LMFDB - Embedded newform 64.24.a.l.1.1

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			1.1
Root			$-1404.92$ 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)

$f(q)$	$=$	$q-511823. q^{3} +6.64159e7 q^{5} +8.09952e9 q^{7} +1.67820e11 q^{9} +O(q^{10})$	$q-511823. q^{3} +6.64159e7 q^{5} +8.09952e9 q^{7} +1.67820e11 q^{9} +1.78341e11 q^{11} +1.22676e13 q^{13} -3.39932e13 q^{15} -1.77057e14 q^{17} +7.96082e14 q^{19} -4.14552e15 q^{21} -2.89245e15 q^{23} -7.50986e15 q^{25} -3.77093e16 q^{27} -4.86101e16 q^{29} -1.32601e17 q^{31} -9.12792e16 q^{33} +5.37937e17 q^{35} -5.80321e17 q^{37} -6.27882e18 q^{39} -2.82843e18 q^{41} +1.78175e18 q^{43} +1.11459e19 q^{45} -1.39096e19 q^{47} +3.82334e19 q^{49} +9.06219e19 q^{51} -1.07885e20 q^{53} +1.18447e19 q^{55} -4.07453e20 q^{57} +5.27322e19 q^{59} -1.12839e20 q^{61} +1.35926e21 q^{63} +8.14761e20 q^{65} -1.35907e21 q^{67} +1.48042e21 q^{69} +4.67830e20 q^{71} +2.11183e20 q^{73} +3.84372e21 q^{75} +1.44448e21 q^{77} +1.15151e21 q^{79} +3.50143e21 q^{81} +1.97495e22 q^{83} -1.17594e22 q^{85} +2.48798e22 q^{87} -4.49358e22 q^{89} +9.93613e22 q^{91} +6.78682e22 q^{93} +5.28725e22 q^{95} +7.68411e22 q^{97} +2.99292e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−511823.	−1.66811	−0.834056	−	0.551679i	$-0.813987\pi$
$3$	−511823.	−1.66811	−0.834056	+	0.551679i	$0.813987\pi$
$4$	0	0
$5$	6.64159e7	0.608299	0.304149	−	0.952624i	$-0.401628\pi$
$5$	6.64159e7	0.608299	0.304149	+	0.952624i	$0.401628\pi$
$6$	0	0
$7$	8.09952e9	1.54822	0.774108	−	0.633053i	$-0.218199\pi$
$7$	8.09952e9	1.54822	0.774108	+	0.633053i	$0.218199\pi$
$8$	0	0
$9$	1.67820e11	1.78260
$10$	0	0
$11$	1.78341e11	0.188467	0.0942337	−	0.995550i	$-0.469960\pi$
$11$	1.78341e11	0.188467	0.0942337	+	0.995550i	$0.469960\pi$
$12$	0	0
$13$	1.22676e13	1.89850	0.949248	−	0.314529i	$-0.101846\pi$
$13$	1.22676e13	1.89850	0.949248	+	0.314529i	$0.101846\pi$
$14$	0	0
$15$	−3.39932e13	−1.01471
$16$	0	0
$17$	−1.77057e14	−1.25300	−0.626499	−	0.779422i	$-0.715513\pi$
$17$	−1.77057e14	−1.25300	−0.626499	+	0.779422i	$0.715513\pi$
$18$	0	0
$19$	7.96082e14	1.56780	0.783902	−	0.620885i	$-0.213227\pi$
$19$	7.96082e14	1.56780	0.783902	+	0.620885i	$0.213227\pi$
$20$	0	0
$21$	−4.14552e15	−2.58260
$22$	0	0
$23$	−2.89245e15	−0.632989	−0.316494	−	0.948594i	$-0.602506\pi$
$23$	−2.89245e15	−0.632989	−0.316494	+	0.948594i	$0.602506\pi$
$24$	0	0
$25$	−7.50986e15	−0.629973
$26$	0	0
$27$	−3.77093e16	−1.30547
$28$	0	0
$29$	−4.86101e16	−0.739861	−0.369930	−	0.929060i	$-0.620618\pi$
$29$	−4.86101e16	−0.739861	−0.369930	+	0.929060i	$0.620618\pi$
$30$	0	0
$31$	−1.32601e17	−0.937318	−0.468659	−	0.883379i	$-0.655263\pi$
$31$	−1.32601e17	−0.937318	−0.468659	+	0.883379i	$0.655263\pi$
$32$	0	0
$33$	−9.12792e16	−0.314385
$34$	0	0
$35$	5.37937e17	0.941778
$36$	0	0
$37$	−5.80321e17	−0.536227	−0.268113	−	0.963387i	$-0.586400\pi$
$37$	−5.80321e17	−0.536227	−0.268113	+	0.963387i	$0.586400\pi$
$38$	0	0
$39$	−6.27882e18	−3.16691
$40$	0	0
$41$	−2.82843e18	−0.802658	−0.401329	−	0.915934i	$-0.631452\pi$
$41$	−2.82843e18	−0.802658	−0.401329	+	0.915934i	$0.631452\pi$
$42$	0	0
$43$	1.78175e18	0.292388	0.146194	−	0.989256i	$-0.453298\pi$
$43$	1.78175e18	0.292388	0.146194	+	0.989256i	$0.453298\pi$
$44$	0	0
$45$	1.11459e19	1.08435
$46$	0	0
$47$	−1.39096e19	−0.820709	−0.410354	−	0.911926i	$-0.634595\pi$
$47$	−1.39096e19	−0.820709	−0.410354	+	0.911926i	$0.634595\pi$
$48$	0	0
$49$	3.82334e19	1.39697
$50$	0	0
$51$	9.06219e19	2.09014
$52$	0	0
$53$	−1.07885e20	−1.59878	−0.799389	−	0.600813i	$-0.794843\pi$
$53$	−1.07885e20	−1.59878	−0.799389	+	0.600813i	$0.794843\pi$
$54$	0	0
$55$	1.18447e19	0.114644
$56$	0	0
$57$	−4.07453e20	−2.61527
$58$	0	0
$59$	5.27322e19	0.227656	0.113828	−	0.993500i	$-0.463689\pi$
$59$	5.27322e19	0.227656	0.113828	+	0.993500i	$0.463689\pi$
$60$	0	0
$61$	−1.12839e20	−0.332020	−0.166010	−	0.986124i	$-0.553088\pi$
$61$	−1.12839e20	−0.332020	−0.166010	+	0.986124i	$0.553088\pi$
$62$	0	0
$63$	1.35926e21	2.75985
$64$	0	0
$65$	8.14761e20	1.15485
$66$	0	0
$67$	−1.35907e21	−1.35951	−0.679753	−	0.733441i	$-0.737913\pi$
$67$	−1.35907e21	−1.35951	−0.679753	+	0.733441i	$0.737913\pi$
$68$	0	0
$69$	1.48042e21	1.05590
$70$	0	0
$71$	4.67830e20	0.240225	0.120112	−	0.992760i	$-0.461675\pi$
$71$	4.67830e20	0.240225	0.120112	+	0.992760i	$0.461675\pi$
$72$	0	0
$73$	2.11183e20	0.0787856	0.0393928	−	0.999224i	$-0.487458\pi$
$73$	2.11183e20	0.0787856	0.0393928	+	0.999224i	$0.487458\pi$
$74$	0	0
$75$	3.84372e21	1.05087
$76$	0	0
$77$	1.44448e21	0.291788
$78$	0	0
$79$	1.15151e21	0.173203	0.0866013	−	0.996243i	$-0.472399\pi$
$79$	1.15151e21	0.173203	0.0866013	+	0.996243i	$0.472399\pi$
$80$	0	0
$81$	3.50143e21	0.395064
$82$	0	0
$83$	1.97495e22	1.68329	0.841645	−	0.540031i	$-0.181587\pi$
