LMFDB - Embedded newform 672.2.k.a.545.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(545,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("672.545");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.k (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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 $ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			545.2
Root			$-0.599676 - 1.28078i$ of defining polynomial
Character	$\chi$	$=$	672.545
Dual form			672.2.k.a.545.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.66757 + 0.468213i) q^{3} +3.33513 q^{5} +(1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} +O(q^{10})$	\(q+(-1.66757 + 0.468213i) q^{3} +3.33513 q^{5} +(1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} +4.00000i q^{11} -5.20798i q^{13} +(-5.56155 + 1.56155i) q^{15} -1.87285 q^{17} -0.936426i q^{19} +(-1.60399 + 4.29269i) q^{21} -7.12311i q^{23} +6.12311 q^{25} +(-3.54042 + 3.80335i) q^{27} +7.12311i q^{29} -2.39871i q^{31} +(-1.87285 - 6.67026i) q^{33} +(5.20798 - 7.12311i) q^{35} +8.24621 q^{37} +(2.43845 + 8.68466i) q^{39} -1.87285 q^{41} +8.00000 q^{43} +(8.54312 - 5.20798i) q^{45} +6.67026 q^{47} +(-2.12311 - 6.67026i) q^{49} +(3.12311 - 0.876894i) q^{51} +0.876894i q^{53} +13.3405i q^{55} +(0.438447 + 1.56155i) q^{57} +7.08084 q^{59} +8.13254i q^{61} +(0.664868 - 7.90936i) q^{63} -17.3693i q^{65} -14.2462 q^{67} +(3.33513 + 11.8782i) q^{69} -2.24621i q^{71} +(-10.2107 + 2.86692i) q^{75} +(8.54312 + 6.24621i) q^{77} +3.12311 q^{79} +(4.12311 - 8.00000i) q^{81} +6.25969 q^{83} -6.24621 q^{85} +(-3.33513 - 11.8782i) q^{87} -4.79741 q^{89} +(-11.1231 - 8.13254i) q^{91} +(1.12311 + 4.00000i) q^{93} -3.12311i q^{95} -2.92456i q^{97} +(6.24621 + 10.2462i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^7 + 4 * q^9	$ 8 q - 4 q^{7} + 4 q^{9} - 28 q^{15} + 8 q^{21} + 16 q^{25} + 36 q^{39} + 64 q^{43} + 16 q^{49} - 8 q^{51} + 20 q^{57} + 32 q^{63} - 48 q^{67} - 8 q^{79} + 16 q^{85} - 56 q^{91} - 24 q^{93} - 16 q^{99}+O(q^{100}) $ 8 * q - 4 * q^7 + 4 * q^9 - 28 * q^15 + 8 * q^21 + 16 * q^25 + 36 * q^39 + 64 * q^43 + 16 * q^49 - 8 * q^51 + 20 * q^57 + 32 * q^63 - 48 * q^67 - 8 * q^79 + 16 * q^85 - 56 * q^91 - 24 * q^93 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.66757	+	0.468213i	−0.962770	+	0.270323i
$3$	−1.66757	+	0.468213i	−0.962770	+	0.270323i
$4$		0			0
$5$	3.33513			1.49152			0.745758	−	0.666217i	$-0.232087\pi$
$5$	3.33513			1.49152			0.745758	+	0.666217i	$0.232087\pi$
$6$		0			0
$7$	1.56155	−	2.13578i	0.590211	−	0.807249i
$7$	1.56155	−	2.13578i	0.590211	−	0.807249i
$8$		0			0
$9$	2.56155	−	1.56155i	0.853851	−	0.520518i
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$		−	5.20798i		−	1.44444i	−0.691666	−	0.722218i	$-0.743123\pi$
$13$		−	5.20798i		−	1.44444i	0.691666	−	0.722218i	$-0.256877\pi$
$14$		0			0
$15$	−5.56155	+	1.56155i	−1.43599	+	0.403191i
$16$		0			0
$17$	−1.87285			−0.454234			−0.227117	−	0.973868i	$-0.572930\pi$
$17$	−1.87285			−0.454234			−0.227117	+	0.973868i	$0.572930\pi$
$18$		0			0
$19$		−	0.936426i		−	0.214831i	−0.994214	−	0.107415i	$-0.965742\pi$
$19$		−	0.936426i		−	0.214831i	0.994214	−	0.107415i	$-0.0342575\pi$
$20$		0			0
$21$	−1.60399	+	4.29269i	−0.350020	+	0.936742i
$22$		0			0
$23$		−	7.12311i		−	1.48527i	−0.669696	−	0.742635i	$-0.733576\pi$
$23$		−	7.12311i		−	1.48527i	0.669696	−	0.742635i	$-0.266424\pi$
$24$		0			0
$25$	6.12311			1.22462
$26$		0			0
$27$	−3.54042	+	3.80335i	−0.681354	+	0.731954i
$28$		0			0
$29$			7.12311i			1.32273i	0.750065	+	0.661364i	$0.230022\pi$
$29$			7.12311i			1.32273i	−0.750065	+	0.661364i	$0.769978\pi$
$30$		0			0
$31$		−	2.39871i		−	0.430820i	−0.976524	−	0.215410i	$-0.930891\pi$
$31$		−	2.39871i		−	0.430820i	0.976524	−	0.215410i	$-0.0691088\pi$
$32$		0			0
$33$	−1.87285	−	6.67026i	−0.326022	−	1.16114i
$34$		0			0
$35$	5.20798	−	7.12311i	0.880310	−	1.20402i
$36$		0			0
$37$	8.24621			1.35567			0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621			1.35567			0.677834	+	0.735215i	$0.262919\pi$
$38$		0			0
$39$	2.43845	+	8.68466i	0.390464	+	1.39066i
$40$		0			0
$41$	−1.87285			−0.292490			−0.146245	−	0.989248i	$-0.546719\pi$
$41$	−1.87285			−0.292490			−0.146245	+	0.989248i	$0.546719\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$	8.54312	−	5.20798i	1.27353	−	0.776361i
$46$		0			0
$47$	6.67026			0.972958			0.486479	−	0.873692i	$-0.338281\pi$
$47$	6.67026			0.972958			0.486479	+	0.873692i	$0.338281\pi$
$48$		0			0
$49$	−2.12311	−	6.67026i	−0.303301	−	0.952895i
$50$		0			0
$51$	3.12311	−	0.876894i	0.437322	−	0.122790i
$52$		0			0
$53$			0.876894i			0.120451i	0.998185	+	0.0602254i	$0.0191819\pi$
$53$			0.876894i			0.120451i	−0.998185	+	0.0602254i	$0.980818\pi$
$54$		0			0
$55$			13.3405i			1.79884i
$56$		0			0
$57$	0.438447	+	1.56155i	0.0580737	+	0.206833i
$58$		0			0
$59$	7.08084			0.921847			0.460923	−	0.887440i	$-0.347518\pi$
$59$	7.08084			0.921847			0.460923	+	0.887440i	$0.347518\pi$
$60$		0			0
$61$			8.13254i			1.04127i	0.853781	+	0.520633i	$0.174304\pi$
$61$			8.13254i			1.04127i	−0.853781	+	0.520633i	$0.825696\pi$
$62$		0			0
$63$	0.664868	−	7.90936i	0.0837655	−	0.996485i
$64$		0			0
$65$		−	17.3693i		−	2.15440i
$66$		0			0
$67$	−14.2462			−1.74045			−0.870226	−	0.492653i	$-0.836027\pi$
$67$	−14.2462			−1.74045			−0.870226	+	0.492653i	$0.836027\pi$
$68$		0			0
$69$	3.33513	+	11.8782i	0.401503	+	1.42997i
$70$		0			0
$71$		−	2.24621i		−	0.266576i	−0.991077	−	0.133288i	$-0.957446\pi$
$71$		−	2.24621i		−	0.266576i	0.991077	−	0.133288i	$-0.0425536\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−10.2107	+	2.86692i	−1.17903	+	0.331043i
$76$		0			0
$77$	8.54312	+	6.24621i	0.973579	+	0.711822i
$78$		0			0
$79$	3.12311			0.351377			0.175688	−	0.984446i	$-0.443785\pi$
