LMFDB - Embedded newform 64.24.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,24,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	64.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:	$214.530583901$
Analytic rank:	$1$
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} - 389656213x^{4} + 47522643058672215x^{2} - 1756479932541937262634975 $ x^6 - 389656213x^4 + 47522643058672215x^2 - 1756479932541937262634975
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{71}\cdot 3^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-13375.1$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-90103.3 q^{3} +8.69741e7 q^{5} -2.82191e9 q^{7} -8.60246e10 q^{9} +O(q^{10})$	$q-90103.3 q^{3} +8.69741e7 q^{5} -2.82191e9 q^{7} -8.60246e10 q^{9} -1.78896e12 q^{11} +4.82597e12 q^{13} -7.83665e12 q^{15} -7.88365e13 q^{17} +8.73007e14 q^{19} +2.54263e14 q^{21} +6.19527e15 q^{23} -4.35643e15 q^{25} +1.62337e16 q^{27} +1.10505e17 q^{29} +5.17226e16 q^{31} +1.61191e17 q^{33} -2.45433e17 q^{35} -1.11995e18 q^{37} -4.34836e17 q^{39} -2.03184e18 q^{41} +8.20815e17 q^{43} -7.48191e18 q^{45} +2.81130e19 q^{47} -1.94056e19 q^{49} +7.10342e18 q^{51} +4.33010e18 q^{53} -1.55593e20 q^{55} -7.86608e19 q^{57} -2.42145e20 q^{59} -1.86266e20 q^{61} +2.42753e20 q^{63} +4.19735e20 q^{65} +1.37567e21 q^{67} -5.58214e20 q^{69} +3.62895e20 q^{71} -1.26910e21 q^{73} +3.92529e20 q^{75} +5.04827e21 q^{77} +2.95820e21 q^{79} +6.63592e21 q^{81} -2.85677e21 q^{83} -6.85673e21 q^{85} -9.95690e21 q^{87} +2.88207e22 q^{89} -1.36184e22 q^{91} -4.66038e21 q^{93} +7.59290e22 q^{95} -3.46031e22 q^{97} +1.53894e23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) $ 6 * q - 163121700 * q^5 + 14120212542 * q^9	$ 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) $ 6 * q - 163121700 * q^5 + 14120212542 * q^9 - 1101295489524 * q^13 + 56435517243468 * q^17 - 1447661324841984 * q^21 + 17512304669291850 * q^25 - 87161858953770708 * q^29 + 48462097564970496 * q^33 + 1542766250573484636 * q^37 - 865941451897399716 * q^41 + 9286254082406210700 * q^45 + 52551978665245784886 * q^49 + 10735070383373044668 * q^53 + 155104978099636154880 * q^57 + 432169207960503656364 * q^61 - 316526040451899107400 * q^65 - 695880182828346153984 * q^69 + 1758226166115151758204 * q^73 - 11926085657398731979776 * q^77 + 4010608843245537630678 * q^81 - 20251531460297137542600 * q^85 + 20087131519604301542940 * q^89 - 67310470168488243892224 * q^93 + 254665147475934678043116 * q^97

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
$3$	−90103.3	−0.293661	−0.146830	−	0.989162i	$-0.546907\pi$
$3$	−90103.3	−0.293661	−0.146830	+	0.989162i	$0.546907\pi$
$4$	0	0
$5$	8.69741e7	0.796590	0.398295	−	0.917257i	$-0.369602\pi$
$5$	8.69741e7	0.796590	0.398295	+	0.917257i	$0.369602\pi$
$6$	0	0
$7$	−2.82191e9	−0.539405	−0.269703	−	0.962944i	$-0.586925\pi$
$7$	−2.82191e9	−0.539405	−0.269703	+	0.962944i	$0.586925\pi$
$8$	0	0
$9$	−8.60246e10	−0.913763
$10$	0	0
$11$	−1.78896e12	−1.89053	−0.945266	−	0.326302i	$-0.894197\pi$
$11$	−1.78896e12	−1.89053	−0.945266	+	0.326302i	$0.894197\pi$
$12$	0	0
$13$	4.82597e12	0.746855	0.373428	−	0.927659i	$-0.378182\pi$
$13$	4.82597e12	0.746855	0.373428	+	0.927659i	$0.378182\pi$
$14$	0	0
$15$	−7.83665e12	−0.233927
$16$	0	0
$17$	−7.88365e13	−0.557911	−0.278955	−	0.960304i	$-0.589988\pi$
$17$	−7.88365e13	−0.557911	−0.278955	+	0.960304i	$0.589988\pi$
$18$	0	0
$19$	8.73007e14	1.71930	0.859650	−	0.510884i	$-0.170682\pi$
$19$	8.73007e14	1.71930	0.859650	+	0.510884i	$0.170682\pi$
$20$	0	0
$21$	2.54263e14	0.158402
$22$	0	0
$23$	6.19527e15	1.35578	0.677891	−	0.735162i	$-0.262894\pi$
$23$	6.19527e15	1.35578	0.677891	+	0.735162i	$0.262894\pi$
$24$	0	0
$25$	−4.35643e15	−0.365444
$26$	0	0
$27$	1.62337e16	0.561997
$28$	0	0
$29$	1.10505e17	1.68192	0.840962	−	0.541094i	$-0.181990\pi$
$29$	1.10505e17	1.68192	0.840962	+	0.541094i	$0.181990\pi$
$30$	0	0
$31$	5.17226e16	0.365613	0.182806	−	0.983149i	$-0.441482\pi$
$31$	5.17226e16	0.365613	0.182806	+	0.983149i	$0.441482\pi$
$32$	0	0
$33$	1.61191e17	0.555175
$34$	0	0
$35$	−2.45433e17	−0.429685
$36$	0	0
$37$	−1.11995e18	−1.03485	−0.517427	−	0.855728i	$-0.673110\pi$
$37$	−1.11995e18	−1.03485	−0.517427	+	0.855728i	$0.673110\pi$
$38$	0	0
$39$	−4.34836e17	−0.219322
$40$	0	0
$41$	−2.03184e18	−0.576600	−0.288300	−	0.957540i	$-0.593090\pi$
$41$	−2.03184e18	−0.576600	−0.288300	+	0.957540i	$0.593090\pi$
$42$	0	0
$43$	8.20815e17	0.134697	0.0673485	−	0.997730i	$-0.478546\pi$
$43$	8.20815e17	0.134697	0.0673485	+	0.997730i	$0.478546\pi$
$44$	0	0
$45$	−7.48191e18	−0.727895
$46$	0	0
$47$	2.81130e19	1.65875	0.829377	−	0.558689i	$-0.188695\pi$
$47$	2.81130e19	1.65875	0.829377	+	0.558689i	$0.188695\pi$
$48$	0	0
$49$	−1.94056e19	−0.709042
$50$	0	0
$51$	7.10342e18	0.163836
$52$	0	0
$53$	4.33010e18	0.0641690	0.0320845	−	0.999485i	$-0.489785\pi$
$53$	4.33010e18	0.0641690	0.0320845	+	0.999485i	$0.489785\pi$
$54$	0	0
$55$	−1.55593e20	−1.50598
$56$	0	0
$57$	−7.86608e19	−0.504891
$58$	0	0
