LMFDB - Embedded newform 445.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.a (trivial)

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:	yes
Analytic conductor:	$3.55334288995$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.50065$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50065 q^{2} +1.23408 q^{3} +4.25326 q^{4} -1.00000 q^{5} -3.08600 q^{6} -4.89614 q^{7} -5.63461 q^{8} -1.47705 q^{9} +O(q^{10})$	\(q-2.50065 q^{2} +1.23408 q^{3} +4.25326 q^{4} -1.00000 q^{5} -3.08600 q^{6} -4.89614 q^{7} -5.63461 q^{8} -1.47705 q^{9} +2.50065 q^{10} +5.13396 q^{11} +5.24885 q^{12} +0.293733 q^{13} +12.2435 q^{14} -1.23408 q^{15} +5.58369 q^{16} +4.78896 q^{17} +3.69359 q^{18} +4.79227 q^{19} -4.25326 q^{20} -6.04222 q^{21} -12.8382 q^{22} +5.18988 q^{23} -6.95355 q^{24} +1.00000 q^{25} -0.734525 q^{26} -5.52503 q^{27} -20.8246 q^{28} +8.87657 q^{29} +3.08600 q^{30} -6.58148 q^{31} -2.69363 q^{32} +6.33571 q^{33} -11.9755 q^{34} +4.89614 q^{35} -6.28228 q^{36} +0.840228 q^{37} -11.9838 q^{38} +0.362490 q^{39} +5.63461 q^{40} -2.51195 q^{41} +15.1095 q^{42} +9.39183 q^{43} +21.8361 q^{44} +1.47705 q^{45} -12.9781 q^{46} +8.06738 q^{47} +6.89071 q^{48} +16.9722 q^{49} -2.50065 q^{50} +5.90996 q^{51} +1.24932 q^{52} -11.4348 q^{53} +13.8162 q^{54} -5.13396 q^{55} +27.5879 q^{56} +5.91403 q^{57} -22.1972 q^{58} +1.04378 q^{59} -5.24885 q^{60} +6.77868 q^{61} +16.4580 q^{62} +7.23185 q^{63} -4.43155 q^{64} -0.293733 q^{65} -15.8434 q^{66} -14.0551 q^{67} +20.3687 q^{68} +6.40471 q^{69} -12.2435 q^{70} +2.08800 q^{71} +8.32261 q^{72} +6.37951 q^{73} -2.10112 q^{74} +1.23408 q^{75} +20.3827 q^{76} -25.1366 q^{77} -0.906461 q^{78} +0.566895 q^{79} -5.58369 q^{80} -2.38717 q^{81} +6.28152 q^{82} +6.12572 q^{83} -25.6991 q^{84} -4.78896 q^{85} -23.4857 q^{86} +10.9544 q^{87} -28.9279 q^{88} +1.00000 q^{89} -3.69359 q^{90} -1.43816 q^{91} +22.0739 q^{92} -8.12206 q^{93} -20.1737 q^{94} -4.79227 q^{95} -3.32415 q^{96} -3.68712 q^{97} -42.4416 q^{98} -7.58312 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 6 q^{3} + 7 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{7} + 3 q^{8} + 12 q^{9}+O(q^{10}) $ 8 * q + q^2 + 6 * q^3 + 7 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^7 + 3 * q^8 + 12 * q^9	\( 8 q + q^{2} + 6 q^{3} + 7 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{7} + 3 q^{8} + 12 q^{9} - q^{10} + 14 q^{11} + 17 q^{12} - 7 q^{13} + 15 q^{14} - 6 q^{15} + 9 q^{16} + 17 q^{17} - q^{18} + 17 q^{19} - 7 q^{20} + 2 q^{22} - q^{23} + 8 q^{24} + 8 q^{25} + 3 q^{26} + 21 q^{27} - 29 q^{28} + 10 q^{29} - 6 q^{30} + q^{31} + 2 q^{32} + 10 q^{33} - 16 q^{34} + 6 q^{35} - 17 q^{36} - 11 q^{37} - 30 q^{38} - 5 q^{39} - 3 q^{40} + 15 q^{41} + 14 q^{42} - 5 q^{43} + 7 q^{44} - 12 q^{45} - 12 q^{46} + 12 q^{47} + 3 q^{48} + 4 q^{49} + q^{50} + 35 q^{51} - 14 q^{52} - q^{53} - 29 q^{54} - 14 q^{55} + 3 q^{56} + 15 q^{57} - 37 q^{58} + 26 q^{59} - 17 q^{60} + 13 q^{61} + 22 q^{62} - 16 q^{63} - 15 q^{64} + 7 q^{65} + 4 q^{66} - 25 q^{67} + 23 q^{68} - 5 q^{69} - 15 q^{70} + 28 q^{71} + 22 q^{72} - 17 q^{73} - 5 q^{74} + 6 q^{75} + 8 q^{76} - 56 q^{78} - 7 q^{79} - 9 q^{80} + 24 q^{81} + 5 q^{82} + 44 q^{83} - 57 q^{84} - 17 q^{85} - 13 q^{86} - 12 q^{87} - 66 q^{88} + 8 q^{89} + q^{90} + 27 q^{91} + 15 q^{92} - 38 q^{93} - 27 q^{94} - 17 q^{95} - 20 q^{96} + q^{97} - 34 q^{98} + q^{99}+O(q^{100}) \) 8 * q + q^2 + 6 * q^3 + 7 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^7 + 3 * q^8 + 12 * q^9 - q^10 + 14 * q^11 + 17 * q^12 - 7 * q^13 + 15 * q^14 - 6 * q^15 + 9 * q^16 + 17 * q^17 - q^18 + 17 * q^19 - 7 * q^20 + 2 * q^22 - q^23 + 8 * q^24 + 8 * q^25 + 3 * q^26 + 21 * q^27 - 29 * q^28 + 10 * q^29 - 6 * q^30 + q^31 + 2 * q^32 + 10 * q^33 - 16 * q^34 + 6 * q^35 - 17 * q^36 - 11 * q^37 - 30 * q^38 - 5 * q^39 - 3 * q^40 + 15 * q^41 + 14 * q^42 - 5 * q^43 + 7 * q^44 - 12 * q^45 - 12 * q^46 + 12 * q^47 + 3 * q^48 + 4 * q^49 + q^50 + 35 * q^51 - 14 * q^52 - q^53 - 29 * q^54 - 14 * q^55 + 3 * q^56 + 15 * q^57 - 37 * q^58 + 26 * q^59 - 17 * q^60 + 13 * q^61 + 22 * q^62 - 16 * q^63 - 15 * q^64 + 7 * q^65 + 4 * q^66 - 25 * q^67 + 23 * q^68 - 5 * q^69 - 15 * q^70 + 28 * q^71 + 22 * q^72 - 17 * q^73 - 5 * q^74 + 6 * q^75 + 8 * q^76 - 56 * q^78 - 7 * q^79 - 9 * q^80 + 24 * q^81 + 5 * q^82 + 44 * q^83 - 57 * q^84 - 17 * q^85 - 13 * q^86 - 12 * q^87 - 66 * q^88 + 8 * q^89 + q^90 + 27 * q^91 + 15 * q^92 - 38 * q^93 - 27 * q^94 - 17 * q^95 - 20 * q^96 + q^97 - 34 * q^98 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.50065	−1.76823	−0.884114	−	0.467272i	$-0.845237\pi$
$2$	−2.50065	−1.76823	−0.884114	+	0.467272i	$0.845237\pi$
$3$	1.23408	0.712495	0.356248	−	0.934392i	$-0.384056\pi$
$3$	1.23408	0.712495	0.356248	+	0.934392i	$0.384056\pi$
$4$	4.25326	2.12663
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.08600	−1.25985
$7$	−4.89614	−1.85057	−0.925284	−	0.379275i	$-0.876173\pi$
$7$	−4.89614	−1.85057	−0.925284	+	0.379275i	$0.876173\pi$
$8$	−5.63461	−1.99214
$9$	−1.47705	−0.492350
$10$	2.50065	0.790775
$11$	5.13396	1.54795	0.773974	−	0.633218i	$-0.218266\pi$
$11$	5.13396	1.54795	0.773974	+	0.633218i	$0.218266\pi$
$12$	5.24885	1.51521
$13$	0.293733	0.0814670	0.0407335	−	0.999170i	$-0.487031\pi$
$13$	0.293733	0.0814670	0.0407335	+	0.999170i	$0.487031\pi$
$14$	12.2435	3.27223
$15$	−1.23408	−0.318638
$16$	5.58369	1.39592
$17$	4.78896	1.16149	0.580747	−	0.814084i	$-0.302760\pi$
$17$	4.78896	1.16149	0.580747	+	0.814084i	$0.302760\pi$
$18$	3.69359	0.870587
$19$	4.79227	1.09942	0.549711	−	0.835355i	$-0.314738\pi$
$19$	4.79227	1.09942	0.549711	+	0.835355i	$0.314738\pi$
$20$	−4.25326	−0.951057
$21$	−6.04222	−1.31852
$22$	−12.8382	−2.73712
$23$	5.18988	1.08216	0.541082	−	0.840970i	$-0.318015\pi$
$23$	5.18988	1.08216	0.541082	+	0.840970i	$0.318015\pi$
$24$	−6.95355	−1.41939
$25$	1.00000	0.200000
$26$	−0.734525	−0.144052
$27$	−5.52503	−1.06329
$28$	−20.8246	−3.93547
$29$	8.87657	1.64834	0.824169	−	0.566344i	$-0.191643\pi$
$29$	8.87657	1.64834	0.824169	+	0.566344i	$0.191643\pi$
$30$	3.08600	0.563424
$31$	−6.58148	−1.18207	−0.591034	−	0.806646i	$-0.701280\pi$
