LMFDB - Embedded newform 6045.2.a.z.1.8

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 15x^{10} - x^{9} + 81x^{8} + 9x^{7} - 192x^{6} - 27x^{5} + 197x^{4} + 28x^{3} - 82x^{2} - 10x + 10 $ x^12 - 15x^10 - x^9 + 81x^8 + 9x^7 - 192x^6 - 27x^5 + 197x^4 + 28x^3 - 82x^2 - 10*x + 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.693206$ 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)

$f(q)$	$=$	$q+0.693206 q^{2} -1.00000 q^{3} -1.51947 q^{4} -1.00000 q^{5} -0.693206 q^{6} +0.105900 q^{7} -2.43971 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.693206 q^{2} -1.00000 q^{3} -1.51947 q^{4} -1.00000 q^{5} -0.693206 q^{6} +0.105900 q^{7} -2.43971 q^{8} +1.00000 q^{9} -0.693206 q^{10} -0.116320 q^{11} +1.51947 q^{12} +1.00000 q^{13} +0.0734106 q^{14} +1.00000 q^{15} +1.34771 q^{16} -6.20172 q^{17} +0.693206 q^{18} +4.84790 q^{19} +1.51947 q^{20} -0.105900 q^{21} -0.0806335 q^{22} +2.99423 q^{23} +2.43971 q^{24} +1.00000 q^{25} +0.693206 q^{26} -1.00000 q^{27} -0.160912 q^{28} -6.07754 q^{29} +0.693206 q^{30} +1.00000 q^{31} +5.81367 q^{32} +0.116320 q^{33} -4.29907 q^{34} -0.105900 q^{35} -1.51947 q^{36} -4.01488 q^{37} +3.36059 q^{38} -1.00000 q^{39} +2.43971 q^{40} +3.63055 q^{41} -0.0734106 q^{42} +7.49826 q^{43} +0.176744 q^{44} -1.00000 q^{45} +2.07562 q^{46} +10.2152 q^{47} -1.34771 q^{48} -6.98879 q^{49} +0.693206 q^{50} +6.20172 q^{51} -1.51947 q^{52} +1.49269 q^{53} -0.693206 q^{54} +0.116320 q^{55} -0.258366 q^{56} -4.84790 q^{57} -4.21299 q^{58} +11.5036 q^{59} -1.51947 q^{60} -15.1948 q^{61} +0.693206 q^{62} +0.105900 q^{63} +1.33466 q^{64} -1.00000 q^{65} +0.0806335 q^{66} -4.77007 q^{67} +9.42330 q^{68} -2.99423 q^{69} -0.0734106 q^{70} +9.46871 q^{71} -2.43971 q^{72} -3.75026 q^{73} -2.78314 q^{74} -1.00000 q^{75} -7.36622 q^{76} -0.0123183 q^{77} -0.693206 q^{78} +5.08074 q^{79} -1.34771 q^{80} +1.00000 q^{81} +2.51672 q^{82} +11.0892 q^{83} +0.160912 q^{84} +6.20172 q^{85} +5.19784 q^{86} +6.07754 q^{87} +0.283787 q^{88} +15.7145 q^{89} -0.693206 q^{90} +0.105900 q^{91} -4.54963 q^{92} -1.00000 q^{93} +7.08124 q^{94} -4.84790 q^{95} -5.81367 q^{96} -8.50478 q^{97} -4.84467 q^{98} -0.116320 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9	$ 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9} - 6 q^{11} - 6 q^{12} + 12 q^{13} + 5 q^{14} + 12 q^{15} - 6 q^{16} + 5 q^{17} - 24 q^{19} - 6 q^{20} + 7 q^{21} + q^{22} + 13 q^{23} + 3 q^{24} + 12 q^{25} - 12 q^{27} - 10 q^{28} + 3 q^{29} + 12 q^{31} - 6 q^{32} + 6 q^{33} - 15 q^{34} + 7 q^{35} + 6 q^{36} - 9 q^{37} + 16 q^{38} - 12 q^{39} + 3 q^{40} - 5 q^{41} - 5 q^{42} - 8 q^{43} + 5 q^{44} - 12 q^{45} - 2 q^{46} + 17 q^{47} + 6 q^{48} - 7 q^{49} - 5 q^{51} + 6 q^{52} + 16 q^{53} + 6 q^{55} + 17 q^{56} + 24 q^{57} + 36 q^{58} - 18 q^{59} + 6 q^{60} - 24 q^{61} - 7 q^{63} - 21 q^{64} - 12 q^{65} - q^{66} - 20 q^{67} + 23 q^{68} - 13 q^{69} - 5 q^{70} - 9 q^{71} - 3 q^{72} - 15 q^{73} + 10 q^{74} - 12 q^{75} - 30 q^{76} + 4 q^{77} - 25 q^{79} + 6 q^{80} + 12 q^{81} - 11 q^{82} - 13 q^{83} + 10 q^{84} - 5 q^{85} - 30 q^{86} - 3 q^{87} + 3 q^{88} + 35 q^{89} - 7 q^{91} + 5 q^{92} - 12 q^{93} + 4 q^{94} + 24 q^{95} + 6 q^{96} - 11 q^{97} + 16 q^{98} - 6 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9 - 6 * q^11 - 6 * q^12 + 12 * q^13 + 5 * q^14 + 12 * q^15 - 6 * q^16 + 5 * q^17 - 24 * q^19 - 6 * q^20 + 7 * q^21 + q^22 + 13 * q^23 + 3 * q^24 + 12 * q^25 - 12 * q^27 - 10 * q^28 + 3 * q^29 + 12 * q^31 - 6 * q^32 + 6 * q^33 - 15 * q^34 + 7 * q^35 + 6 * q^36 - 9 * q^37 + 16 * q^38 - 12 * q^39 + 3 * q^40 - 5 * q^41 - 5 * q^42 - 8 * q^43 + 5 * q^44 - 12 * q^45 - 2 * q^46 + 17 * q^47 + 6 * q^48 - 7 * q^49 - 5 * q^51 + 6 * q^52 + 16 * q^53 + 6 * q^55 + 17 * q^56 + 24 * q^57 + 36 * q^58 - 18 * q^59 + 6 * q^60 - 24 * q^61 - 7 * q^63 - 21 * q^64 - 12 * q^65 - q^66 - 20 * q^67 + 23 * q^68 - 13 * q^69 - 5 * q^70 - 9 * q^71 - 3 * q^72 - 15 * q^73 + 10 * q^74 - 12 * q^75 - 30 * q^76 + 4 * q^77 - 25 * q^79 + 6 * q^80 + 12 * q^81 - 11 * q^82 - 13 * q^83 + 10 * q^84 - 5 * q^85 - 30 * q^86 - 3 * q^87 + 3 * q^88 + 35 * q^89 - 7 * q^91 + 5 * q^92 - 12 * q^93 + 4 * q^94 + 24 * q^95 + 6 * q^96 - 11 * q^97 + 16 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.693206	0.490171	0.245085	−	0.969502i	$-0.421184\pi$
$2$	0.693206	0.490171	0.245085	+	0.969502i	$0.421184\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.51947	−0.759733
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.693206	−0.283000
$7$	0.105900	0.0400265	0.0200132	−	0.999800i	$-0.493629\pi$
$7$	0.105900	0.0400265	0.0200132	+	0.999800i	$0.493629\pi$
$8$	−2.43971	−0.862569
$9$	1.00000	0.333333
$10$	−0.693206	−0.219211
$11$	−0.116320	−0.0350717	−0.0175359	−	0.999846i	$-0.505582\pi$
$11$	−0.116320	−0.0350717	−0.0175359	+	0.999846i	$0.505582\pi$
$12$	1.51947	0.438632
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.0734106	0.0196198
$15$	1.00000	0.258199
$16$	1.34771	0.336926
$17$	−6.20172	−1.50414	−0.752069	−	0.659084i	$-0.770944\pi$
$17$	−6.20172	−1.50414	−0.752069	+	0.659084i	$0.770944\pi$
$18$	0.693206	0.163390
$19$	4.84790	1.11218	0.556092	−	0.831121i	$-0.312300\pi$
$19$	4.84790	1.11218	0.556092	+	0.831121i	$0.312300\pi$
$20$	1.51947	0.339763
$21$	−0.105900	−0.0231093
$22$	−0.0806335	−0.0171911
$23$	2.99423	0.624340	0.312170	−	0.950026i	$-0.398944\pi$
$23$	2.99423	0.624340	0.312170	+	0.950026i	$0.398944\pi$
$24$	2.43971	0.498005
$25$	1.00000	0.200000
$26$	0.693206	0.135949
$27$	−1.00000	−0.192450
$28$	−0.160912	−0.0304094
$29$	−6.07754	−1.12857	−0.564285	−	0.825580i	$-0.690848\pi$
$29$	−6.07754	−1.12857	−0.564285	+	0.825580i	$0.690848\pi$
$30$	0.693206	0.126562
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.81367	1.02772
$33$	0.116320	0.0202487
$34$	−4.29907	−0.737285
$35$	−0.105900	−0.0179004
$36$	−1.51947	−0.253244
$37$	−4.01488	−0.660043	−0.330021	−	0.943973i	$-0.607056\pi$
$37$	−4.01488	−0.660043	−0.330021	+	0.943973i	$0.607056\pi$
$38$	3.36059	0.545160
$39$	−1.00000	−0.160128
$40$	2.43971	0.385753
$41$	3.63055	0.566997	0.283499	−	0.958973i	$-0.408505\pi$
$41$	3.63055	0.566997	0.283499	+	0.958973i	$0.408505\pi$
