LMFDB - Embedded newform 64.17.c.d.63.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,17,Mod(63,64)]

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([1, 0]))

N = Newforms(chi, 17, names="a")

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

chi := DirichletCharacter("64.63");

S:= CuspForms(chi, 17);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 17 $
Character orbit:	$[\chi]$	$=$	64.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$103.887708068$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 5152x^{4} + 242526x^{3} + 17329473x^{2} + 402444531x + 64957563630 $ x^6 - x^5 + 5152x^4 + 242526x^3 + 17329473x^2 + 402444531x + 64957563630
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{62}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			63.5
Root			$34.9299 + 57.5840i$ of defining polynomial
Character	$\chi$	$=$	64.63
Dual form			64.17.c.d.63.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8024.44i q^{3} +664115. q^{5} -6.08943e6i q^{7} -2.13450e7 q^{9} +O(q^{10})$	$q+8024.44i q^{3} +664115. q^{5} -6.08943e6i q^{7} -2.13450e7 q^{9} -8.45870e7i q^{11} +2.60742e8 q^{13} +5.32915e9i q^{15} -2.57124e9 q^{17} -5.86632e9i q^{19} +4.88643e10 q^{21} +1.86711e10i q^{23} +2.88461e11 q^{25} +1.74145e11i q^{27} +4.45787e11 q^{29} -1.48090e12i q^{31} +6.78763e11 q^{33} -4.04408e12i q^{35} +1.01639e11 q^{37} +2.09231e12i q^{39} -8.62814e12 q^{41} -4.49657e12i q^{43} -1.41755e13 q^{45} -3.73165e13i q^{47} -3.84825e12 q^{49} -2.06328e13i q^{51} -2.92011e13 q^{53} -5.61755e13i q^{55} +4.70739e13 q^{57} -1.24750e14i q^{59} +3.30102e14 q^{61} +1.29979e14i q^{63} +1.73163e14 q^{65} +1.53934e14i q^{67} -1.49825e14 q^{69} +8.32820e14i q^{71} +2.42877e14 q^{73} +2.31474e15i q^{75} -5.15086e14 q^{77} -1.13362e15i q^{79} -2.31624e15 q^{81} -2.68294e15i q^{83} -1.70760e15 q^{85} +3.57719e15i q^{87} +4.30383e15 q^{89} -1.58777e15i q^{91} +1.18834e16 q^{93} -3.89591e15i q^{95} +2.79195e15 q^{97} +1.80550e15i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 506740 q^{5} - 137574522 q^{9}+O(q^{10}) $ 6 * q + 506740 * q^5 - 137574522 * q^9	$ 6 q + 506740 q^{5} - 137574522 q^{9} - 2544478092 q^{13} + 1579205132 q^{17} + 27228321792 q^{21} + 271424476050 q^{25} + 1158411768436 q^{29} - 767957621760 q^{33} - 8581446019212 q^{37} + 1840369253132 q^{41} + 34166370110580 q^{45} - 5527245758202 q^{49} - 130668269409932 q^{53} - 122486852367360 q^{57} + 429486008315508 q^{61} + 12\!\cdots\!00 q^{65}+ \cdots + 41\!\cdots\!52 q^{97}+O(q^{100}) $ 6 * q + 506740 * q^5 - 137574522 * q^9 - 2544478092 * q^13 + 1579205132 * q^17 + 27228321792 * q^21 + 271424476050 * q^25 + 1158411768436 * q^29 - 767957621760 * q^33 - 8581446019212 * q^37 + 1840369253132 * q^41 + 34166370110580 * q^45 - 5527245758202 * q^49 - 130668269409932 * q^53 - 122486852367360 * q^57 + 429486008315508 * q^61 + 1237661666277400 * q^65 + 265820219762688 * q^69 - 1145601039770868 * q^73 + 665543599687680 * q^77 - 277566121727226 * q^81 + 1934080153645800 * q^85 + 20426758995091724 * q^89 + 20365632048291840 * q^93 + 4182396554403852 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$1$	$-1$

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
$3$	8024.44i	1.22305i	0.791224	+	0.611526i	$0.209444\pi$
$3$	8024.44i	1.22305i	−0.791224	+	0.611526i	$0.790556\pi$
$4$	0	0
$5$	664115.	1.70013	0.850067	−	0.526674i	$-0.176561\pi$
$5$	664115.	1.70013	0.850067	+	0.526674i	$0.176561\pi$
$6$	0	0
$7$	− 6.08943e6i	− 1.05631i	−0.849147	−	0.528156i	$-0.822884\pi$
$7$	− 6.08943e6i	− 1.05631i	0.849147	−	0.528156i	$-0.177116\pi$
$8$	0	0
$9$	−2.13450e7	−0.495855
$10$	0	0
$11$	− 8.45870e7i	− 0.394604i	−0.980343	−	0.197302i	$-0.936782\pi$
$11$	− 8.45870e7i	− 0.394604i	0.980343	−	0.197302i	$-0.0632180\pi$
$12$	0	0
$13$	2.60742e8	0.319642	0.159821	−	0.987146i	$-0.448908\pi$
$13$	2.60742e8	0.319642	0.159821	+	0.987146i	$0.448908\pi$
$14$	0	0
$15$	5.32915e9i	2.07935i
$16$	0	0
$17$	−2.57124e9	−0.368597	−0.184299	−	0.982870i	$-0.559001\pi$
$17$	−2.57124e9	−0.368597	−0.184299	+	0.982870i	$0.559001\pi$
$18$	0	0
$19$	− 5.86632e9i	− 0.345411i	−0.984973	−	0.172706i	$-0.944749\pi$
$19$	− 5.86632e9i	− 0.345411i	0.984973	−	0.172706i	$-0.0552509\pi$
$20$	0	0
$21$	4.88643e10	1.29192
$22$	0	0
$23$	1.86711e10i	0.238422i	0.992869	+	0.119211i	$0.0380365\pi$
$23$	1.86711e10i	0.238422i	−0.992869	+	0.119211i	$0.961963\pi$
$24$	0	0
$25$	2.88461e11	1.89046
$26$	0	0
$27$	1.74145e11i	0.616595i
$28$	0	0
$29$	4.45787e11	0.891135	0.445568	−	0.895248i	$-0.353002\pi$
$29$	4.45787e11	0.891135	0.445568	+	0.895248i	$0.353002\pi$
$30$	0	0
$31$	− 1.48090e12i	− 1.73633i	−0.496274	−	0.868166i	$-0.665299\pi$
$31$	− 1.48090e12i	− 1.73633i	0.496274	−	0.868166i	$-0.334701\pi$
$32$	0	0
$33$	6.78763e11	0.482622
$34$	0	0
$35$	− 4.04408e12i	− 1.79587i
$36$	0	0
$37$	1.01639e11	0.0289366	0.0144683	−	0.999895i	$-0.495394\pi$
$37$	1.01639e11	0.0289366	0.0144683	+	0.999895i	$0.495394\pi$
$38$	0	0
$39$	2.09231e12i	0.390939i
$40$	0	0
$41$	−8.62814e12	−1.08055	−0.540277	−	0.841487i	$-0.681680\pi$
$41$	−8.62814e12	−1.08055	−0.540277	+	0.841487i	$0.681680\pi$
$42$	0	0
$43$	− 4.49657e12i	− 0.384710i	−0.981325	−	0.192355i	$-0.938387\pi$
$43$	− 4.49657e12i	− 0.384710i	0.981325	−	0.192355i	$-0.0616125\pi$
$44$	0	0