$83$	1.97495e22	1.68329	0.841645	+	0.540031i	$0.181587\pi$
$84$	0	0
$85$	−1.17594e22	−0.762197
$86$	0	0
$87$	2.48798e22	1.23417
$88$	0	0
$89$	−4.49358e22	−1.71636	−0.858179	−	0.513351i	$-0.828404\pi$
$89$	−4.49358e22	−1.71636	−0.858179	+	0.513351i	$0.828404\pi$
$90$	0	0
$91$	9.93613e22	2.93928
$92$	0	0
$93$	6.78682e22	1.56355
$94$	0	0
$95$	5.28725e22	0.953692
$96$	0	0
$97$	7.68411e22	1.09073	0.545367	−	0.838197i	$-0.316390\pi$
$97$	7.68411e22	1.09073	0.545367	+	0.838197i	$0.316390\pi$
$98$	0	0
$99$	2.99292e22	0.335962
$100$	0	0
$101$	4.50566e22	0.401849	0.200924	−	0.979607i	$-0.435605\pi$
$101$	4.50566e22	0.401849	0.200924	+	0.979607i	$0.435605\pi$
$102$	0	0
$103$	1.60252e23	1.14071	0.570355	−	0.821399i	$-0.306806\pi$
$103$	1.60252e23	1.14071	0.570355	+	0.821399i	$0.306806\pi$
$104$	0	0
$105$	−2.75328e23	−1.57099
$106$	0	0
$107$	−4.05572e23	−1.86275	−0.931374	−	0.364064i	$-0.881389\pi$
$107$	−4.05572e23	−1.86275	−0.931374	+	0.364064i	$0.881389\pi$
$108$	0	0
$109$	−3.11599e23	−1.15662	−0.578311	−	0.815817i	$-0.696288\pi$
$109$	−3.11599e23	−1.15662	−0.578311	+	0.815817i	$0.696288\pi$
$110$	0	0
$111$	2.97022e23	0.894487
$112$	0	0
$113$	−4.23692e23	−1.03908	−0.519539	−	0.854446i	$-0.673896\pi$
$113$	−4.23692e23	−1.03908	−0.519539	+	0.854446i	$0.673896\pi$
$114$	0	0
$115$	−1.92105e23	−0.385046
$116$	0	0
$117$	2.05874e24	3.38426
$118$	0	0
$119$	−1.43408e24	−1.93991
$120$	0	0
$121$	−8.63625e23	−0.964480
$122$	0	0
$123$	1.44765e24	1.33892
$124$	0	0
$125$	−1.29051e24	−0.991510
$126$	0	0
$127$	−1.40434e24	−0.898939	−0.449469	−	0.893296i	$-0.648387\pi$
$127$	−1.40434e24	−0.898939	−0.449469	+	0.893296i	$0.648387\pi$
$128$	0	0
$129$	−9.11941e23	−0.487736
$130$	0	0
$131$	−2.97700e24	−1.33401	−0.667005	−	0.745054i	$-0.732424\pi$
$131$	−2.97700e24	−1.33401	−0.667005	+	0.745054i	$0.732424\pi$
$132$	0	0
$133$	6.44788e24	2.42730
$134$	0	0
$135$	−2.50450e24	−0.794113
$136$	0	0
$137$	−3.36139e24	−0.899978	−0.449989	−	0.893034i	$-0.648572\pi$
$137$	−3.36139e24	−0.899978	−0.449989	+	0.893034i	$0.648572\pi$
$138$	0	0
$139$	2.03663e24	0.461574	0.230787	−	0.973004i	$-0.425870\pi$
$139$	2.03663e24	0.461574	0.230787	+	0.973004i	$0.425870\pi$
$140$	0	0
$141$	7.11925e24	1.36903
$142$	0	0
$143$	2.18781e24	0.357805
$144$	0	0
$145$	−3.22848e24	−0.450056
$146$	0	0
$147$	−1.95688e25	−2.33031
$148$	0	0
$149$	−1.23101e25	−1.25493	−0.627466	−	0.778644i	$-0.715908\pi$
$149$	−1.23101e25	−1.25493	−0.627466	+	0.778644i	$0.715908\pi$
$150$	0	0
$151$	−1.67993e25	−1.46912	−0.734559	−	0.678545i	$-0.762611\pi$
$151$	−1.67993e25	−1.46912	−0.734559	+	0.678545i	$0.762611\pi$
$152$	0	0
$153$	−2.97137e25	−2.23360
$154$	0	0
$155$	−8.80681e24	−0.570169
$156$	0	0
$157$	1.88995e25	1.05585	0.527927	−	0.849289i	$-0.322969\pi$
$157$	1.88995e25	1.05585	0.527927	+	0.849289i	$0.322969\pi$
$158$	0	0
$159$	5.52180e25	2.66694
$160$	0	0
$161$	−2.34275e25	−0.980004
$162$	0	0
$163$	3.83023e25	1.39017	0.695084	−	0.718929i	$-0.255367\pi$
$163$	3.83023e25	1.39017	0.695084	+	0.718929i	$0.255367\pi$
$164$	0	0
$165$	−6.06239e24	−0.191240
$166$	0	0
$167$	1.63850e25	0.449995	0.224997	−	0.974359i	$-0.427763\pi$
$167$	1.63850e25	0.449995	0.224997	+	0.974359i	$0.427763\pi$
$168$	0	0
$169$	1.08739e26	2.60429
$170$	0	0
$171$	1.33598e26	2.79477
$172$	0	0
$173$	−3.88404e24	−0.0710809	−0.0355405	−	0.999368i	$-0.511315\pi$
$173$	−3.88404e24	−0.0710809	−0.0355405	+	0.999368i	$0.511315\pi$
$174$	0	0
$175$	−6.08263e25	−0.975334
$176$	0	0
$177$	−2.69896e25	−0.379755
$178$	0	0
$179$	3.91325e25	0.483869	0.241935	−	0.970293i	$-0.422218\pi$
$179$	3.91325e25	0.483869	0.241935	+	0.970293i	$0.422218\pi$
$180$	0	0
$181$	−1.96646e25	−0.213984	−0.106992	−	0.994260i	$-0.534122\pi$
$181$	−1.96646e25	−0.213984	−0.106992	+	0.994260i	$0.534122\pi$
$182$	0	0
$183$	5.77534e25	0.553847
$184$	0	0
$185$	−3.85425e25	−0.326186
$186$	0	0
$187$	−3.15766e25	−0.236149
$188$	0	0
$189$	−3.05427e26	−2.02114
$190$	0	0
$191$	1.42069e26	0.832941	0.416471	−	0.909149i	$-0.363267\pi$
$191$	1.42069e26	0.832941	0.416471	+	0.909149i	$0.363267\pi$
$192$	0	0
$193$	−1.20216e26	−0.625247	−0.312624	−	0.949877i	$-0.601208\pi$
$193$	−1.20216e26	−0.625247	−0.312624	+	0.949877i	$0.601208\pi$
$194$	0	0
$195$	−4.17013e26	−1.92642
$196$	0	0
$197$	−2.95513e26	−1.21399	−0.606994	−	0.794706i	$-0.707625\pi$
$197$	−2.95513e26	−1.21399	−0.606994	+	0.794706i	$0.707625\pi$
$198$	0	0
$199$	−9.12024e25	−0.333577	−0.166788	−	0.985993i	$-0.553340\pi$
$199$	−9.12024e25	−0.333577	−0.166788	+	0.985993i	$0.553340\pi$
$200$	0	0
$201$	6.95603e26	2.26781
$202$	0	0
$203$	−3.93719e26	−1.14546
$204$	0	0
$205$	−1.87853e26	−0.488256
$206$	0	0
$207$	−4.85410e26	−1.12837
$208$	0	0
$209$	1.41974e26	0.295480
$210$	0	0
