LMFDB - Embedded newform 6045.2.a.bg.1.7

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			1.7
Root			$0.976688$ 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)

$f(q)$	$=$	$q-0.976688 q^{2} -1.00000 q^{3} -1.04608 q^{4} -1.00000 q^{5} +0.976688 q^{6} +2.96131 q^{7} +2.97507 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.976688 q^{2} -1.00000 q^{3} -1.04608 q^{4} -1.00000 q^{5} +0.976688 q^{6} +2.96131 q^{7} +2.97507 q^{8} +1.00000 q^{9} +0.976688 q^{10} -4.81802 q^{11} +1.04608 q^{12} -1.00000 q^{13} -2.89227 q^{14} +1.00000 q^{15} -0.813553 q^{16} -6.91190 q^{17} -0.976688 q^{18} -7.22567 q^{19} +1.04608 q^{20} -2.96131 q^{21} +4.70570 q^{22} -8.09046 q^{23} -2.97507 q^{24} +1.00000 q^{25} +0.976688 q^{26} -1.00000 q^{27} -3.09777 q^{28} -7.96201 q^{29} -0.976688 q^{30} +1.00000 q^{31} -5.15555 q^{32} +4.81802 q^{33} +6.75077 q^{34} -2.96131 q^{35} -1.04608 q^{36} +7.72439 q^{37} +7.05723 q^{38} +1.00000 q^{39} -2.97507 q^{40} -1.33035 q^{41} +2.89227 q^{42} -0.349928 q^{43} +5.04004 q^{44} -1.00000 q^{45} +7.90185 q^{46} +2.48724 q^{47} +0.813553 q^{48} +1.76933 q^{49} -0.976688 q^{50} +6.91190 q^{51} +1.04608 q^{52} -4.30415 q^{53} +0.976688 q^{54} +4.81802 q^{55} +8.81009 q^{56} +7.22567 q^{57} +7.77640 q^{58} +11.3302 q^{59} -1.04608 q^{60} -6.25757 q^{61} -0.976688 q^{62} +2.96131 q^{63} +6.66247 q^{64} +1.00000 q^{65} -4.70570 q^{66} +4.18639 q^{67} +7.23040 q^{68} +8.09046 q^{69} +2.89227 q^{70} -13.4480 q^{71} +2.97507 q^{72} +9.74053 q^{73} -7.54431 q^{74} -1.00000 q^{75} +7.55864 q^{76} -14.2676 q^{77} -0.976688 q^{78} -15.9488 q^{79} +0.813553 q^{80} +1.00000 q^{81} +1.29934 q^{82} +9.09113 q^{83} +3.09777 q^{84} +6.91190 q^{85} +0.341770 q^{86} +7.96201 q^{87} -14.3340 q^{88} -5.47272 q^{89} +0.976688 q^{90} -2.96131 q^{91} +8.46327 q^{92} -1.00000 q^{93} -2.42926 q^{94} +7.22567 q^{95} +5.15555 q^{96} -1.05163 q^{97} -1.72809 q^{98} -4.81802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.976688	−0.690623	−0.345311	−	0.938488i	$-0.612227\pi$
$2$	−0.976688	−0.690623	−0.345311	+	0.938488i	$0.612227\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.04608	−0.523040
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.976688	0.398731
$7$	2.96131	1.11927	0.559634	−	0.828740i	$-0.310942\pi$
$7$	2.96131	1.11927	0.559634	+	0.828740i	$0.310942\pi$
$8$	2.97507	1.05185
$9$	1.00000	0.333333
$10$	0.976688	0.308856
$11$	−4.81802	−1.45269	−0.726344	−	0.687331i	$-0.758782\pi$
$11$	−4.81802	−1.45269	−0.726344	+	0.687331i	$0.758782\pi$
$12$	1.04608	0.301978
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.89227	−0.772992
$15$	1.00000	0.258199
$16$	−0.813553	−0.203388
$17$	−6.91190	−1.67638	−0.838191	−	0.545377i	$-0.816386\pi$
$17$	−6.91190	−1.67638	−0.838191	+	0.545377i	$0.816386\pi$
$18$	−0.976688	−0.230208
$19$	−7.22567	−1.65768	−0.828841	−	0.559484i	$-0.810999\pi$
$19$	−7.22567	−1.65768	−0.828841	+	0.559484i	$0.810999\pi$
$20$	1.04608	0.233911
$21$	−2.96131	−0.646210
$22$	4.70570	1.00326
$23$	−8.09046	−1.68698	−0.843488	−	0.537147i	$-0.819502\pi$
$23$	−8.09046	−1.68698	−0.843488	+	0.537147i	$0.819502\pi$
$24$	−2.97507	−0.607284
$25$	1.00000	0.200000
$26$	0.976688	0.191544
$27$	−1.00000	−0.192450
$28$	−3.09777	−0.585423
$29$	−7.96201	−1.47851	−0.739254	−	0.673427i	$-0.764822\pi$
$29$	−7.96201	−1.47851	−0.739254	+	0.673427i	$0.764822\pi$
$30$	−0.976688	−0.178318
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−5.15555	−0.911382
$33$	4.81802	0.838710
$34$	6.75077	1.15775
$35$	−2.96131	−0.500552
$36$	−1.04608	−0.174347
$37$	7.72439	1.26988	0.634941	−	0.772561i	$-0.281025\pi$
$37$	7.72439	1.26988	0.634941	+	0.772561i	$0.281025\pi$
$38$	7.05723	1.14483
$39$	1.00000	0.160128
$40$	−2.97507	−0.470400
$41$	−1.33035	−0.207766	−0.103883	−	0.994590i	$-0.533127\pi$
$41$	−1.33035	−0.207766	−0.103883	+	0.994590i	$0.533127\pi$
$42$	2.89227	0.446287
$43$	−0.349928	−0.0533635	−0.0266817	−	0.999644i	$-0.508494\pi$
$43$	−0.349928	−0.0533635	−0.0266817	+	0.999644i	$0.508494\pi$
$44$	5.04004	0.759815
$45$	−1.00000	−0.149071
$46$	7.90185	1.16506
$47$	2.48724	0.362802	0.181401	−	0.983409i	$-0.441937\pi$
$47$	2.48724	0.362802	0.181401	+	0.983409i	$0.441937\pi$
$48$	0.813553	0.117426
$49$	1.76933	0.252762
$50$	−0.976688	−0.138125
$51$	6.91190	0.967859
$52$	1.04608	0.145065
$53$	−4.30415	−0.591221	−0.295610	−	0.955309i	$-0.595523\pi$
$53$	−4.30415	−0.591221	−0.295610	+	0.955309i	$0.595523\pi$
$54$	0.976688	0.132910
$55$	4.81802	0.649662
$56$	8.81009	1.17730
$57$	7.22567	0.957064
$58$	7.77640	1.02109
$59$	11.3302	1.47506	0.737532	−	0.675312i	$-0.235991\pi$
$59$	11.3302	1.47506	0.737532	+	0.675312i	$0.235991\pi$
$60$	−1.04608	−0.135048
$61$	−6.25757	−0.801200	−0.400600	−	0.916253i	$-0.631198\pi$
$61$	−6.25757	−0.801200	−0.400600	+	0.916253i	$0.631198\pi$
$62$	−0.976688	−0.124039
$63$	2.96131	0.373089
$64$	6.66247	0.832809
$65$	1.00000	0.124035
$66$	−4.70570	−0.579232
$67$	4.18639	0.511449	0.255724	−	0.966750i	$-0.417686\pi$
$67$	4.18639	0.511449	0.255724	+	0.966750i	$0.417686\pi$
$68$	7.23040	0.876815
$69$	8.09046	0.973977
$70$	2.89227	0.345693
$71$	−13.4480	−1.59598	−0.797990	−	0.602671i	$-0.794103\pi$
$71$	−13.4480	−1.59598	−0.797990	+	0.602671i	$0.794103\pi$
$72$	2.97507	0.350615
$73$	9.74053	1.14004	0.570021	−	0.821630i	$-0.306935\pi$
$73$	9.74053	1.14004	0.570021	+	0.821630i	$0.306935\pi$
$74$	−7.54431	−0.877009
$75$	−1.00000	−0.115470
$76$	7.55864	0.867035
$77$	−14.2676	−1.62595
$78$	−0.976688	−0.110588