$45$	−1.41755e13	−0.843021
$46$	0	0
$47$	− 3.73165e13i	− 1.56718i	−0.621281	−	0.783588i	$-0.713387\pi$
$47$	− 3.73165e13i	− 1.56718i	0.621281	−	0.783588i	$-0.286613\pi$
$48$	0	0
$49$	−3.84825e12	−0.115796
$50$	0	0
$51$	− 2.06328e13i	− 0.450813i
$52$	0	0
$53$	−2.92011e13	−0.469021	−0.234511	−	0.972114i	$-0.575349\pi$
$53$	−2.92011e13	−0.469021	−0.234511	+	0.972114i	$0.575349\pi$
$54$	0	0
$55$	− 5.61755e13i	− 0.670880i
$56$	0	0
$57$	4.70739e13	0.422456
$58$	0	0
$59$	− 1.24750e14i	− 0.849617i	−0.905283	−	0.424808i	$-0.860341\pi$
$59$	− 1.24750e14i	− 0.849617i	0.905283	−	0.424808i	$-0.139659\pi$
$60$	0	0
$61$	3.30102e14	1.72191	0.860954	−	0.508683i	$-0.169868\pi$
$61$	3.30102e14	1.72191	0.860954	+	0.508683i	$0.169868\pi$
$62$	0	0
$63$	1.29979e14i	0.523778i
$64$	0	0
$65$	1.73163e14	0.543435
$66$	0	0
$67$	1.53934e14i	0.379084i	0.981873	+	0.189542i	$0.0607002\pi$
$67$	1.53934e14i	0.379084i	−0.981873	+	0.189542i	$0.939300\pi$
$68$	0	0
$69$	−1.49825e14	−0.291603
$70$	0	0
$71$	8.32820e14i	1.28969i	0.764315	+	0.644844i	$0.223078\pi$
$71$	8.32820e14i	1.28969i	−0.764315	+	0.644844i	$0.776922\pi$
$72$	0	0
$73$	2.42877e14	0.301164	0.150582	−	0.988598i	$-0.451885\pi$
$73$	2.42877e14	0.301164	0.150582	+	0.988598i	$0.451885\pi$
$74$	0	0
$75$	2.31474e15i	2.31213i
$76$	0	0
$77$	−5.15086e14	−0.416826
$78$	0	0
$79$	− 1.13362e15i	− 0.747225i	−0.927585	−	0.373613i	$-0.878119\pi$
$79$	− 1.13362e15i	− 0.747225i	0.927585	−	0.373613i	$-0.121881\pi$
$80$	0	0
$81$	−2.31624e15	−1.24998
$82$	0	0
$83$	− 2.68294e15i	− 1.19120i	−0.803280	−	0.595602i	$-0.796914\pi$
$83$	− 2.68294e15i	− 1.19120i	0.803280	−	0.595602i	$-0.203086\pi$
$84$	0	0
$85$	−1.70760e15	−0.626665
$86$	0	0
$87$	3.57719e15i	1.08990i
$88$	0	0
$89$	4.30383e15	1.09329	0.546645	−	0.837365i	$-0.315905\pi$
$89$	4.30383e15	1.09329	0.546645	+	0.837365i	$0.315905\pi$
$90$	0	0
$91$	− 1.58777e15i	− 0.337642i
$92$	0	0
$93$	1.18834e16	2.12362
$94$	0	0
$95$	− 3.89591e15i	− 0.587246i
$96$	0	0
$97$	2.79195e15	0.356233	0.178117	−	0.984009i	$-0.443000\pi$
$97$	2.79195e15	0.356233	0.178117	+	0.984009i	$0.443000\pi$
$98$	0	0
$99$	1.80550e15i	0.195667i
$100$	0	0
$101$	5.84071e14	0.0539380	0.0269690	−	0.999636i	$-0.491414\pi$
$101$	5.84071e14	0.0539380	0.0269690	+	0.999636i	$0.491414\pi$
$102$	0	0
$103$	− 1.49407e16i	− 1.17943i	−0.807610	−	0.589717i	$-0.799239\pi$
$103$	− 1.49407e16i	− 1.17943i	0.807610	−	0.589717i	$-0.200761\pi$
$104$	0	0
$105$	3.24515e16	2.19645
$106$	0	0
$107$	− 1.19858e16i	− 0.697585i	−0.937200	−	0.348792i	$-0.886592\pi$
$107$	− 1.19858e16i	− 0.697585i	0.937200	−	0.348792i	$-0.113408\pi$
$108$	0	0
$109$	1.70282e16	0.854587	0.427294	−	0.904113i	$-0.359467\pi$
$109$	1.70282e16	0.854587	0.427294	+	0.904113i	$0.359467\pi$
$110$	0	0
$111$	8.15597e14i	0.0353909i
$112$	0	0
$113$	−2.91540e16	−1.09666	−0.548328	−	0.836263i	$-0.684735\pi$
$113$	−2.91540e16	−1.09666	−0.548328	+	0.836263i	$0.684735\pi$
$114$	0	0
$115$	1.23997e16i	0.405350i
$116$	0	0
$117$	−5.56553e15	−0.158496
$118$	0	0
$119$	1.56574e16i	0.389354i
$120$	0	0
$121$	3.87948e16	0.844287
$122$	0	0
$123$	− 6.92360e16i	− 1.32157i
$124$	0	0
$125$	9.02352e16	1.51390
$126$	0	0
$127$	1.73245e15i	0.0255994i	0.999918	+	0.0127997i	$0.00407438\pi$
$127$	1.73245e15i	0.0255994i	−0.999918	+	0.0127997i	$0.995926\pi$
$128$	0	0
$129$	3.60825e16	0.470520
$130$	0	0
$131$	1.23331e17i	1.42201i	0.703186	+	0.711006i	$0.251760\pi$
$131$	1.23331e17i	1.42201i	−0.703186	+	0.711006i	$0.748240\pi$
$132$	0	0
$133$	−3.57225e16	−0.364862
$134$	0	0
$135$	1.15652e17i	1.04829i
$136$	0	0
$137$	1.88996e17	1.52295	0.761477	−	0.648192i	$-0.224474\pi$
$137$	1.88996e17	1.52295	0.761477	+	0.648192i	$0.224474\pi$
$138$	0	0
$139$	2.34634e17i	1.68373i	0.539690	+	0.841864i	$0.318541\pi$
$139$	2.34634e17i	1.68373i	−0.539690	+	0.841864i	$0.681459\pi$
$140$	0	0
$141$	2.99444e17	1.91674
$142$	0	0
$143$	− 2.20554e16i	− 0.126132i
$144$	0	0
$145$	2.96054e17	1.51505
$146$	0	0
$147$	− 3.08800e16i	− 0.141625i
$148$	0	0
$149$	3.99792e17	1.64567	0.822836	−	0.568278i	$-0.192390\pi$
$149$	3.99792e17	1.64567	0.822836	+	0.568278i	$0.192390\pi$
$150$	0	0
$151$	− 2.53944e16i	− 0.0939554i	−0.998896	−	0.0469777i	$-0.985041\pi$
$151$	− 2.53944e16i	− 0.0939554i	0.998896	−	0.0469777i	$-0.0149590\pi$
$152$	0	0
$153$	5.48831e16	0.182771
$154$	0	0
$155$	− 9.83489e17i	− 2.95200i
$156$	0	0
$157$	−2.58280e17	−0.699670	−0.349835	−	0.936811i	$-0.613762\pi$
$157$	−2.58280e17	−0.699670	−0.349835	+	0.936811i	$0.613762\pi$
$158$	0	0
$159$	− 2.34323e17i	− 0.573637i
$160$	0	0
$161$	1.13696e17	0.251848
$162$	0	0
$163$	5.51294e17i	1.10632i	0.833073	+	0.553162i	$0.186579\pi$
$163$	5.51294e17i	1.10632i	−0.833073	+	0.553162i	$0.813421\pi$
$164$	0	0
$165$	4.50777e17	0.820521
$166$	0	0
$167$	6.90794e16i	0.114187i	0.998369	+	0.0570935i	$0.0181833\pi$
$167$	6.90794e16i	0.114187i	−0.998369	+	0.0570935i	$0.981817\pi$
$168$	0	0
$169$	−5.97430e17	−0.897829
$170$	0	0
$171$	1.25216e17i	0.171274i
$172$	0	0
$173$	−8.49495e17	−1.05875	−0.529373	−	0.848389i	$-0.677573\pi$
$173$	−8.49495e17	−1.05875	−0.529373	+	0.848389i	$0.677573\pi$
$174$	0	0
$175$	− 1.75656e18i	− 1.99691i
$176$	0	0
$177$	1.00105e18	1.03913
$178$	0	0
$179$	− 3.56080e17i	− 0.337850i	−0.985629	−	0.168925i	$-0.945970\pi$
$179$	− 3.56080e17i	− 0.337850i	0.985629	−	0.168925i	$-0.0540295\pi$