$211$	3.21972e26	0.600579	0.300290	−	0.953848i	$-0.402917\pi$
$211$	3.21972e26	0.600579	0.300290	+	0.953848i	$0.402917\pi$
$212$	0	0
$213$	−2.39446e26	−0.400722
$214$	0	0
$215$	1.18337e26	0.177859
$216$	0	0
$217$	−1.07400e27	−1.45117
$218$	0	0
$219$	−1.08088e26	−0.131423
$220$	0	0
$221$	−2.17206e27	−2.37881
$222$	0	0
$223$	−8.15025e26	−0.804757	−0.402379	−	0.915473i	$-0.631816\pi$
$223$	−8.15025e26	−0.804757	−0.402379	+	0.915473i	$0.631816\pi$
$224$	0	0
$225$	−1.26030e27	−1.12299
$226$	0	0
$227$	6.87425e26	0.553258	0.276629	−	0.960977i	$-0.410783\pi$
$227$	6.87425e26	0.553258	0.276629	+	0.960977i	$0.410783\pi$
$228$	0	0
$229$	3.89037e26	0.283063	0.141531	−	0.989934i	$-0.454797\pi$
$229$	3.89037e26	0.283063	0.141531	+	0.989934i	$0.454797\pi$
$230$	0	0
$231$	−7.39318e26	−0.486736
$232$	0	0
$233$	−5.53309e26	−0.329894	−0.164947	−	0.986302i	$-0.552745\pi$
$233$	−5.53309e26	−0.329894	−0.164947	+	0.986302i	$0.552745\pi$
$234$	0	0
$235$	−9.23817e26	−0.499236
$236$	0	0
$237$	−5.89367e26	−0.288921
$238$	0	0
$239$	−3.69232e27	−1.64332	−0.821662	−	0.569975i	$-0.806953\pi$
$239$	−3.69232e27	−1.64332	−0.821662	+	0.569975i	$0.806953\pi$
$240$	0	0
$241$	2.03986e26	0.0824905	0.0412453	−	0.999149i	$-0.486868\pi$
$241$	2.03986e26	0.0824905	0.0412453	+	0.999149i	$0.486868\pi$
$242$	0	0
$243$	1.75796e27	0.646454
$244$	0	0
$245$	2.53931e27	0.849778
$246$	0	0
$247$	9.76599e27	2.97647
$248$	0	0
$249$	−1.01083e28	−2.80792
$250$	0	0
$251$	−1.77927e27	−0.450810	−0.225405	−	0.974265i	$-0.572371\pi$
$251$	−1.77927e27	−0.450810	−0.225405	+	0.974265i	$0.572371\pi$
$252$	0	0
$253$	−5.15844e26	−0.119298
$254$	0	0
$255$	6.01873e27	1.27143
$256$	0	0
$257$	−1.39139e26	−0.0268670	−0.0134335	−	0.999910i	$-0.504276\pi$
$257$	−1.39139e26	−0.0268670	−0.0134335	+	0.999910i	$0.504276\pi$
$258$	0	0
$259$	−4.70032e27	−0.830195
$260$	0	0
$261$	−8.15774e27	−1.31888
$262$	0	0
$263$	4.55841e27	0.675029	0.337515	−	0.941320i	$-0.390414\pi$
$263$	4.55841e27	0.675029	0.337515	+	0.941320i	$0.390414\pi$
$264$	0	0
$265$	−7.16527e27	−0.972535
$266$	0	0
$267$	2.29992e28	2.86308
$268$	0	0
$269$	4.63126e27	0.529112	0.264556	−	0.964370i	$-0.414775\pi$
$269$	4.63126e27	0.529112	0.264556	+	0.964370i	$0.414775\pi$
$270$	0	0
$271$	1.54452e27	0.162049	0.0810243	−	0.996712i	$-0.474181\pi$
$271$	1.54452e27	0.162049	0.0810243	+	0.996712i	$0.474181\pi$
$272$	0	0
$273$	−5.08554e28	−4.90306
$274$	0	0
$275$	−1.33932e27	−0.118729
$276$	0	0
$277$	−6.82794e27	−0.556894	−0.278447	−	0.960452i	$-0.589820\pi$
$277$	−6.82794e27	−0.556894	−0.278447	+	0.960452i	$0.589820\pi$
$278$	0	0
$279$	−2.22530e28	−1.67086
$280$	0	0
$281$	2.20737e28	1.52670	0.763348	−	0.645988i	$-0.223554\pi$
$281$	2.20737e28	1.52670	0.763348	+	0.645988i	$0.223554\pi$
$282$	0	0
$283$	1.22052e28	0.778040	0.389020	−	0.921229i	$-0.372814\pi$
$283$	1.22052e28	0.778040	0.389020	+	0.921229i	$0.372814\pi$
$284$	0	0
$285$	−2.70614e28	−1.59087
$286$	0	0
$287$	−2.29089e28	−1.24269
$288$	0	0
$289$	1.13816e28	0.570005
$290$	0	0
$291$	−3.93291e28	−1.81947
$292$	0	0
$293$	−8.46114e27	−0.361785	−0.180893	−	0.983503i	$-0.557899\pi$
$293$	−8.46114e27	−0.361785	−0.180893	+	0.983503i	$0.557899\pi$
$294$	0	0
$295$	3.50226e27	0.138483
$296$	0	0
$297$	−6.72514e27	−0.246038
$298$	0	0
$299$	−3.54833e28	−1.20173
$300$	0	0
$301$	1.44313e28	0.452680
$302$	0	0
$303$	−2.30610e28	−0.670329
$304$	0	0
$305$	−7.49428e27	−0.201968
$306$	0	0
$307$	−5.87474e28	−1.46858	−0.734290	−	0.678836i	$-0.762485\pi$
$307$	−5.87474e28	−1.46858	−0.734290	+	0.678836i	$0.762485\pi$
$308$	0	0
$309$	−8.20206e28	−1.90283
$310$	0	0
$311$	2.06607e28	0.445043	0.222521	−	0.974928i	$-0.428571\pi$
$311$	2.06607e28	0.445043	0.222521	+	0.974928i	$0.428571\pi$
$312$	0	0
$313$	−5.57381e28	−1.11530	−0.557650	−	0.830076i	$-0.688297\pi$
$313$	−5.57381e28	−1.11530	−0.557650	+	0.830076i	$0.688297\pi$
$314$	0	0
$315$	9.02763e28	1.67881
$316$	0	0
$317$	−1.67234e28	−0.289163	−0.144582	−	0.989493i	$-0.546184\pi$
$317$	−1.67234e28	−0.289163	−0.144582	+	0.989493i	$0.546184\pi$
$318$	0	0
$319$	−8.66920e27	−0.139440
$320$	0	0
$321$	2.07581e29	3.10727
$322$	0	0
$323$	−1.40952e29	−1.96445
$324$	0	0
$325$	−9.21277e28	−1.19600
$326$	0	0
$327$	1.59483e29	1.92937
$328$	0	0
$329$	−1.12661e29	−1.27063
$330$	0	0
$331$	1.87213e29	1.96931	0.984654	−	0.174518i	$-0.0558368\pi$
$331$	1.87213e29	1.96931	0.984654	+	0.174518i	$0.0558368\pi$
$332$	0	0
$333$	−9.73893e28	−0.955878
$334$	0	0
$335$	−9.02638e28	−0.826986
$336$	0	0
$337$	−4.16278e28	−0.356156	−0.178078	−	0.984016i	$-0.556988\pi$
$337$	−4.16278e28	−0.356156	−0.178078	+	0.984016i	$0.556988\pi$
$338$	0	0
$339$	2.16856e29	1.73330
$340$	0	0
$341$	−2.36482e28	−0.176654
$342$	0	0
$343$	8.79988e28	0.614602
$344$	0	0
$345$	9.83236e28	0.642301
$346$	0	0