$79$	3.12311			0.351377			0.175688	+	0.984446i	$0.443785\pi$
$80$		0			0
$81$	4.12311	−	8.00000i	0.458123	−	0.888889i
$82$		0			0
$83$	6.25969			0.687090			0.343545	−	0.939136i	$-0.388372\pi$
$83$	6.25969			0.687090			0.343545	+	0.939136i	$0.388372\pi$
$84$		0			0
$85$	−6.24621			−0.677497
$86$		0			0
$87$	−3.33513	−	11.8782i	−0.357564	−	1.27348i
$88$		0			0
$89$	−4.79741			−0.508525			−0.254262	−	0.967135i	$-0.581833\pi$
$89$	−4.79741			−0.508525			−0.254262	+	0.967135i	$0.581833\pi$
$90$		0			0
$91$	−11.1231	−	8.13254i	−1.16602	−	0.852522i
$92$		0			0
$93$	1.12311	+	4.00000i	0.116461	+	0.414781i
$94$		0			0
$95$		−	3.12311i		−	0.320424i
$96$		0			0
$97$		−	2.92456i		−	0.296944i	−0.988917	−	0.148472i	$-0.952565\pi$
$97$		−	2.92456i		−	0.296944i	0.988917	−	0.148472i	$-0.0474355\pi$
$98$		0			0
$99$	6.24621	+	10.2462i	0.627768	+	1.02978i
$100$		0			0
$101$	−10.0054			−0.995574			−0.497787	−	0.867299i	$-0.665854\pi$
$101$	−10.0054			−0.995574			−0.497787	+	0.867299i	$0.665854\pi$
$102$		0			0
$103$			14.6875i			1.44721i	0.690217	+	0.723603i	$0.257515\pi$
$103$			14.6875i			1.44721i	−0.690217	+	0.723603i	$0.742485\pi$
$104$		0			0
$105$	−5.34953	+	14.3167i	−0.522060	+	1.39717i
$106$		0			0
$107$		−	2.24621i		−	0.217149i	−0.994088	−	0.108575i	$-0.965371\pi$
$107$		−	2.24621i		−	0.217149i	0.994088	−	0.108575i	$-0.0346287\pi$
$108$		0			0
$109$	−16.2462			−1.55610			−0.778052	−	0.628199i	$-0.783792\pi$
$109$	−16.2462			−1.55610			−0.778052	+	0.628199i	$0.783792\pi$
$110$		0			0
$111$	−13.7511	+	3.86098i	−1.30520	+	0.366468i
$112$		0			0
$113$			3.12311i			0.293797i	0.989152	+	0.146899i	$0.0469291\pi$
$113$			3.12311i			0.293797i	−0.989152	+	0.146899i	$0.953071\pi$
$114$		0			0
$115$		−	23.7565i		−	2.21530i
$116$		0			0
$117$	−8.13254	−	13.3405i	−0.751854	−	1.23333i
$118$		0			0
$119$	−2.92456	+	4.00000i	−0.268094	+	0.366679i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	3.12311	−	0.876894i	0.281601	−	0.0790669i
$124$		0			0
$125$	3.74571			0.335026
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	−13.3405	+	3.74571i	−1.17457	+	0.329791i
$130$		0			0
$131$	−10.0054			−0.874176			−0.437088	−	0.899419i	$-0.643990\pi$
$131$	−10.0054			−0.874176			−0.437088	+	0.899419i	$0.643990\pi$
$132$		0			0
$133$	−2.00000	−	1.46228i	−0.173422	−	0.126796i
$134$		0			0
$135$	−11.8078	+	12.6847i	−1.01625	+	1.09172i
$136$		0			0
$137$			14.2462i			1.21714i	0.793502	+	0.608568i	$0.208256\pi$
$137$			14.2462i			1.21714i	−0.793502	+	0.608568i	$0.791744\pi$
$138$		0			0
$139$			20.9472i			1.77672i	0.459148	+	0.888360i	$0.348155\pi$
$139$			20.9472i			1.77672i	−0.459148	+	0.888360i	$0.651845\pi$
$140$		0			0
$141$	−11.1231	+	3.12311i	−0.936734	+	0.263013i
$142$		0			0
$143$	20.8319			1.74205
$144$		0			0
$145$			23.7565i			1.97287i
$146$		0			0
$147$	6.66352	+	10.1290i	0.549598	+	0.835429i
$148$		0			0
$149$		−	8.87689i		−	0.727224i	−0.931551	−	0.363612i	$-0.881543\pi$
$149$		−	8.87689i		−	0.727224i	0.931551	−	0.363612i	$-0.118457\pi$
$150$		0			0
$151$	20.4924			1.66765			0.833825	−	0.552029i	$-0.186146\pi$
$151$	20.4924			1.66765			0.833825	+	0.552029i	$0.186146\pi$
$152$		0			0
$153$	−4.79741	+	2.92456i	−0.387848	+	0.236437i
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$			18.5485i			1.48033i	0.672424	+	0.740166i	$0.265253\pi$
$157$			18.5485i			1.48033i	−0.672424	+	0.740166i	$0.734747\pi$
$158$		0			0
$159$	−0.410574	−	1.46228i	−0.0325606	−	0.115966i
$160$		0			0
$161$	−15.2134	−	11.1231i	−1.19898	−	0.876624i
$162$		0			0
$163$	1.75379			0.137367			0.0686837	−	0.997638i	$-0.478120\pi$
$163$	1.75379			0.137367			0.0686837	+	0.997638i	$0.478120\pi$
$164$		0			0
$165$	−6.24621	−	22.2462i	−0.486267	−	1.73187i
$166$		0			0
$167$	−14.1617			−1.09586			−0.547932	−	0.836523i	$-0.684585\pi$
$167$	−14.1617			−1.09586			−0.547932	+	0.836523i	$0.684585\pi$
$168$		0			0
$169$	−14.1231			−1.08639
$170$		0			0
$171$	−1.46228	−	2.39871i	−0.111823	−	0.183434i
$172$		0			0
$173$	3.33513			0.253565			0.126783	−	0.991931i	$-0.459535\pi$
$173$	3.33513			0.253565			0.126783	+	0.991931i	$0.459535\pi$
$174$		0			0
$175$	9.56155	−	13.0776i	0.722785	−	0.988574i
$176$		0			0
$177$	−11.8078	+	3.31534i	−0.887526	+	0.249196i
$178$		0			0
$179$		−	24.4924i		−	1.83065i	−0.402716	−	0.915325i	$-0.631934\pi$
$179$		−	24.4924i		−	1.83065i	0.402716	−	0.915325i	$-0.368066\pi$
$180$		0			0
$181$		−	8.13254i		−	0.604487i	−0.953231	−	0.302244i	$-0.902264\pi$
$181$		−	8.13254i		−	0.604487i	0.953231	−	0.302244i	$-0.0977356\pi$
$182$		0			0
$183$	−3.80776	−	13.5616i	−0.281478	−	1.00250i
$184$		0			0
$185$	27.5022			2.02200
$186$		0			0
$187$		−	7.49141i		−	0.547826i
$188$		0			0
$189$	2.59455	+	13.5007i	0.188726	+	0.982030i
$190$		0			0
$191$			10.2462i			0.741390i	0.928755	+	0.370695i	$0.120880\pi$
$191$			10.2462i			0.741390i	−0.928755	+	0.370695i	$0.879120\pi$
$192$		0			0
$193$	9.12311			0.656696			0.328348	−	0.944557i	$-0.393508\pi$
$193$	9.12311			0.656696			0.328348	+	0.944557i	$0.393508\pi$
$194$		0			0
$195$	8.13254	+	28.9645i	0.582384	+	2.07419i
$196$		0			0
$197$		−	13.3693i		−	0.952524i	−0.879303	−	0.476262i	$-0.841991\pi$
$197$		−	13.3693i		−	0.952524i	0.879303	−	0.476262i	$-0.158009\pi$
$198$		0			0
$199$		−	7.19612i		−	0.510119i	−0.966925	−	0.255060i	$-0.917905\pi$
$199$		−	7.19612i		−	0.510119i	0.966925	−	0.255060i	$-0.0820951\pi$
$200$		0			0
$201$	23.7565	−	6.67026i	1.67565	−	0.470484i
$202$		0			0
$203$	15.2134	+	11.1231i	1.06777	+	0.780689i
$204$		0			0
$205$	−6.24621			−0.436254
$206$		0			0
$207$	−11.1231	−	18.2462i	−0.773109	−	1.26820i
$208$		0			0
$209$	3.74571			0.259096
$210$		0			0
$211$	−14.2462			−0.980750			−0.490375	−	0.871512i	$-0.663140\pi$
$211$	−14.2462			−0.980750			−0.490375	+	0.871512i	$0.663140\pi$
