LMFDB - Embedded newform 6045.2.a.bg.1.5

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.69426$ 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)

$f(q)$	$=$	$q-1.69426 q^{2} -1.00000 q^{3} +0.870532 q^{4} -1.00000 q^{5} +1.69426 q^{6} +0.135304 q^{7} +1.91362 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.69426 q^{2} -1.00000 q^{3} +0.870532 q^{4} -1.00000 q^{5} +1.69426 q^{6} +0.135304 q^{7} +1.91362 q^{8} +1.00000 q^{9} +1.69426 q^{10} +3.31446 q^{11} -0.870532 q^{12} -1.00000 q^{13} -0.229240 q^{14} +1.00000 q^{15} -4.98324 q^{16} -5.60978 q^{17} -1.69426 q^{18} +3.42857 q^{19} -0.870532 q^{20} -0.135304 q^{21} -5.61558 q^{22} -5.03004 q^{23} -1.91362 q^{24} +1.00000 q^{25} +1.69426 q^{26} -1.00000 q^{27} +0.117786 q^{28} +5.80401 q^{29} -1.69426 q^{30} +1.00000 q^{31} +4.61569 q^{32} -3.31446 q^{33} +9.50445 q^{34} -0.135304 q^{35} +0.870532 q^{36} +4.66515 q^{37} -5.80891 q^{38} +1.00000 q^{39} -1.91362 q^{40} -0.962088 q^{41} +0.229240 q^{42} +2.42800 q^{43} +2.88535 q^{44} -1.00000 q^{45} +8.52223 q^{46} +2.36267 q^{47} +4.98324 q^{48} -6.98169 q^{49} -1.69426 q^{50} +5.60978 q^{51} -0.870532 q^{52} -0.135907 q^{53} +1.69426 q^{54} -3.31446 q^{55} +0.258920 q^{56} -3.42857 q^{57} -9.83353 q^{58} -0.413716 q^{59} +0.870532 q^{60} +0.964304 q^{61} -1.69426 q^{62} +0.135304 q^{63} +2.14628 q^{64} +1.00000 q^{65} +5.61558 q^{66} -8.12944 q^{67} -4.88349 q^{68} +5.03004 q^{69} +0.229240 q^{70} +8.29872 q^{71} +1.91362 q^{72} +5.88176 q^{73} -7.90401 q^{74} -1.00000 q^{75} +2.98468 q^{76} +0.448459 q^{77} -1.69426 q^{78} +8.11715 q^{79} +4.98324 q^{80} +1.00000 q^{81} +1.63003 q^{82} +0.397111 q^{83} -0.117786 q^{84} +5.60978 q^{85} -4.11368 q^{86} -5.80401 q^{87} +6.34262 q^{88} +11.9542 q^{89} +1.69426 q^{90} -0.135304 q^{91} -4.37881 q^{92} -1.00000 q^{93} -4.00298 q^{94} -3.42857 q^{95} -4.61569 q^{96} +10.6103 q^{97} +11.8288 q^{98} +3.31446 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * 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 $

$2$	−1.69426	−1.19803	−0.599013	−	0.800739i	$-0.704440\pi$
$2$	−1.69426	−1.19803	−0.599013	+	0.800739i	$0.704440\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.870532	0.435266
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.69426	0.691681
$7$	0.135304	0.0511400	0.0255700	−	0.999673i	$-0.491860\pi$
$7$	0.135304	0.0511400	0.0255700	+	0.999673i	$0.491860\pi$
$8$	1.91362	0.676566
$9$	1.00000	0.333333
$10$	1.69426	0.535773
$11$	3.31446	0.999348	0.499674	−	0.866213i	$-0.333453\pi$
$11$	3.31446	0.999348	0.499674	+	0.866213i	$0.333453\pi$
$12$	−0.870532	−0.251301
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.229240	−0.0612671
$15$	1.00000	0.258199
$16$	−4.98324	−1.24581
$17$	−5.60978	−1.36057	−0.680285	−	0.732947i	$-0.738144\pi$
$17$	−5.60978	−1.36057	−0.680285	+	0.732947i	$0.738144\pi$
$18$	−1.69426	−0.399342
$19$	3.42857	0.786569	0.393285	−	0.919417i	$-0.371339\pi$
$19$	3.42857	0.786569	0.393285	+	0.919417i	$0.371339\pi$
$20$	−0.870532	−0.194657
$21$	−0.135304	−0.0295257
$22$	−5.61558	−1.19724
$23$	−5.03004	−1.04884	−0.524418	−	0.851461i	$-0.675717\pi$
$23$	−5.03004	−1.04884	−0.524418	+	0.851461i	$0.675717\pi$
$24$	−1.91362	−0.390616
$25$	1.00000	0.200000
$26$	1.69426	0.332273
$27$	−1.00000	−0.192450
$28$	0.117786	0.0222595
$29$	5.80401	1.07778	0.538889	−	0.842377i	$-0.318844\pi$
$29$	5.80401	1.07778	0.538889	+	0.842377i	$0.318844\pi$
$30$	−1.69426	−0.309329
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	4.61569	0.815946
$33$	−3.31446	−0.576974
$34$	9.50445	1.63000
$35$	−0.135304	−0.0228705
$36$	0.870532	0.145089
$37$	4.66515	0.766947	0.383473	−	0.923552i	$-0.374728\pi$
$37$	4.66515	0.766947	0.383473	+	0.923552i	$0.374728\pi$
$38$	−5.80891	−0.942330
$39$	1.00000	0.160128
$40$	−1.91362	−0.302570
$41$	−0.962088	−0.150253	−0.0751265	−	0.997174i	$-0.523936\pi$
$41$	−0.962088	−0.150253	−0.0751265	+	0.997174i	$0.523936\pi$
$42$	0.229240	0.0353726
$43$	2.42800	0.370267	0.185133	−	0.982713i	$-0.440728\pi$
$43$	2.42800	0.370267	0.185133	+	0.982713i	$0.440728\pi$
$44$	2.88535	0.434982
$45$	−1.00000	−0.149071
$46$	8.52223	1.25653
$47$	2.36267	0.344630	0.172315	−	0.985042i	$-0.444875\pi$
$47$	2.36267	0.344630	0.172315	+	0.985042i	$0.444875\pi$
$48$	4.98324	0.719268
$49$	−6.98169	−0.997385
$50$	−1.69426	−0.239605
$51$	5.60978	0.785526
$52$	−0.870532	−0.120721
$53$	−0.135907	−0.0186683	−0.00933416	−	0.999956i	$-0.502971\pi$
$53$	−0.135907	−0.0186683	−0.00933416	+	0.999956i	$0.502971\pi$
$54$	1.69426	0.230560
$55$	−3.31446	−0.446922
$56$	0.258920	0.0345996
$57$	−3.42857	−0.454126
$58$	−9.83353	−1.29121
$59$	−0.413716	−0.0538613	−0.0269306	−	0.999637i	$-0.508573\pi$
$59$	−0.413716	−0.0538613	−0.0269306	+	0.999637i	$0.508573\pi$
$60$	0.870532	0.112385
$61$	0.964304	0.123466	0.0617332	−	0.998093i	$-0.480337\pi$
$61$	0.964304	0.123466	0.0617332	+	0.998093i	$0.480337\pi$
$62$	−1.69426	−0.215172
$63$	0.135304	0.0170467
$64$	2.14628	0.268285
$65$	1.00000	0.124035
$66$	5.61558	0.691230
$67$	−8.12944	−0.993170	−0.496585	−	0.867988i	$-0.665413\pi$
$67$	−8.12944	−0.993170	−0.496585	+	0.867988i	$0.665413\pi$
$68$	−4.88349	−0.592210
$69$	5.03004	0.605546
$70$	0.229240	0.0273995
$71$	8.29872	0.984877	0.492438	−	0.870347i	$-0.336106\pi$
$71$	8.29872	0.984877	0.492438	+	0.870347i	$0.336106\pi$
$72$	1.91362	0.225522
$73$	5.88176	0.688408	0.344204	−	0.938895i	$-0.388149\pi$
$73$	5.88176	0.688408	0.344204	+	0.938895i	$0.388149\pi$
$74$	−7.90401	−0.918822
$75$	−1.00000	−0.115470
$76$	2.98468	0.342367
$77$	0.448459	0.0511067
$78$	−1.69426	−0.191838
