LMFDB - Embedded newform 6045.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} +1.00000 q^{5} -1.81361 q^{6} +4.81361 q^{7} +1.28917 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} +1.00000 q^{5} -1.81361 q^{6} +4.81361 q^{7} +1.28917 q^{8} +1.00000 q^{9} -1.81361 q^{10} +5.62721 q^{11} +1.28917 q^{12} +1.00000 q^{13} -8.72999 q^{14} +1.00000 q^{15} -4.91638 q^{16} +2.00000 q^{17} -1.81361 q^{18} +1.47556 q^{19} +1.28917 q^{20} +4.81361 q^{21} -10.2056 q^{22} -1.62721 q^{23} +1.28917 q^{24} +1.00000 q^{25} -1.81361 q^{26} +1.00000 q^{27} +6.20555 q^{28} -3.39194 q^{29} -1.81361 q^{30} -1.00000 q^{31} +6.33804 q^{32} +5.62721 q^{33} -3.62721 q^{34} +4.81361 q^{35} +1.28917 q^{36} -8.20555 q^{37} -2.67609 q^{38} +1.00000 q^{39} +1.28917 q^{40} +1.76473 q^{41} -8.72999 q^{42} +8.81361 q^{43} +7.25443 q^{44} +1.00000 q^{45} +2.95112 q^{46} +0.729988 q^{47} -4.91638 q^{48} +16.1708 q^{49} -1.81361 q^{50} +2.00000 q^{51} +1.28917 q^{52} +1.68111 q^{53} -1.81361 q^{54} +5.62721 q^{55} +6.20555 q^{56} +1.47556 q^{57} +6.15165 q^{58} -5.39194 q^{59} +1.28917 q^{60} +7.88666 q^{61} +1.81361 q^{62} +4.81361 q^{63} -1.66196 q^{64} +1.00000 q^{65} -10.2056 q^{66} -4.23527 q^{67} +2.57834 q^{68} -1.62721 q^{69} -8.72999 q^{70} +11.2544 q^{71} +1.28917 q^{72} -6.25945 q^{73} +14.8816 q^{74} +1.00000 q^{75} +1.90225 q^{76} +27.0872 q^{77} -1.81361 q^{78} -5.30833 q^{79} -4.91638 q^{80} +1.00000 q^{81} -3.20053 q^{82} +4.66196 q^{83} +6.20555 q^{84} +2.00000 q^{85} -15.9844 q^{86} -3.39194 q^{87} +7.25443 q^{88} -4.20555 q^{89} -1.81361 q^{90} +4.81361 q^{91} -2.09775 q^{92} -1.00000 q^{93} -1.32391 q^{94} +1.47556 q^{95} +6.33804 q^{96} -14.0680 q^{97} -29.3275 q^{98} +5.62721 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} + 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 + 8 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} + 8 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 4 q^{11} + 3 q^{12} + 3 q^{13} - 6 q^{14} + 3 q^{15} - q^{16} + 6 q^{17} + q^{18} + 10 q^{19} + 3 q^{20} + 8 q^{21} - 16 q^{22} + 8 q^{23} + 3 q^{24} + 3 q^{25} + q^{26} + 3 q^{27} + 4 q^{28} - 2 q^{29} + q^{30} - 3 q^{31} + 7 q^{32} + 4 q^{33} + 2 q^{34} + 8 q^{35} + 3 q^{36} - 10 q^{37} + 16 q^{38} + 3 q^{39} + 3 q^{40} + 10 q^{41} - 6 q^{42} + 20 q^{43} - 4 q^{44} + 3 q^{45} + 20 q^{46} - 18 q^{47} - q^{48} + 9 q^{49} + q^{50} + 6 q^{51} + 3 q^{52} - 4 q^{53} + q^{54} + 4 q^{55} + 4 q^{56} + 10 q^{57} - 8 q^{59} + 3 q^{60} - q^{62} + 8 q^{63} - 17 q^{64} + 3 q^{65} - 16 q^{66} - 8 q^{67} + 6 q^{68} + 8 q^{69} - 6 q^{70} + 8 q^{71} + 3 q^{72} - 8 q^{73} + 6 q^{74} + 3 q^{75} + 28 q^{76} + 28 q^{77} + q^{78} + 6 q^{79} - q^{80} + 3 q^{81} + 20 q^{82} + 26 q^{83} + 4 q^{84} + 6 q^{85} - 2 q^{86} - 2 q^{87} - 4 q^{88} + 2 q^{89} + q^{90} + 8 q^{91} + 16 q^{92} - 3 q^{93} - 28 q^{94} + 10 q^{95} + 7 q^{96} - 10 q^{97} - 45 q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 + 8 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 4 * q^11 + 3 * q^12 + 3 * q^13 - 6 * q^14 + 3 * q^15 - q^16 + 6 * q^17 + q^18 + 10 * q^19 + 3 * q^20 + 8 * q^21 - 16 * q^22 + 8 * q^23 + 3 * q^24 + 3 * q^25 + q^26 + 3 * q^27 + 4 * q^28 - 2 * q^29 + q^30 - 3 * q^31 + 7 * q^32 + 4 * q^33 + 2 * q^34 + 8 * q^35 + 3 * q^36 - 10 * q^37 + 16 * q^38 + 3 * q^39 + 3 * q^40 + 10 * q^41 - 6 * q^42 + 20 * q^43 - 4 * q^44 + 3 * q^45 + 20 * q^46 - 18 * q^47 - q^48 + 9 * q^49 + q^50 + 6 * q^51 + 3 * q^52 - 4 * q^53 + q^54 + 4 * q^55 + 4 * q^56 + 10 * q^57 - 8 * q^59 + 3 * q^60 - q^62 + 8 * q^63 - 17 * q^64 + 3 * q^65 - 16 * q^66 - 8 * q^67 + 6 * q^68 + 8 * q^69 - 6 * q^70 + 8 * q^71 + 3 * q^72 - 8 * q^73 + 6 * q^74 + 3 * q^75 + 28 * q^76 + 28 * q^77 + q^78 + 6 * q^79 - q^80 + 3 * q^81 + 20 * q^82 + 26 * q^83 + 4 * q^84 + 6 * q^85 - 2 * q^86 - 2 * q^87 - 4 * q^88 + 2 * q^89 + q^90 + 8 * q^91 + 16 * q^92 - 3 * q^93 - 28 * q^94 + 10 * q^95 + 7 * q^96 - 10 * q^97 - 45 * q^98 + 4 * 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 $

$2$	−1.81361	−1.28241	−0.641207	−	0.767368i	$-0.721566\pi$
$2$	−1.81361	−1.28241	−0.641207	+	0.767368i	$0.721566\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.28917	0.644584
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.81361	−0.740402
$7$	4.81361	1.81937	0.909686	−	0.415296i	$-0.136322\pi$
$7$	4.81361	1.81937	0.909686	+	0.415296i	$0.136322\pi$
$8$	1.28917	0.455790
$9$	1.00000	0.333333
$10$	−1.81361	−0.573513
$11$	5.62721	1.69667	0.848334	−	0.529461i	$-0.177606\pi$
$11$	5.62721	1.69667	0.848334	+	0.529461i	$0.177606\pi$
$12$	1.28917	0.372151
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−8.72999	−2.33319
$15$	1.00000	0.258199
$16$	−4.91638	−1.22910
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.81361	−0.427471
$19$	1.47556	0.338517	0.169259	−	0.985572i	$-0.445863\pi$
$19$	1.47556	0.338517	0.169259	+	0.985572i	$0.445863\pi$
$20$	1.28917	0.288267
$21$	4.81361	1.05042
$22$	−10.2056	−2.17583
$23$	−1.62721	−0.339297	−0.169649	−	0.985505i	$-0.554263\pi$
$23$	−1.62721	−0.339297	−0.169649	+	0.985505i	$0.554263\pi$
$24$	1.28917	0.263150
$25$	1.00000	0.200000
$26$	−1.81361	−0.355677
$27$	1.00000	0.192450
$28$	6.20555	1.17274
$29$	−3.39194	−0.629868	−0.314934	−	0.949114i	$-0.601982\pi$
$29$	−3.39194	−0.629868	−0.314934	+	0.949114i	$0.601982\pi$
$30$	−1.81361	−0.331118
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	6.33804	1.12042
$33$	5.62721	0.979572
$34$	−3.62721	−0.622062
$35$	4.81361	0.813648
$36$	1.28917	0.214861
$37$	−8.20555	−1.34898	−0.674492	−	0.738282i	$-0.735637\pi$
$37$	−8.20555	−1.34898	−0.674492	+	0.738282i	$0.735637\pi$
$38$	−2.67609	−0.434119
$39$	1.00000	0.160128
$40$	1.28917	0.203835
$41$	1.76473	0.275605	0.137802	−	0.990460i	$-0.455996\pi$
$41$	1.76473	0.275605	0.137802	+	0.990460i	$0.455996\pi$
$42$	−8.72999	−1.34707
$43$	8.81361	1.34406	0.672031	−	0.740523i	$-0.265422\pi$
$43$	8.81361	1.34406	0.672031	+	0.740523i	$0.265422\pi$
$44$	7.25443	1.09365
$45$	1.00000	0.149071
$46$	2.95112	0.435120