$212$		0			0
$213$	1.05171	+	3.74571i	0.0720617	+	0.256652i
$214$		0			0
$215$	26.6811			1.81963
$216$		0			0
$217$	−5.12311	−	3.74571i	−0.347779	−	0.254275i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			9.75379i			0.656111i
$222$		0			0
$223$			18.6638i			1.24982i	0.780697	+	0.624910i	$0.214864\pi$
$223$			18.6638i			1.24982i	−0.780697	+	0.624910i	$0.785136\pi$
$224$		0			0
$225$	15.6847	−	9.56155i	1.04564	−	0.637437i
$226$		0			0
$227$	−16.6757			−1.10680			−0.553401	−	0.832915i	$-0.686670\pi$
$227$	−16.6757			−1.10680			−0.553401	+	0.832915i	$0.686670\pi$
$228$		0			0
$229$			2.28343i			0.150893i	0.997150	+	0.0754465i	$0.0240382\pi$
$229$			2.28343i			0.150893i	−0.997150	+	0.0754465i	$0.975962\pi$
$230$		0			0
$231$	−17.1708	−	6.41597i	−1.12975	−	0.422140i
$232$		0			0
$233$			12.4924i			0.818406i	0.912443	+	0.409203i	$0.134193\pi$
$233$			12.4924i			0.818406i	−0.912443	+	0.409203i	$0.865807\pi$
$234$		0			0
$235$	22.2462			1.45118
$236$		0			0
$237$	−5.20798	+	1.46228i	−0.338295	+	0.0949852i
$238$		0			0
$239$		−	0.876894i		−	0.0567216i	−0.999598	−	0.0283608i	$-0.990971\pi$
$239$		−	0.876894i		−	0.0567216i	0.999598	−	0.0283608i	$-0.00902873\pi$
$240$		0			0
$241$		−	10.4160i		−	0.670952i	−0.942049	−	0.335476i	$-0.891103\pi$
$241$		−	10.4160i		−	0.670952i	0.942049	−	0.335476i	$-0.108897\pi$
$242$		0			0
$243$	−3.12985	+	15.2710i	−0.200780	+	0.979636i
$244$		0			0
$245$	−7.08084	−	22.2462i	−0.452378	−	1.42126i
$246$		0			0
$247$	−4.87689			−0.310309
$248$		0			0
$249$	−10.4384	+	2.93087i	−0.661510	+	0.185736i
$250$		0			0
$251$	6.25969			0.395108			0.197554	−	0.980292i	$-0.436700\pi$
$251$	6.25969			0.395108			0.197554	+	0.980292i	$0.436700\pi$
$252$		0			0
$253$	28.4924			1.79130
$254$		0			0
$255$	10.4160	−	2.92456i	0.652273	−	0.183143i
$256$		0			0
$257$	−12.2888			−0.766556			−0.383278	−	0.923633i	$-0.625205\pi$
$257$	−12.2888			−0.766556			−0.383278	+	0.923633i	$0.625205\pi$
$258$		0			0
$259$	12.8769	−	17.6121i	0.800131	−	1.09436i
$260$		0			0
$261$	11.1231	+	18.2462i	0.688503	+	1.12941i
$262$		0			0
$263$		−	5.75379i		−	0.354794i	−0.984139	−	0.177397i	$-0.943232\pi$
$263$		−	5.75379i		−	0.354794i	0.984139	−	0.177397i	$-0.0567676\pi$
$264$		0			0
$265$			2.92456i			0.179654i
$266$		0			0
$267$	8.00000	−	2.24621i	0.489592	−	0.137466i
$268$		0			0
$269$	−10.0054			−0.610040			−0.305020	−	0.952346i	$-0.598663\pi$
$269$	−10.0054			−0.610040			−0.305020	+	0.952346i	$0.598663\pi$
$270$		0			0
$271$			8.01726i			0.487014i	0.969899	+	0.243507i	$0.0782979\pi$
$271$			8.01726i			0.487014i	−0.969899	+	0.243507i	$0.921702\pi$
$272$		0			0
$273$	22.3563	+	8.35357i	1.35306	+	0.505581i
$274$		0			0
$275$			24.4924i			1.47695i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	−3.74571	−	6.14441i	−0.224250	−	0.367856i
$280$		0			0
$281$			16.0000i			0.954480i	0.878773	+	0.477240i	$0.158363\pi$
$281$			16.0000i			0.954480i	−0.878773	+	0.477240i	$0.841637\pi$
$282$		0			0
$283$		−	10.5312i		−	0.626018i	−0.949750	−	0.313009i	$-0.898663\pi$
$283$		−	10.5312i		−	0.626018i	0.949750	−	0.313009i	$-0.101337\pi$
$284$		0			0
$285$	1.46228	+	5.20798i	0.0866179	+	0.308494i
$286$		0			0
$287$	−2.92456	+	4.00000i	−0.172631	+	0.236113i
$288$		0			0
$289$	−13.4924			−0.793672
$290$		0			0
$291$	1.36932	+	4.87689i	0.0802708	+	0.285889i
$292$		0			0
$293$	−27.9128			−1.63068			−0.815341	−	0.578981i	$-0.803450\pi$
$293$	−27.9128			−1.63068			−0.815341	+	0.578981i	$0.803450\pi$
$294$		0			0
$295$	23.6155			1.37495
$296$		0			0
$297$	−15.2134	−	14.1617i	−0.882770	−	0.821744i
$298$		0			0
$299$	−37.0970			−2.14538
$300$		0			0
$301$	12.4924	−	17.0862i	0.720051	−	0.984834i
$302$		0			0
$303$	16.6847	−	4.68466i	0.958509	−	0.269127i
$304$		0			0
$305$			27.1231i			1.55306i
$306$		0			0
$307$			19.8955i			1.13550i	0.823202	+	0.567748i	$0.192185\pi$
$307$			19.8955i			1.13550i	−0.823202	+	0.567748i	$0.807815\pi$
$308$		0			0
$309$	−6.87689	−	24.4924i	−0.391213	−	1.39333i
$310$		0			0
$311$	−13.3405			−0.756472			−0.378236	−	0.925709i	$-0.623469\pi$
$311$	−13.3405			−0.756472			−0.378236	+	0.925709i	$0.623469\pi$
$312$		0			0
$313$		−	17.9074i		−	1.01219i	−0.862479	−	0.506093i	$-0.831089\pi$
$313$		−	17.9074i		−	1.01219i	0.862479	−	0.506093i	$-0.168911\pi$
$314$		0			0
$315$	2.21742	−	26.3788i	0.124938	−	1.48627i
$316$		0			0
$317$		−	0.876894i		−	0.0492513i	−0.999697	−	0.0246256i	$-0.992161\pi$
$317$		−	0.876894i		−	0.0492513i	0.999697	−	0.0246256i	$-0.00783938\pi$
$318$		0			0
$319$	−28.4924			−1.59527
$320$		0			0
$321$	1.05171	+	3.74571i	0.0587005	+	0.209065i
$322$		0			0
$323$			1.75379i			0.0975834i
$324$		0			0
$325$		−	31.8890i		−	1.76889i
$326$		0			0
$327$	27.0916	−	7.60669i	1.49817	−	0.420651i
$328$		0			0
$329$	10.4160	−	14.2462i	0.574251	−	0.785419i
$330$		0			0
$331$	−30.2462			−1.66248			−0.831241	−	0.555912i	$-0.812369\pi$
$331$	−30.2462			−1.66248			−0.831241	+	0.555912i	$0.812369\pi$
$332$		0			0
$333$	21.1231	−	12.8769i	1.15754	−	0.705649i
$334$		0			0
$335$	−47.5130			−2.59591
$336$		0			0
$337$	9.61553			0.523791			0.261896	−	0.965096i	$-0.415652\pi$
$337$	9.61553			0.523791			0.261896	+	0.965096i	$0.415652\pi$
$338$		0			0
$339$	−1.46228	−	5.20798i	−0.0794201	−	0.282859i
$340$		0			0
$341$	9.59482			0.519589
$342$		0			0
$343$	−17.5616	−	5.88148i	−0.948235	−	0.317570i
$344$		0			0
$345$	11.1231	+	39.6155i	0.598848	+	2.13283i
$346$		0			0
$347$			22.7386i			1.22067i	0.792142	+	0.610337i	$0.208966\pi$
$347$			22.7386i			1.22067i	−0.792142	+	0.610337i	$0.791034\pi$
$348$		0			0
$349$			5.20798i			0.278777i	0.990238	+	0.139389i	$0.0445137\pi$
$349$			5.20798i			0.278777i	−0.990238	+	0.139389i	$0.955486\pi$
$350$		0			0
$351$	19.8078	+	18.4384i	1.05726	+	0.984172i
$352$		0			0
