LMFDB - Embedded newform 855.2.k.j.406.2

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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 13 x^{10} - 10 x^{9} + 44 x^{8} - 20 x^{7} + 119 x^{6} + 13 x^{5} + 83 x^{4} + 14 x^{3} + 46 x^{2} + 12 x + 4 $ x^12 - 3x^11 + 13x^10 - 10x^9 + 44x^8 - 20x^7 + 119x^6 + 13x^5 + 83x^4 + 14x^3 + 46x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			406.2
Root			$-0.398236 - 0.689765i$ of defining polynomial
Character	$\chi$	$=$	855.406
Dual form			855.2.k.j.676.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.898236 - 1.55579i) q^{2} +(-0.613656 + 1.06288i) q^{4} +(-0.500000 - 0.866025i) q^{5} -3.41478 q^{7} -1.38811 q^{8} +O(q^{10})$	\(q+(-0.898236 - 1.55579i) q^{2} +(-0.613656 + 1.06288i) q^{4} +(-0.500000 - 0.866025i) q^{5} -3.41478 q^{7} -1.38811 q^{8} +(-0.898236 + 1.55579i) q^{10} +4.35669 q^{11} +(-2.49135 + 4.31514i) q^{13} +(3.06728 + 5.31268i) q^{14} +(2.47416 + 4.28538i) q^{16} +(-0.0290433 - 0.0503046i) q^{17} +(-1.10304 + 4.21703i) q^{19} +1.22731 q^{20} +(-3.91334 - 6.77810i) q^{22} +(0.216041 - 0.374194i) q^{23} +(-0.500000 + 0.866025i) q^{25} +8.95127 q^{26} +(2.09550 - 3.62951i) q^{28} +(2.74750 - 4.75882i) q^{29} -0.592945 q^{31} +(3.05666 - 5.29428i) q^{32} +(-0.0521756 + 0.0903707i) q^{34} +(1.70739 + 2.95728i) q^{35} +6.62060 q^{37} +(7.55160 - 2.07179i) q^{38} +(0.694056 + 1.20214i) q^{40} +(0.0818718 + 0.141806i) q^{41} +(3.00242 + 5.20034i) q^{43} +(-2.67351 + 4.63066i) q^{44} -0.776222 q^{46} +(-5.54529 + 9.60473i) q^{47} +4.66070 q^{49} +1.79647 q^{50} +(-3.05766 - 5.29603i) q^{52} +(-2.69971 + 4.67603i) q^{53} +(-2.17835 - 3.77300i) q^{55} +4.74009 q^{56} -9.87163 q^{58} +(-1.72248 - 2.98342i) q^{59} +(-2.15698 + 3.73600i) q^{61} +(0.532604 + 0.922498i) q^{62} -1.08574 q^{64} +4.98270 q^{65} +(3.39217 - 5.87541i) q^{67} +0.0712905 q^{68} +(3.06728 - 5.31268i) q^{70} +(4.20501 + 7.28329i) q^{71} +(5.84804 + 10.1291i) q^{73} +(-5.94686 - 10.3003i) q^{74} +(-3.80532 - 3.76021i) q^{76} -14.8771 q^{77} +(-3.64303 - 6.30991i) q^{79} +(2.47416 - 4.28538i) q^{80} +(0.147080 - 0.254751i) q^{82} +16.1185 q^{83} +(-0.0290433 + 0.0503046i) q^{85} +(5.39377 - 9.34228i) q^{86} -6.04757 q^{88} +(-5.59024 + 9.68258i) q^{89} +(8.50740 - 14.7352i) q^{91} +(0.265150 + 0.459252i) q^{92} +19.9239 q^{94} +(4.20357 - 1.15325i) q^{95} +(-4.47355 - 7.74842i) q^{97} +(-4.18641 - 7.25107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8	$ 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} - 3 q^{10} - 8 q^{13} - 10 q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + 10 q^{20} + 2 q^{23} - 6 q^{25} + 40 q^{26} - 26 q^{28} - 4 q^{29} + 24 q^{31} - 15 q^{32} + 7 q^{34} - 2 q^{35} + 29 q^{38} - 6 q^{40} - 12 q^{41} - 4 q^{43} - 6 q^{44} + 48 q^{46} - 6 q^{47} + 32 q^{49} + 6 q^{50} - 20 q^{52} - 26 q^{53} + 44 q^{56} - 20 q^{58} - 16 q^{59} + 20 q^{61} + 25 q^{62} + 28 q^{64} + 16 q^{65} - 12 q^{67} - 54 q^{68} - 10 q^{70} + 8 q^{71} - 4 q^{73} + 16 q^{74} - 66 q^{76} - 48 q^{77} - 12 q^{79} - 3 q^{80} + 26 q^{82} + 44 q^{83} - 4 q^{85} + 44 q^{86} - 32 q^{88} + 8 q^{89} + 2 q^{91} - 36 q^{92} - 14 q^{94} + 6 q^{95} + 30 q^{97} - 49 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8 - 3 * q^10 - 8 * q^13 - 10 * q^14 - 3 * q^16 - 4 * q^17 - 6 * q^19 + 10 * q^20 + 2 * q^23 - 6 * q^25 + 40 * q^26 - 26 * q^28 - 4 * q^29 + 24 * q^31 - 15 * q^32 + 7 * q^34 - 2 * q^35 + 29 * q^38 - 6 * q^40 - 12 * q^41 - 4 * q^43 - 6 * q^44 + 48 * q^46 - 6 * q^47 + 32 * q^49 + 6 * q^50 - 20 * q^52 - 26 * q^53 + 44 * q^56 - 20 * q^58 - 16 * q^59 + 20 * q^61 + 25 * q^62 + 28 * q^64 + 16 * q^65 - 12 * q^67 - 54 * q^68 - 10 * q^70 + 8 * q^71 - 4 * q^73 + 16 * q^74 - 66 * q^76 - 48 * q^77 - 12 * q^79 - 3 * q^80 + 26 * q^82 + 44 * q^83 - 4 * q^85 + 44 * q^86 - 32 * q^88 + 8 * q^89 + 2 * q^91 - 36 * q^92 - 14 * q^94 + 6 * q^95 + 30 * q^97 - 49 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	−0.898236	−	1.55579i	−0.635149	−	1.10011i	−0.986484	−	0.163860i	$-0.947605\pi$
$2$	−0.898236	−	1.55579i	−0.635149	−	1.10011i	0.351335	−	0.936250i	$-0.385728\pi$
$3$		0			0
$3$		0			0
$4$	−0.613656	+	1.06288i	−0.306828	+	0.531442i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	−3.41478			−1.29066			−0.645332	−	0.763902i	$-0.723281\pi$
$7$	−3.41478			−1.29066			−0.645332	+	0.763902i	$0.723281\pi$
$8$	−1.38811			−0.490771
$9$		0			0
$10$	−0.898236	+	1.55579i	−0.284047	+	0.491984i
$11$	4.35669			1.31359			0.656796	−	0.754069i	$-0.271911\pi$
$11$	4.35669			1.31359			0.656796	+	0.754069i	$0.271911\pi$
$12$		0			0
$13$	−2.49135	+	4.31514i	−0.690976	+	1.19680i	0.280543	+	0.959841i	$0.409485\pi$
$13$	−2.49135	+	4.31514i	−0.690976	+	1.19680i	−0.971519	+	0.236963i	$0.923848\pi$
$14$	3.06728	+	5.31268i	0.819764	+	1.41987i
$15$		0			0
$16$	2.47416	+	4.28538i	0.618541	+	1.07134i
$17$	−0.0290433	−	0.0503046i	−0.00704405	−	0.0122006i	0.862482	−	0.506088i	$-0.168909\pi$
$17$	−0.0290433	−	0.0503046i	−0.00704405	−	0.0122006i	−0.869526	+	0.493887i	$0.835576\pi$
$18$		0			0
$19$	−1.10304	+	4.21703i	−0.253054	+	0.967452i
$19$	−1.10304	+	4.21703i	−0.253054	+	0.967452i
$20$	1.22731			0.274435
$21$		0			0
$22$	−3.91334	−	6.77810i	−0.834326	−	1.44510i
$23$	0.216041	−	0.374194i	0.0450476	−	0.0780247i	−0.842622	−	0.538505i	$-0.818989\pi$
$23$	0.216041	−	0.374194i	0.0450476	−	0.0780247i	0.887670	+	0.460480i	$0.152323\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	8.95127			1.75549
$27$		0			0
$28$	2.09550	−	3.62951i	0.396012	−	0.685913i
$29$	2.74750	−	4.75882i	0.510199	−	0.883690i	−0.489731	−	0.871873i	$-0.662905\pi$
$29$	2.74750	−	4.75882i	0.510199	−	0.883690i	0.999930	−	0.0118169i	$-0.00376151\pi$
$30$		0			0
$31$	−0.592945			−0.106496			−0.0532480	−	0.998581i	$-0.516957\pi$
$31$	−0.592945			−0.106496			−0.0532480	+	0.998581i	$0.516957\pi$
$32$	3.05666	−	5.29428i	0.540346	−	0.935906i
$33$		0			0
$34$	−0.0521756	+	0.0903707i	−0.00894804	+	0.0154985i
