LMFDB - Embedded newform 6046.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$1$
Dimension:	$55$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.961298 q^{3} +1.00000 q^{4} +0.560610 q^{5} +0.961298 q^{6} -3.48762 q^{7} -1.00000 q^{8} -2.07591 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.961298 q^{3} +1.00000 q^{4} +0.560610 q^{5} +0.961298 q^{6} -3.48762 q^{7} -1.00000 q^{8} -2.07591 q^{9} -0.560610 q^{10} +0.213691 q^{11} -0.961298 q^{12} -0.408531 q^{13} +3.48762 q^{14} -0.538913 q^{15} +1.00000 q^{16} -0.126556 q^{17} +2.07591 q^{18} -0.298296 q^{19} +0.560610 q^{20} +3.35264 q^{21} -0.213691 q^{22} +5.85618 q^{23} +0.961298 q^{24} -4.68572 q^{25} +0.408531 q^{26} +4.87946 q^{27} -3.48762 q^{28} +9.00193 q^{29} +0.538913 q^{30} -10.2650 q^{31} -1.00000 q^{32} -0.205421 q^{33} +0.126556 q^{34} -1.95519 q^{35} -2.07591 q^{36} +0.145445 q^{37} +0.298296 q^{38} +0.392720 q^{39} -0.560610 q^{40} +7.51094 q^{41} -3.35264 q^{42} +1.41323 q^{43} +0.213691 q^{44} -1.16377 q^{45} -5.85618 q^{46} +9.20709 q^{47} -0.961298 q^{48} +5.16347 q^{49} +4.68572 q^{50} +0.121658 q^{51} -0.408531 q^{52} -0.196877 q^{53} -4.87946 q^{54} +0.119798 q^{55} +3.48762 q^{56} +0.286751 q^{57} -9.00193 q^{58} +7.44726 q^{59} -0.538913 q^{60} -8.85264 q^{61} +10.2650 q^{62} +7.23997 q^{63} +1.00000 q^{64} -0.229027 q^{65} +0.205421 q^{66} -1.96628 q^{67} -0.126556 q^{68} -5.62953 q^{69} +1.95519 q^{70} +2.96065 q^{71} +2.07591 q^{72} -2.04440 q^{73} -0.145445 q^{74} +4.50437 q^{75} -0.298296 q^{76} -0.745274 q^{77} -0.392720 q^{78} +2.47295 q^{79} +0.560610 q^{80} +1.53711 q^{81} -7.51094 q^{82} +12.9310 q^{83} +3.35264 q^{84} -0.0709485 q^{85} -1.41323 q^{86} -8.65354 q^{87} -0.213691 q^{88} -2.32543 q^{89} +1.16377 q^{90} +1.42480 q^{91} +5.85618 q^{92} +9.86776 q^{93} -9.20709 q^{94} -0.167228 q^{95} +0.961298 q^{96} -6.57966 q^{97} -5.16347 q^{98} -0.443603 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 55 q - 55 q^{2} - 4 q^{3} + 55 q^{4} - 7 q^{5} + 4 q^{6} + 17 q^{7} - 55 q^{8} + 29 q^{9}+O(q^{10}) $ 55 * q - 55 * q^2 - 4 * q^3 + 55 * q^4 - 7 * q^5 + 4 * q^6 + 17 * q^7 - 55 * q^8 + 29 * q^9	\( 55 q - 55 q^{2} - 4 q^{3} + 55 q^{4} - 7 q^{5} + 4 q^{6} + 17 q^{7} - 55 q^{8} + 29 q^{9} + 7 q^{10} - 28 q^{11} - 4 q^{12} + q^{13} - 17 q^{14} - 8 q^{15} + 55 q^{16} - 32 q^{17} - 29 q^{18} - 3 q^{19} - 7 q^{20} - 25 q^{21} + 28 q^{22} - 27 q^{23} + 4 q^{24} + 30 q^{25} - q^{26} - q^{27} + 17 q^{28} - 69 q^{29} + 8 q^{30} - 13 q^{31} - 55 q^{32} - 18 q^{33} + 32 q^{34} - 23 q^{35} + 29 q^{36} + 3 q^{37} + 3 q^{38} - 28 q^{39} + 7 q^{40} - 51 q^{41} + 25 q^{42} + 23 q^{43} - 28 q^{44} - 28 q^{45} + 27 q^{46} - 27 q^{47} - 4 q^{48} + 8 q^{49} - 30 q^{50} - 42 q^{51} + q^{52} - 61 q^{53} + q^{54} + 5 q^{55} - 17 q^{56} - 52 q^{57} + 69 q^{58} - 71 q^{59} - 