LMFDB - Embedded newform 64.24.a.l.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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1530688x^{2} + 602279195x - 31243663925 $ x^4 - x^3 - 1530688x^2 + 602279195x - 31243663925
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{7}\cdot 5 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$972.231$ 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)

$f(q)$	$=$	$q+262305. q^{3} -7.64109e7 q^{5} +4.80108e9 q^{7} -2.53392e10 q^{9} +O(q^{10})$	$q+262305. q^{3} -7.64109e7 q^{5} +4.80108e9 q^{7} -2.53392e10 q^{9} +1.05546e11 q^{11} -1.70127e12 q^{13} -2.00430e13 q^{15} +7.28382e12 q^{17} +4.59662e14 q^{19} +1.25935e15 q^{21} -1.23699e15 q^{23} -6.08230e15 q^{25} -3.13408e16 q^{27} +4.27534e16 q^{29} +1.80035e16 q^{31} +2.76852e16 q^{33} -3.66855e17 q^{35} -1.83193e17 q^{37} -4.46251e17 q^{39} +2.74051e18 q^{41} -1.08229e19 q^{43} +1.93619e18 q^{45} +1.91565e19 q^{47} -4.31838e18 q^{49} +1.91058e18 q^{51} +7.00380e19 q^{53} -8.06485e18 q^{55} +1.20572e20 q^{57} +1.32590e20 q^{59} -3.29879e18 q^{61} -1.21656e20 q^{63} +1.29995e20 q^{65} -1.55017e20 q^{67} -3.24468e20 q^{69} -3.06650e21 q^{71} -1.76877e21 q^{73} -1.59542e21 q^{75} +5.06734e20 q^{77} -2.27214e21 q^{79} -5.83535e21 q^{81} -1.79926e22 q^{83} -5.56563e20 q^{85} +1.12144e22 q^{87} +5.43740e21 q^{89} -8.16792e21 q^{91} +4.72242e21 q^{93} -3.51232e22 q^{95} -3.68694e22 q^{97} -2.67445e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 19990040 q^{5} + 284960925396 q^{9}+O(q^{10}) $ 4 * q - 19990040 * q^5 + 284960925396 * q^9	$ 4 q - 19990040 q^{5} + 284960925396 q^{9} + 21132585986568 q^{13} - 339546388528760 q^{17} - 57\!\cdots\!04 q^{21}+ \cdots + 79\!\cdots\!56 q^{97}+O(q^{100}) $ 4 * q - 19990040 * q^5 + 284960925396 * q^9 + 21132585986568 * q^13 - 339546388528760 * q^17 - 5772344793698304 * q^21 - 27184332341841700 * q^25 - 11713498653208888 * q^29 - 127188060952651776 * q^33 - 1527028203255925528 * q^37 - 175841616529668376 * q^41 + 26164165807508085000 * q^45 + 67830117546567696932 * q^49 - 75693842547307242712 * q^53 - 573763276821861734400 * q^57 - 232274843679744391736 * q^61 + 1889512171511171217360 * q^65 + 2311912054144728354816 * q^69 - 3115180773570839678040 * q^73 + 3902426738355338366976 * q^77 - 4667833569712391401500 * q^81 - 24631918485707096544560 * q^85 - 78996853740551396469400 * q^89 + 145181262233476826578944 * q^93 + 79943563283557612080456 * 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
$3$	262305.	0.854894	0.427447	−	0.904040i	$-0.359413\pi$
$3$	262305.	0.854894	0.427447	+	0.904040i	$0.359413\pi$
$4$	0	0
$5$	−7.64109e7	−0.699842	−0.349921	−	0.936779i	$-0.613792\pi$
$5$	−7.64109e7	−0.699842	−0.349921	+	0.936779i	$0.613792\pi$
$6$	0	0
$7$	4.80108e9	0.917723	0.458861	−	0.888508i	$-0.348257\pi$
$7$	4.80108e9	0.917723	0.458861	+	0.888508i	$0.348257\pi$
$8$	0	0
$9$	−2.53392e10	−0.269156
$10$	0	0
$11$	1.05546e11	0.111539	0.0557693	−	0.998444i	$-0.482239\pi$
$11$	1.05546e11	0.111539	0.0557693	+	0.998444i	$0.482239\pi$
$12$	0	0
$13$	−1.70127e12	−0.263284	−0.131642	−	0.991297i	$-0.542025\pi$
$13$	−1.70127e12	−0.263284	−0.131642	+	0.991297i	$0.542025\pi$
$14$	0	0
$15$	−2.00430e13	−0.598291
$16$	0	0
$17$	7.28382e12	0.0515462	0.0257731	−	0.999668i	$-0.491795\pi$
$17$	7.28382e12	0.0515462	0.0257731	+	0.999668i	$0.491795\pi$
$18$	0	0
$19$	4.59662e14	0.905257	0.452629	−	0.891699i	$-0.350486\pi$
$19$	4.59662e14	0.905257	0.452629	+	0.891699i	$0.350486\pi$
$20$	0	0
$21$	1.25935e15	0.784556
$22$	0	0
$23$	−1.23699e15	−0.270704	−0.135352	−	0.990798i	$-0.543217\pi$
$23$	−1.23699e15	−0.270704	−0.135352	+	0.990798i	$0.543217\pi$
$24$	0	0
$25$	−6.08230e15	−0.510221
$26$	0	0
$27$	−3.13408e16	−1.08499
$28$	0	0
$29$	4.27534e16	0.650719	0.325360	−	0.945590i	$-0.394515\pi$
$29$	4.27534e16	0.650719	0.325360	+	0.945590i	$0.394515\pi$
$30$	0	0
$31$	1.80035e16	0.127262	0.0636310	−	0.997973i	$-0.479732\pi$
$31$	1.80035e16	0.127262	0.0636310	+	0.997973i	$0.479732\pi$
$32$	0	0
$33$	2.76852e16	0.0953537
$34$	0	0
$35$	−3.66855e17	−0.642261
$36$	0	0
$37$	−1.83193e17	−0.169273	−0.0846367	−	0.996412i	$-0.526973\pi$
$37$	−1.83193e17	−0.169273	−0.0846367	+	0.996412i	$0.526973\pi$
$38$	0	0
$39$	−4.46251e17	−0.225080
$40$	0	0
$41$	2.74051e18	0.777708	0.388854	−	0.921299i	$-0.372871\pi$
$41$	2.74051e18	0.777708	0.388854	+	0.921299i	$0.372871\pi$
$42$	0	0
$43$	−1.08229e19	−1.77605	−0.888025	−	0.459795i	$-0.847923\pi$
$43$	−1.08229e19	−1.77605	−0.888025	+	0.459795i	$0.847923\pi$
$44$	0	0
$45$	1.93619e18	0.188367
$46$	0	0
$47$	1.91565e19	1.13029	0.565145	−	0.824991i	$-0.308820\pi$
$47$	1.91565e19	1.13029	0.565145	+	0.824991i	$0.308820\pi$
$48$	0	0
$49$	−4.31838e18	−0.157785
$50$	0	0
$51$	1.91058e18	0.0440666
$52$	0	0
$53$	7.00380e19	1.03791	0.518957	−	0.854800i	$-0.326320\pi$
$53$	7.00380e19	1.03791	0.518957	+	0.854800i	$0.326320\pi$
$54$	0	0
$55$	−8.06485e18	−0.0780595
$56$	0	0
$57$	1.20572e20	0.773899
$58$	0	0
$59$	1.32590e20	0.572417	0.286208	−	0.958167i	$-0.407605\pi$
