LMFDB - Embedded newform 847.2.b.d.846.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(846,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("847.846");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.77720518656.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 161x^{4} - 220x^{3} + 232x^{2} - 132x + 33 $ x^8 - 4x^7 + 26x^6 - 64x^5 + 161x^4 - 220x^3 + 232x^2 - 132*x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			846.5
Root			$0.500000 - 2.17595i$ of defining polynomial
Character	$\chi$	$=$	847.846
Dual form			847.2.b.d.846.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.517638i q^{2} +1.73205 q^{4} -3.31662i q^{5} +(2.34521 - 1.22474i) q^{7} +1.93185i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+0.517638i q^{2} +1.73205 q^{4} -3.31662i q^{5} +(2.34521 - 1.22474i) q^{7} +1.93185i q^{8} +3.00000 q^{9} +1.71681 q^{10} -1.71681 q^{13} +(0.633975 + 1.21397i) q^{14} +2.46410 q^{16} -6.40723 q^{17} +1.55291i q^{18} +4.69042 q^{19} -5.74456i q^{20} -1.26795 q^{23} -6.00000 q^{25} -0.888687i q^{26} +(4.06202 - 2.12132i) q^{28} +6.17449i q^{29} -9.06119i q^{31} +5.13922i q^{32} -3.31662i q^{34} +(-4.06202 - 7.77817i) q^{35} +5.19615 q^{36} -8.46410 q^{37} +2.42794i q^{38} +6.40723 q^{40} +1.71681 q^{41} -5.93426i q^{43} -9.94987i q^{45} -0.656339i q^{46} +9.06119i q^{47} +(4.00000 - 5.74456i) q^{49} -3.10583i q^{50} -2.97360 q^{52} +3.92820 q^{53} +(2.36603 + 4.53059i) q^{56} -3.19615 q^{58} +6.63325i q^{59} +8.12404 q^{61} +4.69042 q^{62} +(7.03562 - 3.67423i) q^{63} +2.26795 q^{64} +5.69402i q^{65} +7.66025 q^{67} -11.0976 q^{68} +(4.02628 - 2.10266i) q^{70} -7.46410 q^{71} +5.79555i q^{72} -4.69042 q^{73} -4.38134i q^{74} +8.12404 q^{76} +9.14162i q^{79} -8.17250i q^{80} +9.00000 q^{81} +0.888687i q^{82} -9.38083 q^{83} +21.2504i q^{85} +3.07180 q^{86} +5.74456i q^{89} +5.15043 q^{90} +(-4.02628 + 2.10266i) q^{91} -2.19615 q^{92} -4.69042 q^{94} -15.5563i q^{95} +8.17250i q^{97} +(2.97360 + 2.07055i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 12 q^{14} - 8 q^{16} - 24 q^{23} - 48 q^{25} - 40 q^{37} + 32 q^{49} - 24 q^{53} + 12 q^{56} + 16 q^{58} + 32 q^{64} - 8 q^{67} - 44 q^{70} - 32 q^{71} + 72 q^{81} + 80 q^{86} + 44 q^{91} + 24 q^{92}+O(q^{100}) $ 8 * q + 24 * q^9 + 12 * q^14 - 8 * q^16 - 24 * q^23 - 48 * q^25 - 40 * q^37 + 32 * q^49 - 24 * q^53 + 12 * q^56 + 16 * q^58 + 32 * q^64 - 8 * q^67 - 44 * q^70 - 32 * q^71 + 72 * q^81 + 80 * q^86 + 44 * q^91 + 24 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

$n$	$122$	$365$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$			0.517638i			0.366025i	0.983111	+	0.183013i	$0.0585849\pi$
$2$			0.517638i			0.366025i	−0.983111	+	0.183013i	$0.941415\pi$
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	1.73205			0.866025
$5$		−	3.31662i		−	1.48324i	−0.670820	−	0.741620i	$-0.734058\pi$
$5$		−	3.31662i		−	1.48324i	0.670820	−	0.741620i	$-0.265942\pi$
$6$		0			0
$7$	2.34521	−	1.22474i	0.886405	−	0.462910i
$7$	2.34521	−	1.22474i	0.886405	−	0.462910i
$8$			1.93185i			0.683013i
$9$	3.00000			1.00000
$10$	1.71681			0.542903
$11$		0			0
$11$		0			0
$12$		0			0
$13$	−1.71681			−0.476158			−0.238079	−	0.971246i	$-0.576518\pi$
$13$	−1.71681			−0.476158			−0.238079	+	0.971246i	$0.576518\pi$
$14$	0.633975	+	1.21397i	0.169437	+	0.324447i
$15$		0			0
$16$	2.46410			0.616025
$17$	−6.40723			−1.55398			−0.776990	−	0.629512i	$-0.783255\pi$
$17$	−6.40723			−1.55398			−0.776990	+	0.629512i	$0.783255\pi$
$18$			1.55291i			0.366025i
$19$	4.69042			1.07606			0.538028	−	0.842927i	$-0.319170\pi$
$19$	4.69042			1.07606			0.538028	+	0.842927i	$0.319170\pi$
$20$		−	5.74456i		−	1.28452i
$21$		0			0
$22$		0			0
$23$	−1.26795			−0.264386			−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795			−0.264386			−0.132193	+	0.991224i	$0.542202\pi$
$24$		0			0
$25$	−6.00000			−1.20000
$26$		−	0.888687i		−	0.174286i
$27$		0			0
$28$	4.06202	−	2.12132i	0.767649	−	0.400892i
$29$			6.17449i			1.14657i	0.819354	+	0.573287i	$0.194332\pi$
$29$			6.17449i			1.14657i	−0.819354	+	0.573287i	$0.805668\pi$
$30$		0			0
$31$		−	9.06119i		−	1.62744i	−0.581259	−	0.813719i	$-0.697440\pi$
$31$		−	9.06119i		−	1.62744i	0.581259	−	0.813719i	$-0.302560\pi$
$32$			5.13922i			0.908494i
$33$		0			0
$34$		−	3.31662i		−	0.568796i
$35$	−4.06202	−	7.77817i	−0.686607	−	1.31475i
$36$	5.19615			0.866025
$37$	−8.46410			−1.39149			−0.695745	−	0.718289i	$-0.744926\pi$
$37$	−8.46410			−1.39149			−0.695745	+	0.718289i	$0.744926\pi$
$38$			2.42794i			0.393864i
$39$		0			0
$40$	6.40723			1.01307
$41$	1.71681			0.268121			0.134060	−	0.990973i	$-0.457198\pi$
$41$	1.71681			0.268121			0.134060	+	0.990973i	$0.457198\pi$
$42$		0			0
$43$		−	5.93426i		−	0.904966i	−0.891773	−	0.452483i	$-0.850538\pi$
$43$		−	5.93426i		−	0.904966i	0.891773	−	0.452483i	$-0.149462\pi$
$44$		0			0
$45$		−	9.94987i		−	1.48324i
$46$		−	0.656339i		−	0.0967719i
$47$			9.06119i			1.32171i	0.750514	+	0.660855i	$0.229806\pi$
$47$			9.06119i			1.32171i	−0.750514	+	0.660855i	$0.770194\pi$
$48$		0			0
$49$	4.00000	−	5.74456i	0.571429	−	0.820652i
$50$		−	3.10583i		−	0.439230i
$51$		0			0
$52$	−2.97360			−0.412365
$53$	3.92820			0.539580			0.269790	−	0.962919i	$-0.413046\pi$
$53$	3.92820			0.539580			0.269790	+	0.962919i	$0.413046\pi$
$54$		0			0
$55$		0			0
$56$	2.36603	+	4.53059i	0.316173	+	0.605426i
$57$		0			0
$58$	−3.19615			−0.419675
$59$			6.63325i			0.863576i	0.901975	+	0.431788i	$0.142117\pi$
$59$			6.63325i			0.863576i	−0.901975	+	0.431788i	$0.857883\pi$
$60$		0			0
$61$	8.12404			1.04018			0.520088	−	0.854113i	$-0.325899\pi$
$61$	8.12404			1.04018			0.520088	+	0.854113i	$0.325899\pi$
$62$	4.69042			0.595683
$63$	7.03562	−	3.67423i	0.886405	−	0.462910i
$64$	2.26795			0.283494
$65$			5.69402i			0.706256i
$66$		0			0
$67$	7.66025			0.935849			0.467924	−	0.883768i	$-0.345002\pi$
$67$	7.66025			0.935849			0.467924	+	0.883768i	$0.345002\pi$
$68$	−11.0976			−1.34579
$69$		0			0
$70$	4.02628	−	2.10266i	0.481232	−	0.251315i
