LMFDB - Embedded newform 968.2.k.a.403.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(403,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 7]))

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

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

chi := DirichletCharacter("968.403");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.k (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			403.1
Root			$-1.34500 - 0.437016i$ of defining polynomial
Character	$\chi$	$=$	968.403
Dual form			968.2.k.a.723.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.831254 + 1.14412i) q^{2} +(0.618034 - 1.90211i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(1.66251 + 2.28825i) q^{6} +(2.68999 + 0.874032i) q^{8} +(-0.809017 - 0.587785i) q^{9} -4.00000 q^{12} +(-3.23607 + 2.35114i) q^{16} +(3.32502 + 4.57649i) q^{17} +(1.34500 - 0.437016i) q^{18} +(8.06998 + 2.62210i) q^{19} +(3.32502 - 4.57649i) q^{24} +(1.54508 - 4.75528i) q^{25} +(3.23607 - 2.35114i) q^{27} -5.65685i q^{32} -8.00000 q^{34} +(-0.618034 + 1.90211i) q^{36} +(-9.70820 + 7.05342i) q^{38} +(-10.7600 - 3.49613i) q^{41} -8.48528i q^{43} +(2.47214 + 7.60845i) q^{48} +(5.66312 - 4.11450i) q^{49} +(4.15627 + 5.72061i) q^{50} +(10.7600 - 3.49613i) q^{51} +5.65685i q^{54} +(9.97505 - 13.7295i) q^{57} +(-1.85410 - 5.70634i) q^{59} +(6.47214 + 4.70228i) q^{64} +14.0000 q^{67} +(6.65003 - 9.15298i) q^{68} +(-1.66251 - 2.28825i) q^{72} +(-16.1400 + 5.24419i) q^{73} +(-8.09017 - 5.87785i) q^{75} -16.9706i q^{76} +(-3.39919 - 10.4616i) q^{81} +(12.9443 - 9.40456i) q^{82} +(1.66251 + 2.28825i) q^{83} +(9.70820 + 7.05342i) q^{86} +18.0000 q^{89} +(-10.7600 - 3.49613i) q^{96} +(-8.09017 - 5.87785i) q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 4 q^{4} - 2 q^{9} - 32 q^{12} - 8 q^{16} - 10 q^{25} + 8 q^{27} - 64 q^{34} + 4 q^{36} - 24 q^{38} - 16 q^{48} + 14 q^{49} + 12 q^{59} + 16 q^{64} + 112 q^{67} - 20 q^{75} + 22 q^{81} + 32 q^{82}+ \cdots - 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 + 4 * q^4 - 2 * q^9 - 32 * q^12 - 8 * q^16 - 10 * q^25 + 8 * q^27 - 64 * q^34 + 4 * q^36 - 24 * q^38 - 16 * q^48 + 14 * q^49 + 12 * q^59 + 16 * q^64 + 112 * q^67 - 20 * q^75 + 22 * q^81 + 32 * q^82 + 24 * q^86 + 144 * q^89 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{10}\right)$

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 $

$2$	−0.831254	+	1.14412i	−0.587785	+	0.809017i
$2$	−0.831254	+	1.14412i	−0.587785	+	0.809017i
$3$	0.618034	−	1.90211i	0.356822	−	1.09819i	−0.598123	−	0.801404i	$-0.704087\pi$
$3$	0.618034	−	1.90211i	0.356822	−	1.09819i	0.954945	−	0.296781i	$-0.0959133\pi$
$4$	−0.618034	−	1.90211i	−0.309017	−	0.951057i
$5$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$5$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$6$	1.66251	+	2.28825i	0.678716	+	0.934172i
$7$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$7$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$8$	2.68999	+	0.874032i	0.951057	+	0.309017i
$9$	−0.809017	−	0.587785i	−0.269672	−	0.195928i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−4.00000			−1.15470
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$		0			0
$15$		0			0
$16$	−3.23607	+	2.35114i	−0.809017	+	0.587785i
$17$	3.32502	+	4.57649i	0.806435	+	1.10996i	0.991864	+	0.127304i	$0.0406325\pi$
$17$	3.32502	+	4.57649i	0.806435	+	1.10996i	−0.185429	+	0.982658i	$0.559367\pi$
$18$	1.34500	−	0.437016i	0.317019	−	0.103006i
$19$	8.06998	+	2.62210i	1.85138	+	0.601550i	0.996584	+	0.0825877i	$0.0263185\pi$
$19$	8.06998	+	2.62210i	1.85138	+	0.601550i	0.854797	+	0.518962i	$0.173682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	3.32502	−	4.57649i	0.678716	−	0.934172i
$25$	1.54508	−	4.75528i	0.309017	−	0.951057i
$26$		0			0
$27$	3.23607	−	2.35114i	0.622782	−	0.452477i
$28$		0			0
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$	−8.00000			−1.37199
$35$		0			0
$36$	−0.618034	+	1.90211i	−0.103006	+	0.317019i
$37$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$37$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$38$	−9.70820	+	7.05342i	−1.57488	+	1.14422i
$39$		0			0
$40$		0			0
$41$	−10.7600	−	3.49613i	−1.68043	−	0.546003i	−0.695432	−	0.718592i	$-0.744787\pi$
$41$	−10.7600	−	3.49613i	−1.68043	−	0.546003i	−0.984994	+	0.172588i	$0.944787\pi$
$42$		0			0
$43$		−	8.48528i		−	1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$		−	8.48528i		−	1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$	2.47214	+	7.60845i	0.356822	+	1.09819i
$49$	5.66312	−	4.11450i	0.809017	−	0.587785i
$50$	4.15627	+	5.72061i	0.587785	+	0.809017i
$51$	10.7600	−	3.49613i	1.50670	−	0.489556i
$52$		0			0
$53$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$53$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$54$			5.65685i			0.769800i
$55$		0			0
$56$		0			0
$57$	9.97505	−	13.7295i	1.32123	−	1.81851i
$58$		0			0
$59$	−1.85410	−	5.70634i	−0.241384	−	0.742902i	−0.996210	−	0.0869778i	$-0.972279\pi$
$59$	−1.85410	−	5.70634i	−0.241384	−	0.742902i	0.754827	−	0.655924i	$-0.227721\pi$