$353$	14.3922			0.766021			0.383011	−	0.923744i	$-0.374887\pi$
$353$	14.3922			0.766021			0.383011	+	0.923744i	$0.374887\pi$
$354$		0			0
$355$		−	7.49141i		−	0.397603i
$356$		0			0
$357$	3.00404	−	8.03958i	0.158991	−	0.425500i
$358$		0			0
$359$		−	13.3693i		−	0.705606i	−0.935698	−	0.352803i	$-0.885229\pi$
$359$		−	13.3693i		−	0.705606i	0.935698	−	0.352803i	$-0.114771\pi$
$360$		0			0
$361$	18.1231			0.953848
$362$		0			0
$363$	8.33783	−	2.34107i	0.437623	−	0.122874i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	0.525853i		−	0.0274493i	−0.999906	−	0.0137246i	$-0.995631\pi$
$367$		−	0.525853i		−	0.0274493i	0.999906	−	0.0137246i	$-0.00436883\pi$
$368$		0			0
$369$	−4.79741	+	2.92456i	−0.249743	+	0.152246i
$370$		0			0
$371$	1.87285	+	1.36932i	0.0972337	+	0.0710914i
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$	−6.24621	+	1.75379i	−0.322553	+	0.0905653i
$376$		0			0
$377$	37.0970			1.91059
$378$		0			0
$379$	−24.0000			−1.23280			−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000			−1.23280			−0.616399	+	0.787434i	$0.711409\pi$
$380$		0			0
$381$	13.3405	−	3.74571i	0.683456	−	0.191898i
$382$		0			0
$383$	−0.821147			−0.0419587			−0.0209793	−	0.999780i	$-0.506678\pi$
$383$	−0.821147			−0.0419587			−0.0209793	+	0.999780i	$0.506678\pi$
$384$		0			0
$385$	28.4924	+	20.8319i	1.45211	+	1.06169i
$386$		0			0
$387$	20.4924	−	12.4924i	1.04169	−	0.635026i
$388$		0			0
$389$			15.1231i			0.766772i	0.923588	+	0.383386i	$0.125242\pi$
$389$			15.1231i			0.766772i	−0.923588	+	0.383386i	$0.874758\pi$
$390$		0			0
$391$			13.3405i			0.674660i
$392$		0			0
$393$	16.6847	−	4.68466i	0.841630	−	0.236310i
$394$		0			0
$395$	10.4160			0.524084
$396$		0			0
$397$		−	15.6240i		−	0.784144i	−0.919935	−	0.392072i	$-0.871758\pi$
$397$		−	15.6240i		−	0.784144i	0.919935	−	0.392072i	$-0.128242\pi$
$398$		0			0
$399$	4.01979	+	1.50202i	0.201241	+	0.0751951i
$400$		0			0
$401$		−	4.87689i		−	0.243540i	−0.992558	−	0.121770i	$-0.961143\pi$
$401$		−	4.87689i		−	0.243540i	0.992558	−	0.121770i	$-0.0388571\pi$
$402$		0			0
$403$	−12.4924			−0.622292
$404$		0			0
$405$	13.7511	−	26.6811i	0.683298	−	1.32579i
$406$		0			0
$407$			32.9848i			1.63500i
$408$		0			0
$409$			37.0970i			1.83433i	0.398508	+	0.917165i	$0.369528\pi$
$409$			37.0970i			1.83433i	−0.398508	+	0.917165i	$0.630472\pi$
$410$		0			0
$411$	−6.67026	−	23.7565i	−0.329020	−	1.17182i
$412$		0			0
$413$	11.0571	−	15.1231i	0.544084	−	0.744159i
$414$		0			0
$415$	20.8769			1.02481
$416$		0			0
$417$	−9.80776	−	34.9309i	−0.480288	−	1.71057i
$418$		0			0
$419$	19.6002			0.957533			0.478767	−	0.877942i	$-0.341084\pi$
$419$	19.6002			0.957533			0.478767	+	0.877942i	$0.341084\pi$
$420$		0			0
$421$	18.0000			0.877266			0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000			0.877266			0.438633	+	0.898666i	$0.355463\pi$
$422$		0			0
$423$	17.0862	−	10.4160i	0.830761	−	0.506442i
$424$		0			0
$425$	−11.4677			−0.556264
$426$		0			0
$427$	17.3693	+	12.6994i	0.840560	+	0.614567i
$428$		0			0
$429$	−34.7386	+	9.75379i	−1.67720	+	0.470917i
$430$		0			0
$431$			24.8769i			1.19828i	0.800645	+	0.599139i	$0.204490\pi$
$431$			24.8769i			1.19828i	−0.800645	+	0.599139i	$0.795510\pi$
$432$		0			0
$433$		−	17.9074i		−	0.860574i	−0.902692	−	0.430287i	$-0.858412\pi$
$433$		−	17.9074i		−	0.860574i	0.902692	−	0.430287i	$-0.141588\pi$
$434$		0			0
$435$	−11.1231	−	39.6155i	−0.533312	−	1.89942i
$436$		0			0
$437$	−6.67026			−0.319082
$438$		0			0
$439$			24.0518i			1.14793i	0.818880	+	0.573965i	$0.194595\pi$
$439$			24.0518i			1.14793i	−0.818880	+	0.573965i	$0.805405\pi$
$440$		0			0
$441$	−15.8544	−	13.7709i	−0.754972	−	0.655757i
$442$		0			0
$443$		−	21.7538i		−	1.03355i	−0.856120	−	0.516777i	$-0.827132\pi$
$443$		−	21.7538i		−	1.03355i	0.856120	−	0.516777i	$-0.172868\pi$
$444$		0			0
$445$	−16.0000			−0.758473
$446$		0			0
$447$	4.15628	+	14.8028i	0.196585	+	0.700149i
$448$		0			0
$449$		−	14.2462i		−	0.672320i	−0.941805	−	0.336160i	$-0.890872\pi$
$449$		−	14.2462i		−	0.672320i	0.941805	−	0.336160i	$-0.109128\pi$
$450$		0			0
$451$		−	7.49141i		−	0.352757i
$452$		0			0
$453$	−34.1725	+	9.59482i	−1.60556	+	0.450804i
$454$		0			0
$455$	−37.0970	−	27.1231i	−1.73914	−	1.27155i
$456$		0			0
$457$	11.3693			0.531834			0.265917	−	0.963996i	$-0.414325\pi$
$457$	11.3693			0.531834			0.265917	+	0.963996i	$0.414325\pi$
$458$		0			0
$459$	6.63068	−	7.12311i	0.309494	−	0.332478i
$460$		0			0
$461$	27.0916			1.26178			0.630891	−	0.775871i	$-0.282689\pi$
$461$	27.0916			1.26178			0.630891	+	0.775871i	$0.282689\pi$
$462$		0			0
$463$	6.63068			0.308154			0.154077	−	0.988059i	$-0.450760\pi$
$463$	6.63068			0.308154			0.154077	+	0.988059i	$0.450760\pi$
$464$		0			0
$465$	3.74571	+	13.3405i	0.173703	+	0.618652i
$466$		0			0
$467$	−16.6757			−0.771658			−0.385829	−	0.922570i	$-0.626084\pi$
$467$	−16.6757			−0.771658			−0.385829	+	0.922570i	$0.626084\pi$
$468$		0			0
$469$	−22.2462	+	30.4268i	−1.02723	+	1.40498i
$470$		0			0
$471$	−8.68466	−	30.9309i	−0.400168	−	1.42522i
$472$		0			0
$473$			32.0000i			1.47136i
$474$		0			0
$475$		−	5.73384i		−	0.263087i
$476$		0			0
$477$	1.36932	+	2.24621i	0.0626967	+	0.102847i
$478$		0			0
$479$	−0.821147			−0.0375192			−0.0187596	−	0.999824i	$-0.505972\pi$
$479$	−0.821147			−0.0375192			−0.0187596	+	0.999824i	$0.505972\pi$
$480$		0			0
$481$		−	42.9461i		−	1.95818i
$482$		0			0
$483$	30.5773	+	11.4254i	1.39132	+	0.519874i
$484$		0			0
$485$		−	9.75379i		−	0.442897i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$		0			0
$489$	−2.92456	+	0.821147i	−0.132253	+	0.0371336i
$490$		0			0
$491$			16.4924i			0.744293i	0.928174	+	0.372146i	$0.121378\pi$
$491$			16.4924i			0.744293i	−0.928174	+	0.372146i	$0.878622\pi$
$492$		0			0
$493$		−	13.3405i		−	0.600827i
$494$		0			0
