LMFDB - Embedded newform 855.2.k.c.676.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.k (of order $3$, degree $2$, 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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			676.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	855.676
Dual form			855.2.k.c.406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{4} +(0.500000 - 0.866025i) q^{5} +2.00000 q^{7} +3.00000 q^{11} +(2.00000 + 3.46410i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-3.50000 - 2.59808i) q^{19} +2.00000 q^{20} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.00000 + 3.46410i) q^{28} +(1.50000 + 2.59808i) q^{29} +5.00000 q^{31} +(1.00000 - 1.73205i) q^{35} +8.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(2.00000 - 3.46410i) q^{43} +(3.00000 + 5.19615i) q^{44} +(3.00000 + 5.19615i) q^{47} -3.00000 q^{49} +(-4.00000 + 6.92820i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(1.50000 - 2.59808i) q^{55} +(-4.50000 + 7.79423i) q^{59} +(3.50000 + 6.06218i) q^{61} -8.00000 q^{64} +4.00000 q^{65} +(-1.00000 - 1.73205i) q^{67} +(-4.50000 + 7.79423i) q^{71} +(2.00000 - 3.46410i) q^{73} +(1.00000 - 8.66025i) q^{76} +6.00000 q^{77} +(3.50000 - 6.06218i) q^{79} +(2.00000 + 3.46410i) q^{80} +(1.50000 + 2.59808i) q^{89} +(4.00000 + 6.92820i) q^{91} +(6.00000 - 10.3923i) q^{92} +(-4.00000 + 1.73205i) q^{95} +(5.00000 - 8.66025i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{13} - 4 q^{16} - 7 q^{19} + 4 q^{20} - 6 q^{23} - q^{25} + 4 q^{28} + 3 q^{29} + 10 q^{31} + 2 q^{35} + 16 q^{37} - 6 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{47}+ \cdots + 10 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + q^5 + 4 * q^7 + 6 * q^11 + 4 * q^13 - 4 * q^16 - 7 * q^19 + 4 * q^20 - 6 * q^23 - q^25 + 4 * q^28 + 3 * q^29 + 10 * q^31 + 2 * q^35 + 16 * q^37 - 6 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^47 - 6 * q^49 - 8 * q^52 - 6 * q^53 + 3 * q^55 - 9 * q^59 + 7 * q^61 - 16 * q^64 + 8 * q^65 - 2 * q^67 - 9 * q^71 + 4 * q^73 + 2 * q^76 + 12 * q^77 + 7 * q^79 + 4 * q^80 + 3 * q^89 + 8 * q^91 + 12 * q^92 - 8 * q^95 + 10 * q^97

Character values

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

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	−0.443227	+	0.896410i	$0.646166\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$20$	2.00000			0.447214
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$	2.00000	+	3.46410i	0.377964	+	0.654654i
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	−	1.73205i	0.169031	−	0.292770i
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$	3.00000	+	5.19615i	0.452267	+	0.783349i
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$	−4.00000	+	6.92820i	−0.554700	+	0.960769i
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	1.50000	−	2.59808i	0.202260	−	0.350325i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.50000	+	7.79423i	−0.585850	+	1.01472i	0.408919	+	0.912571i	$0.365906\pi$
$59$	−4.50000	+	7.79423i	−0.585850	+	1.01472i	−0.994769	+	0.102151i	$0.967427\pi$
$60$		0			0
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	0.998264	−	0.0588933i	$-0.0187572\pi$
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	−0.550135	+	0.835076i	$0.685424\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	4.00000			0.496139
$66$		0			0
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	0.798454	−	0.602056i	$-0.205652\pi$
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	−0.920623	+	0.390453i	$0.872318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.50000	+	7.79423i	−0.534052	+	0.925005i	0.465157	+	0.885228i	$0.345998\pi$
$71$	−4.50000	+	7.79423i	−0.534052	+	0.925005i	−0.999209	+	0.0397765i	$0.987335\pi$
$72$		0			0
$73$	2.00000	−	3.46410i	0.234082	−	0.405442i	−0.724923	−	0.688830i	$-0.758125\pi$
$73$	2.00000	−	3.46410i	0.234082	−	0.405442i	0.959006	+	0.283387i	$0.0914581\pi$
$74$		0			0
$75$		0			0
$76$	1.00000	−	8.66025i	0.114708	−	0.993399i
$77$	6.00000			0.683763
$78$		0			0
$79$	3.50000	−	6.06218i	0.393781	−	0.682048i	−0.599164	−	0.800626i	$-0.704500\pi$
$79$	3.50000	−	6.06218i	0.393781	−	0.682048i	0.992945	+	0.118578i	$0.0378336\pi$
$80$	2.00000	+	3.46410i	0.223607	+	0.387298i
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.50000	+	2.59808i	0.159000	+	0.275396i	0.934508	−	0.355942i	$-0.115840\pi$
$89$	1.50000	+	2.59808i	0.159000	+	0.275396i	−0.775509	+	0.631337i	$0.782506\pi$
$90$		0			0
$91$	4.00000	+	6.92820i	0.419314	+	0.726273i
$92$	6.00000	−	10.3923i	0.625543	−	1.08347i
$93$		0			0
$94$		0			0
$95$	−4.00000	+	1.73205i	−0.410391	+	0.177705i
$96$		0			0
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	−0.492287	−	0.870433i	$-0.663839\pi$
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	0.999961	−	0.00888289i	$-0.00282755\pi$
$98$		0			0
$99$		0			0
$100$	1.00000	−	1.73205i	0.100000	−	0.173205i
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	−5.50000	+	9.52628i	−0.526804	+	0.912452i	0.472708	+	0.881219i	$0.343277\pi$
