LMFDB - Embedded newform 1035.2.b.e.829.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,2,Mod(829,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1035.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.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:	$8.26451660920$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.527896576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 2x^5 + 7x^4 - 10x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.1
Root			$1.47984 + 1.47984i$ of defining polynomial
Character	$\chi$	$=$	1035.829
Dual form			1035.2.b.e.829.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37988i q^{2} -3.66382 q^{4} +(-0.479844 + 2.18398i) q^{5} +2.28394i q^{7} +3.95969i q^{8} +O(q^{10})$	\(q-2.37988i q^{2} -3.66382 q^{4} +(-0.479844 + 2.18398i) q^{5} +2.28394i q^{7} +3.95969i q^{8} +(5.19760 + 1.14197i) q^{10} -1.12432 q^{11} -5.95969i q^{13} +5.43550 q^{14} +2.09594 q^{16} -5.80007i q^{17} +4.08401 q^{19} +(1.75806 - 8.00169i) q^{20} +2.67575i q^{22} -1.00000i q^{23} +(-4.53950 - 2.09594i) q^{25} -14.1833 q^{26} -8.36795i q^{28} -0.408263 q^{29} -3.19187 q^{31} +2.93130i q^{32} -13.8035 q^{34} +(-4.98807 - 1.09594i) q^{35} -9.80345i q^{37} -9.71944i q^{38} +(-8.64786 - 1.90003i) q^{40} -6.27087 q^{41} -7.75474i q^{43} +4.11931 q^{44} -2.37988 q^{46} +6.40020i q^{47} +1.78361 q^{49} +(-4.98807 + 10.8035i) q^{50} +21.8352i q^{52} -6.73590i q^{53} +(0.539499 - 2.45549i) q^{55} -9.04370 q^{56} +0.971615i q^{58} -4.75976 q^{59} -6.33265 q^{61} +7.59627i q^{62} +11.1680 q^{64} +(13.0158 + 2.85972i) q^{65} +0.283942i q^{67} +21.2504i q^{68} +(-2.60819 + 11.8710i) q^{70} +13.9516 q^{71} -9.61659i q^{73} -23.3310 q^{74} -14.9631 q^{76} -2.56788i q^{77} -4.48387 q^{79} +(-1.00572 + 4.57747i) q^{80} +14.9239i q^{82} -10.8223i q^{83} +(12.6672 + 2.78313i) q^{85} -18.4553 q^{86} -4.45196i q^{88} +5.68414 q^{89} +13.6116 q^{91} +3.66382i q^{92} +15.2317 q^{94} +(-1.95969 + 8.91938i) q^{95} +11.0676i q^{97} -4.24477i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} + 6 q^{5}+O(q^{10}) $ 8 * q - 4 * q^4 + 6 * q^5	$ 8 q - 4 q^{4} + 6 q^{5} + 6 q^{10} - 4 q^{11} - 8 q^{14} + 4 q^{16} + 8 q^{19} + 8 q^{20} - 16 q^{25} - 12 q^{26} + 8 q^{29} - 28 q^{34} - 28 q^{35} - 10 q^{40} + 16 q^{41} + 12 q^{44} - 28 q^{50} - 16 q^{55} - 28 q^{56} - 16 q^{61} + 40 q^{64} + 14 q^{65} - 28 q^{70} + 48 q^{71} - 72 q^{74} - 36 q^{76} - 48 q^{79} + 2 q^{80} + 12 q^{85} - 28 q^{86} - 16 q^{89} + 52 q^{91} + 84 q^{94} + 4 q^{95}+O(q^{100}) $ 8 * q - 4 * q^4 + 6 * q^5 + 6 * q^10 - 4 * q^11 - 8 * q^14 + 4 * q^16 + 8 * q^19 + 8 * q^20 - 16 * q^25 - 12 * q^26 + 8 * q^29 - 28 * q^34 - 28 * q^35 - 10 * q^40 + 16 * q^41 + 12 * q^44 - 28 * q^50 - 16 * q^55 - 28 * q^56 - 16 * q^61 + 40 * q^64 + 14 * q^65 - 28 * q^70 + 48 * q^71 - 72 * q^74 - 36 * q^76 - 48 * q^79 + 2 * q^80 + 12 * q^85 - 28 * q^86 - 16 * q^89 + 52 * q^91 + 84 * q^94 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1035\mathbb{Z}\right)^\times$.

$n$	$461$	$622$	$856$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	2.37988i		−	1.68283i	−0.540391	−	0.841414i	$-0.681724\pi$
$2$		−	2.37988i		−	1.68283i	0.540391	−	0.841414i	$-0.318276\pi$
$3$		0			0
$3$		0			0
$4$	−3.66382			−1.83191
$5$	−0.479844	+	2.18398i	−0.214593	+	0.976704i
$5$	−0.479844	+	2.18398i	−0.214593	+	0.976704i
$6$		0			0
$7$			2.28394i			0.863249i	0.902053	+	0.431624i	$0.142059\pi$
$7$			2.28394i			0.863249i	−0.902053	+	0.431624i	$0.857941\pi$
$8$			3.95969i			1.39996i
$9$		0			0
$10$	5.19760	+	1.14197i	1.64362	+	0.361123i
$11$	−1.12432			−0.338996			−0.169498	−	0.985531i	$-0.554215\pi$
$11$	−1.12432			−0.338996			−0.169498	+	0.985531i	$0.554215\pi$
$12$		0			0
$13$		−	5.95969i		−	1.65292i	−0.562995	−	0.826460i	$-0.690351\pi$
$13$		−	5.95969i		−	1.65292i	0.562995	−	0.826460i	$-0.309649\pi$
$14$	5.43550			1.45270
$15$		0			0
$16$	2.09594			0.523984
$17$		−	5.80007i		−	1.40672i	−0.710832	−	0.703362i	$-0.751682\pi$
$17$		−	5.80007i		−	1.40672i	0.710832	−	0.703362i	$-0.248318\pi$
$18$		0			0
$19$	4.08401			0.936936			0.468468	−	0.883480i	$-0.344806\pi$
$19$	4.08401			0.936936			0.468468	+	0.883480i	$0.344806\pi$
$20$	1.75806	−	8.00169i	0.393115	−	1.78923i
$21$		0			0
$22$			2.67575i			0.570471i
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.53950	−	2.09594i	−0.907900	−	0.419187i
$26$	−14.1833			−2.78158
$27$		0			0
$28$		−	8.36795i		−	1.58139i
$29$	−0.408263			−0.0758125			−0.0379062	−	0.999281i	$-0.512069\pi$
$29$	−0.408263			−0.0758125			−0.0379062	+	0.999281i	$0.512069\pi$
$30$		0			0
$31$	−3.19187			−0.573277			−0.286639	−	0.958039i	$-0.592538\pi$
$31$	−3.19187			−0.573277			−0.286639	+	0.958039i	$0.592538\pi$
$32$			2.93130i			0.518186i
$33$		0			0
$34$	−13.8035			−2.36727
$35$	−4.98807	−	1.09594i	−0.843138	−	0.185247i
$36$		0			0
$37$		−	9.80345i		−	1.61168i	−0.592135	−	0.805839i	$-0.701715\pi$
$37$		−	9.80345i		−	1.61168i	0.592135	−	0.805839i	$-0.298285\pi$
$38$		−	9.71944i		−	1.57670i
$39$		0			0
$40$	−8.64786	−	1.90003i	−1.36735	−	0.300422i
$41$	−6.27087			−0.979345			−0.489673	−	0.871906i	$-0.662884\pi$
$41$	−6.27087			−0.979345			−0.489673	+	0.871906i	$0.662884\pi$
$42$		0			0
$43$		−	7.75474i		−	1.18259i	−0.806456	−	0.591294i	$-0.798617\pi$
$43$		−	7.75474i		−	1.18259i	0.806456	−	0.591294i	$-0.201383\pi$
$44$	4.11931			0.621009
$45$		0			0
$46$	−2.37988			−0.350894
$47$			6.40020i			0.933566i	0.884372	+	0.466783i	$0.154587\pi$
$47$			6.40020i			0.933566i	−0.884372	+	0.466783i	$0.845413\pi$
$48$		0			0
$49$	1.78361			0.254801
$50$	−4.98807	+	10.8035i	−0.705420	+	1.52784i
$51$		0			0
$52$			21.8352i			3.02800i
$53$		−	6.73590i		−	0.925247i	−0.886555	−	0.462624i	$-0.846908\pi$
$53$		−	6.73590i		−	0.925247i	0.886555	−	0.462624i	$-0.153092\pi$
$54$		0			0
$55$	0.539499	−	2.45549i	0.0727460	−	0.331098i
$56$	−9.04370			−1.20851
$57$		0			0
$58$			0.971615i			0.127579i
$59$	−4.75976			−0.619667			−0.309834	−	0.950791i	$-0.600273\pi$
$59$	−4.75976			−0.619667			−0.309834	+	0.950791i	$0.600273\pi$
$60$		0			0
$61$	−6.33265			−0.810813			−0.405406	−	0.914137i	$-0.632870\pi$
