LMFDB - Embedded newform 6045.2.a.v.1.9

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 12x^{8} + 59x^{7} + 38x^{6} - 302x^{5} + 13x^{4} + 626x^{3} - 167x^{2} - 457x + 135 $ x^10 - 4x^9 - 12x^8 + 59x^7 + 38x^6 - 302x^5 + 13x^4 + 626x^3 - 167x^2 - 457*x + 135
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.55963$ 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)

$f(q)$	$=$	$q+2.55963 q^{2} +1.00000 q^{3} +4.55168 q^{4} -1.00000 q^{5} +2.55963 q^{6} -3.20072 q^{7} +6.53135 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.55963 q^{2} +1.00000 q^{3} +4.55168 q^{4} -1.00000 q^{5} +2.55963 q^{6} -3.20072 q^{7} +6.53135 q^{8} +1.00000 q^{9} -2.55963 q^{10} +0.992055 q^{11} +4.55168 q^{12} +1.00000 q^{13} -8.19264 q^{14} -1.00000 q^{15} +7.61443 q^{16} -0.420040 q^{17} +2.55963 q^{18} +5.10940 q^{19} -4.55168 q^{20} -3.20072 q^{21} +2.53929 q^{22} -0.227351 q^{23} +6.53135 q^{24} +1.00000 q^{25} +2.55963 q^{26} +1.00000 q^{27} -14.5686 q^{28} +9.44327 q^{29} -2.55963 q^{30} +1.00000 q^{31} +6.42741 q^{32} +0.992055 q^{33} -1.07514 q^{34} +3.20072 q^{35} +4.55168 q^{36} +8.32001 q^{37} +13.0781 q^{38} +1.00000 q^{39} -6.53135 q^{40} +2.75060 q^{41} -8.19264 q^{42} +2.19253 q^{43} +4.51552 q^{44} -1.00000 q^{45} -0.581933 q^{46} -2.07137 q^{47} +7.61443 q^{48} +3.24459 q^{49} +2.55963 q^{50} -0.420040 q^{51} +4.55168 q^{52} -5.70075 q^{53} +2.55963 q^{54} -0.992055 q^{55} -20.9050 q^{56} +5.10940 q^{57} +24.1712 q^{58} +14.0150 q^{59} -4.55168 q^{60} +9.55954 q^{61} +2.55963 q^{62} -3.20072 q^{63} +1.22288 q^{64} -1.00000 q^{65} +2.53929 q^{66} +1.83497 q^{67} -1.91189 q^{68} -0.227351 q^{69} +8.19264 q^{70} +7.75930 q^{71} +6.53135 q^{72} +0.113172 q^{73} +21.2961 q^{74} +1.00000 q^{75} +23.2564 q^{76} -3.17529 q^{77} +2.55963 q^{78} -12.6275 q^{79} -7.61443 q^{80} +1.00000 q^{81} +7.04050 q^{82} -7.35068 q^{83} -14.5686 q^{84} +0.420040 q^{85} +5.61205 q^{86} +9.44327 q^{87} +6.47946 q^{88} -1.43859 q^{89} -2.55963 q^{90} -3.20072 q^{91} -1.03483 q^{92} +1.00000 q^{93} -5.30194 q^{94} -5.10940 q^{95} +6.42741 q^{96} -9.76108 q^{97} +8.30494 q^{98} +0.992055 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 10 q^{3} + 20 q^{4} - 10 q^{5} + 4 q^{6} - q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q + 4 * q^2 + 10 * q^3 + 20 * q^4 - 10 * q^5 + 4 * q^6 - q^7 + 15 * q^8 + 10 * q^9	\( 10 q + 4 q^{2} + 10 q^{3} + 20 q^{4} - 10 q^{5} + 4 q^{6} - q^{7} + 15 q^{8} + 10 q^{9} - 4 q^{10} + 6 q^{11} + 20 q^{12} + 10 q^{13} + 11 q^{14} - 10 q^{15} + 16 q^{16} - q^{17} + 4 q^{18} + 2 q^{19} - 20 q^{20} - q^{21} - 21 q^{22} + q^{23} + 15 q^{24} + 10 q^{25} + 4 q^{26} + 10 q^{27} + 10 q^{28} + 15 q^{29} - 4 q^{30} + 10 q^{31} + 34 q^{32} + 6 q^{33} + 3 q^{34} + q^{35} + 20 q^{36} - 3 q^{37} - 6 q^{38} + 10 q^{39} - 15 q^{40} + 43 q^{41} + 11 q^{42} - 8 q^{43} + 53 q^{44} - 10 q^{45} + 22 q^{46} - 11 q^{47} + 16 q^{48} + 31 q^{49} + 4 q^{50} - q^{51} + 20 q^{52} + 10 q^{53} + 4 q^{54} - 6 q^{55} + 17 q^{56} + 2 q^{57} - 16 q^{58} + 48 q^{59} - 20 q^{60} + 6 q^{61} + 4 q^{62} - q^{63} + 29 q^{64} - 10 q^{65} - 21 q^{66} - 16 q^{67} + 19 q^{68} + q^{69} - 11 q^{70} + 53 q^{71} + 15 q^{72} + 7 q^{73} + 24 q^{74} + 10 q^{75} + 30 q^{76} + 4 q^{78} - q^{79} - 16 q^{80} + 10 q^{81} + 53 q^{82} + 41 q^{83} + 10 q^{84} + q^{85} + 14 q^{86} + 15 q^{87} + 9 q^{88} + 31 q^{89} - 4 q^{90} - q^{91} + 23 q^{92} + 10 q^{93} - 18 q^{94} - 2 q^{95} + 34 q^{96} - 37 q^{97} + 28 q^{98} + 6 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 10 * q^3 + 20 * q^4 - 10 * q^5 + 4 * q^6 - q^7 + 15 * q^8 + 10 * q^9 - 4 * q^10 + 6 * q^11 + 20 * q^12 + 10 * q^13 + 11 * q^14 - 10 * q^15 + 16 * q^16 - q^17 + 4 * q^18 + 2 * q^19 - 20 * q^20 - q^21 - 21 * q^22 + q^23 + 15 * q^24 + 10 * q^25 + 4 * q^26 + 10 * q^27 + 10 * q^28 + 15 * q^29 - 4 * q^30 + 10 * q^31 + 34 * q^32 + 6 * q^33 + 3 * q^34 + q^35 + 20 * q^36 - 3 * q^37 - 6 * q^38 + 10 * q^39 - 15 * q^40 + 43 * q^41 + 11 * q^42 - 8 * q^43 + 53 * q^44 - 10 * q^45 + 22 * q^46 - 11 * q^47 + 16 * q^48 + 31 * q^49 + 4 * q^50 - q^51 + 20 * q^52 + 10 * q^53 + 4 * q^54 - 6 * q^55 + 17 * q^56 + 2 * q^57 - 16 * q^58 + 48 * q^59 - 20 * q^60 + 6 * q^61 + 4 * q^62 - q^63 + 29 * q^64 - 10 * q^65 - 21 * q^66 - 16 * q^67 + 19 * q^68 + q^69 - 11 * q^70 + 53 * q^71 + 15 * q^72 + 7 * q^73 + 24 * q^74 + 10 * q^75 + 30 * q^76 + 4 * q^78 - q^79 - 16 * q^80 + 10 * q^81 + 53 * q^82 + 41 * q^83 + 10 * q^84 + q^85 + 14 * q^86 + 15 * q^87 + 9 * q^88 + 31 * q^89 - 4 * q^90 - q^91 + 23 * q^92 + 10 * q^93 - 18 * q^94 - 2 * q^95 + 34 * q^96 - 37 * q^97 + 28 * q^98 + 6 * 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 $

$2$	2.55963	1.80993	0.904964	−	0.425488i	$-0.139897\pi$
$2$	2.55963	1.80993	0.904964	+	0.425488i	$0.139897\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.55168	2.27584
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.55963	1.04496
$7$	−3.20072	−1.20976	−0.604879	−	0.796318i	$-0.706778\pi$
$7$	−3.20072	−1.20976	−0.604879	+	0.796318i	$0.706778\pi$
$8$	6.53135	2.30918
$9$	1.00000	0.333333
$10$	−2.55963	−0.809425
$11$	0.992055	0.299116	0.149558	−	0.988753i	$-0.452215\pi$
$11$	0.992055	0.299116	0.149558	+	0.988753i	$0.452215\pi$
$12$	4.55168	1.31396
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−8.19264	−2.18957
$15$	−1.00000	−0.258199
$16$	7.61443	1.90361
$17$	−0.420040	−0.101875	−0.0509373	−	0.998702i	$-0.516221\pi$
$17$	−0.420040	−0.101875	−0.0509373	+	0.998702i	$0.516221\pi$
$18$	2.55963	0.603309
$19$	5.10940	1.17218	0.586088	−	0.810247i	$-0.300667\pi$
$19$	5.10940	1.17218	0.586088	+	0.810247i	$0.300667\pi$
$20$	−4.55168	−1.01779
$21$	−3.20072	−0.698454
$22$	2.53929	0.541378
$23$	−0.227351	−0.0474060	−0.0237030	−	0.999719i	$-0.507546\pi$
$23$	−0.227351	−0.0474060	−0.0237030	+	0.999719i	$0.507546\pi$
$24$	6.53135	1.33321
$25$	1.00000	0.200000
$26$	2.55963	0.501984
$27$	1.00000	0.192450
$28$	−14.5686	−2.75321
$29$	9.44327	1.75357	0.876786	−	0.480882i	$-0.159683\pi$
$29$	9.44327	1.75357	0.876786	+	0.480882i	$0.159683\pi$
$30$	−2.55963	−0.467321
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	6.42741	1.13622
$33$	0.992055	0.172695