$60$		0			0
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$		0			0
$64$	6.47214	+	4.70228i	0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	14.0000			1.71037			0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000			1.71037			0.855186	+	0.518321i	$0.173443\pi$
$68$	6.65003	−	9.15298i	0.806435	−	1.10996i
$69$		0			0
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$	−1.66251	−	2.28825i	−0.195928	−	0.269672i
$73$	−16.1400	+	5.24419i	−1.88904	+	0.613786i	−0.908352	+	0.418206i	$0.862659\pi$
$73$	−16.1400	+	5.24419i	−1.88904	+	0.613786i	−0.980688	+	0.195580i	$0.937341\pi$
$74$		0			0
$75$	−8.09017	−	5.87785i	−0.934172	−	0.678716i
$76$		−	16.9706i		−	1.94666i
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$		0			0
$81$	−3.39919	−	10.4616i	−0.377687	−	1.16240i
$82$	12.9443	−	9.40456i	1.42946	−	1.03856i
$83$	1.66251	+	2.28825i	0.182484	+	0.251168i	0.890452	−	0.455077i	$-0.150388\pi$
$83$	1.66251	+	2.28825i	0.182484	+	0.251168i	−0.707968	+	0.706244i	$0.750388\pi$
$84$		0			0
$85$		0			0
$86$	9.70820	+	7.05342i	1.04686	+	0.760590i
$87$		0			0
$88$		0			0
$89$	18.0000			1.90800			0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000			1.90800			0.953998	+	0.299813i	$0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−10.7600	−	3.49613i	−1.09819	−	0.356822i
$97$	−8.09017	−	5.87785i	−0.821432	−	0.596806i	0.0956901	−	0.995411i	$-0.469494\pi$
$97$	−8.09017	−	5.87785i	−0.821432	−	0.596806i	−0.917122	+	0.398606i	$0.869494\pi$
$98$			9.89949i			1.00000i
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$	−4.94427	+	15.2169i	−0.489556	+	1.50670i
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.8300	+	6.11822i	1.82036	+	0.591471i	0.999802	+	0.0199092i	$0.00633772\pi$
$107$	18.8300	+	6.11822i	1.82036	+	0.591471i	0.820559	+	0.571562i	$0.193662\pi$
$108$	−6.47214	−	4.70228i	−0.622782	−	0.452477i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.56231	+	17.1190i	−0.523258	+	1.61042i	0.244478	+	0.969655i	$0.421383\pi$
$113$	−5.56231	+	17.1190i	−0.523258	+	1.61042i	−0.767736	+	0.640767i	$0.778617\pi$
$114$	7.41641	+	22.8254i	0.694610	+	2.13779i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	8.06998	+	2.62210i	0.742902	+	0.241384i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−13.3001	+	18.3060i	−1.19923	+	1.65059i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$127$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$128$	−10.7600	+	3.49613i	−0.951057	+	0.309017i
$129$	−16.1400	−	5.24419i	−1.42104	−	0.461725i
$130$		0			0
$131$			14.1421i			1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		0			0
$134$	−11.6376	+	16.0177i	−1.00533	+	1.38372i
$135$		0			0
$136$	4.94427	+	15.2169i	0.423968	+	1.30484i
$137$	−4.85410	+	3.52671i	−0.414714	+	0.301307i	−0.775507	−	0.631338i	$-0.782506\pi$
$137$	−4.85410	+	3.52671i	−0.414714	+	0.301307i	0.360794	+	0.932646i	$0.382506\pi$
$138$		0			0
$139$	8.06998	−	2.62210i	0.684487	−	0.222403i	0.0539282	−	0.998545i	$-0.482826\pi$
$139$	8.06998	−	2.62210i	0.684487	−	0.222403i	0.630559	+	0.776142i	$0.282826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.00000			0.333333
$145$		0			0
$146$	7.41641	−	22.8254i	0.613786	−	1.88904i
$147$	−4.32624	−	13.3148i	−0.356822	−	1.09819i
$148$		0			0
$149$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$149$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$150$	13.4500	−	4.37016i	1.09819	−	0.356822i
$151$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$151$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$152$	19.4164	+	14.1068i	1.57488	+	1.14422i
$153$		−	5.65685i		−	0.457330i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	14.7950	+	4.80718i	1.16240	+	0.377687i
$163$	1.61803	+	1.17557i	0.126734	+	0.0920778i	0.649347	−	0.760493i	$-0.275042\pi$
$163$	1.61803	+	1.17557i	0.126734	+	0.0920778i	−0.522612	+	0.852570i	$0.675042\pi$
$164$			22.6274i			1.76690i
$165$		0			0
$166$	−4.00000			−0.310460
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−4.01722	−	12.3637i	−0.309017	−	0.951057i
$170$		0			0
$171$	−4.98752	−	6.86474i	−0.381405	−	0.524960i
$172$	−16.1400	+	5.24419i	−1.23066	+	0.399866i
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0000			−0.901975
$178$	−14.9626	+	20.5942i	−1.12149	+	1.54360i
$179$	−5.56231	+	17.1190i	−0.415746	+	1.27954i	0.495835	+	0.868416i	$0.334862\pi$
$179$	−5.56231	+	17.1190i	−0.415746	+	1.27954i	−0.911582	+	0.411119i	$0.865138\pi$
$180$		0			0
$181$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$181$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$	12.9443	−	9.40456i	0.934172	−	0.678716i
$193$	−9.97505	−	13.7295i	−0.718020	−	0.988269i	−0.999587	−	0.0287278i	$-0.990854\pi$
$193$	−9.97505	−	13.7295i	−0.718020	−	0.988269i	0.281568	−	0.959541i	$-0.409146\pi$
$194$	13.4500	−	4.37016i	0.965652	−	0.313759i
$195$		0			0
$196$	−11.3262	−	8.22899i	−0.809017	−	0.587785i
$197$		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			$\pi$
$200$	8.31254	−	11.4412i	0.587785	−	0.809017i
$201$	8.65248	−	26.6296i	0.610299	−	1.87831i
