LMFDB - Embedded newform 64.17.c.d.63.1

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			63.1
Root			$11.8147 + 63.8186i$ of defining polynomial
Character	$\chi$	$=$	64.63
Dual form			64.17.c.d.63.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11183.9i q^{3} -389860. q^{5} -1.06777e6i q^{7} -8.20322e7 q^{9} +O(q^{10})$	$q-11183.9i q^{3} -389860. q^{5} -1.06777e6i q^{7} -8.20322e7 q^{9} -1.26927e8i q^{11} -1.12108e9 q^{13} +4.36015e9i q^{15} -7.43906e9 q^{17} -1.76792e10i q^{19} -1.19418e10 q^{21} +8.72327e9i q^{23} -5.96851e8 q^{25} +4.36008e11i q^{27} -2.42638e11 q^{29} -6.45199e10i q^{31} -1.41954e12 q^{33} +4.16282e11i q^{35} -2.41678e12 q^{37} +1.25380e13i q^{39} +5.45949e12 q^{41} +1.67228e13i q^{43} +3.19811e13 q^{45} +1.33646e13i q^{47} +3.20928e13 q^{49} +8.31975e13i q^{51} -8.96899e13 q^{53} +4.94840e13i q^{55} -1.97722e14 q^{57} -9.54083e13i q^{59} +7.10026e13 q^{61} +8.75916e13i q^{63} +4.37066e14 q^{65} -3.55095e14i q^{67} +9.75599e13 q^{69} -8.97021e14i q^{71} -5.31684e14 q^{73} +6.67511e12i q^{75} -1.35530e14 q^{77} -2.18899e15i q^{79} +1.34504e15 q^{81} -2.86414e15i q^{83} +2.90019e15 q^{85} +2.71364e15i q^{87} +2.96991e15 q^{89} +1.19706e15i q^{91} -7.21583e14 q^{93} +6.89241e15i q^{95} -2.04301e15 q^{97} +1.04121e16i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 11183.9i	− 1.70460i	−0.523055	−	0.852299i	$-0.675208\pi$
$3$	− 11183.9i	− 1.70460i	0.523055	−	0.852299i	$-0.324792\pi$
$4$	0	0
$5$	−389860.	−0.998042	−0.499021	−	0.866590i	$-0.666307\pi$
$5$	−389860.	−0.998042	−0.499021	+	0.866590i	$0.666307\pi$
$6$	0	0
$7$	− 1.06777e6i	− 0.185223i	−0.995702	−	0.0926113i	$-0.970479\pi$
$7$	− 1.06777e6i	− 0.185223i	0.995702	−	0.0926113i	$-0.0295214\pi$
$8$	0	0
$9$	−8.20322e7	−1.90565
$10$	0	0
$11$	− 1.26927e8i	− 0.592126i	−0.955168	−	0.296063i	$-0.904326\pi$
$11$	− 1.26927e8i	− 0.592126i	0.955168	−	0.296063i	$-0.0956739\pi$
$12$	0	0
$13$	−1.12108e9	−1.37433	−0.687165	−	0.726501i	$-0.741145\pi$
$13$	−1.12108e9	−1.37433	−0.687165	+	0.726501i	$0.741145\pi$
$14$	0	0
$15$	4.36015e9i	1.70126i
$16$	0	0
$17$	−7.43906e9	−1.06642	−0.533208	−	0.845984i	$-0.679014\pi$
$17$	−7.43906e9	−1.06642	−0.533208	+	0.845984i	$0.679014\pi$
$18$	0	0
$19$	− 1.76792e10i	− 1.04096i	−0.853874	−	0.520479i	$-0.825753\pi$
$19$	− 1.76792e10i	− 1.04096i	0.853874	−	0.520479i	$-0.174247\pi$
$20$	0	0
$21$	−1.19418e10	−0.315730
$22$	0	0
$23$	8.72327e9i	0.111393i	0.998448	+	0.0556963i	$0.0177379\pi$
$23$	8.72327e9i	0.111393i	−0.998448	+	0.0556963i	$0.982262\pi$
$24$	0	0
$25$	−5.96851e8	−0.00391153
$26$	0	0
$27$	4.36008e11i	1.54378i
$28$	0	0
$29$	−2.42638e11	−0.485038	−0.242519	−	0.970147i	$-0.577974\pi$
$29$	−2.42638e11	−0.485038	−0.242519	+	0.970147i	$0.577974\pi$
$30$	0	0
$31$	− 6.45199e10i	− 0.0756485i	−0.999284	−	0.0378243i	$-0.987957\pi$
$31$	− 6.45199e10i	− 0.0756485i	0.999284	−	0.0378243i	$-0.0120427\pi$
$32$	0	0
$33$	−1.41954e12	−1.00934
$34$	0	0
$35$	4.16282e11i	0.184860i
$36$	0	0
$37$	−2.41678e12	−0.688055	−0.344028	−	0.938960i	$-0.611791\pi$
$37$	−2.41678e12	−0.688055	−0.344028	+	0.938960i	$0.611791\pi$
$38$	0	0
$39$	1.25380e13i	2.34268i
$40$	0	0
$41$	5.45949e12	0.683725	0.341862	−	0.939750i	$-0.388942\pi$
$41$	5.45949e12	0.683725	0.341862	+	0.939750i	$0.388942\pi$
$42$	0	0
$43$	1.67228e13i	1.43075i	0.698743	+	0.715373i	$0.253743\pi$
$43$	1.67228e13i	1.43075i	−0.698743	+	0.715373i	$0.746257\pi$
$44$	0	0
$45$	3.19811e13	1.90192
$46$	0	0
$47$	1.33646e13i	0.561272i	0.959814	+	0.280636i	$0.0905454\pi$
$47$	1.33646e13i	0.561272i	−0.959814	+	0.280636i	$0.909455\pi$
$48$	0	0
$49$	3.20928e13	0.965693
$50$	0	0
$51$	8.31975e13i	1.81781i
$52$	0	0
$53$	−8.96899e13	−1.44058	−0.720289	−	0.693675i	$-0.755991\pi$
$53$	−8.96899e13	−1.44058	−0.720289	+	0.693675i	$0.755991\pi$
$54$	0	0
$55$	4.94840e13i	0.590967i
$56$	0	0
$57$	−1.97722e14	−1.77442
$58$	0	0
$59$	− 9.54083e13i	− 0.649785i	−0.945751	−	0.324893i	$-0.894672\pi$
$59$	− 9.54083e13i	− 0.649785i	0.945751	−	0.324893i	$-0.105328\pi$
$60$	0	0
$61$	7.10026e13	0.370370	0.185185	−	0.982704i	$-0.440712\pi$
$61$	7.10026e13	0.370370	0.185185	+	0.982704i	$0.440712\pi$
$62$	0	0
$63$	8.75916e13i	0.352970i
$64$	0	0
$65$	4.37066e14	1.37164
$66$	0	0
$67$	− 3.55095e14i	− 0.874473i	−0.899347	−	0.437236i	$-0.855957\pi$
$67$	− 3.55095e14i	− 0.874473i	0.899347	−	0.437236i	$-0.144043\pi$
$68$	0	0
$69$	9.75599e13	0.189880
$70$	0	0
$71$	− 8.97021e14i	− 1.38911i	−0.719441	−	0.694553i	$-0.755602\pi$
$71$	− 8.97021e14i	− 1.38911i	0.719441	−	0.694553i	$-0.244398\pi$
$72$	0	0
$73$	−5.31684e14	−0.659281	−0.329641	−	0.944107i	$-0.606928\pi$
$73$	−5.31684e14	−0.659281	−0.329641	+	0.944107i	$0.606928\pi$
$74$	0	0
$75$	6.67511e12i	0.00666758i
$76$	0	0
$77$	−1.35530e14	−0.109675
$78$	0	0
$79$	− 2.18899e15i	− 1.44287i	−0.692484	−	0.721433i	$-0.743484\pi$
$79$	− 2.18899e15i	− 1.44287i	0.692484	−	0.721433i	$-0.256516\pi$
$80$	0	0
$81$	1.34504e15	0.725864
