LMFDB - Embedded newform 847.2.b.d.846.1

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			846.1
Root			$0.500000 + 0.273539i$ of defining polynomial
Character	$\chi$	$=$	847.846
Dual form			847.2.b.d.846.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93185i q^{2} -1.73205 q^{4} -3.31662i q^{5} +(2.34521 + 1.22474i) q^{7} -0.517638i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.93185i q^{2} -1.73205 q^{4} -3.31662i q^{5} +(2.34521 + 1.22474i) q^{7} -0.517638i q^{8} +3.00000 q^{9} -6.40723 q^{10} +6.40723 q^{13} +(2.36603 - 4.53059i) q^{14} -4.46410 q^{16} +1.71681 q^{17} -5.79555i q^{18} +4.69042 q^{19} +5.74456i q^{20} -4.73205 q^{23} -6.00000 q^{25} -12.3778i q^{26} +(-4.06202 - 2.12132i) q^{28} +3.72500i q^{29} +2.42794i q^{31} +7.58871i q^{32} -3.31662i q^{34} +(4.06202 - 7.77817i) q^{35} -5.19615 q^{36} -1.53590 q^{37} -9.06119i q^{38} -1.71681 q^{40} -6.40723 q^{41} +8.76268i q^{43} -9.94987i q^{45} +9.14162i q^{46} -2.42794i q^{47} +(4.00000 + 5.74456i) q^{49} +11.5911i q^{50} -11.0976 q^{52} -9.92820 q^{53} +(0.633975 - 1.21397i) q^{56} +7.19615 q^{58} +6.63325i q^{59} -8.12404 q^{61} +4.69042 q^{62} +(7.03562 + 3.67423i) q^{63} +5.73205 q^{64} -21.2504i q^{65} -9.66025 q^{67} -2.97360 q^{68} +(-15.0263 - 7.84722i) q^{70} -0.535898 q^{71} -1.55291i q^{72} -4.69042 q^{73} +2.96713i q^{74} -8.12404 q^{76} -0.656339i q^{79} +14.8058i q^{80} +9.00000 q^{81} +12.3778i q^{82} -9.38083 q^{83} -5.69402i q^{85} +16.9282 q^{86} -5.74456i q^{89} -19.2217 q^{90} +(15.0263 + 7.84722i) q^{91} +8.19615 q^{92} -4.69042 q^{94} -15.5563i q^{95} -14.8058i q^{97} +(11.0976 - 7.72741i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.93185i		−	1.36603i	−0.730406	−	0.683013i	$-0.760669\pi$
$2$		−	1.93185i		−	1.36603i	0.730406	−	0.683013i	$-0.239331\pi$
$3$		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
$8$		−	0.517638i		−	0.183013i
$9$	3.00000			1.00000
$10$	−6.40723			−2.02614
$11$		0			0
$11$		0			0
$12$		0			0
$13$	6.40723			1.77705			0.888523	−	0.458833i	$-0.151732\pi$
$13$	6.40723			1.77705			0.888523	+	0.458833i	$0.151732\pi$
$14$	2.36603	−	4.53059i	0.632347	−	1.21085i
$15$		0			0
$16$	−4.46410			−1.11603
$17$	1.71681			0.416388			0.208194	−	0.978088i	$-0.433241\pi$
$17$	1.71681			0.416388			0.208194	+	0.978088i	$0.433241\pi$
$18$		−	5.79555i		−	1.36603i
$19$	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
$23$	−4.73205			−0.986701			−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205			−0.986701			−0.493350	+	0.869831i	$0.664228\pi$
$24$		0			0
$25$	−6.00000			−1.20000
$26$		−	12.3778i		−	2.42749i
$27$		0			0
$28$	−4.06202	−	2.12132i	−0.767649	−	0.400892i
$29$			3.72500i			0.691716i	0.938287	+	0.345858i	$0.112412\pi$
$29$			3.72500i			0.691716i	−0.938287	+	0.345858i	$0.887588\pi$
$30$		0			0
$31$			2.42794i			0.436071i	0.975941	+	0.218035i	$0.0699648\pi$
$31$			2.42794i			0.436071i	−0.975941	+	0.218035i	$0.930035\pi$
$32$			7.58871i			1.34151i
$33$		0			0
$34$		−	3.31662i		−	0.568796i
$35$	4.06202	−	7.77817i	0.686607	−	1.31475i
$36$	−5.19615			−0.866025
$37$	−1.53590			−0.252500			−0.126250	−	0.991998i	$-0.540294\pi$
$37$	−1.53590			−0.252500			−0.126250	+	0.991998i	$0.540294\pi$
$38$		−	9.06119i		−	1.46992i
$39$		0			0
$40$	−1.71681			−0.271452
$41$	−6.40723			−1.00064			−0.500320	−	0.865840i	$-0.666784\pi$
$41$	−6.40723			−1.00064			−0.500320	+	0.865840i	$0.666784\pi$
$42$		0			0
$43$			8.76268i			1.33630i	0.744028	+	0.668148i	$0.232913\pi$
$43$			8.76268i			1.33630i	−0.744028	+	0.668148i	$0.767087\pi$
$44$		0			0
$45$		−	9.94987i		−	1.48324i
$46$			9.14162i			1.34786i
$47$		−	2.42794i		−	0.354151i	−0.984197	−	0.177076i	$-0.943336\pi$
$47$		−	2.42794i		−	0.354151i	0.984197	−	0.177076i	$-0.0566637\pi$
$48$		0			0
$49$	4.00000	+	5.74456i	0.571429	+	0.820652i
$50$			11.5911i			1.63923i
$51$		0			0
$52$	−11.0976			−1.53897
$53$	−9.92820			−1.36374			−0.681872	−	0.731472i	$-0.738834\pi$
$53$	−9.92820			−1.36374			−0.681872	+	0.731472i	$0.738834\pi$
$54$		0			0
$55$		0			0
$56$	0.633975	−	1.21397i	0.0847184	−	0.162223i
$57$		0			0
$58$	7.19615			0.944901
$59$			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	$-0.674101\pi$
$61$	−8.12404			−1.04018			−0.520088	+	0.854113i	$0.674101\pi$
$62$	4.69042			0.595683
$63$	7.03562	+	3.67423i	0.886405	+	0.462910i
$64$	5.73205			0.716506
$65$		−	21.2504i		−	2.63578i
$66$		0			0
$67$	−9.66025			−1.18019			−0.590094	−	0.807335i	$-0.700909\pi$
$67$	−9.66025			−1.18019			−0.590094	+	0.807335i	$0.700909\pi$
$68$	−2.97360			−0.360603
$69$		0			0
$70$	−15.0263	−	7.84722i	−1.79598	−	0.937922i
$71$	−0.535898			−0.0635994			−0.0317997	−	0.999494i	$-0.510124\pi$
$71$	−0.535898			−0.0635994			−0.0317997	+	0.999494i	$0.510124\pi$
$72$		−	1.55291i		−	0.183013i
$73$	−4.69042			−0.548972			−0.274486	−	0.961591i	$-0.588508\pi$
$73$	−4.69042			−0.548972			−0.274486	+	0.961591i	$0.588508\pi$
$74$			2.96713i			0.344922i
$75$		0			0
$76$	−8.12404			−0.931891
$77$		0			0
$78$		0			0
$79$		−	0.656339i		−	0.0738439i	−0.999318	−	0.0369219i	$-0.988245\pi$
$79$		−	0.656339i		−	0.0738439i	0.999318	−	0.0369219i	$-0.0117553\pi$
$80$			14.8058i			1.65533i
$81$	9.00000			1.00000
$82$			12.3778i			1.36690i
$83$	−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
$85$		−	5.69402i		−	0.617603i
$86$	16.9282			1.82542
$87$		0			0
$88$		0			0
$89$		−	5.74456i		−	0.608922i	−0.952525	−	0.304461i	$-0.901524\pi$
$89$		−	5.74456i		−	0.608922i	0.952525	−	0.304461i	$-0.0984764\pi$
$90$	−19.2217			−2.02614
$91$	15.0263	+	7.84722i	1.57518	+	0.822612i
$92$	8.19615			0.854508
$93$		0			0
$94$	−4.69042			−0.483779
$95$		−	15.5563i		−	1.59605i
$96$		0			0