$59$	−2.42145e20	−1.04539	−0.522694	−	0.852520i	$-0.675073\pi$
$59$	−2.42145e20	−1.04539	−0.522694	+	0.852520i	$0.675073\pi$
$60$	0	0
$61$	−1.86266e20	−0.548077	−0.274038	−	0.961719i	$-0.588359\pi$
$61$	−1.86266e20	−0.548077	−0.274038	+	0.961719i	$0.588359\pi$
$62$	0	0
$63$	2.42753e20	0.492889
$64$	0	0
$65$	4.19735e20	0.594937
$66$	0	0
$67$	1.37567e21	1.37612	0.688058	−	0.725656i	$-0.258463\pi$
$67$	1.37567e21	1.37612	0.688058	+	0.725656i	$0.258463\pi$
$68$	0	0
$69$	−5.58214e20	−0.398140
$70$	0	0
$71$	3.62895e20	0.186342	0.0931708	−	0.995650i	$-0.470300\pi$
$71$	3.62895e20	0.186342	0.0931708	+	0.995650i	$0.470300\pi$
$72$	0	0
$73$	−1.26910e21	−0.473459	−0.236730	−	0.971576i	$-0.576076\pi$
$73$	−1.26910e21	−0.473459	−0.236730	+	0.971576i	$0.576076\pi$
$74$	0	0
$75$	3.92529e20	0.107317
$76$	0	0
$77$	5.04827e21	1.01976
$78$	0	0
$79$	2.95820e21	0.444955	0.222478	−	0.974938i	$-0.428586\pi$
$79$	2.95820e21	0.444955	0.222478	+	0.974938i	$0.428586\pi$
$80$	0	0
$81$	6.63592e21	0.748727
$82$	0	0
$83$	−2.85677e21	−0.243488	−0.121744	−	0.992562i	$-0.538849\pi$
$83$	−2.85677e21	−0.243488	−0.121744	+	0.992562i	$0.538849\pi$
$84$	0	0
$85$	−6.85673e21	−0.444426
$86$	0	0
$87$	−9.95690e21	−0.493915
$88$	0	0
$89$	2.88207e22	1.10083	0.550415	−	0.834891i	$-0.314470\pi$
$89$	2.88207e22	1.10083	0.550415	+	0.834891i	$0.314470\pi$
$90$	0	0
$91$	−1.36184e22	−0.402858
$92$	0	0
$93$	−4.66038e21	−0.107366
$94$	0	0
$95$	7.59290e22	1.36958
$96$	0	0
$97$	−3.46031e22	−0.491179	−0.245590	−	0.969374i	$-0.578982\pi$
$97$	−3.46031e22	−0.491179	−0.245590	+	0.969374i	$0.578982\pi$
$98$	0	0
$99$	1.53894e23	1.72750
$100$	0	0
$101$	−1.78521e23	−1.59219	−0.796093	−	0.605175i	$-0.793103\pi$
$101$	−1.78521e23	−1.59219	−0.796093	+	0.605175i	$0.793103\pi$
$102$	0	0
$103$	−1.48014e23	−1.05359	−0.526797	−	0.849991i	$-0.676607\pi$
$103$	−1.48014e23	−1.05359	−0.526797	+	0.849991i	$0.676607\pi$
$104$	0	0
$105$	2.21143e22	0.126182
$106$	0	0
$107$	−3.47140e23	−1.59438	−0.797189	−	0.603730i	$-0.793680\pi$
$107$	−3.47140e23	−1.59438	−0.797189	+	0.603730i	$0.793680\pi$
$108$	0	0
$109$	−1.12833e23	−0.418824	−0.209412	−	0.977827i	$-0.567155\pi$
$109$	−1.12833e23	−0.418824	−0.209412	+	0.977827i	$0.567155\pi$
$110$	0	0
$111$	1.00911e23	0.303896
$112$	0	0
$113$	2.91296e23	0.714384	0.357192	−	0.934031i	$-0.383734\pi$
$113$	2.91296e23	0.714384	0.357192	+	0.934031i	$0.383734\pi$
$114$	0	0
$115$	5.38828e23	1.08000
$116$	0	0
$117$	−4.15152e23	−0.682449
$118$	0	0
$119$	2.22469e23	0.300940
$120$	0	0
$121$	2.30493e24	2.57411
$122$	0	0
$123$	1.83075e23	0.169325
$124$	0	0
$125$	−1.41571e24	−1.08770
$126$	0	0
$127$	−8.52481e23	−0.545685	−0.272842	−	0.962059i	$-0.587964\pi$
$127$	−8.52481e23	−0.545685	−0.272842	+	0.962059i	$0.587964\pi$
$128$	0	0
$129$	−7.39581e22	−0.0395553
$130$	0	0
$131$	−3.70915e24	−1.66209	−0.831045	−	0.556205i	$-0.812257\pi$
$131$	−3.70915e24	−1.66209	−0.831045	+	0.556205i	$0.812257\pi$
$132$	0	0
$133$	−2.46355e24	−0.927399
$134$	0	0
$135$	1.41191e24	0.447682
$136$	0	0
$137$	3.00512e24	0.804592	0.402296	−	0.915510i	$-0.368212\pi$
$137$	3.00512e24	0.804592	0.402296	+	0.915510i	$0.368212\pi$
$138$	0	0
$139$	2.45075e24	0.555429	0.277714	−	0.960664i	$-0.410423\pi$
$139$	2.45075e24	0.555429	0.277714	+	0.960664i	$0.410423\pi$
$140$	0	0
$141$	−2.53308e24	−0.487111
$142$	0	0
$143$	−8.63345e24	−1.41195
$144$	0	0
$145$	9.61111e24	1.33980
$146$	0	0
$147$	1.74851e24	0.208218
$148$	0	0
$149$	−1.36232e25	−1.38880	−0.694398	−	0.719591i	$-0.744329\pi$
$149$	−1.36232e25	−1.38880	−0.694398	+	0.719591i	$0.744329\pi$
$150$	0	0
$151$	1.53502e25	1.34239	0.671195	−	0.741281i	$-0.265782\pi$
$151$	1.53502e25	1.34239	0.671195	+	0.741281i	$0.265782\pi$
$152$	0	0
$153$	6.78187e24	0.509798
$154$	0	0
$155$	4.49853e24	0.291243
$156$	0	0
$157$	−1.98294e25	−1.10780	−0.553902	−	0.832582i	$-0.686862\pi$
$157$	−1.98294e25	−1.10780	−0.553902	+	0.832582i	$0.686862\pi$
$158$	0	0
$159$	−3.90156e23	−0.0188439
$160$	0	0
$161$	−1.74825e25	−0.731316
$162$	0	0
$163$	1.50672e25	0.546860	0.273430	−	0.961892i	$-0.411842\pi$
$163$	1.50672e25	0.546860	0.273430	+	0.961892i	$0.411842\pi$
$164$	0	0
$165$	1.40194e25	0.442247
$166$	0	0
$167$	−2.03414e25	−0.558653	−0.279326	−	0.960196i	$-0.590111\pi$
$167$	−2.03414e25	−0.558653	−0.279326	+	0.960196i	$0.590111\pi$
$168$	0	0
$169$	−1.84639e25	−0.442208
$170$	0	0
$171$	−7.51001e25	−1.57103
$172$	0	0
$173$	−9.87310e24	−0.180686	−0.0903428	−	0.995911i	$-0.528796\pi$
$173$	−9.87310e24	−0.180686	−0.0903428	+	0.995911i	$0.528796\pi$
$174$	0	0
$175$	1.22935e25	0.197123
$176$	0	0
$177$	2.18181e25	0.306990
$178$	0	0
$179$	4.70539e25	0.581816	0.290908	−	0.956751i	$-0.406043\pi$
$179$	4.70539e25	0.581816	0.290908	+	0.956751i	$0.406043\pi$
$180$	0	0
$181$	1.16446e26	1.26713	0.633565	−	0.773689i	$-0.281591\pi$
$181$	1.16446e26	1.26713	0.633565	+	0.773689i	$0.281591\pi$
$182$	0	0
$183$	1.67832e25	0.160949
$184$	0	0
$185$	−9.74067e25	−0.824354
$186$	0	0
$187$	1.41035e26	1.05475
$188$	0	0
$189$	−4.58100e25	−0.303144
$190$	0	0