$109$	−5.50000	+	9.52628i	−0.526804	+	0.912452i	−0.999512	+	0.0312328i	$0.990057\pi$
$110$		0			0
$111$		0			0
$112$	−4.00000	+	6.92820i	−0.377964	+	0.654654i
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	−6.00000			−0.559503
$116$	−3.00000	+	5.19615i	−0.278543	+	0.482451i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$	5.00000	+	8.66025i	0.449013	+	0.777714i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−1.00000	−	1.73205i	−0.0887357	−	0.153695i	0.818241	−	0.574875i	$-0.194949\pi$
$127$	−1.00000	−	1.73205i	−0.0887357	−	0.153695i	−0.906977	+	0.421180i	$0.861616\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	−0.475380	−	0.879781i	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		0			0
$133$	−7.00000	−	5.19615i	−0.606977	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	−10.0000	−	17.3205i	−0.848189	−	1.46911i	−0.882823	−	0.469706i	$-0.844360\pi$
$139$	−10.0000	−	17.3205i	−0.848189	−	1.46911i	0.0346338	−	0.999400i	$-0.488974\pi$
$140$	4.00000			0.338062
$141$		0			0
$142$		0			0
$143$	6.00000	+	10.3923i	0.501745	+	0.869048i
$144$		0			0
$145$	3.00000			0.249136
$146$		0			0
$147$		0			0
$148$	8.00000	+	13.8564i	0.657596	+	1.13899i
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	−0.620701	−	0.784047i	$-0.713152\pi$
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	0.989355	+	0.145519i	$0.0464853\pi$
$150$		0			0
$151$	−19.0000			−1.54620			−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000			−1.54620			−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.50000	−	4.33013i	0.200805	−	0.347804i
$156$		0			0
$157$	8.00000	−	13.8564i	0.638470	−	1.10586i	−0.347299	−	0.937754i	$-0.612901\pi$
$157$	8.00000	−	13.8564i	0.638470	−	1.10586i	0.985769	−	0.168107i	$-0.0537655\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00000	−	10.3923i	−0.472866	−	0.819028i
$162$		0			0
$163$	−22.0000			−1.72317			−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000			−1.72317			−0.861586	+	0.507611i	$0.830529\pi$
$164$	−12.0000			−0.937043
$165$		0			0
$166$		0			0
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	0.726293	−	0.687386i	$-0.241242\pi$
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	−0.958440	+	0.285295i	$0.907908\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	−0.729155	−	0.684349i	$-0.760087\pi$
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	0.957241	+	0.289292i	$0.0934200\pi$
$174$		0			0
$175$	−1.00000	−	1.73205i	−0.0755929	−	0.130931i
$176$	−6.00000	+	10.3923i	−0.452267	+	0.783349i
$177$		0			0
$178$		0			0
$179$	−21.0000			−1.56961			−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000			−1.56961			−0.784807	+	0.619740i	$0.787238\pi$
$180$		0			0
$181$	−1.00000	−	1.73205i	−0.0743294	−	0.128742i	0.826465	−	0.562988i	$-0.190348\pi$
$181$	−1.00000	−	1.73205i	−0.0743294	−	0.128742i	−0.900794	+	0.434246i	$0.857015\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.00000	−	6.92820i	0.294086	−	0.509372i
$186$		0			0
$187$		0			0
$188$	−6.00000	+	10.3923i	−0.437595	+	0.757937i
$189$		0			0
$190$		0			0
$191$	21.0000			1.51951			0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000			1.51951			0.759753	+	0.650211i	$0.225320\pi$
$192$		0			0
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$		0			0
$195$		0			0
$196$	−3.00000	−	5.19615i	−0.214286	−	0.371154i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−2.50000	−	4.33013i	−0.177220	−	0.306955i	0.763707	−	0.645563i	$-0.223377\pi$
$199$	−2.50000	−	4.33013i	−0.177220	−	0.306955i	−0.940927	+	0.338608i	$0.890044\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.00000	+	5.19615i	0.210559	+	0.364698i
$204$		0			0
$205$	3.00000	+	5.19615i	0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$	−16.0000			−1.10940
$209$	−10.5000	−	7.79423i	−0.726300	−	0.539138i
$210$		0			0
$211$	−8.50000	+	14.7224i	−0.585164	+	1.01353i	0.409691	+	0.912224i	$0.365637\pi$
$211$	−8.50000	+	14.7224i	−0.585164	+	1.01353i	−0.994855	+	0.101310i	$0.967697\pi$
$212$	6.00000	−	10.3923i	0.412082	−	0.713746i
$213$		0			0
$214$		0			0
$215$	−2.00000	−	3.46410i	−0.136399	−	0.236250i
$216$		0			0
$217$	10.0000			0.678844
$218$		0			0
$219$		0			0
$220$	6.00000			0.404520
$221$		0			0
$222$		0			0
$223$	5.00000	−	8.66025i	0.334825	−	0.579934i	−0.648626	−	0.761107i	$-0.724656\pi$
$223$	5.00000	−	8.66025i	0.334825	−	0.579934i	0.983451	+	0.181173i	$0.0579895\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$	−19.0000			−1.25556			−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000			−1.25556			−0.627778	+	0.778393i	$0.716035\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.0000	+	20.7846i	−0.786146	+	1.36165i	0.142166	+	0.989843i	$0.454593\pi$
$233$	−12.0000	+	20.7846i	−0.786146	+	1.36165i	−0.928312	+	0.371802i	$0.878740\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$	−18.0000			−1.17170
$237$		0			0
$238$		0			0
$239$	−15.0000			−0.970269			−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000			−0.970269			−0.485135	+	0.874439i	$0.661229\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$		0			0