$35$	1.70739	+	2.95728i	0.288601	+	0.499872i
$36$		0			0
$37$	6.62060			1.08842			0.544210	−	0.838949i	$-0.316830\pi$
$37$	6.62060			1.08842			0.544210	+	0.838949i	$0.316830\pi$
$38$	7.55160	−	2.07179i	1.22503	−	0.336089i
$39$		0			0
$40$	0.694056	+	1.20214i	0.109740	+	0.190075i
$41$	0.0818718	+	0.141806i	0.0127862	+	0.0221464i	0.872348	−	0.488886i	$-0.162597\pi$
$41$	0.0818718	+	0.141806i	0.0127862	+	0.0221464i	−0.859561	+	0.511032i	$0.829263\pi$
$42$		0			0
$43$	3.00242	+	5.20034i	0.457865	+	0.793045i	0.998848	−	0.0479883i	$-0.0152810\pi$
$43$	3.00242	+	5.20034i	0.457865	+	0.793045i	−0.540983	+	0.841033i	$0.681948\pi$
$44$	−2.67351	+	4.63066i	−0.403047	+	0.698098i
$45$		0			0
$46$	−0.776222			−0.114448
$47$	−5.54529	+	9.60473i	−0.808864	+	1.40099i	0.104788	+	0.994495i	$0.466584\pi$
$47$	−5.54529	+	9.60473i	−0.808864	+	1.40099i	−0.913651	+	0.406499i	$0.866750\pi$
$48$		0			0
$49$	4.66070			0.665814
$50$	1.79647			0.254060
$51$		0			0
$52$	−3.05766	−	5.29603i	−0.424022	−	0.734427i
$53$	−2.69971	+	4.67603i	−0.370834	+	0.642303i	−0.989694	−	0.143198i	$-0.954261\pi$
$53$	−2.69971	+	4.67603i	−0.370834	+	0.642303i	0.618860	+	0.785501i	$0.287595\pi$
$54$		0			0
$55$	−2.17835	−	3.77300i	−0.293728	−	0.508752i
$56$	4.74009			0.633421
$57$		0			0
$58$	−9.87163			−1.29621
$59$	−1.72248	−	2.98342i	−0.224248	−	0.388408i	0.731846	−	0.681470i	$-0.238659\pi$
$59$	−1.72248	−	2.98342i	−0.224248	−	0.388408i	−0.956093	+	0.293062i	$0.905326\pi$
$60$		0			0
$61$	−2.15698	+	3.73600i	−0.276173	+	0.478346i	−0.970430	−	0.241381i	$-0.922400\pi$
$61$	−2.15698	+	3.73600i	−0.276173	+	0.478346i	0.694257	+	0.719727i	$0.255733\pi$
$62$	0.532604	+	0.922498i	0.0676408	+	0.117157i
$63$		0			0
$64$	−1.08574			−0.135718
$65$	4.98270			0.618027
$66$		0			0
$67$	3.39217	−	5.87541i	0.414420	−	0.717796i	−0.580948	−	0.813941i	$-0.697318\pi$
$67$	3.39217	−	5.87541i	0.414420	−	0.717796i	0.995367	+	0.0961450i	$0.0306513\pi$
$68$	0.0712905			0.00864525
$69$		0			0
$70$	3.06728	−	5.31268i	0.366610	−	0.634986i
$71$	4.20501	+	7.28329i	0.499043	+	0.864368i	0.999999	−	0.00110477i	$-0.000351659\pi$
$71$	4.20501	+	7.28329i	0.499043	+	0.864368i	−0.500956	+	0.865472i	$0.667018\pi$
$72$		0			0
$73$	5.84804	+	10.1291i	0.684461	+	1.18552i	0.973606	+	0.228236i	$0.0732958\pi$
$73$	5.84804	+	10.1291i	0.684461	+	1.18552i	−0.289145	+	0.957285i	$0.593371\pi$
$74$	−5.94686	−	10.3003i	−0.691309	−	1.19738i
$75$		0			0
$76$	−3.80532	−	3.76021i	−0.436501	−	0.431325i
$77$	−14.8771			−1.69541
$78$		0			0
$79$	−3.64303	−	6.30991i	−0.409873	−	0.709920i	0.585003	−	0.811031i	$-0.301094\pi$
$79$	−3.64303	−	6.30991i	−0.409873	−	0.709920i	−0.994875	+	0.101111i	$0.967760\pi$
$80$	2.47416	−	4.28538i	0.276620	−	0.479120i
$81$		0			0
$82$	0.147080	−	0.254751i	0.0162423	−	0.0281325i
$83$	16.1185			1.76923			0.884617	−	0.466318i	$-0.154420\pi$
$83$	16.1185			1.76923			0.884617	+	0.466318i	$0.154420\pi$
$84$		0			0
$85$	−0.0290433	+	0.0503046i	−0.00315019	+	0.00545629i
$86$	5.39377	−	9.34228i	0.581625	−	1.00740i
$87$		0			0
$88$	−6.04757			−0.644673
$89$	−5.59024	+	9.68258i	−0.592564	+	1.02635i	0.401321	+	0.915937i	$0.368551\pi$
$89$	−5.59024	+	9.68258i	−0.592564	+	1.02635i	−0.993886	+	0.110414i	$0.964782\pi$
$90$		0			0
$91$	8.50740	−	14.7352i	0.891817	−	1.54467i
$92$	0.265150	+	0.459252i	0.0276438	+	0.0478804i
$93$		0			0
$94$	19.9239			2.05500
$95$	4.20357	−	1.15325i	0.431277	−	0.118321i
$96$		0			0
$97$	−4.47355	−	7.74842i	−0.454221	−	0.786733i	0.544422	−	0.838811i	$-0.316749\pi$
$97$	−4.47355	−	7.74842i	−0.454221	−	0.786733i	−0.998643	+	0.0520780i	$0.983416\pi$
$98$	−4.18641	−	7.25107i	−0.422891	−	0.732469i
$99$		0			0
$100$	−0.613656	−	1.06288i	−0.0613656	−	0.106288i
$101$	1.09523	−	1.89700i	0.108980	−	0.188758i	−0.806377	−	0.591401i	$-0.798575\pi$
$101$	1.09523	−	1.89700i	0.108980	−	0.188758i	0.915357	+	0.402643i	$0.131908\pi$
$102$		0			0
$103$	10.4903			1.03364			0.516818	−	0.856095i	$-0.327117\pi$
$103$	10.4903			1.03364			0.516818	+	0.856095i	$0.327117\pi$
$104$	3.45827	−	5.98990i	0.339111	−	0.587358i
$105$		0			0
$106$	9.69991			0.942138
$107$	−13.6911			−1.32357			−0.661783	−	0.749696i	$-0.730200\pi$
$107$	−13.6911			−1.32357			−0.661783	+	0.749696i	$0.730200\pi$
$108$		0			0
$109$	7.07679	+	12.2574i	0.677834	+	1.17404i	0.975632	+	0.219413i	$0.0704144\pi$
$109$	7.07679	+	12.2574i	0.677834	+	1.17404i	−0.297798	+	0.954629i	$0.596252\pi$
$110$	−3.91334	+	6.77810i	−0.373122	+	0.646266i
$111$		0			0
$112$	−8.44872	−	14.6336i	−0.798329	−	1.38275i
$113$	−13.2851			−1.24976			−0.624879	−	0.780722i	$-0.714852\pi$
$113$	−13.2851			−1.24976			−0.624879	+	0.780722i	$0.714852\pi$
$114$		0			0
$115$	−0.432081			−0.0402918
$116$	3.37205	+	5.84056i	0.313087	+	0.542282i
$117$		0			0
$118$	−3.09438	+	5.35963i	−0.284861	+	0.493394i
$119$	0.0991765	+	0.171779i	0.00909150	+	0.0157469i
$120$		0			0
$121$	7.98075			0.725523
$122$	7.74991			0.701644
$123$		0			0
$124$	0.363864	−	0.630231i	0.0326760	−	0.0565964i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−7.84813	+	13.5934i	−0.696409	+	1.20622i	0.273295	+	0.961930i	$0.411887\pi$
$127$	−7.84813	+	13.5934i	−0.696409	+	1.20622i	−0.969703	+	0.244285i	$0.921447\pi$
$128$	−5.13806	−	8.89938i	−0.454145	−	0.786602i
$129$		0			0
$130$	−4.47564	−	7.75203i	−0.392539	−	0.679898i
$131$	5.86776	+	10.1633i	0.512669	+	0.887968i	0.999892	+	0.0146910i	$0.00467646\pi$
$131$	5.86776	+	10.1633i	0.512669	+	0.887968i	−0.487223	+	0.873277i	$0.661990\pi$
$132$		0			0
$133$	3.76663	−	14.4002i	0.326608	−	1.24866i
$134$	−12.1879			−1.05287
$135$		0			0
$136$	0.0403154	+	0.0698283i	0.00345702	+	0.00598773i
$137$	5.12860	−	8.88300i	0.438166	−	0.758926i	−0.559382	−	0.828910i	$-0.688962\pi$
$137$	5.12860	−	8.88300i	0.438166	−	0.758926i	0.997548	+	0.0699840i	$0.0222948\pi$
$138$		0			0
$139$	−10.9044	+	18.8870i	−0.924899	+	1.60197i	−0.133176	+	0.991092i	$0.542518\pi$
$139$	−10.9044	+	18.8870i	−0.924899	+	1.60197i	−0.791723	+	0.610880i	$0.790816\pi$
$140$	−4.19100			−0.354204