8 q^{60} - 16 q^{61} + 13 q^{62} + 14 q^{63} + 55 q^{64} - 82 q^{65} + 18 q^{66} + 32 q^{67} - 32 q^{68} - 44 q^{69} + 23 q^{70} - 84 q^{71} - 29 q^{72} - 43 q^{73} - 3 q^{74} - 37 q^{75} - 3 q^{76} - 47 q^{77} + 28 q^{78} - 20 q^{79} - 7 q^{80} - 33 q^{81} + 51 q^{82} + 17 q^{83} - 25 q^{84} + 10 q^{85} - 23 q^{86} - q^{87} + 28 q^{88} - 92 q^{89} + 28 q^{90} - 34 q^{91} - 27 q^{92} - 13 q^{93} + 27 q^{94} - 60 q^{95} + 4 q^{96} - 45 q^{97} - 8 q^{98} - 73 q^{99}+O(q^{100}) \) 55 * q - 55 * q^2 - 4 * q^3 + 55 * q^4 - 7 * q^5 + 4 * q^6 + 17 * q^7 - 55 * q^8 + 29 * q^9 + 7 * q^10 - 28 * q^11 - 4 * q^12 + q^13 - 17 * q^14 - 8 * q^15 + 55 * q^16 - 32 * q^17 - 29 * q^18 - 3 * q^19 - 7 * q^20 - 25 * q^21 + 28 * q^22 - 27 * q^23 + 4 * q^24 + 30 * q^25 - q^26 - q^27 + 17 * q^28 - 69 * q^29 + 8 * q^30 - 13 * q^31 - 55 * q^32 - 18 * q^33 + 32 * q^34 - 23 * q^35 + 29 * q^36 + 3 * q^37 + 3 * q^38 - 28 * q^39 + 7 * q^40 - 51 * q^41 + 25 * q^42 + 23 * q^43 - 28 * q^44 - 28 * q^45 + 27 * q^46 - 27 * q^47 - 4 * q^48 + 8 * q^49 - 30 * q^50 - 42 * q^51 + q^52 - 61 * q^53 + q^54 + 5 * q^55 - 17 * q^56 - 52 * q^57 + 69 * q^58 - 71 * q^59 - 8 * q^60 - 16 * q^61 + 13 * q^62 + 14 * q^63 + 55 * q^64 - 82 * q^65 + 18 * q^66 + 32 * q^67 - 32 * q^68 - 44 * q^69 + 23 * q^70 - 84 * q^71 - 29 * q^72 - 43 * q^73 - 3 * q^74 - 37 * q^75 - 3 * q^76 - 47 * q^77 + 28 * q^78 - 20 * q^79 - 7 * q^80 - 33 * q^81 + 51 * q^82 + 17 * q^83 - 25 * q^84 + 10 * q^85 - 23 * q^86 - q^87 + 28 * q^88 - 92 * q^89 + 28 * q^90 - 34 * q^91 - 27 * q^92 - 13 * q^93 + 27 * q^94 - 60 * q^95 + 4 * q^96 - 45 * q^97 - 8 * q^98 - 73 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.961298	−0.555005	−0.277503	−	0.960725i	$-0.589507\pi$
$3$	−0.961298	−0.555005	−0.277503	+	0.960725i	$0.589507\pi$
$4$	1.00000	0.500000
$5$	0.560610	0.250713	0.125356	−	0.992112i	$-0.459993\pi$
$5$	0.560610	0.250713	0.125356	+	0.992112i	$0.459993\pi$
$6$	0.961298	0.392448
$7$	−3.48762	−1.31820	−0.659098	−	0.752057i	$-0.729062\pi$
$7$	−3.48762	−1.31820	−0.659098	+	0.752057i	$0.729062\pi$
$8$	−1.00000	−0.353553
$9$	−2.07591	−0.691969
$10$	−0.560610	−0.177281
$11$	0.213691	0.0644304	0.0322152	−	0.999481i	$-0.489744\pi$
$11$	0.213691	0.0644304	0.0322152	+	0.999481i	$0.489744\pi$
$12$	−0.961298	−0.277503
$13$	−0.408531	−0.113306	−0.0566531	−	0.998394i	$-0.518043\pi$
$13$	−0.408531	−0.113306	−0.0566531	+	0.998394i	$0.518043\pi$
$14$	3.48762	0.932105
$15$	−0.538913	−0.139147
$16$	1.00000	0.250000
$17$	−0.126556	−0.0306943	−0.0153471	−	0.999882i	$-0.504885\pi$
$17$	−0.126556	−0.0306943	−0.0153471	+	0.999882i	$0.504885\pi$
$18$	2.07591	0.489296
$19$	−0.298296	−0.0684337	−0.0342168	−	0.999414i	$-0.510894\pi$
$19$	−0.298296	−0.0684337	−0.0342168	+	0.999414i	$0.510894\pi$
$20$	0.560610	0.125356
$21$	3.35264	0.731605
$22$	−0.213691	−0.0455592