$34$	−1.07514	−0.184386
$35$	3.20072	0.541020
$36$	4.55168	0.758613
$37$	8.32001	1.36780	0.683900	−	0.729576i	$-0.260282\pi$
$37$	8.32001	1.36780	0.683900	+	0.729576i	$0.260282\pi$
$38$	13.0781	2.12156
$39$	1.00000	0.160128
$40$	−6.53135	−1.03270
$41$	2.75060	0.429571	0.214786	−	0.976661i	$-0.431095\pi$
$41$	2.75060	0.429571	0.214786	+	0.976661i	$0.431095\pi$
$42$	−8.19264	−1.26415
$43$	2.19253	0.334358	0.167179	−	0.985927i	$-0.446534\pi$
$43$	2.19253	0.334358	0.167179	+	0.985927i	$0.446534\pi$
$44$	4.51552	0.680740
$45$	−1.00000	−0.149071
$46$	−0.581933	−0.0858014
$47$	−2.07137	−0.302141	−0.151071	−	0.988523i	$-0.548272\pi$
$47$	−2.07137	−0.302141	−0.151071	+	0.988523i	$0.548272\pi$
$48$	7.61443	1.09905
$49$	3.24459	0.463513
$50$	2.55963	0.361986
$51$	−0.420040	−0.0588173
$52$	4.55168	0.631205
$53$	−5.70075	−0.783059	−0.391529	−	0.920166i	$-0.628054\pi$
$53$	−5.70075	−0.783059	−0.391529	+	0.920166i	$0.628054\pi$
$54$	2.55963	0.348321
$55$	−0.992055	−0.133769
$56$	−20.9050	−2.79355
$57$	5.10940	0.676757
$58$	24.1712	3.17384
$59$	14.0150	1.82459	0.912296	−	0.409530i	$-0.134307\pi$
$59$	14.0150	1.82459	0.912296	+	0.409530i	$0.134307\pi$
$60$	−4.55168	−0.587619
$61$	9.55954	1.22397	0.611987	−	0.790868i	$-0.290371\pi$
$61$	9.55954	1.22397	0.611987	+	0.790868i	$0.290371\pi$
$62$	2.55963	0.325073
$63$	−3.20072	−0.403252
$64$	1.22288	0.152860
$65$	−1.00000	−0.124035
$66$	2.53929	0.312565
$67$	1.83497	0.224178	0.112089	−	0.993698i	$-0.464246\pi$
$67$	1.83497	0.224178	0.112089	+	0.993698i	$0.464246\pi$
$68$	−1.91189	−0.231850
$69$	−0.227351	−0.0273698
$70$	8.19264	0.979207
$71$	7.75930	0.920859	0.460430	−	0.887696i	$-0.347695\pi$
$71$	7.75930	0.920859	0.460430	+	0.887696i	$0.347695\pi$
$72$	6.53135	0.769726
$73$	0.113172	0.0132457	0.00662287	−	0.999978i	$-0.497892\pi$
$73$	0.113172	0.0132457	0.00662287	+	0.999978i	$0.497892\pi$
$74$	21.2961	2.47562
$75$	1.00000	0.115470
$76$	23.2564	2.66769
$77$	−3.17529	−0.361858
$78$	2.55963	0.289820
$79$	−12.6275	−1.42071	−0.710354	−	0.703844i	$-0.751465\pi$
$79$	−12.6275	−1.42071	−0.710354	+	0.703844i	$0.751465\pi$
$80$	−7.61443	−0.851320
$81$	1.00000	0.111111
$82$	7.04050	0.777493
$83$	−7.35068	−0.806842	−0.403421	−	0.915014i	$-0.632179\pi$
$83$	−7.35068	−0.806842	−0.403421	+	0.915014i	$0.632179\pi$
$84$	−14.5686	−1.58957
$85$	0.420040	0.0455597
$86$	5.61205	0.605163
$87$	9.44327	1.01242
$88$	6.47946	0.690712
$89$	−1.43859	−0.152491	−0.0762454	−	0.997089i	$-0.524293\pi$
$89$	−1.43859	−0.152491	−0.0762454	+	0.997089i	$0.524293\pi$
$90$	−2.55963	−0.269808
$91$	−3.20072	−0.335526
$92$	−1.03483	−0.107888
$93$	1.00000	0.103695
$94$	−5.30194	−0.546854
$95$	−5.10940	−0.524213
$96$	6.42741	0.655994
$97$	−9.76108	−0.991087	−0.495544	−	0.868583i	$-0.665031\pi$
$97$	−9.76108	−0.991087	−0.495544	+	0.868583i	$0.665031\pi$
$98$	8.30494	0.838925
$99$	0.992055	0.0997053
$100$	4.55168	0.455168
$101$	10.0532	1.00033	0.500163	−	0.865931i	$-0.333273\pi$
$101$	10.0532	1.00033	0.500163	+	0.865931i	$0.333273\pi$
$102$	−1.07514	−0.106455
$103$	−2.77892	−0.273815	−0.136908	−	0.990584i	$-0.543716\pi$
$103$	−2.77892	−0.273815	−0.136908	+	0.990584i	$0.543716\pi$
$104$	6.53135	0.640451
$105$	3.20072	0.312358
$106$	−14.5918	−1.41728
$107$	−4.87103	−0.470900	−0.235450	−	0.971886i	$-0.575656\pi$
$107$	−4.87103	−0.470900	−0.235450	+	0.971886i	$0.575656\pi$
$108$	4.55168	0.437986
$109$	−9.74763	−0.933654	−0.466827	−	0.884349i	$-0.654603\pi$
$109$	−9.74763	−0.933654	−0.466827	+	0.884349i	$0.654603\pi$
$110$	−2.53929	−0.242112
$111$	8.32001	0.789700
$112$	−24.3717	−2.30290
$113$	−16.1736	−1.52149	−0.760744	−	0.649052i	$-0.775166\pi$
$113$	−16.1736	−1.52149	−0.760744	+	0.649052i	$0.775166\pi$
$114$	13.0781	1.22488
$115$	0.227351	0.0212006
$116$	42.9827	3.99085
$117$	1.00000	0.0924500
$118$	35.8731	3.30238
$119$	1.34443	0.123244
$120$	−6.53135	−0.596228
$121$	−10.0158	−0.910530
$122$	24.4688	2.21530
$123$	2.75060	0.248013
$124$	4.55168	0.408753
$125$	−1.00000	−0.0894427
$126$	−8.19264	−0.729858
$127$	18.0653	1.60303	0.801516	−	0.597973i	$-0.204027\pi$
$127$	18.0653	1.60303	0.801516	+	0.597973i	$0.204027\pi$
$128$	−9.72469	−0.859550
$129$	2.19253	0.193042
$130$	−2.55963	−0.224494
$131$	0.323531	0.0282670	0.0141335	−	0.999900i	$-0.495501\pi$
$131$	0.323531	0.0282670	0.0141335	+	0.999900i	$0.495501\pi$
$132$	4.51552	0.393026
$133$	−16.3537	−1.41805
$134$	4.69685	0.405746
$135$	−1.00000	−0.0860663
$136$	−2.74343	−0.235247
$137$	5.34692	0.456818	0.228409	−	0.973565i	$-0.426648\pi$
$137$	5.34692	0.456818	0.228409	+	0.973565i	$0.426648\pi$
$138$	−0.581933	−0.0495374
$139$	11.6081	0.984588	0.492294	−	0.870429i	$-0.336159\pi$
$139$	11.6081	0.984588	0.492294	+	0.870429i	$0.336159\pi$
$140$	14.5686	1.23128
$141$	−2.07137	−0.174441
$142$	19.8609	1.66669
$143$	0.992055	0.0829598
$144$	7.61443	0.634536
$145$	−9.44327	−0.784221
$146$	0.289677	0.0239738
$147$	3.24459	0.267609
$148$	37.8700	3.11290
$149$	3.98054	0.326099	0.163049	−	0.986618i	$-0.447867\pi$
$149$	3.98054	0.326099	0.163049	+	0.986618i	$0.447867\pi$
$150$	2.55963	0.208993
$151$	−22.1827	−1.80521	−0.902603	−	0.430474i	$-0.858347\pi$
$151$	−22.1827	−1.80521	−0.902603	+	0.430474i	$0.858347\pi$
$152$	33.3713	2.70677
$153$	−0.420040	−0.0339582
$154$	−8.12755	−0.654937
$155$	−1.00000	−0.0803219
$156$	4.55168	0.364426
$157$	−16.4605	−1.31369	−0.656844	−	0.754027i	$-0.728109\pi$
$157$	−16.4605	−1.31369	−0.656844	+	0.754027i	$0.728109\pi$
$158$	−32.3218	−2.57138
$159$	−5.70075	−0.452099
$160$	−6.42741	−0.508131
$161$	0.727686	0.0573497
$162$	2.55963	0.201103
$163$	−20.7891	−1.62832	−0.814162	−	0.580638i	$-0.802803\pi$
$163$	−20.7891	−1.62832	−0.814162	+	0.580638i	$0.802803\pi$
$164$	12.5198	0.977635
$165$	−0.992055	−0.0772314
$166$	−18.8150	−1.46033
$167$	0.438068	0.0338987	0.0169494	−	0.999856i	$-0.494605\pi$
$167$	0.438068	0.0338987	0.0169494	+	0.999856i	$0.494605\pi$
$168$	−20.9050	−1.61286
$169$	1.00000	0.0769231
$170$	1.07514	0.0824598
$171$	5.10940	0.390726
$172$	9.97970	0.760945
$173$	−13.8954	−1.05645	−0.528223	−	0.849106i	$-0.677141\pi$
$173$	−13.8954	−1.05645	−0.528223	+	0.849106i	$0.677141\pi$
$174$	24.1712	1.83242