$61$	−6.33265			−0.810813			−0.405406	+	0.914137i	$0.632870\pi$
$62$			7.59627i			0.964727i
$63$		0			0
$64$	11.1680			1.39600
$65$	13.0158	+	2.85972i	1.61441	+	0.354705i
$66$		0			0
$67$			0.283942i			0.0346890i	0.999850	+	0.0173445i	$0.00552120\pi$
$67$			0.283942i			0.0346890i	−0.999850	+	0.0173445i	$0.994479\pi$
$68$			21.2504i			2.57699i
$69$		0			0
$70$	−2.60819	+	11.8710i	−0.311739	+	1.41886i
$71$	13.9516			1.65575			0.827877	−	0.560910i	$-0.189549\pi$
$71$	13.9516			1.65575			0.827877	+	0.560910i	$0.189549\pi$
$72$		0			0
$73$		−	9.61659i		−	1.12554i	−0.826615	−	0.562769i	$-0.809736\pi$
$73$		−	9.61659i		−	1.12554i	0.826615	−	0.562769i	$-0.190264\pi$
$74$	−23.3310			−2.71218
$75$		0			0
$76$	−14.9631			−1.71638
$77$		−	2.56788i		−	0.292637i
$78$		0			0
$79$	−4.48387			−0.504475			−0.252238	−	0.967665i	$-0.581166\pi$
$79$	−4.48387			−0.504475			−0.252238	+	0.967665i	$0.581166\pi$
$80$	−1.00572	+	4.57747i	−0.112443	+	0.511777i
$81$		0			0
$82$			14.9239i			1.64807i
$83$		−	10.8223i		−	1.18790i	−0.804501	−	0.593951i	$-0.797567\pi$
$83$		−	10.8223i		−	1.18790i	0.804501	−	0.593951i	$-0.202433\pi$
$84$		0			0
$85$	12.6672	+	2.78313i	1.37395	+	0.301873i
$86$	−18.4553			−1.99009
$87$		0			0
$88$		−	4.45196i		−	0.474581i
$89$	5.68414			0.602518			0.301259	−	0.953542i	$-0.402593\pi$
$89$	5.68414			0.602518			0.301259	+	0.953542i	$0.402593\pi$
$90$		0			0
$91$	13.6116			1.42688
$92$			3.66382i			0.381980i
$93$		0			0
$94$	15.2317			1.57103
$95$	−1.95969	+	8.91938i	−0.201060	+	0.915109i
$96$		0			0
$97$			11.0676i			1.12374i	0.827226	+	0.561870i	$0.189918\pi$
$97$			11.0676i			1.12374i	−0.827226	+	0.561870i	$0.810082\pi$
$98$		−	4.24477i		−	0.428787i
$99$		0			0
$100$	16.6319	+	7.67913i	1.66319	+	0.767913i
$101$	−1.60014			−0.159219			−0.0796097	−	0.996826i	$-0.525367\pi$
$101$	−1.60014			−0.159219			−0.0796097	+	0.996826i	$0.525367\pi$
$102$		0			0
$103$		−	1.23218i		−	0.121411i	−0.998156	−	0.0607054i	$-0.980665\pi$
$103$		−	1.23218i		−	0.121411i	0.998156	−	0.0607054i	$-0.0193350\pi$
$104$	23.5985			2.31402
$105$		0			0
$106$	−16.0306			−1.55703
$107$			0.235232i			0.0227408i	0.999935	+	0.0113704i	$0.00361938\pi$
$107$			0.235232i			0.0227408i	−0.999935	+	0.0113704i	$0.996381\pi$
$108$		0			0
$109$	−7.43550			−0.712192			−0.356096	−	0.934449i	$-0.615892\pi$
$109$	−7.43550			−0.712192			−0.356096	+	0.934449i	$0.615892\pi$
$110$	−5.84377	−	1.28394i	−0.557181	−	0.122419i
$111$		0			0
$112$			4.78700i			0.452329i
$113$		−	2.28394i		−	0.214855i	−0.994213	−	0.107428i	$-0.965739\pi$
$113$		−	2.28394i		−	0.214855i	0.994213	−	0.107428i	$-0.0342614\pi$
$114$		0			0
$115$	2.18398	+	0.479844i	0.203657	+	0.0447457i
$116$	1.49580			0.138882
$117$		0			0
$118$			11.3276i			1.04279i
$119$	13.2470			1.21435
$120$		0			0
$121$	−9.73590			−0.885082
$122$			15.0709i			1.36446i
$123$		0			0
$124$	11.6944			1.05019
$125$	6.75573	−	8.90843i	0.604251	−	0.796794i
$126$		0			0
$127$		−	10.2151i		−	0.906444i	−0.891398	−	0.453222i	$-0.850275\pi$
$127$		−	10.2151i		−	0.906444i	0.891398	−	0.453222i	$-0.149725\pi$
$128$		−	20.7159i		−	1.83105i
$129$		0			0
$130$	6.80579	−	30.9761i	0.596907	−	2.71678i
$131$	16.2232			1.41742			0.708712	−	0.705498i	$-0.249276\pi$
$131$	16.2232			1.41742			0.708712	+	0.705498i	$0.249276\pi$
$132$		0			0
$133$			9.32764i			0.808809i
$134$	0.675747			0.0583756
$135$		0			0
$136$	22.9665			1.96936
$137$			4.89715i			0.418392i	0.977874	+	0.209196i	$0.0670846\pi$
$137$			4.89715i			0.418392i	−0.977874	+	0.209196i	$0.932915\pi$
$138$		0			0
$139$	−4.43889			−0.376502			−0.188251	−	0.982121i	$-0.560282\pi$
$139$	−4.43889			−0.376502			−0.188251	+	0.982121i	$0.560282\pi$
$140$	18.2754	+	4.01531i	1.54455	+	0.339356i
$141$		0			0
$142$		−	33.2032i		−	2.78635i
$143$			6.70060i			0.560333i
$144$		0			0
$145$	0.195903	−	0.891636i	0.0162688	−	0.0740463i
$146$	−22.8863			−1.89409
$147$		0			0
$148$			35.9181i			2.95245i
$149$	−2.78361			−0.228042			−0.114021	−	0.993478i	$-0.536373\pi$
$149$	−2.78361			−0.228042			−0.114021	+	0.993478i	$0.536373\pi$
$150$		0			0
$151$	11.9278			0.970669			0.485334	−	0.874329i	$-0.338698\pi$
$151$	11.9278			0.970669			0.485334	+	0.874329i	$0.338698\pi$
$152$			16.1714i			1.31167i
$153$		0			0
$154$	−6.11125			−0.492459
$155$	1.53160	−	6.97097i	0.123021	−	0.559922i
$156$		0			0
$157$			22.9550i			1.83201i	0.401167	+	0.916005i	$0.368605\pi$
$157$			22.9550i			1.83201i	−0.401167	+	0.916005i	$0.631395\pi$
$158$			10.6711i			0.848945i
$159$		0			0
$160$	−6.40190	−	1.40657i	−0.506114	−	0.111199i
$161$	2.28394			0.180000
$162$		0			0
$163$			10.2083i			0.799578i	0.916607	+	0.399789i	$0.130917\pi$
$163$			10.2083i			0.799578i	−0.916607	+	0.399789i	$0.869083\pi$
$164$	22.9753			1.79407
$165$		0			0
$166$	−25.7557			−1.99903
$167$		−	6.84715i		−	0.529849i	−0.964269	−	0.264924i	$-0.914653\pi$
$167$		−	6.84715i		−	0.529849i	0.964269	−	0.264924i	$-0.0853470\pi$
$168$		0			0
$169$	−22.5179			−1.73215
$170$	6.62351	−	30.1464i	0.508000	−	2.31212i
$171$		0			0
$172$			28.4120i			2.16639i
$173$			4.29539i			0.326572i	0.986579	+	0.163286i	$0.0522094\pi$
$173$			4.29539i			0.326572i	−0.986579	+	0.163286i	$0.947791\pi$
$174$		0			0
$175$	4.78700	−	10.3680i	0.361863	−	0.783743i
$176$	−2.35651			−0.177628
$177$		0			0
$178$		−	13.5276i		−	1.01393i
$179$	−14.8626			−1.11088			−0.555442	−	0.831555i	$-0.687451\pi$
$179$	−14.8626			−1.11088			−0.555442	+	0.831555i	$0.687451\pi$
$180$		0			0
$181$	13.1311			0.976027			0.488013	−	0.872836i	$-0.337722\pi$
$181$	13.1311			0.976027			0.488013	+	0.872836i	$0.337722\pi$
$182$		−	32.3939i		−	2.40120i
$183$		0			0
$184$	3.95969			0.291912
$185$	21.4105	+	4.70413i	1.57413	+	0.345855i
$186$		0			0
$187$			6.52114i			0.476873i
$188$		−	23.4492i		−	1.71021i
$189$		0			0
$190$	21.2270	+	4.66382i	1.53997	+	0.338349i
$191$	−13.9819			−1.01170			−0.505848	−	0.862623i	$-0.668820\pi$
$191$	−13.9819			−1.01170			−0.505848	+	0.862623i	$0.668820\pi$
$192$		0			0
$193$		−	8.71267i		−	0.627152i	−0.949563	−	0.313576i	$-0.898473\pi$
$193$		−	8.71267i		−	0.627152i	0.949563	−	0.313576i	$-0.101527\pi$
$194$	26.3394			1.89106
