LMFDB - Embedded newform 64.17.c.d.63.4

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			63.4
Root			$-46.2446 + 35.5107i$ of defining polynomial
Character	$\chi$	$=$	64.63
Dual form			64.17.c.d.63.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2908.07i q^{3} -20884.7 q^{5} +8.01505e6i q^{7} +3.45899e7 q^{9} +O(q^{10})$	$q+2908.07i q^{3} -20884.7 q^{5} +8.01505e6i q^{7} +3.45899e7 q^{9} -1.22693e8i q^{11} -4.11898e8 q^{13} -6.07341e7i q^{15} +1.07999e10 q^{17} -3.07436e10i q^{19} -2.33083e10 q^{21} -6.36764e10i q^{23} -1.52152e11 q^{25} +2.25772e11i q^{27} +3.76057e11 q^{29} +3.36655e11i q^{31} +3.56798e11 q^{33} -1.67392e11i q^{35} -1.97558e12 q^{37} -1.19783e12i q^{39} +4.08883e12 q^{41} -1.53830e13i q^{43} -7.22399e11 q^{45} -2.29977e13i q^{47} -3.10082e13 q^{49} +3.14069e13i q^{51} +5.35569e13 q^{53} +2.56240e12i q^{55} +8.94043e13 q^{57} -6.70947e13i q^{59} -1.86362e14 q^{61} +2.77240e14i q^{63} +8.60236e12 q^{65} +1.58513e14i q^{67} +1.85175e14 q^{69} -8.90997e14i q^{71} -2.83994e14 q^{73} -4.42467e14i q^{75} +9.83388e14 q^{77} +1.41787e15i q^{79} +8.32419e14 q^{81} +2.82485e15i q^{83} -2.25553e14 q^{85} +1.09360e15i q^{87} +2.93963e15 q^{89} -3.30139e15i q^{91} -9.79016e14 q^{93} +6.42070e14i q^{95} +1.34225e15 q^{97} -4.24392e15i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2908.07i	0.443235i	0.975134	+	0.221618i	$0.0711337\pi$
$3$	2908.07i	0.443235i	−0.975134	+	0.221618i	$0.928866\pi$
$4$	0	0
$5$	−20884.7	−0.0534648	−0.0267324	−	0.999643i	$-0.508510\pi$
$5$	−20884.7	−0.0534648	−0.0267324	+	0.999643i	$0.508510\pi$
$6$	0	0
$7$	8.01505e6i	1.39034i	0.718844	+	0.695172i	$0.244672\pi$
$7$	8.01505e6i	1.39034i	−0.718844	+	0.695172i	$0.755328\pi$
$8$	0	0
$9$	3.45899e7	0.803542
$10$	0	0
$11$	− 1.22693e8i	− 0.572370i	−0.958174	−	0.286185i	$-0.907613\pi$
$11$	− 1.22693e8i	− 0.572370i	0.958174	−	0.286185i	$-0.0923872\pi$
$12$	0	0
$13$	−4.11898e8	−0.504944	−0.252472	−	0.967604i	$-0.581244\pi$
$13$	−4.11898e8	−0.504944	−0.252472	+	0.967604i	$0.581244\pi$
$14$	0	0
$15$	− 6.07341e7i	− 0.0236975i
$16$	0	0
$17$	1.07999e10	1.54821	0.774103	−	0.633060i	$-0.218201\pi$
$17$	1.07999e10	1.54821	0.774103	+	0.633060i	$0.218201\pi$
$18$	0	0
$19$	− 3.07436e10i	− 1.81019i	−0.425205	−	0.905097i	$-0.639798\pi$
$19$	− 3.07436e10i	− 1.81019i	0.425205	−	0.905097i	$-0.360202\pi$
$20$	0	0
$21$	−2.33083e10	−0.616249
$22$	0	0
$23$	− 6.36764e10i	− 0.813122i	−0.913624	−	0.406561i	$-0.866728\pi$
$23$	− 6.36764e10i	− 0.813122i	0.913624	−	0.406561i	$-0.133272\pi$
$24$	0	0
$25$	−1.52152e11	−0.997142
$26$	0	0
$27$	2.25772e11i	0.799394i
$28$	0	0
$29$	3.76057e11	0.751744	0.375872	−	0.926672i	$-0.377343\pi$
$29$	3.76057e11	0.751744	0.375872	+	0.926672i	$0.377343\pi$
$30$	0	0
$31$	3.36655e11i	0.394722i	0.980331	+	0.197361i	$0.0632372\pi$
$31$	3.36655e11i	0.394722i	−0.980331	+	0.197361i	$0.936763\pi$
$32$	0	0
$33$	3.56798e11	0.253695
$34$	0	0
$35$	− 1.67392e11i	− 0.0743345i
$36$	0	0
$37$	−1.97558e12	−0.562446	−0.281223	−	0.959642i	$-0.590740\pi$
$37$	−1.97558e12	−0.562446	−0.281223	+	0.959642i	$0.590740\pi$
$38$	0	0
$39$	− 1.19783e12i	− 0.223809i
$40$	0	0
$41$	4.08883e12	0.512069	0.256035	−	0.966668i	$-0.417584\pi$
$41$	4.08883e12	0.512069	0.256035	+	0.966668i	$0.417584\pi$
$42$	0	0
$43$	− 1.53830e13i	− 1.31611i	−0.752968	−	0.658057i	$-0.771379\pi$
$43$	− 1.53830e13i	− 1.31611i	0.752968	−	0.658057i	$-0.228621\pi$
$44$	0	0
$45$	−7.22399e11	−0.0429612
$46$	0	0
$47$	− 2.29977e13i	− 0.965830i	−0.875667	−	0.482915i	$-0.839578\pi$
$47$	− 2.29977e13i	− 0.965830i	0.875667	−	0.482915i	$-0.160422\pi$
$48$	0	0
$49$	−3.10082e13	−0.933056
$50$	0	0
$51$	3.14069e13i	0.686220i
$52$	0	0
$53$	5.35569e13	0.860218	0.430109	−	0.902777i	$-0.358475\pi$
$53$	5.35569e13	0.860218	0.430109	+	0.902777i	$0.358475\pi$
$54$	0	0
$55$	2.56240e12i	0.0306017i
$56$	0	0
$57$	8.94043e13	0.802342
$58$	0	0
$59$	− 6.70947e13i	− 0.456954i	−0.973549	−	0.228477i	$-0.926625\pi$
$59$	− 6.70947e13i	− 0.456954i	0.973549	−	0.228477i	$-0.0733745\pi$
$60$	0	0
$61$	−1.86362e14	−0.972117	−0.486058	−	0.873926i	$-0.661566\pi$
$61$	−1.86362e14	−0.972117	−0.486058	+	0.873926i	$0.661566\pi$
$62$	0	0
$63$	2.77240e14i	1.11720i
$64$	0	0
$65$	8.60236e12	0.0269967
$66$	0	0
$67$	1.58513e14i	0.390362i	0.980767	+	0.195181i	$0.0625294\pi$
$67$	1.58513e14i	0.390362i	−0.980767	+	0.195181i	$0.937471\pi$
$68$	0	0
$69$	1.85175e14	0.360404
$70$	0	0
$71$	− 8.90997e14i	− 1.37978i	−0.723915	−	0.689890i	$-0.757659\pi$
$71$	− 8.90997e14i	− 1.37978i	0.723915	−	0.689890i	$-0.242341\pi$
$72$	0	0
$73$	−2.83994e14	−0.352148	−0.176074	−	0.984377i	$-0.556340\pi$
$73$	−2.83994e14	−0.352148	−0.176074	+	0.984377i	$0.556340\pi$
$74$	0	0
$75$	− 4.42467e14i	− 0.441968i
$76$	0	0
$77$	9.83388e14	0.795791
$78$	0	0
$79$	1.41787e15i	0.934590i	0.884102	+	0.467295i	$0.154771\pi$
$79$	1.41787e15i	0.934590i	−0.884102	+	0.467295i	$0.845229\pi$
$80$	0	0
$81$	8.32419e14	0.449223
$82$	0	0
