LMFDB - Embedded newform 20.8.c.a.9.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,8,Mod(9,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("20.9");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	20.c (of order $2$, degree $1$, 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:	$6.24770050968$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1348x^{2} + 93051 $ x^4 + 1348*x^2 + 93051
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.4
Root			$35.7074i$ of defining polynomial
Character	$\chi$	$=$	20.9
Dual form			20.8.c.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+71.4148i q^{3} +(-279.408 + 7.49513i) q^{5} -860.515i q^{7} -2913.08 q^{9} +O(q^{10})$	$q+71.4148i q^{3} +(-279.408 + 7.49513i) q^{5} -860.515i q^{7} -2913.08 q^{9} -5468.16 q^{11} +14343.8i q^{13} +(-535.264 - 19953.9i) q^{15} -8677.25i q^{17} +5340.48 q^{19} +61453.5 q^{21} +74802.5i q^{23} +(78012.6 - 4188.40i) q^{25} -51852.9i q^{27} -91625.0 q^{29} -103909. q^{31} -390508. i q^{33} +(6449.67 + 240435. i) q^{35} +467238. i q^{37} -1.02436e6 q^{39} +385741. q^{41} -257098. i q^{43} +(813938. - 21833.9i) q^{45} -762177. i q^{47} +83057.6 q^{49} +619684. q^{51} +1.14032e6i q^{53} +(1.52785e6 - 40984.6i) q^{55} +381389. i q^{57} -2.31663e6 q^{59} +1.89127e6 q^{61} +2.50675e6i q^{63} +(-107509. - 4.00777e6i) q^{65} -1.89718e6i q^{67} -5.34201e6 q^{69} -2.84022e6 q^{71} -2.59537e6i q^{73} +(299114. + 5.57126e6i) q^{75} +4.70543e6i q^{77} +5.49695e6 q^{79} -2.66784e6 q^{81} -3.79174e6i q^{83} +(65037.1 + 2.42449e6i) q^{85} -6.54339e6i q^{87} +1.95833e6 q^{89} +1.23430e7 q^{91} -7.42063e6i q^{93} +(-1.49217e6 + 40027.6i) q^{95} +1.19937e7i q^{97} +1.59292e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 156 q^{5} - 2036 q^{9}+O(q^{10}) $ 4 * q - 156 * q^5 - 2036 * q^9	$ 4 q - 156 q^{5} - 2036 q^{9} - 2640 q^{11} + 5552 q^{15} - 36336 q^{19} + 63104 q^{21} + 162036 q^{25} - 481896 q^{29} + 161344 q^{31} + 691248 q^{35} - 2366496 q^{39} + 1129464 q^{41} + 2391244 q^{45} - 4312452 q^{49} + 1978688 q^{51} + 4726640 q^{55} - 7747152 q^{59} + 4305128 q^{61} + 3389568 q^{65} - 9357248 q^{69} - 1206048 q^{71} + 3265888 q^{75} + 5716992 q^{79} + 570116 q^{81} - 8256064 q^{85} + 13641576 q^{89} - 7864224 q^{91} - 12453936 q^{95} + 47580560 q^{99}+O(q^{100}) $ 4 * q - 156 * q^5 - 2036 * q^9 - 2640 * q^11 + 5552 * q^15 - 36336 * q^19 + 63104 * q^21 + 162036 * q^25 - 481896 * q^29 + 161344 * q^31 + 691248 * q^35 - 2366496 * q^39 + 1129464 * q^41 + 2391244 * q^45 - 4312452 * q^49 + 1978688 * q^51 + 4726640 * q^55 - 7747152 * q^59 + 4305128 * q^61 + 3389568 * q^65 - 9357248 * q^69 - 1206048 * q^71 + 3265888 * q^75 + 5716992 * q^79 + 570116 * q^81 - 8256064 * q^85 + 13641576 * q^89 - 7864224 * q^91 - 12453936 * q^95 + 47580560 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$-1$

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
$3$			71.4148i			1.52709i	0.645756	+	0.763544i	$0.276542\pi$
$3$			71.4148i			1.52709i	−0.645756	+	0.763544i	$0.723458\pi$
$4$		0			0
$5$	−279.408	+	7.49513i	−0.999640	+	0.0268154i
$5$	−279.408	+	7.49513i	−0.999640	+	0.0268154i
$6$		0			0
$7$		−	860.515i		−	0.948233i	−0.880462	−	0.474117i	$-0.842768\pi$
$7$		−	860.515i		−	0.948233i	0.880462	−	0.474117i	$-0.157232\pi$
$8$		0			0
$9$	−2913.08			−1.33200
$10$		0			0
$11$	−5468.16			−1.23870			−0.619351	−	0.785114i	$-0.712604\pi$
$11$	−5468.16			−1.23870			−0.619351	+	0.785114i	$0.712604\pi$
$12$		0			0
$13$			14343.8i			1.81076i	0.424599	+	0.905382i	$0.360415\pi$
$13$			14343.8i			1.81076i	−0.424599	+	0.905382i	$0.639585\pi$
$14$		0			0
$15$	−535.264	−	19953.9i	−0.0409495	−	1.52654i
$16$		0			0
$17$		−	8677.25i		−	0.428362i	−0.976794	−	0.214181i	$-0.931292\pi$
$17$		−	8677.25i		−	0.428362i	0.976794	−	0.214181i	$-0.0687082\pi$
$18$		0			0
$19$	5340.48			0.178625			0.0893126	−	0.996004i	$-0.471533\pi$
$19$	5340.48			0.178625			0.0893126	+	0.996004i	$0.471533\pi$
$20$		0			0
$21$	61453.5			1.44804
$22$		0			0
$23$			74802.5i			1.28194i	0.767565	+	0.640971i	$0.221468\pi$
$23$			74802.5i			1.28194i	−0.767565	+	0.640971i	$0.778532\pi$
$24$		0			0
$25$	78012.6	−	4188.40i	0.998562	−	0.0536115i
$26$		0			0
$27$		−	51852.9i		−	0.506990i
$28$		0			0
$29$	−91625.0			−0.697624			−0.348812	−	0.937193i	$-0.613415\pi$
$29$	−91625.0			−0.697624			−0.348812	+	0.937193i	$0.613415\pi$
$30$		0			0
$31$	−103909.			−0.626450			−0.313225	−	0.949679i	$-0.601409\pi$
$31$	−103909.			−0.626450			−0.313225	+	0.949679i	$0.601409\pi$
$32$		0			0
$33$		−	390508.i		−	1.89161i
$34$		0			0
$35$	6449.67	+	240435.i	0.0254273	+	0.947892i
$36$		0			0
$37$			467238.i			1.51647i	0.651984	+	0.758233i	$0.273937\pi$
$37$			467238.i			1.51647i	−0.651984	+	0.758233i	$0.726063\pi$
$38$		0			0
$39$	−1.02436e6			−2.76520
$40$		0			0
$41$	385741.			0.874083			0.437042	−	0.899441i	$-0.356026\pi$
$41$	385741.			0.874083			0.437042	+	0.899441i	$0.356026\pi$
$42$		0			0
$43$		−	257098.i		−	0.493128i	−0.969126	−	0.246564i	$-0.920698\pi$
$43$		−	257098.i		−	0.493128i	0.969126	−	0.246564i	$-0.0793016\pi$
$44$		0			0
$45$	813938.	−	21833.9i	1.33152	−	0.0357181i
$46$		0			0
$47$		−	762177.i		−	1.07081i	−0.844595	−	0.535406i	$-0.820159\pi$
$47$		−	762177.i		−	1.07081i	0.844595	−	0.535406i	$-0.179841\pi$
$48$		0			0
$49$	83057.6			0.100854
$50$		0			0
$51$	619684.			0.654146
$52$		0			0
$53$			1.14032e6i			1.05211i	0.850450	+	0.526055i	$0.176330\pi$
$53$			1.14032e6i			1.05211i	−0.850450	+	0.526055i	$0.823670\pi$
$54$		0			0
$55$	1.52785e6	−	40984.6i	1.23826	−	0.0332163i
$56$		0			0
$57$			381389.i			0.272776i
$58$		0			0
$59$	−2.31663e6			−1.46850			−0.734252	−	0.678877i	$-0.762467\pi$
$59$	−2.31663e6			−1.46850			−0.734252	+	0.678877i	$0.762467\pi$
$60$		0			0
$61$	1.89127e6			1.06684			0.533418	−	0.845852i	$-0.320907\pi$
$61$	1.89127e6			1.06684			0.533418	+	0.845852i	$0.320907\pi$
$62$		0			0
$63$			2.50675e6i			1.26304i
$64$		0			0