$195$		0			0
$196$	−6.53483			−0.466773
$197$			22.4876i			1.60218i	0.598547	+	0.801088i	$0.295745\pi$
$197$			22.4876i			1.60218i	−0.598547	+	0.801088i	$0.704255\pi$
$198$		0			0
$199$	−9.01078			−0.638757			−0.319379	−	0.947627i	$-0.603474\pi$
$199$	−9.01078			−0.638757			−0.319379	+	0.947627i	$0.603474\pi$
$200$	8.29926	−	17.9750i	0.586846	−	1.27102i
$201$		0			0
$202$			3.80813i			0.267939i
$203$		−	0.932448i		−	0.0654450i
$204$		0			0
$205$	3.00904	−	13.6954i	0.210161	−	0.956530i
$206$	−2.93245			−0.204313
$207$		0			0
$208$		−	12.4911i		−	0.866104i
$209$	−4.59174			−0.317617
$210$		0			0
$211$	5.60014			0.385529			0.192765	−	0.981245i	$-0.438255\pi$
$211$	5.60014			0.385529			0.192765	+	0.981245i	$0.438255\pi$
$212$			24.6791i			1.69497i
$213$		0			0
$214$	0.559824			0.0382688
$215$	16.9362	+	3.72107i	1.15504	+	0.253775i
$216$		0			0
$217$		−	7.29005i		−	0.494881i
$218$			17.6956i			1.19850i
$219$		0			0
$220$	−1.97663	+	8.99647i	−0.133264	+	0.606542i
$221$	−34.5666			−2.32520
$222$		0			0
$223$			7.06517i			0.473119i	0.971617	+	0.236559i	$0.0760198\pi$
$223$			7.06517i			0.473119i	−0.971617	+	0.236559i	$0.923980\pi$
$224$	−6.69493			−0.447324
$225$		0			0
$226$	−5.43550			−0.361564
$227$		−	25.1072i		−	1.66643i	−0.552952	−	0.833213i	$-0.686499\pi$
$227$		−	25.1072i		−	1.66643i	0.552952	−	0.833213i	$-0.313501\pi$
$228$		0			0
$229$	3.20366			0.211704			0.105852	−	0.994382i	$-0.466243\pi$
$229$	3.20366			0.211704			0.105852	+	0.994382i	$0.466243\pi$
$230$	1.14197	−	5.19760i	0.0752993	−	0.342719i
$231$		0			0
$232$		−	1.61659i		−	0.106135i
$233$			18.6388i			1.22107i	0.791989	+	0.610535i	$0.209046\pi$
$233$			18.6388i			1.22107i	−0.791989	+	0.610535i	$0.790954\pi$
$234$		0			0
$235$	−13.9779	−	3.07110i	−0.911817	−	0.200337i
$236$	17.4389			1.13518
$237$		0			0
$238$		−	31.5263i		−	2.04355i
$239$	3.18185			0.205817			0.102908	−	0.994691i	$-0.467185\pi$
$239$	3.18185			0.205817			0.102908	+	0.994691i	$0.467185\pi$
$240$		0			0
$241$	4.79844			0.309095			0.154547	−	0.987985i	$-0.450608\pi$
$241$	4.79844			0.309095			0.154547	+	0.987985i	$0.450608\pi$
$242$			23.1703i			1.48944i
$243$		0			0
$244$	23.2017			1.48534
$245$	−0.855855	+	3.89536i	−0.0546786	+	0.248865i
$246$		0			0
$247$		−	24.3394i		−	1.54868i
$248$		−	12.6388i		−	0.802566i
$249$		0			0
$250$	−21.2010	−	16.0778i	−1.34087	−	1.01685i
$251$	2.24668			0.141809			0.0709045	−	0.997483i	$-0.477411\pi$
$251$	2.24668			0.141809			0.0709045	+	0.997483i	$0.477411\pi$
$252$		0			0
$253$			1.12432i			0.0706855i
$254$	−24.3107			−1.52539
$255$		0			0
$256$	−26.9653			−1.68533
$257$		−	11.0148i		−	0.687086i	−0.939137	−	0.343543i	$-0.888373\pi$
$257$		−	11.0148i		−	0.687086i	0.939137	−	0.343543i	$-0.111627\pi$
$258$		0			0
$259$	22.3905			1.39128
$260$	−47.6876	−	10.4775i	−2.95746	−	0.649787i
$261$		0			0
$262$		−	38.6092i		−	2.38528i
$263$		−	11.0585i		−	0.681898i	−0.940082	−	0.340949i	$-0.889252\pi$
$263$		−	11.0585i		−	0.681898i	0.940082	−	0.340949i	$-0.110748\pi$
$264$		0			0
$265$	14.7110	+	3.23218i	0.903692	+	0.198551i
$266$	22.1986			1.36109
$267$		0			0
$268$		−	1.04031i		−	0.0635471i
$269$	−9.90392			−0.603853			−0.301926	−	0.953331i	$-0.597630\pi$
$269$	−9.90392			−0.603853			−0.301926	+	0.953331i	$0.597630\pi$
$270$		0			0
$271$	4.82655			0.293192			0.146596	−	0.989196i	$-0.453168\pi$
$271$	4.82655			0.293192			0.146596	+	0.989196i	$0.453168\pi$
$272$		−	12.1566i		−	0.737100i
$273$		0			0
$274$	11.6546			0.704081
$275$	5.10385	+	2.35651i	0.307774	+	0.142103i
$276$		0			0
$277$			7.54303i			0.453217i	0.973986	+	0.226608i	$0.0727637\pi$
$277$			7.54303i			0.453217i	−0.973986	+	0.226608i	$0.927236\pi$
$278$			10.5640i			0.633588i
$279$		0			0
$280$	4.33957	−	19.7512i	0.259339	−	1.18036i
$281$	−6.01145			−0.358613			−0.179306	−	0.983793i	$-0.557385\pi$
$281$	−6.01145			−0.358613			−0.179306	+	0.983793i	$0.557385\pi$
$282$		0			0
$283$			15.6388i			0.929631i	0.885407	+	0.464816i	$0.153879\pi$
$283$			15.6388i			0.929631i	−0.885407	+	0.464816i	$0.846121\pi$
$284$	−51.1163			−3.03319
$285$		0			0
$286$	15.9466			0.942943
$287$		−	14.3223i		−	0.845419i
$288$		0			0
$289$	−16.6408			−0.978870
$290$	−2.12198	−	0.466224i	−0.124607	−	0.0273776i
$291$		0			0
$292$			35.2335i			2.06188i
$293$		−	3.21639i		−	0.187904i	−0.995577	−	0.0939518i	$-0.970050\pi$
$293$		−	3.21639i		−	0.187904i	0.995577	−	0.0939518i	$-0.0299499\pi$
$294$		0			0
$295$	2.28394	−	10.3952i	0.132976	−	0.605231i
$296$	38.8186			2.25629
$297$		0			0
$298$			6.62465i			0.383756i
$299$	−5.95969			−0.344658
$300$		0			0
$301$	17.7114			1.02087
$302$		−	28.3867i		−	1.63347i
$303$		0			0
$304$	8.55982			0.490940
$305$	3.03869	−	13.8304i	0.173995	−	0.791924i
$306$		0			0
$307$		−	34.4702i		−	1.96732i	−0.180044	−	0.983659i	$-0.557624\pi$
$307$		−	34.4702i		−	1.96732i	0.180044	−	0.983659i	$-0.442376\pi$
$308$			9.40826i			0.536086i
$309$		0			0
$310$	−16.5901	−	3.64503i	−0.942252	−	0.207024i
$311$	18.7443			1.06289			0.531446	−	0.847092i	$-0.321649\pi$
$311$	18.7443			1.06289			0.531446	+	0.847092i	$0.321649\pi$
$312$		0			0
$313$		−	9.91260i		−	0.560294i	−0.959957	−	0.280147i	$-0.909617\pi$
$313$		−	9.91260i		−	0.560294i	0.959957	−	0.280147i	$-0.0903831\pi$
$314$	54.6301			3.08296
$315$		0			0
$316$	16.4281			0.924153
$317$			9.24024i			0.518984i	0.965745	+	0.259492i	$0.0835551\pi$
$317$			9.24024i			0.518984i	−0.965745	+	0.259492i	$0.916445\pi$
$318$		0			0
$319$	0.459018			0.0257001
$320$	−5.35891	+	24.3907i	−0.299572	+	1.36348i
$321$		0			0
$322$		−	5.43550i		−	0.302909i
$323$		−	23.6875i		−	1.31801i
$324$		0			0
$325$	−12.4911	+	27.0540i	−0.692883	+	1.50069i
$326$	24.2946			1.34555
$327$		0			0
$328$		−	24.8307i		−	1.37105i
$329$	−14.6177			−0.805899
$330$		0			0
$331$	−23.9684			−1.31742			−0.658712	−	0.752395i	$-0.728898\pi$
$331$	−23.9684			−1.31742			−0.658712	+	0.752395i	$0.728898\pi$
$332$			39.6509i			2.17613i
$333$		0			0
$334$	−16.2954			−0.891644
$335$	−0.620122	−	0.136248i	−0.0338809	−	0.00744401i
$336$		0			0
$337$			9.76477i			0.531921i	0.963984	+	0.265960i	$0.0856890\pi$
$337$			9.76477i			0.531921i	−0.963984	+	0.265960i	$0.914311\pi$