$79$	8.11715	0.913251	0.456625	−	0.889659i	$-0.349058\pi$
$79$	8.11715	0.913251	0.456625	+	0.889659i	$0.349058\pi$
$80$	4.98324	0.557143
$81$	1.00000	0.111111
$82$	1.63003	0.180007
$83$	0.397111	0.0435886	0.0217943	−	0.999762i	$-0.493062\pi$
$83$	0.397111	0.0435886	0.0217943	+	0.999762i	$0.493062\pi$
$84$	−0.117786	−0.0128515
$85$	5.60978	0.608466
$86$	−4.11368	−0.443589
$87$	−5.80401	−0.622255
$88$	6.34262	0.676125
$89$	11.9542	1.26714	0.633570	−	0.773685i	$-0.281589\pi$
$89$	11.9542	1.26714	0.633570	+	0.773685i	$0.281589\pi$
$90$	1.69426	0.178591
$91$	−0.135304	−0.0141837
$92$	−4.37881	−0.456523
$93$	−1.00000	−0.103695
$94$	−4.00298	−0.412876
$95$	−3.42857	−0.351764
$96$	−4.61569	−0.471087
$97$	10.6103	1.07731	0.538657	−	0.842525i	$-0.318932\pi$
$97$	10.6103	1.07731	0.538657	+	0.842525i	$0.318932\pi$
$98$	11.8288	1.19489
$99$	3.31446	0.333116
$100$	0.870532	0.0870532
$101$	−6.95733	−0.692281	−0.346140	−	0.938183i	$-0.612508\pi$
$101$	−6.95733	−0.692281	−0.346140	+	0.938183i	$0.612508\pi$
$102$	−9.50445	−0.941080
$103$	2.48748	0.245099	0.122549	−	0.992462i	$-0.460893\pi$
$103$	2.48748	0.245099	0.122549	+	0.992462i	$0.460893\pi$
$104$	−1.91362	−0.187646
$105$	0.135304	0.0132043
$106$	0.230263	0.0223651
$107$	−11.1920	−1.08197	−0.540985	−	0.841032i	$-0.681948\pi$
$107$	−11.1920	−1.08197	−0.540985	+	0.841032i	$0.681948\pi$
$108$	−0.870532	−0.0837670
$109$	−0.918456	−0.0879721	−0.0439861	−	0.999032i	$-0.514006\pi$
$109$	−0.918456	−0.0879721	−0.0439861	+	0.999032i	$0.514006\pi$
$110$	5.61558	0.535424
$111$	−4.66515	−0.442797
$112$	−0.674251	−0.0637107
$113$	−11.9544	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$113$	−11.9544	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$114$	5.80891	0.544054
$115$	5.03004	0.469054
$116$	5.05258	0.469120
$117$	−1.00000	−0.0924500
$118$	0.700945	0.0645272
$119$	−0.759024	−0.0695796
$120$	1.91362	0.174689
$121$	−0.0143340	−0.00130309
$122$	−1.63379	−0.147916
$123$	0.962088	0.0867486
$124$	0.870532	0.0781761
$125$	−1.00000	−0.0894427
$126$	−0.229240	−0.0204224
$127$	−0.266382	−0.0236376	−0.0118188	−	0.999930i	$-0.503762\pi$
$127$	−0.266382	−0.0236376	−0.0118188	+	0.999930i	$0.503762\pi$
$128$	−12.8677	−1.13736
$129$	−2.42800	−0.213774
$130$	−1.69426	−0.148597
$131$	−21.9492	−1.91771	−0.958855	−	0.283895i	$-0.908373\pi$
$131$	−21.9492	−1.91771	−0.958855	+	0.283895i	$0.908373\pi$
$132$	−2.88535	−0.251137
$133$	0.463899	0.0402252
$134$	13.7734	1.18984
$135$	1.00000	0.0860663
$136$	−10.7350	−0.920516
$137$	−5.66977	−0.484401	−0.242201	−	0.970226i	$-0.577869\pi$
$137$	−5.66977	−0.484401	−0.242201	+	0.970226i	$0.577869\pi$
$138$	−8.52223	−0.725460
$139$	−17.4721	−1.48197	−0.740983	−	0.671523i	$-0.765640\pi$
$139$	−17.4721	−1.48197	−0.740983	+	0.671523i	$0.765640\pi$
$140$	−0.117786	−0.00995475
$141$	−2.36267	−0.198972
$142$	−14.0602	−1.17991
$143$	−3.31446	−0.277169
$144$	−4.98324	−0.415270
$145$	−5.80401	−0.481997
$146$	−9.96526	−0.824731
$147$	6.98169	0.575840
$148$	4.06116	0.333826
$149$	5.74895	0.470972	0.235486	−	0.971878i	$-0.424332\pi$
$149$	5.74895	0.470972	0.235486	+	0.971878i	$0.424332\pi$
$150$	1.69426	0.138336
$151$	3.40428	0.277036	0.138518	−	0.990360i	$-0.455766\pi$
$151$	3.40428	0.277036	0.138518	+	0.990360i	$0.455766\pi$
$152$	6.56098	0.532166
$153$	−5.60978	−0.453524
$154$	−0.759809	−0.0612271
$155$	−1.00000	−0.0803219
$156$	0.870532	0.0696983
$157$	14.5706	1.16286	0.581429	−	0.813597i	$-0.302494\pi$
$157$	14.5706	1.16286	0.581429	+	0.813597i	$0.302494\pi$
$158$	−13.7526	−1.09410
$159$	0.135907	0.0107782
$160$	−4.61569	−0.364902
$161$	−0.680584	−0.0536375
$162$	−1.69426	−0.133114
$163$	−24.0845	−1.88644	−0.943221	−	0.332166i	$-0.892220\pi$
$163$	−24.0845	−1.88644	−0.943221	+	0.332166i	$0.892220\pi$
$164$	−0.837528	−0.0654000
$165$	3.31446	0.258031
$166$	−0.672812	−0.0522203
$167$	0.946142	0.0732147	0.0366073	−	0.999330i	$-0.488345\pi$
$167$	0.946142	0.0732147	0.0366073	+	0.999330i	$0.488345\pi$
$168$	−0.258920	−0.0199761
$169$	1.00000	0.0769231
$170$	−9.50445	−0.728958
$171$	3.42857	0.262190
$172$	2.11365	0.161164
$173$	−10.4838	−0.797070	−0.398535	−	0.917153i	$-0.630481\pi$
$173$	−10.4838	−0.797070	−0.398535	+	0.917153i	$0.630481\pi$
$174$	9.83353	0.745478
$175$	0.135304	0.0102280
$176$	−16.5168	−1.24500
$177$	0.413716	0.0310968
$178$	−20.2535	−1.51807
$179$	8.81528	0.658885	0.329443	−	0.944176i	$-0.393139\pi$
$179$	8.81528	0.658885	0.329443	+	0.944176i	$0.393139\pi$
$180$	−0.870532	−0.0648856
$181$	−5.28207	−0.392613	−0.196307	−	0.980543i	$-0.562895\pi$
$181$	−5.28207	−0.392613	−0.196307	+	0.980543i	$0.562895\pi$
$182$	0.229240	0.0169924
$183$	−0.964304	−0.0712834
$184$	−9.62558	−0.709607
$185$	−4.66515	−0.342989
$186$	1.69426	0.124229
$187$	−18.5934	−1.35968
$188$	2.05678	0.150006
$189$	−0.135304	−0.00984190
$190$	5.80891	0.421423
$191$	−5.33531	−0.386050	−0.193025	−	0.981194i	$-0.561830\pi$
$191$	−5.33531	−0.386050	−0.193025	+	0.981194i	$0.561830\pi$
$192$	−2.14628	−0.154895
$193$	11.9621	0.861047	0.430524	−	0.902579i	$-0.358329\pi$
$193$	11.9621	0.861047	0.430524	+	0.902579i	$0.358329\pi$
$194$	−17.9767	−1.29065
$195$	−1.00000	−0.0716115
$196$	−6.07778	−0.434127
$197$	11.5010	0.819410	0.409705	−	0.912218i	$-0.365632\pi$
$197$	11.5010	0.819410	0.409705	+	0.912218i	$0.365632\pi$
$198$	−5.61558	−0.399082
$199$	11.2253	0.795743	0.397872	−	0.917441i	$-0.369749\pi$
$199$	11.2253	0.795743	0.397872	+	0.917441i	$0.369749\pi$
$200$	1.91362	0.135313
$201$	8.12944	0.573407
$202$	11.7876	0.829370
$203$	0.785305	0.0551176
$204$	4.88349	0.341913
