LMFDB - Embedded newform 64.22.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,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, 22, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
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:	$178.865500344$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5201320x^{3} - 466399708x^{2} + 4990572086304x - 1473608896916400 $ x^5 - 2x^4 - 5201320x^3 - 466399708x^2 + 4990572086304x - 1473608896916400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{2}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1518.36$ 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)

$f(q)$	$=$	$q+53009.9 q^{3} +2.95789e7 q^{5} +3.32722e8 q^{7} -7.65031e9 q^{9} +O(q^{10})$	$q+53009.9 q^{3} +2.95789e7 q^{5} +3.32722e8 q^{7} -7.65031e9 q^{9} -6.27044e10 q^{11} +1.52750e11 q^{13} +1.56798e12 q^{15} -1.24833e12 q^{17} -1.80843e13 q^{19} +1.76376e13 q^{21} +3.13338e14 q^{23} +3.98076e14 q^{25} -9.60044e14 q^{27} -5.19001e14 q^{29} -6.73042e15 q^{31} -3.32395e15 q^{33} +9.84158e15 q^{35} -5.29964e16 q^{37} +8.09726e15 q^{39} -4.78851e16 q^{41} -6.97865e16 q^{43} -2.26288e17 q^{45} +2.25682e17 q^{47} -4.47842e17 q^{49} -6.61738e16 q^{51} +1.23702e17 q^{53} -1.85473e18 q^{55} -9.58644e17 q^{57} +2.43181e18 q^{59} +1.66446e18 q^{61} -2.54543e18 q^{63} +4.51818e18 q^{65} -2.07498e19 q^{67} +1.66100e19 q^{69} -2.24426e18 q^{71} +3.73065e19 q^{73} +2.11020e19 q^{75} -2.08632e19 q^{77} -1.44576e20 q^{79} +2.91331e19 q^{81} +1.23634e20 q^{83} -3.69243e19 q^{85} -2.75122e19 q^{87} -5.06510e20 q^{89} +5.08234e19 q^{91} -3.56779e20 q^{93} -5.34913e20 q^{95} +9.26865e18 q^{97} +4.79708e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 23144 q^{3} - 18311174 q^{5} - 63978640 q^{7} + 2839988161 q^{9}+O(q^{10}) $ 5 * q + 23144 * q^3 - 18311174 * q^5 - 63978640 * q^7 + 2839988161 * q^9	$ 5 q + 23144 q^{3} - 18311174 q^{5} - 63978640 q^{7} + 2839988161 q^{9} + 25629588280 q^{11} - 26739996110 q^{13} - 850898706352 q^{15} - 88104593910 q^{17} + 13998239618440 q^{19} - 11868255565952 q^{21} - 191593435978416 q^{23} - 41142227484149 q^{25} + 436694294703248 q^{27} - 229446229587518 q^{29} + 30\!\cdots\!60 q^{31}+ \cdots + 45\!\cdots\!40 q^{99}+O(q^{100}) $ 5 * q + 23144 * q^3 - 18311174 * q^5 - 63978640 * q^7 + 2839988161 * q^9 + 25629588280 * q^11 - 26739996110 * q^13 - 850898706352 * q^15 - 88104593910 * q^17 + 13998239618440 * q^19 - 11868255565952 * q^21 - 191593435978416 * q^23 - 41142227484149 * q^25 + 436694294703248 * q^27 - 229446229587518 * q^29 + 3073873710228160 * q^31 + 5433145544904640 * q^33 - 8500692544521248 * q^35 + 11689624248645370 * q^37 - 5755218754185200 * q^39 + 73561781960295890 * q^41 + 59173471060274168 * q^43 + 110307182017064626 * q^45 - 189587882278171232 * q^47 + 198072470828678205 * q^49 + 60381606670165200 * q^51 + 448473144842566250 * q^53 + 276218022333741296 * q^55 - 419697547444918720 * q^57 - 580240954920275560 * q^59 - 60707231290899774 * q^61 - 7329049347459246928 * q^63 - 4215936008336579644 * q^65 + 20784270726401480552 * q^67 - 13151945925537436032 * q^69 - 48791036649307600400 * q^71 - 7687983545663740350 * q^73 + 63701791920635021848 * q^75 - 14991304956050303360 * q^77 - 150036082528549569440 * q^79 + 6961005137522997901 * q^81 + 320036481947166592392 * q^83 + 55351169878564568052 * q^85 - 845870037277169219440 * q^87 + 157308716989748010706 * q^89 + 1517228856976775217760 * q^91 + 285928010618243909120 * q^93 - 2644313286399100711792 * q^95 - 165820666835121495910 * q^97 + 4544456840208531002840 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	53009.9	0.518303	0.259151	−	0.965837i	$-0.416557\pi$
$3$	53009.9	0.518303	0.259151	+	0.965837i	$0.416557\pi$
$4$	0	0
$5$	2.95789e7	1.35456	0.677279	−	0.735726i	$-0.263159\pi$
$5$	2.95789e7	1.35456	0.677279	+	0.735726i	$0.263159\pi$
$6$	0	0
$7$	3.32722e8	0.445197	0.222599	−	0.974910i	$-0.428546\pi$
$7$	3.32722e8	0.445197	0.222599	+	0.974910i	$0.428546\pi$
$8$	0	0
$9$	−7.65031e9	−0.731362
$10$	0	0
$11$	−6.27044e10	−0.728912	−0.364456	−	0.931221i	$-0.618745\pi$
$11$	−6.27044e10	−0.728912	−0.364456	+	0.931221i	$0.618745\pi$
$12$	0	0
$13$	1.52750e11	0.307310	0.153655	−	0.988125i	$-0.450896\pi$
$13$	1.52750e11	0.307310	0.153655	+	0.988125i	$0.450896\pi$
$14$	0	0
$15$	1.56798e12	0.702071
$16$	0	0
$17$	−1.24833e12	−0.150181	−0.0750906	−	0.997177i	$-0.523925\pi$
$17$	−1.24833e12	−0.150181	−0.0750906	+	0.997177i	$0.523925\pi$
$18$	0	0
$19$	−1.80843e13	−0.676687	−0.338344	−	0.941023i	$-0.609867\pi$
$19$	−1.80843e13	−0.676687	−0.338344	+	0.941023i	$0.609867\pi$
$20$	0	0
$21$	1.76376e13	0.230747
$22$	0	0
$23$	3.13338e14	1.57714	0.788571	−	0.614944i	$-0.210822\pi$
$23$	3.13338e14	1.57714	0.788571	+	0.614944i	$0.210822\pi$
$24$	0	0
$25$	3.98076e14	0.834827
$26$	0	0
$27$	−9.60044e14	−0.897370
$28$	0	0
$29$	−5.19001e14	−0.229081	−0.114541	−	0.993419i	$-0.536540\pi$
$29$	−5.19001e14	−0.229081	−0.114541	+	0.993419i	$0.536540\pi$
$30$	0	0
$31$	−6.73042e15	−1.47484	−0.737419	−	0.675435i	$-0.763956\pi$
$31$	−6.73042e15	−1.47484	−0.737419	+	0.675435i	$0.763956\pi$
$32$	0	0
$33$	−3.32395e15	−0.377797
$34$	0	0
$35$	9.84158e15	0.603046
$36$	0	0
$37$	−5.29964e16	−1.81187	−0.905937	−	0.423412i	$-0.860832\pi$