$180$	0	0
$181$	1.95526e17	0.169736	0.0848682	−	0.996392i	$-0.472953\pi$
$181$	1.95526e17	0.169736	0.0848682	+	0.996392i	$0.472953\pi$
$182$	0	0
$183$	2.64889e18i	2.10598i
$184$	0	0
$185$	6.75000e16	0.0491960
$186$	0	0
$187$	2.17494e17i	0.145450i
$188$	0	0
$189$	1.06044e18	0.651317
$190$	0	0
$191$	− 9.47555e17i	− 0.534980i	−0.963561	−	0.267490i	$-0.913806\pi$
$191$	− 9.47555e17i	− 0.534980i	0.963561	−	0.267490i	$-0.0861943\pi$
$192$	0	0
$193$	−8.70387e16	−0.0452120	−0.0226060	−	0.999744i	$-0.507196\pi$
$193$	−8.70387e16	−0.0452120	−0.0226060	+	0.999744i	$0.507196\pi$
$194$	0	0
$195$	1.38953e18i	0.664649i
$196$	0	0
$197$	2.62309e18	1.15634	0.578168	−	0.815918i	$-0.303768\pi$
$197$	2.62309e18	1.15634	0.578168	+	0.815918i	$0.303768\pi$
$198$	0	0
$199$	2.73939e18i	1.11386i	0.830561	+	0.556928i	$0.188020\pi$
$199$	2.73939e18i	1.11386i	−0.830561	+	0.556928i	$0.811980\pi$
$200$	0	0
$201$	−1.23523e18	−0.463639
$202$	0	0
$203$	− 2.71459e18i	− 0.941317i
$204$	0	0
$205$	−5.73008e18	−1.83709
$206$	0	0
$207$	− 3.98533e17i	− 0.118223i
$208$	0	0
$209$	−4.96214e17	−0.136301
$210$	0	0
$211$	5.05635e18i	1.28700i	0.765448	+	0.643498i	$0.222517\pi$
$211$	5.05635e18i	1.28700i	−0.765448	+	0.643498i	$0.777483\pi$
$212$	0	0
$213$	−6.68292e18	−1.57735
$214$	0	0
$215$	− 2.98624e18i	− 0.654059i
$216$	0	0
$217$	−9.01785e18	−1.83411
$218$	0	0
$219$	1.94895e18i	0.368339i
$220$	0	0
$221$	−6.70432e17	−0.117819
$222$	0	0
$223$	4.13766e18i	0.676576i	0.941043	+	0.338288i	$0.109848\pi$
$223$	4.13766e18i	0.676576i	−0.941043	+	0.338288i	$0.890152\pi$
$224$	0	0
$225$	−6.15718e18	−0.937393
$226$	0	0
$227$	4.40118e18i	0.624255i	0.950040	+	0.312128i	$0.101042\pi$
$227$	4.40118e18i	0.624255i	−0.950040	+	0.312128i	$0.898958\pi$
$228$	0	0
$229$	−1.16200e19	−1.53646	−0.768230	−	0.640174i	$-0.778862\pi$
$229$	−1.16200e19	−1.53646	−0.768230	+	0.640174i	$0.778862\pi$
$230$	0	0
$231$	− 4.13328e18i	− 0.509799i
$232$	0	0
$233$	4.37523e18	0.503679	0.251839	−	0.967769i	$-0.418965\pi$
$233$	4.37523e18	0.503679	0.251839	+	0.967769i	$0.418965\pi$
$234$	0	0
$235$	− 2.47824e19i	− 2.66441i
$236$	0	0
$237$	9.09668e18	0.913895
$238$	0	0
$239$	− 4.45683e18i	− 0.418643i	−0.977847	−	0.209321i	$-0.932875\pi$
$239$	− 4.45683e18i	− 0.418643i	0.977847	−	0.209321i	$-0.0671254\pi$
$240$	0	0
$241$	−1.72872e19	−1.51910	−0.759552	−	0.650447i	$-0.774582\pi$
$241$	−1.72872e19	−1.51910	−0.759552	+	0.650447i	$0.774582\pi$
$242$	0	0
$243$	− 1.10902e19i	− 0.912199i
$244$	0	0
$245$	−2.55568e18	−0.196869
$246$	0	0
$247$	− 1.52960e18i	− 0.110408i
$248$	0	0
$249$	2.15291e19	1.45690
$250$	0	0
$251$	2.17982e19i	1.38366i	0.722058	+	0.691832i	$0.243196\pi$
$251$	2.17982e19i	1.38366i	−0.722058	+	0.691832i	$0.756804\pi$
$252$	0	0
$253$	1.57933e18	0.0940825
$254$	0	0
$255$	− 1.37026e19i	− 0.766443i
$256$	0	0
$257$	−1.14341e18	−0.0600810	−0.0300405	−	0.999549i	$-0.509564\pi$
$257$	−1.14341e18	−0.0600810	−0.0300405	+	0.999549i	$0.509564\pi$
$258$	0	0
$259$	− 6.18924e17i	− 0.0305660i
$260$	0	0
$261$	−9.51531e18	−0.441874
$262$	0	0
$263$	3.52616e18i	0.154048i	0.997029	+	0.0770239i	$0.0245418\pi$
$263$	3.52616e18i	0.154048i	−0.997029	+	0.0770239i	$0.975458\pi$
$264$	0	0
$265$	−1.93929e19	−0.797399
$266$	0	0
$267$	3.45359e19i	1.33715i
$268$	0	0
$269$	4.06262e17	0.0148179	0.00740897	−	0.999973i	$-0.497642\pi$
$269$	4.06262e17	0.0148179	0.00740897	+	0.999973i	$0.497642\pi$
$270$	0	0
$271$	2.57916e19i	0.886594i	0.896375	+	0.443297i	$0.146191\pi$
$271$	2.57916e19i	0.886594i	−0.896375	+	0.443297i	$0.853809\pi$
$272$	0	0
$273$	1.27410e19	0.412954
$274$	0	0
$275$	− 2.44000e19i	− 0.745983i
$276$	0	0
$277$	−1.55367e19	−0.448251	−0.224126	−	0.974560i	$-0.571953\pi$
$277$	−1.55367e19	−0.448251	−0.224126	+	0.974560i	$0.571953\pi$
$278$	0	0
$279$	3.16098e19i	0.860970i
$280$	0	0
$281$	1.11583e19	0.287044	0.143522	−	0.989647i	$-0.454157\pi$
$281$	1.11583e19	0.287044	0.143522	+	0.989647i	$0.454157\pi$
$282$	0	0
$283$	1.36478e19i	0.331720i	0.986149	+	0.165860i	$0.0530400\pi$
$283$	1.36478e19i	0.331720i	−0.986149	+	0.165860i	$0.946960\pi$
$284$	0	0
$285$	3.12625e19	0.718232
$286$	0	0
$287$	5.25405e19i	1.14140i
$288$	0	0
$289$	−4.20499e19	−0.864136
$290$	0	0
$291$	2.24039e19i	0.435692i
$292$	0	0
$293$	5.25381e19	0.967238	0.483619	−	0.875279i	$-0.339322\pi$
$293$	5.25381e19	0.967238	0.483619	+	0.875279i	$0.339322\pi$
$294$	0	0
$295$	− 8.28481e19i	− 1.44446i
$296$	0	0
$297$	1.47304e19	0.243311
$298$	0	0
$299$	4.86834e18i	0.0762098i
$300$	0	0
$301$	−2.73815e19	−0.406374
$302$	0	0
$303$	4.68685e18i	0.0659690i
$304$	0	0
$305$	2.19226e20	2.92747
$306$	0	0
$307$	− 1.37042e20i	− 1.73679i	−0.495874	−	0.868394i	$-0.665152\pi$
$307$	− 1.37042e20i	− 1.73679i	0.495874	−	0.868394i	$-0.334848\pi$
$308$	0	0
$309$	1.19891e20	1.44251
$310$	0	0
$311$	− 1.20103e20i	− 1.37237i	−0.727427	−	0.686185i	$-0.759284\pi$
$311$	− 1.20103e20i	− 1.37237i	0.727427	−	0.686185i	$-0.240716\pi$
$312$	0	0
$313$	4.49999e19	0.488491	0.244246	−	0.969713i	$-0.421460\pi$
$313$	4.49999e19	0.488491	0.244246	+	0.969713i	$0.421460\pi$
$314$	0	0
$315$	8.63208e19i	0.890494i
$316$	0	0
$317$	9.42515e19	0.924302	0.462151	−	0.886801i	$-0.347078\pi$
$317$	9.42515e19	0.924302	0.462151	+	0.886801i	$0.347078\pi$
$318$	0	0
$319$	− 3.77078e19i	− 0.351646i
$320$	0	0
$321$	9.61794e19	0.853182