$23$	5.85618	1.22110	0.610549	−	0.791978i	$-0.290949\pi$
$23$	5.85618	1.22110	0.610549	+	0.791978i	$0.290949\pi$
$24$	0.961298	0.196224
$25$	−4.68572	−0.937143
$26$	0.408531	0.0801196
$27$	4.87946	0.939052
$28$	−3.48762	−0.659098
$29$	9.00193	1.67162	0.835808	−	0.549021i	$-0.184999\pi$
$29$	9.00193	1.67162	0.835808	+	0.549021i	$0.184999\pi$
$30$	0.538913	0.0983917
$31$	−10.2650	−1.84365	−0.921827	−	0.387600i	$-0.873304\pi$
$31$	−10.2650	−1.84365	−0.921827	+	0.387600i	$0.873304\pi$
$32$	−1.00000	−0.176777
$33$	−0.205421	−0.0357592
$34$	0.126556	0.0217041
$35$	−1.95519	−0.330488
$36$	−2.07591	−0.345984
$37$	0.145445	0.0239110	0.0119555	−	0.999929i	$-0.496194\pi$
$37$	0.145445	0.0239110	0.0119555	+	0.999929i	$0.496194\pi$
$38$	0.298296	0.0483899
$39$	0.392720	0.0628856
$40$	−0.560610	−0.0886403
$41$	7.51094	1.17301	0.586506	−	0.809945i	$-0.300503\pi$
$41$	7.51094	1.17301	0.586506	+	0.809945i	$0.300503\pi$
$42$	−3.35264	−0.517323
$43$	1.41323	0.215515	0.107758	−	0.994177i	$-0.465633\pi$
$43$	1.41323	0.215515	0.107758	+	0.994177i	$0.465633\pi$
$44$	0.213691	0.0322152
$45$	−1.16377	−0.173485
$46$	−5.85618	−0.863447
$47$	9.20709	1.34299	0.671496	−	0.741008i	$-0.265652\pi$
$47$	9.20709	1.34299	0.671496	+	0.741008i	$0.265652\pi$
$48$	−0.961298	−0.138751
$49$	5.16347	0.737638
$50$	4.68572	0.662660
$51$	0.121658	0.0170355
$52$	−0.408531	−0.0566531
$53$	−0.196877	−0.0270431	−0.0135215	−	0.999909i	$-0.504304\pi$
$53$	−0.196877	−0.0270431	−0.0135215	+	0.999909i	$0.504304\pi$
$54$	−4.87946	−0.664010
$55$	0.119798	0.0161535
$56$	3.48762	0.466052
$57$	0.286751	0.0379811
$58$	−9.00193	−1.18201
$59$	7.44726	0.969550	0.484775	−	0.874639i	$-0.338901\pi$
$59$	7.44726	0.969550	0.484775	+	0.874639i	$0.338901\pi$
$60$	−0.538913	−0.0695734
$61$	−8.85264	−1.13346	−0.566732	−	0.823902i	$-0.691793\pi$
$61$	−8.85264	−1.13346	−0.566732	+	0.823902i	$0.691793\pi$
$62$	10.2650	1.30366
$63$	7.23997	0.912150
$64$	1.00000	0.125000
$65$	−0.229027	−0.0284073
$66$	0.205421	0.0252856
$67$	−1.96628	−0.240219	−0.120110	−	0.992761i	$-0.538325\pi$
$67$	−1.96628	−0.240219	−0.120110	+	0.992761i	$0.538325\pi$
$68$	−0.126556	−0.0153471
$69$	−5.62953	−0.677716
$70$	1.95519	0.233690
$71$	2.96065	0.351364	0.175682	−	0.984447i	$-0.443787\pi$
$71$	2.96065	0.351364	0.175682	+	0.984447i	$0.443787\pi$
$72$	2.07591	0.244648
$73$	−2.04440	−0.239279	−0.119639	−	0.992817i	$-0.538174\pi$
$73$	−2.04440	−0.239279	−0.119639	+	0.992817i	$0.538174\pi$
$74$	−0.145445	−0.0169076
$75$	4.50437	0.520120
$76$	−0.298296	−0.0342168
$77$	−0.745274	−0.0849318
$78$	−0.392720	−0.0444668
$79$	2.47295	0.278229	0.139114	−	0.990276i	$-0.455574\pi$
$79$	2.47295	0.278229	0.139114	+	0.990276i	$0.455574\pi$
$80$	0.560610	0.0626781
$81$	1.53711	0.170790
$82$	−7.51094	−0.829444
$83$	12.9310	1.41936	0.709679	−	0.704525i	$-0.248840\pi$