$65$	−107509.	−	4.00777e6i	−0.0485564	−	1.81011i
$66$		0			0
$67$		−	1.89718e6i		−	0.770630i	−0.922785	−	0.385315i	$-0.874093\pi$
$67$		−	1.89718e6i		−	0.770630i	0.922785	−	0.385315i	$-0.125907\pi$
$68$		0			0
$69$	−5.34201e6			−1.95764
$70$		0			0
$71$	−2.84022e6			−0.941776			−0.470888	−	0.882193i	$-0.656066\pi$
$71$	−2.84022e6			−0.941776			−0.470888	+	0.882193i	$0.656066\pi$
$72$		0			0
$73$		−	2.59537e6i		−	0.780854i	−0.920634	−	0.390427i	$-0.872327\pi$
$73$		−	2.59537e6i		−	0.780854i	0.920634	−	0.390427i	$-0.127673\pi$
$74$		0			0
$75$	299114.	+	5.57126e6i	0.0818695	+	1.52489i
$76$		0			0
$77$			4.70543e6i			1.17458i
$78$		0			0
$79$	5.49695e6			1.25437			0.627187	−	0.778869i	$-0.284206\pi$
$79$	5.49695e6			1.25437			0.627187	+	0.778869i	$0.284206\pi$
$80$		0			0
$81$	−2.66784e6			−0.557779
$82$		0			0
$83$		−	3.79174e6i		−	0.727889i	−0.931421	−	0.363945i	$-0.881430\pi$
$83$		−	3.79174e6i		−	0.727889i	0.931421	−	0.363945i	$-0.118570\pi$
$84$		0			0
$85$	65037.1	+	2.42449e6i	0.0114867	+	0.428208i
$86$		0			0
$87$		−	6.54339e6i		−	1.06533i
$88$		0			0
$89$	1.95833e6			0.294456			0.147228	−	0.989103i	$-0.452965\pi$
$89$	1.95833e6			0.294456			0.147228	+	0.989103i	$0.452965\pi$
$90$		0			0
$91$	1.23430e7			1.71703
$92$		0			0
$93$		−	7.42063e6i		−	0.956644i
$94$		0			0
$95$	−1.49217e6	+	40027.6i	−0.178561	+	0.00478991i
$96$		0			0
$97$			1.19937e7i			1.33429i	0.744927	+	0.667146i	$0.232484\pi$
$97$			1.19937e7i			1.33429i	−0.744927	+	0.667146i	$0.767516\pi$
$98$		0			0
$99$	1.59292e7			1.64995
$100$		0			0
$101$	−6.86841e6			−0.663333			−0.331666	−	0.943397i	$-0.607611\pi$
$101$	−6.86841e6			−0.663333			−0.331666	+	0.943397i	$0.607611\pi$
$102$		0			0
$103$			4.12684e6i			0.372124i	0.982538	+	0.186062i	$0.0595725\pi$
$103$			4.12684e6i			0.372124i	−0.982538	+	0.186062i	$0.940427\pi$
$104$		0			0
$105$	−1.71706e7	+	460602.i	−1.44751	+	0.0388297i
$106$		0			0
$107$			3.40370e6i			0.268601i	0.990941	+	0.134301i	$0.0428788\pi$
$107$			3.40370e6i			0.268601i	−0.990941	+	0.134301i	$0.957121\pi$
$108$		0			0
$109$	1.28461e7			0.950123			0.475062	−	0.879953i	$-0.342426\pi$
$109$	1.28461e7			0.950123			0.475062	+	0.879953i	$0.342426\pi$
$110$		0			0
$111$	−3.33678e7			−2.31578
$112$		0			0
$113$			1.61131e7i			1.05052i	0.850942	+	0.525260i	$0.176032\pi$
$113$			1.61131e7i			1.05052i	−0.850942	+	0.525260i	$0.823968\pi$
$114$		0			0
$115$	−560655.	−	2.09004e7i	−0.0343758	−	1.28148i
$116$		0			0
$117$		−	4.17846e7i		−	2.41193i
$118$		0			0
$119$	−7.46690e6			−0.406187
$120$		0			0
$121$	1.04136e7			0.534382
$122$		0			0
$123$			2.75477e7i			1.33480i
$124$		0			0
$125$	−2.17660e7	+	1.75499e6i	−0.996765	+	0.0803691i
$126$		0			0
$127$			2.17289e7i			0.941295i	0.882321	+	0.470647i	$0.155980\pi$
$127$			2.17289e7i			0.941295i	−0.882321	+	0.470647i	$0.844020\pi$
$128$		0			0
$129$	1.83606e7			0.753050
$130$		0			0
$131$	−4.22973e6			−0.164385			−0.0821927	−	0.996616i	$-0.526192\pi$
$131$	−4.22973e6			−0.164385			−0.0821927	+	0.996616i	$0.526192\pi$
$132$		0			0
$133$		−	4.59556e6i		−	0.169378i
$134$		0			0
$135$	388644.	+	1.44881e7i	0.0135952	+	0.506808i
$136$		0			0
$137$			1.51787e7i			0.504327i	0.967685	+	0.252163i	$0.0811421\pi$
$137$			1.51787e7i			0.504327i	−0.967685	+	0.252163i	$0.918858\pi$
$138$		0			0
$139$	3.87908e7			1.22512			0.612558	−	0.790426i	$-0.290141\pi$
$139$	3.87908e7			1.22512			0.612558	+	0.790426i	$0.290141\pi$
$140$		0			0
$141$	5.44307e7			1.63522
$142$		0			0
$143$		−	7.84340e7i		−	2.24300i
$144$		0			0
$145$	2.56008e7	−	686742.i	0.697373	−	0.0187071i
$146$		0			0
$147$			5.93154e6i			0.154013i
$148$		0			0
$149$	−1.06288e6			−0.0263228			−0.0131614	−	0.999913i	$-0.504190\pi$
$149$	−1.06288e6			−0.0263228			−0.0131614	+	0.999913i	$0.504190\pi$
$150$		0			0
$151$	−7.03412e7			−1.66261			−0.831305	−	0.555816i	$-0.812406\pi$
$151$	−7.03412e7			−1.66261			−0.831305	+	0.555816i	$0.812406\pi$
$152$		0			0
$153$			2.52775e7i			0.570577i
$154$		0			0
$155$	2.90329e7	−	778810.i	0.626225	−	0.0167985i
$156$		0			0
$157$		−	886457.i		−	0.0182814i	−0.999958	−	0.00914068i	$-0.997090\pi$
$157$		−	886457.i		−	0.0182814i	0.999958	−	0.00914068i	$-0.00290961\pi$
$158$		0			0
$159$	−8.14358e7			−1.60667
$160$		0			0
$161$	6.43686e7			1.21558
$162$		0			0
$163$		−	4.82444e7i		−	0.872549i	−0.899814	−	0.436275i	$-0.856298\pi$
$163$		−	4.82444e7i		−	0.872549i	0.899814	−	0.436275i	$-0.143702\pi$
$164$		0			0
$165$	2.92691e6	+	1.09111e8i	0.0507242	+	1.89093i
$166$		0			0
$167$			6.45878e7i			1.07311i	0.843867	+	0.536553i	$0.180274\pi$
$167$			6.45878e7i			1.07311i	−0.843867	+	0.536553i	$0.819726\pi$
$168$		0			0
$169$	−1.42995e8			−2.27886
$170$		0			0
$171$	−1.55572e7			−0.237928
$172$		0			0
$173$		−	1.46835e7i		−	0.215610i	−0.994172	−	0.107805i	$-0.965618\pi$
$173$		−	1.46835e7i		−	0.215610i	0.994172	−	0.107805i	$-0.0343823\pi$
$174$		0			0
$175$	−3.60418e6	−	6.71310e7i	−0.0508362	−	0.946869i
$176$		0			0
$177$		−	1.65442e8i		−	2.24253i
$178$		0			0
$179$	4.60914e6			0.0600667			0.0300334	−	0.999549i	$-0.490439\pi$
$179$	4.60914e6			0.0600667			0.0300334	+	0.999549i	$0.490439\pi$
$180$		0			0
$181$	−1.97056e6			−0.0247011			−0.0123505	−	0.999924i	$-0.503931\pi$
$181$	−1.97056e6			−0.0247011			−0.0123505	+	0.999924i	$0.503931\pi$
$182$		0			0
$183$			1.35064e8i			1.62915i
$184$		0			0
$185$	−3.50202e6	−	1.30550e8i	−0.0406646	−	1.51592i
$186$		0			0
$187$			4.74486e7i			0.530612i
$188$		0			0
$189$	−4.46202e7			−0.480745
$190$		0			0
$191$	4.09793e7			0.425547			0.212774	−	0.977102i	$-0.431750\pi$
$191$	4.09793e7			0.425547			0.212774	+	0.977102i	$0.431750\pi$
$192$		0			0
$193$			8.19388e7i			0.820425i	0.911990	+	0.410213i	$0.134546\pi$
$193$			8.19388e7i			0.820425i	−0.911990	+	0.410213i	$0.865454\pi$
$194$		0			0
$195$	2.86214e8	−	7.67770e6i	2.76420	−	0.0741498i
$196$		0			0
$197$			1.46698e8i			1.36708i	0.729914	+	0.683539i	$0.239560\pi$
$197$			1.46698e8i			1.36708i	−0.729914	+	0.683539i	$0.760440\pi$