$175$	−3.20072	−0.241951
$176$	7.55394	0.569400
$177$	14.0150	1.05343
$178$	−3.68226	−0.275997
$179$	0.537343	0.0401629	0.0200814	−	0.999798i	$-0.493607\pi$
$179$	0.537343	0.0401629	0.0200814	+	0.999798i	$0.493607\pi$
$180$	−4.55168	−0.339262
$181$	−14.7335	−1.09513	−0.547565	−	0.836763i	$-0.684445\pi$
$181$	−14.7335	−1.09513	−0.547565	+	0.836763i	$0.684445\pi$
$182$	−8.19264	−0.607279
$183$	9.55954	0.706661
$184$	−1.48491	−0.109469
$185$	−8.32001	−0.611699
$186$	2.55963	0.187681
$187$	−0.416703	−0.0304723
$188$	−9.42824	−0.687625
$189$	−3.20072	−0.232818
$190$	−13.0781	−0.948789
$191$	3.64522	0.263759	0.131879	−	0.991266i	$-0.457899\pi$
$191$	3.64522	0.263759	0.131879	+	0.991266i	$0.457899\pi$
$192$	1.22288	0.0882539
$193$	12.0225	0.865401	0.432701	−	0.901538i	$-0.357561\pi$
$193$	12.0225	0.865401	0.432701	+	0.901538i	$0.357561\pi$
$194$	−24.9847	−1.79380
$195$	−1.00000	−0.0716115
$196$	14.7683	1.05488
$197$	−15.2124	−1.08384	−0.541919	−	0.840430i	$-0.682302\pi$
$197$	−15.2124	−1.08384	−0.541919	+	0.840430i	$0.682302\pi$
$198$	2.53929	0.180459
$199$	26.2872	1.86345	0.931724	−	0.363167i	$-0.118305\pi$
$199$	26.2872	1.86345	0.931724	+	0.363167i	$0.118305\pi$
$200$	6.53135	0.461836
$201$	1.83497	0.129429
$202$	25.7323	1.81052
$203$	−30.2252	−2.12140
$204$	−1.91189	−0.133859
$205$	−2.75060	−0.192110
$206$	−7.11300	−0.495586
$207$	−0.227351	−0.0158020
$208$	7.61443	0.527966
$209$	5.06881	0.350617
$210$	8.19264	0.565346
$211$	18.3246	1.26152	0.630758	−	0.775979i	$-0.282744\pi$
$211$	18.3246	1.26152	0.630758	+	0.775979i	$0.282744\pi$
$212$	−25.9480	−1.78212
$213$	7.75930	0.531658
$214$	−12.4680	−0.852295
$215$	−2.19253	−0.149529
$216$	6.53135	0.444402
$217$	−3.20072	−0.217279
$218$	−24.9503	−1.68985
$219$	0.113172	0.00764743
$220$	−4.51552	−0.304436
$221$	−0.420040	−0.0282549
$222$	21.2961	1.42930
$223$	9.00331	0.602906	0.301453	−	0.953481i	$-0.402528\pi$
$223$	9.00331	0.602906	0.301453	+	0.953481i	$0.402528\pi$
$224$	−20.5723	−1.37455
$225$	1.00000	0.0666667
$226$	−41.3985	−2.75378
$227$	7.64102	0.507152	0.253576	−	0.967315i	$-0.418393\pi$
$227$	7.64102	0.507152	0.253576	+	0.967315i	$0.418393\pi$
$228$	23.2564	1.54019
$229$	0.411746	0.0272089	0.0136045	−	0.999907i	$-0.495669\pi$
$229$	0.411746	0.0272089	0.0136045	+	0.999907i	$0.495669\pi$
$230$	0.581933	0.0383715
$231$	−3.17529	−0.208919
$232$	61.6773	4.04931
$233$	14.2029	0.930464	0.465232	−	0.885189i	$-0.345971\pi$
$233$	14.2029	0.930464	0.465232	+	0.885189i	$0.345971\pi$
$234$	2.55963	0.167328
$235$	2.07137	0.135122
$236$	63.7916	4.15248
$237$	−12.6275	−0.820246
$238$	3.44123	0.223062
$239$	−19.2162	−1.24300	−0.621498	−	0.783416i	$-0.713475\pi$
$239$	−19.2162	−1.24300	−0.621498	+	0.783416i	$0.713475\pi$
$240$	−7.61443	−0.491510
$241$	3.76980	0.242834	0.121417	−	0.992602i	$-0.461256\pi$
$241$	3.76980	0.242834	0.121417	+	0.992602i	$0.461256\pi$
$242$	−25.6368	−1.64799
$243$	1.00000	0.0641500
$244$	43.5120	2.78557
$245$	−3.24459	−0.207289
$246$	7.04050	0.448886
$247$	5.10940	0.325103
$248$	6.53135	0.414741
$249$	−7.35068	−0.465831
$250$	−2.55963	−0.161885
$251$	−19.2130	−1.21272	−0.606358	−	0.795192i	$-0.707370\pi$
$251$	−19.2130	−1.21272	−0.606358	+	0.795192i	$0.707370\pi$
$252$	−14.5686	−0.917738
$253$	−0.225545	−0.0141799
$254$	46.2403	2.90137
$255$	0.420040	0.0263039
$256$	−27.3373	−1.70858
$257$	2.03640	0.127027	0.0635137	−	0.997981i	$-0.479769\pi$
$257$	2.03640	0.127027	0.0635137	+	0.997981i	$0.479769\pi$
$258$	5.61205	0.349391
$259$	−26.6300	−1.65471
$260$	−4.55168	−0.282283
$261$	9.44327	0.584524
$262$	0.828117	0.0511612
$263$	−10.3420	−0.637718	−0.318859	−	0.947802i	$-0.603300\pi$
$263$	−10.3420	−0.637718	−0.318859	+	0.947802i	$0.603300\pi$
$264$	6.47946	0.398783
$265$	5.70075	0.350194
$266$	−41.8595	−2.56657
$267$	−1.43859	−0.0880406
$268$	8.35222	0.510193
$269$	−30.6795	−1.87056	−0.935282	−	0.353905i	$-0.884854\pi$
$269$	−30.6795	−1.87056	−0.935282	+	0.353905i	$0.884854\pi$
$270$	−2.55963	−0.155774
$271$	1.36209	0.0827413	0.0413707	−	0.999144i	$-0.486828\pi$
$271$	1.36209	0.0827413	0.0413707	+	0.999144i	$0.486828\pi$
$272$	−3.19837	−0.193929
$273$	−3.20072	−0.193716
$274$	13.6861	0.826808
$275$	0.992055	0.0598232
$276$	−1.03483	−0.0622894
$277$	6.67022	0.400774	0.200387	−	0.979717i	$-0.435780\pi$
$277$	6.67022	0.400774	0.200387	+	0.979717i	$0.435780\pi$
$278$	29.7124	1.78203
$279$	1.00000	0.0598684
$280$	20.9050	1.24931
$281$	−8.68789	−0.518276	−0.259138	−	0.965840i	$-0.583438\pi$
$281$	−8.68789	−0.518276	−0.259138	+	0.965840i	$0.583438\pi$
$282$	−5.30194	−0.315726
$283$	3.71684	0.220943	0.110472	−	0.993879i	$-0.464764\pi$
$283$	3.71684	0.220943	0.110472	+	0.993879i	$0.464764\pi$
$284$	35.3179	2.09573
$285$	−5.10940	−0.302655
$286$	2.53929	0.150151
$287$	−8.80388	−0.519677
$288$	6.42741	0.378739
$289$	−16.8236	−0.989622
$290$	−24.1712	−1.41938
$291$	−9.76108	−0.572205
$292$	0.515121	0.0301452
$293$	19.5400	1.14154	0.570768	−	0.821111i	$-0.306645\pi$
$293$	19.5400	1.14154	0.570768	+	0.821111i	$0.306645\pi$
$294$	8.30494	0.484354
$295$	−14.0150	−0.815983
$296$	54.3408	3.15850
$297$	0.992055	0.0575649
$298$	10.1887	0.590216
$299$	−0.227351	−0.0131480
$300$	4.55168	0.262791
$301$	−7.01767	−0.404492
$302$	−56.7795	−3.26729
$303$	10.0532	0.577539
$304$	38.9052	2.23137
$305$	−9.55954	−0.547378
$306$	−1.07514	−0.0614619
$307$	−3.50550	−0.200069	−0.100035	−	0.994984i	$-0.531895\pi$
$307$	−3.50550	−0.200069	−0.100035	+	0.994984i	$0.531895\pi$
$308$	−14.4529	−0.823530
$309$	−2.77892	−0.158087
$310$	−2.55963	−0.145377
$311$	10.7211	0.607940	0.303970	−	0.952682i	$-0.401688\pi$
$311$	10.7211	0.607940	0.303970	+	0.952682i	$0.401688\pi$
$312$	6.53135	0.369765
$313$	−9.44756	−0.534008	−0.267004	−	0.963695i	$-0.586034\pi$
$313$	−9.44756	−0.534008	−0.267004	+	0.963695i	$0.586034\pi$
$314$	−42.1326	−2.37768
$315$	3.20072	0.180340
$316$	−57.4765	−3.23331
$317$	10.1364	0.569318	0.284659	−	0.958629i	$-0.408120\pi$
$317$	10.1364	0.569318	0.284659	+	0.958629i	$0.408120\pi$
$318$	−14.5918	−0.818267
$319$	9.36825	0.524521
$320$	−1.22288	−0.0683612
$321$	−4.87103	−0.271874
$322$	1.86260	0.103799
$323$	−2.14615	−0.119415
$324$	4.55168	0.252871