$83$	12.9310	1.41936	0.709679	+	0.704525i	$0.248840\pi$
$84$	3.35264	0.365803
$85$	−0.0709485	−0.00769544
$86$	−1.41323	−0.152392
$87$	−8.65354	−0.927757
$88$	−0.213691	−0.0227796
$89$	−2.32543	−0.246496	−0.123248	−	0.992376i	$-0.539331\pi$
$89$	−2.32543	−0.246496	−0.123248	+	0.992376i	$0.539331\pi$
$90$	1.16377	0.122673
$91$	1.42480	0.149360
$92$	5.85618	0.610549
$93$	9.86776	1.02324
$94$	−9.20709	−0.949639
$95$	−0.167228	−0.0171572
$96$	0.961298	0.0981120
$97$	−6.57966	−0.668063	−0.334032	−	0.942562i	$-0.608409\pi$
$97$	−6.57966	−0.668063	−0.334032	+	0.942562i	$0.608409\pi$
$98$	−5.16347	−0.521589
$99$	−0.443603	−0.0445838
$100$	−4.68572	−0.468572
$101$	−13.6338	−1.35661	−0.678305	−	0.734780i	$-0.737285\pi$
$101$	−13.6338	−1.35661	−0.678305	+	0.734780i	$0.737285\pi$
$102$	−0.121658	−0.0120459
$103$	3.61051	0.355754	0.177877	−	0.984053i	$-0.443077\pi$
$103$	3.61051	0.355754	0.177877	+	0.984053i	$0.443077\pi$
$104$	0.408531	0.0400598
$105$	1.87952	0.183423
$106$	0.196877	0.0191224
$107$	−7.62707	−0.737337	−0.368668	−	0.929561i	$-0.620186\pi$
$107$	−7.62707	−0.737337	−0.368668	+	0.929561i	$0.620186\pi$
$108$	4.87946	0.469526
$109$	12.0772	1.15679	0.578393	−	0.815759i	$-0.303680\pi$
$109$	12.0772	1.15679	0.578393	+	0.815759i	$0.303680\pi$
$110$	−0.119798	−0.0114223
$111$	−0.139816	−0.0132707
$112$	−3.48762	−0.329549
$113$	11.4559	1.07768	0.538840	−	0.842408i	$-0.318863\pi$
$113$	11.4559	1.07768	0.538840	+	0.842408i	$0.318863\pi$
$114$	−0.286751	−0.0268567
$115$	3.28304	0.306145
$116$	9.00193	0.835808
$117$	0.848073	0.0784044
$118$	−7.44726	−0.685576
$119$	0.441378	0.0404611
$120$	0.538913	0.0491958
$121$	−10.9543	−0.995849
$122$	8.85264	0.801480
$123$	−7.22025	−0.651028
$124$	−10.2650	−0.921827
$125$	−5.42991	−0.485666
$126$	−7.23997	−0.644987
$127$	−7.67527	−0.681070	−0.340535	−	0.940232i	$-0.610608\pi$
$127$	−7.67527	−0.681070	−0.340535	+	0.940232i	$0.610608\pi$
$128$	−1.00000	−0.0883883
$129$	−1.35853	−0.119612
$130$	0.229027	0.0200870
$131$	−10.9517	−0.956853	−0.478426	−	0.878128i	$-0.658793\pi$
$131$	−10.9517	−0.956853	−0.478426	+	0.878128i	$0.658793\pi$
$132$	−0.205421	−0.0178796
$133$	1.04034	0.0902089
$134$	1.96628	0.169861
$135$	2.73547	0.235432
$136$	0.126556	0.0108521
$137$	−8.72351	−0.745300	−0.372650	−	0.927972i	$-0.621551\pi$
$137$	−8.72351	−0.745300	−0.372650	+	0.927972i	$0.621551\pi$
$138$	5.62953	0.479218
$139$	2.63051	0.223117	0.111559	−	0.993758i	$-0.464416\pi$
$139$	2.63051	0.223117	0.111559	+	0.993758i	$0.464416\pi$
$140$	−1.95519	−0.165244
$141$	−8.85075	−0.745368
$142$	−2.96065	−0.248452
$143$	−0.0872996	−0.00730036
$144$	−2.07591	−0.172992
$145$	5.04658	0.419095
$146$	2.04440	0.169196
$147$	−4.96363	−0.409393
$148$	0.145445	0.0119555
$149$	−13.7875	−1.12952	−0.564758	−	0.825257i	$-0.691030\pi$