$347$	−7.39463e28	−0.451989	−0.225994	−	0.974129i	$-0.572563\pi$
$347$	−7.39463e28	−0.451989	−0.225994	+	0.974129i	$0.572563\pi$
$348$	0	0
$349$	7.01500e27	0.0401361	0.0200680	−	0.999799i	$-0.493612\pi$
$349$	7.01500e27	0.0401361	0.0200680	+	0.999799i	$0.493612\pi$
$350$	0	0
$351$	−4.62602e29	−2.47842
$352$	0	0
$353$	1.05627e29	0.530108	0.265054	−	0.964234i	$-0.414610\pi$
$353$	1.05627e29	0.530108	0.265054	+	0.964234i	$0.414610\pi$
$354$	0	0
$355$	3.10714e28	0.146128
$356$	0	0
$357$	7.33993e29	3.23599
$358$	0	0
$359$	2.85888e29	1.18198	0.590991	−	0.806678i	$-0.298737\pi$
$359$	2.85888e29	1.18198	0.590991	+	0.806678i	$0.298737\pi$
$360$	0	0
$361$	3.75917e29	1.45801
$362$	0	0
$363$	4.42023e29	1.60886
$364$	0	0
$365$	1.40259e28	0.0479251
$366$	0	0
$367$	4.88292e29	1.56682	0.783412	−	0.621503i	$-0.213477\pi$
$367$	4.88292e29	1.56682	0.783412	+	0.621503i	$0.213477\pi$
$368$	0	0
$369$	−4.74666e29	−1.43082
$370$	0	0
$371$	−8.73816e29	−2.47526
$372$	0	0
$373$	−2.49697e29	−0.664909	−0.332454	−	0.943119i	$-0.607877\pi$
$373$	−2.49697e29	−0.664909	−0.332454	+	0.943119i	$0.607877\pi$
$374$	0	0
$375$	6.60514e29	1.65395
$376$	0	0
$377$	−5.96328e29	−1.40462
$378$	0	0
$379$	−2.20712e29	−0.489187	−0.244594	−	0.969626i	$-0.578655\pi$
$379$	−2.20712e29	−0.489187	−0.244594	+	0.969626i	$0.578655\pi$
$380$	0	0
$381$	7.18775e29	1.49953
$382$	0	0
$383$	−6.94526e29	−1.36428	−0.682138	−	0.731223i	$-0.738950\pi$
$383$	−6.94526e29	−1.36428	−0.682138	+	0.731223i	$0.738950\pi$
$384$	0	0
$385$	9.59363e28	0.177494
$386$	0	0
$387$	2.99013e29	0.521211
$388$	0	0
$389$	8.61479e29	1.41522	0.707611	−	0.706603i	$-0.249773\pi$
$389$	8.61479e29	1.41522	0.707611	+	0.706603i	$0.249773\pi$
$390$	0	0
$391$	5.12129e29	0.793134
$392$	0	0
$393$	1.52370e30	2.22528
$394$	0	0
$395$	7.64782e28	0.105359
$396$	0	0
$397$	1.44976e30	1.88454	0.942268	−	0.334859i	$-0.108689\pi$
$397$	1.44976e30	1.88454	0.942268	+	0.334859i	$0.108689\pi$
$398$	0	0
$399$	−3.30017e30	−4.04901
$400$	0	0
$401$	7.75694e29	0.898525	0.449263	−	0.893400i	$-0.351687\pi$
$401$	7.75694e29	0.898525	0.449263	+	0.893400i	$0.351687\pi$
$402$	0	0
$403$	−1.62669e30	−1.77949
$404$	0	0
$405$	2.32551e29	0.240317
$406$	0	0
$407$	−1.03495e29	−0.101061
$408$	0	0
$409$	−1.06372e30	−0.981771	−0.490885	−	0.871224i	$-0.663327\pi$
$409$	−1.06372e30	−0.981771	−0.490885	+	0.871224i	$0.663327\pi$
$410$	0	0
$411$	1.72044e30	1.50127
$412$	0	0
$413$	4.27106e29	0.352460
$414$	0	0
$415$	1.31168e30	1.02394
$416$	0	0
$417$	−1.04239e30	−0.769957
$418$	0	0
$419$	−8.04610e29	−0.562502	−0.281251	−	0.959634i	$-0.590749\pi$
$419$	−8.04610e29	−0.562502	−0.281251	+	0.959634i	$0.590749\pi$
$420$	0	0
$421$	−2.65647e29	−0.175817	−0.0879085	−	0.996129i	$-0.528018\pi$
$421$	−2.65647e29	−0.175817	−0.0879085	+	0.996129i	$0.528018\pi$
$422$	0	0
$423$	−2.33430e30	−1.46300
$424$	0	0
$425$	1.32967e30	0.789355
$426$	0	0
$427$	−9.13939e29	−0.514039
$428$	0	0
$429$	−1.11977e30	−0.596859
$430$	0	0
$431$	2.39185e30	1.20850	0.604248	−	0.796796i	$-0.293474\pi$
$431$	2.39185e30	1.20850	0.604248	+	0.796796i	$0.293474\pi$
$432$	0	0
$433$	9.92622e29	0.475525	0.237762	−	0.971323i	$-0.423586\pi$
$433$	9.92622e29	0.475525	0.237762	+	0.971323i	$0.423586\pi$
$434$	0	0
$435$	1.65241e30	0.750745
$436$	0	0
$437$	−2.30263e30	−0.992402
$438$	0	0
$439$	1.20991e30	0.494779	0.247389	−	0.968916i	$-0.420427\pi$
$439$	1.20991e30	0.494779	0.247389	+	0.968916i	$0.420427\pi$
$440$	0	0
$441$	6.41632e30	2.49025
$442$	0	0
$443$	1.85839e30	0.684689	0.342345	−	0.939574i	$-0.388779\pi$
$443$	1.85839e30	0.684689	0.342345	+	0.939574i	$0.388779\pi$
$444$	0	0
$445$	−2.98445e30	−1.04406
$446$	0	0
$447$	6.30060e30	2.09337
$448$	0	0
$449$	3.30462e30	1.04301	0.521505	−	0.853248i	$-0.325371\pi$
$449$	3.30462e30	1.04301	0.521505	+	0.853248i	$0.325371\pi$
$450$	0	0
$451$	−5.04426e29	−0.151275
$452$	0	0
$453$	8.59828e30	2.45065
$454$	0	0
$455$	6.59917e30	1.78796
$456$	0	0
$457$	−4.15869e30	−1.07132	−0.535662	−	0.844433i	$-0.679938\pi$
$457$	−4.15869e30	−1.07132	−0.535662	+	0.844433i	$0.679938\pi$
$458$	0	0
$459$	6.67670e30	1.63575
$460$	0	0
$461$	2.22605e30	0.518769	0.259385	−	0.965774i	$-0.416480\pi$
$461$	2.22605e30	0.518769	0.259385	+	0.965774i	$0.416480\pi$
$462$	0	0
$463$	−5.72903e29	−0.127028	−0.0635140	−	0.997981i	$-0.520231\pi$
$463$	−5.72903e29	−0.127028	−0.0635140	+	0.997981i	$0.520231\pi$
$464$	0	0
$465$	4.50753e30	0.951107
$466$	0	0
$467$	−5.48593e30	−1.10181	−0.550904	−	0.834568i	$-0.685717\pi$
$467$	−5.48593e30	−1.10181	−0.550904	+	0.834568i	$0.685717\pi$
$468$	0	0
$469$	−1.10078e31	−2.10481
$470$	0	0
$471$	−9.67320e30	−1.76129
$472$	0	0
$473$	3.17760e29	0.0551056
$474$	0	0
$475$	−5.97847e30	−0.987673
$476$	0	0
$477$	−1.81052e31	−2.84998
$478$	0	0
$479$	−8.66235e30	−1.29950	−0.649751	−	0.760147i	$-0.725127\pi$