$37$	−5.29964e16	−1.81187	−0.905937	+	0.423412i	$0.860832\pi$
$38$	0	0
$39$	8.09726e15	0.159279
$40$	0	0
$41$	−4.78851e16	−0.557147	−0.278573	−	0.960415i	$-0.589862\pi$
$41$	−4.78851e16	−0.557147	−0.278573	+	0.960415i	$0.589862\pi$
$42$	0	0
$43$	−6.97865e16	−0.492439	−0.246220	−	0.969214i	$-0.579188\pi$
$43$	−6.97865e16	−0.492439	−0.246220	+	0.969214i	$0.579188\pi$
$44$	0	0
$45$	−2.26288e17	−0.990672
$46$	0	0
$47$	2.25682e17	0.625850	0.312925	−	0.949778i	$-0.398691\pi$
$47$	2.25682e17	0.625850	0.312925	+	0.949778i	$0.398691\pi$
$48$	0	0
$49$	−4.47842e17	−0.801799
$50$	0	0
$51$	−6.61738e16	−0.0778393
$52$	0	0
$53$	1.23702e17	0.0971587	0.0485793	−	0.998819i	$-0.484531\pi$
$53$	1.23702e17	0.0971587	0.0485793	+	0.998819i	$0.484531\pi$
$54$	0	0
$55$	−1.85473e18	−0.987353
$56$	0	0
$57$	−9.58644e17	−0.350729
$58$	0	0
$59$	2.43181e18	0.619419	0.309709	−	0.950831i	$-0.399768\pi$
$59$	2.43181e18	0.619419	0.309709	+	0.950831i	$0.399768\pi$
$60$	0	0
$61$	1.66446e18	0.298752	0.149376	−	0.988780i	$-0.452274\pi$
$61$	1.66446e18	0.298752	0.149376	+	0.988780i	$0.452274\pi$
$62$	0	0
$63$	−2.54543e18	−0.325601
$64$	0	0
$65$	4.51818e18	0.416269
$66$	0	0
$67$	−2.07498e19	−1.39069	−0.695344	−	0.718677i	$-0.744748\pi$
$67$	−2.07498e19	−1.39069	−0.695344	+	0.718677i	$0.744748\pi$
$68$	0	0
$69$	1.66100e19	0.817437
$70$	0	0
$71$	−2.24426e18	−0.0818202	−0.0409101	−	0.999163i	$-0.513026\pi$
$71$	−2.24426e18	−0.0818202	−0.0409101	+	0.999163i	$0.513026\pi$
$72$	0	0
$73$	3.73065e19	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$73$	3.73065e19	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$74$	0	0
$75$	2.11020e19	0.432693
$76$	0	0
$77$	−2.08632e19	−0.324510
$78$	0	0
$79$	−1.44576e20	−1.71795	−0.858977	−	0.512014i	$-0.828899\pi$
$79$	−1.44576e20	−1.71795	−0.858977	+	0.512014i	$0.828899\pi$
$80$	0	0
$81$	2.91331e19	0.266253
$82$	0	0
$83$	1.23634e20	0.874617	0.437308	−	0.899312i	$-0.355932\pi$
$83$	1.23634e20	0.874617	0.437308	+	0.899312i	$0.355932\pi$
$84$	0	0
$85$	−3.69243e19	−0.203429
$86$	0	0
$87$	−2.75122e19	−0.118733
$88$	0	0
$89$	−5.06510e20	−1.72184	−0.860920	−	0.508740i	$-0.830112\pi$
$89$	−5.06510e20	−1.72184	−0.860920	+	0.508740i	$0.830112\pi$
$90$	0	0
$91$	5.08234e19	0.136813
$92$	0	0
$93$	−3.56779e20	−0.764413
$94$	0	0
$95$	−5.34913e20	−0.916612
$96$	0	0
$97$	9.26865e18	0.0127618	0.00638092	−	0.999980i	$-0.497969\pi$
$97$	9.26865e18	0.0127618	0.00638092	+	0.999980i	$0.497969\pi$
$98$	0	0
$99$	4.79708e20	0.533099
$100$	0	0
$101$	5.13116e20	0.462212	0.231106	−	0.972929i	$-0.425766\pi$
$101$	5.13116e20	0.462212	0.231106	+	0.972929i	$0.425766\pi$
$102$	0	0
$103$	−1.61132e21	−1.18138	−0.590692	−	0.806897i	$-0.701145\pi$
$103$	−1.61132e21	−1.18138	−0.590692	+	0.806897i	$0.701145\pi$
$104$	0	0
$105$	5.21701e20	0.312560
$106$	0	0
$107$	3.75411e21	1.84492	0.922461	−	0.386091i	$-0.126175\pi$
$107$	3.75411e21	1.84492	0.922461	+	0.386091i	$0.126175\pi$
$108$	0	0
$109$	1.29915e21	0.525630	0.262815	−	0.964846i	$-0.415349\pi$
$109$	1.29915e21	0.525630	0.262815	+	0.964846i	$0.415349\pi$
$110$	0	0
$111$	−2.80933e21	−0.939100
$112$	0	0
$113$	4.13155e21	1.14496	0.572479	−	0.819919i	$-0.305982\pi$
$113$	4.13155e21	1.14496	0.572479	+	0.819919i	$0.305982\pi$
$114$	0	0
$115$	9.26820e21	2.13633
$116$	0	0
$117$	−1.16858e21	−0.224755
$118$	0	0
$119$	−4.15347e20	−0.0668603
$120$	0	0
$121$	−3.46840e21	−0.468687
$122$	0	0
$123$	−2.53838e21	−0.288771
$124$	0	0
$125$	−2.32966e21	−0.223737
$126$	0	0
$127$	−7.00077e21	−0.569124	−0.284562	−	0.958658i	$-0.591848\pi$
$127$	−7.00077e21	−0.569124	−0.284562	+	0.958658i	$0.591848\pi$
$128$	0	0
$129$	−3.69937e21	−0.255233
$130$	0	0
$131$	1.04027e22	0.610658	0.305329	−	0.952247i	$-0.401234\pi$
$131$	1.04027e22	0.610658	0.305329	+	0.952247i	$0.401234\pi$
$132$	0	0
$133$	−6.01704e21	−0.301259
$134$	0	0
$135$	−2.83971e22	−1.21554
$136$	0	0
$137$	−2.71282e22	−0.995074	−0.497537	−	0.867443i	$-0.665762\pi$
$137$	−2.71282e22	−0.995074	−0.497537	+	0.867443i	$0.665762\pi$
$138$	0	0
$139$	2.65027e22	0.834900	0.417450	−	0.908700i	$-0.362924\pi$
$139$	2.65027e22	0.834900	0.417450	+	0.908700i	$0.362924\pi$
$140$	0	0
$141$	1.19634e22	0.324380
$142$	0	0
$143$	−9.57810e21	−0.224002
$144$	0	0
$145$	−1.53515e22	−0.310304
$146$	0	0
$147$	−2.37400e22	−0.415575
$148$	0	0
$149$	−1.07285e23	−1.62960	−0.814801	−	0.579740i	$-0.803154\pi$
$149$	−1.07285e23	−1.62960	−0.814801	+	0.579740i	$0.803154\pi$
$150$	0	0
$151$	7.58410e22	1.00149	0.500744	−	0.865595i	$-0.333060\pi$
$151$	7.58410e22	1.00149	0.500744	+	0.865595i	$0.333060\pi$
$152$	0	0
$153$	9.55010e21	0.109837
$154$	0	0
$155$	−1.99079e23	−1.99775
$156$	0	0
$157$	9.71280e22	0.851919	0.425959	−	0.904742i	$-0.359937\pi$
$157$	9.71280e22	0.851919	0.425959	+	0.904742i	$0.359937\pi$
$158$	0	0
$159$	6.55745e21	0.0503576
$160$	0	0
$161$	1.04255e23	0.702139
$162$	0	0
$163$	−5.91920e22	−0.350181	−0.175091	−	0.984552i	$-0.556022\pi$
$163$	−5.91920e22	−0.350181	−0.175091	+	0.984552i	$0.556022\pi$
$164$	0	0
$165$	−9.83190e22	−0.511748
$166$	0	0
$167$	2.51685e23	1.15434	0.577171	−	0.816623i	$-0.304156\pi$
$167$	2.51685e23	1.15434	0.577171	+	0.816623i	$0.304156\pi$
$168$	0	0
$169$	−2.23732e23	−0.905561
$170$	0	0
$171$	1.38350e23	0.494903
$172$	0	0
$173$	1.44685e23	0.458077	0.229038	−	0.973417i	$-0.426442\pi$
$173$	1.44685e23	0.458077	0.229038	+	0.973417i	$0.426442\pi$