$325$	1.00000	0.0554700
$326$	−53.2122	−2.94715
$327$	−9.74763	−0.539045
$328$	17.9651	0.991957
$329$	6.62989	0.365517
$330$	−2.53929	−0.139783
$331$	27.8093	1.52854	0.764270	−	0.644896i	$-0.223099\pi$
$331$	27.8093	1.52854	0.764270	+	0.644896i	$0.223099\pi$
$332$	−33.4580	−1.83624
$333$	8.32001	0.455933
$334$	1.12129	0.0613542
$335$	−1.83497	−0.100255
$336$	−24.3717	−1.32958
$337$	21.1785	1.15367	0.576833	−	0.816862i	$-0.304288\pi$
$337$	21.1785	1.15367	0.576833	+	0.816862i	$0.304288\pi$
$338$	2.55963	0.139225
$339$	−16.1736	−0.878432
$340$	1.91189	0.103687
$341$	0.992055	0.0537228
$342$	13.0781	0.707185
$343$	12.0200	0.649019
$344$	14.3202	0.772092
$345$	0.227351	0.0122402
$346$	−35.5669	−1.91209
$347$	−35.8294	−1.92342	−0.961712	−	0.274061i	$-0.911633\pi$
$347$	−35.8294	−1.92342	−0.961712	+	0.274061i	$0.911633\pi$
$348$	42.9827	2.30412
$349$	10.1633	0.544029	0.272014	−	0.962293i	$-0.412310\pi$
$349$	10.1633	0.544029	0.272014	+	0.962293i	$0.412310\pi$
$350$	−8.19264	−0.437915
$351$	1.00000	0.0533761
$352$	6.37634	0.339860
$353$	32.8762	1.74982	0.874912	−	0.484281i	$-0.160919\pi$
$353$	32.8762	1.74982	0.874912	+	0.484281i	$0.160919\pi$
$354$	35.8731	1.90663
$355$	−7.75930	−0.411821
$356$	−6.54802	−0.347045
$357$	1.34443	0.0711547
$358$	1.37540	0.0726919
$359$	−19.6559	−1.03740	−0.518699	−	0.854957i	$-0.673583\pi$
$359$	−19.6559	−1.03740	−0.518699	+	0.854957i	$0.673583\pi$
$360$	−6.53135	−0.344232
$361$	7.10597	0.373998
$362$	−37.7121	−1.98210
$363$	−10.0158	−0.525695
$364$	−14.5686	−0.763604
$365$	−0.113172	−0.00592368
$366$	24.4688	1.27901
$367$	18.2555	0.952930	0.476465	−	0.879193i	$-0.341918\pi$
$367$	18.2555	0.952930	0.476465	+	0.879193i	$0.341918\pi$
$368$	−1.73115	−0.0902424
$369$	2.75060	0.143190
$370$	−21.2961	−1.10713
$371$	18.2465	0.947311
$372$	4.55168	0.235994
$373$	−34.1046	−1.76587	−0.882935	−	0.469495i	$-0.844436\pi$
$373$	−34.1046	−1.76587	−0.882935	+	0.469495i	$0.844436\pi$
$374$	−1.06660	−0.0551527
$375$	−1.00000	−0.0516398
$376$	−13.5289	−0.697698
$377$	9.44327	0.486353
$378$	−8.19264	−0.421384
$379$	−12.6899	−0.651836	−0.325918	−	0.945398i	$-0.605673\pi$
$379$	−12.6899	−0.651836	−0.325918	+	0.945398i	$0.605673\pi$
$380$	−23.2564	−1.19303
$381$	18.0653	0.925512
$382$	9.33039	0.477384
$383$	−16.8867	−0.862872	−0.431436	−	0.902144i	$-0.641993\pi$
$383$	−16.8867	−0.862872	−0.431436	+	0.902144i	$0.641993\pi$
$384$	−9.72469	−0.496261
$385$	3.17529	0.161828
$386$	30.7732	1.56631
$387$	2.19253	0.111453
$388$	−44.4293	−2.25556
$389$	0.419707	0.0212800	0.0106400	−	0.999943i	$-0.496613\pi$
$389$	0.419707	0.0212800	0.0106400	+	0.999943i	$0.496613\pi$
$390$	−2.55963	−0.129612
$391$	0.0954965	0.00482946
$392$	21.1916	1.07033
$393$	0.323531	0.0163200
$394$	−38.9380	−1.96167
$395$	12.6275	0.635360
$396$	4.51552	0.226913
$397$	−31.3968	−1.57576	−0.787880	−	0.615829i	$-0.788821\pi$
$397$	−31.3968	−1.57576	−0.787880	+	0.615829i	$0.788821\pi$
$398$	67.2853	3.37271
$399$	−16.3537	−0.818711
$400$	7.61443	0.380722
$401$	20.4922	1.02333	0.511665	−	0.859185i	$-0.329029\pi$
$401$	20.4922	1.02333	0.511665	+	0.859185i	$0.329029\pi$
$402$	4.69685	0.234257
$403$	1.00000	0.0498135
$404$	45.7588	2.27658
$405$	−1.00000	−0.0496904
$406$	−77.3653	−3.83957
$407$	8.25391	0.409131
$408$	−2.74343	−0.135820
$409$	16.3578	0.808842	0.404421	−	0.914573i	$-0.367473\pi$
$409$	16.3578	0.808842	0.404421	+	0.914573i	$0.367473\pi$
$410$	−7.04050	−0.347705
$411$	5.34692	0.263744
$412$	−12.6488	−0.623160
$413$	−44.8579	−2.20731
$414$	−0.581933	−0.0286005
$415$	7.35068	0.360831
$416$	6.42741	0.315130
$417$	11.6081	0.568452
$418$	12.9742	0.634591
$419$	−5.55008	−0.271139	−0.135569	−	0.990768i	$-0.543286\pi$
$419$	−5.55008	−0.271139	−0.135569	+	0.990768i	$0.543286\pi$
$420$	14.5686	0.710877
$421$	−21.8797	−1.06635	−0.533175	−	0.846005i	$-0.679001\pi$
$421$	−21.8797	−1.06635	−0.533175	+	0.846005i	$0.679001\pi$
$422$	46.9041	2.28326
$423$	−2.07137	−0.100714
$424$	−37.2336	−1.80822
$425$	−0.420040	−0.0203749
$426$	19.8609	0.962264
$427$	−30.5974	−1.48071
$428$	−22.1714	−1.07169
$429$	0.992055	0.0478969
$430$	−5.61205	−0.270637
$431$	37.0270	1.78353	0.891763	−	0.452503i	$-0.149469\pi$
$431$	37.0270	1.78353	0.891763	+	0.452503i	$0.149469\pi$
$432$	7.61443	0.366350
$433$	−15.6168	−0.750495	−0.375248	−	0.926925i	$-0.622442\pi$
$433$	−15.6168	−0.750495	−0.375248	+	0.926925i	$0.622442\pi$
$434$	−8.19264	−0.393259
$435$	−9.44327	−0.452770
$436$	−44.3681	−2.12485
$437$	−1.16163	−0.0555682
$438$	0.289677	0.0138413
$439$	27.1801	1.29724	0.648618	−	0.761114i	$-0.275347\pi$
$439$	27.1801	1.29724	0.648618	+	0.761114i	$0.275347\pi$
$440$	−6.47946	−0.308896
$441$	3.24459	0.154504
$442$	−1.07514	−0.0511394
$443$	22.5102	1.06949	0.534746	−	0.845013i	$-0.320407\pi$
$443$	22.5102	1.06949	0.534746	+	0.845013i	$0.320407\pi$
$444$	37.8700	1.79723
$445$	1.43859	0.0681959
$446$	23.0451	1.09122
$447$	3.98054	0.188273
$448$	−3.91410	−0.184924
$449$	31.8368	1.50247	0.751237	−	0.660033i	$-0.229458\pi$
$449$	31.8368	1.50247	0.751237	+	0.660033i	$0.229458\pi$
$450$	2.55963	0.120662
$451$	2.72874	0.128492
$452$	−73.6172	−3.46266
$453$	−22.1827	−1.04224
$454$	19.5581	0.917909
$455$	3.20072	0.150052
$456$	33.3713	1.56275
$457$	7.28085	0.340584	0.170292	−	0.985394i	$-0.445529\pi$
$457$	7.28085	0.340584	0.170292	+	0.985394i	$0.445529\pi$
$458$	1.05391	0.0492462
$459$	−0.420040	−0.0196058
$460$	1.03483	0.0482491
$461$	−0.576789	−0.0268637	−0.0134319	−	0.999910i	$-0.504276\pi$
$461$	−0.576789	−0.0268637	−0.0134319	+	0.999910i	$0.504276\pi$
$462$	−8.12755	−0.378128
$463$	−21.3605	−0.992706	−0.496353	−	0.868121i	$-0.665328\pi$
$463$	−21.3605	−0.992706	−0.496353	+	0.868121i	$0.665328\pi$
$464$	71.9052	3.33811
$465$	−1.00000	−0.0463739
$466$	36.3541	1.68407
$467$	−7.03828	−0.325693	−0.162846	−	0.986651i	$-0.552068\pi$
$467$	−7.03828	−0.325693	−0.162846	+	0.986651i	$0.552068\pi$
$468$	4.55168	0.210402
$469$	−5.87324	−0.271201
$470$	5.30194	0.244560
$471$	−16.4605	−0.758458
$472$	91.5366	4.21331
$473$	2.17511	0.100012
$474$	−32.3218	−1.48459
$475$	5.10940	0.234435
$476$	6.11941	0.280483
$477$	−5.70075	−0.261020
$478$	−49.1864	−2.24973