$202$		0			0
$203$		0			0
$204$	−13.3001	−	18.3060i	−0.931191	−	1.28167i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−14.9626	+	20.5942i	−1.03007	+	1.41776i	−0.125168	+	0.992136i	$0.539947\pi$
$211$	−14.9626	+	20.5942i	−1.03007	+	1.41776i	−0.904898	+	0.425628i	$0.860053\pi$
$212$		0			0
$213$		0			0
$214$	−22.6525	+	16.4580i	−1.54849	+	1.12504i
$215$		0			0
$216$	10.7600	−	3.49613i	0.732124	−	0.237881i
$217$		0			0
$218$		0			0
$219$			33.9411i			2.29353i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$223$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$224$		0			0
$225$	−4.04508	+	2.93893i	−0.269672	+	0.195928i
$226$	−14.9626	−	20.5942i	−0.995295	−	1.36991i
$227$	−2.68999	+	0.874032i	−0.178541	+	0.0580115i	−0.396923	−	0.917852i	$-0.629922\pi$
$227$	−2.68999	+	0.874032i	−0.178541	+	0.0580115i	0.218382	+	0.975863i	$0.429922\pi$
$228$	−32.2799	−	10.4884i	−2.13779	−	0.694610i
$229$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$229$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.32502	−	4.57649i	0.217829	−	0.299816i	−0.686092	−	0.727514i	$-0.740675\pi$
$233$	3.32502	−	4.57649i	0.217829	−	0.299816i	0.903921	+	0.427698i	$0.140675\pi$
$234$		0			0
$235$		0			0
$236$	−9.70820	+	7.05342i	−0.631950	+	0.459139i
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$		−	16.9706i		−	1.09317i	−0.837404	−	0.546585i	$-0.815928\pi$
$241$		−	16.9706i		−	1.09317i	0.837404	−	0.546585i	$-0.184072\pi$
$242$		0			0
$243$	−10.0000			−0.641500
$244$		0			0
$245$		0			0
$246$	−9.88854	−	30.4338i	−0.630471	−	1.94039i
$247$		0			0
$248$		0			0
$249$	5.37999	−	1.74806i	0.340943	−	0.110779i
$250$		0			0
$251$	−4.85410	−	3.52671i	−0.306388	−	0.222604i	0.423957	−	0.905682i	$-0.360641\pi$
$251$	−4.85410	−	3.52671i	−0.306388	−	0.222604i	−0.730345	+	0.683078i	$0.760641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	9.27051	+	28.5317i	0.578279	+	1.77976i	0.624734	+	0.780838i	$0.285208\pi$
$257$	9.27051	+	28.5317i	0.578279	+	1.77976i	−0.0464552	+	0.998920i	$0.514792\pi$
$258$	19.4164	−	14.1068i	1.20881	−	0.878254i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−16.1803	−	11.7557i	−0.999625	−	0.726270i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	11.1246	−	34.2380i	0.680815	−	2.09533i
$268$	−8.65248	−	26.6296i	−0.528534	−	1.62666i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$	−21.5200	−	6.99226i	−1.30484	−	0.423968i
$273$		0			0
$274$		−	8.48528i		−	0.512615i
$275$		0			0
$276$		0			0
$277$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$277$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$278$	−3.70820	+	11.4127i	−0.222403	+	0.684487i
$279$		0			0
$280$		0			0
$281$	−16.6251	−	22.8825i	−0.991769	−	1.36505i	−0.930242	−	0.366947i	$-0.880403\pi$
$281$	−16.6251	−	22.8825i	−0.991769	−	1.36505i	−0.0615273	−	0.998105i	$-0.519597\pi$
$282$		0			0
$283$	−24.2099	−	7.86629i	−1.43913	−	0.467602i	−0.517505	−	0.855680i	$-0.673139\pi$
$283$	−24.2099	−	7.86629i	−1.43913	−	0.467602i	−0.921627	+	0.388078i	$0.873139\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−3.32502	+	4.57649i	−0.195928	+	0.269672i
$289$	−4.63525	+	14.2658i	−0.272662	+	0.839168i
$290$		0			0
$291$	−16.1803	+	11.7557i	−0.948508	+	0.689132i
$292$	19.9501	+	27.4589i	1.16749	+	1.60691i
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$	18.8300	+	6.11822i	1.09819	+	0.356822i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−6.18034	+	19.0211i	−0.356822	+	1.09819i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−32.2799	+	10.4884i	−1.85138	+	0.601550i
$305$		0			0
$306$	6.47214	+	4.70228i	0.369987	+	0.268812i
$307$		−	8.48528i		−	0.484281i	−0.970241	−	0.242140i	$-0.922151\pi$
$307$		−	8.48528i		−	0.484281i	0.970241	−	0.242140i	$-0.0778494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	8.09017	−	5.87785i	0.457283	−	0.332236i	−0.335181	−	0.942154i	$-0.608798\pi$
$313$	8.09017	−	5.87785i	0.457283	−	0.332236i	0.792465	+	0.609918i	$0.208798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$317$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	23.2751	−	32.0354i	1.29909	−	1.78804i
$322$		0			0
$323$	14.8328	+	45.6507i	0.825320	+	2.54007i
$324$	−17.7984	+	12.9313i	−0.988799	+	0.718404i
$325$		0			0
$326$	−2.68999	+	0.874032i	−0.148985	+	0.0484082i
$327$		0			0
$328$	−25.8885	−	18.8091i	−1.42946	−	1.03856i
$329$		0			0
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$	3.32502	−	4.57649i	0.182484	−	0.251168i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−32.2799	+	10.4884i	−1.75840	+	0.571339i	−0.997032	−	0.0769821i	$-0.975472\pi$
$337$	−32.2799	+	10.4884i	−1.75840	+	0.571339i	−0.761367	+	0.648321i	$0.775472\pi$
$338$	17.4850	+	5.68121i	0.951057	+	0.309017i
$339$	29.1246	+	21.1603i	1.58183	+	1.14927i
$340$		0			0
$341$		0			0
$342$	12.0000			0.648886
$343$		0			0
$344$	7.41641	−	22.8254i	0.399866	−	1.23066i
$345$		0			0
$346$		0			0
$347$	21.6126	+	29.7472i	1.16023	+	1.59691i	0.710404	+	0.703795i	$0.248512\pi$