$244$	−7.00000	+	12.1244i	−0.448129	+	0.776182i
$245$	−1.50000	+	2.59808i	−0.0958315	+	0.165985i
$246$		0			0
$247$	2.00000	−	17.3205i	0.127257	−	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.50000	+	12.9904i	0.473396	+	0.819946i	0.999536	−	0.0304521i	$-0.00969471\pi$
$251$	7.50000	+	12.9904i	0.473396	+	0.819946i	−0.526140	+	0.850398i	$0.676361\pi$
$252$		0			0
$253$	−9.00000	−	15.5885i	−0.565825	−	0.980038i
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	0.948523	+	0.316709i	$0.102578\pi$
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	−0.199983	+	0.979799i	$0.564089\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$	4.00000	+	6.92820i	0.248069	+	0.429669i
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$		0			0
$268$	2.00000	−	3.46410i	0.122169	−	0.211604i
$269$	−13.5000	+	23.3827i	−0.823110	+	1.42567i	0.0802460	+	0.996775i	$0.474429\pi$
$269$	−13.5000	+	23.3827i	−0.823110	+	1.42567i	−0.903356	+	0.428892i	$0.858904\pi$
$270$		0			0
$271$	12.5000	−	21.6506i	0.759321	−	1.31518i	−0.183876	−	0.982949i	$-0.558865\pi$
$271$	12.5000	−	21.6506i	0.759321	−	1.31518i	0.943197	−	0.332233i	$-0.107802\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	−28.0000			−1.68236			−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000			−1.68236			−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$	−18.0000			−1.06810
$285$		0			0
$286$		0			0
$287$	−6.00000	+	10.3923i	−0.354169	+	0.613438i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	8.00000			0.468165
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	4.50000	+	7.79423i	0.262000	+	0.453798i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	20.7846i	0.693978	−	1.20201i
$300$		0			0
$301$	4.00000	−	6.92820i	0.230556	−	0.399335i
$302$		0			0
$303$		0			0
$304$	16.0000	−	6.92820i	0.917663	−	0.397360i
$305$	7.00000			0.400819
$306$		0			0
$307$	8.00000	−	13.8564i	0.456584	−	0.790827i	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	8.00000	−	13.8564i	0.456584	−	0.790827i	0.998778	+	0.0494267i	$0.0157394\pi$
$308$	6.00000	+	10.3923i	0.341882	+	0.592157i
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$		0			0
$315$		0			0
$316$	14.0000			0.787562
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$	4.50000	+	7.79423i	0.251952	+	0.436393i
$320$	−4.00000	+	6.92820i	−0.223607	+	0.387298i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.00000	−	3.46410i	0.110940	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000	+	10.3923i	0.330791	+	0.572946i
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.00000			−0.109272
$336$		0			0
$337$	−7.00000	+	12.1244i	−0.381314	+	0.660456i	−0.991250	−	0.131995i	$-0.957862\pi$
$337$	−7.00000	+	12.1244i	−0.381314	+	0.660456i	0.609936	+	0.792451i	$0.291195\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	15.0000			0.812296
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	−0.110885	−	0.993833i	$-0.535369\pi$
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	0.916127	−	0.400887i	$-0.131298\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	4.50000	+	7.79423i	0.238835	+	0.413675i
$356$	−3.00000	+	5.19615i	−0.159000	+	0.275396i
$357$		0			0
$358$		0			0
$359$	−12.0000	+	20.7846i	−0.633336	+	1.09697i	0.353529	+	0.935423i	$0.384981\pi$
$359$	−12.0000	+	20.7846i	−0.633336	+	1.09697i	−0.986865	+	0.161546i	$0.948352\pi$
$360$		0			0
$361$	5.50000	+	18.1865i	0.289474	+	0.957186i
$362$		0			0
$363$		0			0
$364$	−8.00000	+	13.8564i	−0.419314	+	0.726273i
$365$	−2.00000	−	3.46410i	−0.104685	−	0.181319i
$366$		0			0
$367$	2.00000	+	3.46410i	0.104399	+	0.180825i	0.913493	−	0.406855i	$-0.133375\pi$
$367$	2.00000	+	3.46410i	0.104399	+	0.180825i	−0.809093	+	0.587680i	$0.800041\pi$
$368$	24.0000			1.25109
$369$		0			0
$370$		0			0
$371$	−6.00000	−	10.3923i	−0.311504	−	0.539542i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.00000	+	10.3923i	−0.309016	+	0.535231i
$378$		0			0
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$	−7.00000	−	5.19615i	−0.359092	−	0.266557i
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$	3.00000	−	5.19615i	0.152894	−	0.264820i
$386$		0			0
$387$		0			0
$388$	20.0000			1.01535
$389$	−13.5000	−	23.3827i	−0.684477	−	1.18555i	−0.973601	−	0.228257i	$-0.926697\pi$
$389$	−13.5000	−	23.3827i	−0.684477	−	1.18555i	0.289124	−	0.957292i	$-0.406636\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.50000	−	6.06218i	−0.176104	−	0.305021i
$396$		0			0
$397$	−10.0000	+	17.3205i	−0.501886	+	0.869291i	0.498112	+	0.867113i	$0.334027\pi$
$397$	−10.0000	+	17.3205i	−0.501886	+	0.869291i	−0.999998	+	0.00217869i	$0.999307\pi$
$398$		0			0
$399$		0			0
$400$	4.00000			0.200000
$401$	4.50000	−	7.79423i	0.224719	−	0.389225i	−0.731516	−	0.681824i	$-0.761187\pi$
$401$	4.50000	−	7.79423i	0.224719	−	0.389225i	0.956235	+	0.292599i	$0.0945202\pi$
$402$		0			0
$403$	10.0000	+	17.3205i	0.498135	+	0.862796i
$404$	15.0000	−	25.9808i	0.746278	−	1.29259i
