LMFDB - Embedded newform 855.2.c.d.514.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(514,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.514");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.c (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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.16516096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			514.3
Root			$-0.285442i$ of defining polynomial
Character	$\chi$	$=$	855.514
Dual form			855.2.c.d.514.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.906968i q^{2} +1.17741 q^{4} +(0.370556 + 2.20515i) q^{5} -2.59637i q^{7} -2.88181i q^{8} +O(q^{10})$	\(q-0.906968i q^{2} +1.17741 q^{4} +(0.370556 + 2.20515i) q^{5} -2.59637i q^{7} -2.88181i q^{8} +(2.00000 - 0.336083i) q^{10} -0.741113 q^{11} -3.78878i q^{13} -2.35482 q^{14} -0.258887 q^{16} -3.16725i q^{17} +1.00000 q^{19} +(0.436297 + 2.59637i) q^{20} +0.672165i q^{22} -0.570885i q^{23} +(-4.72538 + 1.63427i) q^{25} -3.43630 q^{26} -3.05699i q^{28} +6.00000 q^{29} +5.83705 q^{31} -5.52881i q^{32} -2.87259 q^{34} +(5.72538 - 0.962100i) q^{35} +1.40396i q^{37} -0.906968i q^{38} +(6.35482 - 1.06787i) q^{40} +3.83705 q^{41} -2.59637i q^{43} -0.872594 q^{44} -0.517774 q^{46} +5.08247i q^{47} +0.258887 q^{49} +(1.48223 + 4.28576i) q^{50} -4.46094i q^{52} +0.160905i q^{53} +(-0.274624 - 1.63427i) q^{55} -7.48223 q^{56} -5.44181i q^{58} +8.35482 q^{59} -8.57816 q^{61} -5.29401i q^{62} -5.53223 q^{64} +(8.35482 - 1.40396i) q^{65} +14.8464i q^{67} -3.72915i q^{68} +(-0.872594 - 5.19273i) q^{70} -3.64518 q^{71} -10.8461i q^{73} +1.27334 q^{74} +1.17741 q^{76} +1.92420i q^{77} -1.83705 q^{79} +(-0.0959323 - 0.570885i) q^{80} -3.48008i q^{82} +4.19876i q^{83} +(6.98426 - 1.17365i) q^{85} -2.35482 q^{86} +2.13574i q^{88} -16.9015 q^{89} -9.83705 q^{91} -0.672165i q^{92} +4.60963 q^{94} +(0.370556 + 2.20515i) q^{95} +3.78878i q^{97} -0.234802i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{4} + q^{5}+O(q^{10}) $ 6 * q - 8 * q^4 + q^5	$ 6 q - 8 q^{4} + q^{5} + 12 q^{10} - 2 q^{11} + 16 q^{14} - 4 q^{16} + 6 q^{19} - 10 q^{20} + 3 q^{25} - 8 q^{26} + 36 q^{29} + 8 q^{34} + 3 q^{35} + 8 q^{40} - 12 q^{41} + 20 q^{44} - 8 q^{46} + 4 q^{49} + 4 q^{50} - 33 q^{55} - 40 q^{56} + 20 q^{59} - 14 q^{61} + 12 q^{64} + 20 q^{65} + 20 q^{70} - 52 q^{71} - 40 q^{74} - 8 q^{76} + 24 q^{79} + 32 q^{80} + 13 q^{85} + 16 q^{86} + 24 q^{89} - 24 q^{91} + 48 q^{94} + q^{95}+O(q^{100}) $ 6 * q - 8 * q^4 + q^5 + 12 * q^10 - 2 * q^11 + 16 * q^14 - 4 * q^16 + 6 * q^19 - 10 * q^20 + 3 * q^25 - 8 * q^26 + 36 * q^29 + 8 * q^34 + 3 * q^35 + 8 * q^40 - 12 * q^41 + 20 * q^44 - 8 * q^46 + 4 * q^49 + 4 * q^50 - 33 * q^55 - 40 * q^56 + 20 * q^59 - 14 * q^61 + 12 * q^64 + 20 * q^65 + 20 * q^70 - 52 * q^71 - 40 * q^74 - 8 * q^76 + 24 * q^79 + 32 * q^80 + 13 * q^85 + 16 * q^86 + 24 * q^89 - 24 * q^91 + 48 * q^94 + q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	0.906968i		−	0.641323i	−0.947194	−	0.320661i	$-0.896095\pi$
$2$		−	0.906968i		−	0.641323i	0.947194	−	0.320661i	$-0.103905\pi$
$3$		0			0
$3$		0			0
$4$	1.17741			0.588705
$5$	0.370556	+	2.20515i	0.165718	+	0.986173i
$5$	0.370556	+	2.20515i	0.165718	+	0.986173i
$6$		0			0
$7$		−	2.59637i		−	0.981334i	−0.871347	−	0.490667i	$-0.836753\pi$
$7$		−	2.59637i		−	0.981334i	0.871347	−	0.490667i	$-0.163247\pi$
$8$		−	2.88181i		−	1.01887i
$9$		0			0
$10$	2.00000	−	0.336083i	0.632456	−	0.106279i
$11$	−0.741113			−0.223454			−0.111727	−	0.993739i	$-0.535638\pi$
$11$	−0.741113			−0.223454			−0.111727	+	0.993739i	$0.535638\pi$
$12$		0			0
$13$		−	3.78878i		−	1.05082i	−0.850850	−	0.525409i	$-0.823912\pi$
$13$		−	3.78878i		−	1.05082i	0.850850	−	0.525409i	$-0.176088\pi$
$14$	−2.35482			−0.629352
$15$		0			0
$16$	−0.258887			−0.0647218
$17$		−	3.16725i		−	0.768171i	−0.923298	−	0.384086i	$-0.874517\pi$
$17$		−	3.16725i		−	0.768171i	0.923298	−	0.384086i	$-0.125483\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	0.436297	+	2.59637i	0.0975589	+	0.580565i
$21$		0			0
$22$			0.672165i			0.143306i
$23$		−	0.570885i		−	0.119038i	−0.998227	−	0.0595189i	$-0.981043\pi$
$23$		−	0.570885i		−	0.119038i	0.998227	−	0.0595189i	$-0.0189566\pi$
$24$		0			0
$25$	−4.72538	+	1.63427i	−0.945075	+	0.326853i
$26$	−3.43630			−0.673913
$27$		0			0
$28$		−	3.05699i		−	0.577716i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	5.83705			1.04836			0.524182	−	0.851606i	$-0.324371\pi$
$31$	5.83705			1.04836			0.524182	+	0.851606i	$0.324371\pi$
$32$		−	5.52881i		−	0.977365i
$33$		0			0
$34$	−2.87259			−0.492646
$35$	5.72538	−	0.962100i	0.967765	−	0.162625i
$36$		0			0
$37$			1.40396i			0.230809i	0.993319	+	0.115404i	$0.0368164\pi$
$37$			1.40396i			0.230809i	−0.993319	+	0.115404i	$0.963184\pi$
$38$		−	0.906968i		−	0.147130i
$39$		0			0
$40$	6.35482	−	1.06787i	1.00479	−	0.168845i
$41$	3.83705			0.599246			0.299623	−	0.954058i	$-0.403139\pi$
$41$	3.83705			0.599246			0.299623	+	0.954058i	$0.403139\pi$
$42$		0			0
$43$		−	2.59637i		−	0.395942i	−0.980208	−	0.197971i	$-0.936565\pi$
$43$		−	2.59637i		−	0.395942i	0.980208	−	0.197971i	$-0.0634352\pi$
$44$	−0.872594			−0.131548
$45$		0			0
$46$	−0.517774			−0.0763416
$47$			5.08247i			0.741354i	0.928762	+	0.370677i	$0.120874\pi$
$47$			5.08247i			0.741354i	−0.928762	+	0.370677i	$0.879126\pi$
$48$		0			0
$49$	0.258887			0.0369839
$50$	1.48223	+	4.28576i	0.209618	+	0.606098i
$51$		0			0
$52$		−	4.46094i		−	0.618621i
$53$			0.160905i			0.0221020i	0.999939	+	0.0110510i	$0.00351771\pi$
$53$			0.160905i			0.0221020i	−0.999939	+	0.0110510i	$0.996482\pi$
$54$		0			0
$55$	−0.274624	−	1.63427i	−0.0370303	−	0.220364i
$56$	−7.48223			−0.999854
$57$		0			0
$58$		−	5.44181i		−	0.714544i
$59$	8.35482			1.08770			0.543852	−	0.839181i	$-0.316965\pi$
$59$	8.35482			1.08770			0.543852	+	0.839181i	$0.316965\pi$
$60$		0			0
$61$	−8.57816			−1.09832			−0.549160	−	0.835717i	$-0.685052\pi$
$61$	−8.57816			−1.09832			−0.549160	+	0.835717i	$0.685052\pi$
$62$		−	5.29401i		−	0.672340i
$63$		0			0
$64$	−5.53223			−0.691529
$65$	8.35482	−	1.40396i	1.03629	−	0.174139i
$66$		0			0
$67$			14.8464i			1.81378i	0.421371	+	0.906888i	$0.361549\pi$