$149$	−13.7875	−1.12952	−0.564758	+	0.825257i	$0.691030\pi$
$150$	−4.50437	−0.367780
$151$	18.1931	1.48053	0.740266	−	0.672314i	$-0.234699\pi$
$151$	18.1931	1.48053	0.740266	+	0.672314i	$0.234699\pi$
$152$	0.298296	0.0241950
$153$	0.262718	0.0212395
$154$	0.745274	0.0600559
$155$	−5.75469	−0.462227
$156$	0.392720	0.0314428
$157$	−2.96816	−0.236885	−0.118442	−	0.992961i	$-0.537790\pi$
$157$	−2.96816	−0.236885	−0.118442	+	0.992961i	$0.537790\pi$
$158$	−2.47295	−0.196738
$159$	0.189257	0.0150091
$160$	−0.560610	−0.0443201
$161$	−20.4241	−1.60965
$162$	−1.53711	−0.120767
$163$	4.16697	0.326383	0.163191	−	0.986594i	$-0.447821\pi$
$163$	4.16697	0.326383	0.163191	+	0.986594i	$0.447821\pi$
$164$	7.51094	0.586506
$165$	−0.115161	−0.00896528
$166$	−12.9310	−1.00364
$167$	−12.3639	−0.956746	−0.478373	−	0.878157i	$-0.658773\pi$
$167$	−12.3639	−0.956746	−0.478373	+	0.878157i	$0.658773\pi$
$168$	−3.35264	−0.258662
$169$	−12.8331	−0.987162
$170$	0.0709485	0.00544150
$171$	0.619234	0.0473540
$172$	1.41323	0.107758
$173$	−10.4002	−0.790713	−0.395356	−	0.918528i	$-0.629379\pi$
$173$	−10.4002	−0.790713	−0.395356	+	0.918528i	$0.629379\pi$
$174$	8.65354	0.656023
$175$	16.3420	1.23534
$176$	0.213691	0.0161076
$177$	−7.15903	−0.538106
$178$	2.32543	0.174299
$179$	−10.7319	−0.802141	−0.401070	−	0.916047i	$-0.631362\pi$
$179$	−10.7319	−0.802141	−0.401070	+	0.916047i	$0.631362\pi$
$180$	−1.16377	−0.0867427
$181$	17.4364	1.29604	0.648018	−	0.761625i	$-0.275598\pi$
$181$	17.4364	1.29604	0.648018	+	0.761625i	$0.275598\pi$
$182$	−1.42480	−0.105613
$183$	8.51002	0.629079
$184$	−5.85618	−0.431723
$185$	0.0815379	0.00599478
$186$	−9.86776	−0.723539
$187$	−0.0270439	−0.00197764
$188$	9.20709	0.671496
$189$	−17.0177	−1.23785
$190$	0.167228	0.0121320
$191$	−9.36677	−0.677756	−0.338878	−	0.940830i	$-0.610047\pi$
$191$	−9.36677	−0.677756	−0.338878	+	0.940830i	$0.610047\pi$
$192$	−0.961298	−0.0693757
$193$	−5.58452	−0.401982	−0.200991	−	0.979593i	$-0.564416\pi$
$193$	−5.58452	−0.401982	−0.200991	+	0.979593i	$0.564416\pi$
$194$	6.57966	0.472392
$195$	0.220163	0.0157662
$196$	5.16347	0.368819
$197$	−17.5246	−1.24858	−0.624288	−	0.781194i	$-0.714611\pi$
$197$	−17.5246	−1.24858	−0.624288	+	0.781194i	$0.714611\pi$
$198$	0.443603	0.0315255
$199$	7.99383	0.566668	0.283334	−	0.959021i	$-0.408560\pi$
$199$	7.99383	0.566668	0.283334	+	0.959021i	$0.408560\pi$
$200$	4.68572	0.331330
$201$	1.89018	0.133323
$202$	13.6338	0.959269
$203$	−31.3953	−2.20352
$204$	0.121658	0.00851775
$205$	4.21071	0.294089
$206$	−3.61051	−0.251556
$207$	−12.1569	−0.844962
$208$	−0.408531	−0.0283265
$209$	−0.0637432	−0.00440921
$210$	−1.87952	−0.129699
$211$	12.1262	0.834805	0.417403	−	0.908722i	$-0.362940\pi$
$211$	12.1262	0.834805	0.417403	+	0.908722i	$0.362940\pi$
$212$	−0.196877	−0.0135215
$213$	−2.84606	−0.195009
$214$	7.62707	0.521376
