LMFDB - Embedded newform 80.20.c.a.49.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(49,80)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("80.49");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.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:	$183.053357245$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 726881x^{6} + 160513523376x^{4} + 10607307647230976x^{2} + 32429098232548950016 $ x^8 + 726881x^6 + 160513523376x^4 + 10607307647230976*x^2 + 32429098232548950016
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{34}\cdot 3^{8}\cdot 5^{13} $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.6
Root			$56.6657i$ of defining polynomial
Character	$\chi$	$=$	80.49
Dual form			80.20.c.a.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+33157.6i q^{3} +(2.22206e6 + 3.75978e6i) q^{5} +2.70208e7i q^{7} +6.28322e7 q^{9} +O(q^{10})$	\(q+33157.6i q^{3} +(2.22206e6 + 3.75978e6i) q^{5} +2.70208e7i q^{7} +6.28322e7 q^{9} -5.56241e9 q^{11} -4.21072e10i q^{13} +(-1.24665e11 + 7.36781e10i) q^{15} -5.95087e11i q^{17} +1.45552e12 q^{19} -8.95945e11 q^{21} -9.69934e12i q^{23} +(-9.19842e12 + 1.67089e13i) q^{25} +4.06212e13i q^{27} +9.46880e13 q^{29} -6.27411e12 q^{31} -1.84437e14i q^{33} +(-1.01592e14 + 6.00417e13i) q^{35} +1.41692e15i q^{37} +1.39617e15 q^{39} -9.33298e14 q^{41} -2.15163e13i q^{43} +(1.39617e14 + 2.36235e14i) q^{45} +4.64662e15i q^{47} +1.06688e16 q^{49} +1.97317e16 q^{51} +3.42854e16i q^{53} +(-1.23600e16 - 2.09135e16i) q^{55} +4.82616e16i q^{57} +3.91436e16 q^{59} -1.44319e17 q^{61} +1.69777e15i q^{63} +(1.58314e17 - 9.35644e16i) q^{65} -3.79832e16i q^{67} +3.21607e17 q^{69} +7.32379e17 q^{71} +5.16836e17i q^{73} +(-5.54027e17 - 3.04998e17i) q^{75} -1.50301e17i q^{77} +5.46910e16 q^{79} -1.27388e18 q^{81} +5.15297e17i q^{83} +(2.23740e18 - 1.32232e18i) q^{85} +3.13963e18i q^{87} +3.08156e18 q^{89} +1.13777e18 q^{91} -2.08035e17i q^{93} +(3.23425e18 + 5.47244e18i) q^{95} +5.49443e18i q^{97} -3.49498e17 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 147000 q^{5} + 345358584 q^{9}+O(q^{10}) $ 8 * q + 147000 * q^5 + 345358584 * q^9	$ 8 q + 147000 q^{5} + 345358584 q^{9} + 3379575264 q^{11} + 242324628000 q^{15} - 4547188380640 q^{19} - 2983154334624 q^{21} + 17715709625000 q^{25} - 188222300345040 q^{29} - 72115006686976 q^{31} + 299115755916000 q^{35} - 30\!\cdots\!28 q^{39}+ \cdots + 16\!\cdots\!72 q^{99}+O(q^{100}) $ 8 * q + 147000 * q^5 + 345358584 * q^9 + 3379575264 * q^11 + 242324628000 * q^15 - 4547188380640 * q^19 - 2983154334624 * q^21 + 17715709625000 * q^25 - 188222300345040 * q^29 - 72115006686976 * q^31 + 299115755916000 * q^35 - 3011653788049728 * q^39 - 3314269428857904 * q^41 + 10198157211261000 * q^45 - 11129258011816744 * q^49 + 46867071411404544 * q^51 - 69821226326124000 * q^55 - 259648032778012320 * q^59 - 582893562753226064 * q^61 + 747545324689032000 * q^65 + 302144302523815008 * q^69 + 2617007600602926144 * q^71 - 2877932428209000000 * q^75 - 3935092091109861760 * q^79 - 8328177293660704152 * q^81 + 3752387009219584000 * q^85 + 2120141101636921680 * q^89 + 3358638219973817664 * q^91 - 6106065789884820000 * q^95 + 16256944625235082272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			33157.6i			0.972594i	0.873793	+	0.486297i	$0.161653\pi$
$3$			33157.6i			0.972594i	−0.873793	+	0.486297i	$0.838347\pi$
$4$		0			0
$5$	2.22206e6	+	3.75978e6i	0.508792	+	0.860890i
$5$	2.22206e6	+	3.75978e6i	0.508792	+	0.860890i
$6$		0			0
$7$			2.70208e7i			0.253085i	0.991961	+	0.126542i	$0.0403880\pi$
$7$			2.70208e7i			0.253085i	−0.991961	+	0.126542i	$0.959612\pi$
$8$		0			0
$9$	6.28322e7			0.0540603
$10$		0			0
$11$	−5.56241e9			−0.711267			−0.355634	−	0.934625i	$-0.615735\pi$
$11$	−5.56241e9			−0.711267			−0.355634	+	0.934625i	$0.615735\pi$
$12$		0			0
$13$		−	4.21072e10i		−	1.10127i	−0.834746	−	0.550635i	$-0.814385\pi$
$13$		−	4.21072e10i		−	1.10127i	0.834746	−	0.550635i	$-0.185615\pi$
$14$		0			0
$15$	−1.24665e11	+	7.36781e10i	−0.837296	+	0.494848i
$16$		0			0
$17$		−	5.95087e11i		−	1.21707i	−0.793527	−	0.608535i	$-0.791757\pi$
$17$		−	5.95087e11i		−	1.21707i	0.793527	−	0.608535i	$-0.208243\pi$
$18$		0			0
$19$	1.45552e12			1.03481			0.517403	−	0.855742i	$-0.326899\pi$
$19$	1.45552e12			1.03481			0.517403	+	0.855742i	$0.326899\pi$
$20$		0			0
$21$	−8.95945e11			−0.246149
$22$		0			0
$23$		−	9.69934e12i		−	1.12287i	−0.827522	−	0.561433i	$-0.810250\pi$
$23$		−	9.69934e12i		−	1.12287i	0.827522	−	0.561433i	$-0.189750\pi$
$24$		0			0
$25$	−9.19842e12	+	1.67089e13i	−0.482262	+	0.876027i
$26$		0			0
$27$			4.06212e13i			1.02517i
$28$		0			0
$29$	9.46880e13			1.21203			0.606016	−	0.795452i	$-0.292767\pi$
$29$	9.46880e13			1.21203			0.606016	+	0.795452i	$0.292767\pi$
$30$		0			0
$31$	−6.27411e12			−0.0426203			−0.0213101	−	0.999773i	$-0.506784\pi$
$31$	−6.27411e12			−0.0426203			−0.0213101	+	0.999773i	$0.506784\pi$
$32$		0			0
$33$		−	1.84437e14i		−	0.691774i
$34$		0			0
$35$	−1.01592e14	+	6.00417e13i	−0.217878	+	0.128767i
$36$		0			0
$37$			1.41692e15i			1.79237i	0.443679	+	0.896186i	$0.353673\pi$
$37$			1.41692e15i			1.79237i	−0.443679	+	0.896186i	$0.646327\pi$
$38$		0			0
$39$	1.39617e15			1.07109
$40$		0			0
$41$	−9.33298e14			−0.445219			−0.222610	−	0.974908i	$-0.571458\pi$
$41$	−9.33298e14			−0.445219			−0.222610	+	0.974908i	$0.571458\pi$
$42$		0			0
$43$		−	2.15163e13i		−	0.00652856i	−0.999995	−	0.00326428i	$-0.998961\pi$
$43$		−	2.15163e13i		−	0.00652856i	0.999995	−	0.00326428i	$-0.00103905\pi$
$44$		0			0
$45$	1.39617e14	+	2.36235e14i	0.0275054	+	0.0465399i
$46$		0			0
$47$			4.64662e15i			0.605630i	0.953049	+	0.302815i	$0.0979265\pi$
$47$			4.64662e15i			0.605630i	−0.953049	+	0.302815i	$0.902074\pi$
$48$		0			0
$49$	1.06688e16			0.935948
$50$		0			0
$51$	1.97317e16			1.18372
$52$		0			0
$53$			3.42854e16i			1.42721i	0.700547	+	0.713606i	$0.252939\pi$
$53$			3.42854e16i			1.42721i	−0.700547	+	0.713606i	$0.747061\pi$
$54$		0			0
$55$	−1.23600e16	−	2.09135e16i	−0.361887	−	0.612323i
$56$		0			0
$57$			4.82616e16i			1.00645i