$141$		0			0
$142$	7.55418	−	13.0842i	0.633933	−	1.09800i
$143$	−10.8540	+	18.7997i	−0.907660	+	1.57211i
$144$		0			0
$145$	−5.49501			−0.456336
$146$	10.5058	−	18.1966i	0.869469	−	1.50597i
$147$		0			0
$148$	−4.06277	+	7.03693i	−0.333958	+	0.578432i
$149$	−10.3765	−	17.9727i	−0.850079	−	1.47238i	−0.881137	−	0.472862i	$-0.843221\pi$
$149$	−10.3765	−	17.9727i	−0.850079	−	1.47238i	0.0310582	−	0.999518i	$-0.490112\pi$
$150$		0			0
$151$	−14.9510			−1.21669			−0.608346	−	0.793672i	$-0.708167\pi$
$151$	−14.9510			−1.21669			−0.608346	+	0.793672i	$0.708167\pi$
$152$	1.53114	−	5.85370i	0.124192	−	0.474798i
$153$		0			0
$154$	13.3632	+	23.1457i	1.07683	+	1.86513i
$155$	0.296472	+	0.513505i	0.0238132	+	0.0412457i
$156$		0			0
$157$	0.0399391	+	0.0691765i	0.00318749	+	0.00552089i	0.867615	−	0.497237i	$-0.165652\pi$
$157$	0.0399391	+	0.0691765i	0.00318749	+	0.00552089i	−0.864427	+	0.502758i	$0.832319\pi$
$158$	−6.54460	+	11.3356i	−0.520660	+	0.901810i
$159$		0			0
$160$	−6.11331			−0.483300
$161$	−0.737731	+	1.27779i	−0.0581413	+	0.100704i
$162$		0			0
$163$	18.4845			1.44782			0.723909	−	0.689896i	$-0.242344\pi$
$163$	18.4845			1.44782			0.723909	+	0.689896i	$0.242344\pi$
$164$	−0.200965			−0.0156927
$165$		0			0
$166$	−14.4782	−	25.0770i	−1.12373	−	1.94635i
$167$	−9.34180	+	16.1805i	−0.722890	+	1.25208i	0.236946	+	0.971523i	$0.423853\pi$
$167$	−9.34180	+	16.1805i	−0.722890	+	1.25208i	−0.959837	+	0.280560i	$0.909480\pi$
$168$		0			0
$169$	−5.91363	−	10.2427i	−0.454894	−	0.787900i
$170$	0.104351			0.00800337
$171$		0			0
$172$	−7.36982			−0.561943
$173$	−4.29860	−	7.44540i	−0.326817	−	0.566063i	0.655062	−	0.755575i	$-0.272643\pi$
$173$	−4.29860	−	7.44540i	−0.326817	−	0.566063i	−0.981878	+	0.189512i	$0.939309\pi$
$174$		0			0
$175$	1.70739	−	2.95728i	0.129066	−	0.223550i
$176$	10.7792	+	18.6701i	0.812510	+	1.40731i
$177$		0			0
$178$	20.0854			1.50547
$179$	6.16587			0.460859			0.230429	−	0.973089i	$-0.425987\pi$
$179$	6.16587			0.460859			0.230429	+	0.973089i	$0.425987\pi$
$180$		0			0
$181$	−10.0294	+	17.3715i	−0.745481	+	1.29121i	0.204488	+	0.978869i	$0.434447\pi$
$181$	−10.0294	+	17.3715i	−0.745481	+	1.29121i	−0.949970	+	0.312343i	$0.898886\pi$
$182$	−30.5666			−2.26575
$183$		0			0
$184$	−0.299889	+	0.519422i	−0.0221081	+	0.0382923i
$185$	−3.31030	−	5.73361i	−0.243378	−	0.421543i
$186$		0			0
$187$	−0.126533	−	0.219161i	−0.00925300	−	0.0160267i
$188$	−6.80581	−	11.7880i	−0.496364	−	0.859728i
$189$		0			0
$190$	−5.57002	−	5.50398i	−0.404092	−	0.399301i
$191$	−17.0701			−1.23515			−0.617573	−	0.786513i	$-0.711884\pi$
$191$	−17.0701			−1.23515			−0.617573	+	0.786513i	$0.711884\pi$
$192$		0			0
$193$	−3.63576	−	6.29733i	−0.261708	−	0.453292i	0.704988	−	0.709219i	$-0.250952\pi$
$193$	−3.63576	−	6.29733i	−0.261708	−	0.453292i	−0.966696	+	0.255928i	$0.917619\pi$
$194$	−8.03662	+	13.9198i	−0.576995	+	0.999385i
$195$		0			0
$196$	−2.86007	+	4.95378i	−0.204291	+	0.353842i
$197$	−3.59854			−0.256385			−0.128193	−	0.991749i	$-0.540918\pi$
$197$	−3.59854			−0.256385			−0.128193	+	0.991749i	$0.540918\pi$
$198$		0			0
$199$	−10.7918	+	18.6919i	−0.765008	+	1.32503i	0.175234	+	0.984527i	$0.443932\pi$
$199$	−10.7918	+	18.6919i	−0.765008	+	1.32503i	−0.940242	+	0.340506i	$0.889402\pi$
$200$	0.694056	−	1.20214i	0.0490771	−	0.0850041i
$201$		0			0
$202$	−3.93511			−0.276873
$203$	−9.38211	+	16.2503i	−0.658495	+	1.14055i
$204$		0			0
$205$	0.0818718	−	0.141806i	0.00571817	−	0.00990417i
$206$	−9.42272	−	16.3206i	−0.656512	−	1.13711i
$207$		0			0
$208$	−24.6560			−1.70959
$209$	−4.80559	+	18.3723i	−0.332410	+	1.27084i
$210$		0			0
$211$	−0.0351916	−	0.0609536i	−0.00242269	−	0.00419622i	0.864812	−	0.502097i	$-0.167438\pi$
$211$	−0.0351916	−	0.0609536i	−0.00242269	−	0.00419622i	−0.867234	+	0.497900i	$0.834105\pi$
$212$	−3.31339	−	5.73896i	−0.227564	−	0.394153i
$213$		0			0
$214$	12.2978	+	21.3004i	0.840661	+	1.45607i
$215$	3.00242	−	5.20034i	0.204763	−	0.354661i
$216$		0			0
$217$	2.02477			0.137451
$218$	12.7133	−	22.0200i	0.861051	−	1.49138i
$219$		0			0
$220$	5.34702			0.360496
$221$	0.289428			0.0194691
$222$		0			0
$223$	6.52703	+	11.3051i	0.437082	+	0.757049i	0.997463	−	0.0711864i	$-0.0226785\pi$
$223$	6.52703	+	11.3051i	0.437082	+	0.757049i	−0.560381	+	0.828235i	$0.689345\pi$
$224$	−10.4378	+	18.0788i	−0.697405	+	1.20794i
$225$		0			0
$226$	11.9332	+	20.6689i	0.793783	+	1.37487i
$227$	7.75546			0.514748			0.257374	−	0.966312i	$-0.417143\pi$
$227$	7.75546			0.514748			0.257374	+	0.966312i	$0.417143\pi$
$228$		0			0
$229$	19.0597			1.25950			0.629751	−	0.776797i	$-0.283157\pi$
$229$	19.0597			1.25950			0.629751	+	0.776797i	$0.283157\pi$
$230$	0.388111	+	0.672228i	0.0255913	+	0.0443254i
$231$		0			0
$232$	−3.81384	+	6.60577i	−0.250391	+	0.433690i
$233$	6.26723	+	10.8552i	0.410580	+	0.711145i	0.994953	−	0.100340i	$-0.0319930\pi$
$233$	6.26723	+	10.8552i	0.410580	+	0.711145i	−0.584373	+	0.811485i	$0.698660\pi$
$234$		0			0
$235$	11.0906			0.723470
$236$	4.22804			0.275222
$237$		0			0
$238$	0.178168	−	0.308596i	0.0115489	−	0.0200033i
$239$	15.7879			1.02123			0.510617	−	0.859808i	$-0.329417\pi$
$239$	15.7879			1.02123			0.510617	+	0.859808i	$0.329417\pi$
$240$		0			0
$241$	−12.4166	+	21.5061i	−0.799822	+	1.38533i	0.119910	+	0.992785i	$0.461739\pi$
$241$	−12.4166	+	21.5061i	−0.799822	+	1.38533i	−0.919732	+	0.392547i	$0.871594\pi$
$242$	−7.16860	−	12.4164i	−0.460815	−	0.798155i
$243$		0			0
$244$	−2.64729	−	4.58524i	−0.169475	−	0.293540i
$245$	−2.33035	−	4.03628i	−0.148881	−	0.257869i
$246$		0			0
$247$	−15.4490	−	15.2658i	−0.982997	−	0.971342i
$248$	0.823073			0.0522652
$249$		0			0
$250$	−0.898236	−	1.55579i	−0.0568094	−	0.0983968i
$251$	−1.65676	+	2.86958i	−0.104573	+	0.181127i	−0.913564	−	0.406695i	$-0.866681\pi$
$251$	−1.65676	+	2.86958i	−0.104573	+	0.181127i	0.808990	+	0.587822i	$0.200014\pi$
$252$		0			0
$253$	0.941223	−	1.63025i	0.0591742	−	0.102493i
$254$	28.1979			1.76929
$255$		0			0