$322$	0	0
$323$	1.50837e19i	0.127318i
$324$	0	0
$325$	7.52139e19	0.604270
$326$	0	0
$327$	1.36642e20i	1.04520i
$328$	0	0
$329$	−2.27236e20	−1.65543
$330$	0	0
$331$	1.63722e19i	0.113627i	0.998385	+	0.0568137i	$0.0180941\pi$
$331$	1.63722e19i	0.113627i	−0.998385	+	0.0568137i	$0.981906\pi$
$332$	0	0
$333$	−2.16948e18	−0.0143483
$334$	0	0
$335$	1.02230e20i	0.644493i
$336$	0	0
$337$	1.96434e20	1.18080	0.590401	−	0.807110i	$-0.298970\pi$
$337$	1.96434e20	1.18080	0.590401	+	0.807110i	$0.298970\pi$
$338$	0	0
$339$	− 2.33944e20i	− 1.34127i
$340$	0	0
$341$	−1.25265e20	−0.685164
$342$	0	0
$343$	− 1.78936e20i	− 0.933996i
$344$	0	0
$345$	−9.95010e19	−0.495764
$346$	0	0
$347$	− 2.62919e20i	− 1.25080i	−0.780306	−	0.625399i	$-0.784936\pi$
$347$	− 2.62919e20i	− 1.25080i	0.780306	−	0.625399i	$-0.215064\pi$
$348$	0	0
$349$	−1.15127e20	−0.523085	−0.261543	−	0.965192i	$-0.584231\pi$
$349$	−1.15127e20	−0.523085	−0.261543	+	0.965192i	$0.584231\pi$
$350$	0	0
$351$	4.54068e19i	0.197090i
$352$	0	0
$353$	−3.70243e20	−1.53564	−0.767820	−	0.640666i	$-0.778658\pi$
$353$	−3.70243e20	−1.53564	−0.767820	+	0.640666i	$0.778658\pi$
$354$	0	0
$355$	5.53088e20i	2.19264i
$356$	0	0
$357$	−1.25642e20	−0.476200
$358$	0	0
$359$	− 1.72643e20i	− 0.625740i	−0.949796	−	0.312870i	$-0.898710\pi$
$359$	− 1.72643e20i	− 0.625740i	0.949796	−	0.312870i	$-0.101290\pi$
$360$	0	0
$361$	2.54028e20	0.880691
$362$	0	0
$363$	3.11306e20i	1.03261i
$364$	0	0
$365$	1.61298e20	0.512020
$366$	0	0
$367$	1.03804e20i	0.315419i	0.987486	+	0.157709i	$0.0504109\pi$
$367$	1.03804e20i	0.315419i	−0.987486	+	0.157709i	$0.949589\pi$
$368$	0	0
$369$	1.84167e20	0.535798
$370$	0	0
$371$	1.77818e20i	0.495433i
$372$	0	0
$373$	5.77895e19	0.154233	0.0771167	−	0.997022i	$-0.475429\pi$
$373$	5.77895e19	0.154233	0.0771167	+	0.997022i	$0.475429\pi$
$374$	0	0
$375$	7.24087e20i	1.85157i
$376$	0	0
$377$	1.16235e20	0.284844
$378$	0	0
$379$	− 2.68660e20i	− 0.631087i	−0.948911	−	0.315543i	$-0.897813\pi$
$379$	− 2.68660e20i	− 0.631087i	0.948911	−	0.315543i	$-0.102187\pi$
$380$	0	0
$381$	−1.39019e19	−0.0313094
$382$	0	0
$383$	− 7.67660e19i	− 0.165798i	−0.996558	−	0.0828989i	$-0.973582\pi$
$383$	− 7.67660e19i	− 0.165798i	0.996558	−	0.0828989i	$-0.0264179\pi$
$384$	0	0
$385$	−3.42077e20	−0.708659
$386$	0	0
$387$	9.59790e19i	0.190761i
$388$	0	0
$389$	−7.10660e20	−1.35539	−0.677696	−	0.735342i	$-0.737021\pi$
$389$	−7.10660e20	−1.35539	−0.677696	+	0.735342i	$0.737021\pi$
$390$	0	0
$391$	− 4.80079e19i	− 0.0878817i
$392$	0	0
$393$	−9.89665e20	−1.73919
$394$	0	0
$395$	− 7.52855e20i	− 1.27038i
$396$	0	0
$397$	9.64334e20	1.56280	0.781400	−	0.624030i	$-0.214506\pi$
$397$	9.64334e20	1.56280	0.781400	+	0.624030i	$0.214506\pi$
$398$	0	0
$399$	− 2.86653e20i	− 0.446246i
$400$	0	0
$401$	1.99272e20	0.298051	0.149025	−	0.988833i	$-0.452386\pi$
$401$	1.99272e20	0.298051	0.149025	+	0.988833i	$0.452386\pi$
$402$	0	0
$403$	− 3.86134e20i	− 0.555005i
$404$	0	0
$405$	−1.53825e21	−2.12514
$406$	0	0
$407$	− 8.59734e18i	− 0.0114185i
$408$	0	0
$409$	2.42630e19	0.0309855	0.0154927	−	0.999880i	$-0.495068\pi$
$409$	2.42630e19	0.0309855	0.0154927	+	0.999880i	$0.495068\pi$
$410$	0	0
$411$	1.51658e21i	1.86265i
$412$	0	0
$413$	−7.59654e20	−0.897461
$414$	0	0
$415$	− 1.78178e21i	− 2.02521i
$416$	0	0
$417$	−1.88280e21	−2.05929
$418$	0	0
$419$	− 1.25395e21i	− 1.31999i	−0.751272	−	0.659993i	$-0.770559\pi$
$419$	− 1.25395e21i	− 1.31999i	0.751272	−	0.659993i	$-0.229441\pi$
$420$	0	0
$421$	−1.19519e21	−1.21110	−0.605551	−	0.795807i	$-0.707047\pi$
$421$	−1.19519e21	−1.21110	−0.605551	+	0.795807i	$0.707047\pi$
$422$	0	0
$423$	7.96519e20i	0.777093i
$424$	0	0
$425$	−7.41703e20	−0.696817
$426$	0	0
$427$	− 2.01014e21i	− 1.81887i
$428$	0	0
$429$	1.76982e20	0.154266
$430$	0	0
$431$	− 1.98473e21i	− 1.66680i	−0.552668	−	0.833402i	$-0.686390\pi$
$431$	− 1.98473e21i	− 1.66680i	0.552668	−	0.833402i	$-0.313610\pi$
$432$	0	0
$433$	2.25900e21	1.82815	0.914076	−	0.405543i	$-0.132918\pi$
$433$	2.25900e21	1.82815	0.914076	+	0.405543i	$0.132918\pi$
$434$	0	0
$435$	2.37567e21i	1.85298i
$436$	0	0
$437$	1.09530e20	0.0823537
$438$	0	0
$439$	1.75260e21i	1.27048i	0.772316	+	0.635239i	$0.219098\pi$
$439$	1.75260e21i	1.27048i	−0.772316	+	0.635239i	$0.780902\pi$
$440$	0	0
$441$	8.21406e19	0.0574182
$442$	0	0
$443$	− 4.11987e19i	− 0.0277749i	−0.999904	−	0.0138875i	$-0.995579\pi$
$443$	− 4.11987e19i	− 0.0277749i	0.999904	−	0.0138875i	$-0.00442066\pi$
$444$	0	0
$445$	2.85824e21	1.85874
$446$	0	0
$447$	3.20810e21i	2.01274i
$448$	0	0
$449$	−8.40539e20	−0.508847	−0.254423	−	0.967093i	$-0.581886\pi$
$449$	−8.40539e20	−0.508847	−0.254423	+	0.967093i	$0.581886\pi$
$450$	0	0
$451$	7.29828e20i	0.426391i
$452$	0	0
$453$	2.03776e20	0.114912
$454$	0	0
$455$	− 1.05446e21i	− 0.574037i
$456$	0	0
$457$	−2.31848e21	−1.21863	−0.609317	−	0.792926i	$-0.708556\pi$
$457$	−2.31848e21	−1.21863	−0.609317	+	0.792926i	$0.708556\pi$
$458$	0	0
$459$	− 4.47768e20i	− 0.227275i
$460$	0	0
$461$	−4.22089e20	−0.206917	−0.103459	−	0.994634i	$-0.532991\pi$
$461$	−4.22089e20	−0.206917	−0.103459	+	0.994634i	$0.532991\pi$
$462$	0	0
$463$	2.67396e21i	1.26621i	0.774064	+	0.633107i	$0.218221\pi$
$463$	2.67396e21i	1.26621i	−0.774064	+	0.633107i	$0.781779\pi$
$464$	0	0
$465$	7.89195e21	3.61045
$466$	0	0