$191$	1.69434e26	0.993383	0.496691	−	0.867927i	$-0.334548\pi$
$191$	1.69434e26	0.993383	0.496691	+	0.867927i	$0.334548\pi$
$192$	0	0
$193$	−2.36483e26	−1.22996	−0.614980	−	0.788543i	$-0.710836\pi$
$193$	−2.36483e26	−1.22996	−0.614980	+	0.788543i	$0.710836\pi$
$194$	0	0
$195$	−3.78195e25	−0.174710
$196$	0	0
$197$	2.56818e26	1.05503	0.527514	−	0.849546i	$-0.323124\pi$
$197$	2.56818e26	1.05503	0.527514	+	0.849546i	$0.323124\pi$
$198$	0	0
$199$	−1.77225e26	−0.648209	−0.324104	−	0.946021i	$-0.605063\pi$
$199$	−1.77225e26	−0.648209	−0.324104	+	0.946021i	$0.605063\pi$
$200$	0	0
$201$	−1.23953e26	−0.404111
$202$	0	0
$203$	−3.11836e26	−0.907239
$204$	0	0
$205$	−1.76717e26	−0.459314
$206$	0	0
$207$	−5.32945e26	−1.23886
$208$	0	0
$209$	−1.56177e27	−3.25039
$210$	0	0
$211$	1.62488e26	0.303091	0.151545	−	0.988450i	$-0.451575\pi$
$211$	1.62488e26	0.303091	0.151545	+	0.988450i	$0.451575\pi$
$212$	0	0
$213$	−3.26980e25	−0.0547212
$214$	0	0
$215$	7.13897e25	0.107298
$216$	0	0
$217$	−1.45956e26	−0.197213
$218$	0	0
$219$	1.14350e26	0.139036
$220$	0	0
$221$	−3.80463e26	−0.416678
$222$	0	0
$223$	1.43722e27	1.41911	0.709555	−	0.704650i	$-0.248896\pi$
$223$	1.43722e27	1.41911	0.709555	+	0.704650i	$0.248896\pi$
$224$	0	0
$225$	3.74760e26	0.333929
$226$	0	0
$227$	−1.31561e27	−1.05884	−0.529418	−	0.848361i	$-0.677590\pi$
$227$	−1.31561e27	−1.05884	−0.529418	+	0.848361i	$0.677590\pi$
$228$	0	0
$229$	−1.53550e27	−1.11723	−0.558614	−	0.829428i	$-0.688667\pi$
$229$	−1.53550e27	−1.11723	−0.558614	+	0.829428i	$0.688667\pi$
$230$	0	0
$231$	−4.54865e26	−0.299464
$232$	0	0
$233$	1.06378e25	0.00634249	0.00317124	−	0.999995i	$-0.498991\pi$
$233$	1.06378e25	0.00634249	0.00317124	+	0.999995i	$0.498991\pi$
$234$	0	0
$235$	2.44511e27	1.32135
$236$	0	0
$237$	−2.66544e26	−0.130666
$238$	0	0
$239$	−3.18970e27	−1.41963	−0.709813	−	0.704390i	$-0.751221\pi$
$239$	−3.18970e27	−1.41963	−0.709813	+	0.704390i	$0.751221\pi$
$240$	0	0
$241$	−1.78297e27	−0.721022	−0.360511	−	0.932755i	$-0.617398\pi$
$241$	−1.78297e27	−0.721022	−0.360511	+	0.932755i	$0.617398\pi$
$242$	0	0
$243$	−2.12621e27	−0.781869
$244$	0	0
$245$	−1.68778e27	−0.564816
$246$	0	0
$247$	4.21311e27	1.28407
$248$	0	0
$249$	2.57404e26	0.0715028
$250$	0	0
$251$	1.39005e27	0.352194	0.176097	−	0.984373i	$-0.443653\pi$
$251$	1.39005e27	0.352194	0.176097	+	0.984373i	$0.443653\pi$
$252$	0	0
$253$	−1.10831e28	−2.56315
$254$	0	0
$255$	6.17814e26	0.130511
$256$	0	0
$257$	−2.59576e27	−0.501227	−0.250613	−	0.968087i	$-0.580632\pi$
$257$	−2.59576e27	−0.501227	−0.250613	+	0.968087i	$0.580632\pi$
$258$	0	0
$259$	3.16040e27	0.558206
$260$	0	0
$261$	−9.50618e27	−1.53688
$262$	0	0
$263$	−3.26577e27	−0.483610	−0.241805	−	0.970325i	$-0.577739\pi$
$263$	−3.26577e27	−0.483610	−0.241805	+	0.970325i	$0.577739\pi$
$264$	0	0
$265$	3.76606e26	0.0511164
$266$	0	0
$267$	−2.59684e27	−0.323271
$268$	0	0
$269$	−1.12793e28	−1.28864	−0.644321	−	0.764755i	$-0.722860\pi$
$269$	−1.12793e28	−1.28864	−0.644321	+	0.764755i	$0.722860\pi$
$270$	0	0
$271$	4.19102e27	0.439716	0.219858	−	0.975532i	$-0.429441\pi$
$271$	4.19102e27	0.439716	0.219858	+	0.975532i	$0.429441\pi$
$272$	0	0
$273$	1.22707e27	0.118304
$274$	0	0
$275$	7.79347e27	0.690884
$276$	0	0
$277$	−9.09091e27	−0.741464	−0.370732	−	0.928740i	$-0.620893\pi$
$277$	−9.09091e27	−0.741464	−0.370732	+	0.928740i	$0.620893\pi$
$278$	0	0
$279$	−4.44942e27	−0.334083
$280$	0	0
$281$	−1.76605e28	−1.22146	−0.610730	−	0.791839i	$-0.709124\pi$
$281$	−1.76605e28	−1.22146	−0.610730	+	0.791839i	$0.709124\pi$
$282$	0	0
$283$	−2.60360e28	−1.65971	−0.829853	−	0.557982i	$-0.811576\pi$
$283$	−2.60360e28	−1.65971	−0.829853	+	0.557982i	$0.811576\pi$
$284$	0	0
$285$	−6.84145e27	−0.402191
$286$	0	0
$287$	5.73365e27	0.311021
$288$	0	0
$289$	−1.37524e28	−0.688736
$290$	0	0
$291$	3.11785e27	0.144240
$292$	0	0
$293$	−8.19763e27	−0.350518	−0.175259	−	0.984522i	$-0.556076\pi$
$293$	−8.19763e27	−0.350518	−0.175259	+	0.984522i	$0.556076\pi$
$294$	0	0
$295$	−2.10604e28	−0.832746
$296$	0	0
$297$	−2.90414e28	−1.06247
$298$	0	0
$299$	2.98982e28	1.01257
$300$	0	0
$301$	−2.31626e27	−0.0726563
$302$	0	0
$303$	1.60853e28	0.467562
$304$	0	0
$305$	−1.62003e28	−0.436592
$306$	0	0
$307$	−9.26655e27	−0.231647	−0.115824	−	0.993270i	$-0.536951\pi$
$307$	−9.26655e27	−0.231647	−0.115824	+	0.993270i	$0.536951\pi$
$308$	0	0
$309$	1.33365e28	0.309399
$310$	0	0
$311$	8.46807e28	1.82406	0.912032	−	0.410118i	$-0.134513\pi$
$311$	8.46807e28	1.82406	0.912032	+	0.410118i	$0.134513\pi$
$312$	0	0
$313$	−5.61746e28	−1.12403	−0.562017	−	0.827125i	$-0.689975\pi$
$313$	−5.61746e28	−1.12403	−0.562017	+	0.827125i	$0.689975\pi$
$314$	0	0
$315$	2.11133e28	0.392630
$316$	0	0
$317$	6.02015e28	1.04094	0.520470	−	0.853880i	$-0.325757\pi$
$317$	6.02015e28	1.04094	0.520470	+	0.853880i	$0.325757\pi$
$318$	0	0
$319$	−1.97689e29	−3.17973
$320$	0	0
$321$	3.12784e28	0.468206
$322$	0	0
$323$	−6.88248e28	−0.959215
$324$	0	0
$325$	−2.10240e28	−0.272934
$326$	0	0
$327$	1.01666e28	0.122992
$328$	0	0
$329$	−7.93324e28	−0.894741
$330$	0	0