$215$	0.792271	0.0540324
$216$	−4.87946	−0.332005
$217$	35.8005	2.43030
$218$	−12.0772	−0.817971
$219$	1.96528	0.132801
$220$	0.119798	0.00807675
$221$	0.0517020	0.00347785
$222$	0.139816	0.00938382
$223$	−5.50508	−0.368648	−0.184324	−	0.982866i	$-0.559010\pi$
$223$	−5.50508	−0.368648	−0.184324	+	0.982866i	$0.559010\pi$
$224$	3.48762	0.233026
$225$	9.72711	0.648474
$226$	−11.4559	−0.762036
$227$	3.85998	0.256196	0.128098	−	0.991762i	$-0.459113\pi$
$227$	3.85998	0.256196	0.128098	+	0.991762i	$0.459113\pi$
$228$	0.286751	0.0189905
$229$	−17.1815	−1.13539	−0.567694	−	0.823240i	$-0.692164\pi$
$229$	−17.1815	−1.13539	−0.567694	+	0.823240i	$0.692164\pi$
$230$	−3.28304	−0.216477
$231$	0.716430	0.0471376
$232$	−9.00193	−0.591006
$233$	−5.83824	−0.382476	−0.191238	−	0.981544i	$-0.561250\pi$
$233$	−5.83824	−0.382476	−0.191238	+	0.981544i	$0.561250\pi$
$234$	−0.848073	−0.0554403
$235$	5.16159	0.336705
$236$	7.44726	0.484775
$237$	−2.37724	−0.154419
$238$	−0.441378	−0.0286103
$239$	20.6068	1.33294	0.666471	−	0.745531i	$-0.267804\pi$
$239$	20.6068	1.33294	0.666471	+	0.745531i	$0.267804\pi$
$240$	−0.538913	−0.0347867
$241$	−13.2149	−0.851248	−0.425624	−	0.904900i	$-0.639945\pi$
$241$	−13.2149	−0.851248	−0.425624	+	0.904900i	$0.639945\pi$
$242$	10.9543	0.704171
$243$	−16.1160	−1.03384
$244$	−8.85264	−0.566732
$245$	2.89469	0.184935
$246$	7.22025	0.460346
$247$	0.121863	0.00775396
$248$	10.2650	0.651830
$249$	−12.4305	−0.787752
$250$	5.42991	0.343418
$251$	14.3087	0.903155	0.451577	−	0.892232i	$-0.350861\pi$
$251$	14.3087	0.903155	0.451577	+	0.892232i	$0.350861\pi$
$252$	7.23997	0.456075
$253$	1.25142	0.0786758
$254$	7.67527	0.481589
$255$	0.0682026	0.00427101
$256$	1.00000	0.0625000
$257$	22.0433	1.37502	0.687511	−	0.726174i	$-0.258703\pi$
$257$	22.0433	1.37502	0.687511	+	0.726174i	$0.258703\pi$
$258$	1.35853	0.0845786
$259$	−0.507256	−0.0315193
$260$	−0.229027	−0.0142036
$261$	−18.6872	−1.15671
$262$	10.9517	0.676597
$263$	−10.9804	−0.677079	−0.338539	−	0.940952i	$-0.609933\pi$
$263$	−10.9804	−0.677079	−0.338539	+	0.940952i	$0.609933\pi$
$264$	0.205421	0.0126428
$265$	−0.110371	−0.00678005
$266$	−1.04034	−0.0637874
$267$	2.23543	0.136806
$268$	−1.96628	−0.120110
$269$	−5.45407	−0.332541	−0.166270	−	0.986080i	$-0.553172\pi$
$269$	−5.45407	−0.332541	−0.166270	+	0.986080i	$0.553172\pi$
$270$	−2.73547	−0.166476
$271$	2.69639	0.163794	0.0818970	−	0.996641i	$-0.473902\pi$
$271$	2.69639	0.163794	0.0818970	+	0.996641i	$0.473902\pi$
$272$	−0.126556	−0.00767357
$273$	−1.36966	−0.0828954
$274$	8.72351	0.527006
$275$	−1.00130	−0.0603805
$276$	−5.62953	−0.338858
$277$	−20.5534	−1.23494	−0.617468	−	0.786596i	$-0.711841\pi$
$277$	−20.5534	−1.23494	−0.617468	+	0.786596i	$0.711841\pi$
$278$	−2.63051	−0.157768
$279$	21.3093	1.27575
$280$	1.95519	0.116845
$281$	5.41389	0.322966	0.161483	−	0.986876i	$-0.448372\pi$