$83$	2.82485e15i	1.25421i	0.778935	+	0.627105i	$0.215760\pi$
$83$	2.82485e15i	1.25421i	−0.778935	+	0.627105i	$0.784240\pi$
$84$	0	0
$85$	−2.25553e14	−0.0827745
$86$	0	0
$87$	1.09360e15i	0.333199i
$88$	0	0
$89$	2.93963e15	0.746746	0.373373	−	0.927681i	$-0.378201\pi$
$89$	2.93963e15	0.746746	0.373373	+	0.927681i	$0.378201\pi$
$90$	0	0
$91$	− 3.30139e15i	− 0.702045i
$92$	0	0
$93$	−9.79016e14	−0.174955
$94$	0	0
$95$	6.42070e14i	0.0967817i
$96$	0	0
$97$	1.34225e15	0.171261	0.0856307	−	0.996327i	$-0.472709\pi$
$97$	1.34225e15	0.171261	0.0856307	+	0.996327i	$0.472709\pi$
$98$	0	0
$99$	− 4.24392e15i	− 0.459924i
$100$	0	0
$101$	9.59476e15	0.886060	0.443030	−	0.896507i	$-0.353904\pi$
$101$	9.59476e15	0.886060	0.443030	+	0.896507i	$0.353904\pi$
$102$	0	0
$103$	− 1.77633e16i	− 1.40225i	−0.713039	−	0.701124i	$-0.752682\pi$
$103$	− 1.77633e16i	− 1.40225i	0.713039	−	0.701124i	$-0.247318\pi$
$104$	0	0
$105$	4.86787e14	0.0329477
$106$	0	0
$107$	1.40865e15i	0.0819844i	0.999159	+	0.0409922i	$0.0130519\pi$
$107$	1.40865e15i	0.0819844i	−0.999159	+	0.0409922i	$0.986948\pi$
$108$	0	0
$109$	3.24531e16	1.62871	0.814355	−	0.580367i	$-0.197091\pi$
$109$	3.24531e16	1.62871	0.814355	+	0.580367i	$0.197091\pi$
$110$	0	0
$111$	− 5.74512e15i	− 0.249296i
$112$	0	0
$113$	3.76957e16	1.41796	0.708980	−	0.705229i	$-0.249156\pi$
$113$	3.76957e16	1.41796	0.708980	+	0.705229i	$0.249156\pi$
$114$	0	0
$115$	1.32986e15i	0.0434734i
$116$	0	0
$117$	−1.42475e16	−0.405744
$118$	0	0
$119$	8.65619e16i	2.15254i
$120$	0	0
$121$	3.08963e16	0.672393
$122$	0	0
$123$	1.18906e16i	0.226967i
$124$	0	0
$125$	6.36439e15	0.106777
$126$	0	0
$127$	− 8.61032e16i	− 1.27230i	−0.771565	−	0.636150i	$-0.780526\pi$
$127$	− 8.61032e16i	− 1.27230i	0.771565	−	0.636150i	$-0.219474\pi$
$128$	0	0
$129$	4.47348e16	0.583348
$130$	0	0
$131$	− 9.74192e15i	− 0.112324i	−0.998422	−	0.0561622i	$-0.982114\pi$
$131$	− 9.74192e15i	− 0.112324i	0.998422	−	0.0561622i	$-0.0178864\pi$
$132$	0	0
$133$	2.46411e17	2.51679
$134$	0	0
$135$	− 4.71519e15i	− 0.0427394i
$136$	0	0
$137$	1.41062e17	1.13670	0.568348	−	0.822788i	$-0.307583\pi$
$137$	1.41062e17	1.13670	0.568348	+	0.822788i	$0.307583\pi$
$138$	0	0
$139$	6.63698e16i	0.476269i	0.971232	+	0.238134i	$0.0765359\pi$
$139$	6.63698e16i	0.476269i	−0.971232	+	0.238134i	$0.923464\pi$
$140$	0	0
$141$	6.68788e16	0.428090
$142$	0	0
$143$	5.05368e16i	0.289015i
$144$	0	0
$145$	−7.85384e15	−0.0401918
$146$	0	0
$147$	− 9.01738e16i	− 0.413563i
$148$	0	0
$149$	−5.12770e15	−0.0211073	−0.0105537	−	0.999944i	$-0.503359\pi$
$149$	−5.12770e15	−0.0211073	−0.0105537	+	0.999944i	$0.503359\pi$
$150$	0	0
$151$	− 2.37125e16i	− 0.0877329i	−0.999037	−	0.0438664i	$-0.986032\pi$
$151$	− 2.37125e16i	− 0.0877329i	0.999037	−	0.0438664i	$-0.0139676\pi$
$152$	0	0
$153$	3.73567e17	1.24405
$154$	0	0
$155$	− 7.03094e15i	− 0.0211038i
$156$	0	0
$157$	3.07726e17	0.833619	0.416809	−	0.908994i	$-0.363148\pi$
$157$	3.07726e17	0.833619	0.416809	+	0.908994i	$0.363148\pi$
$158$	0	0
$159$	1.55747e17i	0.381279i
$160$	0	0
$161$	5.10370e17	1.13052
$162$	0	0
$163$	2.88371e17i	0.578696i	0.957224	+	0.289348i	$0.0934385\pi$
$163$	2.88371e17i	0.578696i	−0.957224	+	0.289348i	$0.906561\pi$
$164$	0	0
$165$	−7.45162e15	−0.0135637
$166$	0	0
$167$	1.16307e17i	0.192254i	0.995369	+	0.0961271i	$0.0306455\pi$
$167$	1.16307e17i	0.192254i	−0.995369	+	0.0961271i	$0.969354\pi$
$168$	0	0
$169$	−4.95757e17	−0.745032
$170$	0	0
$171$	− 1.06342e18i	− 1.45457i
$172$	0	0
$173$	7.68393e17	0.957667	0.478834	−	0.877906i	$-0.341060\pi$
$173$	7.68393e17	0.957667	0.478834	+	0.877906i	$0.341060\pi$
$174$	0	0
$175$	− 1.21950e18i	− 1.38637i
$176$	0	0
$177$	1.95116e17	0.202538
$178$	0	0
$179$	1.34720e18i	1.27823i	0.769112	+	0.639114i	$0.220699\pi$
$179$	1.34720e18i	1.27823i	−0.769112	+	0.639114i	$0.779301\pi$
$180$	0	0
$181$	5.40539e17	0.469244	0.234622	−	0.972087i	$-0.424615\pi$
$181$	5.40539e17	0.469244	0.234622	+	0.972087i	$0.424615\pi$
$182$	0	0
$183$	− 5.41953e17i	− 0.430877i
$184$	0	0
$185$	4.12594e16	0.0300711
$186$	0	0
$187$	− 1.32507e18i	− 0.886147i
$188$	0	0
$189$	−1.80958e18	−1.11143
$190$	0	0
$191$	2.63366e18i	1.48694i	0.668769	+	0.743470i	$0.266822\pi$
$191$	2.63366e18i	1.48694i	−0.668769	+	0.743470i	$0.733178\pi$
$192$	0	0
$193$	−3.18650e17	−0.165522	−0.0827608	−	0.996569i	$-0.526374\pi$
$193$	−3.18650e17	−0.165522	−0.0827608	+	0.996569i	$0.526374\pi$
$194$	0	0
$195$	2.50163e16i	0.0119659i
$196$	0	0
$197$	−2.60436e18	−1.14808	−0.574040	−	0.818827i	$-0.694625\pi$
$197$	−2.60436e18	−1.14808	−0.574040	+	0.818827i	$0.694625\pi$
$198$	0	0
$199$	1.69903e18i	0.690839i	0.938448	+	0.345420i	$0.112263\pi$
$199$	1.69903e18i	0.690839i	−0.938448	+	0.345420i	$0.887737\pi$
$200$	0	0
$201$	−4.60967e17	−0.173022
$202$	0	0
$203$	3.01412e18i	1.04518i
$204$	0	0
$205$	−8.53940e16	−0.0273777
$206$	0	0
$207$	− 2.20256e18i	− 0.653378i
$208$	0	0
$209$	−3.77201e18	−1.03610
$210$	0	0
$211$	8.88977e17i	0.226272i	0.993580	+	0.113136i	$0.0360896\pi$