$82$	0	0
$83$	− 2.86414e15i	− 1.27166i	−0.771831	−	0.635828i	$-0.780659\pi$
$83$	− 2.86414e15i	− 1.27166i	0.771831	−	0.635828i	$-0.219341\pi$
$84$	0	0
$85$	2.90019e15	1.06433
$86$	0	0
$87$	2.71364e15i	0.826794i
$88$	0	0
$89$	2.96991e15	0.754439	0.377219	−	0.926124i	$-0.376880\pi$
$89$	2.96991e15	0.754439	0.377219	+	0.926124i	$0.376880\pi$
$90$	0	0
$91$	1.19706e15i	0.254557i
$92$	0	0
$93$	−7.21583e14	−0.128950
$94$	0	0
$95$	6.89241e15i	1.03892i
$96$	0	0
$97$	−2.04301e15	−0.260673	−0.130336	−	0.991470i	$-0.541606\pi$
$97$	−2.04301e15	−0.260673	−0.130336	+	0.991470i	$0.541606\pi$
$98$	0	0
$99$	1.04121e16i	1.12839i
$100$	0	0
$101$	8.70720e15	0.804095	0.402048	−	0.915619i	$-0.368299\pi$
$101$	8.70720e15	0.804095	0.402048	+	0.915619i	$0.368299\pi$
$102$	0	0
$103$	− 6.28728e15i	− 0.496324i	−0.968719	−	0.248162i	$-0.920174\pi$
$103$	− 6.28728e15i	− 0.496324i	0.968719	−	0.248162i	$-0.0798265\pi$
$104$	0	0
$105$	4.65564e15	0.315112
$106$	0	0
$107$	1.99151e16i	1.15908i	0.814945	+	0.579538i	$0.196767\pi$
$107$	1.99151e16i	1.15908i	−0.814945	+	0.579538i	$0.803233\pi$
$108$	0	0
$109$	−1.86420e16	−0.935579	−0.467790	−	0.883840i	$-0.654949\pi$
$109$	−1.86420e16	−0.935579	−0.467790	+	0.883840i	$0.654949\pi$
$110$	0	0
$111$	2.70290e16i	1.17286i
$112$	0	0
$113$	2.85508e16	1.07397	0.536983	−	0.843593i	$-0.319564\pi$
$113$	2.85508e16	1.07397	0.536983	+	0.843593i	$0.319564\pi$
$114$	0	0
$115$	− 3.40086e15i	− 0.111175i
$116$	0	0
$117$	9.19649e16	2.61900
$118$	0	0
$119$	7.94322e15i	0.197524i
$120$	0	0
$121$	2.98391e16	0.649387
$122$	0	0
$123$	− 6.10582e16i	− 1.16548i
$124$	0	0
$125$	5.97206e16	1.00195
$126$	0	0
$127$	− 6.87377e16i	− 1.01570i	−0.861446	−	0.507849i	$-0.830441\pi$
$127$	− 6.87377e16i	− 1.01570i	0.861446	−	0.507849i	$-0.169559\pi$
$128$	0	0
$129$	1.87026e17	2.43885
$130$	0	0
$131$	− 1.10790e17i	− 1.27741i	−0.769454	−	0.638703i	$-0.779471\pi$
$131$	− 1.10790e17i	− 1.27741i	0.769454	−	0.638703i	$-0.220529\pi$
$132$	0	0
$133$	−1.88773e16	−0.192809
$134$	0	0
$135$	− 1.69982e17i	− 1.54075i
$136$	0	0
$137$	−1.61324e17	−1.29997	−0.649985	−	0.759947i	$-0.725225\pi$
$137$	−1.61324e17	−1.29997	−0.649985	+	0.759947i	$0.725225\pi$
$138$	0	0
$139$	2.52370e17i	1.81100i	0.424342	+	0.905502i	$0.360505\pi$
$139$	2.52370e17i	1.81100i	−0.424342	+	0.905502i	$0.639495\pi$
$140$	0	0
$141$	1.49468e17	0.956744
$142$	0	0
$143$	1.42296e17i	0.813777i
$144$	0	0
$145$	9.45951e16	0.484088
$146$	0	0
$147$	− 3.58922e17i	− 1.64612i
$148$	0	0
$149$	−4.48178e16	−0.184485	−0.0922424	−	0.995737i	$-0.529403\pi$
$149$	−4.48178e16	−0.184485	−0.0922424	+	0.995737i	$0.529403\pi$
$150$	0	0
$151$	− 2.63140e17i	− 0.973580i	−0.873519	−	0.486790i	$-0.838168\pi$
$151$	− 2.63140e17i	− 0.973580i	0.873519	−	0.486790i	$-0.161832\pi$
$152$	0	0
$153$	6.10242e17	2.03222
$154$	0	0
$155$	2.51538e16i	0.0755004i
$156$	0	0
$157$	6.63199e17	1.79658	0.898291	−	0.439402i	$-0.144810\pi$
$157$	6.63199e17	1.79658	0.898291	+	0.439402i	$0.144810\pi$
$158$	0	0
$159$	1.00308e18i	2.45561i
$160$	0	0
$161$	9.31446e15	0.0206324
$162$	0	0
$163$	4.53009e17i	0.909089i	0.890724	+	0.454545i	$0.150198\pi$
$163$	4.53009e17i	0.909089i	−0.890724	+	0.454545i	$0.849802\pi$
$164$	0	0
$165$	5.53422e17	1.00736
$166$	0	0
$167$	− 1.30678e17i	− 0.216008i	−0.994150	−	0.108004i	$-0.965554\pi$
$167$	− 1.30678e17i	− 0.216008i	0.994150	−	0.108004i	$-0.0344460\pi$
$168$	0	0
$169$	5.91411e17	0.888782
$170$	0	0
$171$	1.45026e18i	1.98371i
$172$	0	0
$173$	−8.96392e17	−1.11720	−0.558598	−	0.829439i	$-0.688660\pi$
$173$	−8.96392e17	−1.11720	−0.558598	+	0.829439i	$0.688660\pi$
$174$	0	0
$175$	6.37301e14i	0 0.000724503i
$176$	0	0
$177$	−1.06703e18	−1.10762
$178$	0	0
$179$	− 1.25142e18i	− 1.18735i	−0.804706	−	0.593673i	$-0.797677\pi$
$179$	− 1.25142e18i	− 1.18735i	0.804706	−	0.593673i	$-0.202323\pi$
$180$	0	0
$181$	1.13448e17	0.0984850	0.0492425	−	0.998787i	$-0.484319\pi$
$181$	1.13448e17	0.0984850	0.0492425	+	0.998787i	$0.484319\pi$
$182$	0	0
$183$	− 7.94084e17i	− 0.631332i
$184$	0	0
$185$	9.42207e17	0.686708
$186$	0	0
$187$	9.44221e17i	0.631453i
$188$	0	0
$189$	4.65557e17	0.285942
$190$	0	0
$191$	1.56832e18i	0.885456i	0.896656	+	0.442728i	$0.145989\pi$
$191$	1.56832e18i	0.885456i	−0.896656	+	0.442728i	$0.854011\pi$
$192$	0	0
$193$	−4.81559e16	−0.0250145	−0.0125072	−	0.999922i	$-0.503981\pi$
$193$	−4.81559e16	−0.0250145	−0.0125072	+	0.999922i	$0.503981\pi$
$194$	0	0
$195$	− 4.88809e18i	− 2.33809i
$196$	0	0
$197$	−7.87391e17	−0.347105	−0.173552	−	0.984825i	$-0.555525\pi$
$197$	−7.87391e17	−0.347105	−0.173552	+	0.984825i	$0.555525\pi$
$198$	0	0
$199$	− 3.27310e18i	− 1.33087i	−0.746458	−	0.665433i	$-0.768247\pi$
$199$	− 3.27310e18i	− 1.33087i	0.746458	−	0.665433i	$-0.231753\pi$
$200$	0	0
$201$	−3.97134e18	−1.49062
$202$	0	0
$203$	2.59082e17i	0.0898399i
$204$	0	0
$205$	−2.12844e18	−0.682386
$206$	0	0
$207$	− 7.15589e17i	− 0.212276i
$208$	0	0
$209$	−2.24397e18	−0.616379