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	6.50000	+	11.2583i	0.321404	+	0.556689i	0.980778	−	0.195127i	$-0.0625118\pi$
$409$	6.50000	+	11.2583i	0.321404	+	0.556689i	−0.659374	+	0.751815i	$0.729178\pi$
$410$		0			0
$411$		0			0
$412$	8.00000	+	13.8564i	0.394132	+	0.682656i
$413$	−9.00000	+	15.5885i	−0.442861	+	0.767058i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	−0.995351	−	0.0963145i	$-0.969295\pi$
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	0.581086	+	0.813842i	$0.302628\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7.00000	+	12.1244i	0.338754	+	0.586739i
$428$	18.0000	+	31.1769i	0.870063	+	1.50699i
$429$		0			0
$430$		0			0
$431$	7.50000	+	12.9904i	0.361262	+	0.625725i	0.988169	−	0.153370i	$-0.0490126\pi$
$431$	7.50000	+	12.9904i	0.361262	+	0.625725i	−0.626907	+	0.779094i	$0.715679\pi$
$432$		0			0
$433$	−13.0000	−	22.5167i	−0.624740	−	1.08208i	−0.988591	−	0.150624i	$-0.951872\pi$
$433$	−13.0000	−	22.5167i	−0.624740	−	1.08208i	0.363851	−	0.931457i	$-0.381462\pi$
$434$		0			0
$435$		0			0
$436$	−22.0000			−1.05361
$437$	−3.00000	+	25.9808i	−0.143509	+	1.24283i
$438$		0			0
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	−0.853847	−	0.520524i	$-0.825737\pi$
$439$	0.500000	−	0.866025i	0.0238637	−	0.0413331i	0.877711	+	0.479191i	$0.159070\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.0000	−	31.1769i	−0.855206	−	1.48126i	−0.876454	−	0.481486i	$-0.840097\pi$
$443$	−18.0000	−	31.1769i	−0.855206	−	1.48126i	0.0212481	−	0.999774i	$-0.493236\pi$
$444$		0			0
$445$	3.00000			0.142214
$446$		0			0
$447$		0			0
$448$	−16.0000			−0.755929
$449$	15.0000			0.707894			0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000			0.707894			0.353947	+	0.935266i	$0.384839\pi$
$450$		0			0
$451$	−9.00000	+	15.5885i	−0.423793	+	0.734032i
$452$	−18.0000	−	31.1769i	−0.846649	−	1.46644i
$453$		0			0
$454$		0			0
$455$	8.00000			0.375046
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$	−6.00000	−	10.3923i	−0.279751	−	0.484544i
$461$	−7.50000	+	12.9904i	−0.349310	+	0.605022i	−0.986127	−	0.165992i	$-0.946917\pi$
$461$	−7.50000	+	12.9904i	−0.349310	+	0.605022i	0.636817	+	0.771015i	$0.280251\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$	−12.0000			−0.557086
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−2.00000	−	3.46410i	−0.0923514	−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000	−	10.3923i	0.275880	−	0.477839i
$474$		0			0
$475$	−0.500000	+	4.33013i	−0.0229416	+	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.50000	−	2.59808i	−0.0685367	−	0.118709i	0.829721	−	0.558179i	$-0.188500\pi$
$479$	−1.50000	−	2.59808i	−0.0685367	−	0.118709i	−0.898257	+	0.439470i	$0.855166\pi$
$480$		0			0
$481$	16.0000	+	27.7128i	0.729537	+	1.26360i
$482$		0			0
$483$		0			0
$484$	−2.00000	−	3.46410i	−0.0909091	−	0.157459i
$485$	−5.00000	−	8.66025i	−0.227038	−	0.393242i
$486$		0			0
$487$	20.0000			0.906287			0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000			0.906287			0.453143	+	0.891438i	$0.350303\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16.5000	+	28.5788i	−0.744635	+	1.28974i	0.205731	+	0.978609i	$0.434043\pi$
$491$	−16.5000	+	28.5788i	−0.744635	+	1.28974i	−0.950365	+	0.311136i	$0.899290\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−10.0000	+	17.3205i	−0.449013	+	0.777714i
$497$	−9.00000	+	15.5885i	−0.403705	+	0.699238i
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$	−1.00000	−	1.73205i	−0.0447214	−	0.0774597i
$501$		0			0
$502$		0			0
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	0.791361	−	0.611349i	$-0.209373\pi$
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	−0.925124	+	0.379664i	$0.876040\pi$
$504$		0			0
$505$	−15.0000			−0.667491
$506$		0			0
$507$		0			0
$508$	2.00000	−	3.46410i	0.0887357	−	0.153695i
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	0.993593	−	0.113020i	$-0.0360525\pi$
$509$	9.00000	+	15.5885i	0.398918	+	0.690946i	−0.594675	+	0.803966i	$0.702719\pi$
$510$		0			0
$511$	4.00000	−	6.92820i	0.176950	−	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.00000	−	6.92820i	0.176261	−	0.305293i
$516$		0			0
$517$	9.00000	+	15.5885i	0.395820	+	0.685580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	11.0000	+	19.0526i	0.480996	+	0.833110i	0.999762	−	0.0218062i	$-0.00694167\pi$
$523$	11.0000	+	19.0526i	0.480996	+	0.833110i	−0.518766	+	0.854916i	$0.673608\pi$
$524$	24.0000			1.04844
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$	2.00000	−	17.3205i	0.0867110	−	0.750939i
$533$	−24.0000			−1.03956
$534$		0			0
$535$	9.00000	−	15.5885i	0.389104	−	0.673948i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.00000			−0.387657
$540$		0			0
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	0.999042	+	0.0437584i	$0.0139332\pi$
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	−0.461625	+	0.887075i	$0.652733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.50000	+	9.52628i	0.235594	+	0.408061i
$546$		0			0
