} +1.71699e18 q^{13} -5.92657e19 q^{15} +1.90455e20 q^{17} +1.58873e21 q^{19} -7.87333e21 q^{21} +6.96105e21 q^{23} +7.70902e22 q^{25} +9.47601e23 q^{27} -1.30633e24 q^{29} +5.31348e24 q^{31} +3.21574e25 q^{33} +2.57068e25 q^{35} -8.25335e25 q^{37} +2.31327e26 q^{39} -4.09102e26 q^{41} +5.28980e26 q^{43} -5.53935e27 q^{45} -8.51099e26 q^{47} -4.31590e27 q^{49} +2.56596e28 q^{51} +2.58983e28 q^{53} -1.04995e29 q^{55} +2.14047e29 q^{57} +2.29414e29 q^{59} -8.85755e28 q^{61} -7.35891e29 q^{63} -7.55293e29 q^{65} -1.09971e30 q^{67} +9.37847e29 q^{69} +1.24605e30 q^{71} -2.58039e30 q^{73} +1.03862e31 q^{75} -1.39484e31 q^{77} +3.55401e31 q^{79} +5.76657e31 q^{81} -5.14661e31 q^{83} -8.37798e31 q^{85} -1.75999e32 q^{87} +5.16507e31 q^{89} -1.00339e32 q^{91} +7.15874e32 q^{93} -6.98872e32 q^{95} -1.10473e33 q^{97} +3.00564e33 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) $ 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9	$ 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots + 22\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9 + 227617657302449700 * q^11 + 272970442217358762 * q^13 - 32142473481241105848 * q^15 + 93037188311816716854 * q^17 + 1324887806175081735228 * q^19 - 2289418149258037352352 * q^21 + 11680621362017694733896 * q^23 + 258183723295544823174621 * q^25 + 1166092519364338103061624 * q^27 + 2899576251477878827068378 * q^29 + 15816960746665911900076128 * q^31 + 51019828271446575126952080 * q^33 + 104388880678377160773832848 * q^35 + 34785435467211797617907442 * q^37 + 135170503164419330229696168 * q^39 - 673854719392272123121298562 * q^41 - 1557799443962179540641188220 * q^43 - 8295378544987729028249800086 * q^45 - 7226966919683191159587179376 * q^47 - 4755447639624414087245018709 * q^49 + 24102421034763142313708267736 * q^51 + 54068957931214078433925305634 * q^53 + 72116739028709103312702411480 * q^55 + 276172301449046303471104708848 * q^57 + 201006239114458347216752503476 * q^59 + 59158715139739534320593297466 * q^61 - 1234485289426293666626080288968 * q^63 - 2252651168064734881763466967812 * q^65 - 2904325260543190867603068997428 * q^67 - 1526403599797234628609099310816 * q^69 + 1693346400885181878584585353944 * q^71 + 1318409001451717739930563879902 * q^73 + 20178451712790450065114527827828 * q^75 + 20835723633799560796766684178720 * q^77 + 43046757284148836281181573950704 * q^79 + 60385181780035992188351168542827 * q^81 - 71356083882380783036145089288964 * q^83 - 137360972548514627995666767037884 * q^85 - 253975679679758553189305216266392 * q^87 - 194457621978564019473800095872498 * q^89 - 384667165850137479369424211871408 * q^91 + 554725843432039554369917205602688 * q^93 - 205626706338774416034575296820376 * q^95 + 566218664019390927229117474480422 * q^97 + 2255131944887948953089434155026900 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.34728e8	1.80699	0.903496	−	0.428596i	$-0.140991\pi$
$3$	1.34728e8	1.80699	0.903496	+	0.428596i	$0.140991\pi$
$4$	0	0
$5$	−4.39893e11	−1.28926	−0.644632	−	0.764493i	$-0.722989\pi$
$5$	−4.39893e11	−1.28926	−0.644632	+	0.764493i	$0.722989\pi$
$6$	0	0
$7$	−5.84388e13	−0.664635	−0.332318	−	0.943168i	$-0.607831\pi$
$7$	−5.84388e13	−0.664635	−0.332318	+	0.943168i	$0.607831\pi$
$8$	0	0
$9$	1.25925e16	2.26522
$10$	0	0
$11$	2.38684e17	1.56619	0.783095	−	0.621902i	$-0.213640\pi$
$11$	2.38684e17	1.56619	0.783095	+	0.621902i	$0.213640\pi$
$12$	0	0
$13$	1.71699e18	0.715654	0.357827	−	0.933788i	$-0.383518\pi$
$13$	1.71699e18	0.715654	0.357827	+	0.933788i	$0.383518\pi$
$14$	0	0
$15$	−5.92657e19	−2.32969
$16$	0	0
$17$	1.90455e20	0.949261	0.474631	−	0.880185i	$-0.342582\pi$
$17$	1.90455e20	0.949261	0.474631	+	0.880185i	$0.342582\pi$
$18$	0	0
$19$	1.58873e21	1.26362	0.631810	−	0.775123i	$-0.282312\pi$
$19$	1.58873e21	1.26362	0.631810	+	0.775123i	$0.282312\pi$
$20$	0	0
$21$	−7.87333e21	−1.20099
$22$	0	0
$23$	6.96105e21	0.236682	0.118341	−	0.992973i	$-0.462242\pi$
$23$	6.96105e21	0.236682	0.118341	+	0.992973i	$0.462242\pi$
$24$	0	0
$25$	7.70902e22	0.662199
$26$	0	0
$27$	9.47601e23	2.28625
$28$	0	0
$29$	−1.30633e24	−0.969362	−0.484681	−	0.874691i	$-0.661064\pi$
$29$	−1.30633e24	−0.969362	−0.484681	+	0.874691i	$0.661064\pi$
$30$	0	0
$31$	5.31348e24	1.31193	0.655966	−	0.754790i	$-0.272261\pi$
$31$	5.31348e24	1.31193	0.655966	+	0.754790i	$0.272261\pi$
$32$	0	0
$33$	3.21574e25	2.83009
$34$	0	0
$35$	2.57068e25	0.856890
$36$	0	0
$37$	−8.25335e25	−1.09977	−0.549885	−	0.835240i	$-0.685328\pi$
$37$	−8.25335e25	−1.09977	−0.549885	+	0.835240i	$0.685328\pi$
$38$	0	0
$39$	2.31327e26	1.29318
$40$	0	0
$41$	−4.09102e26	−1.00207	−0.501035	−	0.865427i	$-0.667047\pi$
$41$	−4.09102e26	−1.00207	−0.501035	+	0.865427i	$0.667047\pi$
$42$	0	0
$43$	5.28980e26	0.590486	0.295243	−	0.955422i	$-0.404599\pi$
$43$	5.28980e26	0.590486	0.295243	+	0.955422i	$0.404599\pi$
$44$	0	0
$45$	−5.53935e27	−2.92047
$46$	0	0
$47$	−8.51099e26	−0.218960	−0.109480	−	0.993989i	$-0.534919\pi$
$47$	−8.51099e26	−0.218960	−0.109480	+	0.993989i	$0.534919\pi$
$48$	0	0
$49$	−4.31590e27	−0.558260
$50$	0	0
$51$	2.56596e28	1.71531
$52$	0	0
$53$	2.58983e28	0.917738	0.458869	−	0.888504i	$-0.348255\pi$