$210$	0	0
$211$	− 1.22766e18i	− 0.312476i	−0.987719	−	0.156238i	$-0.950063\pi$
$211$	− 1.22766e18i	− 0.312476i	0.987719	−	0.156238i	$-0.0499367\pi$
$212$	0	0
$213$	−1.00322e19	−2.36787
$214$	0	0
$215$	− 6.51957e18i	− 1.42795i
$216$	0	0
$217$	−6.88926e16	−0.0140118
$218$	0	0
$219$	5.94628e18i	1.12381i
$220$	0	0
$221$	8.33981e18	1.46561
$222$	0	0
$223$	9.94725e18i	1.62654i	0.581889	+	0.813268i	$0.302314\pi$
$223$	9.94725e18i	1.62654i	−0.581889	+	0.813268i	$0.697686\pi$
$224$	0	0
$225$	4.89610e16	0.00745402
$226$	0	0
$227$	3.06798e18i	0.435157i	0.976043	+	0.217578i	$0.0698158\pi$
$227$	3.06798e18i	0.435157i	−0.976043	+	0.217578i	$0.930184\pi$
$228$	0	0
$229$	6.70151e18	0.886112	0.443056	−	0.896494i	$-0.353894\pi$
$229$	6.70151e18	0.886112	0.443056	+	0.896494i	$0.353894\pi$
$230$	0	0
$231$	1.51574e18i	0.186952i
$232$	0	0
$233$	−7.17629e18	−0.826138	−0.413069	−	0.910700i	$-0.635543\pi$
$233$	−7.17629e18	−0.826138	−0.413069	+	0.910700i	$0.635543\pi$
$234$	0	0
$235$	− 5.21033e18i	− 0.560173i
$236$	0	0
$237$	−2.44813e19	−2.45951
$238$	0	0
$239$	1.01782e19i	0.956067i	0.878342	+	0.478034i	$0.158650\pi$
$239$	1.01782e19i	0.956067i	−0.878342	+	0.478034i	$0.841350\pi$
$240$	0	0
$241$	2.09501e18	0.184098	0.0920492	−	0.995754i	$-0.470658\pi$
$241$	2.09501e18	0.184098	0.0920492	+	0.995754i	$0.470658\pi$
$242$	0	0
$243$	3.72596e18i	0.306470i
$244$	0	0
$245$	−1.25117e19	−0.963802
$246$	0	0
$247$	1.98198e19i	1.43062i
$248$	0	0
$249$	−3.20322e19	−2.16766
$250$	0	0
$251$	1.97213e19i	1.25183i	0.779890	+	0.625916i	$0.215275\pi$
$251$	1.97213e19i	1.25183i	−0.779890	+	0.625916i	$0.784725\pi$
$252$	0	0
$253$	1.10722e18	0.0659585
$254$	0	0
$255$	− 3.24354e19i	− 1.81425i
$256$	0	0
$257$	−4.81863e18	−0.253197	−0.126598	−	0.991954i	$-0.540406\pi$
$257$	−4.81863e18	−0.253197	−0.126598	+	0.991954i	$0.540406\pi$
$258$	0	0
$259$	2.58057e18i	0.127443i
$260$	0	0
$261$	1.99042e19	0.924314
$262$	0	0
$263$	− 5.80322e18i	− 0.253526i	−0.991933	−	0.126763i	$-0.959541\pi$
$263$	− 5.80322e18i	− 0.253526i	0.991933	−	0.126763i	$-0.0404588\pi$
$264$	0	0
$265$	3.49665e19	1.43776
$266$	0	0
$267$	− 3.32151e19i	− 1.28601i
$268$	0	0
$269$	−3.64173e19	−1.32828	−0.664139	−	0.747609i	$-0.731202\pi$
$269$	−3.64173e19	−1.32828	−0.664139	+	0.747609i	$0.731202\pi$
$270$	0	0
$271$	− 2.22307e19i	− 0.764184i	−0.924124	−	0.382092i	$-0.875204\pi$
$271$	− 2.22307e19i	− 0.764184i	0.924124	−	0.382092i	$-0.124796\pi$
$272$	0	0
$273$	1.33878e19	0.433917
$274$	0	0
$275$	7.57569e16i	0.00231612i
$276$	0	0
$277$	5.13041e19	1.48018	0.740089	−	0.672509i	$-0.234783\pi$
$277$	5.13041e19	1.48018	0.740089	+	0.672509i	$0.234783\pi$
$278$	0	0
$279$	5.29271e18i	0.144160i
$280$	0	0
$281$	1.48805e19	0.382797	0.191398	−	0.981512i	$-0.438698\pi$
$281$	1.48805e19	0.382797	0.191398	+	0.981512i	$0.438698\pi$
$282$	0	0
$283$	− 7.34321e19i	− 1.78482i	−0.451225	−	0.892410i	$-0.649013\pi$
$283$	− 7.34321e19i	− 1.78482i	0.451225	−	0.892410i	$-0.350987\pi$
$284$	0	0
$285$	7.70838e19	1.77094
$286$	0	0
$287$	− 5.82949e18i	− 0.126641i
$288$	0	0
$289$	6.67845e18	0.137244
$290$	0	0
$291$	2.28487e19i	0.444343i
$292$	0	0
$293$	−4.26888e19	−0.785911	−0.392955	−	0.919557i	$-0.628547\pi$
$293$	−4.26888e19	−0.785911	−0.392955	+	0.919557i	$0.628547\pi$
$294$	0	0
$295$	3.71959e19i	0.648513i
$296$	0	0
$297$	5.53414e19	0.914110
$298$	0	0
$299$	− 9.77951e18i	− 0.153090i
$300$	0	0
$301$	1.78562e19	0.265007
$302$	0	0
$303$	− 9.73802e19i	− 1.37066i
$304$	0	0
$305$	−2.76811e19	−0.369645
$306$	0	0
$307$	1.27750e20i	1.61903i	0.587100	+	0.809514i	$0.300269\pi$
$307$	1.27750e20i	1.61903i	−0.587100	+	0.809514i	$0.699731\pi$
$308$	0	0
$309$	−7.03161e19	−0.846032
$310$	0	0
$311$	6.03449e19i	0.689537i	0.938688	+	0.344768i	$0.112043\pi$
$311$	6.03449e19i	0.689537i	−0.938688	+	0.344768i	$0.887957\pi$
$312$	0	0
$313$	−2.85722e19	−0.310163	−0.155081	−	0.987902i	$-0.549564\pi$
$313$	−2.85722e19	−0.310163	−0.155081	+	0.987902i	$0.549564\pi$
$314$	0	0
$315$	− 3.41485e19i	− 0.352279i
$316$	0	0
$317$	1.99139e20	1.95291	0.976457	−	0.215712i	$-0.0692072\pi$
$317$	1.99139e20	1.95291	0.976457	+	0.215712i	$0.0692072\pi$
$318$	0	0
$319$	3.07975e19i	0.287204i
$320$	0	0
$321$	2.22728e20	1.97576
$322$	0	0
$323$	1.31517e20i	1.11010i
$324$	0	0
$325$	6.69120e17	0.00537573
$326$	0	0
$327$	2.08490e20i	1.59479i
$328$	0	0
$329$	1.42704e19	0.103960
$330$	0	0
$331$	1.56745e20i	1.08785i	0.839133	+	0.543926i	$0.183063\pi$
$331$	1.56745e20i	1.08785i	−0.839133	+	0.543926i	$0.816937\pi$
$332$	0	0
$333$	1.98254e20	1.31120
$334$	0	0
$335$	1.38438e20i	0.872761i
$336$	0	0
$337$	3.17497e19	0.190854	0.0954268	−	0.995436i	$-0.469578\pi$
$337$	3.17497e19	0.190854	0.0954268	+	0.995436i	$0.469578\pi$
$338$	0	0
$339$	− 3.19308e20i	− 1.83068i
$340$	0	0
$341$	−8.18935e18	−0.0447935
$342$	0	0
$343$	− 6.97529e19i	− 0.364091i
$344$	0	0
$345$	−3.80347e19	−0.189508
$346$	0	0
$347$	− 2.06802e20i	− 0.983827i	−0.870644	−	0.491914i	$-0.836298\pi$
