LMFDB - Embedded newform 315.2.z.a.209.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(104,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("315.104");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.z (of order $6$, degree $2$, 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:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.31116960000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 $ x^8 + x^6 - 8x^4 + 9x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			209.1
Root			$-0.306808 + 1.70466i$ of defining polynomial
Character	$\chi$	$=$	315.209
Dual form			315.2.z.a.104.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 - 1.11803i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(1.32288 - 2.29129i) q^{7} +(0.500000 + 2.95804i) q^{9} +O(q^{10})$	\(q+(-1.32288 - 1.11803i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(1.32288 - 2.29129i) q^{7} +(0.500000 + 2.95804i) q^{9} +(-0.311738 - 0.179982i) q^{11} +(0.613616 - 3.40932i) q^{12} +(3.56618 + 6.17680i) q^{13} +(3.81174 + 0.686044i) q^{15} +(-2.00000 + 3.46410i) q^{16} +5.75583i q^{17} +(-3.87298 - 2.23607i) q^{20} +(-4.31174 + 1.55207i) q^{21} +(2.50000 - 4.33013i) q^{25} +(2.64575 - 4.47214i) q^{27} +5.29150 q^{28} +(5.12348 + 2.95804i) q^{29} +(0.211164 + 0.586627i) q^{33} +5.91608i q^{35} +(-4.62348 + 3.82407i) q^{36} +(2.18826 - 12.1582i) q^{39} -0.719927i q^{44} +(-4.27543 - 5.16920i) q^{45} +(-10.7942 - 6.23202i) q^{47} +(6.51873 - 2.34651i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(6.43521 - 7.61425i) q^{51} +(-7.13235 + 12.3536i) q^{52} +0.804903 q^{55} +(2.62348 + 7.28817i) q^{60} +(7.43916 + 2.76748i) q^{63} -8.00000 q^{64} +(-13.8117 - 7.97421i) q^{65} +(-9.96939 + 5.75583i) q^{68} -16.3084i q^{71} +6.32745 q^{73} +(-8.14842 + 2.93313i) q^{75} +(-0.824780 + 0.476187i) q^{77} +(-7.93521 + 13.7442i) q^{79} -8.94427i q^{80} +(-8.50000 + 2.95804i) q^{81} +(15.7789 + 9.10993i) q^{83} +(-7.00000 - 5.91608i) q^{84} +(-6.43521 - 11.1461i) q^{85} +(-3.47053 - 9.64134i) q^{87} +18.8704 q^{91} +(7.53480 - 13.0507i) q^{97} +(0.376525 - 1.01212i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} + 4 q^{9}+O(q^{10}) $ 8 * q + 8 * q^4 + 4 * q^9	$ 8 q + 8 q^{4} + 4 q^{9} + 18 q^{11} + 10 q^{15} - 16 q^{16} - 14 q^{21} + 20 q^{25} + 4 q^{36} + 38 q^{39} - 28 q^{49} - 10 q^{51} - 20 q^{60} - 64 q^{64} - 90 q^{65} - 2 q^{79} - 68 q^{81} - 56 q^{84} + 10 q^{85} + 28 q^{91} + 44 q^{99}+O(q^{100}) $ 8 * q + 8 * q^4 + 4 * q^9 + 18 * q^11 + 10 * q^15 - 16 * q^16 - 14 * q^21 + 20 * q^25 + 4 * q^36 + 38 * q^39 - 28 * q^49 - 10 * q^51 - 20 * q^60 - 64 * q^64 - 90 * q^65 - 2 * q^79 - 68 * q^81 - 56 * q^84 + 10 * q^85 + 28 * q^91 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$	−1.32288	−	1.11803i	−0.763763	−	0.645497i
$3$	−1.32288	−	1.11803i	−0.763763	−	0.645497i
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	−1.93649	+	1.11803i	−0.866025	+	0.500000i
$5$	−1.93649	+	1.11803i	−0.866025	+	0.500000i
$6$		0			0
$7$	1.32288	−	2.29129i	0.500000	−	0.866025i
$7$	1.32288	−	2.29129i	0.500000	−	0.866025i
$8$		0			0
$9$	0.500000	+	2.95804i	0.166667	+	0.986013i
$10$		0			0
$11$	−0.311738	−	0.179982i	−0.0939925	−	0.0542666i	0.452267	−	0.891883i	$-0.350615\pi$
$11$	−0.311738	−	0.179982i	−0.0939925	−	0.0542666i	−0.546259	+	0.837616i	$0.683949\pi$
$12$	0.613616	−	3.40932i	0.177136	−	0.984186i
$13$	3.56618	+	6.17680i	0.989079	+	1.71314i	0.622179	+	0.782875i	$0.286247\pi$
$13$	3.56618	+	6.17680i	0.989079	+	1.71314i	0.366900	+	0.930261i	$0.380419\pi$
$14$		0			0
$15$	3.81174	+	0.686044i	0.984186	+	0.177136i
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$			5.75583i			1.39599i	0.716101	+	0.697997i	$0.245925\pi$
$17$			5.75583i			1.39599i	−0.716101	+	0.697997i	$0.754075\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−3.87298	−	2.23607i	−0.866025	−	0.500000i
$21$	−4.31174	+	1.55207i	−0.940898	+	0.338689i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$	2.64575	−	4.47214i	0.509175	−	0.860663i
$28$	5.29150			1.00000
$29$	5.12348	+	2.95804i	0.951405	+	0.549294i	0.893517	−	0.449029i	$-0.148230\pi$
$29$	5.12348	+	2.95804i	0.951405	+	0.549294i	0.0578882	+	0.998323i	$0.481563\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$	0.211164	+	0.586627i	0.0367590	+	0.102119i
$34$		0			0
$35$			5.91608i			1.00000i
$36$	−4.62348	+	3.82407i	−0.770579	+	0.637344i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	2.18826	−	12.1582i	0.350402	−	1.94688i
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		−	0.719927i		−	0.108533i
$45$	−4.27543	−	5.16920i	−0.637344	−	0.770579i
$46$		0			0
$47$	−10.7942	−	6.23202i	−1.57449	−	0.909033i	−0.995608	−	0.0936230i	$-0.970155\pi$
$47$	−10.7942	−	6.23202i	−1.57449	−	0.909033i	−0.578884	−	0.815410i	$-0.696511\pi$
$48$	6.51873	−	2.34651i	0.940898	−	0.338689i