$59$	1.32590e20	0.572417	0.286208	+	0.958167i	$0.407605\pi$
$60$	0	0
$61$	−3.29879e18	−0.00970647	−0.00485323	−	0.999988i	$-0.501545\pi$
$61$	−3.29879e18	−0.00970647	−0.00485323	+	0.999988i	$0.501545\pi$
$62$	0	0
$63$	−1.21656e20	−0.247011
$64$	0	0
$65$	1.29995e20	0.184257
$66$	0	0
$67$	−1.55017e20	−0.155067	−0.0775337	−	0.996990i	$-0.524705\pi$
$67$	−1.55017e20	−0.155067	−0.0775337	+	0.996990i	$0.524705\pi$
$68$	0	0
$69$	−3.24468e20	−0.231423
$70$	0	0
$71$	−3.06650e21	−1.57461	−0.787304	−	0.616565i	$-0.788524\pi$
$71$	−3.06650e21	−1.57461	−0.787304	+	0.616565i	$0.788524\pi$
$72$	0	0
$73$	−1.76877e21	−0.659871	−0.329936	−	0.944003i	$-0.607027\pi$
$73$	−1.76877e21	−0.659871	−0.329936	+	0.944003i	$0.607027\pi$
$74$	0	0
$75$	−1.59542e21	−0.436185
$76$	0	0
$77$	5.06734e20	0.102362
$78$	0	0
$79$	−2.27214e21	−0.341762	−0.170881	−	0.985292i	$-0.554661\pi$
$79$	−2.27214e21	−0.341762	−0.170881	+	0.985292i	$0.554661\pi$
$80$	0	0
$81$	−5.83535e21	−0.658399
$82$	0	0
$83$	−1.79926e22	−1.53354	−0.766770	−	0.641922i	$-0.778137\pi$
$83$	−1.79926e22	−1.53354	−0.766770	+	0.641922i	$0.778137\pi$
$84$	0	0
$85$	−5.56563e20	−0.0360742
$86$	0	0
$87$	1.12144e22	0.556296
$88$	0	0
$89$	5.43740e21	0.207686	0.103843	−	0.994594i	$-0.466886\pi$
$89$	5.43740e21	0.207686	0.103843	+	0.994594i	$0.466886\pi$
$90$	0	0
$91$	−8.16792e21	−0.241621
$92$	0	0
$93$	4.72242e21	0.108795
$94$	0	0
$95$	−3.51232e22	−0.633537
$96$	0	0
$97$	−3.68694e22	−0.523348	−0.261674	−	0.965156i	$-0.584275\pi$
$97$	−3.68694e22	−0.523348	−0.261674	+	0.965156i	$0.584275\pi$
$98$	0	0
$99$	−2.67445e21	−0.0300213
$100$	0	0
$101$	1.28969e23	1.15024	0.575122	−	0.818068i	$-0.304955\pi$
$101$	1.28969e23	1.15024	0.575122	+	0.818068i	$0.304955\pi$
$102$	0	0
$103$	−1.53923e23	−1.09566	−0.547830	−	0.836590i	$-0.684546\pi$
$103$	−1.53923e23	−1.09566	−0.547830	+	0.836590i	$0.684546\pi$
$104$	0	0
$105$	−9.62279e22	−0.549065
$106$	0	0
$107$	−7.47133e22	−0.343150	−0.171575	−	0.985171i	$-0.554886\pi$
$107$	−7.47133e22	−0.343150	−0.171575	+	0.985171i	$0.554886\pi$
$108$	0	0
$109$	3.46584e23	1.28648	0.643242	−	0.765663i	$-0.277589\pi$
$109$	3.46584e23	1.28648	0.643242	+	0.765663i	$0.277589\pi$
$110$	0	0
$111$	−4.80524e22	−0.144711
$112$	0	0
$113$	−2.46609e23	−0.604794	−0.302397	−	0.953182i	$-0.597787\pi$
$113$	−2.46609e23	−0.604794	−0.302397	+	0.953182i	$0.597787\pi$
$114$	0	0
$115$	9.45193e22	0.189450
$116$	0	0
$117$	4.31088e22	0.0708645
$118$	0	0
$119$	3.49702e22	0.0473051
$120$	0	0
$121$	−8.84290e23	−0.987559
$122$	0	0
$123$	7.18849e23	0.664858
$124$	0	0
$125$	1.37564e24	1.05692
$126$	0	0
$127$	2.37333e24	1.51920	0.759599	−	0.650391i	$-0.225395\pi$
$127$	2.37333e24	1.51920	0.759599	+	0.650391i	$0.225395\pi$
$128$	0	0
$129$	−2.83889e24	−1.51833
$130$	0	0
$131$	−1.31715e24	−0.590221	−0.295110	−	0.955463i	$-0.595356\pi$
$131$	−1.31715e24	−0.590221	−0.295110	+	0.955463i	$0.595356\pi$
$132$	0	0
$133$	2.20687e24	0.830775
$134$	0	0
$135$	2.39478e24	0.759325
$136$	0	0
$137$	−5.36712e24	−1.43699	−0.718496	−	0.695531i	$-0.755169\pi$
$137$	−5.36712e24	−1.43699	−0.718496	+	0.695531i	$0.755169\pi$
$138$	0	0
$139$	5.78955e24	1.31212	0.656062	−	0.754707i	$-0.272221\pi$
$139$	5.78955e24	1.31212	0.656062	+	0.754707i	$0.272221\pi$
$140$	0	0
$141$	5.02484e24	0.966279
$142$	0	0
$143$	−1.79562e23	−0.0293663
$144$	0	0
$145$	−3.26682e24	−0.455401
$146$	0	0
$147$	−1.13273e24	−0.134890
$148$	0	0
$149$	8.31636e24	0.847796	0.423898	−	0.905710i	$-0.360662\pi$
$149$	8.31636e24	0.847796	0.423898	+	0.905710i	$0.360662\pi$
$150$	0	0
$151$	−2.17654e25	−1.90341	−0.951703	−	0.307020i	$-0.900668\pi$
$151$	−2.17654e25	−1.90341	−0.951703	+	0.307020i	$0.900668\pi$
$152$	0	0
$153$	−1.84566e23	−0.0138740
$154$	0	0
$155$	−1.37567e24	−0.0890633
$156$	0	0
$157$	7.51451e24	0.419812	0.209906	−	0.977722i	$-0.432684\pi$
$157$	7.51451e24	0.419812	0.209906	+	0.977722i	$0.432684\pi$
$158$	0	0
$159$	1.83713e25	0.887306
$160$	0	0
$161$	−5.93887e24	−0.248431
$162$	0	0
$163$	−9.10634e24	−0.330511	−0.165256	−	0.986251i	$-0.552845\pi$
$163$	−9.10634e24	−0.330511	−0.165256	+	0.986251i	$0.552845\pi$
$164$	0	0
$165$	−2.11545e24	−0.0667326
$166$	0	0
$167$	5.77548e24	0.158617	0.0793083	−	0.996850i	$-0.474729\pi$
$167$	5.77548e24	0.158617	0.0793083	+	0.996850i	$0.474729\pi$
$168$	0	0
$169$	−3.88596e25	−0.930682
$170$	0	0
$171$	−1.16475e25	−0.243656
$172$	0	0
$173$	−1.11178e25	−0.203464	−0.101732	−	0.994812i	$-0.532438\pi$
$173$	−1.11178e25	−0.203464	−0.101732	+	0.994812i	$0.532438\pi$
$174$	0	0
$175$	−2.92016e25	−0.468241
$176$	0	0
$177$	3.47790e25	0.489356
$178$	0	0
$179$	−7.03549e25	−0.869930	−0.434965	−	0.900447i	$-0.643239\pi$
$179$	−7.03549e25	−0.869930	−0.434965	+	0.900447i	$0.643239\pi$
$180$	0	0
$181$	−1.54732e26	−1.68375	−0.841875	−	0.539673i	$-0.818548\pi$
$181$	−1.54732e26	−1.68375	−0.841875	+	0.539673i	$0.818548\pi$
$182$	0	0
$183$	−8.65288e23	−0.00829800
$184$	0	0
$185$	1.39979e25	0.118465
$186$	0	0
$187$	7.68777e23	0.00574939
$188$	0	0
$189$	−1.50470e26	−0.995724
$190$	0	0
$191$	−1.73010e25	−0.101435	−0.0507175	−	0.998713i	$-0.516151\pi$