$347$	− 2.06802e20i	− 0.983827i	0.870644	−	0.491914i	$-0.163702\pi$
$348$	0	0
$349$	−3.23582e20	−1.47021	−0.735107	−	0.677951i	$-0.762868\pi$
$349$	−3.23582e20	−1.47021	−0.735107	+	0.677951i	$0.762868\pi$
$350$	0	0
$351$	− 4.88801e20i	− 2.12166i
$352$	0	0
$353$	4.02153e19	0.166799	0.0833996	−	0.996516i	$-0.473422\pi$
$353$	4.02153e19	0.166799	0.0833996	+	0.996516i	$0.473422\pi$
$354$	0	0
$355$	3.49713e20i	1.38639i
$356$	0	0
$357$	8.88359e19	0.336700
$358$	0	0
$359$	− 7.71450e19i	− 0.279610i	−0.990179	−	0.139805i	$-0.955352\pi$
$359$	− 7.71450e19i	− 0.279610i	0.990179	−	0.139805i	$-0.0446475\pi$
$360$	0	0
$361$	−2.41121e19	−0.0835945
$362$	0	0
$363$	− 3.33717e20i	− 1.10694i
$364$	0	0
$365$	2.07282e20	0.657990
$366$	0	0
$367$	8.69548e19i	0.264220i	0.991235	+	0.132110i	$0.0421752\pi$
$367$	8.69548e19i	0.264220i	−0.991235	+	0.132110i	$0.957825\pi$
$368$	0	0
$369$	−4.47854e20	−1.30294
$370$	0	0
$371$	9.57683e19i	0.266827i
$372$	0	0
$373$	1.47147e19	0.0392717	0.0196358	−	0.999807i	$-0.493749\pi$
$373$	1.47147e19	0.0392717	0.0196358	+	0.999807i	$0.493749\pi$
$374$	0	0
$375$	− 6.67908e20i	− 1.70792i
$376$	0	0
$377$	2.72018e20	0.666602
$378$	0	0
$379$	1.42139e20i	0.333888i	0.985966	+	0.166944i	$0.0533899\pi$
$379$	1.42139e20i	0.333888i	−0.985966	+	0.166944i	$0.946610\pi$
$380$	0	0
$381$	−7.68753e20	−1.73136
$382$	0	0
$383$	7.65100e20i	1.65245i	0.563340	+	0.826225i	$0.309516\pi$
$383$	7.65100e20i	1.65245i	−0.563340	+	0.826225i	$0.690484\pi$
$384$	0	0
$385$	5.28376e19	0.109460
$386$	0	0
$387$	− 1.37181e21i	− 2.72651i
$388$	0	0
$389$	−6.56368e20	−1.25184	−0.625922	−	0.779886i	$-0.715277\pi$
$389$	−6.56368e20	−1.25184	−0.625922	+	0.779886i	$0.715277\pi$
$390$	0	0
$391$	− 6.48929e19i	− 0.118791i
$392$	0	0
$393$	−1.23906e21	−2.17746
$394$	0	0
$395$	8.53399e20i	1.44004i
$396$	0	0
$397$	3.26185e20	0.528616	0.264308	−	0.964438i	$-0.414857\pi$
$397$	3.26185e20	0.528616	0.264308	+	0.964438i	$0.414857\pi$
$398$	0	0
$399$	2.11122e20i	0.328662i
$400$	0	0
$401$	−3.88647e20	−0.581299	−0.290650	−	0.956830i	$-0.593871\pi$
$401$	−3.88647e20	−0.581299	−0.290650	+	0.956830i	$0.593871\pi$
$402$	0	0
$403$	7.23322e19i	0.103966i
$404$	0	0
$405$	−5.24378e20	−0.724443
$406$	0	0
$407$	3.06756e20i	0.407416i
$408$	0	0
$409$	−1.37019e21	−1.74983	−0.874915	−	0.484276i	$-0.839083\pi$
$409$	−1.37019e21	−1.74983	−0.874915	+	0.484276i	$0.839083\pi$
$410$	0	0
$411$	1.80422e21i	2.21593i
$412$	0	0
$413$	−1.01874e20	−0.120355
$414$	0	0
$415$	1.11661e21i	1.26917i
$416$	0	0
$417$	2.82247e21	3.08703
$418$	0	0
$419$	3.51169e20i	0.369661i	0.982770	+	0.184831i	$0.0591736\pi$
$419$	3.51169e20i	0.369661i	−0.982770	+	0.184831i	$0.940826\pi$
$420$	0	0
$421$	−5.99942e20	−0.607928	−0.303964	−	0.952683i	$-0.598310\pi$
$421$	−5.99942e20	−0.607928	−0.303964	+	0.952683i	$0.598310\pi$
$422$	0	0
$423$	− 1.09633e21i	− 1.06959i
$424$	0	0
$425$	4.44001e18	0.00417132
$426$	0	0
$427$	− 7.58146e19i	− 0.0686009i
$428$	0	0
$429$	1.59142e21	1.38716
$430$	0	0
$431$	− 4.16237e20i	− 0.349561i	−0.984607	−	0.174780i	$-0.944078\pi$
$431$	− 4.16237e20i	− 0.349561i	0.984607	−	0.174780i	$-0.0559216\pi$
$432$	0	0
$433$	−2.60242e20	−0.210607	−0.105304	−	0.994440i	$-0.533581\pi$
$433$	−2.60242e20	−0.210607	−0.105304	+	0.994440i	$0.533581\pi$
$434$	0	0
$435$	− 1.05794e21i	− 0.825176i
$436$	0	0
$437$	1.54220e20	0.115955
$438$	0	0
$439$	9.76131e20i	0.707607i	0.935320	+	0.353803i	$0.115112\pi$
$439$	9.76131e20i	0.707607i	−0.935320	+	0.353803i	$0.884888\pi$
$440$	0	0
$441$	−2.63264e21	−1.84028
$442$	0	0
$443$	2.08187e20i	0.140354i	0.997535	+	0.0701770i	$0.0223564\pi$
$443$	2.08187e20i	0.140354i	−0.997535	+	0.0701770i	$0.977644\pi$
$444$	0	0
$445$	−1.15785e21	−0.752962
$446$	0	0
$447$	5.01237e20i	0.314473i
$448$	0	0
$449$	−2.37764e20	−0.143938	−0.0719690	−	0.997407i	$-0.522928\pi$
$449$	−2.37764e20	−0.143938	−0.0719690	+	0.997407i	$0.522928\pi$
$450$	0	0
$451$	− 6.92959e20i	− 0.404851i
$452$	0	0
$453$	−2.94292e21	−1.65956
$454$	0	0
$455$	− 4.66686e20i	− 0.254059i
$456$	0	0
$457$	6.36224e20	0.334411	0.167206	−	0.985922i	$-0.446526\pi$
$457$	6.36224e20	0.334411	0.167206	+	0.985922i	$0.446526\pi$
$458$	0	0
$459$	− 3.24349e21i	− 1.64631i
$460$	0	0
$461$	1.40752e21	0.689998	0.344999	−	0.938603i	$-0.387879\pi$
$461$	1.40752e21	0.689998	0.344999	+	0.938603i	$0.387879\pi$
$462$	0	0
$463$	2.63989e20i	0.125008i	0.998045	+	0.0625040i	$0.0199086\pi$
$463$	2.63989e20i	0.125008i	−0.998045	+	0.0625040i	$0.980091\pi$
$464$	0	0
$465$	2.81316e20	0.128698
$466$	0	0
$467$	3.55842e20i	0.157298i	0.996902	+	0.0786488i	$0.0250606\pi$
$467$	3.55842e20i	0.157298i	−0.996902	+	0.0786488i	$0.974939\pi$
$468$	0	0
$469$	−3.79160e20	−0.161972
$470$	0	0
$471$	− 7.41713e21i	− 3.06245i
$472$	0	0
$473$	2.12259e21	0.847182
$474$	0	0
$475$	1.05518e19i	0.00407174i
$476$	0	0
$477$	7.35746e21	2.74524
$478$	0	0
$479$	− 5.80813e20i	− 0.209581i	−0.994494	−	0.104791i	$-0.966583\pi$