$53$	2.58983e28	0.917738	0.458869	+	0.888504i	$0.348255\pi$
$54$	0	0
$55$	−1.04995e29	−2.01923
$56$	0	0
$57$	2.14047e29	2.28335
$58$	0	0
$59$	2.29414e29	1.38536	0.692679	−	0.721246i	$-0.256430\pi$
$59$	2.29414e29	1.38536	0.692679	+	0.721246i	$0.256430\pi$
$60$	0	0
$61$	−8.85755e28	−0.308583	−0.154291	−	0.988025i	$-0.549309\pi$
$61$	−8.85755e28	−0.308583	−0.154291	+	0.988025i	$0.549309\pi$
$62$	0	0
$63$	−7.35891e29	−1.50555
$64$	0	0
$65$	−7.55293e29	−0.922667
$66$	0	0
$67$	−1.09971e30	−0.814789	−0.407394	−	0.913252i	$-0.633563\pi$
$67$	−1.09971e30	−0.814789	−0.407394	+	0.913252i	$0.633563\pi$
$68$	0	0
$69$	9.37847e29	0.427683
$70$	0	0
$71$	1.24605e30	0.354627	0.177314	−	0.984154i	$-0.443259\pi$
$71$	1.24605e30	0.354627	0.177314	+	0.984154i	$0.443259\pi$
$72$	0	0
$73$	−2.58039e30	−0.464363	−0.232182	−	0.972672i	$-0.574586\pi$
$73$	−2.58039e30	−0.464363	−0.232182	+	0.972672i	$0.574586\pi$
$74$	0	0
$75$	1.03862e31	1.19659
$76$	0	0
$77$	−1.39484e31	−1.04095
$78$	0	0
$79$	3.55401e31	1.73729	0.868643	−	0.495439i	$-0.164993\pi$
$79$	3.55401e31	1.73729	0.868643	+	0.495439i	$0.164993\pi$
$80$	0	0
$81$	5.76657e31	1.86601
$82$	0	0
$83$	−5.14661e31	−1.11361	−0.556804	−	0.830644i	$-0.687973\pi$
$83$	−5.14661e31	−1.11361	−0.556804	+	0.830644i	$0.687973\pi$
$84$	0	0
$85$	−8.37798e31	−1.22385
$86$	0	0
$87$	−1.75999e32	−1.75163
$88$	0	0
$89$	5.16507e31	0.353298	0.176649	−	0.984274i	$-0.443474\pi$
$89$	5.16507e31	0.353298	0.176649	+	0.984274i	$0.443474\pi$
$90$	0	0
$91$	−1.00339e32	−0.475649
$92$	0	0
$93$	7.15874e32	2.37065
$94$	0	0
$95$	−6.98872e32	−1.62914
$96$	0	0
$97$	−1.10473e33	−1.82609	−0.913043	−	0.407863i	$-0.866274\pi$
$97$	−1.10473e33	−1.82609	−0.913043	+	0.407863i	$0.866274\pi$
$98$	0	0
$99$	3.00564e33	3.54777
$100$	0	0
$101$	−4.29066e31	−0.0364100	−0.0182050	−	0.999834i	$-0.505795\pi$
$101$	−4.29066e31	−0.0364100	−0.0182050	+	0.999834i	$0.505795\pi$
$102$	0	0
$103$	−5.10178e32	−0.313262	−0.156631	−	0.987657i	$-0.550063\pi$
$103$	−5.10178e32	−0.313262	−0.156631	+	0.987657i	$0.550063\pi$
$104$	0	0
$105$	3.46342e33	1.54839
$106$	0	0
$107$	−2.05277e33	−0.672215	−0.336107	−	0.941824i	$-0.609110\pi$
$107$	−2.05277e33	−0.672215	−0.336107	+	0.941824i	$0.609110\pi$
$108$	0	0
$109$	−4.05659e33	−0.978642	−0.489321	−	0.872104i	$-0.662755\pi$
$109$	−4.05659e33	−0.978642	−0.489321	+	0.872104i	$0.662755\pi$
$110$	0	0
$111$	−1.11196e34	−1.98728
$112$	0	0
$113$	8.02059e32	0.106761	0.0533805	−	0.998574i	$-0.483000\pi$
$113$	8.02059e32	0.106761	0.0533805	+	0.998574i	$0.483000\pi$
$114$	0	0
$115$	−3.06212e33	−0.305146
$116$	0	0
$117$	2.16213e34	1.62112
$118$	0	0
$119$	−1.11300e34	−0.630912
$120$	0	0
$121$	3.37450e34	1.45295
$122$	0	0
$123$	−5.51175e34	−1.81073
$124$	0	0
$125$	1.72988e34	0.435514
$126$	0	0
$127$	7.39847e34	1.43344	0.716721	−	0.697360i	$-0.245642\pi$
$127$	7.39847e34	1.43344	0.716721	+	0.697360i	$0.245642\pi$
$128$	0	0
$129$	7.12683e34	1.06700
$130$	0	0
$131$	4.87568e34	0.566316	0.283158	−	0.959073i	$-0.408618\pi$
$131$	4.87568e34	0.566316	0.283158	+	0.959073i	$0.408618\pi$
$132$	0	0
$133$	−9.28436e34	−0.839847
$134$	0	0
$135$	−4.16843e35	−2.94758
$136$	0	0
$137$	9.18664e34	0.509645	0.254822	−	0.966988i	$-0.417983\pi$
$137$	9.18664e34	0.509645	0.254822	+	0.966988i	$0.417983\pi$
$138$	0	0
$139$	−1.56271e35	−0.682552	−0.341276	−	0.939963i	$-0.610859\pi$
$139$	−1.56271e35	−0.682552	−0.341276	+	0.939963i	$0.610859\pi$
$140$	0	0
$141$	−1.14667e35	−0.395660
$142$	0	0
$143$	4.09819e35	1.12085
$144$	0	0
$145$	5.74645e35	1.24976
$146$	0	0
$147$	−5.81472e35	−1.00877
$148$	0	0
$149$	3.61831e35	0.502264	0.251132	−	0.967953i	$-0.419197\pi$
$149$	3.61831e35	0.502264	0.251132	+	0.967953i	$0.419197\pi$
$150$	0	0
$151$	−1.33622e35	−0.148854	−0.0744269	−	0.997226i	$-0.523713\pi$
$151$	−1.33622e35	−0.148854	−0.0744269	+	0.997226i	$0.523713\pi$
$152$	0	0
$153$	2.39831e36	2.15029
$154$	0	0
$155$	−2.33736e36	−1.69143
$156$	0	0
$157$	−3.98805e35	−0.233570	−0.116785	−	0.993157i	$-0.537259\pi$
$157$	−3.98805e35	−0.233570	−0.116785	+	0.993157i	$0.537259\pi$
$158$	0	0
$159$	3.48922e36	1.65835
$160$	0	0
$161$	−4.06795e35	−0.157307
$162$	0	0
$163$	6.70557e35	0.211515	0.105757	−	0.994392i	$-0.466273\pi$
$163$	6.70557e35	0.211515	0.105757	+	0.994392i	$0.466273\pi$
$164$	0	0
$165$	−1.41458e37	−3.64874
$166$	0	0
$167$	−3.15724e36	−0.667553	−0.333777	−	0.942652i	$-0.608323\pi$
$167$	−3.15724e36	−0.667553	−0.333777	+	0.942652i	$0.608323\pi$
$168$	0	0
$169$	−2.80806e36	−0.487839
$170$	0	0
$171$	2.00061e37	2.86238
$172$	0	0
$173$	−1.21773e36	−0.143810	−0.0719051	−	0.997411i	$-0.522908\pi$
$173$	−1.21773e36	−0.143810	−0.0719051	+	0.997411i	$0.522908\pi$
$174$	0	0
$175$	−4.50505e36	−0.440121
$176$	0	0
$177$	3.09085e37	2.50333
$178$	0	0
$179$	3.07461e35	0.0206879	0.0103439	−	0.999947i	$-0.496707\pi$
$179$	3.07461e35	0.0206879	0.0103439	+	0.999947i	$0.496707\pi$
$180$	0	0
$181$	−3.42619e37	−1.91918	−0.959589	−	0.281407i	$-0.909199\pi$
$181$	−3.42619e37	−1.91918	−0.959589	+	0.281407i	$0.909199\pi$
$182$	0	0
$183$	−1.19336e37	−0.557607
$184$	0	0
$185$	3.63059e37	1.41789
$186$	0	0
$187$	4.54587e37	1.48672
$188$	0	0
$189$	−5.53767e37	−1.51952
$190$	0	0