$97$		−	14.8058i		−	1.50330i	−0.659564	−	0.751648i	$-0.729259\pi$
$97$		−	14.8058i		−	1.50330i	0.659564	−	0.751648i	$-0.270741\pi$
$98$	11.0976	−	7.72741i	1.12103	−	0.780586i
$99$		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
$103$			15.6944i			1.54642i	0.634151	+	0.773209i	$0.281350\pi$
$103$			15.6944i			1.54642i	−0.634151	+	0.773209i	$0.718650\pi$
$104$		−	3.31662i		−	0.325222i
$105$		0			0
$106$			19.1798i			1.86291i
$107$			3.86370i			0.373518i	0.982406	+	0.186759i	$0.0597984\pi$
$107$			3.86370i			0.373518i	−0.982406	+	0.186759i	$0.940202\pi$
$108$		0			0
$109$			13.5230i			1.29526i	0.761953	+	0.647632i	$0.224241\pi$
$109$			13.5230i			1.29526i	−0.761953	+	0.647632i	$0.775759\pi$
$110$		0			0
$111$		0			0
$112$	−10.4692	−	5.46739i	−0.989251	−	0.516619i
$113$	0.803848			0.0756196			0.0378098	−	0.999285i	$-0.487962\pi$
$113$	0.803848			0.0756196			0.0378098	+	0.999285i	$0.487962\pi$
$114$		0			0
$115$			15.6944i			1.46351i
$116$		−	6.45189i		−	0.599043i
$117$	19.2217			1.77705
$118$	12.8145			1.17967
$119$	4.02628	+	2.10266i	0.369088	+	0.192750i
$120$		0			0
$121$		0			0
$122$			15.6944i			1.42091i
$123$		0			0
$124$		−	4.20531i		−	0.377648i
$125$			3.31662i			0.296648i
$126$	7.09808	−	13.5918i	0.632347	−	1.21085i
$127$			11.5911i			1.02854i	0.857627	+	0.514272i	$0.171938\pi$
$127$			11.5911i			1.02854i	−0.857627	+	0.514272i	$0.828062\pi$
$128$			4.10394i			0.362740i
$129$		0			0
$130$	−41.0526			−3.60055
$131$	17.5049			1.52941			0.764704	−	0.644382i	$-0.222885\pi$
$131$	17.5049			1.52941			0.764704	+	0.644382i	$0.222885\pi$
$132$		0			0
$133$	11.0000	+	5.74456i	0.953821	+	0.498117i
$134$			18.6622i			1.61217i
$135$		0			0
$136$		−	0.888687i		−	0.0762043i
$137$	3.07180			0.262441			0.131221	−	0.991353i	$-0.458110\pi$
$137$	3.07180			0.262441			0.131221	+	0.991353i	$0.458110\pi$
$138$		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
$142$			1.03528i			0.0868784i
$143$		0			0
$144$	−13.3923			−1.11603
$145$	12.3544			1.02598
$146$			9.06119i			0.749909i
$147$		0			0
$148$	2.66025			0.218672
$149$			0.795040i			0.0651322i	0.999470	+	0.0325661i	$0.0103679\pi$
$149$			0.795040i			0.0651322i	−0.999470	+	0.0325661i	$0.989632\pi$
$150$		0			0
$151$		−	11.6926i		−	0.951534i	−0.879571	−	0.475767i	$-0.842171\pi$
$151$		−	11.6926i		−	0.951534i	0.879571	−	0.475767i	$-0.157829\pi$
$152$		−	2.42794i		−	0.196932i
$153$	5.15043			0.416388
$154$		0			0
$155$	8.05256			0.646797
$156$		0			0
$157$		−	11.4891i		−	0.916932i	−0.888712	−	0.458466i	$-0.848399\pi$
$157$		−	11.4891i		−	0.916932i	0.888712	−	0.458466i	$-0.151601\pi$
$158$	−1.26795			−0.100873
$159$		0			0
$160$	25.1689			1.98978
$161$	−11.0976	−	5.79555i	−0.874617	−	0.456754i
$162$		−	17.3867i		−	1.36603i
$163$	15.8564			1.24197			0.620985	−	0.783823i	$-0.286733\pi$
$163$	15.8564			1.24197			0.620985	+	0.783823i	$0.286733\pi$
$164$	11.0976			0.866580
$165$		0			0
$166$			18.1224i			1.40657i
$167$	8.12404			0.628657			0.314328	−	0.949314i	$-0.398221\pi$
$167$	8.12404			0.628657			0.314328	+	0.949314i	$0.398221\pi$
$168$		0			0
$169$	28.0526			2.15789
$170$	−11.0000			−0.843661
$171$	14.0712			1.07606
$172$		−	15.1774i		−	1.15727i
$173$	−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
$178$	−11.0976			−0.831803
$179$	−23.6603			−1.76845			−0.884225	−	0.467061i	$-0.845313\pi$
$179$	−23.6603			−1.76845			−0.884225	+	0.467061i	$0.845313\pi$
$180$			17.2337i			1.28452i
$181$			0.888687i			0.0660556i	0.999454	+	0.0330278i	$0.0105150\pi$
$181$			0.888687i			0.0660556i	−0.999454	+	0.0330278i	$0.989485\pi$
$182$	15.1597	−	29.0285i	1.12371	−	2.15174i
$183$		0			0
$184$			2.44949i			0.180579i
$185$			5.09400i			0.374518i
$186$		0			0
$187$		0			0
$188$			4.20531i			0.306704i
$189$		0			0
$190$	−30.0526			−2.18024
$191$	−10.7321			−0.776544			−0.388272	−	0.921545i	$-0.626928\pi$
$191$	−10.7321			−0.776544			−0.388272	+	0.921545i	$0.626928\pi$
$192$		0			0
$193$		−	13.1440i		−	0.946128i	−0.881028	−	0.473064i	$-0.843148\pi$
$193$		−	13.1440i		−	0.946128i	0.881028	−	0.473064i	$-0.156852\pi$
$194$	−28.6025			−2.05354
$195$		0			0
$196$	−6.92820	−	9.94987i	−0.494872	−	0.710705i
$197$		−	2.03339i		−	0.144873i	−0.997373	−	0.0724364i	$-0.976923\pi$
$197$		−	2.03339i		−	0.144873i	0.997373	−	0.0724364i	$-0.0230774\pi$
$198$		0			0
$199$			1.77737i			0.125995i	0.998014	+	0.0629973i	$0.0200660\pi$
$199$			1.77737i			0.125995i	−0.998014	+	0.0629973i	$0.979934\pi$
$200$			3.10583i			0.219615i
$201$		0			0
$202$		−	9.06119i		−	0.637543i
$203$	−4.56218	+	8.73591i	−0.320202	+	0.613140i
$204$		0			0
$205$			21.2504i			1.48419i
$206$	30.3193			2.11245
$207$	−14.1962			−0.986701
$208$	−28.6025			−1.98323
$209$		0			0
$210$		0			0
$211$		−	7.62587i		−	0.524987i	−0.964934	−	0.262493i	$-0.915455\pi$
$211$		−	7.62587i		−	0.524987i	0.964934	−	0.262493i	$-0.0845448\pi$
$212$	17.1962			1.18104
$213$		0			0
$214$	7.46410			0.510235
$215$	29.0625			1.98205
$216$		0			0
$217$	−2.97360	+	5.69402i	−0.201861	+	0.386535i
$218$	26.1244			1.76936
$219$		0			0
$220$		0			0
$221$	11.0000			0.739940
$222$		0			0
$223$		−	18.1224i		−	1.21356i	−0.794868	−	0.606782i	$-0.792460\pi$
$223$		−	18.1224i		−	1.21356i	0.794868	−	0.606782i	$-0.207540\pi$
$224$	−9.29423	+	17.7971i	−0.620997	+	1.18912i
$225$	−18.0000			−1.20000
$226$		−	1.55291i		−	0.103298i
$227$	22.1953			1.47315			0.736576	−	0.676354i	$-0.236441\pi$
$227$	22.1953			1.47315			0.736576	+	0.676354i	$0.236441\pi$
$228$		0			0
$229$			5.74456i			0.379611i	0.981822	+	0.189806i	$0.0607858\pi$
$229$			5.74456i			0.379611i	−0.981822	+	0.189806i	$0.939214\pi$
$230$	30.3193			1.99920
$231$		0			0
$232$	1.92820			0.126593