$479$	−7.00514	−0.320073	−0.160036	−	0.987111i	$-0.551161\pi$
$479$	−7.00514	−0.320073	−0.160036	+	0.987111i	$0.551161\pi$
$480$	−6.42741	−0.293370
$481$	8.32001	0.379360
$482$	9.64926	0.439512
$483$	0.727686	0.0331109
$484$	−45.5888	−2.07222
$485$	9.76108	0.443228
$486$	2.55963	0.116107
$487$	−35.3491	−1.60182	−0.800911	−	0.598784i	$-0.795651\pi$
$487$	−35.3491	−1.60182	−0.800911	+	0.598784i	$0.795651\pi$
$488$	62.4366	2.82637
$489$	−20.7891	−0.940114
$490$	−8.30494	−0.375179
$491$	−6.79228	−0.306531	−0.153266	−	0.988185i	$-0.548979\pi$
$491$	−6.79228	−0.306531	−0.153266	+	0.988185i	$0.548979\pi$
$492$	12.5198	0.564438
$493$	−3.96655	−0.178644
$494$	13.0781	0.588414
$495$	−0.992055	−0.0445896
$496$	7.61443	0.341898
$497$	−24.8353	−1.11402
$498$	−18.8150	−0.843120
$499$	−10.3806	−0.464701	−0.232350	−	0.972632i	$-0.574642\pi$
$499$	−10.3806	−0.464701	−0.232350	+	0.972632i	$0.574642\pi$
$500$	−4.55168	−0.203557
$501$	0.438068	0.0195714
$502$	−49.1782	−2.19493
$503$	19.1211	0.852567	0.426284	−	0.904590i	$-0.359823\pi$
$503$	19.1211	0.852567	0.426284	+	0.904590i	$0.359823\pi$
$504$	−20.9050	−0.931182
$505$	−10.0532	−0.447360
$506$	−0.577310	−0.0256646
$507$	1.00000	0.0444116
$508$	82.2273	3.64825
$509$	−9.01904	−0.399762	−0.199881	−	0.979820i	$-0.564056\pi$
$509$	−9.01904	−0.399762	−0.199881	+	0.979820i	$0.564056\pi$
$510$	1.07514	0.0476082
$511$	−0.362231	−0.0160241
$512$	−50.5239	−2.23286
$513$	5.10940	0.225586
$514$	5.21243	0.229911
$515$	2.77892	0.122454
$516$	9.97970	0.439332
$517$	−2.05492	−0.0903752
$518$	−68.1628	−2.99490
$519$	−13.8954	−0.609939
$520$	−6.53135	−0.286418
$521$	43.0867	1.88766	0.943831	−	0.330428i	$-0.107193\pi$
$521$	43.0867	1.88766	0.943831	+	0.330428i	$0.107193\pi$
$522$	24.1712	1.05795
$523$	−4.29109	−0.187636	−0.0938181	−	0.995589i	$-0.529907\pi$
$523$	−4.29109	−0.187636	−0.0938181	+	0.995589i	$0.529907\pi$
$524$	1.47261	0.0643312
$525$	−3.20072	−0.139691
$526$	−26.4718	−1.15422
$527$	−0.420040	−0.0182972
$528$	7.55394	0.328743
$529$	−22.9483	−0.997753
$530$	14.5918	0.633827
$531$	14.0150	0.608198
$532$	−74.4370	−3.22725
$533$	2.75060	0.119142
$534$	−3.68226	−0.159347
$535$	4.87103	0.210593
$536$	11.9849	0.517667
$537$	0.537343	0.0231880
$538$	−78.5281	−3.38558
$539$	3.21881	0.138644
$540$	−4.55168	−0.195873
$541$	8.47171	0.364227	0.182114	−	0.983277i	$-0.441706\pi$
$541$	8.47171	0.364227	0.182114	+	0.983277i	$0.441706\pi$
$542$	3.48645	0.149756
$543$	−14.7335	−0.632273
$544$	−2.69977	−0.115752
$545$	9.74763	0.417543
$546$	−8.19264	−0.350612
$547$	−44.5217	−1.90361	−0.951805	−	0.306703i	$-0.900774\pi$
$547$	−44.5217	−1.90361	−0.951805	+	0.306703i	$0.900774\pi$
$548$	24.3375	1.03964
$549$	9.55954	0.407991
$550$	2.53929	0.108276
$551$	48.2494	2.05550
$552$	−1.48491	−0.0632019
$553$	40.4172	1.71871
$554$	17.0732	0.725373
$555$	−8.32001	−0.353165
$556$	52.8364	2.24076
$557$	−25.5303	−1.08175	−0.540877	−	0.841102i	$-0.681907\pi$
$557$	−25.5303	−1.08175	−0.540877	+	0.841102i	$0.681907\pi$
$558$	2.55963	0.108358
$559$	2.19253	0.0927341
$560$	24.3717	1.02989
$561$	−0.416703	−0.0175932
$562$	−22.2377	−0.938043
$563$	28.1906	1.18809	0.594045	−	0.804432i	$-0.297530\pi$
$563$	28.1906	1.18809	0.594045	+	0.804432i	$0.297530\pi$
$564$	−9.42824	−0.397000
$565$	16.1736	0.680430
$566$	9.51373	0.399892
$567$	−3.20072	−0.134417
$568$	50.6787	2.12643
$569$	−9.76969	−0.409567	−0.204783	−	0.978807i	$-0.565649\pi$
$569$	−9.76969	−0.409567	−0.204783	+	0.978807i	$0.565649\pi$
$570$	−13.0781	−0.547783
$571$	28.0447	1.17364	0.586818	−	0.809719i	$-0.300380\pi$
$571$	28.0447	1.17364	0.586818	+	0.809719i	$0.300380\pi$
$572$	4.51552	0.188803
$573$	3.64522	0.152281
$574$	−22.5346	−0.940578
$575$	−0.227351	−0.00948119
$576$	1.22288	0.0509534
$577$	−31.1263	−1.29580	−0.647902	−	0.761724i	$-0.724353\pi$
$577$	−31.1263	−1.29580	−0.647902	+	0.761724i	$0.724353\pi$
$578$	−43.0620	−1.79114
$579$	12.0225	0.499640
$580$	−42.9827	−1.78476
$581$	23.5275	0.976084
$582$	−24.9847	−1.03565
$583$	−5.65546	−0.234225
$584$	0.739163	0.0305868
$585$	−1.00000	−0.0413449
$586$	50.0150	2.06610
$587$	45.8821	1.89376	0.946878	−	0.321593i	$-0.104218\pi$
$587$	45.8821	1.89376	0.946878	+	0.321593i	$0.104218\pi$
$588$	14.7683	0.609036
$589$	5.10940	0.210529
$590$	−35.8731	−1.47687
$591$	−15.2124	−0.625755
$592$	63.3521	2.60376
$593$	3.53014	0.144966	0.0724828	−	0.997370i	$-0.476908\pi$
$593$	3.53014	0.144966	0.0724828	+	0.997370i	$0.476908\pi$
$594$	2.53929	0.104188
$595$	−1.34443	−0.0551162
$596$	18.1182	0.742149
$597$	26.2872	1.07586
$598$	−0.581933	−0.0237970
$599$	−40.3885	−1.65023	−0.825115	−	0.564965i	$-0.808890\pi$
$599$	−40.3885	−1.65023	−0.825115	+	0.564965i	$0.808890\pi$
$600$	6.53135	0.266641
$601$	−36.9404	−1.50683	−0.753415	−	0.657545i	$-0.771595\pi$
$601$	−36.9404	−1.50683	−0.753415	+	0.657545i	$0.771595\pi$
$602$	−17.9626	−0.732101
$603$	1.83497	0.0747260
$604$	−100.969	−4.10836
$605$	10.0158	0.407201
$606$	25.7323	1.04530
$607$	12.5550	0.509592	0.254796	−	0.966995i	$-0.417992\pi$
$607$	12.5550	0.509592	0.254796	+	0.966995i	$0.417992\pi$
$608$	32.8402	1.33185
$609$	−30.2252	−1.22479
$610$	−24.4688	−0.990714
$611$	−2.07137	−0.0837989
$612$	−1.91189	−0.0772835
$613$	−10.0335	−0.405251	−0.202625	−	0.979256i	$-0.564947\pi$
$613$	−10.0335	−0.405251	−0.202625	+	0.979256i	$0.564947\pi$
$614$	−8.97276	−0.362111
$615$	−2.75060	−0.110915
$616$	−20.7389	−0.835594
$617$	−19.5937	−0.788812	−0.394406	−	0.918936i	$-0.629050\pi$
$617$	−19.5937	−0.788812	−0.394406	+	0.918936i	$0.629050\pi$
$618$	−7.11300	−0.286127
$619$	−5.98607	−0.240600	−0.120300	−	0.992738i	$-0.538386\pi$
$619$	−5.98607	−0.240600	−0.120300	+	0.992738i	$0.538386\pi$
$620$	−4.55168	−0.182800
$621$	−0.227351	−0.00912328
$622$	27.4421	1.10033
$623$	4.60454	0.184477
$624$	7.61443	0.304821
$625$	1.00000	0.0400000
$626$	−24.1822	−0.966516
$627$	5.06881	0.202429
$628$	−74.9227	−2.98974
$629$	−3.49473	−0.139344
$630$	8.19264	0.326402
$631$	−13.7344	−0.546760	−0.273380	−	0.961906i	$-0.588142\pi$
$631$	−13.7344	−0.546760	−0.273380	+	0.961906i	$0.588142\pi$
$632$	−82.4748	−3.28067
$633$	18.3246	0.728337
$634$	25.9454	1.03043
$635$	−18.0653	−0.716898