$79$	−15.9488	−1.79439	−0.897193	−	0.441639i	$-0.854397\pi$
$79$	−15.9488	−1.79439	−0.897193	+	0.441639i	$0.854397\pi$
$80$	0.813553	0.0909580
$81$	1.00000	0.111111
$82$	1.29934	0.143488
$83$	9.09113	0.997881	0.498940	−	0.866636i	$-0.333723\pi$
$83$	9.09113	0.997881	0.498940	+	0.866636i	$0.333723\pi$
$84$	3.09777	0.337994
$85$	6.91190	0.749700
$86$	0.341770	0.0368540
$87$	7.96201	0.853617
$88$	−14.3340	−1.52800
$89$	−5.47272	−0.580107	−0.290053	−	0.957010i	$-0.593673\pi$
$89$	−5.47272	−0.580107	−0.290053	+	0.957010i	$0.593673\pi$
$90$	0.976688	0.102952
$91$	−2.96131	−0.310429
$92$	8.46327	0.882357
$93$	−1.00000	−0.103695
$94$	−2.42926	−0.250559
$95$	7.22567	0.741338
$96$	5.15555	0.526186
$97$	−1.05163	−0.106777	−0.0533884	−	0.998574i	$-0.517002\pi$
$97$	−1.05163	−0.106777	−0.0533884	+	0.998574i	$0.517002\pi$
$98$	−1.72809	−0.174563
$99$	−4.81802	−0.484230
$100$	−1.04608	−0.104608
$101$	−5.21248	−0.518661	−0.259330	−	0.965789i	$-0.583502\pi$
$101$	−5.21248	−0.518661	−0.259330	+	0.965789i	$0.583502\pi$
$102$	−6.75077	−0.668425
$103$	−11.2781	−1.11126	−0.555631	−	0.831429i	$-0.687523\pi$
$103$	−11.2781	−1.11126	−0.555631	+	0.831429i	$0.687523\pi$
$104$	−2.97507	−0.291730
$105$	2.96131	0.288994
$106$	4.20381	0.408311
$107$	1.36309	0.131775	0.0658877	−	0.997827i	$-0.479012\pi$
$107$	1.36309	0.131775	0.0658877	+	0.997827i	$0.479012\pi$
$108$	1.04608	0.100659
$109$	−9.15273	−0.876673	−0.438336	−	0.898811i	$-0.644432\pi$
$109$	−9.15273	−0.876673	−0.438336	+	0.898811i	$0.644432\pi$
$110$	−4.70570	−0.448671
$111$	−7.72439	−0.733166
$112$	−2.40918	−0.227646
$113$	−5.60226	−0.527017	−0.263508	−	0.964657i	$-0.584880\pi$
$113$	−5.60226	−0.527017	−0.263508	+	0.964657i	$0.584880\pi$
$114$	−7.05723	−0.660970
$115$	8.09046	0.754439
$116$	8.32891	0.773320
$117$	−1.00000	−0.0924500
$118$	−11.0661	−1.01871
$119$	−20.4682	−1.87632
$120$	2.97507	0.271586
$121$	12.2133	1.11030
$122$	6.11170	0.553327
$123$	1.33035	0.119954
$124$	−1.04608	−0.0939408
$125$	−1.00000	−0.0894427
$126$	−2.89227	−0.257664
$127$	−12.7647	−1.13268	−0.566342	−	0.824170i	$-0.691642\pi$
$127$	−12.7647	−1.13268	−0.566342	+	0.824170i	$0.691642\pi$
$128$	3.80395	0.336225
$129$	0.349928	0.0308094
$130$	−0.976688	−0.0856612
$131$	17.0087	1.48606	0.743029	−	0.669259i	$-0.233388\pi$
$131$	17.0087	1.48606	0.743029	+	0.669259i	$0.233388\pi$
$132$	−5.04004	−0.438679
$133$	−21.3974	−1.85539
$134$	−4.08879	−0.353218
$135$	1.00000	0.0860663
$136$	−20.5634	−1.76330
$137$	1.17042	0.0999960	0.0499980	−	0.998749i	$-0.484079\pi$
$137$	1.17042	0.0999960	0.0499980	+	0.998749i	$0.484079\pi$
$138$	−7.90185	−0.672650
$139$	−3.63744	−0.308523	−0.154262	−	0.988030i	$-0.549300\pi$
$139$	−3.63744	−0.308523	−0.154262	+	0.988030i	$0.549300\pi$
$140$	3.09777	0.261809
$141$	−2.48724	−0.209464
$142$	13.1345	1.10222
$143$	4.81802	0.402903
$144$	−0.813553	−0.0677961
$145$	7.96201	0.661209
$146$	−9.51346	−0.787339
$147$	−1.76933	−0.145932
$148$	−8.08033	−0.664199
$149$	−9.36325	−0.767067	−0.383533	−	0.923527i	$-0.625293\pi$
$149$	−9.36325	−0.767067	−0.383533	+	0.923527i	$0.625293\pi$
$150$	0.976688	0.0797462
$151$	6.15047	0.500518	0.250259	−	0.968179i	$-0.419484\pi$
$151$	6.15047	0.500518	0.250259	+	0.968179i	$0.419484\pi$
$152$	−21.4969	−1.74363
$153$	−6.91190	−0.558794
$154$	13.9350	1.12292
$155$	−1.00000	−0.0803219
$156$	−1.04608	−0.0837535
$157$	21.7111	1.73274	0.866368	−	0.499405i	$-0.166448\pi$
$157$	21.7111	1.73274	0.866368	+	0.499405i	$0.166448\pi$
$158$	15.5770	1.23924
$159$	4.30415	0.341342
$160$	5.15555	0.407582
$161$	−23.9583	−1.88818
$162$	−0.976688	−0.0767358
$163$	22.7667	1.78323	0.891613	−	0.452798i	$-0.149574\pi$
$163$	22.7667	1.78323	0.891613	+	0.452798i	$0.149574\pi$
$164$	1.39165	0.108670
$165$	−4.81802	−0.375083
$166$	−8.87919	−0.689159
$167$	−16.3866	−1.26803	−0.634017	−	0.773319i	$-0.718595\pi$
$167$	−16.3866	−1.26803	−0.634017	+	0.773319i	$0.718595\pi$
$168$	−8.81009	−0.679713
$169$	1.00000	0.0769231
$170$	−6.75077	−0.517760
$171$	−7.22567	−0.552561
$172$	0.366052	0.0279112
$173$	−24.0541	−1.82880	−0.914400	−	0.404813i	$-0.867337\pi$
$173$	−24.0541	−1.82880	−0.914400	+	0.404813i	$0.867337\pi$
$174$	−7.77640	−0.589527
$175$	2.96131	0.223854
$176$	3.91972	0.295460
$177$	−11.3302	−0.851629
$178$	5.34514	0.400635
$179$	25.3022	1.89118	0.945589	−	0.325364i	$-0.105487\pi$
$179$	25.3022	1.89118	0.945589	+	0.325364i	$0.105487\pi$
$180$	1.04608	0.0779703
$181$	2.92490	0.217406	0.108703	−	0.994074i	$-0.465330\pi$
$181$	2.92490	0.217406	0.108703	+	0.994074i	$0.465330\pi$
$182$	2.89227	0.214389
$183$	6.25757	0.462573
$184$	−24.0697	−1.77444
$185$	−7.72439	−0.567908
$186$	0.976688	0.0716142
$187$	33.3017	2.43526
$188$	−2.60186	−0.189760
$189$	−2.96131	−0.215403
$190$	−7.05723	−0.511985
$191$	10.6354	0.769547	0.384774	−	0.923011i	$-0.374280\pi$
$191$	10.6354	0.769547	0.384774	+	0.923011i	$0.374280\pi$
$192$	−6.66247	−0.480823
$193$	1.29366	0.0931194	0.0465597	−	0.998916i	$-0.485174\pi$
$193$	1.29366	0.0931194	0.0465597	+	0.998916i	$0.485174\pi$
$194$	1.02711	0.0737425
$195$	−1.00000	−0.0716115
$196$	−1.85087	−0.132205
$197$	14.5346	1.03555	0.517774	−	0.855517i	$-0.326761\pi$
$197$	14.5346	1.03555	0.517774	+	0.855517i	$0.326761\pi$
$198$	4.70570	0.334420
$199$	0.626115	0.0443841	0.0221920	−	0.999754i	$-0.492935\pi$
$199$	0.626115	0.0443841	0.0221920	+	0.999754i	$0.492935\pi$
$200$	2.97507	0.210369
$201$	−4.18639	−0.295285
$202$	5.09096	0.358199
$203$	−23.5780	−1.65485
$204$	−7.23040	−0.506229
$205$	1.33035	0.0929157
$206$	11.0152	0.767462
$207$	−8.09046	−0.562326