$547$	−13.0000	−	22.5167i	−0.555840	−	0.962743i	−0.997838	−	0.0657267i	$-0.979063\pi$
$547$	−13.0000	−	22.5167i	−0.555840	−	0.962743i	0.441998	−	0.897016i	$-0.354270\pi$
$548$	18.0000	−	31.1769i	0.768922	−	1.33181i
$549$		0			0
$550$		0			0
$551$	1.50000	−	12.9904i	0.0639021	−	0.553409i
$552$		0			0
$553$	7.00000	−	12.1244i	0.297670	−	0.515580i
$554$		0			0
$555$		0			0
$556$	20.0000	−	34.6410i	0.848189	−	1.46911i
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$	4.00000	+	6.92820i	0.169031	+	0.292770i
$561$		0			0
$562$		0			0
$563$	18.0000			0.758610			0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000			0.758610			0.379305	+	0.925272i	$0.376163\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.00000			−0.377300			−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000			−0.377300			−0.188650	+	0.982044i	$0.560411\pi$
$570$		0			0
$571$	−31.0000			−1.29731			−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000			−1.29731			−0.648655	+	0.761083i	$0.724668\pi$
$572$	−12.0000	+	20.7846i	−0.501745	+	0.869048i
$573$		0			0
$574$		0			0
$575$	−3.00000	+	5.19615i	−0.125109	+	0.216695i
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$		0			0
$580$	3.00000	+	5.19615i	0.124568	+	0.215758i
$581$		0			0
$582$		0			0
$583$	−9.00000	−	15.5885i	−0.372742	−	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.00000	−	5.19615i	0.123823	−	0.214468i	−0.797449	−	0.603386i	$-0.793818\pi$
$587$	3.00000	−	5.19615i	0.123823	−	0.214468i	0.921272	+	0.388918i	$0.127151\pi$
$588$		0			0
$589$	−17.5000	−	12.9904i	−0.721075	−	0.535259i
$590$		0			0
$591$		0			0
$592$	−16.0000	+	27.7128i	−0.657596	+	1.13899i
$593$	−9.00000	−	15.5885i	−0.369586	−	0.640141i	0.619915	−	0.784669i	$-0.287167\pi$
$593$	−9.00000	−	15.5885i	−0.369586	−	0.640141i	−0.989501	+	0.144528i	$0.953834\pi$
$594$		0			0
$595$		0			0
$596$	18.0000			0.737309
$597$		0			0
$598$		0			0
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	0.954521	+	0.298143i	$0.0963673\pi$
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	−0.219061	+	0.975711i	$0.570299\pi$
$600$		0			0
$601$	5.00000			0.203954			0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000			0.203954			0.101977	+	0.994787i	$0.467483\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	−	32.9090i	−0.773099	−	1.33905i
$605$	−1.00000	+	1.73205i	−0.0406558	+	0.0704179i
$606$		0			0
$607$	20.0000			0.811775			0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000			0.811775			0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	+	20.7846i	−0.485468	+	0.840855i
$612$		0			0
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	−0.885514	−	0.464614i	$-0.846193\pi$
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	0.845124	+	0.534570i	$0.179527\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.00000	+	15.5885i	0.362326	+	0.627568i	0.988343	−	0.152242i	$-0.0486493\pi$
$617$	9.00000	+	15.5885i	0.362326	+	0.627568i	−0.626017	+	0.779809i	$0.715316\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$	10.0000			0.401610
$621$		0			0
$622$		0			0
$623$	3.00000	+	5.19615i	0.120192	+	0.208179i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$	32.0000			1.27694
$629$		0			0
$630$		0			0
$631$	−2.50000	−	4.33013i	−0.0995234	−	0.172380i	0.811964	−	0.583707i	$-0.198398\pi$
$631$	−2.50000	−	4.33013i	−0.0995234	−	0.172380i	−0.911487	+	0.411328i	$0.865065\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.00000			−0.0793676
$636$		0			0
$637$	−6.00000	−	10.3923i	−0.237729	−	0.411758i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.50000	−	2.59808i	0.0592464	−	0.102618i	−0.834881	−	0.550431i	$-0.814464\pi$
$641$	1.50000	−	2.59808i	0.0592464	−	0.102618i	0.894127	+	0.447813i	$0.147797\pi$
$642$		0			0
$643$	−1.00000	+	1.73205i	−0.0394362	+	0.0683054i	−0.885070	−	0.465458i	$-0.845890\pi$
$643$	−1.00000	+	1.73205i	−0.0394362	+	0.0683054i	0.845634	+	0.533764i	$0.179223\pi$
$644$	12.0000	−	20.7846i	0.472866	−	0.819028i
$645$		0			0
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$	−13.5000	+	23.3827i	−0.529921	+	0.917851i
$650$		0			0
$651$		0			0
$652$	−22.0000	−	38.1051i	−0.861586	−	1.49231i
$653$	−12.0000			−0.469596			−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000			−0.469596			−0.234798	+	0.972044i	$0.575443\pi$
$654$		0			0
$655$	−6.00000	−	10.3923i	−0.234439	−	0.406061i
$656$	−12.0000	−	20.7846i	−0.468521	−	0.811503i
$657$		0			0
$658$		0			0
$659$	18.0000	+	31.1769i	0.701180	+	1.21448i	0.968052	+	0.250748i	$0.0806766\pi$
$659$	18.0000	+	31.1769i	0.701180	+	1.21448i	−0.266872	+	0.963732i	$0.585990\pi$
$660$		0			0
$661$	−20.5000	−	35.5070i	−0.797358	−	1.38106i	−0.921331	−	0.388778i	$-0.872897\pi$
$661$	−20.5000	−	35.5070i	−0.797358	−	1.38106i	0.123974	−	0.992286i	$-0.460436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.00000	+	3.46410i	−0.310227	+	0.134332i
$666$		0			0
$667$	9.00000	−	15.5885i	0.348481	−	0.603587i
$668$	6.00000	−	10.3923i	0.232147	−	0.402090i
$669$		0			0
$670$		0			0
$671$	10.5000	+	18.1865i	0.405348	+	0.702083i