$233$		−	11.4524i		−	0.750272i	−0.926970	−	0.375136i	$-0.877596\pi$
$233$		−	11.4524i		−	0.750272i	0.926970	−	0.375136i	$-0.122404\pi$
$234$		−	37.1334i		−	2.42749i
$235$	−8.05256			−0.525291
$236$		−	11.4891i		−	0.747878i
$237$		0			0
$238$	4.06202	−	7.77817i	0.263302	−	0.504184i
$239$		−	10.0754i		−	0.651721i	−0.945418	−	0.325860i	$-0.894346\pi$
$239$		−	10.0754i		−	0.651721i	0.945418	−	0.325860i	$-0.105654\pi$
$240$		0			0
$241$	11.5577			0.744494			0.372247	−	0.928134i	$-0.378587\pi$
$241$	11.5577			0.744494			0.372247	+	0.928134i	$0.378587\pi$
$242$		0			0
$243$		0			0
$244$	14.0712			0.900819
$245$	19.0526	−	13.2665i	1.21722	−	0.847566i
$246$		0			0
$247$	30.0526			1.91220
$248$	1.25679			0.0798064
$249$		0			0
$250$	6.40723			0.405229
$251$			22.3277i			1.40931i	0.709550	+	0.704656i	$0.248898\pi$
$251$			22.3277i			1.40931i	−0.709550	+	0.704656i	$0.751102\pi$
$252$	−12.1861	−	6.36396i	−0.767649	−	0.400892i
$253$		0			0
$254$	22.3923			1.40502
$255$		0			0
$256$	19.3923			1.21202
$257$		−	10.6004i		−	0.661237i	−0.943764	−	0.330619i	$-0.892743\pi$
$257$		−	10.6004i		−	0.661237i	0.943764	−	0.330619i	$-0.107257\pi$
$258$		0			0
$259$	−3.60200	−	1.88108i	−0.223817	−	0.116885i
$260$			36.8067i			2.28266i
$261$			11.1750i			0.691716i
$262$		−	33.8168i		−	2.08921i
$263$			8.76268i			0.540330i	0.962814	+	0.270165i	$0.0870783\pi$
$263$			8.76268i			0.540330i	−0.962814	+	0.270165i	$0.912922\pi$
$264$		0			0
$265$			32.9281i			2.02276i
$266$	11.0976	−	21.2504i	0.680440	−	1.30294i
$267$		0			0
$268$	16.7321			1.02207
$269$		−	14.1552i		−	0.863057i	−0.902099	−	0.431528i	$-0.857974\pi$
$269$		−	14.1552i		−	0.863057i	0.902099	−	0.431528i	$-0.142026\pi$
$270$		0			0
$271$	−1.25679			−0.0763447			−0.0381724	−	0.999271i	$-0.512154\pi$
$271$	−1.25679			−0.0763447			−0.0381724	+	0.999271i	$0.512154\pi$
$272$	−7.66402			−0.464699
$273$		0			0
$274$		−	5.93426i		−	0.358501i
$275$		0			0
$276$		0			0
$277$		−	1.93185i		−	0.116074i	−0.998314	−	0.0580369i	$-0.981516\pi$
$277$		−	1.93185i		−	0.116074i	0.998314	−	0.0580369i	$-0.0184841\pi$
$278$			27.1836i			1.63036i
$279$			7.28381i			0.436071i
$280$	−4.02628	−	2.10266i	−0.240616	−	0.125658i
$281$			8.10634i			0.483584i	0.970328	+	0.241792i	$0.0777352\pi$
$281$			8.10634i			0.483584i	−0.970328	+	0.241792i	$0.922265\pi$
$282$		0			0
$283$	9.38083			0.557633			0.278816	−	0.960344i	$-0.410058\pi$
$283$	9.38083			0.557633			0.278816	+	0.960344i	$0.410058\pi$
$284$	0.928203			0.0550787
$285$		0			0
$286$		0			0
$287$	−15.0263	−	7.84722i	−0.886973	−	0.463207i
$288$			22.7661i			1.34151i
$289$	−14.0526			−0.826621
$290$		−	23.8669i		−	1.40151i
$291$		0			0
$292$	8.12404			0.475423
$293$	17.0449			0.995771			0.497885	−	0.867243i	$-0.334110\pi$
$293$	17.0449			0.995771			0.497885	+	0.867243i	$0.334110\pi$
$294$		0			0
$295$	22.0000			1.28089
$296$			0.795040i			0.0462107i
$297$		0			0
$298$	1.53590			0.0889722
$299$	−30.3193			−1.75341
$300$		0			0
$301$	−10.7321	+	20.5503i	−0.618585	+	1.18450i
$302$	−22.5885			−1.29982
$303$		0			0
$304$	−20.9385			−1.20090
$305$			26.9444i			1.54283i
$306$		−	9.94987i		−	0.568796i
$307$	−24.3721			−1.39099			−0.695495	−	0.718531i	$-0.744815\pi$
$307$	−24.3721			−1.39099			−0.695495	+	0.718531i	$0.744815\pi$
$308$		0			0
$309$		0			0
$310$		−	15.5563i		−	0.883541i
$311$			15.6944i			0.889950i	0.895543	+	0.444975i	$0.146787\pi$
$311$			15.6944i			0.889950i	−0.895543	+	0.444975i	$0.853213\pi$
$312$		0			0
$313$		−	23.8669i		−	1.34904i	−0.738257	−	0.674520i	$-0.764351\pi$
$313$		−	23.8669i		−	1.34904i	0.738257	−	0.674520i	$-0.235649\pi$
$314$	−22.1953			−1.25255
$315$	12.1861	−	23.3345i	0.686607	−	1.31475i
$316$			1.13681i			0.0639507i
$317$	−20.9282			−1.17544			−0.587722	−	0.809063i	$-0.699975\pi$
$317$	−20.9282			−1.17544			−0.587722	+	0.809063i	$0.699975\pi$
$318$		0			0
$319$		0			0
$320$		−	19.0111i		−	1.06275i
$321$		0			0
$322$	−11.1962	+	21.4390i	−0.623937	+	1.19475i
$323$	8.05256			0.448056
$324$	−15.5885			−0.866025
$325$	−38.4434			−2.13245
$326$		−	30.6322i		−	1.69656i
$327$		0			0
$328$			3.31662i			0.183130i
$329$	2.97360	−	5.69402i	0.163940	−	0.313921i
$330$		0			0
$331$	−4.73205			−0.260097			−0.130049	−	0.991508i	$-0.541513\pi$
$331$	−4.73205			−0.260097			−0.130049	+	0.991508i	$0.541513\pi$
$332$	16.2481			0.891729
$333$	−4.60770			−0.252500
$334$		−	15.6944i		−	0.858761i
$335$			32.0394i			1.75050i
$336$		0			0
$337$		−	32.7399i		−	1.78346i	−0.452569	−	0.891729i	$-0.649493\pi$
$337$		−	32.7399i		−	1.78346i	0.452569	−	0.891729i	$-0.350507\pi$
$338$		−	54.1934i		−	2.94773i
$339$		0			0
$340$			9.86233i			0.534860i
$341$		0			0
$342$		−	27.1836i		−	1.46992i
$343$	2.34521	+	18.3712i	0.126629	+	0.991950i
$344$	4.53590			0.244559
$345$		0			0
$346$			27.1836i			1.46140i
$347$		−	23.0064i		−	1.23505i	−0.786553	−	0.617523i	$-0.788136\pi$
$347$		−	23.0064i		−	1.23505i	0.786553	−	0.617523i	$-0.211864\pi$
$348$		0			0
$349$	−3.89364			−0.208422			−0.104211	−	0.994555i	$-0.533232\pi$
$349$	−3.89364			−0.208422			−0.104211	+	0.994555i	$0.533232\pi$
$350$	−14.1962	+	27.1836i	−0.758816	+	1.45302i
$351$		0			0
$352$		0			0
$353$			23.8669i			1.27031i	0.772385	+	0.635154i	$0.219064\pi$
$353$			23.8669i			1.27031i	−0.772385	+	0.635154i	$0.780936\pi$
$354$		0			0
$355$			1.77737i			0.0943332i
$356$			9.94987i			0.527342i
$357$		0			0
$358$			45.7081i			2.41575i
$359$			31.0112i			1.63671i	0.574716	+	0.818353i	$0.305113\pi$
$359$			31.0112i			1.63671i	−0.574716	+	0.818353i	$0.694887\pi$
$360$	−5.15043			−0.271452
$361$	3.00000			0.157895
$362$	1.71681			0.0902336
$363$		0			0
$364$	−26.0263	−	13.5918i	−1.36415	−	0.712403i