$331$	−2.25189e28	−0.236878	−0.118439	−	0.992961i	$-0.537789\pi$
$331$	−2.25189e28	−0.236878	−0.118439	+	0.992961i	$0.537789\pi$
$332$	0	0
$333$	9.63432e28	0.945611
$334$	0	0
$335$	1.19648e29	1.09620
$336$	0	0
$337$	−2.19157e29	−1.87504	−0.937522	−	0.347926i	$-0.886886\pi$
$337$	−2.19157e29	−1.87504	−0.937522	+	0.347926i	$0.886886\pi$
$338$	0	0
$339$	−2.62467e28	−0.209787
$340$	0	0
$341$	−9.25295e28	−0.691202
$342$	0	0
$343$	1.31993e29	0.921866
$344$	0	0
$345$	−4.85501e28	−0.317155
$346$	0	0
$347$	2.59779e29	1.58787	0.793936	−	0.608002i	$-0.208029\pi$
$347$	2.59779e29	1.58787	0.793936	+	0.608002i	$0.208029\pi$
$348$	0	0
$349$	6.64807e28	0.380367	0.190184	−	0.981749i	$-0.439092\pi$
$349$	6.64807e28	0.380367	0.190184	+	0.981749i	$0.439092\pi$
$350$	0	0
$351$	7.83434e28	0.419731
$352$	0	0
$353$	2.81339e29	1.41196	0.705979	−	0.708233i	$-0.250507\pi$
$353$	2.81339e29	1.41196	0.705979	+	0.708233i	$0.250507\pi$
$354$	0	0
$355$	3.15625e28	0.148438
$356$	0	0
$357$	−2.00452e28	−0.0883743
$358$	0	0
$359$	−3.81676e29	−1.57801	−0.789003	−	0.614390i	$-0.789402\pi$
$359$	−3.81676e29	−1.57801	−0.789003	+	0.614390i	$0.789402\pi$
$360$	0	0
$361$	5.04312e29	1.95599
$362$	0	0
$363$	−2.07682e29	−0.755915
$364$	0	0
$365$	−1.10379e29	−0.377153
$366$	0	0
$367$	2.38307e29	0.764676	0.382338	−	0.924023i	$-0.375119\pi$
$367$	2.38307e29	0.764676	0.382338	+	0.924023i	$0.375119\pi$
$368$	0	0
$369$	1.74788e29	0.526876
$370$	0	0
$371$	−1.22191e28	−0.0346131
$372$	0	0
$373$	4.61469e29	1.22883	0.614414	−	0.788984i	$-0.289393\pi$
$373$	4.61469e29	1.22883	0.614414	+	0.788984i	$0.289393\pi$
$374$	0	0
$375$	1.27560e29	0.319415
$376$	0	0
$377$	5.33296e29	1.25615
$378$	0	0
$379$	−1.10567e29	−0.245062	−0.122531	−	0.992465i	$-0.539101\pi$
$379$	−1.10567e29	−0.245062	−0.122531	+	0.992465i	$0.539101\pi$
$380$	0	0
$381$	7.68113e28	0.160246
$382$	0	0
$383$	−4.81688e29	−0.946194	−0.473097	−	0.881010i	$-0.656864\pi$
$383$	−4.81688e29	−0.946194	−0.473097	+	0.881010i	$0.656864\pi$
$384$	0	0
$385$	4.39069e29	0.812333
$386$	0	0
$387$	−7.06103e28	−0.123081
$388$	0	0
$389$	3.77138e28	0.0619554	0.0309777	−	0.999520i	$-0.490138\pi$
$389$	3.77138e28	0.0619554	0.0309777	+	0.999520i	$0.490138\pi$
$390$	0	0
$391$	−4.88413e29	−0.756405
$392$	0	0
$393$	3.34207e29	0.488091
$394$	0	0
$395$	2.57287e29	0.354447
$396$	0	0
$397$	−7.83655e29	−1.01867	−0.509335	−	0.860568i	$-0.670109\pi$
$397$	−7.83655e29	−1.01867	−0.509335	+	0.860568i	$0.670109\pi$
$398$	0	0
$399$	2.21974e29	0.272341
$400$	0	0
$401$	1.29030e30	1.49462	0.747308	−	0.664478i	$-0.231346\pi$
$401$	1.29030e30	1.49462	0.747308	+	0.664478i	$0.231346\pi$
$402$	0	0
$403$	2.49612e29	0.273060
$404$	0	0
$405$	5.77153e29	0.596428
$406$	0	0
$407$	2.00354e30	1.95642
$408$	0	0
$409$	1.83360e29	0.169234	0.0846168	−	0.996414i	$-0.473033\pi$
$409$	1.83360e29	0.169234	0.0846168	+	0.996414i	$0.473033\pi$
$410$	0	0
$411$	−2.70771e29	−0.236277
$412$	0	0
$413$	6.83312e29	0.563888
$414$	0	0
$415$	−2.48465e29	−0.193960
$416$	0	0
$417$	−2.20820e29	−0.163108
$418$	0	0
$419$	−9.89810e29	−0.691975	−0.345988	−	0.938239i	$-0.612456\pi$
$419$	−9.89810e29	−0.691975	−0.345988	+	0.938239i	$0.612456\pi$
$420$	0	0
$421$	−1.53464e28	−0.0101569	−0.00507846	−	0.999987i	$-0.501617\pi$
$421$	−1.53464e28	−0.0101569	−0.00507846	+	0.999987i	$0.501617\pi$
$422$	0	0
$423$	−2.41841e30	−1.51571
$424$	0	0
$425$	3.43446e29	0.203885
$426$	0	0
$427$	5.25626e29	0.295635
$428$	0	0
$429$	7.77902e29	0.414635
$430$	0	0
$431$	−1.68886e30	−0.853308	−0.426654	−	0.904415i	$-0.640308\pi$
$431$	−1.68886e30	−0.853308	−0.426654	+	0.904415i	$0.640308\pi$
$432$	0	0
$433$	−1.08936e30	−0.521868	−0.260934	−	0.965357i	$-0.584031\pi$
$433$	−1.08936e30	−0.521868	−0.260934	+	0.965357i	$0.584031\pi$
$434$	0	0
$435$	−8.65992e29	−0.393448
$436$	0	0
$437$	5.40851e30	2.33100
$438$	0	0
$439$	3.81556e30	1.56033	0.780165	−	0.625574i	$-0.215135\pi$
$439$	3.81556e30	1.56033	0.780165	+	0.625574i	$0.215135\pi$
$440$	0	0
$441$	1.66936e30	0.647896
$442$	0	0
$443$	1.66233e30	0.612455	0.306228	−	0.951958i	$-0.400933\pi$
$443$	1.66233e30	0.612455	0.306228	+	0.951958i	$0.400933\pi$
$444$	0	0
$445$	2.50666e30	0.876910
$446$	0	0
$447$	1.22750e30	0.407835
$448$	0	0
$449$	−3.38378e30	−1.06800	−0.533998	−	0.845486i	$-0.679311\pi$
$449$	−3.38378e30	−1.06800	−0.533998	+	0.845486i	$0.679311\pi$
$450$	0	0
$451$	3.63487e30	1.09008
$452$	0	0
$453$	−1.38310e30	−0.394207
$454$	0	0
$455$	−1.18445e30	−0.320912
$456$	0	0
$457$	−2.84835e30	−0.733766	−0.366883	−	0.930267i	$-0.619575\pi$
$457$	−2.84835e30	−0.733766	−0.366883	+	0.930267i	$0.619575\pi$
$458$	0	0
$459$	−1.27981e30	−0.313544
$460$	0	0
$461$	2.61694e30	0.609864	0.304932	−	0.952374i	$-0.401366\pi$
$461$	2.61694e30	0.609864	0.304932	+	0.952374i	$0.401366\pi$
$462$	0	0
$463$	−2.01719e30	−0.447266	−0.223633	−	0.974673i	$-0.571792\pi$
$463$	−2.01719e30	−0.447266	−0.223633	+	0.974673i	$0.571792\pi$
$464$	0	0
$465$	−4.05332e29	−0.0855268
$466$	0	0
$467$	−2.74839e30	−0.551995	−0.275997	−	0.961158i	$-0.589008\pi$