$495$	20.8319	+	34.1725i	0.936326	+	1.53594i
$496$		0			0
$497$	−4.79741	−	3.50758i	−0.215193	−	0.157336i
$498$		0			0
$499$	20.4924			0.917367			0.458683	−	0.888600i	$-0.348321\pi$
$499$	20.4924			0.917367			0.458683	+	0.888600i	$0.348321\pi$
$500$		0			0
$501$	23.6155	−	6.63068i	1.05506	−	0.296237i
$502$		0			0
$503$	20.8319			0.928850			0.464425	−	0.885612i	$-0.346261\pi$
$503$	20.8319			0.928850			0.464425	+	0.885612i	$0.346261\pi$
$504$		0			0
$505$	−33.3693			−1.48492
$506$		0			0
$507$	23.5512	−	6.61262i	1.04595	−	0.293677i
$508$		0			0
$509$	−2.51398			−0.111430			−0.0557152	−	0.998447i	$-0.517744\pi$
$509$	−2.51398			−0.111430			−0.0557152	+	0.998447i	$0.517744\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	3.56155	+	3.31534i	0.157246	+	0.146376i
$514$		0			0
$515$			48.9848i			2.15853i
$516$		0			0
$517$			26.6811i			1.17343i
$518$		0			0
$519$	−5.56155	+	1.56155i	−0.244125	+	0.0685446i
$520$		0			0
$521$	−36.0453			−1.57917			−0.789587	−	0.613639i	$-0.789705\pi$
$521$	−36.0453			−1.57917			−0.789587	+	0.613639i	$0.789705\pi$
$522$		0			0
$523$			10.5312i			0.460499i	0.973132	+	0.230250i	$0.0739543\pi$
$523$			10.5312i			0.460499i	−0.973132	+	0.230250i	$0.926046\pi$
$524$		0			0
$525$	−9.82141	+	26.2846i	−0.428642	+	1.14715i
$526$		0			0
$527$			4.49242i			0.195693i
$528$		0			0
$529$	−27.7386			−1.20603
$530$		0			0
$531$	18.1379	−	11.0571i	0.787120	−	0.479837i
$532$		0			0
$533$			9.75379i			0.422483i
$534$		0			0
$535$		−	7.49141i		−	0.323882i
$536$		0			0
$537$	11.4677	+	40.8427i	0.494867	+	1.76249i
$538$		0			0
$539$	26.6811	−	8.49242i	1.14923	−	0.365795i
$540$		0			0
$541$	−18.4924			−0.795051			−0.397526	−	0.917591i	$-0.630131\pi$
$541$	−18.4924			−0.795051			−0.397526	+	0.917591i	$0.630131\pi$
$542$		0			0
$543$	3.80776	+	13.5616i	0.163407	+	0.581982i
$544$		0			0
$545$	−54.1833			−2.32096
$546$		0			0
$547$	−10.7386			−0.459151			−0.229575	−	0.973291i	$-0.573734\pi$
$547$	−10.7386			−0.459151			−0.229575	+	0.973291i	$0.573734\pi$
$548$		0			0
$549$	12.6994	+	20.8319i	0.541997	+	0.889085i
$550$		0			0
$551$	6.67026			0.284163
$552$		0			0
$553$	4.87689	−	6.67026i	0.207387	−	0.283648i
$554$		0			0
$555$	−45.8617	+	12.8769i	−1.94672	+	0.546594i
$556$		0			0
$557$			43.6155i			1.84805i	0.382333	+	0.924025i	$0.375121\pi$
$557$			43.6155i			1.84805i	−0.382333	+	0.924025i	$0.624879\pi$
$558$		0			0
$559$		−	41.6639i		−	1.76219i
$560$		0			0
$561$	3.50758	+	12.4924i	0.148090	+	0.527430i
$562$		0			0
$563$	−23.3459			−0.983913			−0.491957	−	0.870620i	$-0.663718\pi$
$563$	−23.3459			−0.983913			−0.491957	+	0.870620i	$0.663718\pi$
$564$		0			0
$565$			10.4160i			0.438203i
$566$		0			0
$567$	−10.6478	−	21.2985i	−0.447165	−	0.894451i
$568$		0			0
$569$		−	14.6307i		−	0.613350i	−0.951814	−	0.306675i	$-0.900784\pi$
$569$		−	14.6307i		−	0.613350i	0.951814	−	0.306675i	$-0.0992165\pi$
$570$		0			0
$571$	26.7386			1.11898			0.559488	−	0.828838i	$-0.310998\pi$
$571$	26.7386			1.11898			0.559488	+	0.828838i	$0.310998\pi$
$572$		0			0
$573$	−4.79741	−	17.0862i	−0.200415	−	0.713788i
$574$		0			0
$575$		−	43.6155i		−	1.81889i
$576$		0			0
$577$		−	19.1896i		−	0.798875i	−0.916760	−	0.399438i	$-0.869205\pi$
$577$		−	19.1896i		−	0.798875i	0.916760	−	0.399438i	$-0.130795\pi$
$578$		0			0
$579$	−15.2134	+	4.27156i	−0.632247	+	0.177520i
$580$		0			0
$581$	9.77484	−	13.3693i	0.405529	−	0.554653i
$582$		0			0
$583$	−3.50758			−0.145269
$584$		0			0
$585$	−27.1231	−	44.4924i	−1.12140	−	1.83954i
$586$		0			0
$587$	−33.7619			−1.39350			−0.696751	−	0.717313i	$-0.745372\pi$
$587$	−33.7619			−1.39350			−0.696751	+	0.717313i	$0.745372\pi$
$588$		0			0
$589$	−2.24621			−0.0925535
$590$		0			0
$591$	6.25969	+	22.2942i	0.257489	+	0.917062i
$592$		0			0
$593$	42.7156			1.75412			0.877059	−	0.480382i	$-0.159502\pi$
$593$	42.7156			1.75412			0.877059	+	0.480382i	$0.159502\pi$
$594$		0			0
$595$	−9.75379	+	13.3405i	−0.399866	+	0.546908i
$596$		0			0
$597$	3.36932	+	12.0000i	0.137897	+	0.491127i
$598$		0			0
$599$			10.2462i			0.418649i	0.977846	+	0.209324i	$0.0671265\pi$
$599$			10.2462i			0.418649i	−0.977846	+	0.209324i	$0.932874\pi$
$600$		0			0
$601$			20.8319i			0.849753i	0.905251	+	0.424876i	$0.139682\pi$
$601$			20.8319i			0.849753i	−0.905251	+	0.424876i	$0.860318\pi$
$602$		0			0
$603$	−36.4924	+	22.2462i	−1.48609	+	0.905936i
$604$		0			0
$605$	−16.6757			−0.677962
$606$		0			0
$607$		−	34.6983i		−	1.40836i	−0.710021	−	0.704181i	$-0.751314\pi$
$607$		−	34.6983i		−	1.40836i	0.710021	−	0.704181i	$-0.248686\pi$
$608$		0			0
$609$	−30.5773	−	11.4254i	−1.23905	−	0.462981i
$610$		0			0
$611$		−	34.7386i		−	1.40537i
$612$		0			0
$613$	10.4924			0.423785			0.211892	−	0.977293i	$-0.432037\pi$
$613$	10.4924			0.423785			0.211892	+	0.977293i	$0.432037\pi$
$614$		0			0
$615$	10.4160	−	2.92456i	0.420012	−	0.117930i
$616$		0			0
$617$		−	31.6155i		−	1.27279i	−0.771362	−	0.636397i	$-0.780424\pi$
$617$		−	31.6155i		−	1.27279i	0.771362	−	0.636397i	$-0.219576\pi$
$618$		0			0
$619$			31.3632i			1.26059i	0.776355	+	0.630296i	$0.217067\pi$
$619$			31.3632i			1.26059i	−0.776355	+	0.630296i	$0.782933\pi$
$620$		0			0
$621$	27.0916	+	25.2188i	1.08715	+	1.01199i
$622$		0			0
$623$	−7.49141	+	10.2462i	−0.300137	+	0.410506i
$624$		0			0
$625$	−18.1231			−0.724924
$626$		0			0
$627$	−6.24621	+	1.75379i	−0.249450	+	0.0700396i
$628$		0			0
$629$	−15.4439			−0.615790
$630$		0			0
$631$	−3.12311			−0.124329			−0.0621644	−	0.998066i	$-0.519800\pi$
$631$	−3.12311			−0.124329			−0.0621644	+	0.998066i	$0.519800\pi$
$632$		0			0
$633$	23.7565	−	6.67026i	0.944236	−	0.265119i
$634$		0			0
$635$	−26.6811			−1.05881
$636$		0			0
$637$	−34.7386	+	11.0571i	−1.37639	+	0.438098i
$638$		0			0
$639$	−3.50758	−	5.75379i	−0.138758	−	0.227616i
$640$		0			0