$467$	1.83871e21i	0.812790i	0.913698	+	0.406395i	$0.133214\pi$
$467$	1.83871e21i	0.812790i	−0.913698	+	0.406395i	$0.866786\pi$
$468$	0	0
$469$	9.37368e20	0.400431
$470$	0	0
$471$	− 2.07255e21i	− 0.855732i
$472$	0	0
$473$	−3.80351e20	−0.151808
$474$	0	0
$475$	− 1.69220e21i	− 0.652985i
$476$	0	0
$477$	6.23297e20	0.232567
$478$	0	0
$479$	− 2.06301e21i	− 0.744418i	−0.928149	−	0.372209i	$-0.878600\pi$
$479$	− 2.06301e21i	− 0.744418i	0.928149	−	0.372209i	$-0.121400\pi$
$480$	0	0
$481$	2.65016e19	0.00924935
$482$	0	0
$483$	9.12349e20i	0.308024i
$484$	0	0
$485$	1.85418e21	0.605644
$486$	0	0
$487$	− 2.36076e21i	− 0.746141i	−0.927803	−	0.373071i	$-0.878305\pi$
$487$	− 2.36076e21i	− 0.746141i	0.927803	−	0.373071i	$-0.121695\pi$
$488$	0	0
$489$	−4.42383e21	−1.35309
$490$	0	0
$491$	5.96542e21i	1.76599i	0.469379	+	0.882997i	$0.344478\pi$
$491$	5.96542e21i	1.76599i	−0.469379	+	0.882997i	$0.655522\pi$
$492$	0	0
$493$	−1.14623e21	−0.328470
$494$	0	0
$495$	1.19906e21i	0.332660i
$496$	0	0
$497$	5.07140e21	1.36231
$498$	0	0
$499$	− 4.45444e21i	− 1.15875i	−0.815062	−	0.579374i	$-0.803297\pi$
$499$	− 4.45444e21i	− 1.15875i	0.815062	−	0.579374i	$-0.196703\pi$
$500$	0	0
$501$	−5.54324e20	−0.139657
$502$	0	0
$503$	6.08849e21i	1.48582i	0.669393	+	0.742909i	$0.266554\pi$
$503$	6.08849e21i	1.48582i	−0.669393	+	0.742909i	$0.733446\pi$
$504$	0	0
$505$	3.87891e20	0.0917019
$506$	0	0
$507$	− 4.79404e21i	− 1.09809i
$508$	0	0
$509$	3.83099e21	0.850292	0.425146	−	0.905125i	$-0.360223\pi$
$509$	3.83099e21	0.850292	0.425146	+	0.905125i	$0.360223\pi$
$510$	0	0
$511$	− 1.47898e21i	− 0.318123i
$512$	0	0
$513$	1.02159e21	0.212979
$514$	0	0
$515$	− 9.92236e21i	− 2.00520i
$516$	0	0
$517$	−3.15649e21	−0.618415
$518$	0	0
$519$	− 6.81672e21i	− 1.29490i
$520$	0	0
$521$	−1.28870e21	−0.237384	−0.118692	−	0.992931i	$-0.537870\pi$
$521$	−1.28870e21	−0.237384	−0.118692	+	0.992931i	$0.537870\pi$
$522$	0	0
$523$	7.97914e21i	1.42542i	0.701459	+	0.712710i	$0.252532\pi$
$523$	7.97914e21i	1.42542i	−0.701459	+	0.712710i	$0.747468\pi$
$524$	0	0
$525$	1.40954e22	2.44233
$526$	0	0
$527$	3.80776e21i	0.640007i
$528$	0	0
$529$	5.78400e21	0.943155
$530$	0	0
$531$	2.66277e21i	0.421287i
$532$	0	0
$533$	−2.24972e21	−0.345391
$534$	0	0
$535$	− 7.95995e21i	− 1.18599i
$536$	0	0
$537$	2.85735e21	0.413208
$538$	0	0
$539$	3.25511e20i	0.0456937i
$540$	0	0
$541$	2.86142e21	0.389945	0.194973	−	0.980809i	$-0.437538\pi$
$541$	2.86142e21	0.389945	0.194973	+	0.980809i	$0.437538\pi$
$542$	0	0
$543$	1.56898e21i	0.207596i
$544$	0	0
$545$	1.13087e22	1.45291
$546$	0	0
$547$	− 8.36994e20i	− 0.104430i	−0.998636	−	0.0522148i	$-0.983372\pi$
$547$	− 8.36994e20i	− 0.104430i	0.998636	−	0.0522148i	$-0.0166281\pi$
$548$	0	0
$549$	−7.04602e21	−0.853817
$550$	0	0
$551$	− 2.61513e21i	− 0.307808i
$552$	0	0
$553$	−6.90311e21	−0.789303
$554$	0	0
$555$	5.41650e20i	0.0601693i
$556$	0	0
$557$	6.03136e21	0.650989	0.325494	−	0.945544i	$-0.394469\pi$
$557$	6.03136e21	0.650989	0.325494	+	0.945544i	$0.394469\pi$
$558$	0	0
$559$	− 1.17244e21i	− 0.122970i
$560$	0	0
$561$	−1.74527e21	−0.177893
$562$	0	0
$563$	− 1.91894e21i	− 0.190105i	−0.995472	−	0.0950526i	$-0.969698\pi$
$563$	− 1.91894e21i	− 0.190105i	0.995472	−	0.0950526i	$-0.0303019\pi$
$564$	0	0
$565$	−1.93616e22	−1.86446
$566$	0	0
$567$	1.41046e22i	1.32037i
$568$	0	0
$569$	−1.43845e22	−1.30918	−0.654588	−	0.755986i	$-0.727158\pi$
$569$	−1.43845e22	−1.30918	−0.654588	+	0.755986i	$0.727158\pi$
$570$	0	0
$571$	6.94578e21i	0.614656i	0.951604	+	0.307328i	$0.0994348\pi$
$571$	6.94578e21i	0.614656i	−0.951604	+	0.307328i	$0.900565\pi$
$572$	0	0
$573$	7.60360e21	0.654308
$574$	0	0
$575$	5.38587e21i	0.450727i
$576$	0	0
$577$	−2.26314e22	−1.84206	−0.921031	−	0.389490i	$-0.872651\pi$
$577$	−2.26314e22	−1.84206	−0.921031	+	0.389490i	$0.872651\pi$
$578$	0	0
$579$	− 6.98437e20i	− 0.0552966i
$580$	0	0
$581$	−1.63376e22	−1.25828
$582$	0	0
$583$	2.47003e21i	0.185078i
$584$	0	0
$585$	−3.69615e21	−0.269465
$586$	0	0
$587$	9.74513e21i	0.691326i	0.938359	+	0.345663i	$0.112346\pi$
$587$	9.74513e21i	0.691326i	−0.938359	+	0.345663i	$0.887654\pi$
$588$	0	0
$589$	−8.68744e21	−0.599749
$590$	0	0
$591$	2.10489e22i	1.41426i
$592$	0	0
$593$	−4.66780e21	−0.305264	−0.152632	−	0.988283i	$-0.548775\pi$
$593$	−4.66780e21	−0.305264	−0.152632	+	0.988283i	$0.548775\pi$
$594$	0	0
$595$	1.03983e22i	0.661954i
$596$	0	0
$597$	−2.19821e22	−1.36230
$598$	0	0
$599$	8.39163e21i	0.506328i	0.967423	+	0.253164i	$0.0814711\pi$
$599$	8.39163e21i	0.506328i	−0.967423	+	0.253164i	$0.918529\pi$
$600$	0	0
$601$	−6.96241e21	−0.409038	−0.204519	−	0.978863i	$-0.565563\pi$
$601$	−6.96241e21	−0.409038	−0.204519	+	0.978863i	$0.565563\pi$
$602$	0	0
$603$	− 3.28570e21i	− 0.187971i
$604$	0	0
$605$	2.57642e22	1.43540
$606$	0	0
$607$	− 2.63904e22i	− 1.43198i	−0.698112	−	0.715989i	$-0.745976\pi$
$607$	− 2.63904e22i	− 1.43198i	0.698112	−	0.715989i	$-0.254024\pi$
$608$	0	0
$609$	2.17831e22	1.15128
$610$	0	0
$611$	− 9.72998e21i	− 0.500936i
$612$	0	0
$613$	−1.20227e22	−0.603001	−0.301501	−	0.953466i	$-0.597488\pi$
$613$	−1.20227e22	−0.603001	−0.301501	+	0.953466i	$0.597488\pi$
$614$	0	0
$615$	− 4.59807e22i	− 2.24685i
$616$	0	0
$617$	−4.08842e21	−0.194659	−0.0973294	−	0.995252i	$-0.531030\pi$
$617$	−4.08842e21	−0.194659	−0.0973294	+	0.995252i	$0.531030\pi$
$618$	0	0