$205$	0.962088	0.0671951
$206$	−4.21445	−0.293635
$207$	−5.03004	−0.349612
$208$	4.98324	0.345525
$209$	11.3639	0.786056
$210$	−0.229240	−0.0158191
$211$	12.2881	0.845951	0.422975	−	0.906141i	$-0.360986\pi$
$211$	12.2881	0.845951	0.422975	+	0.906141i	$0.360986\pi$
$212$	−0.118312	−0.00812568
$213$	−8.29872	−0.568619
$214$	18.9622	1.29623
$215$	−2.42800	−0.165588
$216$	−1.91362	−0.130205
$217$	0.135304	0.00918502
$218$	1.55611	0.105393
$219$	−5.88176	−0.397453
$220$	−2.88535	−0.194530
$221$	5.60978	0.377354
$222$	7.90401	0.530482
$223$	17.2829	1.15735	0.578675	−	0.815558i	$-0.303570\pi$
$223$	17.2829	1.15735	0.578675	+	0.815558i	$0.303570\pi$
$224$	0.624520	0.0417275
$225$	1.00000	0.0666667
$226$	20.2539	1.34727
$227$	24.0355	1.59529	0.797647	−	0.603125i	$-0.206078\pi$
$227$	24.0355	1.59529	0.797647	+	0.603125i	$0.206078\pi$
$228$	−2.98468	−0.197665
$229$	13.1784	0.870854	0.435427	−	0.900224i	$-0.356597\pi$
$229$	13.1784	0.870854	0.435427	+	0.900224i	$0.356597\pi$
$230$	−8.52223	−0.561939
$231$	−0.448459	−0.0295065
$232$	11.1067	0.729188
$233$	−21.2071	−1.38932	−0.694660	−	0.719338i	$-0.744445\pi$
$233$	−21.2071	−1.38932	−0.694660	+	0.719338i	$0.744445\pi$
$234$	1.69426	0.110758
$235$	−2.36267	−0.154123
$236$	−0.360153	−0.0234440
$237$	−8.11715	−0.527266
$238$	1.28599	0.0833582
$239$	−17.0347	−1.10188	−0.550942	−	0.834544i	$-0.685731\pi$
$239$	−17.0347	−1.10188	−0.550942	+	0.834544i	$0.685731\pi$
$240$	−4.98324	−0.321667
$241$	11.0460	0.711534	0.355767	−	0.934575i	$-0.384220\pi$
$241$	11.0460	0.711534	0.355767	+	0.934575i	$0.384220\pi$
$242$	0.0242856	0.00156114
$243$	−1.00000	−0.0641500
$244$	0.839457	0.0537407
$245$	6.98169	0.446044
$246$	−1.63003	−0.103927
$247$	−3.42857	−0.218155
$248$	1.91362	0.121515
$249$	−0.397111	−0.0251659
$250$	1.69426	0.107155
$251$	−16.1565	−1.01979	−0.509894	−	0.860237i	$-0.670316\pi$
$251$	−16.1565	−1.01979	−0.509894	+	0.860237i	$0.670316\pi$
$252$	0.117786	0.00741983
$253$	−16.6719	−1.04815
$254$	0.451322	0.0283184
$255$	−5.60978	−0.351298
$256$	17.5088	1.09430
$257$	−18.8794	−1.17767	−0.588833	−	0.808254i	$-0.700413\pi$
$257$	−18.8794	−1.17767	−0.588833	+	0.808254i	$0.700413\pi$
$258$	4.11368	0.256106
$259$	0.631213	0.0392217
$260$	0.870532	0.0539881
$261$	5.80401	0.359259
$262$	37.1878	2.29747
$263$	−24.0609	−1.48366	−0.741829	−	0.670590i	$-0.766041\pi$
$263$	−24.0609	−1.48366	−0.741829	+	0.670590i	$0.766041\pi$
$264$	−6.34262	−0.390361
$265$	0.135907	0.00834873
$266$	−0.785968	−0.0481908
$267$	−11.9542	−0.731584
$268$	−7.07694	−0.432293
$269$	9.65463	0.588653	0.294326	−	0.955705i	$-0.404905\pi$
$269$	9.65463	0.588653	0.294326	+	0.955705i	$0.404905\pi$
$270$	−1.69426	−0.103110
$271$	27.2842	1.65740	0.828698	−	0.559696i	$-0.189082\pi$
$271$	27.2842	1.65740	0.828698	+	0.559696i	$0.189082\pi$
$272$	27.9549	1.69501
$273$	0.135304	0.00818896
$274$	9.60609	0.580325
$275$	3.31446	0.199870
$276$	4.37881	0.263574
$277$	1.81420	0.109004	0.0545022	−	0.998514i	$-0.482643\pi$
$277$	1.81420	0.109004	0.0545022	+	0.998514i	$0.482643\pi$
$278$	29.6024	1.77543
$279$	1.00000	0.0598684
$280$	−0.258920	−0.0154734
$281$	16.9490	1.01109	0.505545	−	0.862800i	$-0.331292\pi$
$281$	16.9490	1.01109	0.505545	+	0.862800i	$0.331292\pi$
$282$	4.00298	0.238374
$283$	16.6802	0.991535	0.495767	−	0.868455i	$-0.334887\pi$
$283$	16.6802	0.991535	0.495767	+	0.868455i	$0.334887\pi$
$284$	7.22430	0.428683
$285$	3.42857	0.203091
$286$	5.61558	0.332056
$287$	−0.130174	−0.00768394
$288$	4.61569	0.271982
$289$	14.4696	0.851153
$290$	9.83353	0.577445
$291$	−10.6103	−0.621988
$292$	5.12026	0.299641
$293$	−16.5327	−0.965848	−0.482924	−	0.875662i	$-0.660425\pi$
$293$	−16.5327	−0.965848	−0.482924	+	0.875662i	$0.660425\pi$
$294$	−11.8288	−0.689872
$295$	0.413716	0.0240875
$296$	8.92732	0.518890
$297$	−3.31446	−0.192325
$298$	−9.74024	−0.564237
$299$	5.03004	0.290895
$300$	−0.870532	−0.0502602
$301$	0.328518	0.0189354
$302$	−5.76775	−0.331897
$303$	6.95733	0.399688
$304$	−17.0854	−0.979915
$305$	−0.964304	−0.0552159
$306$	9.50445	0.543333
$307$	28.5203	1.62774	0.813870	−	0.581048i	$-0.197357\pi$
$307$	28.5203	1.62774	0.813870	+	0.581048i	$0.197357\pi$
$308$	0.390398	0.0222450
$309$	−2.48748	−0.141508
$310$	1.69426	0.0962277
$311$	−5.50139	−0.311955	−0.155978	−	0.987761i	$-0.549853\pi$
$311$	−5.50139	−0.311955	−0.155978	+	0.987761i	$0.549853\pi$
$312$	1.91362	0.108337
$313$	20.3126	1.14813	0.574067	−	0.818808i	$-0.305365\pi$
$313$	20.3126	1.14813	0.574067	+	0.818808i	$0.305365\pi$
$314$	−24.6864	−1.39313
$315$	−0.135304	−0.00762350
$316$	7.06624	0.397507
$317$	−24.4259	−1.37190	−0.685949	−	0.727650i	$-0.740613\pi$
$317$	−24.4259	−1.37190	−0.685949	+	0.727650i	$0.740613\pi$
$318$	−0.230263	−0.0129125
$319$	19.2372	1.07708
$320$	−2.14628	−0.119981
$321$	11.1920	0.624675
$322$	1.15309	0.0642592
$323$	−19.2335	−1.07018
$324$	0.870532	0.0483629
$325$	−1.00000	−0.0554700
$326$	40.8055	2.26001
$327$	0.918456	0.0507907
$328$	−1.84107	−0.101656
$329$	0.319678	0.0176244
$330$	−5.61558	−0.309127
$331$	−17.3011	−0.950956	−0.475478	−	0.879728i	$-0.657725\pi$
$331$	−17.3011	−0.950956	−0.475478	+	0.879728i	$0.657725\pi$
$332$	0.345698	0.0189726
$333$	4.66515	0.255649
$334$	−1.60302	−0.0877131
$335$	8.12944	0.444159
$336$	0.674251	0.0367834
$337$	23.6285	1.28713	0.643564	−	0.765393i	$-0.277455\pi$
$337$	23.6285	1.28713	0.643564	+	0.765393i	$0.277455\pi$
$338$	−1.69426	−0.0921558
$339$	11.9544	0.649275
$340$	4.88349	0.264844
$341$	3.31446	0.179488
$342$	−5.80891	−0.314110
$343$	−1.89178	−0.102146
$344$	4.64627	0.250510
$345$	−5.03004	−0.270809