$365$			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	$-0.112546\pi$
$368$	21.1244			1.10118
$369$	−19.2217			−1.00064
$370$	9.84085			0.511601
$371$	−23.2837	−	12.1595i	−1.20883	−	0.631291i
$372$		0			0
$373$			12.2474i			0.634149i	0.948401	+	0.317074i	$0.102701\pi$
$373$			12.2474i			0.634149i	−0.948401	+	0.317074i	$0.897299\pi$
$374$		0			0
$375$		0			0
$376$	−1.25679			−0.0648142
$377$			23.8669i			1.22921i
$378$		0			0
$379$	11.1244			0.571420			0.285710	−	0.958316i	$-0.407771\pi$
$379$	11.1244			0.571420			0.285710	+	0.958316i	$0.407771\pi$
$380$			26.9444i			1.38222i
$381$		0			0
$382$			20.7327i			1.06078i
$383$			20.5503i			1.05007i	0.851080	+	0.525036i	$0.175948\pi$
$383$			20.5503i			1.05007i	−0.851080	+	0.525036i	$0.824052\pi$
$384$		0			0
$385$		0			0
$386$	−25.3923			−1.29243
$387$			26.2880i			1.33630i
$388$			25.6443i			1.30189i
$389$	−31.5885			−1.60160			−0.800799	−	0.598933i	$-0.795592\pi$
$389$	−31.5885			−1.60160			−0.800799	+	0.598933i	$0.795592\pi$
$390$		0			0
$391$	−8.12404			−0.410850
$392$	2.97360	−	2.07055i	0.150190	−	0.104579i
$393$		0			0
$394$	−3.92820			−0.197900
$395$	−2.17683			−0.109528
$396$		0			0
$397$		−	16.5831i		−	0.832283i	−0.909300	−	0.416142i	$-0.863382\pi$
$397$		−	16.5831i		−	0.832283i	0.909300	−	0.416142i	$-0.136618\pi$
$398$	3.43362			0.172112
$399$		0			0
$400$	26.7846			1.33923
$401$	22.1244			1.10484			0.552419	−	0.833567i	$-0.313705\pi$
$401$	22.1244			1.10484			0.552419	+	0.833567i	$0.313705\pi$
$402$		0			0
$403$			15.5563i			0.774917i
$404$	−8.12404			−0.404186
$405$		−	29.8496i		−	1.48324i
$406$	16.8765	+	8.81345i	0.837565	+	0.437404i
$407$		0			0
$408$		0			0
$409$	−19.2217			−0.950451			−0.475225	−	0.879864i	$-0.657634\pi$
$409$	−19.2217			−0.950451			−0.475225	+	0.879864i	$0.657634\pi$
$410$	41.0526			2.02744
$411$		0			0
$412$		−	27.1836i		−	1.33924i
$413$	−8.12404	+	15.5563i	−0.399758	+	0.765478i
$414$			27.4249i			1.34786i
$415$			31.1127i			1.52726i
$416$			48.6226i			2.38392i
$417$		0			0
$418$		0			0
$419$		−	33.8168i		−	1.65206i	−0.563627	−	0.826030i	$-0.690594\pi$
$419$		−	33.8168i		−	1.65206i	0.563627	−	0.826030i	$-0.309406\pi$
$420$		0			0
$421$	−12.6603			−0.617023			−0.308512	−	0.951221i	$-0.599831\pi$
$421$	−12.6603			−0.617023			−0.308512	+	0.951221i	$0.599831\pi$
$422$	−14.7321			−0.717145
$423$		−	7.28381i		−	0.354151i
$424$			5.13922i			0.249582i
$425$	−10.3009			−0.499665
$426$		0			0
$427$	−19.0526	−	9.94987i	−0.922018	−	0.481508i
$428$		−	6.69213i		−	0.323476i
$429$		0			0
$430$		−	56.1445i		−	2.70753i
$431$		−	11.7942i		−	0.568106i	−0.958809	−	0.284053i	$-0.908321\pi$
$431$		−	11.7942i		−	0.568106i	0.958809	−	0.284053i	$-0.0916791\pi$
$432$		0			0
$433$		−	13.0284i		−	0.626104i	−0.949736	−	0.313052i	$-0.898649\pi$
$433$		−	13.0284i		−	0.626104i	0.949736	−	0.313052i	$-0.101351\pi$
$434$	11.0000	+	5.74456i	0.528017	+	0.275748i
$435$		0			0
$436$		−	23.4225i		−	1.12173i
$437$	−22.1953			−1.06174
$438$		0			0
$439$	−6.86725			−0.327756			−0.163878	−	0.986481i	$-0.552400\pi$
$439$	−6.86725			−0.327756			−0.163878	+	0.986481i	$0.552400\pi$
$440$		0			0
$441$	12.0000	+	17.2337i	0.571429	+	0.820652i
$442$		−	21.2504i		−	1.01078i
$443$	−10.1962			−0.484434			−0.242217	−	0.970222i	$-0.577875\pi$
$443$	−10.1962			−0.484434			−0.242217	+	0.970222i	$0.577875\pi$
$444$		0			0
$445$	−19.0526			−0.903178
$446$	−35.0097			−1.65776
$447$		0			0
$448$	13.4429	+	7.02030i	0.635115	+	0.331678i
$449$	26.8564			1.26743			0.633716	−	0.773566i	$-0.281529\pi$
$449$	26.8564			1.26743			0.633716	+	0.773566i	$0.281529\pi$
$450$			34.7733i			1.63923i
$451$		0			0
$452$	−1.39230			−0.0654885
$453$		0			0
$454$		−	42.8780i		−	2.01236i
$455$	26.0263	−	49.8365i	1.22013	−	2.33637i
$456$		0			0
$457$			21.4534i			1.00355i	0.864998	+	0.501775i	$0.167319\pi$
$457$			21.4534i			1.00355i	−0.864998	+	0.501775i	$0.832681\pi$
$458$	11.0976			0.518559
$459$		0			0
$460$		−	27.1836i		−	1.26744i
$461$	−7.66402			−0.356949			−0.178475	−	0.983945i	$-0.557116\pi$
$461$	−7.66402			−0.356949			−0.178475	+	0.983945i	$0.557116\pi$
$462$		0			0
$463$	20.3923			0.947711			0.473855	−	0.880603i	$-0.342862\pi$
$463$	20.3923			0.947711			0.473855	+	0.880603i	$0.342862\pi$
$464$		−	16.6288i		−	0.771972i
$465$		0			0
$466$	−22.1244			−1.02489
$467$			32.0394i			1.48261i	0.671169	+	0.741304i	$0.265792\pi$
$467$			32.0394i			1.48261i	−0.671169	+	0.741304i	$0.734208\pi$
$468$	−33.2929			−1.53897
$469$	−22.6553	−	11.8313i	−1.04612	−	0.546321i
$470$			15.5563i			0.717561i
$471$		0			0
$472$	3.43362			0.158045
$473$		0			0
$474$		0			0
$475$	−28.1425			−1.29127
$476$	−6.97372	−	3.64191i	−0.319640	−	0.166927i
$477$	−29.7846			−1.36374
$478$	−19.4641			−0.890267
$479$	10.3009			0.470659			0.235329	−	0.971916i	$-0.424383\pi$
$479$	10.3009			0.470659			0.235329	+	0.971916i	$0.424383\pi$
$480$		0			0
$481$	−9.84085			−0.448704
$482$		−	22.3277i		−	1.01700i
$483$		0			0
$484$		0			0
$485$	−49.1051			−2.22975
$486$		0			0
$487$	19.8564			0.899780			0.449890	−	0.893084i	$-0.351463\pi$
$487$	19.8564			0.899780			0.449890	+	0.893084i	$0.351463\pi$
$488$			4.20531i			0.190366i
$489$		0			0
$490$	−25.6289	−	36.8067i	−1.15780	−	1.66276i
$491$			18.0058i			0.812592i	0.913741	+	0.406296i	$0.133180\pi$
$491$			18.0058i			0.812592i	−0.913741	+	0.406296i	$0.866820\pi$
$492$		0			0
$493$			6.39513i			0.288022i
$494$		−	58.0571i		−	2.61211i
$495$		0			0
$496$		−	10.8386i		−	0.486666i
$497$	−1.25679	−	0.656339i	−0.0563749	−	0.0294408i
$498$		0			0
$499$	11.2679			0.504423			0.252211	−	0.967672i	$-0.418842\pi$
$499$	11.2679			0.504423			0.252211	+	0.967672i	$0.418842\pi$
$500$		−	5.74456i		−	0.256905i