$49$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$50$		0			0
$51$	6.43521	−	7.61425i	0.901110	−	1.06621i
$52$	−7.13235	+	12.3536i	−0.989079	+	1.71314i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	0.804903			0.108533
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$	2.62348	+	7.28817i	0.338689	+	0.940898i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	7.43916	+	2.76748i	0.937246	+	0.348669i
$64$	−8.00000			−1.00000
$65$	−13.8117	−	7.97421i	−1.71314	−	0.989079i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−9.96939	+	5.75583i	−1.20897	+	0.697997i
$69$		0			0
$70$		0			0
$71$		−	16.3084i		−	1.93545i	−0.252010	−	0.967725i	$-0.581092\pi$
$71$		−	16.3084i		−	1.93545i	0.252010	−	0.967725i	$-0.418908\pi$
$72$		0			0
$73$	6.32745			0.740572			0.370286	−	0.928918i	$-0.379260\pi$
$73$	6.32745			0.740572			0.370286	+	0.928918i	$0.379260\pi$
$74$		0			0
$75$	−8.14842	+	2.93313i	−0.940898	+	0.338689i
$76$		0			0
$77$	−0.824780	+	0.476187i	−0.0939925	+	0.0542666i
$78$		0			0
$79$	−7.93521	+	13.7442i	−0.892781	+	1.54634i	−0.0562544	+	0.998416i	$0.517916\pi$
$79$	−7.93521	+	13.7442i	−0.892781	+	1.54634i	−0.836527	+	0.547926i	$0.815418\pi$
$80$		−	8.94427i		−	1.00000i
$81$	−8.50000	+	2.95804i	−0.944444	+	0.328671i
$82$		0			0
$83$	15.7789	+	9.10993i	1.73196	+	0.999945i	0.871227	+	0.490881i	$0.163325\pi$
$83$	15.7789	+	9.10993i	1.73196	+	0.999945i	0.860729	+	0.509064i	$0.170008\pi$
$84$	−7.00000	−	5.91608i	−0.763763	−	0.645497i
$85$	−6.43521	−	11.1461i	−0.697997	−	1.20897i
$86$		0			0
$87$	−3.47053	−	9.64134i	−0.372080	−	1.03366i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	18.8704			1.97816
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.53480	−	13.0507i	0.765043	−	1.32509i	−0.175180	−	0.984536i	$-0.556051\pi$
$97$	7.53480	−	13.0507i	0.765043	−	1.32509i	0.940224	−	0.340557i	$-0.110616\pi$
$98$		0			0
$99$	0.376525	−	1.01212i	0.0378421	−	0.101722i
$100$	10.0000			1.00000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−1.32288	−	2.29129i	−0.130347	−	0.225767i	0.793463	−	0.608618i	$-0.208276\pi$
$103$	−1.32288	−	2.29129i	−0.130347	−	0.225767i	−0.923810	+	0.382851i	$0.874942\pi$
$104$		0			0
$105$	6.61438	−	7.82624i	0.645497	−	0.763763i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	10.3917	+	0.110440i	0.999944	+	0.0106271i
$109$	9.87043			0.945415			0.472708	−	0.881219i	$-0.343277\pi$
$109$	9.87043			0.945415			0.472708	+	0.881219i	$0.343277\pi$
$110$		0			0
$111$		0			0
$112$	5.29150	+	9.16515i	0.500000	+	0.866025i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$			11.8322i			1.09859i
$117$	−16.4881	+	13.6373i	−1.52433	+	1.26077i
$118$		0			0
$119$	13.1883	+	7.61425i	1.20897	+	0.697997i
$120$		0			0
$121$	−5.43521	−	9.41407i	−0.494110	−	0.855824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$	−0.804903	+	0.952374i	−0.0700578	+	0.0828935i
$133$		0			0
$134$		0			0
$135$	−0.123475	+	11.6183i	−0.0106271	+	0.999944i
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$	−10.2470	+	5.91608i	−0.866025	+	0.500000i
$141$	7.31174	+	20.3124i	0.615759	+	1.71062i
$142$		0			0
$143$		−	2.56739i		−	0.214696i
$144$	−11.2470	−	4.18403i	−0.937246	−	0.348669i
$145$	−13.2288			−1.09859
$146$		0			0
$147$	−2.14766	+	11.9326i	−0.177136	+	0.984186i
$148$		0			0
$149$	14.7470	−	8.51416i	1.20812	−	0.697507i	0.245770	−	0.969328i	$-0.420959\pi$
$149$	14.7470	−	8.51416i	1.20812	−	0.697507i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	−3.43521	+	5.94996i	−0.279554	+	0.484201i	−0.971274	−	0.237964i	$-0.923520\pi$
$151$	−3.43521	+	5.94996i	−0.279554	+	0.484201i	0.691720	+	0.722166i	$0.256853\pi$
$152$		0			0
$153$	−17.0260	+	2.87791i	−1.37647	+	0.232666i
$154$		0			0
$155$		0			0
$156$	23.2470	−	8.36806i	1.86125	−	0.669981i
$157$	−11.1010	−	19.2275i	−0.885954	−	1.53452i	−0.844616	−	0.535373i	$-0.820171\pi$
$157$	−11.1010	−	19.2275i	−0.885954	−	1.53452i	−0.0413387	−	0.999145i	$-0.513162\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−1.06479	−	0.899909i	−0.0828935	−	0.0700578i
$166$		0			0
$167$	16.6036	−	9.58612i	1.28483	−	0.741796i	0.307102	−	0.951677i	$-0.400641\pi$
$167$	16.6036	−	9.58612i	1.28483	−	0.741796i	0.977727	+	0.209881i	$0.0673075\pi$
$168$		0			0
$169$	−18.9352	+	32.7968i	−1.45655	+	2.52283i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.68246	+	5.59017i	0.736144	+	0.425013i	0.820666	−	0.571409i	$-0.193603\pi$
$173$	9.68246	+	5.59017i	0.736144	+	0.425013i	−0.0845218	+	0.996422i	$0.526936\pi$
$174$		0			0
$175$	−6.61438	−	11.4564i	−0.500000	−	0.866025i
$176$	1.24695	−	0.719927i	0.0939925	−	0.0542666i
$177$		0			0
$178$		0			0
$179$			14.8685i			1.11133i	0.831408	+	0.555663i	$0.187536\pi$
$179$			14.8685i			1.11133i	−0.831408	+	0.555663i	$0.812464\pi$
$180$	4.67789	−	12.5745i	0.348669	−	0.937246i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.03594	−	1.79431i	0.0757558	−	0.131213i