$636$	−25.9480	−1.02891
$637$	3.24459	0.128555
$638$	23.9792	0.949346
$639$	7.75930	0.306953
$640$	9.72469	0.384402
$641$	−20.6354	−0.815050	−0.407525	−	0.913194i	$-0.633608\pi$
$641$	−20.6354	−0.815050	−0.407525	+	0.913194i	$0.633608\pi$
$642$	−12.4680	−0.492073
$643$	27.2980	1.07653	0.538264	−	0.842776i	$-0.319080\pi$
$643$	27.2980	1.07653	0.538264	+	0.842776i	$0.319080\pi$
$644$	3.31219	0.130519
$645$	−2.19253	−0.0863308
$646$	−5.49334	−0.216133
$647$	−34.2184	−1.34526	−0.672632	−	0.739977i	$-0.734836\pi$
$647$	−34.2184	−1.34526	−0.672632	+	0.739977i	$0.734836\pi$
$648$	6.53135	0.256575
$649$	13.9036	0.545765
$650$	2.55963	0.100397
$651$	−3.20072	−0.125446
$652$	−94.6251	−3.70581
$653$	−16.6134	−0.650132	−0.325066	−	0.945691i	$-0.605387\pi$
$653$	−16.6134	−0.650132	−0.325066	+	0.945691i	$0.605387\pi$
$654$	−24.9503	−0.975633
$655$	−0.323531	−0.0126414
$656$	20.9442	0.817735
$657$	0.113172	0.00441525
$658$	16.9700	0.661560
$659$	−28.7124	−1.11848	−0.559239	−	0.829006i	$-0.688907\pi$
$659$	−28.7124	−1.11848	−0.559239	+	0.829006i	$0.688907\pi$
$660$	−4.51552	−0.175766
$661$	−26.3952	−1.02666	−0.513328	−	0.858193i	$-0.671588\pi$
$661$	−26.3952	−1.02666	−0.513328	+	0.858193i	$0.671588\pi$
$662$	71.1815	2.76655
$663$	−0.420040	−0.0163130
$664$	−48.0099	−1.86314
$665$	16.3537	0.634171
$666$	21.2961	0.825207
$667$	−2.14694	−0.0831297
$668$	1.99395	0.0771481
$669$	9.00331	0.348088
$670$	−4.69685	−0.181455
$671$	9.48359	0.366110
$672$	−20.5723	−0.793594
$673$	18.4088	0.709607	0.354803	−	0.934941i	$-0.384548\pi$
$673$	18.4088	0.709607	0.354803	+	0.934941i	$0.384548\pi$
$674$	54.2090	2.08805
$675$	1.00000	0.0384900
$676$	4.55168	0.175065
$677$	−18.5534	−0.713067	−0.356533	−	0.934283i	$-0.616041\pi$
$677$	−18.5534	−0.713067	−0.356533	+	0.934283i	$0.616041\pi$
$678$	−41.3985	−1.58990
$679$	31.2425	1.19898
$680$	2.74343	0.105206
$681$	7.64102	0.292804
$682$	2.53929	0.0972344
$683$	32.8077	1.25535	0.627675	−	0.778476i	$-0.284007\pi$
$683$	32.8077	1.25535	0.627675	+	0.778476i	$0.284007\pi$
$684$	23.2564	0.889229
$685$	−5.34692	−0.204295
$686$	30.7667	1.17468
$687$	0.411746	0.0157091
$688$	16.6949	0.636486
$689$	−5.70075	−0.217181
$690$	0.581933	0.0221538
$691$	7.93942	0.302030	0.151015	−	0.988531i	$-0.451746\pi$
$691$	7.93942	0.302030	0.151015	+	0.988531i	$0.451746\pi$
$692$	−63.2473	−2.40430
$693$	−3.17529	−0.120619
$694$	−91.7099	−3.48126
$695$	−11.6081	−0.440321
$696$	61.6773	2.33787
$697$	−1.15536	−0.0437624
$698$	26.0142	0.984653
$699$	14.2029	0.537203
$700$	−14.5686	−0.550643
$701$	3.05128	0.115245	0.0576227	−	0.998338i	$-0.481648\pi$
$701$	3.05128	0.115245	0.0576227	+	0.998338i	$0.481648\pi$
$702$	2.55963	0.0966068
$703$	42.5102	1.60330
$704$	1.21317	0.0457229
$705$	2.07137	0.0780125
$706$	84.1508	3.16706
$707$	−32.1773	−1.21015
$708$	63.7916	2.39744
$709$	−13.7324	−0.515729	−0.257865	−	0.966181i	$-0.583019\pi$
$709$	−13.7324	−0.515729	−0.257865	+	0.966181i	$0.583019\pi$
$710$	−19.8609	−0.745366
$711$	−12.6275	−0.473570
$712$	−9.39596	−0.352129
$713$	−0.227351	−0.00851436
$714$	3.44123	0.128785
$715$	−0.992055	−0.0371008
$716$	2.44581	0.0914043
$717$	−19.2162	−0.717644
$718$	−50.3117	−1.87761
$719$	−47.1719	−1.75921	−0.879607	−	0.475701i	$-0.842194\pi$
$719$	−47.1719	−1.75921	−0.879607	+	0.475701i	$0.842194\pi$
$720$	−7.61443	−0.283773
$721$	8.89454	0.331250
$722$	18.1886	0.676910
$723$	3.76980	0.140200
$724$	−67.0620	−2.49234
$725$	9.44327	0.350714
$726$	−25.6368	−0.951469
$727$	−21.0419	−0.780402	−0.390201	−	0.920730i	$-0.627594\pi$
$727$	−21.0419	−0.780402	−0.390201	+	0.920730i	$0.627594\pi$
$728$	−20.9050	−0.774791
$729$	1.00000	0.0370370
$730$	−0.289677	−0.0107214
$731$	−0.920950	−0.0340626
$732$	43.5120	1.60825
$733$	19.0048	0.701959	0.350979	−	0.936383i	$-0.385849\pi$
$733$	19.0048	0.701959	0.350979	+	0.936383i	$0.385849\pi$
$734$	46.7273	1.72474
$735$	−3.24459	−0.119679
$736$	−1.46128	−0.0538634
$737$	1.82040	0.0670552
$738$	7.04050	0.259164
$739$	−18.6402	−0.685691	−0.342845	−	0.939392i	$-0.611391\pi$
$739$	−18.6402	−0.685691	−0.342845	+	0.939392i	$0.611391\pi$
$740$	−37.8700	−1.39213
$741$	5.10940	0.187699
$742$	46.7042	1.71456
$743$	−1.50831	−0.0553345	−0.0276672	−	0.999617i	$-0.508808\pi$
$743$	−1.50831	−0.0553345	−0.0276672	+	0.999617i	$0.508808\pi$
$744$	6.53135	0.239451
$745$	−3.98054	−0.145836
$746$	−87.2950	−3.19610
$747$	−7.35068	−0.268947
$748$	−1.89670	−0.0693501
$749$	15.5908	0.569675
$750$	−2.55963	−0.0934643
$751$	27.9896	1.02135	0.510677	−	0.859773i	$-0.329395\pi$
$751$	27.9896	1.02135	0.510677	+	0.859773i	$0.329395\pi$
$752$	−15.7723	−0.575158
$753$	−19.2130	−0.700162
$754$	24.1712	0.880264
$755$	22.1827	0.807313
$756$	−14.5686	−0.529856
$757$	−34.4461	−1.25196	−0.625982	−	0.779838i	$-0.715302\pi$
$757$	−34.4461	−1.25196	−0.625982	+	0.779838i	$0.715302\pi$
$758$	−32.4814	−1.17978
$759$	−0.225545	−0.00818676
$760$	−33.3713	−1.21050
$761$	−4.16404	−0.150946	−0.0754732	−	0.997148i	$-0.524047\pi$
$761$	−4.16404	−0.150946	−0.0754732	+	0.997148i	$0.524047\pi$
$762$	46.2403	1.67511
$763$	31.1994	1.12949
$764$	16.5919	0.600273
$765$	0.420040	0.0151866
$766$	−43.2237	−1.56174
$767$	14.0150	0.506051
$768$	−27.3373	−0.986451
$769$	45.9812	1.65813	0.829063	−	0.559155i	$-0.188874\pi$
$769$	45.9812	1.65813	0.829063	+	0.559155i	$0.188874\pi$
$770$	8.12755	0.292897
$771$	2.03640	0.0733393
$772$	54.7228	1.96952
$773$	−40.6899	−1.46351	−0.731757	−	0.681566i	$-0.761299\pi$
$773$	−40.6899	−1.46351	−0.731757	+	0.681566i	$0.761299\pi$
$774$	5.61205	0.201721
$775$	1.00000	0.0359211
$776$	−63.7530	−2.28860
$777$	−26.6300	−0.955345
$778$	1.07429	0.0385152
$779$	14.0539	0.503533
$780$	−4.55168	−0.162976
$781$	7.69766	0.275444
$782$	0.244435	0.00874098
$783$	9.44327	0.337475
$784$	24.7057	0.882348
$785$	16.4605	0.587499
$786$	0.828117	0.0295380
$787$	−20.2876	−0.723174	−0.361587	−	0.932338i	$-0.617765\pi$
$787$	−20.2876	−0.723174	−0.361587	+	0.932338i	$0.617765\pi$
$788$	−69.2420	−2.46664
$789$	−10.3420	−0.368186
$790$	32.3218	1.14996
$791$	51.7673	1.84063
$792$	6.47946	0.230237
$793$	9.55954	0.339469
$794$	−80.3640	−2.85201
$795$	5.70075	0.202185
$796$	119.651	4.24091