$347$	21.6126	+	29.7472i	1.16023	+	1.59691i	0.449822	+	0.893118i	$0.351488\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$	9.97505	−	13.7295i	0.530168	−	0.729713i
$355$		0			0
$356$	−11.1246	−	34.2380i	−0.589603	−	1.81461i
$357$		0			0
$358$	−14.9626	−	20.5942i	−0.790796	−	1.08844i
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	42.8779	+	31.1526i	2.25673	+	1.63961i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$367$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$368$		0			0
$369$	6.65003	+	9.15298i	0.346187	+	0.476485i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	30.7426	−	22.3358i	1.57914	−	1.14732i	0.661476	−	0.749966i	$-0.269930\pi$
$379$	30.7426	−	22.3358i	1.57914	−	1.14732i	0.917668	−	0.397349i	$-0.130070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$383$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$384$			22.6274i			1.15470i
$385$		0			0
$386$	24.0000			1.22157
$387$	−4.98752	+	6.86474i	−0.253530	+	0.348954i
$388$	−6.18034	+	19.0211i	−0.313759	+	0.965652i
$389$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$389$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$390$		0			0
$391$		0			0
$392$	18.8300	−	6.11822i	0.951057	−	0.309017i
$393$	26.8999	+	8.74032i	1.35692	+	0.440891i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	6.18034	+	19.0211i	0.309017	+	0.951057i
$401$	−4.85410	+	3.52671i	−0.242402	+	0.176116i	−0.702353	−	0.711829i	$-0.747867\pi$
$401$	−4.85410	+	3.52671i	−0.242402	+	0.176116i	0.459951	+	0.887945i	$0.347867\pi$
$402$	23.2751	+	32.0354i	1.16086	+	1.59778i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	32.0000			1.58424
$409$	−19.9501	+	27.4589i	−0.986469	+	1.35776i	−0.0531978	+	0.998584i	$0.516941\pi$
$409$	−19.9501	+	27.4589i	−0.986469	+	1.35776i	−0.933271	+	0.359174i	$0.883059\pi$
$410$		0			0
$411$	3.70820	+	11.4127i	0.182912	+	0.562946i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	16.9706i		−	0.831052i
$418$		0			0
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$421$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$422$	−11.1246	−	34.2380i	−0.541538	−	1.66668i
$423$		0			0
$424$		0			0
$425$	26.8999	−	8.74032i	1.30484	−	0.423968i
$426$		0			0
$427$		0			0
$428$		−	39.5980i		−	1.91404i
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$	−4.94427	+	15.2169i	−0.237881	+	0.732124i
$433$	11.7426	+	36.1401i	0.564316	+	1.73678i	0.669976	+	0.742382i	$0.266304\pi$
$433$	11.7426	+	36.1401i	0.564316	+	1.73678i	−0.105661	+	0.994402i	$0.533696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−38.8328	−	28.2137i	−1.85550	−	1.34810i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−12.9787	+	39.9444i	−0.616637	+	1.89781i	−0.244416	+	0.969670i	$0.578596\pi$
$443$	−12.9787	+	39.9444i	−0.616637	+	1.89781i	−0.372221	+	0.928144i	$0.621404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.9787	−	24.6870i	−1.60355	−	1.16505i	−0.880236	−	0.474536i	$-0.842616\pi$
$449$	−33.9787	−	24.6870i	−1.60355	−	1.16505i	−0.723319	−	0.690514i	$-0.757384\pi$
$450$		−	7.07107i		−	0.333333i
$451$		0			0
$452$	36.0000			1.69330
$453$		0			0
$454$	1.23607	−	3.80423i	0.0580115	−	0.178541i
$455$		0			0
$456$	38.8328	−	28.2137i	1.81851	−	1.32123i
$457$	19.9501	+	27.4589i	0.933226	+	1.28448i	0.958588	+	0.284797i	$0.0919262\pi$
$457$	19.9501	+	27.4589i	0.933226	+	1.28448i	−0.0253618	+	0.999678i	$0.508074\pi$
$458$		0			0
$459$	21.5200	+	6.99226i	1.00447	+	0.326371i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	2.47214	+	7.60845i	0.114519	+	0.352455i
$467$	−24.2705	+	17.6336i	−1.12311	+	0.815984i	−0.984677	−	0.174389i	$-0.944205\pi$
$467$	−24.2705	+	17.6336i	−1.12311	+	0.815984i	−0.138428	+	0.990372i	$0.544205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	16.9706i		−	0.781133i
$473$		0			0
$474$		0			0
$475$	24.9376	−	34.3237i	1.14422	−	1.57488i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$		0			0
$482$	19.4164	+	14.1068i	0.884393	+	0.642549i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	8.31254	−	11.4412i	0.377064	−	0.518985i
$487$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$487$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$488$		0			0
$489$	3.23607	−	2.35114i	0.146340	−	0.106322i
$490$		0			0
$491$	13.4500	−	4.37016i	0.606989	−	0.197223i	0.0106338	−	0.999943i	$-0.496615\pi$
$491$	13.4500	−	4.37016i	0.606989	−	0.197223i	0.596355	+	0.802721i	$0.296615\pi$
$492$	43.0399	+	13.9845i	1.94039	+	0.630471i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−2.47214	+	7.60845i	−0.110779	+	0.340943i
$499$	−4.32624	−	13.3148i	−0.193669	−	0.596052i	−0.999990	−	0.00457310i	$-0.998544\pi$
$499$	−4.32624	−	13.3148i	−0.193669	−	0.596052i	0.806321	−	0.591479i	$-0.201456\pi$
$500$		0			0
$501$		0			0
$502$	8.06998	−	2.62210i	0.360181	−	0.117030i
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−26.0000			−1.15470
$508$		0			0
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$	13.3001	+	18.3060i	0.587785	+	0.809017i
$513$	32.2799	−	10.4884i	1.42519	−	0.463073i
$514$	−40.3499	−	13.1105i	−1.77976	−	0.578279i
$515$		0			0