$188$		−	24.9281i		−	1.81807i
$189$	−6.74695	−	11.9783i	−0.490768	−	0.871290i
$190$		0			0
$191$	5.12348	+	2.95804i	0.370722	+	0.214036i	0.673774	−	0.738938i	$-0.264672\pi$
$191$	5.12348	+	2.95804i	0.370722	+	0.214036i	−0.303052	+	0.952974i	$0.598006\pi$
$192$	10.5830	+	8.94427i	0.763763	+	0.645497i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$	9.35577	+	25.9909i	0.669981	+	1.86125i
$196$	7.00000	−	12.1244i	0.500000	−	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.5554	−	7.82624i	0.951405	−	0.549294i
$204$	19.6235	+	3.53187i	1.37392	+	0.247280i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−28.5294			−1.97816
$209$		0			0
$210$		0			0
$211$	−1.93521	−	3.35189i	−0.133226	−	0.230753i	0.791693	−	0.610920i	$-0.209200\pi$
$211$	−1.93521	−	3.35189i	−0.133226	−	0.230753i	−0.924918	+	0.380166i	$0.875867\pi$
$212$		0			0
$213$	−18.2333	+	21.5740i	−1.24933	+	1.47822i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−8.37043	−	7.07430i	−0.565621	−	0.478037i
$220$	0.804903	+	1.39413i	0.0542666	+	0.0939925i
$221$	−35.5526	+	20.5263i	−2.39153	+	1.38075i
$222$		0			0
$223$	−4.37108	+	7.57093i	−0.292709	+	0.506987i	−0.974449	−	0.224607i	$-0.927890\pi$
$223$	−4.37108	+	7.57093i	−0.292709	+	0.506987i	0.681740	+	0.731594i	$0.261223\pi$
$224$		0			0
$225$	14.0587	+	5.23004i	0.937246	+	0.348669i
$226$		0			0
$227$	6.63426	+	3.83029i	0.440331	+	0.254225i	0.703738	−	0.710460i	$-0.251513\pi$
$227$	6.63426	+	3.83029i	0.440331	+	0.254225i	−0.263407	+	0.964685i	$0.584846\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$	1.62348	+	0.292196i	0.106817	+	0.0192251i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	27.8704			1.81807
$236$		0			0
$237$	25.8638	−	9.31002i	1.68003	−	0.604751i
$238$		0			0
$239$	25.6174	−	14.7902i	1.65705	−	0.956698i	0.682985	−	0.730433i	$-0.260682\pi$
$239$	25.6174	−	14.7902i	1.65705	−	0.956698i	0.974066	−	0.226266i	$-0.0726518\pi$
$240$	−10.0000	+	11.8322i	−0.645497	+	0.763763i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	14.5516	+	5.59017i	0.933488	+	0.358610i
$244$		0			0
$245$	13.5554	+	7.82624i	0.866025	+	0.500000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−10.6883	−	29.6926i	−0.677341	−	1.88169i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	2.64575	+	15.6525i	0.166667	+	0.986013i
$253$		0			0
$254$		0			0
$255$	−3.94875	+	21.9397i	−0.247280	+	1.37392i
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	25.7483	−	14.8658i	1.60613	−	0.927301i	0.615907	−	0.787819i	$-0.288790\pi$
$257$	25.7483	−	14.8658i	1.60613	−	0.927301i	0.990225	−	0.139482i	$-0.0445438\pi$
$258$		0			0
$259$		0			0
$260$		−	31.8968i		−	1.97816i
$261$	−6.18826	+	16.6345i	−0.383044	+	1.02965i
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−19.9388	−	11.5117i	−1.20897	−	0.697997i
$273$	−24.9632	−	21.0978i	−1.51084	−	1.27690i
$274$		0			0
$275$	−1.55869	+	0.899909i	−0.0939925	+	0.0542666i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−27.3117	−	15.7684i	−1.62928	−	0.940666i	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−27.3117	−	15.7684i	−1.62928	−	0.940666i	−0.644974	−	0.764204i	$-0.723132\pi$
$282$		0			0
$283$	−13.8622	−	24.0101i	−0.824025	−	1.42725i	−0.902662	−	0.430350i	$-0.858390\pi$
$283$	−13.8622	−	24.0101i	−0.824025	−	1.42725i	0.0786368	−	0.996903i	$-0.474943\pi$
$284$	28.2470	−	16.3084i	1.67615	−	0.967725i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.1296			−0.948798
$290$		0			0
$291$	−24.5587	+	8.84024i	−1.43966	+	0.518224i
$292$	6.32745	+	10.9595i	0.370286	+	0.641354i
$293$	−21.3014	+	12.2984i	−1.24444	+	0.718479i	−0.969995	−	0.243124i	$-0.921828\pi$
$293$	−21.3014	+	12.2984i	−1.24444	+	0.718479i	−0.274446	+	0.961602i	$0.588495\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.62968	+	0.917946i	−0.0945638	+	0.0532646i
$298$		0			0
$299$		0			0
$300$	−13.2288	−	11.1803i	−0.763763	−	0.645497i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−23.0069			−1.31307			−0.656535	−	0.754295i	$-0.727979\pi$
$307$	−23.0069			−1.31307			−0.656535	+	0.754295i	$0.727979\pi$
$308$	−1.64956	−	0.952374i	−0.0939925	−	0.0542666i
$309$	−0.811738	+	4.51011i	−0.0461781	+	0.256571i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	14.5516	−	25.2042i	0.822507	−	1.42462i	−0.0813030	−	0.996689i	$-0.525908\pi$
$313$	14.5516	−	25.2042i	0.822507	−	1.42462i	0.903810	−	0.427934i	$-0.140759\pi$
$314$		0			0
$315$	−17.5000	+	2.95804i	−0.986013	+	0.166667i
$316$	−31.7409			−1.78556
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$	−1.06479	−	1.84427i	−0.0596166	−	0.103259i
$320$	15.4919	−	8.94427i	0.866025	−	0.500000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−13.6235	−	11.7644i	−0.756860	−	0.653577i
$325$	35.6618			1.97816
$326$		0			0
$327$	−13.0573	−	11.0355i	−0.722073	−	0.610263i