$467$	−2.74839e30	−0.551995	−0.275997	+	0.961158i	$0.589008\pi$
$468$	0	0
$469$	−3.88202e30	−0.742285
$470$	0	0
$471$	1.78669e30	0.325319
$472$	0	0
$473$	−1.46840e30	−0.254649
$474$	0	0
$475$	−3.80320e30	−0.628308
$476$	0	0
$477$	−3.72495e29	−0.0586353
$478$	0	0
$479$	−4.27382e28	−0.00641148	−0.00320574	−	0.999995i	$-0.501020\pi$
$479$	−4.27382e28	−0.00641148	−0.00320574	+	0.999995i	$0.501020\pi$
$480$	0	0
$481$	−5.40485e30	−0.772885
$482$	0	0
$483$	1.57523e30	0.214759
$484$	0	0
$485$	−3.00957e30	−0.391268
$486$	0	0
$487$	−3.55583e30	−0.440919	−0.220459	−	0.975396i	$-0.570756\pi$
$487$	−3.55583e30	−0.440919	−0.220459	+	0.975396i	$0.570756\pi$
$488$	0	0
$489$	−1.35761e30	−0.160591
$490$	0	0
$491$	2.85757e29	0.0322522	0.0161261	−	0.999870i	$-0.494867\pi$
$491$	2.85757e29	0.0322522	0.0161261	+	0.999870i	$0.494867\pi$
$492$	0	0
$493$	−8.71186e30	−0.938364
$494$	0	0
$495$	1.33848e31	1.37611
$496$	0	0
$497$	−1.02406e30	−0.100514
$498$	0	0
$499$	8.75575e30	0.820610	0.410305	−	0.911948i	$-0.365422\pi$
$499$	8.75575e30	0.820610	0.410305	+	0.911948i	$0.365422\pi$
$500$	0	0
$501$	1.83283e30	0.164054
$502$	0	0
$503$	−1.50987e31	−1.29095	−0.645473	−	0.763783i	$-0.723340\pi$
$503$	−1.50987e31	−1.29095	−0.645473	+	0.763783i	$0.723340\pi$
$504$	0	0
$505$	−1.55267e31	−1.26832
$506$	0	0
$507$	1.66366e30	0.129859
$508$	0	0
$509$	−1.63647e31	−1.22082	−0.610412	−	0.792084i	$-0.708996\pi$
$509$	−1.63647e31	−1.22082	−0.610412	+	0.792084i	$0.708996\pi$
$510$	0	0
$511$	3.58128e30	0.255386
$512$	0	0
$513$	1.41721e31	0.966242
$514$	0	0
$515$	−1.28733e31	−0.839283
$516$	0	0
$517$	−5.02930e31	−3.13593
$518$	0	0
$519$	8.89599e29	0.0530603
$520$	0	0
$521$	1.05597e30	0.0602581	0.0301291	−	0.999546i	$-0.490408\pi$
$521$	1.05597e30	0.0602581	0.0301291	+	0.999546i	$0.490408\pi$
$522$	0	0
$523$	−2.55054e31	−1.39272	−0.696358	−	0.717694i	$-0.745198\pi$
$523$	−2.55054e31	−1.39272	−0.696358	+	0.717694i	$0.745198\pi$
$524$	0	0
$525$	−1.10768e30	−0.0578872
$526$	0	0
$527$	−4.07763e30	−0.203979
$528$	0	0
$529$	1.75009e31	0.838145
$530$	0	0
$531$	2.08304e31	0.955238
$532$	0	0
$533$	−9.80559e30	−0.430636
$534$	0	0
$535$	−3.01922e31	−1.27007
$536$	0	0
$537$	−4.23971e30	−0.170857
$538$	0	0
$539$	3.47157e31	1.34047
$540$	0	0
$541$	1.13793e31	0.421065	0.210532	−	0.977587i	$-0.432480\pi$
$541$	1.13793e31	0.421065	0.210532	+	0.977587i	$0.432480\pi$
$542$	0	0
$543$	−1.04922e31	−0.372106
$544$	0	0
$545$	−9.81355e30	−0.333631
$546$	0	0
$547$	−2.52231e31	−0.822138	−0.411069	−	0.911604i	$-0.634845\pi$
$547$	−2.52231e31	−0.822138	−0.411069	+	0.911604i	$0.634845\pi$
$548$	0	0
$549$	1.60235e31	0.500812
$550$	0	0
$551$	9.64720e31	2.89173
$552$	0	0
$553$	−8.34778e30	−0.240011
$554$	0	0
$555$	8.77666e30	0.242080
$556$	0	0
$557$	−4.49591e31	−1.18982	−0.594912	−	0.803791i	$-0.702813\pi$
$557$	−4.49591e31	−1.18982	−0.594912	+	0.803791i	$0.702813\pi$
$558$	0	0
$559$	3.96123e30	0.100599
$560$	0	0
$561$	−1.27077e31	−0.309738
$562$	0	0
$563$	−2.40725e31	−0.563215	−0.281608	−	0.959530i	$-0.590868\pi$
$563$	−2.40725e31	−0.563215	−0.281608	+	0.959530i	$0.590868\pi$
$564$	0	0
$565$	2.53352e31	0.569071
$566$	0	0
$567$	−1.87259e31	−0.403867
$568$	0	0
$569$	9.55859e30	0.197971	0.0989857	−	0.995089i	$-0.468440\pi$
$569$	9.55859e30	0.197971	0.0989857	+	0.995089i	$0.468440\pi$
$570$	0	0
$571$	4.16090e31	0.827699	0.413850	−	0.910345i	$-0.364184\pi$
$571$	4.16090e31	0.827699	0.413850	+	0.910345i	$0.364184\pi$
$572$	0	0
$573$	−1.52665e31	−0.291718
$574$	0	0
$575$	−2.69893e31	−0.495463
$576$	0	0
$577$	−8.74502e31	−1.54255	−0.771275	−	0.636503i	$-0.780380\pi$
$577$	−8.74502e31	−1.54255	−0.771275	+	0.636503i	$0.780380\pi$
$578$	0	0
$579$	2.13079e31	0.361191
$580$	0	0
$581$	8.06154e30	0.131339
$582$	0	0
$583$	−7.74635e30	−0.121313
$584$	0	0
$585$	−3.61075e31	−0.543632
$586$	0	0
$587$	−6.84982e31	−0.991610	−0.495805	−	0.868434i	$-0.665127\pi$
$587$	−6.84982e31	−0.991610	−0.495805	+	0.868434i	$0.665127\pi$
$588$	0	0
$589$	4.51542e31	0.628598
$590$	0	0
$591$	−2.31402e31	−0.309821
$592$	0	0
$593$	−2.31830e31	−0.298566	−0.149283	−	0.988795i	$-0.547697\pi$
$593$	−2.31830e31	−0.298566	−0.149283	+	0.988795i	$0.547697\pi$
$594$	0	0
$595$	1.93491e31	0.239726
$596$	0	0
$597$	1.59686e31	0.190354
$598$	0	0
$599$	−1.28806e32	−1.47750	−0.738751	−	0.673978i	$-0.764584\pi$
$599$	−1.28806e32	−1.47750	−0.738751	+	0.673978i	$0.764584\pi$
$600$	0	0
$601$	4.32836e31	0.477823	0.238911	−	0.971041i	$-0.423209\pi$
$601$	4.32836e31	0.477823	0.238911	+	0.971041i	$0.423209\pi$
$602$	0	0
$603$	−1.18342e32	−1.25744
$604$	0	0
$605$	2.00470e32	2.05051
$606$	0	0
$607$	1.85178e32	1.82356	0.911780	−	0.410680i	$-0.134708\pi$
$607$	1.85178e32	1.82356	0.911780	+	0.410680i	$0.134708\pi$
$608$	0	0
$609$	2.80974e31	0.266421
$610$	0	0
$611$	1.35673e32	1.23885
$612$	0	0
$613$	1.52395e32	1.34022	0.670110	−	0.742262i	$-0.266247\pi$
$613$	1.52395e32	1.34022	0.670110	+	0.742262i	$0.266247\pi$
$614$	0	0
$615$	1.59228e31	0.134882
$616$	0	0
$617$	1.52649e32	1.24570	0.622850	−	0.782342i	$-0.285975\pi$