$67$			14.8464i			1.81378i	−0.421371	+	0.906888i	$0.638451\pi$
$68$		−	3.72915i		−	0.452226i
$69$		0			0
$70$	−0.872594	−	5.19273i	−0.104295	−	0.620650i
$71$	−3.64518			−0.432603			−0.216302	−	0.976327i	$-0.569399\pi$
$71$	−3.64518			−0.432603			−0.216302	+	0.976327i	$0.569399\pi$
$72$		0			0
$73$		−	10.8461i		−	1.26944i	−0.772743	−	0.634719i	$-0.781116\pi$
$73$		−	10.8461i		−	1.26944i	0.772743	−	0.634719i	$-0.218884\pi$
$74$	1.27334			0.148023
$75$		0			0
$76$	1.17741			0.135058
$77$			1.92420i			0.219283i
$78$		0			0
$79$	−1.83705			−0.206684			−0.103342	−	0.994646i	$-0.532954\pi$
$79$	−1.83705			−0.206684			−0.103342	+	0.994646i	$0.532954\pi$
$80$	−0.0959323	−	0.570885i	−0.0107256	−	0.0638269i
$81$		0			0
$82$		−	3.48008i		−	0.384310i
$83$			4.19876i			0.460873i	0.973087	+	0.230437i	$0.0740154\pi$
$83$			4.19876i			0.460873i	−0.973087	+	0.230437i	$0.925985\pi$
$84$		0			0
$85$	6.98426	−	1.17365i	0.757550	−	0.127300i
$86$	−2.35482			−0.253927
$87$		0			0
$88$			2.13574i			0.227671i
$89$	−16.9015			−1.79156			−0.895778	−	0.444502i	$-0.853381\pi$
$89$	−16.9015			−1.79156			−0.895778	+	0.444502i	$0.853381\pi$
$90$		0			0
$91$	−9.83705			−1.03120
$92$		−	0.672165i		−	0.0700781i
$93$		0			0
$94$	4.60963			0.475447
$95$	0.370556	+	2.20515i	0.0380183	+	0.226244i
$96$		0			0
$97$			3.78878i			0.384692i	0.981327	+	0.192346i	$0.0616096\pi$
$97$			3.78878i			0.384692i	−0.981327	+	0.192346i	$0.938390\pi$
$98$		−	0.234802i		−	0.0237186i
$99$		0			0
$100$	−5.56370	+	1.92420i	−0.556370	+	0.192420i
$101$	−8.35482			−0.831336			−0.415668	−	0.909517i	$-0.636452\pi$
$101$	−8.35482			−0.831336			−0.415668	+	0.909517i	$0.636452\pi$
$102$		0			0
$103$			2.07612i			0.204566i	0.994755	+	0.102283i	$0.0326148\pi$
$103$			2.07612i			0.204566i	−0.994755	+	0.102283i	$0.967385\pi$
$104$	−10.9185			−1.07065
$105$		0			0
$106$	0.145935			0.0141745
$107$		−	5.70399i		−	0.551426i	−0.961240	−	0.275713i	$-0.911086\pi$
$107$		−	5.70399i		−	0.551426i	0.961240	−	0.275713i	$-0.0889139\pi$
$108$		0			0
$109$	−1.64518			−0.157580			−0.0787899	−	0.996891i	$-0.525106\pi$
$109$	−1.64518			−0.157580			−0.0787899	+	0.996891i	$0.525106\pi$
$110$	−1.48223	+	0.249075i	−0.141325	+	0.0237484i
$111$		0			0
$112$			0.672165i			0.0635137i
$113$			3.89006i			0.365946i	0.983118	+	0.182973i	$0.0585720\pi$
$113$			3.89006i			0.365946i	−0.983118	+	0.182973i	$0.941428\pi$
$114$		0			0
$115$	1.25889	−	0.211545i	0.117392	−	0.0197267i
$116$	7.06446			0.655918
$117$		0			0
$118$		−	7.57755i		−	0.697570i
$119$	−8.22334			−0.753832
$120$		0			0
$121$	−10.4508			−0.950068
$122$			7.78011i			0.704378i
$123$		0			0
$124$	6.87259			0.617177
$125$	−5.35482	−	9.81458i	−0.478950	−	0.877842i
$126$		0			0
$127$			14.4233i			1.27986i	0.768432	+	0.639931i	$0.221037\pi$
$127$			14.4233i			1.27986i	−0.768432	+	0.639931i	$0.778963\pi$
$128$		−	6.04007i		−	0.533872i
$129$		0			0
$130$	−1.27334	−	7.57755i	−0.111679	−	0.664595i
$131$	−9.96853			−0.870954			−0.435477	−	0.900200i	$-0.643420\pi$
$131$	−9.96853			−0.870954			−0.435477	+	0.900200i	$0.643420\pi$
$132$		0			0
$133$		−	2.59637i		−	0.225133i
$134$	13.4652			1.16322
$135$		0			0
$136$	−9.12741			−0.782669
$137$		−	9.70431i		−	0.829095i	−0.910028	−	0.414548i	$-0.863940\pi$
$137$		−	9.70431i		−	0.829095i	0.910028	−	0.414548i	$-0.136060\pi$
$138$		0			0
$139$	13.4508			1.14088			0.570439	−	0.821340i	$-0.306773\pi$
$139$	13.4508			1.14088			0.570439	+	0.821340i	$0.306773\pi$
$140$	6.74111	−	1.13279i	0.569728	−	0.0957379i
$141$		0			0
$142$			3.30606i			0.277438i
$143$			2.80791i			0.234809i
$144$		0			0
$145$	2.22334	+	13.2309i	0.184638	+	1.09877i
$146$	−9.83705			−0.814120
$147$		0			0
$148$			1.65303i			0.135878i
$149$	15.0959			1.23671			0.618353	−	0.785900i	$-0.287800\pi$
$149$	15.0959			1.23671			0.618353	+	0.785900i	$0.287800\pi$
$150$		0			0
$151$	14.1919			1.15492			0.577459	−	0.816420i	$-0.304044\pi$
$151$	14.1919			1.15492			0.577459	+	0.816420i	$0.304044\pi$
$152$		−	2.88181i		−	0.233745i
$153$		0			0
$154$	1.74519			0.140631
$155$	2.16295	+	12.8716i	0.173733	+	1.03387i
$156$		0			0
$157$			7.57755i			0.604754i	0.953188	+	0.302377i	$0.0977802\pi$
$157$			7.57755i			0.604754i	−0.953188	+	0.302377i	$0.902220\pi$
$158$			1.66614i			0.132551i
$159$		0			0
$160$	12.1919	−	2.04874i	0.963852	−	0.161967i
$161$	−1.48223			−0.116816
$162$		0			0
$163$			19.6757i			1.54112i	0.637369	+	0.770559i	$0.280023\pi$
$163$			19.6757i			1.54112i	−0.637369	+	0.770559i	$0.719977\pi$
$164$	4.51777			0.352779
$165$		0			0
$166$	3.80814			0.295569
$167$			10.7954i			0.835376i	0.908590	+	0.417688i	$0.137160\pi$
$167$			10.7954i			0.835376i	−0.908590	+	0.417688i	$0.862840\pi$
$168$		0			0
$169$	−1.35482			−0.104217
$170$	−1.06446	−	6.33450i	−0.0816402	−	0.485834i
$171$		0			0
$172$		−	3.05699i		−	0.233093i
$173$			20.3895i			1.55018i	0.631848	+	0.775092i	$0.282297\pi$
$173$			20.3895i			1.55018i	−0.631848	+	0.775092i	$0.717703\pi$
$174$		0			0
$175$	4.24315	+	12.2688i	0.320752	+	0.927434i
$176$	0.191865			0.0144623
$177$		0			0
$178$			15.3291i			1.14897i
$179$	25.0645			1.87341			0.936703	−	0.350126i	$-0.113861\pi$
$179$	25.0645			1.87341			0.936703	+	0.350126i	$0.113861\pi$
$180$		0			0
$181$	−19.4193			−1.44342			−0.721712	−	0.692194i	$-0.756644\pi$
$181$	−19.4193			−1.44342			−0.721712	+	0.692194i	$0.756644\pi$
$182$			8.92188i			0.661334i
$183$		0			0
$184$	−1.64518			−0.121284
$185$	−3.09593	+	0.520245i	−0.227617	+	0.0382491i
$186$		0			0
$187$			2.34729i			0.171651i
$188$			5.98414i			0.436439i
$189$		0			0
$190$	2.00000	−	0.336083i	0.145095	−	0.0243820i
$191$	−11.4508			−0.828547			−0.414274	−	0.910152i	$-0.635964\pi$
$191$	−11.4508			−0.828547			−0.414274	+	0.910152i	$0.635964\pi$
$192$		0			0
$193$		−	3.78878i		−	0.272722i	−0.990659	−	0.136361i	$-0.956459\pi$
$193$		−	3.78878i		−	0.272722i	0.990659	−	0.136361i	$-0.0435407\pi$
$194$	3.43630			0.246712
$195$		0			0
$196$	0.304816			0.0217726
$197$		−	2.28354i		−	0.162695i	−0.996686	−	0.0813477i	$-0.974078\pi$
$197$		−	2.28354i		−	0.162695i	0.996686	−	0.0813477i	$-0.0259224\pi$
$198$		0			0
$199$	19.4508			1.37883			0.689414	−	0.724368i	$-0.257868\pi$