$346$	17.7624	0.954910
$347$	−15.7504	−0.845527	−0.422763	−	0.906240i	$-0.638940\pi$
$347$	−15.7504	−0.845527	−0.422763	+	0.906240i	$0.638940\pi$
$348$	−5.05258	−0.270847
$349$	−19.2647	−1.03122	−0.515608	−	0.856825i	$-0.672434\pi$
$349$	−19.2647	−1.03122	−0.515608	+	0.856825i	$0.672434\pi$
$350$	−0.229240	−0.0122534
$351$	1.00000	0.0533761
$352$	15.2985	0.815414
$353$	−29.2785	−1.55834	−0.779169	−	0.626814i	$-0.784359\pi$
$353$	−29.2785	−1.55834	−0.779169	+	0.626814i	$0.784359\pi$
$354$	−0.700945	−0.0372548
$355$	−8.29872	−0.440450
$356$	10.4065	0.551543
$357$	0.759024	0.0401718
$358$	−14.9354	−0.789361
$359$	0.427159	0.0225446	0.0112723	−	0.999936i	$-0.496412\pi$
$359$	0.427159	0.0225446	0.0112723	+	0.999936i	$0.496412\pi$
$360$	−1.91362	−0.100857
$361$	−7.24487	−0.381309
$362$	8.94922	0.470361
$363$	0.0143340	0.000752341 0
$364$	−0.117786	−0.00617368
$365$	−5.88176	−0.307866
$366$	1.63379	0.0853994
$367$	5.04849	0.263529	0.131765	−	0.991281i	$-0.457936\pi$
$367$	5.04849	0.263529	0.131765	+	0.991281i	$0.457936\pi$
$368$	25.0659	1.30665
$369$	−0.962088	−0.0500843
$370$	7.90401	0.410910
$371$	−0.0183888	−0.000954698 0
$372$	−0.870532	−0.0451350
$373$	34.0489	1.76299	0.881493	−	0.472198i	$-0.156539\pi$
$373$	34.0489	1.76299	0.881493	+	0.472198i	$0.156539\pi$
$374$	31.5021	1.62894
$375$	1.00000	0.0516398
$376$	4.52124	0.233165
$377$	−5.80401	−0.298922
$378$	0.229240	0.0117909
$379$	9.07197	0.465996	0.232998	−	0.972477i	$-0.425146\pi$
$379$	9.07197	0.465996	0.232998	+	0.972477i	$0.425146\pi$
$380$	−2.98468	−0.153111
$381$	0.266382	0.0136472
$382$	9.03943	0.462497
$383$	34.3852	1.75700	0.878502	−	0.477739i	$-0.158543\pi$
$383$	34.3852	1.75700	0.878502	+	0.477739i	$0.158543\pi$
$384$	12.8677	0.656654
$385$	−0.448459	−0.0228556
$386$	−20.2669	−1.03156
$387$	2.42800	0.123422
$388$	9.23661	0.468918
$389$	21.4959	1.08989	0.544943	−	0.838473i	$-0.316551\pi$
$389$	21.4959	1.08989	0.544943	+	0.838473i	$0.316551\pi$
$390$	1.69426	0.0857924
$391$	28.2174	1.42702
$392$	−13.3603	−0.674797
$393$	21.9492	1.10719
$394$	−19.4857	−0.981674
$395$	−8.11715	−0.408418
$396$	2.88535	0.144994
$397$	32.4179	1.62701	0.813503	−	0.581561i	$-0.197558\pi$
$397$	32.4179	1.62701	0.813503	+	0.581561i	$0.197558\pi$
$398$	−19.0187	−0.953321
$399$	−0.463899	−0.0232240
$400$	−4.98324	−0.249162
$401$	7.52409	0.375735	0.187867	−	0.982194i	$-0.439842\pi$
$401$	7.52409	0.375735	0.187867	+	0.982194i	$0.439842\pi$
$402$	−13.7734	−0.686956
$403$	−1.00000	−0.0498135
$404$	−6.05658	−0.301326
$405$	−1.00000	−0.0496904
$406$	−1.33051	−0.0660323
$407$	15.4625	0.766447
$408$	10.7350	0.531460
$409$	12.6007	0.623064	0.311532	−	0.950236i	$-0.399158\pi$
$409$	12.6007	0.623064	0.311532	+	0.950236i	$0.399158\pi$
$410$	−1.63003	−0.0805015
$411$	5.66977	0.279669
$412$	2.16543	0.106683
$413$	−0.0559774	−0.00275447
$414$	8.52223	0.418845
$415$	−0.397111	−0.0194934
$416$	−4.61569	−0.226303
$417$	17.4721	0.855614
$418$	−19.2534	−0.941716
$419$	13.1433	0.642092	0.321046	−	0.947064i	$-0.395966\pi$
$419$	13.1433	0.642092	0.321046	+	0.947064i	$0.395966\pi$
$420$	0.117786	0.00574738
$421$	9.51090	0.463533	0.231766	−	0.972771i	$-0.425549\pi$
$421$	9.51090	0.463533	0.231766	+	0.972771i	$0.425549\pi$
$422$	−20.8194	−1.01347
$423$	2.36267	0.114877
$424$	−0.260075	−0.0126304
$425$	−5.60978	−0.272114
$426$	14.0602	0.681220
$427$	0.130474	0.00631408
$428$	−9.74297	−0.470944
$429$	3.31446	0.160024
$430$	4.11368	0.198379
$431$	−16.3869	−0.789329	−0.394665	−	0.918825i	$-0.629139\pi$
$431$	−16.3869	−0.789329	−0.394665	+	0.918825i	$0.629139\pi$
$432$	4.98324	0.239756
$433$	−19.5990	−0.941867	−0.470934	−	0.882169i	$-0.656083\pi$
$433$	−19.5990	−0.941867	−0.470934	+	0.882169i	$0.656083\pi$
$434$	−0.229240	−0.0110039
$435$	5.80401	0.278281
$436$	−0.799545	−0.0382913
$437$	−17.2459	−0.824983
$438$	9.96526	0.476159
$439$	5.41694	0.258536	0.129268	−	0.991610i	$-0.458737\pi$
$439$	5.41694	0.258536	0.129268	+	0.991610i	$0.458737\pi$
$440$	−6.34262	−0.302372
$441$	−6.98169	−0.332462
$442$	−9.50445	−0.452080
$443$	−3.59667	−0.170883	−0.0854415	−	0.996343i	$-0.527230\pi$
$443$	−3.59667	−0.170883	−0.0854415	+	0.996343i	$0.527230\pi$
$444$	−4.06116	−0.192734
$445$	−11.9542	−0.566682
$446$	−29.2818	−1.38654
$447$	−5.74895	−0.271916
$448$	0.290400	0.0137201
$449$	24.0376	1.13440	0.567202	−	0.823578i	$-0.308026\pi$
$449$	24.0376	1.13440	0.567202	+	0.823578i	$0.308026\pi$
$450$	−1.69426	−0.0798684
$451$	−3.18881	−0.150155
$452$	−10.4067	−0.489490
$453$	−3.40428	−0.159947
$454$	−40.7225	−1.91120
$455$	0.135304	0.00634314
$456$	−6.56098	−0.307246
$457$	−6.76152	−0.316291	−0.158145	−	0.987416i	$-0.550551\pi$
$457$	−6.76152	−0.316291	−0.158145	+	0.987416i	$0.550551\pi$
$458$	−22.3277	−1.04331
$459$	5.60978	0.261842
$460$	4.37881	0.204163
$461$	7.42971	0.346036	0.173018	−	0.984919i	$-0.444648\pi$
$461$	7.42971	0.346036	0.173018	+	0.984919i	$0.444648\pi$
$462$	0.759809	0.0353495
$463$	3.56196	0.165538	0.0827692	−	0.996569i	$-0.473624\pi$
$463$	3.56196	0.165538	0.0827692	+	0.996569i	$0.473624\pi$
$464$	−28.9228	−1.34271
$465$	1.00000	0.0463739
$466$	35.9304	1.66444
$467$	29.5253	1.36627	0.683134	−	0.730293i	$-0.260617\pi$
$467$	29.5253	1.36627	0.683134	+	0.730293i	$0.260617\pi$
$468$	−0.870532	−0.0402403
$469$	−1.09994	−0.0507907
$470$	4.00298	0.184644
$471$	−14.5706	−0.671376
$472$	−0.791695	−0.0364407
$473$	8.04752	0.370025
$474$	13.7526	0.631678
$475$	3.42857	0.157314
$476$	−0.660754	−0.0302856
$477$	−0.135907	−0.00622277
$478$	28.8613	1.32008
$479$	5.46541	0.249721	0.124860	−	0.992174i	$-0.460152\pi$