$47$	0.729988	0.106480	0.0532399	−	0.998582i	$-0.483045\pi$
$47$	0.729988	0.106480	0.0532399	+	0.998582i	$0.483045\pi$
$48$	−4.91638	−0.709619
$49$	16.1708	2.31012
$50$	−1.81361	−0.256483
$51$	2.00000	0.280056
$52$	1.28917	0.178776
$53$	1.68111	0.230919	0.115459	−	0.993312i	$-0.463166\pi$
$53$	1.68111	0.230919	0.115459	+	0.993312i	$0.463166\pi$
$54$	−1.81361	−0.246801
$55$	5.62721	0.758773
$56$	6.20555	0.829252
$57$	1.47556	0.195443
$58$	6.15165	0.807751
$59$	−5.39194	−0.701971	−0.350986	−	0.936381i	$-0.614153\pi$
$59$	−5.39194	−0.701971	−0.350986	+	0.936381i	$0.614153\pi$
$60$	1.28917	0.166431
$61$	7.88666	1.00978	0.504892	−	0.863183i	$-0.331532\pi$
$61$	7.88666	1.00978	0.504892	+	0.863183i	$0.331532\pi$
$62$	1.81361	0.230328
$63$	4.81361	0.606457
$64$	−1.66196	−0.207744
$65$	1.00000	0.124035
$66$	−10.2056	−1.25622
$67$	−4.23527	−0.517421	−0.258710	−	0.965955i	$-0.583297\pi$
$67$	−4.23527	−0.517421	−0.258710	+	0.965955i	$0.583297\pi$
$68$	2.57834	0.312669
$69$	−1.62721	−0.195893
$70$	−8.72999	−1.04343
$71$	11.2544	1.33565	0.667827	−	0.744316i	$-0.267224\pi$
$71$	11.2544	1.33565	0.667827	+	0.744316i	$0.267224\pi$
$72$	1.28917	0.151930
$73$	−6.25945	−0.732613	−0.366307	−	0.930494i	$-0.619378\pi$
$73$	−6.25945	−0.732613	−0.366307	+	0.930494i	$0.619378\pi$
$74$	14.8816	1.72996
$75$	1.00000	0.115470
$76$	1.90225	0.218203
$77$	27.0872	3.08687
$78$	−1.81361	−0.205350
$79$	−5.30833	−0.597233	−0.298617	−	0.954373i	$-0.596525\pi$
$79$	−5.30833	−0.597233	−0.298617	+	0.954373i	$0.596525\pi$
$80$	−4.91638	−0.549668
$81$	1.00000	0.111111
$82$	−3.20053	−0.353439
$83$	4.66196	0.511716	0.255858	−	0.966714i	$-0.417642\pi$
$83$	4.66196	0.511716	0.255858	+	0.966714i	$0.417642\pi$
$84$	6.20555	0.677081
$85$	2.00000	0.216930
$86$	−15.9844	−1.72364
$87$	−3.39194	−0.363655
$88$	7.25443	0.773324
$89$	−4.20555	−0.445787	−0.222894	−	0.974843i	$-0.571550\pi$
$89$	−4.20555	−0.445787	−0.222894	+	0.974843i	$0.571550\pi$
$90$	−1.81361	−0.191171
$91$	4.81361	0.504603
$92$	−2.09775	−0.218706
$93$	−1.00000	−0.103695
$94$	−1.32391	−0.136551
$95$	1.47556	0.151389
$96$	6.33804	0.646874
$97$	−14.0680	−1.42839	−0.714196	−	0.699946i	$-0.753208\pi$
$97$	−14.0680	−1.42839	−0.714196	+	0.699946i	$0.753208\pi$
$98$	−29.3275	−2.96252
$99$	5.62721	0.565556
$100$	1.28917	0.128917
$101$	−1.10278	−0.109730	−0.0548651	−	0.998494i	$-0.517473\pi$
$101$	−1.10278	−0.109730	−0.0548651	+	0.998494i	$0.517473\pi$
$102$	−3.62721	−0.359148
$103$	−5.47556	−0.539523	−0.269762	−	0.962927i	$-0.586945\pi$
$103$	−5.47556	−0.539523	−0.269762	+	0.962927i	$0.586945\pi$
$104$	1.28917	0.126413
$105$	4.81361	0.469760
$106$	−3.04888	−0.296133
$107$	6.28917	0.607997	0.303998	−	0.952673i	$-0.401678\pi$
$107$	6.28917	0.607997	0.303998	+	0.952673i	$0.401678\pi$
$108$	1.28917	0.124050
$109$	−0.935538	−0.0896083	−0.0448042	−	0.998996i	$-0.514266\pi$
$109$	−0.935538	−0.0896083	−0.0448042	+	0.998996i	$0.514266\pi$
$110$	−10.2056	−0.973061
$111$	−8.20555	−0.778836
$112$	−23.6655	−2.23618
$113$	12.7597	1.20033	0.600166	−	0.799875i	$-0.295101\pi$
$113$	12.7597	1.20033	0.600166	+	0.799875i	$0.295101\pi$
$114$	−2.67609	−0.250639
$115$	−1.62721	−0.151738
$116$	−4.37279	−0.406003
$117$	1.00000	0.0924500
$118$	9.77886	0.900217
$119$	9.62721	0.882525
$120$	1.28917	0.117684
$121$	20.6655	1.87868
$122$	−14.3033	−1.29496
$123$	1.76473	0.159120
$124$	−1.28917	−0.115771
$125$	1.00000	0.0894427
$126$	−8.72999	−0.777729
$127$	3.18639	0.282747	0.141373	−	0.989956i	$-0.454848\pi$
$127$	3.18639	0.282747	0.141373	+	0.989956i	$0.454848\pi$
$128$	−9.66196	−0.854004
$129$	8.81361	0.775995
$130$	−1.81361	−0.159064
$131$	6.52444	0.570043	0.285021	−	0.958521i	$-0.407999\pi$
$131$	6.52444	0.570043	0.285021	+	0.958521i	$0.407999\pi$
$132$	7.25443	0.631417
$133$	7.10278	0.615889
$134$	7.68111	0.663547
$135$	1.00000	0.0860663
$136$	2.57834	0.221091
$137$	0.372787	0.0318493	0.0159247	−	0.999873i	$-0.494931\pi$
$137$	0.372787	0.0318493	0.0159247	+	0.999873i	$0.494931\pi$
$138$	2.95112	0.251216
$139$	−1.47556	−0.125156	−0.0625778	−	0.998040i	$-0.519932\pi$
$139$	−1.47556	−0.125156	−0.0625778	+	0.998040i	$0.519932\pi$
$140$	6.20555	0.524465
$141$	0.729988	0.0614761
$142$	−20.4111	−1.71286
$143$	5.62721	0.470571
$144$	−4.91638	−0.409698
$145$	−3.39194	−0.281686
$146$	11.3522	0.939513
$147$	16.1708	1.33375
$148$	−10.5783	−0.869534
$149$	12.4408	1.01919	0.509596	−	0.860414i	$-0.329795\pi$
$149$	12.4408	1.01919	0.509596	+	0.860414i	$0.329795\pi$
$150$	−1.81361	−0.148080
$151$	−9.13249	−0.743192	−0.371596	−	0.928395i	$-0.621189\pi$
$151$	−9.13249	−0.743192	−0.371596	+	0.928395i	$0.621189\pi$
$152$	1.90225	0.154293
$153$	2.00000	0.161690
$154$	−49.1255	−3.95865
$155$	−1.00000	−0.0803219
$156$	1.28917	0.103216
$157$	−11.1567	−0.890400	−0.445200	−	0.895431i	$-0.646867\pi$
$157$	−11.1567	−0.890400	−0.445200	+	0.895431i	$0.646867\pi$
$158$	9.62721	0.765900
$159$	1.68111	0.133321
$160$	6.33804	0.501066
$161$	−7.83276	−0.617308
$162$	−1.81361	−0.142490
$163$	−19.3622	−1.51657	−0.758283	−	0.651925i	$-0.773962\pi$
$163$	−19.3622	−1.51657	−0.758283	+	0.651925i	$0.773962\pi$
$164$	2.27504	0.177650
$165$	5.62721	0.438078
$166$	−8.45495	−0.656232
$167$	1.62721	0.125918	0.0629588	−	0.998016i	$-0.479946\pi$
$167$	1.62721	0.125918	0.0629588	+	0.998016i	$0.479946\pi$
$168$	6.20555	0.478769
$169$	1.00000	0.0769231
$170$	−3.62721	−0.278195
$171$	1.47556	0.112839
$172$	11.3622	0.866361
$173$	−13.2786	−1.00955	−0.504777	−	0.863250i	$-0.668425\pi$
$173$	−13.2786	−1.00955	−0.504777	+	0.863250i	$0.668425\pi$
$174$	6.15165	0.466355
$175$	4.81361	0.363874
$176$	−27.6655	−2.08537
$177$	−5.39194	−0.405283
$178$	7.62721	0.571684
$179$	−18.9114	−1.41350	−0.706751	−	0.707463i	$-0.749840\pi$
$179$	−18.9114	−1.41350	−0.706751	+	0.707463i	$0.749840\pi$
$180$	1.28917	0.0960890
$181$	−22.8816	−1.70078	−0.850389	−	0.526154i	$-0.823634\pi$
$181$	−22.8816	−1.70078	−0.850389	+	0.526154i	$0.823634\pi$
$182$	−8.72999	−0.647110
$183$	7.88666	0.582999
$184$	−2.09775	−0.154648
$185$	−8.20555	−0.603284