$191$	7.22927e36	0.166741	0.0833707	−	0.996519i	$-0.473431\pi$
$191$	7.22927e36	0.166741	0.0833707	+	0.996519i	$0.473431\pi$
$192$	0	0
$193$	−1.78372e37	−0.346442	−0.173221	−	0.984883i	$-0.555417\pi$
$193$	−1.78372e37	−0.346442	−0.173221	+	0.984883i	$0.555417\pi$
$194$	0	0
$195$	−1.01759e38	−1.66725
$196$	0	0
$197$	−7.82760e37	−1.08377	−0.541884	−	0.840454i	$-0.682289\pi$
$197$	−7.82760e37	−1.08377	−0.541884	+	0.840454i	$0.682289\pi$
$198$	0	0
$199$	1.29050e38	1.51245	0.756227	−	0.654309i	$-0.227040\pi$
$199$	1.29050e38	1.51245	0.756227	+	0.654309i	$0.227040\pi$
$200$	0	0
$201$	−1.48162e38	−1.47232
$202$	0	0
$203$	7.63403e37	0.644272
$204$	0	0
$205$	1.79961e38	1.29193
$206$	0	0
$207$	8.76572e37	0.536138
$208$	0	0
$209$	3.79206e38	1.97907
$210$	0	0
$211$	−2.33739e38	−1.04249	−0.521243	−	0.853408i	$-0.674531\pi$
$211$	−2.33739e38	−1.04249	−0.521243	+	0.853408i	$0.674531\pi$
$212$	0	0
$213$	1.67877e38	0.640809
$214$	0	0
$215$	−2.32694e38	−0.761292
$216$	0	0
$217$	−3.10513e38	−0.871956
$218$	0	0
$219$	−3.47650e38	−0.839101
$220$	0	0
$221$	3.27010e38	0.679343
$222$	0	0
$223$	−3.29898e38	−0.590677	−0.295338	−	0.955393i	$-0.595432\pi$
$223$	−3.29898e38	−0.590677	−0.295338	+	0.955393i	$0.595432\pi$
$224$	0	0
$225$	9.70759e38	1.50003
$226$	0	0
$227$	−2.00649e38	−0.267924	−0.133962	−	0.990986i	$-0.542770\pi$
$227$	−2.00649e38	−0.267924	−0.133962	+	0.990986i	$0.542770\pi$
$228$	0	0
$229$	−7.20397e38	−0.832313	−0.416156	−	0.909293i	$-0.636623\pi$
$229$	−7.20397e38	−0.832313	−0.416156	+	0.909293i	$0.636623\pi$
$230$	0	0
$231$	−1.87924e39	−1.88098
$232$	0	0
$233$	−1.44309e39	−1.25291	−0.626455	−	0.779458i	$-0.715495\pi$
$233$	−1.44309e39	−1.25291	−0.626455	+	0.779458i	$0.715495\pi$
$234$	0	0
$235$	3.74392e38	0.282298
$236$	0	0
$237$	4.78823e39	3.13926
$238$	0	0
$239$	3.12394e39	1.78294	0.891472	−	0.453076i	$-0.149673\pi$
$239$	3.12394e39	1.78294	0.891472	+	0.453076i	$0.149673\pi$
$240$	0	0
$241$	−3.54631e39	−1.76399	−0.881993	−	0.471263i	$-0.843798\pi$
$241$	−3.54631e39	−1.76399	−0.881993	+	0.471263i	$0.843798\pi$
$242$	0	0
$243$	2.50140e39	1.08562
$244$	0	0
$245$	1.89853e39	0.719744
$246$	0	0
$247$	2.72784e39	0.904315
$248$	0	0
$249$	−6.93391e39	−2.01228
$250$	0	0
$251$	−3.92288e39	−0.997674	−0.498837	−	0.866696i	$-0.666239\pi$
$251$	−3.92288e39	−0.997674	−0.498837	+	0.866696i	$0.666239\pi$
$252$	0	0
$253$	1.66149e39	0.370689
$254$	0	0
$255$	−1.12875e40	−2.21148
$256$	0	0
$257$	1.13099e40	1.94789	0.973944	−	0.226789i	$-0.0728227\pi$
$257$	1.13099e40	1.94789	0.973944	+	0.226789i	$0.0728227\pi$
$258$	0	0
$259$	4.82316e39	0.730946
$260$	0	0
$261$	−1.64500e40	−2.19582
$262$	0	0
$263$	8.46897e39	0.996691	0.498346	−	0.866979i	$-0.333941\pi$
$263$	8.46897e39	0.996691	0.498346	+	0.866979i	$0.333941\pi$
$264$	0	0
$265$	−1.13925e40	−1.18321
$266$	0	0
$267$	6.95878e39	0.638407
$268$	0	0
$269$	2.01049e40	1.63076	0.815379	−	0.578927i	$-0.196528\pi$
$269$	2.01049e40	1.63076	0.815379	+	0.578927i	$0.196528\pi$
$270$	0	0
$271$	−1.70128e40	−1.22119	−0.610593	−	0.791945i	$-0.709069\pi$
$271$	−1.70128e40	−1.22119	−0.610593	+	0.791945i	$0.709069\pi$
$272$	0	0
$273$	−1.35184e40	−0.859494
$274$	0	0
$275$	1.84002e40	1.03713
$276$	0	0
$277$	−2.24387e39	−0.112223	−0.0561116	−	0.998425i	$-0.517870\pi$
$277$	−2.24387e39	−0.112223	−0.0561116	+	0.998425i	$0.517870\pi$
$278$	0	0
$279$	6.69101e40	2.97182
$280$	0	0
$281$	1.68371e40	0.664679	0.332339	−	0.943160i	$-0.392162\pi$
$281$	1.68371e40	0.664679	0.332339	+	0.943160i	$0.392162\pi$
$282$	0	0
$283$	8.97402e39	0.315145	0.157573	−	0.987507i	$-0.449633\pi$
$283$	8.97402e39	0.315145	0.157573	+	0.987507i	$0.449633\pi$
$284$	0	0
$285$	−9.41575e40	−2.94384
$286$	0	0
$287$	2.39074e40	0.666012
$288$	0	0
$289$	−3.98130e39	−0.0989032
$290$	0	0
$291$	−1.48837e41	−3.29973
$292$	0	0
$293$	2.29366e40	0.454164	0.227082	−	0.973876i	$-0.427081\pi$
$293$	2.29366e40	0.454164	0.227082	+	0.973876i	$0.427081\pi$
$294$	0	0
$295$	−1.00918e41	−1.78609
$296$	0	0
$297$	2.26178e41	3.58070
$298$	0	0
$299$	1.19521e40	0.169383
$300$	0	0
$301$	−3.09129e40	−0.392458
$302$	0	0
$303$	−5.78070e39	−0.0657927
$304$	0	0
$305$	3.89637e40	0.397844
$306$	0	0
$307$	−4.83147e40	−0.442890	−0.221445	−	0.975173i	$-0.571077\pi$
$307$	−4.83147e40	−0.442890	−0.221445	+	0.975173i	$0.571077\pi$
$308$	0	0
$309$	−6.87352e40	−0.566062
$310$	0	0
$311$	3.69262e40	0.273394	0.136697	−	0.990613i	$-0.456351\pi$
$311$	3.69262e40	0.273394	0.136697	+	0.990613i	$0.456351\pi$
$312$	0	0
$313$	5.08926e40	0.338979	0.169490	−	0.985532i	$-0.445788\pi$
$313$	5.08926e40	0.338979	0.169490	+	0.985532i	$0.445788\pi$
$314$	0	0
$315$	3.23713e41	1.94105
$316$	0	0
$317$	2.50845e41	1.35496	0.677481	−	0.735540i	$-0.263072\pi$
$317$	2.50845e41	1.35496	0.677481	+	0.735540i	$0.263072\pi$
$318$	0	0
$319$	−3.11800e41	−1.51820
$320$	0	0
$321$	−2.76565e41	−1.21469
$322$	0	0
$323$	3.02583e41	1.19951
$324$	0	0
$325$	1.32363e41	0.473906
$326$	0	0
$327$	−5.46535e41	−1.76840
$328$	0	0
$329$	4.97372e40	0.145529
$330$	0	0
$331$	−4.23320e41	−1.12075	−0.560373	−	0.828240i	$-0.689342\pi$
$331$	−4.23320e41	−1.12075	−0.560373	+	0.828240i	$0.689342\pi$
$332$	0	0
$333$	−1.03930e42	−2.49122
$334$	0	0
$335$	4.83756e41	1.05048
$336$	0	0