$328$		0			0
$329$	−28.5587	+	16.4884i	−1.57449	+	0.909033i
$330$		0			0
$331$	17.3704	−	30.0865i	0.954765	−	1.65370i	0.219860	−	0.975531i	$-0.429440\pi$
$331$	17.3704	−	30.0865i	0.954765	−	1.65370i	0.734905	−	0.678170i	$-0.237227\pi$
$332$			36.4397i			1.99989i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	3.24695	−	18.0404i	0.177136	−	0.984186i
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$		0			0
$340$	12.8704	−	22.2922i	0.697997	−	1.20897i
$341$		0			0
$342$		0			0
$343$	−18.5203			−1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$	13.2288	−	15.6525i	0.709136	−	0.839061i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$	37.0587	+	0.393847i	1.97805	+	0.0210220i
$352$		0			0
$353$	−25.1744	−	14.5344i	−1.33990	−	0.773590i	−0.353106	−	0.935583i	$-0.614874\pi$
$353$	−25.1744	−	14.5344i	−1.33990	−	0.773590i	−0.986792	+	0.161993i	$0.948208\pi$
$354$		0			0
$355$	18.2333	+	31.5811i	0.967725	+	1.67615i
$356$		0			0
$357$	−8.93344	−	24.8176i	−0.472808	−	1.31349i
$358$		0			0
$359$			11.8322i			0.624477i	0.950004	+	0.312239i	$0.101079\pi$
$359$			11.8322i			0.624477i	−0.950004	+	0.312239i	$0.898921\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	−3.33513	+	18.5304i	−0.175049	+	0.972593i
$364$	18.8704	+	32.6845i	0.989079	+	1.71314i
$365$	−12.2530	+	7.07430i	−0.641354	+	0.370286i
$366$		0			0
$367$	−9.89362	+	17.1363i	−0.516443	+	0.894505i	0.483375	+	0.875413i	$0.339411\pi$
$367$	−9.89362	+	17.1363i	−0.516443	+	0.894505i	−0.999818	+	0.0190919i	$0.993923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	12.5000	−	14.7902i	0.645497	−	0.763763i
$376$		0			0
$377$			42.1956i			2.17318i
$378$		0			0
$379$	−22.7409			−1.16812			−0.584060	−	0.811711i	$-0.698537\pi$
$379$	−22.7409			−1.16812			−0.584060	+	0.811711i	$0.698537\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−27.3978	+	15.8181i	−1.39996	+	0.808269i	−0.994388	−	0.105793i	$-0.966262\pi$
$383$	−27.3978	+	15.8181i	−1.39996	+	0.808269i	−0.405575	+	0.914062i	$0.632929\pi$
$384$		0			0
$385$	1.06479	−	1.84427i	0.0542666	−	0.0939925i
$386$		0			0
$387$		0			0
$388$	30.1392			1.53009
$389$	26.6883	+	15.4085i	1.35315	+	0.781241i	0.988689	−	0.149979i	$-0.0479205\pi$
$389$	26.6883	+	15.4085i	1.35315	+	0.781241i	0.364459	+	0.931219i	$0.381254\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	35.4874i		−	1.78556i
$396$	2.12957	−	0.359964i	0.107015	−	0.0180889i
$397$	34.3948			1.72622			0.863112	−	0.505013i	$-0.168512\pi$
$397$	34.3948			1.72622			0.863112	+	0.505013i	$0.168512\pi$
$398$		0			0
$399$		0			0
$400$	10.0000	+	17.3205i	0.500000	+	0.866025i
$401$	25.6174	−	14.7902i	1.27927	−	0.738587i	0.302556	−	0.953131i	$-0.402160\pi$
$401$	25.6174	−	14.7902i	1.27927	−	0.738587i	0.976714	+	0.214544i	$0.0688266\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	13.1530	−	15.2315i	0.653577	−	0.756860i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$	2.64575	−	4.58258i	0.130347	−	0.225767i
$413$		0			0
$414$		0			0
$415$	−40.7409			−1.99989
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$	20.1698	+	3.63020i	0.984186	+	0.177136i
$421$	−16.9352	+	29.3326i	−0.825372	+	1.42959i	0.0762630	+	0.997088i	$0.475701\pi$
$421$	−16.9352	+	29.3326i	−0.825372	+	1.42959i	−0.901635	+	0.432498i	$0.857632\pi$
$422$		0			0
$423$	13.0375	−	35.0456i	0.633903	−	1.70398i
$424$		0			0
$425$	24.9235	+	14.3896i	1.20897	+	0.697997i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2.87043	+	3.39633i	−0.138585	+	0.163977i
$430$		0			0
$431$			18.1082i			0.872241i	0.899888	+	0.436121i	$0.143648\pi$
$431$			18.1082i			0.872241i	−0.899888	+	0.436121i	$0.856352\pi$
$432$	10.2004	+	18.1094i	0.490768	+	0.871290i
$433$	10.5830			0.508587			0.254293	−	0.967127i	$-0.418157\pi$
$433$	10.5830			0.508587			0.254293	+	0.967127i	$0.418157\pi$
$434$		0			0
$435$	17.5000	+	14.7902i	0.839061	+	0.709136i
$436$	9.87043	+	17.0961i	0.472708	+	0.818754i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	16.1822	−	13.3842i	0.770579	−	0.637344i
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−29.0275	−	5.22442i	−1.37295	−	0.247107i
$448$	−10.5830	+	18.3303i	−0.500000	+	0.866025i
$449$		−	41.4126i		−	1.95438i	−0.212368	−	0.977190i	$-0.568118\pi$
$449$		−	41.4126i		−	1.95438i	0.212368	−	0.977190i	$-0.431882\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	11.1966	−	4.03038i	0.526063	−	0.189364i
$454$		0			0
$455$	−36.5424	+	21.0978i	−1.71314	+	0.989079i
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$	25.7409	+	15.2285i	1.20148	+	0.710805i
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−20.4939	+	11.8322i	−0.951405	+	0.549294i
$465$		0			0
$466$		0			0
$467$		−	14.3688i		−	0.664908i	−0.943119	−	0.332454i	$-0.892123\pi$
$467$		−	14.3688i		−	0.664908i	0.943119	−	0.332454i	$-0.107877\pi$