$31$	−6.58148	−1.18207	−0.591034	+	0.806646i	$0.701280\pi$
$32$	−2.69363	−0.476171
$33$	6.33571	1.10291
$34$	−11.9755	−2.05379
$35$	4.89614	0.827599
$36$	−6.28228	−1.04705
$37$	0.840228	0.138133	0.0690663	−	0.997612i	$-0.477998\pi$
$37$	0.840228	0.138133	0.0690663	+	0.997612i	$0.477998\pi$
$38$	−11.9838	−1.94403
$39$	0.362490	0.0580449
$40$	5.63461	0.890910
$41$	−2.51195	−0.392301	−0.196150	−	0.980574i	$-0.562844\pi$
$41$	−2.51195	−0.392301	−0.196150	+	0.980574i	$0.562844\pi$
$42$	15.1095	2.33145
$43$	9.39183	1.43224	0.716120	−	0.697977i	$-0.245916\pi$
$43$	9.39183	1.43224	0.716120	+	0.697977i	$0.245916\pi$
$44$	21.8361	3.29191
$45$	1.47705	0.220186
$46$	−12.9781	−1.91351
$47$	8.06738	1.17675	0.588374	−	0.808589i	$-0.299768\pi$
$47$	8.06738	1.17675	0.588374	+	0.808589i	$0.299768\pi$
$48$	6.89071	0.994588
$49$	16.9722	2.42460
$50$	−2.50065	−0.353646
$51$	5.90996	0.827559
$52$	1.24932	0.173250
$53$	−11.4348	−1.57070	−0.785348	−	0.619055i	$-0.787516\pi$
$53$	−11.4348	−1.57070	−0.785348	+	0.619055i	$0.787516\pi$
$54$	13.8162	1.88014
$55$	−5.13396	−0.692263
$56$	27.5879	3.68658
$57$	5.91403	0.783333
$58$	−22.1972	−2.91464
$59$	1.04378	0.135888	0.0679441	−	0.997689i	$-0.478356\pi$
$59$	1.04378	0.135888	0.0679441	+	0.997689i	$0.478356\pi$
$60$	−5.24885	−0.677624
$61$	6.77868	0.867921	0.433961	−	0.900932i	$-0.357116\pi$
$61$	6.77868	0.867921	0.433961	+	0.900932i	$0.357116\pi$
$62$	16.4580	2.09017
$63$	7.23185	0.911128
$64$	−4.43155	−0.553943
$65$	−0.293733	−0.0364331
$66$	−15.8434	−1.95019
$67$	−14.0551	−1.71710	−0.858551	−	0.512727i	$-0.828635\pi$
$67$	−14.0551	−1.71710	−0.858551	+	0.512727i	$0.828635\pi$
$68$	20.3687	2.47007
$69$	6.40471	0.771037
$70$	−12.2435	−1.46338
$71$	2.08800	0.247800	0.123900	−	0.992295i	$-0.460460\pi$
$71$	2.08800	0.247800	0.123900	+	0.992295i	$0.460460\pi$
$72$	8.32261	0.980829
$73$	6.37951	0.746665	0.373332	−	0.927698i	$-0.378215\pi$
$73$	6.37951	0.746665	0.373332	+	0.927698i	$0.378215\pi$
$74$	−2.10112	−0.244250
$75$	1.23408	0.142499
$76$	20.3827	2.33806
$77$	−25.1366	−2.86458
$78$	−0.906461	−0.102637
$79$	0.566895	0.0637807	0.0318903	−	0.999491i	$-0.489847\pi$
$79$	0.566895	0.0637807	0.0318903	+	0.999491i	$0.489847\pi$
$80$	−5.58369	−0.624275
$81$	−2.38717	−0.265241
$82$	6.28152	0.693677
$83$	6.12572	0.672385	0.336192	−	0.941793i	$-0.390861\pi$
$83$	6.12572	0.672385	0.336192	+	0.941793i	$0.390861\pi$
$84$	−25.6991	−2.80400
$85$	−4.78896	−0.519436
$86$	−23.4857	−2.53253
$87$	10.9544	1.17443
$88$	−28.9279	−3.08372
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	−3.69359	−0.389339
$91$	−1.43816	−0.150760
$92$	22.0739	2.30136
$93$	−8.12206	−0.842219
$94$	−20.1737	−2.08076
$95$	−4.79227	−0.491676
$96$	−3.32415	−0.339270
$97$	−3.68712	−0.374370	−0.187185	−	0.982325i	$-0.559936\pi$
$97$	−3.68712	−0.374370	−0.187185	+	0.982325i	$0.559936\pi$
$98$	−42.4416	−4.28725
$99$	−7.58312	−0.762132
$100$	4.25326	0.425326
$101$	−6.79147	−0.675777	−0.337888	−	0.941186i	$-0.609713\pi$
$101$	−6.79147	−0.675777	−0.337888	+	0.941186i	$0.609713\pi$
$102$	−14.7787	−1.46331
$103$	10.1620	1.00129	0.500646	−	0.865652i	$-0.333096\pi$
$103$	10.1620	1.00129	0.500646	+	0.865652i	$0.333096\pi$
$104$	−1.65507	−0.162293
$105$	6.04222	0.589661
$106$	28.5945	2.77735
$107$	7.95873	0.769399	0.384700	−	0.923042i	$-0.374305\pi$
$107$	7.95873	0.769399	0.384700	+	0.923042i	$0.374305\pi$
$108$	−23.4994	−2.26123
$109$	−3.91220	−0.374721	−0.187360	−	0.982291i	$-0.559993\pi$
$109$	−3.91220	−0.374721	−0.187360	+	0.982291i	$0.559993\pi$
$110$	12.8382	1.22408
$111$	1.03691	0.0984188
$112$	−27.3385	−2.58325
$113$	−8.18970	−0.770423	−0.385211	−	0.922828i	$-0.625871\pi$
$113$	−8.18970	−0.770423	−0.385211	+	0.922828i	$0.625871\pi$
$114$	−14.7889	−1.38511
$115$	−5.18988	−0.483958
$116$	37.7543	3.50540
$117$	−0.433859	−0.0401103
$118$	−2.61012	−0.240281
$119$	−23.4475	−2.14942
$120$	6.95355	0.634770
$121$	15.3576	1.39614
$122$	−16.9511	−1.53468
$123$	−3.09995	−0.279513
$124$	−27.9927	−2.51382
$125$	−1.00000	−0.0894427
$126$	−18.0843	−1.61108
$127$	−2.03100	−0.180222	−0.0901110	−	0.995932i	$-0.528722\pi$
$127$	−2.03100	−0.180222	−0.0901110	+	0.995932i	$0.528722\pi$
$128$	16.4690	1.45567
$129$	11.5903	1.02046
$130$	0.734525	0.0644221
$131$	9.71902	0.849155	0.424578	−	0.905392i	$-0.360423\pi$
$131$	9.71902	0.849155	0.424578	+	0.905392i	$0.360423\pi$
$132$	26.9474	2.34547
$133$	−23.4636	−2.03455
$134$	35.1469	3.03623
$135$	5.52503	0.475519
$136$	−26.9840	−2.31386
$137$	21.4520	1.83277	0.916386	−	0.400296i	$-0.131093\pi$
$137$	21.4520	1.83277	0.916386	+	0.400296i	$0.131093\pi$
$138$	−16.0160	−1.36337
$139$	−0.595475	−0.0505075	−0.0252538	−	0.999681i	$-0.508039\pi$
$139$	−0.595475	−0.0505075	−0.0252538	+	0.999681i	$0.508039\pi$
$140$	20.8246	1.76000
$141$	9.95578	0.838428
$142$	−5.22136	−0.438166
$143$	1.50802	0.126107
$144$	−8.24739	−0.687283
$145$	−8.87657	−0.737159
$146$	−15.9529	−1.32027
$147$	20.9450	1.72752
$148$	3.57371	0.293757
$149$	−16.4598	−1.34844	−0.674220	−	0.738531i	$-0.735520\pi$
$149$	−16.4598	−1.34844	−0.674220	+	0.738531i	$0.735520\pi$
$150$	−3.08600	−0.251971
$151$	1.01236	0.0823849	0.0411924	−	0.999151i	$-0.486884\pi$
$151$	1.01236	0.0823849	0.0411924	+	0.999151i	$0.486884\pi$
$152$	−27.0026	−2.19020
$153$	−7.07354	−0.571862
$154$	62.8579	5.06523
$155$	6.58148	0.528637
$156$	1.54176	0.123440
$157$	−3.66401	−0.292420	−0.146210	−	0.989254i	$-0.546707\pi$
$157$	−3.66401	−0.292420	−0.146210	+	0.989254i	$0.546707\pi$
$158$	−1.41761	−0.112779
$159$	−14.1115	−1.11911
$160$	2.69363	0.212950
$161$	−25.4104	−2.00262
$162$	5.96947	0.469006
$163$	−0.396340	−0.0310437	−0.0155219	−	0.999880i	$-0.504941\pi$
$163$	−0.396340	−0.0310437	−0.0155219	+	0.999880i	$0.504941\pi$
$164$	−10.6840	−0.834279
$165$	−6.33571	−0.493234
$166$	−15.3183	−1.18893
$167$	−8.91272	−0.689687	−0.344843	−	0.938660i	$-0.612068\pi$
$167$	−8.91272	−0.689687	−0.344843	+	0.938660i	$0.612068\pi$
$168$	34.0456	2.62667
$169$	−12.9137	−0.993363
$170$	11.9755	0.918481
$171$	−7.07842	−0.541300
$172$	39.9459	3.04584
$173$	20.4096	1.55171	0.775855	−	0.630911i	$-0.217319\pi$
$173$	20.4096	1.55171	0.775855	+	0.630911i	$0.217319\pi$
$174$	−27.3931	−2.07666