$71$	−7.46410			−0.885826			−0.442913	−	0.896565i	$-0.646055\pi$
$71$	−7.46410			−0.885826			−0.442913	+	0.896565i	$0.646055\pi$
$72$			5.79555i			0.683013i
$73$	−4.69042			−0.548972			−0.274486	−	0.961591i	$-0.588508\pi$
$73$	−4.69042			−0.548972			−0.274486	+	0.961591i	$0.588508\pi$
$74$		−	4.38134i		−	0.509321i
$75$		0			0
$76$	8.12404			0.931891
$77$		0			0
$78$		0			0
$79$			9.14162i			1.02851i	0.857637	+	0.514256i	$0.171932\pi$
$79$			9.14162i			1.02851i	−0.857637	+	0.514256i	$0.828068\pi$
$80$		−	8.17250i		−	0.913713i
$81$	9.00000			1.00000
$82$			0.888687i			0.0981391i
$83$	−9.38083			−1.02968			−0.514840	−	0.857286i	$-0.672149\pi$
$83$	−9.38083			−1.02968			−0.514840	+	0.857286i	$0.672149\pi$
$84$		0			0
$85$			21.2504i			2.30493i
$86$	3.07180			0.331240
$87$		0			0
$88$		0			0
$89$			5.74456i			0.608922i	0.952525	+	0.304461i	$0.0984764\pi$
$89$			5.74456i			0.608922i	−0.952525	+	0.304461i	$0.901524\pi$
$90$	5.15043			0.542903
$91$	−4.02628	+	2.10266i	−0.422069	+	0.220418i
$92$	−2.19615			−0.228965
$93$		0			0
$94$	−4.69042			−0.483779
$95$		−	15.5563i		−	1.59605i
$96$		0			0
$97$			8.17250i			0.829792i	0.909869	+	0.414896i	$0.136182\pi$
$97$			8.17250i			0.829792i	−0.909869	+	0.414896i	$0.863818\pi$
$98$	2.97360	+	2.07055i	0.300379	+	0.209157i
$99$		0			0
$100$	−10.3923			−1.03923
$101$	4.69042			0.466714			0.233357	−	0.972391i	$-0.425029\pi$
$101$	4.69042			0.466714			0.233357	+	0.972391i	$0.425029\pi$
$102$		0			0
$103$			4.20531i			0.414362i	0.978303	+	0.207181i	$0.0664289\pi$
$103$			4.20531i			0.414362i	−0.978303	+	0.207181i	$0.933571\pi$
$104$		−	3.31662i		−	0.325222i
$105$		0			0
$106$			2.03339i			0.197500i
$107$		−	1.03528i		−	0.100084i	−0.998747	−	0.0500420i	$-0.984064\pi$
$107$		−	1.03528i		−	0.100084i	0.998747	−	0.0500420i	$-0.0159355\pi$
$108$		0			0
$109$		−	3.62347i		−	0.347065i	−0.984828	−	0.173533i	$-0.944482\pi$
$109$		−	3.62347i		−	0.347065i	0.984828	−	0.173533i	$-0.0555182\pi$
$110$		0			0
$111$		0			0
$112$	5.77883	−	3.01790i	0.546048	−	0.285164i
$113$	11.1962			1.05325			0.526623	−	0.850099i	$-0.323458\pi$
$113$	11.1962			1.05325			0.526623	+	0.850099i	$0.323458\pi$
$114$		0			0
$115$			4.20531i			0.392147i
$116$			10.6945i			0.992963i
$117$	−5.15043			−0.476158
$118$	−3.43362			−0.316091
$119$	−15.0263	+	7.84722i	−1.37746	+	0.719353i
$120$		0			0
$121$		0			0
$122$			4.20531i			0.380731i
$123$		0			0
$124$		−	15.6944i		−	1.40940i
$125$			3.31662i			0.296648i
$126$	1.90192	+	3.64191i	0.169437	+	0.324447i
$127$		−	3.10583i		−	0.275598i	−0.990460	−	0.137799i	$-0.955997\pi$
$127$		−	3.10583i		−	0.275598i	0.990460	−	0.137799i	$-0.0440028\pi$
$128$			11.4524i			1.01226i
$129$		0			0
$130$	−2.94744			−0.258508
$131$	1.25679			0.109807			0.0549033	−	0.998492i	$-0.482515\pi$
$131$	1.25679			0.109807			0.0549033	+	0.998492i	$0.482515\pi$
$132$		0			0
$133$	11.0000	−	5.74456i	0.953821	−	0.498117i
$134$			3.96524i			0.342545i
$135$		0			0
$136$		−	12.3778i		−	1.06139i
$137$	16.9282			1.44627			0.723137	−	0.690705i	$-0.242700\pi$
$137$	16.9282			1.44627			0.723137	+	0.690705i	$0.242700\pi$
$138$		0			0
$139$	−14.0712			−1.19351			−0.596754	−	0.802424i	$-0.703543\pi$
$139$	−14.0712			−1.19351			−0.596754	+	0.802424i	$0.703543\pi$
$140$	−7.03562	−	13.4722i	−0.594619	−	1.13861i
$141$		0			0
$142$		−	3.86370i		−	0.324235i
$143$		0			0
$144$	7.39230			0.616025
$145$	20.4785			1.70064
$146$		−	2.42794i		−	0.200938i
$147$		0			0
$148$	−14.6603			−1.20507
$149$		−	16.3514i		−	1.33956i	−0.742561	−	0.669779i	$-0.766389\pi$
$149$		−	16.3514i		−	1.33956i	0.742561	−	0.669779i	$-0.233611\pi$
$150$		0			0
$151$		−	16.5916i		−	1.35021i	−0.737723	−	0.675104i	$-0.764099\pi$
$151$		−	16.5916i		−	1.35021i	0.737723	−	0.675104i	$-0.235901\pi$
$152$			9.06119i			0.734959i
$153$	−19.2217			−1.55398
$154$		0			0
$155$	−30.0526			−2.41388
$156$		0			0
$157$			11.4891i			0.916932i	0.888712	+	0.458466i	$0.151601\pi$
$157$			11.4891i			0.916932i	−0.888712	+	0.458466i	$0.848399\pi$
$158$	−4.73205			−0.376462
$159$		0			0
$160$	17.0449			1.34751
$161$	−2.97360	+	1.55291i	−0.234353	+	0.122387i
$162$			4.65874i			0.366025i
$163$	−11.8564			−0.928665			−0.464333	−	0.885661i	$-0.653706\pi$
$163$	−11.8564			−0.928665			−0.464333	+	0.885661i	$0.653706\pi$
$164$	2.97360			0.232199
$165$		0			0
$166$		−	4.85588i		−	0.376889i
$167$	−8.12404			−0.628657			−0.314328	−	0.949314i	$-0.601779\pi$
$167$	−8.12404			−0.628657			−0.314328	+	0.949314i	$0.601779\pi$
$168$		0			0
$169$	−10.0526			−0.773274
$170$	−11.0000			−0.843661
$171$	14.0712			1.07606
$172$		−	10.2784i		−	0.783723i
$173$	−14.0712			−1.06982			−0.534909	−	0.844910i	$-0.679654\pi$
$173$	−14.0712			−1.06982			−0.534909	+	0.844910i	$0.679654\pi$
$174$		0			0
$175$	−14.0712	+	7.34847i	−1.06369	+	0.555492i
$176$		0			0
$177$		0			0
$178$	−2.97360			−0.222881
$179$	−6.33975			−0.473855			−0.236927	−	0.971527i	$-0.576140\pi$
$179$	−6.33975			−0.473855			−0.236927	+	0.971527i	$0.576140\pi$
$180$		−	17.2337i		−	1.28452i
$181$			12.3778i			0.920036i	0.887910	+	0.460018i	$0.152157\pi$
$181$			12.3778i			0.920036i	−0.887910	+	0.460018i	$0.847843\pi$
$182$	−1.08841	−	2.08416i	−0.0806787	−	0.154488i
$183$		0			0
$184$		−	2.44949i		−	0.180579i
$185$			28.0722i			2.06391i
$186$		0			0
$187$		0			0
$188$			15.6944i			1.14463i
$189$		0			0
$190$	8.05256			0.584194
$191$	−7.26795			−0.525890			−0.262945	−	0.964811i	$-0.584694\pi$
$191$	−7.26795			−0.525890			−0.262945	+	0.964811i	$0.584694\pi$
$192$		0			0
$193$			8.90138i			0.640736i	0.947293	+	0.320368i	$0.103806\pi$
$193$			8.90138i			0.640736i	−0.947293	+	0.320368i	$0.896194\pi$
$194$	−4.23040			−0.303725
$195$		0			0
$196$	6.92820	−	9.94987i	0.494872	−	0.710705i
$197$		−	19.1798i		−	1.36651i	−0.730182	−	0.683253i	$-0.760565\pi$
$197$		−	19.1798i		−	1.36651i	0.730182	−	0.683253i	$-0.239435\pi$
$198$		0			0
$199$			24.7556i			1.75488i	0.479687	+	0.877440i	$0.340750\pi$
$199$			24.7556i			1.75488i	−0.479687	+	0.877440i	$0.659250\pi$
$200$		−	11.5911i		−	0.819615i
$201$		0			0
$202$			2.42794i			0.170829i