$174$	0	0
$175$	1.32449e23	0.371663
$176$	0	0
$177$	1.28910e23	0.321046
$178$	0	0
$179$	−8.03318e23	−1.77800	−0.888998	−	0.457912i	$-0.848598\pi$
$179$	−8.03318e23	−1.77800	−0.888998	+	0.457912i	$0.848598\pi$
$180$	0	0
$181$	−7.78569e23	−1.53346	−0.766730	−	0.641970i	$-0.778117\pi$
$181$	−7.78569e23	−1.53346	−0.766730	+	0.641970i	$0.778117\pi$
$182$	0	0
$183$	8.82330e22	0.154844
$184$	0	0
$185$	−1.56758e24	−2.45429
$186$	0	0
$187$	7.82758e22	0.109469
$188$	0	0
$189$	−3.19428e23	−0.399507
$190$	0	0
$191$	1.64699e24	1.84434	0.922170	−	0.386784i	$-0.126414\pi$
$191$	1.64699e24	1.84434	0.922170	+	0.386784i	$0.126414\pi$
$192$	0	0
$193$	−5.10555e23	−0.512497	−0.256248	−	0.966611i	$-0.582487\pi$
$193$	−5.10555e23	−0.512497	−0.256248	+	0.966611i	$0.582487\pi$
$194$	0	0
$195$	2.39508e23	0.215753
$196$	0	0
$197$	−1.76803e24	−1.43085	−0.715423	−	0.698691i	$-0.753766\pi$
$197$	−1.76803e24	−1.43085	−0.715423	+	0.698691i	$0.753766\pi$
$198$	0	0
$199$	−3.60621e23	−0.262478	−0.131239	−	0.991351i	$-0.541896\pi$
$199$	−3.60621e23	−0.262478	−0.131239	+	0.991351i	$0.541896\pi$
$200$	0	0
$201$	−1.09995e24	−0.720797
$202$	0	0
$203$	−1.72683e23	−0.101986
$204$	0	0
$205$	−1.41639e24	−0.754688
$206$	0	0
$207$	−2.39713e24	−1.15346
$208$	0	0
$209$	1.13396e24	0.493245
$210$	0	0
$211$	4.22345e24	1.66227	0.831134	−	0.556072i	$-0.187692\pi$
$211$	4.22345e24	1.66227	0.831134	+	0.556072i	$0.187692\pi$
$212$	0	0
$213$	−1.18968e23	−0.0424077
$214$	0	0
$215$	−2.06421e24	−0.667038
$216$	0	0
$217$	−2.23936e24	−0.656594
$218$	0	0
$219$	1.97761e24	0.526596
$220$	0	0
$221$	−1.90682e23	−0.0461521
$222$	0	0
$223$	−4.74166e24	−1.04407	−0.522035	−	0.852924i	$-0.674827\pi$
$223$	−4.74166e24	−1.04407	−0.522035	+	0.852924i	$0.674827\pi$
$224$	0	0
$225$	−3.04541e24	−0.610561
$226$	0	0
$227$	−7.49257e24	−1.36886	−0.684430	−	0.729078i	$-0.739949\pi$
$227$	−7.49257e24	−1.36886	−0.684430	+	0.729078i	$0.739949\pi$
$228$	0	0
$229$	−5.72142e24	−0.953304	−0.476652	−	0.879092i	$-0.658150\pi$
$229$	−5.72142e24	−0.953304	−0.476652	+	0.879092i	$0.658150\pi$
$230$	0	0
$231$	−1.10595e24	−0.168194
$232$	0	0
$233$	−9.45292e24	−1.31319	−0.656597	−	0.754242i	$-0.728005\pi$
$233$	−9.45292e24	−1.31319	−0.656597	+	0.754242i	$0.728005\pi$
$234$	0	0
$235$	6.67545e24	0.847750
$236$	0	0
$237$	−7.66395e24	−0.890420
$238$	0	0
$239$	−7.10010e24	−0.755243	−0.377621	−	0.925960i	$-0.623258\pi$
$239$	−7.10010e24	−0.755243	−0.377621	+	0.925960i	$0.623258\pi$
$240$	0	0
$241$	−1.80854e25	−1.76258	−0.881292	−	0.472571i	$-0.843326\pi$
$241$	−1.80854e25	−1.76258	−0.881292	+	0.472571i	$0.843326\pi$
$242$	0	0
$243$	1.15867e25	1.03537
$244$	0	0
$245$	−1.32467e25	−1.08608
$246$	0	0
$247$	−2.76237e24	−0.207952
$248$	0	0
$249$	6.55382e24	0.453317
$250$	0	0
$251$	1.14492e25	0.728114	0.364057	−	0.931377i	$-0.381391\pi$
$251$	1.14492e25	0.728114	0.364057	+	0.931377i	$0.381391\pi$
$252$	0	0
$253$	−1.96477e25	−1.14960
$254$	0	0
$255$	−1.95735e24	−0.105438
$256$	0	0
$257$	−2.05124e25	−1.01793	−0.508967	−	0.860786i	$-0.669973\pi$
$257$	−2.05124e25	−1.01793	−0.508967	+	0.860786i	$0.669973\pi$
$258$	0	0
$259$	−1.76331e25	−0.806642
$260$	0	0
$261$	3.97052e24	0.167541
$262$	0	0
$263$	3.17067e25	1.23485	0.617426	−	0.786629i	$-0.288175\pi$
$263$	3.17067e25	1.23485	0.617426	+	0.786629i	$0.288175\pi$
$264$	0	0
$265$	3.65899e24	0.131607
$266$	0	0
$267$	−2.68500e25	−0.892435
$268$	0	0
$269$	−1.18093e25	−0.362933	−0.181467	−	0.983397i	$-0.558084\pi$
$269$	−1.18093e25	−0.362933	−0.181467	+	0.983397i	$0.558084\pi$
$270$	0	0
$271$	2.13816e25	0.607942	0.303971	−	0.952681i	$-0.401687\pi$
$271$	2.13816e25	0.607942	0.303971	+	0.952681i	$0.401687\pi$
$272$	0	0
$273$	2.69414e24	0.0709108
$274$	0	0
$275$	−2.49612e25	−0.608515
$276$	0	0
$277$	2.02955e25	0.458525	0.229262	−	0.973365i	$-0.426369\pi$
$277$	2.02955e25	0.458525	0.229262	+	0.973365i	$0.426369\pi$
$278$	0	0
$279$	5.14898e25	1.07864
$280$	0	0
$281$	2.51217e25	0.488239	0.244120	−	0.969745i	$-0.421501\pi$
$281$	2.51217e25	0.488239	0.244120	+	0.969745i	$0.421501\pi$
$282$	0	0
$283$	−2.14920e25	−0.387722	−0.193861	−	0.981029i	$-0.562101\pi$
$283$	−2.14920e25	−0.387722	−0.193861	+	0.981029i	$0.562101\pi$
$284$	0	0
$285$	−2.83557e25	−0.475083
$286$	0	0
$287$	−1.59324e25	−0.248040
$288$	0	0
$289$	−6.75336e25	−0.977446
$290$	0	0
$291$	4.91330e23	0.00661450
$292$	0	0
$293$	1.26861e26	1.58934	0.794672	−	0.607039i	$-0.207643\pi$
$293$	1.26861e26	1.58934	0.794672	+	0.607039i	$0.207643\pi$
$294$	0	0
$295$	7.19305e25	0.839038
$296$	0	0
$297$	6.01990e25	0.654104
$298$	0	0
$299$	4.78624e25	0.484671
$300$	0	0
$301$	−2.32195e25	−0.219233
$302$	0	0
$303$	2.72002e25	0.239566
$304$	0	0
$305$	4.92330e25	0.404677
$306$	0	0
$307$	−1.70099e26	−1.30541	−0.652707	−	0.757610i	$-0.726367\pi$
$307$	−1.70099e26	−1.30541	−0.652707	+	0.757610i	$0.726367\pi$
$308$	0	0
$309$	−8.54159e25	−0.612314
$310$	0	0
$311$	−1.72297e26	−1.15423	−0.577116	−	0.816662i	$-0.695822\pi$
$311$	−1.72297e26	−1.15423	−0.577116	+	0.816662i	$0.695822\pi$
$312$	0	0
$313$	−6.58864e25	−0.412648	−0.206324	−	0.978484i	$-0.566150\pi$
$313$	−6.58864e25	−0.412648	−0.206324	+	0.978484i	$0.566150\pi$
$314$	0	0
$315$	−7.52911e25	−0.441045
$316$	0	0
$317$	−2.60230e25	−0.142638	−0.0713188	−	0.997454i	$-0.522721\pi$