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		0			0
$675$		0			0
$676$	−6.00000			−0.230769
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$	10.0000	−	17.3205i	0.383765	−	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−30.0000			−1.14792			−0.573959	−	0.818884i	$-0.694593\pi$
$683$	−30.0000			−1.14792			−0.573959	+	0.818884i	$0.694593\pi$
$684$		0			0
$685$	−18.0000			−0.687745
$686$		0			0
$687$		0			0
$688$	8.00000	+	13.8564i	0.304997	+	0.528271i
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	35.0000			1.33146			0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000			1.33146			0.665731	+	0.746191i	$0.268120\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$		0			0
$695$	−20.0000			−0.758643
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.00000	−	3.46410i	0.0755929	−	0.130931i
$701$	−15.0000	+	25.9808i	−0.566542	+	0.981280i	0.430362	+	0.902656i	$0.358386\pi$
$701$	−15.0000	+	25.9808i	−0.566542	+	0.981280i	−0.996904	+	0.0786236i	$0.974947\pi$
$702$		0			0
$703$	−28.0000	−	20.7846i	−1.05604	−	0.783906i
$704$	−24.0000			−0.904534
$705$		0			0
$706$		0			0
$707$	−15.0000	−	25.9808i	−0.564133	−	0.977107i
$708$		0			0
$709$	−20.5000	−	35.5070i	−0.769894	−	1.33349i	−0.937620	−	0.347661i	$-0.886976\pi$
$709$	−20.5000	−	35.5070i	−0.769894	−	1.33349i	0.167727	−	0.985834i	$-0.446357\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.0000	−	25.9808i	−0.561754	−	0.972987i
$714$		0			0
$715$	12.0000			0.448775
$716$	−21.0000	−	36.3731i	−0.784807	−	1.35933i
$717$		0			0
$718$		0			0
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	−0.601075	−	0.799193i	$-0.705261\pi$
$719$	10.5000	−	18.1865i	0.391584	−	0.678243i	0.992659	+	0.120950i	$0.0385939\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$	2.00000	−	3.46410i	0.0743294	−	0.128742i
$725$	1.50000	−	2.59808i	0.0557086	−	0.0964901i
$726$		0			0
$727$	−19.0000	+	32.9090i	−0.704671	+	1.22053i	0.262139	+	0.965030i	$0.415572\pi$
$727$	−19.0000	+	32.9090i	−0.704671	+	1.22053i	−0.966810	+	0.255496i	$0.917761\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	32.0000			1.18195			0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000			1.18195			0.590973	+	0.806691i	$0.298744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.00000	−	5.19615i	−0.110506	−	0.191403i
$738$		0			0
$739$	−5.50000	+	9.52628i	−0.202321	+	0.350430i	−0.949276	−	0.314445i	$-0.898182\pi$
$739$	−5.50000	+	9.52628i	−0.202321	+	0.350430i	0.746955	+	0.664875i	$0.231515\pi$
$740$	16.0000			0.588172
$741$		0			0
$742$		0			0
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	−0.557470	−	0.830197i	$-0.688228\pi$
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	0.997707	+	0.0676840i	$0.0215610\pi$
$744$		0			0
$745$	−4.50000	−	7.79423i	−0.164867	−	0.285558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.0000			1.31541
$750$		0			0
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	0.922792	−	0.385299i	$-0.125902\pi$
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	−0.795075	+	0.606511i	$0.792568\pi$
$752$	−24.0000			−0.875190
$753$		0			0
$754$		0			0
$755$	−9.50000	+	16.4545i	−0.345740	+	0.598840i
$756$		0			0
$757$	−16.0000	+	27.7128i	−0.581530	+	1.00724i	0.413768	+	0.910382i	$0.364212\pi$
$757$	−16.0000	+	27.7128i	−0.581530	+	1.00724i	−0.995298	+	0.0968571i	$0.969121\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	−11.0000	+	19.0526i	−0.398227	+	0.689749i
$764$	21.0000	+	36.3731i	0.759753	+	1.31593i
$765$		0			0
$766$		0			0
$767$	−36.0000			−1.29988
$768$		0			0
$769$	−20.5000	−	35.5070i	−0.739249	−	1.28042i	−0.952834	−	0.303492i	$-0.901847\pi$
$769$	−20.5000	−	35.5070i	−0.739249	−	1.28042i	0.213585	−	0.976924i	$-0.431486\pi$
$770$		0			0
$771$		0			0
$772$	−28.0000			−1.00774
$773$	−9.00000	−	15.5885i	−0.323708	−	0.560678i	0.657542	−	0.753418i	$-0.271596\pi$
$773$	−9.00000	−	15.5885i	−0.323708	−	0.560678i	−0.981250	+	0.192740i	$0.938263\pi$
$774$		0			0
$775$	−2.50000	−	4.33013i	−0.0898027	−	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000	−	10.3923i	0.859889	−	0.372343i
$780$		0			0
$781$	−13.5000	+	23.3827i	−0.483068	+	0.836698i
$782$		0			0
$783$		0			0
$784$	6.00000	−	10.3923i	0.214286	−	0.371154i
$785$	−8.00000	−	13.8564i	−0.285532	−	0.494556i
$786$		0			0
$787$	−46.0000			−1.63972			−0.819861	−	0.572562i	$-0.805950\pi$
$787$	−46.0000			−1.63972			−0.819861	+	0.572562i	$0.805950\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.0000			−1.28001
$792$		0			0
$793$	−14.0000	+	24.2487i	−0.497155	+	0.861097i
$794$		0			0
$795$		0			0
$796$	5.00000	−	8.66025i	0.177220	−	0.306955i
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.00000	−	10.3923i	0.211735	−	0.366736i
$804$		0			0
$805$	−12.0000			−0.422944
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.0000			1.58212			0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000			1.58212			0.791058	+	0.611741i	$0.209531\pi$
$810$		0			0
$811$	3.50000	+	6.06218i	0.122902	+	0.212872i	0.920911	−	0.389774i	$-0.127447\pi$