$42$	−0.0734106	−0.0113275
$43$	7.49826	1.14347	0.571737	−	0.820437i	$-0.306270\pi$
$43$	7.49826	1.14347	0.571737	+	0.820437i	$0.306270\pi$
$44$	0.176744	0.0266451
$45$	−1.00000	−0.149071
$46$	2.07562	0.306033
$47$	10.2152	1.49004	0.745021	−	0.667041i	$-0.232440\pi$
$47$	10.2152	1.49004	0.745021	+	0.667041i	$0.232440\pi$
$48$	−1.34771	−0.194525
$49$	−6.98879	−0.998398
$50$	0.693206	0.0980341
$51$	6.20172	0.868415
$52$	−1.51947	−0.210712
$53$	1.49269	0.205036	0.102518	−	0.994731i	$-0.467310\pi$
$53$	1.49269	0.205036	0.102518	+	0.994731i	$0.467310\pi$
$54$	−0.693206	−0.0943334
$55$	0.116320	0.0156845
$56$	−0.258366	−0.0345256
$57$	−4.84790	−0.642120
$58$	−4.21299	−0.553192
$59$	11.5036	1.49765	0.748824	−	0.662769i	$-0.230619\pi$
$59$	11.5036	1.49765	0.748824	+	0.662769i	$0.230619\pi$
$60$	−1.51947	−0.196162
$61$	−15.1948	−1.94549	−0.972746	−	0.231871i	$-0.925515\pi$
$61$	−15.1948	−1.94549	−0.972746	+	0.231871i	$0.925515\pi$
$62$	0.693206	0.0880373
$63$	0.105900	0.0133422
$64$	1.33466	0.166832
$65$	−1.00000	−0.124035
$66$	0.0806335	0.00992530
$67$	−4.77007	−0.582757	−0.291378	−	0.956608i	$-0.594114\pi$
$67$	−4.77007	−0.582757	−0.291378	+	0.956608i	$0.594114\pi$
$68$	9.42330	1.14274
$69$	−2.99423	−0.360463
$70$	−0.0734106	−0.00877425
$71$	9.46871	1.12373	0.561864	−	0.827229i	$-0.310084\pi$
$71$	9.46871	1.12373	0.561864	+	0.827229i	$0.310084\pi$
$72$	−2.43971	−0.287523
$73$	−3.75026	−0.438934	−0.219467	−	0.975620i	$-0.570432\pi$
$73$	−3.75026	−0.438934	−0.219467	+	0.975620i	$0.570432\pi$
$74$	−2.78314	−0.323534
$75$	−1.00000	−0.115470
$76$	−7.36622	−0.844963
$77$	−0.0123183	−0.00140380
$78$	−0.693206	−0.0784901
$79$	5.08074	0.571628	0.285814	−	0.958285i	$-0.407736\pi$
$79$	5.08074	0.571628	0.285814	+	0.958285i	$0.407736\pi$
$80$	−1.34771	−0.150678
$81$	1.00000	0.111111
$82$	2.51672	0.277925
$83$	11.0892	1.21720	0.608600	−	0.793477i	$-0.291732\pi$
$83$	11.0892	1.21720	0.608600	+	0.793477i	$0.291732\pi$
$84$	0.160912	0.0175569
$85$	6.20172	0.672671
$86$	5.19784	0.560497
$87$	6.07754	0.651581
$88$	0.283787	0.0302518
$89$	15.7145	1.66574	0.832868	−	0.553472i	$-0.186697\pi$
$89$	15.7145	1.66574	0.832868	+	0.553472i	$0.186697\pi$
$90$	−0.693206	−0.0730703
$91$	0.105900	0.0111014
$92$	−4.54963	−0.474331
$93$	−1.00000	−0.103695
$94$	7.08124	0.730375
$95$	−4.84790	−0.497384
$96$	−5.81367	−0.593355
$97$	−8.50478	−0.863529	−0.431765	−	0.901986i	$-0.642109\pi$
$97$	−8.50478	−0.863529	−0.431765	+	0.901986i	$0.642109\pi$
$98$	−4.84467	−0.489385
$99$	−0.116320	−0.0116906
$100$	−1.51947	−0.151947
$101$	−18.5244	−1.84325	−0.921625	−	0.388081i	$-0.873138\pi$
$101$	−18.5244	−1.84325	−0.921625	+	0.388081i	$0.873138\pi$
$102$	4.29907	0.425671
$103$	−14.3257	−1.41155	−0.705777	−	0.708434i	$-0.749402\pi$
$103$	−14.3257	−1.41155	−0.705777	+	0.708434i	$0.749402\pi$
$104$	−2.43971	−0.239234
$105$	0.105900	0.0103348
$106$	1.03474	0.100503
$107$	−4.29598	−0.415308	−0.207654	−	0.978202i	$-0.566583\pi$
$107$	−4.29598	−0.415308	−0.207654	+	0.978202i	$0.566583\pi$
$108$	1.51947	0.146211
$109$	−2.82812	−0.270885	−0.135443	−	0.990785i	$-0.543246\pi$
$109$	−2.82812	−0.270885	−0.135443	+	0.990785i	$0.543246\pi$
$110$	0.0806335	0.00768810
$111$	4.01488	0.381076
$112$	0.142722	0.0134860
$113$	2.39963	0.225738	0.112869	−	0.993610i	$-0.463996\pi$
$113$	2.39963	0.225738	0.112869	+	0.993610i	$0.463996\pi$
$114$	−3.36059	−0.314749
$115$	−2.99423	−0.279213
$116$	9.23461	0.857412
$117$	1.00000	0.0924500
$118$	7.97440	0.734103
$119$	−0.656763	−0.0602054
$120$	−2.43971	−0.222714
$121$	−10.9865	−0.998770
$122$	−10.5331	−0.953624
$123$	−3.63055	−0.327356
$124$	−1.51947	−0.136452
$125$	−1.00000	−0.0894427
$126$	0.0734106	0.00653994
$127$	21.7160	1.92698	0.963490	−	0.267743i	$-0.0862777\pi$
$127$	21.7160	1.92698	0.963490	+	0.267743i	$0.0862777\pi$
$128$	−10.7021	−0.945945
$129$	−7.49826	−0.660185
$130$	−0.693206	−0.0607982
$131$	−17.8748	−1.56173	−0.780863	−	0.624702i	$-0.785220\pi$
$131$	−17.8748	−1.56173	−0.780863	+	0.624702i	$0.785220\pi$
$132$	−0.176744	−0.0153836
$133$	0.513393	0.0445169
$134$	−3.30664	−0.285650
$135$	1.00000	0.0860663
$136$	15.1304	1.29742
$137$	7.40541	0.632687	0.316344	−	0.948645i	$-0.397545\pi$
$137$	7.40541	0.632687	0.316344	+	0.948645i	$0.397545\pi$
$138$	−2.07562	−0.176688
$139$	−4.03853	−0.342543	−0.171272	−	0.985224i	$-0.554788\pi$
$139$	−4.03853	−0.342543	−0.171272	+	0.985224i	$0.554788\pi$
$140$	0.160912	0.0135995
$141$	−10.2152	−0.860276
$142$	6.56377	0.550819
$143$	−0.116320	−0.00972714
$144$	1.34771	0.112309
$145$	6.07754	0.504712
$146$	−2.59970	−0.215153
$147$	6.98879	0.576425
$148$	6.10047	0.501456
$149$	0.409750	0.0335680	0.0167840	−	0.999859i	$-0.494657\pi$
$149$	0.409750	0.0335680	0.0167840	+	0.999859i	$0.494657\pi$
$150$	−0.693206	−0.0566000
$151$	−11.8143	−0.961434	−0.480717	−	0.876876i	$-0.659624\pi$
$151$	−11.8143	−0.961434	−0.480717	+	0.876876i	$0.659624\pi$
$152$	−11.8275	−0.959337
$153$	−6.20172	−0.501379
$154$	−0.00853910	−0.000688100 0
$155$	−1.00000	−0.0803219
$156$	1.51947	0.121655
$157$	20.9283	1.67026	0.835131	−	0.550051i	$-0.185392\pi$
$157$	20.9283	1.67026	0.835131	+	0.550051i	$0.185392\pi$
$158$	3.52200	0.280195
$159$	−1.49269	−0.118378
$160$	−5.81367	−0.459611
$161$	0.317089	0.0249901
$162$	0.693206	0.0544634
$163$	−6.87233	−0.538282	−0.269141	−	0.963101i	$-0.586740\pi$
$163$	−6.87233	−0.538282	−0.269141	+	0.963101i	$0.586740\pi$
$164$	−5.51650	−0.430766
$165$	−0.116320	−0.00905547
$166$	7.68711	0.596636
$167$	2.34432	0.181409	0.0907043	−	0.995878i	$-0.471088\pi$
$167$	2.34432	0.181409	0.0907043	+	0.995878i	$0.471088\pi$
$168$	0.258366	0.0199334
$169$	1.00000	0.0769231
$170$	4.29907	0.329724
$171$	4.84790	0.370728
$172$	−11.3933	−0.868734
$173$	−8.29576	−0.630715	−0.315357	−	0.948973i	$-0.602124\pi$
$173$	−8.29576	−0.630715	−0.315357	+	0.948973i	$0.602124\pi$
$174$	4.21299	0.319386
$175$	0.105900	0.00800530
$176$	−0.156765	−0.0118166
$177$	−11.5036	−0.864667
$178$	10.8934	0.816495
$179$	−11.1364	−0.832376	−0.416188	−	0.909279i	$-0.636634\pi$
$179$	−11.1364	−0.832376	−0.416188	+	0.909279i	$0.636634\pi$
$180$	1.51947	0.113254