$479$	−8.66235e30	−1.29950	−0.649751	+	0.760147i	$0.725127\pi$
$480$	0	0
$481$	−7.11913e30	−1.01802
$482$	0	0
$483$	1.19907e31	1.63476
$484$	0	0
$485$	5.10347e30	0.663492
$486$	0	0
$487$	−1.21554e31	−1.50726	−0.753628	−	0.657302i	$-0.771698\pi$
$487$	−1.21554e31	−1.50726	−0.753628	+	0.657302i	$0.771698\pi$
$488$	0	0
$489$	−1.96040e31	−2.31896
$490$	0	0
$491$	−3.29482e28	−0.00371873	−0.00185937	−	0.999998i	$-0.500592\pi$
$491$	−3.29482e28	−0.00371873	−0.00185937	+	0.999998i	$0.500592\pi$
$492$	0	0
$493$	8.60677e30	0.927044
$494$	0	0
$495$	1.98777e30	0.204365
$496$	0	0
$497$	3.78920e30	0.371920
$498$	0	0
$499$	9.61289e30	0.900944	0.450472	−	0.892791i	$-0.351256\pi$
$499$	9.61289e30	0.900944	0.450472	+	0.892791i	$0.351256\pi$
$500$	0	0
$501$	−8.38623e30	−0.750642
$502$	0	0
$503$	9.02710e30	0.771821	0.385910	−	0.922536i	$-0.373887\pi$
$503$	9.02710e30	0.771821	0.385910	+	0.922536i	$0.373887\pi$
$504$	0	0
$505$	2.99248e30	0.244444
$506$	0	0
$507$	−5.56552e31	−4.34424
$508$	0	0
$509$	1.58741e30	0.118422	0.0592111	−	0.998245i	$-0.481141\pi$
$509$	1.58741e30	0.118422	0.0592111	+	0.998245i	$0.481141\pi$
$510$	0	0
$511$	1.71048e30	0.121977
$512$	0	0
$513$	−3.00197e31	−2.04671
$514$	0	0
$515$	1.06433e31	0.693892
$516$	0	0
$517$	−2.48066e30	−0.154677
$518$	0	0
$519$	1.98794e30	0.118571
$520$	0	0
$521$	−1.23432e31	−0.704359	−0.352180	−	0.935932i	$-0.614559\pi$
$521$	−1.23432e31	−0.704359	−0.352180	+	0.935932i	$0.614559\pi$
$522$	0	0
$523$	−4.25303e30	−0.232236	−0.116118	−	0.993235i	$-0.537045\pi$
$523$	−4.25303e30	−0.232236	−0.116118	+	0.993235i	$0.537045\pi$
$524$	0	0
$525$	3.11323e31	1.62697
$526$	0	0
$527$	2.34779e31	1.17446
$528$	0	0
$529$	−1.25142e31	−0.599325
$530$	0	0
$531$	8.84951e30	0.405819
$532$	0	0
$533$	−3.46979e31	−1.52384
$534$	0	0
$535$	−2.69364e31	−1.13311
$536$	0	0
$537$	−2.00289e31	−0.807148
$538$	0	0
$539$	6.81860e30	0.263284
$540$	0	0
$541$	3.41841e31	1.26490	0.632449	−	0.774602i	$-0.282050\pi$
$541$	3.41841e31	1.26490	0.632449	+	0.774602i	$0.282050\pi$
$542$	0	0
$543$	1.00648e31	0.356949
$544$	0	0
$545$	−2.06951e31	−0.703571
$546$	0	0
$547$	1.59831e31	0.520964	0.260482	−	0.965479i	$-0.416119\pi$
$547$	1.59831e31	0.520964	0.260482	+	0.965479i	$0.416119\pi$
$548$	0	0
$549$	−1.89365e31	−0.591860
$550$	0	0
$551$	−3.86977e31	−1.15996
$552$	0	0
$553$	9.32664e30	0.268155
$554$	0	0
$555$	1.97270e31	0.544115
$556$	0	0
$557$	−3.92140e31	−1.03778	−0.518892	−	0.854840i	$-0.673655\pi$
$557$	−3.92140e31	−1.03778	−0.518892	+	0.854840i	$0.673655\pi$
$558$	0	0
$559$	2.18577e31	0.555098
$560$	0	0
$561$	1.61616e31	0.393924
$562$	0	0
$563$	4.60230e31	1.07678	0.538392	−	0.842694i	$-0.319032\pi$
$563$	4.60230e31	1.07678	0.538392	+	0.842694i	$0.319032\pi$
$564$	0	0
$565$	−2.81399e31	−0.632070
$566$	0	0
$567$	2.83599e31	0.611645
$568$	0	0
$569$	−3.03202e31	−0.627973	−0.313987	−	0.949427i	$-0.601665\pi$
$569$	−3.03202e31	−0.627973	−0.313987	+	0.949427i	$0.601665\pi$
$570$	0	0
$571$	−2.94262e31	−0.585355	−0.292678	−	0.956211i	$-0.594546\pi$
$571$	−2.94262e31	−0.585355	−0.292678	+	0.956211i	$0.594546\pi$
$572$	0	0
$573$	−7.27140e31	−1.38944
$574$	0	0
$575$	2.17219e31	0.398766
$576$	0	0
$577$	2.29264e31	0.404402	0.202201	−	0.979344i	$-0.435191\pi$
$577$	2.29264e31	0.404402	0.202201	+	0.979344i	$0.435191\pi$
$578$	0	0
$579$	6.15291e31	1.04298
$580$	0	0
$581$	1.59962e32	2.60610
$582$	0	0
$583$	−1.92404e31	−0.301318
$584$	0	0
$585$	1.36733e32	2.05864
$586$	0	0
$587$	1.21386e32	1.75723	0.878617	−	0.477528i	$-0.158467\pi$
$587$	1.21386e32	1.75723	0.878617	+	0.477528i	$0.158467\pi$
$588$	0	0
$589$	−1.05561e32	−1.46953
$590$	0	0
$591$	1.51250e32	2.02507
$592$	0	0
$593$	−2.23484e31	−0.287818	−0.143909	−	0.989591i	$-0.545967\pi$
$593$	−2.23484e31	−0.287818	−0.143909	+	0.989591i	$0.545967\pi$
$594$	0	0
$595$	−9.52454e31	−1.18005
$596$	0	0
$597$	4.66795e31	0.556444
$598$	0	0
$599$	−5.92710e31	−0.679882	−0.339941	−	0.940447i	$-0.610407\pi$
$599$	−5.92710e31	−0.679882	−0.339941	+	0.940447i	$0.610407\pi$
$600$	0	0
$601$	−1.80233e31	−0.198965	−0.0994826	−	0.995039i	$-0.531719\pi$
$601$	−1.80233e31	−0.198965	−0.0994826	+	0.995039i	$0.531719\pi$
$602$	0	0
$603$	−2.28079e32	−2.42346
$604$	0	0
$605$	−5.73584e31	−0.586692
$606$	0	0
$607$	−7.97871e30	−0.0785711	−0.0392855	−	0.999228i	$-0.512508\pi$
$607$	−7.97871e30	−0.0785711	−0.0392855	+	0.999228i	$0.512508\pi$
$608$	0	0
$609$	2.01514e32	1.91076
$610$	0	0
$611$	−1.70637e32	−1.55811
$612$	0	0
$613$	−1.68391e32	−1.48089	−0.740444	−	0.672118i	$-0.765385\pi$
$613$	−1.68391e32	−1.48089	−0.740444	+	0.672118i	$0.765385\pi$
$614$	0	0
$615$	9.61472e31	0.814466
$616$	0	0
$617$	2.12959e32	1.73787	0.868935	−	0.494926i	$-0.164805\pi$
$617$	2.12959e32	1.73787	0.868935	+	0.494926i	$0.164805\pi$