$479$	5.46541	0.249721	0.124860	+	0.992174i	$0.460152\pi$
$480$	4.61569	0.210676
$481$	−4.66515	−0.212713
$482$	−18.7148	−0.852436
$483$	0.680584	0.0309676
$484$	−0.0124782	−0.000567192 0
$485$	−10.6103	−0.481789
$486$	1.69426	0.0768534
$487$	−33.4186	−1.51434	−0.757171	−	0.653217i	$-0.773419\pi$
$487$	−33.4186	−1.51434	−0.757171	+	0.653217i	$0.773419\pi$
$488$	1.84531	0.0835332
$489$	24.0845	1.08914
$490$	−11.8288	−0.534372
$491$	9.77967	0.441350	0.220675	−	0.975347i	$-0.429174\pi$
$491$	9.77967	0.441350	0.220675	+	0.975347i	$0.429174\pi$
$492$	0.837528	0.0377587
$493$	−32.5592	−1.46639
$494$	5.80891	0.261355
$495$	−3.31446	−0.148974
$496$	−4.98324	−0.223754
$497$	1.12285	0.0503666
$498$	0.672812	0.0301494
$499$	−12.0247	−0.538299	−0.269149	−	0.963098i	$-0.586743\pi$
$499$	−12.0247	−0.538299	−0.269149	+	0.963098i	$0.586743\pi$
$500$	−0.870532	−0.0389314
$501$	−0.946142	−0.0422705
$502$	27.3734	1.22173
$503$	10.8180	0.482353	0.241176	−	0.970481i	$-0.422467\pi$
$503$	10.8180	0.482353	0.241176	+	0.970481i	$0.422467\pi$
$504$	0.258920	0.0115332
$505$	6.95733	0.309597
$506$	28.2466	1.25571
$507$	−1.00000	−0.0444116
$508$	−0.231894	−0.0102886
$509$	20.0554	0.888938	0.444469	−	0.895794i	$-0.353392\pi$
$509$	20.0554	0.888938	0.444469	+	0.895794i	$0.353392\pi$
$510$	9.50445	0.420864
$511$	0.795825	0.0352052
$512$	−3.92904	−0.173641
$513$	−3.42857	−0.151375
$514$	31.9868	1.41088
$515$	−2.48748	−0.109612
$516$	−2.11365	−0.0930483
$517$	7.83097	0.344406
$518$	−1.06944	−0.0469886
$519$	10.4838	0.460188
$520$	1.91362	0.0839177
$521$	1.78538	0.0782191	0.0391095	−	0.999235i	$-0.487548\pi$
$521$	1.78538	0.0782191	0.0391095	+	0.999235i	$0.487548\pi$
$522$	−9.83353	−0.430402
$523$	−0.722559	−0.0315953	−0.0157977	−	0.999875i	$-0.505029\pi$
$523$	−0.722559	−0.0315953	−0.0157977	+	0.999875i	$0.505029\pi$
$524$	−19.1075	−0.834714
$525$	−0.135304	−0.00590514
$526$	40.7655	1.77746
$527$	−5.60978	−0.244366
$528$	16.5168	0.718800
$529$	2.30135	0.100059
$530$	−0.230263	−0.0100020
$531$	−0.413716	−0.0179538
$532$	0.403839	0.0175086
$533$	0.962088	0.0416727
$534$	20.2535	0.876456
$535$	11.1920	0.483871
$536$	−15.5566	−0.671945
$537$	−8.81528	−0.380407
$538$	−16.3575	−0.705221
$539$	−23.1406	−0.996735
$540$	0.870532	0.0374617
$541$	−21.0352	−0.904372	−0.452186	−	0.891924i	$-0.649356\pi$
$541$	−21.0352	−0.904372	−0.452186	+	0.891924i	$0.649356\pi$
$542$	−46.2266	−1.98560
$543$	5.28207	0.226675
$544$	−25.8930	−1.11015
$545$	0.918456	0.0393423
$546$	−0.229240	−0.00981058
$547$	33.5866	1.43606	0.718030	−	0.696012i	$-0.245044\pi$
$547$	33.5866	1.43606	0.718030	+	0.696012i	$0.245044\pi$
$548$	−4.93571	−0.210843
$549$	0.964304	0.0411555
$550$	−5.61558	−0.239449
$551$	19.8995	0.847747
$552$	9.62558	0.409692
$553$	1.09828	0.0467037
$554$	−3.07373	−0.130590
$555$	4.66515	0.198025
$556$	−15.2100	−0.645049
$557$	−7.79404	−0.330244	−0.165122	−	0.986273i	$-0.552802\pi$
$557$	−7.79404	−0.330244	−0.165122	+	0.986273i	$0.552802\pi$
$558$	−1.69426	−0.0717239
$559$	−2.42800	−0.102694
$560$	0.674251	0.0284923
$561$	18.5934	0.785014
$562$	−28.7160	−1.21131
$563$	13.9601	0.588347	0.294174	−	0.955752i	$-0.404956\pi$
$563$	13.9601	0.588347	0.294174	+	0.955752i	$0.404956\pi$
$564$	−2.05678	−0.0866059
$565$	11.9544	0.502926
$566$	−28.2607	−1.18788
$567$	0.135304	0.00568222
$568$	15.8806	0.666334
$569$	31.7987	1.33307	0.666535	−	0.745473i	$-0.267777\pi$
$569$	31.7987	1.33307	0.666535	+	0.745473i	$0.267777\pi$
$570$	−5.80891	−0.243309
$571$	28.0455	1.17367	0.586833	−	0.809708i	$-0.300375\pi$
$571$	28.0455	1.17367	0.586833	+	0.809708i	$0.300375\pi$
$572$	−2.88535	−0.120642
$573$	5.33531	0.222886
$574$	0.220549	0.00920556
$575$	−5.03004	−0.209767
$576$	2.14628	0.0894284
$577$	9.75524	0.406116	0.203058	−	0.979167i	$-0.434912\pi$
$577$	9.75524	0.406116	0.203058	+	0.979167i	$0.434912\pi$
$578$	−24.5153	−1.01970
$579$	−11.9621	−0.497126
$580$	−5.05258	−0.209797
$581$	0.0537307	0.00222912
$582$	17.9767	0.745157
$583$	−0.450460	−0.0186562
$584$	11.2554	0.465754
$585$	1.00000	0.0413449
$586$	28.0107	1.15711
$587$	3.34096	0.137896	0.0689480	−	0.997620i	$-0.478036\pi$
$587$	3.34096	0.137896	0.0689480	+	0.997620i	$0.478036\pi$
$588$	6.07778	0.250644
$589$	3.42857	0.141272
$590$	−0.700945	−0.0288575
$591$	−11.5010	−0.473087
$592$	−23.2476	−0.955469
$593$	17.9300	0.736297	0.368149	−	0.929767i	$-0.379992\pi$
$593$	17.9300	0.736297	0.368149	+	0.929767i	$0.379992\pi$
$594$	5.61558	0.230410
$595$	0.759024	0.0311170
$596$	5.00464	0.204998
$597$	−11.2253	−0.459422
$598$	−8.52223	−0.348500
$599$	33.4322	1.36600	0.683001	−	0.730418i	$-0.260675\pi$
$599$	33.4322	1.36600	0.683001	+	0.730418i	$0.260675\pi$
$600$	−1.91362	−0.0781231
$601$	31.1457	1.27046	0.635231	−	0.772322i	$-0.280905\pi$
$601$	31.1457	1.27046	0.635231	+	0.772322i	$0.280905\pi$
$602$	−0.556596	−0.0226852
$603$	−8.12944	−0.331057
$604$	2.96353	0.120584
$605$	0.0143340	0.000582761 0
$606$	−11.7876	−0.478837
$607$	4.31564	0.175166	0.0875831	−	0.996157i	$-0.472086\pi$
$607$	4.31564	0.175166	0.0875831	+	0.996157i	$0.472086\pi$
$608$	15.8252	0.641798
$609$	−0.785305	−0.0318222
$610$	1.63379	0.0661501
$611$	−2.36267	−0.0955832
$612$	−4.88349	−0.197403
$613$	−35.2888	−1.42530	−0.712650	−	0.701519i	$-0.752505\pi$
$613$	−35.2888	−1.42530	−0.712650	+	0.701519i	$0.752505\pi$
$614$	−48.3209	−1.95007
$615$	−0.962088	−0.0387951
$616$	0.858180	0.0345771
$617$	16.2925	0.655912	0.327956	−	0.944693i	$-0.393640\pi$
$617$	16.2925	0.655912	0.327956	+	0.944693i	$0.393640\pi$
$618$	4.21445	0.169530
$619$	38.8689	1.56227	0.781137	−	0.624359i	$-0.214640\pi$