$181$	−22.5690	−1.67754	−0.838770	−	0.544485i	$-0.816725\pi$
$181$	−22.5690	−1.67754	−0.838770	+	0.544485i	$0.816725\pi$
$182$	0.0734106	0.00544156
$183$	15.1948	1.12323
$184$	−7.30507	−0.538537
$185$	4.01488	0.295180
$186$	−0.693206	−0.0508283
$187$	0.721382	0.0527527
$188$	−15.5217	−1.13203
$189$	−0.105900	−0.00770310
$190$	−3.36059	−0.243803
$191$	−18.5591	−1.34289	−0.671443	−	0.741056i	$-0.734325\pi$
$191$	−18.5591	−1.34289	−0.671443	+	0.741056i	$0.734325\pi$
$192$	−1.33466	−0.0963207
$193$	−24.5896	−1.77000	−0.884999	−	0.465593i	$-0.845841\pi$
$193$	−24.5896	−1.77000	−0.884999	+	0.465593i	$0.845841\pi$
$194$	−5.89556	−0.423277
$195$	1.00000	0.0716115
$196$	10.6192	0.758515
$197$	−1.21894	−0.0868456	−0.0434228	−	0.999057i	$-0.513826\pi$
$197$	−1.21894	−0.0868456	−0.0434228	+	0.999057i	$0.513826\pi$
$198$	−0.0806335	−0.00573037
$199$	−11.2075	−0.794480	−0.397240	−	0.917715i	$-0.630032\pi$
$199$	−11.2075	−0.794480	−0.397240	+	0.917715i	$0.630032\pi$
$200$	−2.43971	−0.172514
$201$	4.77007	0.336455
$202$	−12.8413	−0.903507
$203$	−0.643612	−0.0451727
$204$	−9.42330	−0.659763
$205$	−3.63055	−0.253569
$206$	−9.93067	−0.691902
$207$	2.99423	0.208113
$208$	1.34771	0.0934466
$209$	−0.563906	−0.0390062
$210$	0.0734106	0.00506581
$211$	−20.5594	−1.41537	−0.707685	−	0.706528i	$-0.750260\pi$
$211$	−20.5594	−1.41537	−0.707685	+	0.706528i	$0.750260\pi$
$212$	−2.26808	−0.155773
$213$	−9.46871	−0.648785
$214$	−2.97800	−0.203572
$215$	−7.49826	−0.511377
$216$	2.43971	0.166002
$217$	0.105900	0.00718897
$218$	−1.96047	−0.132780
$219$	3.75026	0.253419
$220$	−0.176744	−0.0119161
$221$	−6.20172	−0.417173
$222$	2.78314	0.186792
$223$	0.657340	0.0440187	0.0220094	−	0.999758i	$-0.492994\pi$
$223$	0.657340	0.0440187	0.0220094	+	0.999758i	$0.492994\pi$
$224$	0.615668	0.0411361
$225$	1.00000	0.0666667
$226$	1.66344	0.110650
$227$	11.2447	0.746338	0.373169	−	0.927763i	$-0.378271\pi$
$227$	11.2447	0.746338	0.373169	+	0.927763i	$0.378271\pi$
$228$	7.36622	0.487840
$229$	−19.5969	−1.29500	−0.647498	−	0.762067i	$-0.724185\pi$
$229$	−19.5969	−1.29500	−0.647498	+	0.762067i	$0.724185\pi$
$230$	−2.07562	−0.136862
$231$	0.0123183	0.000810483 0
$232$	14.8275	0.973471
$233$	−15.2550	−0.999391	−0.499695	−	0.866201i	$-0.666555\pi$
$233$	−15.2550	−0.999391	−0.499695	+	0.866201i	$0.666555\pi$
$234$	0.693206	0.0453163
$235$	−10.2152	−0.666367
$236$	−17.4794	−1.13781
$237$	−5.08074	−0.330030
$238$	−0.455272	−0.0295109
$239$	18.2259	1.17894	0.589469	−	0.807791i	$-0.299337\pi$
$239$	18.2259	1.17894	0.589469	+	0.807791i	$0.299337\pi$
$240$	1.34771	0.0869940
$241$	7.40484	0.476988	0.238494	−	0.971144i	$-0.423346\pi$
$241$	7.40484	0.476988	0.238494	+	0.971144i	$0.423346\pi$
$242$	−7.61589	−0.489568
$243$	−1.00000	−0.0641500
$244$	23.0879	1.47805
$245$	6.98879	0.446497
$246$	−2.51672	−0.160460
$247$	4.84790	0.308465
$248$	−2.43971	−0.154922
$249$	−11.0892	−0.702750
$250$	−0.693206	−0.0438422
$251$	2.61704	0.165186	0.0825930	−	0.996583i	$-0.473680\pi$
$251$	2.61704	0.165186	0.0825930	+	0.996583i	$0.473680\pi$
$252$	−0.160912	−0.0101365
$253$	−0.348288	−0.0218967
$254$	15.0536	0.944550
$255$	−6.20172	−0.388367
$256$	−10.0881	−0.630507
$257$	−3.01966	−0.188361	−0.0941806	−	0.995555i	$-0.530023\pi$
$257$	−3.01966	−0.188361	−0.0941806	+	0.995555i	$0.530023\pi$
$258$	−5.19784	−0.323603
$259$	−0.425177	−0.0264192
$260$	1.51947	0.0942332
$261$	−6.07754	−0.376190
$262$	−12.3909	−0.765513
$263$	10.5842	0.652653	0.326326	−	0.945257i	$-0.394189\pi$
$263$	10.5842	0.652653	0.326326	+	0.945257i	$0.394189\pi$
$264$	−0.283787	−0.0174659
$265$	−1.49269	−0.0916949
$266$	0.355887	0.0218209
$267$	−15.7145	−0.961713
$268$	7.24796	0.442739
$269$	0.955152	0.0582366	0.0291183	−	0.999576i	$-0.490730\pi$
$269$	0.955152	0.0582366	0.0291183	+	0.999576i	$0.490730\pi$
$270$	0.693206	0.0421872
$271$	5.28414	0.320989	0.160494	−	0.987037i	$-0.448691\pi$
$271$	5.28414	0.320989	0.160494	+	0.987037i	$0.448691\pi$
$272$	−8.35809	−0.506784
$273$	−0.105900	−0.00640937
$274$	5.13348	0.310125
$275$	−0.116320	−0.00701434
$276$	4.54963	0.273855
$277$	4.11450	0.247216	0.123608	−	0.992331i	$-0.460553\pi$
$277$	4.11450	0.247216	0.123608	+	0.992331i	$0.460553\pi$
$278$	−2.79953	−0.167905
$279$	1.00000	0.0598684
$280$	0.258366	0.0154403
$281$	−27.2489	−1.62553	−0.812767	−	0.582589i	$-0.802040\pi$
$281$	−27.2489	−1.62553	−0.812767	+	0.582589i	$0.802040\pi$
$282$	−7.08124	−0.421682
$283$	−2.50257	−0.148762	−0.0743812	−	0.997230i	$-0.523698\pi$
$283$	−2.50257	−0.148762	−0.0743812	+	0.997230i	$0.523698\pi$
$284$	−14.3874	−0.853734
$285$	4.84790	0.287165
$286$	−0.0806335	−0.00476796
$287$	0.384476	0.0226949
$288$	5.81367	0.342574
$289$	21.4614	1.26243
$290$	4.21299	0.247395
$291$	8.50478	0.498559
$292$	5.69838	0.333473
$293$	21.1259	1.23419	0.617094	−	0.786890i	$-0.288310\pi$
$293$	21.1259	1.23419	0.617094	+	0.786890i	$0.288310\pi$
$294$	4.84467	0.282547
$295$	−11.5036	−0.669768
$296$	9.79517	0.569333
$297$	0.116320	0.00674955
$298$	0.284041	0.0164541
$299$	2.99423	0.173161
$300$	1.51947	0.0877264
$301$	0.794066	0.0457692
$302$	−8.18974	−0.471267
$303$	18.5244	1.06420
$304$	6.53354	0.374724
$305$	15.1948	0.870051
$306$	−4.29907	−0.245762
$307$	−29.4989	−1.68359	−0.841795	−	0.539797i	$-0.818501\pi$
$307$	−29.4989	−1.68359	−0.841795	+	0.539797i	$0.818501\pi$
$308$	0.0187172	0.00106651
$309$	14.3257	0.814961
$310$	−0.693206	−0.0393715
$311$	−19.4304	−1.10180	−0.550898	−	0.834573i	$-0.685715\pi$
$311$	−19.4304	−1.10180	−0.550898	+	0.834573i	$0.685715\pi$
$312$	2.43971	0.138122
$313$	−4.13982	−0.233996	−0.116998	−	0.993132i	$-0.537327\pi$
$313$	−4.13982	−0.233996	−0.116998	+	0.993132i	$0.537327\pi$
$314$	14.5076	0.818713
$315$	−0.105900	−0.00596680
$316$	−7.72001	−0.434285
$317$	24.5139	1.37684	0.688420	−	0.725312i	$-0.258305\pi$
$317$	24.5139	1.37684	0.688420	+	0.725312i	$0.258305\pi$
$318$	−1.03474	−0.0580252
$319$	0.706937	0.0395809
$320$	−1.33466	−0.0746097
$321$	4.29598	0.239778
$322$	0.219808	0.0122494
$323$	−30.0653	−1.67288
$324$	−1.51947	−0.0844147
$325$	1.00000	0.0554700
$326$	−4.76394	−0.263850
$327$	2.82812	0.156396
$328$	−8.85751	−0.489074