$256$	−10.3161	+	17.8681i	−0.644758	+	1.11675i
$257$	12.4325	−	21.5337i	0.775516	−	1.34323i	−0.158988	−	0.987280i	$-0.550823\pi$
$257$	12.4325	−	21.5337i	0.775516	−	1.34323i	0.934504	−	0.355952i	$-0.115843\pi$
$258$		0			0
$259$	−22.6079			−1.40478
$260$	−3.05766	+	5.29603i	−0.189628	+	0.328446i
$261$		0			0
$262$	10.5413	−	18.2580i	0.651242	−	1.12798i
$263$	0.342686	+	0.593549i	0.0211309	+	0.0365998i	0.876397	−	0.481588i	$-0.159940\pi$
$263$	0.342686	+	0.593549i	0.0211309	+	0.0365998i	−0.855267	+	0.518188i	$0.826607\pi$
$264$		0			0
$265$	5.39942			0.331684
$266$	−25.7870	+	7.07470i	−1.58110	+	0.433778i
$267$		0			0
$268$	4.16325	+	7.21097i	0.254311	+	0.440480i
$269$	−5.58704	−	9.67704i	−0.340648	−	0.590019i	0.643905	−	0.765105i	$-0.277313\pi$
$269$	−5.58704	−	9.67704i	−0.340648	−	0.590019i	−0.984553	+	0.175086i	$0.943980\pi$
$270$		0			0
$271$	7.16308	+	12.4068i	0.435126	+	0.753661i	0.997306	−	0.0733542i	$-0.0233704\pi$
$271$	7.16308	+	12.4068i	0.435126	+	0.753661i	−0.562180	+	0.827015i	$0.690037\pi$
$272$	0.143716	−	0.248923i	0.00871406	−	0.0150932i
$273$		0			0
$274$	−18.4268			−1.11320
$275$	−2.17835	+	3.77300i	−0.131359	+	0.227521i
$276$		0			0
$277$	−15.3518			−0.922397			−0.461199	−	0.887297i	$-0.652580\pi$
$277$	−15.3518			−0.922397			−0.461199	+	0.887297i	$0.652580\pi$
$278$	39.1789			2.34979
$279$		0			0
$280$	−2.37004	−	4.10504i	−0.141637	−	0.245323i
$281$	−4.80640	+	8.32493i	−0.286726	+	0.496624i	−0.973026	−	0.230694i	$-0.925900\pi$
$281$	−4.80640	+	8.32493i	−0.286726	+	0.496624i	0.686300	+	0.727318i	$0.259234\pi$
$282$		0			0
$283$	−2.86255	−	4.95809i	−0.170161	−	0.294728i	0.768315	−	0.640072i	$-0.221095\pi$
$283$	−2.86255	−	4.95809i	−0.170161	−	0.294728i	−0.938476	+	0.345344i	$0.887762\pi$
$284$	−10.3217			−0.612482
$285$		0			0
$286$	38.9979			2.30600
$287$	−0.279574	−	0.484236i	−0.0165027	−	0.0285836i
$288$		0			0
$289$	8.49831	−	14.7195i	0.499901	−	0.865854i
$290$	4.93582	+	8.54908i	0.289841	+	0.502019i
$291$		0			0
$292$	−14.3547			−0.840048
$293$	−23.3141			−1.36203			−0.681013	−	0.732271i	$-0.738460\pi$
$293$	−23.3141			−1.36203			−0.681013	+	0.732271i	$0.738460\pi$
$294$		0			0
$295$	−1.72248	+	2.98342i	−0.100287	+	0.173701i
$296$	−9.19013			−0.534165
$297$		0			0
$298$	−18.6412	+	32.2874i	−1.07985	+	1.87036i
$299$	1.07647	+	1.86449i	0.0622536	+	0.107826i
$300$		0			0
$301$	−10.2526	−	17.7580i	−0.590950	−	1.02356i
$302$	13.4295	+	23.2606i	0.772781	+	1.33850i
$303$		0			0
$304$	−20.8006	+	5.70668i	−1.19300	+	0.327301i
$305$	4.31396			0.247017
$306$		0			0
$307$	−4.24686	−	7.35577i	−0.242381	−	0.419816i	0.719011	−	0.694999i	$-0.244595\pi$
$307$	−4.24686	−	7.35577i	−0.242381	−	0.419816i	−0.961392	+	0.275183i	$0.911262\pi$
$308$	9.12944	−	15.8127i	0.520198	−	0.901010i
$309$		0			0
$310$	0.532604	−	0.922498i	0.0302499	−	0.0523943i
$311$	−4.54326			−0.257625			−0.128812	−	0.991669i	$-0.541116\pi$
$311$	−4.54326			−0.257625			−0.128812	+	0.991669i	$0.541116\pi$
$312$		0			0
$313$	9.43148	−	16.3358i	0.533099	−	0.923354i	−0.466154	−	0.884703i	$-0.654361\pi$
$313$	9.43148	−	16.3358i	0.533099	−	0.923354i	0.999253	−	0.0386503i	$-0.0123058\pi$
$314$	0.0717495	−	0.124274i	0.00404906	−	0.00701317i
$315$		0			0
$316$	8.94227			0.503042
$317$	9.61822	−	16.6592i	0.540213	−	0.935677i	−0.458678	−	0.888602i	$-0.651677\pi$
$317$	9.61822	−	16.6592i	0.540213	−	0.935677i	0.998891	−	0.0470741i	$-0.0149897\pi$
$318$		0			0
$319$	11.9700	−	20.7327i	0.670193	−	1.16081i
$320$	0.542870	+	0.940279i	0.0303474	+	0.0525632i
$321$		0			0
$322$	2.65063			0.147714
$323$	0.244172	−	0.0669888i	0.0135861	−	0.00372735i
$324$		0			0
$325$	−2.49135	−	4.31514i	−0.138195	−	0.239361i
$326$	−16.6034	−	28.7580i	−0.919579	−	1.59276i
$327$		0			0
$328$	−0.113647	−	0.196843i	−0.00627511	−	0.0108688i
$329$	18.9359	−	32.7980i	1.04397	−	1.80821i
$330$		0			0
$331$	9.49306			0.521786			0.260893	−	0.965368i	$-0.415983\pi$
$331$	9.49306			0.521786			0.260893	+	0.965368i	$0.415983\pi$
$332$	−9.89121	+	17.1321i	−0.542851	+	0.940245i
$333$		0			0
$334$	33.5646			1.83657
$335$	−6.78434			−0.370668
$336$		0			0
$337$	−11.8854	−	20.5860i	−0.647437	−	1.12139i	−0.983733	−	0.179637i	$-0.942508\pi$
$337$	−11.8854	−	20.5860i	−0.647437	−	1.12139i	0.336296	−	0.941756i	$-0.390826\pi$
$338$	−10.6237	+	18.4007i	−0.577851	+	1.00087i
$339$		0			0
$340$	−0.0356453	−	0.0617394i	−0.00193314	−	0.00334829i
$341$	−2.58328			−0.139892
$342$		0			0
$343$	7.98819			0.431322
$344$	−4.16769	−	7.21866i	−0.224707	−	0.389204i
$345$		0			0
$346$	−7.72232	+	13.3755i	−0.415155	+	0.719069i
$347$	14.0069	+	24.2607i	0.751932	+	1.30238i	0.946885	+	0.321572i	$0.104211\pi$
$347$	14.0069	+	24.2607i	0.751932	+	1.30238i	−0.194953	+	0.980813i	$0.562455\pi$
$348$		0			0
$349$	0.00959659			0.000513694			0.000256847	−	1.00000i	$-0.499918\pi$
$349$	0.00959659			0.000513694			0.000256847		1.00000i	$0.499918\pi$
$350$	−6.13455			−0.327906
$351$		0			0
$352$	13.3169	−	23.0656i	0.709793	−	1.22940i
$353$	−6.08582			−0.323915			−0.161958	−	0.986798i	$-0.551781\pi$
$353$	−6.08582			−0.323915			−0.161958	+	0.986798i	$0.551781\pi$
$354$		0			0
$355$	4.20501	−	7.28329i	0.223179	−	0.386557i
$356$	−6.86097	−	11.8836i	−0.363631	−	0.629827i
$357$		0			0
$358$	−5.53841	−	9.59281i	−0.292714	−	0.506996i
$359$	−8.91042	−	15.4333i	−0.470274	−	0.814538i	0.529148	−	0.848529i	$-0.322512\pi$
$359$	−8.91042	−	15.4333i	−0.470274	−	0.814538i	−0.999422	+	0.0339910i	$0.989178\pi$
$360$		0			0
$361$	−16.5666	−	9.30307i	−0.871927	−	0.489635i
$362$	36.0352			1.89397
$363$		0			0
$364$	10.4412	+	18.0848i	0.547269	+	0.947898i
$365$	5.84804	−	10.1291i	0.306100	−	0.530181i
$366$		0			0
$367$	2.45095	−	4.24517i	0.127938	−	0.221596i	−0.794939	−	0.606689i	$-0.792497\pi$
$367$	2.45095	−	4.24517i	0.127938	−	0.221596i	0.922878	+	0.385093i	$0.125831\pi$
$368$	2.13808			0.111455
$369$		0			0
$370$	−5.94686	+	10.3003i	−0.309163	+	0.535485i
$371$	9.21891	−	15.9676i	0.478622	−	0.828997i
$372$		0			0