$191$	−1.73010e25	−0.101435	−0.0507175	+	0.998713i	$0.516151\pi$
$192$	0	0
$193$	2.16906e26	1.12814	0.564070	−	0.825727i	$-0.309235\pi$
$193$	2.16906e26	1.12814	0.564070	+	0.825727i	$0.309235\pi$
$194$	0	0
$195$	3.40984e25	0.157520
$196$	0	0
$197$	−2.06934e26	−0.850100	−0.425050	−	0.905170i	$-0.639743\pi$
$197$	−2.06934e26	−0.850100	−0.425050	+	0.905170i	$0.639743\pi$
$198$	0	0
$199$	−1.76115e26	−0.644150	−0.322075	−	0.946714i	$-0.604380\pi$
$199$	−1.76115e26	−0.644150	−0.322075	+	0.946714i	$0.604380\pi$
$200$	0	0
$201$	−4.06619e25	−0.132566
$202$	0	0
$203$	2.05262e26	0.597180
$204$	0	0
$205$	−2.09405e26	−0.544273
$206$	0	0
$207$	3.13443e25	0.0728617
$208$	0	0
$209$	4.85154e25	0.100971
$210$	0	0
$211$	6.15155e26	1.14746	0.573728	−	0.819046i	$-0.305496\pi$
$211$	6.15155e26	1.14746	0.573728	+	0.819046i	$0.305496\pi$
$212$	0	0
$213$	−8.04359e26	−1.34612
$214$	0	0
$215$	8.26985e26	1.24296
$216$	0	0
$217$	8.64365e25	0.116791
$218$	0	0
$219$	−4.63958e26	−0.564120
$220$	0	0
$221$	−1.23917e25	−0.0135713
$222$	0	0
$223$	1.00353e27	0.990885	0.495443	−	0.868641i	$-0.335006\pi$
$223$	1.00353e27	0.990885	0.495443	+	0.868641i	$0.335006\pi$
$224$	0	0
$225$	1.54121e26	0.137329
$226$	0	0
$227$	7.60686e25	0.0612221	0.0306110	−	0.999531i	$-0.490255\pi$
$227$	7.60686e25	0.0612221	0.0306110	+	0.999531i	$0.490255\pi$
$228$	0	0
$229$	−1.86047e27	−1.35368	−0.676838	−	0.736132i	$-0.736650\pi$
$229$	−1.86047e27	−1.35368	−0.676838	+	0.736132i	$0.736650\pi$
$230$	0	0
$231$	1.32919e26	0.0875083
$232$	0	0
$233$	−3.18797e27	−1.90073	−0.950365	−	0.311137i	$-0.899290\pi$
$233$	−3.18797e27	−1.90073	−0.950365	+	0.311137i	$0.899290\pi$
$234$	0	0
$235$	−1.46376e27	−0.791025
$236$	0	0
$237$	−5.95993e26	−0.292170
$238$	0	0
$239$	−3.47537e27	−1.54677	−0.773383	−	0.633939i	$-0.781437\pi$
$239$	−3.47537e27	−1.54677	−0.773383	+	0.633939i	$0.781437\pi$
$240$	0	0
$241$	−3.47395e27	−1.40484	−0.702422	−	0.711761i	$-0.747898\pi$
$241$	−3.47395e27	−1.40484	−0.702422	+	0.711761i	$0.747898\pi$
$242$	0	0
$243$	1.41989e27	0.522133
$244$	0	0
$245$	3.29971e26	0.110425
$246$	0	0
$247$	−7.82008e26	−0.238340
$248$	0	0
$249$	−4.71954e27	−1.31101
$250$	0	0
$251$	−1.61461e27	−0.409091	−0.204545	−	0.978857i	$-0.565572\pi$
$251$	−1.61461e27	−0.409091	−0.204545	+	0.978857i	$0.565572\pi$
$252$	0	0
$253$	−1.30559e26	−0.0301940
$254$	0	0
$255$	−1.45989e26	−0.0308396
$256$	0	0
$257$	−8.61701e27	−1.66389	−0.831947	−	0.554854i	$-0.812774\pi$
$257$	−8.61701e27	−1.66389	−0.831947	+	0.554854i	$0.812774\pi$
$258$	0	0
$259$	−8.79524e26	−0.155346
$260$	0	0
$261$	−1.08334e27	−0.175145
$262$	0	0
$263$	−5.54498e27	−0.821125	−0.410562	−	0.911833i	$-0.634668\pi$
$263$	−5.54498e27	−0.821125	−0.410562	+	0.911833i	$0.634668\pi$
$264$	0	0
$265$	−5.35167e27	−0.726376
$266$	0	0
$267$	1.42626e27	0.177549
$268$	0	0
$269$	−8.13459e26	−0.0929361	−0.0464680	−	0.998920i	$-0.514797\pi$
$269$	−8.13459e26	−0.0929361	−0.0464680	+	0.998920i	$0.514797\pi$
$270$	0	0
$271$	8.30363e27	0.871207	0.435603	−	0.900139i	$-0.356535\pi$
$271$	8.30363e27	0.871207	0.435603	+	0.900139i	$0.356535\pi$
$272$	0	0
$273$	−2.14249e27	−0.206561
$274$	0	0
$275$	−6.41962e26	−0.0569093
$276$	0	0
$277$	−2.72637e27	−0.222365	−0.111183	−	0.993800i	$-0.535464\pi$
$277$	−2.72637e27	−0.222365	−0.111183	+	0.993800i	$0.535464\pi$
$278$	0	0
$279$	−4.56196e26	−0.0342534
$280$	0	0
$281$	−1.94166e28	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$281$	−1.94166e28	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$282$	0	0
$283$	−3.28398e27	−0.209342	−0.104671	−	0.994507i	$-0.533379\pi$
$283$	−3.28398e27	−0.209342	−0.104671	+	0.994507i	$0.533379\pi$
$284$	0	0
$285$	−9.21298e27	−0.541607
$286$	0	0
$287$	1.31574e28	0.713720
$288$	0	0
$289$	−1.99145e28	−0.997343
$290$	0	0
$291$	−9.67102e27	−0.447407
$292$	0	0
$293$	−1.66990e28	−0.714026	−0.357013	−	0.934099i	$-0.616205\pi$
$293$	−1.66990e28	−0.714026	−0.357013	+	0.934099i	$0.616205\pi$
$294$	0	0
$295$	−1.01313e28	−0.400601
$296$	0	0
$297$	−3.30790e27	−0.121019
$298$	0	0
$299$	2.10445e27	0.0712720
$300$	0	0
$301$	−5.19615e28	−1.62992
$302$	0	0
$303$	3.38293e28	0.983337
$304$	0	0
$305$	2.52063e26	0.00679300
$306$	0	0
$307$	6.61733e28	1.65421	0.827107	−	0.562045i	$-0.189985\pi$
$307$	6.61733e28	1.65421	0.827107	+	0.562045i	$0.189985\pi$
$308$	0	0
$309$	−4.03748e28	−0.936674
$310$	0	0
$311$	−6.13772e28	−1.32209	−0.661047	−	0.750345i	$-0.729888\pi$
$311$	−6.13772e28	−1.32209	−0.661047	+	0.750345i	$0.729888\pi$
$312$	0	0
$313$	4.60758e28	0.921961	0.460981	−	0.887410i	$-0.347498\pi$
$313$	4.60758e28	0.921961	0.460981	+	0.887410i	$0.347498\pi$
$314$	0	0
$315$	9.29582e27	0.172869
$316$	0	0
$317$	−8.55184e28	−1.47869	−0.739346	−	0.673325i	$-0.764865\pi$
$317$	−8.55184e28	−1.47869	−0.739346	+	0.673325i	$0.764865\pi$
$318$	0	0
$319$	4.51244e27	0.0725803
$320$	0	0
$321$	−1.95977e28	−0.293357
$322$	0	0
$323$	3.34809e27	0.0466626
$324$	0	0
$325$	1.03476e28	0.134333
$326$	0	0
$327$	9.09108e28	1.09981
$328$	0	0
$329$	9.19717e28	1.03729
$330$	0	0