$337$	2.87768e40	0.0566434	0.0283217	−	0.999599i	$-0.490984\pi$
$337$	2.87768e40	0.0566434	0.0283217	+	0.999599i	$0.490984\pi$
$338$	0	0
$339$	1.08060e41	0.192916
$340$	0	0
$341$	1.26825e42	2.05474
$342$	0	0
$343$	7.04006e41	1.03567
$344$	0	0
$345$	−4.12552e41	−0.551396
$346$	0	0
$347$	−7.94303e41	−0.965051	−0.482525	−	0.875882i	$-0.660280\pi$
$347$	−7.94303e41	−0.965051	−0.482525	+	0.875882i	$0.660280\pi$
$348$	0	0
$349$	1.38764e42	1.53341	0.766705	−	0.641999i	$-0.221895\pi$
$349$	1.38764e42	1.53341	0.766705	+	0.641999i	$0.221895\pi$
$350$	0	0
$351$	1.62703e42	1.63616
$352$	0	0
$353$	−1.94987e42	−1.78534	−0.892672	−	0.450707i	$-0.851172\pi$
$353$	−1.94987e42	−1.78534	−0.892672	+	0.450707i	$0.851172\pi$
$354$	0	0
$355$	−5.48127e41	−0.457208
$356$	0	0
$357$	−1.49952e42	−1.14005
$358$	0	0
$359$	−6.33120e40	−0.0438963	−0.0219481	−	0.999759i	$-0.506987\pi$
$359$	−6.33120e40	−0.0438963	−0.0219481	+	0.999759i	$0.506987\pi$
$360$	0	0
$361$	9.43304e41	0.596737
$362$	0	0
$363$	4.54639e42	2.62547
$364$	0	0
$365$	1.13509e42	0.598686
$366$	0	0
$367$	−3.11073e42	−1.49924	−0.749621	−	0.661867i	$-0.769764\pi$
$367$	−3.11073e42	−1.49924	−0.749621	+	0.661867i	$0.769764\pi$
$368$	0	0
$369$	−5.15163e42	−2.26991
$370$	0	0
$371$	−1.51346e42	−0.609961
$372$	0	0
$373$	−3.97777e42	−1.46705	−0.733524	−	0.679663i	$-0.762126\pi$
$373$	−3.97777e42	−1.46705	−0.733524	+	0.679663i	$0.762126\pi$
$374$	0	0
$375$	2.33063e42	0.786970
$376$	0	0
$377$	−2.24296e42	−0.693728
$378$	0	0
$379$	−4.00371e41	−0.113479	−0.0567395	−	0.998389i	$-0.518070\pi$
$379$	−4.00371e41	−0.113479	−0.0567395	+	0.998389i	$0.518070\pi$
$380$	0	0
$381$	9.96779e42	2.59022
$382$	0	0
$383$	−4.12729e42	−0.983751	−0.491875	−	0.870666i	$-0.663688\pi$
$383$	−4.12729e42	−0.983751	−0.491875	+	0.870666i	$0.663688\pi$
$384$	0	0
$385$	6.13580e42	1.34205
$386$	0	0
$387$	6.66119e42	1.33758
$388$	0	0
$389$	−5.15355e41	−0.0950470	−0.0475235	−	0.998870i	$-0.515133\pi$
$389$	−5.15355e41	−0.0950470	−0.0475235	+	0.998870i	$0.515133\pi$
$390$	0	0
$391$	1.32577e42	0.224673
$392$	0	0
$393$	6.56890e42	1.02333
$394$	0	0
$395$	−1.56338e43	−2.23982
$396$	0	0
$397$	3.89760e42	0.513752	0.256876	−	0.966444i	$-0.417307\pi$
$397$	3.89760e42	0.513752	0.256876	+	0.966444i	$0.417307\pi$
$398$	0	0
$399$	−1.25086e43	−1.51760
$400$	0	0
$401$	1.58967e43	1.77593	0.887965	−	0.459911i	$-0.152119\pi$
$401$	1.58967e43	1.77593	0.887965	+	0.459911i	$0.152119\pi$
$402$	0	0
$403$	9.12322e42	0.938890
$404$	0	0
$405$	−2.53667e43	−2.40578
$406$	0	0
$407$	−1.96995e43	−1.72245
$408$	0	0
$409$	1.48346e43	1.19630	0.598150	−	0.801384i	$-0.295903\pi$
$409$	1.48346e43	1.19630	0.598150	+	0.801384i	$0.295903\pi$
$410$	0	0
$411$	1.23770e43	0.920924
$412$	0	0
$413$	−1.34067e43	−0.920758
$414$	0	0
$415$	2.26396e43	1.43573
$416$	0	0
$417$	−2.10541e43	−1.23337
$418$	0	0
$419$	−6.90718e41	−0.0373912	−0.0186956	−	0.999825i	$-0.505951\pi$
$419$	−6.90718e41	−0.0373912	−0.0186956	+	0.999825i	$0.505951\pi$
$420$	0	0
$421$	−2.87526e43	−1.43887	−0.719437	−	0.694558i	$-0.755600\pi$
$421$	−2.87526e43	−1.43887	−0.719437	+	0.694558i	$0.755600\pi$
$422$	0	0
$423$	−1.07175e43	−0.495994
$424$	0	0
$425$	1.46822e43	0.628600
$426$	0	0
$427$	5.17624e42	0.205095
$428$	0	0
$429$	5.52140e43	2.02537
$430$	0	0
$431$	−1.01416e43	−0.344535	−0.172267	−	0.985050i	$-0.555109\pi$
$431$	−1.01416e43	−0.344535	−0.172267	+	0.985050i	$0.555109\pi$
$432$	0	0
$433$	3.96098e43	1.24667	0.623336	−	0.781954i	$-0.285777\pi$
$433$	3.96098e43	1.24667	0.623336	+	0.781954i	$0.285777\pi$
$434$	0	0
$435$	7.74206e43	2.25831
$436$	0	0
$437$	1.10593e43	0.299077
$438$	0	0
$439$	−4.74746e43	−1.19068	−0.595342	−	0.803472i	$-0.702984\pi$
$439$	−4.74746e43	−1.19068	−0.595342	+	0.803472i	$0.702984\pi$
$440$	0	0
$441$	−5.43481e43	−1.26458
$442$	0	0
$443$	−3.78302e43	−0.816915	−0.408458	−	0.912777i	$-0.633933\pi$
$443$	−3.78302e43	−0.816915	−0.408458	+	0.912777i	$0.633933\pi$
$444$	0	0
$445$	−2.27208e43	−0.455494
$446$	0	0
$447$	4.87486e43	0.907587
$448$	0	0
$449$	−9.87539e43	−1.70801	−0.854004	−	0.520267i	$-0.825832\pi$
$449$	−9.87539e43	−1.70801	−0.854004	+	0.520267i	$0.825832\pi$
$450$	0	0
$451$	−9.76463e43	−1.56943
$452$	0	0
$453$	−1.80027e43	−0.268978
$454$	0	0
$455$	4.41384e43	0.613237
$456$	0	0
$457$	1.20565e44	1.55814	0.779069	−	0.626939i	$-0.215692\pi$
$457$	1.20565e44	1.55814	0.779069	+	0.626939i	$0.215692\pi$
$458$	0	0
$459$	1.80476e44	2.17025
$460$	0	0
$461$	−9.88539e43	−1.10644	−0.553221	−	0.833035i	$-0.686601\pi$
$461$	−9.88539e43	−1.10644	−0.553221	+	0.833035i	$0.686601\pi$
$462$	0	0
$463$	−4.50681e43	−0.469660	−0.234830	−	0.972036i	$-0.575453\pi$
$463$	−4.50681e43	−0.469660	−0.234830	+	0.972036i	$0.575453\pi$
$464$	0	0
$465$	−3.14908e44	−3.05639
$466$	0	0
$467$	1.47361e44	1.33246	0.666228	−	0.745748i	$-0.267908\pi$
$467$	1.47361e44	1.33246	0.666228	+	0.745748i	$0.267908\pi$
$468$	0	0
$469$	6.42659e43	0.541537
$470$	0	0
$471$	−5.37301e43	−0.422058
$472$	0	0
$473$	1.26259e44	0.924814
$474$	0	0
$475$	1.22476e44	0.836769
$476$	0	0
$477$	3.26124e44	2.07888
$478$	0	0
$479$	−1.57341e44	−0.936067	−0.468034	−	0.883711i	$-0.655037\pi$
$479$	−1.57341e44	−0.936067	−0.468034	+	0.883711i	$0.655037\pi$