$317$	−2.60230e25	−0.142638	−0.0713188	+	0.997454i	$0.522721\pi$
$318$	0	0
$319$	3.25437e25	0.166980
$320$	0	0
$321$	1.99005e26	0.956228
$322$	0	0
$323$	2.25751e25	0.101626
$324$	0	0
$325$	6.08062e25	0.256550
$326$	0	0
$327$	6.88676e25	0.272435
$328$	0	0
$329$	7.50896e25	0.278627
$330$	0	0
$331$	4.25686e26	1.48216	0.741081	−	0.671416i	$-0.234314\pi$
$331$	4.25686e26	1.48216	0.741081	+	0.671416i	$0.234314\pi$
$332$	0	0
$333$	4.05439e26	1.32514
$334$	0	0
$335$	−6.13758e26	−1.88377
$336$	0	0
$337$	2.48118e26	0.715393	0.357697	−	0.933838i	$-0.383562\pi$
$337$	2.48118e26	0.715393	0.357697	+	0.933838i	$0.383562\pi$
$338$	0	0
$339$	2.19013e26	0.593435
$340$	0	0
$341$	4.22027e26	1.07503
$342$	0	0
$343$	−3.34848e26	−0.802156
$344$	0	0
$345$	4.91306e26	1.10727
$346$	0	0
$347$	5.54944e26	1.17703	0.588517	−	0.808485i	$-0.299712\pi$
$347$	5.54944e26	1.17703	0.588517	+	0.808485i	$0.299712\pi$
$348$	0	0
$349$	6.93278e26	1.38433	0.692167	−	0.721738i	$-0.256656\pi$
$349$	6.93278e26	1.38433	0.692167	+	0.721738i	$0.256656\pi$
$350$	0	0
$351$	−1.46647e26	−0.275770
$352$	0	0
$353$	6.89384e26	1.22131	0.610656	−	0.791896i	$-0.290906\pi$
$353$	6.89384e26	1.22131	0.610656	+	0.791896i	$0.290906\pi$
$354$	0	0
$355$	−6.63828e25	−0.110830
$356$	0	0
$357$	−2.20175e25	−0.0346539
$358$	0	0
$359$	6.70652e26	0.995419	0.497709	−	0.867344i	$-0.334175\pi$
$359$	6.70652e26	0.995419	0.497709	+	0.867344i	$0.334175\pi$
$360$	0	0
$361$	−3.87169e26	−0.542094
$362$	0	0
$363$	−1.83860e26	−0.242922
$364$	0	0
$365$	1.10349e27	1.37623
$366$	0	0
$367$	3.20467e26	0.377390	0.188695	−	0.982036i	$-0.439574\pi$
$367$	3.20467e26	0.377390	0.188695	+	0.982036i	$0.439574\pi$
$368$	0	0
$369$	3.66336e26	0.407476
$370$	0	0
$371$	4.11586e25	0.0432548
$372$	0	0
$373$	−1.79883e27	−1.78668	−0.893341	−	0.449379i	$-0.851645\pi$
$373$	−1.79883e27	−1.78668	−0.893341	+	0.449379i	$0.851645\pi$
$374$	0	0
$375$	−1.23495e26	−0.115963
$376$	0	0
$377$	−7.92774e25	−0.0703988
$378$	0	0
$379$	6.23107e26	0.523421	0.261710	−	0.965146i	$-0.415713\pi$
$379$	6.23107e26	0.523421	0.261710	+	0.965146i	$0.415713\pi$
$380$	0	0
$381$	−3.71110e26	−0.294978
$382$	0	0
$383$	2.80212e26	0.210814	0.105407	−	0.994429i	$-0.466385\pi$
$383$	2.80212e26	0.210814	0.105407	+	0.994429i	$0.466385\pi$
$384$	0	0
$385$	−6.17111e26	−0.439567
$386$	0	0
$387$	5.33888e26	0.360152
$388$	0	0
$389$	1.19882e27	0.766098	0.383049	−	0.923728i	$-0.374874\pi$
$389$	1.19882e27	0.766098	0.383049	+	0.923728i	$0.374874\pi$
$390$	0	0
$391$	−3.91149e26	−0.236857
$392$	0	0
$393$	5.51446e26	0.316506
$394$	0	0
$395$	−4.27640e27	−2.32707
$396$	0	0
$397$	−1.42805e27	−0.736958	−0.368479	−	0.929636i	$-0.620121\pi$
$397$	−1.42805e27	−0.736958	−0.368479	+	0.929636i	$0.620121\pi$
$398$	0	0
$399$	−3.18962e26	−0.156144
$400$	0	0
$401$	1.10892e27	0.515091	0.257546	−	0.966266i	$-0.417086\pi$
$401$	1.10892e27	0.515091	0.257546	+	0.966266i	$0.417086\pi$
$402$	0	0
$403$	−1.02807e27	−0.453232
$404$	0	0
$405$	8.61726e26	0.360655
$406$	0	0
$407$	3.32311e27	1.32070
$408$	0	0
$409$	−2.54008e27	−0.958853	−0.479427	−	0.877582i	$-0.659155\pi$
$409$	−2.54008e27	−0.958853	−0.479427	+	0.877582i	$0.659155\pi$
$410$	0	0
$411$	−1.43806e27	−0.515749
$412$	0	0
$413$	8.09119e26	0.275764
$414$	0	0
$415$	3.65696e27	1.18472
$416$	0	0
$417$	1.40490e27	0.432731
$418$	0	0
$419$	−1.79759e27	−0.526554	−0.263277	−	0.964720i	$-0.584803\pi$
$419$	−1.79759e27	−0.526554	−0.263277	+	0.964720i	$0.584803\pi$
$420$	0	0
$421$	1.56180e27	0.435174	0.217587	−	0.976041i	$-0.430181\pi$
$421$	1.56180e27	0.435174	0.217587	+	0.976041i	$0.430181\pi$
$422$	0	0
$423$	−1.72654e27	−0.457723
$424$	0	0
$425$	−4.96931e26	−0.125375
$426$	0	0
$427$	5.53804e26	0.133004
$428$	0	0
$429$	−5.07734e26	−0.116101
$430$	0	0
$431$	2.73052e27	0.594613	0.297306	−	0.954782i	$-0.403912\pi$
$431$	2.73052e27	0.594613	0.297306	+	0.954782i	$0.403912\pi$
$432$	0	0
$433$	−7.86036e27	−1.63050	−0.815248	−	0.579112i	$-0.803399\pi$
$433$	−7.86036e27	−1.63050	−0.815248	+	0.579112i	$0.803399\pi$
$434$	0	0
$435$	−8.13781e26	−0.160831
$436$	0	0
$437$	−5.66648e27	−1.06723
$438$	0	0
$439$	−6.41517e27	−1.15168	−0.575839	−	0.817563i	$-0.695324\pi$
$439$	−6.41517e27	−1.15168	−0.575839	+	0.817563i	$0.695324\pi$
$440$	0	0
$441$	3.42613e27	0.586406
$442$	0	0
$443$	5.36336e27	0.875382	0.437691	−	0.899126i	$-0.355796\pi$
$443$	5.36336e27	0.875382	0.437691	+	0.899126i	$0.355796\pi$
$444$	0	0
$445$	−1.49820e28	−2.33233
$446$	0	0
$447$	−5.68714e27	−0.844628
$448$	0	0
$449$	−4.38024e27	−0.620742	−0.310371	−	0.950615i	$-0.600453\pi$
$449$	−4.38024e27	−0.620742	−0.310371	+	0.950615i	$0.600453\pi$
$450$	0	0
$451$	3.00261e27	0.406111
$452$	0	0
$453$	4.02032e27	0.519074
$454$	0	0
$455$	1.50330e27	0.185322
$456$	0	0
$457$	−8.26672e27	−0.973224	−0.486612	−	0.873618i	$-0.661767\pi$
$457$	−8.26672e27	−0.973224	−0.486612	+	0.873618i	$0.661767\pi$
$458$	0	0
$459$	1.19845e27	0.134768
$460$	0	0
$461$	−6.75450e27	−0.725660	−0.362830	−	0.931855i	$-0.618189\pi$
$461$	−6.75450e27	−0.725660	−0.362830	+	0.931855i	$0.618189\pi$
$462$	0	0
$463$	−1.69592e27	−0.174102	−0.0870512	−	0.996204i	$-0.527744\pi$
$463$	−1.69592e27	−0.174102	−0.0870512	+	0.996204i	$0.527744\pi$
$464$	0	0
$465$	−1.05531e28	−1.03544
$466$	0	0
$467$	1.48387e28	1.39177	0.695887	−	0.718151i	$-0.255011\pi$