$211$	8.88977e17i	0.226272i	−0.993580	+	0.113136i	$0.963910\pi$
$212$	0	0
$213$	2.59108e18	0.611567
$214$	0	0
$215$	3.21269e17i	0.0703658i
$216$	0	0
$217$	−2.69831e18	−0.548800
$218$	0	0
$219$	− 8.25872e17i	− 0.156085i
$220$	0	0
$221$	−4.44846e18	−0.781757
$222$	0	0
$223$	− 5.81069e18i	− 0.950142i	−0.879947	−	0.475071i	$-0.842422\pi$
$223$	− 5.81069e18i	− 0.950142i	0.879947	−	0.475071i	$-0.157578\pi$
$224$	0	0
$225$	−5.26291e18	−0.801246
$226$	0	0
$227$	− 9.78862e18i	− 1.38840i	−0.719782	−	0.694200i	$-0.755758\pi$
$227$	− 9.78862e18i	− 1.38840i	0.719782	−	0.694200i	$-0.244242\pi$
$228$	0	0
$229$	−1.27212e19	−1.68206	−0.841032	−	0.540985i	$-0.818052\pi$
$229$	−1.27212e19	−1.68206	−0.841032	+	0.540985i	$0.818052\pi$
$230$	0	0
$231$	2.85976e18i	0.352723i
$232$	0	0
$233$	−4.21991e18	−0.485798	−0.242899	−	0.970052i	$-0.578098\pi$
$233$	−4.21991e18	−0.485798	−0.242899	+	0.970052i	$0.578098\pi$
$234$	0	0
$235$	4.80299e17i	0.0516379i
$236$	0	0
$237$	−4.12327e18	−0.414243
$238$	0	0
$239$	− 8.47899e17i	− 0.0796454i	−0.999207	−	0.0398227i	$-0.987321\pi$
$239$	− 8.47899e17i	− 0.0796454i	0.999207	−	0.0398227i	$-0.0126793\pi$
$240$	0	0
$241$	4.06546e18	0.357251	0.178626	−	0.983917i	$-0.442835\pi$
$241$	4.06546e18	0.357251	0.178626	+	0.983917i	$0.442835\pi$
$242$	0	0
$243$	1.21395e19i	0.998505i
$244$	0	0
$245$	6.47596e17	0.0498856
$246$	0	0
$247$	1.26632e19i	0.914046i
$248$	0	0
$249$	−8.21484e18	−0.555910
$250$	0	0
$251$	8.96061e17i	0.0568785i	0.999596	+	0.0284392i	$0.00905371\pi$
$251$	8.96061e17i	0.0568785i	−0.999596	+	0.0284392i	$0.990946\pi$
$252$	0	0
$253$	−7.81262e18	−0.465407
$254$	0	0
$255$	− 6.55923e17i	− 0.0366886i
$256$	0	0
$257$	−1.01410e19	−0.532862	−0.266431	−	0.963854i	$-0.585844\pi$
$257$	−1.01410e19	−0.532862	−0.266431	+	0.963854i	$0.585844\pi$
$258$	0	0
$259$	− 1.58344e19i	− 0.781994i
$260$	0	0
$261$	1.30078e19	0.604058
$262$	0	0
$263$	− 3.75483e19i	− 1.64038i	−0.572092	−	0.820190i	$-0.693868\pi$
$263$	− 3.75483e19i	− 1.64038i	0.572092	−	0.820190i	$-0.306132\pi$
$264$	0	0
$265$	−1.11852e18	−0.0459914
$266$	0	0
$267$	8.54865e18i	0.330984i
$268$	0	0
$269$	2.11943e19	0.773039	0.386519	−	0.922281i	$-0.373677\pi$
$269$	2.11943e19	0.773039	0.386519	+	0.922281i	$0.373677\pi$
$270$	0	0
$271$	1.85665e19i	0.638227i	0.947716	+	0.319114i	$0.103385\pi$
$271$	1.85665e19i	0.638227i	−0.947716	+	0.319114i	$0.896615\pi$
$272$	0	0
$273$	9.60065e18	0.311171
$274$	0	0
$275$	1.86679e19i	0.570734i
$276$	0	0
$277$	−3.27126e19	−0.943793	−0.471896	−	0.881654i	$-0.656430\pi$
$277$	−3.27126e19	−0.943793	−0.471896	+	0.881654i	$0.656430\pi$
$278$	0	0
$279$	1.16449e19i	0.317176i
$280$	0	0
$281$	8.62462e18	0.221865	0.110933	−	0.993828i	$-0.464616\pi$
$281$	8.62462e18	0.221865	0.110933	+	0.993828i	$0.464616\pi$
$282$	0	0
$283$	− 3.29539e19i	− 0.800968i	−0.916304	−	0.400484i	$-0.868842\pi$
$283$	− 3.29539e19i	− 0.800968i	0.916304	−	0.400484i	$-0.131158\pi$
$284$	0	0
$285$	−1.86718e18	−0.0428971
$286$	0	0
$287$	3.27722e19i	0.711952i
$288$	0	0
$289$	6.79768e19	1.39694
$290$	0	0
$291$	3.90335e18i	0.0759091i
$292$	0	0
$293$	3.10886e19	0.572348	0.286174	−	0.958178i	$-0.407616\pi$
$293$	3.10886e19	0.572348	0.286174	+	0.958178i	$0.407616\pi$
$294$	0	0
$295$	1.40125e18i	0.0244309i
$296$	0	0
$297$	2.77006e19	0.457549
$298$	0	0
$299$	2.62282e19i	0.410581i
$300$	0	0
$301$	1.23296e20	1.82985
$302$	0	0
$303$	2.79022e19i	0.392733i
$304$	0	0
$305$	3.89211e18	0.0519741
$306$	0	0
$307$	8.30079e19i	1.05199i	0.850487	+	0.525996i	$0.176307\pi$
$307$	8.30079e19i	1.05199i	−0.850487	+	0.525996i	$0.823693\pi$
$308$	0	0
$309$	5.16568e19	0.621526
$310$	0	0
$311$	− 1.27201e20i	− 1.45347i	−0.686918	−	0.726735i	$-0.741037\pi$
$311$	− 1.27201e20i	− 1.45347i	0.686918	−	0.726735i	$-0.258963\pi$
$312$	0	0
$313$	−8.09185e17	−0.00878402	−0.00439201	−	0.999990i	$-0.501398\pi$
$313$	−8.09185e17	−0.00878402	−0.00439201	+	0.999990i	$0.501398\pi$
$314$	0	0
$315$	− 5.79007e18i	− 0.0597309i
$316$	0	0
$317$	−3.42169e19	−0.335557	−0.167779	−	0.985825i	$-0.553659\pi$
$317$	−3.42169e19	−0.335557	−0.167779	+	0.985825i	$0.553659\pi$
$318$	0	0
$319$	− 4.61394e19i	− 0.430276i
$320$	0	0
$321$	−4.09643e18	−0.0363384
$322$	0	0
$323$	− 3.32028e20i	− 2.80255i
$324$	0	0
$325$	6.26710e19	0.503500
$326$	0	0
$327$	9.43757e19i	0.721902i
$328$	0	0
$329$	1.84328e20	1.34284
$330$	0	0
$331$	1.28124e20i	0.889218i	0.895725	+	0.444609i	$0.146657\pi$
$331$	1.28124e20i	0.889218i	−0.895725	+	0.444609i	$0.853343\pi$
$332$	0	0
$333$	−6.83351e19	−0.451950
$334$	0	0
$335$	− 3.31050e18i	− 0.0208706i
$336$	0	0
$337$	9.12598e19	0.548580	0.274290	−	0.961647i	$-0.411557\pi$
$337$	9.12598e19	0.548580	0.274290	+	0.961647i	$0.411557\pi$
$338$	0	0
$339$	1.09622e20i	0.628490i
$340$	0	0
$341$	4.13051e19	0.225927
$342$	0	0
$343$	1.78316e19i	0.0930757i
$344$	0	0
$345$	−3.86733e18	−0.0192690
$346$	0	0
$347$	− 4.44288e18i	− 0.0211363i	−0.999944	−	0.0105681i	$-0.996636\pi$