$619$	38.8689	1.56227	0.781137	+	0.624359i	$0.214640\pi$
$620$	−0.870532	−0.0349614
$621$	5.03004	0.201849
$622$	9.32081	0.373730
$623$	1.61745	0.0648016
$624$	−4.98324	−0.199489
$625$	1.00000	0.0400000
$626$	−34.4149	−1.37549
$627$	−11.3639	−0.453830
$628$	12.6841	0.506152
$629$	−26.1705	−1.04349
$630$	0.229240	0.00913315
$631$	23.3433	0.929283	0.464641	−	0.885499i	$-0.346183\pi$
$631$	23.3433	0.929283	0.464641	+	0.885499i	$0.346183\pi$
$632$	15.5331	0.617874
$633$	−12.2881	−0.488410
$634$	41.3840	1.64357
$635$	0.266382	0.0105710
$636$	0.118312	0.00469136
$637$	6.98169	0.276625
$638$	−32.5929	−1.29036
$639$	8.29872	0.328292
$640$	12.8677	0.508642
$641$	14.8924	0.588215	0.294107	−	0.955772i	$-0.404978\pi$
$641$	14.8924	0.588215	0.294107	+	0.955772i	$0.404978\pi$
$642$	−18.9622	−0.748377
$643$	−20.4279	−0.805596	−0.402798	−	0.915289i	$-0.631962\pi$
$643$	−20.4279	−0.805596	−0.402798	+	0.915289i	$0.631962\pi$
$644$	−0.592470	−0.0233466
$645$	2.42800	0.0956025
$646$	32.5867	1.28211
$647$	−27.4264	−1.07824	−0.539122	−	0.842227i	$-0.681244\pi$
$647$	−27.4264	−1.07824	−0.539122	+	0.842227i	$0.681244\pi$
$648$	1.91362	0.0751740
$649$	−1.37125	−0.0538262
$650$	1.69426	0.0664545
$651$	−0.135304	−0.00530297
$652$	−20.9663	−0.821104
$653$	0.824258	0.0322557	0.0161279	−	0.999870i	$-0.494866\pi$
$653$	0.824258	0.0322557	0.0161279	+	0.999870i	$0.494866\pi$
$654$	−1.55611	−0.0608486
$655$	21.9492	0.857626
$656$	4.79431	0.187187
$657$	5.88176	0.229469
$658$	−0.541618	−0.0211145
$659$	44.7866	1.74464	0.872320	−	0.488936i	$-0.162615\pi$
$659$	44.7866	1.74464	0.872320	+	0.488936i	$0.162615\pi$
$660$	2.88535	0.112312
$661$	29.8787	1.16215	0.581074	−	0.813851i	$-0.302633\pi$
$661$	29.8787	1.16215	0.581074	+	0.813851i	$0.302633\pi$
$662$	29.3127	1.13927
$663$	−5.60978	−0.217866
$664$	0.759919	0.0294906
$665$	−0.463899	−0.0179892
$666$	−7.90401	−0.306274
$667$	−29.1944	−1.13041
$668$	0.823647	0.0318679
$669$	−17.2829	−0.668196
$670$	−13.7734	−0.532114
$671$	3.19615	0.123386
$672$	−0.624520	−0.0240914
$673$	−30.1728	−1.16308	−0.581539	−	0.813519i	$-0.697549\pi$
$673$	−30.1728	−1.16308	−0.581539	+	0.813519i	$0.697549\pi$
$674$	−40.0330	−1.54201
$675$	−1.00000	−0.0384900
$676$	0.870532	0.0334820
$677$	1.41822	0.0545065	0.0272532	−	0.999629i	$-0.491324\pi$
$677$	1.41822	0.0545065	0.0272532	+	0.999629i	$0.491324\pi$
$678$	−20.2539	−0.777848
$679$	1.43562	0.0550939
$680$	10.7350	0.411667
$681$	−24.0355	−0.921043
$682$	−5.61558	−0.215032
$683$	−18.8430	−0.721008	−0.360504	−	0.932758i	$-0.617395\pi$
$683$	−18.8430	−0.721008	−0.360504	+	0.932758i	$0.617395\pi$
$684$	2.98468	0.114122
$685$	5.66977	0.216631
$686$	3.20517	0.122374
$687$	−13.1784	−0.502788
$688$	−12.0993	−0.461282
$689$	0.135907	0.00517766
$690$	8.52223	0.324436
$691$	−47.4005	−1.80320	−0.901600	−	0.432570i	$-0.857607\pi$
$691$	−47.4005	−1.80320	−0.901600	+	0.432570i	$0.857607\pi$
$692$	−9.12649	−0.346937
$693$	0.448459	0.0170356
$694$	26.6854	1.01296
$695$	17.4721	0.662756
$696$	−11.1067	−0.420997
$697$	5.39710	0.204430
$698$	32.6395	1.23542
$699$	21.2071	0.802125
$700$	0.117786	0.00445190
$701$	43.3041	1.63557	0.817787	−	0.575521i	$-0.195201\pi$
$701$	43.3041	1.63557	0.817787	+	0.575521i	$0.195201\pi$
$702$	−1.69426	−0.0639459
$703$	15.9948	0.603256
$704$	7.11377	0.268110
$705$	2.36267	0.0889831
$706$	49.6055	1.86693
$707$	−0.941354	−0.0354032
$708$	0.360153	0.0135354
$709$	−5.17349	−0.194294	−0.0971472	−	0.995270i	$-0.530972\pi$
$709$	−5.17349	−0.194294	−0.0971472	+	0.995270i	$0.530972\pi$
$710$	14.0602	0.527671
$711$	8.11715	0.304417
$712$	22.8757	0.857304
$713$	−5.03004	−0.188377
$714$	−1.28599	−0.0481269
$715$	3.31446	0.123954
$716$	7.67398	0.286790
$717$	17.0347	0.636173
$718$	−0.723720	−0.0270090
$719$	41.4354	1.54528	0.772641	−	0.634844i	$-0.218935\pi$
$719$	41.4354	1.54528	0.772641	+	0.634844i	$0.218935\pi$
$720$	4.98324	0.185714
$721$	0.336566	0.0125344
$722$	12.2747	0.456818
$723$	−11.0460	−0.410804
$724$	−4.59821	−0.170891
$725$	5.80401	0.215556
$726$	−0.0242856	−0.000901324 0
$727$	39.2788	1.45677	0.728386	−	0.685167i	$-0.240271\pi$
$727$	39.2788	1.45677	0.728386	+	0.685167i	$0.240271\pi$
$728$	−0.258920	−0.00959620
$729$	1.00000	0.0370370
$730$	9.96526	0.368831
$731$	−13.6205	−0.503774
$732$	−0.839457	−0.0310272
$733$	−0.640124	−0.0236435	−0.0118218	−	0.999930i	$-0.503763\pi$
$733$	−0.640124	−0.0236435	−0.0118218	+	0.999930i	$0.503763\pi$
$734$	−8.55348	−0.315715
$735$	−6.98169	−0.257524
$736$	−23.2171	−0.855794
$737$	−26.9447	−0.992522
$738$	1.63003	0.0600023
$739$	−14.9240	−0.548989	−0.274494	−	0.961589i	$-0.588510\pi$
$739$	−14.9240	−0.548989	−0.274494	+	0.961589i	$0.588510\pi$
$740$	−4.06116	−0.149291
$741$	3.42857	0.125952
$742$	0.0311555	0.00114375
$743$	−46.0218	−1.68838	−0.844189	−	0.536046i	$-0.819917\pi$
$743$	−46.0218	−1.68838	−0.844189	+	0.536046i	$0.819917\pi$
$744$	−1.91362	−0.0701566
$745$	−5.74895	−0.210625
$746$	−57.6878	−2.11210
$747$	0.397111	0.0145295
$748$	−16.1861	−0.591824
$749$	−1.51432	−0.0553319
$750$	−1.69426	−0.0618658
$751$	−5.60574	−0.204556	−0.102278	−	0.994756i	$-0.532613\pi$
$751$	−5.60574	−0.204556	−0.102278	+	0.994756i	$0.532613\pi$
$752$	−11.7737	−0.429344
$753$	16.1565	0.588775
$754$	9.83353	0.358116
$755$	−3.40428	−0.123894
$756$	−0.117786	−0.00428384
$757$	−19.7362	−0.717325	−0.358663	−	0.933467i	$-0.616767\pi$
$757$	−19.7362	−0.717325	−0.358663	+	0.933467i	$0.616767\pi$
$758$	−15.3703	−0.558275
$759$	16.6719	0.605152
$760$	−6.56098	−0.237992
$761$	−32.8156	−1.18956	−0.594782	−	0.803887i	$-0.702762\pi$