$186$	1.81361	0.132980
$187$	11.2544	0.823005
$188$	0.941078	0.0686351
$189$	4.81361	0.350138
$190$	−2.67609	−0.194144
$191$	−25.2927	−1.83012	−0.915059	−	0.403320i	$-0.867856\pi$
$191$	−25.2927	−1.83012	−0.915059	+	0.403320i	$0.867856\pi$
$192$	−1.66196	−0.119941
$193$	−10.4791	−0.754304	−0.377152	−	0.926151i	$-0.623097\pi$
$193$	−10.4791	−0.754304	−0.377152	+	0.926151i	$0.623097\pi$
$194$	25.5139	1.83179
$195$	1.00000	0.0716115
$196$	20.8469	1.48906
$197$	12.2297	0.871332	0.435666	−	0.900108i	$-0.356513\pi$
$197$	12.2297	0.871332	0.435666	+	0.900108i	$0.356513\pi$
$198$	−10.2056	−0.725277
$199$	−6.20555	−0.439900	−0.219950	−	0.975511i	$-0.570589\pi$
$199$	−6.20555	−0.439900	−0.219950	+	0.975511i	$0.570589\pi$
$200$	1.28917	0.0911580
$201$	−4.23527	−0.298733
$202$	2.00000	0.140720
$203$	−16.3275	−1.14596
$204$	2.57834	0.180520
$205$	1.76473	0.123254
$206$	9.93051	0.691892
$207$	−1.62721	−0.113099
$208$	−4.91638	−0.340890
$209$	8.30330	0.574351
$210$	−8.72999	−0.602426
$211$	−13.6514	−0.939801	−0.469900	−	0.882719i	$-0.655710\pi$
$211$	−13.6514	−0.939801	−0.469900	+	0.882719i	$0.655710\pi$
$212$	2.16724	0.148846
$213$	11.2544	0.771141
$214$	−11.4061	−0.779703
$215$	8.81361	0.601083
$216$	1.28917	0.0877168
$217$	−4.81361	−0.326769
$218$	1.69670	0.114915
$219$	−6.25945	−0.422974
$220$	7.25443	0.489093
$221$	2.00000	0.134535
$222$	14.8816	0.998790
$223$	−23.5733	−1.57859	−0.789293	−	0.614017i	$-0.789553\pi$
$223$	−23.5733	−1.57859	−0.789293	+	0.614017i	$0.789553\pi$
$224$	30.5089	2.03846
$225$	1.00000	0.0666667
$226$	−23.1411	−1.53932
$227$	18.6761	1.23958	0.619788	−	0.784769i	$-0.287219\pi$
$227$	18.6761	1.23958	0.619788	+	0.784769i	$0.287219\pi$
$228$	1.90225	0.125979
$229$	13.8328	0.914095	0.457047	−	0.889442i	$-0.348907\pi$
$229$	13.8328	0.914095	0.457047	+	0.889442i	$0.348907\pi$
$230$	2.95112	0.194591
$231$	27.0872	1.78221
$232$	−4.37279	−0.287088
$233$	10.3275	0.676576	0.338288	−	0.941043i	$-0.390152\pi$
$233$	10.3275	0.676576	0.338288	+	0.941043i	$0.390152\pi$
$234$	−1.81361	−0.118559
$235$	0.729988	0.0476192
$236$	−6.95112	−0.452480
$237$	−5.30833	−0.344813
$238$	−17.4600	−1.13176
$239$	11.6811	0.755588	0.377794	−	0.925890i	$-0.376683\pi$
$239$	11.6811	0.755588	0.377794	+	0.925890i	$0.376683\pi$
$240$	−4.91638	−0.317351
$241$	−13.0036	−0.837634	−0.418817	−	0.908071i	$-0.637555\pi$
$241$	−13.0036	−0.837634	−0.418817	+	0.908071i	$0.637555\pi$
$242$	−37.4791	−2.40925
$243$	1.00000	0.0641500
$244$	10.1672	0.650891
$245$	16.1708	1.03311
$246$	−3.20053	−0.204058
$247$	1.47556	0.0938878
$248$	−1.28917	−0.0818623
$249$	4.66196	0.295439
$250$	−1.81361	−0.114703
$251$	24.9497	1.57481	0.787405	−	0.616436i	$-0.211424\pi$
$251$	24.9497	1.57481	0.787405	+	0.616436i	$0.211424\pi$
$252$	6.20555	0.390913
$253$	−9.15667	−0.575675
$254$	−5.77886	−0.362598
$255$	2.00000	0.125245
$256$	20.8469	1.30293
$257$	20.4494	1.27560	0.637800	−	0.770202i	$-0.279845\pi$
$257$	20.4494	1.27560	0.637800	+	0.770202i	$0.279845\pi$
$258$	−15.9844	−0.995146
$259$	−39.4983	−2.45430
$260$	1.28917	0.0799508
$261$	−3.39194	−0.209956
$262$	−11.8328	−0.731031
$263$	−6.93554	−0.427664	−0.213832	−	0.976870i	$-0.568594\pi$
$263$	−6.93554	−0.427664	−0.213832	+	0.976870i	$0.568594\pi$
$264$	7.25443	0.446479
$265$	1.68111	0.103270
$266$	−12.8816	−0.789824
$267$	−4.20555	−0.257375
$268$	−5.45998	−0.333521
$269$	−11.1567	−0.680234	−0.340117	−	0.940383i	$-0.610467\pi$
$269$	−11.1567	−0.680234	−0.340117	+	0.940383i	$0.610467\pi$
$270$	−1.81361	−0.110373
$271$	17.2686	1.04899	0.524495	−	0.851413i	$-0.324254\pi$
$271$	17.2686	1.04899	0.524495	+	0.851413i	$0.324254\pi$
$272$	−9.83276	−0.596199
$273$	4.81361	0.291333
$274$	−0.676089	−0.0408440
$275$	5.62721	0.339334
$276$	−2.09775	−0.126270
$277$	−0.304754	−0.0183109	−0.00915546	−	0.999958i	$-0.502914\pi$
$277$	−0.304754	−0.0183109	−0.00915546	+	0.999958i	$0.502914\pi$
$278$	2.67609	0.160501
$279$	−1.00000	−0.0598684
$280$	6.20555	0.370853
$281$	13.6569	0.814704	0.407352	−	0.913271i	$-0.366452\pi$
$281$	13.6569	0.814704	0.407352	+	0.913271i	$0.366452\pi$
$282$	−1.32391	−0.0788378
$283$	15.9844	0.950175	0.475087	−	0.879939i	$-0.342416\pi$
$283$	15.9844	0.950175	0.475087	+	0.879939i	$0.342416\pi$
$284$	14.5089	0.860942
$285$	1.47556	0.0874048
$286$	−10.2056	−0.603467
$287$	8.49472	0.501427
$288$	6.33804	0.373473
$289$	−13.0000	−0.764706
$290$	6.15165	0.361237
$291$	−14.0680	−0.824683
$292$	−8.06949	−0.472231
$293$	5.40608	0.315826	0.157913	−	0.987453i	$-0.449523\pi$
$293$	5.40608	0.315826	0.157913	+	0.987453i	$0.449523\pi$
$294$	−29.3275	−1.71041
$295$	−5.39194	−0.313931
$296$	−10.5783	−0.614853
$297$	5.62721	0.326524
$298$	−22.5628	−1.30703
$299$	−1.62721	−0.0941042
$300$	1.28917	0.0744302
$301$	42.4252	2.44535
$302$	16.5628	0.953079
$303$	−1.10278	−0.0633528
$304$	−7.25443	−0.416070
$305$	7.88666	0.451589
$306$	−3.62721	−0.207354
$307$	−20.3517	−1.16153	−0.580765	−	0.814071i	$-0.697247\pi$
$307$	−20.3517	−1.16153	−0.580765	+	0.814071i	$0.697247\pi$
$308$	34.9200	1.98975
$309$	−5.47556	−0.311494
$310$	1.81361	0.103006
$311$	−11.2544	−0.638180	−0.319090	−	0.947724i	$-0.603377\pi$
$311$	−11.2544	−0.638180	−0.319090	+	0.947724i	$0.603377\pi$
$312$	1.28917	0.0729848
$313$	30.6464	1.73224	0.866118	−	0.499840i	$-0.166608\pi$
$313$	30.6464	1.73224	0.866118	+	0.499840i	$0.166608\pi$
$314$	20.2338	1.14186
$315$	4.81361	0.271216
$316$	−6.84333	−0.384967
$317$	−26.8972	−1.51070	−0.755349	−	0.655322i	$-0.772533\pi$
$317$	−26.8972	−1.51070	−0.755349	+	0.655322i	$0.772533\pi$
$318$	−3.04888	−0.170972
$319$	−19.0872	−1.06868
$320$	−1.66196	−0.0929061
$321$	6.28917	0.351027
$322$	14.2056	0.791644
$323$	2.95112	0.164205
$324$	1.28917	0.0716205
$325$	1.00000	0.0554700
$326$	35.1155	1.94487
$327$	−0.935538	−0.0517354
$328$	2.27504	0.125618
$329$	3.51388	0.193726
$330$	−10.2056	−0.561797
$331$	14.5925	0.802075	0.401037	−	0.916062i	$-0.368650\pi$
$331$	14.5925	0.802075	0.401037	+	0.916062i	$0.368650\pi$
$332$	6.01005	0.329844
$333$	−8.20555	−0.449661