$617$	1.52649e32	1.24570	0.622850	+	0.782342i	$0.285975\pi$
$618$	0	0
$619$	2.34135e31	0.184087	0.0920436	−	0.995755i	$-0.470660\pi$
$619$	2.34135e31	0.184087	0.0920436	+	0.995755i	$0.470660\pi$
$620$	0	0
$621$	1.00572e32	0.761946
$622$	0	0
$623$	−8.13295e31	−0.593793
$624$	0	0
$625$	−7.11973e31	−0.501006
$626$	0	0
$627$	1.40721e32	0.954512
$628$	0	0
$629$	8.82929e31	0.577356
$630$	0	0
$631$	1.15959e32	0.731083	0.365541	−	0.930795i	$-0.380884\pi$
$631$	1.15959e32	0.731083	0.365541	+	0.930795i	$0.380884\pi$
$632$	0	0
$633$	−1.46407e31	−0.0890060
$634$	0	0
$635$	−7.41437e31	−0.434687
$636$	0	0
$637$	−9.36508e31	−0.529551
$638$	0	0
$639$	−3.12179e31	−0.170272
$640$	0	0
$641$	1.77959e32	0.936380	0.468190	−	0.883628i	$-0.344906\pi$
$641$	1.77959e32	0.936380	0.468190	+	0.883628i	$0.344906\pi$
$642$	0	0
$643$	−3.08542e32	−1.56635	−0.783175	−	0.621802i	$-0.786401\pi$
$643$	−3.08542e32	−1.56635	−0.783175	+	0.621802i	$0.786401\pi$
$644$	0	0
$645$	−6.43244e30	−0.0315093
$646$	0	0
$647$	−2.33984e31	−0.110608	−0.0553041	−	0.998470i	$-0.517613\pi$
$647$	−2.33984e31	−0.110608	−0.0553041	+	0.998470i	$0.517613\pi$
$648$	0	0
$649$	4.33187e32	1.97634
$650$	0	0
$651$	1.31512e31	0.0579139
$652$	0	0
$653$	−6.52634e31	−0.277439	−0.138720	−	0.990332i	$-0.544299\pi$
$653$	−6.52634e31	−0.277439	−0.138720	+	0.990332i	$0.544299\pi$
$654$	0	0
$655$	−3.22600e32	−1.32401
$656$	0	0
$657$	1.09174e32	0.432629
$658$	0	0
$659$	−2.89998e32	−1.10972	−0.554858	−	0.831945i	$-0.687227\pi$
$659$	−2.89998e32	−1.10972	−0.554858	+	0.831945i	$0.687227\pi$
$660$	0	0
$661$	−1.71093e32	−0.632290	−0.316145	−	0.948711i	$-0.602389\pi$
$661$	−1.71093e32	−0.632290	−0.316145	+	0.948711i	$0.602389\pi$
$662$	0	0
$663$	3.42809e31	0.122362
$664$	0	0
$665$	−2.14265e32	−0.738757
$666$	0	0
$667$	6.84611e32	2.28032
$668$	0	0
$669$	−1.29498e32	−0.416737
$670$	0	0
$671$	3.33222e32	1.03616
$672$	0	0
$673$	4.95286e32	1.48827	0.744136	−	0.668028i	$-0.232861\pi$
$673$	4.95286e32	1.48827	0.744136	+	0.668028i	$0.232861\pi$
$674$	0	0
$675$	−7.07210e31	−0.205379
$676$	0	0
$677$	1.74662e32	0.490262	0.245131	−	0.969490i	$-0.421169\pi$
$677$	1.74662e32	0.490262	0.245131	+	0.969490i	$0.421169\pi$
$678$	0	0
$679$	9.76467e31	0.264945
$680$	0	0
$681$	1.18540e32	0.310939
$682$	0	0
$683$	6.92071e31	0.175514	0.0877572	−	0.996142i	$-0.472030\pi$
$683$	6.92071e31	0.175514	0.0877572	+	0.996142i	$0.472030\pi$
$684$	0	0
$685$	2.61368e32	0.640930
$686$	0	0
$687$	1.38354e32	0.328086
$688$	0	0
$689$	2.08969e31	0.0479249
$690$	0	0
$691$	−6.47375e31	−0.143601	−0.0718006	−	0.997419i	$-0.522875\pi$
$691$	−6.47375e31	−0.143601	−0.0718006	+	0.997419i	$0.522875\pi$
$692$	0	0
$693$	−4.34275e32	−0.931822
$694$	0	0
$695$	2.13152e32	0.442449
$696$	0	0
$697$	1.60183e32	0.321691
$698$	0	0
$699$	−9.58502e29	−0.00186254
$700$	0	0
$701$	−5.15230e32	−0.968821	−0.484410	−	0.874841i	$-0.660966\pi$
$701$	−5.15230e32	−0.968821	−0.484410	+	0.874841i	$0.660966\pi$
$702$	0	0
$703$	−9.77725e32	−1.77922
$704$	0	0
$705$	−2.20312e32	−0.388028
$706$	0	0
$707$	5.03770e32	0.858833
$708$	0	0
$709$	5.65407e30	0.00933102	0.00466551	−	0.999989i	$-0.498515\pi$
$709$	5.65407e30	0.00933102	0.00466551	+	0.999989i	$0.498515\pi$
$710$	0	0
$711$	−2.54478e32	−0.406584
$712$	0	0
$713$	3.20436e32	0.495691
$714$	0	0
$715$	−7.50887e32	−1.12475
$716$	0	0
$717$	2.87402e32	0.416888
$718$	0	0
$719$	9.36983e32	1.31628	0.658142	−	0.752894i	$-0.271343\pi$
$719$	9.36983e32	1.31628	0.658142	+	0.752894i	$0.271343\pi$
$720$	0	0
$721$	4.17681e32	0.568314
$722$	0	0
$723$	1.60652e32	0.211736
$724$	0	0
$725$	−4.81410e32	−0.614650
$726$	0	0
$727$	5.32159e32	0.658258	0.329129	−	0.944285i	$-0.393245\pi$
$727$	5.32159e32	0.658258	0.329129	+	0.944285i	$0.393245\pi$
$728$	0	0
$729$	−4.33148e32	−0.519122
$730$	0	0
$731$	−6.47102e31	−0.0751489
$732$	0	0
$733$	1.06080e33	1.19382	0.596908	−	0.802310i	$-0.296396\pi$
$733$	1.06080e33	1.19382	0.596908	+	0.802310i	$0.296396\pi$
$734$	0	0
$735$	1.52075e32	0.165864
$736$	0	0
$737$	−2.46102e33	−2.60159
$738$	0	0
$739$	−1.10630e33	−1.13360	−0.566801	−	0.823854i	$-0.691819\pi$
$739$	−1.10630e33	−1.13360	−0.566801	+	0.823854i	$0.691819\pi$
$740$	0	0
$741$	−3.79615e32	−0.377080
$742$	0	0
$743$	−1.51839e33	−1.46222	−0.731110	−	0.682259i	$-0.760998\pi$
$743$	−1.51839e33	−1.46222	−0.731110	+	0.682259i	$0.760998\pi$
$744$	0	0
$745$	−1.18487e33	−1.10630
$746$	0	0
$747$	2.45752e32	0.222490
$748$	0	0
$749$	9.79597e32	0.860016
$750$	0	0
$751$	−9.12621e31	−0.0777018	−0.0388509	−	0.999245i	$-0.512370\pi$
$751$	−9.12621e31	−0.0777018	−0.0388509	+	0.999245i	$0.512370\pi$
$752$	0	0
$753$	−1.25248e32	−0.103425
$754$	0	0
$755$	1.33507e33	1.06933
$756$	0	0
$757$	−1.65020e33	−1.28214	−0.641069	−	0.767483i	$-0.721509\pi$
$757$	−1.65020e33	−1.28214	−0.641069	+	0.767483i	$0.721509\pi$
$758$	0	0
$759$	9.98620e32	0.752696
$760$	0	0
$761$	−1.33807e33	−0.978485	−0.489243	−	0.872148i	$-0.662727\pi$
$761$	−1.33807e33	−0.978485	−0.489243	+	0.872148i	$0.662727\pi$
$762$	0	0
$763$	3.18404e32	0.225916
$764$	0	0