$175$	−4.89614	−0.370114
$176$	28.6664	2.16081
$177$	1.28810	0.0968197
$178$	−2.50065	−0.187432
$179$	−3.85954	−0.288476	−0.144238	−	0.989543i	$-0.546073\pi$
$179$	−3.85954	−0.288476	−0.144238	+	0.989543i	$0.546073\pi$
$180$	6.28228	0.468253
$181$	−4.85674	−0.360998	−0.180499	−	0.983575i	$-0.557771\pi$
$181$	−4.85674	−0.360998	−0.180499	+	0.983575i	$0.557771\pi$
$182$	3.59634	0.266578
$183$	8.36542	0.618390
$184$	−29.2429	−2.15582
$185$	−0.840228	−0.0617748
$186$	20.3104	1.48923
$187$	24.5864	1.79793
$188$	34.3126	2.50251
$189$	27.0513	1.96770
$190$	11.9838	0.869395
$191$	7.44680	0.538831	0.269416	−	0.963024i	$-0.413169\pi$
$191$	7.44680	0.538831	0.269416	+	0.963024i	$0.413169\pi$
$192$	−5.46887	−0.394682
$193$	−7.68593	−0.553245	−0.276623	−	0.960979i	$-0.589215\pi$
$193$	−7.68593	−0.553245	−0.276623	+	0.960979i	$0.589215\pi$
$194$	9.22021	0.661972
$195$	−0.362490	−0.0259585
$196$	72.1872	5.15623
$197$	−26.9802	−1.92226	−0.961131	−	0.276092i	$-0.910961\pi$
$197$	−26.9802	−1.92226	−0.961131	+	0.276092i	$0.910961\pi$
$198$	18.9627	1.34762
$199$	5.24242	0.371625	0.185813	−	0.982585i	$-0.440508\pi$
$199$	5.24242	0.371625	0.185813	+	0.982585i	$0.440508\pi$
$200$	−5.63461	−0.398427
$201$	−17.3451	−1.22343
$202$	16.9831	1.19493
$203$	−43.4609	−3.05036
$204$	25.1366	1.75991
$205$	2.51195	0.175442
$206$	−25.4116	−1.77051
$207$	−7.66571	−0.532804
$208$	1.64012	0.113722
$209$	24.6033	1.70185
$210$	−15.1095	−1.04265
$211$	−0.509470	−0.0350734	−0.0175367	−	0.999846i	$-0.505582\pi$
$211$	−0.509470	−0.0350734	−0.0175367	+	0.999846i	$0.505582\pi$
$212$	−48.6353	−3.34029
$213$	2.57675	0.176556
$214$	−19.9020	−1.36047
$215$	−9.39183	−0.640518
$216$	31.1314	2.11822
$217$	32.2239	2.18750
$218$	9.78305	0.662592
$219$	7.87281	0.531995
$220$	−21.8361	−1.47219
$221$	1.40668	0.0946235
$222$	−2.59294	−0.174027
$223$	11.1310	0.745384	0.372692	−	0.927955i	$-0.378435\pi$
$223$	11.1310	0.745384	0.372692	+	0.927955i	$0.378435\pi$
$224$	13.1884	0.881186
$225$	−1.47705	−0.0984701
$226$	20.4796	1.36228
$227$	−1.50278	−0.0997427	−0.0498713	−	0.998756i	$-0.515881\pi$
$227$	−1.50278	−0.0997427	−0.0498713	+	0.998756i	$0.515881\pi$
$228$	25.1539	1.66586
$229$	−8.07817	−0.533820	−0.266910	−	0.963721i	$-0.586003\pi$
$229$	−8.07817	−0.533820	−0.266910	+	0.963721i	$0.586003\pi$
$230$	12.9781	0.855749
$231$	−31.0205	−2.04100
$232$	−50.0160	−3.28371
$233$	9.45410	0.619359	0.309679	−	0.950841i	$-0.399778\pi$
$233$	9.45410	0.619359	0.309679	+	0.950841i	$0.399778\pi$
$234$	1.08493	0.0709241
$235$	−8.06738	−0.526258
$236$	4.43945	0.288984
$237$	0.699593	0.0454434
$238$	58.6339	3.80067
$239$	−9.38903	−0.607326	−0.303663	−	0.952780i	$-0.598210\pi$
$239$	−9.38903	−0.607326	−0.303663	+	0.952780i	$0.598210\pi$
$240$	−6.89071	−0.444793
$241$	−1.14337	−0.0736510	−0.0368255	−	0.999322i	$-0.511725\pi$
$241$	−1.14337	−0.0736510	−0.0368255	+	0.999322i	$0.511725\pi$
$242$	−38.4039	−2.46870
$243$	13.6291	0.874310
$244$	28.8315	1.84575
$245$	−16.9722	−1.08431
$246$	7.75188	0.494242
$247$	1.40765	0.0895665
$248$	37.0841	2.35484
$249$	7.55962	0.479071
$250$	2.50065	0.158155
$251$	−16.7613	−1.05796	−0.528981	−	0.848634i	$-0.677426\pi$
$251$	−16.7613	−1.05796	−0.528981	+	0.848634i	$0.677426\pi$
$252$	30.7589	1.93763
$253$	26.6446	1.67513
$254$	5.07882	0.318674
$255$	−5.90996	−0.370096
$256$	−32.3202	−2.02001
$257$	17.2697	1.07725	0.538626	−	0.842545i	$-0.318944\pi$
$257$	17.2697	1.07725	0.538626	+	0.842545i	$0.318944\pi$
$258$	−28.9832	−1.80441
$259$	−4.11387	−0.255624
$260$	−1.24932	−0.0774798
$261$	−13.1111	−0.811560
$262$	−24.3039	−1.50150
$263$	27.3161	1.68438	0.842192	−	0.539177i	$-0.181265\pi$
$263$	27.3161	1.68438	0.842192	+	0.539177i	$0.181265\pi$
$264$	−35.6993	−2.19714
$265$	11.4348	0.702437
$266$	58.6743	3.59755
$267$	1.23408	0.0755244
$268$	−59.7799	−3.65164
$269$	−8.56066	−0.521953	−0.260976	−	0.965345i	$-0.584044\pi$
$269$	−8.56066	−0.521953	−0.260976	+	0.965345i	$0.584044\pi$
$270$	−13.8162	−0.840826
$271$	−6.04443	−0.367173	−0.183587	−	0.983004i	$-0.558771\pi$
$271$	−6.04443	−0.367173	−0.183587	+	0.983004i	$0.558771\pi$
$272$	26.7401	1.62136
$273$	−1.77480	−0.107416
$274$	−53.6441	−3.24076
$275$	5.13396	0.309590
$276$	27.2409	1.63971
$277$	1.06300	0.0638694	0.0319347	−	0.999490i	$-0.489833\pi$
$277$	1.06300	0.0638694	0.0319347	+	0.999490i	$0.489833\pi$
$278$	1.48908	0.0893088
$279$	9.72118	0.581992
$280$	−27.5879	−1.64869
$281$	−0.545199	−0.0325239	−0.0162619	−	0.999868i	$-0.505177\pi$
$281$	−0.545199	−0.0325239	−0.0162619	+	0.999868i	$0.505177\pi$
$282$	−24.8959	−1.48253
$283$	14.1051	0.838461	0.419230	−	0.907880i	$-0.362300\pi$
$283$	14.1051	0.838461	0.419230	+	0.907880i	$0.362300\pi$
$284$	8.88079	0.526978
$285$	−5.91403	−0.350317
$286$	−3.77102	−0.222985
$287$	12.2989	0.725980
$288$	3.97863	0.234443
$289$	5.93418	0.349069
$290$	22.1972	1.30346
$291$	−4.55020	−0.266737
$292$	27.1337	1.58788
$293$	−17.0930	−0.998586	−0.499293	−	0.866433i	$-0.666407\pi$
$293$	−17.0930	−0.998586	−0.499293	+	0.866433i	$0.666407\pi$
$294$	−52.3762	−3.05464
$295$	−1.04378	−0.0607711
$296$	−4.73436	−0.275179
$297$	−28.3653	−1.64592
$298$	41.1602	2.38435
$299$	1.52444	0.0881606
$300$	5.24885	0.303043
$301$	−45.9837	−2.65046
$302$	−2.53157	−0.145675
$303$	−8.38121	−0.481488
$304$	26.7585	1.53471
$305$	−6.77868	−0.388146
$306$	17.6885	1.01118
$307$	−0.280920	−0.0160329	−0.00801646	−	0.999968i	$-0.502552\pi$
$307$	−0.280920	−0.0160329	−0.00801646	+	0.999968i	$0.502552\pi$
$308$	−106.912	−6.09190
$309$	12.5407	0.713416
$310$	−16.4580	−0.934751
$311$	−12.9003	−0.731507	−0.365753	−	0.930712i	$-0.619189\pi$
$311$	−12.9003	−0.731507	−0.365753	+	0.930712i	$0.619189\pi$
$312$	−2.04249	−0.115633
$313$	−3.04911	−0.172346	−0.0861729	−	0.996280i	$-0.527464\pi$
$313$	−3.04911	−0.172346	−0.0861729	+	0.996280i	$0.527464\pi$
$314$	9.16241	0.517065
$315$	−7.23185	−0.407469
$316$	2.41115	0.135638
$317$	1.96859	0.110567	0.0552835	−	0.998471i	$-0.482394\pi$
$317$	1.96859	0.110567	0.0552835	+	0.998471i	$0.482394\pi$
$318$	35.2879	1.97885
$319$	45.5720	2.55154
$320$	4.43155	0.247731
$321$	9.82169	0.548193
$322$	63.5425	3.54108
$323$	22.9500	1.27697
$324$	−10.1532	−0.564069