$618$	0	0
$619$	9.74772e31	0.766409	0.383204	−	0.923664i	$-0.374821\pi$
$619$	9.74772e31	0.766409	0.383204	+	0.923664i	$0.374821\pi$
$620$	0	0
$621$	1.09072e32	0.826346
$622$	0	0
$623$	−3.63959e32	−2.65729
$624$	0	0
$625$	3.81399e30	0.0268386
$626$	0	0
$627$	−7.26658e31	−0.492894
$628$	0	0
$629$	1.02750e32	0.671891
$630$	0	0
$631$	−2.94906e32	−1.85929	−0.929643	−	0.368461i	$-0.879885\pi$
$631$	−2.94906e32	−1.85929	−0.929643	+	0.368461i	$0.879885\pi$
$632$	0	0
$633$	−1.64793e32	−1.00183
$634$	0	0
$635$	−9.32706e31	−0.546823
$636$	0	0
$637$	4.69031e32	2.65215
$638$	0	0
$639$	7.85111e31	0.428224
$640$	0	0
$641$	1.19392e32	0.628217	0.314108	−	0.949387i	$-0.398294\pi$
$641$	1.19392e32	0.628217	0.314108	+	0.949387i	$0.398294\pi$
$642$	0	0
$643$	1.99156e32	1.01104	0.505518	−	0.862816i	$-0.331301\pi$
$643$	1.99156e32	1.01104	0.505518	+	0.862816i	$0.331301\pi$
$644$	0	0
$645$	−6.05674e31	−0.296689
$646$	0	0
$647$	−1.92622e32	−0.910556	−0.455278	−	0.890349i	$-0.650460\pi$
$647$	−1.92622e32	−0.910556	−0.455278	+	0.890349i	$0.650460\pi$
$648$	0	0
$649$	9.40434e30	0.0429056
$650$	0	0
$651$	5.49700e32	2.42072
$652$	0	0
$653$	4.16657e31	0.177124	0.0885619	−	0.996071i	$-0.471773\pi$
$653$	4.16657e31	0.177124	0.0885619	+	0.996071i	$0.471773\pi$
$654$	0	0
$655$	−1.97720e32	−0.811476
$656$	0	0
$657$	3.54407e31	0.140443
$658$	0	0
$659$	−4.24683e32	−1.62511	−0.812555	−	0.582884i	$-0.801924\pi$
$659$	−4.24683e32	−1.62511	−0.812555	+	0.582884i	$0.801924\pi$
$660$	0	0
$661$	3.11799e32	1.15228	0.576141	−	0.817351i	$-0.304558\pi$
$661$	3.11799e32	1.15228	0.576141	+	0.817351i	$0.304558\pi$
$662$	0	0
$663$	1.11171e33	3.96813
$664$	0	0
$665$	4.28242e32	1.47652
$666$	0	0
$667$	1.40603e32	0.468324
$668$	0	0
$669$	4.17149e32	1.34243
$670$	0	0
$671$	−2.01238e31	−0.0625750
$672$	0	0
$673$	−3.88818e32	−1.16835	−0.584175	−	0.811628i	$-0.698582\pi$
$673$	−3.88818e32	−1.16835	−0.584175	+	0.811628i	$0.698582\pi$
$674$	0	0
$675$	2.83192e32	0.822408
$676$	0	0
$677$	1.63425e31	0.0458722	0.0229361	−	0.999737i	$-0.492699\pi$
$677$	1.63425e31	0.0458722	0.0229361	+	0.999737i	$0.492699\pi$
$678$	0	0
$679$	6.22376e32	1.68869
$680$	0	0
$681$	−3.51840e32	−0.922897
$682$	0	0
$683$	−3.62303e32	−0.918828	−0.459414	−	0.888222i	$-0.651941\pi$
$683$	−3.62303e32	−0.918828	−0.459414	+	0.888222i	$0.651941\pi$
$684$	0	0
$685$	−2.23249e32	−0.547456
$686$	0	0
$687$	−1.99118e32	−0.472181
$688$	0	0
$689$	−1.32349e33	−3.03527
$690$	0	0
$691$	−2.07306e32	−0.459849	−0.229924	−	0.973208i	$-0.573848\pi$
$691$	−2.07306e32	−0.459849	−0.229924	+	0.973208i	$0.573848\pi$
$692$	0	0
$693$	2.42412e32	0.520142
$694$	0	0
$695$	1.35264e32	0.280775
$696$	0	0
$697$	5.00793e32	1.00573
$698$	0	0
$699$	2.83196e32	0.550300
$700$	0	0
$701$	−8.02325e32	−1.50867	−0.754333	−	0.656492i	$-0.772040\pi$
$701$	−8.02325e32	−1.50867	−0.754333	+	0.656492i	$0.772040\pi$
$702$	0	0
$703$	−4.61983e32	−0.840698
$704$	0	0
$705$	4.72831e32	0.832782
$706$	0	0
$707$	3.64937e32	0.622149
$708$	0	0
$709$	−8.91866e31	−0.147186	−0.0735932	−	0.997288i	$-0.523447\pi$
$709$	−8.91866e31	−0.147186	−0.0735932	+	0.997288i	$0.523447\pi$
$710$	0	0
$711$	1.93245e32	0.308751
$712$	0	0
$713$	3.83542e32	0.593312
$714$	0	0
$715$	1.45306e32	0.217652
$716$	0	0
$717$	1.88981e33	2.74125
$718$	0	0
$719$	−5.03762e30	−0.00707691	−0.00353845	−	0.999994i	$-0.501126\pi$
$719$	−5.03762e30	−0.00707691	−0.00353845	+	0.999994i	$0.501126\pi$
$720$	0	0
$721$	1.29796e33	1.76606
$722$	0	0
$723$	−1.04405e32	−0.137603
$724$	0	0
$725$	3.65055e32	0.466092
$726$	0	0
$727$	−5.51919e32	−0.682700	−0.341350	−	0.939936i	$-0.610884\pi$
$727$	−5.51919e32	−0.682700	−0.341350	+	0.939936i	$0.610884\pi$
$728$	0	0
$729$	−1.22940e33	−1.47342
$730$	0	0
$731$	−3.15471e32	−0.366362
$732$	0	0
$733$	7.03898e32	0.792162	0.396081	−	0.918216i	$-0.370370\pi$
$733$	7.03898e32	0.792162	0.396081	+	0.918216i	$0.370370\pi$
$734$	0	0
$735$	−1.29968e33	−1.41752
$736$	0	0
$737$	−2.42378e32	−0.256223
$738$	0	0
$739$	−1.44259e33	−1.47820	−0.739100	−	0.673596i	$-0.764749\pi$
$739$	−1.44259e33	−1.47820	−0.739100	+	0.673596i	$0.764749\pi$
$740$	0	0
$741$	−4.99846e33	−4.96508
$742$	0	0
$743$	−5.64689e32	−0.543799	−0.271899	−	0.962326i	$-0.587652\pi$
$743$	−5.64689e32	−0.543799	−0.271899	+	0.962326i	$0.587652\pi$
$744$	0	0
$745$	−8.17587e32	−0.763373
$746$	0	0
$747$	3.31436e33	3.00064
$748$	0	0
$749$	−3.28493e33	−2.88394
$750$	0	0
$751$	−4.11304e31	−0.0350190	−0.0175095	−	0.999847i	$-0.505574\pi$
$751$	−4.11304e31	−0.0350190	−0.0175095	+	0.999847i	$0.505574\pi$
$752$	0	0
$753$	9.10671e32	0.752003
$754$	0	0
$755$	−1.11574e33	−0.893662
$756$	0	0
$757$	6.53195e32	0.507505	0.253752	−	0.967269i	$-0.418335\pi$
$757$	6.53195e32	0.507505	0.253752	+	0.967269i	$0.418335\pi$
$758$	0	0