$198$		0			0
$199$	−1.53885e7			−0.138423			−0.0692116	−	0.997602i	$-0.522048\pi$
$199$	−1.53885e7			−0.138423			−0.0692116	+	0.997602i	$0.522048\pi$
$200$		0			0
$201$	1.35486e8			1.17682
$202$		0			0
$203$			7.88447e7i			0.661510i
$204$		0			0
$205$	−1.07779e8	+	2.89118e6i	−0.873769	+	0.0234389i
$206$		0			0
$207$		−	2.17906e8i		−	1.70755i
$208$		0			0
$209$	−2.92026e7			−0.221263
$210$		0			0
$211$	−4.37621e7			−0.320708			−0.160354	−	0.987060i	$-0.551264\pi$
$211$	−4.37621e7			−0.320708			−0.160354	+	0.987060i	$0.551264\pi$
$212$		0			0
$213$		−	2.02834e8i		−	1.43818i
$214$		0			0
$215$	1.92699e6	+	7.18354e7i	0.0132234	+	0.492951i
$216$		0			0
$217$			8.94150e7i			0.594021i
$218$		0			0
$219$	1.85348e8			1.19243
$220$		0			0
$221$	1.24464e8			0.775662
$222$		0			0
$223$		−	4.33168e7i		−	0.261571i	−0.991411	−	0.130785i	$-0.958250\pi$
$223$		−	4.33168e7i		−	0.261571i	0.991411	−	0.130785i	$-0.0417499\pi$
$224$		0			0
$225$	−2.27257e8	+	1.22011e7i	−1.33008	+	0.0714105i
$226$		0			0
$227$		−	1.64636e8i		−	0.934186i	−0.884208	−	0.467093i	$-0.845301\pi$
$227$		−	1.64636e8i		−	0.934186i	0.884208	−	0.467093i	$-0.154699\pi$
$228$		0			0
$229$	−1.98853e8			−1.09423			−0.547114	−	0.837058i	$-0.684273\pi$
$229$	−1.98853e8			−1.09423			−0.547114	+	0.837058i	$0.684273\pi$
$230$		0			0
$231$	−3.36038e8			−1.79368
$232$		0			0
$233$			6.39144e7i			0.331019i	0.986208	+	0.165510i	$0.0529269\pi$
$233$			6.39144e7i			0.331019i	−0.986208	+	0.165510i	$0.947073\pi$
$234$		0			0
$235$	5.71262e6	+	2.12958e8i	0.0287143	+	1.07043i
$236$		0			0
$237$			3.92564e8i			1.91554i
$238$		0			0
$239$	6.82882e7			0.323559			0.161779	−	0.986827i	$-0.448277\pi$
$239$	6.82882e7			0.323559			0.161779	+	0.986827i	$0.448277\pi$
$240$		0			0
$241$	−2.19558e8			−1.01039			−0.505195	−	0.863005i	$-0.668579\pi$
$241$	−2.19558e8			−1.01039			−0.505195	+	0.863005i	$0.668579\pi$
$242$		0			0
$243$		−	3.03926e8i		−	1.35877i
$244$		0			0
$245$	−2.32069e7	+	622528.i	−0.100818	+	0.00270444i
$246$		0			0
$247$			7.66026e7i			0.323448i
$248$		0			0
$249$	2.70786e8			1.11155
$250$		0			0
$251$	−4.20435e7			−0.167819			−0.0839094	−	0.996473i	$-0.526741\pi$
$251$	−4.20435e7			−0.167819			−0.0839094	+	0.996473i	$0.526741\pi$
$252$		0			0
$253$		−	4.09032e8i		−	1.58794i
$254$		0			0
$255$	−1.73145e8	+	4.64462e6i	−0.653911	+	0.0175412i
$256$		0			0
$257$		−	2.47945e8i		−	0.911151i	−0.890197	−	0.455575i	$-0.849434\pi$
$257$		−	2.47945e8i		−	0.911151i	0.890197	−	0.455575i	$-0.150566\pi$
$258$		0			0
$259$	4.02066e8			1.43796
$260$		0			0
$261$	2.66911e8			0.929233
$262$		0			0
$263$			2.95042e8i			1.00009i	0.866000	+	0.500044i	$0.166683\pi$
$263$			2.95042e8i			1.00009i	−0.866000	+	0.500044i	$0.833317\pi$
$264$		0			0
$265$	−8.54686e6	−	3.18615e8i	−0.0282128	−	1.05173i
$266$		0			0
$267$			1.39854e8i			0.449661i
$268$		0			0
$269$	3.39709e8			1.06408			0.532040	−	0.846719i	$-0.321426\pi$
$269$	3.39709e8			1.06408			0.532040	+	0.846719i	$0.321426\pi$
$270$		0			0
$271$	−8.43577e7			−0.257473			−0.128737	−	0.991679i	$-0.541092\pi$
$271$	−8.43577e7			−0.257473			−0.128737	+	0.991679i	$0.541092\pi$
$272$		0			0
$273$			8.81475e8i			2.62205i
$274$		0			0
$275$	−4.26586e8	+	2.29028e7i	−1.23692	+	0.0664087i
$276$		0			0
$277$		−	3.20428e8i		−	0.905840i	−0.891551	−	0.452920i	$-0.850382\pi$
$277$		−	3.20428e8i		−	0.905840i	0.891551	−	0.452920i	$-0.149618\pi$
$278$		0			0
$279$	3.02695e8			0.834430
$280$		0			0
$281$	5.84239e8			1.57079			0.785395	−	0.618994i	$-0.212460\pi$
$281$	5.84239e8			1.57079			0.785395	+	0.618994i	$0.212460\pi$
$282$		0			0
$283$			4.39110e8i			1.15165i	0.817573	+	0.575826i	$0.195319\pi$
$283$			4.39110e8i			1.15165i	−0.817573	+	0.575826i	$0.804681\pi$
$284$		0			0
$285$	−2.85857e6	−	1.06563e8i	−0.00731461	−	0.272678i
$286$		0			0
$287$		−	3.31936e8i		−	0.828835i
$288$		0			0
$289$	3.35044e8			0.816506
$290$		0			0
$291$	−8.56526e8			−2.03758
$292$		0			0
$293$		−	5.26225e8i		−	1.22218i	−0.791561	−	0.611090i	$-0.790731\pi$
$293$		−	5.26225e8i		−	1.22218i	0.791561	−	0.611090i	$-0.209269\pi$
$294$		0			0
$295$	6.47286e8	−	1.73635e7i	1.46798	−	0.0393785i
$296$		0			0
$297$			2.83540e8i			0.628010i
$298$		0			0
$299$	−1.07295e9			−2.32129
$300$		0			0
$301$	−2.21237e8			−0.467601
$302$		0			0
$303$		−	4.90507e8i		−	1.01297i
$304$		0			0
$305$	−5.28435e8	+	1.41753e7i	−1.06645	+	0.0286077i
$306$		0			0
$307$		−	2.65685e8i		−	0.524062i	−0.965059	−	0.262031i	$-0.915608\pi$
$307$		−	2.65685e8i		−	0.524062i	0.965059	−	0.262031i	$-0.0843922\pi$
$308$		0			0
$309$	−2.94718e8			−0.568266
$310$		0			0
$311$	−2.87733e7			−0.0542410			−0.0271205	−	0.999632i	$-0.508634\pi$
$311$	−2.87733e7			−0.0542410			−0.0271205	+	0.999632i	$0.508634\pi$
$312$		0			0
$313$			6.64576e8i			1.22501i	0.790467	+	0.612505i	$0.209838\pi$
$313$			6.64576e8i			1.22501i	−0.790467	+	0.612505i	$0.790162\pi$
$314$		0			0
$315$	−1.87884e7	−	7.00405e8i	−0.0338691	−	1.26259i
$316$		0			0
$317$			6.01423e8i			1.06041i	0.847871	+	0.530203i	$0.177884\pi$
$317$			6.01423e8i			1.06041i	−0.847871	+	0.530203i	$0.822116\pi$
$318$		0			0
$319$	5.01020e8			0.864148
$320$		0			0
$321$	−2.43075e8			−0.410178
$322$		0			0
$323$		−	4.63407e7i		−	0.0765162i
$324$		0			0
$325$	6.00775e7	+	1.11900e9i	0.0970778	+	1.80816i
$326$		0			0
$327$			9.17405e8i			1.45092i
$328$		0			0
$329$	−6.55864e8			−1.01538
$330$		0			0
$331$	1.07797e9			1.63383			0.816916	−	0.576757i	$-0.195682\pi$
$331$	1.07797e9			1.63383			0.816916	+	0.576757i	$0.195682\pi$
$332$		0			0
$333$		−	1.36110e9i		−	2.01993i
$334$		0			0
$335$	1.42196e7	+	5.30086e8i	0.0206647	+	0.770352i
$336$		0			0
$337$		−	1.17944e9i		−	1.67869i	−0.543602	−	0.839343i	$-0.682940\pi$
$337$		−	1.17944e9i		−	1.67869i	0.543602	−	0.839343i	$-0.317060\pi$
$338$		0			0
$339$	−1.15071e9			−1.60424
$340$		0			0
$341$	5.68190e8			0.775985
$342$		0			0
$343$		−	7.80143e8i		−	1.04387i
$344$		0			0