$329$	1.08179	0.0596411
$330$	−0.0806335	−0.00443873
$331$	9.30883	0.511659	0.255830	−	0.966722i	$-0.417651\pi$
$331$	9.30883	0.511659	0.255830	+	0.966722i	$0.417651\pi$
$332$	−16.8497	−0.924746
$333$	−4.01488	−0.220014
$334$	1.62509	0.0889212
$335$	4.77007	0.260617
$336$	−0.142722	−0.00778613
$337$	−17.1222	−0.932706	−0.466353	−	0.884599i	$-0.654432\pi$
$337$	−17.1222	−0.932706	−0.466353	+	0.884599i	$0.654432\pi$
$338$	0.693206	0.0377054
$339$	−2.39963	−0.130330
$340$	−9.42330	−0.511050
$341$	−0.116320	−0.00629906
$342$	3.36059	0.181720
$343$	−1.48141	−0.0799889
$344$	−18.2936	−0.986325
$345$	2.99423	0.161204
$346$	−5.75067	−0.309158
$347$	−23.4419	−1.25843	−0.629215	−	0.777232i	$-0.716623\pi$
$347$	−23.4419	−1.25843	−0.629215	+	0.777232i	$0.716623\pi$
$348$	−9.23461	−0.495027
$349$	34.8045	1.86304	0.931521	−	0.363688i	$-0.118483\pi$
$349$	34.8045	1.86304	0.931521	+	0.363688i	$0.118483\pi$
$350$	0.0734106	0.00392396
$351$	−1.00000	−0.0533761
$352$	−0.676244	−0.0360439
$353$	4.05898	0.216038	0.108019	−	0.994149i	$-0.465549\pi$
$353$	4.05898	0.216038	0.108019	+	0.994149i	$0.465549\pi$
$354$	−7.97440	−0.423834
$355$	−9.46871	−0.502547
$356$	−23.8777	−1.26551
$357$	0.656763	0.0347596
$358$	−7.71985	−0.408006
$359$	−13.1047	−0.691641	−0.345821	−	0.938301i	$-0.612399\pi$
$359$	−13.1047	−0.691641	−0.345821	+	0.938301i	$0.612399\pi$
$360$	2.43971	0.128584
$361$	4.50214	0.236955
$362$	−15.6450	−0.822281
$363$	10.9865	0.576640
$364$	−0.160912	−0.00843406
$365$	3.75026	0.196297
$366$	10.5331	0.550575
$367$	−25.3671	−1.32415	−0.662075	−	0.749438i	$-0.730324\pi$
$367$	−25.3671	−1.32415	−0.662075	+	0.749438i	$0.730324\pi$
$368$	4.03534	0.210357
$369$	3.63055	0.188999
$370$	2.78314	0.144689
$371$	0.158076	0.00820687
$372$	1.51947	0.0787806
$373$	−27.1760	−1.40712	−0.703560	−	0.710636i	$-0.748407\pi$
$373$	−27.1760	−1.40712	−0.703560	+	0.710636i	$0.748407\pi$
$374$	0.500067	0.0258578
$375$	1.00000	0.0516398
$376$	−24.9222	−1.28526
$377$	−6.07754	−0.313009
$378$	−0.0734106	−0.00377584
$379$	−2.20906	−0.113472	−0.0567359	−	0.998389i	$-0.518069\pi$
$379$	−2.20906	−0.113472	−0.0567359	+	0.998389i	$0.518069\pi$
$380$	7.36622	0.377879
$381$	−21.7160	−1.11254
$382$	−12.8652	−0.658243
$383$	−28.2919	−1.44565	−0.722825	−	0.691031i	$-0.757157\pi$
$383$	−28.2919	−1.44565	−0.722825	+	0.691031i	$0.757157\pi$
$384$	10.7021	0.546141
$385$	0.0123183	0.000627797 0
$386$	−17.0457	−0.867601
$387$	7.49826	0.381158
$388$	12.9227	0.656052
$389$	31.5879	1.60157	0.800785	−	0.598952i	$-0.204416\pi$
$389$	31.5879	1.60157	0.800785	+	0.598952i	$0.204416\pi$
$390$	0.693206	0.0351019
$391$	−18.5694	−0.939094
$392$	17.0506	0.861188
$393$	17.8748	0.901663
$394$	−0.844974	−0.0425692
$395$	−5.08074	−0.255640
$396$	0.176744	0.00888171
$397$	−28.7913	−1.44499	−0.722496	−	0.691375i	$-0.757005\pi$
$397$	−28.7913	−1.44499	−0.722496	+	0.691375i	$0.757005\pi$
$398$	−7.76912	−0.389431
$399$	−0.513393	−0.0257018
$400$	1.34771	0.0673853
$401$	−17.4305	−0.870437	−0.435218	−	0.900325i	$-0.643329\pi$
$401$	−17.4305	−0.870437	−0.435218	+	0.900325i	$0.643329\pi$
$402$	3.30664	0.164920
$403$	1.00000	0.0498135
$404$	28.1472	1.40038
$405$	−1.00000	−0.0496904
$406$	−0.446156	−0.0221423
$407$	0.467010	0.0231488
$408$	−15.1304	−0.749068
$409$	−6.03867	−0.298593	−0.149297	−	0.988792i	$-0.547701\pi$
$409$	−6.03867	−0.298593	−0.149297	+	0.988792i	$0.547701\pi$
$410$	−2.51672	−0.124292
$411$	−7.40541	−0.365282
$412$	21.7674	1.07240
$413$	1.21824	0.0599456
$414$	2.07562	0.102011
$415$	−11.0892	−0.544348
$416$	5.81367	0.285038
$417$	4.03853	0.197768
$418$	−0.390903	−0.0191197
$419$	17.8322	0.871161	0.435581	−	0.900150i	$-0.356543\pi$
$419$	17.8322	0.871161	0.435581	+	0.900150i	$0.356543\pi$
$420$	−0.160912	−0.00785168
$421$	6.28762	0.306440	0.153220	−	0.988192i	$-0.451036\pi$
$421$	6.28762	0.306440	0.153220	+	0.988192i	$0.451036\pi$
$422$	−14.2519	−0.693773
$423$	10.2152	0.496680
$424$	−3.64173	−0.176858
$425$	−6.20172	−0.300828
$426$	−6.56377	−0.318016
$427$	−1.60913	−0.0778713
$428$	6.52759	0.315523
$429$	0.116320	0.00561597
$430$	−5.19784	−0.250662
$431$	18.1595	0.874711	0.437356	−	0.899289i	$-0.355915\pi$
$431$	18.1595	0.874711	0.437356	+	0.899289i	$0.355915\pi$
$432$	−1.34771	−0.0648415
$433$	38.2050	1.83601	0.918007	−	0.396564i	$-0.129798\pi$
$433$	38.2050	1.83601	0.918007	+	0.396564i	$0.129798\pi$
$434$	0.0734106	0.00352382
$435$	−6.07754	−0.291396
$436$	4.29724	0.205800
$437$	14.5157	0.694381
$438$	2.59970	0.124218
$439$	8.62662	0.411726	0.205863	−	0.978581i	$-0.434000\pi$
$439$	8.62662	0.411726	0.205863	+	0.978581i	$0.434000\pi$
$440$	−0.283787	−0.0135290
$441$	−6.98879	−0.332799
$442$	−4.29907	−0.204486
$443$	−26.2745	−1.24834	−0.624170	−	0.781288i	$-0.714563\pi$
$443$	−26.2745	−1.24834	−0.624170	+	0.781288i	$0.714563\pi$
$444$	−6.10047	−0.289516
$445$	−15.7145	−0.744940
$446$	0.455672	0.0215767
$447$	−0.409750	−0.0193805
$448$	0.141341	0.00667771
$449$	9.24992	0.436531	0.218265	−	0.975889i	$-0.429960\pi$
$449$	9.24992	0.436531	0.218265	+	0.975889i	$0.429960\pi$
$450$	0.693206	0.0326780
$451$	−0.422305	−0.0198856
$452$	−3.64616	−0.171501
$453$	11.8143	0.555084
$454$	7.79491	0.365833
$455$	−0.105900	−0.00496468
$456$	11.8275	0.553873
$457$	14.9266	0.698238	0.349119	−	0.937078i	$-0.386481\pi$
$457$	14.9266	0.698238	0.349119	+	0.937078i	$0.386481\pi$
$458$	−13.5847	−0.634770
$459$	6.20172	0.289472
$460$	4.54963	0.212127
$461$	24.6783	1.14938	0.574692	−	0.818370i	$-0.305122\pi$
$461$	24.6783	1.14938	0.574692	+	0.818370i	$0.305122\pi$
$462$	0.00853910	0.000397275 0
$463$	20.6292	0.958720	0.479360	−	0.877618i	$-0.340869\pi$
$463$	20.6292	0.958720	0.479360	+	0.877618i	$0.340869\pi$
$464$	−8.19073	−0.380245
$465$	1.00000	0.0463739
$466$	−10.5749	−0.489872
$467$	−36.0714	−1.66919	−0.834593	−	0.550867i	$-0.814297\pi$
$467$	−36.0714	−1.66919	−0.834593	+	0.550867i	$0.814297\pi$
$468$	−1.51947	−0.0702373
$469$	−0.505151	−0.0233257
$470$	−7.08124	−0.326633
$471$	−20.9283	−0.964326
$472$	−28.0656	−1.29182
$473$	−0.872195	−0.0401036
$474$	−3.52200	−0.161771
$475$	4.84790	0.222437
$476$	0.997929	0.0457400