$331$	−3.02049e28	−0.317728	−0.158864	−	0.987300i	$-0.550783\pi$
$331$	−3.02049e28	−0.317728	−0.158864	+	0.987300i	$0.550783\pi$
$332$	0	0
$333$	4.64197e27	0.0455610
$334$	0	0
$335$	1.18450e28	0.108523
$336$	0	0
$337$	3.05026e28	0.260971	0.130486	−	0.991450i	$-0.458346\pi$
$337$	3.05026e28	0.260971	0.130486	+	0.991450i	$0.458346\pi$
$338$	0	0
$339$	−6.46869e28	−0.517034
$340$	0	0
$341$	1.90020e27	0.0141946
$342$	0	0
$343$	−1.52132e29	−1.06253
$344$	0	0
$345$	2.47929e28	0.161960
$346$	0	0
$347$	−3.11656e29	−1.90497	−0.952483	−	0.304592i	$-0.901480\pi$
$347$	−3.11656e29	−1.90497	−0.952483	+	0.304592i	$0.901480\pi$
$348$	0	0
$349$	8.37571e28	0.479213	0.239607	−	0.970870i	$-0.422982\pi$
$349$	8.37571e28	0.479213	0.239607	+	0.970870i	$0.422982\pi$
$350$	0	0
$351$	5.33192e28	0.285661
$352$	0	0
$353$	8.25912e28	0.414500	0.207250	−	0.978288i	$-0.433549\pi$
$353$	8.25912e28	0.414500	0.207250	+	0.978288i	$0.433549\pi$
$354$	0	0
$355$	2.34314e29	1.10198
$356$	0	0
$357$	9.17287e27	0.0404409
$358$	0	0
$359$	−3.54134e29	−1.46414	−0.732068	−	0.681231i	$-0.761445\pi$
$359$	−3.54134e29	−1.46414	−0.732068	+	0.681231i	$0.761445\pi$
$360$	0	0
$361$	−4.65407e28	−0.180510
$362$	0	0
$363$	−2.31954e29	−0.844258
$364$	0	0
$365$	1.35154e29	0.461806
$366$	0	0
$367$	1.52670e29	0.489886	0.244943	−	0.969538i	$-0.421231\pi$
$367$	1.52670e29	0.489886	0.244943	+	0.969538i	$0.421231\pi$
$368$	0	0
$369$	−6.94423e28	−0.209325
$370$	0	0
$371$	3.36258e29	0.952517
$372$	0	0
$373$	5.44896e29	1.45098	0.725491	−	0.688231i	$-0.241613\pi$
$373$	5.44896e29	1.45098	0.725491	+	0.688231i	$0.241613\pi$
$374$	0	0
$375$	3.60838e29	0.903551
$376$	0	0
$377$	−7.27350e28	−0.171324
$378$	0	0
$379$	2.44910e29	0.542819	0.271409	−	0.962464i	$-0.412510\pi$
$379$	2.44910e29	0.542819	0.271409	+	0.962464i	$0.412510\pi$
$380$	0	0
$381$	6.22535e29	1.29875
$382$	0	0
$383$	−2.28310e29	−0.448476	−0.224238	−	0.974534i	$-0.571989\pi$
$383$	−2.28310e29	−0.448476	−0.224238	+	0.974534i	$0.571989\pi$
$384$	0	0
$385$	−3.87200e28	−0.0716369
$386$	0	0
$387$	2.74243e29	0.478035
$388$	0	0
$389$	4.32727e29	0.710876	0.355438	−	0.934700i	$-0.384332\pi$
$389$	4.32727e29	0.710876	0.355438	+	0.934700i	$0.384332\pi$
$390$	0	0
$391$	−9.00999e27	−0.0139538
$392$	0	0
$393$	−3.45494e29	−0.504576
$394$	0	0
$395$	1.73616e29	0.239179
$396$	0	0
$397$	9.56104e29	1.24284	0.621419	−	0.783479i	$-0.286557\pi$
$397$	9.56104e29	1.24284	0.621419	+	0.783479i	$0.286557\pi$
$398$	0	0
$399$	5.78874e29	0.710225
$400$	0	0
$401$	5.41841e29	0.627642	0.313821	−	0.949482i	$-0.398391\pi$
$401$	5.41841e29	0.627642	0.313821	+	0.949482i	$0.398391\pi$
$402$	0	0
$403$	−3.06289e28	−0.0335060
$404$	0	0
$405$	4.45884e29	0.460775
$406$	0	0
$407$	−1.93353e28	−0.0188805
$408$	0	0
$409$	1.20749e30	1.11446	0.557231	−	0.830357i	$-0.311864\pi$
$409$	1.20749e30	1.11446	0.557231	+	0.830357i	$0.311864\pi$
$410$	0	0
$411$	−1.40782e30	−1.22848
$412$	0	0
$413$	6.36575e29	0.525320
$414$	0	0
$415$	1.37483e30	1.07324
$416$	0	0
$417$	1.51863e30	1.12173
$418$	0	0
$419$	5.45599e29	0.381428	0.190714	−	0.981646i	$-0.438920\pi$
$419$	5.45599e29	0.381428	0.190714	+	0.981646i	$0.438920\pi$
$420$	0	0
$421$	3.59144e29	0.237698	0.118849	−	0.992912i	$-0.462080\pi$
$421$	3.59144e29	0.237698	0.118849	+	0.992912i	$0.462080\pi$
$422$	0	0
$423$	−4.85410e29	−0.304225
$424$	0	0
$425$	−4.43024e28	−0.0262999
$426$	0	0
$427$	−1.58377e28	−0.00890784
$428$	0	0
$429$	−4.71000e28	−0.0251051
$430$	0	0
$431$	−1.05845e30	−0.534790	−0.267395	−	0.963587i	$-0.586163\pi$
$431$	−1.05845e30	−0.534790	−0.267395	+	0.963587i	$0.586163\pi$
$432$	0	0
$433$	2.00974e30	0.962783	0.481391	−	0.876506i	$-0.340132\pi$
$433$	2.00974e30	0.962783	0.481391	+	0.876506i	$0.340132\pi$
$434$	0	0
$435$	−8.56905e29	−0.389319
$436$	0	0
$437$	−5.68596e29	−0.245057
$438$	0	0
$439$	−3.82164e30	−1.56281	−0.781407	−	0.624021i	$-0.785498\pi$
$439$	−3.82164e30	−1.56281	−0.781407	+	0.624021i	$0.785498\pi$
$440$	0	0
$441$	1.09424e29	0.0424688
$442$	0	0
$443$	2.71272e30	0.999451	0.499725	−	0.866184i	$-0.333434\pi$
$443$	2.71272e30	0.999451	0.499725	+	0.866184i	$0.333434\pi$
$444$	0	0
$445$	−4.15477e29	−0.145347
$446$	0	0
$447$	2.18142e30	0.724776
$448$	0	0
$449$	5.81492e28	0.0183532	0.00917658	−	0.999958i	$-0.497079\pi$
$449$	5.81492e28	0.0183532	0.00917658	+	0.999958i	$0.497079\pi$
$450$	0	0
$451$	2.89249e29	0.0867445
$452$	0	0
$453$	−5.70917e30	−1.62721
$454$	0	0
$455$	6.24118e29	0.169097
$456$	0	0
$457$	4.38331e30	1.12919	0.564594	−	0.825369i	$-0.309033\pi$
$457$	4.38331e30	1.12919	0.564594	+	0.825369i	$0.309033\pi$
$458$	0	0
$459$	−2.28281e29	−0.0559273
$460$	0	0
$461$	−4.02809e30	−0.938725	−0.469362	−	0.883006i	$-0.655516\pi$
$461$	−4.02809e30	−0.938725	−0.469362	+	0.883006i	$0.655516\pi$
$462$	0	0
$463$	−2.95393e30	−0.654967	−0.327483	−	0.944857i	$-0.606201\pi$
$463$	−2.95393e30	−0.654967	−0.327483	+	0.944857i	$0.606201\pi$
$464$	0	0
$465$	−3.60844e29	−0.0761397
$466$	0	0
$467$	−6.56729e30	−1.31899	−0.659496	−	0.751708i	$-0.729230\pi$
$467$	−6.56729e30	−1.31899	−0.659496	+	0.751708i	$0.729230\pi$