$345$	1.49260e9	−	4.00391e7i	1.95694	−	0.0524949i
$346$		0			0
$347$			3.42964e8i			0.440651i	0.975426	+	0.220326i	$0.0707120\pi$
$347$			3.42964e8i			0.440651i	−0.975426	+	0.220326i	$0.929288\pi$
$348$		0			0
$349$	8.51245e7			0.107193			0.0535964	−	0.998563i	$-0.482932\pi$
$349$	8.51245e7			0.107193			0.0535964	+	0.998563i	$0.482932\pi$
$350$		0			0
$351$	7.43766e8			0.918040
$352$		0			0
$353$		−	7.46737e7i		−	0.0903558i	−0.998979	−	0.0451779i	$-0.985615\pi$
$353$		−	7.46737e7i		−	0.0903558i	0.998979	−	0.0451779i	$-0.0143855\pi$
$354$		0			0
$355$	7.93580e8	−	2.12878e7i	0.941438	−	0.0252541i
$356$		0			0
$357$		−	5.33247e8i		−	0.620283i
$358$		0			0
$359$	6.59926e8			0.752774			0.376387	−	0.926463i	$-0.377166\pi$
$359$	6.59926e8			0.752774			0.376387	+	0.926463i	$0.377166\pi$
$360$		0			0
$361$	−8.65351e8			−0.968093
$362$		0			0
$363$			7.43686e8i			0.816049i
$364$		0			0
$365$	1.94527e7	+	7.25168e8i	0.0209389	+	0.780573i
$366$		0			0
$367$			1.14445e9i			1.20855i	0.796774	+	0.604277i	$0.206538\pi$
$367$			1.14445e9i			1.20855i	−0.796774	+	0.604277i	$0.793462\pi$
$368$		0			0
$369$	−1.12370e9			−1.16428
$370$		0			0
$371$	9.81263e8			0.997646
$372$		0			0
$373$			1.92846e9i			1.92411i	0.272861	+	0.962053i	$0.412030\pi$
$373$			1.92846e9i			1.92411i	−0.272861	+	0.962053i	$0.587970\pi$
$374$		0			0
$375$	−1.25332e8	−	1.55441e9i	−0.122731	−	1.52215i
$376$		0			0
$377$		−	1.31425e9i		−	1.26323i
$378$		0			0
$379$	−4.70589e8			−0.444022			−0.222011	−	0.975044i	$-0.571262\pi$
$379$	−4.70589e8			−0.444022			−0.222011	+	0.975044i	$0.571262\pi$
$380$		0			0
$381$	−1.55177e9			−1.43744
$382$		0			0
$383$			4.05122e8i			0.368460i	0.982883	+	0.184230i	$0.0589791\pi$
$383$			4.05122e8i			0.368460i	−0.982883	+	0.184230i	$0.941021\pi$
$384$		0			0
$385$	−3.52678e7	−	1.31474e9i	−0.0314968	−	1.17416i
$386$		0			0
$387$			7.48948e8i			0.656846i
$388$		0			0
$389$	−1.58753e9			−1.36741			−0.683706	−	0.729758i	$-0.739633\pi$
$389$	−1.58753e9			−1.36741			−0.683706	+	0.729758i	$0.739633\pi$
$390$		0			0
$391$	6.49080e8			0.549135
$392$		0			0
$393$		−	3.02065e8i		−	0.251031i
$394$		0			0
$395$	−1.53589e9	+	4.12004e7i	−1.25392	+	0.0336366i
$396$		0			0
$397$			3.89890e7i			0.0312734i	0.999878	+	0.0156367i	$0.00497752\pi$
$397$			3.89890e7i			0.0312734i	−0.999878	+	0.0156367i	$0.995022\pi$
$398$		0			0
$399$	3.28191e8			0.258656
$400$		0			0
$401$	7.43635e8			0.575910			0.287955	−	0.957644i	$-0.407025\pi$
$401$	7.43635e8			0.575910			0.287955	+	0.957644i	$0.407025\pi$
$402$		0			0
$403$		−	1.49044e9i		−	1.13435i
$404$		0			0
$405$	7.45416e8	−	1.99958e7i	0.557579	−	0.0149571i
$406$		0			0
$407$		−	2.55493e9i		−	1.87845i
$408$		0			0
$409$	−5.04931e7			−0.0364922			−0.0182461	−	0.999834i	$-0.505808\pi$
$409$	−5.04931e7			−0.0364922			−0.0182461	+	0.999834i	$0.505808\pi$
$410$		0			0
$411$	−1.08398e9			−0.770152
$412$		0			0
$413$			1.99350e9i			1.39248i
$414$		0			0
$415$	2.84196e7	+	1.05944e9i	0.0195186	+	0.727627i
$416$		0			0
$417$			2.77024e9i			1.87086i
$418$		0			0
$419$	−9.65771e8			−0.641394			−0.320697	−	0.947182i	$-0.603917\pi$
$419$	−9.65771e8			−0.641394			−0.320697	+	0.947182i	$0.603917\pi$
$420$		0			0
$421$	−2.22814e9			−1.45531			−0.727655	−	0.685943i	$-0.759390\pi$
$421$	−2.22814e9			−1.45531			−0.727655	+	0.685943i	$0.759390\pi$
$422$		0			0
$423$			2.22028e9i			1.42632i
$424$		0			0
$425$	−3.63438e7	−	6.76935e8i	−0.0229651	−	0.427746i
$426$		0			0
$427$		−	1.62746e9i		−	1.01161i
$428$		0			0
$429$	5.60136e9			3.42525
$430$		0			0
$431$	2.66758e9			1.60490			0.802449	−	0.596721i	$-0.203530\pi$
$431$	2.66758e9			1.60490			0.802449	+	0.596721i	$0.203530\pi$
$432$		0			0
$433$		−	1.53592e9i		−	0.909201i	−0.890695	−	0.454601i	$-0.849782\pi$
$433$		−	1.53592e9i		−	0.909201i	0.890695	−	0.454601i	$-0.150218\pi$
$434$		0			0
$435$	4.90436e7	+	1.82827e9i	0.0285673	+	1.06495i
$436$		0			0
$437$			3.99481e8i			0.228987i
$438$		0			0
$439$	1.31907e9			0.744119			0.372060	−	0.928209i	$-0.378652\pi$
$439$	1.31907e9			0.744119			0.372060	+	0.928209i	$0.378652\pi$
$440$		0			0
$441$	−2.41953e8			−0.134337
$442$		0			0
$443$		−	2.85200e9i		−	1.55860i	−0.626649	−	0.779302i	$-0.715574\pi$
$443$		−	2.85200e9i		−	1.55860i	0.626649	−	0.779302i	$-0.284426\pi$
$444$		0			0
$445$	−5.47173e8	+	1.46779e7i	−0.294350	+	0.00789597i
$446$		0			0
$447$		−	7.59054e7i		−	0.0401973i
$448$		0			0
$449$	3.27931e9			1.70970			0.854851	−	0.518874i	$-0.173649\pi$
$449$	3.27931e9			1.70970			0.854851	+	0.518874i	$0.173649\pi$
$450$		0			0
$451$	−2.10930e9			−1.08273
$452$		0			0
$453$		−	5.02341e9i		−	2.53895i
$454$		0			0
$455$	−3.44874e9	+	9.25127e7i	−1.71641	+	0.0460428i
$456$		0			0
$457$			1.70952e9i			0.837852i	0.908020	+	0.418926i	$0.137593\pi$
$457$			1.70952e9i			0.837852i	−0.908020	+	0.418926i	$0.862407\pi$
$458$		0			0
$459$	−4.49940e8			−0.217175
$460$		0			0
$461$	3.36612e9			1.60021			0.800105	−	0.599861i	$-0.204777\pi$
$461$	3.36612e9			1.60021			0.800105	+	0.599861i	$0.204777\pi$
$462$		0			0
$463$		−	3.88893e7i		−	0.0182095i	−0.999959	−	0.00910473i	$-0.997102\pi$
$463$		−	3.88893e7i		−	0.0182095i	0.999959	−	0.00910473i	$-0.00289817\pi$
$464$		0			0
$465$	5.56186e7	+	2.07338e9i	0.0256528	+	0.956300i
$466$		0			0
$467$			4.12767e9i			1.87541i	0.347436	+	0.937704i	$0.387052\pi$
$467$			4.12767e9i			1.87541i	−0.347436	+	0.937704i	$0.612948\pi$
$468$		0			0
$469$	−1.63255e9			−0.730736
$470$		0			0
$471$	6.33062e7			0.0279173
$472$		0			0
$473$			1.40586e9i			0.610839i
$474$		0			0
$475$	4.16625e8	−	2.23681e7i	0.178368	−	0.00957637i
$476$		0			0
$477$		−	3.32185e9i		−	1.40141i
$478$		0			0
$479$	−1.52474e9			−0.633900			−0.316950	−	0.948442i	$-0.602659\pi$
$479$	−1.52474e9			−0.633900			−0.316950	+	0.948442i	$0.602659\pi$
$480$		0			0
$481$	−6.70196e9			−2.74596
$482$		0			0
$483$			4.59688e9i			1.85630i
$484$		0			0
$485$	−8.98942e7	−	3.35113e9i	−0.0357796	−	1.33381i
$486$		0			0