$325$	0.293733	0.0162934
$326$	0.991107	0.0548923
$327$	−4.82796	−0.266987
$328$	14.1539	0.781517
$329$	−39.4990	−2.17765
$330$	15.8434	0.872151
$331$	19.6542	1.08029	0.540147	−	0.841571i	$-0.318369\pi$
$331$	19.6542	1.08029	0.540147	+	0.841571i	$0.318369\pi$
$332$	26.0543	1.42991
$333$	−1.24106	−0.0680096
$334$	22.2876	1.21952
$335$	14.0551	0.767912
$336$	−33.7379	−1.84055
$337$	−19.5743	−1.06628	−0.533140	−	0.846027i	$-0.678988\pi$
$337$	−19.5743	−1.06628	−0.533140	+	0.846027i	$0.678988\pi$
$338$	32.2927	1.75649
$339$	−10.1067	−0.548922
$340$	−20.3687	−1.10465
$341$	−33.7891	−1.82978
$342$	17.7007	0.957142
$343$	−48.8254	−2.63632
$344$	−52.9193	−2.85322
$345$	−6.40471	−0.344818
$346$	−51.0372	−2.74378
$347$	13.2711	0.712431	0.356215	−	0.934404i	$-0.384067\pi$
$347$	13.2711	0.712431	0.356215	+	0.934404i	$0.384067\pi$
$348$	46.5918	2.49758
$349$	25.7299	1.37729	0.688646	−	0.725097i	$-0.258205\pi$
$349$	25.7299	1.37729	0.688646	+	0.725097i	$0.258205\pi$
$350$	12.2435	0.654445
$351$	−1.62289	−0.0866233
$352$	−13.8290	−0.737088
$353$	33.4472	1.78022	0.890108	−	0.455750i	$-0.150629\pi$
$353$	33.4472	1.78022	0.890108	+	0.455750i	$0.150629\pi$
$354$	−3.22110	−0.171199
$355$	−2.08800	−0.110819
$356$	4.25326	0.225422
$357$	−28.9360	−1.53145
$358$	9.65137	0.510091
$359$	2.49760	0.131818	0.0659090	−	0.997826i	$-0.479005\pi$
$359$	2.49760	0.131818	0.0659090	+	0.997826i	$0.479005\pi$
$360$	−8.32261	−0.438640
$361$	3.96581	0.208727
$362$	12.1450	0.638327
$363$	18.9524	0.994744
$364$	−6.11687	−0.320611
$365$	−6.37951	−0.333919
$366$	−20.9190	−1.09345
$367$	−31.2573	−1.63162	−0.815810	−	0.578320i	$-0.803708\pi$
$367$	−31.2573	−1.63162	−0.815810	+	0.578320i	$0.803708\pi$
$368$	28.9786	1.51062
$369$	3.71028	0.193150
$370$	2.10112	0.109232
$371$	55.9866	2.90668
$372$	−34.5452	−1.79109
$373$	−35.2179	−1.82351	−0.911757	−	0.410730i	$-0.865274\pi$
$373$	−35.2179	−1.82351	−0.911757	+	0.410730i	$0.865274\pi$
$374$	−61.4819	−3.17915
$375$	−1.23408	−0.0637275
$376$	−45.4566	−2.34424
$377$	2.60735	0.134285
$378$	−67.6460	−3.47933
$379$	−3.50658	−0.180121	−0.0900605	−	0.995936i	$-0.528706\pi$
$379$	−3.50658	−0.180121	−0.0900605	+	0.995936i	$0.528706\pi$
$380$	−20.3827	−1.04561
$381$	−2.50641	−0.128407
$382$	−18.6218	−0.952776
$383$	21.9576	1.12198	0.560991	−	0.827822i	$-0.310420\pi$
$383$	21.9576	1.12198	0.560991	+	0.827822i	$0.310420\pi$
$384$	20.3240	1.03716
$385$	25.1366	1.28108
$386$	19.2198	0.978263
$387$	−13.8722	−0.705164
$388$	−15.6823	−0.796147
$389$	−13.3115	−0.674918	−0.337459	−	0.941340i	$-0.609567\pi$
$389$	−13.3115	−0.674918	−0.337459	+	0.941340i	$0.609567\pi$
$390$	0.906461	0.0459004
$391$	24.8541	1.25693
$392$	−95.6318	−4.83014
$393$	11.9940	0.605019
$394$	67.4682	3.39900
$395$	−0.566895	−0.0285236
$396$	−32.2530	−1.62077
$397$	17.8358	0.895155	0.447578	−	0.894245i	$-0.352287\pi$
$397$	17.8358	0.895155	0.447578	+	0.894245i	$0.352287\pi$
$398$	−13.1095	−0.657118
$399$	−28.9559	−1.44961
$400$	5.58369	0.279184
$401$	−3.19387	−0.159494	−0.0797471	−	0.996815i	$-0.525411\pi$
$401$	−3.19387	−0.159494	−0.0797471	+	0.996815i	$0.525411\pi$
$402$	43.3740	2.16330
$403$	−1.93320	−0.0962996
$404$	−28.8859	−1.43713
$405$	2.38717	0.118619
$406$	108.681	5.39373
$407$	4.31370	0.213822
$408$	−33.3003	−1.64861
$409$	−24.5244	−1.21266	−0.606328	−	0.795215i	$-0.707358\pi$
$409$	−24.5244	−1.21266	−0.606328	+	0.795215i	$0.707358\pi$
$410$	−6.28152	−0.310222
$411$	26.4735	1.30584
$412$	43.2216	2.12938
$413$	−5.11048	−0.251470
$414$	19.1693	0.942118
$415$	−6.12572	−0.300700
$416$	−0.791209	−0.0387922
$417$	−0.734863	−0.0359864
$418$	−61.5243	−3.00925
$419$	4.89589	0.239180	0.119590	−	0.992823i	$-0.461842\pi$
$419$	4.89589	0.239180	0.119590	+	0.992823i	$0.461842\pi$
$420$	25.6991	1.25399
$421$	4.48965	0.218812	0.109406	−	0.993997i	$-0.465105\pi$
$421$	4.48965	0.218812	0.109406	+	0.993997i	$0.465105\pi$
$422$	1.27401	0.0620177
$423$	−11.9159	−0.579372
$424$	64.4309	3.12904
$425$	4.78896	0.232299
$426$	−6.44356	−0.312192
$427$	−33.1894	−1.60615
$428$	33.8505	1.63623
$429$	1.86101	0.0898504
$430$	23.4857	1.13258
$431$	35.8073	1.72477	0.862387	−	0.506249i	$-0.168968\pi$
$431$	35.8073	1.72477	0.862387	+	0.506249i	$0.168968\pi$
$432$	−30.8500	−1.48427
$433$	−18.7569	−0.901399	−0.450700	−	0.892676i	$-0.648825\pi$
$433$	−18.7569	−0.901399	−0.450700	+	0.892676i	$0.648825\pi$
$434$	−80.5807	−3.86800
$435$	−10.9544	−0.525222
$436$	−16.6396	−0.796892
$437$	24.8713	1.18975
$438$	−19.6871	−0.940688
$439$	31.4002	1.49865	0.749325	−	0.662202i	$-0.230378\pi$
$439$	31.4002	1.49865	0.749325	+	0.662202i	$0.230378\pi$
$440$	28.9279	1.37908
$441$	−25.0688	−1.19375
$442$	−3.51761	−0.167316
$443$	−25.6079	−1.21667	−0.608334	−	0.793681i	$-0.708162\pi$
$443$	−25.6079	−1.21667	−0.608334	+	0.793681i	$0.708162\pi$
$444$	4.41023	0.209300
$445$	−1.00000	−0.0474045
$446$	−27.8347	−1.31801
$447$	−20.3127	−0.960757
$448$	21.6975	1.02511
$449$	−21.3542	−1.00777	−0.503883	−	0.863772i	$-0.668096\pi$
$449$	−21.3542	−1.00777	−0.503883	+	0.863772i	$0.668096\pi$
$450$	3.69359	0.174117
$451$	−12.8963	−0.607261
$452$	−34.8329	−1.63840
$453$	1.24933	0.0586988
$454$	3.75792	0.176368
$455$	1.43816	0.0674220
$456$	−33.3233	−1.56051
$457$	−27.2671	−1.27550	−0.637750	−	0.770243i	$-0.720135\pi$
$457$	−27.2671	−1.27550	−0.637750	+	0.770243i	$0.720135\pi$
$458$	20.2007	0.943916
$459$	−26.4592	−1.23501
$460$	−22.0739	−1.02920
$461$	−18.9331	−0.881801	−0.440900	−	0.897556i	$-0.645341\pi$
$461$	−18.9331	−0.881801	−0.440900	+	0.897556i	$0.645341\pi$
$462$	77.5715	3.60895
$463$	30.1659	1.40193	0.700965	−	0.713196i	$-0.252753\pi$
$463$	30.1659	1.40193	0.700965	+	0.713196i	$0.252753\pi$
$464$	49.5640	2.30095
$465$	8.12206	0.376652
$466$	−23.6414	−1.09517
$467$	34.1721	1.58130	0.790649	−	0.612270i	$-0.209743\pi$
$467$	34.1721	1.58130	0.790649	+	0.612270i	$0.209743\pi$
$468$	−1.84532	−0.0852997
$469$	68.8157	3.17762
$470$	20.1737	0.930544
$471$	−4.52168	−0.208348
$472$	−5.88128	−0.270708
$473$	48.2173	2.21703
$474$	−1.74944	−0.0803544
$475$	4.79227	0.219884
$476$	−99.7281	−4.57103
$477$	16.8898	0.773333
$478$	23.4787	1.07389
$479$	19.0555	0.870666	0.435333	−	0.900269i	$-0.356631\pi$