$199$	19.4508			1.37883			0.689414	+	0.724368i	$0.257868\pi$
$200$	4.70964	+	13.6176i	0.333022	+	0.962911i
$201$		0			0
$202$			7.57755i			0.533155i
$203$		−	15.5782i		−	1.09337i
$204$		0			0
$205$	1.42184	+	8.46126i	0.0993057	+	0.590960i
$206$	1.88297			0.131193
$207$		0			0
$208$			0.980865i			0.0680108i
$209$	−0.741113			−0.0512639
$210$		0			0
$211$	11.2274			0.772927			0.386463	−	0.922305i	$-0.373697\pi$
$211$	11.2274			0.772927			0.386463	+	0.922305i	$0.373697\pi$
$212$			0.189451i			0.0130115i
$213$		0			0
$214$	−5.17334			−0.353642
$215$	5.72538	−	0.962100i	0.390467	−	0.0656147i
$216$		0			0
$217$		−	15.1551i		−	1.02880i
$218$			1.49213i			0.101059i
$219$		0			0
$220$	−0.323345	−	1.92420i	−0.0217999	−	0.129730i
$221$	−12.0000			−0.807207
$222$		0			0
$223$			4.03785i			0.270394i	0.990819	+	0.135197i	$0.0431668\pi$
$223$			4.03785i			0.270394i	−0.990819	+	0.135197i	$0.956833\pi$
$224$	−14.3548			−0.959122
$225$		0			0
$226$	3.52815			0.234689
$227$			11.2185i			0.744600i	0.928112	+	0.372300i	$0.121431\pi$
$227$			11.2185i			0.744600i	−0.928112	+	0.372300i	$0.878569\pi$
$228$		0			0
$229$	−16.1315			−1.06600			−0.532999	−	0.846116i	$-0.678935\pi$
$229$	−16.1315			−1.06600			−0.532999	+	0.846116i	$0.678935\pi$
$230$	−0.191865	−	1.14177i	−0.0126512	−	0.0752861i
$231$		0			0
$232$		−	17.2908i		−	1.13520i
$233$		−	2.12676i		−	0.139329i	−0.997570	−	0.0696644i	$-0.977807\pi$
$233$		−	2.12676i		−	0.139329i	0.997570	−	0.0696644i	$-0.0221928\pi$
$234$		0			0
$235$	−11.2076	+	1.88334i	−0.731103	+	0.122856i
$236$	9.83705			0.640337
$237$		0			0
$238$			7.45830i			0.483450i
$239$	−14.4152			−0.932442			−0.466221	−	0.884668i	$-0.654385\pi$
$239$	−14.4152			−0.932442			−0.466221	+	0.884668i	$0.654385\pi$
$240$		0			0
$241$	−0.162955			−0.0104968			−0.00524842	−	0.999986i	$-0.501671\pi$
$241$	−0.162955			−0.0104968			−0.00524842	+	0.999986i	$0.501671\pi$
$242$			9.47849i			0.609301i
$243$		0			0
$244$	−10.1000			−0.646587
$245$	0.0959323	+	0.570885i	0.00612889	+	0.0364725i
$246$		0			0
$247$		−	3.78878i		−	0.241074i
$248$		−	16.8212i		−	1.06815i
$249$		0			0
$250$	−8.90150	+	4.85665i	−0.562980	+	0.307161i
$251$	−12.9330			−0.816322			−0.408161	−	0.912910i	$-0.633830\pi$
$251$	−12.9330			−0.816322			−0.408161	+	0.912910i	$0.633830\pi$
$252$		0			0
$253$			0.423090i			0.0265995i
$254$	13.0815			0.820805
$255$		0			0
$256$	−16.5426			−1.03391
$257$			11.0445i			0.688938i	0.938798	+	0.344469i	$0.111941\pi$
$257$			11.0445i			0.688938i	−0.938798	+	0.344469i	$0.888059\pi$
$258$		0			0
$259$	3.64518			0.226501
$260$	9.83705	−	1.65303i	0.610068	−	0.102517i
$261$		0			0
$262$			9.04113i			0.558563i
$263$			17.8527i			1.10085i	0.834885	+	0.550424i	$0.185534\pi$
$263$			17.8527i			1.10085i	−0.834885	+	0.550424i	$0.814466\pi$
$264$		0			0
$265$	−0.354819	+	0.0596243i	−0.0217964	+	0.00366269i
$266$	−2.35482			−0.144383
$267$		0			0
$268$			17.4803i			1.06778i
$269$	24.9934			1.52387			0.761936	−	0.647652i	$-0.224249\pi$
$269$	24.9934			1.52387			0.761936	+	0.647652i	$0.224249\pi$
$270$		0			0
$271$	−23.8660			−1.44975			−0.724877	−	0.688879i	$-0.758103\pi$
$271$	−23.8660			−1.44975			−0.724877	+	0.688879i	$0.758103\pi$
$272$			0.819960i			0.0497174i
$273$		0			0
$274$	−8.80150			−0.531718
$275$	3.50204	−	1.21118i	0.211181	−	0.0730366i
$276$		0			0
$277$		−	21.2315i		−	1.27568i	−0.770169	−	0.637840i	$-0.779828\pi$
$277$		−	21.2315i		−	1.27568i	0.770169	−	0.637840i	$-0.220172\pi$
$278$		−	12.1994i		−	0.731671i
$279$		0			0
$280$	−2.77259	−	16.4994i	−0.165694	−	0.986030i
$281$	−3.83705			−0.228899			−0.114449	−	0.993429i	$-0.536510\pi$
$281$	−3.83705			−0.228899			−0.114449	+	0.993429i	$0.536510\pi$
$282$		0			0
$283$		−	0.211545i		−	0.0125751i	−0.999980	−	0.00628753i	$-0.997999\pi$
$283$		−	0.211545i		−	0.0125751i	0.999980	−	0.00628753i	$-0.00200139\pi$
$284$	−4.29187			−0.254676
$285$		0			0
$286$	2.54668			0.150589
$287$		−	9.96237i		−	0.588060i
$288$		0			0
$289$	6.96853			0.409913
$290$	12.0000	−	2.01650i	0.704664	−	0.118413i
$291$		0			0
$292$		−	12.7703i		−	0.747324i
$293$		−	14.9942i		−	0.875970i	−0.898982	−	0.437985i	$-0.855692\pi$
$293$		−	14.9942i		−	0.875970i	0.898982	−	0.437985i	$-0.144308\pi$
$294$		0			0
$295$	3.09593	+	18.4236i	0.180252	+	1.07267i
$296$	4.04593			0.235165
$297$		0			0
$298$		−	13.6915i		−	0.793129i
$299$	−2.16295			−0.125087
$300$		0			0
$301$	−6.74111			−0.388551
$302$		−	12.8716i		−	0.740675i
$303$		0			0
$304$	−0.258887			−0.0148482
$305$	−3.17869	−	18.9161i	−0.182011	−	1.08313i
$306$		0			0
$307$			1.65303i			0.0943434i	0.998887	+	0.0471717i	$0.0150208\pi$
$307$			1.65303i			0.0943434i	−0.998887	+	0.0471717i	$0.984979\pi$
$308$			2.26557i			0.129093i
$309$		0			0
$310$	11.6741	−	1.96173i	0.663044	−	0.111419i
$311$	0.741113			0.0420247			0.0210123	−	0.999779i	$-0.493311\pi$
$311$	0.741113			0.0420247			0.0210123	+	0.999779i	$0.493311\pi$
$312$		0			0
$313$			26.8849i			1.51962i	0.650143	+	0.759812i	$0.274709\pi$
$313$			26.8849i			1.51962i	−0.650143	+	0.759812i	$0.725291\pi$
$314$	6.87259			0.387843
$315$		0			0
$316$	−2.16295			−0.121676
$317$			8.16155i			0.458398i	0.973380	+	0.229199i	$0.0736107\pi$
$317$			8.16155i			0.458398i	−0.973380	+	0.229199i	$0.926389\pi$
$318$		0			0
$319$	−4.44668			−0.248966
$320$	−2.05000	−	12.1994i	−0.114599	−	0.681967i
$321$		0			0
$322$			1.34433i			0.0749166i
$323$		−	3.16725i		−	0.176231i
$324$		0			0
$325$	6.19186	+	17.9034i	0.343463	+	0.993101i
$326$	17.8452			0.988354
$327$		0			0
$328$		−	11.0576i		−	0.610555i
$329$	13.1959			0.727516
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			4.94366i			0.271318i
$333$		0			0
$334$	9.79112			0.535746
$335$	−32.7385	+	5.50143i	−1.78870	+	0.300575i
$336$		0			0
$337$		−	9.90275i		−	0.539437i	−0.962939	−	0.269718i	$-0.913069\pi$
$337$		−	9.90275i		−	0.539437i	0.962939	−	0.269718i	$-0.0869306\pi$
$338$			1.22878i			0.0668367i
$339$		0			0
$340$	8.22334	−	1.38186i	0.445973	−	0.0749419i
$341$	−4.32591			−0.234261
$342$		0			0
$343$		−	18.8467i		−	1.01763i
$344$	−7.48223			−0.403415
$345$		0			0
$346$	18.4926			0.994169
$347$			21.2781i			1.14227i	0.820858	+	0.571133i	$0.193496\pi$