$487$		−	1.13272e9i		−	0.444399i	−0.975001	−	0.222199i	$-0.928676\pi$
$487$		−	1.13272e9i		−	0.444399i	0.975001	−	0.222199i	$-0.0713236\pi$
$488$		0			0
$489$	3.44536e9			1.33246
$490$		0			0
$491$	2.26065e9			0.861884			0.430942	−	0.902380i	$-0.358181\pi$
$491$	2.26065e9			0.861884			0.430942	+	0.902380i	$0.358181\pi$
$492$		0			0
$493$			7.95053e8i			0.298835i
$494$		0			0
$495$	−4.45074e9	+	1.19391e8i	−1.64936	+	0.0442440i
$496$		0			0
$497$			2.44405e9i			0.893024i
$498$		0			0
$499$	4.60721e8			0.165992			0.0829958	−	0.996550i	$-0.473551\pi$
$499$	4.60721e8			0.165992			0.0829958	+	0.996550i	$0.473551\pi$
$500$		0			0
$501$	−4.61252e9			−1.63873
$502$		0			0
$503$		−	8.49112e8i		−	0.297493i	−0.988875	−	0.148747i	$-0.952476\pi$
$503$		−	8.49112e8i		−	0.297493i	0.988875	−	0.148747i	$-0.0475239\pi$
$504$		0			0
$505$	1.91909e9	−	5.14797e7i	0.663094	−	0.0177875i
$506$		0			0
$507$		−	1.02120e10i		−	3.48003i
$508$		0			0
$509$	−8.54063e8			−0.287063			−0.143532	−	0.989646i	$-0.545846\pi$
$509$	−8.54063e8			−0.287063			−0.143532	+	0.989646i	$0.545846\pi$
$510$		0			0
$511$	−2.23336e9			−0.740432
$512$		0			0
$513$		−	2.76919e8i		−	0.0905612i
$514$		0			0
$515$	−3.09312e7	−	1.15307e9i	−0.00997866	−	0.371990i
$516$		0			0
$517$			4.16771e9i			1.32642i
$518$		0			0
$519$	1.04862e9			0.329256
$520$		0			0
$521$	5.64909e9			1.75003			0.875016	−	0.484093i	$-0.160851\pi$
$521$	5.64909e9			1.75003			0.875016	+	0.484093i	$0.160851\pi$
$522$		0			0
$523$			3.30599e9i			1.01052i	0.862967	+	0.505260i	$0.168604\pi$
$523$			3.30599e9i			1.01052i	−0.862967	+	0.505260i	$0.831396\pi$
$524$		0			0
$525$	4.79415e9	−	2.57392e8i	1.44595	−	0.0776314i
$526$		0			0
$527$			9.01642e8i			0.268347i
$528$		0			0
$529$	−2.19059e9			−0.643377
$530$		0			0
$531$	6.74854e9			1.95604
$532$		0			0
$533$			5.53299e9i			1.58276i
$534$		0			0
$535$	−2.55112e7	−	9.51021e8i	−0.00720266	−	0.268505i
$536$		0			0
$537$			3.29161e8i			0.0917272i
$538$		0			0
$539$	−4.54172e8			−0.124928
$540$		0			0
$541$	2.97074e9			0.806629			0.403314	−	0.915061i	$-0.367858\pi$
$541$	2.97074e9			0.806629			0.403314	+	0.915061i	$0.367858\pi$
$542$		0			0
$543$		−	1.40727e8i		−	0.0377207i
$544$		0			0
$545$	−3.58931e9	+	9.62835e7i	−0.949781	+	0.0254779i
$546$		0			0
$547$			4.68111e9i			1.22291i	0.791281	+	0.611453i	$0.209415\pi$
$547$			4.68111e9i			1.22291i	−0.791281	+	0.611453i	$0.790585\pi$
$548$		0			0
$549$	−5.50941e9			−1.42102
$550$		0			0
$551$	−4.89322e8			−0.124613
$552$		0			0
$553$		−	4.73021e9i		−	1.18944i
$554$		0			0
$555$	9.32322e9	−	2.50096e8i	2.31494	−	0.0620985i
$556$		0			0
$557$		−	3.64883e9i		−	0.894666i	−0.894367	−	0.447333i	$-0.852374\pi$
$557$		−	3.64883e9i		−	0.894666i	0.894367	−	0.447333i	$-0.147626\pi$
$558$		0			0
$559$	3.68776e9			0.892939
$560$		0			0
$561$	−3.38853e9			−0.810292
$562$		0			0
$563$		−	7.10106e9i		−	1.67704i	−0.544869	−	0.838521i	$-0.683421\pi$
$563$		−	7.10106e9i		−	1.67704i	0.544869	−	0.838521i	$-0.316579\pi$
$564$		0			0
$565$	−1.20770e8	−	4.50213e9i	−0.0281701	−	1.05014i
$566$		0			0
$567$			2.29572e9i			0.528905i
$568$		0			0
$569$	−4.60359e9			−1.04762			−0.523810	−	0.851835i	$-0.675490\pi$
$569$	−4.60359e9			−1.04762			−0.523810	+	0.851835i	$0.675490\pi$
$570$		0			0
$571$	4.37732e9			0.983971			0.491985	−	0.870603i	$-0.336271\pi$
$571$	4.37732e9			0.983971			0.491985	+	0.870603i	$0.336271\pi$
$572$		0			0
$573$			2.92653e9i			0.649848i
$574$		0			0
$575$	3.13303e8	+	5.83554e9i	0.0687269	+	1.28010i
$576$		0			0
$577$		−	5.10061e9i		−	1.10537i	−0.833391	−	0.552684i	$-0.813604\pi$
$577$		−	5.10061e9i		−	1.10537i	0.833391	−	0.552684i	$-0.186396\pi$
$578$		0			0
$579$	−5.85165e9			−1.25286
$580$		0			0
$581$	−3.26285e9			−0.690208
$582$		0			0
$583$		−	6.23546e9i		−	1.30325i
$584$		0			0
$585$	3.13181e8	+	1.16749e10i	0.0646770	+	2.41107i
$586$		0			0
$587$			7.99745e9i			1.63199i	0.578057	+	0.815996i	$0.303811\pi$
$587$			7.99745e9i			1.63199i	−0.578057	+	0.815996i	$0.696189\pi$
$588$		0			0
$589$	−5.54923e8			−0.111900
$590$		0			0
$591$	−1.04764e10			−2.08765
$592$		0			0
$593$		−	8.40837e9i		−	1.65585i	−0.560841	−	0.827924i	$-0.689522\pi$
$593$		−	8.40837e9i		−	1.65585i	0.560841	−	0.827924i	$-0.310478\pi$
$594$		0			0
$595$	2.08631e9	−	5.59654e7i	0.406041	−	0.0108921i
$596$		0			0
$597$		−	1.09896e9i		−	0.211385i
$598$		0			0
$599$	−7.34084e9			−1.39557			−0.697785	−	0.716307i	$-0.745831\pi$
$599$	−7.34084e9			−1.39557			−0.697785	+	0.716307i	$0.745831\pi$
$600$		0			0
$601$	2.46145e9			0.462520			0.231260	−	0.972892i	$-0.425715\pi$
$601$	2.46145e9			0.462520			0.231260	+	0.972892i	$0.425715\pi$
$602$		0			0
$603$			5.52662e9i			1.02648i
$604$		0			0
$605$	−2.90964e9	+	7.80513e7i	−0.534190	+	0.0143297i
$606$		0			0
$607$			3.52629e9i			0.639967i	0.947423	+	0.319984i	$0.103677\pi$
$607$			3.52629e9i			0.639967i	−0.947423	+	0.319984i	$0.896323\pi$
$608$		0			0
$609$	−5.63068e9			−1.01018
$610$		0			0
$611$	1.09325e10			1.93899
$612$		0			0
$613$			9.77585e9i			1.71413i	0.515211	+	0.857064i	$0.327714\pi$
$613$			9.77585e9i			1.71413i	−0.515211	+	0.857064i	$0.672286\pi$
$614$		0			0
$615$	−2.06473e8	−	7.69704e9i	−0.0357933	−	1.33432i
$616$		0			0
$617$		−	8.58671e9i		−	1.47173i	−0.677127	−	0.735866i	$-0.736775\pi$
$617$		−	8.58671e9i		−	1.47173i	0.677127	−	0.735866i	$-0.263225\pi$
$618$		0			0
$619$	−3.59570e9			−0.609349			−0.304674	−	0.952457i	$-0.598548\pi$
$619$	−3.59570e9			−0.609349			−0.304674	+	0.952457i	$0.598548\pi$
$620$		0			0
$621$	3.87872e9			0.649933
$622$		0			0
$623$		−	1.68517e9i		−	0.279213i
$624$		0			0
$625$	6.06843e9	−	6.53497e8i	0.994252	−	0.107069i
$626$		0			0
$627$		−	2.08550e9i		−	0.337889i
$628$		0			0
$629$	4.05434e9			0.649596
$630$		0			0
$631$	−1.07303e9			−0.170024			−0.0850121	−	0.996380i	$-0.527093\pi$
$631$	−1.07303e9			−0.170024			−0.0850121	+	0.996380i	$0.527093\pi$
$632$		0			0
$633$		−	3.12526e9i		−	0.489749i
$634$		0			0