$479$	19.0555	0.870666	0.435333	+	0.900269i	$0.356631\pi$
$480$	3.32415	0.151726
$481$	0.246803	0.0112532
$482$	2.85917	0.130232
$483$	−31.3584	−1.42686
$484$	65.3196	2.96907
$485$	3.68712	0.167424
$486$	−34.0817	−1.54598
$487$	25.7093	1.16500	0.582501	−	0.812830i	$-0.302074\pi$
$487$	25.7093	1.16500	0.582501	+	0.812830i	$0.302074\pi$
$488$	−38.1952	−1.72902
$489$	−0.489114	−0.0221185
$490$	42.4416	1.91732
$491$	21.1649	0.955158	0.477579	−	0.878589i	$-0.341514\pi$
$491$	21.1649	0.955158	0.477579	+	0.878589i	$0.341514\pi$
$492$	−13.1849	−0.594420
$493$	42.5096	1.91454
$494$	−3.52004	−0.158374
$495$	7.58312	0.340836
$496$	−36.7489	−1.65008
$497$	−10.2231	−0.458570
$498$	−18.9040	−0.847107
$499$	25.0315	1.12056	0.560281	−	0.828302i	$-0.310693\pi$
$499$	25.0315	1.12056	0.560281	+	0.828302i	$0.310693\pi$
$500$	−4.25326	−0.190211
$501$	−10.9990	−0.491399
$502$	41.9141	1.87072
$503$	5.82665	0.259797	0.129899	−	0.991527i	$-0.458535\pi$
$503$	5.82665	0.259797	0.129899	+	0.991527i	$0.458535\pi$
$504$	−40.7487	−1.81509
$505$	6.79147	0.302217
$506$	−66.6289	−2.96202
$507$	−15.9365	−0.707767
$508$	−8.63836	−0.383265
$509$	−13.7961	−0.611503	−0.305752	−	0.952111i	$-0.598908\pi$
$509$	−13.7961	−0.611503	−0.305752	+	0.952111i	$0.598908\pi$
$510$	14.7787	0.654414
$511$	−31.2350	−1.38175
$512$	47.8834	2.11617
$513$	−26.4774	−1.16901
$514$	−43.1854	−1.90483
$515$	−10.1620	−0.447792
$516$	49.2963	2.17015
$517$	41.4176	1.82154
$518$	10.2874	0.452001
$519$	25.1870	1.10559
$520$	1.65507	0.0725798
$521$	34.4027	1.50721	0.753604	−	0.657329i	$-0.228314\pi$
$521$	34.4027	1.50721	0.753604	+	0.657329i	$0.228314\pi$
$522$	32.7864	1.43502
$523$	−25.0955	−1.09735	−0.548674	−	0.836036i	$-0.684867\pi$
$523$	−25.0955	−1.09735	−0.548674	+	0.836036i	$0.684867\pi$
$524$	41.3375	1.80584
$525$	−6.04222	−0.263704
$526$	−68.3081	−2.97838
$527$	−31.5185	−1.37297
$528$	35.3766	1.53957
$529$	3.93481	0.171079
$530$	−28.5945	−1.24207
$531$	−1.54171	−0.0669046
$532$	−99.7968	−4.32674
$533$	−0.737844	−0.0319596
$534$	−3.08600	−0.133544
$535$	−7.95873	−0.344086
$536$	79.1950	3.42070
$537$	−4.76298	−0.205538
$538$	21.4072	0.922931
$539$	87.1347	3.75316
$540$	23.4994	1.01125
$541$	16.2525	0.698750	0.349375	−	0.936983i	$-0.386394\pi$
$541$	16.2525	0.698750	0.349375	+	0.936983i	$0.386394\pi$
$542$	15.1150	0.649246
$543$	−5.99359	−0.257210
$544$	−12.8997	−0.553070
$545$	3.91220	0.167580
$546$	4.43816	0.189936
$547$	17.3751	0.742905	0.371452	−	0.928452i	$-0.378860\pi$
$547$	17.3751	0.742905	0.371452	+	0.928452i	$0.378860\pi$
$548$	91.2411	3.89762
$549$	−10.0125	−0.427321
$550$	−12.8382	−0.547425
$551$	42.5389	1.81222
$552$	−36.0881	−1.53601
$553$	−2.77560	−0.118030
$554$	−2.65819	−0.112936
$555$	−1.03691	−0.0440142
$556$	−2.53271	−0.107411
$557$	−43.3855	−1.83830	−0.919152	−	0.393904i	$-0.871124\pi$
$557$	−43.3855	−1.83830	−0.919152	+	0.393904i	$0.871124\pi$
$558$	−24.3093	−1.02909
$559$	2.75869	0.116680
$560$	27.3385	1.15526
$561$	30.3415	1.28102
$562$	1.36335	0.0575096
$563$	14.5950	0.615108	0.307554	−	0.951531i	$-0.400490\pi$
$563$	14.5950	0.615108	0.307554	+	0.951531i	$0.400490\pi$
$564$	42.3445	1.78302
$565$	8.18970	0.344543
$566$	−35.2719	−1.48259
$567$	11.6879	0.490846
$568$	−11.7651	−0.493651
$569$	−25.4644	−1.06752	−0.533762	−	0.845635i	$-0.679222\pi$
$569$	−25.4644	−1.06752	−0.533762	+	0.845635i	$0.679222\pi$
$570$	14.7889	0.619440
$571$	3.34046	0.139794	0.0698969	−	0.997554i	$-0.477733\pi$
$571$	3.34046	0.139794	0.0698969	+	0.997554i	$0.477733\pi$
$572$	6.41398	0.268182
$573$	9.18993	0.383915
$574$	−30.7552	−1.28370
$575$	5.18988	0.216433
$576$	6.54562	0.272734
$577$	−28.9214	−1.20401	−0.602007	−	0.798491i	$-0.705632\pi$
$577$	−28.9214	−1.20401	−0.602007	+	0.798491i	$0.705632\pi$
$578$	−14.8393	−0.617234
$579$	−9.48503	−0.394185
$580$	−37.7543	−1.56766
$581$	−29.9924	−1.24429
$582$	11.3785	0.471652
$583$	−58.7060	−2.43135
$584$	−35.9460	−1.48746
$585$	0.433859	0.0179379
$586$	42.7437	1.76573
$587$	16.5407	0.682706	0.341353	−	0.939935i	$-0.389115\pi$
$587$	16.5407	0.682706	0.341353	+	0.939935i	$0.389115\pi$
$588$	89.0846	3.67379
$589$	−31.5402	−1.29959
$590$	2.61012	0.107457
$591$	−33.2957	−1.36960
$592$	4.69157	0.192822
$593$	−27.0667	−1.11150	−0.555748	−	0.831351i	$-0.687568\pi$
$593$	−27.0667	−1.11150	−0.555748	+	0.831351i	$0.687568\pi$
$594$	70.9317	2.91036
$595$	23.4475	0.961252
$596$	−70.0078	−2.86763
$597$	6.46955	0.264781
$598$	−3.81209	−0.155888
$599$	7.11352	0.290651	0.145325	−	0.989384i	$-0.453577\pi$
$599$	7.11352	0.290651	0.145325	+	0.989384i	$0.453577\pi$
$600$	−6.95355	−0.283878
$601$	−42.8950	−1.74972	−0.874861	−	0.484374i	$-0.839047\pi$
$601$	−42.8950	−1.74972	−0.874861	+	0.484374i	$0.839047\pi$
$602$	114.989	4.68661
$603$	20.7601	0.845416
$604$	4.30584	0.175202
$605$	−15.3576	−0.624373
$606$	20.9585	0.851380
$607$	−27.9087	−1.13278	−0.566390	−	0.824137i	$-0.691660\pi$
$607$	−27.9087	−1.13278	−0.566390	+	0.824137i	$0.691660\pi$
$608$	−12.9086	−0.523512
$609$	−53.6342	−2.17337
$610$	16.9511	0.686331
$611$	2.36966	0.0958662
$612$	−30.0856	−1.21614
$613$	11.2291	0.453538	0.226769	−	0.973949i	$-0.427184\pi$
$613$	11.2291	0.453538	0.226769	+	0.973949i	$0.427184\pi$
$614$	0.702482	0.0283499
$615$	3.09995	0.125002
$616$	141.635	5.70664
$617$	−29.7394	−1.19726	−0.598631	−	0.801025i	$-0.704288\pi$
$617$	−29.7394	−1.19726	−0.598631	+	0.801025i	$0.704288\pi$
$618$	−31.3600	−1.26148
$619$	−47.0184	−1.88983	−0.944915	−	0.327315i	$-0.893856\pi$
$619$	−47.0184	−1.88983	−0.944915	+	0.327315i	$0.893856\pi$
$620$	27.9927	1.12422
$621$	−28.6742	−1.15066
$622$	32.2591	1.29347
$623$	−4.89614	−0.196160
$624$	2.02403	0.0810261
$625$	1.00000	0.0400000
$626$	7.62476	0.304747
$627$	30.3624	1.21256
$628$	−15.5840	−0.621869
$629$	4.02382	0.160440
$630$	18.0843	0.720497
$631$	−4.61993	−0.183917	−0.0919583	−	0.995763i	$-0.529313\pi$
$631$	−4.61993	−0.183917	−0.0919583	+	0.995763i	$0.529313\pi$
$632$	−3.19423	−0.127060
$633$	−0.628726	−0.0249896
$634$	−4.92276	−0.195508
$635$	2.03100	0.0805978
$636$	−60.0198	−2.37994
$637$	4.98531	0.197525
$638$	−113.960	−4.51170
$639$	−3.08408	−0.122004
$640$	−16.4690	−0.650995