$347$			21.2781i			1.14227i	−0.820858	+	0.571133i	$0.806504\pi$
$348$		0			0
$349$	16.4152			0.878686			0.439343	−	0.898319i	$-0.355211\pi$
$349$	16.4152			0.878686			0.439343	+	0.898319i	$0.355211\pi$
$350$	11.1274	−	3.84840i	0.594785	−	0.205706i
$351$		0			0
$352$			4.09748i			0.218396i
$353$			23.8744i			1.27071i	0.772221	+	0.635354i	$0.219146\pi$
$353$			23.8744i			1.27071i	−0.772221	+	0.635354i	$0.780854\pi$
$354$		0			0
$355$	−1.35075	−	8.03817i	−0.0716901	−	0.426622i
$356$	−19.9000			−1.05470
$357$		0			0
$358$		−	22.7327i		−	1.20146i
$359$	2.22334			0.117343			0.0586717	−	0.998277i	$-0.481313\pi$
$359$	2.22334			0.117343			0.0586717	+	0.998277i	$0.481313\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$			17.6127i			0.925701i
$363$		0			0
$364$	−11.5822			−0.607074
$365$	23.9172	−	4.01909i	1.25189	−	0.210369i
$366$		0			0
$367$		−	4.52057i		−	0.235972i	−0.993015	−	0.117986i	$-0.962356\pi$
$367$		−	4.52057i		−	0.235972i	0.993015	−	0.117986i	$-0.0376437\pi$
$368$			0.147795i			0.00770433i
$369$		0			0
$370$	0.471845	+	2.80791i	0.0245301	+	0.145976i
$371$	0.417768			0.0216894
$372$		0			0
$373$		−	15.5186i		−	0.803521i	−0.915745	−	0.401760i	$-0.868398\pi$
$373$		−	15.5186i		−	0.803521i	0.915745	−	0.401760i	$-0.131602\pi$
$374$	2.12892			0.110084
$375$		0			0
$376$	14.6467			0.755345
$377$		−	22.7327i		−	1.17079i
$378$		0			0
$379$	18.9015			0.970905			0.485453	−	0.874263i	$-0.338655\pi$
$379$	18.9015			0.970905			0.485453	+	0.874263i	$0.338655\pi$
$380$	0.436297	+	2.59637i	0.0223816	+	0.133191i
$381$		0			0
$382$			10.3855i			0.531366i
$383$		−	13.7046i		−	0.700274i	−0.936699	−	0.350137i	$-0.886135\pi$
$383$		−	13.7046i		−	0.700274i	0.936699	−	0.350137i	$-0.113865\pi$
$384$		0			0
$385$	−4.24315	+	0.713025i	−0.216251	+	0.0363391i
$386$	−3.43630			−0.174903
$387$		0			0
$388$			4.46094i			0.226470i
$389$	12.7411			0.646000			0.323000	−	0.946399i	$-0.395309\pi$
$389$	12.7411			0.646000			0.323000	+	0.946399i	$0.395309\pi$
$390$		0			0
$391$	−1.80814			−0.0914413
$392$		−	0.746063i		−	0.0376819i
$393$		0			0
$394$	−2.07110			−0.104340
$395$	−0.680729	−	4.05096i	−0.0342512	−	0.203826i
$396$		0			0
$397$		−	38.6522i		−	1.93990i	−0.243306	−	0.969950i	$-0.578232\pi$
$397$		−	38.6522i		−	1.93990i	0.243306	−	0.969950i	$-0.421768\pi$
$398$		−	17.6412i		−	0.884274i
$399$		0			0
$400$	1.22334	−	0.423090i	0.0611669	−	0.0211545i
$401$	−31.8660			−1.59131			−0.795655	−	0.605750i	$-0.792873\pi$
$401$	−31.8660			−1.59131			−0.795655	+	0.605750i	$0.792873\pi$
$402$		0			0
$403$		−	22.1153i		−	1.10164i
$404$	−9.83705			−0.489411
$405$		0			0
$406$	−14.1289			−0.701206
$407$		−	1.04049i		−	0.0515751i
$408$		0			0
$409$	−11.0645			−0.547102			−0.273551	−	0.961857i	$-0.588198\pi$
$409$	−11.0645			−0.547102			−0.273551	+	0.961857i	$0.588198\pi$
$410$	7.67409	−	1.28956i	0.378996	−	0.0636871i
$411$		0			0
$412$			2.44444i			0.120429i
$413$		−	21.6922i		−	1.06740i
$414$		0			0
$415$	−9.25889	+	1.55588i	−0.454501	+	0.0763749i
$416$	−20.9474			−1.02703
$417$		0			0
$418$			0.672165i			0.0328767i
$419$	−25.7452			−1.25773			−0.628867	−	0.777513i	$-0.716481\pi$
$419$	−25.7452			−1.25773			−0.628867	+	0.777513i	$0.716481\pi$
$420$		0			0
$421$	27.4482			1.33774			0.668871	−	0.743378i	$-0.266778\pi$
$421$	27.4482			1.33774			0.668871	+	0.743378i	$0.266778\pi$
$422$		−	10.1829i		−	0.495696i
$423$		0			0
$424$	0.463697			0.0225191
$425$	5.17613	+	14.9664i	0.251079	+	0.725979i
$426$		0			0
$427$			22.2720i			1.07782i
$428$		−	6.71593i		−	0.324627i
$429$		0			0
$430$	−0.872594	−	5.19273i	−0.0420802	−	0.250416i
$431$	−1.74519			−0.0840627			−0.0420314	−	0.999116i	$-0.513383\pi$
$431$	−1.74519			−0.0840627			−0.0420314	+	0.999116i	$0.513383\pi$
$432$		0			0
$433$			18.5208i			0.890052i	0.895518	+	0.445026i	$0.146806\pi$
$433$			18.5208i			0.890052i	−0.895518	+	0.445026i	$0.853194\pi$
$434$	−13.7452			−0.659790
$435$		0			0
$436$	−1.93705			−0.0927679
$437$		−	0.570885i		−	0.0273091i
$438$		0			0
$439$	−29.4482			−1.40549			−0.702743	−	0.711444i	$-0.748041\pi$
$439$	−29.4482			−1.40549			−0.702743	+	0.711444i	$0.748041\pi$
$440$	−4.70964	+	0.791414i	−0.224523	+	0.0377292i
$441$		0			0
$442$			10.8836i			0.517681i
$443$			11.7388i			0.557726i	0.960331	+	0.278863i	$0.0899576\pi$
$443$			11.7388i			0.557726i	−0.960331	+	0.278863i	$0.910042\pi$
$444$		0			0
$445$	−6.26296	−	37.2704i	−0.296893	−	1.76678i
$446$	3.66220			0.173410
$447$		0			0
$448$			14.3637i			0.678620i
$449$	−7.06446			−0.333392			−0.166696	−	0.986008i	$-0.553310\pi$
$449$	−7.06446			−0.333392			−0.166696	+	0.986008i	$0.553310\pi$
$450$		0			0
$451$	−2.84368			−0.133904
$452$			4.58019i			0.215434i
$453$		0			0
$454$	10.1748			0.477529
$455$	−3.64518	−	21.6922i	−0.170889	−	1.01694i
$456$		0			0
$457$		−	34.5000i		−	1.61384i	−0.590660	−	0.806920i	$-0.701133\pi$
$457$		−	34.5000i		−	1.61384i	0.590660	−	0.806920i	$-0.298867\pi$
$458$			14.6307i			0.683649i
$459$		0			0
$460$	1.48223	−	0.249075i	0.0691091	−	0.0116132i
$461$	−8.03147			−0.374063			−0.187032	−	0.982354i	$-0.559887\pi$
$461$	−8.03147			−0.374063			−0.187032	+	0.982354i	$0.559887\pi$
$462$		0			0
$463$			25.3290i			1.17714i	0.808447	+	0.588570i	$0.200309\pi$
$463$			25.3290i			1.17714i	−0.808447	+	0.588570i	$0.799691\pi$
$464$	−1.55332			−0.0721112
$465$		0			0
$466$	−1.92890			−0.0893547
$467$		−	26.8759i		−	1.24367i	−0.783149	−	0.621834i	$-0.786388\pi$
$467$		−	26.8759i		−	1.24367i	0.783149	−	0.621834i	$-0.213612\pi$
$468$		0			0
$469$	38.5467			1.77992
$470$	1.70813	+	10.1649i	0.0787901	+	0.468873i
$471$		0			0
$472$		−	24.0770i		−	1.10823i
$473$			1.92420i			0.0884748i
$474$		0			0
$475$	−4.72538	+	1.63427i	−0.216815	+	0.0749852i
$476$	−9.68224			−0.443785
$477$		0			0
$478$			13.0741i			0.597996i
$479$	−28.9015			−1.32054			−0.660272	−	0.751027i	$-0.729559\pi$
$479$	−28.9015			−1.32054			−0.660272	+	0.751027i	$0.729559\pi$
$480$		0			0
$481$	5.31927			0.242538
$482$			0.147795i			0.00673187i
$483$		0			0
$484$	−12.3048			−0.559310
$485$	−8.35482	+	1.40396i	−0.379373	+	0.0637503i
$486$		0			0
$487$		−	17.7294i		−	0.803395i	−0.915773	−	0.401697i	$-0.868420\pi$
$487$		−	17.7294i		−	0.803395i	0.915773	−	0.401697i	$-0.131580\pi$
