LMFDB - Embedded newform 855.2.c.d.514.5

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			514.5
Root			$2.68667i$ of defining polynomial
Character	$\chi$	$=$	855.514
Dual form			855.2.c.d.514.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82254i q^{2} -1.32164 q^{4} +(-1.94827 - 1.09737i) q^{5} -1.45033i q^{7} +1.23634i q^{8} +O(q^{10})$	\(q+1.82254i q^{2} -1.32164 q^{4} +(-1.94827 - 1.09737i) q^{5} -1.45033i q^{7} +1.23634i q^{8} +(2.00000 - 3.55080i) q^{10} +3.89655 q^{11} +3.05888i q^{13} +2.64327 q^{14} -4.89655 q^{16} +3.92301i q^{17} +1.00000 q^{19} +(2.57491 + 1.45033i) q^{20} +7.10160i q^{22} +5.37334i q^{23} +(2.59155 + 4.27596i) q^{25} -5.57491 q^{26} +1.91681i q^{28} +6.00000 q^{29} -8.43637 q^{31} -6.45146i q^{32} -7.14982 q^{34} +(-1.59155 + 2.82564i) q^{35} +5.95953i q^{37} +1.82254i q^{38} +(1.35673 - 2.40873i) q^{40} -10.4364 q^{41} -1.45033i q^{43} -5.14982 q^{44} -9.79310 q^{46} +4.90686i q^{47} +4.89655 q^{49} +(-7.79310 + 4.72319i) q^{50} -4.04272i q^{52} +4.23127i q^{53} +(-7.59155 - 4.27596i) q^{55} +1.79310 q^{56} +10.9352i q^{58} +3.35673 q^{59} +10.3329 q^{61} -15.3756i q^{62} +1.96491 q^{64} +(3.35673 - 5.95953i) q^{65} +9.84404i q^{67} -5.18479i q^{68} +(-5.14982 - 2.90066i) q^{70} -8.64327 q^{71} -2.43418i q^{73} -10.8615 q^{74} -1.32164 q^{76} -5.65127i q^{77} +12.4364 q^{79} +(9.53982 + 5.37334i) q^{80} -19.0207i q^{82} -12.6635i q^{83} +(4.30500 - 7.64310i) q^{85} +2.64327 q^{86} +4.81746i q^{88} +12.3662 q^{89} +4.43637 q^{91} -7.10160i q^{92} -8.94292 q^{94} +(-1.94827 - 1.09737i) q^{95} -3.05888i q^{97} +8.92414i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			1.82254i			1.28873i	0.764719	+	0.644364i	$0.222878\pi$
$2$			1.82254i			1.28873i	−0.764719	+	0.644364i	$0.777122\pi$
$3$		0			0
$3$		0			0
$4$	−1.32164			−0.660819
$5$	−1.94827	−	1.09737i	−0.871295	−	0.490760i
$5$	−1.94827	−	1.09737i	−0.871295	−	0.490760i
$6$		0			0
$7$		−	1.45033i		−	0.548172i	−0.961705	−	0.274086i	$-0.911625\pi$
$7$		−	1.45033i		−	0.548172i	0.961705	−	0.274086i	$-0.0883754\pi$
$8$			1.23634i			0.437112i
$9$		0			0
$10$	2.00000	−	3.55080i	0.632456	−	1.12286i
$11$	3.89655			1.17485			0.587427	−	0.809277i	$-0.300141\pi$
$11$	3.89655			1.17485			0.587427	+	0.809277i	$0.300141\pi$
$12$		0			0
$13$			3.05888i			0.848380i	0.905573	+	0.424190i	$0.139441\pi$
$13$			3.05888i			0.848380i	−0.905573	+	0.424190i	$0.860559\pi$
$14$	2.64327			0.706445
$15$		0			0
$16$	−4.89655			−1.22414
$17$			3.92301i			0.951469i	0.879589	+	0.475735i	$0.157818\pi$
$17$			3.92301i			0.951469i	−0.879589	+	0.475735i	$0.842182\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	2.57491	+	1.45033i	0.575768	+	0.324303i
$21$		0			0
$22$			7.10160i			1.51407i
$23$			5.37334i			1.12042i	0.828351	+	0.560209i	$0.189279\pi$
$23$			5.37334i			1.12042i	−0.828351	+	0.560209i	$0.810721\pi$
$24$		0			0
$25$	2.59155	+	4.27596i	0.518310	+	0.855193i
$26$	−5.57491			−1.09333
$27$		0			0
$28$			1.91681i			0.362242i
$29$	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
$31$	−8.43637			−1.51522			−0.757609	−	0.652709i	$-0.773632\pi$
$31$	−8.43637			−1.51522			−0.757609	+	0.652709i	$0.773632\pi$
$32$		−	6.45146i		−	1.14047i
$33$		0			0
$34$	−7.14982			−1.22618
$35$	−1.59155	+	2.82564i	−0.269021	+	0.477620i
$36$		0			0
$37$			5.95953i			0.979741i	0.871795	+	0.489871i	$0.162956\pi$
$37$			5.95953i			0.979741i	−0.871795	+	0.489871i	$0.837044\pi$
$38$			1.82254i			0.295654i
$39$		0			0
$40$	1.35673	−	2.40873i	0.214517	−	0.380854i
$41$	−10.4364			−1.62989			−0.814944	−	0.579540i	$-0.803232\pi$
$41$	−10.4364			−1.62989			−0.814944	+	0.579540i	$0.803232\pi$
$42$		0			0
$43$		−	1.45033i		−	0.221173i	−0.993867	−	0.110586i	$-0.964727\pi$
$43$		−	1.45033i		−	0.221173i	0.993867	−	0.110586i	$-0.0352729\pi$
$44$	−5.14982			−0.776365
$45$		0			0
$46$	−9.79310			−1.44391
$47$			4.90686i			0.715739i	0.933772	+	0.357869i	$0.116497\pi$
$47$			4.90686i			0.715739i	−0.933772	+	0.357869i	$0.883503\pi$
$48$		0			0
$49$	4.89655			0.699507
$50$	−7.79310	+	4.72319i	−1.10211	+	0.667960i
$51$		0			0
$52$		−	4.04272i		−	0.560625i
$53$			4.23127i			0.581209i	0.956843	+	0.290605i	$0.0938564\pi$
$53$			4.23127i			0.581209i	−0.956843	+	0.290605i	$0.906144\pi$
$54$		0			0
$55$	−7.59155	−	4.27596i	−1.02364	−	0.576571i
$56$	1.79310			0.239613
$57$		0			0
$58$			10.9352i			1.43586i
$59$	3.35673			0.437008			0.218504	−	0.975836i	$-0.429882\pi$
$59$	3.35673			0.437008			0.218504	+	0.975836i	$0.429882\pi$
$60$		0			0
$61$	10.3329			1.32300			0.661498	−	0.749947i	$-0.269921\pi$
$61$	10.3329			1.32300			0.661498	+	0.749947i	$0.269921\pi$
$62$		−	15.3756i		−	1.95270i
$63$		0			0
$64$	1.96491			0.245614
$65$	3.35673	−	5.95953i	0.416351	−	0.739189i
$66$		0			0
$67$			9.84404i			1.20264i	0.799008	+	0.601320i	$0.205358\pi$
$67$			9.84404i			1.20264i	−0.799008	+	0.601320i	$0.794642\pi$
$68$		−	5.18479i		−	0.628749i
$69$		0			0
$70$	−5.14982	−	2.90066i	−0.615522	−	0.346695i
$71$	−8.64327			−1.02577			−0.512884	−	0.858458i	$-0.671423\pi$
$71$	−8.64327			−1.02577			−0.512884	+	0.858458i	$0.671423\pi$
$72$		0			0
$73$		−	2.43418i		−	0.284899i	−0.989802	−	0.142449i	$-0.954502\pi$
$73$		−	2.43418i		−	0.284899i	0.989802	−	0.142449i	$-0.0454978\pi$
$74$	−10.8615			−1.26262
$75$		0			0
$76$	−1.32164			−0.151602
$77$		−	5.65127i		−	0.644022i
$78$		0			0
$79$	12.4364			1.39920			0.699601	−	0.714534i	$-0.253361\pi$
$79$	12.4364			1.39920			0.699601	+	0.714534i	$0.253361\pi$
$80$	9.53982	+	5.37334i	1.06658	+	0.600757i
$81$		0			0
$82$		−	19.0207i		−	2.10048i
$83$		−	12.6635i		−	1.39000i	−0.719011	−	0.694999i	$-0.755405\pi$
$83$		−	12.6635i		−	1.39000i	0.719011	−	0.694999i	$-0.244595\pi$
$84$		0			0
$85$	4.30500	−	7.64310i	0.466943	−	0.829011i
$86$	2.64327			0.285032
$87$		0			0
$88$			4.81746i			0.513543i
$89$	12.3662			1.31081			0.655407	−	0.755276i	$-0.272497\pi$