$468$	0	0
$469$	−7.44251e29	−0.142309
$470$	0	0
$471$	1.97109e30	0.358895
$472$	0	0
$473$	−1.14231e30	−0.198098
$474$	0	0
$475$	−2.79580e30	−0.461881
$476$	0	0
$477$	−1.77471e30	−0.279361
$478$	0	0
$479$	−2.64308e30	−0.396509	−0.198254	−	0.980151i	$-0.563527\pi$
$479$	−2.64308e30	−0.396509	−0.198254	+	0.980151i	$0.563527\pi$
$480$	0	0
$481$	3.11660e29	0.0445670
$482$	0	0
$483$	−1.55780e30	−0.212382
$484$	0	0
$485$	2.81722e30	0.366261
$486$	0	0
$487$	4.79532e30	0.594613	0.297307	−	0.954782i	$-0.403912\pi$
$487$	4.79532e30	0.594613	0.297307	+	0.954782i	$0.403912\pi$
$488$	0	0
$489$	−2.38864e30	−0.282552
$490$	0	0
$491$	9.02420e30	1.01852	0.509262	−	0.860611i	$-0.329918\pi$
$491$	9.02420e30	1.01852	0.509262	+	0.860611i	$0.329918\pi$
$492$	0	0
$493$	3.11408e29	0.0335421
$494$	0	0
$495$	2.04357e29	0.0210102
$496$	0	0
$497$	−1.47225e31	−1.44505
$498$	0	0
$499$	8.38918e30	0.786255	0.393127	−	0.919484i	$-0.371393\pi$
$499$	8.38918e30	0.786255	0.393127	+	0.919484i	$0.371393\pi$
$500$	0	0
$501$	1.51494e30	0.135600
$502$	0	0
$503$	1.65557e31	1.41552	0.707759	−	0.706454i	$-0.249706\pi$
$503$	1.65557e31	1.41552	0.707759	+	0.706454i	$0.249706\pi$
$504$	0	0
$505$	−9.85464e30	−0.804989
$506$	0	0
$507$	−1.01931e31	−0.795634
$508$	0	0
$509$	−6.26982e30	−0.467736	−0.233868	−	0.972268i	$-0.575138\pi$
$509$	−6.26982e30	−0.467736	−0.233868	+	0.972268i	$0.575138\pi$
$510$	0	0
$511$	−8.49202e30	−0.605579
$512$	0	0
$513$	−1.44062e31	−0.982199
$514$	0	0
$515$	1.17614e31	0.766790
$516$	0	0
$517$	2.02189e30	0.126071
$518$	0	0
$519$	−2.91625e30	−0.173940
$520$	0	0
$521$	−1.42153e31	−0.811188	−0.405594	−	0.914053i	$-0.632935\pi$
$521$	−1.42153e31	−0.811188	−0.405594	+	0.914053i	$0.632935\pi$
$522$	0	0
$523$	2.20163e31	1.20220	0.601098	−	0.799176i	$-0.294730\pi$
$523$	2.20163e31	1.20220	0.601098	+	0.799176i	$0.294730\pi$
$524$	0	0
$525$	−7.65974e30	−0.400297
$526$	0	0
$527$	1.31135e29	0.00655987
$528$	0	0
$529$	−1.93503e31	−0.926719
$530$	0	0
$531$	−3.35972e30	−0.154070
$532$	0	0
$533$	−4.66234e30	−0.204758
$534$	0	0
$535$	5.70891e30	0.240151
$536$	0	0
$537$	−1.84544e31	−0.743698
$538$	0	0
$539$	−4.55787e29	−0.0175991
$540$	0	0
$541$	−1.96305e30	−0.0726377	−0.0363188	−	0.999340i	$-0.511563\pi$
$541$	−1.96305e30	−0.0726377	−0.0363188	+	0.999340i	$0.511563\pi$
$542$	0	0
$543$	−4.05871e31	−1.43943
$544$	0	0
$545$	−2.64828e31	−0.900335
$546$	0	0
$547$	−5.16100e31	−1.68221	−0.841104	−	0.540873i	$-0.818094\pi$
$547$	−5.16100e31	−1.68221	−0.841104	+	0.540873i	$0.818094\pi$
$548$	0	0
$549$	8.35887e28	0.00261256
$550$	0	0
$551$	1.96521e31	0.589068
$552$	0	0
$553$	−1.09087e31	−0.313642
$554$	0	0
$555$	3.67173e30	0.101275
$556$	0	0
$557$	−1.12249e31	−0.297063	−0.148531	−	0.988908i	$-0.547455\pi$
$557$	−1.12249e31	−0.297063	−0.148531	+	0.988908i	$0.547455\pi$
$558$	0	0
$559$	1.84126e31	0.467605
$560$	0	0
$561$	2.01654e29	0.00491512
$562$	0	0
$563$	−4.74535e31	−1.11025	−0.555127	−	0.831766i	$-0.687330\pi$
$563$	−4.74535e31	−1.11025	−0.555127	+	0.831766i	$0.687330\pi$
$564$	0	0
$565$	1.88436e31	0.423260
$566$	0	0
$567$	−2.80160e31	−0.604227
$568$	0	0
$569$	5.54838e31	1.14915	0.574573	−	0.818453i	$-0.305168\pi$
$569$	5.54838e31	1.14915	0.574573	+	0.818453i	$0.305168\pi$
$570$	0	0
$571$	−4.29136e31	−0.853650	−0.426825	−	0.904334i	$-0.640368\pi$
$571$	−4.29136e31	−0.853650	−0.426825	+	0.904334i	$0.640368\pi$
$572$	0	0
$573$	−4.53815e30	−0.0867162
$574$	0	0
$575$	7.52373e30	0.138119
$576$	0	0
$577$	6.13314e31	1.08183	0.540917	−	0.841076i	$-0.318077\pi$
$577$	6.13314e31	1.08183	0.540917	+	0.841076i	$0.318077\pi$
$578$	0	0
$579$	5.68956e31	0.964440
$580$	0	0
$581$	−8.63837e31	−1.40736
$582$	0	0
$583$	7.39222e30	0.115768
$584$	0	0
$585$	−3.29398e30	−0.0495940
$586$	0	0
$587$	1.07372e32	1.55437	0.777186	−	0.629271i	$-0.216647\pi$
$587$	1.07372e32	1.55437	0.777186	+	0.629271i	$0.216647\pi$
$588$	0	0
$589$	8.27554e30	0.115205
$590$	0	0
$591$	−5.42798e31	−0.726745
$592$	0	0
$593$	1.02967e32	1.32608	0.663039	−	0.748585i	$-0.269266\pi$
$593$	1.02967e32	1.32608	0.663039	+	0.748585i	$0.269266\pi$
$594$	0	0
$595$	−2.67211e30	−0.0331061
$596$	0	0
$597$	−4.61960e31	−0.550680
$598$	0	0
$599$	4.35682e30	0.0499760	0.0249880	−	0.999688i	$-0.492045\pi$
$599$	4.35682e30	0.0499760	0.0249880	+	0.999688i	$0.492045\pi$
$600$	0	0
$601$	7.05839e31	0.779200	0.389600	−	0.920984i	$-0.372613\pi$
$601$	7.05839e31	0.779200	0.389600	+	0.920984i	$0.372613\pi$
$602$	0	0
$603$	3.92802e30	0.0417374
$604$	0	0
$605$	6.75694e31	0.691136
$606$	0	0
$607$	9.10387e31	0.896512	0.448256	−	0.893905i	$-0.352045\pi$
$607$	9.10387e31	0.896512	0.448256	+	0.893905i	$0.352045\pi$
$608$	0	0
$609$	5.38414e31	0.510525
$610$	0	0
$611$	−3.25903e31	−0.297587
$612$	0	0
$613$	7.80589e30	0.0686479	0.0343239	−	0.999411i	$-0.489072\pi$
$613$	7.80589e30	0.0686479	0.0343239	+	0.999411i	$0.489072\pi$
$614$	0	0
$615$	−5.49279e31	−0.465296
$616$	0	0
$617$	1.10274e31	0.0899899	0.0449949	−	0.998987i	$-0.485673\pi$
$617$	1.10274e31	0.0899899	0.0449949	+	0.998987i	$0.485673\pi$