$641$		−	36.1080i		−	1.42618i	−0.701073	−	0.713089i	$-0.747295\pi$
$641$		−	36.1080i		−	1.42618i	0.701073	−	0.713089i	$-0.252705\pi$
$642$		0			0
$643$		−	20.1261i		−	0.793695i	−0.917885	−	0.396847i	$-0.870104\pi$
$643$		−	20.1261i		−	0.793695i	0.917885	−	0.396847i	$-0.129896\pi$
$644$		0			0
$645$	−44.4924	+	12.4924i	−1.75189	+	0.491889i
$646$		0			0
$647$	6.67026			0.262235			0.131118	−	0.991367i	$-0.458143\pi$
$647$	6.67026			0.262235			0.131118	+	0.991367i	$0.458143\pi$
$648$		0			0
$649$			28.3234i			1.11179i
$650$		0			0
$651$	10.2969	+	3.84751i	0.403568	+	0.150796i
$652$		0			0
$653$		−	48.1080i		−	1.88261i	−0.337558	−	0.941305i	$-0.609601\pi$
$653$		−	48.1080i		−	1.88261i	0.337558	−	0.941305i	$-0.390399\pi$
$654$		0			0
$655$	−33.3693			−1.30385
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.2462i			1.02241i	0.859460	+	0.511204i	$0.170800\pi$
$659$			26.2462i			1.02241i	−0.859460	+	0.511204i	$0.829200\pi$
$660$		0			0
$661$			37.7382i			1.46784i	0.679234	+	0.733922i	$0.262312\pi$
$661$			37.7382i			1.46784i	−0.679234	+	0.733922i	$0.737688\pi$
$662$		0			0
$663$	−4.56685	−	16.2651i	−0.177362	−	0.631684i
$664$		0			0
$665$	−6.67026	−	4.87689i	−0.258662	−	0.189118i
$666$		0			0
$667$	50.7386			1.96461
$668$		0			0
$669$	−8.73863	−	31.1231i	−0.337855	−	1.20329i
$670$		0			0
$671$	−32.5302			−1.25581
$672$		0			0
$673$	−24.2462			−0.934623			−0.467311	−	0.884093i	$-0.654777\pi$
$673$	−24.2462			−0.934623			−0.467311	+	0.884093i	$0.654777\pi$
$674$		0			0
$675$	−21.6784	+	23.2883i	−0.834400	+	0.896366i
$676$		0			0
$677$	27.0916			1.04122			0.520608	−	0.853796i	$-0.325705\pi$
$677$	27.0916			1.04122			0.520608	+	0.853796i	$0.325705\pi$
$678$		0			0
$679$	−6.24621	−	4.56685i	−0.239708	−	0.175260i
$680$		0			0
$681$	27.8078	−	7.80776i	1.06560	−	0.299194i
$682$		0			0
$683$		−	36.9848i		−	1.41519i	−0.706620	−	0.707593i	$-0.749781\pi$
$683$		−	36.9848i		−	1.41519i	0.706620	−	0.707593i	$-0.250219\pi$
$684$		0			0
$685$			47.5130i			1.81538i
$686$		0			0
$687$	−1.06913	−	3.80776i	−0.0407899	−	0.145275i
$688$		0			0
$689$	4.56685			0.173983
$690$		0			0
$691$		−	22.8201i		−	0.868116i	−0.900885	−	0.434058i	$-0.857081\pi$
$691$		−	22.8201i		−	0.868116i	0.900885	−	0.434058i	$-0.142919\pi$
$692$		0			0
$693$	31.6374	+	2.65947i	1.20181	+	0.101025i
$694$		0			0
$695$			69.8617i			2.65001i
$696$		0			0
$697$	3.50758			0.132859
$698$		0			0
$699$	−5.84912	−	20.8319i	−0.221234	−	0.787936i
$700$		0			0
$701$			46.3542i			1.75077i	0.483424	+	0.875386i	$0.339393\pi$
$701$			46.3542i			1.75077i	−0.483424	+	0.875386i	$0.660607\pi$
$702$		0			0
$703$		−	7.72197i		−	0.291240i
$704$		0			0
$705$	−37.0970	+	10.4160i	−1.39715	+	0.392288i
$706$		0			0
$707$	−15.6240	+	21.3693i	−0.587599	+	0.803676i
$708$		0			0
$709$	−11.7538			−0.441423			−0.220711	−	0.975339i	$-0.570838\pi$
$709$	−11.7538			−0.441423			−0.220711	+	0.975339i	$0.570838\pi$
$710$		0			0
$711$	8.00000	−	4.87689i	0.300023	−	0.182898i
$712$		0			0
$713$	−17.0862			−0.639884
$714$		0			0
$715$	69.4773			2.59830
$716$		0			0
$717$	0.410574	+	1.46228i	0.0153331	+	0.0546098i
$718$		0			0
$719$	48.3341			1.80256			0.901280	−	0.433238i	$-0.142629\pi$
$719$	48.3341			1.80256			0.901280	+	0.433238i	$0.142629\pi$
$720$		0			0
$721$	31.3693	+	22.9354i	1.16825	+	0.854157i
$722$		0			0
$723$	4.87689	+	17.3693i	0.181374	+	0.645972i
$724$		0			0
$725$			43.6155i			1.61984i
$726$		0			0
$727$		−	4.50212i		−	0.166974i	−0.996509	−	0.0834871i	$-0.973394\pi$
$727$		−	4.50212i		−	0.166974i	0.996509	−	0.0834871i	$-0.0266058\pi$
$728$		0			0
$729$	−1.93087	−	26.9309i	−0.0715137	−	0.997440i
$730$		0			0
$731$	−14.9828			−0.554160
$732$		0			0
$733$		−	48.1541i		−	1.77861i	−0.457311	−	0.889307i	$-0.651188\pi$
$733$		−	48.1541i		−	1.77861i	0.457311	−	0.889307i	$-0.348812\pi$
$734$		0			0
$735$	22.2237	+	33.7817i	0.819735	+	1.24606i
$736$		0			0
$737$		−	56.9848i		−	2.09906i
$738$		0			0
$739$	−24.0000			−0.882854			−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000			−0.882854			−0.441427	+	0.897297i	$0.645528\pi$
$740$		0			0
$741$	8.13254	−	2.28343i	0.298756	−	0.0838838i
$742$		0			0
$743$			8.87689i			0.325662i	0.986654	+	0.162831i	$0.0520625\pi$
$743$			8.87689i			0.325662i	−0.986654	+	0.162831i	$0.947938\pi$
$744$		0			0
$745$		−	29.6056i		−	1.08467i
$746$		0			0
$747$	16.0345	−	9.77484i	0.586673	−	0.357643i
$748$		0			0
$749$	−4.79741	−	3.50758i	−0.175294	−	0.128164i
$750$		0			0
$751$	−32.9848			−1.20363			−0.601817	−	0.798634i	$-0.705556\pi$
$751$	−32.9848			−1.20363			−0.601817	+	0.798634i	$0.705556\pi$
$752$		0			0
$753$	−10.4384	+	2.93087i	−0.380398	+	0.106807i
$754$		0			0
$755$	68.3449			2.48733
$756$		0			0
$757$	1.50758			0.0547938			0.0273969	−	0.999625i	$-0.491278\pi$
$757$	1.50758			0.0547938			0.0273969	+	0.999625i	$0.491278\pi$
$758$		0			0
$759$	−47.5130	+	13.3405i	−1.72461	+	0.484230i
$760$		0			0
$761$	−36.0453			−1.30664			−0.653321	−	0.757081i	$-0.726625\pi$
$761$	−36.0453			−1.30664			−0.653321	+	0.757081i	$0.726625\pi$
$762$		0			0
$763$	−25.3693	+	34.6983i	−0.918431	+	1.25616i
$764$		0			0
$765$	−16.0000	+	9.75379i	−0.578481	+	0.352649i
$766$		0			0
$767$		−	36.8769i		−	1.33155i
$768$		0			0
$769$			31.2479i			1.12683i	0.826175	+	0.563414i	$0.190512\pi$
$769$			31.2479i			1.12683i	−0.826175	+	0.563414i	$0.809488\pi$
$770$		0			0
$771$	20.4924	−	5.75379i	0.738017	−	0.207218i
$772$		0			0
$773$	43.3567			1.55943			0.779716	−	0.626133i	$-0.215363\pi$
$773$	43.3567			1.55943			0.779716	+	0.626133i	$0.215363\pi$
$774$		0			0
$775$		−	14.6875i		−	0.527592i
$776$		0			0
$777$	−13.2269	+	35.3984i	−0.474511	+	1.26991i
$778$		0			0
$779$			1.75379i			0.0628360i
$780$		0			0
$781$	8.98485			0.321503
$782$		0			0
$783$	−27.0916	−	25.2188i	−0.968176	−	0.901246i