$338$			53.5898i			2.91490i
$339$		0			0
$340$	−46.4104	−	10.1969i	−2.51696	−	0.553004i
$341$	3.58869			0.194338
$342$		0			0
$343$			20.0613i			1.08321i
$344$	30.7064			1.65558
$345$		0			0
$346$	10.2225			0.549565
$347$		−	1.38441i		−	0.0743190i	−0.999309	−	0.0371595i	$-0.988169\pi$
$347$		−	1.38441i		−	0.0743190i	0.999309	−	0.0371595i	$-0.0118310\pi$
$348$		0			0
$349$	32.3106			1.72954			0.864772	−	0.502164i	$-0.167463\pi$
$349$	32.3106			1.72954			0.864772	+	0.502164i	$0.167463\pi$
$350$	−24.6745	−	11.3925i	−1.31891	−	0.608953i
$351$		0			0
$352$		−	3.29573i		−	0.175663i
$353$			11.4386i			0.608813i	0.952542	+	0.304406i	$0.0984581\pi$
$353$			11.4386i			0.608813i	−0.952542	+	0.304406i	$0.901542\pi$
$354$		0			0
$355$	−6.69461	+	30.4700i	−0.355313	+	1.61718i
$356$	−20.8257			−1.10376
$357$		0			0
$358$			35.3712i			1.86943i
$359$	9.95568			0.525441			0.262720	−	0.964872i	$-0.415380\pi$
$359$	9.95568			0.525441			0.262720	+	0.964872i	$0.415380\pi$
$360$		0			0
$361$	−2.32087			−0.122151
$362$		−	31.2504i		−	1.64248i
$363$		0			0
$364$	−49.8704			−2.61392
$365$	21.0024	+	4.61447i	1.09932	+	0.241532i
$366$		0			0
$367$			27.6785i			1.44481i	0.691472	+	0.722403i	$0.256963\pi$
$367$			27.6785i			1.44481i	−0.691472	+	0.722403i	$0.743037\pi$
$368$		−	2.09594i		−	0.109258i
$369$		0			0
$370$	11.1953	−	50.9544i	0.582014	−	2.64899i
$371$	15.3844			0.798719
$372$		0			0
$373$		−	28.7356i		−	1.48787i	−0.668251	−	0.743936i	$-0.732957\pi$
$373$		−	28.7356i		−	1.48787i	0.668251	−	0.743936i	$-0.267043\pi$
$374$	15.5195			0.802495
$375$		0			0
$376$	−25.3428			−1.30696
$377$			2.43312i			0.125312i
$378$		0			0
$379$	18.2715			0.938546			0.469273	−	0.883053i	$-0.344516\pi$
$379$	18.2715			0.938546			0.469273	+	0.883053i	$0.344516\pi$
$380$	7.17995	−	32.6790i	0.368323	−	1.67640i
$381$		0			0
$382$			33.2753i			1.70251i
$383$		−	29.0356i		−	1.48365i	−0.670593	−	0.741826i	$-0.733960\pi$
$383$		−	29.0356i		−	1.48365i	0.670593	−	0.741826i	$-0.266040\pi$
$384$		0			0
$385$	5.60819	+	1.23218i	0.285820	+	0.0627979i
$386$	−20.7351			−1.05539
$387$		0			0
$388$		−	40.5495i		−	2.05859i
$389$	2.86423			0.145222			0.0726112	−	0.997360i	$-0.476867\pi$
$389$	2.86423			0.145222			0.0726112	+	0.997360i	$0.476867\pi$
$390$		0			0
$391$	−5.80007			−0.293322
$392$			7.06254i			0.356712i
$393$		0			0
$394$	53.5177			2.69619
$395$	2.15156	−	9.79267i	0.108257	−	0.492723i
$396$		0			0
$397$			16.5592i			0.831080i	0.909575	+	0.415540i	$0.136407\pi$
$397$			16.5592i			0.831080i	−0.909575	+	0.415540i	$0.863593\pi$
$398$			21.4446i			1.07492i
$399$		0			0
$400$	−9.51450	−	4.39295i	−0.475725	−	0.219647i
$401$	3.09479			0.154547			0.0772733	−	0.997010i	$-0.475379\pi$
$401$	3.09479			0.154547			0.0772733	+	0.997010i	$0.475379\pi$
$402$		0			0
$403$			19.0226i			0.947582i
$404$	5.86261			0.291676
$405$		0			0
$406$	−2.21911			−0.110133
$407$			11.0222i			0.546352i
$408$		0			0
$409$	−8.93617			−0.441865			−0.220933	−	0.975289i	$-0.570910\pi$
$409$	−8.93617			−0.441865			−0.220933	+	0.975289i	$0.570910\pi$
$410$	−32.5935	−	7.16115i	−1.60968	−	0.353664i
$411$		0			0
$412$			4.51450i			0.222413i
$413$		−	10.8710i		−	0.534927i
$414$		0			0
$415$	23.6356	+	5.19302i	1.16023	+	0.254915i
$416$	17.4697			0.856520
$417$		0			0
$418$			10.9278i			0.534495i
$419$	24.1237			1.17852			0.589260	−	0.807944i	$-0.299419\pi$
$419$	24.1237			1.17852			0.589260	+	0.807944i	$0.299419\pi$
$420$		0			0
$421$	23.8602			1.16288			0.581438	−	0.813591i	$-0.302490\pi$
$421$	23.8602			1.16288			0.581438	+	0.813591i	$0.302490\pi$
$422$		−	13.3276i		−	0.648779i
$423$		0			0
$424$	26.6721			1.29531
$425$	−12.1566	+	26.3294i	−0.589680	+	1.27716i
$426$		0			0
$427$		−	14.4634i		−	0.699933i
$428$		−	0.861848i		−	0.0416590i
$429$		0			0
$430$	8.85569	−	40.3060i	0.427059	−	1.94373i
$431$	−4.45096			−0.214395			−0.107198	−	0.994238i	$-0.534188\pi$
$431$	−4.45096			−0.214395			−0.107198	+	0.994238i	$0.534188\pi$
$432$		0			0
$433$		−	8.10929i		−	0.389707i	−0.980832	−	0.194854i	$-0.937577\pi$
$433$		−	8.10929i		−	0.389707i	0.980832	−	0.194854i	$-0.0624232\pi$
$434$	−17.3494			−0.832799
$435$		0			0
$436$	27.2423			1.30467
$437$		−	4.08401i		−	0.195365i
$438$		0			0
$439$	−4.47180			−0.213428			−0.106714	−	0.994290i	$-0.534033\pi$
$439$	−4.47180			−0.213428			−0.106714	+	0.994290i	$0.534033\pi$
$440$	9.72297	+	2.13625i	0.463525	+	0.101842i
$441$		0			0
$442$			82.2643i			3.91291i
$443$			9.60047i			0.456132i	0.973646	+	0.228066i	$0.0732402\pi$
$443$			9.60047i			0.456132i	−0.973646	+	0.228066i	$0.926760\pi$
$444$		0			0
$445$	−2.72750	+	12.4140i	−0.129296	+	0.588482i
$446$	16.8142			0.796177
$447$		0			0
$448$			25.5071i			1.20510i
$449$	−6.79944			−0.320886			−0.160443	−	0.987045i	$-0.551292\pi$
$449$	−6.79944			−0.320886			−0.160443	+	0.987045i	$0.551292\pi$
$450$		0			0
$451$	7.05047			0.331994
$452$			8.36795i			0.393595i
$453$		0			0
$454$	−59.7522			−2.80431
$455$	−6.53144	+	29.7274i	−0.306199	+	1.39364i
$456$		0			0
$457$		−	19.5582i		−	0.914894i	−0.889237	−	0.457447i	$-0.848764\pi$
$457$		−	19.5582i		−	0.914894i	0.889237	−	0.457447i	$-0.151236\pi$
$458$		−	7.62431i		−	0.356261i
$459$		0			0
$460$	−8.00169	−	1.75806i	−0.373081	−	0.0819701i
$461$	42.7081			1.98912			0.994558	−	0.104184i	$-0.0332230\pi$
$461$	42.7081			1.98912			0.994558	+	0.104184i	$0.0332230\pi$
$462$		0			0
$463$			7.42209i			0.344934i	0.985015	+	0.172467i	$0.0551738\pi$
$463$			7.42209i			0.344934i	−0.985015	+	0.172467i	$0.944826\pi$
$464$	−0.855693			−0.0397245
$465$		0			0
$466$	44.3581			2.05485
$467$			22.5041i			1.04136i	0.853751	+	0.520682i	$0.174322\pi$
$467$			22.5041i			1.04136i	−0.853751	+	0.520682i	$0.825678\pi$
$468$		0			0
$469$	−0.648506			−0.0299452
$470$	−7.30885	+	33.2657i	−0.337132	+	1.53443i
$471$		0			0
$472$		−	18.8472i		−	0.867510i
$473$			8.71882i			0.400892i
$474$		0			0
$475$	−18.5394	−	8.55982i	−0.850644	−	0.392752i
$476$	−48.5347			−2.22458
$477$		0			0
$478$		−	7.57241i		−	0.346354i
$479$	−10.1667			−0.464530			−0.232265	−	0.972653i	$-0.574614\pi$
$479$	−10.1667			−0.464530			−0.232265	+	0.972653i	$0.574614\pi$