$635$	−1.62861e8	−	6.07124e9i	−0.0252412	−	0.940956i
$636$		0			0
$637$			1.19136e9i			0.182623i
$638$		0			0
$639$	8.27379e9			1.25444
$640$		0			0
$641$	6.55897e8			0.0983632			0.0491816	−	0.998790i	$-0.484339\pi$
$641$	6.55897e8			0.0983632			0.0491816	+	0.998790i	$0.484339\pi$
$642$		0			0
$643$		−	2.80837e9i		−	0.416596i	−0.978065	−	0.208298i	$-0.933208\pi$
$643$		−	2.80837e9i		−	0.416596i	0.978065	−	0.208298i	$-0.0667925\pi$
$644$		0			0
$645$	−5.13011e9	+	1.37616e8i	−0.752779	+	0.0201934i
$646$		0			0
$647$			3.26212e9i			0.473516i	0.971569	+	0.236758i	$0.0760848\pi$
$647$			3.26212e9i			0.473516i	−0.971569	+	0.236758i	$0.923915\pi$
$648$		0			0
$649$	1.26677e10			1.81904
$650$		0			0
$651$	−6.38556e9			−0.907122
$652$		0			0
$653$			1.08084e10i			1.51902i	0.650494	+	0.759511i	$0.274562\pi$
$653$			1.08084e10i			1.51902i	−0.650494	+	0.759511i	$0.725438\pi$
$654$		0			0
$655$	1.18182e9	−	3.17024e7i	0.164326	−	0.00440806i
$656$		0			0
$657$			7.56053e9i			1.04010i
$658$		0			0
$659$	3.98310e9			0.542153			0.271077	−	0.962558i	$-0.412620\pi$
$659$	3.98310e9			0.542153			0.271077	+	0.962558i	$0.412620\pi$
$660$		0			0
$661$	4.29255e9			0.578109			0.289055	−	0.957313i	$-0.406659\pi$
$661$	4.29255e9			0.578109			0.289055	+	0.957313i	$0.406659\pi$
$662$		0			0
$663$			8.88861e9i			1.18450i
$664$		0			0
$665$	3.44443e7	+	1.28404e9i	0.00454195	+	0.169317i
$666$		0			0
$667$		−	6.85378e9i		−	0.894314i
$668$		0			0
$669$	3.09346e9			0.399442
$670$		0			0
$671$	−1.03417e10			−1.32149
$672$		0			0
$673$			3.87778e9i			0.490377i	0.969475	+	0.245189i	$0.0788499\pi$
$673$			3.87778e9i			0.490377i	−0.969475	+	0.245189i	$0.921150\pi$
$674$		0			0
$675$	−2.17181e8	−	4.04518e9i	−0.0271805	−	0.506261i
$676$		0			0
$677$		−	7.11448e8i		−	0.0881217i	−0.999029	−	0.0440609i	$-0.985970\pi$
$677$		−	7.11448e8i		−	0.0881217i	0.999029	−	0.0440609i	$-0.0140295\pi$
$678$		0			0
$679$	1.03207e10			1.26522
$680$		0			0
$681$	1.17574e10			1.42658
$682$		0			0
$683$			5.88562e9i			0.706838i	0.935465	+	0.353419i	$0.114981\pi$
$683$			5.88562e9i			0.706838i	−0.935465	+	0.353419i	$0.885019\pi$
$684$		0			0
$685$	−1.13766e8	−	4.24105e9i	−0.0135237	−	0.504146i
$686$		0			0
$687$		−	1.42010e10i		−	1.67098i
$688$		0			0
$689$	−1.63565e10			−1.90512
$690$		0			0
$691$	−1.37685e10			−1.58750			−0.793750	−	0.608244i	$-0.791874\pi$
$691$	−1.37685e10			−1.58750			−0.793750	+	0.608244i	$0.791874\pi$
$692$		0			0
$693$		−	1.37073e10i		−	1.56454i
$694$		0			0
$695$	−1.08385e10	+	2.90742e8i	−1.22467	+	0.0328520i
$696$		0			0
$697$		−	3.34717e9i		−	0.374424i
$698$		0			0
$699$	−4.56444e9			−0.505495
$700$		0			0
$701$	4.72222e9			0.517766			0.258883	−	0.965909i	$-0.416646\pi$
$701$	4.72222e9			0.517766			0.258883	+	0.965909i	$0.416646\pi$
$702$		0			0
$703$			2.49528e9i			0.270879i
$704$		0			0
$705$	−1.52084e10	+	4.07966e8i	−1.63464	+	0.0438492i
$706$		0			0
$707$			5.91037e9i			0.628994i
$708$		0			0
$709$	7.89205e9			0.831626			0.415813	−	0.909450i	$-0.363497\pi$
$709$	7.89205e9			0.831626			0.415813	+	0.909450i	$0.363497\pi$
$710$		0			0
$711$	−1.60131e10			−1.67082
$712$		0			0
$713$		−	7.77264e9i		−	0.803073i
$714$		0			0
$715$	5.87874e8	+	2.19151e10i	0.0601469	+	2.24219i
$716$		0			0
$717$			4.87679e9i			0.494102i
$718$		0			0
$719$	1.46373e10			1.46862			0.734311	−	0.678813i	$-0.237505\pi$
$719$	1.46373e10			1.46862			0.734311	+	0.678813i	$0.237505\pi$
$720$		0			0
$721$	3.55121e9			0.352860
$722$		0			0
$723$		−	1.56797e10i		−	1.54295i
$724$		0			0
$725$	−7.14791e9	+	3.83762e8i	−0.696620	+	0.0374007i
$726$		0			0
$727$			1.36906e10i			1.32146i	0.750624	+	0.660729i	$0.229753\pi$
$727$			1.36906e10i			1.32146i	−0.750624	+	0.660729i	$0.770247\pi$
$728$		0			0
$729$	1.58702e10			1.51718
$730$		0			0
$731$	−2.23091e9			−0.211237
$732$		0			0
$733$		−	7.39509e9i		−	0.693553i	−0.937948	−	0.346776i	$-0.887276\pi$
$733$		−	7.39509e9i		−	0.693553i	0.937948	−	0.346776i	$-0.112724\pi$
$734$		0			0
$735$	−4.44577e7	−	1.65732e9i	−0.00412992	−	0.153958i
$736$		0			0
$737$			1.03741e10i			0.954580i
$738$		0			0
$739$	−1.61194e10			−1.46924			−0.734620	−	0.678479i	$-0.762639\pi$
$739$	−1.61194e10			−1.46924			−0.734620	+	0.678479i	$0.762639\pi$
$740$		0			0
$741$	−5.47056e9			−0.493933
$742$		0			0
$743$			4.86465e9i			0.435101i	0.976049	+	0.217551i	$0.0698067\pi$
$743$			4.86465e9i			0.435101i	−0.976049	+	0.217551i	$0.930193\pi$
$744$		0			0
$745$	2.96977e8	−	7.96643e6i	0.0263134	−	0.000705857i
$746$		0			0
$747$			1.10456e10i			0.969547i
$748$		0			0
$749$	2.92893e9			0.254697
$750$		0			0
$751$	−2.68210e9			−0.231065			−0.115533	−	0.993304i	$-0.536858\pi$
$751$	−2.68210e9			−0.231065			−0.115533	+	0.993304i	$0.536858\pi$
$752$		0			0
$753$		−	3.00253e9i		−	0.256274i
$754$		0			0
$755$	1.96539e10	−	5.27217e8i	1.66201	−	0.0445836i
$756$		0			0
$757$		−	1.25746e10i		−	1.05356i	−0.850002	−	0.526779i	$-0.823400\pi$
$757$		−	1.25746e10i		−	1.05356i	0.850002	−	0.526779i	$-0.176600\pi$
$758$		0			0
$759$	2.92110e10			2.42493
$760$		0			0
$761$	−8.02931e8			−0.0660438			−0.0330219	−	0.999455i	$-0.510513\pi$
$761$	−8.02931e8			−0.0660438			−0.0330219	+	0.999455i	$0.510513\pi$
$762$		0			0
$763$		−	1.10543e10i		−	0.900938i
$764$		0			0
$765$	−1.89458e8	−	7.06274e9i	−0.0153003	−	0.570372i
$766$		0			0
$767$		−	3.32293e10i		−	2.65911i
$768$		0			0
$769$	−9.44845e9			−0.749236			−0.374618	−	0.927179i	$-0.622226\pi$
$769$	−9.44845e9			−0.749236			−0.374618	+	0.927179i	$0.622226\pi$
$770$		0			0
$771$	1.77070e10			1.39141
$772$		0			0
$773$			8.28701e9i			0.645312i	0.946516	+	0.322656i	$0.104576\pi$
$773$			8.28701e9i			0.645312i	−0.946516	+	0.322656i	$0.895424\pi$
$774$		0			0
$775$	−8.10620e9	+	4.35212e8i	−0.625549	+	0.0335849i
$776$		0			0
$777$			2.87134e10i			2.19590i
$778$		0			0
$779$	2.06004e9			0.156133
$780$		0			0
$781$	1.55308e10			1.16658
$782$		0			0
$783$			4.75102e9i			0.353688i
$784$		0			0