$89$	12.3662			1.31081			0.655407	+	0.755276i	$0.272497\pi$
$90$		0			0
$91$	4.43637			0.465058
$92$		−	7.10160i		−	0.740393i
$93$		0			0
$94$	−8.94292			−0.922392
$95$	−1.94827	−	1.09737i	−0.199889	−	0.112588i
$96$		0			0
$97$		−	3.05888i		−	0.310582i	−0.987869	−	0.155291i	$-0.950369\pi$
$97$		−	3.05888i		−	0.310582i	0.987869	−	0.155291i	$-0.0496315\pi$
$98$			8.92414i			0.901474i
$99$		0			0
$100$	−3.42509	−	5.65127i	−0.342509	−	0.565127i
$101$	−3.35673			−0.334007			−0.167003	−	0.985956i	$-0.553409\pi$
$101$	−3.35673			−0.334007			−0.167003	+	0.985956i	$0.553409\pi$
$102$		0			0
$103$			13.0611i			1.28695i	0.765466	+	0.643476i	$0.222508\pi$
$103$			13.0611i			1.28695i	−0.765466	+	0.643476i	$0.777492\pi$
$104$	−3.78181			−0.370837
$105$		0			0
$106$	−7.71164			−0.749020
$107$		−	5.77099i		−	0.557903i	−0.960305	−	0.278951i	$-0.910013\pi$
$107$		−	5.77099i		−	0.557903i	0.960305	−	0.278951i	$-0.0899868\pi$
$108$		0			0
$109$	−6.64327			−0.636310			−0.318155	−	0.948039i	$-0.603063\pi$
$109$	−6.64327			−0.636310			−0.318155	+	0.948039i	$0.603063\pi$
$110$	7.79310	−	13.8359i	0.743043	−	1.31920i
$111$		0			0
$112$			7.10160i			0.671038i
$113$			9.41606i			0.885789i	0.896574	+	0.442894i	$0.146048\pi$
$113$			9.41606i			0.885789i	−0.896574	+	0.442894i	$0.853952\pi$
$114$		0			0
$115$	5.89655	−	10.4687i	0.549856	−	0.976215i
$116$	−7.92982			−0.736266
$117$		0			0
$118$			6.11775i			0.563185i
$119$	5.68965			0.521569
$120$		0			0
$121$	4.18310			0.380282
$122$			18.8321i			1.70498i
$123$		0			0
$124$	11.1498			1.00128
$125$	−0.356726	−	11.1746i	−0.0319065	−	0.999491i
$126$		0			0
$127$		−	11.0934i		−	0.984383i	−0.870487	−	0.492192i	$-0.836196\pi$
$127$		−	11.0934i		−	0.984383i	0.870487	−	0.492192i	$-0.163804\pi$
$128$		−	9.32179i		−	0.823938i
$129$		0			0
$130$	10.8615	+	6.11775i	0.952613	+	0.536562i
$131$	−4.61000			−0.402778			−0.201389	−	0.979511i	$-0.564545\pi$
$131$	−4.61000			−0.402778			−0.201389	+	0.979511i	$0.564545\pi$
$132$		0			0
$133$		−	1.45033i		−	0.125759i
$134$	−17.9411			−1.54988
$135$		0			0
$136$	−4.85018			−0.415899
$137$		−	13.1808i		−	1.12612i	−0.826417	−	0.563058i	$-0.809625\pi$
$137$		−	13.1808i		−	1.12612i	0.826417	−	0.563058i	$-0.190375\pi$
$138$		0			0
$139$	−1.18310			−0.100349			−0.0501745	−	0.998740i	$-0.515978\pi$
$139$	−1.18310			−0.100349			−0.0501745	+	0.998740i	$0.515978\pi$
$140$	2.10345	−	3.73447i	0.177774	−	0.315620i
$141$		0			0
$142$		−	15.7527i		−	1.32194i
$143$			11.9191i			0.996722i
$144$		0			0
$145$	−11.6896	−	6.58423i	−0.970772	−	0.546791i
$146$	4.43637			0.367157
$147$		0			0
$148$		−	7.87634i		−	0.647431i
$149$	5.46018			0.447315			0.223658	−	0.974668i	$-0.428200\pi$
$149$	5.46018			0.447315			0.223658	+	0.974668i	$0.428200\pi$
$150$		0			0
$151$	−5.07965			−0.413376			−0.206688	−	0.978407i	$-0.566268\pi$
$151$	−5.07965			−0.413376			−0.206688	+	0.978407i	$0.566268\pi$
$152$			1.23634i			0.100280i
$153$		0			0
$154$	10.2996			0.829969
$155$	16.4364	+	9.25784i	1.32020	+	0.743608i
$156$		0			0
$157$		−	6.11775i		−	0.488250i	−0.969744	−	0.244125i	$-0.921499\pi$
$157$		−	6.11775i		−	0.488250i	0.969744	−	0.244125i	$-0.0785007\pi$
$158$			22.6657i			1.80319i
$159$		0			0
$160$	−7.07965	+	12.5692i	−0.559695	+	0.993683i
$161$	7.79310			0.614182
$162$		0			0
$163$		−	16.4365i		−	1.28740i	−0.765277	−	0.643701i	$-0.777398\pi$
$163$		−	16.4365i		−	1.28740i	0.765277	−	0.643701i	$-0.222602\pi$
$164$	13.7931			1.07706
$165$		0			0
$166$	23.0796			1.79133
$167$		−	3.80329i		−	0.294308i	−0.989114	−	0.147154i	$-0.952989\pi$
$167$		−	3.80329i		−	0.294308i	0.989114	−	0.147154i	$-0.0470112\pi$
$168$		0			0
$169$	3.64327			0.280252
$170$	13.9298	+	7.84602i	1.06837	+	0.601762i
$171$		0			0
$172$			1.91681i			0.146155i
$173$			11.3838i			0.865491i	0.901516	+	0.432746i	$0.142455\pi$
$173$			11.3838i			0.865491i	−0.901516	+	0.432746i	$0.857545\pi$
$174$		0			0
$175$	6.20155	−	3.75860i	0.468793	−	0.284123i
$176$	−19.0796			−1.43818
$177$		0			0
$178$			22.5378i			1.68928i
$179$	10.0702			0.752680			0.376340	−	0.926482i	$-0.377182\pi$
$179$	10.0702			0.752680			0.376340	+	0.926482i	$0.377182\pi$
$180$		0			0
$181$	0.573097			0.0425980			0.0212990	−	0.999773i	$-0.493220\pi$
$181$	0.573097			0.0425980			0.0212990	+	0.999773i	$0.493220\pi$
$182$			8.08545i			0.599333i
$183$		0			0
$184$	−6.64327			−0.489749
$185$	6.53982	−	11.6108i	0.480817	−	0.853643i
$186$		0			0
$187$			15.2862i			1.11784i
$188$		−	6.48509i		−	0.472973i
$189$		0			0
$190$	2.00000	−	3.55080i	0.145095	−	0.257602i
$191$	3.18310			0.230321			0.115160	−	0.993347i	$-0.463262\pi$
$191$	3.18310			0.230321			0.115160	+	0.993347i	$0.463262\pi$
$192$		0			0
$193$			3.05888i			0.220183i	0.993921	+	0.110091i	$0.0351143\pi$
$193$			3.05888i			0.220183i	−0.993921	+	0.110091i	$0.964886\pi$
$194$	5.57491			0.400255
$195$		0			0
$196$	−6.47146			−0.462247
$197$			21.4933i			1.53134i	0.643235	+	0.765669i	$0.277592\pi$
$197$			21.4933i			1.53134i	−0.643235	+	0.765669i	$0.722408\pi$
$198$		0			0
$199$	4.81690			0.341461			0.170731	−	0.985318i	$-0.445387\pi$
$199$	4.81690			0.341461			0.170731	+	0.985318i	$0.445387\pi$
$200$	−5.28655	+	3.20404i	−0.373815	+	0.226560i
$201$		0			0
$202$		−	6.11775i		−	0.430444i
$203$		−	8.70197i		−	0.610758i
$204$		0			0
$205$	20.3329	+	11.4526i	1.42011	+	0.799883i
$206$	−23.8044			−1.65853
$207$		0			0
$208$		−	14.9779i		−	1.03853i
$209$	3.89655			0.269530
$210$		0			0
$211$	10.5066			0.723301			0.361650	−	0.932314i	$-0.382213\pi$
$211$	10.5066			0.723301			0.361650	+	0.932314i	$0.382213\pi$
$212$		−	5.59220i		−	0.384074i
$213$		0			0
$214$	10.5178			0.718984
$215$	−1.59155	+	2.82564i	−0.108543	+	0.192707i
$216$		0			0
$217$			12.2355i			0.830600i
$218$		−	12.1076i		−	0.820031i
$219$		0			0
$220$	10.0333	+	5.65127i	0.676443	+	0.381009i
$221$	−12.0000			−0.807207