$480$	0	0
$481$	−1.41709e44	−0.787055
$482$	0	0
$483$	−5.48066e43	−0.284253
$484$	0	0
$485$	4.85961e44	2.35431
$486$	0	0
$487$	−3.18979e44	−1.44389	−0.721945	−	0.691950i	$-0.756752\pi$
$487$	−3.18979e44	−1.44389	−0.721945	+	0.691950i	$0.756752\pi$
$488$	0	0
$489$	9.03427e43	0.382205
$490$	0	0
$491$	3.97831e44	1.57345	0.786727	−	0.617301i	$-0.211774\pi$
$491$	3.97831e44	1.57345	0.786727	+	0.617301i	$0.211774\pi$
$492$	0	0
$493$	−2.48797e44	−0.920177
$494$	0	0
$495$	−1.32216e45	−4.57401
$496$	0	0
$497$	−7.28175e43	−0.235698
$498$	0	0
$499$	3.71007e44	1.12389	0.561945	−	0.827175i	$-0.310053\pi$
$499$	3.71007e44	1.12389	0.561945	+	0.827175i	$0.310053\pi$
$500$	0	0
$501$	−4.25368e44	−1.20626
$502$	0	0
$503$	3.58796e44	0.952743	0.476371	−	0.879244i	$-0.341952\pi$
$503$	3.58796e44	0.952743	0.476371	+	0.879244i	$0.341952\pi$
$504$	0	0
$505$	1.88743e43	0.0469421
$506$	0	0
$507$	−3.78324e44	−0.881522
$508$	0	0
$509$	2.88589e44	0.630139	0.315070	−	0.949069i	$-0.397972\pi$
$509$	2.88589e44	0.630139	0.315070	+	0.949069i	$0.397972\pi$
$510$	0	0
$511$	1.50795e44	0.308632
$512$	0	0
$513$	1.50549e45	2.88895
$514$	0	0
$515$	2.24424e44	0.403877
$516$	0	0
$517$	−2.03144e44	−0.342934
$518$	0	0
$519$	−1.64062e44	−0.259864
$520$	0	0
$521$	−4.02237e44	−0.597943	−0.298971	−	0.954262i	$-0.596644\pi$
$521$	−4.02237e44	−0.597943	−0.298971	+	0.954262i	$0.596644\pi$
$522$	0	0
$523$	−6.82047e44	−0.951780	−0.475890	−	0.879505i	$-0.657874\pi$
$523$	−6.82047e44	−0.951780	−0.475890	+	0.879505i	$0.657874\pi$
$524$	0	0
$525$	−6.06956e44	−0.795296
$526$	0	0
$527$	1.01198e45	1.24537
$528$	0	0
$529$	−8.16549e44	−0.943981
$530$	0	0
$531$	2.88890e45	3.13815
$532$	0	0
$533$	−7.02426e44	−0.717136
$534$	0	0
$535$	9.02998e44	0.866662
$536$	0	0
$537$	4.14236e43	0.0373828
$538$	0	0
$539$	−1.03014e45	−0.874341
$540$	0	0
$541$	2.04672e44	0.163420	0.0817098	−	0.996656i	$-0.473962\pi$
$541$	2.04672e44	0.163420	0.0817098	+	0.996656i	$0.473962\pi$
$542$	0	0
$543$	−4.61603e45	−3.46794
$544$	0	0
$545$	1.78446e45	1.26173
$546$	0	0
$547$	−5.26369e44	−0.350348	−0.175174	−	0.984538i	$-0.556049\pi$
$547$	−5.26369e44	−0.350348	−0.175174	+	0.984538i	$0.556049\pi$
$548$	0	0
$549$	−1.11539e45	−0.699009
$550$	0	0
$551$	−2.07541e45	−1.22491
$552$	0	0
$553$	−2.07692e45	−1.15466
$554$	0	0
$555$	4.89141e45	2.56212
$556$	0	0
$557$	−3.31476e45	−1.63622	−0.818109	−	0.575064i	$-0.804977\pi$
$557$	−3.31476e45	−1.63622	−0.818109	+	0.575064i	$0.804977\pi$
$558$	0	0
$559$	9.08255e44	0.422584
$560$	0	0
$561$	6.12455e45	2.68650
$562$	0	0
$563$	4.46448e45	1.84664	0.923321	−	0.384028i	$-0.125464\pi$
$563$	4.46448e45	1.84664	0.923321	+	0.384028i	$0.125464\pi$
$564$	0	0
$565$	−3.52820e44	−0.137643
$566$	0	0
$567$	−3.36991e45	−1.24022
$568$	0	0
$569$	−1.55500e45	−0.539981	−0.269991	−	0.962863i	$-0.587021\pi$
$569$	−1.55500e45	−0.539981	−0.269991	+	0.962863i	$0.587021\pi$
$570$	0	0
$571$	4.45599e45	1.46032	0.730161	−	0.683275i	$-0.239445\pi$
$571$	4.45599e45	1.46032	0.730161	+	0.683275i	$0.239445\pi$
$572$	0	0
$573$	9.73983e44	0.301301
$574$	0	0
$575$	5.36629e44	0.156731
$576$	0	0
$577$	−4.82092e45	−1.32963	−0.664813	−	0.747010i	$-0.731489\pi$
$577$	−4.82092e45	−1.32963	−0.664813	+	0.747010i	$0.731489\pi$
$578$	0	0
$579$	−2.40316e45	−0.626017
$580$	0	0
$581$	3.00762e45	0.740144
$582$	0	0
$583$	6.18151e45	1.43735
$584$	0	0
$585$	−9.51103e45	−2.09005
$586$	0	0
$587$	−3.94339e45	−0.819108	−0.409554	−	0.912286i	$-0.634316\pi$
$587$	−3.94339e45	−0.819108	−0.409554	+	0.912286i	$0.634316\pi$
$588$	0	0
$589$	8.44171e45	1.65778
$590$	0	0
$591$	−1.05460e46	−1.95836
$592$	0	0
$593$	−5.04718e45	−0.886434	−0.443217	−	0.896414i	$-0.646163\pi$
$593$	−5.04718e45	−0.886434	−0.443217	+	0.896414i	$0.646163\pi$
$594$	0	0
$595$	4.89599e45	0.813412
$596$	0	0
$597$	1.73866e46	2.73300
$598$	0	0
$599$	−7.97373e45	−1.18609	−0.593047	−	0.805168i	$-0.702075\pi$
$599$	−7.97373e45	−1.18609	−0.593047	+	0.805168i	$0.702075\pi$
$600$	0	0
$601$	4.22349e45	0.594624	0.297312	−	0.954780i	$-0.403910\pi$
$601$	4.22349e45	0.594624	0.297312	+	0.954780i	$0.403910\pi$
$602$	0	0
$603$	−1.38482e46	−1.84568
$604$	0	0
$605$	−1.48442e46	−1.87324
$606$	0	0
$607$	−1.34096e46	−1.60252	−0.801258	−	0.598320i	$-0.795835\pi$
$607$	−1.34096e46	−1.60252	−0.801258	+	0.598320i	$0.795835\pi$
$608$	0	0
$609$	1.02852e46	1.16419
$610$	0	0
$611$	−1.46133e45	−0.156700
$612$	0	0
$613$	1.30841e46	1.32937	0.664687	−	0.747122i	$-0.268565\pi$
$613$	1.30841e46	1.32937	0.664687	+	0.747122i	$0.268565\pi$
$614$	0	0
$615$	2.42458e46	2.33451
$616$	0	0
$617$	1.63656e46	1.49358	0.746788	−	0.665062i	$-0.231595\pi$
$617$	1.63656e46	1.49358	0.746788	+	0.665062i	$0.231595\pi$
$618$	0	0
$619$	−2.40690e45	−0.208239	−0.104120	−	0.994565i	$-0.533202\pi$
$619$	−2.40690e45	−0.208239	−0.104120	+	0.994565i	$0.533202\pi$
$620$	0	0
$621$	6.59631e45	0.541115
$622$	0	0
$623$	−3.01840e45	−0.234814
$624$	0	0
$625$	−1.65841e46	−1.22369
$626$	0	0
$627$	5.10895e46	3.57617
$628$	0	0
$629$	−1.57189e46	−1.04397
$630$	0	0
$631$	−1.53069e46	−0.964722	−0.482361	−	0.875972i	$-0.660221\pi$
$631$	−1.53069e46	−0.964722	−0.482361	+	0.875972i	$0.660221\pi$
$632$	0	0
$633$	−3.14911e46	−1.88376