$501$		0			0
$502$	43.1338			1.92515
$503$	27.8057			1.23980			0.619898	−	0.784682i	$-0.287174\pi$
$503$	27.8057			1.23980			0.619898	+	0.784682i	$0.287174\pi$
$504$	1.90192	−	3.64191i	0.0847184	−	0.162223i
$505$		−	15.5563i		−	0.692248i
$506$		0			0
$507$		0			0
$508$		−	20.0764i		−	0.890746i
$509$		−	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
$512$		−	29.2552i		−	1.29291i
$513$		0			0
$514$	−20.4785			−0.903267
$515$	52.0526			2.29371
$516$		0			0
$517$		0			0
$518$	−3.63397	+	6.95853i	−0.159668	+	0.305740i
$519$		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	$0.906142\pi$
$522$	21.5885			0.944901
$523$	−6.86725			−0.300284			−0.150142	−	0.988664i	$-0.547973\pi$
$523$	−6.86725			−0.300284			−0.150142	+	0.988664i	$0.547973\pi$
$524$	−30.3193			−1.32451
$525$		0			0
$526$	16.9282			0.738105
$527$			4.16831i			0.181574i
$528$		0			0
$529$	−0.607695			−0.0264215
$530$	63.6123			2.76314
$531$			19.8997i			0.863576i
$532$	−19.0526	−	9.94987i	−0.826033	−	0.431382i
$533$	−41.0526			−1.77818
$534$		0			0
$535$	12.8145			0.554017
$536$			5.00052i			0.215989i
$537$		0			0
$538$	−27.3457			−1.17896
$539$		0			0
$540$		0			0
$541$			34.8749i			1.49939i	0.661785	+	0.749694i	$0.269799\pi$
$541$			34.8749i			1.49939i	−0.661785	+	0.749694i	$0.730201\pi$
$542$			2.42794i			0.104289i
$543$		0			0
$544$			13.0284i			0.558587i
$545$	44.8506			1.92119
$546$		0			0
$547$		−	35.1523i		−	1.50300i	−0.659732	−	0.751501i	$-0.729330\pi$
$547$		−	35.1523i		−	1.50300i	0.659732	−	0.751501i	$-0.270670\pi$
$548$	−5.32051			−0.227281
$549$	−24.3721			−1.04018
$550$		0			0
$551$			17.4718i			0.744324i
$552$		0			0
$553$	0.803848	−	1.53925i	0.0341831	−	0.0654556i
$554$	−3.73205			−0.158560
$555$		0			0
$556$	24.3721			1.03361
$557$			9.41902i			0.399097i	0.979888	+	0.199548i	$0.0639475\pi$
$557$			9.41902i			0.399097i	−0.979888	+	0.199548i	$0.936052\pi$
$558$	14.0712			0.595683
$559$			56.1445i			2.37466i
$560$	−18.1333	+	34.7226i	−0.766270	+	1.46730i
$561$		0			0
$562$	15.6603			0.660588
$563$	25.6289			1.08013			0.540065	−	0.841623i	$-0.318400\pi$
$563$	25.6289			1.08013			0.540065	+	0.841623i	$0.318400\pi$
$564$		0			0
$565$		−	2.66606i		−	0.112162i
$566$		−	18.1224i		−	0.761740i
$567$	21.1069	+	11.0227i	0.886405	+	0.462910i
$568$			0.277401i			0.0116395i
$569$			7.34847i			0.308064i	0.988066	+	0.154032i	$0.0492259\pi$
$569$			7.34847i			0.308064i	−0.988066	+	0.154032i	$0.950774\pi$
$570$		0			0
$571$		−	23.2838i		−	0.974395i	−0.873292	−	0.487197i	$-0.838019\pi$
$571$		−	23.2838i		−	0.974395i	0.873292	−	0.487197i	$-0.161981\pi$
$572$		0			0
$573$		0			0
$574$	−15.1597	+	29.0285i	−0.632752	+	1.21163i
$575$	28.3923			1.18404
$576$	17.1962			0.716506
$577$		−	34.7055i		−	1.44481i	−0.691471	−	0.722404i	$-0.743037\pi$
$577$		−	34.7055i		−	1.44481i	0.691471	−	0.722404i	$-0.256963\pi$
$578$			27.1475i			1.12919i
$579$		0			0
$580$	−21.3985			−0.888525
$581$	−22.0000	−	11.4891i	−0.912714	−	0.476649i
$582$		0			0
$583$		0			0
$584$			2.42794i			0.100469i
$585$		−	63.7511i		−	2.63578i
$586$		−	32.9281i		−	1.36025i
$587$		−	28.9609i		−	1.19535i	−0.801740	−	0.597673i	$-0.796092\pi$
$587$		−	28.9609i		−	1.19535i	0.801740	−	0.597673i	$-0.203908\pi$
$588$		0			0
$589$			11.3880i			0.469236i
$590$		−	42.5007i		−	1.74973i
$591$		0			0
$592$	6.85641			0.281797
$593$	9.84085			0.404115			0.202058	−	0.979374i	$-0.435237\pi$
$593$	9.84085			0.404115			0.202058	+	0.979374i	$0.435237\pi$
$594$		0			0
$595$	6.97372	−	13.3537i	0.285895	−	0.547447i
$596$		−	1.37705i		−	0.0564061i
$597$		0			0
$598$			58.5724i			2.39521i
$599$	−4.98076			−0.203508			−0.101754	−	0.994810i	$-0.532446\pi$
$599$	−4.98076			−0.203508			−0.101754	+	0.994810i	$0.532446\pi$
$600$		0			0
$601$	7.66402			0.312622			0.156311	−	0.987708i	$-0.450040\pi$
$601$	7.66402			0.312622			0.156311	+	0.987708i	$0.450040\pi$
$602$	39.7002	+	20.7327i	1.61806	+	0.845003i
$603$	−28.9808			−1.18019
$604$			20.2523i			0.824053i
$605$		0			0
$606$		0			0
$607$	6.86725			0.278733			0.139366	−	0.990241i	$-0.455493\pi$
$607$	6.86725			0.278733			0.139366	+	0.990241i	$0.455493\pi$
$608$			35.5942i			1.44353i
$609$		0			0
$610$	52.0526			2.10755
$611$		−	15.5563i		−	0.629343i
$612$	−8.92081			−0.360603
$613$		−	35.4940i		−	1.43359i	−0.697284	−	0.716795i	$-0.745608\pi$
$613$		−	35.4940i		−	1.43359i	0.697284	−	0.716795i	$-0.254392\pi$
$614$			47.0833i			1.90013i
$615$		0			0
$616$		0			0
$617$	39.4449			1.58799			0.793995	−	0.607924i	$-0.207997\pi$
$617$	39.4449			1.58799			0.793995	+	0.607924i	$0.207997\pi$
$618$		0			0
$619$			3.07850i			0.123735i	0.998084	+	0.0618677i	$0.0197057\pi$
$619$			3.07850i			0.123735i	−0.998084	+	0.0618677i	$0.980294\pi$
$620$	−13.9474			−0.560143
$621$		0			0
$622$	30.3193			1.21569
$623$	7.03562	−	13.4722i	0.281876	−	0.539752i
$624$		0			0
$625$	−19.0000			−0.760000
$626$	−46.1074			−1.84282
$627$		0			0
$628$			19.8997i			0.794086i
$629$	−2.63685			−0.105138
$630$	−45.0788	−	23.5417i	−1.79598	−	0.937922i
$631$	9.46410			0.376760			0.188380	−	0.982096i	$-0.439676\pi$
$631$	9.46410			0.376760			0.188380	+	0.982096i	$0.439676\pi$
$632$	−0.339746			−0.0135144
$633$		0			0
$634$			40.4302i			1.60569i
$635$	38.4434			1.52558
$636$		0			0
$637$	25.6289	+	36.8067i	1.01545	+	1.45834i
$638$		0			0
$639$	−1.60770			−0.0635994
$640$	13.6112			0.538031
$641$	−7.24871			−0.286307			−0.143153	−	0.989701i	$-0.545724\pi$
$641$	−7.24871			−0.286307			−0.143153	+	0.989701i	$0.545724\pi$
$642$		0			0
$643$		−	41.1006i		−	1.62085i	−0.585842	−	0.810425i	$-0.699236\pi$
$643$		−	41.1006i		−	1.62085i	0.585842	−	0.810425i	$-0.300764\pi$
$644$	19.2217	+	10.0382i	0.757440	+	0.395560i