$58$		0			0
$59$	3.91436e16			0.588256			0.294128	−	0.955766i	$-0.404971\pi$
$59$	3.91436e16			0.588256			0.294128	+	0.955766i	$0.404971\pi$
$60$		0			0
$61$	−1.44319e17			−1.58011			−0.790057	−	0.613033i	$-0.789949\pi$
$61$	−1.44319e17			−1.58011			−0.790057	+	0.613033i	$0.789949\pi$
$62$		0			0
$63$			1.69777e15i			0.0136818i
$64$		0			0
$65$	1.58314e17	−	9.35644e16i	0.948073	−	0.560317i
$66$		0			0
$67$		−	3.79832e16i		−	0.170562i	−0.996357	−	0.0852808i	$-0.972821\pi$
$67$		−	3.79832e16i		−	0.170562i	0.996357	−	0.0852808i	$-0.0271787\pi$
$68$		0			0
$69$	3.21607e17			1.09209
$70$		0			0
$71$	7.32379e17			1.89575			0.947877	−	0.318637i	$-0.103225\pi$
$71$	7.32379e17			1.89575			0.947877	+	0.318637i	$0.103225\pi$
$72$		0			0
$73$			5.16836e17i			1.02751i	0.857937	+	0.513754i	$0.171746\pi$
$73$			5.16836e17i			1.02751i	−0.857937	+	0.513754i	$0.828254\pi$
$74$		0			0
$75$	−5.54027e17	−	3.04998e17i	−0.852019	−	0.469046i
$76$		0			0
$77$		−	1.50301e17i		−	0.180011i
$78$		0			0
$79$	5.46910e16			0.0513403			0.0256702	−	0.999670i	$-0.491828\pi$
$79$	5.46910e16			0.0513403			0.0256702	+	0.999670i	$0.491828\pi$
$80$		0			0
$81$	−1.27388e18			−0.943017
$82$		0			0
$83$			5.15297e17i			0.302563i	0.988491	+	0.151281i	$0.0483399\pi$
$83$			5.15297e17i			0.302563i	−0.988491	+	0.151281i	$0.951660\pi$
$84$		0			0
$85$	2.23740e18	−	1.32232e18i	1.04776	−	0.619235i
$86$		0			0
$87$			3.13963e18i			1.17882i
$88$		0			0
$89$	3.08156e18			0.932321			0.466160	−	0.884700i	$-0.345637\pi$
$89$	3.08156e18			0.932321			0.466160	+	0.884700i	$0.345637\pi$
$90$		0			0
$91$	1.13777e18			0.278715
$92$		0			0
$93$		−	2.08035e17i		−	0.0414522i
$94$		0			0
$95$	3.23425e18	+	5.47244e18i	0.526501	+	0.890854i
$96$		0			0
$97$			5.49443e18i			0.733823i	0.930256	+	0.366912i	$0.119585\pi$
$97$			5.49443e18i			0.733823i	−0.930256	+	0.366912i	$0.880415\pi$
$98$		0			0
$99$	−3.49498e17			−0.0384513
$100$		0			0
$101$	8.81595e18			0.802077			0.401038	−	0.916061i	$-0.368649\pi$
$101$	8.81595e18			0.802077			0.401038	+	0.916061i	$0.368649\pi$
$102$		0			0
$103$		−	1.69018e19i		−	1.27638i	−0.769878	−	0.638191i	$-0.779683\pi$
$103$		−	1.69018e19i		−	1.27638i	0.769878	−	0.638191i	$-0.220317\pi$
$104$		0			0
$105$	−1.99084e18	−	3.36856e18i	−0.125239	−	0.211907i
$106$		0			0
$107$			2.55780e19i			1.34500i	0.740099	+	0.672498i	$0.234779\pi$
$107$			2.55780e19i			1.34500i	−0.740099	+	0.672498i	$0.765221\pi$
$108$		0			0
$109$	−2.26374e19			−0.998330			−0.499165	−	0.866507i	$-0.666360\pi$
$109$	−2.26374e19			−0.998330			−0.499165	+	0.866507i	$0.666360\pi$
$110$		0			0
$111$	−4.69816e19			−1.74325
$112$		0			0
$113$		−	2.99224e18i		−	0.0937025i	−0.998902	−	0.0468512i	$-0.985081\pi$
$113$		−	2.99224e18i		−	0.0937025i	0.998902	−	0.0468512i	$-0.0149187\pi$
$114$		0			0
$115$	3.64674e19	−	2.15525e19i	0.966664	−	0.571305i
$116$		0			0
$117$		−	2.64568e18i		−	0.0595350i
$118$		0			0
$119$	1.60797e19			0.308022
$120$		0			0
$121$	−3.02187e19			−0.494099
$122$		0			0
$123$		−	3.09460e19i		−	0.433018i
$124$		0			0
$125$	−8.32612e19	+	2.54401e18i	−0.999534	+	0.0305404i
$126$		0			0
$127$		−	1.73870e20i		−	1.79510i	−0.440914	−	0.897549i	$-0.645346\pi$
$127$		−	1.73870e20i		−	1.79510i	0.440914	−	0.897549i	$-0.354654\pi$
$128$		0			0
$129$	7.13430e17			0.00634964
$130$		0			0
$131$	−1.18459e20			−0.910943			−0.455472	−	0.890250i	$-0.650529\pi$
$131$	−1.18459e20			−0.910943			−0.455472	+	0.890250i	$0.650529\pi$
$132$		0			0
$133$			3.93293e19i			0.261894i
$134$		0			0
$135$	−1.52727e20	+	9.02626e19i	−0.882561	+	0.521599i
$136$		0			0
$137$		−	1.76822e20i		−	0.888570i	−0.895885	−	0.444285i	$-0.853458\pi$
$137$		−	1.76822e20i		−	0.888570i	0.895885	−	0.444285i	$-0.146542\pi$
$138$		0			0
$139$	3.90655e19			0.171062			0.0855309	−	0.996336i	$-0.472741\pi$
$139$	3.90655e19			0.171062			0.0855309	+	0.996336i	$0.472741\pi$
$140$		0			0
$141$	−1.54071e20			−0.589033
$142$		0			0
$143$			2.34217e20i			0.783298i
$144$		0			0
$145$	2.10402e20	+	3.56006e20i	0.616672	+	1.04343i
$146$		0			0
$147$			3.53751e20i			0.910298i
$148$		0			0
$149$	2.39602e20			0.542277			0.271139	−	0.962540i	$-0.412600\pi$
$149$	2.39602e20			0.542277			0.271139	+	0.962540i	$0.412600\pi$
$150$		0			0
$151$	9.81901e20			1.95788			0.978941	−	0.204142i	$-0.0654403\pi$
$151$	9.81901e20			1.95788			0.978941	+	0.204142i	$0.0654403\pi$
$152$		0			0
$153$		−	3.73906e19i		−	0.0657952i
$154$		0			0
$155$	−1.39414e19	−	2.35893e19i	−0.0216848	−	0.0366914i
$156$		0			0
$157$			5.26858e20i			0.725515i	0.931884	+	0.362758i	$0.118165\pi$
$157$			5.26858e20i			0.725515i	−0.931884	+	0.362758i	$0.881835\pi$
$158$		0			0
$159$	−1.13682e21			−1.38810
$160$		0			0
$161$	2.62084e20			0.284180
$162$		0			0
$163$			1.47793e21i			1.42519i	0.701576	+	0.712595i	$0.252480\pi$
$163$			1.47793e21i			1.42519i	−0.701576	+	0.712595i	$0.747520\pi$
$164$		0			0
$165$	6.93441e20	−	4.09828e20i	0.595541	−	0.351969i
$166$		0			0
$167$			4.73685e20i			0.362814i	0.983408	+	0.181407i	$0.0580651\pi$
$167$			4.73685e20i			0.362814i	−0.983408	+	0.181407i	$0.941935\pi$
$168$		0			0
$169$	−3.11092e20			−0.212797
$170$		0			0
$171$	9.14535e19			0.0559419
$172$		0			0
$173$			2.66872e20i			0.146172i	0.997326	+	0.0730862i	$0.0232848\pi$
$173$			2.66872e20i			0.146172i	−0.997326	+	0.0730862i	$0.976715\pi$
$174$		0			0
$175$	−4.51487e20	−	2.48548e20i	−0.221709	−	0.122053i
$176$		0			0
$177$			1.29791e21i			0.572134i
$178$		0			0
$179$	1.71409e21			0.679095			0.339547	−	0.940589i	$-0.389726\pi$
$179$	1.71409e21			0.679095			0.339547	+	0.940589i	$0.389726\pi$
$180$		0			0
$181$	−3.11223e20			−0.110949			−0.0554747	−	0.998460i	$-0.517667\pi$
$181$	−3.11223e20			−0.110949			−0.0554747	+	0.998460i	$0.517667\pi$
$182$		0			0
$183$		−	4.78526e21i		−	1.53681i
$184$		0			0
$185$	−5.32730e21	+	3.14847e21i	−1.54303	+	0.911943i
$186$		0			0
$187$			3.31012e21i			0.865662i
$188$		0			0
$189$	−1.09762e21			−0.259456
$190$		0			0
$191$	−6.54783e21			−1.40049			−0.700246	−	0.713902i	$-0.746926\pi$