$479$	− 5.80813e20i	− 0.209581i	0.994494	−	0.104791i	$-0.0334172\pi$
$480$	0	0
$481$	2.70941e21	0.945615
$482$	0	0
$483$	− 1.04172e20i	− 0.0351700i
$484$	0	0
$485$	7.96487e20	0.260163
$486$	0	0
$487$	6.05794e21i	1.91467i	0.288988	+	0.957333i	$0.406681\pi$
$487$	6.05794e21i	1.91467i	−0.288988	+	0.957333i	$0.593319\pi$
$488$	0	0
$489$	5.06640e21	1.54963
$490$	0	0
$491$	3.77866e21i	1.11863i	0.828956	+	0.559314i	$0.188935\pi$
$491$	3.77866e21i	1.11863i	−0.828956	+	0.559314i	$0.811065\pi$
$492$	0	0
$493$	1.80500e21	0.517252
$494$	0	0
$495$	− 4.05928e21i	− 1.12618i
$496$	0	0
$497$	−9.57813e20	−0.257294
$498$	0	0
$499$	1.85736e21i	0.483162i	0.970381	+	0.241581i	$0.0776659\pi$
$499$	1.85736e21i	0.483162i	−0.970381	+	0.241581i	$0.922334\pi$
$500$	0	0
$501$	−1.46148e21	−0.368207
$502$	0	0
$503$	5.73319e21i	1.39911i	0.714578	+	0.699556i	$0.246619\pi$
$503$	5.73319e21i	1.39911i	−0.714578	+	0.699556i	$0.753381\pi$
$504$	0	0
$505$	−3.39459e21	−0.802521
$506$	0	0
$507$	− 6.61426e21i	− 1.51502i
$508$	0	0
$509$	2.45518e21	0.544930	0.272465	−	0.962166i	$-0.412161\pi$
$509$	2.45518e21	0.544930	0.272465	+	0.962166i	$0.412161\pi$
$510$	0	0
$511$	5.67717e20i	0.122114i
$512$	0	0
$513$	7.70827e21	1.60701
$514$	0	0
$515$	2.45116e21i	0.495352i
$516$	0	0
$517$	1.69634e21	0.332344
$518$	0	0
$519$	1.00251e22i	1.90437i
$520$	0	0
$521$	−2.93027e21	−0.539767	−0.269884	−	0.962893i	$-0.586985\pi$
$521$	−2.93027e21	−0.539767	−0.269884	+	0.962893i	$0.586985\pi$
$522$	0	0
$523$	4.71391e21i	0.842109i	0.907035	+	0.421054i	$0.138340\pi$
$523$	4.71391e21i	0.842109i	−0.907035	+	0.421054i	$0.861660\pi$
$524$	0	0
$525$	7.12749e18	0.00123499
$526$	0	0
$527$	4.79968e20i	0.0806728i
$528$	0	0
$529$	6.05651e21	0.987592
$530$	0	0
$531$	7.82655e21i	1.23827i
$532$	0	0
$533$	−6.12054e21	−0.939663
$534$	0	0
$535$	− 7.76411e21i	− 1.15681i
$536$	0	0
$537$	−1.39957e22	−2.02395
$538$	0	0
$539$	− 4.07346e21i	− 0.571812i
$540$	0	0
$541$	1.23122e22	1.67787	0.838933	−	0.544235i	$-0.183180\pi$
$541$	1.23122e22	1.67787	0.838933	+	0.544235i	$0.183180\pi$
$542$	0	0
$543$	− 1.26879e21i	− 0.167877i
$544$	0	0
$545$	7.26778e21	0.933748
$546$	0	0
$547$	9.06413e21i	1.13091i	0.824780	+	0.565454i	$0.191299\pi$
$547$	9.06413e21i	1.13091i	−0.824780	+	0.565454i	$0.808701\pi$
$548$	0	0
$549$	−5.82450e21	−0.705797
$550$	0	0
$551$	4.28965e21i	0.504904i
$552$	0	0
$553$	−2.33734e21	−0.267251
$554$	0	0
$555$	− 1.05375e22i	− 1.17056i
$556$	0	0
$557$	−6.93670e21	−0.748706	−0.374353	−	0.927286i	$-0.622135\pi$
$557$	−6.93670e21	−0.748706	−0.374353	+	0.927286i	$0.622135\pi$
$558$	0	0
$559$	− 1.87477e22i	− 1.96632i
$560$	0	0
$561$	1.05600e22	1.07637
$562$	0	0
$563$	− 9.45591e21i	− 0.936777i	−0.883523	−	0.468388i	$-0.844835\pi$
$563$	− 9.45591e21i	− 0.936777i	0.883523	−	0.468388i	$-0.155165\pi$
$564$	0	0
$565$	−1.11308e22	−1.07186
$566$	0	0
$567$	− 1.43620e21i	− 0.134446i
$568$	0	0
$569$	−1.07736e22	−0.980536	−0.490268	−	0.871572i	$-0.663101\pi$
$569$	−1.07736e22	−0.980536	−0.490268	+	0.871572i	$0.663101\pi$
$570$	0	0
$571$	− 1.49259e22i	− 1.32084i	−0.750895	−	0.660421i	$-0.770378\pi$
$571$	− 1.49259e22i	− 1.32084i	0.750895	−	0.660421i	$-0.229622\pi$
$572$	0	0
$573$	1.75399e22	1.50935
$574$	0	0
$575$	− 5.20650e18i	0 0.000435715i
$576$	0	0
$577$	4.37719e21	0.356278	0.178139	−	0.984005i	$-0.442992\pi$
$577$	4.37719e21	0.356278	0.178139	+	0.984005i	$0.442992\pi$
$578$	0	0
$579$	5.38570e20i	0.0426396i
$580$	0	0
$581$	−3.05825e21	−0.235539
$582$	0	0
$583$	1.13841e22i	0.853003i
$584$	0	0
$585$	−3.58535e22	−2.61387
$586$	0	0
$587$	− 3.64181e21i	− 0.258353i	−0.991622	−	0.129176i	$-0.958767\pi$
$587$	− 3.64181e21i	− 0.258353i	0.991622	−	0.129176i	$-0.0412333\pi$
$588$	0	0
$589$	−1.14066e21	−0.0787470
$590$	0	0
$591$	8.80607e21i	0.591674i
$592$	0	0
$593$	−2.66186e21	−0.174080	−0.0870399	−	0.996205i	$-0.527741\pi$
$593$	−2.66186e21	−0.174080	−0.0870399	+	0.996205i	$0.527741\pi$
$594$	0	0
$595$	− 3.09674e21i	− 0.197138i
$596$	0	0
$597$	−3.66059e22	−2.26859
$598$	0	0
$599$	− 2.49736e22i	− 1.50684i	−0.657540	−	0.753420i	$-0.728403\pi$
$599$	− 2.49736e22i	− 1.50684i	0.657540	−	0.753420i	$-0.271597\pi$
$600$	0	0
$601$	2.22580e22	1.30765	0.653823	−	0.756648i	$-0.273164\pi$
$601$	2.22580e22	1.30765	0.653823	+	0.756648i	$0.273164\pi$
$602$	0	0
$603$	2.91292e22i	1.66644i
$604$	0	0
$605$	−1.16331e22	−0.648115
$606$	0	0
$607$	− 8.65974e21i	− 0.469889i	−0.972009	−	0.234945i	$-0.924509\pi$
$607$	− 8.65974e21i	− 0.469889i	0.972009	−	0.234945i	$-0.0754909\pi$
$608$	0	0
$609$	2.89754e21	0.153141
$610$	0	0
$611$	− 1.49828e22i	− 0.771373i
$612$	0	0
$613$	1.90666e22	0.956287	0.478144	−	0.878282i	$-0.341310\pi$
$613$	1.90666e22	0.956287	0.478144	+	0.878282i	$0.341310\pi$
$614$	0	0
$615$	2.38042e22i	1.16319i
$616$	0	0
$617$	−1.95858e22	−0.932526	−0.466263	−	0.884646i	$-0.654400\pi$
$617$	−1.95858e22	−0.932526	−0.466263	+	0.884646i	$0.654400\pi$
$618$	0	0