$480$		0			0
$481$	−58.4255			−2.66398
$482$		−	11.4197i		−	0.520153i
$483$		0			0
$484$	35.6706			1.62139
$485$	−24.1713	−	5.31070i	−1.09756	−	0.241147i
$486$		0			0
$487$			34.9917i			1.58562i	0.609467	+	0.792812i	$0.291384\pi$
$487$			34.9917i			1.58562i	−0.609467	+	0.792812i	$0.708616\pi$
$488$		−	25.0753i		−	1.13511i
$489$		0			0
$490$	9.27048	+	2.03683i	0.418798	+	0.0920146i
$491$	2.27087			0.102483			0.0512415	−	0.998686i	$-0.483682\pi$
$491$	2.27087			0.102483			0.0512415	+	0.998686i	$0.483682\pi$
$492$		0			0
$493$			2.36795i			0.106647i
$494$	−57.9249			−2.60616
$495$		0			0
$496$	−6.68996			−0.300388
$497$			31.8647i			1.42933i
$498$		0			0
$499$	−20.9929			−0.939773			−0.469887	−	0.882727i	$-0.655705\pi$
$499$	−20.9929			−0.939773			−0.469887	+	0.882727i	$0.655705\pi$
$500$	−24.7518	+	32.6389i	−1.10693	+	1.45966i
$501$		0			0
$502$		−	5.34682i		−	0.238640i
$503$		−	18.5041i		−	0.825055i	−0.910945	−	0.412528i	$-0.864646\pi$
$503$		−	18.5041i		−	0.825055i	0.910945	−	0.412528i	$-0.135354\pi$
$504$		0			0
$505$	0.767816	−	3.49466i	0.0341674	−	0.155510i
$506$	2.67575			0.118951
$507$		0			0
$508$			37.4263i			1.66052i
$509$	−41.8472			−1.85484			−0.927421	−	0.374019i	$-0.877980\pi$
$509$	−41.8472			−1.85484			−0.927421	+	0.374019i	$0.877980\pi$
$510$		0			0
$511$	21.9637			0.971619
$512$			22.7423i			1.00508i
$513$		0			0
$514$	−26.2140			−1.15625
$515$	2.69106	+	0.591257i	0.118582	+	0.0260539i
$516$		0			0
$517$		−	7.19588i		−	0.316475i
$518$		−	53.2867i		−	2.34128i
$519$		0			0
$520$	−11.3236	+	51.5386i	−0.496573	+	2.26012i
$521$	−34.1103			−1.49440			−0.747199	−	0.664600i	$-0.768602\pi$
$521$	−34.1103			−1.49440			−0.747199	+	0.664600i	$0.768602\pi$
$522$		0			0
$523$		−	36.4755i		−	1.59496i	−0.603343	−	0.797482i	$-0.706165\pi$
$523$		−	36.4755i		−	1.59496i	0.603343	−	0.797482i	$-0.293835\pi$
$524$	−59.4387			−2.59659
$525$		0			0
$526$	−26.3180			−1.14752
$527$			18.5131i			0.806442i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$	7.69220	−	35.0105i	0.334128	−	1.52076i
$531$		0			0
$532$		−	34.1748i		−	1.48167i
$533$			37.3724i			1.61878i
$534$		0			0
$535$	−0.513741	−	0.112875i	−0.0222110	−	0.00488000i
$536$	−1.12432			−0.0485633
$537$		0			0
$538$			23.5701i			1.01618i
$539$	−2.00535			−0.0863765
$540$		0			0
$541$	28.2171			1.21315			0.606573	−	0.795028i	$-0.292544\pi$
$541$	28.2171			1.21315			0.606573	+	0.795028i	$0.292544\pi$
$542$		−	11.4866i		−	0.493392i
$543$		0			0
$544$	17.0018			0.728945
$545$	3.56788	−	16.2390i	0.152831	−	0.695601i
$546$		0			0
$547$			15.1638i			0.648356i	0.945996	+	0.324178i	$0.105088\pi$
$547$			15.1638i			0.648356i	−0.945996	+	0.324178i	$0.894912\pi$
$548$		−	17.9423i		−	0.766456i
$549$		0			0
$550$	5.60819	−	12.1465i	0.239134	−	0.517931i
$551$	−1.66735			−0.0710314
$552$		0			0
$553$		−	10.2409i		−	0.435488i
$554$	17.9515			0.762686
$555$		0			0
$556$	16.2633			0.689717
$557$			9.52222i			0.403469i	0.979440	+	0.201735i	$0.0646579\pi$
$557$			9.52222i			0.403469i	−0.979440	+	0.201735i	$0.935342\pi$
$558$		0			0
$559$	−46.2159			−1.95472
$560$	−10.4547	−	2.29701i	−0.441791	−	0.0970665i
$561$		0			0
$562$			14.3065i			0.603484i
$563$			32.7494i			1.38022i	0.723702	+	0.690112i	$0.242439\pi$
$563$			32.7494i			1.38022i	−0.723702	+	0.690112i	$0.757561\pi$
$564$		0			0
$565$	4.98807	+	1.09594i	0.209850	+	0.0461064i
$566$	37.2185			1.56441
$567$		0			0
$568$			55.2441i			2.31799i
$569$	−26.8926			−1.12740			−0.563698	−	0.825981i	$-0.690622\pi$
$569$	−26.8926			−1.12740			−0.563698	+	0.825981i	$0.690622\pi$
$570$		0			0
$571$	0.920000			0.0385008			0.0192504	−	0.999815i	$-0.493872\pi$
$571$	0.920000			0.0385008			0.0192504	+	0.999815i	$0.493872\pi$
$572$		−	24.5498i		−	1.02648i
$573$		0			0
$574$	−34.0853			−1.42269
$575$	−2.09594	+	4.53950i	−0.0874066	+	0.189310i
$576$		0			0
$577$			42.5888i			1.77300i	0.462732	+	0.886498i	$0.346869\pi$
$577$			42.5888i			1.77300i	−0.462732	+	0.886498i	$0.653131\pi$
$578$			39.6030i			1.64727i
$579$		0			0
$580$	−0.717752	+	3.26679i	−0.0298030	+	0.135646i
$581$	24.7175			1.02545
$582$		0			0
$583$			7.57332i			0.313655i
$584$	38.0787			1.57571
$585$		0			0
$586$	−7.65462			−0.316209
$587$		−	9.51212i		−	0.392607i	−0.980543	−	0.196304i	$-0.937106\pi$
$587$		−	9.51212i		−	0.392607i	0.980543	−	0.196304i	$-0.0628938\pi$
$588$		0			0
$589$	−13.0356			−0.537124
$590$	−24.7393	−	5.43550i	−1.01850	−	0.223776i
$591$		0			0
$592$		−	20.5474i		−	0.844494i
$593$		−	31.0719i		−	1.27597i	−0.770048	−	0.637986i	$-0.779768\pi$
$593$		−	31.0719i		−	1.27597i	0.770048	−	0.637986i	$-0.220232\pi$
$594$		0			0
$595$	−6.35651	+	28.9312i	−0.260591	+	1.18606i
$596$	10.1986			0.417753
$597$		0			0
$598$			14.1833i			0.580000i
$599$	2.04065			0.0833787			0.0416893	−	0.999131i	$-0.486726\pi$
$599$	2.04065			0.0833787			0.0416893	+	0.999131i	$0.486726\pi$
$600$		0			0
$601$	−25.2377			−1.02947			−0.514733	−	0.857351i	$-0.672109\pi$
$601$	−25.2377			−1.02947			−0.514733	+	0.857351i	$0.672109\pi$
$602$		−	42.1509i		−	1.71794i
$603$		0			0
$604$	−43.7012			−1.77818
$605$	4.67172	−	21.2630i	0.189932	−	0.864463i
$606$		0			0
$607$			10.4305i			0.423361i	0.977339	+	0.211680i	$0.0678936\pi$
$607$			10.4305i			0.423361i	−0.977339	+	0.211680i	$0.932106\pi$
$608$			11.9715i			0.485507i
$609$		0			0
$610$	−32.9146	−	7.23170i	−1.33267	−	0.292803i
$611$	38.1432			1.54311
$612$		0			0
$613$			4.97745i			0.201037i	0.994935	+	0.100519i	$0.0320502\pi$
$613$			4.97745i			0.201037i	−0.994935	+	0.100519i	$0.967950\pi$
$614$	−82.0348			−3.31066
$615$		0			0
$616$	10.1680			0.409681
$617$		−	30.5881i		−	1.23143i	−0.787969	−	0.615715i	$-0.788867\pi$
$617$		−	30.5881i		−	1.23143i	0.787969	−	0.615715i	$-0.211133\pi$
$618$		0			0
$619$	48.0404			1.93091			0.965453	−	0.260579i	$-0.0839133\pi$
$619$	48.0404			1.93091			0.965453	+	0.260579i	$0.0839133\pi$
$620$	−5.61151	+	25.5404i	−0.225364	+	1.02573i
$621$		0			0
$622$		−	44.6092i		−	1.78866i
$623$			12.9823i			0.520123i
$624$		0			0
$625$	16.2141	+	19.0290i	0.648564	+	0.761160i
$626$	−23.5908			−0.942878
$627$		0			0