$203$	7.56218	+	14.4805i	0.530761	+	1.01633i
$204$		0			0
$205$		−	5.69402i		−	0.397688i
$206$	−2.17683			−0.151667
$207$	−3.80385			−0.264386
$208$	−4.23040			−0.293325
$209$		0			0
$210$		0			0
$211$			21.7680i			1.49857i	0.662247	+	0.749286i	$0.269603\pi$
$211$			21.7680i			1.49857i	−0.662247	+	0.749286i	$0.730397\pi$
$212$	6.80385			0.467290
$213$		0			0
$214$	0.535898			0.0366333
$215$	−19.6817			−1.34228
$216$		0			0
$217$	−11.0976	−	21.2504i	−0.753357	−	1.44257i
$218$	1.87564			0.127035
$219$		0			0
$220$		0			0
$221$	11.0000			0.739940
$222$		0			0
$223$			4.85588i			0.325173i	0.986694	+	0.162587i	$0.0519837\pi$
$223$			4.85588i			0.325173i	−0.986694	+	0.162587i	$0.948016\pi$
$224$	6.29423	+	12.0525i	0.420551	+	0.805294i
$225$	−18.0000			−1.20000
$226$			5.79555i			0.385515i
$227$	5.94721			0.394730			0.197365	−	0.980330i	$-0.436762\pi$
$227$	5.94721			0.394730			0.197365	+	0.980330i	$0.436762\pi$
$228$		0			0
$229$		−	5.74456i		−	0.379611i	−0.981822	−	0.189806i	$-0.939214\pi$
$229$		−	5.74456i		−	0.379611i	0.981822	−	0.189806i	$-0.0607858\pi$
$230$	−2.17683			−0.143536
$231$		0			0
$232$	−11.9282			−0.783125
$233$		−	4.10394i		−	0.268858i	−0.990923	−	0.134429i	$-0.957080\pi$
$233$		−	4.10394i		−	0.268858i	0.990923	−	0.134429i	$-0.0429200\pi$
$234$		−	2.66606i		−	0.174286i
$235$	30.0526			1.96041
$236$			11.4891i			0.747878i
$237$		0			0
$238$	−4.06202	−	7.77817i	−0.263302	−	0.504184i
$239$			24.2175i			1.56650i	0.621707	+	0.783250i	$0.286439\pi$
$239$			24.2175i			1.56650i	−0.621707	+	0.783250i	$0.713561\pi$
$240$		0			0
$241$	−20.9385			−1.34877			−0.674383	−	0.738381i	$-0.735590\pi$
$241$	−20.9385			−1.34877			−0.674383	+	0.738381i	$0.735590\pi$
$242$		0			0
$243$		0			0
$244$	14.0712			0.900819
$245$	−19.0526	−	13.2665i	−1.21722	−	0.847566i
$246$		0			0
$247$	−8.05256			−0.512372
$248$	17.5049			1.11156
$249$		0			0
$250$	−1.71681			−0.108581
$251$			10.8386i			0.684124i	0.939677	+	0.342062i	$0.111125\pi$
$251$			10.8386i			0.684124i	−0.939677	+	0.342062i	$0.888875\pi$
$252$	12.1861	−	6.36396i	0.767649	−	0.400892i
$253$		0			0
$254$	1.60770			0.100876
$255$		0			0
$256$	−1.39230			−0.0870191
$257$			23.8669i			1.48878i	0.667746	+	0.744389i	$0.267259\pi$
$257$			23.8669i			1.48878i	−0.667746	+	0.744389i	$0.732741\pi$
$258$		0			0
$259$	−19.8501	+	10.3664i	−1.23342	+	0.644135i
$260$			9.86233i			0.611636i
$261$			18.5235i			1.14657i
$262$			0.650564i			0.0401920i
$263$		−	5.93426i		−	0.365922i	−0.983120	−	0.182961i	$-0.941432\pi$
$263$		−	5.93426i		−	0.365922i	0.983120	−	0.182961i	$-0.0585682\pi$
$264$		0			0
$265$		−	13.0284i		−	0.800327i
$266$	2.97360	+	5.69402i	0.182323	+	0.349123i
$267$		0			0
$268$	13.2679			0.810469
$269$		−	25.6443i		−	1.56356i	−0.623554	−	0.781781i	$-0.714312\pi$
$269$		−	25.6443i		−	1.56356i	0.623554	−	0.781781i	$-0.285688\pi$
$270$		0			0
$271$	−17.5049			−1.06335			−0.531673	−	0.846950i	$-0.678436\pi$
$271$	−17.5049			−1.06335			−0.531673	+	0.846950i	$0.678436\pi$
$272$	−15.7881			−0.957292
$273$		0			0
$274$			8.76268i			0.529373i
$275$		0			0
$276$		0			0
$277$			0.517638i			0.0311019i	0.999879	+	0.0155509i	$0.00495021\pi$
$277$			0.517638i			0.0311019i	−0.999879	+	0.0155509i	$0.995050\pi$
$278$		−	7.28381i		−	0.436854i
$279$		−	27.1836i		−	1.62744i
$280$	15.0263	−	7.84722i	0.897992	−	0.468961i
$281$			3.20736i			0.191335i	0.995413	+	0.0956677i	$0.0304986\pi$
$281$			3.20736i			0.191335i	−0.995413	+	0.0956677i	$0.969501\pi$
$282$		0			0
$283$	9.38083			0.557633			0.278816	−	0.960344i	$-0.410058\pi$
$283$	9.38083			0.557633			0.278816	+	0.960344i	$0.410058\pi$
$284$	−12.9282			−0.767148
$285$		0			0
$286$		0			0
$287$	4.02628	−	2.10266i	0.237664	−	0.124116i
$288$			15.4176i			0.908494i
$289$	24.0526			1.41486
$290$			10.6004i			0.622479i
$291$		0			0
$292$	−8.12404			−0.475423
$293$	25.1689			1.47038			0.735191	−	0.677860i	$-0.237092\pi$
$293$	25.1689			1.47038			0.735191	+	0.677860i	$0.237092\pi$
$294$		0			0
$295$	22.0000			1.28089
$296$		−	16.3514i		−	0.950405i
$297$		0			0
$298$	8.46410			0.490312
$299$	2.17683			0.125889
$300$		0			0
$301$	−7.26795	−	13.9171i	−0.418918	−	0.802166i
$302$	8.58846			0.494210
$303$		0			0
$304$	11.5577			0.662877
$305$		−	26.9444i		−	1.54283i
$306$		−	9.94987i		−	0.568796i
$307$	24.3721			1.39099			0.695495	−	0.718531i	$-0.255185\pi$
$307$	24.3721			1.39099			0.695495	+	0.718531i	$0.255185\pi$
$308$		0			0
$309$		0			0
$310$		−	15.5563i		−	0.883541i
$311$			4.20531i			0.238461i	0.992867	+	0.119231i	$0.0380428\pi$
$311$			4.20531i			0.238461i	−0.992867	+	0.119231i	$0.961957\pi$
$312$		0			0
$313$			10.6004i			0.599172i	0.954069	+	0.299586i	$0.0968486\pi$
$313$			10.6004i			0.599172i	−0.954069	+	0.299586i	$0.903151\pi$
$314$	−5.94721			−0.335620
$315$	−12.1861	−	23.3345i	−0.686607	−	1.31475i
$316$			15.8338i			0.890718i
$317$	−7.07180			−0.397192			−0.198596	−	0.980081i	$-0.563638\pi$
$317$	−7.07180			−0.397192			−0.198596	+	0.980081i	$0.563638\pi$
$318$		0			0
$319$		0			0
$320$		−	7.52194i		−	0.420489i
$321$		0			0
$322$	−0.803848	−	1.53925i	−0.0447967	−	0.0857791i
$323$	−30.0526			−1.67217
$324$	15.5885			0.866025
$325$	10.3009			0.571389
$326$		−	6.13733i		−	0.339915i
$327$		0			0
$328$			3.31662i			0.183130i
$329$	11.0976	+	21.2504i	0.611833	+	1.17157i
$330$		0			0
$331$	−1.26795			−0.0696928			−0.0348464	−	0.999393i	$-0.511094\pi$
$331$	−1.26795			−0.0696928			−0.0348464	+	0.999393i	$0.511094\pi$
$332$	−16.2481			−0.891729
$333$	−25.3923			−1.39149
$334$		−	4.20531i		−	0.230104i
$335$		−	25.4062i		−	1.38809i
$336$		0			0
$337$			28.4973i			1.55235i	0.630519	+	0.776173i	$0.282842\pi$
$337$			28.4973i			1.55235i	−0.630519	+	0.776173i	$0.717158\pi$
$338$		−	5.20359i		−	0.283038i
$339$		0			0
$340$			36.8067i			1.99612i
$341$		0			0
$342$			7.28381i			0.393864i
$343$	2.34521	−	18.3712i	0.126629	−	0.991950i
$344$	11.4641			0.618103
$345$		0			0
$346$		−	7.28381i		−	0.391580i
$347$		−	27.9053i		−	1.49804i	−0.662549	−	0.749018i	$-0.730525\pi$