$641$	−0.763355	−0.0301507	−0.0150754	−	0.999886i	$-0.504799\pi$
$641$	−0.763355	−0.0301507	−0.0150754	+	0.999886i	$0.504799\pi$
$642$	−24.5606	−0.969331
$643$	−24.5483	−0.968091	−0.484046	−	0.875043i	$-0.660833\pi$
$643$	−24.5483	−0.968091	−0.484046	+	0.875043i	$0.660833\pi$
$644$	−108.077	−4.25882
$645$	−11.5903	−0.456366
$646$	−57.3899	−2.25798
$647$	−20.9739	−0.824567	−0.412284	−	0.911056i	$-0.635269\pi$
$647$	−20.9739	−0.824567	−0.412284	+	0.911056i	$0.635269\pi$
$648$	13.4508	0.528396
$649$	5.35871	0.210348
$650$	−0.734525	−0.0288104
$651$	39.7668	1.55858
$652$	−1.68573	−0.0660185
$653$	−0.521815	−0.0204202	−0.0102101	−	0.999948i	$-0.503250\pi$
$653$	−0.521815	−0.0204202	−0.0102101	+	0.999948i	$0.503250\pi$
$654$	12.0730	0.472093
$655$	−9.71902	−0.379754
$656$	−14.0260	−0.547621
$657$	−9.42285	−0.367621
$658$	98.7733	3.85059
$659$	1.63679	0.0637603	0.0318802	−	0.999492i	$-0.489851\pi$
$659$	1.63679	0.0637603	0.0318802	+	0.999492i	$0.489851\pi$
$660$	−26.9474	−1.04893
$661$	25.0711	0.975151	0.487576	−	0.873081i	$-0.337881\pi$
$661$	25.0711	0.975151	0.487576	+	0.873081i	$0.337881\pi$
$662$	−49.1484	−1.91021
$663$	1.73595	0.0674188
$664$	−34.5161	−1.33948
$665$	23.4636	0.909880
$666$	3.10346	0.120256
$667$	46.0683	1.78377
$668$	−37.9081	−1.46671
$669$	13.7365	0.531083
$670$	−35.1469	−1.35784
$671$	34.8015	1.34350
$672$	16.2755	0.627841
$673$	−33.2875	−1.28314	−0.641569	−	0.767065i	$-0.721716\pi$
$673$	−33.2875	−1.28314	−0.641569	+	0.767065i	$0.721716\pi$
$674$	48.9485	1.88542
$675$	−5.52503	−0.212659
$676$	−54.9254	−2.11251
$677$	17.5066	0.672834	0.336417	−	0.941713i	$-0.390785\pi$
$677$	17.5066	0.672834	0.336417	+	0.941713i	$0.390785\pi$
$678$	25.2734	0.970620
$679$	18.0527	0.692798
$680$	26.9840	1.03479
$681$	−1.85454	−0.0710662
$682$	84.4947	3.23547
$683$	−46.2972	−1.77151	−0.885757	−	0.464150i	$-0.846360\pi$
$683$	−46.2972	−1.77151	−0.885757	+	0.464150i	$0.846360\pi$
$684$	−30.1064	−1.15115
$685$	−21.4520	−0.819640
$686$	122.095	4.66162
$687$	−9.96909	−0.380345
$688$	52.4410	1.99930
$689$	−3.35879	−0.127960
$690$	16.0160	0.609717
$691$	4.46210	0.169746	0.0848732	−	0.996392i	$-0.472951\pi$
$691$	4.46210	0.169746	0.0848732	+	0.996392i	$0.472951\pi$
$692$	86.8071	3.29991
$693$	37.1280	1.41038
$694$	−33.1864	−1.25974
$695$	0.595475	0.0225877
$696$	−61.7237	−2.33963
$697$	−12.0296	−0.455655
$698$	−64.3416	−2.43537
$699$	11.6671	0.441290
$700$	−20.8246	−0.787094
$701$	−5.13544	−0.193963	−0.0969815	−	0.995286i	$-0.530919\pi$
$701$	−5.13544	−0.193963	−0.0969815	+	0.995286i	$0.530919\pi$
$702$	4.05827	0.153170
$703$	4.02659	0.151866
$704$	−22.7514	−0.857475
$705$	−9.95578	−0.374956
$706$	−83.6398	−3.14783
$707$	33.2520	1.25057
$708$	5.47863	0.205900
$709$	4.95502	0.186090	0.0930449	−	0.995662i	$-0.470340\pi$
$709$	4.95502	0.186090	0.0930449	+	0.995662i	$0.470340\pi$
$710$	5.22136	0.195954
$711$	−0.837333	−0.0314024
$712$	−5.63461	−0.211166
$713$	−34.1571	−1.27919
$714$	72.3588	2.70796
$715$	−1.50802	−0.0563966
$716$	−16.4156	−0.613481
$717$	−11.5868	−0.432717
$718$	−6.24562	−0.233084
$719$	−32.2241	−1.20176	−0.600878	−	0.799341i	$-0.705182\pi$
$719$	−32.2241	−1.20176	−0.600878	+	0.799341i	$0.705182\pi$
$720$	8.24739	0.307362
$721$	−49.7546	−1.85296
$722$	−9.91712	−0.369077
$723$	−1.41101	−0.0524760
$724$	−20.6570	−0.767710
$725$	8.87657	0.329668
$726$	−47.3934	−1.75893
$727$	−31.3167	−1.16147	−0.580735	−	0.814093i	$-0.697235\pi$
$727$	−31.3167	−1.16147	−0.580735	+	0.814093i	$0.697235\pi$
$728$	8.10348	0.300335
$729$	23.9809	0.888183
$730$	15.9529	0.590444
$731$	44.9771	1.66354
$732$	35.5803	1.31509
$733$	−1.84406	−0.0681118	−0.0340559	−	0.999420i	$-0.510842\pi$
$733$	−1.84406	−0.0681118	−0.0340559	+	0.999420i	$0.510842\pi$
$734$	78.1637	2.88508
$735$	−20.9450	−0.772569
$736$	−13.9796	−0.515295
$737$	−72.1583	−2.65799
$738$	−9.27812	−0.341532
$739$	23.1432	0.851338	0.425669	−	0.904879i	$-0.360039\pi$
$739$	23.1432	0.851338	0.425669	+	0.904879i	$0.360039\pi$
$740$	−3.57371	−0.131372
$741$	1.73715	0.0638158
$742$	−140.003	−5.13967
$743$	−2.13174	−0.0782059	−0.0391029	−	0.999235i	$-0.512450\pi$
$743$	−2.13174	−0.0782059	−0.0391029	+	0.999235i	$0.512450\pi$
$744$	45.7647	1.67781
$745$	16.4598	0.603041
$746$	88.0677	3.22439
$747$	−9.04800	−0.331049
$748$	104.572	3.82354
$749$	−38.9671	−1.42383
$750$	3.08600	0.112685
$751$	9.83764	0.358981	0.179490	−	0.983760i	$-0.442555\pi$
$751$	9.83764	0.358981	0.179490	+	0.983760i	$0.442555\pi$
$752$	45.0457	1.64265
$753$	−20.6847	−0.753793
$754$	−6.52006	−0.237447
$755$	−1.01236	−0.0368436
$756$	115.056	4.18456
$757$	−38.2109	−1.38880	−0.694400	−	0.719590i	$-0.744330\pi$
$757$	−38.2109	−1.38880	−0.694400	+	0.719590i	$0.744330\pi$
$758$	8.76874	0.318495
$759$	32.8815	1.19352
$760$	27.0026	0.979486
$761$	−5.67543	−0.205734	−0.102867	−	0.994695i	$-0.532802\pi$
$761$	−5.67543	−0.205734	−0.102867	+	0.994695i	$0.532802\pi$
$762$	6.26766	0.227053
$763$	19.1547	0.693446
$764$	31.6731	1.14589
$765$	7.07354	0.255745
$766$	−54.9083	−1.98392
$767$	0.306592	0.0110704
$768$	−39.8856	−1.43925
$769$	19.1784	0.691590	0.345795	−	0.938310i	$-0.387609\pi$
$769$	19.1784	0.691590	0.345795	+	0.938310i	$0.387609\pi$
$770$	−62.8579	−2.26524
$771$	21.3121	0.767537
$772$	−32.6902	−1.17655
$773$	26.8742	0.966598	0.483299	−	0.875455i	$-0.339438\pi$
$773$	26.8742	0.966598	0.483299	+	0.875455i	$0.339438\pi$
$774$	34.6896	1.24689
$775$	−6.58148	−0.236414
$776$	20.7755	0.745797
$777$	−5.07684	−0.182131
$778$	33.2873	1.19341
$779$	−12.0379	−0.431304
$780$	−1.54176	−0.0552040
$781$	10.7197	0.383581
$782$	−62.1515	−2.22253
$783$	−49.0433	−1.75267
$784$	94.7675	3.38455
$785$	3.66401	0.130774
$786$	−29.9929	−1.06981
$787$	11.4331	0.407546	0.203773	−	0.979018i	$-0.434680\pi$
$787$	11.4331	0.407546	0.203773	+	0.979018i	$0.434680\pi$
$788$	−114.754	−4.08794
$789$	33.7102	1.20012
$790$	1.41761	0.0504362
$791$	40.0980	1.42572
$792$	42.7280	1.51827
$793$	1.99113	0.0707069
$794$	−44.6012	−1.58284
$795$	14.1115	0.500483
$796$	22.2974	0.790309
$797$	−38.2874	−1.35621	−0.678105	−	0.734965i	$-0.737199\pi$
$797$	−38.2874	−1.35621	−0.678105	+	0.734965i	$0.737199\pi$
$798$	72.4087	2.56324