$619$	− 2.56470e22i	− 1.18990i	−0.803763	−	0.594950i	$-0.797172\pi$
$619$	− 2.56470e22i	− 1.18990i	0.803763	−	0.594950i	$-0.202828\pi$
$620$	0	0
$621$	−3.25147e21	−0.147010
$622$	0	0
$623$	− 2.62079e22i	− 1.15486i
$624$	0	0
$625$	1.59109e22	0.683370
$626$	0	0
$627$	− 3.98184e21i	− 0.166703i
$628$	0	0
$629$	−2.61339e20	−0.0106659
$630$	0	0
$631$	4.40682e22i	1.75344i	0.481003	+	0.876719i	$0.340273\pi$
$631$	4.40682e22i	1.75344i	−0.481003	+	0.876719i	$0.659727\pi$
$632$	0	0
$633$	−4.05744e22	−1.57406
$634$	0	0
$635$	1.15054e21i	0.0435224i
$636$	0	0
$637$	−1.00340e21	−0.0370134
$638$	0	0
$639$	− 1.77765e22i	− 0.639498i
$640$	0	0
$641$	1.23073e22	0.431814	0.215907	−	0.976414i	$-0.430729\pi$
$641$	1.23073e22	0.431814	0.215907	+	0.976414i	$0.430729\pi$
$642$	0	0
$643$	2.61752e22i	0.895782i	0.894088	+	0.447891i	$0.147825\pi$
$643$	2.61752e22i	0.895782i	−0.894088	+	0.447891i	$0.852175\pi$
$644$	0	0
$645$	2.39629e22	0.799948
$646$	0	0
$647$	− 1.14100e22i	− 0.371579i	−0.982590	−	0.185789i	$-0.940516\pi$
$647$	− 1.14100e22i	− 0.371579i	0.982590	−	0.185789i	$-0.0594842\pi$
$648$	0	0
$649$	−1.05522e22	−0.335262
$650$	0	0
$651$	− 7.23632e22i	− 2.24321i
$652$	0	0
$653$	−2.99192e22	−0.904991	−0.452495	−	0.891767i	$-0.649466\pi$
$653$	−2.99192e22	−0.904991	−0.452495	+	0.891767i	$0.649466\pi$
$654$	0	0
$655$	8.19062e22i	2.41761i
$656$	0	0
$657$	−5.18420e21	−0.149334
$658$	0	0
$659$	1.96568e22i	0.552625i	0.961068	+	0.276312i	$0.0891124\pi$
$659$	1.96568e22i	0.552625i	−0.961068	+	0.276312i	$0.910888\pi$
$660$	0	0
$661$	1.52272e22	0.417838	0.208919	−	0.977933i	$-0.433006\pi$
$661$	1.52272e22	0.417838	0.208919	+	0.977933i	$0.433006\pi$
$662$	0	0
$663$	− 5.37984e21i	− 0.144099i
$664$	0	0
$665$	−2.37239e22	−0.620315
$666$	0	0
$667$	8.32333e21i	0.212466i
$668$	0	0
$669$	−3.32025e22	−0.827487
$670$	0	0
$671$	− 2.79223e22i	− 0.679472i
$672$	0	0
$673$	5.86300e22	1.39316	0.696578	−	0.717481i	$-0.254705\pi$
$673$	5.86300e22	1.39316	0.696578	+	0.717481i	$0.254705\pi$
$674$	0	0
$675$	5.02339e22i	1.16565i
$676$	0	0
$677$	−5.47078e22	−1.23977	−0.619883	−	0.784694i	$-0.712820\pi$
$677$	−5.47078e22	−1.23977	−0.619883	+	0.784694i	$0.712820\pi$
$678$	0	0
$679$	− 1.70014e22i	− 0.376294i
$680$	0	0
$681$	−3.53170e22	−0.763496
$682$	0	0
$683$	3.15143e22i	0.665490i	0.943017	+	0.332745i	$0.107975\pi$
$683$	3.15143e22i	0.665490i	−0.943017	+	0.332745i	$0.892025\pi$
$684$	0	0
$685$	1.25515e23	2.58923
$686$	0	0
$687$	− 9.32437e22i	− 1.87917i
$688$	0	0
$689$	−7.61396e21	−0.149919
$690$	0	0
$691$	2.30302e21i	0.0443072i	0.999755	+	0.0221536i	$0.00705229\pi$
$691$	2.30302e21i	0.0443072i	−0.999755	+	0.0221536i	$0.992948\pi$
$692$	0	0
$693$	1.09945e22	0.206685
$694$	0	0
$695$	1.55824e23i	2.86256i
$696$	0	0
$697$	2.21851e22	0.398289
$698$	0	0
$699$	3.51088e22i	0.616025i
$700$	0	0
$701$	−2.73224e22	−0.468570	−0.234285	−	0.972168i	$-0.575275\pi$
$701$	−2.73224e22	−0.468570	−0.234285	+	0.972168i	$0.575275\pi$
$702$	0	0
$703$	− 5.96247e20i	− 0.00999501i
$704$	0	0
$705$	1.98865e23	3.25871
$706$	0	0
$707$	− 3.55666e21i	− 0.0569754i
$708$	0	0
$709$	5.28953e22	0.828414	0.414207	−	0.910183i	$-0.364059\pi$
$709$	5.28953e22	0.828414	0.414207	+	0.910183i	$0.364059\pi$
$710$	0	0
$711$	2.41971e22i	0.370516i
$712$	0	0
$713$	2.76500e22	0.413980
$714$	0	0
$715$	− 1.46473e22i	− 0.214442i
$716$	0	0
$717$	3.57636e22	0.512021
$718$	0	0
$719$	− 1.93249e22i	− 0.270573i	−0.990806	−	0.135287i	$-0.956804\pi$
$719$	− 1.93249e22i	− 0.270573i	0.990806	−	0.135287i	$-0.0431955\pi$
$720$	0	0
$721$	−9.09805e22	−1.24585
$722$	0	0
$723$	− 1.38720e23i	− 1.85794i
$724$	0	0
$725$	1.28592e23	1.68465
$726$	0	0
$727$	7.74416e21i	0.0992428i	0.998768	+	0.0496214i	$0.0158015\pi$
$727$	7.74416e21i	0.0992428i	−0.998768	+	0.0496214i	$0.984199\pi$
$728$	0	0
$729$	−1.07139e22	−0.134316
$730$	0	0
$731$	1.15618e22i	0.141803i
$732$	0	0
$733$	7.60409e22	0.912464	0.456232	−	0.889861i	$-0.349199\pi$
$733$	7.60409e22	0.912464	0.456232	+	0.889861i	$0.349199\pi$
$734$	0	0
$735$	− 2.05079e22i	− 0.240781i
$736$	0	0
$737$	1.30208e22	0.149588
$738$	0	0
$739$	5.80220e22i	0.652285i	0.945321	+	0.326143i	$0.105749\pi$
$739$	5.80220e22i	0.652285i	−0.945321	+	0.326143i	$0.894251\pi$
$740$	0	0
$741$	1.22741e22	0.135035
$742$	0	0
$743$	− 1.22235e23i	− 1.31609i	−0.752979	−	0.658044i	$-0.771384\pi$
$743$	− 1.22235e23i	− 1.31609i	0.752979	−	0.658044i	$-0.228616\pi$
$744$	0	0
$745$	2.65508e23	2.79786
$746$	0	0
$747$	5.72672e22i	0.590665i
$748$	0	0
$749$	−7.29867e22	−0.736867
$750$	0	0
$751$	1.22983e23i	1.21542i	0.794158	+	0.607711i	$0.207912\pi$
$751$	1.22983e23i	1.21542i	−0.794158	+	0.607711i	$0.792088\pi$
$752$	0	0
$753$	−1.74918e23	−1.69229
$754$	0	0
$755$	− 1.68648e22i	− 0.159737i
$756$	0	0
$757$	−1.82196e23	−1.68956	−0.844778	−	0.535117i	$-0.820267\pi$
$757$	−1.82196e23	−1.68956	−0.844778	+	0.535117i	$0.820267\pi$
$758$	0	0
$759$	1.26732e22i	0.115068i
$760$	0	0
$761$	−5.44886e22	−0.484427	−0.242214	−	0.970223i	$-0.577873\pi$
$761$	−5.44886e22	−0.484427	−0.242214	+	0.970223i	$0.577873\pi$
$762$	0	0
$763$	− 1.03692e23i	− 0.902711i
$764$	0	0
$765$	3.64487e22	0.310735
$766$	0	0
$767$	− 3.25275e22i	− 0.271573i
$768$	0	0
$769$	−2.69231e22	−0.220148	−0.110074	−	0.993923i	$-0.535109\pi$
$769$	−2.69231e22	−0.220148	−0.110074	+	0.993923i	$0.535109\pi$
$770$	0	0