$347$		−	27.9053i		−	1.49804i	0.662549	−	0.749018i	$-0.269475\pi$
$348$		0			0
$349$	36.7266			1.96593			0.982964	−	0.183800i	$-0.0588400\pi$
$349$	36.7266			1.96593			0.982964	+	0.183800i	$0.0588400\pi$
$350$	−3.80385	−	7.28381i	−0.203324	−	0.389336i
$351$		0			0
$352$		0			0
$353$		−	10.6004i		−	0.564204i	−0.959384	−	0.282102i	$-0.908968\pi$
$353$		−	10.6004i		−	0.564204i	0.959384	−	0.282102i	$-0.0910317\pi$
$354$		0			0
$355$			24.7556i			1.31389i
$356$			9.94987i			0.527342i
$357$		0			0
$358$		−	3.28169i		−	0.173443i
$359$			11.4152i			0.602474i	0.953549	+	0.301237i	$0.0973995\pi$
$359$			11.4152i			0.602474i	−0.953549	+	0.301237i	$0.902601\pi$
$360$	19.2217			1.01307
$361$	3.00000			0.157895
$362$	−6.40723			−0.336756
$363$		0			0
$364$	−6.97372	+	3.64191i	−0.365522	+	0.190888i
$365$			15.5563i			0.814257i
$366$		0			0
$367$		−	13.2665i		−	0.692506i	−0.938141	−	0.346253i	$-0.887454\pi$
$367$		−	13.2665i		−	0.692506i	0.938141	−	0.346253i	$-0.112546\pi$
$368$	−3.12436			−0.162868
$369$	5.15043			0.268121
$370$	−14.5313			−0.755445
$371$	9.21245	−	4.81105i	0.478287	−	0.249777i
$372$		0			0
$373$		−	12.2474i		−	0.634149i	−0.948401	−	0.317074i	$-0.897299\pi$
$373$		−	12.2474i		−	0.634149i	0.948401	−	0.317074i	$-0.102701\pi$
$374$		0			0
$375$		0			0
$376$	−17.5049			−0.902745
$377$		−	10.6004i		−	0.545950i
$378$		0			0
$379$	−13.1244			−0.674153			−0.337076	−	0.941477i	$-0.609438\pi$
$379$	−13.1244			−0.674153			−0.337076	+	0.941477i	$0.609438\pi$
$380$		−	26.9444i		−	1.38222i
$381$		0			0
$382$		−	3.76217i		−	0.192489i
$383$		−	13.9171i		−	0.711129i	−0.934652	−	0.355564i	$-0.884289\pi$
$383$		−	13.9171i		−	0.711129i	0.934652	−	0.355564i	$-0.115711\pi$
$384$		0			0
$385$		0			0
$386$	−4.60770			−0.234526
$387$		−	17.8028i		−	0.904966i
$388$			14.1552i			0.718621i
$389$	−0.411543			−0.0208660			−0.0104330	−	0.999946i	$-0.503321\pi$
$389$	−0.411543			−0.0208660			−0.0104330	+	0.999946i	$0.503321\pi$
$390$		0			0
$391$	8.12404			0.410850
$392$	11.0976	+	7.72741i	0.560516	+	0.390293i
$393$		0			0
$394$	9.92820			0.500176
$395$	30.3193			1.52553
$396$		0			0
$397$		−	16.5831i		−	0.832283i	−0.909300	−	0.416142i	$-0.863382\pi$
$397$		−	16.5831i		−	0.832283i	0.909300	−	0.416142i	$-0.136618\pi$
$398$	−12.8145			−0.642331
$399$		0			0
$400$	−14.7846			−0.739230
$401$	−2.12436			−0.106085			−0.0530426	−	0.998592i	$-0.516892\pi$
$401$	−2.12436			−0.106085			−0.0530426	+	0.998592i	$0.516892\pi$
$402$		0			0
$403$			15.5563i			0.774917i
$404$	8.12404			0.404186
$405$		−	29.8496i		−	1.48324i
$406$	−7.49564	+	3.91447i	−0.372002	+	0.194272i
$407$		0			0
$408$		0			0
$409$	5.15043			0.254673			0.127336	−	0.991860i	$-0.459357\pi$
$409$	5.15043			0.254673			0.127336	+	0.991860i	$0.459357\pi$
$410$	2.94744			0.145564
$411$		0			0
$412$			7.28381i			0.358848i
$413$	8.12404	+	15.5563i	0.399758	+	0.765478i
$414$		−	1.96902i		−	0.0967719i
$415$			31.1127i			1.52726i
$416$		−	8.82306i		−	0.432586i
$417$		0			0
$418$		0			0
$419$			0.650564i			0.0317821i	0.999874	+	0.0158911i	$0.00505850\pi$
$419$			0.650564i			0.0317821i	−0.999874	+	0.0158911i	$0.994942\pi$
$420$		0			0
$421$	4.66025			0.227127			0.113563	−	0.993531i	$-0.463773\pi$
$421$	4.66025			0.227127			0.113563	+	0.993531i	$0.463773\pi$
$422$	−11.2679			−0.548515
$423$			27.1836i			1.32171i
$424$			7.58871i			0.368540i
$425$	38.4434			1.86478
$426$		0			0
$427$	19.0526	−	9.94987i	0.922018	−	0.481508i
$428$		−	1.79315i		−	0.0866752i
$429$		0			0
$430$		−	10.1880i		−	0.491309i
$431$		−	36.2891i		−	1.74798i	−0.485941	−	0.873992i	$-0.661523\pi$
$431$		−	36.2891i		−	1.74798i	0.485941	−	0.873992i	$-0.338477\pi$
$432$		0			0
$433$			32.9281i			1.58242i	0.611542	+	0.791212i	$0.290550\pi$
$433$			32.9281i			1.58242i	−0.611542	+	0.791212i	$0.709450\pi$
$434$	11.0000	−	5.74456i	0.528017	−	0.275748i
$435$		0			0
$436$		−	6.27603i		−	0.300567i
$437$	−5.94721			−0.284494
$438$		0			0
$439$	25.6289			1.22320			0.611601	−	0.791167i	$-0.290526\pi$
$439$	25.6289			1.22320			0.611601	+	0.791167i	$0.290526\pi$
$440$		0			0
$441$	12.0000	−	17.2337i	0.571429	−	0.820652i
$442$			5.69402i			0.270837i
$443$	0.196152			0.00931948			0.00465974	−	0.999989i	$-0.498517\pi$
$443$	0.196152			0.00931948			0.00465974	+	0.999989i	$0.498517\pi$
$444$		0			0
$445$	19.0526			0.903178
$446$	−2.51359			−0.119022
$447$		0			0
$448$	5.31881	−	2.77766i	0.251290	−	0.131232i
$449$	−0.856406			−0.0404163			−0.0202082	−	0.999796i	$-0.506433\pi$
$449$	−0.856406			−0.0404163			−0.0202082	+	0.999796i	$0.506433\pi$
$450$		−	9.31749i		−	0.439230i
$451$		0			0
$452$	19.3923			0.912137
$453$		0			0
$454$			3.07850i			0.144481i
$455$	6.97372	+	13.3537i	0.326933	+	0.626029i
$456$		0			0
$457$			33.7009i			1.57646i	0.615380	+	0.788231i	$0.289003\pi$
$457$			33.7009i			1.57646i	−0.615380	+	0.788231i	$0.710997\pi$
$458$	2.97360			0.138947
$459$		0			0
$460$			7.28381i			0.339610i
$461$	−15.7881			−0.735323			−0.367662	−	0.929960i	$-0.619842\pi$
$461$	−15.7881			−0.735323			−0.367662	+	0.929960i	$0.619842\pi$
$462$		0			0
$463$	−0.392305			−0.0182320			−0.00911598	−	0.999958i	$-0.502902\pi$
$463$	−0.392305			−0.0182320			−0.00911598	+	0.999958i	$0.502902\pi$
$464$			15.2146i			0.706319i
$465$		0			0
$466$	2.12436			0.0984089
$467$		−	25.4062i		−	1.17566i	−0.808985	−	0.587829i	$-0.799983\pi$
$467$		−	25.4062i		−	1.17566i	0.808985	−	0.587829i	$-0.200017\pi$
$468$	−8.92081			−0.412365
$469$	17.9649	−	9.38186i	0.829541	−	0.433214i
$470$			15.5563i			0.717561i
$471$		0			0
$472$	−12.8145			−0.589833
$473$		0			0
$474$		0			0
$475$	−28.1425			−1.29127
$476$	−26.0263	+	13.5918i	−1.19291	+	0.622978i
$477$	11.7846			0.539580
$478$	−12.5359			−0.573379
$479$	−38.4434			−1.75652			−0.878261	−	0.478182i	$-0.841296\pi$
$479$	−38.4434			−1.75652			−0.878261	+	0.478182i	$0.841296\pi$
$480$		0			0
$481$	14.5313			0.662569
$482$		−	10.8386i		−	0.493683i
$483$		0			0
$484$		0			0
$485$	27.1051			1.23078
$486$		0			0
$487$	−7.85641			−0.356008			−0.178004	−	0.984030i	$-0.556964\pi$