$645$		0			0
$646$		−	15.5563i		−	0.612056i
$647$		−	38.0221i		−	1.49480i	−0.664372	−	0.747402i	$-0.731301\pi$
$647$		−	38.0221i		−	1.49480i	0.664372	−	0.747402i	$-0.268699\pi$
$648$		−	4.65874i		−	0.183013i
$649$		0			0
$650$			74.2669i			2.91299i
$651$		0			0
$652$	−27.4641			−1.07558
$653$	9.46410			0.370359			0.185179	−	0.982705i	$-0.440713\pi$
$653$	9.46410			0.370359			0.185179	+	0.982705i	$0.440713\pi$
$654$		0			0
$655$		−	58.0571i		−	2.26848i
$656$	28.6025			1.11674
$657$	−14.0712			−0.548972
$658$	−11.0000	−	5.74456i	−0.428825	−	0.223946i
$659$			43.3601i			1.68907i	0.535499	+	0.844536i	$0.320124\pi$
$659$			43.3601i			1.68907i	−0.535499	+	0.844536i	$0.679876\pi$
$660$		0			0
$661$			13.0284i			0.506745i	0.967369	+	0.253373i	$0.0815398\pi$
$661$			13.0284i			0.506745i	−0.967369	+	0.253373i	$0.918460\pi$
$662$			9.14162i			0.355299i
$663$		0			0
$664$			4.85588i			0.188445i
$665$	19.0526	−	36.4829i	0.738827	−	1.41475i
$666$			8.90138i			0.344922i
$667$		−	17.6269i		−	0.682516i
$668$	−14.0712			−0.544433
$669$		0			0
$670$	61.8954			2.39123
$671$		0			0
$672$		0			0
$673$		−	31.7690i		−	1.22461i	−0.790623	−	0.612303i	$-0.790243\pi$
$673$		−	31.7690i		−	1.22461i	0.790623	−	0.612303i	$-0.209757\pi$
$674$	−63.2487			−2.43625
$675$		0			0
$676$	−48.5885			−1.86879
$677$	17.0449			0.655087			0.327543	−	0.944836i	$-0.393779\pi$
$677$	17.0449			0.655087			0.327543	+	0.944836i	$0.393779\pi$
$678$		0			0
$679$	18.1333	−	34.7226i	0.695891	−	1.33253i
$680$	−2.94744			−0.113029
$681$		0			0
$682$		0			0
$683$	33.3731			1.27698			0.638492	−	0.769628i	$-0.279558\pi$
$683$	33.3731			1.27698			0.638492	+	0.769628i	$0.279558\pi$
$684$	−24.3721			−0.931891
$685$		−	10.1880i		−	0.389263i
$686$	35.4904	−	4.53059i	1.35503	−	0.172979i
$687$		0			0
$688$		−	39.1175i		−	1.49134i
$689$	−63.6123			−2.42343
$690$		0			0
$691$		−	4.85588i		−	0.184726i	−0.995725	−	0.0923631i	$-0.970558\pi$
$691$		−	4.85588i		−	0.184726i	0.995725	−	0.0923631i	$-0.0294421\pi$
$692$	24.3721			0.926489
$693$		0			0
$694$	−44.4449			−1.68710
$695$			46.6690i			1.77026i
$696$		0			0
$697$	−11.0000			−0.416655
$698$			7.52194i			0.284709i
$699$		0			0
$700$	24.3721	+	12.7279i	0.921179	+	0.481070i
$701$			9.55772i			0.360990i	0.983576	+	0.180495i	$0.0577700\pi$
$701$			9.55772i			0.360990i	−0.983576	+	0.180495i	$0.942230\pi$
$702$		0			0
$703$	−7.20400			−0.271704
$704$		0			0
$705$		0			0
$706$	46.1074			1.73527
$707$	11.0000	+	5.74456i	0.413698	+	0.216047i
$708$		0			0
$709$	−38.6410			−1.45119			−0.725597	−	0.688120i	$-0.758436\pi$
$709$	−38.6410			−1.45119			−0.725597	+	0.688120i	$0.758436\pi$
$710$	3.43362			0.128862
$711$		−	1.96902i		−	0.0738439i
$712$	−2.97360			−0.111441
$713$		−	11.4891i		−	0.430271i
$714$		0			0
$715$		0			0
$716$	40.9808			1.53152
$717$		0			0
$718$	59.9090			2.23578
$719$		−	1.77737i		−	0.0662849i	−0.999451	−	0.0331424i	$-0.989449\pi$
$719$		−	1.77737i		−	0.0662849i	0.999451	−	0.0331424i	$-0.0105515\pi$
$720$			44.4173i			1.65533i
$721$	−19.2217	+	36.8067i	−0.715853	+	1.37075i
$722$		−	5.79555i		−	0.215688i
$723$		0			0
$724$		−	1.53925i		−	0.0572058i
$725$		−	22.3500i		−	0.830059i
$726$		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
$730$	30.0526			1.11230
$731$			15.0439i			0.556418i
$732$		0			0
$733$	−11.0976			−0.409901			−0.204950	−	0.978772i	$-0.565703\pi$
$733$	−11.0976			−0.409901			−0.204950	+	0.978772i	$0.565703\pi$
$734$	−25.6289			−0.945980
$735$		0			0
$736$		−	35.9101i		−	1.32367i
$737$		0			0
$738$			37.1334i			1.36690i
$739$		−	0.656339i		−	0.0241438i	−0.999927	−	0.0120719i	$-0.996157\pi$
$739$		−	0.656339i		−	0.0241438i	0.999927	−	0.0120719i	$-0.00384270\pi$
$740$		−	8.82306i		−	0.324342i
$741$		0			0
$742$	−23.4904	+	44.9807i	−0.862359	+	1.65129i
$743$		−	18.8380i		−	0.691101i	−0.938400	−	0.345550i	$-0.887692\pi$
$743$		−	18.8380i		−	0.691101i	0.938400	−	0.345550i	$-0.112308\pi$
$744$		0			0
$745$	2.63685			0.0966066
$746$	23.6603			0.866263
$747$	−28.1425			−1.02968
$748$		0			0
$749$	−4.73205	+	9.06119i	−0.172905	+	0.331089i
$750$		0			0
$751$	−15.6077			−0.569533			−0.284766	−	0.958597i	$-0.591916\pi$
$751$	−15.6077			−0.569533			−0.284766	+	0.958597i	$0.591916\pi$
$752$			10.8386i			0.395242i
$753$		0			0
$754$	46.1074			1.67913
$755$	−38.7801			−1.41135
$756$		0			0
$757$	−1.87564			−0.0681715			−0.0340857	−	0.999419i	$-0.510852\pi$
$757$	−1.87564			−0.0681715			−0.0340857	+	0.999419i	$0.510852\pi$
$758$		−	21.4906i		−	0.780574i
$759$		0			0
$760$	−8.05256			−0.292097
$761$	−2.97360			−0.107793			−0.0538965	−	0.998547i	$-0.517164\pi$
$761$	−2.97360			−0.107793			−0.0538965	+	0.998547i	$0.517164\pi$
$762$		0			0
$763$	−16.5622	+	31.7142i	−0.599591	+	1.14813i
$764$	18.5885			0.672507
$765$		−	17.0821i		−	0.617603i
$766$	39.7002			1.43442
$767$			42.5007i			1.53461i
$768$		0			0
$769$	42.6738			1.53886			0.769428	−	0.638734i	$-0.220542\pi$
$769$	42.6738			1.53886			0.769428	+	0.638734i	$0.220542\pi$
$770$		0			0
$771$		0			0
$772$			22.7661i			0.819371i
$773$		−	24.7556i		−	0.890398i	−0.895432	−	0.445199i	$-0.853133\pi$
$773$		−	24.7556i		−	0.890398i	0.895432	−	0.445199i	$-0.146867\pi$
$774$	50.7846			1.82542
$775$		−	14.5676i		−	0.523285i
$776$	−7.66402			−0.275122
$777$		0			0
$778$			61.0242i			2.18782i
$779$	−30.0526			−1.07674
$780$		0			0
$781$		0			0
$782$			15.6944i			0.561232i
$783$		0			0
$784$	−17.8564	−	25.6443i	−0.637729	−	0.915868i
$785$	−38.1051			−1.36003
$786$		0			0
$787$	35.0097			1.24796			0.623981	−	0.781439i	$-0.285514\pi$
$787$	35.0097			1.24796			0.623981	+	0.781439i	$0.285514\pi$
$788$			3.52193i			0.125464i
$789$		0			0
$790$			4.20531i			0.149618i