$281$	5.41389	0.322966	0.161483	+	0.986876i	$0.448372\pi$
$282$	8.85075	0.527055
$283$	−4.10829	−0.244212	−0.122106	−	0.992517i	$-0.538965\pi$
$283$	−4.10829	−0.244212	−0.122106	+	0.992517i	$0.538965\pi$
$284$	2.96065	0.175682
$285$	0.160755	0.00952233
$286$	0.0872996	0.00516213
$287$	−26.1953	−1.54626
$288$	2.07591	0.122324
$289$	−16.9840	−0.999058
$290$	−5.04658	−0.296345
$291$	6.32501	0.370779
$292$	−2.04440	−0.119639
$293$	33.2092	1.94010	0.970051	−	0.242902i	$-0.0780994\pi$
$293$	33.2092	1.94010	0.970051	+	0.242902i	$0.0780994\pi$
$294$	4.96363	0.289485
$295$	4.17501	0.243078
$296$	−0.145445	−0.00845381
$297$	1.04270	0.0605035
$298$	13.7875	0.798688
$299$	−2.39243	−0.138358
$300$	4.50437	0.260060
$301$	−4.92880	−0.284091
$302$	−18.1931	−1.04689
$303$	13.1061	0.752926
$304$	−0.298296	−0.0171084
$305$	−4.96288	−0.284174
$306$	−0.262718	−0.0150186
$307$	−1.42629	−0.0814027	−0.0407014	−	0.999171i	$-0.512959\pi$
$307$	−1.42629	−0.0814027	−0.0407014	+	0.999171i	$0.512959\pi$
$308$	−0.745274	−0.0424659
$309$	−3.47077	−0.197445
$310$	5.75469	0.326844
$311$	11.5613	0.655582	0.327791	−	0.944750i	$-0.393696\pi$
$311$	11.5613	0.655582	0.327791	+	0.944750i	$0.393696\pi$
$312$	−0.392720	−0.0222334
$313$	23.8970	1.35074	0.675369	−	0.737480i	$-0.263985\pi$
$313$	23.8970	1.35074	0.675369	+	0.737480i	$0.263985\pi$
$314$	2.96816	0.167503
$315$	4.05880	0.228687
$316$	2.47295	0.139114
$317$	9.21583	0.517613	0.258806	−	0.965929i	$-0.416671\pi$
$317$	9.21583	0.517613	0.258806	+	0.965929i	$0.416671\pi$
$318$	−0.189257	−0.0106130
$319$	1.92364	0.107703
$320$	0.560610	0.0313391
$321$	7.33188	0.409226
$322$	20.4241	1.13819
$323$	0.0377510	0.00210052
$324$	1.53711	0.0853950
$325$	1.91426	0.106184
$326$	−4.16697	−0.230787
$327$	−11.6098	−0.642022
$328$	−7.51094	−0.414722
$329$	−32.1108	−1.77033
$330$	0.115161	0.00633941
$331$	−29.4587	−1.61919	−0.809597	−	0.586986i	$-0.800314\pi$
$331$	−29.4587	−1.61919	−0.809597	+	0.586986i	$0.800314\pi$
$332$	12.9310	0.709679
$333$	−0.301930	−0.0165457
$334$	12.3639	0.676521
$335$	−1.10232	−0.0602260
$336$	3.35264	0.182901
$337$	10.6837	0.581976	0.290988	−	0.956727i	$-0.406016\pi$
$337$	10.6837	0.581976	0.290988	+	0.956727i	$0.406016\pi$
$338$	12.8331	0.698029
$339$	−11.0125	−0.598119
$340$	−0.0709485	−0.00384772
$341$	−2.19355	−0.118787
$342$	−0.619234	−0.0334843
$343$	6.40512	0.345844
$344$	−1.41323	−0.0761962
$345$	−3.15598	−0.169912
$346$	10.4002	0.559118
$347$	10.7497	0.577075	0.288538	−	0.957469i	$-0.406831\pi$
$347$	10.7497	0.577075	0.288538	+	0.957469i	$0.406831\pi$
$348$	−8.65354	−0.463878
$349$	−25.2992	−1.35424	−0.677118	−	0.735874i	$-0.736771\pi$
$349$	−25.2992	−1.35424	−0.677118	+	0.735874i	$0.736771\pi$
$350$	−16.3420	−0.873516
$351$	−1.99341	−0.106400
$352$	−0.213691	−0.0113898