$334$	−2.95112	−0.161478
$335$	−4.23527	−0.231397
$336$	−23.6655	−1.29106
$337$	19.4600	1.06005	0.530026	−	0.847981i	$-0.322182\pi$
$337$	19.4600	1.06005	0.530026	+	0.847981i	$0.322182\pi$
$338$	−1.81361	−0.0986472
$339$	12.7597	0.693012
$340$	2.57834	0.139830
$341$	−5.62721	−0.304731
$342$	−2.67609	−0.144706
$343$	44.1447	2.38359
$344$	11.3622	0.612610
$345$	−1.62721	−0.0876062
$346$	24.0822	1.29467
$347$	−33.1411	−1.77911	−0.889553	−	0.456831i	$-0.848984\pi$
$347$	−33.1411	−1.77911	−0.889553	+	0.456831i	$0.848984\pi$
$348$	−4.37279	−0.234406
$349$	−12.6167	−0.675354	−0.337677	−	0.941262i	$-0.609641\pi$
$349$	−12.6167	−0.675354	−0.337677	+	0.941262i	$0.609641\pi$
$350$	−8.72999	−0.466637
$351$	1.00000	0.0533761
$352$	35.6655	1.90098
$353$	2.66196	0.141682	0.0708408	−	0.997488i	$-0.477432\pi$
$353$	2.66196	0.141682	0.0708408	+	0.997488i	$0.477432\pi$
$354$	9.77886	0.519741
$355$	11.2544	0.597323
$356$	−5.42166	−0.287348
$357$	9.62721	0.509526
$358$	34.2978	1.81269
$359$	−10.3814	−0.547908	−0.273954	−	0.961743i	$-0.588332\pi$
$359$	−10.3814	−0.547908	−0.273954	+	0.961743i	$0.588332\pi$
$360$	1.28917	0.0679451
$361$	−16.8227	−0.885406
$362$	41.4983	2.18110
$363$	20.6655	1.08466
$364$	6.20555	0.325259
$365$	−6.25945	−0.327635
$366$	−14.3033	−0.747646
$367$	−15.1950	−0.793172	−0.396586	−	0.917998i	$-0.629805\pi$
$367$	−15.1950	−0.793172	−0.396586	+	0.917998i	$0.629805\pi$
$368$	8.00000	0.417029
$369$	1.76473	0.0918682
$370$	14.8816	0.773660
$371$	8.09221	0.420127
$372$	−1.28917	−0.0668403
$373$	4.20555	0.217755	0.108878	−	0.994055i	$-0.465274\pi$
$373$	4.20555	0.217755	0.108878	+	0.994055i	$0.465274\pi$
$374$	−20.4111	−1.05543
$375$	1.00000	0.0516398
$376$	0.941078	0.0485324
$377$	−3.39194	−0.174694
$378$	−8.72999	−0.449022
$379$	−26.7683	−1.37500	−0.687498	−	0.726187i	$-0.741291\pi$
$379$	−26.7683	−1.37500	−0.687498	+	0.726187i	$0.741291\pi$
$380$	1.90225	0.0975833
$381$	3.18639	0.163244
$382$	45.8711	2.34697
$383$	32.4635	1.65881	0.829405	−	0.558648i	$-0.188680\pi$
$383$	32.4635	1.65881	0.829405	+	0.558648i	$0.188680\pi$
$384$	−9.66196	−0.493060
$385$	27.0872	1.38049
$386$	19.0050	0.967330
$387$	8.81361	0.448021
$388$	−18.1361	−0.920719
$389$	−21.5381	−1.09202	−0.546011	−	0.837778i	$-0.683854\pi$
$389$	−21.5381	−1.09202	−0.546011	+	0.837778i	$0.683854\pi$
$390$	−1.81361	−0.0918355
$391$	−3.25443	−0.164583
$392$	20.8469	1.05293
$393$	6.52444	0.329114
$394$	−22.1799	−1.11741
$395$	−5.30833	−0.267091
$396$	7.25443	0.364549
$397$	−4.31335	−0.216481	−0.108240	−	0.994125i	$-0.534522\pi$
$397$	−4.31335	−0.216481	−0.108240	+	0.994125i	$0.534522\pi$
$398$	11.2544	0.564133
$399$	7.10278	0.355584
$400$	−4.91638	−0.245819
$401$	−2.31889	−0.115800	−0.0578999	−	0.998322i	$-0.518440\pi$
$401$	−2.31889	−0.115800	−0.0578999	+	0.998322i	$0.518440\pi$
$402$	7.68111	0.383099
$403$	−1.00000	−0.0498135
$404$	−1.42166	−0.0707304
$405$	1.00000	0.0496904
$406$	29.6116	1.46960
$407$	−46.1744	−2.28878
$408$	2.57834	0.127647
$409$	−0.759707	−0.0375651	−0.0187826	−	0.999824i	$-0.505979\pi$
$409$	−0.759707	−0.0375651	−0.0187826	+	0.999824i	$0.505979\pi$
$410$	−3.20053	−0.158063
$411$	0.372787	0.0183882
$412$	−7.05892	−0.347768
$413$	−25.9547	−1.27715
$414$	2.95112	0.145040
$415$	4.66196	0.228846
$416$	6.33804	0.310748
$417$	−1.47556	−0.0722586
$418$	−15.0589	−0.736556
$419$	−36.9583	−1.80553	−0.902765	−	0.430135i	$-0.858466\pi$
$419$	−36.9583	−1.80553	−0.902765	+	0.430135i	$0.858466\pi$
$420$	6.20555	0.302800
$421$	4.67609	0.227899	0.113949	−	0.993487i	$-0.463650\pi$
$421$	4.67609	0.227899	0.113949	+	0.993487i	$0.463650\pi$
$422$	24.7583	1.20521
$423$	0.729988	0.0354932
$424$	2.16724	0.105250
$425$	2.00000	0.0970143
$426$	−20.4111	−0.988921
$427$	37.9633	1.83717
$428$	8.10780	0.391905
$429$	5.62721	0.271684
$430$	−15.9844	−0.770837
$431$	−16.0680	−0.773970	−0.386985	−	0.922086i	$-0.626483\pi$
$431$	−16.0680	−0.773970	−0.386985	+	0.922086i	$0.626483\pi$
$432$	−4.91638	−0.236540
$433$	−7.22471	−0.347197	−0.173599	−	0.984816i	$-0.555540\pi$
$433$	−7.22471	−0.347197	−0.173599	+	0.984816i	$0.555540\pi$
$434$	8.72999	0.419053
$435$	−3.39194	−0.162631
$436$	−1.20607	−0.0577601
$437$	−2.40105	−0.114858
$438$	11.3522	0.542428
$439$	−31.6897	−1.51247	−0.756234	−	0.654302i	$-0.772963\pi$
$439$	−31.6897	−1.51247	−0.756234	+	0.654302i	$0.772963\pi$
$440$	7.25443	0.345841
$441$	16.1708	0.770038
$442$	−3.62721	−0.172529
$443$	3.97582	0.188897	0.0944485	−	0.995530i	$-0.469891\pi$
$443$	3.97582	0.188897	0.0944485	+	0.995530i	$0.469891\pi$
$444$	−10.5783	−0.502026
$445$	−4.20555	−0.199362
$446$	42.7527	2.02440
$447$	12.4408	0.588431
$448$	−8.00000	−0.377964
$449$	−11.9461	−0.563771	−0.281886	−	0.959448i	$-0.590960\pi$
$449$	−11.9461	−0.563771	−0.281886	+	0.959448i	$0.590960\pi$
$450$	−1.81361	−0.0854942
$451$	9.93051	0.467610
$452$	16.4494	0.773715
$453$	−9.13249	−0.429082
$454$	−33.8711	−1.58965
$455$	4.81361	0.225665
$456$	1.90225	0.0890809
$457$	21.3466	0.998554	0.499277	−	0.866443i	$-0.333599\pi$
$457$	21.3466	0.998554	0.499277	+	0.866443i	$0.333599\pi$
$458$	−25.0872	−1.17225
$459$	2.00000	0.0933520
$460$	−2.09775	−0.0978082
$461$	28.6761	1.33558	0.667789	−	0.744350i	$-0.267241\pi$
$461$	28.6761	1.33558	0.667789	+	0.744350i	$0.267241\pi$
$462$	−49.1255	−2.28553
$463$	−26.6167	−1.23698	−0.618490	−	0.785792i	$-0.712255\pi$
$463$	−26.6167	−1.23698	−0.618490	+	0.785792i	$0.712255\pi$
$464$	16.6761	0.774168
$465$	−1.00000	−0.0463739
$466$	−18.7300	−0.867650
$467$	16.2680	0.752795	0.376398	−	0.926458i	$-0.377163\pi$
$467$	16.2680	0.752795	0.376398	+	0.926458i	$0.377163\pi$
$468$	1.28917	0.0595918
$469$	−20.3869	−0.941381
$470$	−1.32391	−0.0610675
$471$	−11.1567	−0.514072
$472$	−6.95112	−0.319951
$473$	49.5960	2.28043
$474$	9.62721	0.442193
$475$	1.47556	0.0677034
$476$	12.4111	0.568862
$477$	1.68111	0.0769728
$478$	−21.1849	−0.968977
$479$	10.0297	0.458270	0.229135	−	0.973395i	$-0.426410\pi$
$479$	10.0297	0.458270	0.229135	+	0.973395i	$0.426410\pi$
$480$	6.33804	0.289291
$481$	−8.20555	−0.374141