$619$	1.14734e22i	0.532314i	0.963930	+	0.266157i	$0.0857539\pi$
$619$	1.14734e22i	0.532314i	−0.963930	+	0.266157i	$0.914246\pi$
$620$	0	0
$621$	−3.80342e21	−0.171965
$622$	0	0
$623$	− 3.17119e21i	− 0.139739i
$624$	0	0
$625$	−2.31916e22	−0.996073
$626$	0	0
$627$	2.50963e22i	1.05068i
$628$	0	0
$629$	1.79786e22	0.733753
$630$	0	0
$631$	− 2.17142e22i	− 0.863991i	−0.901876	−	0.431995i	$-0.857810\pi$
$631$	− 2.17142e22i	− 0.863991i	0.901876	−	0.431995i	$-0.142190\pi$
$632$	0	0
$633$	−1.37299e22	−0.532646
$634$	0	0
$635$	2.67981e22i	1.01371i
$636$	0	0
$637$	−3.59787e22	−1.32718
$638$	0	0
$639$	7.35846e22i	2.64716i
$640$	0	0
$641$	−1.08805e22	−0.381753	−0.190877	−	0.981614i	$-0.561133\pi$
$641$	−1.08805e22	−0.381753	−0.190877	+	0.981614i	$0.561133\pi$
$642$	0	0
$643$	− 2.77111e22i	− 0.948345i	−0.880432	−	0.474172i	$-0.842747\pi$
$643$	− 2.77111e22i	− 0.948345i	0.880432	−	0.474172i	$-0.157253\pi$
$644$	0	0
$645$	−7.29141e22	−2.43407
$646$	0	0
$647$	2.15902e22i	0.703109i	0.936167	+	0.351555i	$0.114347\pi$
$647$	2.15902e22i	0.703109i	−0.936167	+	0.351555i	$0.885653\pi$
$648$	0	0
$649$	−1.21099e22	−0.384755
$650$	0	0
$651$	7.70485e20i	0.0238845i
$652$	0	0
$653$	3.41825e22	1.03395	0.516973	−	0.856002i	$-0.327059\pi$
$653$	3.41825e22	1.03395	0.516973	+	0.856002i	$0.327059\pi$
$654$	0	0
$655$	4.31925e22i	1.27490i
$656$	0	0
$657$	4.36152e22	1.25636
$658$	0	0
$659$	− 1.24387e22i	− 0.349696i	−0.984595	−	0.174848i	$-0.944057\pi$
$659$	− 1.24387e22i	− 0.349696i	0.984595	−	0.174848i	$-0.0559434\pi$
$660$	0	0
$661$	1.37447e22	0.377157	0.188578	−	0.982058i	$-0.439612\pi$
$661$	1.37447e22	0.377157	0.188578	+	0.982058i	$0.439612\pi$
$662$	0	0
$663$	− 9.32713e22i	− 2.49827i
$664$	0	0
$665$	7.35952e21	0.192432
$666$	0	0
$667$	− 2.11660e21i	− 0.0540297i
$668$	0	0
$669$	1.11249e23	2.77259
$670$	0	0
$671$	− 9.01219e21i	− 0.219306i
$672$	0	0
$673$	1.81248e22	0.430679	0.215339	−	0.976539i	$-0.430914\pi$
$673$	1.81248e22	0.430679	0.215339	+	0.976539i	$0.430914\pi$
$674$	0	0
$675$	− 2.60232e20i	− 0.00603852i
$676$	0	0
$677$	−2.73580e22	−0.619977	−0.309988	−	0.950740i	$-0.600325\pi$
$677$	−2.73580e22	−0.619977	−0.309988	+	0.950740i	$0.600325\pi$
$678$	0	0
$679$	2.18146e21i	0.0482825i
$680$	0	0
$681$	3.43119e22	0.741767
$682$	0	0
$683$	− 5.92941e21i	− 0.125212i	−0.998038	−	0.0626060i	$-0.980059\pi$
$683$	− 5.92941e21i	− 0.125212i	0.998038	−	0.0626060i	$-0.0199411\pi$
$684$	0	0
$685$	6.28937e22	1.29743
$686$	0	0
$687$	− 7.49488e22i	− 1.51046i
$688$	0	0
$689$	1.00550e23	1.97983
$690$	0	0
$691$	− 8.61448e22i	− 1.65731i	−0.559757	−	0.828657i	$-0.689106\pi$
$691$	− 8.61448e22i	− 1.65731i	0.559757	−	0.828657i	$-0.310894\pi$
$692$	0	0
$693$	1.11178e22	0.209003
$694$	0	0
$695$	− 9.83891e22i	− 1.80746i
$696$	0	0
$697$	−4.06135e22	−0.729135
$698$	0	0
$699$	8.02586e22i	1.40823i
$700$	0	0
$701$	−4.35956e22	−0.747649	−0.373825	−	0.927499i	$-0.621954\pi$
$701$	−4.35956e22	−0.747649	−0.373825	+	0.927499i	$0.621954\pi$
$702$	0	0
$703$	4.27267e22i	0.716237i
$704$	0	0
$705$	−5.82717e22	−0.954871
$706$	0	0
$707$	− 9.29730e21i	− 0.148937i
$708$	0	0
$709$	−8.06491e22	−1.26308	−0.631539	−	0.775344i	$-0.717576\pi$
$709$	−8.06491e22	−1.26308	−0.631539	+	0.775344i	$0.717576\pi$
$710$	0	0
$711$	1.79567e23i	2.74961i
$712$	0	0
$713$	5.62825e20	0.00842669
$714$	0	0
$715$	− 5.54757e22i	− 0.812183i
$716$	0	0
$717$	1.13832e23	1.62971
$718$	0	0
$719$	− 1.00543e23i	− 1.40774i	−0.710329	−	0.703870i	$-0.751454\pi$
$719$	− 1.00543e23i	− 1.40774i	0.710329	−	0.703870i	$-0.248546\pi$
$720$	0	0
$721$	−6.71338e21	−0.0919303
$722$	0	0
$723$	− 2.34303e22i	− 0.313814i
$724$	0	0
$725$	1.44819e20	0.00189724
$726$	0	0
$727$	− 1.04861e23i	− 1.34381i	−0.740637	−	0.671905i	$-0.765476\pi$
$727$	− 1.04861e23i	− 1.34381i	0.740637	−	0.671905i	$-0.234524\pi$
$728$	0	0
$729$	9.95702e22	1.24827
$730$	0	0
$731$	− 1.24402e23i	− 1.52577i
$732$	0	0
$733$	−9.82469e22	−1.17893	−0.589464	−	0.807795i	$-0.700661\pi$
$733$	−9.82469e22	−1.17893	−0.589464	+	0.807795i	$0.700661\pi$
$734$	0	0
$735$	1.39929e23i	1.64290i
$736$	0	0
$737$	−4.50713e22	−0.517798
$738$	0	0
$739$	6.82761e22i	0.767561i	0.923424	+	0.383781i	$0.125378\pi$
$739$	6.82761e22i	0.767561i	−0.923424	+	0.383781i	$0.874622\pi$
$740$	0	0
$741$	2.21662e23	2.43863
$742$	0	0
$743$	1.19418e23i	1.28575i	0.765970	+	0.642876i	$0.222259\pi$
$743$	1.19418e23i	1.28575i	−0.765970	+	0.642876i	$0.777741\pi$
$744$	0	0
$745$	1.74727e22	0.184124
$746$	0	0
$747$	2.34952e23i	2.42334i
$748$	0	0
$749$	2.12648e22	0.214687
$750$	0	0
$751$	− 4.88585e22i	− 0.482859i	−0.970418	−	0.241430i	$-0.922384\pi$
$751$	− 4.88585e22i	− 0.482859i	0.970418	−	0.241430i	$-0.0776163\pi$
$752$	0	0
$753$	2.20561e23	2.13387
$754$	0	0
$755$	1.02588e23i	0.971674i
$756$	0	0
$757$	1.86534e23	1.72978	0.864892	−	0.501958i	$-0.167387\pi$
$757$	1.86534e23	1.72978	0.864892	+	0.501958i	$0.167387\pi$
$758$	0	0
$759$	− 1.23830e22i	− 0.112433i
$760$	0	0