$191$	−6.54783e21			−1.40049			−0.700246	+	0.713902i	$0.746926\pi$
$192$		0			0
$193$		−	7.76020e20i		−	0.150341i	−0.997171	−	0.0751706i	$-0.976050\pi$
$193$		−	7.76020e20i		−	0.150341i	0.997171	−	0.0751706i	$-0.0239501\pi$
$194$		0			0
$195$	3.10238e21	+	5.24931e21i	0.544961	+	0.922090i
$196$		0			0
$197$			5.13756e21i			0.819082i	0.912292	+	0.409541i	$0.134311\pi$
$197$			5.13756e21i			0.819082i	−0.912292	+	0.409541i	$0.865689\pi$
$198$		0			0
$199$	−4.15884e21			−0.602376			−0.301188	−	0.953565i	$-0.597383\pi$
$199$	−4.15884e21			−0.602376			−0.301188	+	0.953565i	$0.597383\pi$
$200$		0			0
$201$	1.25943e21			0.165887
$202$		0			0
$203$			2.55854e21i			0.306747i
$204$		0			0
$205$	−2.07384e21	−	3.50900e21i	−0.226524	−	0.383285i
$206$		0			0
$207$		−	6.09431e20i		−	0.0607024i
$208$		0			0
$209$	−8.09620e21			−0.736024
$210$		0			0
$211$	−5.95290e21			−0.494362			−0.247181	−	0.968969i	$-0.579504\pi$
$211$	−5.95290e21			−0.494362			−0.247181	+	0.968969i	$0.579504\pi$
$212$		0			0
$213$			2.42840e22i			1.84380i
$214$		0			0
$215$	8.08966e19	−	4.78104e19i	0.00562037	−	0.00332167i
$216$		0			0
$217$		−	1.69531e20i		−	0.0107865i
$218$		0			0
$219$	−1.71371e22			−0.999349
$220$		0			0
$221$	−2.50574e22			−1.34032
$222$		0			0
$223$			3.76323e22i			1.84784i	0.382584	+	0.923921i	$0.375034\pi$
$223$			3.76323e22i			1.84784i	−0.382584	+	0.923921i	$0.624966\pi$
$224$		0			0
$225$	−5.77957e20	+	1.04986e21i	−0.0260712	+	0.0473582i
$226$		0			0
$227$			3.15704e22i			1.30929i	0.755938	+	0.654644i	$0.227181\pi$
$227$			3.15704e22i			1.30929i	−0.755938	+	0.654644i	$0.772819\pi$
$228$		0			0
$229$	2.99352e22			1.14221			0.571104	−	0.820878i	$-0.306515\pi$
$229$	2.99352e22			1.14221			0.571104	+	0.820878i	$0.306515\pi$
$230$		0			0
$231$	4.98362e21			0.175078
$232$		0			0
$233$		−	9.59429e20i		−	0.0310550i	−0.999879	−	0.0155275i	$-0.995057\pi$
$233$		−	9.59429e20i		−	0.0310550i	0.999879	−	0.0155275i	$-0.00494275\pi$
$234$		0			0
$235$	−1.74703e22	+	1.03250e22i	−0.521381	+	0.308140i
$236$		0			0
$237$			1.81342e21i			0.0499333i
$238$		0			0
$239$	−4.17690e22			−1.06188			−0.530938	−	0.847411i	$-0.678160\pi$
$239$	−4.17690e22			−1.06188			−0.530938	+	0.847411i	$0.678160\pi$
$240$		0			0
$241$	−4.86489e22			−1.14264			−0.571322	−	0.820726i	$-0.693569\pi$
$241$	−4.86489e22			−1.14264			−0.571322	+	0.820726i	$0.693569\pi$
$242$		0			0
$243$			4.97374e21i			0.108000i
$244$		0			0
$245$	2.37066e22	+	4.01123e22i	0.476202	+	0.805748i
$246$		0			0
$247$		−	6.12878e22i		−	1.13960i
$248$		0			0
$249$	−1.70860e22			−0.294271
$250$		0			0
$251$	4.95879e22			0.791545			0.395772	−	0.918349i	$-0.370477\pi$
$251$	4.95879e22			0.791545			0.395772	+	0.918349i	$0.370477\pi$
$252$		0			0
$253$			5.39517e22i			0.798657i
$254$		0			0
$255$	4.38449e22	+	7.41868e22i	0.602265	+	1.01905i
$256$		0			0
$257$			6.03408e22i			0.769568i	0.923007	+	0.384784i	$0.125724\pi$
$257$			6.03408e22i			0.769568i	−0.923007	+	0.384784i	$0.874276\pi$
$258$		0			0
$259$	−3.82862e22			−0.453622
$260$		0			0
$261$	5.94945e21			0.0655228
$262$		0			0
$263$			1.28674e23i			1.31798i	0.752151	+	0.658991i	$0.229017\pi$
$263$			1.28674e23i			1.31798i	−0.752151	+	0.658991i	$0.770983\pi$
$264$		0			0
$265$	−1.28906e23	+	7.61841e22i	−1.22867	+	0.726153i
$266$		0			0
$267$			1.02177e23i			0.906770i
$268$		0			0
$269$	9.53759e22			0.788482			0.394241	−	0.919007i	$-0.371007\pi$
$269$	9.53759e22			0.788482			0.394241	+	0.919007i	$0.371007\pi$
$270$		0			0
$271$	−5.86695e22			−0.452068			−0.226034	−	0.974119i	$-0.572576\pi$
$271$	−5.86695e22			−0.452068			−0.226034	+	0.974119i	$0.572576\pi$
$272$		0			0
$273$			3.77257e22i			0.271077i
$274$		0			0
$275$	5.11654e22	−	9.29417e22i	0.343017	−	0.623089i
$276$		0			0
$277$			8.11673e22i			0.507953i	0.967210	+	0.253976i	$0.0817386\pi$
$277$			8.11673e22i			0.507953i	−0.967210	+	0.253976i	$0.918261\pi$
$278$		0			0
$279$	−3.94216e20			−0.00230406
$280$		0			0
$281$	9.85616e22			0.538267			0.269134	−	0.963103i	$-0.413263\pi$
$281$	9.85616e22			0.538267			0.269134	+	0.963103i	$0.413263\pi$
$282$		0			0
$283$		−	8.61893e22i		−	0.440030i	−0.975497	−	0.220015i	$-0.929389\pi$
$283$		−	8.61893e22i		−	0.440030i	0.975497	−	0.220015i	$-0.0706106\pi$
$284$		0			0
$285$	−1.81453e23	+	1.07240e23i	−0.866440	+	0.512072i
$286$		0			0
$287$		−	2.52184e22i		−	0.112678i
$288$		0			0
$289$	−1.15056e23			−0.481261
$290$		0			0
$291$	−1.82182e23			−0.713712
$292$		0			0
$293$			5.65789e22i			0.207688i	0.994594	+	0.103844i	$0.0331143\pi$
$293$			5.65789e22i			0.207688i	−0.994594	+	0.103844i	$0.966886\pi$
$294$		0			0
$295$	8.69792e22	+	1.47171e23i	0.299300	+	0.506424i
$296$		0			0
$297$		−	2.25952e23i		−	0.729172i
$298$		0			0
$299$	−4.08412e23			−1.23658
$300$		0			0
$301$	5.81387e20			0.00165228
$302$		0			0
$303$			2.92316e23i			0.780095i
$304$		0			0
$305$	−3.20684e23	−	5.42606e23i	−0.803949	−	1.36030i
$306$		0			0
$307$			8.75128e21i			0.0206185i	0.999947	+	0.0103092i	$0.00328159\pi$
$307$			8.75128e21i			0.0206185i	−0.999947	+	0.0103092i	$0.996718\pi$
$308$		0			0
$309$	5.60425e23			1.24140
$310$		0			0
$311$	3.00525e23			0.626119			0.313060	−	0.949733i	$-0.398646\pi$
$311$	3.00525e23			0.626119			0.313060	+	0.949733i	$0.398646\pi$
$312$		0			0
$313$			3.65350e22i			0.0716206i	0.999359	+	0.0358103i	$0.0114012\pi$
$313$			3.65350e22i			0.0716206i	−0.999359	+	0.0358103i	$0.988599\pi$
$314$		0			0
$315$	−6.38326e21	+	3.77255e21i	−0.0117786	+	0.00696120i
$316$		0			0
$317$		−	4.43734e23i		−	0.771009i	−0.922706	−	0.385504i	$-0.874027\pi$
$317$		−	4.43734e23i		−	0.771009i	0.922706	−	0.385504i	$-0.125973\pi$
$318$		0			0
$319$	−5.26694e23			−0.862079
$320$		0			0
$321$	−8.48107e23			−1.30814
$322$		0			0
$323$		−	8.66161e23i		−	1.25943i
$324$		0			0
$325$	7.03564e23	+	3.87319e23i	0.964743	+	0.531101i
$326$		0			0
$327$		−	7.50602e23i		−	0.970970i
$328$		0			0
$329$	−1.25555e23			−0.153276
$330$		0			0
$331$	5.86541e23			0.675978			0.337989	−	0.941150i	$-0.390253\pi$
$331$	5.86541e23			0.675978			0.337989	+	0.941150i	$0.390253\pi$
$332$		0			0