$618$	0	0
$619$	5.38453e31	0.423356	0.211678	−	0.977339i	$-0.432107\pi$
$619$	5.38453e31	0.423356	0.211678	+	0.977339i	$0.432107\pi$
$620$	0	0
$621$	3.87682e31	0.293712
$622$	0	0
$623$	2.61054e31	0.190598
$624$	0	0
$625$	−3.26074e31	−0.229454
$626$	0	0
$627$	1.27258e31	0.0863196
$628$	0	0
$629$	−1.33435e30	−0.00872541
$630$	0	0
$631$	4.33640e31	0.273396	0.136698	−	0.990613i	$-0.456351\pi$
$631$	4.33640e31	0.273396	0.136698	+	0.990613i	$0.456351\pi$
$632$	0	0
$633$	1.61358e32	0.980954
$634$	0	0
$635$	−1.81348e32	−1.06320
$636$	0	0
$637$	7.34672e30	0.0415422
$638$	0	0
$639$	7.77028e31	0.423816
$640$	0	0
$641$	8.87077e31	0.466760	0.233380	−	0.972386i	$-0.425021\pi$
$641$	8.87077e31	0.466760	0.233380	+	0.972386i	$0.425021\pi$
$642$	0	0
$643$	1.61105e32	0.817869	0.408935	−	0.912564i	$-0.365900\pi$
$643$	1.61105e32	0.817869	0.408935	+	0.912564i	$0.365900\pi$
$644$	0	0
$645$	2.16922e32	1.06259
$646$	0	0
$647$	−4.44418e30	−0.0210084	−0.0105042	−	0.999945i	$-0.503344\pi$
$647$	−4.44418e30	−0.0210084	−0.0105042	+	0.999945i	$0.503344\pi$
$648$	0	0
$649$	1.39943e31	0.0638466
$650$	0	0
$651$	2.26727e31	0.0998441
$652$	0	0
$653$	2.63154e32	1.11869	0.559343	−	0.828936i	$-0.311053\pi$
$653$	2.63154e32	1.11869	0.559343	+	0.828936i	$0.311053\pi$
$654$	0	0
$655$	1.00644e32	0.413062
$656$	0	0
$657$	4.48193e31	0.177609
$658$	0	0
$659$	−1.12234e32	−0.429481	−0.214740	−	0.976671i	$-0.568891\pi$
$659$	−1.12234e32	−0.429481	−0.214740	+	0.976671i	$0.568891\pi$
$660$	0	0
$661$	1.28423e32	0.474597	0.237299	−	0.971437i	$-0.423738\pi$
$661$	1.28423e32	0.474597	0.237299	+	0.971437i	$0.423738\pi$
$662$	0	0
$663$	−3.25041e30	−0.0116020
$664$	0	0
$665$	−1.68629e32	−0.581411
$666$	0	0
$667$	−5.28854e31	−0.176152
$668$	0	0
$669$	2.63230e32	0.847102
$670$	0	0
$671$	−3.48173e29	−0.00108265
$672$	0	0
$673$	−1.04191e32	−0.313080	−0.156540	−	0.987672i	$-0.550034\pi$
$673$	−1.04191e32	−0.313080	−0.156540	+	0.987672i	$0.550034\pi$
$674$	0	0
$675$	1.90625e32	0.553586
$676$	0	0
$677$	4.07220e32	1.14304	0.571518	−	0.820590i	$-0.306355\pi$
$677$	4.07220e32	1.14304	0.571518	+	0.820590i	$0.306355\pi$
$678$	0	0
$679$	−1.77013e32	−0.480288
$680$	0	0
$681$	1.99532e31	0.0523384
$682$	0	0
$683$	−1.74621e32	−0.442853	−0.221426	−	0.975177i	$-0.571071\pi$
$683$	−1.74621e32	−0.442853	−0.221426	+	0.975177i	$0.571071\pi$
$684$	0	0
$685$	4.10106e32	1.00567
$686$	0	0
$687$	−4.88011e32	−1.15725
$688$	0	0
$689$	−1.19153e32	−0.273266
$690$	0	0
$691$	2.76949e32	0.614330	0.307165	−	0.951656i	$-0.400620\pi$
$691$	2.76949e32	0.614330	0.307165	+	0.951656i	$0.400620\pi$
$692$	0	0
$693$	−1.28402e31	−0.0275512
$694$	0	0
$695$	−4.42385e32	−0.918280
$696$	0	0
$697$	1.99614e31	0.0400879
$698$	0	0
$699$	−8.36220e32	−1.62492
$700$	0	0
$701$	7.04451e31	0.132463	0.0662313	−	0.997804i	$-0.478902\pi$
$701$	7.04451e31	0.132463	0.0662313	+	0.997804i	$0.478902\pi$
$702$	0	0
$703$	−8.42068e31	−0.153236
$704$	0	0
$705$	−3.83952e32	−0.676243
$706$	0	0
$707$	6.19191e32	1.05560
$708$	0	0
$709$	−5.16891e32	−0.853036	−0.426518	−	0.904479i	$-0.640260\pi$
$709$	−5.16891e32	−0.853036	−0.426518	+	0.904479i	$0.640260\pi$
$710$	0	0
$711$	5.75742e31	0.0919872
$712$	0	0
$713$	−2.22702e31	−0.0344503
$714$	0	0
$715$	1.37205e31	0.0205518
$716$	0	0
$717$	−9.11606e32	−1.32232
$718$	0	0
$719$	1.05484e33	1.48185	0.740924	−	0.671588i	$-0.234388\pi$
$719$	1.05484e33	1.48185	0.740924	+	0.671588i	$0.234388\pi$
$720$	0	0
$721$	−7.38998e32	−1.00551
$722$	0	0
$723$	−9.11236e32	−1.20099
$724$	0	0
$725$	−2.60039e32	−0.332010
$726$	0	0
$727$	3.58399e32	0.443324	0.221662	−	0.975124i	$-0.428852\pi$
$727$	3.58399e32	0.443324	0.221662	+	0.975124i	$0.428852\pi$
$728$	0	0
$729$	9.21801e32	1.10477
$730$	0	0
$731$	−7.88319e31	−0.0915487
$732$	0	0
$733$	−9.18126e32	−1.03325	−0.516626	−	0.856211i	$-0.672812\pi$
$733$	−9.18126e32	−1.03325	−0.516626	+	0.856211i	$0.672812\pi$
$734$	0	0
$735$	8.65531e31	0.0944014
$736$	0	0
$737$	−1.63615e31	−0.0172960
$738$	0	0
$739$	3.69430e32	0.378548	0.189274	−	0.981924i	$-0.439387\pi$
$739$	3.69430e32	0.378548	0.189274	+	0.981924i	$0.439387\pi$
$740$	0	0
$741$	−2.05125e32	−0.203755
$742$	0	0
$743$	−1.18771e32	−0.114378	−0.0571888	−	0.998363i	$-0.518214\pi$
$743$	−1.18771e32	−0.114378	−0.0571888	+	0.998363i	$0.518214\pi$
$744$	0	0
$745$	−6.35461e32	−0.593323
$746$	0	0
$747$	4.55917e32	0.412762
$748$	0	0
$749$	−3.58705e32	−0.314917
$750$	0	0
$751$	8.99060e32	0.765472	0.382736	−	0.923858i	$-0.374982\pi$
$751$	8.99060e32	0.765472	0.382736	+	0.923858i	$0.374982\pi$
$752$	0	0
$753$	−4.23520e32	−0.349729
$754$	0	0
$755$	1.66311e33	1.33208
$756$	0	0
$757$	−1.38514e33	−1.07620	−0.538098	−	0.842883i	$-0.680857\pi$
$757$	−1.38514e33	−1.07620	−0.538098	+	0.842883i	$0.680857\pi$
$758$	0	0
$759$	−3.42462e31	−0.0258126
$760$	0	0
$761$	1.68497e33	1.23216	0.616082	−	0.787682i	$-0.288719\pi$
$761$	1.68497e33	1.23216	0.616082	+	0.787682i	$0.288719\pi$
$762$	0	0
$763$	1.66398e33	1.18063
$764$	0	0
$765$	1.41029e31	0.00970960
$766$	0	0
$767$	−2.25571e32	−0.150708
$768$	0	0