$482$	23.5834	1.07419
$483$	−7.83276	−0.356403
$484$	26.6413	1.21097
$485$	−14.0680	−0.638796
$486$	−1.81361	−0.0822669
$487$	−16.8972	−0.765686	−0.382843	−	0.923813i	$-0.625055\pi$
$487$	−16.8972	−0.765686	−0.382843	+	0.923813i	$0.625055\pi$
$488$	10.1672	0.460249
$489$	−19.3622	−0.875590
$490$	−29.3275	−1.32488
$491$	−1.66698	−0.0752297	−0.0376148	−	0.999292i	$-0.511976\pi$
$491$	−1.66698	−0.0752297	−0.0376148	+	0.999292i	$0.511976\pi$
$492$	2.27504	0.102567
$493$	−6.78389	−0.305531
$494$	−2.67609	−0.120403
$495$	5.62721	0.252924
$496$	4.91638	0.220752
$497$	54.1744	2.43005
$498$	−8.45495	−0.378875
$499$	24.4353	1.09387	0.546937	−	0.837174i	$-0.315794\pi$
$499$	24.4353	1.09387	0.546937	+	0.837174i	$0.315794\pi$
$500$	1.28917	0.0576534
$501$	1.62721	0.0726985
$502$	−45.2489	−2.01956
$503$	9.65139	0.430334	0.215167	−	0.976577i	$-0.430970\pi$
$503$	9.65139	0.430334	0.215167	+	0.976577i	$0.430970\pi$
$504$	6.20555	0.276417
$505$	−1.10278	−0.0490728
$506$	16.6066	0.738254
$507$	1.00000	0.0444116
$508$	4.10780	0.182254
$509$	−32.5527	−1.44287	−0.721437	−	0.692480i	$-0.756518\pi$
$509$	−32.5527	−1.44287	−0.721437	+	0.692480i	$0.756518\pi$
$510$	−3.62721	−0.160616
$511$	−30.1305	−1.33290
$512$	−18.4842	−0.816892
$513$	1.47556	0.0651477
$514$	−37.0872	−1.63585
$515$	−5.47556	−0.241282
$516$	11.3622	0.500194
$517$	4.10780	0.180661
$518$	71.6344	3.14743
$519$	−13.2786	−0.582866
$520$	1.28917	0.0565338
$521$	−11.8272	−0.518160	−0.259080	−	0.965856i	$-0.583419\pi$
$521$	−11.8272	−0.518160	−0.259080	+	0.965856i	$0.583419\pi$
$522$	6.15165	0.269250
$523$	29.3919	1.28522	0.642610	−	0.766193i	$-0.277852\pi$
$523$	29.3919	1.28522	0.642610	+	0.766193i	$0.277852\pi$
$524$	8.41110	0.367441
$525$	4.81361	0.210083
$526$	12.5783	0.548442
$527$	−2.00000	−0.0871214
$528$	−27.6655	−1.20399
$529$	−20.3522	−0.884877
$530$	−3.04888	−0.132435
$531$	−5.39194	−0.233990
$532$	9.15667	0.396992
$533$	1.76473	0.0764390
$534$	7.62721	0.330062
$535$	6.28917	0.271904
$536$	−5.45998	−0.235835
$537$	−18.9114	−0.816085
$538$	20.2338	0.872342
$539$	90.9966	3.91950
$540$	1.28917	0.0554770
$541$	32.9200	1.41534	0.707670	−	0.706543i	$-0.249747\pi$
$541$	32.9200	1.41534	0.707670	+	0.706543i	$0.249747\pi$
$542$	−31.3184	−1.34524
$543$	−22.8816	−0.981945
$544$	12.6761	0.543483
$545$	−0.935538	−0.0400741
$546$	−8.72999	−0.373609
$547$	−1.25997	−0.0538722	−0.0269361	−	0.999637i	$-0.508575\pi$
$547$	−1.25997	−0.0538722	−0.0269361	+	0.999637i	$0.508575\pi$
$548$	0.480585	0.0205296
$549$	7.88666	0.336595
$550$	−10.2056	−0.435166
$551$	−5.00502	−0.213221
$552$	−2.09775	−0.0892862
$553$	−25.5522	−1.08659
$554$	0.552705	0.0234822
$555$	−8.20555	−0.348306
$556$	−1.90225	−0.0806733
$557$	33.7250	1.42897	0.714486	−	0.699649i	$-0.246660\pi$
$557$	33.7250	1.42897	0.714486	+	0.699649i	$0.246660\pi$
$558$	1.81361	0.0767761
$559$	8.81361	0.372776
$560$	−23.6655	−1.00005
$561$	11.2544	0.475162
$562$	−24.7683	−1.04479
$563$	3.86393	0.162845	0.0814227	−	0.996680i	$-0.474054\pi$
$563$	3.86393	0.162845	0.0814227	+	0.996680i	$0.474054\pi$
$564$	0.941078	0.0396265
$565$	12.7597	0.536805
$566$	−28.9894	−1.21852
$567$	4.81361	0.202152
$568$	14.5089	0.608778
$569$	31.5194	1.32136	0.660681	−	0.750667i	$-0.270267\pi$
$569$	31.5194	1.32136	0.660681	+	0.750667i	$0.270267\pi$
$570$	−2.67609	−0.112089
$571$	5.67557	0.237515	0.118758	−	0.992923i	$-0.462109\pi$
$571$	5.67557	0.237515	0.118758	+	0.992923i	$0.462109\pi$
$572$	7.25443	0.303323
$573$	−25.2927	−1.05662
$574$	−15.4061	−0.643037
$575$	−1.62721	−0.0678595
$576$	−1.66196	−0.0692481
$577$	−36.5089	−1.51988	−0.759942	−	0.649991i	$-0.774773\pi$
$577$	−36.5089	−1.51988	−0.759942	+	0.649991i	$0.774773\pi$
$578$	23.5769	0.980669
$579$	−10.4791	−0.435498
$580$	−4.37279	−0.181570
$581$	22.4408	0.931002
$582$	25.5139	1.05758
$583$	9.45998	0.391792
$584$	−8.06949	−0.333918
$585$	1.00000	0.0413449
$586$	−9.80450	−0.405020
$587$	4.49472	0.185517	0.0927584	−	0.995689i	$-0.470432\pi$
$587$	4.49472	0.185517	0.0927584	+	0.995689i	$0.470432\pi$
$588$	20.8469	0.859712
$589$	−1.47556	−0.0607995
$590$	9.77886	0.402589
$591$	12.2297	0.503064
$592$	40.3416	1.65803
$593$	−4.24940	−0.174502	−0.0872510	−	0.996186i	$-0.527808\pi$
$593$	−4.24940	−0.174502	−0.0872510	+	0.996186i	$0.527808\pi$
$594$	−10.2056	−0.418739
$595$	9.62721	0.394677
$596$	16.0383	0.656955
$597$	−6.20555	−0.253976
$598$	2.95112	0.120680
$599$	−26.4011	−1.07872	−0.539359	−	0.842076i	$-0.681333\pi$
$599$	−26.4011	−1.07872	−0.539359	+	0.842076i	$0.681333\pi$
$600$	1.28917	0.0526301
$601$	46.6394	1.90246	0.951230	−	0.308483i	$-0.0998212\pi$
$601$	46.6394	1.90246	0.951230	+	0.308483i	$0.0998212\pi$
$602$	−76.9427	−3.13595
$603$	−4.23527	−0.172474
$604$	−11.7733	−0.479050
$605$	20.6655	0.840173
$606$	2.00000	0.0812444
$607$	45.4005	1.84275	0.921375	−	0.388674i	$-0.127067\pi$
$607$	45.4005	1.84275	0.921375	+	0.388674i	$0.127067\pi$
$608$	9.35218	0.379281
$609$	−16.3275	−0.661623
$610$	−14.3033	−0.579124
$611$	0.729988	0.0295322
$612$	2.57834	0.104223
$613$	21.6655	0.875062	0.437531	−	0.899203i	$-0.355853\pi$
$613$	21.6655	0.875062	0.437531	+	0.899203i	$0.355853\pi$
$614$	36.9099	1.48956
$615$	1.76473	0.0711608
$616$	34.9200	1.40696
$617$	−5.52946	−0.222608	−0.111304	−	0.993786i	$-0.535503\pi$
$617$	−5.52946	−0.222608	−0.111304	+	0.993786i	$0.535503\pi$
$618$	9.93051	0.399464
$619$	−28.1602	−1.13186	−0.565928	−	0.824455i	$-0.691482\pi$
$619$	−28.1602	−1.13186	−0.565928	+	0.824455i	$0.691482\pi$
$620$	−1.28917	−0.0517743
$621$	−1.62721	−0.0652978
$622$	20.4111	0.818411
$623$	−20.2439	−0.811053
$624$	−4.91638	−0.196813
$625$	1.00000	0.0400000
$626$	−55.5805	−2.22144
$627$	8.30330	0.331602
$628$	−14.3828	−0.573938
$629$	−16.4111	−0.654353
$630$	−8.72999	−0.347811
$631$	−26.8716	−1.06974	−0.534871	−	0.844934i	$-0.679640\pi$
$631$	−26.8716	−1.06974	−0.534871	+	0.844934i	$0.679640\pi$
$632$	−6.84333	−0.272213
$633$	−13.6514	−0.542594
$634$	48.7810	1.93734
$635$	3.18639	0.126448
$636$	2.16724	0.0859365