$468$	−40.1086	−	14.9210i	−1.85402	−	0.689723i
$469$		0			0
$470$		0			0
$471$	−6.81174	+	37.8468i	−0.313868	+	1.74389i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$			30.4570i			1.39599i
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	10.8704	−	18.8281i	0.494110	−	0.855824i
$485$			33.6967i			1.53009i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.9413	−	6.89432i	0.538904	−	0.311136i	−0.205731	−	0.978609i	$-0.565957\pi$
$491$	11.9413	−	6.89432i	0.538904	−	0.311136i	0.744635	+	0.667472i	$0.232624\pi$
$492$		0			0
$493$	−17.0260	+	29.4899i	−0.766811	+	1.32816i
$494$		0			0
$495$	0.402452	+	2.38094i	0.0180889	+	0.107015i
$496$		0			0
$497$	−37.3672	−	21.5740i	−1.67615	−	0.967725i
$498$		0			0
$499$	2.56479	+	4.44234i	0.114816	+	0.198867i	0.917706	−	0.397260i	$-0.130039\pi$
$499$	2.56479	+	4.44234i	0.114816	+	0.198867i	−0.802890	+	0.596127i	$0.796706\pi$
$500$	−19.3649	+	11.1803i	−0.866025	+	0.500000i
$501$	−32.6822	−	5.88220i	−1.46013	−	0.262797i
$502$		0			0
$503$		−	38.0132i		−	1.69492i	−0.530857	−	0.847461i	$-0.678130\pi$
$503$		−	38.0132i		−	1.69492i	0.530857	−	0.847461i	$-0.321870\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	61.7168	−	22.2158i	2.74094	−	0.986639i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	8.37043	−	14.4980i	0.370286	−	0.641354i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.12348	+	2.95804i	0.225767	+	0.130347i
$516$		0			0
$517$	2.24330	+	3.88551i	0.0986602	+	0.170884i
$518$		0			0
$519$	−6.55869	−	18.2204i	−0.287894	−	0.799788i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−4.71764			−0.206288			−0.103144	−	0.994666i	$-0.532890\pi$
$523$	−4.71764			−0.206288			−0.103144	+	0.994666i	$0.532890\pi$
$524$		0			0
$525$	−4.05869	+	22.5505i	−0.177136	+	0.984186i
$526$		0			0
$527$		0			0
$528$	−2.45446	−	0.441759i	−0.106817	−	0.0192251i
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	16.6235	−	19.6692i	0.717358	−	0.848789i
$538$		0			0
$539$			2.51975i			0.108533i
$540$	−20.2470	+	11.4044i	−0.871290	+	0.490768i
$541$	36.8704			1.58518			0.792592	−	0.609753i	$-0.208731\pi$
$541$	36.8704			1.58518			0.792592	+	0.609753i	$0.208731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.1140	+	11.0355i	−0.818754	+	0.472708i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	20.9946	+	36.3637i	0.892781	+	1.54634i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−20.4939	−	11.8322i	−0.866025	−	0.500000i
$561$	−3.37652	+	1.21543i	−0.142557	+	0.0513153i
$562$		0			0
$563$	7.45904	−	4.30648i	0.314361	−	0.181496i	−0.334515	−	0.942390i	$-0.608573\pi$
$563$	7.45904	−	4.30648i	0.314361	−	0.181496i	0.648876	+	0.760894i	$0.275239\pi$
$564$	−27.8704	+	32.9767i	−1.17356	+	1.38857i
$565$		0			0
$566$		0			0
$567$	−4.46672	+	23.3891i	−0.187585	+	0.982248i
$568$		0			0
$569$	15.9939	+	9.23408i	0.670499	+	0.387113i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	15.9939	+	9.23408i	0.670499	+	0.387113i	−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	23.3704	+	40.4788i	0.978022	+	1.69398i	0.669579	+	0.742741i	$0.266474\pi$
$571$	23.3704	+	40.4788i	0.978022	+	1.69398i	0.308443	+	0.951243i	$0.400192\pi$
$572$	4.44685	−	2.56739i	0.185932	−	0.107348i
$573$	−3.47053	−	9.64134i	−0.144984	−	0.402773i
$574$		0			0
$575$		0			0
$576$	−4.00000	−	23.6643i	−0.166667	−	0.986013i
$577$	11.8500			0.493322			0.246661	−	0.969102i	$-0.420667\pi$
$577$	11.8500			0.493322			0.246661	+	0.969102i	$0.420667\pi$
$578$		0			0
$579$		0			0
$580$	−13.2288	−	22.9129i	−0.549294	−	0.951405i
$581$	41.7470	−	24.1026i	1.73196	−	0.999945i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	16.6822	−	44.8428i	0.689723	−	1.85402i
$586$		0			0
$587$	−7.74597	−	4.47214i	−0.319710	−	0.184585i	0.331553	−	0.943437i	$-0.392427\pi$
$587$	−7.74597	−	4.47214i	−0.319710	−	0.184585i	−0.651263	+	0.758852i	$0.725761\pi$
$588$	−22.8156	+	8.21278i	−0.940898	+	0.338689i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			42.4853i			1.74466i	0.488916	+	0.872331i	$0.337392\pi$
$593$			42.4853i			1.74466i	−0.488916	+	0.872331i	$0.662608\pi$
$594$		0			0
$595$	−34.0519			−1.39599
$596$	29.4939	+	17.0283i	1.20812	+	0.697507i
$597$		0			0
$598$		0			0
$599$	−42.0587	+	24.2826i	−1.71847	+	0.992160i	−0.796748	+	0.604311i	$0.793448\pi$
$599$	−42.0587	+	24.2826i	−1.71847	+	0.992160i	−0.921723	+	0.387849i	$0.873218\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	−13.7409			−0.559107
$605$	21.0505	+	12.1535i	0.855824	+	0.494110i
$606$		0			0
$607$	22.4889	+	38.9519i	0.912796	+	1.58101i	0.810097	+	0.586296i	$0.199414\pi$
$607$	22.4889	+	38.9519i	0.912796	+	1.58101i	0.102699	+	0.994712i	$0.467252\pi$
$608$		0			0
$609$	−26.6822	−	4.80230i	−1.08122	−	0.194599i
$610$		0			0
$611$		−	88.8979i		−	3.59642i
$612$	−22.0107	−	26.6119i	−0.889729	−	1.07572i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	37.7409	+	31.8968i	1.51084	+	1.27690i