$333$			8.90279e22i			0.0968961i
$334$		0			0
$335$	1.42809e23	−	8.44009e22i	0.146835	−	0.0867803i
$336$		0			0
$337$		−	1.61676e24i		−	1.57094i	−0.618897	−	0.785472i	$-0.712420\pi$
$337$		−	1.61676e24i		−	1.57094i	0.618897	−	0.785472i	$-0.287580\pi$
$338$		0			0
$339$	9.92156e22			0.0911345
$340$		0			0
$341$	3.48992e22			0.0303144
$342$		0			0
$343$			5.96285e23i			0.489959i
$344$		0			0
$345$	7.14629e23	+	1.20917e24i	0.555648	+	0.940171i
$346$		0			0
$347$		−	1.75971e24i		−	1.29512i	−0.762014	−	0.647560i	$-0.775789\pi$
$347$		−	1.75971e24i		−	1.29512i	0.762014	−	0.647560i	$-0.224211\pi$
$348$		0			0
$349$	2.36703e24			1.64954			0.824768	−	0.565472i	$-0.191306\pi$
$349$	2.36703e24			1.64954			0.824768	+	0.565472i	$0.191306\pi$
$350$		0			0
$351$	1.71044e24			1.12899
$352$		0			0
$353$		−	1.43847e23i		−	0.0899585i	−0.998988	−	0.0449793i	$-0.985678\pi$
$353$		−	1.43847e23i		−	0.0899585i	0.998988	−	0.0449793i	$-0.0143222\pi$
$354$		0			0
$355$	1.62739e24	+	2.75359e24i	0.964543	+	1.63203i
$356$		0			0
$357$			5.33165e23i			0.299581i
$358$		0			0
$359$	1.16835e24			0.622550			0.311275	−	0.950320i	$-0.399244\pi$
$359$	1.16835e24			0.622550			0.311275	+	0.950320i	$0.399244\pi$
$360$		0			0
$361$	1.40119e23			0.0708238
$362$		0			0
$363$		−	1.00198e24i		−	0.480558i
$364$		0			0
$365$	−1.94319e24	+	1.14844e24i	−0.884571	+	0.522788i
$366$		0			0
$367$		−	2.80054e24i		−	1.21036i	−0.796089	−	0.605180i	$-0.793101\pi$
$367$		−	2.80054e24i		−	1.21036i	0.796089	−	0.605180i	$-0.206899\pi$
$368$		0			0
$369$	−5.86412e22			−0.0240687
$370$		0			0
$371$	−9.26418e23			−0.361206
$372$		0			0
$373$		−	1.42432e24i		−	0.527682i	−0.964566	−	0.263841i	$-0.915011\pi$
$373$		−	1.42432e24i		−	0.527682i	0.964566	−	0.263841i	$-0.0849894\pi$
$374$		0			0
$375$	−8.43535e22	−	2.76074e24i	−0.0297034	−	0.972141i
$376$		0			0
$377$		−	3.98704e24i		−	1.33478i
$378$		0			0
$379$	2.87049e24			0.913868			0.456934	−	0.889501i	$-0.348948\pi$
$379$	2.87049e24			0.913868			0.456934	+	0.889501i	$0.348948\pi$
$380$		0			0
$381$	5.76511e24			1.74590
$382$		0			0
$383$			5.69586e23i			0.164124i	0.996627	+	0.0820618i	$0.0261505\pi$
$383$			5.69586e23i			0.164124i	−0.996627	+	0.0820618i	$0.973850\pi$
$384$		0			0
$385$	5.65098e23	−	3.33976e23i	0.154970	−	0.0915881i
$386$		0			0
$387$		−	1.35192e21i		−	0.000352936i
$388$		0			0
$389$	4.66196e23			0.115890			0.0579452	−	0.998320i	$-0.481545\pi$
$389$	4.66196e23			0.115890			0.0579452	+	0.998320i	$0.481545\pi$
$390$		0			0
$391$	−5.77195e24			−1.36661
$392$		0			0
$393$		−	3.92783e24i		−	0.885978i
$394$		0			0
$395$	1.21526e23	+	2.05626e23i	0.0261215	+	0.0441983i
$396$		0			0
$397$		−	4.81447e24i		−	0.986367i	−0.869925	−	0.493184i	$-0.835833\pi$
$397$		−	4.81447e24i		−	0.986367i	0.869925	−	0.493184i	$-0.164167\pi$
$398$		0			0
$399$	−1.30407e24			−0.254717
$400$		0			0
$401$	−1.07502e24			−0.200237			−0.100118	−	0.994976i	$-0.531922\pi$
$401$	−1.07502e24			−0.200237			−0.100118	+	0.994976i	$0.531922\pi$
$402$		0			0
$403$			2.64185e23i			0.0469365i
$404$		0			0
$405$	−2.83062e24	−	4.78950e24i	−0.479799	−	0.811834i
$406$		0			0
$407$		−	7.88148e24i		−	1.27485i
$408$		0			0
$409$	−6.08999e24			−0.940254			−0.470127	−	0.882599i	$-0.655792\pi$
$409$	−6.08999e24			−0.940254			−0.470127	+	0.882599i	$0.655792\pi$
$410$		0			0
$411$	5.86301e24			0.864218
$412$		0			0
$413$			1.05769e24i			0.148879i
$414$		0			0
$415$	−1.93740e24	+	1.14502e24i	−0.260473	+	0.153941i
$416$		0			0
$417$			1.29532e24i			0.166374i
$418$		0			0
$419$	−6.17193e24			−0.757509			−0.378754	−	0.925497i	$-0.623648\pi$
$419$	−6.17193e24			−0.757509			−0.378754	+	0.925497i	$0.623648\pi$
$420$		0			0
$421$	−2.02089e24			−0.237062			−0.118531	−	0.992950i	$-0.537818\pi$
$421$	−2.02089e24			−0.237062			−0.118531	+	0.992950i	$0.537818\pi$
$422$		0			0
$423$			2.91957e23i			0.0327405i
$424$		0			0
$425$	9.94324e24	+	5.47386e24i	1.06619	+	0.586947i
$426$		0			0
$427$		−	3.89960e24i		−	0.399903i
$428$		0			0
$429$	−7.76610e24			−0.761831
$430$		0			0
$431$	5.28591e24			0.496119			0.248059	−	0.968745i	$-0.420207\pi$
$431$	5.28591e24			0.496119			0.248059	+	0.968745i	$0.420207\pi$
$432$		0			0
$433$			8.30646e24i			0.746073i	0.927817	+	0.373036i	$0.121683\pi$
$433$			8.30646e24i			0.746073i	−0.927817	+	0.373036i	$0.878317\pi$
$434$		0			0
$435$	−1.18043e25	+	6.97644e24i	−1.01483	+	0.599771i
$436$		0			0
$437$		−	1.41176e25i		−	1.16195i
$438$		0			0
$439$	8.06653e24			0.635732			0.317866	−	0.948136i	$-0.397034\pi$
$439$	8.06653e24			0.635732			0.317866	+	0.948136i	$0.397034\pi$
$440$		0			0
$441$	6.70342e23			0.0505976
$442$		0			0
$443$			2.08283e24i			0.150597i	0.997161	+	0.0752987i	$0.0239910\pi$
$443$			2.08283e24i			0.150597i	−0.997161	+	0.0752987i	$0.976009\pi$
$444$		0			0
$445$	6.84739e24	+	1.15860e25i	0.474357	+	0.802625i
$446$		0			0
$447$			7.94464e24i			0.527416i
$448$		0			0
$449$	6.94790e24			0.442093			0.221046	−	0.975263i	$-0.429053\pi$
$449$	6.94790e24			0.442093			0.221046	+	0.975263i	$0.429053\pi$
$450$		0			0
$451$	5.19139e24			0.316670
$452$		0			0
$453$			3.25575e25i			1.90423i
$454$		0			0
$455$	2.52818e24	+	4.27776e24i	0.141808	+	0.239943i
$456$		0			0
$457$			8.71540e24i			0.468904i	0.972128	+	0.234452i	$0.0753295\pi$
$457$			8.71540e24i			0.468904i	−0.972128	+	0.234452i	$0.924671\pi$
$458$		0			0
$459$	2.41732e25			1.24771
$460$		0			0
$461$	−2.94818e25			−1.46014			−0.730072	−	0.683371i	$-0.760513\pi$
$461$	−2.94818e25			−1.46014			−0.730072	+	0.683371i	$0.760513\pi$
$462$		0			0
$463$			1.35704e25i			0.645019i	0.946566	+	0.322509i	$0.104526\pi$
$463$			1.35704e25i			0.645019i	−0.946566	+	0.322509i	$0.895474\pi$
$464$		0			0
$465$	7.82165e23	−	4.62265e23i	0.0356858	−	0.0210905i
$466$		0			0
$467$		−	2.43556e25i		−	1.06681i	−0.845859	−	0.533407i	$-0.820911\pi$
$467$		−	2.43556e25i		−	1.06681i	0.845859	−	0.533407i	$-0.179089\pi$
$468$		0			0
$469$	1.02634e24			0.0431666
$470$		0			0
$471$	−1.74694e25			−0.705632
$472$		0			0
$473$			1.19683e23i			0.00464355i
$474$		0			0