$761$	−1.86105e22	−0.165456	−0.0827279	−	0.996572i	$-0.526363\pi$
$761$	−1.86105e22	−0.165456	−0.0827279	+	0.996572i	$0.526363\pi$
$762$	0	0
$763$	1.99054e22i	0.173290i
$764$	0	0
$765$	−2.37909e23	−2.02824
$766$	0	0
$767$	1.06961e23i	0.893019i
$768$	0	0
$769$	7.64403e21	0.0625046	0.0312523	−	0.999512i	$-0.490050\pi$
$769$	7.64403e21	0.0625046	0.0312523	+	0.999512i	$0.490050\pi$
$770$	0	0
$771$	5.38909e22i	0.431599i
$772$	0	0
$773$	2.77389e22	0.217597	0.108799	−	0.994064i	$-0.465300\pi$
$773$	2.77389e22	0.217597	0.108799	+	0.994064i	$0.465300\pi$
$774$	0	0
$775$	3.85088e19i	0 0.000295901i
$776$	0	0
$777$	2.88607e22	0.217240
$778$	0	0
$779$	− 9.65193e22i	− 0.711729i
$780$	0	0
$781$	−1.13857e23	−0.822526
$782$	0	0
$783$	− 1.05792e23i	− 0.748790i
$784$	0	0
$785$	−2.58555e23	−1.79306
$786$	0	0
$787$	1.33486e23i	0.907063i	0.891240	+	0.453531i	$0.149836\pi$
$787$	1.33486e23i	0.907063i	−0.891240	+	0.453531i	$0.850164\pi$
$788$	0	0
$789$	−6.49024e22	−0.432160
$790$	0	0
$791$	− 3.04857e22i	− 0.198923i
$792$	0	0
$793$	−7.95998e22	−0.509010
$794$	0	0
$795$	− 3.91061e23i	− 2.45080i
$796$	0	0
$797$	1.00556e23	0.617647	0.308824	−	0.951119i	$-0.400065\pi$
$797$	1.00556e23	0.617647	0.308824	+	0.951119i	$0.400065\pi$
$798$	0	0
$799$	− 9.94202e22i	− 0.598550i
$800$	0	0
$801$	−2.43629e23	−1.43770
$802$	0	0
$803$	6.74853e22i	0.390378i
$804$	0	0
$805$	−3.63134e21	−0.0205920
$806$	0	0
$807$	4.07286e23i	2.26418i
$808$	0	0
$809$	3.21093e22	0.175002	0.0875010	−	0.996164i	$-0.472112\pi$
$809$	3.21093e22	0.175002	0.0875010	+	0.996164i	$0.472112\pi$
$810$	0	0
$811$	− 1.74376e23i	− 0.931792i	−0.884839	−	0.465896i	$-0.845732\pi$
$811$	− 1.74376e23i	− 0.931792i	0.884839	−	0.465896i	$-0.154268\pi$
$812$	0	0
$813$	−2.48625e23	−1.30263
$814$	0	0
$815$	− 1.76610e23i	− 0.907309i
$816$	0	0
$817$	2.95646e23	1.48935
$818$	0	0
$819$	− 9.81975e22i	− 0.485098i
$820$	0	0
$821$	−2.77373e23	−1.34375	−0.671875	−	0.740664i	$-0.734511\pi$
$821$	−2.77373e23	−1.34375	−0.671875	+	0.740664i	$0.734511\pi$
$822$	0	0
$823$	− 1.57320e23i	− 0.747456i	−0.927538	−	0.373728i	$-0.878079\pi$
$823$	− 1.57320e23i	− 0.747456i	0.927538	−	0.373728i	$-0.121921\pi$
$824$	0	0
$825$	8.47255e20	0.00394805
$826$	0	0
$827$	8.52321e22i	0.389546i	0.980848	+	0.194773i	$0.0623970\pi$
$827$	8.52321e22i	0.389546i	−0.980848	+	0.194773i	$0.937603\pi$
$828$	0	0
$829$	1.30428e23	0.584704	0.292352	−	0.956311i	$-0.405562\pi$
$829$	1.30428e23	0.584704	0.292352	+	0.956311i	$0.405562\pi$
$830$	0	0
$831$	− 5.73778e23i	− 2.52311i
$832$	0	0
$833$	−2.38740e23	−1.02983
$834$	0	0
$835$	5.09461e22i	0.215585i
$836$	0	0
$837$	2.81312e22	0.116784
$838$	0	0
$839$	3.04979e23i	1.24215i	0.783751	+	0.621075i	$0.213304\pi$
$839$	3.04979e23i	1.24215i	−0.783751	+	0.621075i	$0.786696\pi$
$840$	0	0
$841$	−1.91373e23	−0.764738
$842$	0	0
$843$	− 1.66422e23i	− 0.652514i
$844$	0	0
$845$	−2.30568e23	−0.887043
$846$	0	0
$847$	− 3.18614e22i	− 0.120281i
$848$	0	0
$849$	−8.21255e23	−3.04240
$850$	0	0
$851$	− 2.10822e22i	− 0.0766443i
$852$	0	0
$853$	6.07340e22	0.216691	0.108345	−	0.994113i	$-0.465445\pi$
$853$	6.07340e22	0.216691	0.108345	+	0.994113i	$0.465445\pi$
$854$	0	0
$855$	− 5.65399e23i	− 1.97982i
$856$	0	0
$857$	4.13905e23	1.42251	0.711254	−	0.702935i	$-0.248128\pi$
$857$	4.13905e23	1.42251	0.711254	+	0.702935i	$0.248128\pi$
$858$	0	0
$859$	4.51382e23i	1.52264i	0.648374	+	0.761322i	$0.275449\pi$
$859$	4.51382e23i	1.52264i	−0.648374	+	0.761322i	$0.724551\pi$
$860$	0	0
$861$	−6.51962e22	−0.215872
$862$	0	0
$863$	4.39182e23i	1.42744i	0.700430	+	0.713721i	$0.252991\pi$
$863$	4.39182e23i	1.42744i	−0.700430	+	0.713721i	$0.747009\pi$
$864$	0	0
$865$	3.49468e23	1.11501
$866$	0	0
$867$	− 7.46909e22i	− 0.233946i
$868$	0	0
$869$	−2.77842e23	−0.854359
$870$	0	0
$871$	3.98091e23i	1.20181i
$872$	0	0
$873$	1.67592e23	0.496753
$874$	0	0
$875$	− 6.37680e22i	− 0.185583i
$876$	0	0
$877$	−1.92199e23	−0.549230	−0.274615	−	0.961554i	$-0.588550\pi$
$877$	−1.92199e23	−0.549230	−0.274615	+	0.961554i	$0.588550\pi$
$878$	0	0
$879$	4.77426e23i	1.33966i
$880$	0	0
$881$	−8.58923e22	−0.236672	−0.118336	−	0.992974i	$-0.537756\pi$
$881$	−8.58923e22	−0.236672	−0.118336	+	0.992974i	$0.537756\pi$
$882$	0	0
$883$	1.84130e23i	0.498240i	0.968473	+	0.249120i	$0.0801414\pi$
$883$	1.84130e23i	0.498240i	−0.968473	+	0.249120i	$0.919859\pi$
$884$	0	0
$885$	4.15994e23	1.10545
$886$	0	0
$887$	− 3.79305e22i	− 0.0989920i	−0.998774	−	0.0494960i	$-0.984238\pi$
$887$	− 3.79305e22i	− 0.0989920i	0.998774	−	0.0494960i	$-0.0157615\pi$
$888$	0	0
$889$	−7.33961e22	−0.188130
$890$	0	0
$891$	− 1.70723e23i	− 0.429803i
$892$	0	0
$893$	2.36275e23	0.584261
$894$	0	0
$895$	4.87878e23i	1.18502i
$896$	0	0
$897$	−1.09373e23	−0.260957
$898$	0	0
$899$	1.56550e22i	0.0366924i
$900$	0	0
$901$	6.67209e23	1.53626
$902$	0	0
$903$	− 1.99701e23i	− 0.451730i
$904$	0	0
$905$	−4.42290e22	−0.0982922
$906$	0	0
$907$	1.13073e23i	0.246887i	0.992352	+	0.123444i	$0.0393938\pi$