$488$			24.7206i			1.11905i
$489$		0			0
$490$	0.517774	−	0.0870075i	0.0233907	−	0.00393060i
$491$	35.1645			1.58695			0.793475	−	0.608603i	$-0.208270\pi$
$491$	35.1645			1.58695			0.793475	+	0.608603i	$0.208270\pi$
$492$		0			0
$493$		−	19.0035i		−	0.855875i
$494$	−3.43630			−0.154606
$495$		0			0
$496$	−1.51114			−0.0678520
$497$			9.46422i			0.424528i
$498$		0			0
$499$	−21.4508			−0.960268			−0.480134	−	0.877195i	$-0.659412\pi$
$499$	−21.4508			−0.960268			−0.480134	+	0.877195i	$0.659412\pi$
$500$	−6.30482	−	11.5558i	−0.281960	−	0.516790i
$501$		0			0
$502$			11.7298i			0.523526i
$503$		−	5.34053i		−	0.238122i	−0.992887	−	0.119061i	$-0.962012\pi$
$503$		−	5.34053i		−	0.238122i	0.992887	−	0.119061i	$-0.0379884\pi$
$504$		0			0
$505$	−3.09593	−	18.4236i	−0.137767	−	0.819841i
$506$	0.383729			0.0170588
$507$		0			0
$508$			16.9821i			0.753461i
$509$	36.1919			1.60418			0.802088	−	0.597206i	$-0.203722\pi$
$509$	36.1919			1.60418			0.802088	+	0.597206i	$0.203722\pi$
$510$		0			0
$511$	−28.1604			−1.24574
$512$			2.92346i			0.129200i
$513$		0			0
$514$	10.0170			0.441832
$515$	−4.57816	+	0.769320i	−0.201738	+	0.0339003i
$516$		0			0
$517$		−	3.76668i		−	0.165658i
$518$		−	3.30606i		−	0.145260i
$519$		0			0
$520$	−4.04593	−	24.0770i	−0.177426	−	1.05585i
$521$	2.77259			0.121469			0.0607346	−	0.998154i	$-0.480656\pi$
$521$	2.77259			0.121469			0.0607346	+	0.998154i	$0.480656\pi$
$522$		0			0
$523$			20.5373i			0.898033i	0.893524	+	0.449016i	$0.148226\pi$
$523$			20.5373i			0.898033i	−0.893524	+	0.449016i	$0.851774\pi$
$524$	−11.7370			−0.512735
$525$		0			0
$526$	16.1919			0.705999
$527$		−	18.4874i		−	0.805323i
$528$		0			0
$529$	22.6741			0.985830
$530$	0.0540773	+	0.321810i	0.00234897	+	0.0139785i
$531$		0			0
$532$		−	3.05699i		−	0.132537i
$533$		−	14.5377i		−	0.629698i
$534$		0			0
$535$	12.5782	−	2.11365i	0.543801	−	0.0913811i
$536$	42.7845			1.84801
$537$		0			0
$538$		−	22.6682i		−	0.977294i
$539$	−0.191865			−0.00826419
$540$		0			0
$541$	35.4797			1.52539			0.762695	−	0.646758i	$-0.223876\pi$
$541$	35.4797			1.52539			0.762695	+	0.646758i	$0.223876\pi$
$542$			21.6456i			0.929760i
$543$		0			0
$544$	−17.5111			−0.750784
$545$	−0.609632	−	3.62787i	−0.0261138	−	0.155401i
$546$		0			0
$547$			43.0756i			1.84178i	0.389822	+	0.920890i	$0.372537\pi$
$547$			43.0756i			1.84178i	−0.389822	+	0.920890i	$0.627463\pi$
$548$		−	11.4260i		−	0.488092i
$549$		0			0
$550$	−1.09850	−	3.17623i	−0.0468401	−	0.135435i
$551$	6.00000			0.255609
$552$		0			0
$553$			4.76964i			0.202826i
$554$	−19.2563			−0.818123
$555$		0			0
$556$	15.8370			0.671640
$557$		−	40.4376i		−	1.71340i	−0.515818	−	0.856698i	$-0.672512\pi$
$557$		−	40.4376i		−	1.71340i	0.515818	−	0.856698i	$-0.327488\pi$
$558$		0			0
$559$	−9.83705			−0.416063
$560$	−1.48223	+	0.249075i	−0.0626355	+	0.0105254i
$561$		0			0
$562$			3.48008i			0.146798i
$563$			19.7173i			0.830986i	0.909596	+	0.415493i	$0.136391\pi$
$563$			19.7173i			0.830986i	−0.909596	+	0.415493i	$0.863609\pi$
$564$		0			0
$565$	−8.57816	+	1.44149i	−0.360886	+	0.0606437i
$566$	−0.191865			−0.00806467
$567$		0			0
$568$			10.5047i			0.440768i
$569$	18.6807			0.783137			0.391568	−	0.920149i	$-0.371933\pi$
$569$	18.6807			0.783137			0.391568	+	0.920149i	$0.371933\pi$
$570$		0			0
$571$	−29.9371			−1.25283			−0.626413	−	0.779491i	$-0.715478\pi$
$571$	−29.9371			−1.25283			−0.626413	+	0.779491i	$0.715478\pi$
$572$			3.30606i			0.138233i
$573$		0			0
$574$	−9.03555			−0.377137
$575$	0.932977	+	2.69765i	0.0389079	+	0.112500i
$576$		0			0
$577$		−	0.156779i		−	0.00652679i	−0.999995	−	0.00326339i	$-0.998961\pi$
$577$		−	0.156779i		−	0.00652679i	0.999995	−	0.00326339i	$-0.00103877\pi$
$578$		−	6.32023i		−	0.262887i
$579$		0			0
$580$	2.61778	+	15.5782i	0.108697	+	0.646849i
$581$	10.9015			0.452271
$582$		0			0
$583$		−	0.119249i		−	0.00493877i
$584$	−31.2563			−1.29340
$585$		0			0
$586$	−13.5993			−0.561780
$587$		−	31.1474i		−	1.28559i	−0.766038	−	0.642795i	$-0.777775\pi$
$587$		−	31.1474i		−	1.28559i	0.766038	−	0.642795i	$-0.222225\pi$
$588$		0			0
$589$	5.83705			0.240511
$590$	16.7096	−	2.80791i	0.687925	−	0.115600i
$591$		0			0
$592$		−	0.363466i		−	0.0149384i
$593$			28.8728i			1.18567i	0.805326	+	0.592833i	$0.201991\pi$
$593$			28.8728i			1.18567i	−0.805326	+	0.592833i	$0.798009\pi$
$594$		0			0
$595$	−3.04721	−	18.1337i	−0.124923	−	0.743409i
$596$	17.7741			0.728055
$597$		0			0
$598$			1.96173i			0.0802211i
$599$	25.3274			1.03485			0.517425	−	0.855728i	$-0.326891\pi$
$599$	25.3274			1.03485			0.517425	+	0.855728i	$0.326891\pi$
$600$		0			0
$601$	−19.8370			−0.809170			−0.404585	−	0.914500i	$-0.632584\pi$
$601$	−19.8370			−0.809170			−0.404585	+	0.914500i	$0.632584\pi$
$602$			6.11397i			0.249187i
$603$		0			0
$604$	16.7096			0.679906
$605$	−3.87259	−	23.0455i	−0.157443	−	0.936932i
$606$		0			0
$607$		−	2.49921i		−	0.101440i	−0.998713	−	0.0507199i	$-0.983848\pi$
$607$		−	2.49921i		−	0.101440i	0.998713	−	0.0507199i	$-0.0161516\pi$
$608$		−	5.52881i		−	0.224223i
$609$		0			0
$610$	−17.1563	+	2.88297i	−0.694639	+	0.116728i
$611$	19.2563			0.779027
$612$		0			0
$613$			0.883711i			0.0356927i	0.999841	+	0.0178464i	$0.00568098\pi$
$613$			0.883711i			0.0356927i	−0.999841	+	0.0178464i	$0.994319\pi$
$614$	1.49925			0.0605046
$615$		0			0
$616$	5.54517			0.223421
$617$			29.4085i			1.18394i	0.805959	+	0.591971i	$0.201650\pi$
$617$			29.4085i			1.18394i	−0.805959	+	0.591971i	$0.798350\pi$
$618$		0			0
$619$	−30.3208			−1.21870			−0.609348	−	0.792903i	$-0.708569\pi$
$619$	−30.3208			−1.21870			−0.609348	+	0.792903i	$0.708569\pi$
$620$	2.54668	+	15.1551i	0.102277	+	0.608644i
$621$		0			0
$622$		−	0.672165i		−	0.0269514i
$623$			43.8825i			1.75811i
$624$		0			0
$625$	19.6584	−	15.4450i	0.786334	−	0.617801i
$626$	24.3837			0.974570
$627$		0			0
$628$			8.92188i			0.356022i
$629$	4.44668			0.177301
$630$		0			0
$631$	17.7767			0.707678			0.353839	−	0.935306i	$-0.384876\pi$
$631$	17.7767			0.707678			0.353839	+	0.935306i	$0.384876\pi$
$632$			5.29401i			0.210584i
$633$		0			0
$634$	7.40226			0.293981
$635$	−31.8056	+	5.34465i	−1.26217	+	0.212096i
$636$		0			0