$222$		0			0
$223$		−	16.8947i		−	1.13136i	−0.824626	−	0.565678i	$-0.808615\pi$
$223$		−	16.8947i		−	1.13136i	0.824626	−	0.565678i	$-0.191385\pi$
$224$	−9.35673			−0.625173
$225$		0			0
$226$	−17.1611			−1.14154
$227$			17.1342i			1.13724i	0.822602	+	0.568618i	$0.192522\pi$
$227$			17.1342i			1.13724i	−0.822602	+	0.568618i	$0.807478\pi$
$228$		0			0
$229$	−25.0464			−1.65511			−0.827555	−	0.561384i	$-0.810269\pi$
$229$	−25.0464			−1.65511			−0.827555	+	0.561384i	$0.810269\pi$
$230$	19.0796	+	10.7467i	1.25807	+	0.708615i
$231$		0			0
$232$			7.41804i			0.487018i
$233$		−	19.2986i		−	1.26429i	−0.774849	−	0.632147i	$-0.782174\pi$
$233$		−	19.2986i		−	1.26429i	0.774849	−	0.632147i	$-0.217826\pi$
$234$		0			0
$235$	5.38465	−	9.55991i	0.351256	−	0.623620i
$236$	−4.43637			−0.288783
$237$		0			0
$238$			10.3696i			0.672161i
$239$	18.7693			1.21408			0.607042	−	0.794669i	$-0.292356\pi$
$239$	18.7693			1.21408			0.607042	+	0.794669i	$0.292356\pi$
$240$		0			0
$241$	−14.4364			−0.929929			−0.464964	−	0.885329i	$-0.653933\pi$
$241$	−14.4364			−0.929929			−0.464964	+	0.885329i	$0.653933\pi$
$242$			7.62385i			0.490079i
$243$		0			0
$244$	−13.6564			−0.874260
$245$	−9.53982	−	5.37334i	−0.609477	−	0.343290i
$246$		0			0
$247$			3.05888i			0.194632i
$248$		−	10.4302i		−	0.662320i
$249$		0			0
$250$	20.3662	−	0.650145i	1.28807	−	0.0411188i
$251$	10.9762			0.692811			0.346406	−	0.938085i	$-0.387402\pi$
$251$	10.9762			0.692811			0.346406	+	0.938085i	$0.387402\pi$
$252$		0			0
$253$			20.9375i			1.31633i
$254$	20.2182			1.26860
$255$		0			0
$256$	20.9191			1.30745
$257$		−	17.6392i		−	1.10030i	−0.835066	−	0.550150i	$-0.814570\pi$
$257$		−	17.6392i		−	1.10030i	0.835066	−	0.550150i	$-0.185430\pi$
$258$		0			0
$259$	8.64327			0.537067
$260$	−4.43637	+	7.87634i	−0.275132	+	0.488470i
$261$		0			0
$262$		−	8.40189i		−	0.519071i
$263$			1.68976i			0.104195i	0.998642	+	0.0520975i	$0.0165907\pi$
$263$			1.68976i			0.104195i	−0.998642	+	0.0520975i	$0.983409\pi$
$264$		0			0
$265$	4.64327	−	8.24367i	0.285234	−	0.506405i
$266$	2.64327			0.162070
$267$		0			0
$268$		−	13.0102i		−	0.794727i
$269$	−27.1022			−1.65245			−0.826226	−	0.563339i	$-0.809516\pi$
$269$	−27.1022			−1.65245			−0.826226	+	0.563339i	$0.809516\pi$
$270$		0			0
$271$	23.9524			1.45500			0.727502	−	0.686105i	$-0.240681\pi$
$271$	23.9524			1.45500			0.727502	+	0.686105i	$0.240681\pi$
$272$		−	19.2092i		−	1.16473i
$273$		0			0
$274$	24.0226			1.45126
$275$	10.0981	+	16.6615i	0.608938	+	1.00473i
$276$		0			0
$277$		−	8.23549i		−	0.494822i	−0.968911	−	0.247411i	$-0.920420\pi$
$277$		−	8.23549i		−	0.494822i	0.968911	−	0.247411i	$-0.0795799\pi$
$278$		−	2.15624i		−	0.129323i
$279$		0			0
$280$	−3.49345	−	1.96770i	−0.208774	−	0.117592i
$281$	10.4364			0.622582			0.311291	−	0.950315i	$-0.399239\pi$
$281$	10.4364			0.622582			0.311291	+	0.950315i	$0.399239\pi$
$282$		0			0
$283$		−	10.4687i		−	0.622302i	−0.950361	−	0.311151i	$-0.899286\pi$
$283$		−	10.4687i		−	0.622302i	0.950361	−	0.311151i	$-0.100714\pi$
$284$	11.4233			0.677847
$285$		0			0
$286$	−21.7229			−1.28450
$287$			15.1362i			0.893459i
$288$		0			0
$289$	1.61000			0.0947059
$290$	12.0000	−	21.3048i	0.704664	−	1.25106i
$291$		0			0
$292$			3.21710i			0.188266i
$293$			16.4668i			0.961999i	0.876721	+	0.481000i	$0.159726\pi$
$293$			16.4668i			0.961999i	−0.876721	+	0.481000i	$0.840274\pi$
$294$		0			0
$295$	−6.53982	−	3.68358i	−0.380763	−	0.214466i
$296$	−7.36801			−0.428257
$297$		0			0
$298$			9.95137i			0.576467i
$299$	−16.4364			−0.950540
$300$		0			0
$301$	−2.10345			−0.121241
$302$		−	9.25784i		−	0.532729i
$303$		0			0
$304$	−4.89655			−0.280836
$305$	−20.1314	−	11.3391i	−1.15272	−	0.649273i
$306$		0			0
$307$		−	7.87634i		−	0.449526i	−0.974413	−	0.224763i	$-0.927839\pi$
$307$		−	7.87634i		−	0.449526i	0.974413	−	0.224763i	$-0.0721609\pi$
$308$			7.46893i			0.425582i
$309$		0			0
$310$	−16.8727	+	29.9559i	−0.958308	+	1.70138i
$311$	−3.89655			−0.220953			−0.110477	−	0.993879i	$-0.535238\pi$
$311$	−3.89655			−0.220953			−0.110477	+	0.993879i	$0.535238\pi$
$312$		0			0
$313$			7.76901i			0.439130i	0.975598	+	0.219565i	$0.0704638\pi$
$313$			7.76901i			0.439130i	−0.975598	+	0.219565i	$0.929536\pi$
$314$	11.1498			0.629221
$315$		0			0
$316$	−16.4364			−0.924618
$317$			19.0510i			1.07001i	0.844849	+	0.535005i	$0.179690\pi$
$317$			19.0510i			1.07001i	−0.844849	+	0.535005i	$0.820310\pi$
$318$		0			0
$319$	23.3793			1.30899
$320$	−3.82819	−	2.15624i	−0.214002	−	0.120537i
$321$		0			0
$322$			14.2032i			0.791514i
$323$			3.92301i			0.218282i
$324$		0			0
$325$	−13.0796	+	7.92723i	−0.725528	+	0.439724i
$326$	29.9560			1.65911
$327$		0			0
$328$		−	12.9029i		−	0.712444i
$329$	7.11655			0.392348
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			16.7365i			0.918536i
$333$		0			0
$334$	6.93164			0.379282
$335$	10.8026	−	19.1789i	0.590207	−	1.04785i
$336$		0			0
$337$			6.89249i			0.375458i	0.982221	+	0.187729i	$0.0601127\pi$
$337$			6.89249i			0.375458i	−0.982221	+	0.187729i	$0.939887\pi$
$338$			6.64000i			0.361168i
$339$		0			0
$340$	−5.68965	+	10.1014i	−0.308565	+	0.547826i
$341$	−32.8727			−1.78016
$342$		0			0
$343$		−	17.2539i		−	0.931623i
$344$	1.79310			0.0966774
$345$		0			0
$346$	−20.7473			−1.11538
$347$		−	30.5503i		−	1.64002i	−0.572346	−	0.820012i	$-0.693967\pi$
$347$		−	30.5503i		−	1.64002i	0.572346	−	0.820012i	$-0.306033\pi$
$348$		0			0
$349$	−16.7693			−0.897640			−0.448820	−	0.893622i	$-0.648156\pi$
$349$	−16.7693			−0.897640			−0.448820	+	0.893622i	$0.648156\pi$
$350$	6.85018	+	11.3025i	0.366157	+	0.604147i
$351$		0			0
$352$		−	25.1384i		−	1.33988i
$353$		−	29.0999i		−	1.54883i	−0.632676	−	0.774417i	$-0.718044\pi$
$353$		−	29.0999i		−	1.54883i	0.632676	−	0.774417i	$-0.281956\pi$
$354$		0			0
$355$	16.8395	+	9.48489i	0.893746	+	0.503406i
$356$	−16.3436			−0.866210