$487$	−7.85641			−0.356008			−0.178004	+	0.984030i	$0.556964\pi$
$488$			15.6944i			0.710454i
$489$		0			0
$490$	6.86725	−	9.86233i	0.310231	−	0.445535i
$491$			13.1069i			0.591504i	0.955265	+	0.295752i	$0.0955702\pi$
$491$			13.1069i			0.591504i	−0.955265	+	0.295752i	$0.904430\pi$
$492$		0			0
$493$		−	39.5614i		−	1.78175i
$494$		−	4.16831i		−	0.187541i
$495$		0			0
$496$		−	22.3277i		−	1.00254i
$497$	−17.5049	+	9.14162i	−0.785201	+	0.410058i
$498$		0			0
$499$	14.7321			0.659497			0.329749	−	0.944069i	$-0.393036\pi$
$499$	14.7321			0.659497			0.329749	+	0.944069i	$0.393036\pi$
$500$			5.74456i			0.256905i
$501$		0			0
$502$	−5.61045			−0.250407
$503$	−37.1866			−1.65807			−0.829034	−	0.559199i	$-0.811109\pi$
$503$	−37.1866			−1.65807			−0.829034	+	0.559199i	$0.811109\pi$
$504$	7.09808	+	13.5918i	0.316173	+	0.605426i
$505$		−	15.5563i		−	0.692248i
$506$		0			0
$507$		0			0
$508$		−	5.37945i		−	0.238675i
$509$		−	26.5330i		−	1.17605i	−0.808841	−	0.588027i	$-0.799905\pi$
$509$		−	26.5330i		−	1.17605i	0.808841	−	0.588027i	$-0.200095\pi$
$510$		0			0
$511$	−11.0000	+	5.74456i	−0.486611	+	0.254124i
$512$			22.1841i			0.980408i
$513$		0			0
$514$	−12.3544			−0.544931
$515$	13.9474			0.614598
$516$		0			0
$517$		0			0
$518$	−5.36603	−	10.2752i	−0.235770	−	0.451464i
$519$		0			0
$520$	−11.0000			−0.482382
$521$			13.2665i			0.581216i	0.956842	+	0.290608i	$0.0938575\pi$
$521$			13.2665i			0.581216i	−0.956842	+	0.290608i	$0.906142\pi$
$522$	−9.58846			−0.419675
$523$	25.6289			1.12067			0.560337	−	0.828265i	$-0.310672\pi$
$523$	25.6289			1.12067			0.560337	+	0.828265i	$0.310672\pi$
$524$	2.17683			0.0950952
$525$		0			0
$526$	3.07180			0.133937
$527$			58.0571i			2.52901i
$528$		0			0
$529$	−21.3923			−0.930100
$530$	6.74398			0.292940
$531$			19.8997i			0.863576i
$532$	19.0526	−	9.94987i	0.826033	−	0.431382i
$533$	−2.94744			−0.127668
$534$		0			0
$535$	−3.43362			−0.148448
$536$			14.7985i			0.639197i
$537$		0			0
$538$	13.2745			0.572303
$539$		0			0
$540$		0			0
$541$			10.3800i			0.446270i	0.974788	+	0.223135i	$0.0716290\pi$
$541$			10.3800i			0.446270i	−0.974788	+	0.223135i	$0.928371\pi$
$542$		−	9.06119i		−	0.389211i
$543$		0			0
$544$		−	32.9281i		−	1.41178i
$545$	−12.0177			−0.514781
$546$		0			0
$547$			4.03957i			0.172719i	0.996264	+	0.0863597i	$0.0275234\pi$
$547$			4.03957i			0.172719i	−0.996264	+	0.0863597i	$0.972477\pi$
$548$	29.3205			1.25251
$549$	24.3721			1.04018
$550$		0			0
$551$			28.9609i			1.23378i
$552$		0			0
$553$	11.1962	+	21.4390i	0.476109	+	0.911679i
$554$	−0.267949			−0.0113841
$555$		0			0
$556$	−24.3721			−1.03361
$557$		−	15.0759i		−	0.638785i	−0.947622	−	0.319393i	$-0.896521\pi$
$557$		−	15.0759i		−	0.638785i	0.947622	−	0.319393i	$-0.103479\pi$
$558$	14.0712			0.595683
$559$			10.1880i			0.430906i
$560$	−10.0092	−	19.1662i	−0.422967	−	0.809920i
$561$		0			0
$562$	−1.66025			−0.0700336
$563$	−6.86725			−0.289420			−0.144710	−	0.989474i	$-0.546225\pi$
$563$	−6.86725			−0.289420			−0.144710	+	0.989474i	$0.546225\pi$
$564$		0			0
$565$		−	37.1334i		−	1.56222i
$566$			4.85588i			0.204108i
$567$	21.1069	−	11.0227i	0.886405	−	0.462910i
$568$		−	14.4195i		−	0.605030i
$569$		−	7.34847i		−	0.308064i	−0.988066	−	0.154032i	$-0.950774\pi$
$569$		−	7.34847i		−	0.308064i	0.988066	−	0.154032i	$-0.0492259\pi$
$570$		0			0
$571$		−	13.4858i		−	0.564363i	−0.959361	−	0.282182i	$-0.908942\pi$
$571$		−	13.4858i		−	0.564363i	0.959361	−	0.282182i	$-0.0910580\pi$
$572$		0			0
$573$		0			0
$574$	1.08841	+	2.08416i	0.0454296	+	0.0869910i
$575$	7.60770			0.317263
$576$	6.80385			0.283494
$577$		−	11.7272i		−	0.488212i	−0.969749	−	0.244106i	$-0.921506\pi$
$577$		−	11.7272i		−	0.488212i	0.969749	−	0.244106i	$-0.0784944\pi$
$578$			12.4505i			0.517873i
$579$		0			0
$580$	35.4698			1.47280
$581$	−22.0000	+	11.4891i	−0.912714	+	0.476649i
$582$		0			0
$583$		0			0
$584$		−	9.06119i		−	0.374955i
$585$			17.0821i			0.706256i
$586$			13.0284i			0.538197i
$587$		−	17.4718i		−	0.721139i	−0.932732	−	0.360569i	$-0.882582\pi$
$587$		−	17.4718i		−	0.721139i	0.932732	−	0.360569i	$-0.117418\pi$
$588$		0			0
$589$		−	42.5007i		−	1.75121i
$590$			11.3880i			0.468838i
$591$		0			0
$592$	−20.8564			−0.857193
$593$	−14.5313			−0.596728			−0.298364	−	0.954452i	$-0.596441\pi$
$593$	−14.5313			−0.596728			−0.298364	+	0.954452i	$0.596441\pi$
$594$		0			0
$595$	26.0263	+	49.8365i	1.06697	+	2.04310i
$596$		−	28.3214i		−	1.16009i
$597$		0			0
$598$			1.12681i			0.0460787i
$599$	46.9808			1.91958			0.959791	−	0.280716i	$-0.0905718\pi$
$599$	46.9808			1.91958			0.959791	+	0.280716i	$0.0905718\pi$
$600$		0			0
$601$	15.7881			0.644008			0.322004	−	0.946738i	$-0.395643\pi$
$601$	15.7881			0.644008			0.322004	+	0.946738i	$0.395643\pi$
$602$	7.20400	−	3.76217i	0.293613	−	0.153334i
$603$	22.9808			0.935849
$604$		−	28.7375i		−	1.16931i
$605$		0			0
$606$		0			0
$607$	−25.6289			−1.04025			−0.520123	−	0.854092i	$-0.674114\pi$
$607$	−25.6289			−1.04025			−0.520123	+	0.854092i	$0.674114\pi$
$608$			24.1051i			0.977589i
$609$		0			0
$610$	13.9474			0.564715
$611$		−	15.5563i		−	0.629343i
$612$	−33.2929			−1.34579
$613$		−	28.1456i		−	1.13679i	−0.822756	−	0.568394i	$-0.807565\pi$
$613$		−	28.1456i		−	1.13679i	0.822756	−	0.568394i	$-0.192435\pi$
$614$			12.6159i			0.509138i
$615$		0			0
$616$		0			0
$617$	−19.4449			−0.782821			−0.391410	−	0.920216i	$-0.628013\pi$
$617$	−19.4449			−0.782821			−0.391410	+	0.920216i	$0.628013\pi$
$618$		0			0
$619$		−	42.8780i		−	1.72341i	−0.507408	−	0.861706i	$-0.669396\pi$
$619$		−	42.8780i		−	1.72341i	0.507408	−	0.861706i	$-0.330604\pi$
$620$	−52.0526			−2.09048
$621$		0			0
$622$	−2.17683			−0.0872829
$623$	7.03562	+	13.4722i	0.281876	+	0.539752i
$624$		0			0
$625$	−19.0000			−0.760000
$626$	−5.48719			−0.219312
$627$		0			0
$628$			19.8997i			0.794086i
$629$	54.2314			2.16235
$630$	12.0788	−	6.30797i	0.481232	−	0.251315i
$631$	2.53590			0.100953			0.0504763	−	0.998725i	$-0.483926\pi$
$631$	2.53590			0.100953			0.0504763	+	0.998725i	$0.483926\pi$