$791$	1.88519	+	0.984508i	0.0670296	+	0.0350051i
$792$		0			0
$793$	−52.0526			−1.84844
$794$	−32.0361			−1.13692
$795$		0			0
$796$		−	3.07850i		−	0.109115i
$797$			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
$799$		−	4.16831i		−	0.147464i
$800$		−	45.5322i		−	1.60981i
$801$		−	17.2337i		−	0.608922i
$802$		−	42.7410i		−	1.50924i
$803$		0			0
$804$		0			0
$805$	−19.2217	+	36.8067i	−0.677475	+	1.29727i
$806$	30.0526			1.05856
$807$		0			0
$808$		−	2.42794i		−	0.0854146i
$809$		−	3.96524i		−	0.139410i	−0.997568	−	0.0697052i	$-0.977794\pi$
$809$		−	3.96524i		−	0.139410i	0.997568	−	0.0697052i	$-0.0222059\pi$
$810$	−57.6650			−2.02614
$811$	53.4346			1.87634			0.938172	−	0.346169i	$-0.112518\pi$
$811$	53.4346			1.87634			0.938172	+	0.346169i	$0.112518\pi$
$812$	7.90192	−	15.1310i	0.277303	−	0.530995i
$813$		0			0
$814$		0			0
$815$		−	52.5898i		−	1.84214i
$816$		0			0
$817$			41.1006i			1.43793i
$818$			37.1334i			1.29834i
$819$	45.0788	+	23.5417i	1.57518	+	0.822612i
$820$		−	36.8067i		−	1.28535i
$821$			13.8375i			0.482933i	0.970409	+	0.241467i	$0.0776284\pi$
$821$			13.8375i			0.482933i	−0.970409	+	0.241467i	$0.922372\pi$
$822$		0			0
$823$	31.1244			1.08493			0.542463	−	0.840079i	$-0.317492\pi$
$823$	31.1244			1.08493			0.542463	+	0.840079i	$0.317492\pi$
$824$	8.12404			0.283014
$825$		0			0
$826$	30.0526	+	15.6944i	1.04566	+	0.546079i
$827$		−	21.3147i		−	0.741186i	−0.928795	−	0.370593i	$-0.879154\pi$
$827$		−	21.3147i		−	0.741186i	0.928795	−	0.370593i	$-0.120846\pi$
$828$	24.5885			0.854508
$829$			1.53925i			0.0534604i	0.999643	+	0.0267302i	$0.00850950\pi$
$829$			1.53925i			0.0534604i	−0.999643	+	0.0267302i	$0.991491\pi$
$830$	60.1051			2.08628
$831$		0			0
$832$	36.7266			1.27326
$833$	6.86725	+	9.86233i	0.237936	+	0.341709i
$834$		0			0
$835$		−	26.9444i		−	0.932449i
$836$		0			0
$837$		0			0
$838$	−65.3291			−2.25676
$839$		−	9.06119i		−	0.312827i	−0.987692	−	0.156413i	$-0.950007\pi$
$839$		−	9.06119i		−	0.312827i	0.987692	−	0.156413i	$-0.0499932\pi$
$840$		0			0
$841$	15.1244			0.521530
$842$			24.4577i			0.842869i
$843$		0			0
$844$			13.2084i			0.454652i
$845$		−	93.0398i		−	3.20067i
$846$	−14.0712			−0.483779
$847$		0			0
$848$	44.3205			1.52197
$849$		0			0
$850$			19.8997i			0.682556i
$851$	7.26795			0.249142
$852$		0			0
$853$	0.796775			0.0272811			0.0136405	−	0.999907i	$-0.495658\pi$
$853$	0.796775			0.0272811			0.0136405	+	0.999907i	$0.495658\pi$
$854$	−19.2217	+	36.8067i	−0.657752	+	1.25950i
$855$		−	46.6690i		−	1.59605i
$856$	2.00000			0.0683586
$857$	49.0810			1.67657			0.838287	−	0.545229i	$-0.183557\pi$
$857$	49.0810			1.67657			0.838287	+	0.545229i	$0.183557\pi$
$858$		0			0
$859$		−	15.6944i		−	0.535487i	−0.963490	−	0.267744i	$-0.913722\pi$
$859$		−	15.6944i		−	0.535487i	0.963490	−	0.267744i	$-0.0862780\pi$
$860$	−50.3378			−1.71650
$861$		0			0
$862$	−22.7846			−0.776047
$863$	24.1051			0.820548			0.410274	−	0.911962i	$-0.365433\pi$
$863$	24.1051			0.820548			0.410274	+	0.911962i	$0.365433\pi$
$864$		0			0
$865$			46.6690i			1.58680i
$866$	−25.1689			−0.855274
$867$		0			0
$868$	5.15043	−	9.86233i	0.174817	−	0.334749i
$869$		0			0
$870$		0			0
$871$	−61.8954			−2.09725
$872$	7.00000			0.237050
$873$		−	44.4173i		−	1.50330i
$874$			42.8780i			1.45037i
$875$	−4.06202	+	7.77817i	−0.137321	+	0.262950i
$876$		0			0
$877$			11.1750i			0.377353i	0.982039	+	0.188677i	$0.0604198\pi$
$877$			11.1750i			0.377353i	−0.982039	+	0.188677i	$0.939580\pi$
$878$			13.2665i			0.447723i
$879$		0			0
$880$		0			0
$881$		−	49.2731i		−	1.66005i	−0.557723	−	0.830027i	$-0.688325\pi$
$881$		−	49.2731i		−	1.66005i	0.557723	−	0.830027i	$-0.311675\pi$
$882$	33.2929	−	23.1822i	1.12103	−	0.780586i
$883$	18.0526			0.607517			0.303758	−	0.952749i	$-0.401758\pi$
$883$	18.0526			0.607517			0.303758	+	0.952749i	$0.401758\pi$
$884$	−19.0526			−0.640807
$885$		0			0
$886$			19.6975i			0.661749i
$887$	−20.0185			−0.672154			−0.336077	−	0.941835i	$-0.609100\pi$
$887$	−20.0185			−0.672154			−0.336077	+	0.941835i	$0.609100\pi$
$888$		0			0
$889$	−14.1962	+	27.1836i	−0.476124	+	0.911707i
$890$			36.8067i			1.23376i
$891$		0			0
$892$			31.3889i			1.05098i
$893$		−	11.3880i		−	0.381086i
$894$		0			0
$895$			78.4722i			2.62304i
$896$	−5.02628	+	9.62459i	−0.167916	+	0.321535i
$897$		0			0
$898$		−	51.8826i		−	1.73134i
$899$	−9.04407			−0.301637
$900$	31.1769			1.03923
$901$	−17.0449			−0.567846
$902$		0			0
$903$		0			0
$904$		−	0.416102i		−	0.0138394i
$905$	2.94744			0.0979763
$906$		0			0
$907$	−44.5359			−1.47879			−0.739395	−	0.673272i	$-0.764888\pi$
$907$	−44.5359			−1.47879			−0.739395	+	0.673272i	$0.764888\pi$
$908$	−38.4434			−1.27579
$909$	14.0712			0.466714
$910$	−96.2768	−	50.2789i	−3.19154	−	1.66673i
$911$	−46.0526			−1.52579			−0.762895	−	0.646523i	$-0.776223\pi$
$911$	−46.0526			−1.52579			−0.762895	+	0.646523i	$0.776223\pi$
$912$		0			0
$913$		0			0
$914$	41.4449			1.37087
$915$		0			0
$916$		−	9.94987i		−	0.328753i
$917$	41.0526	+	21.4390i	1.35567	+	0.707978i
$918$		0			0
$919$			23.6627i			0.780560i	0.920696	+	0.390280i	$0.127622\pi$
$919$			23.6627i			0.780560i	−0.920696	+	0.390280i	$0.872378\pi$
$920$	8.12404			0.267842
$921$		0			0
$922$			14.8058i			0.487601i
$923$	−3.43362			−0.113019
$924$		0			0
$925$	9.21539			0.303000
$926$		−	39.3949i		−	1.29460i
$927$			47.0833i			1.54642i
$928$	−28.2679			−0.927941
$929$			6.39513i			0.209817i	0.994482	+	0.104909i	$0.0334550\pi$
$929$			6.39513i			0.209817i	−0.994482	+	0.104909i	$0.966545\pi$
$930$		0			0
$931$	18.7617	+	26.9444i	0.614889	+	0.883067i
$932$			19.8362i			0.649755i
$933$		0			0
$934$	61.8954			2.02528
$935$		0			0