$634$	0	0
$635$	−3.25453e46	−1.84808
$636$	0	0
$637$	−7.41038e45	−0.399521
$638$	0	0
$639$	1.56909e46	0.803310
$640$	0	0
$641$	9.03309e45	0.439217	0.219608	−	0.975588i	$-0.429522\pi$
$641$	9.03309e45	0.439217	0.219608	+	0.975588i	$0.429522\pi$
$642$	0	0
$643$	−6.28642e45	−0.290351	−0.145175	−	0.989406i	$-0.546375\pi$
$643$	−6.28642e45	−0.290351	−0.145175	+	0.989406i	$0.546375\pi$
$644$	0	0
$645$	−3.13504e46	−1.37565
$646$	0	0
$647$	4.05377e46	1.69020	0.845100	−	0.534608i	$-0.179541\pi$
$647$	4.05377e46	1.69020	0.845100	+	0.534608i	$0.179541\pi$
$648$	0	0
$649$	5.47576e46	2.16974
$650$	0	0
$651$	−4.18348e46	−1.57562
$652$	0	0
$653$	−1.97437e46	−0.706903	−0.353452	−	0.935453i	$-0.614992\pi$
$653$	−1.97437e46	−0.706903	−0.353452	+	0.935453i	$0.614992\pi$
$654$	0	0
$655$	−2.14478e46	−0.730131
$656$	0	0
$657$	−3.24936e46	−1.05189
$658$	0	0
$659$	1.63443e46	0.503220	0.251610	−	0.967829i	$-0.419040\pi$
$659$	1.63443e46	0.503220	0.251610	+	0.967829i	$0.419040\pi$
$660$	0	0
$661$	5.92845e45	0.173627	0.0868134	−	0.996225i	$-0.472332\pi$
$661$	5.92845e45	0.173627	0.0868134	+	0.996225i	$0.472332\pi$
$662$	0	0
$663$	4.40574e46	1.22757
$664$	0	0
$665$	4.08412e46	1.08278
$666$	0	0
$667$	−9.09343e45	−0.229431
$668$	0	0
$669$	−4.44465e46	−1.06735
$670$	0	0
$671$	−2.11416e46	−0.483299
$672$	0	0
$673$	1.45328e46	0.316302	0.158151	−	0.987415i	$-0.449447\pi$
$673$	1.45328e46	0.316302	0.158151	+	0.987415i	$0.449447\pi$
$674$	0	0
$675$	7.30508e46	1.51395
$676$	0	0
$677$	7.75928e46	1.53147	0.765735	−	0.643156i	$-0.222375\pi$
$677$	7.75928e46	1.53147	0.765735	+	0.643156i	$0.222375\pi$
$678$	0	0
$679$	6.45589e46	1.21368
$680$	0	0
$681$	−2.70331e46	−0.484136
$682$	0	0
$683$	−6.20051e45	−0.105800	−0.0529000	−	0.998600i	$-0.516846\pi$
$683$	−6.20051e45	−0.105800	−0.0529000	+	0.998600i	$0.516846\pi$
$684$	0	0
$685$	−4.04113e46	−0.657066
$686$	0	0
$687$	−9.70575e46	−1.50398
$688$	0	0
$689$	4.44672e46	0.656783
$690$	0	0
$691$	1.81450e46	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$691$	1.81450e46	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$692$	0	0
$693$	−1.75646e47	−2.35797
$694$	0	0
$695$	6.87427e46	0.879989
$696$	0	0
$697$	−7.79157e46	−0.951227
$698$	0	0
$699$	−1.94424e47	−2.26400
$700$	0	0
$701$	1.34726e47	1.49660	0.748298	−	0.663363i	$-0.230871\pi$
$701$	1.34726e47	1.49660	0.748298	+	0.663363i	$0.230871\pi$
$702$	0	0
$703$	−1.31124e47	−1.38969
$704$	0	0
$705$	5.04410e46	0.510110
$706$	0	0
$707$	2.50741e45	0.0241994
$708$	0	0
$709$	−1.04063e47	−0.958594	−0.479297	−	0.877653i	$-0.659108\pi$
$709$	−1.04063e47	−0.958594	−0.479297	+	0.877653i	$0.659108\pi$
$710$	0	0
$711$	4.47539e47	3.93534
$712$	0	0
$713$	3.69875e46	0.310511
$714$	0	0
$715$	−1.80276e47	−1.44507
$716$	0	0
$717$	4.20882e47	3.22177
$718$	0	0
$719$	−6.64979e46	−0.486162	−0.243081	−	0.970006i	$-0.578158\pi$
$719$	−6.64979e46	−0.486162	−0.243081	+	0.970006i	$0.578158\pi$
$720$	0	0
$721$	2.98142e46	0.208205
$722$	0	0
$723$	−4.77786e47	−3.18751
$724$	0	0
$725$	−1.00705e47	−0.641911
$726$	0	0
$727$	7.19404e46	0.438183	0.219091	−	0.975704i	$-0.429691\pi$
$727$	7.19404e46	0.438183	0.219091	+	0.975704i	$0.429691\pi$
$728$	0	0
$729$	1.64405e46	0.0956999
$730$	0	0
$731$	1.00747e47	0.560526
$732$	0	0
$733$	2.85394e47	1.51785	0.758925	−	0.651178i	$-0.225725\pi$
$733$	2.85394e47	1.51785	0.758925	+	0.651178i	$0.225725\pi$
$734$	0	0
$735$	2.55785e47	1.30057
$736$	0	0
$737$	−2.62484e47	−1.27611
$738$	0	0
$739$	1.92837e47	0.896512	0.448256	−	0.893905i	$-0.352045\pi$
$739$	1.92837e47	0.896512	0.448256	+	0.893905i	$0.352045\pi$
$740$	0	0
$741$	3.67516e47	1.63409
$742$	0	0
$743$	−1.73889e47	−0.739532	−0.369766	−	0.929125i	$-0.620562\pi$
$743$	−1.73889e47	−0.739532	−0.369766	+	0.929125i	$0.620562\pi$
$744$	0	0
$745$	−1.59167e47	−0.647550
$746$	0	0
$747$	−6.48087e47	−2.52257
$748$	0	0
$749$	1.19961e47	0.446778
$750$	0	0
$751$	−8.03243e45	−0.0286278	−0.0143139	−	0.999898i	$-0.504556\pi$
$751$	−8.03243e45	−0.0286278	−0.0143139	+	0.999898i	$0.504556\pi$
$752$	0	0
$753$	−5.28521e47	−1.80279
$754$	0	0
$755$	5.87795e46	0.191912
$756$	0	0
$757$	−4.29145e47	−1.34129	−0.670643	−	0.741780i	$-0.733982\pi$
$757$	−4.29145e47	−1.34129	−0.670643	+	0.741780i	$0.733982\pi$
$758$	0	0
$759$	2.23849e47	0.669833
$760$	0	0
$761$	1.16851e47	0.334799	0.167400	−	0.985889i	$-0.446463\pi$
$761$	1.16851e47	0.334799	0.167400	+	0.985889i	$0.446463\pi$
$762$	0	0
$763$	2.37062e47	0.650440
$764$	0	0
$765$	−1.05500e48	−2.77229
$766$	0	0
$767$	3.93903e47	0.991438
$768$	0	0
$769$	3.48880e46	0.0841185	0.0420592	−	0.999115i	$-0.486608\pi$
$769$	3.48880e46	0.0841185	0.0420592	+	0.999115i	$0.486608\pi$
$770$	0	0
$771$	1.52376e48	3.51982
$772$	0	0
$773$	−1.23838e47	−0.274089	−0.137044	−	0.990565i	$-0.543760\pi$
$773$	−1.23838e47	−0.274089	−0.137044	+	0.990565i	$0.543760\pi$
$774$	0	0
$775$	4.09617e47	0.868761
$776$	0	0
$777$	6.49813e47	1.32081
$778$	0	0
$779$	−6.49955e47	−1.26624
$780$	0	0
$781$	2.97412e47	0.555414
$782$	0	0
$783$	−1.23788e48	−2.21620
$784$	0	0
$785$	1.75431e47	0.301133
$786$	0	0
$787$	6.66573e47	1.09715	0.548573	−	0.836102i	$-0.315171\pi$
$787$	6.66573e47	1.09715	0.548573	+	0.836102i	$0.315171\pi$
$788$	0	0