$632$	−17.6603			−0.702487
$633$		0			0
$634$		−	3.66063i		−	0.145382i
$635$	−10.3009			−0.408777
$636$		0			0
$637$	−6.86725	+	9.86233i	−0.272090	+	0.390760i
$638$		0			0
$639$	−22.3923			−0.885826
$640$	37.9833			1.50142
$641$	41.2487			1.62923			0.814613	−	0.580005i	$-0.196949\pi$
$641$	41.2487			1.62923			0.814613	+	0.580005i	$0.196949\pi$
$642$		0			0
$643$			27.8341i			1.09767i	0.835930	+	0.548835i	$0.184929\pi$
$643$			27.8341i			1.09767i	−0.835930	+	0.548835i	$0.815071\pi$
$644$	−5.15043	+	2.68973i	−0.202956	+	0.105990i
$645$		0			0
$646$		−	15.5563i		−	0.612056i
$647$		−	15.0439i		−	0.591436i	−0.955275	−	0.295718i	$-0.904441\pi$
$647$		−	15.0439i		−	0.591436i	0.955275	−	0.295718i	$-0.0955588\pi$
$648$			17.3867i			0.683013i
$649$		0			0
$650$			5.33212i			0.209143i
$651$		0			0
$652$	−20.5359			−0.804248
$653$	2.53590			0.0992374			0.0496187	−	0.998768i	$-0.484199\pi$
$653$	2.53590			0.0992374			0.0496187	+	0.998768i	$0.484199\pi$
$654$		0			0
$655$		−	4.16831i		−	0.162869i
$656$	4.23040			0.165169
$657$	−14.0712			−0.548972
$658$	−11.0000	+	5.74456i	−0.428825	+	0.223946i
$659$			18.8652i			0.734886i	0.930046	+	0.367443i	$0.119767\pi$
$659$			18.8652i			0.734886i	−0.930046	+	0.367443i	$0.880233\pi$
$660$		0			0
$661$		−	32.9281i		−	1.28076i	−0.768060	−	0.640378i	$-0.778778\pi$
$661$		−	32.9281i		−	1.28076i	0.768060	−	0.640378i	$-0.221222\pi$
$662$		−	0.656339i		−	0.0255093i
$663$		0			0
$664$		−	18.1224i		−	0.703285i
$665$	−19.0526	−	36.4829i	−0.738827	−	1.41475i
$666$		−	13.1440i		−	0.509321i
$667$		−	7.82894i		−	0.303138i
$668$	−14.0712			−0.544433
$669$		0			0
$670$	13.1512			0.508076
$671$		0			0
$672$		0			0
$673$		−	21.9711i		−	0.846923i	−0.905914	−	0.423461i	$-0.860815\pi$
$673$		−	21.9711i		−	0.846923i	0.905914	−	0.423461i	$-0.139185\pi$
$674$	−14.7513			−0.568198
$675$		0			0
$676$	−17.4115			−0.669675
$677$	25.1689			0.967319			0.483660	−	0.875256i	$-0.339307\pi$
$677$	25.1689			0.967319			0.483660	+	0.875256i	$0.339307\pi$
$678$		0			0
$679$	10.0092	+	19.1662i	0.384119	+	0.735532i
$680$	−41.0526			−1.57429
$681$		0			0
$682$		0			0
$683$	−39.3731			−1.50657			−0.753284	−	0.657695i	$-0.771532\pi$
$683$	−39.3731			−1.50657			−0.753284	+	0.657695i	$0.771532\pi$
$684$	24.3721			0.931891
$685$		−	56.1445i		−	2.14517i
$686$	9.50962	+	1.21397i	0.363079	+	0.0463495i
$687$		0			0
$688$		−	14.6226i		−	0.557482i
$689$	−6.74398			−0.256925
$690$		0			0
$691$			18.1224i			0.689408i	0.938711	+	0.344704i	$0.112021\pi$
$691$			18.1224i			0.689408i	−0.938711	+	0.344704i	$0.887979\pi$
$692$	−24.3721			−0.926489
$693$		0			0
$694$	14.4449			0.548320
$695$			46.6690i			1.77026i
$696$		0			0
$697$	−11.0000			−0.416655
$698$			19.0111i			0.719579i
$699$		0			0
$700$	−24.3721	+	12.7279i	−0.921179	+	0.481070i
$701$		−	22.2856i		−	0.841717i	−0.907126	−	0.420859i	$-0.861729\pi$
$701$		−	22.2856i		−	0.841717i	0.907126	−	0.420859i	$-0.138271\pi$
$702$		0			0
$703$	−39.7002			−1.49732
$704$		0			0
$705$		0			0
$706$	5.48719			0.206513
$707$	11.0000	−	5.74456i	0.413698	−	0.216047i
$708$		0			0
$709$	30.6410			1.15075			0.575374	−	0.817891i	$-0.304857\pi$
$709$	30.6410			1.15075			0.575374	+	0.817891i	$0.304857\pi$
$710$	−12.8145			−0.480918
$711$			27.4249i			1.02851i
$712$	−11.0976			−0.415902
$713$			11.4891i			0.430271i
$714$		0			0
$715$		0			0
$716$	−10.9808			−0.410370
$717$		0			0
$718$	−5.90897			−0.220521
$719$		−	24.7556i		−	0.923229i	−0.887081	−	0.461615i	$-0.847270\pi$
$719$		−	24.7556i		−	0.923229i	0.887081	−	0.461615i	$-0.152730\pi$
$720$		−	24.5175i		−	0.913713i
$721$	5.15043	+	9.86233i	0.191812	+	0.367292i
$722$			1.55291i			0.0577935i
$723$		0			0
$724$			21.4390i			0.796774i
$725$		−	37.0470i		−	1.37589i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	−4.06202	−	7.77817i	−0.150548	−	0.288278i
$729$	27.0000			1.00000
$730$	−8.05256			−0.298039
$731$			38.0221i			1.40630i
$732$		0			0
$733$	−2.97360			−0.109833			−0.0549163	−	0.998491i	$-0.517489\pi$
$733$	−2.97360			−0.109833			−0.0549163	+	0.998491i	$0.517489\pi$
$734$	6.86725			0.253475
$735$		0			0
$736$		−	6.51626i		−	0.240193i
$737$		0			0
$738$			2.66606i			0.0981391i
$739$			9.14162i			0.336280i	0.985763	+	0.168140i	$0.0537761\pi$
$739$			9.14162i			0.336280i	−0.985763	+	0.168140i	$0.946224\pi$
$740$			48.6226i			1.78740i
$741$		0			0
$742$	2.49038	+	4.76872i	0.0914248	+	0.175065i
$743$			30.1518i			1.10616i	0.833128	+	0.553080i	$0.186548\pi$
$743$			30.1518i			1.10616i	−0.833128	+	0.553080i	$0.813452\pi$
$744$		0			0
$745$	−54.2314			−1.98689
$746$	6.33975			0.232115
$747$	−28.1425			−1.02968
$748$		0			0
$749$	−1.26795	−	2.42794i	−0.0463299	−	0.0887149i
$750$		0			0
$751$	−36.3923			−1.32797			−0.663987	−	0.747744i	$-0.731137\pi$
$751$	−36.3923			−1.32797			−0.663987	+	0.747744i	$0.731137\pi$
$752$			22.3277i			0.814207i
$753$		0			0
$754$	5.48719			0.199832
$755$	−55.0282			−2.00268
$756$		0			0
$757$	−26.1244			−0.949506			−0.474753	−	0.880119i	$-0.657463\pi$
$757$	−26.1244			−0.949506			−0.474753	+	0.880119i	$0.657463\pi$
$758$		−	6.79367i		−	0.246757i
$759$		0			0
$760$	30.0526			1.09012
$761$	−11.0976			−0.402289			−0.201145	−	0.979562i	$-0.564466\pi$
$761$	−11.0976			−0.402289			−0.201145	+	0.979562i	$0.564466\pi$
$762$		0			0
$763$	−4.43782	−	8.49778i	−0.160660	−	0.307640i
$764$	−12.5885			−0.455434
$765$			63.7511i			2.30493i
$766$	7.20400			0.260291
$767$		−	11.3880i		−	0.411198i
$768$		0			0
$769$	18.3016			0.659974			0.329987	−	0.943985i	$-0.392956\pi$
$769$	18.3016			0.659974			0.329987	+	0.943985i	$0.392956\pi$
$770$		0			0
$771$		0			0
$772$			15.4176i			0.554893i
$773$		−	1.77737i		−	0.0639277i	−0.999489	−	0.0319638i	$-0.989824\pi$
$773$		−	1.77737i		−	0.0639277i	0.999489	−	0.0319638i	$-0.0101761\pi$
$774$	9.21539			0.331240
$775$			54.3671i			1.95292i
$776$	−15.7881			−0.566758
$777$		0			0
$778$		−	0.213030i		−	0.00763750i
$779$	8.05256			0.288513
$780$		0			0
$781$		0			0
$782$			4.20531i			0.150382i
$783$		0			0
$784$	9.85641	−	14.1552i	0.352015	−	0.505542i