$799$	38.6344	1.36679
$800$	−2.69363	−0.0952342
$801$	−1.47705	−0.0521890
$802$	7.98676	0.282022
$803$	32.7521	1.15580
$804$	−73.7731	−2.60178
$805$	25.4104	0.895598
$806$	4.83426	0.170280
$807$	−10.5645	−0.371889
$808$	38.2673	1.34624
$809$	−4.98203	−0.175159	−0.0875794	−	0.996158i	$-0.527913\pi$
$809$	−4.98203	−0.175159	−0.0875794	+	0.996158i	$0.527913\pi$
$810$	−5.96947	−0.209746
$811$	2.56566	0.0900924	0.0450462	−	0.998985i	$-0.485656\pi$
$811$	2.56566	0.0900924	0.0450462	+	0.998985i	$0.485656\pi$
$812$	−184.851	−6.48699
$813$	−7.45930	−0.261609
$814$	−10.7871	−0.378086
$815$	0.396340	0.0138832
$816$	32.9993	1.15521
$817$	45.0082	1.57464
$818$	61.3271	2.14425
$819$	2.12424	0.0742268
$820$	10.6840	0.373101
$821$	7.39548	0.258104	0.129052	−	0.991638i	$-0.458807\pi$
$821$	7.39548	0.258104	0.129052	+	0.991638i	$0.458807\pi$
$822$	−66.2010	−2.30902
$823$	14.2404	0.496389	0.248195	−	0.968710i	$-0.420163\pi$
$823$	14.2404	0.496389	0.248195	+	0.968710i	$0.420163\pi$
$824$	−57.2590	−1.99471
$825$	6.33571	0.220581
$826$	12.7795	0.444657
$827$	−22.1225	−0.769273	−0.384637	−	0.923068i	$-0.625673\pi$
$827$	−22.1225	−0.769273	−0.384637	+	0.923068i	$0.625673\pi$
$828$	−32.6042	−1.13308
$829$	43.2604	1.50250	0.751248	−	0.660021i	$-0.229452\pi$
$829$	43.2604	1.50250	0.751248	+	0.660021i	$0.229452\pi$
$830$	15.3183	0.531706
$831$	1.31182	0.0455067
$832$	−1.30169	−0.0451281
$833$	81.2793	2.81616
$834$	1.83764	0.0636321
$835$	8.91272	0.308437
$836$	104.644	3.61920
$837$	36.3629	1.25689
$838$	−12.2429	−0.422925
$839$	−6.71522	−0.231835	−0.115917	−	0.993259i	$-0.536981\pi$
$839$	−6.71522	−0.231835	−0.115917	+	0.993259i	$0.536981\pi$
$840$	−34.0456	−1.17468
$841$	49.7935	1.71702
$842$	−11.2271	−0.386910
$843$	−0.672818	−0.0231731
$844$	−2.16691	−0.0745880
$845$	12.9137	0.444245
$846$	29.7976	1.02446
$847$	−75.1928	−2.58365
$848$	−63.8486	−2.19257
$849$	17.4068	0.597399
$850$	−11.9755	−0.410757
$851$	4.36068	0.149482
$852$	10.9596	0.375470
$853$	−3.93209	−0.134632	−0.0673160	−	0.997732i	$-0.521444\pi$
$853$	−3.93209	−0.134632	−0.0673160	+	0.997732i	$0.521444\pi$
$854$	82.9951	2.84003
$855$	7.07842	0.242077
$856$	−44.8443	−1.53275
$857$	−4.62748	−0.158072	−0.0790358	−	0.996872i	$-0.525184\pi$
$857$	−4.62748	−0.158072	−0.0790358	+	0.996872i	$0.525184\pi$
$858$	−4.65374	−0.158876
$859$	−19.7777	−0.674805	−0.337403	−	0.941360i	$-0.609548\pi$
$859$	−19.7777	−0.674805	−0.337403	+	0.941360i	$0.609548\pi$
$860$	−39.9459	−1.36214
$861$	15.1778	0.517257
$862$	−89.5415	−3.04979
$863$	8.03916	0.273656	0.136828	−	0.990595i	$-0.456309\pi$
$863$	8.03916	0.273656	0.136828	+	0.990595i	$0.456309\pi$
$864$	14.8824	0.506309
$865$	−20.4096	−0.693946
$866$	46.9045	1.59388
$867$	7.32324	0.248710
$868$	137.056	4.65200
$869$	2.91042	0.0987292
$870$	27.3931	0.928713
$871$	−4.12845	−0.139887
$872$	22.0437	0.746495
$873$	5.44607	0.184321
$874$	−62.1944	−2.10376
$875$	4.89614	0.165520
$876$	33.4851	1.13136
$877$	55.5811	1.87684	0.938420	−	0.345496i	$-0.112289\pi$
$877$	55.5811	1.87684	0.938420	+	0.345496i	$0.112289\pi$
$878$	−78.5210	−2.64996
$879$	−21.0941	−0.711488
$880$	−28.6664	−0.966345
$881$	44.8890	1.51235	0.756174	−	0.654370i	$-0.227066\pi$
$881$	44.8890	1.51235	0.756174	+	0.654370i	$0.227066\pi$
$882$	62.6884	2.11083
$883$	−13.5457	−0.455848	−0.227924	−	0.973679i	$-0.573194\pi$
$883$	−13.5457	−0.455848	−0.227924	+	0.973679i	$0.573194\pi$
$884$	5.98297	0.201229
$885$	−1.28810	−0.0432991
$886$	64.0364	2.15135
$887$	36.1333	1.21324	0.606619	−	0.794992i	$-0.292525\pi$
$887$	36.1333	1.21324	0.606619	+	0.794992i	$0.292525\pi$
$888$	−5.84257	−0.196064
$889$	9.94406	0.333513
$890$	2.50065	0.0838220
$891$	−12.2556	−0.410579
$892$	47.3429	1.58516
$893$	38.6610	1.29374
$894$	50.7950	1.69884
$895$	3.85954	0.129010
$896$	−80.6346	−2.69381
$897$	1.88128	0.0628141
$898$	53.3994	1.78196
$899$	−58.4210	−1.94845
$900$	−6.28228	−0.209409
$901$	−54.7610	−1.82435
$902$	32.2491	1.07378
$903$	−56.7475	−1.88844
$904$	46.1458	1.53479
$905$	4.85674	0.161443
$906$	−3.12415	−0.103793
$907$	−41.6649	−1.38346	−0.691730	−	0.722156i	$-0.743151\pi$
$907$	−41.6649	−1.38346	−0.691730	+	0.722156i	$0.743151\pi$
$908$	−6.39169	−0.212116
$909$	10.0314	0.332719
$910$	−3.59634	−0.119217
$911$	−55.3548	−1.83399	−0.916993	−	0.398903i	$-0.869391\pi$
$911$	−55.3548	−1.83399	−0.916993	+	0.398903i	$0.869391\pi$
$912$	33.0221	1.09347
$913$	31.4492	1.04082
$914$	68.1855	2.25537
$915$	−8.36542	−0.276552
$916$	−34.3585	−1.13524
$917$	−47.5857	−1.57142
$918$	66.1652	2.18378
$919$	9.44714	0.311633	0.155816	−	0.987786i	$-0.450199\pi$
$919$	9.44714	0.311633	0.155816	+	0.987786i	$0.450199\pi$
$920$	29.2429	0.964111
$921$	−0.346677	−0.0114234
$922$	47.3450	1.55922
$923$	0.613315	0.0201875
$924$	−131.938	−4.34045
$925$	0.840228	0.0276265
$926$	−75.4345	−2.47893
$927$	−15.0098	−0.492987
$928$	−23.9102	−0.784890
$929$	−23.0801	−0.757234	−0.378617	−	0.925553i	$-0.623600\pi$
$929$	−23.0801	−0.757234	−0.378617	+	0.925553i	$0.623600\pi$
$930$	−20.3104	−0.666006
$931$	81.3353	2.66566
$932$	40.2107	1.31715
$933$	−15.9199	−0.521195
$934$	−85.4526	−2.79609
$935$	−24.5864	−0.804060
$936$	2.44463	0.0799052
$937$	31.4855	1.02859	0.514293	−	0.857615i	$-0.328054\pi$
$937$	31.4855	1.02859	0.514293	+	0.857615i	$0.328054\pi$
$938$	−172.084	−5.61875
$939$	−3.76284	−0.122796
$940$	−34.3126	−1.11916
$941$	16.7640	0.546492	0.273246	−	0.961944i	$-0.411903\pi$
$941$	16.7640	0.546492	0.273246	+	0.961944i	$0.411903\pi$
$942$	11.3071	0.368406
$943$	−13.0367	−0.424534
$944$	5.82812	0.189689
$945$	−27.0513	−0.879980
$946$	−120.575	−3.92022
$947$	−59.7924	−1.94299	−0.971497	−	0.237053i	$-0.923818\pi$
$947$	−59.7924	−1.94299	−0.971497	+	0.237053i	$0.923818\pi$
$948$	2.97555	0.0966413
$949$	1.87387	0.0608285
$950$	−11.9838	−0.388805
$951$	2.42939	0.0787785
$952$	132.117	4.28195
$953$	17.8565	0.578428	0.289214	−	0.957264i	$-0.406606\pi$
$953$	17.8565	0.578428	0.289214	+	0.957264i	$0.406606\pi$
$954$	−42.2356	−1.36743
$955$	−7.44680	−0.240973
$956$	−39.9340	−1.29156
$957$	56.2394	1.81796
$958$	−47.6511	−1.53954
$959$	−105.032	−3.39167
$960$	5.46887	0.176507
$961$	12.3159	0.397287
$962$	−0.617168	−0.0198983