$475$	−1.33885e25	+	2.43201e25i	−0.499048	+	0.906518i
$476$		0			0
$477$			2.15423e24i			0.0771554i
$478$		0			0
$479$	1.71994e23			0.00592005			0.00296003	−	0.999996i	$-0.499058\pi$
$479$	1.71994e23			0.00592005			0.00296003	+	0.999996i	$0.499058\pi$
$480$		0			0
$481$	5.96623e25			1.97389
$482$		0			0
$483$			8.69008e24i			0.276392i
$484$		0			0
$485$	−2.06579e25	+	1.22089e25i	−0.631741	+	0.373363i
$486$		0			0
$487$			5.44742e25i			1.60201i	0.598657	+	0.801006i	$0.295701\pi$
$487$			5.44742e25i			1.60201i	−0.598657	+	0.801006i	$0.704299\pi$
$488$		0			0
$489$	−4.90048e25			−1.38613
$490$		0			0
$491$	−6.23616e25			−1.69685			−0.848425	−	0.529316i	$-0.822449\pi$
$491$	−6.23616e25			−1.69685			−0.848425	+	0.529316i	$0.822449\pi$
$492$		0			0
$493$		−	5.63476e25i		−	1.47513i
$494$		0			0
$495$	−7.76605e23	−	1.31404e24i	−0.0195637	−	0.0331023i
$496$		0			0
$497$			1.97895e25i			0.479787i
$498$		0			0
$499$	−9.50774e24			−0.221882			−0.110941	−	0.993827i	$-0.535386\pi$
$499$	−9.50774e24			−0.221882			−0.110941	+	0.993827i	$0.535386\pi$
$500$		0			0
$501$	−1.57063e25			−0.352870
$502$		0			0
$503$			1.28370e25i			0.277694i	0.990314	+	0.138847i	$0.0443397\pi$
$503$			1.28370e25i			0.277694i	−0.990314	+	0.138847i	$0.955660\pi$
$504$		0			0
$505$	1.95895e25	+	3.31460e25i	0.408090	+	0.690500i
$506$		0			0
$507$		−	1.03151e25i		−	0.206965i
$508$		0			0
$509$	−2.61312e25			−0.505058			−0.252529	−	0.967589i	$-0.581262\pi$
$509$	−2.61312e25			−0.505058			−0.252529	+	0.967589i	$0.581262\pi$
$510$		0			0
$511$	−1.39653e25			−0.260047
$512$		0			0
$513$			5.91250e25i			1.06086i
$514$		0			0
$515$	6.35473e25	−	3.75568e25i	1.09882	−	0.649412i
$516$		0			0
$517$		−	2.58464e25i		−	0.430765i
$518$		0			0
$519$	−8.84885e24			−0.142166
$520$		0			0
$521$	−1.39805e25			−0.216554			−0.108277	−	0.994121i	$-0.534533\pi$
$521$	−1.39805e25			−0.216554			−0.108277	+	0.994121i	$0.534533\pi$
$522$		0			0
$523$			7.37992e25i			1.10226i	0.834418	+	0.551132i	$0.185804\pi$
$523$			7.37992e25i			1.10226i	−0.834418	+	0.551132i	$0.814196\pi$
$524$		0			0
$525$	8.24128e24	−	1.49702e25i	0.118708	−	0.215633i
$526$		0			0
$527$			3.73364e24i			0.0518719i
$528$		0			0
$529$	−1.94618e25			−0.260827
$530$		0			0
$531$	2.45948e24			0.0318013
$532$		0			0
$533$			3.92985e25i			0.490307i
$534$		0			0
$535$	−9.61678e25	+	5.68358e25i	−1.15789	+	0.684323i
$536$		0			0
$537$			5.68353e25i			0.660484i
$538$		0			0
$539$	−5.93441e25			−0.665709
$540$		0			0
$541$	−2.49490e25			−0.270197			−0.135098	−	0.990832i	$-0.543135\pi$
$541$	−2.49490e25			−0.270197			−0.135098	+	0.990832i	$0.543135\pi$
$542$		0			0
$543$		−	1.03194e25i		−	0.107909i
$544$		0			0
$545$	−5.03015e25	−	8.51115e25i	−0.507942	−	0.859452i
$546$		0			0
$547$			5.47088e25i			0.533553i	0.963758	+	0.266776i	$0.0859585\pi$
$547$			5.47088e25i			0.533553i	−0.963758	+	0.266776i	$0.914041\pi$
$548$		0			0
$549$	−9.06784e24			−0.0854214
$550$		0			0
$551$	1.37820e26			1.25422
$552$		0			0
$553$			1.47779e24i			0.0129935i
$554$		0			0
$555$	−1.04396e26	−	1.76641e26i	−0.886951	−	1.50075i
$556$		0			0
$557$			2.16234e26i			1.77541i	0.460412	+	0.887705i	$0.347702\pi$
$557$			2.16234e26i			1.77541i	−0.460412	+	0.887705i	$0.652298\pi$
$558$		0			0
$559$	−9.05990e23			−0.00718971
$560$		0			0
$561$	−1.09756e26			−0.841938
$562$		0			0
$563$		−	1.99170e26i		−	1.47705i	−0.674227	−	0.738524i	$-0.735523\pi$
$563$		−	1.99170e26i		−	1.47705i	0.674227	−	0.738524i	$-0.264477\pi$
$564$		0			0
$565$	1.12502e25	−	6.64892e24i	0.0806675	−	0.0476750i
$566$		0			0
$567$		−	3.44211e25i		−	0.238663i
$568$		0			0
$569$	2.58945e26			1.73636			0.868182	−	0.496246i	$-0.165289\pi$
$569$	2.58945e26			1.73636			0.868182	+	0.496246i	$0.165289\pi$
$570$		0			0
$571$	1.85085e26			1.20040			0.600202	−	0.799849i	$-0.295087\pi$
$571$	1.85085e26			1.20040			0.600202	+	0.799849i	$0.295087\pi$
$572$		0			0
$573$		−	2.17111e26i		−	1.36211i
$574$		0			0
$575$	1.62065e26	+	8.92187e25i	0.983660	+	0.541516i
$576$		0			0
$577$		−	1.24419e26i		−	0.730663i	−0.930878	−	0.365331i	$-0.880956\pi$
$577$		−	1.24419e26i		−	0.730663i	0.930878	−	0.365331i	$-0.119044\pi$
$578$		0			0
$579$	2.57310e25			0.146221
$580$		0			0
$581$	−1.39237e25			−0.0765741
$582$		0			0
$583$		−	1.90710e26i		−	1.01513i
$584$		0			0
$585$	9.94719e24	−	5.87886e24i	0.0512531	−	0.0302909i
$586$		0			0
$587$			3.19668e26i			1.59454i	0.603620	+	0.797272i	$0.293725\pi$
$587$			3.19668e26i			1.59454i	−0.603620	+	0.797272i	$0.706275\pi$
$588$		0			0
$589$	−9.13210e24			−0.0441037
$590$		0			0
$591$	−1.70349e26			−0.796635
$592$		0			0
$593$		−	1.78032e26i		−	0.806265i	−0.915142	−	0.403132i	$-0.867921\pi$
$593$		−	1.78032e26i		−	0.806265i	0.915142	−	0.403132i	$-0.132079\pi$
$594$		0			0
$595$	3.57300e25	+	6.04562e25i	0.156719	+	0.265173i
$596$		0			0
$597$		−	1.37897e26i		−	0.585868i
$598$		0			0
$599$	−1.92332e26			−0.791584			−0.395792	−	0.918340i	$-0.629530\pi$
$599$	−1.92332e26			−0.791584			−0.395792	+	0.918340i	$0.629530\pi$
$600$		0			0
$601$	3.81486e26			1.52114			0.760572	−	0.649253i	$-0.224918\pi$
$601$	3.81486e26			1.52114			0.760572	+	0.649253i	$0.224918\pi$
$602$		0			0
$603$		−	2.38657e24i		−	0.00922061i
$604$		0			0
$605$	−6.71475e25	−	1.13616e26i	−0.251393	−	0.425365i
$606$		0			0
$607$		−	2.42669e26i		−	0.880483i	−0.897879	−	0.440242i	$-0.854893\pi$
$607$		−	2.42669e26i		−	0.880483i	0.897879	−	0.440242i	$-0.145107\pi$
$608$		0			0
$609$	−8.48353e25			−0.298340
$610$		0			0
$611$	1.95656e26			0.666963
$612$		0			0
$613$			2.73712e26i			0.904524i	0.891885	+	0.452262i	$0.149383\pi$
$613$			2.73712e26i			0.904524i	−0.891885	+	0.452262i	$0.850617\pi$
$614$		0			0
$615$	1.16350e26	−	6.87637e25i	0.372780	−	0.220316i
$616$		0			0
$617$			2.32043e26i			0.720873i	0.932784	+	0.360436i	$0.117372\pi$
$617$			2.32043e26i			0.720873i	−0.932784	+	0.360436i	$0.882628\pi$
$618$		0			0
$619$	3.05361e26			0.919924			0.459962	−	0.887939i	$-0.347863\pi$
$619$	3.05361e26			0.919924			0.459962	+	0.887939i	$0.347863\pi$
$620$		0			0
$621$	3.93999e26			1.15113
$622$		0			0