$357$		0			0
$358$			18.3533i			0.970000i
$359$	−11.6896			−0.616956			−0.308478	−	0.951231i	$-0.599820\pi$
$359$	−11.6896			−0.616956			−0.308478	+	0.951231i	$0.599820\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$			1.04449i			0.0548972i
$363$		0			0
$364$	−5.86328			−0.307319
$365$	−2.67120	+	4.74244i	−0.139817	+	0.248231i
$366$		0			0
$367$			4.20095i			0.219288i	0.993971	+	0.109644i	$0.0349710\pi$
$367$			4.20095i			0.219288i	−0.993971	+	0.109644i	$0.965029\pi$
$368$		−	26.3108i		−	1.37155i
$369$		0			0
$370$	21.1611	+	11.9191i	1.10011	+	0.619643i
$371$	6.13672			0.318603
$372$		0			0
$373$		−	16.9456i		−	0.877412i	−0.898631	−	0.438706i	$-0.855437\pi$
$373$		−	16.9456i		−	0.877412i	0.898631	−	0.438706i	$-0.144563\pi$
$374$	−27.8596			−1.44059
$375$		0			0
$376$	−6.06655			−0.312858
$377$			18.3533i			0.945241i
$378$		0			0
$379$	−10.3662			−0.532476			−0.266238	−	0.963907i	$-0.585781\pi$
$379$	−10.3662			−0.532476			−0.266238	+	0.963907i	$0.585781\pi$
$380$	2.57491	+	1.45033i	0.132090	+	0.0744002i
$381$		0			0
$382$			5.80131i			0.296821i
$383$		−	20.5907i		−	1.05214i	−0.850442	−	0.526068i	$-0.823666\pi$
$383$		−	20.5907i		−	1.05214i	0.850442	−	0.526068i	$-0.176334\pi$
$384$		0			0
$385$	−6.20155	+	11.0102i	−0.316060	+	0.561133i
$386$	−5.57491			−0.283756
$387$		0			0
$388$			4.04272i			0.205238i
$389$	8.10345			0.410861			0.205431	−	0.978672i	$-0.434141\pi$
$389$	8.10345			0.410861			0.205431	+	0.978672i	$0.434141\pi$
$390$		0			0
$391$	−21.0796			−1.06604
$392$			6.05380i			0.305763i
$393$		0			0
$394$	−39.1724			−1.97348
$395$	−24.2295	−	13.6473i	−1.21912	−	0.686672i
$396$		0			0
$397$		−	3.46891i		−	0.174100i	−0.996204	−	0.0870499i	$-0.972256\pi$
$397$		−	3.46891i		−	0.174100i	0.996204	−	0.0870499i	$-0.0277440\pi$
$398$			8.77898i			0.440050i
$399$		0			0
$400$	−12.6896	−	20.9375i	−0.634482	−	1.04687i
$401$	15.9524			0.796625			0.398312	−	0.917250i	$-0.369596\pi$
$401$	15.9524			0.796625			0.398312	+	0.917250i	$0.369596\pi$
$402$		0			0
$403$		−	25.8058i		−	1.28548i
$404$	4.43637			0.220718
$405$		0			0
$406$	15.8596			0.787101
$407$			23.2216i			1.15105i
$408$		0			0
$409$	3.92982			0.194317			0.0971586	−	0.995269i	$-0.469025\pi$
$409$	3.92982			0.194317			0.0971586	+	0.995269i	$0.469025\pi$
$410$	−20.8727	+	37.0575i	−1.03083	+	1.83014i
$411$		0			0
$412$		−	17.2621i		−	0.850442i
$413$		−	4.86835i		−	0.239556i
$414$		0			0
$415$	−13.8965	+	24.6719i	−0.682155	+	1.21110i
$416$	19.7342			0.967549
$417$		0			0
$418$			7.10160i			0.347351i
$419$	−34.2996			−1.67565			−0.837824	−	0.545941i	$-0.816172\pi$
$419$	−34.2996			−1.67565			−0.837824	+	0.545941i	$0.816172\pi$
$420$		0			0
$421$	−26.0891			−1.27151			−0.635753	−	0.771893i	$-0.719310\pi$
$421$	−26.0891			−1.27151			−0.635753	+	0.771893i	$0.719310\pi$
$422$			19.1486i			0.932138i
$423$		0			0
$424$	−5.23129			−0.254054
$425$	−16.7746	+	10.1667i	−0.813690	+	0.493156i
$426$		0			0
$427$		−	14.9861i		−	0.725229i
$428$			7.62715i			0.368672i
$429$		0			0
$430$	−5.14982	−	2.90066i	−0.248347	−	0.139882i
$431$	−10.2996			−0.496117			−0.248058	−	0.968745i	$-0.579792\pi$
$431$	−10.2996			−0.496117			−0.248058	+	0.968745i	$0.579792\pi$
$432$		0			0
$433$		−	36.2319i		−	1.74119i	−0.491999	−	0.870596i	$-0.663734\pi$
$433$		−	36.2319i		−	1.74119i	0.491999	−	0.870596i	$-0.336266\pi$
$434$	−22.2996			−1.07042
$435$		0			0
$436$	8.78000			0.420486
$437$			5.37334i			0.257042i
$438$		0			0
$439$	24.0891			1.14971			0.574855	−	0.818255i	$-0.305058\pi$
$439$	24.0891			1.14971			0.574855	+	0.818255i	$0.305058\pi$
$440$	5.28655	−	9.38574i	0.252026	−	0.447448i
$441$		0			0
$442$		−	21.8704i		−	1.04027i
$443$			5.52337i			0.262423i	0.991354	+	0.131212i	$0.0418868\pi$
$443$			5.52337i			0.262423i	−0.991354	+	0.131212i	$0.958113\pi$
$444$		0			0
$445$	−24.0927	−	13.5703i	−1.14211	−	0.643295i
$446$	30.7913			1.45801
$447$		0			0
$448$		−	2.84977i		−	0.134639i
$449$	7.92982			0.374231			0.187116	−	0.982338i	$-0.440086\pi$
$449$	7.92982			0.374231			0.187116	+	0.982338i	$0.440086\pi$
$450$		0			0
$451$	−40.6658			−1.91488
$452$		−	12.4446i		−	0.585346i
$453$		0			0
$454$	−31.2277			−1.46559
$455$	−8.64327	−	4.86835i	−0.405203	−	0.228232i
$456$		0			0
$457$			22.6534i			1.05968i	0.848098	+	0.529840i	$0.177748\pi$
$457$			22.6534i			1.05968i	−0.848098	+	0.529840i	$0.822252\pi$
$458$		−	45.6479i		−	2.13299i
$459$		0			0
$460$	−7.79310	+	13.8359i	−0.363355	+	0.645101i
$461$	−13.3900			−0.623634			−0.311817	−	0.950142i	$-0.600938\pi$
$461$	−13.3900			−0.623634			−0.311817	+	0.950142i	$0.600938\pi$
$462$		0			0
$463$		−	16.9029i		−	0.785546i	−0.919635	−	0.392773i	$-0.871516\pi$
$463$		−	16.9029i		−	0.785546i	0.919635	−	0.392773i	$-0.128484\pi$
$464$	−29.3793			−1.36390
$465$		0			0
$466$	35.1724			1.62933
$467$		−	22.2501i		−	1.02961i	−0.857306	−	0.514807i	$-0.827864\pi$
$467$		−	22.2501i		−	1.02961i	0.857306	−	0.514807i	$-0.172136\pi$
$468$		0			0
$469$	14.2771			0.659254
$470$	17.4233	+	9.81371i	0.803676	+	0.452673i
$471$		0			0
$472$			4.15006i			0.191022i
$473$		−	5.65127i		−	0.259846i
$474$		0			0
$475$	2.59155	+	4.27596i	0.118908	+	0.196195i
$476$	−7.51965			−0.344663
$477$		0			0
$478$			34.2077i			1.56462i
$479$	0.366196			0.0167319			0.00836597	−	0.999965i	$-0.497337\pi$
$479$	0.366196			0.0167319			0.00836597	+	0.999965i	$0.497337\pi$
$480$		0			0
$481$	−18.2295			−0.831192
$482$		−	26.3108i		−	1.19842i
$483$		0			0
$484$	−5.52854			−0.251297
$485$	−3.35673	+	5.95953i	−0.152421	+	0.270608i
$486$		0			0
$487$			26.8461i			1.21651i	0.793740	+	0.608257i	$0.208131\pi$
$487$			26.8461i			1.21651i	−0.793740	+	0.608257i	$0.791869\pi$
$488$			12.7750i			0.578298i
$489$		0			0
$490$	9.79310	−	17.3867i	0.442407	−	0.785450i
$491$	23.7266			1.07076			0.535382	−	0.844610i	$-0.320168\pi$
$491$	23.7266			1.07076			0.535382	+	0.844610i	$0.320168\pi$