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	22.2020	−	38.4549i	0.885954	−	1.53452i
$629$		0			0
$630$		0			0
$631$	−8.12957			−0.323633			−0.161817	−	0.986821i	$-0.551735\pi$
$631$	−8.12957			−0.323633			−0.161817	+	0.986821i	$0.551735\pi$
$632$		0			0
$633$	−1.18748	+	6.59776i	−0.0471980	+	0.262238i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	24.9632	−	43.2376i	0.989079	−	1.71314i
$638$		0			0
$639$	48.2409	−	8.15419i	1.90838	−	0.322575i
$640$		0			0
$641$	−40.9878	−	23.6643i	−1.61892	−	0.934684i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−40.9878	−	23.6643i	−1.61892	−	0.934684i	−0.631721	−	0.775196i	$-0.717651\pi$
$642$		0			0
$643$	15.4721	+	26.7984i	0.610158	+	1.05683i	0.991213	+	0.132273i	$0.0422275\pi$
$643$	15.4721	+	26.7984i	0.610158	+	1.05683i	−0.381055	+	0.924552i	$0.624439\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	17.8885i		−	0.703271i	−0.936137	−	0.351636i	$-0.885626\pi$
$647$		−	17.8885i		−	0.703271i	0.936137	−	0.351636i	$-0.114374\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.16372	+	18.7168i	0.123429	+	0.730214i
$658$		0			0
$659$	−40.8117	−	23.5627i	−1.58980	−	0.917871i	−0.993339	−	0.115229i	$-0.963240\pi$
$659$	−40.8117	−	23.5627i	−1.58980	−	0.917871i	−0.596461	−	0.802642i	$-0.703427\pi$
$660$	0.493902	−	2.74417i	0.0192251	−	0.106817i
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$	69.9808	+	12.5953i	2.71783	+	0.489160i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	33.2073	+	19.1722i	1.28483	+	0.741796i
$669$	14.2470	−	5.12838i	0.550819	−	0.198275i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$	−12.7505	−	22.6368i	−0.490768	−	0.871290i
$676$	−75.7409			−2.91311
$677$	−28.2226	−	16.2943i	−1.08468	−	0.626242i	−0.152527	−	0.988299i	$-0.548741\pi$
$677$	−28.2226	−	16.2943i	−1.08468	−	0.626242i	−0.932156	+	0.362058i	$0.882074\pi$
$678$		0			0
$679$	−19.9352	−	34.5288i	−0.765043	−	1.32509i
$680$		0			0
$681$	−4.49390	−	12.4843i	−0.172207	−	0.478400i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$			22.3607i			0.850026i
$693$	−1.82097	−	2.20164i	−0.0691730	−	0.0836334i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	13.2288	−	22.9129i	0.500000	−	0.866025i
$701$			49.2851i			1.86147i	0.365690	+	0.930737i	$0.380833\pi$
$701$			49.2851i			1.86147i	−0.365690	+	0.930737i	$0.619167\pi$
$702$		0			0
$703$		0			0
$704$	2.49390	+	1.43985i	0.0939925	+	0.0542666i
$705$	−36.8691	−	31.1601i	−1.38857	−	1.17356i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	−0.856484	−	0.516174i	$-0.827356\pi$
$709$	0.500000	−	0.866025i	0.0187779	−	0.0325243i	0.875262	+	0.483650i	$0.160689\pi$
$710$		0			0
$711$	−44.6235	−	16.6006i	−1.67351	−	0.622570i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.87043	+	4.97172i	0.107348	+	0.185932i
$716$	−25.7530	+	14.8685i	−0.962437	+	0.555663i
$717$	−50.4245	−	9.07550i	−1.88314	−	0.338931i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	26.4575	−	4.47214i	0.986013	−	0.166667i
$721$	−7.00000			−0.260694
$722$		0			0
$723$		0			0
$724$		0			0
$725$	25.6174	−	14.7902i	0.951405	−	0.549294i
$726$		0			0
$727$	−24.5608	+	42.5405i	−0.910909	+	1.57774i	−0.0981255	+	0.995174i	$0.531285\pi$
$727$	−24.5608	+	42.5405i	−0.910909	+	1.57774i	−0.812783	+	0.582566i	$0.802049\pi$
$728$		0			0
$729$	−13.0000	−	23.6643i	−0.481481	−	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−25.1346	−	43.5345i	−0.928369	−	1.60798i	−0.786051	−	0.618161i	$-0.787878\pi$
$733$	−25.1346	−	43.5345i	−0.928369	−	1.60798i	−0.142318	−	0.989821i	$-0.545455\pi$
$734$		0			0
$735$	−9.18216	−	25.5086i	−0.338689	−	0.940898i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	11.0000			0.404642			0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000			0.404642			0.202321	+	0.979319i	$0.435152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$	−19.0382	+	32.9752i	−0.697507	+	1.20812i
$746$		0			0
$747$	−19.0581	+	51.2295i	−0.697300	+	1.87439i
$748$	4.14378			0.151512
$749$		0			0
$750$		0			0
$751$	6.50000	+	11.2583i	0.237188	+	0.410822i	0.959906	−	0.280321i	$-0.0904408\pi$
$751$	6.50000	+	11.2583i	0.237188	+	0.410822i	−0.722718	+	0.691143i	$0.757107\pi$
$752$	43.1767	−	24.9281i	1.57449	−	0.909033i
$753$		0			0
$754$		0			0
$755$		−	15.3627i		−	0.559107i
$756$	14.0000	−	23.6643i	0.509175	−	0.860663i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	13.0573	−	22.6160i	0.472708	−	0.818754i
$764$			11.8322i			0.428073i
$765$	29.7530	−	24.6087i	1.07572	−	0.889729i
$766$		0			0
$767$		0			0
$768$	−4.90893	+	27.2746i	−0.177136	+	0.984186i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	−50.6822	−	9.12187i	−1.82527	−	0.328516i
$772$		0			0
$773$			46.9990i			1.69044i	0.534421	+	0.845218i	$0.320530\pi$