$963$	−11.7554	−0.378814
$964$	−4.86305	−0.156628
$965$	7.68593	0.247419
$966$	78.4164	2.52301
$967$	−31.9918	−1.02879	−0.514393	−	0.857554i	$-0.671983\pi$
$967$	−31.9918	−1.02879	−0.514393	+	0.857554i	$0.671983\pi$
$968$	−86.5339	−2.78130
$969$	28.3221	0.909836
$970$	−9.22021	−0.296043
$971$	13.8839	0.445555	0.222778	−	0.974869i	$-0.428488\pi$
$971$	13.8839	0.445555	0.222778	+	0.974869i	$0.428488\pi$
$972$	57.9683	1.85933
$973$	2.91553	0.0934676
$974$	−64.2901	−2.05999
$975$	0.362490	0.0116090
$976$	37.8500	1.21155
$977$	5.04704	0.161469	0.0807346	−	0.996736i	$-0.474273\pi$
$977$	5.04704	0.161469	0.0807346	+	0.996736i	$0.474273\pi$
$978$	1.22310	0.0391105
$979$	5.13396	0.164082
$980$	−72.1872	−2.30593
$981$	5.77852	0.184494
$982$	−52.9260	−1.68894
$983$	−26.6803	−0.850971	−0.425485	−	0.904965i	$-0.639897\pi$
$983$	−26.6803	−0.850971	−0.425485	+	0.904965i	$0.639897\pi$
$984$	17.4670	0.556827
$985$	26.9802	0.859662
$986$	−106.302	−3.38533
$987$	−48.7449	−1.55157
$988$	5.98709	0.190475
$989$	48.7424	1.54992
$990$	−18.9627	−0.602676
$991$	−37.2339	−1.18277	−0.591386	−	0.806388i	$-0.701419\pi$
$991$	−37.2339	−1.18277	−0.591386	+	0.806388i	$0.701419\pi$
$992$	17.7281	0.562867
$993$	24.2549	0.769705
$994$	25.5645	0.810857
$995$	−5.24242	−0.166196
$996$	32.1530	1.01881
$997$	−15.6534	−0.495748	−0.247874	−	0.968792i	$-0.579732\pi$
$997$	−15.6534	−0.495748	−0.247874	+	0.968792i	$0.579732\pi$
$998$	−62.5950	−1.98141
$999$	−4.64228	−0.146875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.g.1.1	✓	8
3.2	odd	2		4005.2.a.p.1.8		8
4.3	odd	2		7120.2.a.bk.1.4		8
5.4	even	2		2225.2.a.l.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.1	✓	8	1.1	even	1	trivial
2225.2.a.l.1.8		8	5.4	even	2
4005.2.a.p.1.8		8	3.2	odd	2
7120.2.a.bk.1.4		8	4.3	odd	2

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:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50065 q^{2} +1.23408 q^{3} +4.25326 q^{4} -1.00000 q^{5} -3.08600 q^{6} -4.89614 q^{7} -5.63461 q^{8} -1.47705 q^{9} +O(q^{10})\)	\(q-2.50065 q^{2} +1.23408 q^{3} +4.25326 q^{4} -1.00000 q^{5} -3.08600 q^{6} -4.89614 q^{7} -5.63461 q^{8} -1.47705 q^{9} +2.50065 q^{10} +5.13396 q^{11} +5.24885 q^{12} +0.293733 q^{13} +12.2435 q^{14} -1.23408 q^{15} +5.58369 q^{16} +4.78896 q^{17} +3.69359 q^{18} +4.79227 q^{19} -4.25326 q^{20} -6.04222 q^{21} -12.8382 q^{22} +5.18988 q^{23} -6.95355 q^{24} +1.00000 q^{25} -0.734525 q^{26} -5.52503 q^{27} -20.8246 q^{28} +8.87657 q^{29} +3.08600 q^{30} -6.58148 q^{31} -2.69363 q^{32} +6.33571 q^{33} -11.9755 q^{34} +4.89614 q^{35} -6.28228 q^{36} +0.840228 q^{37} -11.9838 q^{38} +0.362490 q^{39} +5.63461 q^{40} -2.51195 q^{41} +15.1095 q^{42} +9.39183 q^{43} +21.8361 q^{44} +1.47705 q^{45} -12.9781 q^{46} +8.06738 q^{47} +6.89071 q^{48} +16.9722 q^{49} -2.50065 q^{50} +5.90996 q^{51} +1.24932 q^{52} -11.4348 q^{53} +13.8162 q^{54} -5.13396 q^{55} +27.5879 q^{56} +5.91403 q^{57} -22.1972 q^{58} +1.04378 q^{59} -5.24885 q^{60} +6.77868 q^{61} +16.4580 q^{62} +7.23185 q^{63} -4.43155 q^{64} -0.293733 q^{65} -15.8434 q^{66} -14.0551 q^{67} +20.3687 q^{68} +6.40471 q^{69} -12.2435 q^{70} +2.08800 q^{71} +8.32261 q^{72} +6.37951 q^{73} -2.10112 q^{74} +1.23408 q^{75} +20.3827 q^{76} -25.1366 q^{77} -0.906461 q^{78} +0.566895 q^{79} -5.58369 q^{80} -2.38717 q^{81} +6.28152 q^{82} +6.12572 q^{83} -25.6991 q^{84} -4.78896 q^{85} -23.4857 q^{86} +10.9544 q^{87} -28.9279 q^{88} +1.00000 q^{89} -3.69359 q^{90} -1.43816 q^{91} +22.0739 q^{92} -8.12206 q^{93} -20.1737 q^{94} -4.79227 q^{95} -3.32415 q^{96} -3.68712 q^{97} -42.4416 q^{98} -7.58312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 6 q^{3} + 7 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{7} + 3 q^{8} + 12 q^{9}+O(q^{10}) \) 8 * q + q^2 + 6 * q^3 + 7 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^7 + 3 * q^8 + 12 * q^9	\( 8 q + q^{2} + 6 q^{3} + 7 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{7} + 3 q^{8} + 12 q^{9} - q^{10} + 14 q^{11} + 17 q^{12} - 7 q^{13} + 15 q^{14} - 6 q^{15} + 9 q^{16} + 17 q^{17} - q^{18} + 17 q^{19} - 7 q^{20} + 2 q^{22} - q^{23} + 8 q^{24} + 8 q^{25} + 3 q^{26} + 21 q^{27} - 29 q^{28} + 10 q^{29} - 6 q^{30} + q^{31} + 2 q^{32} + 10 q^{33} - 16 q^{34} + 6 q^{35} - 17 q^{36} - 11 q^{37} - 30 q^{38} - 5 q^{39} - 3 q^{40} + 15 q^{41} + 14 q^{42} - 5 q^{43} + 7 q^{44} - 12 q^{45} - 12 q^{46} + 12 q^{47} + 3 q^{48} + 4 q^{49} + q^{50} + 35 q^{51} - 14 q^{52} - q^{53} - 29 q^{54} - 14 q^{55} + 3 q^{56} + 15 q^{57} - 37 q^{58} + 26 q^{59} - 17 q^{60} + 13 q^{61} + 22 q^{62} - 16 q^{63} - 15 q^{64} + 7 q^{65} + 4 q^{66} - 25 q^{67} + 23 q^{68} - 5 q^{69} - 15 q^{70} + 28 q^{71} + 22 q^{72} - 17 q^{73} - 5 q^{74} + 6 q^{75} + 8 q^{76} - 56 q^{78} - 7 q^{79} - 9 q^{80} + 24 q^{81} + 5 q^{82} + 44 q^{83} - 57 q^{84} - 17 q^{85} - 13 q^{86} - 12 q^{87} - 66 q^{88} + 8 q^{89} + q^{90} + 27 q^{91} + 15 q^{92} - 38 q^{93} - 27 q^{94} - 17 q^{95} - 20 q^{96} + q^{97} - 34 q^{98} + q^{99}+O(q^{100}) \) 8 * q + q^2 + 6 * q^3 + 7 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^7 + 3 * q^8 + 12 * q^9 - q^10 + 14 * q^11 + 17 * q^12 - 7 * q^13 + 15 * q^14 - 6 * q^15 + 9 * q^16 + 17 * q^17 - q^18 + 17 * q^19 - 7 * q^20 + 2 * q^22 - q^23 + 8 * q^24 + 8 * q^25 + 3 * q^26 + 21 * q^27 - 29 * q^28 + 10 * q^29 - 6 * q^30 + q^31 + 2 * q^32 + 10 * q^33 - 16 * q^34 + 6 * q^35 - 17 * q^36 - 11 * q^37 - 30 * q^38 - 5 * q^39 - 3 * q^40 + 15 * q^41 + 14 * q^42 - 5 * q^43 + 7 * q^44 - 12 * q^45 - 12 * q^46 + 12 * q^47 + 3 * q^48 + 4 * q^49 + q^50 + 35 * q^51 - 14 * q^52 - q^53 - 29 * q^54 - 14 * q^55 + 3 * q^56 + 15 * q^57 - 37 * q^58 + 26 * q^59 - 17 * q^60 + 13 * q^61 + 22 * q^62 - 16 * q^63 - 15 * q^64 + 7 * q^65 + 4 * q^66 - 25 * q^67 + 23 * q^68 - 5 * q^69 - 15 * q^70 + 28 * q^71 + 22 * q^72 - 17 * q^73 - 5 * q^74 + 6 * q^75 + 8 * q^76 - 56 * q^78 - 7 * q^79 - 9 * q^80 + 24 * q^81 + 5 * q^82 + 44 * q^83 - 57 * q^84 - 17 * q^85 - 13 * q^86 - 12 * q^87 - 66 * q^88 + 8 * q^89 + q^90 + 27 * q^91 + 15 * q^92 - 38 * q^93 - 27 * q^94 - 17 * q^95 - 20 * q^96 + q^97 - 34 * q^98 + q^99

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