$492$		0			0
$493$			23.5381i			1.06010i
$494$	−5.57491			−0.250827
$495$		0			0
$496$	41.3091			1.85483
$497$			12.5356i			0.562298i
$498$		0			0
$499$	−6.81690			−0.305166			−0.152583	−	0.988291i	$-0.548759\pi$
$499$	−6.81690			−0.305166			−0.152583	+	0.988291i	$0.548759\pi$
$500$	0.471462	+	14.7688i	0.0210844	+	0.660482i
$501$		0			0
$502$			20.0045i			0.892845i
$503$			23.4102i			1.04381i	0.853004	+	0.521904i	$0.174778\pi$
$503$			23.4102i			1.04381i	−0.853004	+	0.521904i	$0.825222\pi$
$504$		0			0
$505$	6.53982	+	3.68358i	0.291018	+	0.163917i
$506$	−38.1593			−1.69639
$507$		0			0
$508$			14.6615i			0.650499i
$509$	16.9204			0.749981			0.374991	−	0.927029i	$-0.377646\pi$
$509$	16.9204			0.749981			0.374991	+	0.927029i	$0.377646\pi$
$510$		0			0
$511$	−3.53035			−0.156174
$512$			19.4823i			0.861003i
$513$		0			0
$514$	32.1480			1.41799
$515$	14.3329	−	25.4467i	0.631584	−	1.12131i
$516$		0			0
$517$			19.1198i			0.840888i
$518$			15.7527i			0.692133i
$519$		0			0
$520$	7.36801	+	4.15006i	0.323109	+	0.181992i
$521$	3.49345			0.153051			0.0765254	−	0.997068i	$-0.475617\pi$
$521$	3.49345			0.153051			0.0765254	+	0.997068i	$0.475617\pi$
$522$		0			0
$523$		−	14.9271i		−	0.652714i	−0.945247	−	0.326357i	$-0.894179\pi$
$523$		−	14.9271i		−	0.652714i	0.945247	−	0.326357i	$-0.105821\pi$
$524$	6.09275			0.266163
$525$		0			0
$526$	−3.07965			−0.134279
$527$		−	33.0960i		−	1.44168i
$528$		0			0
$529$	−5.87275			−0.255337
$530$	15.0244	+	8.46253i	0.652618	+	0.367589i
$531$		0			0
$532$			1.91681i			0.0831041i
$533$		−	31.9236i		−	1.38276i
$534$		0			0
$535$	−6.33292	+	11.2435i	−0.273796	+	0.486098i
$536$	−12.1706			−0.525689
$537$		0			0
$538$		−	49.3948i		−	2.12956i
$539$	19.0796			0.821819
$540$		0			0
$541$	−12.6991			−0.545978			−0.272989	−	0.962017i	$-0.588012\pi$
$541$	−12.6991			−0.545978			−0.272989	+	0.962017i	$0.588012\pi$
$542$			43.6541i			1.87510i
$543$		0			0
$544$	25.3091			1.08512
$545$	12.9429	+	7.29014i	0.554414	+	0.312275i
$546$		0			0
$547$			31.8162i			1.36036i	0.733043	+	0.680182i	$0.238099\pi$
$547$			31.8162i			1.36036i	−0.733043	+	0.680182i	$0.761901\pi$
$548$			17.4203i			0.744158i
$549$		0			0
$550$	−30.3662	+	18.4041i	−1.29482	+	0.784756i
$551$	6.00000			0.255609
$552$		0			0
$553$		−	18.0368i		−	0.767003i
$554$	15.0095			0.637691
$555$		0			0
$556$	1.56363			0.0663125
$557$		−	9.64731i		−	0.408770i	−0.978891	−	0.204385i	$-0.934481\pi$
$557$		−	9.64731i		−	0.408770i	0.978891	−	0.204385i	$-0.0655194\pi$
$558$		0			0
$559$	4.43637			0.187639
$560$	7.79310	−	13.8359i	0.329319	−	0.584672i
$561$		0			0
$562$			19.0207i			0.802338i
$563$			4.28216i			0.180471i	0.995920	+	0.0902357i	$0.0287620\pi$
$563$			4.28216i			0.180471i	−0.995920	+	0.0902357i	$0.971238\pi$
$564$		0			0
$565$	10.3329	−	18.3451i	0.434709	−	0.771783i
$566$	19.0796			0.801977
$567$		0			0
$568$		−	10.6860i		−	0.448376i
$569$	42.2295			1.77035			0.885176	−	0.465257i	$-0.154038\pi$
$569$	42.2295			1.77035			0.885176	+	0.465257i	$0.154038\pi$
$570$		0			0
$571$	−19.2200			−0.804332			−0.402166	−	0.915567i	$-0.631743\pi$
$571$	−19.2200			−0.804332			−0.402166	+	0.915567i	$0.631743\pi$
$572$		−	15.7527i		−	0.658653i
$573$		0			0
$574$	−27.5862			−1.15143
$575$	−22.9762	+	13.9253i	−0.958174	+	0.580724i
$576$		0			0
$577$			40.7919i			1.69819i	0.528239	+	0.849096i	$0.322852\pi$
$577$			40.7919i			1.69819i	−0.528239	+	0.849096i	$0.677148\pi$
$578$			2.93428i			0.122050i
$579$		0			0
$580$	15.4495	+	8.70197i	0.641504	+	0.361329i
$581$	−18.3662			−0.761958
$582$		0			0
$583$			16.4873i			0.682836i
$584$	3.00947			0.124533
$585$		0			0
$586$	−30.0113			−1.23975
$587$		−	31.8851i		−	1.31604i	−0.753001	−	0.658019i	$-0.771395\pi$
$587$		−	31.8851i		−	1.31604i	0.753001	−	0.658019i	$-0.228605\pi$
$588$		0			0
$589$	−8.43637			−0.347615
$590$	6.71345	−	11.9191i	0.276388	−	0.490700i
$591$		0			0
$592$		−	29.1811i		−	1.19934i
$593$			38.8973i			1.59732i	0.601783	+	0.798660i	$0.294457\pi$
$593$			38.8973i			1.59732i	−0.601783	+	0.798660i	$0.705543\pi$
$594$		0			0
$595$	−11.0850	−	6.24366i	−0.454441	−	0.255965i
$596$	−7.21637			−0.295594
$597$		0			0
$598$		−	29.9559i		−	1.22499i
$599$	28.1629			1.15071			0.575353	−	0.817905i	$-0.304865\pi$
$599$	28.1629			1.15071			0.575353	+	0.817905i	$0.304865\pi$
$600$		0			0
$601$	−5.56363			−0.226945			−0.113473	−	0.993541i	$-0.536197\pi$
$601$	−5.56363			−0.226945			−0.113473	+	0.993541i	$0.536197\pi$
$602$		−	3.83361i		−	0.156246i
$603$		0			0
$604$	6.71345			0.273166
$605$	−8.14982	−	4.59042i	−0.331337	−	0.186627i
$606$		0			0
$607$		−	33.9986i		−	1.37996i	−0.723828	−	0.689980i	$-0.757619\pi$
$607$		−	33.9986i		−	1.37996i	0.723828	−	0.689980i	$-0.242381\pi$
$608$		−	6.45146i		−	0.261641i
$609$		0			0
$610$	20.6658	−	36.6902i	0.836736	−	1.48554i
$611$	−15.0095			−0.607218
$612$		0			0
$613$			17.5703i			0.709659i	0.934931	+	0.354830i	$0.115461\pi$
$613$			17.5703i			0.709659i	−0.934931	+	0.354830i	$0.884539\pi$
$614$	14.3549			0.579317
$615$		0			0
$616$	6.98690			0.281510
$617$		−	13.0791i		−	0.526544i	−0.964722	−	0.263272i	$-0.915198\pi$
$617$		−	13.0791i		−	0.526544i	0.964722	−	0.263272i	$-0.0848016\pi$
$618$		0			0
$619$	18.9393			0.761234			0.380617	−	0.924733i	$-0.375712\pi$
$619$	18.9393			0.761234			0.380617	+	0.924733i	$0.375712\pi$
$620$	−21.7229	−	12.2355i	−0.872414	−	0.491390i
$621$		0			0
$622$		−	7.10160i		−	0.284748i
$623$		−	17.9350i		−	0.718552i
$624$		0			0
$625$	−11.5677	+	22.1627i	−0.462710	+	0.886510i
$626$	−14.1593			−0.565919
$627$		0			0
$628$			8.08545i			0.322645i
$629$	−23.3793			−0.932194
$630$		0			0
$631$	31.6896			1.26154			0.630772	−	0.775968i	$-0.282738\pi$
$631$	31.6896			1.26154			0.630772	+	0.775968i	$0.282738\pi$
$632$			15.3756i			0.611608i
$633$		0			0
$634$	−34.7211			−1.37895
$635$	−12.1736	+	21.6131i	−0.483096	+	0.857688i