$208$	0.813553	0.0564098
$209$	34.8134	2.40810
$210$	−2.89227	−0.199586
$211$	−25.9584	−1.78705	−0.893525	−	0.449013i	$-0.851776\pi$
$211$	−25.9584	−1.78705	−0.893525	+	0.449013i	$0.851776\pi$
$212$	4.50249	0.309232
$213$	13.4480	0.921439
$214$	−1.33132	−0.0910070
$215$	0.349928	0.0238649
$216$	−2.97507	−0.202428
$217$	2.96131	0.201027
$218$	8.93936	0.605450
$219$	−9.74053	−0.658204
$220$	−5.04004	−0.339800
$221$	6.91190	0.464945
$222$	7.54431	0.506341
$223$	−17.1646	−1.14943	−0.574714	−	0.818354i	$-0.694887\pi$
$223$	−17.1646	−1.14943	−0.574714	+	0.818354i	$0.694887\pi$
$224$	−15.2672	−1.02008
$225$	1.00000	0.0666667
$226$	5.47166	0.363970
$227$	3.97453	0.263799	0.131899	−	0.991263i	$-0.457892\pi$
$227$	3.97453	0.263799	0.131899	+	0.991263i	$0.457892\pi$
$228$	−7.55864	−0.500583
$229$	−25.5802	−1.69039	−0.845195	−	0.534459i	$-0.820516\pi$
$229$	−25.5802	−1.69039	−0.845195	+	0.534459i	$0.820516\pi$
$230$	−7.90185	−0.521033
$231$	14.2676	0.938742
$232$	−23.6875	−1.55516
$233$	14.4781	0.948492	0.474246	−	0.880392i	$-0.342721\pi$
$233$	14.4781	0.948492	0.474246	+	0.880392i	$0.342721\pi$
$234$	0.976688	0.0638481
$235$	−2.48724	−0.162250
$236$	−11.8523	−0.771519
$237$	15.9488	1.03599
$238$	19.9911	1.29583
$239$	29.7734	1.92588	0.962940	−	0.269716i	$-0.0869297\pi$
$239$	29.7734	1.92588	0.962940	+	0.269716i	$0.0869297\pi$
$240$	−0.813553	−0.0525146
$241$	14.1848	0.913721	0.456860	−	0.889538i	$-0.348974\pi$
$241$	14.1848	0.913721	0.456860	+	0.889538i	$0.348974\pi$
$242$	−11.9286	−0.766801
$243$	−1.00000	−0.0641500
$244$	6.54593	0.419060
$245$	−1.76933	−0.113039
$246$	−1.29934	−0.0828427
$247$	7.22567	0.459758
$248$	2.97507	0.188917
$249$	−9.09113	−0.576127
$250$	0.976688	0.0617712
$251$	1.42495	0.0899421	0.0449711	−	0.998988i	$-0.485680\pi$
$251$	1.42495	0.0899421	0.0449711	+	0.998988i	$0.485680\pi$
$252$	−3.09777	−0.195141
$253$	38.9800	2.45065
$254$	12.4671	0.782257
$255$	−6.91190	−0.432840
$256$	−17.0402	−1.06501
$257$	−29.1468	−1.81813	−0.909063	−	0.416659i	$-0.863201\pi$
$257$	−29.1468	−1.81813	−0.909063	+	0.416659i	$0.863201\pi$
$258$	−0.341770	−0.0212777
$259$	22.8743	1.42134
$260$	−1.04608	−0.0648752
$261$	−7.96201	−0.492836
$262$	−16.6122	−1.02631
$263$	−6.37879	−0.393333	−0.196667	−	0.980470i	$-0.563012\pi$
$263$	−6.37879	−0.393333	−0.196667	+	0.980470i	$0.563012\pi$
$264$	14.3340	0.882194
$265$	4.30415	0.264402
$266$	20.8986	1.28138
$267$	5.47272	0.334925
$268$	−4.37930	−0.267508
$269$	−9.09766	−0.554694	−0.277347	−	0.960770i	$-0.589455\pi$
$269$	−9.09766	−0.554694	−0.277347	+	0.960770i	$0.589455\pi$
$270$	−0.976688	−0.0594393
$271$	0.0988983	0.00600765	0.00300382	−	0.999995i	$-0.499044\pi$
$271$	0.0988983	0.00600765	0.00300382	+	0.999995i	$0.499044\pi$
$272$	5.62320	0.340956
$273$	2.96131	0.179226
$274$	−1.14314	−0.0690595
$275$	−4.81802	−0.290538
$276$	−8.46327	−0.509429
$277$	21.3674	1.28385	0.641923	−	0.766769i	$-0.278137\pi$
$277$	21.3674	1.28385	0.641923	+	0.766769i	$0.278137\pi$
$278$	3.55264	0.213073
$279$	1.00000	0.0598684
$280$	−8.81009	−0.526504
$281$	12.7157	0.758553	0.379277	−	0.925283i	$-0.376173\pi$
$281$	12.7157	0.758553	0.379277	+	0.925283i	$0.376173\pi$
$282$	2.42926	0.144660
$283$	1.49705	0.0889903	0.0444951	−	0.999010i	$-0.485832\pi$
$283$	1.49705	0.0889903	0.0444951	+	0.999010i	$0.485832\pi$
$284$	14.0677	0.834762
$285$	−7.22567	−0.428012
$286$	−4.70570	−0.278254
$287$	−3.93958	−0.232546
$288$	−5.15555	−0.303794
$289$	30.7743	1.81025
$290$	−7.77640	−0.456646
$291$	1.05163	0.0616477
$292$	−10.1894	−0.596288
$293$	−6.46984	−0.377972	−0.188986	−	0.981980i	$-0.560520\pi$
$293$	−6.46984	−0.377972	−0.188986	+	0.981980i	$0.560520\pi$
$294$	1.72809	0.100784
$295$	−11.3302	−0.659669
$296$	22.9806	1.33572
$297$	4.81802	0.279570
$298$	9.14497	0.529754
$299$	8.09046	0.467883
$300$	1.04608	0.0603955
$301$	−1.03624	−0.0597280
$302$	−6.00709	−0.345669
$303$	5.21248	0.299449
$304$	5.87847	0.337153
$305$	6.25757	0.358308
$306$	6.75077	0.385916
$307$	5.47948	0.312731	0.156365	−	0.987699i	$-0.450022\pi$
$307$	5.47948	0.312731	0.156365	+	0.987699i	$0.450022\pi$
$308$	14.9251	0.850437
$309$	11.2781	0.641587
$310$	0.976688	0.0554721
$311$	−2.63594	−0.149470	−0.0747351	−	0.997203i	$-0.523811\pi$
$311$	−2.63594	−0.149470	−0.0747351	+	0.997203i	$0.523811\pi$
$312$	2.97507	0.168430
$313$	−10.9369	−0.618189	−0.309094	−	0.951031i	$-0.600026\pi$
$313$	−10.9369	−0.618189	−0.309094	+	0.951031i	$0.600026\pi$
$314$	−21.2050	−1.19667
$315$	−2.96131	−0.166851
$316$	16.6838	0.938536
$317$	−19.5823	−1.09985	−0.549925	−	0.835214i	$-0.685344\pi$
$317$	−19.5823	−1.09985	−0.549925	+	0.835214i	$0.685344\pi$
$318$	−4.20381	−0.235738
$319$	38.3612	2.14781
$320$	−6.66247	−0.372444
$321$	−1.36309	−0.0760805
$322$	23.3998	1.30402
$323$	49.9431	2.77891
$324$	−1.04608	−0.0581156
$325$	−1.00000	−0.0554700
$326$	−22.2360	−1.23154
$327$	9.15273	0.506147
$328$	−3.95789	−0.218538
$329$	7.36549	0.406073
$330$	4.70570	0.259041
$331$	11.3046	0.621359	0.310680	−	0.950515i	$-0.399443\pi$
$331$	11.3046	0.621359	0.310680	+	0.950515i	$0.399443\pi$
$332$	−9.51005	−0.521932
$333$	7.72439	0.423294
$334$	16.0046	0.875733
$335$	−4.18639	−0.228727
$336$	2.40918	0.131432
$337$	18.6419	1.01549	0.507745	−	0.861507i	$-0.330479\pi$
$337$	18.6419	1.01549	0.507745	+	0.861507i	$0.330479\pi$
$338$	−0.976688	−0.0531248
$339$	5.60226	0.304273
$340$	−7.23040	−0.392124
$341$	−4.81802	−0.260911
$342$	7.05723	0.381611
$343$	−15.4896	−0.836360
$344$	−1.04106	−0.0561301
$345$	−8.09046	−0.435576