$467$	1.48387e28	1.39177	0.695887	+	0.718151i	$0.255011\pi$
$468$	0	0
$469$	−6.90394e27	−0.619130
$470$	0	0
$471$	5.14874e27	0.441552
$472$	0	0
$473$	4.37592e27	0.358945
$474$	0	0
$475$	−7.19892e27	−0.564917
$476$	0	0
$477$	−9.46362e26	−0.0710582
$478$	0	0
$479$	−4.92513e27	−0.353912	−0.176956	−	0.984219i	$-0.556625\pi$
$479$	−4.92513e27	−0.353912	−0.176956	+	0.984219i	$0.556625\pi$
$480$	0	0
$481$	−8.09520e27	−0.556806
$482$	0	0
$483$	5.52652e27	0.363921
$484$	0	0
$485$	2.74157e26	0.0172867
$486$	0	0
$487$	−2.09921e28	−1.26766	−0.633829	−	0.773473i	$-0.718518\pi$
$487$	−2.09921e28	−1.26766	−0.633829	+	0.773473i	$0.718518\pi$
$488$	0	0
$489$	−3.13776e27	−0.181500
$490$	0	0
$491$	2.25505e28	1.24969	0.624843	−	0.780751i	$-0.285163\pi$
$491$	2.25505e28	1.24969	0.624843	+	0.780751i	$0.285163\pi$
$492$	0	0
$493$	6.47884e26	0.0344037
$494$	0	0
$495$	1.41893e28	0.722113
$496$	0	0
$497$	−7.46716e26	−0.0364261
$498$	0	0
$499$	1.54863e28	0.724255	0.362128	−	0.932129i	$-0.382050\pi$
$499$	1.54863e28	0.724255	0.362128	+	0.932129i	$0.382050\pi$
$500$	0	0
$501$	1.33418e28	0.598299
$502$	0	0
$503$	−3.39955e28	−1.46204	−0.731018	−	0.682358i	$-0.760954\pi$
$503$	−3.39955e28	−1.46204	−0.731018	+	0.682358i	$0.760954\pi$
$504$	0	0
$505$	1.51774e28	0.626093
$506$	0	0
$507$	−1.18600e28	−0.469355
$508$	0	0
$509$	1.77751e28	0.674955	0.337477	−	0.941334i	$-0.390426\pi$
$509$	1.77751e28	0.674955	0.337477	+	0.941334i	$0.390426\pi$
$510$	0	0
$511$	1.24127e28	0.452321
$512$	0	0
$513$	1.73617e28	0.607239
$514$	0	0
$515$	−4.76611e28	−1.60025
$516$	0	0
$517$	−1.41513e28	−0.456190
$518$	0	0
$519$	7.66971e27	0.237423
$520$	0	0
$521$	−1.25825e28	−0.374087	−0.187043	−	0.982352i	$-0.559890\pi$
$521$	−1.25825e28	−0.374087	−0.187043	+	0.982352i	$0.559890\pi$
$522$	0	0
$523$	−5.62597e28	−1.60668	−0.803341	−	0.595519i	$-0.796946\pi$
$523$	−5.62597e28	−1.60668	−0.803341	+	0.595519i	$0.796946\pi$
$524$	0	0
$525$	7.02110e27	0.192634
$526$	0	0
$527$	8.40178e27	0.221493
$528$	0	0
$529$	5.87090e28	1.48737
$530$	0	0
$531$	−1.86041e28	−0.453019
$532$	0	0
$533$	−7.31445e27	−0.171217
$534$	0	0
$535$	1.11043e29	2.49905
$536$	0	0
$537$	−4.25838e28	−0.921540
$538$	0	0
$539$	2.80817e28	0.584441
$540$	0	0
$541$	9.96351e27	0.199453	0.0997266	−	0.995015i	$-0.468203\pi$
$541$	9.96351e27	0.199453	0.0997266	+	0.995015i	$0.468203\pi$
$542$	0	0
$543$	−4.12719e28	−0.794796
$544$	0	0
$545$	3.84274e28	0.711996
$546$	0	0
$547$	3.81673e28	0.680494	0.340247	−	0.940336i	$-0.389489\pi$
$547$	3.81673e28	0.680494	0.340247	+	0.940336i	$0.389489\pi$
$548$	0	0
$549$	−1.27336e28	−0.218496
$550$	0	0
$551$	9.38575e27	0.155016
$552$	0	0
$553$	−4.81036e28	−0.764829
$554$	0	0
$555$	−8.30971e28	−1.27207
$556$	0	0
$557$	1.57737e28	0.232516	0.116258	−	0.993219i	$-0.462910\pi$
$557$	1.57737e28	0.232516	0.116258	+	0.993219i	$0.462910\pi$
$558$	0	0
$559$	−1.06599e28	−0.151331
$560$	0	0
$561$	4.14939e27	0.0567380
$562$	0	0
$563$	−3.72768e28	−0.491022	−0.245511	−	0.969394i	$-0.578956\pi$
$563$	−3.72768e28	−0.491022	−0.245511	+	0.969394i	$0.578956\pi$
$564$	0	0
$565$	1.22207e29	1.55091
$566$	0	0
$567$	9.69324e27	0.118535
$568$	0	0
$569$	9.86510e28	1.16258	0.581290	−	0.813697i	$-0.302548\pi$
$569$	9.86510e28	1.16258	0.581290	+	0.813697i	$0.302548\pi$
$570$	0	0
$571$	1.06187e29	1.20612	0.603061	−	0.797695i	$-0.293948\pi$
$571$	1.06187e29	1.20612	0.603061	+	0.797695i	$0.293948\pi$
$572$	0	0
$573$	8.73069e28	0.955927
$574$	0	0
$575$	1.24732e29	1.31664
$576$	0	0
$577$	1.53593e29	1.56323	0.781617	−	0.623758i	$-0.214395\pi$
$577$	1.53593e29	1.56323	0.781617	+	0.623758i	$0.214395\pi$
$578$	0	0
$579$	−2.70645e28	−0.265629
$580$	0	0
$581$	4.11358e28	0.389377
$582$	0	0
$583$	−7.75669e27	−0.0708201
$584$	0	0
$585$	−3.45655e28	−0.304443
$586$	0	0
$587$	2.08326e27	0.0177029	0.00885143	−	0.999961i	$-0.497182\pi$
$587$	2.08326e27	0.0177029	0.00885143	+	0.999961i	$0.497182\pi$
$588$	0	0
$589$	1.21715e29	0.998004
$590$	0	0
$591$	−9.37228e28	−0.741612
$592$	0	0
$593$	6.57932e28	0.502466	0.251233	−	0.967927i	$-0.419164\pi$
$593$	6.57932e28	0.502466	0.251233	+	0.967927i	$0.419164\pi$
$594$	0	0
$595$	−1.22855e28	−0.0905661
$596$	0	0
$597$	−1.91165e28	−0.136043
$598$	0	0
$599$	−1.27367e28	−0.0875137	−0.0437568	−	0.999042i	$-0.513933\pi$
$599$	−1.27367e28	−0.0875137	−0.0437568	+	0.999042i	$0.513933\pi$
$600$	0	0
$601$	2.55976e29	1.69831	0.849157	−	0.528141i	$-0.177111\pi$
$601$	2.55976e29	1.69831	0.849157	+	0.528141i	$0.177111\pi$
$602$	0	0
$603$	1.58743e29	1.01710
$604$	0	0
$605$	−1.02592e29	−0.634864
$606$	0	0
$607$	−1.63390e29	−0.976665	−0.488332	−	0.872658i	$-0.662395\pi$
$607$	−1.63390e29	−0.976665	−0.488332	+	0.872658i	$0.662395\pi$
$608$	0	0
$609$	−9.15392e27	−0.0528598
$610$	0	0
$611$	3.44730e28	0.192330
$612$	0	0
$613$	8.12188e28	0.437846	0.218923	−	0.975742i	$-0.429746\pi$
$613$	8.12188e28	0.437846	0.218923	+	0.975742i	$0.429746\pi$
$614$	0	0
$615$	−7.50827e28	−0.391157
$616$	0	0
$617$	−3.58082e29	−1.80297	−0.901484	−	0.432812i	$-0.857521\pi$
$617$	−3.58082e29	−1.80297	−0.901484	+	0.432812i	$0.857521\pi$
$618$	0	0
$619$	2.81301e29	1.36905	0.684524	−	0.728990i	$-0.260010\pi$
$619$	2.81301e29	1.36905	0.684524	+	0.728990i	$0.260010\pi$
$620$	0	0
$621$	−3.00818e29	−1.41528
$622$	0	0
$623$	−1.68527e29	−0.766559
$624$	0	0
$625$	−2.58726e29	−1.13789