$784$		0			0
$785$			61.8617i			2.20794i
$786$		0			0
$787$			40.9580i			1.46000i	0.683450	+	0.729998i	$0.260479\pi$
$787$			40.9580i			1.46000i	−0.683450	+	0.729998i	$0.739521\pi$
$788$		0			0
$789$	2.69400	+	9.59482i	0.0959089	+	0.341585i
$790$		0			0
$791$	6.67026	+	4.87689i	0.237167	+	0.173402i
$792$		0			0
$793$	42.3542			1.50404
$794$		0			0
$795$	−1.36932	−	4.87689i	−0.0485647	−	0.172966i
$796$		0			0
$797$	19.6002			0.694275			0.347138	−	0.937814i	$-0.387154\pi$
$797$	19.6002			0.694275			0.347138	+	0.937814i	$0.387154\pi$
$798$		0			0
$799$	−12.4924			−0.441950
$800$		0			0
$801$	−12.2888	+	7.49141i	−0.434204	+	0.264696i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−50.7386	−	37.0970i	−1.78830	−	1.30750i
$806$		0			0
$807$	16.6847	−	4.68466i	0.587328	−	0.164908i
$808$		0			0
$809$		−	7.61553i		−	0.267748i	−0.990998	−	0.133874i	$-0.957258\pi$
$809$		−	7.61553i		−	0.267748i	0.990998	−	0.133874i	$-0.0427417\pi$
$810$		0			0
$811$		−	5.73384i		−	0.201342i	−0.994920	−	0.100671i	$-0.967901\pi$
$811$		−	5.73384i		−	0.201342i	0.994920	−	0.100671i	$-0.0320990\pi$
$812$		0			0
$813$	−3.75379	−	13.3693i	−0.131651	−	0.468882i
$814$		0			0
$815$	5.84912			0.204886
$816$		0			0
$817$		−	7.49141i		−	0.262091i
$818$		0			0
$819$	−41.1918	−	3.46262i	−1.43936	−	0.120994i
$820$		0			0
$821$			7.12311i			0.248598i	0.992245	+	0.124299i	$0.0396682\pi$
$821$			7.12311i			0.248598i	−0.992245	+	0.124299i	$0.960332\pi$
$822$		0			0
$823$	19.1231			0.666590			0.333295	−	0.942823i	$-0.391840\pi$
$823$	19.1231			0.666590			0.333295	+	0.942823i	$0.391840\pi$
$824$		0			0
$825$	−11.4677	−	40.8427i	−0.399253	−	1.42196i
$826$		0			0
$827$		−	30.7386i		−	1.06889i	−0.845204	−	0.534444i	$-0.820521\pi$
$827$		−	30.7386i		−	1.06889i	0.845204	−	0.534444i	$-0.179479\pi$
$828$		0			0
$829$		−	18.5485i		−	0.644216i	−0.946703	−	0.322108i	$-0.895608\pi$
$829$		−	18.5485i		−	0.644216i	0.946703	−	0.322108i	$-0.104392\pi$
$830$		0			0
$831$	−3.33513	+	0.936426i	−0.115694	+	0.0324843i
$832$		0			0
$833$	3.97626	+	12.4924i	0.137769	+	0.432837i
$834$		0			0
$835$	−47.2311			−1.63450
$836$		0			0
$837$	9.12311	+	8.49242i	0.315341	+	0.293541i
$838$		0			0
$839$	40.8427			1.41005			0.705024	−	0.709184i	$-0.250936\pi$
$839$	40.8427			1.41005			0.705024	+	0.709184i	$0.250936\pi$
$840$		0			0
$841$	−21.7386			−0.749608
$842$		0			0
$843$	−7.49141	−	26.6811i	−0.258018	−	0.918944i
$844$		0			0
$845$	−47.1024			−1.62037
$846$		0			0
$847$	−7.80776	+	10.6789i	−0.268278	+	0.366931i
$848$		0			0
$849$	4.93087	+	17.5616i	0.169227	+	0.602711i
$850$		0			0
$851$		−	58.7386i		−	2.01353i
$852$		0			0
$853$		−	39.3805i		−	1.34836i	−0.738567	−	0.674180i	$-0.764497\pi$
$853$		−	39.3805i		−	1.34836i	0.738567	−	0.674180i	$-0.235503\pi$
$854$		0			0
$855$	−4.87689	−	8.00000i	−0.166786	−	0.273594i
$856$		0			0
$857$	−15.2134			−0.519679			−0.259840	−	0.965652i	$-0.583670\pi$
$857$	−15.2134			−0.519679			−0.259840	+	0.965652i	$0.583670\pi$
$858$		0			0
$859$			26.5658i			0.906413i	0.891406	+	0.453206i	$0.149720\pi$
$859$			26.5658i			0.906413i	−0.891406	+	0.453206i	$0.850280\pi$
$860$		0			0
$861$	3.00404	−	8.03958i	0.102377	−	0.273988i
$862$		0			0
$863$		−	27.2311i		−	0.926956i	−0.886108	−	0.463478i	$-0.846601\pi$
$863$		−	27.2311i		−	0.926956i	0.886108	−	0.463478i	$-0.153399\pi$
$864$		0			0
$865$	11.1231			0.378197
$866$		0			0
$867$	22.4995	−	6.31733i	0.764123	−	0.214548i
$868$		0			0
$869$			12.4924i			0.423776i
$870$		0			0
$871$			74.1941i			2.51397i
$872$		0			0
$873$	−4.56685	−	7.49141i	−0.154565	−	0.253546i
$874$		0			0
$875$	5.84912	−	8.00000i	0.197736	−	0.270449i
$876$		0			0
$877$	28.2462			0.953807			0.476903	−	0.878956i	$-0.341759\pi$
$877$	28.2462			0.953807			0.476903	+	0.878956i	$0.341759\pi$
$878$		0			0
$879$	46.5464	−	13.0691i	1.56997	−	0.440811i
$880$		0			0
$881$	48.5647			1.63619			0.818093	−	0.575086i	$-0.195031\pi$
$881$	48.5647			1.63619			0.818093	+	0.575086i	$0.195031\pi$
$882$		0			0
$883$	26.7386			0.899827			0.449913	−	0.893072i	$-0.351455\pi$
$883$	26.7386			0.899827			0.449913	+	0.893072i	$0.351455\pi$
$884$		0			0
$885$	−39.3805	+	11.0571i	−1.32376	+	0.371680i
$886$		0			0
$887$	20.0108			0.671897			0.335948	−	0.941880i	$-0.390943\pi$
$887$	20.0108			0.671897			0.335948	+	0.941880i	$0.390943\pi$
$888$		0			0
$889$	−12.4924	+	17.0862i	−0.418982	+	0.573054i
$890$		0			0
$891$	32.0000	+	16.4924i	1.07204	+	0.552517i
$892$		0			0
$893$		−	6.24621i		−	0.209021i
$894$		0			0
$895$		−	81.6855i		−	2.73044i
$896$		0			0
$897$	61.8617	−	17.3693i	2.06550	−	0.579945i
$898$		0			0
$899$	17.0862			0.569858
$900$		0			0
$901$		−	1.64229i		−	0.0547127i
$902$		0			0
$903$	−12.8319	+	34.3415i	−0.427020	+	1.14281i
$904$		0			0
$905$		−	27.1231i		−	0.901603i
$906$		0			0
$907$	−24.0000			−0.796907			−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000			−0.796907			−0.398453	+	0.917189i	$0.630453\pi$
$908$		0			0
$909$	−25.6294	+	15.6240i	−0.850072	+	0.518214i
$910$		0			0
$911$		−	7.12311i		−	0.235999i	−0.993014	−	0.118000i	$-0.962352\pi$
$911$		−	7.12311i		−	0.235999i	0.993014	−	0.118000i	$-0.0376481\pi$
$912$		0			0
$913$			25.0388i			0.828662i
$914$		0			0
$915$	−12.6994	−	45.2296i	−0.419829	−	1.49524i
$916$		0			0
$917$	−15.6240	+	21.3693i	−0.515948	+	0.705677i
$918$		0			0
$919$	0.384472			0.0126826			0.00634128	−	0.999980i	$-0.497981\pi$
$919$	0.384472			0.0126826			0.00634128	+	0.999980i	$0.497981\pi$
$920$		0			0
$921$	−9.31534	−	33.1771i	−0.306951	−	1.09322i
$922$		0			0
$923$	−11.6982			−0.385052
$924$		0			0
$925$	50.4924			1.66018
$926$		0			0
$927$	22.9354	+	37.6229i	0.753296	+	1.23570i
$928$		0			0
$929$	11.4677			0.376242			0.188121	−	0.982146i	$-0.439760\pi$
$929$	11.4677			0.376242			0.188121	+	0.982146i	$0.439760\pi$