$347$	− 4.44288e18i	− 0.0211363i	0.999944	−	0.0105681i	$-0.00336401\pi$
$348$	0	0
$349$	2.74791e20	1.24853	0.624265	−	0.781212i	$-0.285398\pi$
$349$	2.74791e20	1.24853	0.624265	+	0.781212i	$0.285398\pi$
$350$	0	0
$351$	− 9.29952e19i	− 0.403649i
$352$	0	0
$353$	1.04150e20	0.431979	0.215990	−	0.976396i	$-0.430702\pi$
$353$	1.04150e20	0.431979	0.215990	+	0.976396i	$0.430702\pi$
$354$	0	0
$355$	1.86082e19i	0.0737696i
$356$	0	0
$357$	−2.51728e20	−0.954081
$358$	0	0
$359$	3.31485e20i	1.20146i	0.799453	+	0.600728i	$0.205123\pi$
$359$	3.31485e20i	1.20146i	−0.799453	+	0.600728i	$0.794877\pi$
$360$	0	0
$361$	−6.56724e20	−2.27680
$362$	0	0
$363$	8.98484e19i	0.298028i
$364$	0	0
$365$	5.93112e18	0.0188275
$366$	0	0
$367$	− 1.64874e20i	− 0.500985i	−0.968119	−	0.250493i	$-0.919407\pi$
$367$	− 1.64874e20i	− 0.500985i	0.968119	−	0.250493i	$-0.0805925\pi$
$368$	0	0
$369$	1.41432e20	0.411469
$370$	0	0
$371$	4.29261e20i	1.19600i
$372$	0	0
$373$	−3.33516e20	−0.890114	−0.445057	−	0.895502i	$-0.646817\pi$
$373$	−3.33516e20	−0.890114	−0.445057	+	0.895502i	$0.646817\pi$
$374$	0	0
$375$	1.85081e19i	0.0473273i
$376$	0	0
$377$	−1.54897e20	−0.379588
$378$	0	0
$379$	− 3.65318e20i	− 0.858138i	−0.903272	−	0.429069i	$-0.858842\pi$
$379$	− 3.65318e20i	− 0.858138i	0.903272	−	0.429069i	$-0.141158\pi$
$380$	0	0
$381$	2.50394e20	0.563929
$382$	0	0
$383$	4.15911e19i	0.0898277i	0.998991	+	0.0449138i	$0.0143013\pi$
$383$	4.15911e19i	0.0898277i	−0.998991	+	0.0449138i	$0.985699\pi$
$384$	0	0
$385$	−2.05378e19	−0.0425468
$386$	0	0
$387$	− 5.32096e20i	− 1.05755i
$388$	0	0
$389$	−6.84573e20	−1.30564	−0.652820	−	0.757513i	$-0.726414\pi$
$389$	−6.84573e20	−1.30564	−0.652820	+	0.757513i	$0.726414\pi$
$390$	0	0
$391$	− 6.87699e20i	− 1.25888i
$392$	0	0
$393$	2.83301e19	0.0497861
$394$	0	0
$395$	− 2.96119e19i	− 0.0499677i
$396$	0	0
$397$	−8.61557e20	−1.39624	−0.698120	−	0.715981i	$-0.745980\pi$
$397$	−8.61557e20	−1.39624	−0.698120	+	0.715981i	$0.745980\pi$
$398$	0	0
$399$	7.16580e20i	1.11553i
$400$	0	0
$401$	−6.05379e20	−0.905467	−0.452733	−	0.891646i	$-0.649551\pi$
$401$	−6.05379e20	−0.905467	−0.452733	+	0.891646i	$0.649551\pi$
$402$	0	0
$403$	− 1.38668e20i	− 0.199313i
$404$	0	0
$405$	−1.73848e19	−0.0240176
$406$	0	0
$407$	2.42389e20i	0.321927i
$408$	0	0
$409$	−3.22735e20	−0.412154	−0.206077	−	0.978536i	$-0.566070\pi$
$409$	−3.22735e20	−0.412154	−0.206077	+	0.978536i	$0.566070\pi$
$410$	0	0
$411$	4.10217e20i	0.503824i
$412$	0	0
$413$	5.37768e20	0.635323
$414$	0	0
$415$	− 5.89960e19i	− 0.0670561i
$416$	0	0
$417$	−1.93008e20	−0.211099
$418$	0	0
$419$	1.72049e21i	1.81109i	0.424245	+	0.905547i	$0.360540\pi$
$419$	1.72049e21i	1.81109i	−0.424245	+	0.905547i	$0.639460\pi$
$420$	0	0
$421$	1.25573e20	0.127245	0.0636223	−	0.997974i	$-0.479735\pi$
$421$	1.25573e20	0.127245	0.0636223	+	0.997974i	$0.479735\pi$
$422$	0	0
$423$	− 7.95486e20i	− 0.776086i
$424$	0	0
$425$	−1.64322e21	−1.54378
$426$	0	0
$427$	− 1.49370e21i	− 1.35158i
$428$	0	0
$429$	−1.46965e20	−0.128101
$430$	0	0
$431$	− 1.00565e21i	− 0.844557i	−0.906466	−	0.422279i	$-0.861230\pi$
$431$	− 1.00565e21i	− 0.844557i	0.906466	−	0.422279i	$-0.138770\pi$
$432$	0	0
$433$	1.06994e21	0.865875	0.432937	−	0.901424i	$-0.357477\pi$
$433$	1.06994e21	0.865875	0.432937	+	0.901424i	$0.357477\pi$
$434$	0	0
$435$	− 2.28395e19i	− 0.0178144i
$436$	0	0
$437$	−1.95764e21	−1.47191
$438$	0	0
$439$	1.68877e21i	1.22421i	0.790777	+	0.612104i	$0.209677\pi$
$439$	1.68877e21i	1.22421i	−0.790777	+	0.612104i	$0.790323\pi$
$440$	0	0
$441$	−1.07257e21	−0.749750
$442$	0	0
$443$	− 1.63136e21i	− 1.09982i	−0.835225	−	0.549909i	$-0.814662\pi$
$443$	− 1.63136e21i	− 1.09982i	0.835225	−	0.549909i	$-0.185338\pi$
$444$	0	0
$445$	−6.13933e19	−0.0399246
$446$	0	0
$447$	− 1.49117e19i	− 0.00935551i
$448$	0	0
$449$	2.20546e21	1.33515	0.667574	−	0.744544i	$-0.267333\pi$
$449$	2.20546e21	1.33515	0.667574	+	0.744544i	$0.267333\pi$
$450$	0	0
$451$	− 5.01670e20i	− 0.293093i
$452$	0	0
$453$	6.89576e19	0.0388863
$454$	0	0
$455$	6.89484e19i	0.0375347i
$456$	0	0
$457$	1.78288e21	0.937113	0.468557	−	0.883433i	$-0.344774\pi$
$457$	1.78288e21	0.937113	0.468557	+	0.883433i	$0.344774\pi$
$458$	0	0
$459$	2.43832e21i	1.23763i
$460$	0	0
$461$	3.96968e21	1.94602	0.973012	−	0.230755i	$-0.0741194\pi$
$461$	3.96968e21	1.94602	0.973012	+	0.230755i	$0.0741194\pi$
$462$	0	0
$463$	− 1.07770e21i	− 0.510328i	−0.966898	−	0.255164i	$-0.917870\pi$
$463$	− 1.07770e21i	− 0.510328i	0.966898	−	0.255164i	$-0.0821295\pi$
$464$	0	0
$465$	2.04464e19	0.00935393
$466$	0	0
$467$	− 1.75887e21i	− 0.777499i	−0.921344	−	0.388749i	$-0.872907\pi$
$467$	− 1.75887e21i	− 0.777499i	0.921344	−	0.388749i	$-0.127093\pi$
$468$	0	0
$469$	−1.27049e21	−0.542737
$470$	0	0
$471$	8.94889e20i	0.369489i
$472$	0	0
$473$	−1.88738e21	−0.753304
$474$	0	0
$475$	4.67768e21i	1.80502i
$476$	0	0
$477$	1.85253e21	0.691221
$478$	0	0
$479$	1.24125e21i	0.447895i	0.974601	+	0.223947i	$0.0718944\pi$