$346$	23.4933	1.26301
$347$	−6.99334	−0.375422	−0.187711	−	0.982224i	$-0.560107\pi$
$347$	−6.99334	−0.375422	−0.187711	+	0.982224i	$0.560107\pi$
$348$	−8.32891	−0.446476
$349$	3.96415	0.212196	0.106098	−	0.994356i	$-0.466164\pi$
$349$	3.96415	0.212196	0.106098	+	0.994356i	$0.466164\pi$
$350$	−2.89227	−0.154598
$351$	1.00000	0.0533761
$352$	24.8396	1.32395
$353$	6.48003	0.344897	0.172449	−	0.985019i	$-0.444832\pi$
$353$	6.48003	0.344897	0.172449	+	0.985019i	$0.444832\pi$
$354$	11.0661	0.588154
$355$	13.4480	0.713744
$356$	5.72491	0.303419
$357$	20.4682	1.08329
$358$	−24.7124	−1.30609
$359$	−15.2533	−0.805040	−0.402520	−	0.915411i	$-0.631866\pi$
$359$	−15.2533	−0.805040	−0.402520	+	0.915411i	$0.631866\pi$
$360$	−2.97507	−0.156800
$361$	33.2103	1.74791
$362$	−2.85672	−0.150146
$363$	−12.2133	−0.641034
$364$	3.09777	0.162367
$365$	−9.74053	−0.509843
$366$	−6.11170	−0.319463
$367$	−2.06420	−0.107750	−0.0538751	−	0.998548i	$-0.517157\pi$
$367$	−2.06420	−0.107750	−0.0538751	+	0.998548i	$0.517157\pi$
$368$	6.58202	0.343111
$369$	−1.33035	−0.0692553
$370$	7.54431	0.392210
$371$	−12.7459	−0.661735
$372$	1.04608	0.0542368
$373$	−37.3442	−1.93361	−0.966806	−	0.255512i	$-0.917756\pi$
$373$	−37.3442	−1.93361	−0.966806	+	0.255512i	$0.917756\pi$
$374$	−32.5253	−1.68185
$375$	1.00000	0.0516398
$376$	7.39972	0.381612
$377$	7.96201	0.410064
$378$	2.89227	0.148762
$379$	29.9180	1.53678	0.768392	−	0.639980i	$-0.221057\pi$
$379$	29.9180	1.53678	0.768392	+	0.639980i	$0.221057\pi$
$380$	−7.55864	−0.387750
$381$	12.7647	0.653956
$382$	−10.3874	−0.531467
$383$	−20.8021	−1.06294	−0.531469	−	0.847078i	$-0.678360\pi$
$383$	−20.8021	−1.06294	−0.531469	+	0.847078i	$0.678360\pi$
$384$	−3.80395	−0.194119
$385$	14.2676	0.727146
$386$	−1.26350	−0.0643104
$387$	−0.349928	−0.0177878
$388$	1.10009	0.0558486
$389$	14.3129	0.725694	0.362847	−	0.931849i	$-0.381805\pi$
$389$	14.3129	0.725694	0.362847	+	0.931849i	$0.381805\pi$
$390$	0.976688	0.0494565
$391$	55.9204	2.82802
$392$	5.26389	0.265867
$393$	−17.0087	−0.857976
$394$	−14.1958	−0.715173
$395$	15.9488	0.802474
$396$	5.04004	0.253272
$397$	−15.9833	−0.802177	−0.401088	−	0.916039i	$-0.631368\pi$
$397$	−15.9833	−0.802177	−0.401088	+	0.916039i	$0.631368\pi$
$398$	−0.611518	−0.0306527
$399$	21.3974	1.07121
$400$	−0.813553	−0.0406777
$401$	11.2651	0.562552	0.281276	−	0.959627i	$-0.409242\pi$
$401$	11.2651	0.562552	0.281276	+	0.959627i	$0.409242\pi$
$402$	4.08879	0.203930
$403$	−1.00000	−0.0498135
$404$	5.45267	0.271281
$405$	−1.00000	−0.0496904
$406$	23.0283	1.14288
$407$	−37.2163	−1.84474
$408$	20.5634	1.01804
$409$	−33.3806	−1.65056	−0.825282	−	0.564720i	$-0.808984\pi$
$409$	−33.3806	−1.65056	−0.825282	+	0.564720i	$0.808984\pi$
$410$	−1.29934	−0.0641697
$411$	−1.17042	−0.0577327
$412$	11.7978	0.581235
$413$	33.5522	1.65099
$414$	7.90185	0.388355
$415$	−9.09113	−0.446266
$416$	5.15555	0.252772
$417$	3.63744	0.178126
$418$	−34.0019	−1.66309
$419$	24.0804	1.17641	0.588203	−	0.808713i	$-0.299836\pi$
$419$	24.0804	1.17641	0.588203	+	0.808713i	$0.299836\pi$
$420$	−3.09777	−0.151155
$421$	34.1417	1.66397	0.831983	−	0.554802i	$-0.187206\pi$
$421$	34.1417	1.66397	0.831983	+	0.554802i	$0.187206\pi$
$422$	25.3533	1.23418
$423$	2.48724	0.120934
$424$	−12.8052	−0.621873
$425$	−6.91190	−0.335276
$426$	−13.1345	−0.636367
$427$	−18.5306	−0.896758
$428$	−1.42591	−0.0689238
$429$	−4.81802	−0.232616
$430$	−0.341770	−0.0164816
$431$	15.1844	0.731406	0.365703	−	0.930732i	$-0.380829\pi$
$431$	15.1844	0.731406	0.365703	+	0.930732i	$0.380829\pi$
$432$	0.813553	0.0391421
$433$	−29.9153	−1.43764	−0.718818	−	0.695198i	$-0.755317\pi$
$433$	−29.9153	−1.43764	−0.718818	+	0.695198i	$0.755317\pi$
$434$	−2.89227	−0.138833
$435$	−7.96201	−0.381749
$436$	9.57450	0.458535
$437$	58.4590	2.79647
$438$	9.51346	0.454571
$439$	−7.81105	−0.372801	−0.186400	−	0.982474i	$-0.559682\pi$
$439$	−7.81105	−0.372801	−0.186400	+	0.982474i	$0.559682\pi$
$440$	14.3340	0.683345
$441$	1.76933	0.0842540
$442$	−6.75077	−0.321101
$443$	−35.3255	−1.67837	−0.839183	−	0.543849i	$-0.816966\pi$
$443$	−35.3255	−1.67837	−0.839183	+	0.543849i	$0.816966\pi$
$444$	8.08033	0.383476
$445$	5.47272	0.259432
$446$	16.7645	0.793821
$447$	9.36325	0.442866
$448$	19.7296	0.932137
$449$	−14.0205	−0.661667	−0.330834	−	0.943689i	$-0.607330\pi$
$449$	−14.0205	−0.661667	−0.330834	+	0.943689i	$0.607330\pi$
$450$	−0.976688	−0.0460415
$451$	6.40966	0.301819
$452$	5.86042	0.275651
$453$	−6.15047	−0.288974
$454$	−3.88187	−0.182185
$455$	2.96131	0.138828
$456$	21.4969	1.00668
$457$	22.9567	1.07387	0.536933	−	0.843625i	$-0.319583\pi$
$457$	22.9567	1.07387	0.536933	+	0.843625i	$0.319583\pi$
$458$	24.9839	1.16742
$459$	6.91190	0.322620
$460$	−8.46327	−0.394602
$461$	2.99015	0.139265	0.0696325	−	0.997573i	$-0.477817\pi$
$461$	2.99015	0.139265	0.0696325	+	0.997573i	$0.477817\pi$
$462$	−13.9350	−0.648316
$463$	5.01001	0.232835	0.116418	−	0.993200i	$-0.462859\pi$
$463$	5.01001	0.232835	0.116418	+	0.993200i	$0.462859\pi$
$464$	6.47752	0.300711
$465$	1.00000	0.0463739
$466$	−14.1406	−0.655050
$467$	36.7292	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$467$	36.7292	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$468$	1.04608	0.0483551
$469$	12.3972	0.572448
$470$	2.42926	0.112053
$471$	−21.7111	−1.00040
$472$	33.7081	1.55154
$473$	1.68596	0.0775205
$474$	−15.5770	−0.715478
$475$	−7.22567	−0.331537
$476$	21.4114	0.981392
$477$	−4.30415	−0.197074
$478$	−29.0793	−1.33006
$479$	3.95141	0.180545	0.0902724	−	0.995917i	$-0.471226\pi$