$789$	1.14101e48	1.80101
$790$	0	0
$791$	−4.68714e46	−0.0709571
$792$	0	0
$793$	−1.52084e47	−0.220838
$794$	0	0
$795$	−1.53488e48	−2.13805
$796$	0	0
$797$	−1.00743e48	−1.34633	−0.673166	−	0.739492i	$-0.735066\pi$
$797$	−1.00743e48	−1.34633	−0.673166	+	0.739492i	$0.735066\pi$
$798$	0	0
$799$	−1.62096e47	−0.207851
$800$	0	0
$801$	6.50412e47	0.800298
$802$	0	0
$803$	−6.15898e47	−0.727281
$804$	0	0
$805$	1.78946e47	0.202811
$806$	0	0
$807$	2.70869e48	2.94677
$808$	0	0
$809$	−5.01023e47	−0.523247	−0.261624	−	0.965170i	$-0.584258\pi$
$809$	−5.01023e47	−0.523247	−0.261624	+	0.965170i	$0.584258\pi$
$810$	0	0
$811$	−1.96073e46	−0.0196596	−0.00982978	−	0.999952i	$-0.503129\pi$
$811$	−1.96073e46	−0.0196596	−0.00982978	+	0.999952i	$0.503129\pi$
$812$	0	0
$813$	−2.29209e48	−2.20667
$814$	0	0
$815$	−2.94973e47	−0.272698
$816$	0	0
$817$	8.40408e47	0.746151
$818$	0	0
$819$	−1.26352e48	−1.07745
$820$	0	0
$821$	−2.20701e48	−1.80776	−0.903881	−	0.427783i	$-0.859295\pi$
$821$	−2.20701e48	−1.80776	−0.903881	+	0.427783i	$0.859295\pi$
$822$	0	0
$823$	−1.22606e48	−0.964753	−0.482377	−	0.875964i	$-0.660226\pi$
$823$	−1.22606e48	−0.964753	−0.482377	+	0.875964i	$0.660226\pi$
$824$	0	0
$825$	2.47902e48	1.87409
$826$	0	0
$827$	5.69912e47	0.413968	0.206984	−	0.978344i	$-0.433635\pi$
$827$	5.69912e47	0.413968	0.206984	+	0.978344i	$0.433635\pi$
$828$	0	0
$829$	2.85621e47	0.199361	0.0996806	−	0.995019i	$-0.468218\pi$
$829$	2.85621e47	0.199361	0.0996806	+	0.995019i	$0.468218\pi$
$830$	0	0
$831$	−3.02312e47	−0.202786
$832$	0	0
$833$	−8.21987e47	−0.529934
$834$	0	0
$835$	1.38885e48	0.860652
$836$	0	0
$837$	5.03506e48	2.99940
$838$	0	0
$839$	1.07133e48	0.613553	0.306777	−	0.951782i	$-0.400750\pi$
$839$	1.07133e48	0.613553	0.306777	+	0.951782i	$0.400750\pi$
$840$	0	0
$841$	−1.09579e47	−0.0603383
$842$	0	0
$843$	2.26842e48	1.20107
$844$	0	0
$845$	1.23525e48	0.628953
$846$	0	0
$847$	−1.97202e48	−0.965683
$848$	0	0
$849$	1.20905e48	0.569465
$850$	0	0
$851$	−5.74520e47	−0.260296
$852$	0	0
$853$	−3.08488e48	−1.34456	−0.672278	−	0.740299i	$-0.734684\pi$
$853$	−3.08488e48	−1.34456	−0.672278	+	0.740299i	$0.734684\pi$
$854$	0	0
$855$	−8.80056e48	−3.69036
$856$	0	0
$857$	−1.63429e48	−0.659394	−0.329697	−	0.944087i	$-0.606947\pi$
$857$	−1.63429e48	−0.659394	−0.329697	+	0.944087i	$0.606947\pi$
$858$	0	0
$859$	9.43457e47	0.366298	0.183149	−	0.983085i	$-0.441371\pi$
$859$	9.43457e47	0.366298	0.183149	+	0.983085i	$0.441371\pi$
$860$	0	0
$861$	3.22100e48	1.20348
$862$	0	0
$863$	9.06251e47	0.325889	0.162945	−	0.986635i	$-0.447901\pi$
$863$	9.06251e47	0.325889	0.162945	+	0.986635i	$0.447901\pi$
$864$	0	0
$865$	5.35670e47	0.185409
$866$	0	0
$867$	−5.36392e47	−0.178717
$868$	0	0
$869$	8.48285e48	2.72092
$870$	0	0
$871$	−1.88820e48	−0.583107
$872$	0	0
$873$	−1.39113e49	−4.13649
$874$	0	0
$875$	−1.01092e48	−0.289458
$876$	0	0
$877$	1.93817e48	0.534441	0.267220	−	0.963635i	$-0.413895\pi$
$877$	1.93817e48	0.534441	0.267220	+	0.963635i	$0.413895\pi$
$878$	0	0
$879$	3.09019e48	0.820672
$880$	0	0
$881$	4.83488e48	1.23675	0.618377	−	0.785881i	$-0.287790\pi$
$881$	4.83488e48	1.23675	0.618377	+	0.785881i	$0.287790\pi$
$882$	0	0
$883$	5.63759e48	1.38913	0.694563	−	0.719432i	$-0.255598\pi$
$883$	5.63759e48	1.38913	0.694563	+	0.719432i	$0.255598\pi$
$884$	0	0
$885$	−1.35964e49	−3.22746
$886$	0	0
$887$	1.24504e48	0.284737	0.142369	−	0.989814i	$-0.454528\pi$
$887$	1.24504e48	0.284737	0.142369	+	0.989814i	$0.454528\pi$
$888$	0	0
$889$	−4.32357e48	−0.952717
$890$	0	0
$891$	1.37639e49	2.92253
$892$	0	0
$893$	−1.35217e48	−0.276683
$894$	0	0
$895$	−1.35250e47	−0.0266721
$896$	0	0
$897$	1.61028e48	0.306073
$898$	0	0
$899$	−6.94116e48	−1.27174
$900$	0	0
$901$	4.93246e48	0.871173
$902$	0	0
$903$	−4.16483e48	−0.709169
$904$	0	0
$905$	1.50716e49	2.47432
$906$	0	0
$907$	7.99759e48	1.26601	0.633007	−	0.774146i	$-0.281820\pi$
$907$	7.99759e48	1.26601	0.633007	+	0.774146i	$0.281820\pi$
$908$	0	0
$909$	−5.40301e47	−0.0824768
$910$	0	0
$911$	8.86733e48	1.30539	0.652694	−	0.757621i	$-0.273639\pi$
$911$	8.86733e48	1.30539	0.652694	+	0.757621i	$0.273639\pi$
$912$	0	0
$913$	−1.22841e49	−1.74412
$914$	0	0
$915$	5.24950e48	0.718902
$916$	0	0
$917$	−2.84929e48	−0.376394
$918$	0	0
$919$	−1.32606e49	−1.68989	−0.844943	−	0.534856i	$-0.820366\pi$
$919$	−1.32606e49	−1.68989	−0.844943	+	0.534856i	$0.820366\pi$
$920$	0	0
$921$	−6.50933e48	−0.800299
$922$	0	0
$923$	2.13946e48	0.253791
$924$	0	0
$925$	−6.36252e48	−0.728267
$926$	0	0
$927$	−6.42443e48	−0.709608
$928$	0	0
$929$	−7.45085e48	−0.794231	−0.397115	−	0.917769i	$-0.629989\pi$
$929$	−7.45085e48	−0.794231	−0.397115	+	0.917769i	$0.629989\pi$
$930$	0	0
$931$	−6.85682e48	−0.705429
$932$	0	0
$933$	4.97499e48	0.494021
$934$	0	0
$935$	−1.99969e49	−1.91678
$936$	0	0
$937$	4.58685e48	0.424435	0.212218	−	0.977222i	$-0.431931\pi$
$937$	4.58685e48	0.424435	0.212218	+	0.977222i	$0.431931\pi$
$938$	0	0
$939$	6.85665e48	0.612533
$940$	0	0
$941$	2.06166e49	1.77823	0.889117	−	0.457681i	$-0.151320\pi$
$941$	2.06166e49	1.77823	0.889117	+	0.457681i	$0.151320\pi$
$942$	0	0
$943$	−2.84778e48	−0.237172
$944$	0	0
$945$	2.43598e49	1.95906
$946$	0	0