$765$	5.89847e32	0.406100
$766$	0	0
$767$	−1.16859e33	−0.780754
$768$	0	0
$769$	−1.21063e33	−0.784981	−0.392491	−	0.919756i	$-0.628386\pi$
$769$	−1.21063e33	−0.784981	−0.392491	+	0.919756i	$0.628386\pi$
$770$	0	0
$771$	2.33887e32	0.147191
$772$	0	0
$773$	−4.95192e32	−0.302489	−0.151245	−	0.988496i	$-0.548328\pi$
$773$	−4.95192e32	−0.302489	−0.151245	+	0.988496i	$0.548328\pi$
$774$	0	0
$775$	−2.25326e32	−0.133611
$776$	0	0
$777$	−2.84762e32	−0.163923
$778$	0	0
$779$	−1.77381e33	−0.991347
$780$	0	0
$781$	−6.49203e32	−0.352285
$782$	0	0
$783$	1.79391e33	0.945237
$784$	0	0
$785$	−1.72464e33	−0.882466
$786$	0	0
$787$	−1.57068e33	−0.780512	−0.390256	−	0.920706i	$-0.627613\pi$
$787$	−1.57068e33	−0.780512	−0.390256	+	0.920706i	$0.627613\pi$
$788$	0	0
$789$	2.94257e32	0.142017
$790$	0	0
$791$	−8.22009e32	−0.385343
$792$	0	0
$793$	−8.98916e32	−0.409334
$794$	0	0
$795$	−3.39335e31	−0.0150109
$796$	0	0
$797$	−2.81450e33	−1.20957	−0.604784	−	0.796389i	$-0.706741\pi$
$797$	−2.81450e33	−1.20957	−0.604784	+	0.796389i	$0.706741\pi$
$798$	0	0
$799$	−2.21633e33	−0.925437
$800$	0	0
$801$	−2.47929e33	−1.00590
$802$	0	0
$803$	2.27036e33	0.895089
$804$	0	0
$805$	−1.52052e33	−0.582559
$806$	0	0
$807$	1.01630e33	0.378424
$808$	0	0
$809$	5.38141e33	1.94755	0.973773	−	0.227522i	$-0.0730623\pi$
$809$	5.38141e33	1.94755	0.973773	+	0.227522i	$0.0730623\pi$
$810$	0	0
$811$	4.58328e33	1.61227	0.806133	−	0.591734i	$-0.201557\pi$
$811$	4.58328e33	1.61227	0.806133	+	0.591734i	$0.201557\pi$
$812$	0	0
$813$	−3.77624e32	−0.129128
$814$	0	0
$815$	1.31046e33	0.435623
$816$	0	0
$817$	7.16578e32	0.231585
$818$	0	0
$819$	1.17152e33	0.368117
$820$	0	0
$821$	1.22466e33	0.374171	0.187085	−	0.982344i	$-0.440096\pi$
$821$	1.22466e33	0.374171	0.187085	+	0.982344i	$0.440096\pi$
$822$	0	0
$823$	6.40647e33	1.90336	0.951680	−	0.307092i	$-0.0993560\pi$
$823$	6.40647e33	1.90336	0.951680	+	0.307092i	$0.0993560\pi$
$824$	0	0
$825$	−7.02217e32	−0.202885
$826$	0	0
$827$	−5.76179e33	−1.61899	−0.809495	−	0.587127i	$-0.800259\pi$
$827$	−5.76179e33	−1.61899	−0.809495	+	0.587127i	$0.800259\pi$
$828$	0	0
$829$	1.18586e33	0.324083	0.162041	−	0.986784i	$-0.448192\pi$
$829$	1.18586e33	0.324083	0.162041	+	0.986784i	$0.448192\pi$
$830$	0	0
$831$	8.19121e32	0.217739
$832$	0	0
$833$	1.52987e33	0.395582
$834$	0	0
$835$	−1.76918e33	−0.445017
$836$	0	0
$837$	8.39650e32	0.205473
$838$	0	0
$839$	−5.48917e33	−1.30690	−0.653452	−	0.756968i	$-0.726680\pi$
$839$	−5.48917e33	−1.30690	−0.653452	+	0.756968i	$0.726680\pi$
$840$	0	0
$841$	7.89472e33	1.82887
$842$	0	0
$843$	1.59127e33	0.358695
$844$	0	0
$845$	−1.60588e33	−0.352258
$846$	0	0
$847$	−6.50431e33	−1.38849
$848$	0	0
$849$	2.34593e33	0.487391
$850$	0	0
$851$	−6.93839e33	−1.40304
$852$	0	0
$853$	−5.26518e33	−1.03633	−0.518166	−	0.855280i	$-0.673385\pi$
$853$	−5.26518e33	−1.03633	−0.518166	+	0.855280i	$0.673385\pi$
$854$	0	0
$855$	−6.53176e33	−1.25147
$856$	0	0
$857$	4.17673e33	0.779035	0.389518	−	0.921019i	$-0.372642\pi$
$857$	4.17673e33	0.779035	0.389518	+	0.921019i	$0.372642\pi$
$858$	0	0
$859$	5.26384e33	0.955831	0.477916	−	0.878406i	$-0.341392\pi$
$859$	5.26384e33	0.955831	0.477916	+	0.878406i	$0.341392\pi$
$860$	0	0
$861$	−5.16621e32	−0.0913347
$862$	0	0
$863$	1.00762e34	1.73450	0.867249	−	0.497874i	$-0.165886\pi$
$863$	1.00762e34	1.73450	0.867249	+	0.497874i	$0.165886\pi$
$864$	0	0
$865$	−8.58704e32	−0.143932
$866$	0	0
$867$	1.23913e33	0.202255
$868$	0	0
$869$	−5.29210e33	−0.841202
$870$	0	0
$871$	6.63896e33	1.02776
$872$	0	0
$873$	2.97672e33	0.448821
$874$	0	0
$875$	3.99500e33	0.586711
$876$	0	0
$877$	−1.02444e33	−0.146551	−0.0732757	−	0.997312i	$-0.523345\pi$
$877$	−1.02444e33	−0.146551	−0.0732757	+	0.997312i	$0.523345\pi$
$878$	0	0
$879$	7.38633e32	0.102933
$880$	0	0
$881$	−5.47705e33	−0.743574	−0.371787	−	0.928318i	$-0.621255\pi$
$881$	−5.47705e33	−0.743574	−0.371787	+	0.928318i	$0.621255\pi$
$882$	0	0
$883$	9.11500e33	1.20561	0.602807	−	0.797887i	$-0.294049\pi$
$883$	9.11500e33	1.20561	0.602807	+	0.797887i	$0.294049\pi$
$884$	0	0
$885$	1.89761e33	0.244545
$886$	0	0
$887$	−8.55624e33	−1.07439	−0.537194	−	0.843458i	$-0.680516\pi$
$887$	−8.55624e33	−1.07439	−0.537194	+	0.843458i	$0.680516\pi$
$888$	0	0
$889$	2.40562e33	0.294345
$890$	0	0
$891$	−1.18714e34	−1.41549
$892$	0	0
$893$	2.45429e34	2.85190
$894$	0	0
$895$	4.09247e33	0.463469
$896$	0	0
$897$	−2.69392e33	−0.297353
$898$	0	0
$899$	5.71563e33	0.614933
$900$	0	0
$901$	−3.41370e32	−0.0358006
$902$	0	0
$903$	2.08703e32	0.0213363
$904$	0	0
$905$	1.01278e34	1.00938
$906$	0	0
$907$	−1.31084e34	−1.27369	−0.636847	−	0.770991i	$-0.719762\pi$
$907$	−1.31084e34	−1.27369	−0.636847	+	0.770991i	$0.719762\pi$
$908$	0	0
$909$	1.53572e34	1.45488
$910$	0	0
$911$	−6.07445e33	−0.561107	−0.280553	−	0.959838i	$-0.590518\pi$
$911$	−6.07445e33	−0.561107	−0.280553	+	0.959838i	$0.590518\pi$
$912$	0	0
$913$	5.11064e33	0.460321
$914$	0	0
$915$	1.45970e33	0.128210
$916$	0	0
$917$	1.04669e34	0.896541
$918$	0	0
$919$	−1.07738e34	−0.899998	−0.449999	−	0.893029i	$-0.648576\pi$