$636$		0			0
$637$			14.9779i			0.593448i
$638$			42.6096i			1.68693i
$639$		0			0
$640$	−10.2295	+	18.1614i	−0.404355	+	0.717893i
$641$	−47.9750			−1.89490			−0.947449	−	0.319908i	$-0.896348\pi$
$641$	−47.9750			−1.89490			−0.947449	+	0.319908i	$0.896348\pi$
$642$		0			0
$643$		−	0.200927i		−	0.00792378i	−0.999992	−	0.00396189i	$-0.998739\pi$
$643$		−	0.200927i		−	0.00792378i	0.999992	−	0.00396189i	$-0.00126111\pi$
$644$	−10.2996			−0.405863
$645$		0			0
$646$	−7.14982			−0.281306
$647$			1.58798i			0.0624299i	0.999513	+	0.0312150i	$0.00993765\pi$
$647$			1.58798i			0.0624299i	−0.999513	+	0.0312150i	$0.990062\pi$
$648$		0			0
$649$	13.0796			0.513421
$650$	−14.4477	−	23.8381i	−0.566684	−	0.935008i
$651$		0			0
$652$			21.7230i			0.850739i
$653$			33.5624i			1.31340i	0.754152	+	0.656700i	$0.228048\pi$
$653$			33.5624i			1.31340i	−0.754152	+	0.656700i	$0.771952\pi$
$654$		0			0
$655$	8.98155	+	5.05889i	0.350938	+	0.197667i
$656$	51.1022			1.99521
$657$		0			0
$658$			12.9702i			0.505630i
$659$	15.3567			0.598213			0.299107	−	0.954220i	$-0.403311\pi$
$659$	15.3567			0.598213			0.299107	+	0.954220i	$0.403311\pi$
$660$		0			0
$661$	12.8026			0.497962			0.248981	−	0.968508i	$-0.419904\pi$
$661$	12.8026			0.497962			0.248981	+	0.968508i	$0.419904\pi$
$662$			14.5803i			0.566679i
$663$		0			0
$664$	15.6564			0.607585
$665$	−1.59155	+	2.82564i	−0.0617176	+	0.109573i
$666$		0			0
$667$			32.2400i			1.24834i
$668$			5.02657i			0.194484i
$669$		0			0
$670$	34.9542	+	19.6881i	1.35040	+	0.760617i
$671$	40.2627			1.55433
$672$		0			0
$673$			7.82545i			0.301649i	0.988561	+	0.150824i	$0.0481928\pi$
$673$			7.82545i			0.301649i	−0.988561	+	0.150824i	$0.951807\pi$
$674$	−12.5618			−0.483863
$675$		0			0
$676$	−4.81509			−0.185196
$677$			21.6516i			0.832137i	0.909333	+	0.416069i	$0.136592\pi$
$677$			21.6516i			0.832137i	−0.909333	+	0.416069i	$0.863408\pi$
$678$		0			0
$679$	−4.43637			−0.170252
$680$	9.44947	+	5.32245i	0.362371	+	0.204107i
$681$		0			0
$682$		−	59.9118i		−	2.29414i
$683$		−	6.38751i		−	0.244411i	−0.992505	−	0.122206i	$-0.961003\pi$
$683$		−	6.38751i		−	0.244411i	0.992505	−	0.122206i	$-0.0389967\pi$
$684$		0			0
$685$	−14.4643	+	25.6799i	−0.552652	+	0.981179i
$686$	31.4458			1.20061
$687$		0			0
$688$			7.10160i			0.270746i
$689$	−12.9429			−0.493086
$690$		0			0
$691$	44.4958			1.69270			0.846351	−	0.532626i	$-0.178795\pi$
$691$	44.4958			1.69270			0.846351	+	0.532626i	$0.178795\pi$
$692$		−	15.0452i		−	0.571933i
$693$		0			0
$694$	55.6789			2.11354
$695$	2.30500	+	1.29830i	0.0874336	+	0.0492473i
$696$		0			0
$697$		−	40.9420i		−	1.55079i
$698$		−	30.5626i		−	1.15681i
$699$		0			0
$700$	−8.19620	+	4.96750i	−0.309787	+	0.187754i
$701$	−17.5160			−0.661571			−0.330785	−	0.943706i	$-0.607314\pi$
$701$	−17.5160			−0.661571			−0.330785	+	0.943706i	$0.607314\pi$
$702$		0			0
$703$			5.95953i			0.224768i
$704$	7.65638			0.288560
$705$		0			0
$706$	53.0357			1.99602
$707$			4.86835i			0.183093i
$708$		0			0
$709$	−11.4269			−0.429146			−0.214573	−	0.976708i	$-0.568836\pi$
$709$	−11.4269			−0.429146			−0.214573	+	0.976708i	$0.568836\pi$
$710$	−17.2865	+	30.6905i	−0.648753	+	1.15180i
$711$		0			0
$712$			15.2888i			0.572973i
$713$		−	45.3315i		−	1.69768i
$714$		0			0
$715$	13.0796	−	23.2216i	0.489151	−	0.868439i
$716$	−13.3091			−0.497385
$717$		0			0
$718$		−	21.3048i		−	0.795088i
$719$	15.8965			0.592841			0.296421	−	0.955057i	$-0.404207\pi$
$719$	15.8965			0.592841			0.296421	+	0.955057i	$0.404207\pi$
$720$		0			0
$721$	18.9429			0.705471
$722$			1.82254i			0.0678278i
$723$		0			0
$724$	−0.757427			−0.0281495
$725$	15.5493	+	25.6558i	0.577486	+	0.952832i
$726$		0			0
$727$		−	41.9905i		−	1.55734i	−0.627434	−	0.778670i	$-0.715895\pi$
$727$		−	41.9905i		−	1.55734i	0.627434	−	0.778670i	$-0.284105\pi$
$728$			5.48487i			0.203283i
$729$		0			0
$730$	−8.64327	−	4.86835i	−0.319902	−	0.180186i
$731$	5.68965			0.210439
$732$		0			0
$733$		−	0.632884i		−	0.0233761i	−0.999932	−	0.0116881i	$-0.996279\pi$
$733$		−	0.632884i		−	0.0233761i	0.999932	−	0.0116881i	$-0.00372051\pi$
$734$	−7.65638			−0.282602
$735$		0			0
$736$	34.6658			1.27780
$737$			38.3578i			1.41293i
$738$		0			0
$739$	29.5493			1.08699			0.543494	−	0.839413i	$-0.317101\pi$
$739$	29.5493			1.08699			0.543494	+	0.839413i	$0.317101\pi$
$740$	−8.64327	+	15.3453i	−0.317733	+	0.564103i
$741$		0			0
$742$			11.1844i			0.410592i
$743$			31.3374i			1.14966i	0.818274	+	0.574829i	$0.194931\pi$
$743$			31.3374i			1.14966i	−0.818274	+	0.574829i	$0.805069\pi$
$744$		0			0
$745$	−10.6379	−	5.99184i	−0.389743	−	0.219524i
$746$	30.8840			1.13074
$747$		0			0
$748$		−	20.2028i		−	0.738688i
$749$	−8.36983			−0.305827
$750$		0			0
$751$	25.0131			0.912741			0.456371	−	0.889790i	$-0.349149\pi$
$751$	25.0131			0.912741			0.456371	+	0.889790i	$0.349149\pi$
$752$		−	24.0267i		−	0.876163i
$753$		0			0
$754$	−33.4495			−1.21816
$755$	9.89655	+	5.57426i	0.360172	+	0.202868i
$756$		0			0
$757$			32.0900i			1.16633i	0.812354	+	0.583165i	$0.198186\pi$
$757$			32.0900i			1.16633i	−0.812354	+	0.583165i	$0.801814\pi$
$758$		−	18.8928i		−	0.686216i
$759$		0			0
$760$	1.35673	−	2.40873i	0.0492136	−	0.0873739i
$761$	40.4922			1.46784			0.733921	−	0.679235i	$-0.237688\pi$
$761$	40.4922			1.46784			0.733921	+	0.679235i	$0.237688\pi$
$762$		0			0
$763$			9.63492i			0.348808i
$764$	−4.20690			−0.152200
$765$		0			0
$766$	37.5273			1.35592
$767$			10.2678i			0.370749i
$768$		0			0
$769$	−3.09398			−0.111572			−0.0557859	−	0.998443i	$-0.517766\pi$
$769$	−3.09398			−0.111572			−0.0557859	+	0.998443i	$0.517766\pi$
$770$	−20.0665	−	11.3025i	−0.723148	−	0.407316i
$771$		0			0
$772$		−	4.04272i		−	0.145501i
$773$			1.96350i			0.0706220i	0.999376	+	0.0353110i	$0.0112422\pi$
$773$			1.96350i			0.0706220i	−0.999376	+	0.0353110i	$0.988758\pi$
$774$		0			0