$769$	−7.22692e31	−0.0468598	−0.0234299	−	0.999725i	$-0.507459\pi$
$769$	−7.22692e31	−0.0468598	−0.0234299	+	0.999725i	$0.507459\pi$
$770$	0	0
$771$	−2.26029e33	−1.42245
$772$	0	0
$773$	−2.52910e33	−1.54490	−0.772452	−	0.635073i	$-0.780970\pi$
$773$	−2.52910e33	−1.54490	−0.772452	+	0.635073i	$0.780970\pi$
$774$	0	0
$775$	−1.09503e32	−0.0649317
$776$	0	0
$777$	−2.30704e32	−0.132804
$778$	0	0
$779$	1.25971e33	0.704026
$780$	0	0
$781$	−3.23657e32	−0.175630
$782$	0	0
$783$	−1.33993e33	−0.706026
$784$	0	0
$785$	−5.74190e32	−0.293802
$786$	0	0
$787$	−1.40717e33	−0.699258	−0.349629	−	0.936888i	$-0.613692\pi$
$787$	−1.40717e33	−0.699258	−0.349629	+	0.936888i	$0.613692\pi$
$788$	0	0
$789$	−1.45448e33	−0.701974
$790$	0	0
$791$	−1.18399e33	−0.555033
$792$	0	0
$793$	5.61212e30	0.00255555
$794$	0	0
$795$	−1.40377e33	−0.620975
$796$	0	0
$797$	−8.57498e32	−0.368521	−0.184261	−	0.982877i	$-0.558989\pi$
$797$	−8.57498e32	−0.368521	−0.184261	+	0.982877i	$0.558989\pi$
$798$	0	0
$799$	1.39532e32	0.0582622
$800$	0	0
$801$	−1.37780e32	−0.0558999
$802$	0	0
$803$	−1.86687e32	−0.0736012
$804$	0	0
$805$	4.53795e32	0.173863
$806$	0	0
$807$	−2.13374e32	−0.0794505
$808$	0	0
$809$	−2.70419e33	−0.978655	−0.489328	−	0.872100i	$-0.662758\pi$
$809$	−2.70419e33	−0.978655	−0.489328	+	0.872100i	$0.662758\pi$
$810$	0	0
$811$	−1.26399e33	−0.444635	−0.222318	−	0.974974i	$-0.571362\pi$
$811$	−1.26399e33	−0.444635	−0.222318	+	0.974974i	$0.571362\pi$
$812$	0	0
$813$	2.17808e33	0.744789
$814$	0	0
$815$	6.95823e32	0.231306
$816$	0	0
$817$	−4.97486e33	−1.60778
$818$	0	0
$819$	2.06969e32	0.0650339
$820$	0	0
$821$	−1.17036e33	−0.357578	−0.178789	−	0.983887i	$-0.557218\pi$
$821$	−1.17036e33	−0.357578	−0.178789	+	0.983887i	$0.557218\pi$
$822$	0	0
$823$	−5.77458e32	−0.171562	−0.0857812	−	0.996314i	$-0.527339\pi$
$823$	−5.77458e32	−0.171562	−0.0857812	+	0.996314i	$0.527339\pi$
$824$	0	0
$825$	−1.68390e32	−0.0486514
$826$	0	0
$827$	6.61350e32	0.185831	0.0929154	−	0.995674i	$-0.470381\pi$
$827$	6.61350e32	0.185831	0.0929154	+	0.995674i	$0.470381\pi$
$828$	0	0
$829$	4.93400e33	1.34841	0.674205	−	0.738544i	$-0.264486\pi$
$829$	4.93400e33	1.34841	0.674205	+	0.738544i	$0.264486\pi$
$830$	0	0
$831$	−7.15140e32	−0.190099
$832$	0	0
$833$	−3.14543e31	−0.00813322
$834$	0	0
$835$	−4.41309e32	−0.111007
$836$	0	0
$837$	−5.64246e32	−0.138078
$838$	0	0
$839$	7.31981e33	1.74276	0.871380	−	0.490609i	$-0.163226\pi$
$839$	7.31981e33	1.74276	0.871380	+	0.490609i	$0.163226\pi$
$840$	0	0
$841$	−2.48887e33	−0.576565
$842$	0	0
$843$	−5.09308e33	−1.14806
$844$	0	0
$845$	2.96930e33	0.651330
$846$	0	0
$847$	−4.24555e33	−0.906305
$848$	0	0
$849$	−8.61406e32	−0.178966
$850$	0	0
$851$	2.26607e32	0.0458230
$852$	0	0
$853$	3.52017e33	0.692867	0.346434	−	0.938074i	$-0.387393\pi$
$853$	3.52017e33	0.692867	0.346434	+	0.938074i	$0.387393\pi$
$854$	0	0
$855$	8.89994e32	0.170520
$856$	0	0
$857$	5.01795e33	0.935938	0.467969	−	0.883745i	$-0.344986\pi$
$857$	5.01795e33	0.935938	0.467969	+	0.883745i	$0.344986\pi$
$858$	0	0
$859$	9.36061e33	1.69974	0.849871	−	0.526991i	$-0.176680\pi$
$859$	9.36061e33	1.69974	0.849871	+	0.526991i	$0.176680\pi$
$860$	0	0
$861$	3.45125e33	0.610155
$862$	0	0
$863$	−6.68952e33	−1.15152	−0.575759	−	0.817619i	$-0.695293\pi$
$863$	−6.68952e33	−1.15152	−0.575759	+	0.817619i	$0.695293\pi$
$864$	0	0
$865$	8.49520e32	0.142393
$866$	0	0
$867$	−5.22368e33	−0.852623
$868$	0	0
$869$	−2.39815e32	−0.0381196
$870$	0	0
$871$	2.63726e32	0.0408267
$872$	0	0
$873$	9.34241e32	0.140862
$874$	0	0
$875$	6.60457e33	0.969956
$876$	0	0
$877$	1.11822e34	1.59967	0.799836	−	0.600219i	$-0.204920\pi$
$877$	1.11822e34	1.59967	0.799836	+	0.600219i	$0.204920\pi$
$878$	0	0
$879$	−4.38024e33	−0.610416
$880$	0	0
$881$	−6.35806e32	−0.0863181	−0.0431590	−	0.999068i	$-0.513742\pi$
$881$	−6.35806e32	−0.0863181	−0.0431590	+	0.999068i	$0.513742\pi$
$882$	0	0
$883$	5.55784e33	0.735119	0.367560	−	0.930000i	$-0.380193\pi$
$883$	5.55784e33	0.735119	0.367560	+	0.930000i	$0.380193\pi$
$884$	0	0
$885$	−2.65749e33	−0.342472
$886$	0	0
$887$	1.02372e34	1.28547	0.642735	−	0.766089i	$-0.277800\pi$
$887$	1.02372e34	1.28547	0.642735	+	0.766089i	$0.277800\pi$
$888$	0	0
$889$	1.13945e34	1.39420
$890$	0	0
$891$	−6.15897e32	−0.0734369
$892$	0	0
$893$	8.80549e33	1.02320
$894$	0	0
$895$	5.37588e33	0.608814
$896$	0	0
$897$	5.52007e32	0.0609300
$898$	0	0
$899$	7.69713e32	0.0828118
$900$	0	0
$901$	5.10145e32	0.0535005
$902$	0	0
$903$	−1.36298e34	−1.39341
$904$	0	0
$905$	1.18232e34	1.17836
$906$	0	0
$907$	−1.29936e34	−1.26255	−0.631273	−	0.775561i	$-0.717467\pi$
$907$	−1.29936e34	−1.26255	−0.631273	+	0.775561i	$0.717467\pi$
$908$	0	0
$909$	−3.26798e33	−0.309595
$910$	0	0
$911$	1.27059e34	1.17367	0.586833	−	0.809708i	$-0.300375\pi$
$911$	1.27059e34	1.17367	0.586833	+	0.809708i	$0.300375\pi$
$912$	0	0
$913$	−1.89904e33	−0.171049
$914$	0	0
$915$	6.61175e31	0.00580729
$916$	0	0
$917$	−6.32373e33	−0.541659
$918$	0	0
$919$	1.25455e34	1.04800	0.524000	−	0.851718i	$-0.324439\pi$
$919$	1.25455e34	1.04800	0.524000	+	0.851718i	$0.324439\pi$