$479$	1.24125e21i	0.447895i	−0.974601	+	0.223947i	$0.928106\pi$
$480$	0	0
$481$	8.13738e20	0.284004
$482$	0	0
$483$	1.48419e21i	0.501086i
$484$	0	0
$485$	−2.80325e19	−0.00915646
$486$	0	0
$487$	2.00656e21i	0.634191i	0.948394	+	0.317095i	$0.102708\pi$
$487$	2.00656e21i	0.634191i	−0.948394	+	0.317095i	$0.897292\pi$
$488$	0	0
$489$	−8.38602e20	−0.256499
$490$	0	0
$491$	− 2.15312e21i	− 0.637406i	−0.947855	−	0.318703i	$-0.896753\pi$
$491$	− 2.15312e21i	− 0.637406i	0.947855	−	0.318703i	$-0.103247\pi$
$492$	0	0
$493$	4.06138e21	1.16385
$494$	0	0
$495$	8.86330e19i	0.0245897i
$496$	0	0
$497$	7.14139e21	1.91837
$498$	0	0
$499$	− 3.88611e21i	− 1.01090i	−0.862855	−	0.505452i	$-0.831326\pi$
$499$	− 3.88611e21i	− 1.01090i	0.862855	−	0.505452i	$-0.168674\pi$
$500$	0	0
$501$	−3.38230e20	−0.0852139
$502$	0	0
$503$	− 6.23175e21i	− 1.52078i	−0.649468	−	0.760389i	$-0.725008\pi$
$503$	− 6.23175e21i	− 1.52078i	0.649468	−	0.760389i	$-0.274992\pi$
$504$	0	0
$505$	−2.00384e20	−0.0473730
$506$	0	0
$507$	− 1.44169e21i	− 0.330224i
$508$	0	0
$509$	3.00972e21	0.668012	0.334006	−	0.942571i	$-0.391599\pi$
$509$	3.00972e21	0.668012	0.334006	+	0.942571i	$0.391599\pi$
$510$	0	0
$511$	− 2.27622e21i	− 0.489607i
$512$	0	0
$513$	6.94105e21	1.44706
$514$	0	0
$515$	3.70980e20i	0.0749710i
$516$	0	0
$517$	−2.82164e21	−0.552812
$518$	0	0
$519$	2.23454e21i	0.424472i
$520$	0	0
$521$	−1.87599e21	−0.345563	−0.172782	−	0.984960i	$-0.555276\pi$
$521$	−1.87599e21	−0.345563	−0.172782	+	0.984960i	$0.555276\pi$
$522$	0	0
$523$	− 2.42581e21i	− 0.433354i	−0.976243	−	0.216677i	$-0.930478\pi$
$523$	− 2.42581e21i	− 0.433354i	0.976243	−	0.216677i	$-0.0695218\pi$
$524$	0	0
$525$	3.54640e21	0.614488
$526$	0	0
$527$	3.63585e21i	0.611112i
$528$	0	0
$529$	2.07793e21	0.338833
$530$	0	0
$531$	− 2.32080e21i	− 0.367182i
$532$	0	0
$533$	−1.68418e21	−0.258566
$534$	0	0
$535$	− 2.94191e19i	− 0.00438328i
$536$	0	0
$537$	−3.91775e21	−0.566556
$538$	0	0
$539$	3.80447e21i	0.534053i
$540$	0	0
$541$	8.19551e21	1.11686	0.558429	−	0.829552i	$-0.311404\pi$
$541$	8.19551e21	1.11686	0.558429	+	0.829552i	$0.311404\pi$
$542$	0	0
$543$	1.57192e21i	0.207985i
$544$	0	0
$545$	−6.77773e20	−0.0870787
$546$	0	0
$547$	1.21191e22i	1.51207i	0.654534	+	0.756033i	$0.272865\pi$
$547$	1.21191e22i	1.51207i	−0.654534	+	0.756033i	$0.727135\pi$
$548$	0	0
$549$	−6.44623e21	−0.781137
$550$	0	0
$551$	− 1.15613e22i	− 1.36080i
$552$	0	0
$553$	−1.13643e22	−1.29940
$554$	0	0
$555$	1.19985e20i	0.0133286i
$556$	0	0
$557$	4.91878e20	0.0530904	0.0265452	−	0.999648i	$-0.491549\pi$
$557$	4.91878e20	0.0530904	0.0265452	+	0.999648i	$0.491549\pi$
$558$	0	0
$559$	6.33623e21i	0.664563i
$560$	0	0
$561$	3.85339e21	0.392771
$562$	0	0
$563$	− 5.70450e21i	− 0.565133i	−0.959248	−	0.282567i	$-0.908814\pi$
$563$	− 5.70450e21i	− 0.565133i	0.959248	−	0.282567i	$-0.0911858\pi$
$564$	0	0
$565$	−7.87262e20	−0.0758110
$566$	0	0
$567$	6.67188e21i	0.624574i
$568$	0	0
$569$	−1.36181e21	−0.123942	−0.0619709	−	0.998078i	$-0.519739\pi$
$569$	−1.36181e21	−0.123942	−0.0619709	+	0.998078i	$0.519739\pi$
$570$	0	0
$571$	− 2.50144e21i	− 0.221361i	−0.993856	−	0.110680i	$-0.964697\pi$
$571$	− 2.50144e21i	− 0.221361i	0.993856	−	0.110680i	$-0.0353030\pi$
$572$	0	0
$573$	−7.65887e21	−0.659065
$574$	0	0
$575$	9.68847e21i	0.810798i
$576$	0	0
$577$	7.04236e21	0.573207	0.286603	−	0.958049i	$-0.407474\pi$
$577$	7.04236e21	0.573207	0.286603	+	0.958049i	$0.407474\pi$
$578$	0	0
$579$	− 9.26654e20i	− 0.0733651i
$580$	0	0
$581$	−2.26413e22	−1.74378
$582$	0	0
$583$	− 6.57103e21i	− 0.492363i
$584$	0	0
$585$	2.97555e20	0.0216930
$586$	0	0
$587$	− 6.86684e21i	− 0.487138i	−0.969884	−	0.243569i	$-0.921682\pi$
$587$	− 6.86684e21i	− 0.487138i	0.969884	−	0.243569i	$-0.0783182\pi$
$588$	0	0
$589$	1.03500e22	0.714524
$590$	0	0
$591$	− 7.57367e21i	− 0.508869i
$592$	0	0
$593$	−4.47095e21	−0.292390	−0.146195	−	0.989256i	$-0.546703\pi$
$593$	−4.47095e21	−0.292390	−0.146195	+	0.989256i	$0.546703\pi$
$594$	0	0
$595$	− 1.80782e21i	− 0.115085i
$596$	0	0
$597$	−4.94090e21	−0.306204
$598$	0	0
$599$	1.16626e22i	0.703686i	0.936059	+	0.351843i	$0.114445\pi$
$599$	1.16626e22i	0.703686i	−0.936059	+	0.351843i	$0.885555\pi$
$600$	0	0
$601$	−2.93277e22	−1.72299	−0.861495	−	0.507767i	$-0.830471\pi$
$601$	−2.93277e22	−1.72299	−0.861495	+	0.507767i	$0.830471\pi$
$602$	0	0
$603$	5.48295e21i	0.313672i
$604$	0	0
$605$	−6.45259e20	−0.0359493
$606$	0	0
$607$	− 3.02402e22i	− 1.64087i	−0.571738	−	0.820436i	$-0.693731\pi$
$607$	− 3.02402e22i	− 1.64087i	0.571738	−	0.820436i	$-0.306269\pi$
$608$	0	0
$609$	−8.76526e21	−0.463262
$610$	0	0
$611$	9.47269e21i	0.487690i
$612$	0	0
$613$	−2.49845e22	−1.25310	−0.626551	−	0.779381i	$-0.715534\pi$
$613$	−2.49845e22	−1.25310	−0.626551	+	0.779381i	$0.715534\pi$
$614$	0	0
$615$	− 2.48332e20i	− 0.0121348i
$616$	0	0
$617$	−3.75621e22	−1.78842	−0.894209	−	0.447650i	$-0.852261\pi$
$617$	−3.75621e22	−1.78842	−0.894209	+	0.447650i	$0.852261\pi$
$618$	0	0