$477$	1.49269	0.0683454
$478$	12.6343	0.577881
$479$	−27.2545	−1.24529	−0.622644	−	0.782505i	$-0.713942\pi$
$479$	−27.2545	−1.24529	−0.622644	+	0.782505i	$0.713942\pi$
$480$	5.81367	0.265356
$481$	−4.01488	−0.183063
$482$	5.13308	0.233805
$483$	−0.317089	−0.0144281
$484$	16.6936	0.758798
$485$	8.50478	0.386182
$486$	−0.693206	−0.0314445
$487$	−15.7289	−0.712743	−0.356372	−	0.934344i	$-0.615986\pi$
$487$	−15.7289	−0.712743	−0.356372	+	0.934344i	$0.615986\pi$
$488$	37.0709	1.67812
$489$	6.87233	0.310777
$490$	4.84467	0.218860
$491$	14.3999	0.649859	0.324929	−	0.945738i	$-0.394659\pi$
$491$	14.3999	0.649859	0.324929	+	0.945738i	$0.394659\pi$
$492$	5.51650	0.248703
$493$	37.6912	1.69753
$494$	3.36059	0.151200
$495$	0.116320	0.00522818
$496$	1.34771	0.0605138
$497$	1.00274	0.0449789
$498$	−7.68711	−0.344468
$499$	−1.97162	−0.0882618	−0.0441309	−	0.999026i	$-0.514052\pi$
$499$	−1.97162	−0.0882618	−0.0441309	+	0.999026i	$0.514052\pi$
$500$	1.51947	0.0679526
$501$	−2.34432	−0.104736
$502$	1.81415	0.0809693
$503$	2.46431	0.109878	0.0549391	−	0.998490i	$-0.482504\pi$
$503$	2.46431	0.109878	0.0549391	+	0.998490i	$0.482504\pi$
$504$	−0.258366	−0.0115085
$505$	18.5244	0.824327
$506$	−0.241435	−0.0107331
$507$	−1.00000	−0.0444116
$508$	−32.9967	−1.46399
$509$	−3.84336	−0.170354	−0.0851770	−	0.996366i	$-0.527146\pi$
$509$	−3.84336	−0.170354	−0.0851770	+	0.996366i	$0.527146\pi$
$510$	−4.29907	−0.190366
$511$	−0.397153	−0.0175690
$512$	14.4111	0.636889
$513$	−4.84790	−0.214040
$514$	−2.09325	−0.0923292
$515$	14.3257	0.631266
$516$	11.3933	0.501564
$517$	−1.18823	−0.0522583
$518$	−0.294735	−0.0129499
$519$	8.29576	0.364143
$520$	2.43971	0.106989
$521$	−27.2674	−1.19461	−0.597303	−	0.802016i	$-0.703761\pi$
$521$	−27.2674	−1.19461	−0.597303	+	0.802016i	$0.703761\pi$
$522$	−4.21299	−0.184397
$523$	−9.98922	−0.436798	−0.218399	−	0.975860i	$-0.570083\pi$
$523$	−9.98922	−0.436798	−0.218399	+	0.975860i	$0.570083\pi$
$524$	27.1601	1.18649
$525$	−0.105900	−0.00462186
$526$	7.33707	0.319911
$527$	−6.20172	−0.270151
$528$	0.156765	0.00682231
$529$	−14.0346	−0.610200
$530$	−1.03474	−0.0449462
$531$	11.5036	0.499216
$532$	−0.780083	−0.0338209
$533$	3.63055	0.157257
$534$	−10.8934	−0.471404
$535$	4.29598	0.185731
$536$	11.6376	0.502668
$537$	11.1364	0.480573
$538$	0.662117	0.0285459
$539$	0.812933	0.0350155
$540$	−1.51947	−0.0653874
$541$	−28.9030	−1.24264	−0.621318	−	0.783558i	$-0.713402\pi$
$541$	−28.9030	−1.24264	−0.621318	+	0.783558i	$0.713402\pi$
$542$	3.66300	0.157339
$543$	22.5690	0.968529
$544$	−36.0547	−1.54583
$545$	2.82812	0.121144
$546$	−0.0734106	−0.00314168
$547$	−14.2057	−0.607394	−0.303697	−	0.952769i	$-0.598221\pi$
$547$	−14.2057	−0.607394	−0.303697	+	0.952769i	$0.598221\pi$
$548$	−11.2523	−0.480673
$549$	−15.1948	−0.648498
$550$	−0.0806335	−0.00343822
$551$	−29.4633	−1.25518
$552$	7.30507	0.310924
$553$	0.538051	0.0228803
$554$	2.85220	0.121178
$555$	−4.01488	−0.170422
$556$	6.13640	0.260241
$557$	38.9701	1.65122	0.825608	−	0.564243i	$-0.190832\pi$
$557$	38.9701	1.65122	0.825608	+	0.564243i	$0.190832\pi$
$558$	0.693206	0.0293458
$559$	7.49826	0.317142
$560$	−0.142722	−0.00603111
$561$	−0.721382	−0.0304568
$562$	−18.8891	−0.796789
$563$	31.4969	1.32744	0.663719	−	0.747982i	$-0.268977\pi$
$563$	31.4969	1.32744	0.663719	+	0.747982i	$0.268977\pi$
$564$	15.5217	0.653580
$565$	−2.39963	−0.100953
$566$	−1.73480	−0.0729190
$567$	0.105900	0.00444739
$568$	−23.1009	−0.969294
$569$	−7.27505	−0.304986	−0.152493	−	0.988305i	$-0.548730\pi$
$569$	−7.27505	−0.304986	−0.152493	+	0.988305i	$0.548730\pi$
$570$	3.36059	0.140760
$571$	20.3245	0.850553	0.425276	−	0.905064i	$-0.360177\pi$
$571$	20.3245	0.850553	0.425276	+	0.905064i	$0.360177\pi$
$572$	0.176744	0.00739003
$573$	18.5591	0.775316
$574$	0.266521	0.0111244
$575$	2.99423	0.124868
$576$	1.33466	0.0556108
$577$	−11.4319	−0.475915	−0.237958	−	0.971276i	$-0.576478\pi$
$577$	−11.4319	−0.475915	−0.237958	+	0.971276i	$0.576478\pi$
$578$	14.8771	0.618807
$579$	24.5896	1.02191
$580$	−9.23461	−0.383446
$581$	1.17435	0.0487202
$582$	5.89556	0.244379
$583$	−0.173629	−0.00719096
$584$	9.14956	0.378611
$585$	−1.00000	−0.0413449
$586$	14.6446	0.604962
$587$	−14.6987	−0.606682	−0.303341	−	0.952882i	$-0.598102\pi$
$587$	−14.6987	−0.606682	−0.303341	+	0.952882i	$0.598102\pi$
$588$	−10.6192	−0.437929
$589$	4.84790	0.199754
$590$	−7.97440	−0.328301
$591$	1.21894	0.0501404
$592$	−5.41088	−0.222386
$593$	−22.1534	−0.909731	−0.454866	−	0.890560i	$-0.650313\pi$
$593$	−22.1534	−0.909731	−0.454866	+	0.890560i	$0.650313\pi$
$594$	0.0806335	0.00330843
$595$	0.656763	0.0269247
$596$	−0.622601	−0.0255027
$597$	11.2075	0.458693
$598$	2.07562	0.0848783
$599$	−2.43667	−0.0995597	−0.0497799	−	0.998760i	$-0.515852\pi$
$599$	−2.43667	−0.0995597	−0.0497799	+	0.998760i	$0.515852\pi$
$600$	2.43971	0.0996009
$601$	38.3077	1.56260	0.781302	−	0.624153i	$-0.214556\pi$
$601$	38.3077	1.56260	0.781302	+	0.624153i	$0.214556\pi$
$602$	0.550452	0.0224347
$603$	−4.77007	−0.194252
$604$	17.9514	0.730433
$605$	10.9865	0.446664
$606$	12.8413	0.521640
$607$	30.4223	1.23480	0.617401	−	0.786649i	$-0.288186\pi$
$607$	30.4223	1.23480	0.617401	+	0.786649i	$0.288186\pi$
$608$	28.1841	1.14302
$609$	0.643612	0.0260805
$610$	10.5331	0.426473
$611$	10.2152	0.413263
$612$	9.42330	0.380914
$613$	−35.6658	−1.44053	−0.720263	−	0.693701i	$-0.755979\pi$
$613$	−35.6658	−1.44053	−0.720263	+	0.693701i	$0.755979\pi$
$614$	−20.4488	−0.825247
$615$	3.63055	0.146398
$616$	0.0300531	0.00121087
$617$	34.0432	1.37053	0.685263	−	0.728296i	$-0.259687\pi$
$617$	34.0432	1.37053	0.685263	+	0.728296i	$0.259687\pi$
$618$	9.93067	0.399470
$619$	−22.2232	−0.893225	−0.446612	−	0.894727i	$-0.647370\pi$
$619$	−22.2232	−0.893225	−0.446612	+	0.894727i	$0.647370\pi$
$620$	1.51947	0.0610232
$621$	−2.99423	−0.120154
$622$	−13.4693	−0.540068
$623$	1.66417	0.0666736
$624$	−1.34771	−0.0539514
$625$	1.00000	0.0400000
$626$	−2.86975	−0.114698
$627$	0.563906	0.0225202
$628$	−31.7999	−1.26895
$629$	24.8992	0.992795
$630$	−0.0734106	−0.00292475
$631$	−1.58972	−0.0632856	−0.0316428	−	0.999499i	$-0.510074\pi$