$773$			46.9990i			1.69044i	−0.534421	+	0.845218i	$0.679470\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−35.6618	+	42.1956i	−1.27690	+	1.51084i
$781$	−2.93521	+	5.08394i	−0.105030	+	0.181918i
$782$		0			0
$783$	26.7842	−	15.0866i	0.957189	−	0.539153i
$784$	28.0000			1.00000
$785$	42.9939	+	24.8225i	1.53452	+	0.885954i
$786$		0			0
$787$	−25.7681	−	44.6317i	−0.918535	−	1.59095i	−0.801642	−	0.597804i	$-0.796040\pi$
$787$	−25.7681	−	44.6317i	−0.918535	−	1.59095i	−0.116892	−	0.993145i	$-0.537293\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.2532	+	10.5385i	−0.646562	+	0.373293i	−0.787138	−	0.616777i	$-0.788438\pi$
$797$	−18.2532	+	10.5385i	−0.646562	+	0.373293i	0.140576	+	0.990070i	$0.455105\pi$
$798$		0			0
$799$	35.8704	−	62.1294i	1.26900	−	2.19798i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.97250	−	1.13883i	−0.0696081	−	0.0401883i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	13.0687i		−	0.459471i	−0.973253	−	0.229736i	$-0.926214\pi$
$809$		−	13.0687i		−	0.459471i	0.973253	−	0.229736i	$-0.0737862\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	27.1109	+	15.6525i	0.951405	+	0.549294i
$813$		0			0
$814$		0			0
$815$		0			0
$816$	13.5061	+	37.5207i	0.472808	+	1.31349i
$817$		0			0
$818$		0			0
$819$	9.43521	+	55.8195i	0.329693	+	1.95049i
$820$		0			0
$821$	40.1883	+	23.2027i	1.40258	+	0.809780i	0.994657	−	0.103236i	$-0.0329198\pi$
$821$	40.1883	+	23.2027i	1.40258	+	0.809780i	0.407923	+	0.913016i	$0.366253\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	3.06808	+	0.552199i	0.106817	+	0.0192251i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−28.5294	−	49.4144i	−0.989079	−	1.71314i
$833$	34.8929	−	20.1454i	1.20897	−	0.697997i
$834$		0			0
$835$	−21.4352	+	37.1269i	−0.741796	+	1.28483i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	3.00000	+	5.19615i	0.103448	+	0.179178i
$842$		0			0
$843$	18.5004	+	51.3951i	0.637187	+	1.77014i
$844$	3.87043	−	6.70377i	0.133226	−	0.230753i
$845$		−	84.6808i		−	2.91311i
$846$		0			0
$847$	−28.7604			−0.988221
$848$		0			0
$849$	−8.50610	+	47.2609i	−0.291929	+	1.62199i
$850$		0			0
$851$		0			0
$852$	−55.6005	−	10.0071i	−1.90484	−	0.342837i
$853$	−21.1660	+	36.6606i	−0.724710	+	1.25524i	0.234383	+	0.972144i	$0.424693\pi$
$853$	−21.1660	+	36.6606i	−0.724710	+	1.25524i	−0.959093	+	0.283091i	$0.908640\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.51035	−	1.44935i	−0.0857520	−	0.0495090i	0.456511	−	0.889718i	$-0.349099\pi$
$857$	−2.51035	−	1.44935i	−0.0857520	−	0.0495090i	−0.542263	+	0.840209i	$0.682432\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−25.0000			−0.850026
$866$		0			0
$867$	21.3374	+	18.0334i	0.724657	+	0.612447i
$868$		0			0
$869$	4.94741	−	2.85639i	0.167829	−	0.0968963i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	42.3718	+	15.7629i	1.43407	+	0.533494i
$874$		0			0
$875$	25.6174	+	14.7902i	0.866025	+	0.500000i
$876$	3.88262	−	21.5723i	0.131182	−	0.728861i
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	41.9291	+	7.54648i	1.41423	+	0.254536i
$880$	−1.60981	+	2.78827i	−0.0542666	+	0.0939925i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−71.1052	−	41.0526i	−2.39153	−	1.38075i
$885$		0			0
$886$		0			0
$887$	30.9839	−	17.8885i	1.04034	−	0.600639i	0.120408	−	0.992725i	$-0.461580\pi$
$887$	30.9839	−	17.8885i	1.04034	−	0.600639i	0.919929	+	0.392086i	$0.128246\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	3.18216	+	0.607713i	0.106606	+	0.0203592i
$892$	−17.4843			−0.585418
$893$		0			0
$894$		0			0
$895$	−16.6235	−	28.7928i	−0.555663	−	0.962437i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	5.00000	+	29.5804i	0.166667	+	0.986013i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$			15.3212i			0.508450i
$909$		0			0
$910$		0			0
$911$	−16.6174	−	9.59405i	−0.550558	−	0.317865i	0.198789	−	0.980042i	$-0.436299\pi$
$911$	−16.6174	−	9.59405i	−0.550558	−	0.317865i	−0.749347	+	0.662177i	$0.769633\pi$
$912$		0			0
$913$	−3.27924	−	5.67982i	−0.108527	−	0.187975i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	29.0000			0.956622			0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000			0.956622			0.478311	+	0.878191i	$0.341249\pi$
$920$		0			0
$921$	30.4352	+	25.7224i	1.00287	+	0.847584i
$922$		0			0
$923$	100.734	−	58.1586i	3.31569	−	1.91431i
$924$	1.11738	+	3.10414i	0.0367590	+	0.102119i
$925$		0			0
$926$		0			0
$927$	6.11628	−	5.05876i	0.200885	−	0.166152i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$			4.63289i			0.151512i
$936$		0			0
$937$	24.6167			0.804191			0.402096	−	0.915598i	$-0.368282\pi$
$937$	24.6167			0.804191			0.402096	+	0.915598i	$0.368282\pi$
$938$		0			0
$939$	−47.4291	+	17.0728i	−1.54779	+	0.557148i
$940$	27.8704	+	48.2730i	0.909033	+	1.57449i
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	26.4575	+	15.6525i	0.860663	+	0.509175i