$619$	1.74823e22i	0.811098i	0.914073	+	0.405549i	$0.132920\pi$
$619$	1.74823e22i	0.811098i	−0.914073	+	0.405549i	$0.867080\pi$
$620$	0	0
$621$	1.43764e22	0.650005
$622$	0	0
$623$	2.35613e22i	1.03823i
$624$	0	0
$625$	2.30836e22	0.991433
$626$	0	0
$627$	− 1.09692e22i	− 0.459237i
$628$	0	0
$629$	−2.13361e22	−0.870783
$630$	0	0
$631$	2.85630e22i	1.13650i	0.822857	+	0.568249i	$0.192379\pi$
$631$	2.85630e22i	1.13650i	−0.822857	+	0.568249i	$0.807621\pi$
$632$	0	0
$633$	−2.58521e21	−0.100292
$634$	0	0
$635$	1.79824e21i	0.0680233i
$636$	0	0
$637$	1.27722e22	0.471141
$638$	0	0
$639$	− 3.08195e22i	− 1.10871i
$640$	0	0
$641$	3.82378e22	1.34162	0.670809	−	0.741630i	$-0.265947\pi$
$641$	3.82378e22	1.34162	0.670809	+	0.741630i	$0.265947\pi$
$642$	0	0
$643$	− 4.63594e22i	− 1.58654i	−0.608872	−	0.793269i	$-0.708378\pi$
$643$	− 4.63594e22i	− 1.58654i	0.608872	−	0.793269i	$-0.291622\pi$
$644$	0	0
$645$	−9.34273e20	−0.0311886
$646$	0	0
$647$	− 1.48835e22i	− 0.484699i	−0.970189	−	0.242349i	$-0.922082\pi$
$647$	− 1.48835e22i	− 0.484699i	0.970189	−	0.242349i	$-0.0779180\pi$
$648$	0	0
$649$	−8.23203e21	−0.261547
$650$	0	0
$651$	− 7.84687e21i	− 0.243247i
$652$	0	0
$653$	2.67284e21	0.0808477	0.0404238	−	0.999183i	$-0.487129\pi$
$653$	2.67284e21	0.0808477	0.0404238	+	0.999183i	$0.487129\pi$
$654$	0	0
$655$	2.03457e20i	0.00600540i
$656$	0	0
$657$	−9.82330e21	−0.282966
$658$	0	0
$659$	5.31164e22i	1.49329i	0.665221	+	0.746647i	$0.268337\pi$
$659$	5.31164e22i	1.49329i	−0.665221	+	0.746647i	$0.731663\pi$
$660$	0	0
$661$	−5.33611e22	−1.46424	−0.732122	−	0.681173i	$-0.761470\pi$
$661$	−5.33611e22	−1.46424	−0.732122	+	0.681173i	$0.761470\pi$
$662$	0	0
$663$	− 1.29364e22i	− 0.346502i
$664$	0	0
$665$	−5.14622e21	−0.134560
$666$	0	0
$667$	− 2.39460e22i	− 0.611259i
$668$	0	0
$669$	1.68979e22	0.421137
$670$	0	0
$671$	2.28652e22i	0.556411i
$672$	0	0
$673$	8.77020e21	0.208396	0.104198	−	0.994557i	$-0.466772\pi$
$673$	8.77020e21	0.208396	0.104198	+	0.994557i	$0.466772\pi$
$674$	0	0
$675$	− 3.43517e22i	− 0.797109i
$676$	0	0
$677$	8.48321e22	1.92243	0.961216	−	0.275797i	$-0.0889417\pi$
$677$	8.48321e22	1.92243	0.961216	+	0.275797i	$0.0889417\pi$
$678$	0	0
$679$	1.07582e22i	0.238112i
$680$	0	0
$681$	2.84660e22	0.615388
$682$	0	0
$683$	6.16719e21i	0.130233i	0.997878	+	0.0651166i	$0.0207419\pi$
$683$	6.16719e21i	0.130233i	−0.997878	+	0.0651166i	$0.979258\pi$
$684$	0	0
$685$	−2.94603e21	−0.0607733
$686$	0	0
$687$	− 3.69940e22i	− 0.745551i
$688$	0	0
$689$	−2.20600e22	−0.434361
$690$	0	0
$691$	7.44424e22i	1.43218i	0.698010	+	0.716088i	$0.254069\pi$
$691$	7.44424e22i	1.43218i	−0.698010	+	0.716088i	$0.745931\pi$
$692$	0	0
$693$	3.40153e22	0.639452
$694$	0	0
$695$	− 1.38611e21i	− 0.0254636i
$696$	0	0
$697$	4.41590e22	0.792788
$698$	0	0
$699$	− 1.22718e22i	− 0.215323i
$700$	0	0
$701$	3.70247e22	0.634961	0.317481	−	0.948265i	$-0.397163\pi$
$701$	3.70247e22	0.634961	0.317481	+	0.948265i	$0.397163\pi$
$702$	0	0
$703$	6.07364e22i	1.01814i
$704$	0	0
$705$	−1.39674e21	−0.0228878
$706$	0	0
$707$	7.69025e22i	1.23193i
$708$	0	0
$709$	1.37266e22	0.214978	0.107489	−	0.994206i	$-0.465719\pi$
$709$	1.37266e22	0.214978	0.107489	+	0.994206i	$0.465719\pi$
$710$	0	0
$711$	4.90441e22i	0.750982i
$712$	0	0
$713$	2.14370e22	0.320957
$714$	0	0
$715$	− 1.05545e21i	− 0.0154521i
$716$	0	0
$717$	2.46575e21	0.0353017
$718$	0	0
$719$	− 1.84266e22i	− 0.257997i	−0.991645	−	0.128999i	$-0.958824\pi$
$719$	− 1.84266e22i	− 0.257997i	0.991645	−	0.128999i	$-0.0411763\pi$
$720$	0	0
$721$	1.42374e23	1.94961
$722$	0	0
$723$	1.18226e22i	0.158346i
$724$	0	0
$725$	−5.72177e22	−0.749595
$726$	0	0
$727$	1.11665e22i	0.143101i	0.997437	+	0.0715507i	$0.0227948\pi$
$727$	1.11665e22i	0.143101i	−0.997437	+	0.0715507i	$0.977205\pi$
$728$	0	0
$729$	5.30450e20	0.00665004
$730$	0	0
$731$	− 1.66135e23i	− 2.03762i
$732$	0	0
$733$	1.07239e23	1.28683	0.643414	−	0.765518i	$-0.277517\pi$
$733$	1.07239e23	1.28683	0.643414	+	0.765518i	$0.277517\pi$
$734$	0	0
$735$	1.88325e21i	0.0221111i
$736$	0	0
$737$	1.94484e22	0.223431
$738$	0	0
$739$	− 1.55835e23i	− 1.75190i	−0.482402	−	0.875950i	$-0.660236\pi$
$739$	− 1.55835e23i	− 1.75190i	0.482402	−	0.875950i	$-0.339764\pi$
$740$	0	0
$741$	−3.68255e22	−0.405138
$742$	0	0
$743$	− 1.74048e23i	− 1.87395i	−0.349396	−	0.936975i	$-0.613613\pi$
$743$	− 1.74048e23i	− 1.87395i	0.349396	−	0.936975i	$-0.386387\pi$
$744$	0	0
$745$	1.07091e20	0.00112850
$746$	0	0
$747$	9.77110e22i	1.00781i
$748$	0	0
$749$	−1.12904e22	−0.113987
$750$	0	0
$751$	1.69399e22i	0.167414i	0.996490	+	0.0837069i	$0.0266760\pi$
$751$	1.69399e22i	0.167414i	−0.996490	+	0.0837069i	$0.973324\pi$
$752$	0	0
$753$	−2.60581e21	−0.0252105
$754$	0	0
$755$	4.95229e20i	0.00469062i
$756$	0	0
$757$	1.65475e23	1.53449	0.767246	−	0.641352i	$-0.221626\pi$
$757$	1.65475e23	1.53449	0.767246	+	0.641352i	$0.221626\pi$
$758$	0	0
$759$	− 2.27196e22i	− 0.206285i
$760$	0	0
$761$	−6.44028e22	−0.572568	−0.286284	−	0.958145i	$-0.592420\pi$