$631$	−1.58972	−0.0632856	−0.0316428	+	0.999499i	$0.510074\pi$
$632$	−12.3956	−0.493069
$633$	20.5594	0.817164
$634$	16.9932	0.674887
$635$	−21.7160	−0.861772
$636$	2.26808	0.0899353
$637$	−6.98879	−0.276906
$638$	0.490053	0.0194014
$639$	9.46871	0.374576
$640$	10.7021	0.423039
$641$	−11.6315	−0.459417	−0.229709	−	0.973259i	$-0.573777\pi$
$641$	−11.6315	−0.459417	−0.229709	+	0.973259i	$0.573777\pi$
$642$	2.97800	0.117532
$643$	26.4201	1.04191	0.520954	−	0.853585i	$-0.325576\pi$
$643$	26.4201	1.04191	0.520954	+	0.853585i	$0.325576\pi$
$644$	−0.481806	−0.0189858
$645$	7.49826	0.295244
$646$	−20.8415	−0.819997
$647$	−4.41306	−0.173495	−0.0867476	−	0.996230i	$-0.527647\pi$
$647$	−4.41306	−0.173495	−0.0867476	+	0.996230i	$0.527647\pi$
$648$	−2.43971	−0.0958410
$649$	−1.33810	−0.0525250
$650$	0.693206	0.0271898
$651$	−0.105900	−0.00415055
$652$	10.4423	0.408951
$653$	27.1785	1.06358	0.531788	−	0.846878i	$-0.321520\pi$
$653$	27.1785	1.06358	0.531788	+	0.846878i	$0.321520\pi$
$654$	1.96047	0.0766606
$655$	17.8748	0.698425
$656$	4.89292	0.191036
$657$	−3.75026	−0.146311
$658$	0.749905	0.0292343
$659$	22.7003	0.884278	0.442139	−	0.896947i	$-0.354220\pi$
$659$	22.7003	0.884278	0.442139	+	0.896947i	$0.354220\pi$
$660$	0.176744	0.00687974
$661$	−19.5909	−0.761999	−0.380999	−	0.924575i	$-0.624420\pi$
$661$	−19.5909	−0.761999	−0.380999	+	0.924575i	$0.624420\pi$
$662$	6.45293	0.250800
$663$	6.20172	0.240855
$664$	−27.0545	−1.04992
$665$	−0.513393	−0.0199085
$666$	−2.78314	−0.107845
$667$	−18.1975	−0.704612
$668$	−3.56211	−0.137822
$669$	−0.657340	−0.0254142
$670$	3.30664	0.127747
$671$	1.76745	0.0682317
$672$	−0.615668	−0.0237499
$673$	−7.65074	−0.294914	−0.147457	−	0.989068i	$-0.547109\pi$
$673$	−7.65074	−0.294914	−0.147457	+	0.989068i	$0.547109\pi$
$674$	−11.8692	−0.457185
$675$	−1.00000	−0.0384900
$676$	−1.51947	−0.0584410
$677$	−13.1746	−0.506343	−0.253171	−	0.967421i	$-0.581474\pi$
$677$	−13.1746	−0.506343	−0.253171	+	0.967421i	$0.581474\pi$
$678$	−1.66344	−0.0638840
$679$	−0.900657	−0.0345641
$680$	−15.1304	−0.580226
$681$	−11.2447	−0.430899
$682$	−0.0806335	−0.00308762
$683$	23.2658	0.890241	0.445121	−	0.895471i	$-0.353161\pi$
$683$	23.2658	0.890241	0.445121	+	0.895471i	$0.353161\pi$
$684$	−7.36622	−0.281654
$685$	−7.40541	−0.282946
$686$	−1.02693	−0.0392082
$687$	19.5969	0.747667
$688$	10.1054	0.385266
$689$	1.49269	0.0568668
$690$	2.07562	0.0790174
$691$	−9.22566	−0.350961	−0.175480	−	0.984483i	$-0.556148\pi$
$691$	−9.22566	−0.350961	−0.175480	+	0.984483i	$0.556148\pi$
$692$	12.6051	0.479175
$693$	−0.0123183	−0.000467932 0
$694$	−16.2501	−0.616845
$695$	4.03853	0.153190
$696$	−14.8275	−0.562033
$697$	−22.5157	−0.852842
$698$	24.1267	0.913208
$699$	15.2550	0.576998
$700$	−0.160912	−0.00608189
$701$	2.17814	0.0822671	0.0411336	−	0.999154i	$-0.486903\pi$
$701$	2.17814	0.0822671	0.0411336	+	0.999154i	$0.486903\pi$
$702$	−0.693206	−0.0261634
$703$	−19.4638	−0.734089
$704$	−0.155247	−0.00585109
$705$	10.2152	0.384727
$706$	2.81371	0.105895
$707$	−1.96174	−0.0737788
$708$	17.4794	0.656916
$709$	0.104712	0.00393253	0.00196627	−	0.999998i	$-0.499374\pi$
$709$	0.104712	0.00393253	0.00196627	+	0.999998i	$0.499374\pi$
$710$	−6.56377	−0.246334
$711$	5.08074	0.190543
$712$	−38.3390	−1.43681
$713$	2.99423	0.112135
$714$	0.455272	0.0170381
$715$	0.116320	0.00435011
$716$	16.9214	0.632383
$717$	−18.2259	−0.680660
$718$	−9.08428	−0.339022
$719$	10.3444	0.385779	0.192890	−	0.981220i	$-0.438214\pi$
$719$	10.3444	0.385779	0.192890	+	0.981220i	$0.438214\pi$
$720$	−1.34771	−0.0502260
$721$	−1.51709	−0.0564996
$722$	3.12091	0.116148
$723$	−7.40484	−0.275389
$724$	34.2928	1.27448
$725$	−6.07754	−0.225714
$726$	7.61589	0.282652
$727$	20.7946	0.771229	0.385615	−	0.922660i	$-0.373989\pi$
$727$	20.7946	0.771229	0.385615	+	0.922660i	$0.373989\pi$
$728$	−0.258366	−0.00957569
$729$	1.00000	0.0370370
$730$	2.59970	0.0962192
$731$	−46.5021	−1.71994
$732$	−23.0879	−0.853355
$733$	−0.431230	−0.0159278	−0.00796392	−	0.999968i	$-0.502535\pi$
$733$	−0.431230	−0.0159278	−0.00796392	+	0.999968i	$0.502535\pi$
$734$	−17.5846	−0.649060
$735$	−6.98879	−0.257785
$736$	17.4075	0.641647
$737$	0.554853	0.0204383
$738$	2.51672	0.0926418
$739$	−36.7482	−1.35180	−0.675902	−	0.736992i	$-0.736246\pi$
$739$	−36.7482	−1.35180	−0.675902	+	0.736992i	$0.736246\pi$
$740$	−6.10047	−0.224258
$741$	−4.84790	−0.178092
$742$	0.109579	0.00402277
$743$	−53.0881	−1.94762	−0.973808	−	0.227373i	$-0.926986\pi$
$743$	−53.0881	−1.94762	−0.973808	+	0.227373i	$0.926986\pi$
$744$	2.43971	0.0894443
$745$	−0.409750	−0.0150121
$746$	−18.8386	−0.689729
$747$	11.0892	0.405733
$748$	−1.09612	−0.0400779
$749$	−0.454945	−0.0166233
$750$	0.693206	0.0253123
$751$	−18.8749	−0.688754	−0.344377	−	0.938832i	$-0.611910\pi$
$751$	−18.8749	−0.688754	−0.344377	+	0.938832i	$0.611910\pi$
$752$	13.7671	0.502034
$753$	−2.61704	−0.0953701
$754$	−4.21299	−0.153428
$755$	11.8143	0.429966
$756$	0.160912	0.00585230
$757$	34.1764	1.24216	0.621081	−	0.783746i	$-0.286694\pi$
$757$	34.1764	1.24216	0.621081	+	0.783746i	$0.286694\pi$
$758$	−1.53133	−0.0556205
$759$	0.348288	0.0126420
$760$	11.8275	0.429028
$761$	17.9817	0.651835	0.325918	−	0.945398i	$-0.394327\pi$
$761$	17.9817	0.651835	0.325918	+	0.945398i	$0.394327\pi$
$762$	−15.0536	−0.545336
$763$	−0.299499	−0.0108426
$764$	28.1998	1.02023
$765$	6.20172	0.224224
$766$	−19.6121	−0.708615
$767$	11.5036	0.415373
$768$	10.0881	0.364023
$769$	−32.1874	−1.16071	−0.580354	−	0.814364i	$-0.697086\pi$
$769$	−32.1874	−1.16071	−0.580354	+	0.814364i	$0.697086\pi$
$770$	0.00853910	0.000307728 0
$771$	3.01966	0.108750
$772$	37.3630	1.34472
$773$	9.08252	0.326676	0.163338	−	0.986570i	$-0.447774\pi$
$773$	9.08252	0.326676	0.163338	+	0.986570i	$0.447774\pi$
$774$	5.19784	0.186832
$775$	1.00000	0.0359211
$776$	20.7492	0.744854
$777$	0.425177	0.0152531
$778$	21.8969	0.785042
$779$	17.6006	0.630605
$780$	−1.51947	−0.0544056
$781$	−1.10140	−0.0394111
$782$	−12.8724	−0.460316
$783$	6.07754	0.217194
$784$	−9.41882	−0.336387
$785$	−20.9283	−0.746964
$786$	12.3909	0.441969
$787$	12.2940	0.438233	0.219117	−	0.975699i	$-0.429682\pi$