$759$	2.64021e32	0.199002
$760$	0	0
$761$	−1.07363e33	−0.785114	−0.392557	−	0.919728i	$-0.628409\pi$
$761$	−1.07363e33	−0.785114	−0.392557	+	0.919728i	$0.628409\pi$
$762$	0	0
$763$	−2.52380e33	−1.79070
$764$	0	0
$765$	−1.97346e33	−1.35869
$766$	0	0
$767$	6.46896e32	0.432203
$768$	0	0
$769$	−1.39400e33	−0.903880	−0.451940	−	0.892048i	$-0.649268\pi$
$769$	−1.39400e33	−0.903880	−0.451940	+	0.892048i	$0.649268\pi$
$770$	0	0
$771$	7.12148e31	0.0448172
$772$	0	0
$773$	1.65421e33	1.01048	0.505238	−	0.862980i	$-0.331405\pi$
$773$	1.65421e33	1.01048	0.505238	+	0.862980i	$0.331405\pi$
$774$	0	0
$775$	9.95814e32	0.590485
$776$	0	0
$777$	2.40573e33	1.38486
$778$	0	0
$779$	−2.25166e33	−1.25841
$780$	0	0
$781$	8.34335e31	0.0452745
$782$	0	0
$783$	1.83306e33	0.965863
$784$	0	0
$785$	1.25523e33	0.642275
$786$	0	0
$787$	−3.36348e33	−1.67140	−0.835699	−	0.549188i	$-0.814937\pi$
$787$	−3.36348e33	−1.67140	−0.835699	+	0.549188i	$0.814937\pi$
$788$	0	0
$789$	−2.33310e33	−1.12603
$790$	0	0
$791$	−3.43170e33	−1.60872
$792$	0	0
$793$	−1.38425e33	−0.630339
$794$	0	0
$795$	3.66735e33	1.62230
$796$	0	0
$797$	2.33147e33	1.00198	0.500990	−	0.865453i	$-0.332969\pi$
$797$	2.33147e33	1.00198	0.500990	+	0.865453i	$0.332969\pi$
$798$	0	0
$799$	2.46279e33	1.02835
$800$	0	0
$801$	−7.54112e33	−3.05958
$802$	0	0
$803$	3.76627e31	0.0148485
$804$	0	0
$805$	−1.55596e33	−0.596135
$806$	0	0
$807$	−2.37038e33	−0.882618
$808$	0	0
$809$	4.54875e33	1.64621	0.823103	−	0.567893i	$-0.192241\pi$
$809$	4.54875e33	1.64621	0.823103	+	0.567893i	$0.192241\pi$
$810$	0	0
$811$	−3.34564e32	−0.117690	−0.0588449	−	0.998267i	$-0.518742\pi$
$811$	−3.34564e32	−0.117690	−0.0588449	+	0.998267i	$0.518742\pi$
$812$	0	0
$813$	−7.90519e32	−0.270315
$814$	0	0
$815$	2.54388e33	0.845637
$816$	0	0
$817$	1.41842e33	0.458407
$818$	0	0
$819$	1.66748e34	5.23957
$820$	0	0
$821$	−3.36447e32	−0.102794	−0.0513972	−	0.998678i	$-0.516367\pi$
$821$	−3.36447e32	−0.102794	−0.0513972	+	0.998678i	$0.516367\pi$
$822$	0	0
$823$	−2.25997e33	−0.671436	−0.335718	−	0.941962i	$-0.608979\pi$
$823$	−2.25997e33	−0.671436	−0.335718	+	0.941962i	$0.608979\pi$
$824$	0	0
$825$	6.85494e32	0.198054
$826$	0	0
$827$	6.77365e32	0.190331	0.0951654	−	0.995461i	$-0.469662\pi$
$827$	6.77365e32	0.190331	0.0951654	+	0.995461i	$0.469662\pi$
$828$	0	0
$829$	1.84647e33	0.504620	0.252310	−	0.967646i	$-0.418810\pi$
$829$	1.84647e33	0.504620	0.252310	+	0.967646i	$0.418810\pi$
$830$	0	0
$831$	3.49470e33	0.928962
$832$	0	0
$833$	−6.76950e33	−1.75041
$834$	0	0
$835$	1.08823e33	0.273731
$836$	0	0
$837$	5.00029e33	1.22364
$838$	0	0
$839$	5.81074e33	1.38347	0.691734	−	0.722152i	$-0.256847\pi$
$839$	5.81074e33	1.38347	0.691734	+	0.722152i	$0.256847\pi$
$840$	0	0
$841$	−1.95377e33	−0.452606
$842$	0	0
$843$	−1.12978e34	−2.54670
$844$	0	0
$845$	7.22200e33	1.58418
$846$	0	0
$847$	−6.99494e33	−1.49322
$848$	0	0
$849$	−6.24691e33	−1.29786
$850$	0	0
$851$	1.67855e33	0.339426
$852$	0	0
$853$	−5.35584e33	−1.05418	−0.527088	−	0.849811i	$-0.676716\pi$
$853$	−5.35584e33	−1.05418	−0.527088	+	0.849811i	$0.676716\pi$
$854$	0	0
$855$	8.87304e33	1.70005
$856$	0	0
$857$	5.93588e33	1.10715	0.553574	−	0.832800i	$-0.313264\pi$
$857$	5.93588e33	1.10715	0.553574	+	0.832800i	$0.313264\pi$
$858$	0	0
$859$	1.31091e32	0.0238041	0.0119021	−	0.999929i	$-0.496211\pi$
$859$	1.31091e32	0.0238041	0.0119021	+	0.999929i	$0.496211\pi$
$860$	0	0
$861$	1.17253e34	2.07295
$862$	0	0
$863$	9.12023e32	0.156994	0.0784968	−	0.996914i	$-0.474988\pi$
$863$	9.12023e32	0.156994	0.0784968	+	0.996914i	$0.474988\pi$
$864$	0	0
$865$	−2.57962e32	−0.0432384
$866$	0	0
$867$	−5.82538e33	−0.950833
$868$	0	0
$869$	2.05361e32	0.0326430
$870$	0	0
$871$	−1.66725e34	−2.58102
$872$	0	0
$873$	1.28955e34	1.94434
$874$	0	0
$875$	−1.04525e34	−1.53507
$876$	0	0
$877$	3.03947e33	0.434813	0.217406	−	0.976081i	$-0.430240\pi$
$877$	3.03947e33	0.434813	0.217406	+	0.976081i	$0.430240\pi$
$878$	0	0
$879$	4.33060e33	0.603499
$880$	0	0
$881$	−1.71133e33	−0.232332	−0.116166	−	0.993230i	$-0.537061\pi$
$881$	−1.71133e33	−0.232332	−0.116166	+	0.993230i	$0.537061\pi$
$882$	0	0
$883$	−7.11572e33	−0.941175	−0.470588	−	0.882353i	$-0.655958\pi$
$883$	−7.11572e33	−0.941175	−0.470588	+	0.882353i	$0.655958\pi$
$884$	0	0
$885$	−1.79254e33	−0.231004
$886$	0	0
$887$	−3.28147e33	−0.412047	−0.206024	−	0.978547i	$-0.566052\pi$
$887$	−3.28147e33	−0.412047	−0.206024	+	0.978547i	$0.566052\pi$
$888$	0	0
$889$	−1.13745e34	−1.39175
$890$	0	0
$891$	6.24450e32	0.0744567
$892$	0	0
$893$	−1.10732e34	−1.28671
$894$	0	0
$895$	2.59902e33	0.294337
$896$	0	0
$897$	1.81612e34	2.00462
$898$	0	0
$899$	6.44575e33	0.693485
$900$	0	0
$901$	1.91018e34	2.00327
$902$	0	0
$903$	−7.38628e33	−0.755121
$904$	0	0
$905$	−1.30604e33	−0.130166
$906$	0	0