$761$	−6.44028e22	−0.572568	−0.286284	+	0.958145i	$0.592420\pi$
$762$	0	0
$763$	2.60113e23i	2.26447i
$764$	0	0
$765$	−7.80184e21	−0.0665129
$766$	0	0
$767$	2.76362e22i	0.230736i
$768$	0	0
$769$	−1.30870e23	−1.07011	−0.535056	−	0.844817i	$-0.679709\pi$
$769$	−1.30870e23	−1.07011	−0.535056	+	0.844817i	$0.679709\pi$
$770$	0	0
$771$	− 2.94907e22i	− 0.236183i
$772$	0	0
$773$	−5.04534e22	−0.395781	−0.197890	−	0.980224i	$-0.563409\pi$
$773$	−5.04534e22	−0.395781	−0.197890	+	0.980224i	$0.563409\pi$
$774$	0	0
$775$	− 5.12227e22i	− 0.393594i
$776$	0	0
$777$	4.60475e22	0.346607
$778$	0	0
$779$	− 1.25705e23i	− 0.926945i
$780$	0	0
$781$	−1.09319e23	−0.789744
$782$	0	0
$783$	8.49033e22i	0.600939i
$784$	0	0
$785$	−6.42677e21	−0.0445693
$786$	0	0
$787$	− 4.64368e22i	− 0.315547i	−0.987475	−	0.157774i	$-0.949568\pi$
$787$	− 4.64368e22i	− 0.315547i	0.987475	−	0.157774i	$-0.0504316\pi$
$788$	0	0
$789$	1.09193e23	0.727074
$790$	0	0
$791$	3.02133e23i	1.97145i
$792$	0	0
$793$	7.67621e22	0.490864
$794$	0	0
$795$	− 3.25273e21i	− 0.0203850i
$796$	0	0
$797$	−2.20040e23	−1.35156	−0.675779	−	0.737104i	$-0.736193\pi$
$797$	−2.20040e23	−1.35156	−0.675779	+	0.737104i	$0.736193\pi$
$798$	0	0
$799$	− 2.48373e23i	− 1.49530i
$800$	0	0
$801$	1.01681e23	0.600042
$802$	0	0
$803$	3.48439e22i	0.201559i
$804$	0	0
$805$	−1.06589e22	−0.0604430
$806$	0	0
$807$	6.16345e22i	0.342638i
$808$	0	0
$809$	9.10543e22	0.496263	0.248132	−	0.968726i	$-0.420183\pi$
$809$	9.10543e22	0.496263	0.248132	+	0.968726i	$0.420183\pi$
$810$	0	0
$811$	1.32185e23i	0.706344i	0.935558	+	0.353172i	$0.114897\pi$
$811$	1.32185e23i	0.706344i	−0.935558	+	0.353172i	$0.885103\pi$
$812$	0	0
$813$	−5.39926e22	−0.282885
$814$	0	0
$815$	− 6.02254e21i	− 0.0309399i
$816$	0	0
$817$	−4.72928e23	−2.38242
$818$	0	0
$819$	− 1.14194e23i	− 0.564123i
$820$	0	0
$821$	−1.56343e23	−0.757416	−0.378708	−	0.925516i	$-0.623631\pi$
$821$	−1.56343e23	−0.757416	−0.378708	+	0.925516i	$0.623631\pi$
$822$	0	0
$823$	3.70305e22i	0.175939i	0.996123	+	0.0879693i	$0.0280377\pi$
$823$	3.70305e22i	0.175939i	−0.996123	+	0.0879693i	$0.971962\pi$
$824$	0	0
$825$	−5.42875e22	−0.252969
$826$	0	0
$827$	− 5.39578e22i	− 0.246610i	−0.992369	−	0.123305i	$-0.960651\pi$
$827$	− 5.39578e22i	− 0.246610i	0.992369	−	0.123305i	$-0.0393493\pi$
$828$	0	0
$829$	−1.78650e23	−0.800880	−0.400440	−	0.916323i	$-0.631143\pi$
$829$	−1.78650e23	−0.800880	−0.400440	+	0.916323i	$0.631143\pi$
$830$	0	0
$831$	− 9.51304e22i	− 0.418322i
$832$	0	0
$833$	−3.34885e23	−1.44456
$834$	0	0
$835$	− 2.42905e21i	− 0.0102788i
$836$	0	0
$837$	−7.60075e22	−0.315539
$838$	0	0
$839$	− 3.18011e23i	− 1.29523i	−0.761969	−	0.647613i	$-0.775767\pi$
$839$	− 3.18011e23i	− 1.29523i	0.761969	−	0.647613i	$-0.224233\pi$
$840$	0	0
$841$	−1.08827e23	−0.434881
$842$	0	0
$843$	2.50810e22i	0.0983385i
$844$	0	0
$845$	1.03537e22	0.0398330
$846$	0	0
$847$	2.47635e23i	0.934857i
$848$	0	0
$849$	9.58322e22	0.355018
$850$	0	0
$851$	1.25798e23i	0.457338i
$852$	0	0
$853$	2.55227e23	0.910615	0.455307	−	0.890334i	$-0.349529\pi$
$853$	2.55227e23	0.910615	0.455307	+	0.890334i	$0.349529\pi$
$854$	0	0
$855$	2.22091e22i	0.0777682i
$856$	0	0
$857$	1.01988e23	0.350510	0.175255	−	0.984523i	$-0.443925\pi$
$857$	1.01988e23	0.350510	0.175255	+	0.984523i	$0.443925\pi$
$858$	0	0
$859$	− 1.72983e21i	− 0.00583523i	−0.999996	−	0.00291761i	$-0.999071\pi$
$859$	− 1.72983e21i	− 0.00583523i	0.999996	−	0.00291761i	$-0.000928706\pi$
$860$	0	0
$861$	−9.53038e22	−0.315562
$862$	0	0
$863$	2.82882e23i	0.919432i	0.888066	+	0.459716i	$0.152049\pi$
$863$	2.82882e23i	0.919432i	−0.888066	+	0.459716i	$0.847951\pi$
$864$	0	0
$865$	−1.60477e22	−0.0512015
$866$	0	0
$867$	1.97681e23i	0.619174i
$868$	0	0
$869$	1.73963e23	0.534931
$870$	0	0
$871$	− 6.52913e22i	− 0.197111i
$872$	0	0
$873$	4.64283e22	0.137616
$874$	0	0
$875$	5.10110e22i	0.148456i
$876$	0	0
$877$	−5.04775e23	−1.44245	−0.721225	−	0.692700i	$-0.756421\pi$
$877$	−5.04775e23	−1.44245	−0.721225	+	0.692700i	$0.756421\pi$
$878$	0	0
$879$	9.04078e22i	0.253685i
$880$	0	0
$881$	−5.50563e23	−1.51705	−0.758525	−	0.651644i	$-0.774080\pi$
$881$	−5.50563e23	−1.51705	−0.758525	+	0.651644i	$0.774080\pi$
$882$	0	0
$883$	− 3.13488e23i	− 0.848273i	−0.905598	−	0.424136i	$-0.860578\pi$
$883$	− 3.13488e23i	− 0.848273i	0.905598	−	0.424136i	$-0.139422\pi$
$884$	0	0
$885$	−4.07494e21	−0.0108287
$886$	0	0
$887$	− 3.94823e23i	− 1.03042i	−0.857065	−	0.515209i	$-0.827714\pi$
$887$	− 3.94823e23i	− 1.03042i	0.857065	−	0.515209i	$-0.172286\pi$
$888$	0	0
$889$	6.90122e23	1.76893
$890$	0	0
$891$	− 1.02132e23i	− 0.257122i
$892$	0	0
$893$	−7.07030e23	−1.74834
$894$	0	0
$895$	− 2.81359e22i	− 0.0683402i
$896$	0	0
$897$	−7.62733e22	−0.181984
$898$	0	0
$899$	1.26602e23i	0.296730i
$900$	0	0
$901$	5.78409e23	1.33179
$902$	0	0
$903$	3.58552e23i	0.811054i
$904$	0	0
$905$	−1.12890e22	−0.0250880
$906$	0	0
$907$	− 8.29563e23i	− 1.81130i	−0.424022	−	0.905652i	$-0.639382\pi$