$373$	−12.9995			−0.673090			−0.336545	−	0.941667i	$-0.609258\pi$
$373$	−12.9995			−0.673090			−0.336545	+	0.941667i	$0.609258\pi$
$374$	−0.227313	+	0.393717i	−0.0117541	+	0.0203586i
$375$		0			0
$376$	7.69748	−	13.3324i	0.396967	−	0.687567i
$377$	13.6900	+	23.7117i	0.705070	+	1.22122i
$378$		0			0
$379$	16.7313			0.859427			0.429713	−	0.902965i	$-0.358615\pi$
$379$	16.7313			0.859427			0.429713	+	0.902965i	$0.358615\pi$
$380$	−1.35377	+	5.17561i	−0.0694470	+	0.265503i
$381$		0			0
$382$	15.3330	+	26.5575i	0.784502	+	1.35880i
$383$	−14.9461	−	25.8875i	−0.763712	−	1.32279i	−0.940925	−	0.338615i	$-0.890042\pi$
$383$	−14.9461	−	25.8875i	−0.763712	−	1.32279i	0.177213	−	0.984172i	$-0.443292\pi$
$384$		0			0
$385$	7.43856	+	12.8840i	0.379104	+	0.656628i
$386$	−6.53155	+	11.3130i	−0.332447	+	0.575815i
$387$		0			0
$388$	10.9809			0.557471
$389$	−15.8669	+	27.4822i	−0.804483	+	1.39340i	0.112157	+	0.993691i	$0.464224\pi$
$389$	−15.8669	+	27.4822i	−0.804483	+	1.39340i	−0.916640	+	0.399714i	$0.869109\pi$
$390$		0			0
$391$	−0.0250982			−0.00126927
$392$	−6.46957			−0.326763
$393$		0			0
$394$	3.23234	+	5.59857i	0.162843	+	0.282052i
$395$	−3.64303	+	6.30991i	−0.183301	+	0.317486i
$396$		0			0
$397$	−16.9999	−	29.4447i	−0.853202	−	1.47779i	−0.878303	−	0.478104i	$-0.841324\pi$
$397$	−16.9999	−	29.4447i	−0.853202	−	1.47779i	0.0251017	−	0.999685i	$-0.492009\pi$
$398$	38.7742			1.94358
$399$		0			0
$400$	−4.94833			−0.247416
$401$	4.55045	+	7.88161i	0.227239	+	0.393589i	0.956989	−	0.290125i	$-0.0936970\pi$
$401$	4.55045	+	7.88161i	0.227239	+	0.393589i	−0.729750	+	0.683714i	$0.760364\pi$
$402$		0			0
$403$	1.47723	−	2.55864i	0.0735861	−	0.127455i
$404$	1.34419	+	2.32821i	0.0668761	+	0.115833i
$405$		0			0
$406$	33.7094			1.67297
$407$	28.8439			1.42974
$408$		0			0
$409$	13.9425	−	24.1491i	0.689411	−	1.19409i	−0.282618	−	0.959233i	$-0.591203\pi$
$409$	13.9425	−	24.1491i	0.689411	−	1.19409i	0.972029	−	0.234862i	$-0.0754639\pi$
$410$	−0.294161			−0.0145276
$411$		0			0
$412$	−6.43741	+	11.1499i	−0.317148	+	0.549317i
$413$	5.88188	+	10.1877i	0.289428	+	0.501305i
$414$		0			0
$415$	−8.05924	−	13.9590i	−0.395613	−	0.685221i
$416$	15.2304	+	26.3798i	0.746731	+	1.29338i
$417$		0			0
$418$	32.9000	−	9.02615i	1.60919	−	0.441483i
$419$	31.5961			1.54357			0.771785	−	0.635884i	$-0.219364\pi$
$419$	31.5961			1.54357			0.771785	+	0.635884i	$0.219364\pi$
$420$		0			0
$421$	−6.42177	−	11.1228i	−0.312978	−	0.542094i	0.666028	−	0.745927i	$-0.267993\pi$
$421$	−6.42177	−	11.1228i	−0.312978	−	0.542094i	−0.979006	+	0.203833i	$0.934660\pi$
$422$	−0.0632207	+	0.109502i	−0.00307754	+	0.00533045i
$423$		0			0
$424$	3.74750	−	6.49086i	0.181995	−	0.315224i
$425$	0.0580867			0.00281762
$426$		0			0
$427$	7.36561	−	12.7576i	0.356447	−	0.617384i
$428$	8.40161	−	14.5520i	0.406107	−	0.703398i
$429$		0			0
$430$	−10.7875			−0.520221
$431$	−16.6064	+	28.7630i	−0.799900	+	1.38547i	0.119781	+	0.992800i	$0.461781\pi$
$431$	−16.6064	+	28.7630i	−0.799900	+	1.38547i	−0.919681	+	0.392667i	$0.871553\pi$
$432$		0			0
$433$	−3.66146	+	6.34183i	−0.175958	+	0.304769i	−0.940493	−	0.339814i	$-0.889636\pi$
$433$	−3.66146	+	6.34183i	−0.175958	+	0.304769i	0.764534	+	0.644583i	$0.222969\pi$
$434$	−1.81872	−	3.15012i	−0.0873016	−	0.151211i
$435$		0			0
$436$	−17.3709			−0.831914
$437$	1.33968	+	1.32380i	0.0640857	+	0.0633259i
$438$		0			0
$439$	−11.4327	−	19.8020i	−0.545651	−	0.945096i	−0.998566	−	0.0535417i	$-0.982949\pi$
$439$	−11.4327	−	19.8020i	−0.545651	−	0.945096i	0.452914	−	0.891554i	$-0.350384\pi$
$440$	3.02379	+	5.23735i	0.144153	+	0.249681i
$441$		0			0
$442$	−0.259975	−	0.450290i	−0.0123657	−	0.0214181i
$443$	−7.40824	+	12.8315i	−0.351976	+	0.609641i	−0.986596	−	0.163184i	$-0.947824\pi$
$443$	−7.40824	+	12.8315i	−0.351976	+	0.609641i	0.634619	+	0.772825i	$0.281157\pi$
$444$		0			0
$445$	11.1805			0.530006
$446$	11.7256	−	20.3094i	0.555225	−	0.961677i
$447$		0			0
$448$	3.70756			0.175166
$449$	41.8708			1.97600			0.988002	−	0.154441i	$-0.0493576\pi$
$449$	41.8708			1.97600			0.988002	+	0.154441i	$0.0493576\pi$
$450$		0			0
$451$	0.356690	+	0.617805i	0.0167959	+	0.0290913i
$452$	8.15249	−	14.1205i	0.383461	−	0.664174i
$453$		0			0
$454$	−6.96623	−	12.0659i	−0.326942	−	0.566279i
$455$	−17.0148			−0.797666
$456$		0			0
$457$	38.4845			1.80023			0.900114	−	0.435655i	$-0.143483\pi$
$457$	38.4845			1.80023			0.900114	+	0.435655i	$0.143483\pi$
$458$	−17.1201	−	29.6530i	−0.799972	−	1.38559i
$459$		0			0
$460$	0.265150	−	0.459252i	0.0123627	−	0.0214128i
$461$	−13.9830	−	24.2192i	−0.651251	−	1.12800i	−0.982820	−	0.184569i	$-0.940911\pi$
$461$	−13.9830	−	24.2192i	−0.651251	−	1.12800i	0.331568	−	0.943431i	$-0.392422\pi$
$462$		0			0
$463$	12.5403			0.582797			0.291399	−	0.956602i	$-0.405879\pi$
$463$	12.5403			0.582797			0.291399	+	0.956602i	$0.405879\pi$
$464$	27.1911			1.26232
$465$		0			0
$466$	11.2589	−	19.5010i	0.521559	−	0.903366i
$467$	24.4525			1.13153			0.565764	−	0.824567i	$-0.308581\pi$
$467$	24.4525			1.13153			0.565764	+	0.824567i	$0.308581\pi$
$468$		0			0
$469$	−11.5835	+	20.0632i	−0.534877	+	0.926434i
$470$	−9.96196	−	17.2546i	−0.459511	−	0.795896i
$471$		0			0
$472$	2.39099	+	4.14132i	0.110054	+	0.190620i
$473$	13.0806	+	22.6563i	0.601447	+	1.04174i
$474$		0			0
$475$	−3.10053	−	3.06377i	−0.142262	−	0.140575i
$476$	−0.243441			−0.0111581
$477$		0			0
$478$	−14.1813	−	24.5626i	−0.648635	−	1.12347i
$479$	3.27876	−	5.67898i	0.149810	−	0.259479i	−0.781347	−	0.624097i	$-0.785467\pi$
$479$	3.27876	−	5.67898i	0.149810	−	0.259479i	0.931157	+	0.364618i	$0.118800\pi$
$480$		0			0
$481$	−16.4942	+	28.5688i	−0.752071	+	1.30263i
$482$	44.6121			2.03202
$483$		0			0
$484$	−4.89744	+	8.48261i	−0.222611	+	0.385573i
$485$	−4.47355	+	7.74842i	−0.203134	+	0.351838i
$486$		0			0
$487$	−12.6305			−0.572341			−0.286170	−	0.958179i	$-0.592382\pi$
$487$	−12.6305			−0.572341			−0.286170	+	0.958179i	$0.592382\pi$
$488$	2.99413	−	5.18598i	0.135538	−	0.234758i
$489$		0			0