$628$		−	84.1030i		−	3.35608i
$629$	−56.8607			−2.26718
$630$		0			0
$631$	10.6157			0.422605			0.211303	−	0.977421i	$-0.432229\pi$
$631$	10.6157			0.422605			0.211303	+	0.977421i	$0.432229\pi$
$632$		−	17.7547i		−	0.706246i
$633$		0			0
$634$	21.9907			0.873360
$635$	22.3095	+	4.90166i	0.885327	+	0.194516i
$636$		0			0
$637$		−	10.6298i		−	0.421166i
$638$		−	1.09241i		−	0.0432488i
$639$		0			0
$640$	45.2431	+	9.94041i	1.78839	+	0.392929i
$641$	−3.47844			−0.137390			−0.0686951	−	0.997638i	$-0.521884\pi$
$641$	−3.47844			−0.137390			−0.0686951	+	0.997638i	$0.521884\pi$
$642$		0			0
$643$		−	2.05348i		−	0.0809813i	−0.999180	−	0.0404906i	$-0.987108\pi$
$643$		−	2.05348i		−	0.0809813i	0.999180	−	0.0404906i	$-0.0128921\pi$
$644$	−8.36795			−0.329743
$645$		0			0
$646$	−56.3734			−2.21798
$647$			19.3408i			0.760367i	0.924911	+	0.380184i	$0.124139\pi$
$647$			19.3408i			0.760367i	−0.924911	+	0.380184i	$0.875861\pi$
$648$		0			0
$649$	5.35149			0.210064
$650$	64.3852	+	29.7274i	2.52540	+	1.16600i
$651$		0			0
$652$		−	37.4015i		−	1.46476i
$653$			21.9288i			0.858139i	0.903271	+	0.429070i	$0.141159\pi$
$653$			21.9288i			0.858139i	−0.903271	+	0.429070i	$0.858841\pi$
$654$		0			0
$655$	−7.78459	+	35.4310i	−0.304169	+	1.38440i
$656$	−13.1433			−0.513161
$657$		0			0
$658$			34.7883i			1.35619i
$659$	38.1351			1.48553			0.742767	−	0.669550i	$-0.233513\pi$
$659$	38.1351			1.48553			0.742767	+	0.669550i	$0.233513\pi$
$660$		0			0
$661$	−28.9007			−1.12411			−0.562053	−	0.827101i	$-0.689988\pi$
$661$	−28.9007			−1.12411			−0.562053	+	0.827101i	$0.689988\pi$
$662$			57.0419i			2.21700i
$663$		0			0
$664$	42.8529			1.66302
$665$	−20.3713	−	4.47581i	−0.789967	−	0.173565i
$666$		0			0
$667$			0.408263i			0.0158080i
$668$			25.0867i			0.970635i
$669$		0			0
$670$	−0.324253	+	1.47581i	−0.0125270	+	0.0570157i
$671$	7.11993			0.274862
$672$		0			0
$673$		−	49.5051i		−	1.90828i	−0.299361	−	0.954140i	$-0.596774\pi$
$673$		−	49.5051i		−	1.90828i	0.299361	−	0.954140i	$-0.403226\pi$
$674$	23.2390			0.895131
$675$		0			0
$676$	82.5015			3.17313
$677$		−	6.87426i		−	0.264199i	−0.991236	−	0.132100i	$-0.957828\pi$
$677$		−	6.87426i		−	0.264199i	0.991236	−	0.132100i	$-0.0421719\pi$
$678$		0			0
$679$	−25.2776			−0.970067
$680$	−11.0203	+	50.1582i	−0.422610	+	1.92348i
$681$		0			0
$682$		−	8.54064i		−	0.327038i
$683$		−	36.9887i		−	1.41533i	−0.706547	−	0.707666i	$-0.749748\pi$
$683$		−	36.9887i		−	1.41533i	0.706547	−	0.707666i	$-0.250252\pi$
$684$		0			0
$685$	−10.6953	−	2.34987i	−0.408645	−	0.0897839i
$686$	47.7433			1.82285
$687$		0			0
$688$		−	16.2535i		−	0.619657i
$689$	−40.1439			−1.52936
$690$		0			0
$691$	38.5485			1.46645			0.733227	−	0.679984i	$-0.238013\pi$
$691$	38.5485			1.46645			0.733227	+	0.679984i	$0.238013\pi$
$692$		−	15.7375i		−	0.598251i
$693$		0			0
$694$	−3.29472			−0.125066
$695$	2.12998	−	9.69443i	0.0807946	−	0.367731i
$696$		0			0
$697$			36.3715i			1.37767i
$698$		−	76.8952i		−	2.91053i
$699$		0			0
$700$	−17.5387	+	37.9863i	−0.662900	+	1.43575i
$701$	36.3146			1.37158			0.685791	−	0.727798i	$-0.259456\pi$
$701$	36.3146			1.37158			0.685791	+	0.727798i	$0.259456\pi$
$702$		0			0
$703$		−	40.0374i		−	1.51004i
$704$	−12.5564			−0.473239
$705$		0			0
$706$	27.2224			1.02453
$707$		−	3.65462i		−	0.137446i
$708$		0			0
$709$	3.24463			0.121855			0.0609274	−	0.998142i	$-0.480594\pi$
$709$	3.24463			0.121855			0.0609274	+	0.998142i	$0.480594\pi$
$710$	72.5149	+	15.9324i	2.72144	+	0.597931i
$711$		0			0
$712$			22.5074i			0.843502i
$713$			3.19187i			0.119537i
$714$		0			0
$715$	−14.6340	−	3.21525i	−0.547279	−	0.120243i
$716$	54.4539			2.03504
$717$		0			0
$718$		−	23.6933i		−	0.884226i
$719$	10.4214			0.388654			0.194327	−	0.980937i	$-0.437748\pi$
$719$	10.4214			0.388654			0.194327	+	0.980937i	$0.437748\pi$
$720$		0			0
$721$	2.81424			0.104808
$722$			5.52338i			0.205559i
$723$		0			0
$724$	−48.1100			−1.78799
$725$	1.85331	+	0.855693i	0.0688301	+	0.0317796i
$726$		0			0
$727$		−	9.67651i		−	0.358882i	−0.983769	−	0.179441i	$-0.942571\pi$
$727$		−	9.67651i		−	0.358882i	0.983769	−	0.179441i	$-0.0574289\pi$
$728$			53.8976i			1.99758i
$729$		0			0
$730$	10.9819	−	49.9832i	0.406457	−	1.84996i
$731$	−44.9780			−1.66357
$732$		0			0
$733$		−	15.5782i		−	0.575396i	−0.957721	−	0.287698i	$-0.907110\pi$
$733$		−	15.5782i		−	0.575396i	0.957721	−	0.287698i	$-0.0928899\pi$
$734$	65.8715			2.43136
$735$		0			0
$736$	2.93130			0.108049
$737$		−	0.319242i		−	0.0117594i
$738$		0			0
$739$	−14.4328			−0.530919			−0.265459	−	0.964122i	$-0.585524\pi$
$739$	−14.4328			−0.530919			−0.265459	+	0.964122i	$0.585524\pi$
$740$	−78.4442	−	17.2351i	−2.88367	−	0.633575i
$741$		0			0
$742$		−	36.6130i		−	1.34411i
$743$		−	4.50305i		−	0.165201i	−0.996583	−	0.0826005i	$-0.973677\pi$
$743$		−	4.50305i		−	0.165201i	0.996583	−	0.0826005i	$-0.0263226\pi$
$744$		0			0
$745$	1.33570	−	6.07934i	0.0489362	−	0.222730i
$746$	−68.3872			−2.50383
$747$		0			0
$748$		−	23.8923i		−	0.873588i
$749$	−0.537257			−0.0196309
$750$		0			0
$751$	18.5451			0.676721			0.338361	−	0.941017i	$-0.390128\pi$
$751$	18.5451			0.676721			0.338361	+	0.941017i	$0.390128\pi$
$752$			13.4144i			0.489174i
$753$		0			0
$754$	5.79053			0.210878
$755$	−5.72347	+	26.0500i	−0.208299	+	0.948055i
$756$		0			0
$757$		−	27.0366i		−	0.982663i	−0.870973	−	0.491332i	$-0.836510\pi$
$757$		−	27.0366i		−	0.982663i	0.870973	−	0.491332i	$-0.163490\pi$
$758$		−	43.4840i		−	1.57941i
$759$		0			0
$760$	−35.3180	−	7.75976i	−1.28112	−	0.281476i
$761$	−15.4066			−0.558490			−0.279245	−	0.960220i	$-0.590084\pi$
$761$	−15.4066			−0.558490			−0.279245	+	0.960220i	$0.590084\pi$
$762$		0			0
$763$		−	16.9823i		−	0.614799i
$764$	51.2272			1.85334
$765$		0			0
$766$	−69.1013			−2.49673
$767$			28.3667i			1.02426i
$768$		0			0
$769$	−15.1216			−0.545299			−0.272650	−	0.962113i	$-0.587900\pi$
$769$	−15.1216			−0.545299			−0.272650	+	0.962113i	$0.587900\pi$
$770$	2.93245	−	13.3468i	0.105678	−	0.480986i
$771$		0			0
$772$			31.9217i			1.14889i
$773$		−	28.7131i		−	1.03274i	−0.856366	−	0.516370i	$-0.827283\pi$