$626$	0	0
$627$	6.01113e28	0.255651
$628$	0	0
$629$	6.61569e28	0.272109
$630$	0	0
$631$	−4.06289e29	−1.61632	−0.808160	−	0.588963i	$-0.799536\pi$
$631$	−4.06289e29	−1.61632	−0.808160	+	0.588963i	$0.799536\pi$
$632$	0	0
$633$	2.23884e29	0.861559
$634$	0	0
$635$	−2.07075e29	−0.770911
$636$	0	0
$637$	−6.84078e28	−0.246401
$638$	0	0
$639$	1.71693e28	0.0598402
$640$	0	0
$641$	−9.03687e28	−0.304796	−0.152398	−	0.988319i	$-0.548699\pi$
$641$	−9.03687e28	−0.304796	−0.152398	+	0.988319i	$0.548699\pi$
$642$	0	0
$643$	7.07128e28	0.230825	0.115412	−	0.993318i	$-0.463181\pi$
$643$	7.07128e28	0.230825	0.115412	+	0.993318i	$0.463181\pi$
$644$	0	0
$645$	−1.09424e29	−0.345728
$646$	0	0
$647$	4.02238e29	1.23023	0.615117	−	0.788436i	$-0.289109\pi$
$647$	4.02238e29	1.23023	0.615117	+	0.788436i	$0.289109\pi$
$648$	0	0
$649$	−1.52486e29	−0.451502
$650$	0	0
$651$	−1.18708e29	−0.340315
$652$	0	0
$653$	−2.53244e29	−0.702992	−0.351496	−	0.936189i	$-0.614327\pi$
$653$	−2.53244e29	−0.702992	−0.351496	+	0.936189i	$0.614327\pi$
$654$	0	0
$655$	3.07701e29	0.827171
$656$	0	0
$657$	−2.85406e29	−0.743065
$658$	0	0
$659$	−8.42257e28	−0.212397	−0.106198	−	0.994345i	$-0.533868\pi$
$659$	−8.42257e28	−0.212397	−0.106198	+	0.994345i	$0.533868\pi$
$660$	0	0
$661$	4.67727e29	1.14255	0.571277	−	0.820757i	$-0.306448\pi$
$661$	4.67727e29	1.14255	0.571277	+	0.820757i	$0.306448\pi$
$662$	0	0
$663$	−1.01080e28	−0.0239208
$664$	0	0
$665$	−1.77978e29	−0.408073
$666$	0	0
$667$	−1.62623e29	−0.361293
$668$	0	0
$669$	−2.51355e29	−0.541144
$670$	0	0
$671$	−1.04369e29	−0.217764
$672$	0	0
$673$	3.52317e29	0.712485	0.356243	−	0.934393i	$-0.384058\pi$
$673$	3.52317e29	0.712485	0.356243	+	0.934393i	$0.384058\pi$
$674$	0	0
$675$	−3.82171e29	−0.749149
$676$	0	0
$677$	−5.47586e28	−0.104057	−0.0520285	−	0.998646i	$-0.516569\pi$
$677$	−5.47586e28	−0.104057	−0.0520285	+	0.998646i	$0.516569\pi$
$678$	0	0
$679$	3.08389e27	0.00568154
$680$	0	0
$681$	−3.97180e29	−0.709484
$682$	0	0
$683$	2.22046e29	0.384615	0.192308	−	0.981335i	$-0.438403\pi$
$683$	2.22046e29	0.384615	0.192308	+	0.981335i	$0.438403\pi$
$684$	0	0
$685$	−8.02424e29	−1.34788
$686$	0	0
$687$	−3.03292e29	−0.494100
$688$	0	0
$689$	1.88956e28	0.0298578
$690$	0	0
$691$	−7.77439e27	−0.0119165	−0.00595823	−	0.999982i	$-0.501897\pi$
$691$	−7.77439e27	−0.0119165	−0.00595823	+	0.999982i	$0.501897\pi$
$692$	0	0
$693$	1.59610e29	0.237334
$694$	0	0
$695$	7.83922e29	1.13092
$696$	0	0
$697$	5.97764e28	0.0836730
$698$	0	0
$699$	−5.01098e29	−0.680632
$700$	0	0
$701$	−8.62775e29	−1.13726	−0.568628	−	0.822595i	$-0.692526\pi$
$701$	−8.62775e29	−1.13726	−0.568628	+	0.822595i	$0.692526\pi$
$702$	0	0
$703$	9.58401e29	1.22607
$704$	0	0
$705$	3.53865e29	0.439391
$706$	0	0
$707$	1.70725e29	0.205776
$708$	0	0
$709$	4.14287e28	0.0484748	0.0242374	−	0.999706i	$-0.492284\pi$
$709$	4.14287e28	0.0484748	0.0242374	+	0.999706i	$0.492284\pi$
$710$	0	0
$711$	1.10605e30	1.25645
$712$	0	0
$713$	−2.10890e30	−2.32603
$714$	0	0
$715$	−2.83310e29	−0.303423
$716$	0	0
$717$	−3.76375e29	−0.391444
$718$	0	0
$719$	8.02551e29	0.810624	0.405312	−	0.914178i	$-0.367163\pi$
$719$	8.02551e29	0.810624	0.405312	+	0.914178i	$0.367163\pi$
$720$	0	0
$721$	−5.36122e29	−0.525949
$722$	0	0
$723$	−9.58707e29	−0.913553
$724$	0	0
$725$	−2.06602e29	−0.191243
$726$	0	0
$727$	−5.58476e29	−0.502219	−0.251110	−	0.967959i	$-0.580795\pi$
$727$	−5.58476e29	−0.502219	−0.251110	+	0.967959i	$0.580795\pi$
$728$	0	0
$729$	3.09469e29	0.270382
$730$	0	0
$731$	8.71166e28	0.0739551
$732$	0	0
$733$	−3.98057e28	−0.0328363	−0.0164181	−	0.999865i	$-0.505226\pi$
$733$	−3.98057e28	−0.0328363	−0.0164181	+	0.999865i	$0.505226\pi$
$734$	0	0
$735$	−7.02205e29	−0.562920
$736$	0	0
$737$	1.30111e30	1.01369
$738$	0	0
$739$	1.16579e30	0.882784	0.441392	−	0.897314i	$-0.354485\pi$
$739$	1.16579e30	0.882784	0.441392	+	0.897314i	$0.354485\pi$
$740$	0	0
$741$	−1.46433e29	−0.107782
$742$	0	0
$743$	−2.09842e30	−1.50145	−0.750725	−	0.660615i	$-0.770296\pi$
$743$	−2.09842e30	−1.50145	−0.750725	+	0.660615i	$0.770296\pi$
$744$	0	0
$745$	−3.17337e30	−2.20739
$746$	0	0
$747$	−9.45838e29	−0.639662
$748$	0	0
$749$	1.24908e30	0.821354
$750$	0	0
$751$	−9.38811e29	−0.600287	−0.300143	−	0.953894i	$-0.597035\pi$
$751$	−9.38811e29	−0.600287	−0.300143	+	0.953894i	$0.597035\pi$
$752$	0	0
$753$	6.06918e29	0.377384
$754$	0	0
$755$	2.24330e30	1.35657
$756$	0	0
$757$	−1.12891e30	−0.663976	−0.331988	−	0.943284i	$-0.607719\pi$
$757$	−1.12891e30	−0.663976	−0.331988	+	0.943284i	$0.607719\pi$
$758$	0	0
$759$	−1.04152e30	−0.595839
$760$	0	0
$761$	2.29050e30	1.27465	0.637325	−	0.770595i	$-0.280041\pi$
$761$	2.29050e30	1.27465	0.637325	+	0.770595i	$0.280041\pi$
$762$	0	0
$763$	4.32255e29	0.234009
$764$	0	0
$765$	2.82482e29	0.148780
$766$	0	0
$767$	3.71460e29	0.190353
$768$	0	0
$769$	3.09201e30	1.54175	0.770876	−	0.636985i	$-0.219819\pi$
$769$	3.09201e30	1.54175	0.770876	+	0.636985i	$0.219819\pi$
$770$	0	0
$771$	−1.08736e30	−0.527598
$772$	0	0
$773$	3.22557e30	1.52308	0.761539	−	0.648119i	$-0.224444\pi$
$773$	3.22557e30	1.52308	0.761539	+	0.648119i	$0.224444\pi$
$774$	0	0
$775$	−2.67922e30	−1.23123
$776$	0	0
$777$	−9.34728e29	−0.418085
$778$	0	0
$779$	8.65966e29	0.377014
$780$	0	0
$781$	1.40725e29	0.0596397
$782$	0	0
$783$	4.98264e29	0.205570
$784$	0	0