$623$			8.32661e25i			0.235956i
$624$		0			0
$625$	−1.94576e26	−	3.07391e26i	−0.534846	−	0.844949i
$626$		0			0
$627$		−	2.68451e26i		−	0.715852i
$628$		0			0
$629$	8.43189e26			2.18144
$630$		0			0
$631$	−3.28337e26			−0.824216			−0.412108	−	0.911135i	$-0.635207\pi$
$631$	−3.28337e26			−0.824216			−0.412108	+	0.911135i	$0.635207\pi$
$632$		0			0
$633$		−	1.97384e26i		−	0.480814i
$634$		0			0
$635$	6.53712e26	−	3.86348e26i	1.54538	−	0.913331i
$636$		0			0
$637$		−	4.49232e26i		−	1.03073i
$638$		0			0
$639$	4.60170e25			0.102485
$640$		0			0
$641$	−6.76321e26			−1.46218			−0.731091	−	0.682280i	$-0.760988\pi$
$641$	−6.76321e26			−1.46218			−0.731091	+	0.682280i	$0.760988\pi$
$642$		0			0
$643$			3.72363e26i			0.781560i	0.920484	+	0.390780i	$0.127795\pi$
$643$			3.72363e26i			0.781560i	−0.920484	+	0.390780i	$0.872205\pi$
$644$		0			0
$645$	1.58528e24	+	2.68234e24i	0.00323064	+	0.00546634i
$646$		0			0
$647$			5.88512e26i			1.16457i	0.812986	+	0.582284i	$0.197841\pi$
$647$			5.88512e26i			1.16457i	−0.812986	+	0.582284i	$0.802159\pi$
$648$		0			0
$649$	−2.17733e26			−0.418407
$650$		0			0
$651$	5.62126e24			0.0104909
$652$		0			0
$653$		−	1.34507e25i		−	0.0243821i	−0.999926	−	0.0121911i	$-0.996119\pi$
$653$		−	1.34507e25i		−	0.0243821i	0.999926	−	0.0121911i	$-0.00388063\pi$
$654$		0			0
$655$	−2.63223e26	−	4.45381e26i	−0.463480	−	0.784222i
$656$		0			0
$657$			3.24739e25i			0.0555474i
$658$		0			0
$659$	−7.10095e26			−1.18006			−0.590031	−	0.807380i	$-0.700885\pi$
$659$	−7.10095e26			−1.18006			−0.590031	+	0.807380i	$0.700885\pi$
$660$		0			0
$661$	3.17692e26			0.512971			0.256486	−	0.966548i	$-0.417435\pi$
$661$	3.17692e26			0.512971			0.256486	+	0.966548i	$0.417435\pi$
$662$		0			0
$663$		−	8.30845e26i		−	1.30359i
$664$		0			0
$665$	−1.47869e26	+	8.73918e25i	−0.225462	+	0.133249i
$666$		0			0
$667$		−	9.18412e26i		−	1.36095i
$668$		0			0
$669$	−1.24780e27			−1.79720
$670$		0			0
$671$	8.02759e26			1.12388
$672$		0			0
$673$		−	8.95316e26i		−	1.21852i	−0.792970	−	0.609261i	$-0.791466\pi$
$673$		−	8.95316e26i		−	1.21852i	0.792970	−	0.609261i	$-0.208534\pi$
$674$		0			0
$675$	−6.78735e26	−	3.73651e26i	−0.898079	−	0.494402i
$676$		0			0
$677$			1.10734e27i			1.42459i	0.701882	+	0.712293i	$0.252343\pi$
$677$			1.10734e27i			1.42459i	−0.701882	+	0.712293i	$0.747657\pi$
$678$		0			0
$679$	−1.48464e26			−0.185720
$680$		0			0
$681$	−1.04680e27			−1.27341
$682$		0			0
$683$		−	9.58956e26i		−	1.13449i	−0.823548	−	0.567247i	$-0.808009\pi$
$683$		−	9.58956e26i		−	1.13449i	0.823548	−	0.567247i	$-0.191991\pi$
$684$		0			0
$685$	6.64813e26	−	3.92909e26i	0.764961	−	0.452097i
$686$		0			0
$687$			9.92581e26i			1.11090i
$688$		0			0
$689$	1.44366e27			1.57175
$690$		0			0
$691$	5.16103e26			0.546632			0.273316	−	0.961924i	$-0.411880\pi$
$691$	5.16103e26			0.546632			0.273316	+	0.961924i	$0.411880\pi$
$692$		0			0
$693$		−	9.44372e24i		−	0.00973144i
$694$		0			0
$695$	8.68058e25	+	1.46878e26i	0.0870348	+	0.147265i
$696$		0			0
$697$			5.55394e26i			0.541863i
$698$		0			0
$699$	3.18124e25			0.0302039
$700$		0			0
$701$	3.31094e26			0.305936			0.152968	−	0.988231i	$-0.451117\pi$
$701$	3.31094e26			0.305936			0.152968	+	0.988231i	$0.451117\pi$
$702$		0			0
$703$			2.06235e27i			1.85476i
$704$		0			0
$705$	−3.42354e26	−	5.79273e26i	−0.299695	−	0.507092i
$706$		0			0
$707$			2.38214e26i			0.202994i
$708$		0			0
$709$	2.83893e26			0.235513			0.117757	−	0.993042i	$-0.462430\pi$
$709$	2.83893e26			0.235513			0.117757	+	0.993042i	$0.462430\pi$
$710$		0			0
$711$	3.43635e24			0.00277547
$712$		0			0
$713$			6.08548e25i			0.0478568i
$714$		0			0
$715$	−8.80606e26	+	5.20444e26i	−0.674333	+	0.398535i
$716$		0			0
$717$		−	1.38496e27i		−	1.03277i
$718$		0			0
$719$	−3.71131e26			−0.269527			−0.134764	−	0.990878i	$-0.543027\pi$
$719$	−3.71131e26			−0.269527			−0.134764	+	0.990878i	$0.543027\pi$
$720$		0			0
$721$	4.56701e26			0.323033
$722$		0			0
$723$		−	1.61308e27i		−	1.11133i
$724$		0			0
$725$	−8.70981e26	+	1.58213e27i	−0.584517	+	1.06177i
$726$		0			0
$727$			1.58762e27i			1.03793i	0.854795	+	0.518966i	$0.173683\pi$
$727$			1.58762e27i			1.03793i	−0.854795	+	0.518966i	$0.826317\pi$
$728$		0			0
$729$	−1.64549e27			−1.04806
$730$		0			0
$731$	−1.28041e25			−0.00794572
$732$		0			0
$733$			2.28793e27i			1.38342i	0.722174	+	0.691711i	$0.243143\pi$
$733$			2.28793e27i			1.38342i	−0.722174	+	0.691711i	$0.756857\pi$
$734$		0			0
$735$	−1.33003e27	+	7.86055e26i	−0.783666	+	0.463152i
$736$		0			0
$737$			2.11278e26i			0.121315i
$738$		0			0
$739$	3.00171e27			1.67976			0.839878	−	0.542775i	$-0.182626\pi$
$739$	3.00171e27			1.67976			0.839878	+	0.542775i	$0.182626\pi$
$740$		0			0
$741$	2.03216e27			1.10837
$742$		0			0
$743$			2.74458e27i			1.45909i	0.683933	+	0.729545i	$0.260268\pi$
$743$			2.74458e27i			1.45909i	−0.683933	+	0.729545i	$0.739732\pi$
$744$		0			0
$745$	5.32409e26	+	9.00852e26i	0.275906	+	0.466841i
$746$		0			0
$747$			3.23772e25i			0.0163566i
$748$		0			0
$749$	−6.91138e26			−0.340398
$750$		0			0
$751$	−1.30272e27			−0.625562			−0.312781	−	0.949825i	$-0.601261\pi$
$751$	−1.30272e27			−0.625562			−0.312781	+	0.949825i	$0.601261\pi$
$752$		0			0
$753$			1.64422e27i			0.769852i
$754$		0			0
$755$	2.18184e27	+	3.69173e27i	0.996154	+	1.68552i
$756$		0			0
$757$			1.66513e27i			0.741372i	0.928758	+	0.370686i	$0.120877\pi$
$757$			1.66513e27i			0.741372i	−0.928758	+	0.370686i	$0.879123\pi$
$758$		0			0
$759$	−1.78891e27			−0.776770
$760$		0			0
$761$	3.74821e27			1.58734			0.793669	−	0.608349i	$-0.208168\pi$
$761$	3.74821e27			1.58734			0.793669	+	0.608349i	$0.208168\pi$
$762$		0			0
$763$		−	6.11679e26i		−	0.252662i
$764$		0			0
$765$	1.40581e26	−	8.30840e25i	0.0566424	−	0.0334760i
$766$		0			0
$767$		−	1.64822e27i		−	0.647829i
$768$		0			0
$769$	−2.72740e27			−1.04580			−0.522900	−	0.852394i	$-0.675150\pi$
$769$	−2.72740e27			−1.04580			−0.522900	+	0.852394i	$0.675150\pi$
$770$		0			0
$771$	−2.00076e27			−0.748477
$772$		0			0
$773$		−	1.98940e27i		−	0.726135i	−0.931763	−	0.363067i	$-0.881729\pi$