$479$	3.95141	0.180545	0.0902724	+	0.995917i	$0.471226\pi$
$480$	−5.15555	−0.235318
$481$	−7.72439	−0.352202
$482$	−13.8541	−0.631036
$483$	23.9583	1.09014
$484$	−12.7761	−0.580734
$485$	1.05163	0.0477521
$486$	0.976688	0.0443035
$487$	18.9523	0.858809	0.429405	−	0.903112i	$-0.358723\pi$
$487$	18.9523	0.858809	0.429405	+	0.903112i	$0.358723\pi$
$488$	−18.6167	−0.842739
$489$	−22.7667	−1.02955
$490$	1.72809	0.0780670
$491$	−30.7793	−1.38905	−0.694525	−	0.719468i	$-0.744386\pi$
$491$	−30.7793	−1.38905	−0.694525	+	0.719468i	$0.744386\pi$
$492$	−1.39165	−0.0627406
$493$	55.0326	2.47854
$494$	−7.05723	−0.317520
$495$	4.81802	0.216554
$496$	−0.813553	−0.0365296
$497$	−39.8235	−1.78633
$498$	8.87919	0.397886
$499$	−11.9660	−0.535672	−0.267836	−	0.963465i	$-0.586309\pi$
$499$	−11.9660	−0.535672	−0.267836	+	0.963465i	$0.586309\pi$
$500$	1.04608	0.0467822
$501$	16.3866	0.732100
$502$	−1.39173	−0.0621160
$503$	11.0879	0.494385	0.247192	−	0.968966i	$-0.420492\pi$
$503$	11.0879	0.494385	0.247192	+	0.968966i	$0.420492\pi$
$504$	8.81009	0.392433
$505$	5.21248	0.231952
$506$	−38.0713	−1.69248
$507$	−1.00000	−0.0444116
$508$	13.3529	0.592440
$509$	29.9740	1.32857	0.664287	−	0.747478i	$-0.268735\pi$
$509$	29.9740	1.32857	0.664287	+	0.747478i	$0.268735\pi$
$510$	6.75077	0.298929
$511$	28.8447	1.27601
$512$	9.03507	0.399298
$513$	7.22567	0.319021
$514$	28.4673	1.25564
$515$	11.2781	0.496971
$516$	−0.366052	−0.0161146
$517$	−11.9836	−0.527038
$518$	−22.3410	−0.981608
$519$	24.0541	1.05586
$520$	2.97507	0.130465
$521$	24.5023	1.07346	0.536732	−	0.843752i	$-0.319658\pi$
$521$	24.5023	1.07346	0.536732	+	0.843752i	$0.319658\pi$
$522$	7.77640	0.340364
$523$	−9.97694	−0.436261	−0.218130	−	0.975920i	$-0.569996\pi$
$523$	−9.97694	−0.436261	−0.218130	+	0.975920i	$0.569996\pi$
$524$	−17.7925	−0.777269
$525$	−2.96131	−0.129242
$526$	6.23009	0.271645
$527$	−6.91190	−0.301087
$528$	−3.91972	−0.170584
$529$	42.4555	1.84589
$530$	−4.20381	−0.182602
$531$	11.3302	0.491688
$532$	22.3834	0.970445
$533$	1.33035	0.0576239
$534$	−5.34514	−0.231307
$535$	−1.36309	−0.0589317
$536$	12.4548	0.537965
$537$	−25.3022	−1.09187
$538$	8.88557	0.383084
$539$	−8.52469	−0.367184
$540$	−1.04608	−0.0450162
$541$	−16.9922	−0.730553	−0.365277	−	0.930899i	$-0.619026\pi$
$541$	−16.9922	−0.730553	−0.365277	+	0.930899i	$0.619026\pi$
$542$	−0.0965928	−0.00414902
$543$	−2.92490	−0.125520
$544$	35.6346	1.52782
$545$	9.15273	0.392060
$546$	−2.89227	−0.123778
$547$	12.8885	0.551072	0.275536	−	0.961291i	$-0.411145\pi$
$547$	12.8885	0.551072	0.275536	+	0.961291i	$0.411145\pi$
$548$	−1.22436	−0.0523019
$549$	−6.25757	−0.267067
$550$	4.70570	0.200652
$551$	57.5309	2.45090
$552$	24.0697	1.02447
$553$	−47.2294	−2.00840
$554$	−20.8693	−0.886653
$555$	7.72439	0.327882
$556$	3.80505	0.161370
$557$	−18.6181	−0.788874	−0.394437	−	0.918923i	$-0.629060\pi$
$557$	−18.6181	−0.788874	−0.394437	+	0.918923i	$0.629060\pi$
$558$	−0.976688	−0.0413465
$559$	0.349928	0.0148004
$560$	2.40918	0.101806
$561$	−33.3017	−1.40600
$562$	−12.4192	−0.523874
$563$	−29.1019	−1.22650	−0.613250	−	0.789889i	$-0.710138\pi$
$563$	−29.1019	−1.22650	−0.613250	+	0.789889i	$0.710138\pi$
$564$	2.60186	0.109558
$565$	5.60226	0.235689
$566$	−1.46215	−0.0614587
$567$	2.96131	0.124363
$568$	−40.0086	−1.67872
$569$	−32.7907	−1.37466	−0.687328	−	0.726347i	$-0.741217\pi$
$569$	−32.7907	−1.37466	−0.687328	+	0.726347i	$0.741217\pi$
$570$	7.05723	0.295595
$571$	−39.5842	−1.65655	−0.828273	−	0.560325i	$-0.810676\pi$
$571$	−39.5842	−1.65655	−0.828273	+	0.560325i	$0.810676\pi$
$572$	−5.04004	−0.210735
$573$	−10.6354	−0.444298
$574$	3.84774	0.160601
$575$	−8.09046	−0.337395
$576$	6.66247	0.277603
$577$	−15.2497	−0.634855	−0.317428	−	0.948283i	$-0.602819\pi$
$577$	−15.2497	−0.634855	−0.317428	+	0.948283i	$0.602819\pi$
$578$	−30.0569	−1.25020
$579$	−1.29366	−0.0537625
$580$	−8.32891	−0.345839
$581$	26.9216	1.11690
$582$	−1.02711	−0.0425753
$583$	20.7375	0.858860
$584$	28.9788	1.19915
$585$	1.00000	0.0413449
$586$	6.31901	0.261036
$587$	32.9657	1.36064	0.680320	−	0.732915i	$-0.261841\pi$
$587$	32.9657	1.36064	0.680320	+	0.732915i	$0.261841\pi$
$588$	1.85087	0.0763284
$589$	−7.22567	−0.297729
$590$	11.0661	0.455582
$591$	−14.5346	−0.597874
$592$	−6.28420	−0.258279
$593$	−38.4872	−1.58048	−0.790240	−	0.612798i	$-0.790044\pi$
$593$	−38.4872	−1.58048	−0.790240	+	0.612798i	$0.790044\pi$
$594$	−4.70570	−0.193077
$595$	20.4682	0.839116
$596$	9.79471	0.401207
$597$	−0.626115	−0.0256252
$598$	−7.90185	−0.323131
$599$	6.13508	0.250673	0.125336	−	0.992114i	$-0.459999\pi$
$599$	6.13508	0.250673	0.125336	+	0.992114i	$0.459999\pi$
$600$	−2.97507	−0.121457
$601$	−11.3301	−0.462165	−0.231083	−	0.972934i	$-0.574227\pi$
$601$	−11.3301	−0.462165	−0.231083	+	0.972934i	$0.574227\pi$
$602$	1.01209	0.0412495
$603$	4.18639	0.170483
$604$	−6.43389	−0.261791
$605$	−12.2133	−0.496543
$606$	−5.09096	−0.206806
$607$	−3.34266	−0.135674	−0.0678372	−	0.997696i	$-0.521610\pi$
$607$	−3.34266	−0.135674	−0.0678372	+	0.997696i	$0.521610\pi$
$608$	37.2523	1.51078
$609$	23.5780	0.955427
$610$	−6.11170	−0.247455
$611$	−2.48724	−0.100623
$612$	7.23040	0.292272
$613$	9.28443	0.374995	0.187497	−	0.982265i	$-0.439962\pi$
$613$	9.28443	0.374995	0.187497	+	0.982265i	$0.439962\pi$
$614$	−5.35175	−0.215979
$615$	−1.33035	−0.0536449
$616$	−42.4472	−1.71025
$617$	2.36064	0.0950357	0.0475179	−	0.998870i	$-0.484869\pi$
$617$	2.36064	0.0950357	0.0475179	+	0.998870i	$0.484869\pi$
$618$	−11.0152	−0.443095