$797$	11.4623	0.406017	0.203008	−	0.979177i	$-0.434928\pi$
$797$	11.4623	0.406017	0.203008	+	0.979177i	$0.434928\pi$
$798$	−41.8595	−1.48181
$799$	0.870060	0.0307805
$800$	6.42741	0.227243
$801$	−1.43859	−0.0508303
$802$	52.4523	1.85215
$803$	0.112273	0.00396201
$804$	8.35222	0.294560
$805$	−0.727686	−0.0256476
$806$	2.55963	0.0901589
$807$	−30.6795	−1.07997
$808$	65.6607	2.30993
$809$	17.7881	0.625395	0.312698	−	0.949853i	$-0.398767\pi$
$809$	17.7881	0.625395	0.312698	+	0.949853i	$0.398767\pi$
$810$	−2.55963	−0.0899361
$811$	7.51848	0.264010	0.132005	−	0.991249i	$-0.457859\pi$
$811$	7.51848	0.264010	0.132005	+	0.991249i	$0.457859\pi$
$812$	−137.576	−4.82796
$813$	1.36209	0.0477707
$814$	21.1269	0.740498
$815$	20.7891	0.728209
$816$	−3.19837	−0.111965
$817$	11.2025	0.391926
$818$	41.8699	1.46395
$819$	−3.20072	−0.111842
$820$	−12.5198	−0.437212
$821$	25.6035	0.893569	0.446784	−	0.894642i	$-0.352569\pi$
$821$	25.6035	0.893569	0.446784	+	0.894642i	$0.352569\pi$
$822$	13.6861	0.477358
$823$	0.879712	0.0306648	0.0153324	−	0.999882i	$-0.495119\pi$
$823$	0.879712	0.0306648	0.0153324	+	0.999882i	$0.495119\pi$
$824$	−18.1501	−0.632288
$825$	0.992055	0.0345389
$826$	−114.820	−3.99508
$827$	−4.70346	−0.163555	−0.0817776	−	0.996651i	$-0.526060\pi$
$827$	−4.70346	−0.163555	−0.0817776	+	0.996651i	$0.526060\pi$
$828$	−1.03483	−0.0359628
$829$	22.6384	0.786265	0.393133	−	0.919482i	$-0.371391\pi$
$829$	22.6384	0.786265	0.393133	+	0.919482i	$0.371391\pi$
$830$	18.8150	0.653078
$831$	6.67022	0.231387
$832$	1.22288	0.0423958
$833$	−1.36286	−0.0472202
$834$	29.7124	1.02886
$835$	−0.438068	−0.0151600
$836$	23.0716	0.797948
$837$	1.00000	0.0345651
$838$	−14.2061	−0.490742
$839$	22.0917	0.762692	0.381346	−	0.924432i	$-0.375461\pi$
$839$	22.0917	0.762692	0.381346	+	0.924432i	$0.375461\pi$
$840$	20.9050	0.721291
$841$	60.1753	2.07501
$842$	−56.0038	−1.93002
$843$	−8.68789	−0.299227
$844$	83.4077	2.87101
$845$	−1.00000	−0.0344010
$846$	−5.30194	−0.182285
$847$	32.0578	1.10152
$848$	−43.4080	−1.49064
$849$	3.71684	0.127562
$850$	−1.07514	−0.0368772
$851$	−1.89156	−0.0648419
$852$	35.3179	1.20997
$853$	−45.7421	−1.56618	−0.783089	−	0.621909i	$-0.786357\pi$
$853$	−45.7421	−1.56618	−0.783089	+	0.621909i	$0.786357\pi$
$854$	−78.3178	−2.67998
$855$	−5.10940	−0.174738
$856$	−31.8144	−1.08739
$857$	9.85690	0.336705	0.168353	−	0.985727i	$-0.446155\pi$
$857$	9.85690	0.336705	0.168353	+	0.985727i	$0.446155\pi$
$858$	2.53929	0.0866899
$859$	−37.5776	−1.28213	−0.641066	−	0.767486i	$-0.721508\pi$
$859$	−37.5776	−1.28213	−0.641066	+	0.767486i	$0.721508\pi$
$860$	−9.97970	−0.340305
$861$	−8.80388	−0.300036
$862$	94.7751	3.22805
$863$	17.6833	0.601946	0.300973	−	0.953633i	$-0.402689\pi$
$863$	17.6833	0.601946	0.300973	+	0.953633i	$0.402689\pi$
$864$	6.42741	0.218665
$865$	13.8954	0.472457
$866$	−39.9731	−1.35834
$867$	−16.8236	−0.571358
$868$	−14.5686	−0.494492
$869$	−12.5272	−0.424957
$870$	−24.1712	−0.819481
$871$	1.83497	0.0621758
$872$	−63.6652	−2.15597
$873$	−9.76108	−0.330362
$874$	−2.97333	−0.100574
$875$	3.20072	0.108204
$876$	0.515121	0.0174043
$877$	22.5892	0.762783	0.381391	−	0.924414i	$-0.375445\pi$
$877$	22.5892	0.762783	0.381391	+	0.924414i	$0.375445\pi$
$878$	69.5709	2.34790
$879$	19.5400	0.659066
$880$	−7.55394	−0.254643
$881$	−14.1836	−0.477859	−0.238930	−	0.971037i	$-0.576796\pi$
$881$	−14.1836	−0.477859	−0.238930	+	0.971037i	$0.576796\pi$
$882$	8.30494	0.279642
$883$	38.1058	1.28236	0.641181	−	0.767390i	$-0.278445\pi$
$883$	38.1058	1.28236	0.641181	+	0.767390i	$0.278445\pi$
$884$	−1.91189	−0.0643037
$885$	−14.0150	−0.471108
$886$	57.6177	1.93570
$887$	4.12613	0.138542	0.0692710	−	0.997598i	$-0.477933\pi$
$887$	4.12613	0.138542	0.0692710	+	0.997598i	$0.477933\pi$
$888$	54.3408	1.82356
$889$	−57.8218	−1.93928
$890$	3.68226	0.123430
$891$	0.992055	0.0332351
$892$	40.9802	1.37212
$893$	−10.5835	−0.354163
$894$	10.1887	0.340761
$895$	−0.537343	−0.0179614
$896$	31.1260	1.03985
$897$	−0.227351	−0.00759103
$898$	81.4904	2.71937
$899$	9.44327	0.314951
$900$	4.55168	0.151723
$901$	2.39454	0.0797738
$902$	6.98456	0.232561
$903$	−7.01767	−0.233533
$904$	−105.636	−3.51339
$905$	14.7335	0.489757
$906$	−56.7795	−1.88637
$907$	24.3766	0.809412	0.404706	−	0.914447i	$-0.367374\pi$
$907$	24.3766	0.809412	0.404706	+	0.914447i	$0.367374\pi$
$908$	34.7795	1.15420
$909$	10.0532	0.333442
$910$	8.19264	0.271583
$911$	1.27263	0.0421640	0.0210820	−	0.999778i	$-0.493289\pi$
$911$	1.27263	0.0421640	0.0210820	+	0.999778i	$0.493289\pi$
$912$	38.9052	1.28828
$913$	−7.29229	−0.241339
$914$	18.6362	0.616432
$915$	−9.55954	−0.316029
$916$	1.87413	0.0619231
$917$	−1.03553	−0.0341962
$918$	−1.07514	−0.0354851
$919$	−17.4268	−0.574858	−0.287429	−	0.957802i	$-0.592801\pi$
$919$	−17.4268	−0.574858	−0.287429	+	0.957802i	$0.592801\pi$
$920$	1.48491	0.0489560
$921$	−3.50550	−0.115510
$922$	−1.47636	−0.0486214
$923$	7.75930	0.255400
$924$	−14.4529	−0.475466
$925$	8.32001	0.273560
$926$	−54.6748	−1.79673
$927$	−2.77892	−0.0912717
$928$	60.6957	1.99244
$929$	34.2734	1.12447	0.562237	−	0.826976i	$-0.309941\pi$
$929$	34.2734	1.12447	0.562237	+	0.826976i	$0.309941\pi$
$930$	−2.55963	−0.0839334
$931$	16.5779	0.543319
$932$	64.6471	2.11759
$933$	10.7211	0.350994
$934$	−18.0154	−0.589481
$935$	0.416703	0.0136276
$936$	6.53135	0.213484
$937$	−34.4868	−1.12664	−0.563318	−	0.826240i	$-0.690475\pi$
$937$	−34.4868	−1.12664	−0.563318	+	0.826240i	$0.690475\pi$
$938$	−15.0333	−0.490854
$939$	−9.44756	−0.308310
$940$	9.42824	0.307515
$941$	−0.246490	−0.00803534	−0.00401767	−	0.999992i	$-0.501279\pi$
$941$	−0.246490	−0.00803534	−0.00401767	+	0.999992i	$0.501279\pi$
$942$	−42.1326	−1.37275
$943$	−0.625351	−0.0203642
$944$	106.716	3.47331
$945$	3.20072	0.104119
$946$	5.56747	0.181014
$947$	52.1782	1.69557	0.847783	−	0.530344i	$-0.177937\pi$
$947$	52.1782	1.69557	0.847783	+	0.530344i	$0.177937\pi$
$948$	−57.4765	−1.86675
$949$	0.113172	0.00367371
$950$	13.0781	0.424311
$951$	10.1364	0.328696
$952$	8.78093	0.284592
$953$	12.2949	0.398270	0.199135	−	0.979972i	$-0.436187\pi$
$953$	12.2949	0.398270	0.199135	+	0.979972i	$0.436187\pi$
$954$	−14.5918	−0.472427