$920$	0	0
$921$	1.73576e34	1.41418
$922$	0	0
$923$	5.21694e33	0.414569
$924$	0	0
$925$	1.11424e33	0.0863668
$926$	0	0
$927$	3.90029e33	0.294904
$928$	0	0
$929$	−2.95935e33	−0.218281	−0.109141	−	0.994026i	$-0.534810\pi$
$929$	−2.95935e33	−0.218281	−0.109141	+	0.994026i	$0.534810\pi$
$930$	0	0
$931$	−1.98499e33	−0.142836
$932$	0	0
$933$	−1.60995e34	−1.13025
$934$	0	0
$935$	−5.87430e31	−0.00402367
$936$	0	0
$937$	1.37737e34	0.920546	0.460273	−	0.887777i	$-0.347752\pi$
$937$	1.37737e34	0.920546	0.460273	+	0.887777i	$0.347752\pi$
$938$	0	0
$939$	1.20859e34	0.788179
$940$	0	0
$941$	−5.61074e33	−0.357059	−0.178529	−	0.983935i	$-0.557134\pi$
$941$	−5.61074e33	−0.357059	−0.178529	+	0.983935i	$0.557134\pi$
$942$	0	0
$943$	−3.38997e33	−0.210529
$944$	0	0
$945$	1.14975e34	0.696850
$946$	0	0
$947$	−1.27325e34	−0.753161	−0.376581	−	0.926384i	$-0.622900\pi$
$947$	−1.27325e34	−0.753161	−0.376581	+	0.926384i	$0.622900\pi$
$948$	0	0
$949$	3.00916e33	0.173733
$950$	0	0
$951$	−2.24319e34	−1.26413
$952$	0	0
$953$	2.08531e34	1.14710	0.573551	−	0.819170i	$-0.305565\pi$
$953$	2.08531e34	1.14710	0.573551	+	0.819170i	$0.305565\pi$
$954$	0	0
$955$	1.32199e33	0.0709885
$956$	0	0
$957$	1.18364e33	0.0620485
$958$	0	0
$959$	−2.57680e34	−1.31876
$960$	0	0
$961$	−1.96892e34	−0.983804
$962$	0	0
$963$	1.89318e33	0.0923611
$964$	0	0
$965$	−1.65740e34	−0.789520
$966$	0	0
$967$	7.28388e33	0.338812	0.169406	−	0.985546i	$-0.445815\pi$
$967$	7.28388e33	0.338812	0.169406	+	0.985546i	$0.445815\pi$
$968$	0	0
$969$	8.78222e32	0.0398916
$970$	0	0
$971$	−3.99336e34	−1.77140	−0.885702	−	0.464254i	$-0.846323\pi$
$971$	−3.99336e34	−1.77140	−0.885702	+	0.464254i	$0.846323\pi$
$972$	0	0
$973$	2.77961e34	1.20417
$974$	0	0
$975$	2.71424e33	0.114840
$976$	0	0
$977$	−2.75687e34	−1.13928	−0.569638	−	0.821896i	$-0.692916\pi$
$977$	−2.75687e34	−1.13928	−0.569638	+	0.821896i	$0.692916\pi$
$978$	0	0
$979$	5.73896e32	0.0231650
$980$	0	0
$981$	−8.78218e33	−0.346265
$982$	0	0
$983$	−3.92166e34	−1.51044	−0.755222	−	0.655469i	$-0.772471\pi$
$983$	−3.92166e34	−1.51044	−0.755222	+	0.655469i	$0.772471\pi$
$984$	0	0
$985$	1.58120e34	0.594936
$986$	0	0
$987$	2.41247e34	0.886776
$988$	0	0
$989$	1.33878e34	0.480784
$990$	0	0
$991$	2.75962e34	0.968284	0.484142	−	0.874990i	$-0.339132\pi$
$991$	2.75962e34	0.968284	0.484142	+	0.874990i	$0.339132\pi$
$992$	0	0
$993$	−7.92290e33	−0.271624
$994$	0	0
$995$	1.34571e34	0.450804
$996$	0	0
$997$	8.83085e33	0.289073	0.144537	−	0.989499i	$-0.453831\pi$
$997$	8.83085e33	0.289073	0.144537	+	0.989499i	$0.453831\pi$
$998$	0	0
$999$	5.74142e33	0.183661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.l.1.3		4
4.3	odd	2	inner	64.24.a.l.1.2		4
8.3	odd	2		32.24.a.b.1.3	yes	4
8.5	even	2		32.24.a.b.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.b.1.2	✓	4	8.5	even	2
32.24.a.b.1.3	yes	4	8.3	odd	2
64.24.a.l.1.2		4	4.3	odd	2	inner
64.24.a.l.1.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+262305. q^{3} -7.64109e7 q^{5} +4.80108e9 q^{7} -2.53392e10 q^{9} +O(q^{10})\)	\(q+262305. q^{3} -7.64109e7 q^{5} +4.80108e9 q^{7} -2.53392e10 q^{9} +1.05546e11 q^{11} -1.70127e12 q^{13} -2.00430e13 q^{15} +7.28382e12 q^{17} +4.59662e14 q^{19} +1.25935e15 q^{21} -1.23699e15 q^{23} -6.08230e15 q^{25} -3.13408e16 q^{27} +4.27534e16 q^{29} +1.80035e16 q^{31} +2.76852e16 q^{33} -3.66855e17 q^{35} -1.83193e17 q^{37} -4.46251e17 q^{39} +2.74051e18 q^{41} -1.08229e19 q^{43} +1.93619e18 q^{45} +1.91565e19 q^{47} -4.31838e18 q^{49} +1.91058e18 q^{51} +7.00380e19 q^{53} -8.06485e18 q^{55} +1.20572e20 q^{57} +1.32590e20 q^{59} -3.29879e18 q^{61} -1.21656e20 q^{63} +1.29995e20 q^{65} -1.55017e20 q^{67} -3.24468e20 q^{69} -3.06650e21 q^{71} -1.76877e21 q^{73} -1.59542e21 q^{75} +5.06734e20 q^{77} -2.27214e21 q^{79} -5.83535e21 q^{81} -1.79926e22 q^{83} -5.56563e20 q^{85} +1.12144e22 q^{87} +5.43740e21 q^{89} -8.16792e21 q^{91} +4.72242e21 q^{93} -3.51232e22 q^{95} -3.68694e22 q^{97} -2.67445e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 19990040 q^{5} + 284960925396 q^{9}+O(q^{10}) \) 4 * q - 19990040 * q^5 + 284960925396 * q^9	\( 4 q - 19990040 q^{5} + 284960925396 q^{9} + 21132585986568 q^{13} - 339546388528760 q^{17} - 57\!\cdots\!04 q^{21}+ \cdots + 79\!\cdots\!56 q^{97}+O(q^{100}) \) 4 * q - 19990040 * q^5 + 284960925396 * q^9 + 21132585986568 * q^13 - 339546388528760 * q^17 - 5772344793698304 * q^21 - 27184332341841700 * q^25 - 11713498653208888 * q^29 - 127188060952651776 * q^33 - 1527028203255925528 * q^37 - 175841616529668376 * q^41 + 26164165807508085000 * q^45 + 67830117546567696932 * q^49 - 75693842547307242712 * q^53 - 573763276821861734400 * q^57 - 232274843679744391736 * q^61 + 1889512171511171217360 * q^65 + 2311912054144728354816 * q^69 - 3115180773570839678040 * q^73 + 3902426738355338366976 * q^77 - 4667833569712391401500 * q^81 - 24631918485707096544560 * q^85 - 78996853740551396469400 * q^89 + 145181262233476826578944 * q^93 + 79943563283557612080456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more