$773$		−	28.7131i		−	1.03274i	0.856366	−	0.516370i	$-0.172717\pi$
$774$		0			0
$775$	14.4895	+	6.68996i	0.520478	+	0.240311i
$776$	−43.8241			−1.57319
$777$		0			0
$778$		−	6.81653i		−	0.244384i
$779$	−25.6103			−0.917584
$780$		0			0
$781$	−15.6861			−0.561293
$782$			13.8035i			0.493611i
$783$		0			0
$784$	3.73833			0.133512
$785$	−50.1332	−	11.0148i	−1.78933	−	0.393136i
$786$		0			0
$787$		−	16.3131i		−	0.581501i	−0.956799	−	0.290750i	$-0.906095\pi$
$787$		−	16.3131i		−	0.581501i	0.956799	−	0.290750i	$-0.0939049\pi$
$788$		−	82.3905i		−	2.93504i
$789$		0			0
$790$	−23.3054	−	5.12045i	−0.829168	−	0.182178i
$791$	5.21639			0.185473
$792$		0			0
$793$			37.7406i			1.34021i
$794$	39.4088			1.39857
$795$		0			0
$796$	33.0139			1.17015
$797$		−	2.88374i		−	0.102147i	−0.998695	−	0.0510736i	$-0.983736\pi$
$797$		−	2.88374i		−	0.102147i	0.998695	−	0.0510736i	$-0.0162643\pi$
$798$		0			0
$799$	37.1216			1.31327
$800$	6.14383	−	13.3067i	0.217217	−	0.470461i
$801$		0			0
$802$		−	7.36523i		−	0.260075i
$803$			10.8121i			0.381552i
$804$		0			0
$805$	−1.09594	+	4.98807i	−0.0386267	+	0.175806i
$806$	45.2714			1.59462
$807$		0			0
$808$		−	6.33604i		−	0.222901i
$809$	39.7878			1.39886			0.699432	−	0.714699i	$-0.253436\pi$
$809$	39.7878			1.39886			0.699432	+	0.714699i	$0.253436\pi$
$810$		0			0
$811$	38.5454			1.35351			0.676756	−	0.736208i	$-0.263385\pi$
$811$	38.5454			1.35351			0.676756	+	0.736208i	$0.263385\pi$
$812$			3.41632i			0.119889i
$813$		0			0
$814$	26.2316			0.919416
$815$	−22.2947	−	4.89841i	−0.780951	−	0.171584i
$816$		0			0
$817$		−	31.6705i		−	1.10801i
$818$			21.2670i			0.743583i
$819$		0			0
$820$	−11.0246	+	50.1776i	−0.384995	+	1.75228i
$821$	16.3428			0.570368			0.285184	−	0.958473i	$-0.407945\pi$
$821$	16.3428			0.570368			0.285184	+	0.958473i	$0.407945\pi$
$822$		0			0
$823$		−	38.6446i		−	1.34707i	−0.739157	−	0.673534i	$-0.764776\pi$
$823$		−	38.6446i		−	1.34707i	0.739157	−	0.673534i	$-0.235224\pi$
$824$	4.87907			0.169970
$825$		0			0
$826$	−25.8717			−0.900191
$827$		−	21.6969i		−	0.754474i	−0.926117	−	0.377237i	$-0.876874\pi$
$827$		−	21.6969i		−	0.754474i	0.926117	−	0.377237i	$-0.123126\pi$
$828$		0			0
$829$	−13.9429			−0.484259			−0.242129	−	0.970244i	$-0.577846\pi$
$829$	−13.9429			−0.484259			−0.242129	+	0.970244i	$0.577846\pi$
$830$	12.3587	−	56.2499i	0.428978	−	1.95246i
$831$		0			0
$832$		−	66.5579i		−	2.30748i
$833$		−	10.3451i		−	0.358435i
$834$		0			0
$835$	14.9540	+	3.28557i	0.517505	+	0.113702i
$836$	16.8233			0.581846
$837$		0			0
$838$		−	57.4115i		−	1.98325i
$839$	52.4484			1.81072			0.905360	−	0.424644i	$-0.139601\pi$
$839$	52.4484			1.81072			0.905360	+	0.424644i	$0.139601\pi$
$840$		0			0
$841$	−28.8333			−0.994252
$842$		−	56.7844i		−	1.95692i
$843$		0			0
$844$	−20.5179			−0.706255
$845$	10.8051	−	49.1785i	0.371706	−	1.69179i
$846$		0			0
$847$		−	22.2362i		−	0.764046i
$848$		−	14.1180i		−	0.484815i
$849$		0			0
$850$	62.6608	+	28.9312i	2.14925	+	0.992331i
$851$	−9.80345			−0.336058
$852$		0			0
$853$		−	30.4634i		−	1.04305i	−0.853237	−	0.521524i	$-0.825364\pi$
$853$		−	30.4634i		−	1.04305i	0.853237	−	0.521524i	$-0.174636\pi$
$854$	−34.4211			−1.17787
$855$		0			0
$856$	−0.931446			−0.0318362
$857$			42.4911i			1.45147i	0.687975	+	0.725735i	$0.258500\pi$
$857$			42.4911i			1.45147i	−0.687975	+	0.725735i	$0.741500\pi$
$858$		0			0
$859$	15.5856			0.531775			0.265888	−	0.964004i	$-0.414335\pi$
$859$	15.5856			0.531775			0.265888	+	0.964004i	$0.414335\pi$
$860$	−62.0511	−	13.6333i	−2.11592	−	0.464893i
$861$		0			0
$862$			10.5927i			0.360790i
$863$			50.8852i			1.73215i	0.499913	+	0.866076i	$0.333365\pi$
$863$			50.8852i			1.73215i	−0.499913	+	0.866076i	$0.666635\pi$
$864$		0			0
$865$	−9.38102	−	2.06112i	−0.318964	−	0.0700801i
$866$	−19.2991			−0.655811
$867$		0			0
$868$			26.7094i			0.906577i
$869$	5.04131			0.171015
$870$		0			0
$871$	1.69220			0.0573382
$872$		−	29.4423i		−	0.997041i
$873$		0			0
$874$	−9.71944			−0.328765
$875$	20.3463	+	15.4297i	0.687832	+	0.521619i
$876$		0			0
$877$		−	30.1080i		−	1.01667i	−0.861158	−	0.508337i	$-0.830260\pi$
$877$		−	30.1080i		−	1.01667i	0.861158	−	0.508337i	$-0.169740\pi$
$878$			10.6424i			0.359162i
$879$		0			0
$880$	1.13076	−	5.14655i	0.0381178	−	0.173490i
$881$	−20.2245			−0.681379			−0.340690	−	0.940176i	$-0.610661\pi$
$881$	−20.2245			−0.681379			−0.340690	+	0.940176i	$0.610661\pi$
$882$		0			0
$883$			30.6482i			1.03139i	0.856771	+	0.515697i	$0.172467\pi$
$883$			30.6482i			1.03139i	−0.856771	+	0.515697i	$0.827533\pi$
$884$	126.646			4.25956
$885$		0			0
$886$	22.8480			0.767592
$887$		−	14.0064i		−	0.470290i	−0.971960	−	0.235145i	$-0.924443\pi$
$887$		−	14.0064i		−	0.470290i	0.971960	−	0.235145i	$-0.0755565\pi$
$888$		0			0
$889$	23.3307			0.782487
$890$	29.5439	+	6.49113i	0.990313	+	0.217583i
$891$		0			0
$892$		−	25.8855i		−	0.866711i
$893$			26.1385i			0.874691i
$894$		0			0
$895$	7.13174	−	32.4596i	0.238388	−	1.08500i
$896$	47.3139			1.58065
$897$		0			0
$898$			16.1818i			0.539995i
$899$	1.30312			0.0434616
$900$		0			0
$901$	−39.0687			−1.30157
$902$		−	16.7793i		−	0.558688i
$903$		0			0
$904$	9.04370			0.300789
$905$	−6.30088	+	28.6780i	−0.209448	+	0.953289i
$906$		0			0
$907$		−	36.4739i		−	1.21109i	−0.795809	−	0.605547i	$-0.792954\pi$
$907$		−	36.4739i		−	1.21109i	0.795809	−	0.605547i	$-0.207046\pi$
$908$			91.9884i			3.05274i
$909$		0			0
$910$	70.7475	+	15.5440i	2.34526	+	0.515280i
$911$	14.2327			0.471549			0.235775	−	0.971808i	$-0.424237\pi$
$911$	14.2327			0.471549			0.235775	+	0.971808i	$0.424237\pi$
$912$		0			0
$913$			12.1677i			0.402693i
$914$	−46.5461			−1.53961
$915$		0			0
$916$	−11.7376			−0.387822
$917$			37.0528i			1.22359i
$918$		0			0
$919$	−34.3246			−1.13226			−0.566132	−	0.824315i	$-0.691561\pi$
$919$	−34.3246			−1.13226			−0.566132	+	0.824315i	$0.691561\pi$
$920$	−1.90003	+	8.64786i	−0.0626423	+	0.285112i
$921$		0			0
$922$		−	101.640i		−	3.34734i
$923$		−	83.1474i		−	2.73683i
$924$		0			0
$925$	−20.5474	+	44.5028i	−0.675595	+	1.46324i
$926$	17.6637			0.580464
$927$		0			0
$928$		−	1.19674i		−	0.0392850i
$929$	6.14579			0.201637			0.100818	−	0.994905i	$-0.467854\pi$