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$	41.9892	+	35.4874i	1.36375	+	1.15258i
$949$	22.5648	+	39.0834i	0.732484	+	1.26870i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−13.2288			−0.428073
$956$	51.2348	+	29.5804i	1.65705	+	0.956698i
$957$	−0.653370	+	3.63020i	−0.0211205	+	0.117348i
$958$		0			0
$959$		0			0
$960$	−30.4939	−	5.48835i	−0.984186	−	0.177136i
$961$	−15.5000	+	26.8468i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	4.86917	+	30.7943i	0.156179	+	0.987729i
$973$		0			0
$974$		0			0
$975$	−47.1761	−	39.8711i	−1.51084	−	1.27690i
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$			31.3050i			1.00000i
$981$	4.93521	+	29.1971i	0.157569	+	0.932192i
$982$		0			0
$983$	−29.0834	−	16.7913i	−0.927616	−	0.535559i	−0.0415592	−	0.999136i	$-0.513233\pi$
$983$	−29.0834	−	16.7913i	−0.927616	−	0.535559i	−0.886057	+	0.463577i	$0.846566\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	56.2141	+	10.1175i	1.78932	+	0.322044i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−4.74085			−0.150598			−0.0752991	−	0.997161i	$-0.523991\pi$
$991$	−4.74085			−0.150598			−0.0752991	+	0.997161i	$0.523991\pi$
$992$		0			0
$993$	−56.6166	+	20.3799i	−1.79667	+	0.646737i
$994$		0			0
$995$		0			0
$996$	40.7409	−	48.2052i	1.29092	−	1.52744i
$997$	−9.26013	+	16.0390i	−0.293271	+	0.507961i	−0.974581	−	0.224034i	$-0.928077\pi$
$997$	−9.26013	+	16.0390i	−0.293271	+	0.507961i	0.681310	+	0.731995i	$0.261411\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.a.209.1	yes	8
3.2	odd	2		945.2.z.a.629.4		8
5.4	even	2	inner	315.2.z.a.209.4	yes	8
7.6	odd	2	inner	315.2.z.a.209.4	yes	8
9.4	even	3		945.2.z.a.314.4		8
9.5	odd	6	inner	315.2.z.a.104.2	✓	8
15.14	odd	2		945.2.z.a.629.1		8
21.20	even	2		945.2.z.a.629.1		8
35.34	odd	2	CM	315.2.z.a.209.1	yes	8
45.4	even	6		945.2.z.a.314.1		8
45.14	odd	6	inner	315.2.z.a.104.3	yes	8
63.13	odd	6		945.2.z.a.314.1		8
63.41	even	6	inner	315.2.z.a.104.3	yes	8
105.104	even	2		945.2.z.a.629.4		8
315.104	even	6	inner	315.2.z.a.104.2	✓	8
315.139	odd	6		945.2.z.a.314.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.a.104.2	✓	8	9.5	odd	6	inner
315.2.z.a.104.2	✓	8	315.104	even	6	inner
315.2.z.a.104.3	yes	8	45.14	odd	6	inner
315.2.z.a.104.3	yes	8	63.41	even	6	inner
315.2.z.a.209.1	yes	8	1.1	even	1	trivial
315.2.z.a.209.1	yes	8	35.34	odd	2	CM
315.2.z.a.209.4	yes	8	5.4	even	2	inner
315.2.z.a.209.4	yes	8	7.6	odd	2	inner
945.2.z.a.314.1		8	45.4	even	6
945.2.z.a.314.1		8	63.13	odd	6
945.2.z.a.314.4		8	9.4	even	3
945.2.z.a.314.4		8	315.139	odd	6
945.2.z.a.629.1		8	15.14	odd	2
945.2.z.a.629.1		8	21.20	even	2
945.2.z.a.629.4		8	3.2	odd	2
945.2.z.a.629.4		8	105.104	even	2

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.32288 - 1.11803i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(1.32288 - 2.29129i) q^{7} +(0.500000 + 2.95804i) q^{9} +O(q^{10})\)	\(q+(-1.32288 - 1.11803i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-1.93649 + 1.11803i) q^{5} +(1.32288 - 2.29129i) q^{7} +(0.500000 + 2.95804i) q^{9} +(-0.311738 - 0.179982i) q^{11} +(0.613616 - 3.40932i) q^{12} +(3.56618 + 6.17680i) q^{13} +(3.81174 + 0.686044i) q^{15} +(-2.00000 + 3.46410i) q^{16} +5.75583i q^{17} +(-3.87298 - 2.23607i) q^{20} +(-4.31174 + 1.55207i) q^{21} +(2.50000 - 4.33013i) q^{25} +(2.64575 - 4.47214i) q^{27} +5.29150 q^{28} +(5.12348 + 2.95804i) q^{29} +(0.211164 + 0.586627i) q^{33} +5.91608i q^{35} +(-4.62348 + 3.82407i) q^{36} +(2.18826 - 12.1582i) q^{39} -0.719927i q^{44} +(-4.27543 - 5.16920i) q^{45} +(-10.7942 - 6.23202i) q^{47} +(6.51873 - 2.34651i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(6.43521 - 7.61425i) q^{51} +(-7.13235 + 12.3536i) q^{52} +0.804903 q^{55} +(2.62348 + 7.28817i) q^{60} +(7.43916 + 2.76748i) q^{63} -8.00000 q^{64} +(-13.8117 - 7.97421i) q^{65} +(-9.96939 + 5.75583i) q^{68} -16.3084i q^{71} +6.32745 q^{73} +(-8.14842 + 2.93313i) q^{75} +(-0.824780 + 0.476187i) q^{77} +(-7.93521 + 13.7442i) q^{79} -8.94427i q^{80} +(-8.50000 + 2.95804i) q^{81} +(15.7789 + 9.10993i) q^{83} +(-7.00000 - 5.91608i) q^{84} +(-6.43521 - 11.1461i) q^{85} +(-3.47053 - 9.64134i) q^{87} +18.8704 q^{91} +(7.53480 - 13.0507i) q^{97} +(0.376525 - 1.01212i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 4 q^{9}+O(q^{10}) \) 8 * q + 8 * q^4 + 4 * q^9	\( 8 q + 8 q^{4} + 4 q^{9} + 18 q^{11} + 10 q^{15} - 16 q^{16} - 14 q^{21} + 20 q^{25} + 4 q^{36} + 38 q^{39} - 28 q^{49} - 10 q^{51} - 20 q^{60} - 64 q^{64} - 90 q^{65} - 2 q^{79} - 68 q^{81} - 56 q^{84} + 10 q^{85} + 28 q^{91} + 44 q^{99}+O(q^{100}) \) 8 * q + 8 * q^4 + 4 * q^9 + 18 * q^11 + 10 * q^15 - 16 * q^16 - 14 * q^21 + 20 * q^25 + 4 * q^36 + 38 * q^39 - 28 * q^49 - 10 * q^51 - 20 * q^60 - 64 * q^64 - 90 * q^65 - 2 * q^79 - 68 * q^81 - 56 * q^84 + 10 * q^85 + 28 * q^91 + 44 * q^99

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