$619$	14.0291	0.563876	0.281938	−	0.959433i	$-0.409023\pi$
$619$	14.0291	0.563876	0.281938	+	0.959433i	$0.409023\pi$
$620$	1.04608	0.0420116
$621$	8.09046	0.324659
$622$	2.57449	0.103228
$623$	−16.2064	−0.649295
$624$	−0.813553	−0.0325682
$625$	1.00000	0.0400000
$626$	10.6819	0.426935
$627$	−34.8134	−1.39032
$628$	−22.7116	−0.906291
$629$	−53.3902	−2.12880
$630$	2.89227	0.115231
$631$	−35.1377	−1.39881	−0.699404	−	0.714727i	$-0.746551\pi$
$631$	−35.1377	−1.39881	−0.699404	+	0.714727i	$0.746551\pi$
$632$	−47.4489	−1.88742
$633$	25.9584	1.03175
$634$	19.1258	0.759582
$635$	12.7647	0.506552
$636$	−4.50249	−0.178535
$637$	−1.76933	−0.0701035
$638$	−37.4669	−1.48333
$639$	−13.4480	−0.531993
$640$	−3.80395	−0.150364
$641$	−37.5265	−1.48221	−0.741104	−	0.671390i	$-0.765698\pi$
$641$	−37.5265	−1.48221	−0.741104	+	0.671390i	$0.765698\pi$
$642$	1.33132	0.0525429
$643$	−5.44983	−0.214920	−0.107460	−	0.994209i	$-0.534272\pi$
$643$	−5.44983	−0.214920	−0.107460	+	0.994209i	$0.534272\pi$
$644$	25.0623	0.987594
$645$	−0.349928	−0.0137784
$646$	−48.7788	−1.91918
$647$	−20.9631	−0.824146	−0.412073	−	0.911151i	$-0.635195\pi$
$647$	−20.9631	−0.824146	−0.412073	+	0.911151i	$0.635195\pi$
$648$	2.97507	0.116872
$649$	−54.5891	−2.14281
$650$	0.976688	0.0383088
$651$	−2.96131	−0.116063
$652$	−23.8158	−0.932700
$653$	−14.8303	−0.580353	−0.290176	−	0.956973i	$-0.593714\pi$
$653$	−14.8303	−0.580353	−0.290176	+	0.956973i	$0.593714\pi$
$654$	−8.93936	−0.349557
$655$	−17.0087	−0.664586
$656$	1.08231	0.0422572
$657$	9.74053	0.380014
$658$	−7.19378	−0.280443
$659$	3.37057	0.131299	0.0656494	−	0.997843i	$-0.479088\pi$
$659$	3.37057	0.131299	0.0656494	+	0.997843i	$0.479088\pi$
$660$	5.04004	0.196183
$661$	32.7013	1.27193	0.635966	−	0.771717i	$-0.280602\pi$
$661$	32.7013	1.27193	0.635966	+	0.771717i	$0.280602\pi$
$662$	−11.0411	−0.429125
$663$	−6.91190	−0.268436
$664$	27.0467	1.04962
$665$	21.3974	0.829756
$666$	−7.54431	−0.292336
$667$	64.4163	2.49421
$668$	17.1417	0.663233
$669$	17.1646	0.663623
$670$	4.08879	0.157964
$671$	30.1491	1.16389
$672$	15.2672	0.588944
$673$	18.6444	0.718689	0.359345	−	0.933205i	$-0.383000\pi$
$673$	18.6444	0.718689	0.359345	+	0.933205i	$0.383000\pi$
$674$	−18.2073	−0.701321
$675$	−1.00000	−0.0384900
$676$	−1.04608	−0.0402339
$677$	14.9405	0.574212	0.287106	−	0.957899i	$-0.407307\pi$
$677$	14.9405	0.574212	0.287106	+	0.957899i	$0.407307\pi$
$678$	−5.47166	−0.210138
$679$	−3.11420	−0.119512
$680$	20.5634	0.788570
$681$	−3.97453	−0.152304
$682$	4.70570	0.180191
$683$	45.4183	1.73788	0.868942	−	0.494914i	$-0.164800\pi$
$683$	45.4183	1.73788	0.868942	+	0.494914i	$0.164800\pi$
$684$	7.55864	0.289012
$685$	−1.17042	−0.0447196
$686$	15.1285	0.577609
$687$	25.5802	0.975947
$688$	0.284685	0.0108535
$689$	4.30415	0.163975
$690$	7.90185	0.300818
$691$	36.5592	1.39078	0.695389	−	0.718634i	$-0.255232\pi$
$691$	36.5592	1.39078	0.695389	+	0.718634i	$0.255232\pi$
$692$	25.1625	0.956536
$693$	−14.2676	−0.541983
$694$	6.83031	0.259275
$695$	3.63744	0.137976
$696$	23.6875	0.897874
$697$	9.19525	0.348295
$698$	−3.87173	−0.146547
$699$	−14.4781	−0.547612
$700$	−3.09777	−0.117085
$701$	27.9145	1.05432	0.527158	−	0.849767i	$-0.323257\pi$
$701$	27.9145	1.05432	0.527158	+	0.849767i	$0.323257\pi$
$702$	−0.976688	−0.0368627
$703$	−55.8139	−2.10506
$704$	−32.0999	−1.20981
$705$	2.48724	0.0936750
$706$	−6.32897	−0.238194
$707$	−15.4357	−0.580521
$708$	11.8523	0.445436
$709$	31.7824	1.19361	0.596806	−	0.802385i	$-0.296436\pi$
$709$	31.7824	1.19361	0.596806	+	0.802385i	$0.296436\pi$
$710$	−13.1345	−0.492927
$711$	−15.9488	−0.598129
$712$	−16.2817	−0.610183
$713$	−8.09046	−0.302990
$714$	−19.9911	−0.748148
$715$	−4.81802	−0.180184
$716$	−26.4682	−0.989162
$717$	−29.7734	−1.11191
$718$	14.8977	0.555979
$719$	−33.7869	−1.26004	−0.630020	−	0.776579i	$-0.716953\pi$
$719$	−33.7869	−1.26004	−0.630020	+	0.776579i	$0.716953\pi$
$720$	0.813553	0.0303193
$721$	−33.3978	−1.24380
$722$	−32.4361	−1.20715
$723$	−14.1848	−0.527537
$724$	−3.05968	−0.113712
$725$	−7.96201	−0.295702
$726$	11.9286	0.442713
$727$	30.1239	1.11723	0.558617	−	0.829426i	$-0.311332\pi$
$727$	30.1239	1.11723	0.558617	+	0.829426i	$0.311332\pi$
$728$	−8.81009	−0.326524
$729$	1.00000	0.0370370
$730$	9.51346	0.352109
$731$	2.41866	0.0894575
$732$	−6.54593	−0.241944
$733$	−40.9324	−1.51187	−0.755935	−	0.654646i	$-0.772818\pi$
$733$	−40.9324	−1.51187	−0.755935	+	0.654646i	$0.772818\pi$
$734$	2.01608	0.0744147
$735$	1.76933	0.0652628
$736$	41.7108	1.53748
$737$	−20.1701	−0.742975
$738$	1.29934	0.0478293
$739$	40.3225	1.48329	0.741644	−	0.670794i	$-0.234047\pi$
$739$	40.3225	1.48329	0.741644	+	0.670794i	$0.234047\pi$
$740$	8.08033	0.297039
$741$	−7.22567	−0.265442
$742$	12.4488	0.457009
$743$	−49.7712	−1.82593	−0.912965	−	0.408038i	$-0.866213\pi$
$743$	−49.7712	−1.82593	−0.912965	+	0.408038i	$0.866213\pi$
$744$	−2.97507	−0.109071
$745$	9.36325	0.343043
$746$	36.4737	1.33540
$747$	9.09113	0.332627
$748$	−34.8362	−1.27374
$749$	4.03654	0.147492
$750$	−0.976688	−0.0356636
$751$	15.5443	0.567221	0.283610	−	0.958940i	$-0.408468\pi$
$751$	15.5443	0.567221	0.283610	+	0.958940i	$0.408468\pi$
$752$	−2.02351	−0.0737896
$753$	−1.42495	−0.0519281
$754$	−7.77640	−0.283200
$755$	−6.15047	−0.223839
$756$	3.09777	0.112665
$757$	21.7330	0.789899	0.394950	−	0.918703i	$-0.370762\pi$
$757$	21.7330	0.789899	0.394950	+	0.918703i	$0.370762\pi$
$758$	−29.2205	−1.06134
$759$	−38.9800	−1.41488
$760$	21.4969	0.779774
$761$	45.7366	1.65795	0.828976	−	0.559285i	$-0.188924\pi$