$637$		−	0.980865i		−	0.0388633i
$638$			4.03299i			0.159668i
$639$		0			0
$640$	13.3193	−	2.23819i	0.526490	−	0.0884722i
$641$	32.6675			1.29029			0.645143	−	0.764062i	$-0.276798\pi$
$641$	32.6675			1.29029			0.645143	+	0.764062i	$0.276798\pi$
$642$		0			0
$643$		−	31.8661i		−	1.25668i	−0.777941	−	0.628338i	$-0.783736\pi$
$643$		−	31.8661i		−	1.25668i	0.777941	−	0.628338i	$-0.216264\pi$
$644$	−1.74519			−0.0687700
$645$		0			0
$646$	−2.87259			−0.113021
$647$		−	21.2601i		−	0.835820i	−0.908488	−	0.417910i	$-0.862763\pi$
$647$		−	21.2601i		−	0.835820i	0.908488	−	0.417910i	$-0.137237\pi$
$648$		0			0
$649$	−6.19186			−0.243052
$650$	16.2378	−	5.61582i	0.636899	−	0.220271i
$651$		0			0
$652$			23.1663i			0.907263i
$653$			12.8340i			0.502234i	0.967957	+	0.251117i	$0.0807980\pi$
$653$			12.8340i			0.502234i	−0.967957	+	0.251117i	$0.919202\pi$
$654$		0			0
$655$	−3.69390	−	21.9821i	−0.144333	−	0.858912i
$656$	−0.993361			−0.0387842
$657$		0			0
$658$		−	11.9683i		−	0.466573i
$659$	20.3548			0.792911			0.396456	−	0.918054i	$-0.370240\pi$
$659$	20.3548			0.792911			0.396456	+	0.918054i	$0.370240\pi$
$660$		0			0
$661$	−30.7385			−1.19559			−0.597795	−	0.801649i	$-0.703957\pi$
$661$	−30.7385			−1.19559			−0.597795	+	0.801649i	$0.703957\pi$
$662$		−	7.25574i		−	0.282002i
$663$		0			0
$664$	12.1000			0.469571
$665$	5.72538	−	0.962100i	0.222021	−	0.0373086i
$666$		0			0
$667$		−	3.42531i		−	0.132629i
$668$			12.7107i			0.491790i
$669$		0			0
$670$	4.98962	+	29.6928i	0.192766	+	1.14713i
$671$	6.35738			0.245424
$672$		0			0
$673$		−	21.2094i		−	0.817564i	−0.912632	−	0.408782i	$-0.865954\pi$
$673$		−	21.2094i		−	0.817564i	0.912632	−	0.408782i	$-0.134046\pi$
$674$	−8.98147			−0.345953
$675$		0			0
$676$	−1.59518			−0.0613530
$677$		−	11.2650i		−	0.432951i	−0.976288	−	0.216475i	$-0.930544\pi$
$677$		−	11.2650i		−	0.432951i	0.976288	−	0.216475i	$-0.0694561\pi$
$678$		0			0
$679$	9.83705			0.377511
$680$	−3.38222	−	20.1273i	−0.129702	−	0.771847i
$681$		0			0
$682$			3.92346i			0.150237i
$683$		−	12.3603i		−	0.472954i	−0.971637	−	0.236477i	$-0.924007\pi$
$683$		−	12.3603i		−	0.472954i	0.971637	−	0.236477i	$-0.0759928\pi$
$684$		0			0
$685$	21.3995	−	3.59600i	0.817632	−	0.137396i
$686$	−17.0934			−0.652628
$687$		0			0
$688$			0.672165i			0.0256261i
$689$	0.609632			0.0232251
$690$		0			0
$691$	22.7493			0.865423			0.432711	−	0.901533i	$-0.357557\pi$
$691$	22.7493			0.865423			0.432711	+	0.901533i	$0.357557\pi$
$692$			24.0068i			0.912601i
$693$		0			0
$694$	19.2985			0.732561
$695$	4.98426	+	29.6609i	0.189064	+	1.12510i
$696$		0			0
$697$		−	12.1529i		−	0.460323i
$698$		−	14.8881i		−	0.563521i
$699$		0			0
$700$	4.99593	+	14.4454i	0.188828	+	0.545985i
$701$	16.0289			0.605404			0.302702	−	0.953085i	$-0.402111\pi$
$701$	16.0289			0.605404			0.302702	+	0.953085i	$0.402111\pi$
$702$		0			0
$703$			1.40396i			0.0529512i
$704$	4.10001			0.154525
$705$		0			0
$706$	21.6533			0.814934
$707$			21.6922i			0.815818i
$708$		0			0
$709$	−31.4193			−1.17998			−0.589988	−	0.807412i	$-0.700867\pi$
$709$	−31.4193			−1.17998			−0.589988	+	0.807412i	$0.700867\pi$
$710$	−7.29036	+	1.22508i	−0.273602	+	0.0459765i
$711$		0			0
$712$			48.7069i			1.82537i
$713$		−	3.33228i		−	0.124795i
$714$		0			0
$715$	−6.19186	+	1.04049i	−0.231563	+	0.0389121i
$716$	29.5111			1.10288
$717$		0			0
$718$		−	2.01650i		−	0.0752550i
$719$	11.2589			0.419886			0.209943	−	0.977714i	$-0.432672\pi$
$719$	11.2589			0.419886			0.209943	+	0.977714i	$0.432672\pi$
$720$		0			0
$721$	5.39037			0.200748
$722$		−	0.906968i		−	0.0337538i
$723$		0			0
$724$	−22.8644			−0.849750
$725$	−28.3523	+	9.80559i	−1.05298	+	0.364171i
$726$		0			0
$727$			48.9829i			1.81668i	0.418237	+	0.908338i	$0.362648\pi$
$727$			48.9829i			1.81668i	−0.418237	+	0.908338i	$0.637352\pi$
$728$			28.3485i			1.05066i
$729$		0			0
$730$	−3.64518	−	21.6922i	−0.134914	−	0.802863i
$731$	−8.22334			−0.304151
$732$		0			0
$733$			35.9260i			1.32696i	0.748195	+	0.663479i	$0.230921\pi$
$733$			35.9260i			1.32696i	−0.748195	+	0.663479i	$0.769079\pi$
$734$	−4.10001			−0.151334
$735$		0			0
$736$	−3.15632			−0.116343
$737$		−	11.0029i		−	0.405296i
$738$		0			0
$739$	−14.3523			−0.527956			−0.263978	−	0.964529i	$-0.585035\pi$
$739$	−14.3523			−0.527956			−0.263978	+	0.964529i	$0.585035\pi$
$740$	−3.64518	+	0.612541i	−0.134000	+	0.0225175i
$741$		0			0
$742$		−	0.378902i		−	0.0139099i
$743$			12.5629i			0.460887i	0.973086	+	0.230443i	$0.0740176\pi$
$743$			12.5629i			0.460887i	−0.973086	+	0.230443i	$0.925982\pi$
$744$		0			0
$745$	5.59390	+	33.2888i	0.204944	+	1.21961i
$746$	−14.0748			−0.515316
$747$		0			0
$748$			2.76372i			0.101052i
$749$	−14.8096			−0.541133
$750$		0			0
$751$	26.4548			0.965350			0.482675	−	0.875799i	$-0.339665\pi$
$751$	26.4548			0.965350			0.482675	+	0.875799i	$0.339665\pi$
$752$		−	1.31578i		−	0.0479817i
$753$		0			0
$754$	−20.6178			−0.750855
$755$	5.25889	+	31.2952i	0.191390	+	1.13895i
$756$		0			0
$757$		−	15.7350i		−	0.571897i	−0.958245	−	0.285949i	$-0.907691\pi$
$757$		−	15.7350i		−	0.571897i	0.958245	−	0.285949i	$-0.0923087\pi$
$758$		−	17.1431i		−	0.622664i
$759$		0			0
$760$	6.35482	−	1.06787i	0.230514	−	0.0387358i
$761$	−16.9619			−0.614868			−0.307434	−	0.951569i	$-0.599470\pi$
$761$	−16.9619			−0.614868			−0.307434	+	0.951569i	$0.599470\pi$
$762$		0			0
$763$			4.27149i			0.154638i
$764$	−13.4822			−0.487770
$765$		0			0
$766$	−12.4297			−0.449102
$767$		−	31.6545i		−	1.14298i
$768$		0			0
$769$	−41.9974			−1.51447			−0.757233	−	0.653145i	$-0.773449\pi$
$769$	−41.9974			−1.51447			−0.757233	+	0.653145i	$0.773449\pi$
$770$	0.646690	+	3.84840i	0.0233051	+	0.138687i
$771$		0			0
$772$		−	4.46094i		−	0.160553i
$773$		−	40.9579i		−	1.47315i	−0.676355	−	0.736576i	$-0.736441\pi$
$773$		−	40.9579i		−	1.47315i	0.676355	−	0.736576i	$-0.263559\pi$
$774$		0			0
$775$	−27.5822	+	9.53928i	−0.990783	+	0.342661i
$776$	10.9185			0.391952
$777$		0			0
$778$		−	11.5558i		−	0.414295i
$779$	3.83705			0.137476
$780$		0			0
$781$	2.70149			0.0966669
$782$			1.63992i			0.0586434i
$783$		0			0
$784$	−0.0670225			−0.00239366
$785$	−16.7096	+	2.80791i	−0.596393	+	0.100219i
$786$		0			0