$761$	−32.8156	−1.18956	−0.594782	+	0.803887i	$0.702762\pi$
$762$	−0.451322	−0.0163497
$763$	−0.124271	−0.00449890
$764$	−4.64456	−0.168034
$765$	5.60978	0.202822
$766$	−58.2577	−2.10494
$767$	0.413716	0.0149384
$768$	−17.5088	−0.631794
$769$	19.5544	0.705149	0.352575	−	0.935784i	$-0.385306\pi$
$769$	19.5544	0.705149	0.352575	+	0.935784i	$0.385306\pi$
$770$	0.759809	0.0273816
$771$	18.8794	0.679926
$772$	10.4133	0.374785
$773$	−21.5447	−0.774908	−0.387454	−	0.921889i	$-0.626645\pi$
$773$	−21.5447	−0.774908	−0.387454	+	0.921889i	$0.626645\pi$
$774$	−4.11368	−0.147863
$775$	1.00000	0.0359211
$776$	20.3041	0.728874
$777$	−0.631213	−0.0226446
$778$	−36.4197	−1.30571
$779$	−3.29859	−0.118184
$780$	−0.870532	−0.0311700
$781$	27.5058	0.984235
$782$	−47.8078	−1.70960
$783$	−5.80401	−0.207418
$784$	34.7914	1.24255
$785$	−14.5706	−0.520046
$786$	−37.1878	−1.32644
$787$	30.6949	1.09415	0.547077	−	0.837083i	$-0.315741\pi$
$787$	30.6949	1.09415	0.547077	+	0.837083i	$0.315741\pi$
$788$	10.0120	0.356661
$789$	24.0609	0.856590
$790$	13.7526	0.489295
$791$	−1.61748	−0.0575109
$792$	6.34262	0.225375
$793$	−0.964304	−0.0342434
$794$	−54.9244	−1.94920
$795$	−0.135907	−0.00482014
$796$	9.77201	0.346360
$797$	−23.8726	−0.845611	−0.422805	−	0.906220i	$-0.638955\pi$
$797$	−23.8726	−0.845611	−0.422805	+	0.906220i	$0.638955\pi$
$798$	0.785968	0.0278230
$799$	−13.2540	−0.468894
$800$	4.61569	0.163189
$801$	11.9542	0.422380
$802$	−12.7478	−0.450140
$803$	19.4949	0.687960
$804$	7.07694	0.249584
$805$	0.680584	0.0239874
$806$	1.69426	0.0596779
$807$	−9.65463	−0.339859
$808$	−13.3137	−0.468374
$809$	23.1283	0.813146	0.406573	−	0.913618i	$-0.366724\pi$
$809$	23.1283	0.813146	0.406573	+	0.913618i	$0.366724\pi$
$810$	1.69426	0.0595304
$811$	27.5687	0.968070	0.484035	−	0.875049i	$-0.339171\pi$
$811$	27.5687	0.968070	0.484035	+	0.875049i	$0.339171\pi$
$812$	0.683633	0.0239908
$813$	−27.2842	−0.956898
$814$	−26.1975	−0.918223
$815$	24.0845	0.843642
$816$	−27.9549	−0.978616
$817$	8.32458	0.291240
$818$	−21.3489	−0.746447
$819$	−0.135304	−0.00472790
$820$	0.837528	0.0292478
$821$	4.88312	0.170422	0.0852111	−	0.996363i	$-0.472844\pi$
$821$	4.88312	0.170422	0.0852111	+	0.996363i	$0.472844\pi$
$822$	−9.60609	−0.335051
$823$	−30.5873	−1.06621	−0.533104	−	0.846050i	$-0.678975\pi$
$823$	−30.5873	−1.06621	−0.533104	+	0.846050i	$0.678975\pi$
$824$	4.76009	0.165826
$825$	−3.31446	−0.115395
$826$	0.0948405	0.00329992
$827$	47.5031	1.65185	0.825923	−	0.563783i	$-0.190655\pi$
$827$	47.5031	1.65185	0.825923	+	0.563783i	$0.190655\pi$
$828$	−4.37881	−0.152174
$829$	2.18382	0.0758473	0.0379237	−	0.999281i	$-0.487926\pi$
$829$	2.18382	0.0758473	0.0379237	+	0.999281i	$0.487926\pi$
$830$	0.672812	0.0233536
$831$	−1.81420	−0.0629338
$832$	−2.14628	−0.0744090
$833$	39.1657	1.35701
$834$	−29.6024	−1.02505
$835$	−0.946142	−0.0327426
$836$	9.89262	0.342143
$837$	−1.00000	−0.0345651
$838$	−22.2682	−0.769242
$839$	54.5381	1.88286	0.941432	−	0.337202i	$-0.109480\pi$
$839$	54.5381	1.88286	0.941432	+	0.337202i	$0.109480\pi$
$840$	0.258920	0.00893358
$841$	4.68656	0.161606
$842$	−16.1140	−0.555324
$843$	−16.9490	−0.583753
$844$	10.6972	0.368213
$845$	−1.00000	−0.0344010
$846$	−4.00298	−0.137625
$847$	−0.00193945	−6.66402e−5 0
$848$	0.677259	0.0232572
$849$	−16.6802	−0.572463
$850$	9.50445	0.326000
$851$	−23.4659	−0.804402
$852$	−7.22430	−0.247500
$853$	28.2412	0.966961	0.483480	−	0.875355i	$-0.339372\pi$
$853$	28.2412	0.966961	0.483480	+	0.875355i	$0.339372\pi$
$854$	−0.221057	−0.00756443
$855$	−3.42857	−0.117255
$856$	−21.4172	−0.732024
$857$	31.6935	1.08263	0.541315	−	0.840820i	$-0.317927\pi$
$857$	31.6935	1.08263	0.541315	+	0.840820i	$0.317927\pi$
$858$	−5.61558	−0.191713
$859$	19.7805	0.674902	0.337451	−	0.941343i	$-0.390435\pi$
$859$	19.7805	0.674902	0.337451	+	0.941343i	$0.390435\pi$
$860$	−2.11365	−0.0720749
$861$	0.130174	0.00443632
$862$	27.7637	0.945637
$863$	14.6628	0.499127	0.249564	−	0.968358i	$-0.419713\pi$
$863$	14.6628	0.499127	0.249564	+	0.968358i	$0.419713\pi$
$864$	−4.61569	−0.157029
$865$	10.4838	0.356460
$866$	33.2059	1.12838
$867$	−14.4696	−0.491413
$868$	0.117786	0.00399792
$869$	26.9040	0.912656
$870$	−9.83353	−0.333388
$871$	8.12944	0.275456
$872$	−1.75757	−0.0595190
$873$	10.6103	0.359105
$874$	29.2191	0.988350
$875$	−0.135304	−0.00457410
$876$	−5.12026	−0.172998
$877$	−46.1016	−1.55674	−0.778371	−	0.627805i	$-0.783954\pi$
$877$	−46.1016	−1.55674	−0.778371	+	0.627805i	$0.783954\pi$
$878$	−9.17772	−0.309733
$879$	16.5327	0.557632
$880$	16.5168	0.556780
$881$	43.5241	1.46636	0.733181	−	0.680033i	$-0.238035\pi$
$881$	43.5241	1.46636	0.733181	+	0.680033i	$0.238035\pi$
$882$	11.8288	0.398298
$883$	−3.68445	−0.123992	−0.0619958	−	0.998076i	$-0.519747\pi$
$883$	−3.68445	−0.123992	−0.0619958	+	0.998076i	$0.519747\pi$
$884$	4.88349	0.164249
$885$	−0.413716	−0.0139069
$886$	6.09371	0.204722
$887$	5.94918	0.199754	0.0998770	−	0.995000i	$-0.468155\pi$
$887$	5.94918	0.199754	0.0998770	+	0.995000i	$0.468155\pi$
$888$	−8.92732	−0.299581
$889$	−0.0360425	−0.00120883
$890$	20.2535	0.678900
$891$	3.31446	0.111039
$892$	15.0453	0.503755
$893$	8.10058	0.271075
$894$	9.74024	0.325762
$895$	−8.81528	−0.294662
$896$	−1.74105	−0.0581645
$897$	−5.03004	−0.167948
$898$	−40.7261	−1.35905
$899$	5.80401	0.193575
$900$	0.870532	0.0290177
$901$	0.762410	0.0253996
$902$	5.40268	0.179890
$903$	−0.328518	−0.0109324
$904$	−22.8762	−0.760851
$905$	5.28207	0.175582
$906$	5.76775	0.191621
$907$	2.98997	0.0992803	0.0496401	−	0.998767i	$-0.484193\pi$
$907$	2.98997	0.0992803	0.0496401	+	0.998767i	$0.484193\pi$