$936$		−	9.94987i		−	0.325222i
$937$	25.1689			0.822232			0.411116	−	0.911583i	$-0.365139\pi$
$937$	25.1689			0.822232			0.411116	+	0.911583i	$0.365139\pi$
$938$	−22.8564	+	43.7667i	−0.746288	+	1.42903i
$939$		0			0
$940$	13.9474			0.454915
$941$	−7.32726			−0.238862			−0.119431	−	0.992843i	$-0.538107\pi$
$941$	−7.32726			−0.238862			−0.119431	+	0.992843i	$0.538107\pi$
$942$		0			0
$943$	30.3193			0.987333
$944$		−	29.6115i		−	0.963772i
$945$		0			0
$946$		0			0
$947$	−36.1051			−1.17326			−0.586629	−	0.809856i	$-0.699545\pi$
$947$	−36.1051			−1.17326			−0.586629	+	0.809856i	$0.699545\pi$
$948$		0			0
$949$	−30.0526			−0.975547
$950$			54.3671i			1.76390i
$951$		0			0
$952$	1.08841	−	2.08416i	0.0352757	−	0.0675479i
$953$			9.76079i			0.316183i	0.987424	+	0.158092i	$0.0505341\pi$
$953$			9.76079i			0.316183i	−0.987424	+	0.158092i	$0.949466\pi$
$954$			57.5394i			1.86291i
$955$			35.5942i			1.15180i
$956$			17.4510i			0.564407i
$957$		0			0
$958$		−	19.8997i		−	0.642932i
$959$	7.20400	+	3.76217i	0.232629	+	0.121487i
$960$		0			0
$961$	25.1051			0.809843
$962$			19.0111i			0.612941i
$963$			11.5911i			0.373518i
$964$	−20.0185			−0.644751
$965$	−43.5938			−1.40333
$966$		0			0
$967$		−	38.8401i		−	1.24901i	−0.781019	−	0.624507i	$-0.785300\pi$
$967$		−	38.8401i		−	1.24901i	0.781019	−	0.624507i	$-0.214700\pi$
$968$		0			0
$969$		0			0
$970$			94.8638i			3.04589i
$971$			55.0177i			1.76560i	0.469747	+	0.882801i	$0.344345\pi$
$971$			55.0177i			1.76560i	−0.469747	+	0.882801i	$0.655655\pi$
$972$		0			0
$973$	−33.0000	−	17.2337i	−1.05793	−	0.552487i
$974$		−	38.3596i		−	1.22912i
$975$		0			0
$976$	36.2665			1.16086
$977$	18.8038			0.601588			0.300794	−	0.953689i	$-0.402748\pi$
$977$	18.8038			0.601588			0.300794	+	0.953689i	$0.402748\pi$
$978$		0			0
$979$		0			0
$980$	−33.0000	+	22.9783i	−1.05415	+	0.734013i
$981$			40.5689i			1.29526i
$982$	34.7846			1.11002
$983$		−	47.0833i		−	1.50172i	−0.660459	−	0.750862i	$-0.729638\pi$
$983$		−	47.0833i		−	1.50172i	0.660459	−	0.750862i	$-0.270362\pi$
$984$		0			0
$985$	−6.74398			−0.214881
$986$	12.3544			0.393445
$987$		0			0
$988$	−52.0526			−1.65601
$989$		−	41.4655i		−	1.31852i
$990$		0			0
$991$	−11.9474			−0.379523			−0.189761	−	0.981830i	$-0.560771\pi$
$991$	−11.9474			−0.379523			−0.189761	+	0.981830i	$0.560771\pi$
$992$	−18.4249			−0.584991
$993$		0			0
$994$	−1.26795	+	2.42794i	−0.0402169	+	0.0770095i
$995$	5.89488			0.186880
$996$		0			0
$997$	2.97360			0.0941750			0.0470875	−	0.998891i	$-0.485006\pi$
$997$	2.97360			0.0941750			0.0470875	+	0.998891i	$0.485006\pi$
$998$		−	21.7680i		−	0.689054i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.b.d.846.1	✓	8
7.6	odd	2	inner	847.2.b.d.846.2	yes	8
11.2	odd	10		847.2.l.k.524.8		32
11.3	even	5		847.2.l.k.475.2		32
11.4	even	5		847.2.l.k.699.7		32
11.5	even	5		847.2.l.k.118.7		32
11.6	odd	10		847.2.l.k.118.1		32
11.7	odd	10		847.2.l.k.699.1		32
11.8	odd	10		847.2.l.k.475.8		32
11.9	even	5		847.2.l.k.524.2		32
11.10	odd	2	inner	847.2.b.d.846.7	yes	8
77.6	even	10		847.2.l.k.118.2		32
77.13	even	10		847.2.l.k.524.7		32
77.20	odd	10		847.2.l.k.524.1		32
77.27	odd	10		847.2.l.k.118.8		32
77.41	even	10		847.2.l.k.475.7		32
77.48	odd	10		847.2.l.k.699.8		32
77.62	even	10		847.2.l.k.699.2		32
77.69	odd	10		847.2.l.k.475.1		32
77.76	even	2	inner	847.2.b.d.846.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.b.d.846.1	✓	8	1.1	even	1	trivial
847.2.b.d.846.2	yes	8	7.6	odd	2	inner
847.2.b.d.846.7	yes	8	11.10	odd	2	inner
847.2.b.d.846.8	yes	8	77.76	even	2	inner
847.2.l.k.118.1		32	11.6	odd	10
847.2.l.k.118.2		32	77.6	even	10
847.2.l.k.118.7		32	11.5	even	5
847.2.l.k.118.8		32	77.27	odd	10
847.2.l.k.475.1		32	77.69	odd	10
847.2.l.k.475.2		32	11.3	even	5
847.2.l.k.475.7		32	77.41	even	10
847.2.l.k.475.8		32	11.8	odd	10
847.2.l.k.524.1		32	77.20	odd	10
847.2.l.k.524.2		32	11.9	even	5
847.2.l.k.524.7		32	77.13	even	10
847.2.l.k.524.8		32	11.2	odd	10
847.2.l.k.699.1		32	11.7	odd	10
847.2.l.k.699.2		32	77.62	even	10
847.2.l.k.699.7		32	11.4	even	5
847.2.l.k.699.8		32	77.48	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.93185i q^{2} -1.73205 q^{4} -3.31662i q^{5} +(2.34521 + 1.22474i) q^{7} -0.517638i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.93185i q^{2} -1.73205 q^{4} -3.31662i q^{5} +(2.34521 + 1.22474i) q^{7} -0.517638i q^{8} +3.00000 q^{9} -6.40723 q^{10} +6.40723 q^{13} +(2.36603 - 4.53059i) q^{14} -4.46410 q^{16} +1.71681 q^{17} -5.79555i q^{18} +4.69042 q^{19} +5.74456i q^{20} -4.73205 q^{23} -6.00000 q^{25} -12.3778i q^{26} +(-4.06202 - 2.12132i) q^{28} +3.72500i q^{29} +2.42794i q^{31} +7.58871i q^{32} -3.31662i q^{34} +(4.06202 - 7.77817i) q^{35} -5.19615 q^{36} -1.53590 q^{37} -9.06119i q^{38} -1.71681 q^{40} -6.40723 q^{41} +8.76268i q^{43} -9.94987i q^{45} +9.14162i q^{46} -2.42794i q^{47} +(4.00000 + 5.74456i) q^{49} +11.5911i q^{50} -11.0976 q^{52} -9.92820 q^{53} +(0.633975 - 1.21397i) q^{56} +7.19615 q^{58} +6.63325i q^{59} -8.12404 q^{61} +4.69042 q^{62} +(7.03562 + 3.67423i) q^{63} +5.73205 q^{64} -21.2504i q^{65} -9.66025 q^{67} -2.97360 q^{68} +(-15.0263 - 7.84722i) q^{70} -0.535898 q^{71} -1.55291i q^{72} -4.69042 q^{73} +2.96713i q^{74} -8.12404 q^{76} -0.656339i q^{79} +14.8058i q^{80} +9.00000 q^{81} +12.3778i q^{82} -9.38083 q^{83} -5.69402i q^{85} +16.9282 q^{86} -5.74456i q^{89} -19.2217 q^{90} +(15.0263 + 7.84722i) q^{91} +8.19615 q^{92} -4.69042 q^{94} -15.5563i q^{95} -14.8058i q^{97} +(11.0976 - 7.72741i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 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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