$771$	− 9.17524e21i	− 0.0734822i
$772$	0	0
$773$	−9.71909e22	−0.762412	−0.381206	−	0.924490i	$-0.624491\pi$
$773$	−9.71909e22	−0.762412	−0.381206	+	0.924490i	$0.624491\pi$
$774$	0	0
$775$	− 4.27182e23i	− 3.28246i
$776$	0	0
$777$	4.96652e21	0.0373838
$778$	0	0
$779$	5.06154e22i	0.373235i
$780$	0	0
$781$	7.04457e22	0.508916
$782$	0	0
$783$	7.76314e22i	0.549469i
$784$	0	0
$785$	−1.71527e23	−1.18953
$786$	0	0
$787$	− 5.64292e22i	− 0.383448i	−0.981449	−	0.191724i	$-0.938592\pi$
$787$	− 5.64292e22i	− 0.383448i	0.981449	−	0.191724i	$-0.0614078\pi$
$788$	0	0
$789$	−2.82954e22	−0.188408
$790$	0	0
$791$	1.77531e23i	1.15841i
$792$	0	0
$793$	8.60716e22	0.550395
$794$	0	0
$795$	− 1.55617e23i	− 0.975261i
$796$	0	0
$797$	−3.79805e22	−0.233289	−0.116644	−	0.993174i	$-0.537214\pi$
$797$	−3.79805e22	−0.233289	−0.116644	+	0.993174i	$0.537214\pi$
$798$	0	0
$799$	9.59498e22i	0.577657i
$800$	0	0
$801$	−9.18651e22	−0.542114
$802$	0	0
$803$	− 2.05442e22i	− 0.118841i
$804$	0	0
$805$	7.55074e22	0.428176
$806$	0	0
$807$	3.26002e21i	0.0181231i
$808$	0	0
$809$	−8.55762e22	−0.466407	−0.233203	−	0.972428i	$-0.574921\pi$
$809$	−8.55762e22	−0.466407	−0.233203	+	0.972428i	$0.574921\pi$
$810$	0	0
$811$	2.45054e23i	1.30947i	0.755858	+	0.654735i	$0.227220\pi$
$811$	2.45054e23i	1.30947i	−0.755858	+	0.654735i	$0.772780\pi$
$812$	0	0
$813$	−2.06964e23	−1.08435
$814$	0	0
$815$	3.66123e23i	1.88090i
$816$	0	0
$817$	−2.63783e22	−0.132883
$818$	0	0
$819$	3.38909e22i	0.167422i
$820$	0	0
$821$	1.71716e23	0.831891	0.415945	−	0.909390i	$-0.363451\pi$
$821$	1.71716e23	0.831891	0.415945	+	0.909390i	$0.363451\pi$
$822$	0	0
$823$	1.47347e23i	0.700072i	0.936736	+	0.350036i	$0.113831\pi$
$823$	1.47347e23i	0.700072i	−0.936736	+	0.350036i	$0.886169\pi$
$824$	0	0
$825$	1.95797e23	0.912375
$826$	0	0
$827$	2.33798e23i	1.06856i	0.845309	+	0.534278i	$0.179416\pi$
$827$	2.33798e23i	1.06856i	−0.845309	+	0.534278i	$0.820584\pi$
$828$	0	0
$829$	−1.20714e23	−0.541153	−0.270577	−	0.962698i	$-0.587214\pi$
$829$	−1.20714e23	−0.541153	−0.270577	+	0.962698i	$0.587214\pi$
$830$	0	0
$831$	− 1.24674e23i	− 0.548234i
$832$	0	0
$833$	9.89478e21	0.0426821
$834$	0	0
$835$	4.58767e22i	0.194133i
$836$	0	0
$837$	2.57891e23	1.07061
$838$	0	0
$839$	− 1.45067e23i	− 0.590845i	−0.955367	−	0.295422i	$-0.904540\pi$
$839$	− 1.45067e23i	− 0.590845i	0.955367	−	0.295422i	$-0.0954603\pi$
$840$	0	0
$841$	−5.15203e22	−0.205878
$842$	0	0
$843$	8.95392e22i	0.351069i
$844$	0	0
$845$	−3.96762e23	−1.52643
$846$	0	0
$847$	− 2.36238e23i	− 0.891831i
$848$	0	0
$849$	−1.09516e23	−0.405711
$850$	0	0
$851$	1.89771e21i	0.00689912i
$852$	0	0
$853$	−2.66781e23	−0.951836	−0.475918	−	0.879490i	$-0.657884\pi$
$853$	−2.66781e23	−0.951836	−0.475918	+	0.879490i	$0.657884\pi$
$854$	0	0
$855$	8.31580e22i	0.291189i
$856$	0	0
$857$	−3.74252e23	−1.28623	−0.643114	−	0.765771i	$-0.722358\pi$
$857$	−3.74252e23	−1.28623	−0.643114	+	0.765771i	$0.722358\pi$
$858$	0	0
$859$	1.89765e23i	0.640134i	0.947395	+	0.320067i	$0.103705\pi$
$859$	1.89765e23i	0.640134i	−0.947395	+	0.320067i	$0.896295\pi$
$860$	0	0
$861$	−4.21608e23	−1.39599
$862$	0	0
$863$	− 1.83506e23i	− 0.596437i	−0.954498	−	0.298219i	$-0.903608\pi$
$863$	− 1.83506e23i	− 0.596437i	0.954498	−	0.298219i	$-0.0963924\pi$
$864$	0	0
$865$	−5.64162e23	−1.80001
$866$	0	0
$867$	− 3.37427e23i	− 1.05688i
$868$	0	0
$869$	−9.58896e22	−0.294858
$870$	0	0
$871$	4.01370e22i	0.121171i
$872$	0	0
$873$	−5.95941e22	−0.176640
$874$	0	0
$875$	− 5.49481e23i	− 1.59915i
$876$	0	0
$877$	−1.50314e23	−0.429540	−0.214770	−	0.976665i	$-0.568900\pi$
$877$	−1.50314e23	−0.429540	−0.214770	+	0.976665i	$0.568900\pi$
$878$	0	0
$879$	4.21589e23i	1.18298i
$880$	0	0
$881$	3.44866e23	0.950260	0.475130	−	0.879915i	$-0.342401\pi$
$881$	3.44866e23	0.950260	0.475130	+	0.879915i	$0.342401\pi$
$882$	0	0
$883$	− 1.40540e23i	− 0.380289i	−0.981756	−	0.190145i	$-0.939104\pi$
$883$	− 1.40540e23i	− 0.380289i	0.981756	−	0.190145i	$-0.0608957\pi$
$884$	0	0
$885$	6.64810e23	1.76665
$886$	0	0
$887$	− 2.45496e23i	− 0.640701i	−0.947299	−	0.320351i	$-0.896199\pi$
$887$	− 2.45496e23i	− 0.640701i	0.947299	−	0.320351i	$-0.103801\pi$
$888$	0	0
$889$	1.05496e22	0.0270410
$890$	0	0
$891$	1.95924e23i	0.493249i
$892$	0	0
$893$	−2.18910e23	−0.541320
$894$	0	0
$895$	− 2.36478e23i	− 0.574390i
$896$	0	0
$897$	−3.90657e22	−0.0932086
$898$	0	0
$899$	− 6.60167e23i	− 1.54731i
$900$	0	0
$901$	7.50832e22	0.172880
$902$	0	0
$903$	− 2.19722e23i	− 0.497017i
$904$	0	0
$905$	1.29851e23	0.288575
$906$	0	0
$907$	− 7.11816e23i	− 1.55421i	−0.629371	−	0.777105i	$-0.716688\pi$
$907$	− 7.11816e23i	− 1.55421i	0.629371	−	0.777105i	$-0.283312\pi$
$908$	0	0
$909$	−1.24670e22	−0.0267455
$910$	0	0
$911$	− 6.43689e23i	− 1.35684i	−0.734674	−	0.678421i	$-0.762665\pi$
$911$	− 6.43689e23i	− 1.35684i	0.734674	−	0.678421i	$-0.237335\pi$
$912$	0	0
$913$	−2.26942e23	−0.470054
$914$	0	0
$915$	1.75917e24i	3.58045i
$916$	0	0
$917$	7.51018e23	1.50209
$918$	0	0
$919$	6.05075e23i	1.18928i	0.803991	+	0.594641i	$0.202706\pi$
$919$	6.05075e23i	1.18928i	−0.803991	+	0.594641i	$0.797294\pi$
$920$	0	0
$921$	1.09969e24	2.12418
$922$	0	0
$923$	2.17151e23i	0.412239i
$924$	0	0
$925$	2.93189e22	0.0547033
$926$	0	0
$927$	3.18909e23i	0.584829i
$928$	0	0
$929$	7.39327e22	0.133263	0.0666317	−	0.997778i	$-0.478775\pi$