$785$	2.87294e30	1.15397
$786$	0	0
$787$	2.89674e30	1.13286	0.566428	−	0.824111i	$-0.308325\pi$
$787$	2.89674e30	1.13286	0.566428	+	0.824111i	$0.308325\pi$
$788$	0	0
$789$	1.68077e30	0.640028
$790$	0	0
$791$	1.37466e30	0.509732
$792$	0	0
$793$	2.54247e29	0.0918093
$794$	0	0
$795$	1.93962e29	0.0682123
$796$	0	0
$797$	−2.37208e29	−0.0812488	−0.0406244	−	0.999174i	$-0.512935\pi$
$797$	−2.37208e29	−0.0812488	−0.0406244	+	0.999174i	$0.512935\pi$
$798$	0	0
$799$	−2.81726e29	−0.0939909
$800$	0	0
$801$	3.87496e30	1.25929
$802$	0	0
$803$	−2.33928e30	−0.740575
$804$	0	0
$805$	3.08374e30	0.951088
$806$	0	0
$807$	−6.26012e29	−0.188109
$808$	0	0
$809$	−1.86516e30	−0.546081	−0.273041	−	0.962003i	$-0.588029\pi$
$809$	−1.86516e30	−0.546081	−0.273041	+	0.962003i	$0.588029\pi$
$810$	0	0
$811$	3.16326e30	0.902436	0.451218	−	0.892414i	$-0.350990\pi$
$811$	3.16326e30	0.902436	0.451218	+	0.892414i	$0.350990\pi$
$812$	0	0
$813$	1.13344e30	0.315098
$814$	0	0
$815$	−1.75084e30	−0.474341
$816$	0	0
$817$	1.26204e30	0.333227
$818$	0	0
$819$	−3.88814e29	−0.100060
$820$	0	0
$821$	2.16337e30	0.542661	0.271330	−	0.962486i	$-0.412536\pi$
$821$	2.16337e30	0.542661	0.271330	+	0.962486i	$0.412536\pi$
$822$	0	0
$823$	−1.33211e28	−0.00325718	−0.00162859	−	0.999999i	$-0.500518\pi$
$823$	−1.33211e28	−0.00325718	−0.00162859	+	0.999999i	$0.500518\pi$
$824$	0	0
$825$	−1.32319e30	−0.315395
$826$	0	0
$827$	−3.01442e30	−0.700481	−0.350240	−	0.936660i	$-0.613900\pi$
$827$	−3.01442e30	−0.700481	−0.350240	+	0.936660i	$0.613900\pi$
$828$	0	0
$829$	−6.66240e30	−1.50941	−0.754707	−	0.656063i	$-0.772221\pi$
$829$	−6.66240e30	−1.50941	−0.754707	+	0.656063i	$0.772221\pi$
$830$	0	0
$831$	1.07586e30	0.237655
$832$	0	0
$833$	5.59054e29	0.120415
$834$	0	0
$835$	7.44458e30	1.56362
$836$	0	0
$837$	6.46150e30	1.32348
$838$	0	0
$839$	5.61633e30	1.12190	0.560948	−	0.827851i	$-0.310437\pi$
$839$	5.61633e30	1.12190	0.560948	+	0.827851i	$0.310437\pi$
$840$	0	0
$841$	−4.86348e30	−0.947522
$842$	0	0
$843$	1.33170e30	0.253056
$844$	0	0
$845$	−6.61775e30	−1.22663
$846$	0	0
$847$	−1.15402e30	−0.208658
$848$	0	0
$849$	−1.13929e30	−0.200957
$850$	0	0
$851$	−1.66058e31	−2.85758
$852$	0	0
$853$	7.63635e30	1.28210	0.641049	−	0.767500i	$-0.278500\pi$
$853$	7.63635e30	1.28210	0.641049	+	0.767500i	$0.278500\pi$
$854$	0	0
$855$	4.09225e30	0.670375
$856$	0	0
$857$	6.82772e30	1.09138	0.545691	−	0.837986i	$-0.316267\pi$
$857$	6.82772e30	1.09138	0.545691	+	0.837986i	$0.316267\pi$
$858$	0	0
$859$	−3.53359e30	−0.551174	−0.275587	−	0.961276i	$-0.588872\pi$
$859$	−3.53359e30	−0.551174	−0.275587	+	0.961276i	$0.588872\pi$
$860$	0	0
$861$	−8.44577e29	−0.128560
$862$	0	0
$863$	1.21032e31	1.79799	0.898993	−	0.437964i	$-0.144300\pi$
$863$	1.21032e31	1.79799	0.898993	+	0.437964i	$0.144300\pi$
$864$	0	0
$865$	4.27962e30	0.620491
$866$	0	0
$867$	−3.57995e30	−0.506613
$868$	0	0
$869$	9.06555e30	1.25224
$870$	0	0
$871$	−3.16954e30	−0.427371
$872$	0	0
$873$	−7.09080e28	−0.00933353
$874$	0	0
$875$	−7.75129e29	−0.0996069
$876$	0	0
$877$	−9.89341e30	−1.24122	−0.620612	−	0.784118i	$-0.713116\pi$
$877$	−9.89341e30	−1.24122	−0.620612	+	0.784118i	$0.713116\pi$
$878$	0	0
$879$	6.72488e30	0.823762
$880$	0	0
$881$	1.26349e31	1.51120	0.755602	−	0.655031i	$-0.227344\pi$
$881$	1.26349e31	1.51120	0.755602	+	0.655031i	$0.227344\pi$
$882$	0	0
$883$	3.82850e30	0.447138	0.223569	−	0.974688i	$-0.428229\pi$
$883$	3.82850e30	0.447138	0.223569	+	0.974688i	$0.428229\pi$
$884$	0	0
$885$	3.81303e30	0.434876
$886$	0	0
$887$	6.84704e30	0.762614	0.381307	−	0.924448i	$-0.375474\pi$
$887$	6.84704e30	0.762614	0.381307	+	0.924448i	$0.375474\pi$
$888$	0	0
$889$	−2.32931e30	−0.253372
$890$	0	0
$891$	−1.82677e30	−0.194075
$892$	0	0
$893$	−4.08130e30	−0.423505
$894$	0	0
$895$	−2.37613e31	−2.40840
$896$	0	0
$897$	2.53718e30	0.251206
$898$	0	0
$899$	3.49309e30	0.337858
$900$	0	0
$901$	−1.54421e29	−0.0145914
$902$	0	0
$903$	−1.23086e30	−0.113629
$904$	0	0
$905$	−2.30293e31	−2.07716
$906$	0	0
$907$	−5.15695e30	−0.454482	−0.227241	−	0.973839i	$-0.572970\pi$
$907$	−5.15695e30	−0.454482	−0.227241	+	0.973839i	$0.572970\pi$
$908$	0	0
$909$	−3.92550e30	−0.338044
$910$	0	0
$911$	−8.91453e30	−0.750163	−0.375081	−	0.926992i	$-0.622385\pi$
$911$	−8.91453e30	−0.750163	−0.375081	+	0.926992i	$0.622385\pi$
$912$	0	0
$913$	−7.75240e30	−0.637519
$914$	0	0
$915$	2.60984e30	0.209745
$916$	0	0
$917$	3.46121e30	0.271863
$918$	0	0
$919$	−1.72413e30	−0.132360	−0.0661800	−	0.997808i	$-0.521081\pi$
$919$	−1.72413e30	−0.132360	−0.0661800	+	0.997808i	$0.521081\pi$
$920$	0	0
$921$	−9.01692e30	−0.676600
$922$	0	0
$923$	−3.42811e29	−0.0251441
$924$	0	0
$925$	−2.10966e31	−1.51260
$926$	0	0
$927$	1.23271e31	0.864019
$928$	0	0
$929$	−7.63699e30	−0.523308	−0.261654	−	0.965162i	$-0.584268\pi$
$929$	−7.63699e30	−0.523308	−0.261654	+	0.965162i	$0.584268\pi$
$930$	0	0
$931$	8.09889e30	0.542567
$932$	0	0
$933$	−9.13343e30	−0.598242
$934$	0	0
$935$	2.31531e30	0.148282
$936$	0	0
$937$	−3.04557e30	−0.190723	−0.0953615	−	0.995443i	$-0.530401\pi$
$937$	−3.04557e30	−0.190723	−0.0953615	+	0.995443i	$0.530401\pi$
$938$	0	0
$939$	−3.49263e30	−0.213877
$940$	0	0
$941$	−1.92309e31	−1.15162	−0.575809	−	0.817584i	$-0.695313\pi$
$941$	−1.92309e31	−1.15162	−0.575809	+	0.817584i	$0.695313\pi$
$942$	0	0
$943$	−1.50042e31	−0.878699