$787$	12.2940	0.438233	0.219117	+	0.975699i	$0.429682\pi$
$788$	1.85213	0.0659795
$789$	−10.5842	−0.376809
$790$	−3.52200	−0.125307
$791$	0.254121	0.00903551
$792$	0.283787	0.0100839
$793$	−15.1948	−0.539583
$794$	−19.9583	−0.708293
$795$	1.49269	0.0529401
$796$	17.0294	0.603592
$797$	−48.1050	−1.70397	−0.851983	−	0.523570i	$-0.824600\pi$
$797$	−48.1050	−1.70397	−0.851983	+	0.523570i	$0.824600\pi$
$798$	−0.355887	−0.0125983
$799$	−63.3519	−2.24123
$800$	5.81367	0.205544
$801$	15.7145	0.555245
$802$	−12.0829	−0.426663
$803$	0.436229	0.0153942
$804$	−7.24796	−0.255616
$805$	−0.317089	−0.0111759
$806$	0.693206	0.0244171
$807$	−0.955152	−0.0336229
$808$	45.1943	1.58993
$809$	−13.1551	−0.462510	−0.231255	−	0.972893i	$-0.574283\pi$
$809$	−13.1551	−0.462510	−0.231255	+	0.972893i	$0.574283\pi$
$810$	−0.693206	−0.0243568
$811$	44.9492	1.57838	0.789190	−	0.614149i	$-0.210501\pi$
$811$	44.9492	1.57838	0.789190	+	0.614149i	$0.210501\pi$
$812$	0.977946	0.0343192
$813$	−5.28414	−0.185323
$814$	0.323734	0.0113469
$815$	6.87233	0.240727
$816$	8.35809	0.292592
$817$	36.3508	1.27175
$818$	−4.18604	−0.146362
$819$	0.105900	0.00370045
$820$	5.51650	0.192644
$821$	−2.41168	−0.0841683	−0.0420841	−	0.999114i	$-0.513400\pi$
$821$	−2.41168	−0.0841683	−0.0420841	+	0.999114i	$0.513400\pi$
$822$	−5.13348	−0.179051
$823$	37.2822	1.29957	0.649787	−	0.760116i	$-0.274858\pi$
$823$	37.2822	1.29957	0.649787	+	0.760116i	$0.274858\pi$
$824$	34.9506	1.21756
$825$	0.116320	0.00404973
$826$	0.844490	0.0293836
$827$	4.88454	0.169852	0.0849260	−	0.996387i	$-0.472935\pi$
$827$	4.88454	0.169852	0.0849260	+	0.996387i	$0.472935\pi$
$828$	−4.54963	−0.158110
$829$	17.7631	0.616939	0.308470	−	0.951234i	$-0.400183\pi$
$829$	17.7631	0.616939	0.308470	+	0.951234i	$0.400183\pi$
$830$	−7.68711	−0.266824
$831$	−4.11450	−0.142730
$832$	1.33466	0.0462710
$833$	43.3425	1.50173
$834$	2.79953	0.0969398
$835$	−2.34432	−0.0811284
$836$	0.856836	0.0296343
$837$	−1.00000	−0.0345651
$838$	12.3614	0.427018
$839$	−41.4904	−1.43241	−0.716204	−	0.697891i	$-0.754122\pi$
$839$	−41.4904	−1.43241	−0.716204	+	0.697891i	$0.754122\pi$
$840$	−0.258366	−0.00891448
$841$	7.93648	0.273672
$842$	4.35862	0.150208
$843$	27.2489	0.938503
$844$	31.2394	1.07530
$845$	−1.00000	−0.0344010
$846$	7.08124	0.243458
$847$	−1.16347	−0.0399773
$848$	2.01170	0.0690821
$849$	2.50257	0.0858881
$850$	−4.29907	−0.147457
$851$	−12.0215	−0.412091
$852$	14.3874	0.492903
$853$	−31.8633	−1.09098	−0.545489	−	0.838118i	$-0.683656\pi$
$853$	−31.8633	−1.09098	−0.545489	+	0.838118i	$0.683656\pi$
$854$	−1.11546	−0.0381702
$855$	−4.84790	−0.165795
$856$	10.4810	0.358232
$857$	44.7707	1.52934	0.764669	−	0.644424i	$-0.222903\pi$
$857$	44.7707	1.52934	0.764669	+	0.644424i	$0.222903\pi$
$858$	0.0806335	0.00275278
$859$	7.26115	0.247747	0.123874	−	0.992298i	$-0.460468\pi$
$859$	7.26115	0.247747	0.123874	+	0.992298i	$0.460468\pi$
$860$	11.3933	0.388510
$861$	−0.384476	−0.0131029
$862$	12.5883	0.428758
$863$	35.0240	1.19223	0.596116	−	0.802898i	$-0.296710\pi$
$863$	35.0240	1.19223	0.596116	+	0.802898i	$0.296710\pi$
$864$	−5.81367	−0.197785
$865$	8.29576	0.282064
$866$	26.4839	0.899960
$867$	−21.4614	−0.728866
$868$	−0.160912	−0.00546170
$869$	−0.590990	−0.0200480
$870$	−4.21299	−0.142834
$871$	−4.77007	−0.161628
$872$	6.89982	0.233657
$873$	−8.50478	−0.287843
$874$	10.0624	0.340365
$875$	−0.105900	−0.00358008
$876$	−5.69838	−0.192531
$877$	30.7064	1.03688	0.518441	−	0.855114i	$-0.326513\pi$
$877$	30.7064	1.03688	0.518441	+	0.855114i	$0.326513\pi$
$878$	5.98002	0.201816
$879$	−21.1259	−0.712558
$880$	0.156765	0.00528454
$881$	−41.0078	−1.38159	−0.690794	−	0.723051i	$-0.742739\pi$
$881$	−41.0078	−1.38159	−0.690794	+	0.723051i	$0.742739\pi$
$882$	−4.84467	−0.163128
$883$	25.7073	0.865120	0.432560	−	0.901605i	$-0.357610\pi$
$883$	25.7073	0.865120	0.432560	+	0.901605i	$0.357610\pi$
$884$	9.42330	0.316940
$885$	11.5036	0.386691
$886$	−18.2137	−0.611900
$887$	46.1208	1.54858	0.774292	−	0.632828i	$-0.218106\pi$
$887$	46.1208	1.54858	0.774292	+	0.632828i	$0.218106\pi$
$888$	−9.79517	−0.328704
$889$	2.29972	0.0771303
$890$	−10.8934	−0.365148
$891$	−0.116320	−0.00389686
$892$	−0.998805	−0.0334425
$893$	49.5223	1.65720
$894$	−0.284041	−0.00949976
$895$	11.1364	0.372250
$896$	−1.13336	−0.0378628
$897$	−2.99423	−0.0999744
$898$	6.41210	0.213975
$899$	−6.07754	−0.202697
$900$	−1.51947	−0.0506488
$901$	−9.25722	−0.308403
$902$	−0.292744	−0.00974732
$903$	−0.794066	−0.0264249
$904$	−5.85442	−0.194715
$905$	22.5690	0.750219
$906$	8.18974	0.272086
$907$	−38.5198	−1.27903	−0.639514	−	0.768779i	$-0.720864\pi$
$907$	−38.5198	−1.27903	−0.639514	+	0.768779i	$0.720864\pi$
$908$	−17.0860	−0.567018
$909$	−18.5244	−0.614417
$910$	−0.0734106	−0.00243354
$911$	−47.0247	−1.55800	−0.778999	−	0.627025i	$-0.784272\pi$
$911$	−47.0247	−1.55800	−0.778999	+	0.627025i	$0.784272\pi$
$912$	−6.53354	−0.216347
$913$	−1.28989	−0.0426893
$914$	10.3472	0.342256
$915$	−15.1948	−0.502324
$916$	29.7767	0.983851
$917$	−1.89294	−0.0625104
$918$	4.29907	0.141890
$919$	−37.6491	−1.24193	−0.620965	−	0.783839i	$-0.713259\pi$
$919$	−37.6491	−1.24193	−0.620965	+	0.783839i	$0.713259\pi$
$920$	7.30507	0.240841
$921$	29.4989	0.972021
$922$	17.1072	0.563394
$923$	9.46871	0.311666
$924$	−0.0187172	−0.000615750 0
$925$	−4.01488	−0.132009
$926$	14.3003	0.469937
$927$	−14.3257	−0.470518
$928$	−35.3328	−1.15986
$929$	−36.1892	−1.18733	−0.593664	−	0.804713i	$-0.702319\pi$
$929$	−36.1892	−1.18733	−0.593664	+	0.804713i	$0.702319\pi$
$930$	0.693206	0.0227311
$931$	−33.8809	−1.11040
$932$	23.1795	0.759270
$933$	19.4304	0.636122
$934$	−25.0049	−0.818186
$935$	−0.721382	−0.0235917
$936$	−2.43971	−0.0797446
$937$	5.66097	0.184936	0.0924679	−	0.995716i	$-0.470524\pi$
$937$	5.66097	0.184936	0.0924679	+	0.995716i	$0.470524\pi$
$938$	−0.350174	−0.0114336
$939$	4.13982	0.135098
$940$	15.5217	0.506261
$941$	−29.9514	−0.976387	−0.488194	−	0.872735i	$-0.662344\pi$
$941$	−29.9514	−0.976387	−0.488194	+	0.872735i	$0.662344\pi$
$942$	−14.5076	−0.472684
$943$	10.8707	0.353999
$944$	15.5035	0.504597
$945$	0.105900	0.00344493
$946$	−0.604611	−0.0196576