$955$	−3.64522	−0.117956
$956$	−87.4662	−2.82886
$957$	9.36825	0.302832
$958$	−17.9305	−0.579309
$959$	−17.1140	−0.552639
$960$	−1.22288	−0.0394683
$961$	1.00000	0.0322581
$962$	21.2961	0.686614
$963$	−4.87103	−0.156967
$964$	17.1589	0.552651
$965$	−12.0225	−0.387019
$966$	1.86260	0.0599283
$967$	22.2800	0.716476	0.358238	−	0.933630i	$-0.383378\pi$
$967$	22.2800	0.716476	0.358238	+	0.933630i	$0.383378\pi$
$968$	−65.4168	−2.10258
$969$	−2.14615	−0.0689443
$970$	24.9847	0.802210
$971$	32.2886	1.03619	0.518095	−	0.855323i	$-0.326642\pi$
$971$	32.2886	1.03619	0.518095	+	0.855323i	$0.326642\pi$
$972$	4.55168	0.145995
$973$	−37.1543	−1.19111
$974$	−90.4805	−2.89918
$975$	1.00000	0.0320256
$976$	72.7905	2.32997
$977$	26.5408	0.849115	0.424558	−	0.905401i	$-0.360430\pi$
$977$	26.5408	0.849115	0.424558	+	0.905401i	$0.360430\pi$
$978$	−53.2122	−1.70154
$979$	−1.42717	−0.0456124
$980$	−14.7683	−0.471757
$981$	−9.74763	−0.311218
$982$	−17.3857	−0.554800
$983$	42.9597	1.37020	0.685101	−	0.728448i	$-0.259758\pi$
$983$	42.9597	1.37020	0.685101	+	0.728448i	$0.259758\pi$
$984$	17.9651	0.572706
$985$	15.2124	0.484707
$986$	−10.1529	−0.323334
$987$	6.62989	0.211032
$988$	23.2564	0.739883
$989$	−0.498474	−0.0158505
$990$	−2.53929	−0.0807039
$991$	−54.4440	−1.72947	−0.864736	−	0.502227i	$-0.832514\pi$
$991$	−54.4440	−1.72947	−0.864736	+	0.502227i	$0.832514\pi$
$992$	6.42741	0.204070
$993$	27.8093	0.882503
$994$	−63.5691	−2.01629
$995$	−26.2872	−0.833359
$996$	−33.4580	−1.06016
$997$	−1.91641	−0.0606934	−0.0303467	−	0.999539i	$-0.509661\pi$
$997$	−1.91641	−0.0606934	−0.0303467	+	0.999539i	$0.509661\pi$
$998$	−26.5705	−0.841075
$999$	8.32001	0.263233

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.v.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.v.1.9	✓	10	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)

\(f(q)\)	\(=\)	\(q+2.55963 q^{2} +1.00000 q^{3} +4.55168 q^{4} -1.00000 q^{5} +2.55963 q^{6} -3.20072 q^{7} +6.53135 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.55963 q^{2} +1.00000 q^{3} +4.55168 q^{4} -1.00000 q^{5} +2.55963 q^{6} -3.20072 q^{7} +6.53135 q^{8} +1.00000 q^{9} -2.55963 q^{10} +0.992055 q^{11} +4.55168 q^{12} +1.00000 q^{13} -8.19264 q^{14} -1.00000 q^{15} +7.61443 q^{16} -0.420040 q^{17} +2.55963 q^{18} +5.10940 q^{19} -4.55168 q^{20} -3.20072 q^{21} +2.53929 q^{22} -0.227351 q^{23} +6.53135 q^{24} +1.00000 q^{25} +2.55963 q^{26} +1.00000 q^{27} -14.5686 q^{28} +9.44327 q^{29} -2.55963 q^{30} +1.00000 q^{31} +6.42741 q^{32} +0.992055 q^{33} -1.07514 q^{34} +3.20072 q^{35} +4.55168 q^{36} +8.32001 q^{37} +13.0781 q^{38} +1.00000 q^{39} -6.53135 q^{40} +2.75060 q^{41} -8.19264 q^{42} +2.19253 q^{43} +4.51552 q^{44} -1.00000 q^{45} -0.581933 q^{46} -2.07137 q^{47} +7.61443 q^{48} +3.24459 q^{49} +2.55963 q^{50} -0.420040 q^{51} +4.55168 q^{52} -5.70075 q^{53} +2.55963 q^{54} -0.992055 q^{55} -20.9050 q^{56} +5.10940 q^{57} +24.1712 q^{58} +14.0150 q^{59} -4.55168 q^{60} +9.55954 q^{61} +2.55963 q^{62} -3.20072 q^{63} +1.22288 q^{64} -1.00000 q^{65} +2.53929 q^{66} +1.83497 q^{67} -1.91189 q^{68} -0.227351 q^{69} +8.19264 q^{70} +7.75930 q^{71} +6.53135 q^{72} +0.113172 q^{73} +21.2961 q^{74} +1.00000 q^{75} +23.2564 q^{76} -3.17529 q^{77} +2.55963 q^{78} -12.6275 q^{79} -7.61443 q^{80} +1.00000 q^{81} +7.04050 q^{82} -7.35068 q^{83} -14.5686 q^{84} +0.420040 q^{85} +5.61205 q^{86} +9.44327 q^{87} +6.47946 q^{88} -1.43859 q^{89} -2.55963 q^{90} -3.20072 q^{91} -1.03483 q^{92} +1.00000 q^{93} -5.30194 q^{94} -5.10940 q^{95} +6.42741 q^{96} -9.76108 q^{97} +8.30494 q^{98} +0.992055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 10 q^{3} + 20 q^{4} - 10 q^{5} + 4 q^{6} - q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q + 4 * q^2 + 10 * q^3 + 20 * q^4 - 10 * q^5 + 4 * q^6 - q^7 + 15 * q^8 + 10 * q^9	\( 10 q + 4 q^{2} + 10 q^{3} + 20 q^{4} - 10 q^{5} + 4 q^{6} - q^{7} + 15 q^{8} + 10 q^{9} - 4 q^{10} + 6 q^{11} + 20 q^{12} + 10 q^{13} + 11 q^{14} - 10 q^{15} + 16 q^{16} - q^{17} + 4 q^{18} + 2 q^{19} - 20 q^{20} - q^{21} - 21 q^{22} + q^{23} + 15 q^{24} + 10 q^{25} + 4 q^{26} + 10 q^{27} + 10 q^{28} + 15 q^{29} - 4 q^{30} + 10 q^{31} + 34 q^{32} + 6 q^{33} + 3 q^{34} + q^{35} + 20 q^{36} - 3 q^{37} - 6 q^{38} + 10 q^{39} - 15 q^{40} + 43 q^{41} + 11 q^{42} - 8 q^{43} + 53 q^{44} - 10 q^{45} + 22 q^{46} - 11 q^{47} + 16 q^{48} + 31 q^{49} + 4 q^{50} - q^{51} + 20 q^{52} + 10 q^{53} + 4 q^{54} - 6 q^{55} + 17 q^{56} + 2 q^{57} - 16 q^{58} + 48 q^{59} - 20 q^{60} + 6 q^{61} + 4 q^{62} - q^{63} + 29 q^{64} - 10 q^{65} - 21 q^{66} - 16 q^{67} + 19 q^{68} + q^{69} - 11 q^{70} + 53 q^{71} + 15 q^{72} + 7 q^{73} + 24 q^{74} + 10 q^{75} + 30 q^{76} + 4 q^{78} - q^{79} - 16 q^{80} + 10 q^{81} + 53 q^{82} + 41 q^{83} + 10 q^{84} + q^{85} + 14 q^{86} + 15 q^{87} + 9 q^{88} + 31 q^{89} - 4 q^{90} - q^{91} + 23 q^{92} + 10 q^{93} - 18 q^{94} - 2 q^{95} + 34 q^{96} - 37 q^{97} + 28 q^{98} + 6 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 10 * q^3 + 20 * q^4 - 10 * q^5 + 4 * q^6 - q^7 + 15 * q^8 + 10 * q^9 - 4 * q^10 + 6 * q^11 + 20 * q^12 + 10 * q^13 + 11 * q^14 - 10 * q^15 + 16 * q^16 - q^17 + 4 * q^18 + 2 * q^19 - 20 * q^20 - q^21 - 21 * q^22 + q^23 + 15 * q^24 + 10 * q^25 + 4 * q^26 + 10 * q^27 + 10 * q^28 + 15 * q^29 - 4 * q^30 + 10 * q^31 + 34 * q^32 + 6 * q^33 + 3 * q^34 + q^35 + 20 * q^36 - 3 * q^37 - 6 * q^38 + 10 * q^39 - 15 * q^40 + 43 * q^41 + 11 * q^42 - 8 * q^43 + 53 * q^44 - 10 * q^45 + 22 * q^46 - 11 * q^47 + 16 * q^48 + 31 * q^49 + 4 * q^50 - q^51 + 20 * q^52 + 10 * q^53 + 4 * q^54 - 6 * q^55 + 17 * q^56 + 2 * q^57 - 16 * q^58 + 48 * q^59 - 20 * q^60 + 6 * q^61 + 4 * q^62 - q^63 + 29 * q^64 - 10 * q^65 - 21 * q^66 - 16 * q^67 + 19 * q^68 + q^69 - 11 * q^70 + 53 * q^71 + 15 * q^72 + 7 * q^73 + 24 * q^74 + 10 * q^75 + 30 * q^76 + 4 * q^78 - q^79 - 16 * q^80 + 10 * q^81 + 53 * q^82 + 41 * q^83 + 10 * q^84 + q^85 + 14 * q^86 + 15 * q^87 + 9 * q^88 + 31 * q^89 - 4 * q^90 - q^91 + 23 * q^92 + 10 * q^93 - 18 * q^94 - 2 * q^95 + 34 * q^96 - 37 * q^97 + 28 * q^98 + 6 * q^99

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