$907$	−1.23436e34	−1.19938	−0.599690	−	0.800232i	$-0.704710\pi$
$907$	−1.23436e34	−1.19938	−0.599690	+	0.800232i	$0.704710\pi$
$908$	0	0
$909$	7.56139e33	0.716336
$910$	0	0
$911$	−1.89282e34	−1.74843	−0.874217	−	0.485536i	$-0.838624\pi$
$911$	−1.89282e34	−1.74843	−0.874217	+	0.485536i	$0.838624\pi$
$912$	0	0
$913$	3.52216e33	0.317245
$914$	0	0
$915$	3.83574e33	0.336905
$916$	0	0
$917$	−2.41123e34	−2.06533
$918$	0	0
$919$	−2.38841e33	−0.199517	−0.0997585	−	0.995012i	$-0.531807\pi$
$919$	−2.38841e33	−0.199517	−0.0997585	+	0.995012i	$0.531807\pi$
$920$	0	0
$921$	3.00683e34	2.44976
$922$	0	0
$923$	5.73914e33	0.456065
$924$	0	0
$925$	4.35813e33	0.337808
$926$	0	0
$927$	2.68934e34	2.03343
$928$	0	0
$929$	2.25176e34	1.66089	0.830446	−	0.557100i	$-0.188086\pi$
$929$	2.25176e34	1.66089	0.830446	+	0.557100i	$0.188086\pi$
$930$	0	0
$931$	3.04370e34	2.19018
$932$	0	0
$933$	−1.05746e34	−0.742381
$934$	0	0
$935$	−2.09719e33	−0.143649
$936$	0	0
$937$	−5.08802e33	−0.340051	−0.170025	−	0.985440i	$-0.554385\pi$
$937$	−5.08802e33	−0.340051	−0.170025	+	0.985440i	$0.554385\pi$
$938$	0	0
$939$	2.85280e34	1.86045
$940$	0	0
$941$	1.48815e34	0.947036	0.473518	−	0.880784i	$-0.342984\pi$
$941$	1.48815e34	0.947036	0.473518	+	0.880784i	$0.342984\pi$
$942$	0	0
$943$	8.18109e33	0.508074
$944$	0	0
$945$	−2.02852e34	−1.22946
$946$	0	0
$947$	−3.96465e33	−0.234520	−0.117260	−	0.993101i	$-0.537411\pi$
$947$	−3.96465e33	−0.234520	−0.117260	+	0.993101i	$0.537411\pi$
$948$	0	0
$949$	2.59070e33	0.149574
$950$	0	0
$951$	8.55942e33	0.482357
$952$	0	0
$953$	6.43683e33	0.354082	0.177041	−	0.984203i	$-0.443347\pi$
$953$	6.43683e33	0.354082	0.177041	+	0.984203i	$0.443347\pi$
$954$	0	0
$955$	9.43561e33	0.506677
$956$	0	0
$957$	4.43710e33	0.232601
$958$	0	0
$959$	−2.72256e34	−1.39336
$960$	0	0
$961$	−2.43031e33	−0.121435
$962$	0	0
$963$	−6.80629e34	−3.32054
$964$	0	0
$965$	−7.98422e33	−0.380337
$966$	0	0
$967$	2.72016e34	1.26529	0.632646	−	0.774441i	$-0.281969\pi$
$967$	2.72016e34	1.26529	0.632646	+	0.774441i	$0.281969\pi$
$968$	0	0
$969$	7.21425e34	3.27693
$970$	0	0
$971$	−1.62936e34	−0.722762	−0.361381	−	0.932418i	$-0.617695\pi$
$971$	−1.62936e34	−0.722762	−0.361381	+	0.932418i	$0.617695\pi$
$972$	0	0
$973$	1.64957e34	0.714616
$974$	0	0
$975$	4.71531e34	1.99506
$976$	0	0
$977$	−1.48817e34	−0.614987	−0.307493	−	0.951550i	$-0.599490\pi$
$977$	−1.48817e34	−0.614987	−0.307493	+	0.951550i	$0.599490\pi$
$978$	0	0
$979$	−8.01392e33	−0.323477
$980$	0	0
$981$	−5.22924e34	−2.06179
$982$	0	0
$983$	1.58730e34	0.611353	0.305677	−	0.952135i	$-0.401117\pi$
$983$	1.58730e34	0.611353	0.305677	+	0.952135i	$0.401117\pi$
$984$	0	0
$985$	−1.96267e34	−0.738467
$986$	0	0
$987$	5.76625e34	2.11956
$988$	0	0
$989$	−5.15363e33	−0.185078
$990$	0	0
$991$	−1.49367e34	−0.524092	−0.262046	−	0.965055i	$-0.584397\pi$
$991$	−1.49367e34	−0.524092	−0.262046	+	0.965055i	$0.584397\pi$
$992$	0	0
$993$	−9.58198e34	−3.28503
$994$	0	0
$995$	−6.05729e33	−0.202914
$996$	0	0
$997$	−3.34649e34	−1.09546	−0.547729	−	0.836656i	$-0.684507\pi$
$997$	−3.34649e34	−1.09546	−0.547729	+	0.836656i	$0.684507\pi$
$998$	0	0
$999$	2.18835e34	0.700026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.l.1.1		4
4.3	odd	2	inner	64.24.a.l.1.4		4
8.3	odd	2		32.24.a.b.1.1	✓	4
8.5	even	2		32.24.a.b.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.b.1.1	✓	4	8.3	odd	2
32.24.a.b.1.4	yes	4	8.5	even	2
64.24.a.l.1.1		4	1.1	even	1	trivial
64.24.a.l.1.4		4	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)

\(f(q)\)	\(=\)	\(q-511823. q^{3} +6.64159e7 q^{5} +8.09952e9 q^{7} +1.67820e11 q^{9} +O(q^{10})\)	\(q-511823. q^{3} +6.64159e7 q^{5} +8.09952e9 q^{7} +1.67820e11 q^{9} +1.78341e11 q^{11} +1.22676e13 q^{13} -3.39932e13 q^{15} -1.77057e14 q^{17} +7.96082e14 q^{19} -4.14552e15 q^{21} -2.89245e15 q^{23} -7.50986e15 q^{25} -3.77093e16 q^{27} -4.86101e16 q^{29} -1.32601e17 q^{31} -9.12792e16 q^{33} +5.37937e17 q^{35} -5.80321e17 q^{37} -6.27882e18 q^{39} -2.82843e18 q^{41} +1.78175e18 q^{43} +1.11459e19 q^{45} -1.39096e19 q^{47} +3.82334e19 q^{49} +9.06219e19 q^{51} -1.07885e20 q^{53} +1.18447e19 q^{55} -4.07453e20 q^{57} +5.27322e19 q^{59} -1.12839e20 q^{61} +1.35926e21 q^{63} +8.14761e20 q^{65} -1.35907e21 q^{67} +1.48042e21 q^{69} +4.67830e20 q^{71} +2.11183e20 q^{73} +3.84372e21 q^{75} +1.44448e21 q^{77} +1.15151e21 q^{79} +3.50143e21 q^{81} +1.97495e22 q^{83} -1.17594e22 q^{85} +2.48798e22 q^{87} -4.49358e22 q^{89} +9.93613e22 q^{91} +6.78682e22 q^{93} +5.28725e22 q^{95} +7.68411e22 q^{97} +2.99292e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 19990040 q^{5} + 284960925396 q^{9}+O(q^{10}) \) 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

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