$919$	−1.07738e34	−0.899998	−0.449999	+	0.893029i	$0.648576\pi$
$920$	0	0
$921$	8.34946e32	0.0680257
$922$	0	0
$923$	1.75132e33	0.139170
$924$	0	0
$925$	4.87899e33	0.378181
$926$	0	0
$927$	1.27328e34	0.962736
$928$	0	0
$929$	−1.19991e34	−0.885053	−0.442526	−	0.896756i	$-0.645918\pi$
$929$	−1.19991e34	−0.885053	−0.442526	+	0.896756i	$0.645918\pi$
$930$	0	0
$931$	−1.69412e34	−1.21906
$932$	0	0
$933$	−7.63001e33	−0.535656
$934$	0	0
$935$	1.22664e34	0.840201
$936$	0	0
$937$	1.91467e34	1.27964	0.639821	−	0.768524i	$-0.279009\pi$
$937$	1.91467e34	1.27964	0.639821	+	0.768524i	$0.279009\pi$
$938$	0	0
$939$	5.06151e33	0.330085
$940$	0	0
$941$	3.00075e34	1.90963	0.954814	−	0.297205i	$-0.0960545\pi$
$941$	3.00075e34	1.90963	0.954814	+	0.297205i	$0.0960545\pi$
$942$	0	0
$943$	−1.25878e34	−0.781744
$944$	0	0
$945$	−3.98428e33	−0.241482
$946$	0	0
$947$	5.89213e33	0.348536	0.174268	−	0.984698i	$-0.444244\pi$
$947$	5.89213e33	0.348536	0.174268	+	0.984698i	$0.444244\pi$
$948$	0	0
$949$	−6.12463e33	−0.353605
$950$	0	0
$951$	−5.42435e33	−0.305684
$952$	0	0
$953$	−2.16859e34	−1.19292	−0.596458	−	0.802644i	$-0.703426\pi$
$953$	−2.16859e34	−1.19292	−0.596458	+	0.802644i	$0.703426\pi$
$954$	0	0
$955$	1.47364e34	0.791319
$956$	0	0
$957$	1.78125e34	0.933762
$958$	0	0
$959$	−8.48018e33	−0.434001
$960$	0	0
$961$	−1.73381e34	−0.866327
$962$	0	0
$963$	2.98626e34	1.45688
$964$	0	0
$965$	−2.05679e34	−0.979774
$966$	0	0
$967$	3.56101e34	1.65641	0.828207	−	0.560422i	$-0.189361\pi$
$967$	3.56101e34	1.65641	0.828207	+	0.560422i	$0.189361\pi$
$968$	0	0
$969$	6.20134e33	0.281684
$970$	0	0
$971$	1.10041e34	0.488127	0.244063	−	0.969759i	$-0.421520\pi$
$971$	1.10041e34	0.488127	0.244063	+	0.969759i	$0.421520\pi$
$972$	0	0
$973$	−6.91578e33	−0.299601
$974$	0	0
$975$	1.89433e33	0.0801500
$976$	0	0
$977$	−2.41096e34	−0.996330	−0.498165	−	0.867082i	$-0.665993\pi$
$977$	−2.41096e34	−0.996330	−0.498165	+	0.867082i	$0.665993\pi$
$978$	0	0
$979$	−5.15590e34	−2.08115
$980$	0	0
$981$	9.70642e33	0.382706
$982$	0	0
$983$	2.47320e34	0.952562	0.476281	−	0.879293i	$-0.341984\pi$
$983$	2.47320e34	0.952562	0.476281	+	0.879293i	$0.341984\pi$
$984$	0	0
$985$	2.23365e34	0.840425
$986$	0	0
$987$	7.14810e33	0.262751
$988$	0	0
$989$	5.08517e33	0.182620
$990$	0	0
$991$	−4.39629e34	−1.54255	−0.771276	−	0.636501i	$-0.780381\pi$
$991$	−4.39629e34	−1.54255	−0.771276	+	0.636501i	$0.780381\pi$
$992$	0	0
$993$	2.02903e33	0.0695619
$994$	0	0
$995$	−1.54140e34	−0.516357
$996$	0	0
$997$	−5.89142e34	−1.92853	−0.964264	−	0.264943i	$-0.914647\pi$
$997$	−5.89142e34	−1.92853	−0.964264	+	0.264943i	$0.914647\pi$
$998$	0	0
$999$	−1.81809e34	−0.581585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.n.1.3		6
4.3	odd	2	inner	64.24.a.n.1.4		6
8.3	odd	2		32.24.a.d.1.3	✓	6
8.5	even	2		32.24.a.d.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.d.1.3	✓	6	8.3	odd	2
32.24.a.d.1.4	yes	6	8.5	even	2
64.24.a.n.1.3		6	1.1	even	1	trivial
64.24.a.n.1.4		6	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q-90103.3 q^{3} +8.69741e7 q^{5} -2.82191e9 q^{7} -8.60246e10 q^{9} +O(q^{10})\)	\(q-90103.3 q^{3} +8.69741e7 q^{5} -2.82191e9 q^{7} -8.60246e10 q^{9} -1.78896e12 q^{11} +4.82597e12 q^{13} -7.83665e12 q^{15} -7.88365e13 q^{17} +8.73007e14 q^{19} +2.54263e14 q^{21} +6.19527e15 q^{23} -4.35643e15 q^{25} +1.62337e16 q^{27} +1.10505e17 q^{29} +5.17226e16 q^{31} +1.61191e17 q^{33} -2.45433e17 q^{35} -1.11995e18 q^{37} -4.34836e17 q^{39} -2.03184e18 q^{41} +8.20815e17 q^{43} -7.48191e18 q^{45} +2.81130e19 q^{47} -1.94056e19 q^{49} +7.10342e18 q^{51} +4.33010e18 q^{53} -1.55593e20 q^{55} -7.86608e19 q^{57} -2.42145e20 q^{59} -1.86266e20 q^{61} +2.42753e20 q^{63} +4.19735e20 q^{65} +1.37567e21 q^{67} -5.58214e20 q^{69} +3.62895e20 q^{71} -1.26910e21 q^{73} +3.92529e20 q^{75} +5.04827e21 q^{77} +2.95820e21 q^{79} +6.63592e21 q^{81} -2.85677e21 q^{83} -6.85673e21 q^{85} -9.95690e21 q^{87} +2.88207e22 q^{89} -1.36184e22 q^{91} -4.66038e21 q^{93} +7.59290e22 q^{95} -3.46031e22 q^{97} +1.53894e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) \) 6 * q - 163121700 * q^5 + 14120212542 * q^9	\( 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) \) 6 * q - 163121700 * q^5 + 14120212542 * q^9 - 1101295489524 * q^13 + 56435517243468 * q^17 - 1447661324841984 * q^21 + 17512304669291850 * q^25 - 87161858953770708 * q^29 + 48462097564970496 * q^33 + 1542766250573484636 * q^37 - 865941451897399716 * q^41 + 9286254082406210700 * q^45 + 52551978665245784886 * q^49 + 10735070383373044668 * q^53 + 155104978099636154880 * q^57 + 432169207960503656364 * q^61 - 316526040451899107400 * q^65 - 695880182828346153984 * q^69 + 1758226166115151758204 * q^73 - 11926085657398731979776 * q^77 + 4010608843245537630678 * q^81 - 20251531460297137542600 * q^85 + 20087131519604301542940 * q^89 - 67310470168488243892224 * q^93 + 254665147475934678043116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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