$490$	−4.18641	+	7.25107i	−0.189123	+	0.327570i
$491$	6.46604	+	11.1995i	0.291808	+	0.505427i	0.974237	−	0.225525i	$-0.0724098\pi$
$491$	6.46604	+	11.1995i	0.291808	+	0.505427i	−0.682429	+	0.730952i	$0.739076\pi$
$492$		0			0
$493$	−0.319187			−0.0143755
$494$	−9.87359	+	37.7478i	−0.444234	+	1.69835i
$495$		0			0
$496$	−1.46704	−	2.54099i	−0.0658722	−	0.114094i
$497$	−14.3592	−	24.8708i	−0.644097	−	1.11561i
$498$		0			0
$499$	14.9627	+	25.9162i	0.669824	+	1.16017i	0.977953	+	0.208825i	$0.0669639\pi$
$499$	14.9627	+	25.9162i	0.669824	+	1.16017i	−0.308129	+	0.951345i	$0.599703\pi$
$500$	−0.613656	+	1.06288i	−0.0274435	+	0.0475336i
$501$		0			0
$502$	5.95263			0.265679
$503$	−15.1592	+	26.2565i	−0.675914	+	1.17072i	0.300287	+	0.953849i	$0.402918\pi$
$503$	−15.1592	+	26.2565i	−0.675914	+	1.17072i	−0.976201	+	0.216869i	$0.930416\pi$
$504$		0			0
$505$	−2.19046			−0.0974744
$506$	−3.38176			−0.150338
$507$		0			0
$508$	−9.63211	−	16.6833i	−0.427356	−	0.740202i
$509$	13.2252	−	22.9067i	0.586197	−	1.01532i	−0.408528	−	0.912746i	$-0.633958\pi$
$509$	13.2252	−	22.9067i	0.586197	−	1.01532i	0.994725	−	0.102577i	$-0.0327088\pi$
$510$		0			0
$511$	−19.9697	−	34.5886i	−0.883409	−	1.53011i
$512$	16.5130			0.729779
$513$		0			0
$514$	−44.6692			−1.97027
$515$	−5.24513	−	9.08483i	−0.231128	−	0.400325i
$516$		0			0
$517$	−24.1591	+	41.8448i	−1.06252	+	1.84033i
$518$	20.3072	+	35.1731i	0.892247	+	1.54542i
$519$		0			0
$520$	−6.91654			−0.303310
$521$	3.32486			0.145665			0.0728324	−	0.997344i	$-0.476796\pi$
$521$	3.32486			0.145665			0.0728324	+	0.997344i	$0.476796\pi$
$522$		0			0
$523$	−8.88844	+	15.3952i	−0.388664	+	0.673186i	−0.992270	−	0.124097i	$-0.960397\pi$
$523$	−8.88844	+	15.3952i	−0.388664	+	0.673186i	0.603606	+	0.797283i	$0.293730\pi$
$524$	−14.4032			−0.629205
$525$		0			0
$526$	0.615626	−	1.06630i	0.0268426	−	0.0464927i
$527$	0.0172211	+	0.0298278i	0.000750163	+	0.00129932i
$528$		0			0
$529$	11.4067	+	19.7569i	0.495941	+	0.858996i
$530$	−4.84995	−	8.40037i	−0.210669	−	0.364889i
$531$		0			0
$532$	12.9943	+	12.8403i	0.563376	+	0.556696i
$533$	−0.815884			−0.0353399
$534$		0			0
$535$	6.84553	+	11.8568i	0.295958	+	0.512615i
$536$	−4.70871	+	8.15573i	−0.203385	+	0.352274i
$537$		0			0
$538$	−10.0370	+	17.3845i	−0.432724	+	0.749500i
$539$	20.3052			0.874608
$540$		0			0
$541$	17.7241	−	30.6991i	0.762021	−	1.31986i	−0.179787	−	0.983706i	$-0.557541\pi$
$541$	17.7241	−	30.6991i	0.762021	−	1.31986i	0.941808	−	0.336153i	$-0.109126\pi$
$542$	12.8683	−	22.2885i	0.552740	−	0.957374i
$543$		0			0
$544$	−0.355102			−0.0152249
$545$	7.07679	−	12.2574i	0.303136	−	0.525048i
$546$		0			0
$547$	−9.90142	+	17.1498i	−0.423354	+	0.733271i	−0.996265	−	0.0863465i	$-0.972481\pi$
$547$	−9.90142	+	17.1498i	−0.423354	+	0.733271i	0.572911	+	0.819618i	$0.305814\pi$
$548$	6.29440	+	10.9022i	0.268883	+	0.465720i
$549$		0			0
$550$	7.82667			0.333730
$551$	17.0375	+	16.8355i	0.725820	+	0.717214i
$552$		0			0
$553$	12.4401	+	21.5469i	0.529008	+	0.916269i
$554$	13.7895	+	23.8841i	0.585860	+	1.01474i
$555$		0			0
$556$	−13.3831	−	23.1802i	−0.567570	−	0.983061i
$557$	−13.6340	+	23.6148i	−0.577692	+	1.00059i	0.418052	+	0.908423i	$0.362713\pi$
$557$	−13.6340	+	23.6148i	−0.577692	+	1.00059i	−0.995743	+	0.0921684i	$0.970620\pi$
$558$		0			0
$559$	−29.9203			−1.26549
$560$	−8.44872	+	14.6336i	−0.357024	+	0.618383i
$561$		0			0
$562$	17.2691			0.728455
$563$	28.6245			1.20638			0.603190	−	0.797598i	$-0.293896\pi$
$563$	28.6245			1.20638			0.603190	+	0.797598i	$0.293896\pi$
$564$		0			0
$565$	6.64256	+	11.5052i	0.279454	+	0.484029i
$566$	−5.14250	+	8.90706i	−0.216155	+	0.374392i
$567$		0			0
$568$	−5.83702	−	10.1100i	−0.244916	−	0.424207i
$569$	−32.2230			−1.35086			−0.675430	−	0.737424i	$-0.736042\pi$
$569$	−32.2230			−1.35086			−0.675430	+	0.737424i	$0.736042\pi$
$570$		0			0
$571$	−7.39249			−0.309366			−0.154683	−	0.987964i	$-0.549436\pi$
$571$	−7.39249			−0.309366			−0.154683	+	0.987964i	$0.549436\pi$
$572$	−13.3213	−	23.0731i	−0.556991	−	0.964737i
$573$		0			0
$574$	−0.502247	+	0.869917i	−0.0209634	+	0.0363096i
$575$	0.216041	+	0.374194i	0.00900952	+	0.0156049i
$576$		0			0
$577$	3.03682			0.126425			0.0632123	−	0.998000i	$-0.479865\pi$
$577$	3.03682			0.126425			0.0632123	+	0.998000i	$0.479865\pi$
$578$	−30.5340			−1.27005
$579$		0			0
$580$	3.37205	−	5.84056i	0.140017	−	0.242516i
$581$	−55.0410			−2.28349
$582$		0			0
$583$	−11.7618	+	20.3720i	−0.487124	+	0.843723i
$584$	−8.11773	−	14.0603i	−0.335914	−	0.581820i
$585$		0			0
$586$	20.9416	+	36.2719i	0.865089	+	1.49838i
$587$	−7.96968	−	13.8039i	−0.328944	−	0.569748i	0.653359	−	0.757048i	$-0.273359\pi$
$587$	−7.96968	−	13.8039i	−0.328944	−	0.569748i	−0.982303	+	0.187301i	$0.940026\pi$
$588$		0			0
$589$	0.654040	−	2.50046i	0.0269492	−	0.103030i
$590$	6.18877			0.254788
$591$		0			0
$592$	16.3804	+	28.3718i	0.673232	+	1.16607i
$593$	13.2914	−	23.0213i	0.545811	−	0.945373i	−0.452744	−	0.891640i	$-0.649555\pi$
$593$	13.2914	−	23.0213i	0.545811	−	0.945373i	0.998555	−	0.0537322i	$-0.0171117\pi$
$594$		0			0
$595$	0.0991765	−	0.171779i	0.00406584	−	0.00704224i
$596$	25.4705			1.04331
$597$		0			0
$598$	1.93384	−	3.34951i	0.0790806	−	0.136972i
$599$	18.8742	−	32.6911i	0.771181	−	1.33572i	−0.165736	−	0.986170i	$-0.553000\pi$
$599$	18.8742	−	32.6911i	0.771181	−	1.33572i	0.936916	−	0.349554i	$-0.113667\pi$
$600$		0			0
$601$	−16.9222			−0.690269			−0.345134	−	0.938553i	$-0.612167\pi$
$601$	−16.9222			−0.690269			−0.345134	+	0.938553i	$0.612167\pi$
$602$	−18.4185	+	31.9018i	−0.750682	+	1.30022i
$603$		0			0
$604$	9.17476	−	15.8911i	0.373316	−	0.646601i
$605$	−3.99037	−	6.91153i	−0.162232	−	0.280994i
$606$		0			0
$607$	36.3712			1.47626			0.738131	−	0.674658i	$-0.235709\pi$
$607$	36.3712			1.47626			0.738131	+	0.674658i	$0.235709\pi$
$608$	18.9545	+	18.7298i	0.768708	+	0.759593i
$609$		0			0
$610$	−3.87496	−	6.71162i	−0.156892	−	0.271746i
$611$	−27.6305	−	47.8574i	−1.11781	−	1.93610i
$612$		0			0