$516$			33.9411i			1.49417i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.85410	−	5.70634i	−0.0812297	−	0.249999i	0.902191	−	0.431336i	$-0.141958\pi$
$521$	−1.85410	−	5.70634i	−0.0812297	−	0.249999i	−0.983421	+	0.181337i	$0.941958\pi$
$522$		0			0
$523$	−14.9626	−	20.5942i	−0.654267	−	0.900522i	0.345007	−	0.938600i	$-0.387876\pi$
$523$	−14.9626	−	20.5942i	−0.654267	−	0.900522i	−0.999275	+	0.0380781i	$0.987876\pi$
$524$	26.8999	−	8.74032i	1.17513	−	0.381823i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−1.85410	+	5.70634i	−0.0804612	+	0.247634i
$532$		0			0
$533$		0			0
$534$	29.9251	+	41.1884i	1.29499	+	1.78240i
$535$		0			0
$536$	37.6599	+	12.2364i	1.62666	+	0.528534i
$537$	29.1246	+	21.1603i	1.25682	+	0.913133i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$541$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$542$		0			0
$543$		0			0
$544$	25.8885	−	18.8091i	1.10996	−	0.806435i
$545$		0			0
$546$		0			0
$547$	8.06998	+	2.62210i	0.345048	+	0.112113i	0.476414	−	0.879221i	$-0.341936\pi$
$547$	8.06998	+	2.62210i	0.345048	+	0.112113i	−0.131366	+	0.991334i	$0.541936\pi$
$548$	9.70820	+	7.05342i	0.414714	+	0.301307i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−9.97505	−	13.7295i	−0.423036	−	0.582259i
$557$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$557$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	40.0000			1.68730
$563$	21.6126	−	29.7472i	0.910863	−	1.25369i	−0.0560088	−	0.998430i	$-0.517837\pi$
$563$	21.6126	−	29.7472i	0.910863	−	1.25369i	0.966871	−	0.255264i	$-0.0821625\pi$
$564$		0			0
$565$		0			0
$566$	29.1246	−	21.1603i	1.22420	−	0.889432i
$567$		0			0
$568$		0			0
$569$	−21.5200	−	6.99226i	−0.902163	−	0.293131i	−0.179034	−	0.983843i	$-0.557297\pi$
$569$	−21.5200	−	6.99226i	−0.902163	−	0.293131i	−0.723130	+	0.690712i	$0.757297\pi$
$570$		0			0
$571$		−	42.4264i		−	1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$		−	42.4264i		−	1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2.47214	−	7.60845i	−0.103006	−	0.317019i
$577$	−27.5066	+	19.9847i	−1.14511	+	0.831974i	−0.987824	−	0.155579i	$-0.950276\pi$
$577$	−27.5066	+	19.9847i	−1.14511	+	0.831974i	−0.157290	+	0.987552i	$0.550276\pi$
$578$	−12.4688	−	17.1618i	−0.518634	−	0.713839i
$579$	−32.2799	+	10.4884i	−1.34151	+	0.435882i
$580$		0			0
$581$		0			0
$582$		−	28.2843i		−	1.17242i
$583$		0			0
$584$	−48.0000			−1.98625
$585$		0			0
$586$		0			0
$587$	−1.85410	−	5.70634i	−0.0765270	−	0.235526i	0.905474	−	0.424402i	$-0.139516\pi$
$587$	−1.85410	−	5.70634i	−0.0765270	−	0.235526i	−0.982001	+	0.188876i	$0.939516\pi$
$588$	−22.6525	+	16.4580i	−0.934172	+	0.678716i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.2548i			1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$			45.2548i			1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$599$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$600$	−16.6251	−	22.8825i	−0.678716	−	0.934172i
$601$	−16.1400	+	5.24419i	−0.658363	+	0.213915i	−0.619098	−	0.785314i	$-0.712502\pi$
$601$	−16.1400	+	5.24419i	−0.658363	+	0.213915i	−0.0392649	+	0.999229i	$0.512502\pi$
$602$		0			0
$603$	−11.3262	−	8.22899i	−0.461240	−	0.335111i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$	14.8328	−	45.6507i	0.601550	−	1.85138i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−10.7600	+	3.49613i	−0.434946	+	0.141323i
$613$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$613$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$614$	9.70820	+	7.05342i	0.391791	+	0.284653i
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	8.03444	−	24.7275i	0.322931	−	0.993881i	−0.649434	−	0.760418i	$-0.724994\pi$
$619$	8.03444	−	24.7275i	0.322931	−	0.993881i	0.972366	−	0.233463i	$-0.0750058\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.2254	−	14.6946i	−0.809017	−	0.587785i
$626$			14.1421i			0.565233i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$631$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$632$		0			0
$633$	29.9251	+	41.1884i	1.18942	+	1.63709i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.9787	+	39.9444i	−0.512628	+	1.57771i	0.274928	+	0.961465i	$0.411346\pi$
$641$	−12.9787	+	39.9444i	−0.512628	+	1.57771i	−0.787556	+	0.616243i	$0.788654\pi$
$642$	17.3050	+	53.2592i	0.682972	+	2.10197i
$643$	−40.4508	+	29.3893i	−1.59523	+	1.15900i	−0.699269	+	0.714859i	$0.746491\pi$
$643$	−40.4508	+	29.3893i	−1.59523	+	1.15900i	−0.895957	+	0.444140i	$0.853509\pi$
$644$		0			0
$645$		0			0
$646$	−64.5599	−	20.9768i	−2.54007	−	0.825320i
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$		−	31.1127i		−	1.22222i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.23607	−	3.80423i	0.0484082	−	0.148985i
$653$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$653$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$654$		0			0
$655$		0			0
$656$	43.0399	−	13.9845i	1.68043	−	0.546003i
$657$	16.1400	+	5.24419i	0.629680	+	0.204595i
$658$		0			0
$659$		−	48.0833i		−	1.87306i	−0.350590	−	0.936529i	$-0.614019\pi$