$908$	20.9237	0.694377
$909$	−6.95733	−0.230760
$910$	−0.229240	−0.00759924
$911$	−1.06497	−0.0352839	−0.0176420	−	0.999844i	$-0.505616\pi$
$911$	−1.06497	−0.0352839	−0.0176420	+	0.999844i	$0.505616\pi$
$912$	17.0854	0.565754
$913$	1.31621	0.0435602
$914$	11.4558	0.378924
$915$	0.964304	0.0318789
$916$	11.4722	0.379053
$917$	−2.96981	−0.0980718
$918$	−9.50445	−0.313693
$919$	−13.9949	−0.461648	−0.230824	−	0.972995i	$-0.574142\pi$
$919$	−13.9949	−0.461648	−0.230824	+	0.972995i	$0.574142\pi$
$920$	9.62558	0.317346
$921$	−28.5203	−0.939776
$922$	−12.5879	−0.414560
$923$	−8.29872	−0.273156
$924$	−0.390398	−0.0128432
$925$	4.66515	0.153389
$926$	−6.03490	−0.198319
$927$	2.48748	0.0816996
$928$	26.7895	0.879409
$929$	−51.9425	−1.70418	−0.852089	−	0.523397i	$-0.824665\pi$
$929$	−51.9425	−1.70418	−0.852089	+	0.523397i	$0.824665\pi$
$930$	−1.69426	−0.0555571
$931$	−23.9373	−0.784512
$932$	−18.4614	−0.604724
$933$	5.50139	0.180107
$934$	−50.0236	−1.63682
$935$	18.5934	0.608069
$936$	−1.91362	−0.0625486
$937$	50.2202	1.64062	0.820312	−	0.571917i	$-0.193800\pi$
$937$	50.2202	1.64062	0.820312	+	0.571917i	$0.193800\pi$
$938$	1.86360	0.0608486
$939$	−20.3126	−0.662876
$940$	−2.05678	−0.0670846
$941$	0.402561	0.0131231	0.00656156	−	0.999978i	$-0.497911\pi$
$941$	0.402561	0.0131231	0.00656156	+	0.999978i	$0.497911\pi$
$942$	24.6864	0.804326
$943$	4.83935	0.157591
$944$	2.06165	0.0671009
$945$	0.135304	0.00440143
$946$	−13.6346	−0.443300
$947$	2.34743	0.0762812	0.0381406	−	0.999272i	$-0.487857\pi$
$947$	2.34743	0.0762812	0.0381406	+	0.999272i	$0.487857\pi$
$948$	−7.06624	−0.229501
$949$	−5.88176	−0.190930
$950$	−5.80891	−0.188466
$951$	24.4259	0.792065
$952$	−1.45248	−0.0470752
$953$	21.4204	0.693874	0.346937	−	0.937888i	$-0.387222\pi$
$953$	21.4204	0.693874	0.346937	+	0.937888i	$0.387222\pi$
$954$	0.230263	0.00745504
$955$	5.33531	0.172647
$956$	−14.8292	−0.479612
$957$	−19.2372	−0.621850
$958$	−9.25985	−0.299172
$959$	−0.767141	−0.0247723
$960$	2.14628	0.0692710
$961$	1.00000	0.0322581
$962$	7.90401	0.254835
$963$	−11.1920	−0.360656
$964$	9.61587	0.309706
$965$	−11.9621	−0.385072
$966$	−1.15309	−0.0371000
$967$	16.3587	0.526060	0.263030	−	0.964788i	$-0.415278\pi$
$967$	16.3587	0.526060	0.263030	+	0.964788i	$0.415278\pi$
$968$	−0.0274298	−0.000881629 0
$969$	19.2335	0.617870
$970$	17.9767	0.577196
$971$	57.9015	1.85815	0.929073	−	0.369896i	$-0.120607\pi$
$971$	57.9015	1.85815	0.929073	+	0.369896i	$0.120607\pi$
$972$	−0.870532	−0.0279223
$973$	−2.36404	−0.0757878
$974$	56.6200	1.81422
$975$	1.00000	0.0320256
$976$	−4.80536	−0.153816
$977$	21.0970	0.674954	0.337477	−	0.941334i	$-0.390426\pi$
$977$	21.0970	0.674954	0.337477	+	0.941334i	$0.390426\pi$
$978$	−40.8055	−1.30481
$979$	39.6217	1.26631
$980$	6.07778	0.194148
$981$	−0.918456	−0.0293240
$982$	−16.5693	−0.528749
$983$	−10.8035	−0.344577	−0.172289	−	0.985047i	$-0.555116\pi$
$983$	−10.8035	−0.344577	−0.172289	+	0.985047i	$0.555116\pi$
$984$	1.84107	0.0586911
$985$	−11.5010	−0.366451
$986$	55.1639	1.75678
$987$	−0.319678	−0.0101754
$988$	−2.98468	−0.0949554
$989$	−12.2130	−0.388349
$990$	5.61558	0.178475
$991$	−21.2902	−0.676305	−0.338152	−	0.941091i	$-0.609802\pi$
$991$	−21.2902	−0.676305	−0.338152	+	0.941091i	$0.609802\pi$
$992$	4.61569	0.146548
$993$	17.3011	0.549035
$994$	−1.90240	−0.0603405
$995$	−11.2253	−0.355867
$996$	−0.345698	−0.0109539
$997$	−7.31692	−0.231729	−0.115865	−	0.993265i	$-0.536964\pi$
$997$	−7.31692	−0.231729	−0.115865	+	0.993265i	$0.536964\pi$
$998$	20.3730	0.644896
$999$	−4.66515	−0.147599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.5	✓	16	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)

\(f(q)\)	\(=\)	\(q-1.69426 q^{2} -1.00000 q^{3} +0.870532 q^{4} -1.00000 q^{5} +1.69426 q^{6} +0.135304 q^{7} +1.91362 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.69426 q^{2} -1.00000 q^{3} +0.870532 q^{4} -1.00000 q^{5} +1.69426 q^{6} +0.135304 q^{7} +1.91362 q^{8} +1.00000 q^{9} +1.69426 q^{10} +3.31446 q^{11} -0.870532 q^{12} -1.00000 q^{13} -0.229240 q^{14} +1.00000 q^{15} -4.98324 q^{16} -5.60978 q^{17} -1.69426 q^{18} +3.42857 q^{19} -0.870532 q^{20} -0.135304 q^{21} -5.61558 q^{22} -5.03004 q^{23} -1.91362 q^{24} +1.00000 q^{25} +1.69426 q^{26} -1.00000 q^{27} +0.117786 q^{28} +5.80401 q^{29} -1.69426 q^{30} +1.00000 q^{31} +4.61569 q^{32} -3.31446 q^{33} +9.50445 q^{34} -0.135304 q^{35} +0.870532 q^{36} +4.66515 q^{37} -5.80891 q^{38} +1.00000 q^{39} -1.91362 q^{40} -0.962088 q^{41} +0.229240 q^{42} +2.42800 q^{43} +2.88535 q^{44} -1.00000 q^{45} +8.52223 q^{46} +2.36267 q^{47} +4.98324 q^{48} -6.98169 q^{49} -1.69426 q^{50} +5.60978 q^{51} -0.870532 q^{52} -0.135907 q^{53} +1.69426 q^{54} -3.31446 q^{55} +0.258920 q^{56} -3.42857 q^{57} -9.83353 q^{58} -0.413716 q^{59} +0.870532 q^{60} +0.964304 q^{61} -1.69426 q^{62} +0.135304 q^{63} +2.14628 q^{64} +1.00000 q^{65} +5.61558 q^{66} -8.12944 q^{67} -4.88349 q^{68} +5.03004 q^{69} +0.229240 q^{70} +8.29872 q^{71} +1.91362 q^{72} +5.88176 q^{73} -7.90401 q^{74} -1.00000 q^{75} +2.98468 q^{76} +0.448459 q^{77} -1.69426 q^{78} +8.11715 q^{79} +4.98324 q^{80} +1.00000 q^{81} +1.63003 q^{82} +0.397111 q^{83} -0.117786 q^{84} +5.60978 q^{85} -4.11368 q^{86} -5.80401 q^{87} +6.34262 q^{88} +11.9542 q^{89} +1.69426 q^{90} -0.135304 q^{91} -4.37881 q^{92} -1.00000 q^{93} -4.00298 q^{94} -3.42857 q^{95} -4.61569 q^{96} +10.6103 q^{97} +11.8288 q^{98} +3.31446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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