$773$		−	1.98940e27i		−	0.726135i	0.931763	−	0.363067i	$-0.118271\pi$
$774$		0			0
$775$	5.77119e25	−	1.04833e26i	0.0205542	−	0.0373365i
$776$		0			0
$777$		−	1.26948e27i		−	0.441190i
$778$		0			0
$779$	−1.35843e27			−0.460715
$780$		0			0
$781$	−4.07380e27			−1.34839
$782$		0			0
$783$			3.84634e27i			1.24254i
$784$		0			0
$785$	−1.98087e27	+	1.17071e27i	−0.624589	+	0.369136i
$786$		0			0
$787$		−	5.25578e27i		−	1.61762i	−0.588068	−	0.808812i	$-0.700111\pi$
$787$		−	5.25578e27i		−	1.61762i	0.588068	−	0.808812i	$-0.299889\pi$
$788$		0			0
$789$	−4.26651e27			−1.28186
$790$		0			0
$791$	8.08526e25			0.0237147
$792$		0			0
$793$			6.07684e27i			1.74013i
$794$		0			0
$795$	−2.52608e27	−	4.27421e27i	−0.706253	−	1.19500i
$796$		0			0
$797$		−	2.82824e27i		−	0.772081i	−0.922482	−	0.386040i	$-0.873843\pi$
$797$		−	2.82824e27i		−	0.772081i	0.922482	−	0.386040i	$-0.126157\pi$
$798$		0			0
$799$	2.76514e27			0.737095
$800$		0			0
$801$	1.93621e26			0.0504015
$802$		0			0
$803$		−	2.87485e27i		−	0.730833i
$804$		0			0
$805$	5.82365e26	+	9.85377e26i	0.144589	+	0.244648i
$806$		0			0
$807$			3.16244e27i			0.766873i
$808$		0			0
$809$	2.81690e26			0.0667207			0.0333603	−	0.999443i	$-0.489379\pi$
$809$	2.81690e26			0.0667207			0.0333603	+	0.999443i	$0.489379\pi$
$810$		0			0
$811$	2.86731e27			0.663401			0.331700	−	0.943385i	$-0.392378\pi$
$811$	2.86731e27			0.663401			0.331700	+	0.943385i	$0.392378\pi$
$812$		0			0
$813$		−	1.94534e27i		−	0.439679i
$814$		0			0
$815$	−5.55671e27	+	3.28405e27i	−1.22693	+	0.725125i
$816$		0			0
$817$		−	3.13174e25i		−	0.00675579i
$818$		0			0
$819$	7.14884e25			0.0150674
$820$		0			0
$821$	−4.13541e27			−0.851646			−0.425823	−	0.904807i	$-0.640015\pi$
$821$	−4.13541e27			−0.851646			−0.425823	+	0.904807i	$0.640015\pi$
$822$		0			0
$823$		−	4.19567e26i		−	0.0844313i	−0.999109	−	0.0422156i	$-0.986558\pi$
$823$		−	4.19567e26i		−	0.0844313i	0.999109	−	0.0422156i	$-0.0134416\pi$
$824$		0			0
$825$	3.08173e27	+	1.69653e27i	0.606013	+	0.333617i
$826$		0			0
$827$		−	8.90302e26i		−	0.171094i	−0.996334	−	0.0855470i	$-0.972736\pi$
$827$		−	8.90302e26i		−	0.171094i	0.996334	−	0.0855470i	$-0.0272638\pi$
$828$		0			0
$829$	2.26694e27			0.425768			0.212884	−	0.977078i	$-0.431714\pi$
$829$	2.26694e27			0.425768			0.212884	+	0.977078i	$0.431714\pi$
$830$		0			0
$831$	−2.69132e27			−0.494032
$832$		0			0
$833$		−	6.34885e27i		−	1.13911i
$834$		0			0
$835$	−1.78095e27	+	1.05256e27i	−0.312343	+	0.184597i
$836$		0			0
$837$		−	2.54862e26i		−	0.0436932i
$838$		0			0
$839$	−6.09159e27			−1.02092			−0.510461	−	0.859901i	$-0.670525\pi$
$839$	−6.09159e27			−1.02092			−0.510461	+	0.859901i	$0.670525\pi$
$840$		0			0
$841$	2.86256e27			0.469022
$842$		0			0
$843$			3.26807e27i			0.523516i
$844$		0			0
$845$	−6.91265e26	−	1.16964e27i	−0.108269	−	0.183195i
$846$		0			0
$847$		−	8.16531e26i		−	0.125049i
$848$		0			0
$849$	2.85783e27			0.427970
$850$		0			0
$851$	1.37432e28			2.01259
$852$		0			0
$853$		−	1.32363e27i		−	0.189562i	−0.995498	−	0.0947810i	$-0.969785\pi$
$853$		−	1.32363e27i		−	0.189562i	0.995498	−	0.0947810i	$-0.0302151\pi$
$854$		0			0
$855$	2.03215e26	+	3.43845e26i	0.0284628	+	0.0481598i
$856$		0			0
$857$			2.10638e27i			0.288548i	0.989538	+	0.144274i	$0.0460847\pi$
$857$			2.10638e27i			0.288548i	−0.989538	+	0.144274i	$0.953915\pi$
$858$		0			0
$859$	7.62793e27			1.02205			0.511025	−	0.859566i	$-0.329266\pi$
$859$	7.62793e27			1.02205			0.511025	+	0.859566i	$0.329266\pi$
$860$		0			0
$861$	8.36184e26			0.109590
$862$		0			0
$863$			2.70442e27i			0.346714i	0.984859	+	0.173357i	$0.0554615\pi$
$863$			2.70442e27i			0.346714i	−0.984859	+	0.173357i	$0.944539\pi$
$864$		0			0
$865$	−1.00338e27	+	5.93005e26i	−0.125838	+	0.0743713i
$866$		0			0
$867$		−	3.81499e27i		−	0.468072i
$868$		0			0
$869$	−3.04214e26			−0.0365167
$870$		0			0
$871$	−1.59937e27			−0.187835
$872$		0			0
$873$			3.45227e26i			0.0396707i
$874$		0			0
$875$	−6.87412e25	−	2.24978e27i	−0.00772931	−	0.252967i
$876$		0			0
$877$		−	9.43005e27i		−	1.03757i	−0.854905	−	0.518785i	$-0.826384\pi$
$877$		−	9.43005e27i		−	1.03757i	0.854905	−	0.518785i	$-0.173616\pi$
$878$		0			0
$879$	−1.87602e27			−0.201997
$880$		0			0
$881$	1.59431e28			1.67998			0.839988	−	0.542605i	$-0.182562\pi$
$881$	1.59431e28			1.67998			0.839988	+	0.542605i	$0.182562\pi$
$882$		0			0
$883$		−	3.02275e27i		−	0.311728i	−0.987779	−	0.155864i	$-0.950184\pi$
$883$		−	3.02275e27i		−	0.311728i	0.987779	−	0.155864i	$-0.0498162\pi$
$884$		0			0
$885$	−4.87985e27	+	2.88403e27i	−0.492545	+	0.291097i
$886$		0			0
$887$		−	9.50319e26i		−	0.0938847i	−0.998898	−	0.0469424i	$-0.985052\pi$
$887$		−	9.50319e26i		−	0.0938847i	0.998898	−	0.0469424i	$-0.0149477\pi$
$888$		0			0
$889$	4.69809e27			0.454312
$890$		0			0
$891$	7.08583e27			0.670737
$892$		0			0
$893$			6.76325e27i			0.626710i
$894$		0			0
$895$	3.80881e27	+	6.44461e27i	0.345518	+	0.584626i
$896$		0			0
$897$		−	1.35420e28i		−	1.20269i
$898$		0			0
$899$	−5.94083e26			−0.0516571
$900$		0			0
$901$	2.04028e28			1.73702
$902$		0			0
$903$			1.92774e25i			0.00160700i
$904$		0			0
$905$	−6.91555e26	−	1.17013e27i	−0.0564502	−	0.0955153i
$906$		0			0
$907$		−	9.98796e27i		−	0.798376i	−0.916869	−	0.399188i	$-0.869292\pi$
$907$		−	9.98796e27i		−	0.798376i	0.916869	−	0.399188i	$-0.130708\pi$
$908$		0			0
$909$	5.53925e26			0.0433605
$910$		0			0
$911$	2.89381e27			0.221843			0.110921	−	0.993829i	$-0.464620\pi$
$911$	2.89381e27			0.221843			0.110921	+	0.993829i	$0.464620\pi$
$912$		0			0
$913$		−	2.86629e27i		−	0.215203i
$914$		0			0
$915$	1.79915e28	−	1.06331e28i	1.32302	−	0.781916i
$916$		0			0
$917$		−	3.20086e27i		−	0.230546i
$918$		0			0
$919$	−1.36611e28			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$919$	−1.36611e28			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$920$		0			0
$921$	−2.90172e26			−0.0200534
$922$		0			0
$923$		−	3.08384e28i		−	2.08774i
$924$		0			0
$925$	−2.36751e28	−	1.30334e28i	−1.57017	−	0.864393i
$926$		0			0
$927$		−	1.06198e27i		−	0.0690015i
$928$		0			0