$785$	38.1051			1.36003
$786$		0			0
$787$	2.51359			0.0895997			0.0447998	−	0.998996i	$-0.485735\pi$
$787$	2.51359			0.0895997			0.0447998	+	0.998996i	$0.485735\pi$
$788$		−	33.2204i		−	1.18343i
$789$		0			0
$790$			15.6944i			0.558383i
$791$	26.2573	−	13.7124i	0.933602	−	0.487558i
$792$		0			0
$793$	−13.9474			−0.495288
$794$	8.58406			0.304637
$795$		0			0
$796$			42.8780i			1.51977i
$797$			6.63325i			0.234962i	0.993075	+	0.117481i	$0.0374819\pi$
$797$			6.63325i			0.234962i	−0.993075	+	0.117481i	$0.962518\pi$
$798$		0			0
$799$		−	58.0571i		−	2.05391i
$800$		−	30.8353i		−	1.09019i
$801$			17.2337i			0.608922i
$802$		−	1.09965i		−	0.0388299i
$803$		0			0
$804$		0			0
$805$	5.15043	+	9.86233i	0.181529	+	0.347601i
$806$	−8.05256			−0.283639
$807$		0			0
$808$			9.06119i			0.318771i
$809$		−	18.6622i		−	0.656127i	−0.944656	−	0.328064i	$-0.893604\pi$
$809$		−	18.6622i		−	0.656127i	0.944656	−	0.328064i	$-0.106396\pi$
$810$	15.4513			0.542903
$811$	−44.0538			−1.54694			−0.773469	−	0.633834i	$-0.781480\pi$
$811$	−44.0538			−1.54694			−0.773469	+	0.633834i	$0.781480\pi$
$812$	13.0981	+	25.0809i	0.459652	+	0.880167i
$813$		0			0
$814$		0			0
$815$			39.3233i			1.37743i
$816$		0			0
$817$		−	27.8341i		−	0.973793i
$818$			2.66606i			0.0932166i
$819$	−12.0788	+	6.30797i	−0.422069	+	0.220418i
$820$		−	9.86233i		−	0.344408i
$821$		−	44.9502i		−	1.56877i	−0.620272	−	0.784387i	$-0.712978\pi$
$821$		−	44.9502i		−	1.56877i	0.620272	−	0.784387i	$-0.287022\pi$
$822$		0			0
$823$	6.87564			0.239670			0.119835	−	0.992794i	$-0.461763\pi$
$823$	6.87564			0.239670			0.119835	+	0.992794i	$0.461763\pi$
$824$	−8.12404			−0.283014
$825$		0			0
$826$	−8.05256	+	4.20531i	−0.280184	+	0.146322i
$827$		−	40.9107i		−	1.42260i	−0.702887	−	0.711301i	$-0.748106\pi$
$827$		−	40.9107i		−	1.42260i	0.702887	−	0.711301i	$-0.251894\pi$
$828$	−6.58846			−0.228965
$829$		−	21.4390i		−	0.744607i	−0.928111	−	0.372304i	$-0.878568\pi$
$829$		−	21.4390i		−	0.744607i	0.928111	−	0.372304i	$-0.121432\pi$
$830$	−16.1051			−0.559017
$831$		0			0
$832$	−3.89364			−0.134988
$833$	−25.6289	+	36.8067i	−0.887989	+	1.27528i
$834$		0			0
$835$			26.9444i			0.932449i
$836$		0			0
$837$		0			0
$838$	−0.336757			−0.0116331
$839$			2.42794i			0.0838217i	0.999121	+	0.0419109i	$0.0133446\pi$
$839$			2.42794i			0.0838217i	−0.999121	+	0.0419109i	$0.986655\pi$
$840$		0			0
$841$	−9.12436			−0.314633
$842$			2.41233i			0.0831342i
$843$		0			0
$844$			37.7033i			1.29780i
$845$			33.3406i			1.14695i
$846$	−14.0712			−0.483779
$847$		0			0
$848$	9.67949			0.332395
$849$		0			0
$850$			19.8997i			0.682556i
$851$	10.7321			0.367890
$852$		0			0
$853$	41.4170			1.41809			0.709045	−	0.705163i	$-0.249126\pi$
$853$	41.4170			1.41809			0.709045	+	0.705163i	$0.249126\pi$
$854$	5.15043	+	9.86233i	0.176244	+	0.337482i
$855$		−	46.6690i		−	1.59605i
$856$	2.00000			0.0683586
$857$	16.5848			0.566527			0.283264	−	0.959042i	$-0.408583\pi$
$857$	16.5848			0.566527			0.283264	+	0.959042i	$0.408583\pi$
$858$		0			0
$859$		−	4.20531i		−	0.143483i	−0.997423	−	0.0717417i	$-0.977144\pi$
$859$		−	4.20531i		−	0.143483i	0.997423	−	0.0717417i	$-0.0228557\pi$
$860$	−34.0897			−1.16245
$861$		0			0
$862$	18.7846			0.639806
$863$	−52.1051			−1.77368			−0.886839	−	0.462078i	$-0.847104\pi$
$863$	−52.1051			−1.77368			−0.886839	+	0.462078i	$0.847104\pi$
$864$		0			0
$865$			46.6690i			1.58680i
$866$	−17.0449			−0.579208
$867$		0			0
$868$	−19.2217	−	36.8067i	−0.652426	−	1.24930i
$869$		0			0
$870$		0			0
$871$	−13.1512			−0.445612
$872$	7.00000			0.237050
$873$			24.5175i			0.829792i
$874$		−	3.07850i		−	0.104132i
$875$	4.06202	+	7.77817i	0.137321	+	0.262950i
$876$		0			0
$877$			18.5235i			0.625493i	0.949837	+	0.312747i	$0.101249\pi$
$877$			18.5235i			0.625493i	−0.949837	+	0.312747i	$0.898751\pi$
$878$			13.2665i			0.447723i
$879$		0			0
$880$		0			0
$881$			42.6399i			1.43657i	0.695747	+	0.718287i	$0.255073\pi$
$881$			42.6399i			1.43657i	−0.695747	+	0.718287i	$0.744927\pi$
$882$	8.92081	+	6.21166i	0.300379	+	0.209157i
$883$	−20.0526			−0.674822			−0.337411	−	0.941357i	$-0.609551\pi$
$883$	−20.0526			−0.674822			−0.337411	+	0.941357i	$0.609551\pi$
$884$	19.0526			0.640807
$885$		0			0
$886$			0.101536i			0.00341117i
$887$	−36.2665			−1.21771			−0.608856	−	0.793281i	$-0.708371\pi$
$887$	−36.2665			−1.21771			−0.608856	+	0.793281i	$0.708371\pi$
$888$		0			0
$889$	−3.80385	−	7.28381i	−0.127577	−	0.244291i
$890$			9.86233i			0.330586i
$891$		0			0
$892$			8.41062i			0.281609i
$893$			42.5007i			1.42223i
$894$		0			0
$895$			21.0266i			0.702840i
$896$	14.0263	+	26.8583i	0.468585	+	0.897272i
$897$		0			0
$898$		−	0.443309i		−	0.0147934i
$899$	55.9482			1.86598
$900$	−31.1769			−1.03923
$901$	−25.1689			−0.838497
$902$		0			0
$903$		0			0
$904$			21.6293i			0.719380i
$905$	41.0526			1.36463
$906$		0			0
$907$	−51.4641			−1.70884			−0.854419	−	0.519585i	$-0.826087\pi$
$907$	−51.4641			−1.70884			−0.854419	+	0.519585i	$0.826087\pi$
$908$	10.3009			0.341846
$909$	14.0712			0.466714
$910$	−6.91236	+	3.60986i	−0.229143	+	0.119666i
$911$	−7.94744			−0.263310			−0.131655	−	0.991296i	$-0.542029\pi$
$911$	−7.94744			−0.263310			−0.131655	+	0.991296i	$0.542029\pi$
$912$		0			0
$913$		0			0
$914$	−17.4449			−0.577025
$915$		0			0
$916$		−	9.94987i		−	0.328753i
$917$	2.94744	−	1.53925i	0.0973331	−	0.0508305i
$918$		0			0
$919$			18.7637i			0.618958i	0.950906	+	0.309479i	$0.100155\pi$
$919$			18.7637i			0.618958i	−0.950906	+	0.309479i	$0.899845\pi$
$920$	−8.12404			−0.267842
$921$		0			0
$922$		−	8.17250i		−	0.269147i
$923$	12.8145			0.421793
$924$		0			0
$925$	50.7846			1.66979
$926$		−	0.203072i		−	0.00667336i
$927$			12.6159i			0.414362i
$928$	−31.7321			−1.04166
$929$		−	39.5614i		−	1.29797i	−0.760803	−	0.648983i	$-0.775194\pi$
$929$		−	39.5614i		−	1.29797i	0.760803	−	0.648983i	$-0.224806\pi$
$930$		0			0
$931$	18.7617	−	26.9444i	0.614889	−	0.883067i
$932$		−	7.10823i		−	0.232838i
$933$		0			0
$934$	13.1512			0.430321
$935$		0			0
$936$		−	9.94987i		−	0.325222i