$811$	3.50000	+	6.06218i	0.122902	+	0.212872i	−0.798009	+	0.602645i	$0.794113\pi$
$812$	−6.00000	+	10.3923i	−0.210559	+	0.364698i
$813$		0			0
$814$		0			0
$815$	−11.0000	+	19.0526i	−0.385313	+	0.667382i
$816$		0			0
$817$	−16.0000	+	6.92820i	−0.559769	+	0.242387i
$818$		0			0
$819$		0			0
$820$	−6.00000	+	10.3923i	−0.209529	+	0.362915i
$821$	13.5000	+	23.3827i	0.471153	+	0.816061i	0.999456	−	0.0329950i	$-0.0105045\pi$
$821$	13.5000	+	23.3827i	0.471153	+	0.816061i	−0.528302	+	0.849056i	$0.677171\pi$
$822$		0			0
$823$	11.0000	+	19.0526i	0.383436	+	0.664130i	0.991551	−	0.129719i	$-0.0414074\pi$
$823$	11.0000	+	19.0526i	0.383436	+	0.664130i	−0.608115	+	0.793849i	$0.708074\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.0000	−	25.9808i	−0.521601	−	0.903440i	−0.999684	−	0.0251251i	$-0.992002\pi$
$827$	−15.0000	−	25.9808i	−0.521601	−	0.903440i	0.478083	−	0.878315i	$-0.341332\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$	−16.0000	−	27.7128i	−0.554700	−	0.960769i
$833$		0			0
$834$		0			0
$835$	−6.00000			−0.207639
$836$	3.00000	−	25.9808i	0.103757	−	0.898563i
$837$		0			0
$838$		0			0
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	−0.995353	−	0.0962912i	$-0.969302\pi$
$839$	−12.0000	+	20.7846i	−0.414286	+	0.717564i	0.581067	+	0.813856i	$0.302635\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$	−34.0000			−1.17033
$845$	1.50000	+	2.59808i	0.0516016	+	0.0893765i
$846$		0			0
$847$	−4.00000			−0.137442
$848$	24.0000			0.824163
$849$		0			0
$850$		0			0
$851$	−24.0000	−	41.5692i	−0.822709	−	1.42497i
$852$		0			0
$853$	−7.00000	+	12.1244i	−0.239675	+	0.415130i	−0.960621	−	0.277862i	$-0.910374\pi$
$853$	−7.00000	+	12.1244i	−0.239675	+	0.415130i	0.720946	+	0.692992i	$0.243708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.0000	+	41.5692i	−0.819824	+	1.41998i	0.0859870	+	0.996296i	$0.472596\pi$
$857$	−24.0000	+	41.5692i	−0.819824	+	1.41998i	−0.905811	+	0.423681i	$0.860738\pi$
$858$		0			0
$859$	21.5000	+	37.2391i	0.733571	+	1.27058i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	21.5000	+	37.2391i	0.733571	+	1.27058i	−0.221777	+	0.975097i	$0.571186\pi$
$860$	4.00000	−	6.92820i	0.136399	−	0.236250i
$861$		0			0
$862$		0			0
$863$	18.0000			0.612727			0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000			0.612727			0.306364	+	0.951915i	$0.400888\pi$
$864$		0			0
$865$	−3.00000	−	5.19615i	−0.102003	−	0.176674i
$866$		0			0
$867$		0			0
$868$	10.0000	+	17.3205i	0.339422	+	0.587896i
$869$	10.5000	−	18.1865i	0.356188	−	0.616936i
$870$		0			0
$871$	4.00000	−	6.92820i	0.135535	−	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.00000			−0.0676123
$876$		0			0
$877$	−25.0000	+	43.3013i	−0.844190	+	1.46218i	0.0421327	+	0.999112i	$0.486585\pi$
$877$	−25.0000	+	43.3013i	−0.844190	+	1.46218i	−0.886323	+	0.463068i	$0.846749\pi$
$878$		0			0
$879$		0			0
$880$	6.00000	+	10.3923i	0.202260	+	0.350325i
$881$	33.0000			1.11180			0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000			1.11180			0.555899	+	0.831250i	$0.312374\pi$
$882$		0			0
$883$	−22.0000	−	38.1051i	−0.740359	−	1.28234i	−0.952332	−	0.305064i	$-0.901322\pi$
$883$	−22.0000	−	38.1051i	−0.740359	−	1.28234i	0.211973	−	0.977276i	$-0.432011\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	0.747539	−	0.664218i	$-0.231235\pi$
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	−0.948999	+	0.315279i	$0.897902\pi$
$888$		0			0
$889$	−2.00000	−	3.46410i	−0.0670778	−	0.116182i
$890$		0			0
$891$		0			0
$892$	20.0000			0.669650
$893$	3.00000	−	25.9808i	0.100391	−	0.869413i
$894$		0			0
$895$	−10.5000	+	18.1865i	−0.350976	+	0.607909i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7.50000	+	12.9904i	0.250139	+	0.433253i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.00000			−0.0664822
$906$		0			0
$907$	−4.00000	+	6.92820i	−0.132818	+	0.230047i	−0.924762	−	0.380547i	$-0.875736\pi$
$907$	−4.00000	+	6.92820i	−0.132818	+	0.230047i	0.791944	+	0.610594i	$0.209069\pi$
$908$	24.0000	+	41.5692i	0.796468	+	1.37952i
$909$		0			0
$910$		0			0
$911$	45.0000			1.49092			0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000			1.49092			0.745458	+	0.666552i	$0.232231\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−19.0000	−	32.9090i	−0.627778	−	1.08734i
$917$	12.0000	−	20.7846i	0.396275	−	0.686368i
$918$		0			0
$919$	32.0000			1.05558			0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000			1.05558			0.527791	+	0.849374i	$0.323020\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−36.0000			−1.18495
$924$		0			0
$925$	−4.00000	−	6.92820i	−0.131519	−	0.227798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.5000	−	49.3634i	0.935055	−	1.61956i	0.160518	−	0.987033i	$-0.448683\pi$
$929$	28.5000	−	49.3634i	0.935055	−	1.61956i	0.774536	−	0.632529i	$-0.217983\pi$
$930$		0			0
$931$	10.5000	+	7.79423i	0.344124	+	0.255446i
$932$	−48.0000			−1.57229
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	17.0000	+	29.4449i	0.555366	+	0.961922i	0.997875	+	0.0651578i	$0.0207551\pi$