$659$		−	48.0833i		−	1.87306i	0.350590	−	0.936529i	$-0.385981\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	21.6126	−	29.7472i	0.839998	−	1.15616i
$663$		0			0
$664$	2.47214	+	7.60845i	0.0959375	+	0.295265i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.9251	+	41.1884i	−1.15353	+	1.58770i	−0.420838	+	0.907136i	$0.638264\pi$
$673$	−29.9251	+	41.1884i	−1.15353	+	1.58770i	−0.732691	+	0.680561i	$0.761736\pi$
$674$	14.8328	−	45.6507i	0.571339	−	1.75840i
$675$	−6.18034	−	19.0211i	−0.237881	−	0.732124i
$676$	−21.0344	+	15.2824i	−0.809017	+	0.587785i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$	−48.4199	+	15.7326i	−1.85955	+	0.604206i
$679$		0			0
$680$		0			0
$681$			5.65685i			0.216771i
$682$		0			0
$683$	−42.0000			−1.60709			−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000			−1.60709			−0.803543	+	0.595247i	$0.797054\pi$
$684$	−9.97505	+	13.7295i	−0.381405	+	0.524960i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	19.9501	+	27.4589i	0.760590	+	1.04686i
$689$		0			0
$690$		0			0
$691$	37.2148	+	27.0381i	1.41572	+	1.02858i	0.992460	+	0.122569i	$0.0391133\pi$
$691$	37.2148	+	27.0381i	1.41572	+	1.02858i	0.423257	+	0.906010i	$0.360887\pi$
$692$		0			0
$693$		0			0
$694$	−52.0000			−1.97389
$695$		0			0
$696$		0			0
$697$	−19.7771	−	60.8676i	−0.749111	−	2.30553i
$698$		0			0
$699$	−6.65003	−	9.15298i	−0.251527	−	0.346198i
$700$		0			0
$701$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$701$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	24.9376	−	34.3237i	0.938540	−	1.29179i
$707$		0			0
$708$	7.41641	+	22.8254i	0.278726	+	0.857829i
$709$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$709$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$710$		0			0
$711$		0			0
$712$	48.4199	+	15.7326i	1.81461	+	0.589603i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	36.0000			1.34538
$717$		0			0
$718$		0			0
$719$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$719$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$		0			0
$722$	−71.2848	+	23.1618i	−2.65295	+	0.861995i
$723$	−32.2799	−	10.4884i	−1.20050	−	0.390067i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	4.01722	−	12.3637i	0.148786	−	0.457916i
$730$		0			0
$731$	38.8328	−	28.2137i	1.43628	−	1.04352i
$732$		0			0
$733$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$733$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−16.0000			−0.588968
$739$	24.9376	−	34.3237i	0.917345	−	1.26262i	−0.0472504	−	0.998883i	$-0.515046\pi$
$739$	24.9376	−	34.3237i	0.917345	−	1.26262i	0.964595	−	0.263734i	$-0.0849541\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$743$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	2.82843i		−	0.103487i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$751$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$752$		0			0
$753$	−9.70820	+	7.05342i	−0.353787	+	0.257041i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$757$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$758$			53.7401i			1.95193i
$759$		0			0
$760$		0			0
$761$	6.65003	−	9.15298i	0.241063	−	0.331795i	−0.671293	−	0.741192i	$-0.734261\pi$
$761$	6.65003	−	9.15298i	0.241063	−	0.331795i	0.912356	+	0.409397i	$0.134261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−25.8885	−	18.8091i	−0.934172	−	0.678716i
$769$			50.9117i			1.83592i	0.396670	+	0.917961i	$0.370166\pi$
$769$			50.9117i			1.83592i	−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	60.0000			2.16085
$772$	−19.9501	+	27.4589i	−0.718020	+	0.988269i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$	−3.70820	−	11.4127i	−0.133289	−	0.410220i
$775$		0			0
$776$	−16.6251	−	22.8825i	−0.596806	−	0.821432i
$777$		0			0
$778$		0			0
$779$	−77.6656	−	56.4274i	−2.78266	−	2.02172i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−8.65248	+	26.6296i	−0.309017	+	0.951057i
$785$		0			0
$786$	−32.3607	+	23.5114i	−1.15427	+	0.838624i
$787$	−14.9626	−	20.5942i	−0.533358	−	0.734104i	0.454280	−	0.890859i	$-0.349897\pi$
$787$	−14.9626	−	20.5942i	−0.533358	−	0.734104i	−0.987638	+	0.156755i	$0.949897\pi$
$788$		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
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	−26.8999	−	8.74032i	−0.951057	−	0.309017i
$801$	−14.5623	−	10.5801i	−0.514534	−	0.373831i
$802$		−	8.48528i		−	0.299626i
$803$		0			0
$804$	−56.0000			−1.97497
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−33.2502	−	45.7649i	−1.16901	−	1.60901i	−0.669837	−	0.742508i	$-0.733636\pi$
$809$	−33.2502	−	45.7649i	−1.16901	−	1.60901i	−0.499176	−	0.866501i	$-0.666364\pi$
$810$		0			0
$811$	−40.3499	−	13.1105i	−1.41688	−	0.460371i	−0.502268	−	0.864712i	$-0.667501\pi$
$811$	−40.3499	−	13.1105i	−1.41688	−	0.460371i	−0.914609	+	0.404340i	$0.867501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−26.6001	+	36.6119i	−0.931191	+	1.28167i
$817$	22.2492	−	68.4761i	0.778402	−	2.39567i
$818$	−14.8328	−	45.6507i	−0.518617	−	1.59614i
$819$		0			0
$820$		0			0
$821$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$821$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$822$	−16.1400	−	5.24419i	−0.562946	−	0.182912i