$637$	16.1708	0.640711
$638$	34.6167	1.37049
$639$	11.2544	0.445218
$640$	−9.66196	−0.381922
$641$	−11.3325	−0.447607	−0.223804	−	0.974634i	$-0.571847\pi$
$641$	−11.3325	−0.447607	−0.223804	+	0.974634i	$0.571847\pi$
$642$	−11.4061	−0.450162
$643$	−1.25997	−0.0496882	−0.0248441	−	0.999691i	$-0.507909\pi$
$643$	−1.25997	−0.0496882	−0.0248441	+	0.999691i	$0.507909\pi$
$644$	−10.0978	−0.397907
$645$	8.81361	0.347035
$646$	−5.35218	−0.210579
$647$	−21.1411	−0.831142	−0.415571	−	0.909561i	$-0.636418\pi$
$647$	−21.1411	−0.831142	−0.415571	+	0.909561i	$0.636418\pi$
$648$	1.28917	0.0506433
$649$	−30.3416	−1.19101
$650$	−1.81361	−0.0711355
$651$	−4.81361	−0.188660
$652$	−24.9612	−0.977555
$653$	28.1502	1.10160	0.550801	−	0.834636i	$-0.314322\pi$
$653$	28.1502	1.10160	0.550801	+	0.834636i	$0.314322\pi$
$654$	1.69670	0.0663461
$655$	6.52444	0.254931
$656$	−8.67609	−0.338744
$657$	−6.25945	−0.244204
$658$	−6.37279	−0.248437
$659$	11.6217	0.452716	0.226358	−	0.974044i	$-0.427318\pi$
$659$	11.6217	0.452716	0.226358	+	0.974044i	$0.427318\pi$
$660$	7.25443	0.282378
$661$	−4.95112	−0.192576	−0.0962882	−	0.995353i	$-0.530697\pi$
$661$	−4.95112	−0.192576	−0.0962882	+	0.995353i	$0.530697\pi$
$662$	−26.4650	−1.02859
$663$	2.00000	0.0776736
$664$	6.01005	0.233235
$665$	7.10278	0.275434
$666$	14.8816	0.576652
$667$	5.51941	0.213713
$668$	2.09775	0.0811645
$669$	−23.5733	−0.911397
$670$	7.68111	0.296747
$671$	44.3799	1.71327
$672$	30.5089	1.17690
$673$	30.1447	1.16199	0.580996	−	0.813907i	$-0.302663\pi$
$673$	30.1447	1.16199	0.580996	+	0.813907i	$0.302663\pi$
$674$	−35.2927	−1.35943
$675$	1.00000	0.0384900
$676$	1.28917	0.0495834
$677$	−30.1205	−1.15762	−0.578812	−	0.815461i	$-0.696484\pi$
$677$	−30.1205	−1.15762	−0.578812	+	0.815461i	$0.696484\pi$
$678$	−23.1411	−0.888728
$679$	−67.7180	−2.59878
$680$	2.57834	0.0988747
$681$	18.6761	0.715669
$682$	10.2056	0.390791
$683$	39.2716	1.50269	0.751343	−	0.659912i	$-0.229406\pi$
$683$	39.2716	1.50269	0.751343	+	0.659912i	$0.229406\pi$
$684$	1.90225	0.0727343
$685$	0.372787	0.0142435
$686$	−80.0610	−3.05674
$687$	13.8328	0.527753
$688$	−43.3311	−1.65198
$689$	1.68111	0.0640453
$690$	2.95112	0.112347
$691$	16.5472	0.629484	0.314742	−	0.949177i	$-0.398082\pi$
$691$	16.5472	0.629484	0.314742	+	0.949177i	$0.398082\pi$
$692$	−17.1184	−0.650742
$693$	27.0872	1.02896
$694$	60.1049	2.28155
$695$	−1.47556	−0.0559713
$696$	−4.37279	−0.165750
$697$	3.52946	0.133688
$698$	22.8816	0.866083
$699$	10.3275	0.390621
$700$	6.20555	0.234548
$701$	22.8222	0.861983	0.430991	−	0.902356i	$-0.358164\pi$
$701$	22.8222	0.861983	0.430991	+	0.902356i	$0.358164\pi$
$702$	−1.81361	−0.0684502
$703$	−12.1078	−0.456654
$704$	−9.35218	−0.352473
$705$	0.729988	0.0274929
$706$	−4.82774	−0.181694
$707$	−5.30833	−0.199640
$708$	−6.95112	−0.261239
$709$	−42.4705	−1.59501	−0.797507	−	0.603309i	$-0.793848\pi$
$709$	−42.4705	−1.59501	−0.797507	+	0.603309i	$0.793848\pi$
$710$	−20.4111	−0.766015
$711$	−5.30833	−0.199078
$712$	−5.42166	−0.203185
$713$	1.62721	0.0609396
$714$	−17.4600	−0.653423
$715$	5.62721	0.210446
$716$	−24.3799	−0.911121
$717$	11.6811	0.436239
$718$	18.8277	0.702645
$719$	12.2947	0.458515	0.229258	−	0.973366i	$-0.426370\pi$
$719$	12.2947	0.458515	0.229258	+	0.973366i	$0.426370\pi$
$720$	−4.91638	−0.183223
$721$	−26.3572	−0.981593
$722$	30.5098	1.13546
$723$	−13.0036	−0.483608
$724$	−29.4983	−1.09630
$725$	−3.39194	−0.125974
$726$	−37.4791	−1.39098
$727$	7.07160	0.262271	0.131136	−	0.991364i	$-0.458138\pi$
$727$	7.07160	0.262271	0.131136	+	0.991364i	$0.458138\pi$
$728$	6.20555	0.229993
$729$	1.00000	0.0370370
$730$	11.3522	0.420163
$731$	17.6272	0.651966
$732$	10.1672	0.375792
$733$	−12.5003	−0.461708	−0.230854	−	0.972988i	$-0.574152\pi$
$733$	−12.5003	−0.461708	−0.230854	+	0.972988i	$0.574152\pi$
$734$	27.5577	1.01717
$735$	16.1708	0.596469
$736$	−10.3133	−0.380155
$737$	−23.8328	−0.877891
$738$	−3.20053	−0.117813
$739$	−46.8716	−1.72420	−0.862100	−	0.506739i	$-0.830851\pi$
$739$	−46.8716	−1.72420	−0.862100	+	0.506739i	$0.830851\pi$
$740$	−10.5783	−0.388867
$741$	1.47556	0.0542061
$742$	−14.6761	−0.538776
$743$	0.358654	0.0131577	0.00657886	−	0.999978i	$-0.497906\pi$
$743$	0.358654	0.0131577	0.00657886	+	0.999978i	$0.497906\pi$
$744$	−1.28917	−0.0472632
$745$	12.4408	0.455796
$746$	−7.62721	−0.279252
$747$	4.66196	0.170572
$748$	14.5089	0.530496
$749$	30.2736	1.10617
$750$	−1.81361	−0.0662235
$751$	14.8675	0.542523	0.271261	−	0.962506i	$-0.412559\pi$
$751$	14.8675	0.542523	0.271261	+	0.962506i	$0.412559\pi$
$752$	−3.58890	−0.130874
$753$	24.9497	0.909217
$754$	6.15165	0.224030
$755$	−9.13249	−0.332365
$756$	6.20555	0.225694
$757$	47.3608	1.72136	0.860678	−	0.509149i	$-0.170040\pi$
$757$	47.3608	1.72136	0.860678	+	0.509149i	$0.170040\pi$
$758$	48.5472	1.76331
$759$	−9.15667	−0.332366
$760$	1.90225	0.0690018
$761$	42.8661	1.55389	0.776947	−	0.629566i	$-0.216767\pi$
$761$	42.8661	1.55389	0.776947	+	0.629566i	$0.216767\pi$
$762$	−5.77886	−0.209346
$763$	−4.50331	−0.163031
$764$	−32.6066	−1.17967
$765$	2.00000	0.0723102
$766$	−58.8761	−2.12728
$767$	−5.39194	−0.194692
$768$	20.8469	0.752248
$769$	−7.88666	−0.284400	−0.142200	−	0.989838i	$-0.545418\pi$
$769$	−7.88666	−0.284400	−0.142200	+	0.989838i	$0.545418\pi$
$770$	−49.1255	−1.77036
$771$	20.4494	0.736468
$772$	−13.5094	−0.486213
$773$	12.3658	0.444767	0.222383	−	0.974959i	$-0.428616\pi$
$773$	12.3658	0.444767	0.222383	+	0.974959i	$0.428616\pi$
$774$	−15.9844	−0.574548
$775$	−1.00000	−0.0359211
$776$	−18.1361	−0.651047
$777$	−39.4983	−1.41699
$778$	39.0616	1.40042
$779$	2.60397	0.0932969
$780$	1.28917	0.0461596
$781$	63.3311	2.26616
$782$	5.90225	0.211064
$783$	−3.39194	−0.121218
$784$	−79.5019	−2.83935
$785$	−11.1567	−0.398199
$786$	−11.8328	−0.422061
$787$	33.9945	1.21177	0.605886	−	0.795552i	$-0.292819\pi$
$787$	33.9945	1.21177	0.605886	+	0.795552i	$0.292819\pi$
$788$	15.7662	0.561647
$789$	−6.93554	−0.246912
$790$	9.62721	0.342521
$791$	61.4202	2.18385
$792$	7.25443	0.257775
$793$	7.88666	0.280064