$907$	− 8.29563e23i	− 1.81130i	0.424022	−	0.905652i	$-0.360618\pi$
$908$	0	0
$909$	3.31881e23	0.711987
$910$	0	0
$911$	− 1.61506e23i	− 0.340441i	−0.985406	−	0.170220i	$-0.945552\pi$
$911$	− 1.61506e23i	− 0.340441i	0.985406	−	0.170220i	$-0.0544479\pi$
$912$	0	0
$913$	3.46588e23	0.717872
$914$	0	0
$915$	1.13185e22i	0.0230367i
$916$	0	0
$917$	7.80820e22	0.156169
$918$	0	0
$919$	− 1.03389e22i	− 0.0203212i	−0.999948	−	0.0101606i	$-0.996766\pi$
$919$	− 1.03389e22i	− 0.0203212i	0.999948	−	0.0101606i	$-0.00323428\pi$
$920$	0	0
$921$	−2.41393e23	−0.466280
$922$	0	0
$923$	3.67000e23i	0.696711i
$924$	0	0
$925$	3.00588e23	0.560839
$926$	0	0
$927$	− 6.14429e23i	− 1.12677i
$928$	0	0
$929$	−5.06026e23	−0.912110	−0.456055	−	0.889951i	$-0.650738\pi$
$929$	−5.06026e23	−0.912110	−0.456055	+	0.889951i	$0.650738\pi$
$930$	0	0
$931$	9.53301e23i	1.68901i
$932$	0	0
$933$	3.69908e23	0.644229
$934$	0	0
$935$	2.76737e22i	0.0473777i
$936$	0	0
$937$	7.05535e23	1.18741	0.593707	−	0.804681i	$-0.297664\pi$
$937$	7.05535e23	1.18741	0.593707	+	0.804681i	$0.297664\pi$
$938$	0	0
$939$	− 2.35317e21i	− 0.00389339i
$940$	0	0
$941$	5.35920e23	0.871733	0.435866	−	0.900011i	$-0.356442\pi$
$941$	5.35920e23	0.871733	0.435866	+	0.900011i	$0.356442\pi$
$942$	0	0
$943$	− 2.60362e23i	− 0.416375i
$944$	0	0
$945$	3.77925e22	0.0594225
$946$	0	0
$947$	− 7.06321e23i	− 1.09195i	−0.837802	−	0.545974i	$-0.816160\pi$
$947$	− 7.06321e23i	− 1.09195i	0.837802	−	0.545974i	$-0.183840\pi$
$948$	0	0
$949$	1.16976e23	0.177815
$950$	0	0
$951$	− 9.95050e22i	− 0.148731i
$952$	0	0
$953$	4.68997e23	0.689330	0.344665	−	0.938726i	$-0.387993\pi$
$953$	4.68997e23	0.689330	0.344665	+	0.938726i	$0.387993\pi$
$954$	0	0
$955$	− 5.50033e22i	− 0.0794990i
$956$	0	0
$957$	1.34177e23	0.190713
$958$	0	0
$959$	1.13062e24i	1.58040i
$960$	0	0
$961$	6.14086e23	0.844194
$962$	0	0
$963$	4.87248e22i	0.0658780i
$964$	0	0
$965$	6.65490e21	0.00884959
$966$	0	0
$967$	8.25976e23i	1.08033i	0.841560	+	0.540164i	$0.181638\pi$
$967$	8.25976e23i	1.08033i	−0.841560	+	0.540164i	$0.818362\pi$
$968$	0	0
$969$	9.65558e23	1.24219
$970$	0	0
$971$	7.53118e23i	0.953036i	0.879165	+	0.476518i	$0.158101\pi$
$971$	7.53118e23i	0.953036i	−0.879165	+	0.476518i	$0.841899\pi$
$972$	0	0
$973$	−5.31958e23	−0.662177
$974$	0	0
$975$	1.82251e23i	0.223169i
$976$	0	0
$977$	1.47220e23	0.177341	0.0886705	−	0.996061i	$-0.471738\pi$
$977$	1.47220e23	0.177341	0.0886705	+	0.996061i	$0.471738\pi$
$978$	0	0
$979$	− 3.60671e23i	− 0.427415i
$980$	0	0
$981$	1.12255e24	1.30874
$982$	0	0
$983$	8.61926e23i	0.988649i	0.869278	+	0.494324i	$0.164584\pi$
$983$	8.61926e23i	0.988649i	−0.869278	+	0.494324i	$0.835416\pi$
$984$	0	0
$985$	5.43914e22	0.0613819
$986$	0	0
$987$	5.36037e23i	0.595192i
$988$	0	0
$989$	−9.79534e23	−1.07016
$990$	0	0
$991$	− 1.15127e24i	− 1.23762i	−0.785540	−	0.618811i	$-0.787615\pi$
$991$	− 1.15127e24i	− 1.23762i	0.785540	−	0.618811i	$-0.212385\pi$
$992$	0	0
$993$	−3.72594e23	−0.394133
$994$	0	0
$995$	− 3.54838e22i	− 0.0369356i
$996$	0	0
$997$	3.91782e23	0.401313	0.200657	−	0.979662i	$-0.435692\pi$
$997$	3.91782e23	0.401313	0.200657	+	0.979662i	$0.435692\pi$
$998$	0	0
$999$	− 4.46032e23i	− 0.449616i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.17.c.d.63.4		6
4.3	odd	2	inner	64.17.c.d.63.3		6
8.3	odd	2		4.17.b.b.3.1	✓	6
8.5	even	2		4.17.b.b.3.2	yes	6
24.5	odd	2		36.17.d.b.19.5		6
24.11	even	2		36.17.d.b.19.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.17.b.b.3.1	✓	6	8.3	odd	2
4.17.b.b.3.2	yes	6	8.5	even	2
36.17.d.b.19.5		6	24.5	odd	2
36.17.d.b.19.6		6	24.11	even	2
64.17.c.d.63.3		6	4.3	odd	2	inner
64.17.c.d.63.4		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)

\(f(q)\)	\(=\)	\(q+2908.07i q^{3} -20884.7 q^{5} +8.01505e6i q^{7} +3.45899e7 q^{9} +O(q^{10})\)	\(q+2908.07i q^{3} -20884.7 q^{5} +8.01505e6i q^{7} +3.45899e7 q^{9} -1.22693e8i q^{11} -4.11898e8 q^{13} -6.07341e7i q^{15} +1.07999e10 q^{17} -3.07436e10i q^{19} -2.33083e10 q^{21} -6.36764e10i q^{23} -1.52152e11 q^{25} +2.25772e11i q^{27} +3.76057e11 q^{29} +3.36655e11i q^{31} +3.56798e11 q^{33} -1.67392e11i q^{35} -1.97558e12 q^{37} -1.19783e12i q^{39} +4.08883e12 q^{41} -1.53830e13i q^{43} -7.22399e11 q^{45} -2.29977e13i q^{47} -3.10082e13 q^{49} +3.14069e13i q^{51} +5.35569e13 q^{53} +2.56240e12i q^{55} +8.94043e13 q^{57} -6.70947e13i q^{59} -1.86362e14 q^{61} +2.77240e14i q^{63} +8.60236e12 q^{65} +1.58513e14i q^{67} +1.85175e14 q^{69} -8.90997e14i q^{71} -2.83994e14 q^{73} -4.42467e14i q^{75} +9.83388e14 q^{77} +1.41787e15i q^{79} +8.32419e14 q^{81} +2.82485e15i q^{83} -2.25553e14 q^{85} +1.09360e15i q^{87} +2.93963e15 q^{89} -3.30139e15i q^{91} -9.79016e14 q^{93} +6.42070e14i q^{95} +1.34225e15 q^{97} -4.24392e15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 506740 q^{5} - 137574522 q^{9}+O(q^{10}) \) 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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