$823$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$823$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.6376	+	16.0177i	−0.404677	+	0.556991i	−0.961910	−	0.273366i	$-0.911863\pi$
$827$	−11.6376	+	16.0177i	−0.404677	+	0.556991i	0.557233	+	0.830356i	$0.311863\pi$
$828$		0			0
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	37.6599	+	12.2364i	1.30484	+	0.423968i
$834$	19.4164	+	14.1068i	0.672335	+	0.488480i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−14.9626	+	20.5942i	−0.516873	+	0.711415i
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$	−53.7999	+	17.4806i	−1.85297	+	0.602065i
$844$	48.4199	+	15.7326i	1.66668	+	0.541538i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−29.9251	+	41.1884i	−1.02703	+	1.41358i
$850$	−12.3607	+	38.0423i	−0.423968	+	1.30484i
$851$		0			0
$852$		0			0
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$	45.3050	+	32.9160i	1.54849	+	1.12504i
$857$			22.6274i			0.772938i	0.922302	+	0.386469i	$0.126305\pi$
$857$			22.6274i			0.772938i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	58.0000			1.97893			0.989467	−	0.144757i	$-0.0462401\pi$
$859$	58.0000			1.97893			0.989467	+	0.144757i	$0.0462401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$863$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$864$	−13.3001	−	18.3060i	−0.452477	−	0.622782i
$865$		0			0
$866$	−51.1099	−	16.6066i	−1.73678	−	0.564316i
$867$	24.2705	+	17.6336i	0.824270	+	0.598867i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	3.09017	+	9.51057i	0.104586	+	0.321884i
$874$		0			0
$875$		0			0
$876$	64.5599	−	20.9768i	2.18128	−	0.708739i
$877$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$877$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$	5.81878	−	8.00886i	0.195928	−	0.269672i
$883$	0.618034	−	1.90211i	0.0207985	−	0.0640112i	−0.940119	−	0.340848i	$-0.889286\pi$
$883$	0.618034	−	1.90211i	0.0207985	−	0.0640112i	0.960917	+	0.276836i	$0.0892860\pi$
$884$		0			0
$885$		0			0
$886$	−34.9127	−	48.0532i	−1.17291	−	1.61438i
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	56.4899	−	18.3547i	1.88509	−	0.612503i
$899$		0			0
$900$	8.09017	+	5.87785i	0.269672	+	0.195928i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−29.9251	+	41.1884i	−0.995295	+	1.36991i
$905$		0			0
$906$		0			0
$907$	8.09017	−	5.87785i	0.268630	−	0.195171i	−0.445313	−	0.895375i	$-0.646908\pi$
$907$	8.09017	−	5.87785i	0.268630	−	0.195171i	0.713943	+	0.700204i	$0.246908\pi$
$908$	3.32502	+	4.57649i	0.110345	+	0.151876i
$909$		0			0
$910$		0			0
$911$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$911$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$912$			67.8823i			2.24781i
$913$		0			0
$914$	−48.0000			−1.58770
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−25.8885	+	18.8091i	−0.854449	+	0.620794i
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$	−16.1400	−	5.24419i	−0.531830	−	0.172802i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	43.6869	−	31.7404i	1.43332	−	1.04137i	0.443935	−	0.896059i	$-0.353582\pi$
$929$	43.6869	−	31.7404i	1.43332	−	1.04137i	0.989386	−	0.145310i	$-0.0464179\pi$
$930$		0			0
$931$	56.4899	−	18.3547i	1.85138	−	0.601550i
$932$	−10.7600	−	3.49613i	−0.352455	−	0.114519i
$933$		0			0
$934$		−	42.4264i		−	1.38823i
$935$		0			0
$936$		0			0
$937$	−29.9251	+	41.1884i	−0.977612	+	1.34557i	−0.0395055	+	0.999219i	$0.512578\pi$
$937$	−29.9251	+	41.1884i	−0.977612	+	1.34557i	−0.938106	+	0.346348i	$0.887422\pi$
$938$		0			0
$939$	−6.18034	−	19.0211i	−0.201688	−	0.620731i
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$	19.4164	+	14.1068i	0.631950	+	0.459139i
$945$		0			0
$946$		0			0
$947$	−30.0000			−0.974869			−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000			−0.974869			−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$	18.5410	+	57.0634i	0.601550	+	1.85138i
$951$		0			0
$952$		0			0
$953$	43.0399	−	13.9845i	1.39420	−	0.453003i	0.486889	−	0.873464i	$-0.338132\pi$
$953$	43.0399	−	13.9845i	1.39420	−	0.453003i	0.907311	+	0.420461i	$0.138132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	9.57953	+	29.4828i	0.309017	+	0.951057i
$962$		0			0
$963$	−11.6376	−	16.0177i	−0.375015	−	0.516164i
$964$	−32.2799	+	10.4884i	−1.03967	+	0.337808i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$	96.0000			3.08396
$970$		0			0
$971$	16.6869	−	51.3571i	0.535509	−	1.64813i	−0.207039	−	0.978333i	$-0.566383\pi$
$971$	16.6869	−	51.3571i	0.535509	−	1.64813i	0.742547	−	0.669793i	$-0.233617\pi$
$972$	6.18034	+	19.0211i	0.198234	+	0.610103i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.85410	−	3.52671i	−0.155296	−	0.112829i	0.507423	−	0.861697i	$-0.330598\pi$
$977$	−4.85410	−	3.52671i	−0.155296	−	0.112829i	−0.662720	+	0.748867i	$0.730598\pi$
$978$			5.65685i			0.180886i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−6.18034	+	19.0211i	−0.197223	+	0.606989i
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$	−51.7771	+	37.6183i	−1.65059	+	1.19923i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−16.0689	+	49.4549i	−0.509931	+	1.56941i