$794$	7.82272	0.277618
$795$	1.68111	0.0596229
$796$	−8.00000	−0.283552
$797$	1.04334	0.0369569	0.0184784	−	0.999829i	$-0.494118\pi$
$797$	1.04334	0.0369569	0.0184784	+	0.999829i	$0.494118\pi$
$798$	−12.8816	−0.456005
$799$	1.45998	0.0516502
$800$	6.33804	0.224084
$801$	−4.20555	−0.148596
$802$	4.20555	0.148503
$803$	−35.2233	−1.24300
$804$	−5.45998	−0.192559
$805$	−7.83276	−0.276069
$806$	1.81361	0.0638816
$807$	−11.1567	−0.392734
$808$	−1.42166	−0.0500139
$809$	14.3119	0.503179	0.251590	−	0.967834i	$-0.419047\pi$
$809$	14.3119	0.503179	0.251590	+	0.967834i	$0.419047\pi$
$810$	−1.81361	−0.0637236
$811$	−14.2211	−0.499372	−0.249686	−	0.968327i	$-0.580327\pi$
$811$	−14.2211	−0.499372	−0.249686	+	0.968327i	$0.580327\pi$
$812$	−21.0489	−0.738671
$813$	17.2686	0.605635
$814$	83.7422	2.93516
$815$	−19.3622	−0.678229
$816$	−9.83276	−0.344216
$817$	13.0050	0.454988
$818$	1.37781	0.0481740
$819$	4.81361	0.168201
$820$	2.27504	0.0794477
$821$	−24.8122	−0.865950	−0.432975	−	0.901406i	$-0.642536\pi$
$821$	−24.8122	−0.865950	−0.432975	+	0.901406i	$0.642536\pi$
$822$	−0.676089	−0.0235813
$823$	−9.05747	−0.315724	−0.157862	−	0.987461i	$-0.550460\pi$
$823$	−9.05747	−0.315724	−0.157862	+	0.987461i	$0.550460\pi$
$824$	−7.05892	−0.245909
$825$	5.62721	0.195914
$826$	47.0716	1.63783
$827$	−38.7285	−1.34672	−0.673362	−	0.739313i	$-0.735150\pi$
$827$	−38.7285	−1.34672	−0.673362	+	0.739313i	$0.735150\pi$
$828$	−2.09775	−0.0729019
$829$	39.2927	1.36469	0.682347	−	0.731029i	$-0.260960\pi$
$829$	39.2927	1.36469	0.682347	+	0.731029i	$0.260960\pi$
$830$	−8.45495	−0.293476
$831$	−0.304754	−0.0105718
$832$	−1.66196	−0.0576179
$833$	32.3416	1.12057
$834$	2.67609	0.0926654
$835$	1.62721	0.0563120
$836$	10.7044	0.370218
$837$	−1.00000	−0.0345651
$838$	67.0278	2.31544
$839$	−0.569743	−0.0196697	−0.00983486	−	0.999952i	$-0.503131\pi$
$839$	−0.569743	−0.0196697	−0.00983486	+	0.999952i	$0.503131\pi$
$840$	6.20555	0.214112
$841$	−17.4947	−0.603266
$842$	−8.48059	−0.292260
$843$	13.6569	0.470370
$844$	−17.5989	−0.605781
$845$	1.00000	0.0344010
$846$	−1.32391	−0.0455170
$847$	99.4757	3.41803
$848$	−8.26499	−0.283821
$849$	15.9844	0.548584
$850$	−3.62721	−0.124412
$851$	13.3522	0.457707
$852$	14.5089	0.497065
$853$	−8.79248	−0.301049	−0.150524	−	0.988606i	$-0.548096\pi$
$853$	−8.79248	−0.301049	−0.150524	+	0.988606i	$0.548096\pi$
$854$	−68.8505	−2.35601
$855$	1.47556	0.0504632
$856$	8.10780	0.277119
$857$	−26.5783	−0.907899	−0.453949	−	0.891027i	$-0.649985\pi$
$857$	−26.5783	−0.907899	−0.453949	+	0.891027i	$0.649985\pi$
$858$	−10.2056	−0.348412
$859$	−30.7044	−1.04762	−0.523810	−	0.851835i	$-0.675490\pi$
$859$	−30.7044	−1.04762	−0.523810	+	0.851835i	$0.675490\pi$
$860$	11.3622	0.387449
$861$	8.49472	0.289499
$862$	29.1411	0.992549
$863$	25.0136	0.851473	0.425737	−	0.904847i	$-0.360015\pi$
$863$	25.0136	0.851473	0.425737	+	0.904847i	$0.360015\pi$
$864$	6.33804	0.215625
$865$	−13.2786	−0.451486
$866$	13.1028	0.445250
$867$	−13.0000	−0.441503
$868$	−6.20555	−0.210630
$869$	−29.8711	−1.01331
$870$	6.15165	0.208561
$871$	−4.23527	−0.143507
$872$	−1.20607	−0.0408426
$873$	−14.0680	−0.476131
$874$	4.35457	0.147295
$875$	4.81361	0.162730
$876$	−8.06949	−0.272643
$877$	50.3799	1.70121	0.850605	−	0.525806i	$-0.176236\pi$
$877$	50.3799	1.70121	0.850605	+	0.525806i	$0.176236\pi$
$878$	57.4727	1.93961
$879$	5.40608	0.182342
$880$	−27.6655	−0.932605
$881$	9.43026	0.317713	0.158857	−	0.987302i	$-0.449219\pi$
$881$	9.43026	0.317713	0.158857	+	0.987302i	$0.449219\pi$
$882$	−29.3275	−0.987508
$883$	2.22522	0.0748847	0.0374424	−	0.999299i	$-0.488079\pi$
$883$	2.22522	0.0748847	0.0374424	+	0.999299i	$0.488079\pi$
$884$	2.57834	0.0867189
$885$	−5.39194	−0.181248
$886$	−7.21057	−0.242244
$887$	19.7491	0.663111	0.331556	−	0.943436i	$-0.392427\pi$
$887$	19.7491	0.663111	0.331556	+	0.943436i	$0.392427\pi$
$888$	−10.5783	−0.354986
$889$	15.3380	0.514422
$890$	7.62721	0.255665
$891$	5.62721	0.188519
$892$	−30.3900	−1.01753
$893$	1.07714	0.0360452
$894$	−22.5628	−0.754611
$895$	−18.9114	−0.632137
$896$	−46.5089	−1.55375
$897$	−1.62721	−0.0543311
$898$	21.6655	0.722988
$899$	3.39194	0.113128
$900$	1.28917	0.0429723
$901$	3.36222	0.112012
$902$	−18.0100	−0.599669
$903$	42.4252	1.41182
$904$	16.4494	0.547099
$905$	−22.8816	−0.760611
$906$	16.5628	0.550261
$907$	7.89220	0.262056	0.131028	−	0.991379i	$-0.458172\pi$
$907$	7.89220	0.262056	0.131028	+	0.991379i	$0.458172\pi$
$908$	24.0766	0.799011
$909$	−1.10278	−0.0365767
$910$	−8.72999	−0.289396
$911$	−53.1653	−1.76144	−0.880722	−	0.473634i	$-0.842942\pi$
$911$	−53.1653	−1.76144	−0.880722	+	0.473634i	$0.842942\pi$
$912$	−7.25443	−0.240218
$913$	26.2338	0.868213
$914$	−38.7144	−1.28056
$915$	7.88666	0.260725
$916$	17.8328	0.589211
$917$	31.4061	1.03712
$918$	−3.62721	−0.119716
$919$	−37.1608	−1.22582	−0.612910	−	0.790153i	$-0.710001\pi$
$919$	−37.1608	−1.22582	−0.612910	+	0.790153i	$0.710001\pi$
$920$	−2.09775	−0.0691608
$921$	−20.3517	−0.670610
$922$	−52.0071	−1.71276
$923$	11.2544	0.370444
$924$	34.9200	1.14878
$925$	−8.20555	−0.269797
$926$	48.2721	1.58632
$927$	−5.47556	−0.179841
$928$	−21.4983	−0.705716
$929$	21.6499	0.710311	0.355156	−	0.934807i	$-0.384428\pi$
$929$	21.6499	0.710311	0.355156	+	0.934807i	$0.384428\pi$
$930$	1.81361	0.0594705
$931$	23.8610	0.782014
$932$	13.3139	0.436110
$933$	−11.2544	−0.368453
$934$	−29.5038	−0.965395
$935$	11.2544	0.368059
$936$	1.28917	0.0421378
$937$	22.0766	0.721212	0.360606	−	0.932718i	$-0.382570\pi$
$937$	22.0766	0.721212	0.360606	+	0.932718i	$0.382570\pi$
$938$	36.9739	1.20724
$939$	30.6464	1.00011
$940$	0.941078	0.0306946
$941$	−29.8172	−0.972012	−0.486006	−	0.873955i	$-0.661547\pi$
$941$	−29.8172	−0.972012	−0.486006	+	0.873955i	$0.661547\pi$
$942$	20.2338	0.659253
$943$	−2.87159	−0.0935119
$944$	26.5089	0.862790
$945$	4.81361	0.156587
$946$	−89.9477	−2.92445
$947$	3.52946	0.114692	0.0573460	−	0.998354i	$-0.481736\pi$
$947$	3.52946	0.114692	0.0573460	+	0.998354i	$0.481736\pi$
$948$	−6.84333	−0.222261