$613$	−3.66105	−	6.34113i	−0.147869	−	0.256116i	0.782571	−	0.622561i	$-0.213908\pi$
$613$	−3.66105	−	6.34113i	−0.147869	−	0.256116i	−0.930439	+	0.366446i	$0.880575\pi$
$614$	−7.62936	+	13.2144i	−0.307896	+	0.533291i
$615$		0			0
$616$	20.6511			0.832057
$617$	4.91459	−	8.51233i	0.197854	−	0.342693i	−0.749978	−	0.661462i	$-0.769936\pi$
$617$	4.91459	−	8.51233i	0.197854	−	0.342693i	0.947832	+	0.318769i	$0.103269\pi$
$618$		0			0
$619$	8.29932			0.333578			0.166789	−	0.985993i	$-0.446660\pi$
$619$	8.29932			0.333578			0.166789	+	0.985993i	$0.446660\pi$
$620$	−0.727729			−0.0292263
$621$		0			0
$622$	4.08092	+	7.06836i	0.163630	+	0.283415i
$623$	19.0894	−	33.0639i	0.764802	−	1.32468i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	−33.8868			−1.35439
$627$		0			0
$628$	−0.0980355			−0.00391204
$629$	−0.192284	−	0.333046i	−0.00766688	−	0.0132794i
$630$		0			0
$631$	21.4448	−	37.1435i	0.853705	−	1.47866i	−0.0241360	−	0.999709i	$-0.507683\pi$
$631$	21.4448	−	37.1435i	0.853705	−	1.47866i	0.877841	−	0.478952i	$-0.158983\pi$
$632$	5.05693	+	8.75885i	0.201154	+	0.348409i
$633$		0			0
$634$	−34.5577			−1.37246
$635$	15.6963			0.622887
$636$		0			0
$637$	−11.6114	+	20.1116i	−0.460061	+	0.796850i
$638$	−43.0076			−1.70269
$639$		0			0
$640$	−5.13806	+	8.89938i	−0.203100	+	0.351779i
$641$	−11.2034	−	19.4048i	−0.442507	−	0.766444i	0.555368	−	0.831605i	$-0.312577\pi$
$641$	−11.2034	−	19.4048i	−0.442507	−	0.766444i	−0.997875	+	0.0651607i	$0.979244\pi$
$642$		0			0
$643$	−2.71105	−	4.69568i	−0.106914	−	0.185180i	0.807605	−	0.589724i	$-0.200763\pi$
$643$	−2.71105	−	4.69568i	−0.106914	−	0.185180i	−0.914518	+	0.404544i	$0.867430\pi$
$644$	−0.905427	−	1.56824i	−0.0356788	−	0.0617975i
$645$		0			0
$646$	−0.323544	−	0.319708i	−0.0127297	−	0.0125787i
$647$	−25.1940			−0.990477			−0.495238	−	0.868757i	$-0.664919\pi$
$647$	−25.1940			−0.990477			−0.495238	+	0.868757i	$0.664919\pi$
$648$		0			0
$649$	−7.50430	−	12.9978i	−0.294570	−	0.510210i
$650$	−4.47564	+	7.75203i	−0.175549	+	0.304060i
$651$		0			0
$652$	−11.3431	+	19.6469i	−0.444231	+	0.769431i
$653$	−8.11903			−0.317722			−0.158861	−	0.987301i	$-0.550782\pi$
$653$	−8.11903			−0.317722			−0.158861	+	0.987301i	$0.550782\pi$
$654$		0			0
$655$	5.86776	−	10.1633i	0.229272	−	0.397112i
$656$	−0.405129	+	0.701703i	−0.0158176	+	0.0273969i
$657$		0			0
$658$	−68.0357			−2.65231
$659$	−6.73290	+	11.6617i	−0.262277	+	0.454277i	−0.966847	−	0.255358i	$-0.917807\pi$
$659$	−6.73290	+	11.6617i	−0.262277	+	0.454277i	0.704570	+	0.709635i	$0.251140\pi$
$660$		0			0
$661$	−4.51947	+	7.82796i	−0.175787	+	0.304472i	−0.940433	−	0.339978i	$-0.889580\pi$
$661$	−4.51947	+	7.82796i	−0.175787	+	0.304472i	0.764646	+	0.644450i	$0.222914\pi$
$662$	−8.52701	−	14.7692i	−0.331412	−	0.574022i
$663$		0			0
$664$	−22.3743			−0.868289
$665$	−14.3543	+	3.93811i	−0.556634	+	0.152713i
$666$		0			0
$667$	−1.18715	−	2.05620i	−0.0459665	−	0.0796163i
$668$	−11.4653	−	19.8585i	−0.443606	−	0.768348i
$669$		0			0
$670$	6.09394	+	10.5550i	0.235429	+	0.407776i
$671$	−9.39729	+	16.2766i	−0.362779	+	0.628351i
$672$		0			0
$673$	13.6988			0.528048			0.264024	−	0.964516i	$-0.414950\pi$
$673$	13.6988			0.528048			0.264024	+	0.964516i	$0.414950\pi$
$674$	−21.3517	+	36.9823i	−0.822438	+	1.42450i
$675$		0			0
$676$	14.5157			0.558298
$677$	11.6030			0.445941			0.222970	−	0.974825i	$-0.428425\pi$
$677$	11.6030			0.445941			0.222970	+	0.974825i	$0.428425\pi$
$678$		0			0
$679$	15.2762	+	26.4591i	0.586246	+	1.01541i
$680$	0.0403154	−	0.0698283i	0.00154602	−	0.00267779i
$681$		0			0
$682$	2.32039	+	4.01904i	0.0888524	+	0.153897i
$683$	25.5290			0.976840			0.488420	−	0.872609i	$-0.337574\pi$
$683$	25.5290			0.976840			0.488420	+	0.872609i	$0.337574\pi$
$684$		0			0
$685$	−10.2572			−0.391908
$686$	−7.17528	−	12.4279i	−0.273953	−	0.474501i
$687$		0			0
$688$	−14.8570	+	25.7330i	−0.566416	+	0.981062i
$689$	−13.4518	−	23.2993i	−0.512474	−	0.887631i
$690$		0			0
$691$	−46.1415			−1.75530			−0.877652	−	0.479298i	$-0.840891\pi$
$691$	−46.1415			−1.75530			−0.877652	+	0.479298i	$0.840891\pi$
$692$	10.5515			0.401106
$693$		0			0
$694$	25.1631	−	43.5838i	0.955178	−	1.65442i
$695$	21.8088			0.827255
$696$		0			0
$697$	0.00475566	−	0.00823705i	0.000180134	−	0.000312000i
$698$	−0.00862000	−	0.0149303i	−0.000326272	−	0.000565119i
$699$		0			0
$700$	2.09550	+	3.62951i	0.0792024	+	0.137183i
$701$	0.219837	+	0.380768i	0.00830312	+	0.0143814i	0.870147	−	0.492792i	$-0.164024\pi$
$701$	0.219837	+	0.380768i	0.00830312	+	0.0143814i	−0.861844	+	0.507173i	$0.830690\pi$
$702$		0			0
$703$	−7.30277	+	27.9192i	−0.275429	+	1.05299i
$704$	−4.73024			−0.178277
$705$		0			0
$706$	5.46650	+	9.46826i	0.205735	+	0.356343i
$707$	−3.73997	+	6.47782i	−0.140656	+	0.243624i
$708$		0			0
$709$	19.4677	−	33.7191i	0.731125	−	1.26635i	−0.225278	−	0.974295i	$-0.572329\pi$
$709$	19.4677	−	33.7191i	0.731125	−	1.26635i	0.956403	−	0.292051i	$-0.0943377\pi$
$710$	−15.1084			−0.567007
$711$		0			0
$712$	7.75988	−	13.4405i	0.290814	−	0.503704i
$713$	−0.128100	+	0.221876i	−0.00479739	+	0.00830932i
$714$		0			0
$715$	21.7081			0.811835
$716$	−3.78373	+	6.55361i	−0.141404	+	0.244920i
$717$		0			0
$718$	−16.0073	+	27.7255i	−0.597388	+	1.03471i
$719$	−11.5179	−	19.9496i	−0.429546	−	0.743995i	0.567287	−	0.823520i	$-0.307993\pi$
$719$	−11.5179	−	19.9496i	−0.429546	−	0.743995i	−0.996833	+	0.0795252i	$0.974660\pi$
$720$		0			0
$721$	−35.8219			−1.33408
$722$	0.407100	+	34.1305i	0.0151507	+	1.27021i
$723$		0			0
$724$	−12.3092	−	21.3202i	−0.457469	−	0.792360i
$725$	2.74750	+	4.75882i	0.102040	+	0.176738i
$726$		0			0
$727$	22.2700	+	38.5727i	0.825948	+	1.43058i	0.901193	+	0.433419i	$0.142693\pi$
$727$	22.2700	+	38.5727i	0.825948	+	1.43058i	−0.0752445	+	0.997165i	$0.523974\pi$
$728$	−11.8092	+	20.4542i	−0.437678	+	0.758081i
$729$		0			0
$730$	−21.0117			−0.777677
$731$	0.174401	−	0.302071i	0.00645044	−	0.0111725i
$732$		0			0
$733$	34.2392			1.26465			0.632326	−	0.774702i	$-0.282100\pi$