$937$	17.0449			0.556831			0.278416	−	0.960461i	$-0.410191\pi$
$937$	17.0449			0.556831			0.278416	+	0.960461i	$0.410191\pi$
$938$	4.85641	+	9.29931i	0.158567	+	0.303633i
$939$		0			0
$940$	52.0526			1.69777
$941$	49.5410			1.61499			0.807495	−	0.589874i	$-0.200823\pi$
$941$	49.5410			1.61499			0.807495	+	0.589874i	$0.200823\pi$
$942$		0			0
$943$	−2.17683			−0.0708873
$944$			16.3450i			0.531984i
$945$		0			0
$946$		0			0
$947$	40.1051			1.30324			0.651621	−	0.758545i	$-0.274089\pi$
$947$	40.1051			1.30324			0.651621	+	0.758545i	$0.274089\pi$
$948$		0			0
$949$	8.05256			0.261397
$950$		−	14.5676i		−	0.472636i
$951$		0			0
$952$	−15.1597	−	29.0285i	−0.491327	−	0.940820i
$953$			17.1093i			0.554223i	0.960838	+	0.277112i	$0.0893772\pi$
$953$			17.1093i			0.554223i	−0.960838	+	0.277112i	$0.910623\pi$
$954$			6.10016i			0.197500i
$955$			24.1051i			0.780021i
$956$			41.9459i			1.35663i
$957$		0			0
$958$		−	19.8997i		−	0.642932i
$959$	39.7002	−	20.7327i	1.28198	−	0.669495i
$960$		0			0
$961$	−51.1051			−1.64855
$962$			7.52194i			0.242517i
$963$		−	3.10583i		−	0.100084i
$964$	−36.2665			−1.16807
$965$	29.5225			0.950364
$966$		0			0
$967$		−	29.0421i		−	0.933932i	−0.884275	−	0.466966i	$-0.845347\pi$
$967$		−	29.0421i		−	0.933932i	0.884275	−	0.466966i	$-0.154653\pi$
$968$		0			0
$969$		0			0
$970$			14.0306i			0.450497i
$971$		−	48.3844i		−	1.55273i	−0.630283	−	0.776365i	$-0.717061\pi$
$971$		−	48.3844i		−	1.55273i	0.630283	−	0.776365i	$-0.282939\pi$
$972$		0			0
$973$	−33.0000	+	17.2337i	−1.05793	+	0.552487i
$974$		−	4.06678i		−	0.130308i
$975$		0			0
$976$	20.0185			0.640775
$977$	29.1962			0.934068			0.467034	−	0.884239i	$-0.345323\pi$
$977$	29.1962			0.934068			0.467034	+	0.884239i	$0.345323\pi$
$978$		0			0
$979$		0			0
$980$	−33.0000	−	22.9783i	−1.05415	−	0.734013i
$981$		−	10.8704i		−	0.347065i
$982$	−6.78461			−0.216506
$983$		−	12.6159i		−	0.402386i	−0.979552	−	0.201193i	$-0.935518\pi$
$983$		−	12.6159i		−	0.402386i	0.979552	−	0.201193i	$-0.0644818\pi$
$984$		0			0
$985$	−63.6123			−2.02685
$986$	20.4785			0.652167
$987$		0			0
$988$	−13.9474			−0.443727
$989$			7.52433i			0.239260i
$990$		0			0
$991$	−50.0526			−1.58997			−0.794986	−	0.606628i	$-0.792522\pi$
$991$	−50.0526			−1.58997			−0.794986	+	0.606628i	$0.792522\pi$
$992$	46.5674			1.47852
$993$		0			0
$994$	−4.73205	−	9.06119i	−0.150092	−	0.287403i
$995$	82.1051			2.60291
$996$		0			0
$997$	11.0976			0.351466			0.175733	−	0.984438i	$-0.443770\pi$
$997$	11.0976			0.351466			0.175733	+	0.984438i	$0.443770\pi$
$998$			7.62587i			0.241393i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.b.d.846.5	yes	8
7.6	odd	2	inner	847.2.b.d.846.6	yes	8
11.2	odd	10		847.2.l.k.524.4		32
11.3	even	5		847.2.l.k.475.6		32
11.4	even	5		847.2.l.k.699.3		32
11.5	even	5		847.2.l.k.118.3		32
11.6	odd	10		847.2.l.k.118.5		32
11.7	odd	10		847.2.l.k.699.5		32
11.8	odd	10		847.2.l.k.475.4		32
11.9	even	5		847.2.l.k.524.6		32
11.10	odd	2	inner	847.2.b.d.846.3	✓	8
77.6	even	10		847.2.l.k.118.6		32
77.13	even	10		847.2.l.k.524.3		32
77.20	odd	10		847.2.l.k.524.5		32
77.27	odd	10		847.2.l.k.118.4		32
77.41	even	10		847.2.l.k.475.3		32
77.48	odd	10		847.2.l.k.699.4		32
77.62	even	10		847.2.l.k.699.6		32
77.69	odd	10		847.2.l.k.475.5		32
77.76	even	2	inner	847.2.b.d.846.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.b.d.846.3	✓	8	11.10	odd	2	inner
847.2.b.d.846.4	yes	8	77.76	even	2	inner
847.2.b.d.846.5	yes	8	1.1	even	1	trivial
847.2.b.d.846.6	yes	8	7.6	odd	2	inner
847.2.l.k.118.3		32	11.5	even	5
847.2.l.k.118.4		32	77.27	odd	10
847.2.l.k.118.5		32	11.6	odd	10
847.2.l.k.118.6		32	77.6	even	10
847.2.l.k.475.3		32	77.41	even	10
847.2.l.k.475.4		32	11.8	odd	10
847.2.l.k.475.5		32	77.69	odd	10
847.2.l.k.475.6		32	11.3	even	5
847.2.l.k.524.3		32	77.13	even	10
847.2.l.k.524.4		32	11.2	odd	10
847.2.l.k.524.5		32	77.20	odd	10
847.2.l.k.524.6		32	11.9	even	5
847.2.l.k.699.3		32	11.4	even	5
847.2.l.k.699.4		32	77.48	odd	10
847.2.l.k.699.5		32	11.7	odd	10
847.2.l.k.699.6		32	77.62	even	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.517638i q^{2} +1.73205 q^{4} -3.31662i q^{5} +(2.34521 - 1.22474i) q^{7} +1.93185i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+0.517638i q^{2} +1.73205 q^{4} -3.31662i q^{5} +(2.34521 - 1.22474i) q^{7} +1.93185i q^{8} +3.00000 q^{9} +1.71681 q^{10} -1.71681 q^{13} +(0.633975 + 1.21397i) q^{14} +2.46410 q^{16} -6.40723 q^{17} +1.55291i q^{18} +4.69042 q^{19} -5.74456i q^{20} -1.26795 q^{23} -6.00000 q^{25} -0.888687i q^{26} +(4.06202 - 2.12132i) q^{28} +6.17449i q^{29} -9.06119i q^{31} +5.13922i q^{32} -3.31662i q^{34} +(-4.06202 - 7.77817i) q^{35} +5.19615 q^{36} -8.46410 q^{37} +2.42794i q^{38} +6.40723 q^{40} +1.71681 q^{41} -5.93426i q^{43} -9.94987i q^{45} -0.656339i q^{46} +9.06119i q^{47} +(4.00000 - 5.74456i) q^{49} -3.10583i q^{50} -2.97360 q^{52} +3.92820 q^{53} +(2.36603 + 4.53059i) q^{56} -3.19615 q^{58} +6.63325i q^{59} +8.12404 q^{61} +4.69042 q^{62} +(7.03562 - 3.67423i) q^{63} +2.26795 q^{64} +5.69402i q^{65} +7.66025 q^{67} -11.0976 q^{68} +(4.02628 - 2.10266i) q^{70} -7.46410 q^{71} +5.79555i q^{72} -4.69042 q^{73} -4.38134i q^{74} +8.12404 q^{76} +9.14162i q^{79} -8.17250i q^{80} +9.00000 q^{81} +0.888687i q^{82} -9.38083 q^{83} +21.2504i q^{85} +3.07180 q^{86} +5.74456i q^{89} +5.15043 q^{90} +(-4.02628 + 2.10266i) q^{91} -2.19615 q^{92} -4.69042 q^{94} -15.5563i q^{95} +8.17250i q^{97} +(2.97360 + 2.07055i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 12 q^{14} - 8 q^{16} - 24 q^{23} - 48 q^{25} - 40 q^{37} + 32 q^{49} - 24 q^{53} + 12 q^{56} + 16 q^{58} + 32 q^{64} - 8 q^{67} - 44 q^{70} - 32 q^{71} + 72 q^{81} + 80 q^{86} + 44 q^{91} + 24 q^{92}+O(q^{100}) \) 8 * q + 24 * q^9 + 12 * q^14 - 8 * q^16 - 24 * q^23 - 48 * q^25 - 40 * q^37 + 32 * q^49 - 24 * q^53 + 12 * q^56 + 16 * q^58 + 32 * q^64 - 8 * q^67 - 44 * q^70 - 32 * q^71 + 72 * q^81 + 80 * q^86 + 44 * q^91 + 24 * q^92

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