$775$	−21.8633	−	36.0736i	−0.785352	−	1.29580i
$776$	3.78181			0.135759
$777$		0			0
$778$			14.7688i			0.529488i
$779$	−10.4364			−0.373922
$780$		0			0
$781$	−33.6789			−1.20513
$782$		−	38.4184i		−	1.37384i
$783$		0			0
$784$	−23.9762			−0.856293
$785$	−6.71345	+	11.9191i	−0.239613	+	0.425410i
$786$		0			0
$787$			0.107331i			0.00382595i	0.999998	+	0.00191297i	$0.000608919\pi$
$787$			0.107331i			0.00382595i	−0.999998	+	0.00191297i	$0.999391\pi$
$788$		−	28.4064i		−	1.01194i
$789$		0			0
$790$	24.8727	−	44.1591i	0.884933	−	1.57111i
$791$	13.6564			0.485565
$792$		0			0
$793$			31.6071i			1.12240i
$794$	6.32222			0.224367
$795$		0			0
$796$	−6.36620			−0.225644
$797$		−	8.32068i		−	0.294734i	−0.989082	−	0.147367i	$-0.952920\pi$
$797$		−	8.32068i		−	0.294734i	0.989082	−	0.147367i	$-0.0470798\pi$
$798$		0			0
$799$	−19.2496			−0.681003
$800$	27.5862	−	16.7193i	0.975319	−	0.591115i
$801$		0			0
$802$			29.0738i			1.02663i
$803$		−	9.48489i		−	0.334714i
$804$		0			0
$805$	−15.1831	−	8.55193i	−0.535134	−	0.301416i
$806$	47.0320			1.65663
$807$		0			0
$808$		−	4.15006i		−	0.145998i
$809$	−27.6231			−0.971177			−0.485588	−	0.874188i	$-0.661395\pi$
$809$	−27.6231			−0.971177			−0.485588	+	0.874188i	$0.661395\pi$
$810$		0			0
$811$	23.0095			0.807972			0.403986	−	0.914765i	$-0.367624\pi$
$811$	23.0095			0.807972			0.403986	+	0.914765i	$0.367624\pi$
$812$			11.5008i			0.403600i
$813$		0			0
$814$	−42.3222			−1.48339
$815$	−18.0369	+	32.0227i	−0.631805	+	1.12171i
$816$		0			0
$817$		−	1.45033i		−	0.0507405i
$818$			7.16224i			0.250422i
$819$		0			0
$820$	−26.8727	−	15.1362i	−0.938437	−	0.528578i
$821$	−31.1355			−1.08664			−0.543318	−	0.839527i	$-0.682832\pi$
$821$	−31.1355			−1.08664			−0.543318	+	0.839527i	$0.682832\pi$
$822$		0			0
$823$			20.4201i			0.711800i	0.934524	+	0.355900i	$0.115826\pi$
$823$			20.4201i			0.711800i	−0.934524	+	0.355900i	$0.884174\pi$
$824$	−16.1480			−0.562543
$825$		0			0
$826$	8.87275			0.308722
$827$		−	0.902638i		−	0.0313878i	−0.999877	−	0.0156939i	$-0.995004\pi$
$827$		−	0.902638i		−	0.0313878i	0.999877	−	0.0156939i	$-0.00499573\pi$
$828$		0			0
$829$	13.4971			0.468773			0.234386	−	0.972143i	$-0.424692\pi$
$829$	13.4971			0.468773			0.234386	+	0.972143i	$0.424692\pi$
$830$	−44.9655	−	25.3270i	−1.56078	−	0.879112i
$831$		0			0
$832$			6.01042i			0.208374i
$833$			19.2092i			0.665560i
$834$		0			0
$835$	−4.17363	+	7.40986i	−0.144434	+	0.256429i
$836$	−5.14982			−0.178110
$837$		0			0
$838$		−	62.5123i		−	2.15945i
$839$	33.1022			1.14282			0.571408	−	0.820666i	$-0.306397\pi$
$839$	33.1022			1.14282			0.571408	+	0.820666i	$0.306397\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	47.5484i		−	1.63862i
$843$		0			0
$844$	−13.8858			−0.477971
$845$	−7.09810	−	3.99803i	−0.244182	−	0.137536i
$846$		0			0
$847$		−	6.06686i		−	0.208460i
$848$		−	20.7186i		−	0.711480i
$849$		0			0
$850$	−18.5291	−	30.5724i	−0.635544	−	1.04862i
$851$	−32.0226			−1.09772
$852$		0			0
$853$			50.9097i			1.74312i	0.490293	+	0.871558i	$0.336890\pi$
$853$			50.9097i			1.74312i	−0.490293	+	0.871558i	$0.663110\pi$
$854$	27.3128			0.934623
$855$		0			0
$856$	7.13491			0.243866
$857$			21.2333i			0.725317i	0.931922	+	0.362659i	$0.118131\pi$
$857$			21.2333i			0.725317i	−0.931922	+	0.362659i	$0.881869\pi$
$858$		0			0
$859$	−29.4827			−1.00594			−0.502969	−	0.864304i	$-0.667759\pi$
$859$	−29.4827			−1.00594			−0.502969	+	0.864304i	$0.667759\pi$
$860$	2.10345	−	3.73447i	0.0717271	−	0.127344i
$861$		0			0
$862$		−	18.7715i		−	0.639359i
$863$		−	25.7755i		−	0.877408i	−0.898632	−	0.438704i	$-0.855438\pi$
$863$		−	25.7755i		−	0.877408i	0.898632	−	0.438704i	$-0.144562\pi$
$864$		0			0
$865$	12.4922	−	22.1787i	0.424748	−	0.754098i
$866$	66.0339			2.24392
$867$		0			0
$868$		−	16.1709i		−	0.548876i
$869$	48.4589			1.64386
$870$		0			0
$871$	−30.1117			−1.02030
$872$		−	8.21335i		−	0.278139i
$873$		0			0
$874$	−9.79310			−0.331257
$875$	−16.2069	+	0.517369i	−0.547893	+	0.0174903i
$876$		0			0
$877$		−	54.2687i		−	1.83252i	−0.400581	−	0.916261i	$-0.631192\pi$
$877$		−	54.2687i		−	1.83252i	0.400581	−	0.916261i	$-0.368808\pi$
$878$			43.9033i			1.48166i
$879$		0			0
$880$	37.1724	+	20.9375i	1.25308	+	0.705802i
$881$	28.4922			0.959927			0.479964	−	0.877288i	$-0.340650\pi$
$881$	28.4922			0.959927			0.479964	+	0.877288i	$0.340650\pi$
$882$		0			0
$883$			40.5264i			1.36382i	0.731435	+	0.681911i	$0.238851\pi$
$883$			40.5264i			1.36382i	−0.731435	+	0.681911i	$0.761149\pi$
$884$	15.8596			0.533418
$885$		0			0
$886$	−10.0665			−0.338192
$887$		−	11.4705i		−	0.385142i	−0.981283	−	0.192571i	$-0.938317\pi$
$887$		−	11.4705i		−	0.385142i	0.981283	−	0.192571i	$-0.0616826\pi$
$888$		0			0
$889$	−16.0891			−0.539612
$890$	24.7324	−	43.9099i	0.829032	−	1.47186i
$891$		0			0
$892$			22.3287i			0.747621i
$893$			4.90686i			0.164202i
$894$		0			0
$895$	−19.6195	−	11.0507i	−0.655807	−	0.369385i
$896$	−13.5197			−0.451660
$897$		0			0
$898$			14.4524i			0.482282i
$899$	−50.6182			−1.68821
$900$		0			0
$901$	−16.5993			−0.553003
$902$		−	74.1150i		−	2.46776i
$903$		0			0
$904$	−11.6415			−0.387189
$905$	−1.11655	−	0.628901i	−0.0371154	−	0.0209054i
$906$		0			0
$907$			48.1000i			1.59714i	0.601905	+	0.798568i	$0.294409\pi$
$907$			48.1000i			1.59714i	−0.601905	+	0.798568i	$0.705591\pi$
$908$		−	22.6452i		−	0.751506i
$909$		0			0
$910$	8.87275	−	15.7527i	0.294129	−	0.522196i
$911$	12.8062			0.424288			0.212144	−	0.977238i	$-0.431955\pi$
$911$	12.8062			0.424288			0.212144	+	0.977238i	$0.431955\pi$
$912$		0			0
$913$		−	49.3439i		−	1.63304i
$914$	−41.2865			−1.36564
$915$		0			0
$916$	33.1022			1.09373
$917$			6.68601i			0.220792i
$918$		0			0
$919$	38.7135			1.27704			0.638519	−	0.769606i	$-0.279547\pi$
$919$	38.7135			1.27704			0.638519	+	0.769606i	$0.279547\pi$
$920$	12.9429	+	7.29014i	0.426716	+	0.240349i