$733$	34.2392			1.26465			0.632326	+	0.774702i	$0.282100\pi$
$734$	−8.80612			−0.325040
$735$		0			0
$736$	−1.32072	−	2.28756i	−0.0486826	−	0.0843207i
$737$	14.7786	−	25.5974i	0.544378	−	0.942891i
$738$		0			0
$739$	2.42166	+	4.19444i	0.0890823	+	0.154295i	0.907123	−	0.420865i	$-0.138273\pi$
$739$	2.42166	+	4.19444i	0.0890823	+	0.154295i	−0.818041	+	0.575160i	$0.804940\pi$
$740$	8.12555			0.298701
$741$		0			0
$742$	−33.1230			−1.21598
$743$	6.53384	+	11.3169i	0.239703	+	0.415178i	0.960629	−	0.277834i	$-0.0896165\pi$
$743$	6.53384	+	11.3169i	0.239703	+	0.415178i	−0.720926	+	0.693012i	$0.756283\pi$
$744$		0			0
$745$	−10.3765	+	17.9727i	−0.380167	+	0.658468i
$746$	11.6766	+	20.2245i	0.427512	+	0.740473i
$747$		0			0
$748$	0.310591			0.0113563
$749$	46.7519			1.70828
$750$		0			0
$751$	−4.24060	+	7.34493i	−0.154742	+	0.268020i	−0.932965	−	0.359967i	$-0.882788\pi$
$751$	−4.24060	+	7.34493i	−0.154742	+	0.268020i	0.778223	+	0.627988i	$0.216121\pi$
$752$	−54.8798			−2.00126
$753$		0			0
$754$	24.5937	−	42.5975i	0.895648	−	1.55131i
$755$	7.47548	+	12.9479i	0.272061	+	0.471223i
$756$		0			0
$757$	17.6798	+	30.6223i	0.642583	+	1.11299i	0.984854	+	0.173386i	$0.0554707\pi$
$757$	17.6798	+	30.6223i	0.642583	+	1.11299i	−0.342271	+	0.939601i	$0.611196\pi$
$758$	−15.0286	−	26.0303i	−0.545864	−	0.945464i
$759$		0			0
$760$	−5.83502	+	1.60085i	−0.211659	+	0.0580688i
$761$	−19.8496			−0.719547			−0.359773	−	0.933040i	$-0.617146\pi$
$761$	−19.8496			−0.719547			−0.359773	+	0.933040i	$0.617146\pi$
$762$		0			0
$763$	−24.1657	−	41.8562i	−0.874856	−	1.51529i
$764$	10.4752	−	18.1435i	0.378978	−	0.656409i
$765$		0			0
$766$	−26.8503	+	46.5061i	−0.970141	+	1.68033i
$767$	17.1652			0.619798
$768$		0			0
$769$	−2.82265	+	4.88897i	−0.101787	+	0.176301i	−0.912421	−	0.409253i	$-0.865789\pi$
$769$	−2.82265	+	4.88897i	−0.101787	+	0.176301i	0.810634	+	0.585553i	$0.199123\pi$
$770$	13.3632	−	23.1457i	0.481575	−	0.834113i
$771$		0			0
$772$	8.92444			0.321198
$773$	24.6102	−	42.6260i	0.885166	−	1.53315i	0.0396424	−	0.999214i	$-0.487378\pi$
$773$	24.6102	−	42.6260i	0.885166	−	1.53315i	0.845523	−	0.533938i	$-0.179289\pi$
$774$		0			0
$775$	0.296472	−	0.513505i	0.0106496	−	0.0184456i
$776$	6.20979	+	10.7557i	0.222918	+	0.386106i
$777$		0			0
$778$	57.0088			2.04387
$779$	−0.688308	+	0.188838i	−0.0246612	+	0.00676583i
$780$		0			0
$781$	18.3199	+	31.7310i	0.655539	+	1.13543i
$782$	0.0225441	+	0.0390475i	0.000806175	+	0.00139634i
$783$		0			0

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)

\(f(q)\)	\(=\)	\(q+(-0.898236 - 1.55579i) q^{2} +(-0.613656 + 1.06288i) q^{4} +(-0.500000 - 0.866025i) q^{5} -3.41478 q^{7} -1.38811 q^{8} +O(q^{10})\)	\(q+(-0.898236 - 1.55579i) q^{2} +(-0.613656 + 1.06288i) q^{4} +(-0.500000 - 0.866025i) q^{5} -3.41478 q^{7} -1.38811 q^{8} +(-0.898236 + 1.55579i) q^{10} +4.35669 q^{11} +(-2.49135 + 4.31514i) q^{13} +(3.06728 + 5.31268i) q^{14} +(2.47416 + 4.28538i) q^{16} +(-0.0290433 - 0.0503046i) q^{17} +(-1.10304 + 4.21703i) q^{19} +1.22731 q^{20} +(-3.91334 - 6.77810i) q^{22} +(0.216041 - 0.374194i) q^{23} +(-0.500000 + 0.866025i) q^{25} +8.95127 q^{26} +(2.09550 - 3.62951i) q^{28} +(2.74750 - 4.75882i) q^{29} -0.592945 q^{31} +(3.05666 - 5.29428i) q^{32} +(-0.0521756 + 0.0903707i) q^{34} +(1.70739 + 2.95728i) q^{35} +6.62060 q^{37} +(7.55160 - 2.07179i) q^{38} +(0.694056 + 1.20214i) q^{40} +(0.0818718 + 0.141806i) q^{41} +(3.00242 + 5.20034i) q^{43} +(-2.67351 + 4.63066i) q^{44} -0.776222 q^{46} +(-5.54529 + 9.60473i) q^{47} +4.66070 q^{49} +1.79647 q^{50} +(-3.05766 - 5.29603i) q^{52} +(-2.69971 + 4.67603i) q^{53} +(-2.17835 - 3.77300i) q^{55} +4.74009 q^{56} -9.87163 q^{58} +(-1.72248 - 2.98342i) q^{59} +(-2.15698 + 3.73600i) q^{61} +(0.532604 + 0.922498i) q^{62} -1.08574 q^{64} +4.98270 q^{65} +(3.39217 - 5.87541i) q^{67} +0.0712905 q^{68} +(3.06728 - 5.31268i) q^{70} +(4.20501 + 7.28329i) q^{71} +(5.84804 + 10.1291i) q^{73} +(-5.94686 - 10.3003i) q^{74} +(-3.80532 - 3.76021i) q^{76} -14.8771 q^{77} +(-3.64303 - 6.30991i) q^{79} +(2.47416 - 4.28538i) q^{80} +(0.147080 - 0.254751i) q^{82} +16.1185 q^{83} +(-0.0290433 + 0.0503046i) q^{85} +(5.39377 - 9.34228i) q^{86} -6.04757 q^{88} +(-5.59024 + 9.68258i) q^{89} +(8.50740 - 14.7352i) q^{91} +(0.265150 + 0.459252i) q^{92} +19.9239 q^{94} +(4.20357 - 1.15325i) q^{95} +(-4.47355 - 7.74842i) q^{97} +(-4.18641 - 7.25107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8	\( 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} - 3 q^{10} - 8 q^{13} - 10 q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + 10 q^{20} + 2 q^{23} - 6 q^{25} + 40 q^{26} - 26 q^{28} - 4 q^{29} + 24 q^{31} - 15 q^{32} + 7 q^{34} - 2 q^{35} + 29 q^{38} - 6 q^{40} - 12 q^{41} - 4 q^{43} - 6 q^{44} + 48 q^{46} - 6 q^{47} + 32 q^{49} + 6 q^{50} - 20 q^{52} - 26 q^{53} + 44 q^{56} - 20 q^{58} - 16 q^{59} + 20 q^{61} + 25 q^{62} + 28 q^{64} + 16 q^{65} - 12 q^{67} - 54 q^{68} - 10 q^{70} + 8 q^{71} - 4 q^{73} + 16 q^{74} - 66 q^{76} - 48 q^{77} - 12 q^{79} - 3 q^{80} + 26 q^{82} + 44 q^{83} - 4 q^{85} + 44 q^{86} - 32 q^{88} + 8 q^{89} + 2 q^{91} - 36 q^{92} - 14 q^{94} + 6 q^{95} + 30 q^{97} - 49 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8 - 3 * q^10 - 8 * q^13 - 10 * q^14 - 3 * q^16 - 4 * q^17 - 6 * q^19 + 10 * q^20 + 2 * q^23 - 6 * q^25 + 40 * q^26 - 26 * q^28 - 4 * q^29 + 24 * q^31 - 15 * q^32 + 7 * q^34 - 2 * q^35 + 29 * q^38 - 6 * q^40 - 12 * q^41 - 4 * q^43 - 6 * q^44 + 48 * q^46 - 6 * q^47 + 32 * q^49 + 6 * q^50 - 20 * q^52 - 26 * q^53 + 44 * q^56 - 20 * q^58 - 16 * q^59 + 20 * q^61 + 25 * q^62 + 28 * q^64 + 16 * q^65 - 12 * q^67 - 54 * q^68 - 10 * q^70 + 8 * q^71 - 4 * q^73 + 16 * q^74 - 66 * q^76 - 48 * q^77 - 12 * q^79 - 3 * q^80 + 26 * q^82 + 44 * q^83 - 4 * q^85 + 44 * q^86 - 32 * q^88 + 8 * q^89 + 2 * q^91 - 36 * q^92 - 14 * q^94 + 6 * q^95 + 30 * q^97 - 49 * q^98

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