$937$	17.0000	+	29.4449i	0.555366	+	0.961922i	−0.442509	+	0.896764i	$0.645912\pi$
$938$		0			0
$939$		0			0
$940$	6.00000	+	10.3923i	0.195698	+	0.338960i
$941$	−1.50000	−	2.59808i	−0.0488986	−	0.0846949i	0.840540	−	0.541749i	$-0.182238\pi$
$941$	−1.50000	−	2.59808i	−0.0488986	−	0.0846949i	−0.889439	+	0.457054i	$0.848904\pi$
$942$		0			0
$943$	36.0000			1.17232
$944$	−18.0000	−	31.1769i	−0.585850	−	1.01472i
$945$		0			0
$946$		0			0
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	−0.999896	−	0.0144491i	$-0.995401\pi$
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	0.512461	+	0.858710i	$0.328734\pi$
$948$		0			0
$949$	16.0000			0.519382
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.0000	+	41.5692i	−0.777436	+	1.34656i	0.155979	+	0.987760i	$0.450147\pi$
$953$	−24.0000	+	41.5692i	−0.777436	+	1.34656i	−0.933415	+	0.358799i	$0.883186\pi$
$954$		0			0
$955$	10.5000	−	18.1865i	0.339772	−	0.588502i
$956$	−15.0000	−	25.9808i	−0.485135	−	0.840278i
$957$		0			0
$958$		0			0
$959$	−18.0000	−	31.1769i	−0.581250	−	1.00676i
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$	17.0000	−	29.4449i	0.547533	−	0.948355i
$965$	7.00000	+	12.1244i	0.225338	+	0.390297i
$966$		0			0
$967$	5.00000	−	8.66025i	0.160789	−	0.278495i	−0.774363	−	0.632742i	$-0.781929\pi$
$967$	5.00000	−	8.66025i	0.160789	−	0.278495i	0.935152	+	0.354247i	$0.115263\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.0000	−	51.9615i	0.962746	−	1.66752i	0.247193	−	0.968966i	$-0.420492\pi$
$971$	30.0000	−	51.9615i	0.962746	−	1.66752i	0.715553	−	0.698558i	$-0.246175\pi$
$972$		0			0
$973$	−20.0000	−	34.6410i	−0.641171	−	1.11054i
$974$		0			0
$975$		0			0
$976$	−28.0000			−0.896258
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$	4.50000	+	7.79423i	0.143821	+	0.249105i
$980$	−6.00000			−0.191663
$981$		0			0
$982$		0			0
$983$	−9.00000	+	15.5885i	−0.287055	+	0.497195i	−0.973106	−	0.230360i	$-0.926010\pi$
$983$	−9.00000	+	15.5885i	−0.287055	+	0.497195i	0.686050	+	0.727554i	$0.259343\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	32.0000	−	13.8564i	1.01806	−	0.440831i
$989$	−24.0000			−0.763156
$990$		0			0
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	0.491698	+	0.870766i	$0.336377\pi$
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	−0.999954	+	0.00956046i	$0.996957\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.00000			−0.158511
$996$		0			0
$997$	−25.0000	−	43.3013i	−0.791758	−	1.37136i	−0.924878	−	0.380265i	$-0.875833\pi$
$997$	−25.0000	−	43.3013i	−0.791758	−	1.37136i	0.133120	−	0.991100i	$-0.457501\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.k.c.676.1		2
3.2	odd	2		285.2.i.b.106.1	✓	2
19.7	even	3	inner	855.2.k.c.406.1		2
57.8	even	6		5415.2.a.f.1.1		1
57.11	odd	6		5415.2.a.g.1.1		1
57.26	odd	6		285.2.i.b.121.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.i.b.106.1	✓	2	3.2	odd	2
285.2.i.b.121.1	yes	2	57.26	odd	6
855.2.k.c.406.1		2	19.7	even	3	inner
855.2.k.c.676.1		2	1.1	even	1	trivial
5415.2.a.f.1.1		1	57.8	even	6
5415.2.a.g.1.1		1	57.11	odd	6

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.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{4} +(0.500000 - 0.866025i) q^{5} +2.00000 q^{7} +3.00000 q^{11} +(2.00000 + 3.46410i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-3.50000 - 2.59808i) q^{19} +2.00000 q^{20} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.00000 + 3.46410i) q^{28} +(1.50000 + 2.59808i) q^{29} +5.00000 q^{31} +(1.00000 - 1.73205i) q^{35} +8.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(2.00000 - 3.46410i) q^{43} +(3.00000 + 5.19615i) q^{44} +(3.00000 + 5.19615i) q^{47} -3.00000 q^{49} +(-4.00000 + 6.92820i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(1.50000 - 2.59808i) q^{55} +(-4.50000 + 7.79423i) q^{59} +(3.50000 + 6.06218i) q^{61} -8.00000 q^{64} +4.00000 q^{65} +(-1.00000 - 1.73205i) q^{67} +(-4.50000 + 7.79423i) q^{71} +(2.00000 - 3.46410i) q^{73} +(1.00000 - 8.66025i) q^{76} +6.00000 q^{77} +(3.50000 - 6.06218i) q^{79} +(2.00000 + 3.46410i) q^{80} +(1.50000 + 2.59808i) q^{89} +(4.00000 + 6.92820i) q^{91} +(6.00000 - 10.3923i) q^{92} +(-4.00000 + 1.73205i) q^{95} +(5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{13} - 4 q^{16} - 7 q^{19} + 4 q^{20} - 6 q^{23} - q^{25} + 4 q^{28} + 3 q^{29} + 10 q^{31} + 2 q^{35} + 16 q^{37} - 6 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{47}+ \cdots + 10 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + q^5 + 4 * q^7 + 6 * q^11 + 4 * q^13 - 4 * q^16 - 7 * q^19 + 4 * q^20 - 6 * q^23 - q^25 + 4 * q^28 + 3 * q^29 + 10 * q^31 + 2 * q^35 + 16 * q^37 - 6 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^47 - 6 * q^49 - 8 * q^52 - 6 * q^53 + 3 * q^55 - 9 * q^59 + 7 * q^61 - 16 * q^64 + 8 * q^65 - 2 * q^67 - 9 * q^71 + 4 * q^73 + 2 * q^76 + 12 * q^77 + 7 * q^79 + 4 * q^80 + 3 * q^89 + 8 * q^91 + 12 * q^92 - 8 * q^95 + 10 * q^97

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