$921$		0			0
$922$		−	24.4038i		−	0.803695i
$923$		−	26.4387i		−	0.870241i
$924$		0			0
$925$	−25.4827	+	15.4444i	−0.837868	+	0.507809i
$926$	30.8062			1.01235
$927$		0			0
$928$		−	38.7087i		−	1.27068i
$929$	−36.0189			−1.18174			−0.590872	−	0.806766i	$-0.701216\pi$
$929$	−36.0189			−1.18174			−0.590872	+	0.806766i	$0.701216\pi$
$930$		0			0
$931$	4.89655			0.160478
$932$			25.5058i			0.835469i
$933$		0			0
$934$	40.5517			1.32689
$935$	16.7746	−	29.7817i	0.548590	−	0.973966i
$936$		0			0
$937$		−	45.2421i		−	1.47799i	−0.673709	−	0.738997i	$-0.735300\pi$
$937$		−	45.2421i		−	1.47799i	0.673709	−	0.738997i	$-0.264700\pi$
$938$			26.0205i			0.849599i
$939$		0			0
$940$	−7.11655	+	12.6347i	−0.232116	+	0.412099i
$941$	11.7455			0.382892			0.191446	−	0.981503i	$-0.438682\pi$
$941$	11.7455			0.382892			0.191446	+	0.981503i	$0.438682\pi$
$942$		0			0
$943$		−	56.0781i		−	1.82616i
$944$	−16.4364			−0.534958
$945$		0			0
$946$	10.2996			0.334870
$947$			13.7752i			0.447635i	0.974631	+	0.223817i	$0.0718519\pi$
$947$			13.7752i			0.447635i	−0.974631	+	0.223817i	$0.928148\pi$
$948$		0			0
$949$	7.44584			0.241702
$950$	−7.79310	+	4.72319i	−0.252842	+	0.153241i
$951$		0			0
$952$			7.03434i			0.227984i
$953$			9.01421i			0.291999i	0.989285	+	0.145999i	$0.0466398\pi$
$953$			9.01421i			0.291999i	−0.989285	+	0.145999i	$0.953360\pi$
$954$		0			0
$955$	−6.20155	−	3.49304i	−0.200677	−	0.113032i
$956$	−24.8062			−0.802290
$957$		0			0
$958$			0.667406i			0.0215629i
$959$	−19.1166			−0.617306
$960$		0			0
$961$	40.1724			1.29588
$962$		−	33.2239i		−	1.07118i
$963$		0			0
$964$	19.0796			0.614514
$965$	3.35673	−	5.95953i	0.108057	−	0.191844i
$966$		0			0
$967$			30.3232i			0.975129i	0.873087	+	0.487564i	$0.162115\pi$
$967$			30.3232i			0.975129i	−0.873087	+	0.487564i	$0.837885\pi$
$968$			5.17173i			0.166226i
$969$		0			0
$970$	−10.8615	−	6.11775i	−0.348740	−	0.196429i
$971$	31.5636			1.01292			0.506462	−	0.862262i	$-0.330953\pi$
$971$	31.5636			1.01292			0.506462	+	0.862262i	$0.330953\pi$
$972$		0			0
$973$			1.71588i			0.0550086i
$974$	−48.9280			−1.56775
$975$		0			0
$976$	−50.5957			−1.61953
$977$		−	43.1285i		−	1.37980i	−0.723903	−	0.689902i	$-0.757654\pi$
$977$		−	43.1285i		−	1.37980i	0.723903	−	0.689902i	$-0.242346\pi$
$978$		0			0
$979$	48.1855			1.54002
$980$	12.6082	+	7.10160i	0.402754	+	0.226852i
$981$		0			0
$982$			43.2425i			1.37992i
$983$			7.81570i			0.249282i	0.992202	+	0.124641i	$0.0397779\pi$
$983$			7.81570i			0.249282i	−0.992202	+	0.124641i	$0.960222\pi$
$984$		0			0
$985$	23.5862	−	41.8749i	0.751519	−	1.33425i
$986$	−42.8989			−1.36618
$987$		0			0
$988$		−	4.04272i		−	0.128616i
$989$	7.79310			0.247806
$990$		0			0
$991$	−23.5197			−0.747126			−0.373563	−	0.927605i	$-0.621864\pi$
$991$	−23.5197			−0.747126			−0.373563	+	0.927605i	$0.621864\pi$
$992$			54.4269i			1.72806i
$993$		0			0
$994$	−22.8465			−0.724648
$995$	−9.38465	−	5.28593i	−0.297513	−	0.167575i
$996$		0			0
$997$		−	33.6395i		−	1.06537i	−0.846313	−	0.532686i	$-0.821183\pi$
$997$		−	33.6395i		−	1.06537i	0.846313	−	0.532686i	$-0.178817\pi$
$998$		−	12.4240i		−	0.393276i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.c.d.514.5		6
3.2	odd	2		95.2.b.b.39.2	✓	6
5.2	odd	4		4275.2.a.br.1.2		6
5.3	odd	4		4275.2.a.br.1.5		6
5.4	even	2	inner	855.2.c.d.514.2		6
12.11	even	2		1520.2.d.h.609.2		6
15.2	even	4		475.2.a.j.1.5		6
15.8	even	4		475.2.a.j.1.2		6
15.14	odd	2		95.2.b.b.39.5	yes	6
57.56	even	2		1805.2.b.e.1084.5		6
60.23	odd	4		7600.2.a.ck.1.5		6
60.47	odd	4		7600.2.a.ck.1.2		6
60.59	even	2		1520.2.d.h.609.5		6
285.113	odd	4		9025.2.a.bx.1.5		6
285.227	odd	4		9025.2.a.bx.1.2		6
285.284	even	2		1805.2.b.e.1084.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.2	✓	6	3.2	odd	2
95.2.b.b.39.5	yes	6	15.14	odd	2
475.2.a.j.1.2		6	15.8	even	4
475.2.a.j.1.5		6	15.2	even	4
855.2.c.d.514.2		6	5.4	even	2	inner
855.2.c.d.514.5		6	1.1	even	1	trivial
1520.2.d.h.609.2		6	12.11	even	2
1520.2.d.h.609.5		6	60.59	even	2
1805.2.b.e.1084.2		6	285.284	even	2
1805.2.b.e.1084.5		6	57.56	even	2
4275.2.a.br.1.2		6	5.2	odd	4
4275.2.a.br.1.5		6	5.3	odd	4
7600.2.a.ck.1.2		6	60.47	odd	4
7600.2.a.ck.1.5		6	60.23	odd	4
9025.2.a.bx.1.2		6	285.227	odd	4
9025.2.a.bx.1.5		6	285.113	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)

\(f(q)\)	\(=\)	\(q+1.82254i q^{2} -1.32164 q^{4} +(-1.94827 - 1.09737i) q^{5} -1.45033i q^{7} +1.23634i q^{8} +O(q^{10})\)	\(q+1.82254i q^{2} -1.32164 q^{4} +(-1.94827 - 1.09737i) q^{5} -1.45033i q^{7} +1.23634i q^{8} +(2.00000 - 3.55080i) q^{10} +3.89655 q^{11} +3.05888i q^{13} +2.64327 q^{14} -4.89655 q^{16} +3.92301i q^{17} +1.00000 q^{19} +(2.57491 + 1.45033i) q^{20} +7.10160i q^{22} +5.37334i q^{23} +(2.59155 + 4.27596i) q^{25} -5.57491 q^{26} +1.91681i q^{28} +6.00000 q^{29} -8.43637 q^{31} -6.45146i q^{32} -7.14982 q^{34} +(-1.59155 + 2.82564i) q^{35} +5.95953i q^{37} +1.82254i q^{38} +(1.35673 - 2.40873i) q^{40} -10.4364 q^{41} -1.45033i q^{43} -5.14982 q^{44} -9.79310 q^{46} +4.90686i q^{47} +4.89655 q^{49} +(-7.79310 + 4.72319i) q^{50} -4.04272i q^{52} +4.23127i q^{53} +(-7.59155 - 4.27596i) q^{55} +1.79310 q^{56} +10.9352i q^{58} +3.35673 q^{59} +10.3329 q^{61} -15.3756i q^{62} +1.96491 q^{64} +(3.35673 - 5.95953i) q^{65} +9.84404i q^{67} -5.18479i q^{68} +(-5.14982 - 2.90066i) q^{70} -8.64327 q^{71} -2.43418i q^{73} -10.8615 q^{74} -1.32164 q^{76} -5.65127i q^{77} +12.4364 q^{79} +(9.53982 + 5.37334i) q^{80} -19.0207i q^{82} -12.6635i q^{83} +(4.30500 - 7.64310i) q^{85} +2.64327 q^{86} +4.81746i q^{88} +12.3662 q^{89} +4.43637 q^{91} -7.10160i q^{92} -8.94292 q^{94} +(-1.94827 - 1.09737i) q^{95} -3.05888i q^{97} +8.92414i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} + q^{5}+O(q^{10}) \) 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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