$785$	6.64411e6	+	2.47683e8i	0.000490222	+	0.0182748i
$786$		0			0
$787$		−	1.78703e10i		−	1.30684i	−0.756997	−	0.653418i	$-0.773334\pi$
$787$		−	1.78703e10i		−	1.30684i	0.756997	−	0.653418i	$-0.226666\pi$
$788$		0			0
$789$	−2.10703e10			−1.52722
$790$		0			0
$791$	1.38656e10			0.996138
$792$		0			0
$793$			2.71279e10i			1.93179i
$794$		0			0
$795$	2.27538e10	−	6.10372e8i	1.60609	−	0.0430834i
$796$		0			0
$797$		−	9.71398e9i		−	0.679662i	−0.940486	−	0.339831i	$-0.889630\pi$
$797$		−	9.71398e9i		−	0.679662i	0.940486	−	0.339831i	$-0.110370\pi$
$798$		0			0
$799$	−6.61360e9			−0.458695
$800$		0			0
$801$	−5.70477e9			−0.392215
$802$		0			0
$803$			1.41919e10i			0.967246i
$804$		0			0
$805$	−1.79851e10	+	4.82452e8i	−1.21514	+	0.0325963i
$806$		0			0
$807$			2.42603e10i			1.62494i
$808$		0			0
$809$	7.81490e9			0.518924			0.259462	−	0.965753i	$-0.416455\pi$
$809$	7.81490e9			0.518924			0.259462	+	0.965753i	$0.416455\pi$
$810$		0			0
$811$	−1.86034e10			−1.22467			−0.612334	−	0.790599i	$-0.709769\pi$
$811$	−1.86034e10			−1.22467			−0.612334	+	0.790599i	$0.709769\pi$
$812$		0			0
$813$		−	6.02439e9i		−	0.393185i
$814$		0			0
$815$	3.61598e8	+	1.34799e10i	0.0233978	+	0.872235i
$816$		0			0
$817$		−	1.37303e9i		−	0.0880851i
$818$		0			0
$819$	−3.59562e10			−2.28708
$820$		0			0
$821$	−1.49536e10			−0.943072			−0.471536	−	0.881847i	$-0.656300\pi$
$821$	−1.49536e10			−0.943072			−0.471536	+	0.881847i	$0.656300\pi$
$822$		0			0
$823$		−	8.91042e9i		−	0.557184i	−0.960409	−	0.278592i	$-0.910132\pi$
$823$		−	8.91042e9i		−	0.557184i	0.960409	−	0.278592i	$-0.0898678\pi$
$824$		0			0
$825$	−1.63560e9	−	3.04645e10i	−0.101412	−	1.88889i
$826$		0			0
$827$		−	3.03382e10i		−	1.86518i	−0.360935	−	0.932591i	$-0.617543\pi$
$827$		−	3.03382e10i		−	1.86518i	0.360935	−	0.932591i	$-0.382457\pi$
$828$		0			0
$829$	−1.12896e10			−0.688239			−0.344120	−	0.938926i	$-0.611823\pi$
$829$	−1.12896e10			−0.688239			−0.344120	+	0.938926i	$0.611823\pi$
$830$		0			0
$831$	2.28833e10			1.38330
$832$		0			0
$833$		−	7.20711e8i		−	0.0432020i
$834$		0			0
$835$	−4.84094e8	−	1.80463e10i	−0.0287758	−	1.07272i
$836$		0			0
$837$			5.38797e9i			0.317604i
$838$		0			0
$839$	1.92069e10			1.12277			0.561384	−	0.827556i	$-0.310269\pi$
$839$	1.92069e10			1.12277			0.561384	+	0.827556i	$0.310269\pi$
$840$		0			0
$841$	−8.85473e9			−0.513321
$842$		0			0
$843$			4.17233e10i			2.39874i
$844$		0			0
$845$	3.99540e10	−	1.07177e9i	2.27804	−	0.0611087i
$846$		0			0
$847$		−	8.96105e9i		−	0.506719i
$848$		0			0
$849$	−3.13590e10			−1.75867
$850$		0			0
$851$	−3.49506e10			−1.94402
$852$		0			0
$853$		−	8.06518e9i		−	0.444931i	−0.974941	−	0.222465i	$-0.928590\pi$
$853$		−	8.06518e9i		−	0.444931i	0.974941	−	0.222465i	$-0.0714104\pi$
$854$		0			0
$855$	4.34682e9	−	1.16604e8i	0.237843	−	0.00638015i
$856$		0			0
$857$			8.33806e9i			0.452514i	0.974068	+	0.226257i	$0.0726489\pi$
$857$			8.33806e9i			0.452514i	−0.974068	+	0.226257i	$0.927351\pi$
$858$		0			0
$859$	−3.50871e10			−1.88873			−0.944367	−	0.328892i	$-0.893325\pi$
$859$	−3.50871e10			−1.88873			−0.944367	+	0.328892i	$0.893325\pi$
$860$		0			0
$861$	2.37052e10			1.26570
$862$		0			0
$863$		−	2.03137e10i		−	1.07585i	−0.842993	−	0.537925i	$-0.819208\pi$
$863$		−	2.03137e10i		−	1.07585i	0.842993	−	0.537925i	$-0.180792\pi$
$864$		0			0
$865$	1.10055e8	+	4.10270e9i	0.00578168	+	0.215533i
$866$		0			0
$867$			2.39271e10i			1.24688i
$868$		0			0
$869$	−3.00582e10			−1.55380
$870$		0			0
$871$	2.72127e10			1.39543
$872$		0			0
$873$		−	3.49385e10i		−	1.77727i
$874$		0			0
$875$	1.51019e9	+	1.87299e10i	0.0762086	+	0.945166i
$876$		0			0
$877$			1.85549e10i			0.928879i	0.885605	+	0.464440i	$0.153744\pi$
$877$			1.85549e10i			0.928879i	−0.885605	+	0.464440i	$0.846256\pi$
$878$		0			0
$879$	3.75803e10			1.86638
$880$		0			0
$881$	1.77206e10			0.873097			0.436549	−	0.899681i	$-0.356201\pi$
$881$	1.77206e10			0.873097			0.436549	+	0.899681i	$0.356201\pi$
$882$		0			0
$883$		−	4.69416e9i		−	0.229454i	−0.993397	−	0.114727i	$-0.963401\pi$
$883$		−	4.69416e9i		−	0.229454i	0.993397	−	0.114727i	$-0.0365993\pi$
$884$		0			0
$885$	1.24001e9	+	4.62258e10i	0.0601345	+	2.24173i
$886$		0			0
$887$			3.71147e10i			1.78572i	0.450336	+	0.892859i	$0.351304\pi$
$887$			3.71147e10i			1.78572i	−0.450336	+	0.892859i	$0.648696\pi$
$888$		0			0
$889$	1.86981e10			0.892567
$890$		0			0
$891$	1.45882e10			0.690922
$892$		0			0
$893$		−	4.07039e9i		−	0.191274i
$894$		0			0
$895$	−1.28783e9	+	3.45461e7i	−0.0600451	+	0.00161071i
$896$		0			0
$897$		−	7.66246e10i		−	3.54482i
$898$		0			0
$899$	9.52065e9			0.437026
$900$		0			0
$901$	9.89484e9			0.450684
$902$		0			0
$903$		−	1.57996e10i		−	0.714067i
$904$		0			0
$905$	5.50591e8	−	1.47696e7i	0.0246922	−	0.000662369i
$906$		0			0
$907$			9.96301e9i			0.443369i	0.975118	+	0.221684i	$0.0711555\pi$
$907$			9.96301e9i			0.443369i	−0.975118	+	0.221684i	$0.928845\pi$
$908$		0			0
$909$	2.00082e10			0.883558
$910$		0			0
$911$	−1.95417e10			−0.856343			−0.428171	−	0.903698i	$-0.640842\pi$
$911$	−1.95417e10			−0.856343			−0.428171	+	0.903698i	$0.640842\pi$
$912$		0			0
$913$			2.07338e10i			0.901637i
$914$		0			0
$915$	−1.01233e9	−	3.77381e10i	−0.0436864	−	1.62857i
$916$		0			0
$917$			3.63974e9i			0.155876i
$918$		0			0
$919$	−2.22276e10			−0.944687			−0.472343	−	0.881415i	$-0.656592\pi$
$919$	−2.22276e10			−0.944687			−0.472343	+	0.881415i	$0.656592\pi$
$920$		0			0
$921$	1.89738e10			0.800289
$922$		0			0
$923$		−	4.07395e10i		−	1.70533i
$924$		0			0
$925$	1.95698e9	+	3.64505e10i	0.0813000	+	1.51428i
$926$		0			0
$927$		−	1.20218e10i		−	0.495668i
$928$		0			0
$929$	1.77199e10			0.725115			0.362557	−	0.931961i	$-0.381904\pi$
$929$	1.77199e10			0.725115			0.362557	+	0.931961i	$0.381904\pi$
$930$		0			0
$931$	4.43567e8			0.0180151
$932$		0			0
$933$		−	2.05484e9i		−	0.0828308i
$934$		0			0
$935$	−3.55633e8	−	1.32575e10i	−0.0142286	−	0.530422i
$936$		0			0