$761$	45.7366	1.65795	0.828976	+	0.559285i	$0.188924\pi$
$762$	−12.4671	−0.451637
$763$	−27.1040	−0.981232
$764$	−11.1254	−0.402504
$765$	6.91190	0.249900
$766$	20.3172	0.734089
$767$	−11.3302	−0.409109
$768$	17.0402	0.614886
$769$	−0.0503792	−0.00181672	−0.000908360	−	1.00000i	$-0.500289\pi$
$769$	−0.0503792	−0.00181672	−0.000908360		1.00000i	$0.500289\pi$
$770$	−13.9350	−0.502184
$771$	29.1468	1.04970
$772$	−1.35327	−0.0487052
$773$	−10.3051	−0.370649	−0.185324	−	0.982677i	$-0.559334\pi$
$773$	−10.3051	−0.370649	−0.185324	+	0.982677i	$0.559334\pi$
$774$	0.341770	0.0122847
$775$	1.00000	0.0359211
$776$	−3.12867	−0.112313
$777$	−22.8743	−0.820610
$778$	−13.9793	−0.501181
$779$	9.61268	0.344410
$780$	1.04608	0.0374557
$781$	64.7926	2.31846
$782$	−54.6168	−1.95309
$783$	7.96201	0.284539
$784$	−1.43945	−0.0514088
$785$	−21.7111	−0.774903
$786$	16.6122	0.592538
$787$	10.8470	0.386653	0.193326	−	0.981134i	$-0.438072\pi$
$787$	10.8470	0.386653	0.193326	+	0.981134i	$0.438072\pi$
$788$	−15.2044	−0.541633
$789$	6.37879	0.227091
$790$	−15.5770	−0.554206
$791$	−16.5900	−0.589873
$792$	−14.3340	−0.509335
$793$	6.25757	0.222213
$794$	15.6107	0.554001
$795$	−4.30415	−0.152653
$796$	−0.654966	−0.0232147
$797$	−29.5874	−1.04804	−0.524019	−	0.851706i	$-0.675568\pi$
$797$	−29.5874	−1.04804	−0.524019	+	0.851706i	$0.675568\pi$
$798$	−20.8986	−0.739803
$799$	−17.1916	−0.608194
$800$	−5.15555	−0.182276
$801$	−5.47272	−0.193369
$802$	−11.0025	−0.388511
$803$	−46.9301	−1.65613
$804$	4.37930	0.154446
$805$	23.9583	0.844420
$806$	0.976688	0.0344024
$807$	9.09766	0.320253
$808$	−15.5075	−0.545551
$809$	−28.0062	−0.984647	−0.492323	−	0.870412i	$-0.663852\pi$
$809$	−28.0062	−0.984647	−0.492323	+	0.870412i	$0.663852\pi$
$810$	0.976688	0.0343173
$811$	−23.0266	−0.808572	−0.404286	−	0.914633i	$-0.632480\pi$
$811$	−23.0266	−0.808572	−0.404286	+	0.914633i	$0.632480\pi$
$812$	24.6644	0.865552
$813$	−0.0988983	−0.00346852
$814$	36.3487	1.27402
$815$	−22.7667	−0.797483
$816$	−5.62320	−0.196851
$817$	2.52846	0.0884597
$818$	32.6024	1.13992
$819$	−2.96131	−0.103476
$820$	−1.39165	−0.0485987
$821$	−5.30265	−0.185064	−0.0925319	−	0.995710i	$-0.529496\pi$
$821$	−5.30265	−0.185064	−0.0925319	+	0.995710i	$0.529496\pi$
$822$	1.14314	0.0398715
$823$	41.6954	1.45341	0.726706	−	0.686949i	$-0.241050\pi$
$823$	41.6954	1.45341	0.726706	+	0.686949i	$0.241050\pi$
$824$	−33.5531	−1.16888
$825$	4.81802	0.167742
$826$	−32.7700	−1.14021
$827$	20.0844	0.698404	0.349202	−	0.937048i	$-0.386453\pi$
$827$	20.0844	0.698404	0.349202	+	0.937048i	$0.386453\pi$
$828$	8.46327	0.294119
$829$	6.33938	0.220176	0.110088	−	0.993922i	$-0.464887\pi$
$829$	6.33938	0.220176	0.110088	+	0.993922i	$0.464887\pi$
$830$	8.87919	0.308201
$831$	−21.3674	−0.741228
$832$	−6.66247	−0.230980
$833$	−12.2295	−0.423725
$834$	−3.55264	−0.123018
$835$	16.3866	0.567082
$836$	−36.4177	−1.25953
$837$	−1.00000	−0.0345651
$838$	−23.5191	−0.812453
$839$	23.3547	0.806294	0.403147	−	0.915135i	$-0.367916\pi$
$839$	23.3547	0.806294	0.403147	+	0.915135i	$0.367916\pi$
$840$	8.81009	0.303977
$841$	34.3936	1.18599
$842$	−33.3458	−1.14917
$843$	−12.7157	−0.437951
$844$	27.1546	0.934700
$845$	−1.00000	−0.0344010
$846$	−2.42926	−0.0835197
$847$	36.1675	1.24273
$848$	3.50166	0.120247
$849$	−1.49705	−0.0513786
$850$	6.75077	0.231549
$851$	−62.4938	−2.14226
$852$	−14.0677	−0.481950
$853$	26.9181	0.921657	0.460829	−	0.887489i	$-0.347552\pi$
$853$	26.9181	0.921657	0.460829	+	0.887489i	$0.347552\pi$
$854$	18.0986	0.619321
$855$	7.22567	0.247113
$856$	4.05530	0.138607
$857$	22.2074	0.758591	0.379295	−	0.925276i	$-0.376166\pi$
$857$	22.2074	0.758591	0.379295	+	0.925276i	$0.376166\pi$
$858$	4.70570	0.160650
$859$	11.5583	0.394365	0.197182	−	0.980367i	$-0.436821\pi$
$859$	11.5583	0.394365	0.197182	+	0.980367i	$0.436821\pi$
$860$	−0.366052	−0.0124823
$861$	3.93958	0.134260
$862$	−14.8304	−0.505125
$863$	−48.1576	−1.63930	−0.819651	−	0.572863i	$-0.805833\pi$
$863$	−48.1576	−1.63930	−0.819651	+	0.572863i	$0.805833\pi$
$864$	5.15555	0.175395
$865$	24.0541	0.817864
$866$	29.2179	0.992864
$867$	−30.7743	−1.04515
$868$	−3.09777	−0.105145
$869$	76.8419	2.60668
$870$	7.77640	0.263645
$871$	−4.18639	−0.141850
$872$	−27.2300	−0.922125
$873$	−1.05163	−0.0355923
$874$	−57.0962	−1.93131
$875$	−2.96131	−0.100110
$876$	10.1894	0.344267
$877$	−10.4489	−0.352834	−0.176417	−	0.984316i	$-0.556451\pi$
$877$	−10.4489	−0.352834	−0.176417	+	0.984316i	$0.556451\pi$
$878$	7.62895	0.257465
$879$	6.46984	0.218222
$880$	−3.91972	−0.132134
$881$	6.09212	0.205249	0.102624	−	0.994720i	$-0.467276\pi$
$881$	6.09212	0.205249	0.102624	+	0.994720i	$0.467276\pi$
$882$	−1.72809	−0.0581877
$883$	−56.6375	−1.90600	−0.953002	−	0.302963i	$-0.902024\pi$
$883$	−56.6375	−1.90600	−0.953002	+	0.302963i	$0.902024\pi$
$884$	−7.23040	−0.243185
$885$	11.3302	0.380860
$886$	34.5020	1.15912
$887$	23.0567	0.774169	0.387084	−	0.922044i	$-0.373482\pi$
$887$	23.0567	0.774169	0.387084	+	0.922044i	$0.373482\pi$
$888$	−22.9806	−0.771178
$889$	−37.8002	−1.26778
$890$	−5.34514	−0.179169
$891$	−4.81802	−0.161410
$892$	17.9556	0.601197
$893$	−17.9720	−0.601410
$894$	−9.14497	−0.305854
$895$	−25.3022	−0.845760
$896$	11.2647	0.376326
$897$	−8.09046	−0.270132
$898$	13.6936	0.456962
$899$	−7.96201	−0.265548
$900$	−1.04608	−0.0348694
$901$	29.7499	0.991112
$902$	−6.26024	−0.208443
$903$	1.03624	0.0344840
$904$	−16.6671	−0.554341
$905$	−2.92490	−0.0972270
$906$	6.00709	0.199572
$907$	2.87880	0.0955889	0.0477944	−	0.998857i	$-0.484781\pi$