$930$		0			0
$931$	−6.24621	+	1.98813i	−0.204711	+	0.0651584i
$932$		0			0
$933$	22.2462	−	6.24621i	0.728308	−	0.204492i
$934$		0			0
$935$		−	24.9848i		−	0.817092i
$936$		0			0
$937$		−	40.0216i		−	1.30745i	−0.756733	−	0.653724i	$-0.773206\pi$
$937$		−	40.0216i		−	1.30745i	0.756733	−	0.653724i	$-0.226794\pi$
$938$		0			0
$939$	8.38447	+	29.8617i	0.273617	+	0.974501i
$940$		0			0
$941$	19.6002			0.638949			0.319474	−	0.947595i	$-0.396494\pi$
$941$	19.6002			0.638949			0.319474	+	0.947595i	$0.396494\pi$
$942$		0			0
$943$			13.3405i			0.434427i
$944$		0			0
$945$	8.65318	+	45.0265i	0.281488	+	1.46471i
$946$		0			0
$947$			22.7386i			0.738906i	0.929249	+	0.369453i	$0.120455\pi$
$947$			22.7386i			0.738906i	−0.929249	+	0.369453i	$0.879545\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.410574	+	1.46228i	0.0133138	+	0.0474177i
$952$		0			0
$953$			14.2462i			0.461480i	0.973015	+	0.230740i	$0.0741147\pi$
$953$			14.2462i			0.461480i	−0.973015	+	0.230740i	$0.925885\pi$
$954$		0			0
$955$			34.1725i			1.10580i
$956$		0			0
$957$	47.5130	−	13.3405i	1.53588	−	0.431238i
$958$		0			0
$959$	30.4268	+	22.2462i	0.982531	+	0.718368i
$960$		0			0
$961$	25.2462			0.814394
$962$		0			0
$963$	−3.50758	−	5.75379i	−0.113030	−	0.185413i
$964$		0			0
$965$	30.4268			0.979472
$966$		0			0
$967$	−36.4924			−1.17352			−0.586759	−	0.809762i	$-0.699596\pi$
$967$	−36.4924			−1.17352			−0.586759	+	0.809762i	$0.699596\pi$
$968$		0			0
$969$	−0.821147	−	2.92456i	−0.0263790	−	0.0939504i
$970$		0			0
$971$	44.9990			1.44409			0.722043	−	0.691848i	$-0.243203\pi$
$971$	44.9990			1.44409			0.722043	+	0.691848i	$0.243203\pi$
$972$		0			0
$973$	44.7386	+	32.7102i	1.43425	+	1.04864i
$974$		0			0
$975$	14.9309	+	53.1771i	0.478171	+	1.70303i
$976$		0			0
$977$			46.2462i			1.47955i	0.672856	+	0.739774i	$0.265067\pi$
$977$			46.2462i			1.47955i	−0.672856	+	0.739774i	$0.734933\pi$
$978$		0			0
$979$		−	19.1896i		−	0.613304i
$980$		0			0
$981$	−41.6155	+	25.3693i	−1.32868	+	0.809980i
$982$		0			0
$983$	−25.8599			−0.824803			−0.412402	−	0.911002i	$-0.635310\pi$
$983$	−25.8599			−0.824803			−0.412402	+	0.911002i	$0.635310\pi$
$984$		0			0
$985$		−	44.5884i		−	1.42071i
$986$		0			0
$987$	−10.6991	+	28.6334i	−0.340555	+	0.911411i
$988$		0			0
$989$		−	56.9848i		−	1.81201i
$990$		0			0
$991$	28.8769			0.917305			0.458652	−	0.888616i	$-0.348332\pi$
$991$	28.8769			0.917305			0.458652	+	0.888616i	$0.348332\pi$
$992$		0			0
$993$	50.4376	−	14.1617i	1.60059	−	0.449407i
$994$		0			0
$995$		−	24.0000i		−	0.760851i
$996$		0			0
$997$		−	8.13254i		−	0.257560i	−0.991673	−	0.128780i	$-0.958894\pi$
$997$		−	8.13254i		−	0.257560i	0.991673	−	0.128780i	$-0.0411061\pi$
$998$		0			0
$999$	−29.1950	+	31.3632i	−0.923690	+	0.992287i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.k.a.545.2	yes	8
3.2	odd	2	inner	672.2.k.a.545.8	yes	8
4.3	odd	2		672.2.k.d.545.7	yes	8
7.6	odd	2	inner	672.2.k.a.545.7	yes	8
8.3	odd	2		1344.2.k.j.1217.2		8
8.5	even	2		1344.2.k.e.1217.7		8
12.11	even	2		672.2.k.d.545.1	yes	8
21.20	even	2	inner	672.2.k.a.545.1	✓	8
24.5	odd	2		1344.2.k.e.1217.1		8
24.11	even	2		1344.2.k.j.1217.8		8
28.27	even	2		672.2.k.d.545.2	yes	8
56.13	odd	2		1344.2.k.e.1217.2		8
56.27	even	2		1344.2.k.j.1217.7		8
84.83	odd	2		672.2.k.d.545.8	yes	8
168.83	odd	2		1344.2.k.j.1217.1		8
168.125	even	2		1344.2.k.e.1217.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.k.a.545.1	✓	8	21.20	even	2	inner
672.2.k.a.545.2	yes	8	1.1	even	1	trivial
672.2.k.a.545.7	yes	8	7.6	odd	2	inner
672.2.k.a.545.8	yes	8	3.2	odd	2	inner
672.2.k.d.545.1	yes	8	12.11	even	2
672.2.k.d.545.2	yes	8	28.27	even	2
672.2.k.d.545.7	yes	8	4.3	odd	2
672.2.k.d.545.8	yes	8	84.83	odd	2
1344.2.k.e.1217.1		8	24.5	odd	2
1344.2.k.e.1217.2		8	56.13	odd	2
1344.2.k.e.1217.7		8	8.5	even	2
1344.2.k.e.1217.8		8	168.125	even	2
1344.2.k.j.1217.1		8	168.83	odd	2
1344.2.k.j.1217.2		8	8.3	odd	2
1344.2.k.j.1217.7		8	56.27	even	2
1344.2.k.j.1217.8		8	24.11	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.66757 + 0.468213i) q^{3} +3.33513 q^{5} +(1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} +O(q^{10})\)	\(q+(-1.66757 + 0.468213i) q^{3} +3.33513 q^{5} +(1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} +4.00000i q^{11} -5.20798i q^{13} +(-5.56155 + 1.56155i) q^{15} -1.87285 q^{17} -0.936426i q^{19} +(-1.60399 + 4.29269i) q^{21} -7.12311i q^{23} +6.12311 q^{25} +(-3.54042 + 3.80335i) q^{27} +7.12311i q^{29} -2.39871i q^{31} +(-1.87285 - 6.67026i) q^{33} +(5.20798 - 7.12311i) q^{35} +8.24621 q^{37} +(2.43845 + 8.68466i) q^{39} -1.87285 q^{41} +8.00000 q^{43} +(8.54312 - 5.20798i) q^{45} +6.67026 q^{47} +(-2.12311 - 6.67026i) q^{49} +(3.12311 - 0.876894i) q^{51} +0.876894i q^{53} +13.3405i q^{55} +(0.438447 + 1.56155i) q^{57} +7.08084 q^{59} +8.13254i q^{61} +(0.664868 - 7.90936i) q^{63} -17.3693i q^{65} -14.2462 q^{67} +(3.33513 + 11.8782i) q^{69} -2.24621i q^{71} +(-10.2107 + 2.86692i) q^{75} +(8.54312 + 6.24621i) q^{77} +3.12311 q^{79} +(4.12311 - 8.00000i) q^{81} +6.25969 q^{83} -6.24621 q^{85} +(-3.33513 - 11.8782i) q^{87} -4.79741 q^{89} +(-11.1231 - 8.13254i) q^{91} +(1.12311 + 4.00000i) q^{93} -3.12311i q^{95} -2.92456i q^{97} +(6.24621 + 10.2462i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^7 + 4 * q^9	\( 8 q - 4 q^{7} + 4 q^{9} - 28 q^{15} + 8 q^{21} + 16 q^{25} + 36 q^{39} + 64 q^{43} + 16 q^{49} - 8 q^{51} + 20 q^{57} + 32 q^{63} - 48 q^{67} - 8 q^{79} + 16 q^{85} - 56 q^{91} - 24 q^{93} - 16 q^{99}+O(q^{100}) \) 8 * q - 4 * q^7 + 4 * q^9 - 28 * q^15 + 8 * q^21 + 16 * q^25 + 36 * q^39 + 64 * q^43 + 16 * q^49 - 8 * q^51 + 20 * q^57 + 32 * q^63 - 48 * q^67 - 8 * q^79 + 16 * q^85 - 56 * q^91 - 24 * q^93 - 16 * q^99

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