$949$	−6.25945	−0.203190
$950$	−2.67609	−0.0868238
$951$	−26.8972	−0.872202
$952$	12.4111	0.402246
$953$	24.5527	0.795340	0.397670	−	0.917528i	$-0.369819\pi$
$953$	24.5527	0.795340	0.397670	+	0.917528i	$0.369819\pi$
$954$	−3.04888	−0.0987110
$955$	−25.2927	−0.818454
$956$	15.0589	0.487040
$957$	−19.0872	−0.617001
$958$	−18.1900	−0.587691
$959$	1.79445	0.0579458
$960$	−1.66196	−0.0536394
$961$	1.00000	0.0322581
$962$	14.8816	0.479803
$963$	6.28917	0.202666
$964$	−16.7638	−0.539925
$965$	−10.4791	−0.337335
$966$	14.2056	0.457056
$967$	29.6555	0.953656	0.476828	−	0.878997i	$-0.341786\pi$
$967$	29.6555	0.953656	0.476828	+	0.878997i	$0.341786\pi$
$968$	26.6413	0.856285
$969$	2.95112	0.0948038
$970$	25.5139	0.819201
$971$	36.7738	1.18013	0.590064	−	0.807356i	$-0.299102\pi$
$971$	36.7738	1.18013	0.590064	+	0.807356i	$0.299102\pi$
$972$	1.28917	0.0413501
$973$	−7.10278	−0.227705
$974$	30.6449	0.981926
$975$	1.00000	0.0320256
$976$	−38.7738	−1.24112
$977$	35.5083	1.13601	0.568006	−	0.823024i	$-0.307715\pi$
$977$	35.5083	1.13601	0.568006	+	0.823024i	$0.307715\pi$
$978$	35.1155	1.12287
$979$	−23.6655	−0.756353
$980$	20.8469	0.665930
$981$	−0.935538	−0.0298694
$982$	3.02324	0.0964756
$983$	8.66196	0.276273	0.138137	−	0.990413i	$-0.455889\pi$
$983$	8.66196	0.276273	0.138137	+	0.990413i	$0.455889\pi$
$984$	2.27504	0.0725255
$985$	12.2297	0.389672
$986$	12.3033	0.391817
$987$	3.51388	0.111848
$988$	1.90225	0.0605186
$989$	−14.3416	−0.456037
$990$	−10.2056	−0.324354
$991$	−1.53500	−0.0487609	−0.0243805	−	0.999703i	$-0.507761\pi$
$991$	−1.53500	−0.0487609	−0.0243805	+	0.999703i	$0.507761\pi$
$992$	−6.33804	−0.201233
$993$	14.5925	0.463078
$994$	−98.2510	−3.11633
$995$	−6.20555	−0.196729
$996$	6.01005	0.190436
$997$	−41.8766	−1.32625	−0.663123	−	0.748511i	$-0.730769\pi$
$997$	−41.8766	−1.32625	−0.663123	+	0.748511i	$0.730769\pi$
$998$	−44.3160	−1.40280
$999$	−8.20555	−0.259612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.r.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.r.1.1	✓	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} +1.00000 q^{5} -1.81361 q^{6} +4.81361 q^{7} +1.28917 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} +1.00000 q^{5} -1.81361 q^{6} +4.81361 q^{7} +1.28917 q^{8} +1.00000 q^{9} -1.81361 q^{10} +5.62721 q^{11} +1.28917 q^{12} +1.00000 q^{13} -8.72999 q^{14} +1.00000 q^{15} -4.91638 q^{16} +2.00000 q^{17} -1.81361 q^{18} +1.47556 q^{19} +1.28917 q^{20} +4.81361 q^{21} -10.2056 q^{22} -1.62721 q^{23} +1.28917 q^{24} +1.00000 q^{25} -1.81361 q^{26} +1.00000 q^{27} +6.20555 q^{28} -3.39194 q^{29} -1.81361 q^{30} -1.00000 q^{31} +6.33804 q^{32} +5.62721 q^{33} -3.62721 q^{34} +4.81361 q^{35} +1.28917 q^{36} -8.20555 q^{37} -2.67609 q^{38} +1.00000 q^{39} +1.28917 q^{40} +1.76473 q^{41} -8.72999 q^{42} +8.81361 q^{43} +7.25443 q^{44} +1.00000 q^{45} +2.95112 q^{46} +0.729988 q^{47} -4.91638 q^{48} +16.1708 q^{49} -1.81361 q^{50} +2.00000 q^{51} +1.28917 q^{52} +1.68111 q^{53} -1.81361 q^{54} +5.62721 q^{55} +6.20555 q^{56} +1.47556 q^{57} +6.15165 q^{58} -5.39194 q^{59} +1.28917 q^{60} +7.88666 q^{61} +1.81361 q^{62} +4.81361 q^{63} -1.66196 q^{64} +1.00000 q^{65} -10.2056 q^{66} -4.23527 q^{67} +2.57834 q^{68} -1.62721 q^{69} -8.72999 q^{70} +11.2544 q^{71} +1.28917 q^{72} -6.25945 q^{73} +14.8816 q^{74} +1.00000 q^{75} +1.90225 q^{76} +27.0872 q^{77} -1.81361 q^{78} -5.30833 q^{79} -4.91638 q^{80} +1.00000 q^{81} -3.20053 q^{82} +4.66196 q^{83} +6.20555 q^{84} +2.00000 q^{85} -15.9844 q^{86} -3.39194 q^{87} +7.25443 q^{88} -4.20555 q^{89} -1.81361 q^{90} +4.81361 q^{91} -2.09775 q^{92} -1.00000 q^{93} -1.32391 q^{94} +1.47556 q^{95} +6.33804 q^{96} -14.0680 q^{97} -29.3275 q^{98} +5.62721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} + 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 + 8 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} + 8 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 4 q^{11} + 3 q^{12} + 3 q^{13} - 6 q^{14} + 3 q^{15} - q^{16} + 6 q^{17} + q^{18} + 10 q^{19} + 3 q^{20} + 8 q^{21} - 16 q^{22} + 8 q^{23} + 3 q^{24} + 3 q^{25} + q^{26} + 3 q^{27} + 4 q^{28} - 2 q^{29} + q^{30} - 3 q^{31} + 7 q^{32} + 4 q^{33} + 2 q^{34} + 8 q^{35} + 3 q^{36} - 10 q^{37} + 16 q^{38} + 3 q^{39} + 3 q^{40} + 10 q^{41} - 6 q^{42} + 20 q^{43} - 4 q^{44} + 3 q^{45} + 20 q^{46} - 18 q^{47} - q^{48} + 9 q^{49} + q^{50} + 6 q^{51} + 3 q^{52} - 4 q^{53} + q^{54} + 4 q^{55} + 4 q^{56} + 10 q^{57} - 8 q^{59} + 3 q^{60} - q^{62} + 8 q^{63} - 17 q^{64} + 3 q^{65} - 16 q^{66} - 8 q^{67} + 6 q^{68} + 8 q^{69} - 6 q^{70} + 8 q^{71} + 3 q^{72} - 8 q^{73} + 6 q^{74} + 3 q^{75} + 28 q^{76} + 28 q^{77} + q^{78} + 6 q^{79} - q^{80} + 3 q^{81} + 20 q^{82} + 26 q^{83} + 4 q^{84} + 6 q^{85} - 2 q^{86} - 2 q^{87} - 4 q^{88} + 2 q^{89} + q^{90} + 8 q^{91} + 16 q^{92} - 3 q^{93} - 28 q^{94} + 10 q^{95} + 7 q^{96} - 10 q^{97} - 45 q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + q^6 + 8 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 4 * q^11 + 3 * q^12 + 3 * q^13 - 6 * q^14 + 3 * q^15 - q^16 + 6 * q^17 + q^18 + 10 * q^19 + 3 * q^20 + 8 * q^21 - 16 * q^22 + 8 * q^23 + 3 * q^24 + 3 * q^25 + q^26 + 3 * q^27 + 4 * q^28 - 2 * q^29 + q^30 - 3 * q^31 + 7 * q^32 + 4 * q^33 + 2 * q^34 + 8 * q^35 + 3 * q^36 - 10 * q^37 + 16 * q^38 + 3 * q^39 + 3 * q^40 + 10 * q^41 - 6 * q^42 + 20 * q^43 - 4 * q^44 + 3 * q^45 + 20 * q^46 - 18 * q^47 - q^48 + 9 * q^49 + q^50 + 6 * q^51 + 3 * q^52 - 4 * q^53 + q^54 + 4 * q^55 + 4 * q^56 + 10 * q^57 - 8 * q^59 + 3 * q^60 - q^62 + 8 * q^63 - 17 * q^64 + 3 * q^65 - 16 * q^66 - 8 * q^67 + 6 * q^68 + 8 * q^69 - 6 * q^70 + 8 * q^71 + 3 * q^72 - 8 * q^73 + 6 * q^74 + 3 * q^75 + 28 * q^76 + 28 * q^77 + q^78 + 6 * q^79 - q^80 + 3 * q^81 + 20 * q^82 + 26 * q^83 + 4 * q^84 + 6 * q^85 - 2 * q^86 - 2 * q^87 - 4 * q^88 + 2 * q^89 + q^90 + 8 * q^91 + 16 * q^92 - 3 * q^93 - 28 * q^94 + 10 * q^95 + 7 * q^96 - 10 * q^97 - 45 * q^98 + 4 * q^99

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