$929$	6.14579			0.201637			0.100818	+	0.994905i	$0.467854\pi$
$930$		0			0
$931$	7.28428			0.238733
$932$		−	68.2893i		−	2.23689i
$933$		0			0
$934$	53.5569			1.75244
$935$	−14.2420	−	3.12913i	−0.465763	−	0.102334i
$936$		0			0
$937$			18.4581i			0.602998i	0.953466	+	0.301499i	$0.0974871\pi$
$937$			18.4581i			0.602998i	−0.953466	+	0.301499i	$0.902513\pi$
$938$			1.54337i			0.0503927i
$939$		0			0
$940$	51.2125	+	11.2520i	1.67037	+	0.366998i
$941$	−30.7623			−1.00282			−0.501411	−	0.865209i	$-0.667185\pi$
$941$	−30.7623			−1.00282			−0.501411	+	0.865209i	$0.667185\pi$
$942$		0			0
$943$			6.27087i			0.204208i
$944$	−9.97615			−0.324696
$945$		0			0
$946$	20.7497			0.674632
$947$		−	11.4692i		−	0.372698i	−0.982484	−	0.186349i	$-0.940334\pi$
$947$		−	11.4692i		−	0.372698i	0.982484	−	0.186349i	$-0.0596655\pi$
$948$		0			0
$949$	−57.3119			−1.86042
$950$	−20.3713	+	44.1214i	−0.660933	+	1.43149i
$951$		0			0
$952$			52.4541i			1.70005i
$953$			2.40664i			0.0779586i	0.999240	+	0.0389793i	$0.0124106\pi$
$953$			2.40664i			0.0779586i	−0.999240	+	0.0389793i	$0.987589\pi$
$954$		0			0
$955$	6.70914	−	30.5362i	0.217103	−	0.988127i
$956$	−11.6577			−0.377038
$957$		0			0
$958$			24.1956i			0.781724i
$959$	−11.1848			−0.361176
$960$		0			0
$961$	−20.8119			−0.671353
$962$			139.046i			4.48301i
$963$		0			0
$964$	−17.5806			−0.566234
$965$	19.0283	+	4.18073i	0.612541	+	0.134582i
$966$		0			0
$967$			57.4766i			1.84832i	0.382001	+	0.924162i	$0.375235\pi$
$967$			57.4766i			1.84832i	−0.382001	+	0.924162i	$0.624765\pi$
$968$		−	38.5511i		−	1.23908i
$969$		0			0
$970$	−12.6388	+	57.5247i	−0.405808	+	1.84701i
$971$	−53.1909			−1.70698			−0.853489	−	0.521111i	$-0.825518\pi$
$971$	−53.1909			−1.70698			−0.853489	+	0.521111i	$0.825518\pi$
$972$		0			0
$973$		−	10.1382i		−	0.325015i
$974$	83.2759			2.66833
$975$		0			0
$976$	−13.2728			−0.424853
$977$		−	54.4716i		−	1.74270i	−0.490661	−	0.871350i	$-0.663245\pi$
$977$		−	54.4716i		−	1.74270i	0.490661	−	0.871350i	$-0.336755\pi$
$978$		0			0
$979$	−6.39080			−0.204251
$980$	3.13570	−	14.2719i	0.100166	−	0.455899i
$981$		0			0
$982$		−	5.40440i		−	0.172461i
$983$			37.5908i			1.19896i	0.800390	+	0.599480i	$0.204626\pi$
$983$			37.5908i			1.19896i	−0.800390	+	0.599480i	$0.795374\pi$
$984$		0			0
$985$	−49.1124	−	10.7905i	−1.56485	−	0.343815i
$986$	5.63544			0.179469
$987$		0			0
$988$			89.1753i			2.83704i
$989$	−7.75474			−0.246587
$990$		0			0
$991$	14.0545			0.446455			0.223228	−	0.974766i	$-0.428341\pi$
$991$	14.0545			0.446455			0.223228	+	0.974766i	$0.428341\pi$
$992$		−	9.35635i		−	0.297064i
$993$		0			0
$994$	75.8341			2.40531
$995$	4.32377	−	19.6793i	0.137073	−	0.623877i
$996$		0			0
$997$			10.9378i			0.346404i	0.984886	+	0.173202i	$0.0554113\pi$
$997$			10.9378i			0.346404i	−0.984886	+	0.173202i	$0.944589\pi$
$998$			49.9606i			1.58148i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.b.e.829.1		8
3.2	odd	2		115.2.b.b.24.8	yes	8
5.2	odd	4		5175.2.a.bv.1.4		4
5.3	odd	4		5175.2.a.bw.1.1		4
5.4	even	2	inner	1035.2.b.e.829.8		8
12.11	even	2		1840.2.e.d.369.2		8
15.2	even	4		575.2.a.i.1.1		4
15.8	even	4		575.2.a.j.1.4		4
15.14	odd	2		115.2.b.b.24.1	✓	8
60.23	odd	4		9200.2.a.ck.1.4		4
60.47	odd	4		9200.2.a.cq.1.1		4
60.59	even	2		1840.2.e.d.369.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.1	✓	8	15.14	odd	2
115.2.b.b.24.8	yes	8	3.2	odd	2
575.2.a.i.1.1		4	15.2	even	4
575.2.a.j.1.4		4	15.8	even	4
1035.2.b.e.829.1		8	1.1	even	1	trivial
1035.2.b.e.829.8		8	5.4	even	2	inner
1840.2.e.d.369.2		8	12.11	even	2
1840.2.e.d.369.7		8	60.59	even	2
5175.2.a.bv.1.4		4	5.2	odd	4
5175.2.a.bw.1.1		4	5.3	odd	4
9200.2.a.ck.1.4		4	60.23	odd	4
9200.2.a.cq.1.1		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.37988i q^{2} -3.66382 q^{4} +(-0.479844 + 2.18398i) q^{5} +2.28394i q^{7} +3.95969i q^{8} +O(q^{10})\)	\(q-2.37988i q^{2} -3.66382 q^{4} +(-0.479844 + 2.18398i) q^{5} +2.28394i q^{7} +3.95969i q^{8} +(5.19760 + 1.14197i) q^{10} -1.12432 q^{11} -5.95969i q^{13} +5.43550 q^{14} +2.09594 q^{16} -5.80007i q^{17} +4.08401 q^{19} +(1.75806 - 8.00169i) q^{20} +2.67575i q^{22} -1.00000i q^{23} +(-4.53950 - 2.09594i) q^{25} -14.1833 q^{26} -8.36795i q^{28} -0.408263 q^{29} -3.19187 q^{31} +2.93130i q^{32} -13.8035 q^{34} +(-4.98807 - 1.09594i) q^{35} -9.80345i q^{37} -9.71944i q^{38} +(-8.64786 - 1.90003i) q^{40} -6.27087 q^{41} -7.75474i q^{43} +4.11931 q^{44} -2.37988 q^{46} +6.40020i q^{47} +1.78361 q^{49} +(-4.98807 + 10.8035i) q^{50} +21.8352i q^{52} -6.73590i q^{53} +(0.539499 - 2.45549i) q^{55} -9.04370 q^{56} +0.971615i q^{58} -4.75976 q^{59} -6.33265 q^{61} +7.59627i q^{62} +11.1680 q^{64} +(13.0158 + 2.85972i) q^{65} +0.283942i q^{67} +21.2504i q^{68} +(-2.60819 + 11.8710i) q^{70} +13.9516 q^{71} -9.61659i q^{73} -23.3310 q^{74} -14.9631 q^{76} -2.56788i q^{77} -4.48387 q^{79} +(-1.00572 + 4.57747i) q^{80} +14.9239i q^{82} -10.8223i q^{83} +(12.6672 + 2.78313i) q^{85} -18.4553 q^{86} -4.45196i q^{88} +5.68414 q^{89} +13.6116 q^{91} +3.66382i q^{92} +15.2317 q^{94} +(-1.95969 + 8.91938i) q^{95} +11.0676i q^{97} -4.24477i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} + 6 q^{5}+O(q^{10}) \) 8 * q - 4 * q^4 + 6 * q^5	\( 8 q - 4 q^{4} + 6 q^{5} + 6 q^{10} - 4 q^{11} - 8 q^{14} + 4 q^{16} + 8 q^{19} + 8 q^{20} - 16 q^{25} - 12 q^{26} + 8 q^{29} - 28 q^{34} - 28 q^{35} - 10 q^{40} + 16 q^{41} + 12 q^{44} - 28 q^{50} - 16 q^{55} - 28 q^{56} - 16 q^{61} + 40 q^{64} + 14 q^{65} - 28 q^{70} + 48 q^{71} - 72 q^{74} - 36 q^{76} - 48 q^{79} + 2 q^{80} + 12 q^{85} - 28 q^{86} - 16 q^{89} + 52 q^{91} + 84 q^{94} + 4 q^{95}+O(q^{100}) \) 8 * q - 4 * q^4 + 6 * q^5 + 6 * q^10 - 4 * q^11 - 8 * q^14 + 4 * q^16 + 8 * q^19 + 8 * q^20 - 16 * q^25 - 12 * q^26 + 8 * q^29 - 28 * q^34 - 28 * q^35 - 10 * q^40 + 16 * q^41 + 12 * q^44 - 28 * q^50 - 16 * q^55 - 28 * q^56 - 16 * q^61 + 40 * q^64 + 14 * q^65 - 28 * q^70 + 48 * q^71 - 72 * q^74 - 36 * q^76 - 48 * q^79 + 2 * q^80 + 12 * q^85 - 28 * q^86 - 16 * q^89 + 52 * q^91 + 84 * q^94 + 4 * q^95

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