$929$	7.39327e22	0.133263	0.0666317	+	0.997778i	$0.478775\pi$
$930$	0	0
$931$	2.25750e22i	0.0399973i
$932$	0	0
$933$	9.63761e23	1.67848
$934$	0	0
$935$	1.44441e23i	0.247285i
$936$	0	0
$937$	8.95201e23	1.50662	0.753310	−	0.657665i	$-0.228456\pi$
$937$	8.95201e23	1.50662	0.753310	+	0.657665i	$0.228456\pi$
$938$	0	0
$939$	3.61099e23i	0.597450i
$940$	0	0
$941$	−7.53207e23	−1.22517	−0.612586	−	0.790404i	$-0.709871\pi$
$941$	−7.53207e23	−1.22517	−0.612586	+	0.790404i	$0.709871\pi$
$942$	0	0
$943$	− 1.61097e23i	− 0.257628i
$944$	0	0
$945$	7.04255e23	1.10733
$946$	0	0
$947$	− 5.07612e23i	− 0.784751i	−0.919805	−	0.392376i	$-0.871653\pi$
$947$	− 5.07612e23i	− 0.784751i	0.919805	−	0.392376i	$-0.128347\pi$
$948$	0	0
$949$	6.33282e22	0.0962648
$950$	0	0
$951$	7.56315e23i	1.13047i
$952$	0	0
$953$	−1.49025e23	−0.219036	−0.109518	−	0.993985i	$-0.534931\pi$
$953$	−1.49025e23	−0.219036	−0.109518	+	0.993985i	$0.534931\pi$
$954$	0	0
$955$	− 6.29286e23i	− 0.909538i
$956$	0	0
$957$	3.02584e23	0.430081
$958$	0	0
$959$	− 1.15088e24i	− 1.60872i
$960$	0	0
$961$	−1.46565e24	−2.01485
$962$	0	0
$963$	2.55836e23i	0.345901i
$964$	0	0
$965$	−5.78037e22	−0.0768665
$966$	0	0
$967$	2.74682e23i	0.359268i	0.983733	+	0.179634i	$0.0574914\pi$
$967$	2.74682e23i	0.359268i	−0.983733	+	0.179634i	$0.942509\pi$
$968$	0	0
$969$	−1.21039e23	−0.155716
$970$	0	0
$971$	9.78763e23i	1.23858i	0.785163	+	0.619290i	$0.212579\pi$
$971$	9.78763e23i	1.23858i	−0.785163	+	0.619290i	$0.787421\pi$
$972$	0	0
$973$	1.42879e24	1.77854
$974$	0	0
$975$	6.03549e23i	0.739053i
$976$	0	0
$977$	−2.84501e23	−0.342711	−0.171355	−	0.985209i	$-0.554815\pi$
$977$	−2.84501e23	−0.342711	−0.171355	+	0.985209i	$0.554815\pi$
$978$	0	0
$979$	− 3.64048e23i	− 0.431417i
$980$	0	0
$981$	−3.63466e23	−0.423752
$982$	0	0
$983$	2.87568e23i	0.329847i	0.986306	+	0.164923i	$0.0527376\pi$
$983$	2.87568e23i	0.329847i	−0.986306	+	0.164923i	$0.947262\pi$
$984$	0	0
$985$	1.74204e24	1.96593
$986$	0	0
$987$	− 1.82344e24i	− 2.02467i
$988$	0	0
$989$	8.39558e22	0.0917234
$990$	0	0
$991$	1.14612e24i	1.23209i	0.787713	+	0.616043i	$0.211265\pi$
$991$	1.14612e24i	1.23209i	−0.787713	+	0.616043i	$0.788735\pi$
$992$	0	0
$993$	−1.31378e23	−0.138972
$994$	0	0
$995$	1.81927e24i	1.89370i
$996$	0	0
$997$	−7.08801e23	−0.726044	−0.363022	−	0.931781i	$-0.618255\pi$
$997$	−7.08801e23	−0.726044	−0.363022	+	0.931781i	$0.618255\pi$
$998$	0	0
$999$	1.76999e22i	0.0178421i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.17.c.d.63.5		6
4.3	odd	2	inner	64.17.c.d.63.2		6
8.3	odd	2		4.17.b.b.3.5	✓	6
8.5	even	2		4.17.b.b.3.6	yes	6
24.5	odd	2		36.17.d.b.19.1		6
24.11	even	2		36.17.d.b.19.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.17.b.b.3.5	✓	6	8.3	odd	2
4.17.b.b.3.6	yes	6	8.5	even	2
36.17.d.b.19.1		6	24.5	odd	2
36.17.d.b.19.2		6	24.11	even	2
64.17.c.d.63.2		6	4.3	odd	2	inner
64.17.c.d.63.5		6	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:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	64.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8024.44i q^{3} +664115. q^{5} -6.08943e6i q^{7} -2.13450e7 q^{9} +O(q^{10})\)	\(q+8024.44i q^{3} +664115. q^{5} -6.08943e6i q^{7} -2.13450e7 q^{9} -8.45870e7i q^{11} +2.60742e8 q^{13} +5.32915e9i q^{15} -2.57124e9 q^{17} -5.86632e9i q^{19} +4.88643e10 q^{21} +1.86711e10i q^{23} +2.88461e11 q^{25} +1.74145e11i q^{27} +4.45787e11 q^{29} -1.48090e12i q^{31} +6.78763e11 q^{33} -4.04408e12i q^{35} +1.01639e11 q^{37} +2.09231e12i q^{39} -8.62814e12 q^{41} -4.49657e12i q^{43} -1.41755e13 q^{45} -3.73165e13i q^{47} -3.84825e12 q^{49} -2.06328e13i q^{51} -2.92011e13 q^{53} -5.61755e13i q^{55} +4.70739e13 q^{57} -1.24750e14i q^{59} +3.30102e14 q^{61} +1.29979e14i q^{63} +1.73163e14 q^{65} +1.53934e14i q^{67} -1.49825e14 q^{69} +8.32820e14i q^{71} +2.42877e14 q^{73} +2.31474e15i q^{75} -5.15086e14 q^{77} -1.13362e15i q^{79} -2.31624e15 q^{81} -2.68294e15i q^{83} -1.70760e15 q^{85} +3.57719e15i q^{87} +4.30383e15 q^{89} -1.58777e15i q^{91} +1.18834e16 q^{93} -3.89591e15i q^{95} +2.79195e15 q^{97} +1.80550e15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 506740 q^{5} - 137574522 q^{9}+O(q^{10}) \) 6 * q + 506740 * q^5 - 137574522 * q^9	\( 6 q + 506740 q^{5} - 137574522 q^{9} - 2544478092 q^{13} + 1579205132 q^{17} + 27228321792 q^{21} + 271424476050 q^{25} + 1158411768436 q^{29} - 767957621760 q^{33} - 8581446019212 q^{37} + 1840369253132 q^{41} + 34166370110580 q^{45} - 5527245758202 q^{49} - 130668269409932 q^{53} - 122486852367360 q^{57} + 429486008315508 q^{61} + 12\!\cdots\!00 q^{65}+ \cdots + 41\!\cdots\!52 q^{97}+O(q^{100}) \) 6 * q + 506740 * q^5 - 137574522 * q^9 - 2544478092 * q^13 + 1579205132 * q^17 + 27228321792 * q^21 + 271424476050 * q^25 + 1158411768436 * q^29 - 767957621760 * q^33 - 8581446019212 * q^37 + 1840369253132 * q^41 + 34166370110580 * q^45 - 5527245758202 * q^49 - 130668269409932 * q^53 - 122486852367360 * q^57 + 429486008315508 * q^61 + 1237661666277400 * q^65 + 265820219762688 * q^69 - 1145601039770868 * q^73 + 665543599687680 * q^77 - 277566121727226 * q^81 + 1934080153645800 * q^85 + 20426758995091724 * q^89 + 20365632048291840 * q^93 + 4182396554403852 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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