$994$		0			0
$995$		0			0
$996$	−6.65003	−	9.15298i	−0.210714	−	0.290023i
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$	18.8300	+	6.11822i	0.596052	+	0.193669i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.k.a.403.1		8
8.3	odd	2	CM	968.2.k.a.403.1		8
11.2	odd	10	inner	968.2.k.a.699.1		8
11.3	even	5	inner	968.2.k.a.723.2		8
11.4	even	5	inner	968.2.k.a.475.1		8
11.5	even	5		88.2.g.a.43.1	✓	2
11.6	odd	10		88.2.g.a.43.2	yes	2
11.7	odd	10	inner	968.2.k.a.475.2		8
11.8	odd	10	inner	968.2.k.a.723.1		8
11.9	even	5	inner	968.2.k.a.699.2		8
11.10	odd	2	inner	968.2.k.a.403.2		8
33.5	odd	10		792.2.h.b.307.2		2
33.17	even	10		792.2.h.b.307.1		2
44.27	odd	10		352.2.g.a.175.1		2
44.39	even	10		352.2.g.a.175.2		2
88.3	odd	10	inner	968.2.k.a.723.2		8
88.5	even	10		352.2.g.a.175.1		2
88.19	even	10	inner	968.2.k.a.723.1		8
88.27	odd	10		88.2.g.a.43.1	✓	2
88.35	even	10	inner	968.2.k.a.699.1		8
88.43	even	2	inner	968.2.k.a.403.2		8
88.51	even	10	inner	968.2.k.a.475.2		8
88.59	odd	10	inner	968.2.k.a.475.1		8
88.61	odd	10		352.2.g.a.175.2		2
88.75	odd	10	inner	968.2.k.a.699.2		8
88.83	even	10		88.2.g.a.43.2	yes	2
132.71	even	10		3168.2.h.b.2287.2		2
132.83	odd	10		3168.2.h.b.2287.1		2
176.5	even	20		2816.2.e.d.2815.1		4
176.27	odd	20		2816.2.e.d.2815.4		4
176.61	odd	20		2816.2.e.d.2815.3		4
176.83	even	20		2816.2.e.d.2815.2		4
176.93	even	20		2816.2.e.d.2815.4		4
176.115	odd	20		2816.2.e.d.2815.1		4
176.149	odd	20		2816.2.e.d.2815.2		4
176.171	even	20		2816.2.e.d.2815.3		4
264.5	odd	10		3168.2.h.b.2287.2		2
264.83	odd	10		792.2.h.b.307.1		2
264.149	even	10		3168.2.h.b.2287.1		2
264.203	even	10		792.2.h.b.307.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.g.a.43.1	✓	2	11.5	even	5
88.2.g.a.43.1	✓	2	88.27	odd	10
88.2.g.a.43.2	yes	2	11.6	odd	10
88.2.g.a.43.2	yes	2	88.83	even	10
352.2.g.a.175.1		2	44.27	odd	10
352.2.g.a.175.1		2	88.5	even	10
352.2.g.a.175.2		2	44.39	even	10
352.2.g.a.175.2		2	88.61	odd	10
792.2.h.b.307.1		2	33.17	even	10
792.2.h.b.307.1		2	264.83	odd	10
792.2.h.b.307.2		2	33.5	odd	10
792.2.h.b.307.2		2	264.203	even	10
968.2.k.a.403.1		8	1.1	even	1	trivial
968.2.k.a.403.1		8	8.3	odd	2	CM
968.2.k.a.403.2		8	11.10	odd	2	inner
968.2.k.a.403.2		8	88.43	even	2	inner
968.2.k.a.475.1		8	11.4	even	5	inner
968.2.k.a.475.1		8	88.59	odd	10	inner
968.2.k.a.475.2		8	11.7	odd	10	inner
968.2.k.a.475.2		8	88.51	even	10	inner
968.2.k.a.699.1		8	11.2	odd	10	inner
968.2.k.a.699.1		8	88.35	even	10	inner
968.2.k.a.699.2		8	11.9	even	5	inner
968.2.k.a.699.2		8	88.75	odd	10	inner
968.2.k.a.723.1		8	11.8	odd	10	inner
968.2.k.a.723.1		8	88.19	even	10	inner
968.2.k.a.723.2		8	11.3	even	5	inner
968.2.k.a.723.2		8	88.3	odd	10	inner
2816.2.e.d.2815.1		4	176.5	even	20
2816.2.e.d.2815.1		4	176.115	odd	20
2816.2.e.d.2815.2		4	176.83	even	20
2816.2.e.d.2815.2		4	176.149	odd	20
2816.2.e.d.2815.3		4	176.61	odd	20
2816.2.e.d.2815.3		4	176.171	even	20
2816.2.e.d.2815.4		4	176.27	odd	20
2816.2.e.d.2815.4		4	176.93	even	20
3168.2.h.b.2287.1		2	132.83	odd	10
3168.2.h.b.2287.1		2	264.149	even	10
3168.2.h.b.2287.2		2	132.71	even	10
3168.2.h.b.2287.2		2	264.5	odd	10

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.k (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.831254 + 1.14412i) q^{2} +(0.618034 - 1.90211i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(1.66251 + 2.28825i) q^{6} +(2.68999 + 0.874032i) q^{8} +(-0.809017 - 0.587785i) q^{9} -4.00000 q^{12} +(-3.23607 + 2.35114i) q^{16} +(3.32502 + 4.57649i) q^{17} +(1.34500 - 0.437016i) q^{18} +(8.06998 + 2.62210i) q^{19} +(3.32502 - 4.57649i) q^{24} +(1.54508 - 4.75528i) q^{25} +(3.23607 - 2.35114i) q^{27} -5.65685i q^{32} -8.00000 q^{34} +(-0.618034 + 1.90211i) q^{36} +(-9.70820 + 7.05342i) q^{38} +(-10.7600 - 3.49613i) q^{41} -8.48528i q^{43} +(2.47214 + 7.60845i) q^{48} +(5.66312 - 4.11450i) q^{49} +(4.15627 + 5.72061i) q^{50} +(10.7600 - 3.49613i) q^{51} +5.65685i q^{54} +(9.97505 - 13.7295i) q^{57} +(-1.85410 - 5.70634i) q^{59} +(6.47214 + 4.70228i) q^{64} +14.0000 q^{67} +(6.65003 - 9.15298i) q^{68} +(-1.66251 - 2.28825i) q^{72} +(-16.1400 + 5.24419i) q^{73} +(-8.09017 - 5.87785i) q^{75} -16.9706i q^{76} +(-3.39919 - 10.4616i) q^{81} +(12.9443 - 9.40456i) q^{82} +(1.66251 + 2.28825i) q^{83} +(9.70820 + 7.05342i) q^{86} +18.0000 q^{89} +(-10.7600 - 3.49613i) q^{96} +(-8.09017 - 5.87785i) q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 4 q^{4} - 2 q^{9} - 32 q^{12} - 8 q^{16} - 10 q^{25} + 8 q^{27} - 64 q^{34} + 4 q^{36} - 24 q^{38} - 16 q^{48} + 14 q^{49} + 12 q^{59} + 16 q^{64} + 112 q^{67} - 20 q^{75} + 22 q^{81} + 32 q^{82}+ \cdots - 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 + 4 * q^4 - 2 * q^9 - 32 * q^12 - 8 * q^16 - 10 * q^25 + 8 * q^27 - 64 * q^34 + 4 * q^36 - 24 * q^38 - 16 * q^48 + 14 * q^49 + 12 * q^59 + 16 * q^64 + 112 * q^67 - 20 * q^75 + 22 * q^81 + 32 * q^82 + 24 * q^86 + 144 * q^89 - 20 * q^97

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