$947$	−53.4199	−1.73591	−0.867957	−	0.496639i	$-0.834567\pi$
$947$	−53.4199	−1.73591	−0.867957	+	0.496639i	$0.834567\pi$
$948$	7.72001	0.250734
$949$	−3.75026	−0.121738
$950$	3.36059	0.109032
$951$	−24.5139	−0.794919
$952$	1.60231	0.0519313
$953$	−3.65250	−0.118316	−0.0591581	−	0.998249i	$-0.518842\pi$
$953$	−3.65250	−0.118316	−0.0591581	+	0.998249i	$0.518842\pi$
$954$	1.03474	0.0335009
$955$	18.5591	0.600557
$956$	−27.6937	−0.895677
$957$	−0.706937	−0.0228520
$958$	−18.8930	−0.610404
$959$	0.784234	0.0253242
$960$	1.33466	0.0430759
$961$	1.00000	0.0322581
$962$	−2.78314	−0.0897321
$963$	−4.29598	−0.138436
$964$	−11.2514	−0.362383
$965$	24.5896	0.791567
$966$	−0.219808	−0.00707221
$967$	46.3982	1.49206	0.746032	−	0.665910i	$-0.231956\pi$
$967$	46.3982	1.49206	0.746032	+	0.665910i	$0.231956\pi$
$968$	26.8039	0.861508
$969$	30.0653	0.965838
$970$	5.89556	0.189295
$971$	2.93411	0.0941600	0.0470800	−	0.998891i	$-0.485008\pi$
$971$	2.93411	0.0941600	0.0470800	+	0.998891i	$0.485008\pi$
$972$	1.51947	0.0487369
$973$	−0.427681	−0.0137108
$974$	−10.9034	−0.349366
$975$	−1.00000	−0.0320256
$976$	−20.4781	−0.655488
$977$	−6.84295	−0.218925	−0.109463	−	0.993991i	$-0.534913\pi$
$977$	−6.84295	−0.218925	−0.109463	+	0.993991i	$0.534913\pi$
$978$	4.76394	0.152334
$979$	−1.82791	−0.0584202
$980$	−10.6192	−0.339218
$981$	−2.82812	−0.0902951
$982$	9.98210	0.318542
$983$	34.1780	1.09011	0.545054	−	0.838401i	$-0.316509\pi$
$983$	34.1780	1.09011	0.545054	+	0.838401i	$0.316509\pi$
$984$	8.85751	0.282367
$985$	1.21894	0.0388386
$986$	26.1278	0.832078
$987$	−1.08179	−0.0344338
$988$	−7.36622	−0.234351
$989$	22.4515	0.713916
$990$	0.0806335	0.00256270
$991$	−15.1450	−0.481095	−0.240548	−	0.970637i	$-0.577327\pi$
$991$	−15.1450	−0.481095	−0.240548	+	0.970637i	$0.577327\pi$
$992$	5.81367	0.184584
$993$	−9.30883	−0.295407
$994$	0.695104	0.0220474
$995$	11.2075	0.355302
$996$	16.8497	0.533902
$997$	−52.0783	−1.64934	−0.824668	−	0.565617i	$-0.808638\pi$
$997$	−52.0783	−1.64934	−0.824668	+	0.565617i	$0.808638\pi$
$998$	−1.36674	−0.0432634
$999$	4.01488	0.127025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.z.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.z.1.8	✓	12	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)

\(f(q)\)	\(=\)	\(q+0.693206 q^{2} -1.00000 q^{3} -1.51947 q^{4} -1.00000 q^{5} -0.693206 q^{6} +0.105900 q^{7} -2.43971 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.693206 q^{2} -1.00000 q^{3} -1.51947 q^{4} -1.00000 q^{5} -0.693206 q^{6} +0.105900 q^{7} -2.43971 q^{8} +1.00000 q^{9} -0.693206 q^{10} -0.116320 q^{11} +1.51947 q^{12} +1.00000 q^{13} +0.0734106 q^{14} +1.00000 q^{15} +1.34771 q^{16} -6.20172 q^{17} +0.693206 q^{18} +4.84790 q^{19} +1.51947 q^{20} -0.105900 q^{21} -0.0806335 q^{22} +2.99423 q^{23} +2.43971 q^{24} +1.00000 q^{25} +0.693206 q^{26} -1.00000 q^{27} -0.160912 q^{28} -6.07754 q^{29} +0.693206 q^{30} +1.00000 q^{31} +5.81367 q^{32} +0.116320 q^{33} -4.29907 q^{34} -0.105900 q^{35} -1.51947 q^{36} -4.01488 q^{37} +3.36059 q^{38} -1.00000 q^{39} +2.43971 q^{40} +3.63055 q^{41} -0.0734106 q^{42} +7.49826 q^{43} +0.176744 q^{44} -1.00000 q^{45} +2.07562 q^{46} +10.2152 q^{47} -1.34771 q^{48} -6.98879 q^{49} +0.693206 q^{50} +6.20172 q^{51} -1.51947 q^{52} +1.49269 q^{53} -0.693206 q^{54} +0.116320 q^{55} -0.258366 q^{56} -4.84790 q^{57} -4.21299 q^{58} +11.5036 q^{59} -1.51947 q^{60} -15.1948 q^{61} +0.693206 q^{62} +0.105900 q^{63} +1.33466 q^{64} -1.00000 q^{65} +0.0806335 q^{66} -4.77007 q^{67} +9.42330 q^{68} -2.99423 q^{69} -0.0734106 q^{70} +9.46871 q^{71} -2.43971 q^{72} -3.75026 q^{73} -2.78314 q^{74} -1.00000 q^{75} -7.36622 q^{76} -0.0123183 q^{77} -0.693206 q^{78} +5.08074 q^{79} -1.34771 q^{80} +1.00000 q^{81} +2.51672 q^{82} +11.0892 q^{83} +0.160912 q^{84} +6.20172 q^{85} +5.19784 q^{86} +6.07754 q^{87} +0.283787 q^{88} +15.7145 q^{89} -0.693206 q^{90} +0.105900 q^{91} -4.54963 q^{92} -1.00000 q^{93} +7.08124 q^{94} -4.84790 q^{95} -5.81367 q^{96} -8.50478 q^{97} -4.84467 q^{98} -0.116320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9	\( 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9} - 6 q^{11} - 6 q^{12} + 12 q^{13} + 5 q^{14} + 12 q^{15} - 6 q^{16} + 5 q^{17} - 24 q^{19} - 6 q^{20} + 7 q^{21} + q^{22} + 13 q^{23} + 3 q^{24} + 12 q^{25} - 12 q^{27} - 10 q^{28} + 3 q^{29} + 12 q^{31} - 6 q^{32} + 6 q^{33} - 15 q^{34} + 7 q^{35} + 6 q^{36} - 9 q^{37} + 16 q^{38} - 12 q^{39} + 3 q^{40} - 5 q^{41} - 5 q^{42} - 8 q^{43} + 5 q^{44} - 12 q^{45} - 2 q^{46} + 17 q^{47} + 6 q^{48} - 7 q^{49} - 5 q^{51} + 6 q^{52} + 16 q^{53} + 6 q^{55} + 17 q^{56} + 24 q^{57} + 36 q^{58} - 18 q^{59} + 6 q^{60} - 24 q^{61} - 7 q^{63} - 21 q^{64} - 12 q^{65} - q^{66} - 20 q^{67} + 23 q^{68} - 13 q^{69} - 5 q^{70} - 9 q^{71} - 3 q^{72} - 15 q^{73} + 10 q^{74} - 12 q^{75} - 30 q^{76} + 4 q^{77} - 25 q^{79} + 6 q^{80} + 12 q^{81} - 11 q^{82} - 13 q^{83} + 10 q^{84} - 5 q^{85} - 30 q^{86} - 3 q^{87} + 3 q^{88} + 35 q^{89} - 7 q^{91} + 5 q^{92} - 12 q^{93} + 4 q^{94} + 24 q^{95} + 6 q^{96} - 11 q^{97} + 16 q^{98} - 6 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9 - 6 * q^11 - 6 * q^12 + 12 * q^13 + 5 * q^14 + 12 * q^15 - 6 * q^16 + 5 * q^17 - 24 * q^19 - 6 * q^20 + 7 * q^21 + q^22 + 13 * q^23 + 3 * q^24 + 12 * q^25 - 12 * q^27 - 10 * q^28 + 3 * q^29 + 12 * q^31 - 6 * q^32 + 6 * q^33 - 15 * q^34 + 7 * q^35 + 6 * q^36 - 9 * q^37 + 16 * q^38 - 12 * q^39 + 3 * q^40 - 5 * q^41 - 5 * q^42 - 8 * q^43 + 5 * q^44 - 12 * q^45 - 2 * q^46 + 17 * q^47 + 6 * q^48 - 7 * q^49 - 5 * q^51 + 6 * q^52 + 16 * q^53 + 6 * q^55 + 17 * q^56 + 24 * q^57 + 36 * q^58 - 18 * q^59 + 6 * q^60 - 24 * q^61 - 7 * q^63 - 21 * q^64 - 12 * q^65 - q^66 - 20 * q^67 + 23 * q^68 - 13 * q^69 - 5 * q^70 - 9 * q^71 - 3 * q^72 - 15 * q^73 + 10 * q^74 - 12 * q^75 - 30 * q^76 + 4 * q^77 - 25 * q^79 + 6 * q^80 + 12 * q^81 - 11 * q^82 - 13 * q^83 + 10 * q^84 - 5 * q^85 - 30 * q^86 - 3 * q^87 + 3 * q^88 + 35 * q^89 - 7 * q^91 + 5 * q^92 - 12 * q^93 + 4 * q^94 + 24 * q^95 + 6 * q^96 - 11 * q^97 + 16 * q^98 - 6 * q^99

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