$947$	−9.92376e48	−0.770729	−0.385365	−	0.922764i	$-0.625924\pi$
$947$	−9.92376e48	−0.770729	−0.385365	+	0.922764i	$0.625924\pi$
$948$	0	0
$949$	−4.43051e48	−0.332323
$950$	0	0
$951$	3.37957e49	2.44841
$952$	0	0
$953$	5.61698e48	0.393070	0.196535	−	0.980497i	$-0.437031\pi$
$953$	5.61698e48	0.393070	0.196535	+	0.980497i	$0.437031\pi$
$954$	0	0
$955$	−3.18010e48	−0.214974
$956$	0	0
$957$	−4.20082e49	−2.74338
$958$	0	0
$959$	−5.36856e48	−0.338728
$960$	0	0
$961$	1.18296e49	0.721166
$962$	0	0
$963$	−2.58495e49	−1.52272
$964$	0	0
$965$	7.84643e48	0.446654
$966$	0	0
$967$	4.81816e48	0.265060	0.132530	−	0.991179i	$-0.457690\pi$
$967$	4.81816e48	0.265060	0.132530	+	0.991179i	$0.457690\pi$
$968$	0	0
$969$	4.07663e49	2.16750
$970$	0	0
$971$	1.74311e49	0.895793	0.447897	−	0.894085i	$-0.352173\pi$
$971$	1.74311e49	0.895793	0.447897	+	0.894085i	$0.352173\pi$
$972$	0	0
$973$	9.13231e48	0.453648
$974$	0	0
$975$	1.78330e49	0.856345
$976$	0	0
$977$	−2.09476e49	−0.972464	−0.486232	−	0.873830i	$-0.661629\pi$
$977$	−2.09476e49	−0.972464	−0.486232	+	0.873830i	$0.661629\pi$
$978$	0	0
$979$	1.23282e49	0.553332
$980$	0	0
$981$	−5.10827e49	−2.21684
$982$	0	0
$983$	−1.66660e49	−0.699356	−0.349678	−	0.936870i	$-0.613709\pi$
$983$	−1.66660e49	−0.699356	−0.349678	+	0.936870i	$0.613709\pi$
$984$	0	0
$985$	3.44330e49	1.39726
$986$	0	0
$987$	6.70098e48	0.262969
$988$	0	0
$989$	3.68226e48	0.139758
$990$	0	0
$991$	−1.93978e49	−0.712092	−0.356046	−	0.934468i	$-0.615875\pi$
$991$	−1.93978e49	−0.712092	−0.356046	+	0.934468i	$0.615875\pi$
$992$	0	0
$993$	−5.70330e49	−2.02518
$994$	0	0
$995$	−5.67682e49	−1.94995
$996$	0	0
$997$	3.04287e49	1.01114	0.505572	−	0.862784i	$-0.331282\pi$
$997$	3.04287e49	1.01114	0.505572	+	0.862784i	$0.331282\pi$
$998$	0	0
$999$	−7.82089e49	−2.51435

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.34.a.a.1.3	✓	3
4.3	odd	2		16.34.a.d.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.34.a.a.1.3	✓	3	1.1	even	1	trivial
16.34.a.d.1.1		3	4.3	odd	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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34728e8 q^{3} -4.39893e11 q^{5} -5.84388e13 q^{7} +1.25925e16 q^{9} +O(q^{10})\)	\(q+1.34728e8 q^{3} -4.39893e11 q^{5} -5.84388e13 q^{7} +1.25925e16 q^{9} +2.38684e17 q^{11} +1.71699e18 q^{13} -5.92657e19 q^{15} +1.90455e20 q^{17} +1.58873e21 q^{19} -7.87333e21 q^{21} +6.96105e21 q^{23} +7.70902e22 q^{25} +9.47601e23 q^{27} -1.30633e24 q^{29} +5.31348e24 q^{31} +3.21574e25 q^{33} +2.57068e25 q^{35} -8.25335e25 q^{37} +2.31327e26 q^{39} -4.09102e26 q^{41} +5.28980e26 q^{43} -5.53935e27 q^{45} -8.51099e26 q^{47} -4.31590e27 q^{49} +2.56596e28 q^{51} +2.58983e28 q^{53} -1.04995e29 q^{55} +2.14047e29 q^{57} +2.29414e29 q^{59} -8.85755e28 q^{61} -7.35891e29 q^{63} -7.55293e29 q^{65} -1.09971e30 q^{67} +9.37847e29 q^{69} +1.24605e30 q^{71} -2.58039e30 q^{73} +1.03862e31 q^{75} -1.39484e31 q^{77} +3.55401e31 q^{79} +5.76657e31 q^{81} -5.14661e31 q^{83} -8.37798e31 q^{85} -1.75999e32 q^{87} +5.16507e31 q^{89} -1.00339e32 q^{91} +7.15874e32 q^{93} -6.98872e32 q^{95} -1.10473e33 q^{97} +3.00564e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) \) 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9	\( 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots + 22\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9 + 227617657302449700 * q^11 + 272970442217358762 * q^13 - 32142473481241105848 * q^15 + 93037188311816716854 * q^17 + 1324887806175081735228 * q^19 - 2289418149258037352352 * q^21 + 11680621362017694733896 * q^23 + 258183723295544823174621 * q^25 + 1166092519364338103061624 * q^27 + 2899576251477878827068378 * q^29 + 15816960746665911900076128 * q^31 + 51019828271446575126952080 * q^33 + 104388880678377160773832848 * q^35 + 34785435467211797617907442 * q^37 + 135170503164419330229696168 * q^39 - 673854719392272123121298562 * q^41 - 1557799443962179540641188220 * q^43 - 8295378544987729028249800086 * q^45 - 7226966919683191159587179376 * q^47 - 4755447639624414087245018709 * q^49 + 24102421034763142313708267736 * q^51 + 54068957931214078433925305634 * q^53 + 72116739028709103312702411480 * q^55 + 276172301449046303471104708848 * q^57 + 201006239114458347216752503476 * q^59 + 59158715139739534320593297466 * q^61 - 1234485289426293666626080288968 * q^63 - 2252651168064734881763466967812 * q^65 - 2904325260543190867603068997428 * q^67 - 1526403599797234628609099310816 * q^69 + 1693346400885181878584585353944 * q^71 + 1318409001451717739930563879902 * q^73 + 20178451712790450065114527827828 * q^75 + 20835723633799560796766684178720 * q^77 + 43046757284148836281181573950704 * q^79 + 60385181780035992188351168542827 * q^81 - 71356083882380783036145089288964 * q^83 - 137360972548514627995666767037884 * q^85 - 253975679679758553189305216266392 * q^87 - 194457621978564019473800095872498 * q^89 - 384667165850137479369424211871408 * q^91 + 554725843432039554369917205602688 * q^93 - 205626706338774416034575296820376 * q^95 + 566218664019390927229117474480422 * q^97 + 2255131944887948953089434155026900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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