$937$		−	7.62760e9i		−	0.302900i	−0.988465	−	0.151450i	$-0.951606\pi$
$937$		−	7.62760e9i		−	0.302900i	0.988465	−	0.151450i	$-0.0483942\pi$
$938$		0			0
$939$	−4.74606e10			−1.87070
$940$		0			0
$941$	−3.75184e10			−1.46784			−0.733922	−	0.679233i	$-0.762312\pi$
$941$	−3.75184e10			−1.46784			−0.733922	+	0.679233i	$0.762312\pi$
$942$		0			0
$943$			2.88544e10i			1.12052i
$944$		0			0
$945$	1.24672e10	−	3.34434e8i	0.480572	−	0.0128914i
$946$		0			0
$947$			3.60633e10i			1.37988i	0.723868	+	0.689939i	$0.242363\pi$
$947$			3.60633e10i			1.37988i	−0.723868	+	0.689939i	$0.757637\pi$
$948$		0			0
$949$	3.72275e10			1.41394
$950$		0			0
$951$	−4.29505e10			−1.61933
$952$		0			0
$953$			4.82515e10i			1.80587i	0.429782	+	0.902933i	$0.358590\pi$
$953$			4.82515e10i			1.80587i	−0.429782	+	0.902933i	$0.641410\pi$
$954$		0			0
$955$	−1.14499e10	+	3.07146e8i	−0.425394	+	0.0114112i
$956$		0			0
$957$			3.57803e10i			1.31963i
$958$		0			0
$959$	1.30615e10			0.478219
$960$		0			0
$961$	−1.67156e10			−0.607560
$962$		0			0
$963$		−	9.91525e9i		−	0.357777i
$964$		0			0
$965$	−6.14142e8	−	2.28944e10i	−0.0220000	−	0.820130i
$966$		0			0
$967$		−	1.37995e10i		−	0.490763i	−0.969427	−	0.245381i	$-0.921087\pi$
$967$		−	1.37995e10i		−	0.490763i	0.969427	−	0.245381i	$-0.0789132\pi$
$968$		0			0
$969$	3.30941e9			0.116847
$970$		0			0
$971$	3.97135e10			1.39210			0.696051	−	0.717992i	$-0.254939\pi$
$971$	3.97135e10			1.39210			0.696051	+	0.717992i	$0.254939\pi$
$972$		0			0
$973$		−	3.33801e10i		−	1.16169i
$974$		0			0
$975$	−7.99129e10	+	4.29042e9i	−2.76122	+	0.148246i
$976$		0			0
$977$			2.50857e9i			0.0860589i	0.999074	+	0.0430294i	$0.0137009\pi$
$977$			2.50857e9i			0.0860589i	−0.999074	+	0.0430294i	$0.986299\pi$
$978$		0			0
$979$	−1.07085e10			−0.364744
$980$		0			0
$981$	−3.74218e10			−1.26556
$982$		0			0
$983$		−	4.09376e9i		−	0.137463i	−0.997635	−	0.0687313i	$-0.978105\pi$
$983$		−	4.09376e9i		−	0.137463i	0.997635	−	0.0687313i	$-0.0218951\pi$
$984$		0			0
$985$	−1.09952e9	−	4.09887e10i	−0.0366588	−	1.36659i
$986$		0			0
$987$		−	4.68385e10i		−	1.55057i
$988$		0			0
$989$	1.92316e10			0.632162
$990$		0			0
$991$	1.58598e10			0.517653			0.258827	−	0.965924i	$-0.416664\pi$
$991$	1.58598e10			0.517653			0.258827	+	0.965924i	$0.416664\pi$
$992$		0			0
$993$			7.69828e10i			2.49501i
$994$		0			0
$995$	4.29966e9	−	1.15339e8i	0.138374	−	0.00371188i
$996$		0			0
$997$		−	3.86132e10i		−	1.23396i	−0.786977	−	0.616982i	$-0.788355\pi$
$997$		−	3.86132e10i		−	1.23396i	0.786977	−	0.616982i	$-0.211645\pi$
$998$		0			0
$999$	2.42277e10			0.768833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.8.c.a.9.4	yes	4
3.2	odd	2		180.8.d.b.109.3		4
4.3	odd	2		80.8.c.c.49.1		4
5.2	odd	4		100.8.a.e.1.4		4
5.3	odd	4		100.8.a.e.1.1		4
5.4	even	2	inner	20.8.c.a.9.1	✓	4
8.3	odd	2		320.8.c.i.129.4		4
8.5	even	2		320.8.c.j.129.1		4
15.14	odd	2		180.8.d.b.109.4		4
20.3	even	4		400.8.a.bk.1.4		4
20.7	even	4		400.8.a.bk.1.1		4
20.19	odd	2		80.8.c.c.49.4		4
40.19	odd	2		320.8.c.i.129.1		4
40.29	even	2		320.8.c.j.129.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.8.c.a.9.1	✓	4	5.4	even	2	inner
20.8.c.a.9.4	yes	4	1.1	even	1	trivial
80.8.c.c.49.1		4	4.3	odd	2
80.8.c.c.49.4		4	20.19	odd	2
100.8.a.e.1.1		4	5.3	odd	4
100.8.a.e.1.4		4	5.2	odd	4
180.8.d.b.109.3		4	3.2	odd	2
180.8.d.b.109.4		4	15.14	odd	2
320.8.c.i.129.1		4	40.19	odd	2
320.8.c.i.129.4		4	8.3	odd	2
320.8.c.j.129.1		4	8.5	even	2
320.8.c.j.129.4		4	40.29	even	2
400.8.a.bk.1.1		4	20.7	even	4
400.8.a.bk.1.4		4	20.3	even	4

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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	20.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+71.4148i q^{3} +(-279.408 + 7.49513i) q^{5} -860.515i q^{7} -2913.08 q^{9} +O(q^{10})\)	\(q+71.4148i q^{3} +(-279.408 + 7.49513i) q^{5} -860.515i q^{7} -2913.08 q^{9} -5468.16 q^{11} +14343.8i q^{13} +(-535.264 - 19953.9i) q^{15} -8677.25i q^{17} +5340.48 q^{19} +61453.5 q^{21} +74802.5i q^{23} +(78012.6 - 4188.40i) q^{25} -51852.9i q^{27} -91625.0 q^{29} -103909. q^{31} -390508. i q^{33} +(6449.67 + 240435. i) q^{35} +467238. i q^{37} -1.02436e6 q^{39} +385741. q^{41} -257098. i q^{43} +(813938. - 21833.9i) q^{45} -762177. i q^{47} +83057.6 q^{49} +619684. q^{51} +1.14032e6i q^{53} +(1.52785e6 - 40984.6i) q^{55} +381389. i q^{57} -2.31663e6 q^{59} +1.89127e6 q^{61} +2.50675e6i q^{63} +(-107509. - 4.00777e6i) q^{65} -1.89718e6i q^{67} -5.34201e6 q^{69} -2.84022e6 q^{71} -2.59537e6i q^{73} +(299114. + 5.57126e6i) q^{75} +4.70543e6i q^{77} +5.49695e6 q^{79} -2.66784e6 q^{81} -3.79174e6i q^{83} +(65037.1 + 2.42449e6i) q^{85} -6.54339e6i q^{87} +1.95833e6 q^{89} +1.23430e7 q^{91} -7.42063e6i q^{93} +(-1.49217e6 + 40027.6i) q^{95} +1.19937e7i q^{97} +1.59292e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 156 q^{5} - 2036 q^{9}+O(q^{10}) \) 4 * q - 156 * q^5 - 2036 * q^9	\( 4 q - 156 q^{5} - 2036 q^{9} - 2640 q^{11} + 5552 q^{15} - 36336 q^{19} + 63104 q^{21} + 162036 q^{25} - 481896 q^{29} + 161344 q^{31} + 691248 q^{35} - 2366496 q^{39} + 1129464 q^{41} + 2391244 q^{45} - 4312452 q^{49} + 1978688 q^{51} + 4726640 q^{55} - 7747152 q^{59} + 4305128 q^{61} + 3389568 q^{65} - 9357248 q^{69} - 1206048 q^{71} + 3265888 q^{75} + 5716992 q^{79} + 570116 q^{81} - 8256064 q^{85} + 13641576 q^{89} - 7864224 q^{91} - 12453936 q^{95} + 47580560 q^{99}+O(q^{100}) \) 4 * q - 156 * q^5 - 2036 * q^9 - 2640 * q^11 + 5552 * q^15 - 36336 * q^19 + 63104 * q^21 + 162036 * q^25 - 481896 * q^29 + 161344 * q^31 + 691248 * q^35 - 2366496 * q^39 + 1129464 * q^41 + 2391244 * q^45 - 4312452 * q^49 + 1978688 * q^51 + 4726640 * q^55 - 7747152 * q^59 + 4305128 * q^61 + 3389568 * q^65 - 9357248 * q^69 - 1206048 * q^71 + 3265888 * q^75 + 5716992 * q^79 + 570116 * q^81 - 8256064 * q^85 + 13641576 * q^89 - 7864224 * q^91 - 12453936 * q^95 + 47580560 * q^99

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