$907$	2.87880	0.0955889	0.0477944	+	0.998857i	$0.484781\pi$
$908$	−4.15768	−0.137977
$909$	−5.21248	−0.172887
$910$	−2.89227	−0.0958779
$911$	5.81237	0.192572	0.0962861	−	0.995354i	$-0.469304\pi$
$911$	5.81237	0.192572	0.0962861	+	0.995354i	$0.469304\pi$
$912$	−5.87847	−0.194656
$913$	−43.8013	−1.44961
$914$	−22.4215	−0.741637
$915$	−6.25757	−0.206869
$916$	26.7590	0.884142
$917$	50.3680	1.66330
$918$	−6.75077	−0.222808
$919$	19.6834	0.649296	0.324648	−	0.945835i	$-0.394754\pi$
$919$	19.6834	0.649296	0.324648	+	0.945835i	$0.394754\pi$
$920$	24.0697	0.793554
$921$	−5.47948	−0.180555
$922$	−2.92044	−0.0961796
$923$	13.4480	0.442645
$924$	−14.9251	−0.491000
$925$	7.72439	0.253976
$926$	−4.89322	−0.160801
$927$	−11.2781	−0.370421
$928$	41.0486	1.34749
$929$	50.7551	1.66522	0.832611	−	0.553858i	$-0.186845\pi$
$929$	50.7551	1.66522	0.832611	+	0.553858i	$0.186845\pi$
$930$	−0.976688	−0.0320269
$931$	−12.7846	−0.418999
$932$	−15.1453	−0.496100
$933$	2.63594	0.0862967
$934$	−35.8730	−1.17380
$935$	−33.3017	−1.08908
$936$	−2.97507	−0.0972432
$937$	16.8616	0.550844	0.275422	−	0.961323i	$-0.411182\pi$
$937$	16.8616	0.550844	0.275422	+	0.961323i	$0.411182\pi$
$938$	−12.1082	−0.395346
$939$	10.9369	0.356912
$940$	2.60186	0.0848632
$941$	4.22969	0.137884	0.0689420	−	0.997621i	$-0.478038\pi$
$941$	4.22969	0.137884	0.0689420	+	0.997621i	$0.478038\pi$
$942$	21.2050	0.690896
$943$	10.7631	0.350496
$944$	−9.21771	−0.300011
$945$	2.96131	0.0963313
$946$	−1.64666	−0.0535374
$947$	−10.1396	−0.329491	−0.164746	−	0.986336i	$-0.552680\pi$
$947$	−10.1396	−0.329491	−0.164746	+	0.986336i	$0.552680\pi$
$948$	−16.6838	−0.541864
$949$	−9.74053	−0.316191
$950$	7.05723	0.228967
$951$	19.5823	0.634999
$952$	−60.8945	−1.97360
$953$	2.60156	0.0842728	0.0421364	−	0.999112i	$-0.486584\pi$
$953$	2.60156	0.0842728	0.0421364	+	0.999112i	$0.486584\pi$
$954$	4.20381	0.136104
$955$	−10.6354	−0.344152
$956$	−31.1454	−1.00731
$957$	−38.3612	−1.24004
$958$	−3.85930	−0.124688
$959$	3.46598	0.111922
$960$	6.66247	0.215030
$961$	1.00000	0.0322581
$962$	7.54431	0.243238
$963$	1.36309	0.0439251
$964$	−14.8384	−0.477913
$965$	−1.29366	−0.0416443
$966$	−23.3998	−0.752876
$967$	−20.3293	−0.653748	−0.326874	−	0.945068i	$-0.605995\pi$
$967$	−20.3293	−0.653748	−0.326874	+	0.945068i	$0.605995\pi$
$968$	36.3356	1.16787
$969$	−49.9431	−1.60440
$970$	−1.02711	−0.0329787
$971$	15.1626	0.486591	0.243295	−	0.969952i	$-0.421772\pi$
$971$	15.1626	0.486591	0.243295	+	0.969952i	$0.421772\pi$
$972$	1.04608	0.0335531
$973$	−10.7716	−0.345321
$974$	−18.5105	−0.593113
$975$	1.00000	0.0320256
$976$	5.09087	0.162955
$977$	−38.3987	−1.22848	−0.614242	−	0.789118i	$-0.710538\pi$
$977$	−38.3987	−1.22848	−0.614242	+	0.789118i	$0.710538\pi$
$978$	22.2360	0.711028
$979$	26.3677	0.842715
$980$	1.85087	0.0591237
$981$	−9.15273	−0.292224
$982$	30.0618	0.959310
$983$	−27.8397	−0.887950	−0.443975	−	0.896039i	$-0.646432\pi$
$983$	−27.8397	−0.887950	−0.443975	+	0.896039i	$0.646432\pi$
$984$	3.95789	0.126173
$985$	−14.5346	−0.463111
$986$	−53.7497	−1.71174
$987$	−7.36549	−0.234446
$988$	−7.55864	−0.240472
$989$	2.83107	0.0900229
$990$	−4.70570	−0.149557
$991$	−3.58260	−0.113805	−0.0569026	−	0.998380i	$-0.518122\pi$
$991$	−3.58260	−0.113805	−0.0569026	+	0.998380i	$0.518122\pi$
$992$	−5.15555	−0.163689
$993$	−11.3046	−0.358742
$994$	38.8952	1.23368
$995$	−0.626115	−0.0198492
$996$	9.51005	0.301338
$997$	52.4954	1.66255	0.831273	−	0.555864i	$-0.187613\pi$
$997$	52.4954	1.66255	0.831273	+	0.555864i	$0.187613\pi$
$998$	11.6871	0.369947
$999$	−7.72439	−0.244389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-0.976688 q^{2} -1.00000 q^{3} -1.04608 q^{4} -1.00000 q^{5} +0.976688 q^{6} +2.96131 q^{7} +2.97507 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.976688 q^{2} -1.00000 q^{3} -1.04608 q^{4} -1.00000 q^{5} +0.976688 q^{6} +2.96131 q^{7} +2.97507 q^{8} +1.00000 q^{9} +0.976688 q^{10} -4.81802 q^{11} +1.04608 q^{12} -1.00000 q^{13} -2.89227 q^{14} +1.00000 q^{15} -0.813553 q^{16} -6.91190 q^{17} -0.976688 q^{18} -7.22567 q^{19} +1.04608 q^{20} -2.96131 q^{21} +4.70570 q^{22} -8.09046 q^{23} -2.97507 q^{24} +1.00000 q^{25} +0.976688 q^{26} -1.00000 q^{27} -3.09777 q^{28} -7.96201 q^{29} -0.976688 q^{30} +1.00000 q^{31} -5.15555 q^{32} +4.81802 q^{33} +6.75077 q^{34} -2.96131 q^{35} -1.04608 q^{36} +7.72439 q^{37} +7.05723 q^{38} +1.00000 q^{39} -2.97507 q^{40} -1.33035 q^{41} +2.89227 q^{42} -0.349928 q^{43} +5.04004 q^{44} -1.00000 q^{45} +7.90185 q^{46} +2.48724 q^{47} +0.813553 q^{48} +1.76933 q^{49} -0.976688 q^{50} +6.91190 q^{51} +1.04608 q^{52} -4.30415 q^{53} +0.976688 q^{54} +4.81802 q^{55} +8.81009 q^{56} +7.22567 q^{57} +7.77640 q^{58} +11.3302 q^{59} -1.04608 q^{60} -6.25757 q^{61} -0.976688 q^{62} +2.96131 q^{63} +6.66247 q^{64} +1.00000 q^{65} -4.70570 q^{66} +4.18639 q^{67} +7.23040 q^{68} +8.09046 q^{69} +2.89227 q^{70} -13.4480 q^{71} +2.97507 q^{72} +9.74053 q^{73} -7.54431 q^{74} -1.00000 q^{75} +7.55864 q^{76} -14.2676 q^{77} -0.976688 q^{78} -15.9488 q^{79} +0.813553 q^{80} +1.00000 q^{81} +1.29934 q^{82} +9.09113 q^{83} +3.09777 q^{84} +6.91190 q^{85} +0.341770 q^{86} +7.96201 q^{87} -14.3340 q^{88} -5.47272 q^{89} +0.976688 q^{90} -2.96131 q^{91} +8.46327 q^{92} -1.00000 q^{93} -2.42926 q^{94} +7.22567 q^{95} +5.15555 q^{96} -1.05163 q^{97} -1.72809 q^{98} -4.81802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 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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