$787$		−	28.5379i		−	1.01727i	−0.860983	−	0.508634i	$-0.830151\pi$
$787$		−	28.5379i		−	1.01727i	0.860983	−	0.508634i	$-0.169849\pi$
$788$		−	2.68866i		−	0.0957796i
$789$		0			0
$790$	−3.67409	+	0.617399i	−0.130718	+	0.0219661i
$791$	10.1000			0.359115
$792$		0			0
$793$			32.5007i			1.15413i
$794$	−35.0563			−1.24410
$795$		0			0
$796$	22.9015			0.811722
$797$			33.2790i			1.17880i	0.807840	+	0.589402i	$0.200636\pi$
$797$			33.2790i			1.17880i	−0.807840	+	0.589402i	$0.799364\pi$
$798$		0			0
$799$	16.0974			0.569487
$800$	9.03555	+	26.1257i	0.319455	+	0.923684i
$801$		0			0
$802$			28.9014i			1.02054i
$803$			8.03817i			0.283661i
$804$		0			0
$805$	−0.549248	−	3.26853i	−0.0193585	−	0.115201i
$806$	−20.0578			−0.706507
$807$		0			0
$808$			24.0770i			0.847025i
$809$	−34.4234			−1.21026			−0.605130	−	0.796126i	$-0.706879\pi$
$809$	−34.4234			−1.21026			−0.605130	+	0.796126i	$0.706879\pi$
$810$		0			0
$811$	−11.2563			−0.395263			−0.197631	−	0.980276i	$-0.563325\pi$
$811$	−11.2563			−0.395263			−0.197631	+	0.980276i	$0.563325\pi$
$812$		−	18.3419i		−	0.643675i
$813$		0			0
$814$	−0.943690			−0.0330763
$815$	−43.3878	+	7.29095i	−1.51981	+	0.255391i
$816$		0			0
$817$		−	2.59637i		−	0.0908353i
$818$			10.0351i			0.350869i
$819$		0			0
$820$	1.67409	+	9.96237i	0.0584618	+	0.347901i
$821$	31.3167			1.09296			0.546480	−	0.837472i	$-0.315967\pi$
$821$	31.3167			1.09296			0.546480	+	0.837472i	$0.315967\pi$
$822$		0			0
$823$		−	13.4800i		−	0.469882i	−0.972010	−	0.234941i	$-0.924510\pi$
$823$		−	13.4800i		−	0.469882i	0.972010	−	0.234941i	$-0.0754898\pi$
$824$	5.98298			0.208427
$825$		0			0
$826$	−19.6741			−0.684549
$827$			15.9882i			0.555963i	0.960586	+	0.277982i	$0.0896654\pi$
$827$			15.9882i			0.555963i	−0.960586	+	0.277982i	$0.910335\pi$
$828$		0			0
$829$	48.4837			1.68391			0.841955	−	0.539548i	$-0.181405\pi$
$829$	48.4837			1.68391			0.841955	+	0.539548i	$0.181405\pi$
$830$	1.41113	+	8.39751i	0.0489810	+	0.291482i
$831$		0			0
$832$			20.9604i			0.726670i
$833$		−	0.819960i		−	0.0284099i
$834$		0			0
$835$	−23.8056	+	4.00032i	−0.823826	+	0.138437i
$836$	−0.872594			−0.0301793
$837$		0			0
$838$			23.3501i			0.806614i
$839$	−18.9934			−0.655724			−0.327862	−	0.944726i	$-0.606328\pi$
$839$	−18.9934			−0.655724			−0.327862	+	0.944726i	$0.606328\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	24.8946i		−	0.857925i
$843$		0			0
$844$	13.2193			0.455026
$845$	−0.502037	−	2.98758i	−0.0172706	−	0.102776i
$846$		0			0
$847$			27.1340i			0.932334i
$848$		−	0.0416562i		−	0.00143048i
$849$		0			0
$850$	13.5741	−	4.69458i	0.465587	−	0.161023i
$851$	0.801497			0.0274750
$852$		0			0
$853$		−	44.1210i		−	1.51067i	−0.655337	−	0.755337i	$-0.727473\pi$
$853$		−	44.1210i		−	1.51067i	0.655337	−	0.755337i	$-0.272527\pi$
$854$	20.2000			0.691230
$855$		0			0
$856$	−16.4378			−0.561833
$857$		−	32.4149i		−	1.10727i	−0.832759	−	0.553635i	$-0.813240\pi$
$857$		−	32.4149i		−	1.10727i	0.832759	−	0.553635i	$-0.186760\pi$
$858$		0			0
$859$	−6.29444			−0.214763			−0.107382	−	0.994218i	$-0.534247\pi$
$859$	−6.29444			−0.214763			−0.107382	+	0.994218i	$0.534247\pi$
$860$	6.74111	−	1.13279i	0.229870	−	0.0386277i
$861$		0			0
$862$			1.58283i			0.0539113i
$863$		−	17.4338i		−	0.593453i	−0.954963	−	0.296726i	$-0.904105\pi$
$863$		−	17.4338i		−	0.593453i	0.954963	−	0.296726i	$-0.0958950\pi$
$864$		0			0
$865$	−44.9619	+	7.55546i	−1.52875	+	0.256893i
$866$	16.7978			0.570811
$867$		0			0
$868$		−	17.8438i		−	0.605657i
$869$	1.36146			0.0461843
$870$		0			0
$871$	56.2497			1.90595
$872$			4.74109i			0.160554i
$873$		0			0
$874$	−0.517774			−0.0175140
$875$	−25.4822	+	13.9031i	−0.861456	+	0.470009i
$876$		0			0
$877$			23.2904i			0.786462i	0.919440	+	0.393231i	$0.128643\pi$
$877$			23.2904i			0.786462i	−0.919440	+	0.393231i	$0.871357\pi$
$878$			26.7086i			0.901370i
$879$		0			0
$880$	0.0710967	+	0.423090i	0.00239667	+	0.0142624i
$881$	−28.9619			−0.975751			−0.487875	−	0.872913i	$-0.662228\pi$
$881$	−28.9619			−0.975751			−0.487875	+	0.872913i	$0.662228\pi$
$882$		0			0
$883$			37.3627i			1.25735i	0.777667	+	0.628677i	$0.216403\pi$
$883$			37.3627i			1.25735i	−0.777667	+	0.628677i	$0.783597\pi$
$884$	−14.1289			−0.475207
$885$		0			0
$886$	10.6467			0.357683
$887$			23.0234i			0.773050i	0.922279	+	0.386525i	$0.126325\pi$
$887$			23.0234i			0.773050i	−0.922279	+	0.386525i	$0.873675\pi$
$888$		0			0
$889$	37.4482			1.25597
$890$	−33.8030	+	5.68030i	−1.13308	+	0.190404i
$891$		0			0
$892$			4.75420i			0.159183i
$893$			5.08247i			0.170078i
$894$		0			0
$895$	9.28780	+	55.2709i	0.310457	+	1.84750i
$896$	−15.6822			−0.523907
$897$		0			0
$898$			6.40723i			0.213812i
$899$	35.0223			1.16806
$900$		0			0
$901$	0.509626			0.0169781
$902$			2.57913i			0.0858756i
$903$		0			0
$904$	11.2104			0.372852
$905$	−7.19594	−	42.8224i	−0.239201	−	1.42347i
$906$		0			0
$907$		−	35.2693i		−	1.17110i	−0.810637	−	0.585549i	$-0.800879\pi$
$907$		−	35.2693i		−	1.17110i	0.810637	−	0.585549i	$-0.199121\pi$
$908$			13.2088i			0.438350i
$909$		0			0
$910$	−19.6741	+	3.30606i	−0.652190	+	0.109595i
$911$	4.97260			0.164750			0.0823748	−	0.996601i	$-0.473750\pi$
$911$	4.97260			0.164750			0.0823748	+	0.996601i	$0.473750\pi$
$912$		0			0
$913$		−	3.11175i		−	0.102984i
$914$	−31.2904			−1.03499
$915$		0			0
$916$	−18.9934			−0.627558
$917$			25.8819i			0.854697i
$918$		0			0
$919$	48.7096			1.60678			0.803391	−	0.595451i	$-0.203027\pi$
$919$	48.7096			1.60678			0.803391	+	0.595451i	$0.203027\pi$
$920$	−0.609632	−	3.62787i	−0.0200990	−	0.119607i
$921$		0			0
$922$			7.28429i			0.239895i
$923$			13.8108i			0.454587i
$924$		0			0
$925$	−2.29444	−	6.63422i	−0.0754406	−	0.218132i
$926$	22.9726			0.754926
$927$		0			0
$928$		−	33.1729i		−	1.08895i
$929$	32.5126			1.06671			0.533353	−	0.845893i	$-0.320932\pi$
$929$	32.5126			1.06671			0.533353	+	0.845893i	$0.320932\pi$
$930$		0			0
$931$	0.258887			0.00848468
$932$		−	2.50407i		−	0.0820235i
$933$		0			0
$934$	−24.3756			−0.797593
$935$	−5.17613	+	0.869804i	−0.169277	+	0.0284456i
$936$		0			0
$937$		−	0.385560i		−	0.0125957i	−0.999980	−	0.00629785i	$-0.997995\pi$