$907$	1.13073e23i	0.246887i	−0.992352	+	0.123444i	$0.960606\pi$
$908$	0	0
$909$	−7.14271e23	−1.53233
$910$	0	0
$911$	− 6.25689e23i	− 1.31890i	−0.751749	−	0.659449i	$-0.770789\pi$
$911$	− 6.25689e23i	− 1.31890i	0.751749	−	0.659449i	$-0.229211\pi$
$912$	0	0
$913$	−3.63538e23	−0.752981
$914$	0	0
$915$	3.09582e23i	0.630096i
$916$	0	0
$917$	−1.18298e23	−0.236604
$918$	0	0
$919$	3.64131e23i	0.715705i	0.933778	+	0.357852i	$0.116491\pi$
$919$	3.64131e23i	0.715705i	−0.933778	+	0.357852i	$0.883509\pi$
$920$	0	0
$921$	1.42874e24	2.75979
$922$	0	0
$923$	1.00563e24i	1.90909i
$924$	0	0
$925$	1.44246e21	0.00269135
$926$	0	0
$927$	5.15759e23i	0.945821i
$928$	0	0
$929$	−7.59572e23	−1.36913	−0.684563	−	0.728954i	$-0.740007\pi$
$929$	−7.59572e23	−1.36913	−0.684563	+	0.728954i	$0.740007\pi$
$930$	0	0
$931$	− 5.67374e23i	− 1.00525i
$932$	0	0
$933$	6.74889e23	1.17538
$934$	0	0
$935$	− 3.68114e23i	− 0.630217i
$936$	0	0
$937$	6.78799e23	1.14242	0.571208	−	0.820805i	$-0.306475\pi$
$937$	6.78799e23	1.14242	0.571208	+	0.820805i	$0.306475\pi$
$938$	0	0
$939$	3.19548e23i	0.528703i
$940$	0	0
$941$	−6.36693e23	−1.03565	−0.517825	−	0.855487i	$-0.673258\pi$
$941$	−6.36693e23	−1.03565	−0.517825	+	0.855487i	$0.673258\pi$
$942$	0	0
$943$	4.76246e22i	0.0761619i
$944$	0	0
$945$	−1.81502e23	−0.285383
$946$	0	0
$947$	− 6.94373e23i	− 1.07348i	−0.843749	−	0.536738i	$-0.819656\pi$
$947$	− 6.94373e23i	− 1.07348i	0.843749	−	0.536738i	$-0.180344\pi$
$948$	0	0
$949$	5.96062e23	0.906070
$950$	0	0
$951$	− 2.22715e24i	− 3.32893i
$952$	0	0
$953$	−7.36518e23	−1.08253	−0.541265	−	0.840852i	$-0.682054\pi$
$953$	−7.36518e23	−1.08253	−0.541265	+	0.840852i	$0.682054\pi$
$954$	0	0
$955$	− 6.11425e23i	− 0.883723i
$956$	0	0
$957$	3.44435e23	0.489567
$958$	0	0
$959$	1.72257e23i	0.240784i
$960$	0	0
$961$	7.23260e23	0.994277
$962$	0	0
$963$	− 1.63368e24i	− 2.20880i
$964$	0	0
$965$	1.87741e22	0.0249655
$966$	0	0
$967$	7.91831e23i	1.03567i	0.855481	+	0.517834i	$0.173261\pi$
$967$	7.91831e23i	1.03567i	−0.855481	+	0.517834i	$0.826739\pi$
$968$	0	0
$969$	1.47086e24	1.89227
$970$	0	0
$971$	2.22361e23i	0.281387i	0.990053	+	0.140694i	$0.0449333\pi$
$971$	2.22361e23i	0.281387i	−0.990053	+	0.140694i	$0.955067\pi$
$972$	0	0
$973$	2.69474e23	0.335439
$974$	0	0
$975$	− 7.48335e21i	− 0.00916345i
$976$	0	0
$977$	−7.83431e23	−0.943722	−0.471861	−	0.881673i	$-0.656418\pi$
$977$	−7.83431e23	−0.943722	−0.471861	+	0.881673i	$0.656418\pi$
$978$	0	0
$979$	− 3.76964e23i	− 0.446723i
$980$	0	0
$981$	1.52924e24	1.78289
$982$	0	0
$983$	8.24241e23i	0.945423i	0.881217	+	0.472712i	$0.156725\pi$
$983$	8.24241e23i	0.945423i	−0.881217	+	0.472712i	$0.843275\pi$
$984$	0	0
$985$	3.06972e23	0.346425
$986$	0	0
$987$	− 1.59598e23i	− 0.177211i
$988$	0	0
$989$	−1.45878e23	−0.159375
$990$	0	0
$991$	− 7.40279e23i	− 0.795804i	−0.917428	−	0.397902i	$-0.869738\pi$
$991$	− 7.40279e23i	− 0.795804i	0.917428	−	0.397902i	$-0.130262\pi$
$992$	0	0
$993$	1.75302e24	1.85435
$994$	0	0
$995$	1.27605e24i	1.32826i
$996$	0	0
$997$	5.94670e23	0.609137	0.304569	−	0.952490i	$-0.401488\pi$
$997$	5.94670e23	0.609137	0.304569	+	0.952490i	$0.401488\pi$
$998$	0	0
$999$	− 1.05374e24i	− 1.06220i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.17.c.d.63.1		6
4.3	odd	2	inner	64.17.c.d.63.6		6
8.3	odd	2		4.17.b.b.3.3	✓	6
8.5	even	2		4.17.b.b.3.4	yes	6
24.5	odd	2		36.17.d.b.19.3		6
24.11	even	2		36.17.d.b.19.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.17.b.b.3.3	✓	6	8.3	odd	2
4.17.b.b.3.4	yes	6	8.5	even	2
36.17.d.b.19.3		6	24.5	odd	2
36.17.d.b.19.4		6	24.11	even	2
64.17.c.d.63.1		6	1.1	even	1	trivial
64.17.c.d.63.6		6	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-11183.9i q^{3} -389860. q^{5} -1.06777e6i q^{7} -8.20322e7 q^{9} +O(q^{10})\)	\(q-11183.9i q^{3} -389860. q^{5} -1.06777e6i q^{7} -8.20322e7 q^{9} -1.26927e8i q^{11} -1.12108e9 q^{13} +4.36015e9i q^{15} -7.43906e9 q^{17} -1.76792e10i q^{19} -1.19418e10 q^{21} +8.72327e9i q^{23} -5.96851e8 q^{25} +4.36008e11i q^{27} -2.42638e11 q^{29} -6.45199e10i q^{31} -1.41954e12 q^{33} +4.16282e11i q^{35} -2.41678e12 q^{37} +1.25380e13i q^{39} +5.45949e12 q^{41} +1.67228e13i q^{43} +3.19811e13 q^{45} +1.33646e13i q^{47} +3.20928e13 q^{49} +8.31975e13i q^{51} -8.96899e13 q^{53} +4.94840e13i q^{55} -1.97722e14 q^{57} -9.54083e13i q^{59} +7.10026e13 q^{61} +8.75916e13i q^{63} +4.37066e14 q^{65} -3.55095e14i q^{67} +9.75599e13 q^{69} -8.97021e14i q^{71} -5.31684e14 q^{73} +6.67511e12i q^{75} -1.35530e14 q^{77} -2.18899e15i q^{79} +1.34504e15 q^{81} -2.86414e15i q^{83} +2.90019e15 q^{85} +2.71364e15i q^{87} +2.96991e15 q^{89} +1.19706e15i q^{91} -7.21583e14 q^{93} +6.89241e15i q^{95} -2.04301e15 q^{97} +1.04121e16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 506740 q^{5} - 137574522 q^{9}+O(q^{10}) \) 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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