$944$	0	0
$945$	−9.44834e30	−0.541155
$946$	0	0
$947$	4.03228e29	0.0225879	0.0112940	−	0.999936i	$-0.496405\pi$
$947$	4.03228e29	0.0225879	0.0112940	+	0.999936i	$0.496405\pi$
$948$	0	0
$949$	5.69856e30	0.312227
$950$	0	0
$951$	−1.37947e30	−0.0739295
$952$	0	0
$953$	−3.34819e31	−1.75523	−0.877617	−	0.479362i	$-0.840868\pi$
$953$	−3.34819e31	−1.75523	−0.877617	+	0.479362i	$0.840868\pi$
$954$	0	0
$955$	4.87163e31	2.49827
$956$	0	0
$957$	1.72514e30	0.0865462
$958$	0	0
$959$	−9.02616e30	−0.443004
$960$	0	0
$961$	2.44731e31	1.17515
$962$	0	0
$963$	−2.87201e31	−1.34931
$964$	0	0
$965$	−1.51017e31	−0.694207
$966$	0	0
$967$	−2.40427e31	−1.08144	−0.540722	−	0.841201i	$-0.681849\pi$
$967$	−2.40427e31	−1.08144	−0.540722	+	0.841201i	$0.681849\pi$
$968$	0	0
$969$	1.19670e30	0.0526729
$970$	0	0
$971$	2.15389e31	0.927729	0.463865	−	0.885906i	$-0.346462\pi$
$971$	2.15389e31	0.927729	0.463865	+	0.885906i	$0.346462\pi$
$972$	0	0
$973$	8.81804e30	0.371695
$974$	0	0
$975$	3.22333e30	0.132971
$976$	0	0
$977$	3.34661e31	1.35118	0.675589	−	0.737278i	$-0.263889\pi$
$977$	3.34661e31	1.35118	0.675589	+	0.737278i	$0.263889\pi$
$978$	0	0
$979$	3.17604e31	1.25507
$980$	0	0
$981$	−9.93887e30	−0.384426
$982$	0	0
$983$	−2.47550e31	−0.937239	−0.468619	−	0.883400i	$-0.655248\pi$
$983$	−2.47550e31	−0.937239	−0.468619	+	0.883400i	$0.655248\pi$
$984$	0	0
$985$	−5.22963e31	−1.93816
$986$	0	0
$987$	3.98049e30	0.144413
$988$	0	0
$989$	−2.18668e31	−0.776646
$990$	0	0
$991$	2.24076e31	0.779153	0.389576	−	0.920994i	$-0.372621\pi$
$991$	2.24076e31	0.779153	0.389576	+	0.920994i	$0.372621\pi$
$992$	0	0
$993$	2.25656e31	0.768209
$994$	0	0
$995$	−1.06668e31	−0.355542
$996$	0	0
$997$	3.99151e31	1.30268	0.651341	−	0.758785i	$-0.274207\pi$
$997$	3.99151e31	1.30268	0.651341	+	0.758785i	$0.274207\pi$
$998$	0	0
$999$	5.08789e31	1.62592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.a.p.1.4		5
4.3	odd	2		64.22.a.o.1.2		5
8.3	odd	2		32.22.a.d.1.4	yes	5
8.5	even	2		32.22.a.c.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.22.a.c.1.2	✓	5	8.5	even	2
32.22.a.d.1.4	yes	5	8.3	odd	2
64.22.a.o.1.2		5	4.3	odd	2
64.22.a.p.1.4		5	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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+53009.9 q^{3} +2.95789e7 q^{5} +3.32722e8 q^{7} -7.65031e9 q^{9} +O(q^{10})\)	\(q+53009.9 q^{3} +2.95789e7 q^{5} +3.32722e8 q^{7} -7.65031e9 q^{9} -6.27044e10 q^{11} +1.52750e11 q^{13} +1.56798e12 q^{15} -1.24833e12 q^{17} -1.80843e13 q^{19} +1.76376e13 q^{21} +3.13338e14 q^{23} +3.98076e14 q^{25} -9.60044e14 q^{27} -5.19001e14 q^{29} -6.73042e15 q^{31} -3.32395e15 q^{33} +9.84158e15 q^{35} -5.29964e16 q^{37} +8.09726e15 q^{39} -4.78851e16 q^{41} -6.97865e16 q^{43} -2.26288e17 q^{45} +2.25682e17 q^{47} -4.47842e17 q^{49} -6.61738e16 q^{51} +1.23702e17 q^{53} -1.85473e18 q^{55} -9.58644e17 q^{57} +2.43181e18 q^{59} +1.66446e18 q^{61} -2.54543e18 q^{63} +4.51818e18 q^{65} -2.07498e19 q^{67} +1.66100e19 q^{69} -2.24426e18 q^{71} +3.73065e19 q^{73} +2.11020e19 q^{75} -2.08632e19 q^{77} -1.44576e20 q^{79} +2.91331e19 q^{81} +1.23634e20 q^{83} -3.69243e19 q^{85} -2.75122e19 q^{87} -5.06510e20 q^{89} +5.08234e19 q^{91} -3.56779e20 q^{93} -5.34913e20 q^{95} +9.26865e18 q^{97} +4.79708e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 23144 q^{3} - 18311174 q^{5} - 63978640 q^{7} + 2839988161 q^{9}+O(q^{10}) \) 5 * q + 23144 * q^3 - 18311174 * q^5 - 63978640 * q^7 + 2839988161 * q^9	\( 5 q + 23144 q^{3} - 18311174 q^{5} - 63978640 q^{7} + 2839988161 q^{9} + 25629588280 q^{11} - 26739996110 q^{13} - 850898706352 q^{15} - 88104593910 q^{17} + 13998239618440 q^{19} - 11868255565952 q^{21} - 191593435978416 q^{23} - 41142227484149 q^{25} + 436694294703248 q^{27} - 229446229587518 q^{29} + 30\!\cdots\!60 q^{31}+ \cdots + 45\!\cdots\!40 q^{99}+O(q^{100}) \) 5 * q + 23144 * q^3 - 18311174 * q^5 - 63978640 * q^7 + 2839988161 * q^9 + 25629588280 * q^11 - 26739996110 * q^13 - 850898706352 * q^15 - 88104593910 * q^17 + 13998239618440 * q^19 - 11868255565952 * q^21 - 191593435978416 * q^23 - 41142227484149 * q^25 + 436694294703248 * q^27 - 229446229587518 * q^29 + 3073873710228160 * q^31 + 5433145544904640 * q^33 - 8500692544521248 * q^35 + 11689624248645370 * q^37 - 5755218754185200 * q^39 + 73561781960295890 * q^41 + 59173471060274168 * q^43 + 110307182017064626 * q^45 - 189587882278171232 * q^47 + 198072470828678205 * q^49 + 60381606670165200 * q^51 + 448473144842566250 * q^53 + 276218022333741296 * q^55 - 419697547444918720 * q^57 - 580240954920275560 * q^59 - 60707231290899774 * q^61 - 7329049347459246928 * q^63 - 4215936008336579644 * q^65 + 20784270726401480552 * q^67 - 13151945925537436032 * q^69 - 48791036649307600400 * q^71 - 7687983545663740350 * q^73 + 63701791920635021848 * q^75 - 14991304956050303360 * q^77 - 150036082528549569440 * q^79 + 6961005137522997901 * q^81 + 320036481947166592392 * q^83 + 55351169878564568052 * q^85 - 845870037277169219440 * q^87 + 157308716989748010706 * q^89 + 1517228856976775217760 * q^91 + 285928010618243909120 * q^93 - 2644313286399100711792 * q^95 - 165820666835121495910 * q^97 + 4544456840208531002840 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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