$937$		−	0.385560i		−	0.0125957i	0.999980	−	0.00629785i	$-0.00200468\pi$
$938$		−	34.9606i		−	1.14150i
$939$		0			0
$940$	−13.1959	+	2.21746i	−0.430404	+	0.0723257i
$941$	−45.3482			−1.47831			−0.739154	−	0.673536i	$-0.764775\pi$
$941$	−45.3482			−1.47831			−0.739154	+	0.673536i	$0.764775\pi$
$942$		0			0
$943$		−	2.19051i		−	0.0713329i
$944$	−2.16295			−0.0703982
$945$		0			0
$946$	1.74519			0.0567409
$947$		−	9.61202i		−	0.312349i	−0.987730	−	0.156174i	$-0.950084\pi$
$947$		−	9.61202i		−	0.312349i	0.987730	−	0.156174i	$-0.0499161\pi$
$948$		0			0
$949$	−41.0934			−1.33395
$950$	1.48223	+	4.28576i	0.0480898	+	0.139049i
$951$		0			0
$952$			23.6981i			0.768059i
$953$		−	59.8421i		−	1.93848i	−0.246126	−	0.969238i	$-0.579158\pi$
$953$		−	59.8421i		−	1.93848i	0.246126	−	0.969238i	$-0.420842\pi$
$954$		0			0
$955$	−4.24315	−	25.2506i	−0.137305	−	0.817091i
$956$	−16.9726			−0.548933
$957$		0			0
$958$			26.2127i			0.846895i
$959$	−25.1959			−0.813619
$960$		0			0
$961$	3.07110			0.0990676
$962$		−	4.82441i		−	0.155545i
$963$		0			0
$964$	−0.191865			−0.00617954
$965$	8.35482	−	1.40396i	0.268951	−	0.0451949i
$966$		0			0
$967$		−	0.368324i		−	0.0118445i	−0.999982	−	0.00592225i	$-0.998115\pi$
$967$		−	0.368324i		−	0.0118445i	0.999982	−	0.00592225i	$-0.00188512\pi$
$968$			30.1171i			0.967999i
$969$		0			0
$970$	1.27334	+	7.57755i	0.0408845	+	0.243300i
$971$	45.8370			1.47098			0.735490	−	0.677535i	$-0.236952\pi$
$971$	45.8370			1.47098			0.735490	+	0.677535i	$0.236952\pi$
$972$		0			0
$973$		−	34.9231i		−	1.11958i
$974$	−16.0800			−0.515235
$975$		0			0
$976$	2.22077			0.0710853
$977$		−	29.0337i		−	0.928872i	−0.885607	−	0.464436i	$-0.846257\pi$
$977$		−	29.0337i		−	0.928872i	0.885607	−	0.464436i	$-0.153743\pi$
$978$		0			0
$979$	12.5259			0.400330
$980$	0.112952	+	0.672165i	0.00360811	+	0.0214715i
$981$		0			0
$982$		−	31.8930i		−	1.01775i
$983$		−	11.0160i		−	0.351355i	−0.984448	−	0.175677i	$-0.943788\pi$
$983$		−	11.0160i		−	0.351355i	0.984448	−	0.175677i	$-0.0562116\pi$
$984$		0			0
$985$	5.03555	−	0.846180i	0.160446	−	0.0269615i
$986$	−17.2356			−0.548892
$987$		0			0
$988$		−	4.46094i		−	0.141921i
$989$	−1.48223			−0.0471320
$990$		0			0
$991$	−25.6822			−0.815823			−0.407912	−	0.913021i	$-0.633743\pi$
$991$	−25.6822			−0.815823			−0.407912	+	0.913021i	$0.633743\pi$
$992$		−	32.2719i		−	1.02463i
$993$		0			0
$994$	8.58374			0.272260
$995$	7.20760	+	42.8918i	0.228496	+	1.35976i
$996$		0			0
$997$			20.3854i			0.645611i	0.946465	+	0.322805i	$0.104626\pi$
$997$			20.3854i			0.645611i	−0.946465	+	0.322805i	$0.895374\pi$
$998$			19.4551i			0.615842i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.c.d.514.3		6
3.2	odd	2		95.2.b.b.39.4	yes	6
5.2	odd	4		4275.2.a.br.1.4		6
5.3	odd	4		4275.2.a.br.1.3		6
5.4	even	2	inner	855.2.c.d.514.4		6
12.11	even	2		1520.2.d.h.609.1		6
15.2	even	4		475.2.a.j.1.3		6
15.8	even	4		475.2.a.j.1.4		6
15.14	odd	2		95.2.b.b.39.3	✓	6
57.56	even	2		1805.2.b.e.1084.3		6
60.23	odd	4		7600.2.a.ck.1.6		6
60.47	odd	4		7600.2.a.ck.1.1		6
60.59	even	2		1520.2.d.h.609.6		6
285.113	odd	4		9025.2.a.bx.1.3		6
285.227	odd	4		9025.2.a.bx.1.4		6
285.284	even	2		1805.2.b.e.1084.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.3	✓	6	15.14	odd	2
95.2.b.b.39.4	yes	6	3.2	odd	2
475.2.a.j.1.3		6	15.2	even	4
475.2.a.j.1.4		6	15.8	even	4
855.2.c.d.514.3		6	1.1	even	1	trivial
855.2.c.d.514.4		6	5.4	even	2	inner
1520.2.d.h.609.1		6	12.11	even	2
1520.2.d.h.609.6		6	60.59	even	2
1805.2.b.e.1084.3		6	57.56	even	2
1805.2.b.e.1084.4		6	285.284	even	2
4275.2.a.br.1.3		6	5.3	odd	4
4275.2.a.br.1.4		6	5.2	odd	4
7600.2.a.ck.1.1		6	60.47	odd	4
7600.2.a.ck.1.6		6	60.23	odd	4
9025.2.a.bx.1.3		6	285.113	odd	4
9025.2.a.bx.1.4		6	285.227	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.906968i q^{2} +1.17741 q^{4} +(0.370556 + 2.20515i) q^{5} -2.59637i q^{7} -2.88181i q^{8} +O(q^{10})\)	\(q-0.906968i q^{2} +1.17741 q^{4} +(0.370556 + 2.20515i) q^{5} -2.59637i q^{7} -2.88181i q^{8} +(2.00000 - 0.336083i) q^{10} -0.741113 q^{11} -3.78878i q^{13} -2.35482 q^{14} -0.258887 q^{16} -3.16725i q^{17} +1.00000 q^{19} +(0.436297 + 2.59637i) q^{20} +0.672165i q^{22} -0.570885i q^{23} +(-4.72538 + 1.63427i) q^{25} -3.43630 q^{26} -3.05699i q^{28} +6.00000 q^{29} +5.83705 q^{31} -5.52881i q^{32} -2.87259 q^{34} +(5.72538 - 0.962100i) q^{35} +1.40396i q^{37} -0.906968i q^{38} +(6.35482 - 1.06787i) q^{40} +3.83705 q^{41} -2.59637i q^{43} -0.872594 q^{44} -0.517774 q^{46} +5.08247i q^{47} +0.258887 q^{49} +(1.48223 + 4.28576i) q^{50} -4.46094i q^{52} +0.160905i q^{53} +(-0.274624 - 1.63427i) q^{55} -7.48223 q^{56} -5.44181i q^{58} +8.35482 q^{59} -8.57816 q^{61} -5.29401i q^{62} -5.53223 q^{64} +(8.35482 - 1.40396i) q^{65} +14.8464i q^{67} -3.72915i q^{68} +(-0.872594 - 5.19273i) q^{70} -3.64518 q^{71} -10.8461i q^{73} +1.27334 q^{74} +1.17741 q^{76} +1.92420i q^{77} -1.83705 q^{79} +(-0.0959323 - 0.570885i) q^{80} -3.48008i q^{82} +4.19876i q^{83} +(6.98426 - 1.17365i) q^{85} -2.35482 q^{86} +2.13574i q^{88} -16.9015 q^{89} -9.83705 q^{91} -0.672165i q^{92} +4.60963 q^{94} +(0.370556 + 2.20515i) q^{95} +3.78878i q^{97} -0.234802i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} + q^{5}+O(q^{10}) \) 6 * q - 8 * q^4 + q^5	\( 6 q - 8 q^{4} + q^{5} + 12 q^{10} - 2 q^{11} + 16 q^{14} - 4 q^{16} + 6 q^{19} - 10 q^{20} + 3 q^{25} - 8 q^{26} + 36 q^{29} + 8 q^{34} + 3 q^{35} + 8 q^{40} - 12 q^{41} + 20 q^{44} - 8 q^{46} + 4 q^{49} + 4 q^{50} - 33 q^{55} - 40 q^{56} + 20 q^{59} - 14 q^{61} + 12 q^{64} + 20 q^{65} + 20 q^{70} - 52 q^{71} - 40 q^{74} - 8 q^{76} + 24 q^{79} + 32 q^{80} + 13 q^{85} + 16 q^{86} + 24 q^{89} - 24 q^{91} + 48 q^{94} + q^{95}+O(q^{100}) \) 6 * q - 8 * q^4 + q^5 + 12 * q^10 - 2 * q^11 + 16 * q^14 - 4 * q^16 + 6 * q^19 - 10 * q^20 + 3 * q^25 - 8 * q^26 + 36 * q^29 + 8 * q^34 + 3 * q^35 + 8 * q^40 - 12 * q^41 + 20 * q^44 - 8 * q^46 + 4 * q^49 + 4 * q^50 - 33 * q^55 - 40 * q^56 + 20 * q^59 - 14 * q^61 + 12 * q^64 + 20 * q^65 + 20 * q^70 - 52 * q^71 - 40 * q^74 - 8 * q^76 + 24 * q^79 + 32 * q^80 + 13 * q^85 + 16 * q^86 + 24 * q^89 - 24 * q^91 + 48 * q^94 + q^95

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