$929$	−2.48929e28			−1.58462			−0.792311	−	0.610118i	$-0.791122\pi$
$929$	−2.48929e28			−1.58462			−0.792311	+	0.610118i	$0.791122\pi$
$930$		0			0
$931$	1.55286e28			0.968525
$932$		0			0
$933$			9.96471e27i			0.608960i
$934$		0			0
$935$	−1.24453e28	+	7.35527e27i	−0.745240	+	0.440442i
$936$		0			0
$937$			1.77582e28i			1.04201i	0.853553	+	0.521006i	$0.174443\pi$
$937$			1.77582e28i			1.04201i	−0.853553	+	0.521006i	$0.825557\pi$
$938$		0			0
$939$	−1.21141e27			−0.0696578
$940$		0			0
$941$	1.32109e28			0.744439			0.372220	−	0.928145i	$-0.378597\pi$
$941$	1.32109e28			0.744439			0.372220	+	0.928145i	$0.378597\pi$
$942$		0			0
$943$			9.05238e27i			0.499921i
$944$		0			0
$945$	−2.43897e27	−	4.12680e27i	−0.132009	−	0.223363i
$946$		0			0
$947$		−	1.92300e28i		−	1.02013i	−0.860137	−	0.510063i	$-0.829622\pi$
$947$		−	1.92300e28i		−	1.02013i	0.860137	−	0.510063i	$-0.170378\pi$
$948$		0			0
$949$	2.17625e28			1.13156
$950$		0			0
$951$	1.47132e28			0.749879
$952$		0			0
$953$		−	2.59851e28i		−	1.29820i	−0.760704	−	0.649099i	$-0.775146\pi$
$953$		−	2.59851e28i		−	1.29820i	0.760704	−	0.649099i	$-0.224854\pi$
$954$		0			0
$955$	−1.45496e28	−	2.46184e28i	−0.712558	−	1.20567i
$956$		0			0
$957$		−	1.74639e28i		−	0.838453i
$958$		0			0
$959$	4.77788e27			0.224884
$960$		0			0
$961$	−2.16313e28			−0.998184
$962$		0			0
$963$			1.60712e27i			0.0727109i
$964$		0			0
$965$	2.91766e27	−	1.72436e27i	0.129427	−	0.0764924i
$966$		0			0
$967$			2.59918e28i			1.13054i	0.824907	+	0.565269i	$0.191228\pi$
$967$			2.59918e28i			1.13054i	−0.824907	+	0.565269i	$0.808772\pi$
$968$		0			0
$969$	2.87199e28			1.22492
$970$		0			0
$971$	−1.02365e28			−0.428122			−0.214061	−	0.976820i	$-0.568669\pi$
$971$	−1.02365e28			−0.428122			−0.214061	+	0.976820i	$0.568669\pi$
$972$		0			0
$973$			1.05558e27i			0.0432932i
$974$		0			0
$975$	−1.28426e28	+	2.33285e28i	−0.516546	+	0.938303i
$976$		0			0
$977$		−	5.73585e27i		−	0.226256i	−0.993580	−	0.113128i	$-0.963913\pi$
$977$		−	5.73585e27i		−	0.226256i	0.993580	−	0.113128i	$-0.0360870\pi$
$978$		0			0
$979$	−1.71409e28			−0.663129
$980$		0			0
$981$	−1.42235e27			−0.0539700
$982$		0			0
$983$			2.73216e28i			1.01683i	0.861113	+	0.508414i	$0.169768\pi$
$983$			2.73216e28i			1.01683i	−0.861113	+	0.508414i	$0.830232\pi$
$984$		0			0
$985$	−1.93161e28	+	1.14159e28i	−0.705140	+	0.416742i
$986$		0			0
$987$		−	4.16312e27i		−	0.149075i
$988$		0			0
$989$	−2.08694e26			−0.00733069
$990$		0			0
$991$	3.38695e28			1.16710			0.583551	−	0.812077i	$-0.301663\pi$
$991$	3.38695e28			1.16710			0.583551	+	0.812077i	$0.301663\pi$
$992$		0			0
$993$			1.94483e28i			0.657453i
$994$		0			0
$995$	−9.24117e27	−	1.56363e28i	−0.306484	−	0.518580i
$996$		0			0
$997$			4.64239e28i			1.51056i	0.655403	+	0.755279i	$0.272499\pi$
$997$			4.64239e28i			1.51056i	−0.655403	+	0.755279i	$0.727501\pi$
$998$		0			0
$999$	−5.75569e28			−1.83749

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.c.a.49.6		8
4.3	odd	2		5.20.b.a.4.4	✓	8
5.4	even	2	inner	80.20.c.a.49.3		8
12.11	even	2		45.20.b.b.19.5		8
20.3	even	4		25.20.a.f.1.4		8
20.7	even	4		25.20.a.f.1.5		8
20.19	odd	2		5.20.b.a.4.5	yes	8
60.59	even	2		45.20.b.b.19.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.20.b.a.4.4	✓	8	4.3	odd	2
5.20.b.a.4.5	yes	8	20.19	odd	2
25.20.a.f.1.4		8	20.3	even	4
25.20.a.f.1.5		8	20.7	even	4
45.20.b.b.19.4		8	60.59	even	2
45.20.b.b.19.5		8	12.11	even	2
80.20.c.a.49.3		8	5.4	even	2	inner
80.20.c.a.49.6		8	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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+33157.6i q^{3} +(2.22206e6 + 3.75978e6i) q^{5} +2.70208e7i q^{7} +6.28322e7 q^{9} +O(q^{10})\)	\(q+33157.6i q^{3} +(2.22206e6 + 3.75978e6i) q^{5} +2.70208e7i q^{7} +6.28322e7 q^{9} -5.56241e9 q^{11} -4.21072e10i q^{13} +(-1.24665e11 + 7.36781e10i) q^{15} -5.95087e11i q^{17} +1.45552e12 q^{19} -8.95945e11 q^{21} -9.69934e12i q^{23} +(-9.19842e12 + 1.67089e13i) q^{25} +4.06212e13i q^{27} +9.46880e13 q^{29} -6.27411e12 q^{31} -1.84437e14i q^{33} +(-1.01592e14 + 6.00417e13i) q^{35} +1.41692e15i q^{37} +1.39617e15 q^{39} -9.33298e14 q^{41} -2.15163e13i q^{43} +(1.39617e14 + 2.36235e14i) q^{45} +4.64662e15i q^{47} +1.06688e16 q^{49} +1.97317e16 q^{51} +3.42854e16i q^{53} +(-1.23600e16 - 2.09135e16i) q^{55} +4.82616e16i q^{57} +3.91436e16 q^{59} -1.44319e17 q^{61} +1.69777e15i q^{63} +(1.58314e17 - 9.35644e16i) q^{65} -3.79832e16i q^{67} +3.21607e17 q^{69} +7.32379e17 q^{71} +5.16836e17i q^{73} +(-5.54027e17 - 3.04998e17i) q^{75} -1.50301e17i q^{77} +5.46910e16 q^{79} -1.27388e18 q^{81} +5.15297e17i q^{83} +(2.23740e18 - 1.32232e18i) q^{85} +3.13963e18i q^{87} +3.08156e18 q^{89} +1.13777e18 q^{91} -2.08035e17i q^{93} +(3.23425e18 + 5.47244e18i) q^{95} +5.49443e18i q^{97} -3.49498e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 147000 q^{5} + 345358584 q^{9}+O(q^{10}) \) 8 * q + 147000 * q^5 + 345358584 * q^9	\( 8 q + 147000 q^{5} + 345358584 q^{9} + 3379575264 q^{11} + 242324628000 q^{15} - 4547188380640 q^{19} - 2983154334624 q^{21} + 17715709625000 q^{25} - 188222300345040 q^{29} - 72115006686976 q^{31} + 299115755916000 q^{35} - 30\!\cdots\!28 q^{39}+ \cdots + 16\!\cdots\!72 q^{99}+O(q^{100}) \) 8 * q + 147000 * q^5 + 345358584 * q^9 + 3379575264 * q^11 + 242324628000 * q^15 - 4547188380640 * q^19 - 2983154334624 * q^21 + 17715709625000 * q^25 - 188222300345040 * q^29 - 72115006686976 * q^31 + 299115755916000 * q^35 - 3011653788049728 * q^39 - 3314269428857904 * q^41 + 10198157211261000 * q^45 - 11129258011816744 * q^49 + 46867071411404544 * q^51 - 69821226326124000 * q^55 - 259648032778012320 * q^59 - 582893562753226064 * q^61 + 747545324689032000 * q^65 + 302144302523815008 * q^69 + 2617007600602926144 * q^71 - 2877932428209000000 * q^75 - 3935092091109861760 * q^79 - 8328177293660704152 * q^81 + 3752387009219584000 * q^85 + 2120141101636921680 * q^89 + 3358638219973817664 * q^91 - 6106065789884820000 * q^95 + 16256944625235082272 * q^99

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