$353$	16.6217	0.884683	0.442341	−	0.896847i	$-0.354148\pi$
$353$	16.6217	0.884683	0.442341	+	0.896847i	$0.354148\pi$
$354$	7.15903	0.380498
$355$	1.65977	0.0880914
$356$	−2.32543	−0.123248
$357$	−0.424296	−0.0224561
$358$	10.7319	0.567199
$359$	−22.5144	−1.18826	−0.594132	−	0.804368i	$-0.702504\pi$
$359$	−22.5144	−1.18826	−0.594132	+	0.804368i	$0.702504\pi$
$360$	1.16377	0.0613363
$361$	−18.9110	−0.995317
$362$	−17.4364	−0.916435
$363$	10.5304	0.552701
$364$	1.42480	0.0746798
$365$	−1.14611	−0.0599903
$366$	−8.51002	−0.444826
$367$	−20.5883	−1.07470	−0.537351	−	0.843359i	$-0.680575\pi$
$367$	−20.5883	−1.07470	−0.537351	+	0.843359i	$0.680575\pi$
$368$	5.85618	0.305275
$369$	−15.5920	−0.811687
$370$	−0.0815379	−0.00423895
$371$	0.686630	0.0356481
$372$	9.86776	0.511619
$373$	31.2699	1.61909	0.809546	−	0.587056i	$-0.199713\pi$
$373$	31.2699	1.61909	0.809546	+	0.587056i	$0.199713\pi$
$374$	0.0270439	0.00139841
$375$	5.21976	0.269547
$376$	−9.20709	−0.474819
$377$	−3.67757	−0.189405
$378$	17.0177	0.875295
$379$	−36.3973	−1.86960	−0.934801	−	0.355171i	$-0.884423\pi$
$379$	−36.3973	−1.86960	−0.934801	+	0.355171i	$0.884423\pi$
$380$	−0.167228	−0.00857859
$381$	7.37822	0.377998
$382$	9.36677	0.479246
$383$	14.3222	0.731831	0.365915	−	0.930648i	$-0.380756\pi$
$383$	14.3222	0.731831	0.365915	+	0.930648i	$0.380756\pi$
$384$	0.961298	0.0490560
$385$	−0.417808	−0.0212935
$386$	5.58452	0.284244
$387$	−2.93373	−0.149130
$388$	−6.57966	−0.334032
$389$	16.3354	0.828236	0.414118	−	0.910223i	$-0.364090\pi$
$389$	16.3354	0.828236	0.414118	+	0.910223i	$0.364090\pi$
$390$	−0.220163	−0.0111484
$391$	−0.741134	−0.0374807
$392$	−5.16347	−0.260794
$393$	10.5278	0.531059
$394$	17.5246	0.882877
$395$	1.38636	0.0697555
$396$	−0.443603	−0.0222919
$397$	−24.5577	−1.23251	−0.616257	−	0.787545i	$-0.711352\pi$
$397$	−24.5577	−1.23251	−0.616257	+	0.787545i	$0.711352\pi$
$398$	−7.99383	−0.400695
$399$	−1.00008	−0.0500665
$400$	−4.68572	−0.234286
$401$	−37.1183	−1.85360	−0.926800	−	0.375554i	$-0.877452\pi$
$401$	−37.1183	−1.85360	−0.926800	+	0.375554i	$0.877452\pi$
$402$	−1.89018	−0.0942736
$403$	4.19359	0.208898
$404$	−13.6338	−0.678305
$405$	0.861720	0.0428192
$406$	31.3953	1.55812
$407$	0.0310803	0.00154059
$408$	−0.121658	−0.00602296
$409$	−9.74723	−0.481970	−0.240985	−	0.970529i	$-0.577470\pi$
$409$	−9.74723	−0.481970	−0.240985	+	0.970529i	$0.577470\pi$
$410$	−4.21071	−0.207952
$411$	8.38589	0.413645
$412$	3.61051	0.177877
$413$	−25.9732	−1.27806
$414$	12.1569	0.597478
$415$	7.24924	0.355851
$416$	0.408531	0.0200299
$417$	−2.52871	−0.123831
$418$	0.0637432	0.00311778
$419$	−27.1709	−1.32739	−0.663694	−	0.748004i	$-0.731012\pi$
$419$	−27.1709	−1.32739	−0.663694	+	0.748004i	$0.731012\pi$
$420$	1.87952	0.0917113
$421$	−26.1217	−1.27309	−0.636547	−	0.771238i	$-0.719638\pi$
$421$	−26.1217	−1.27309	−0.636547	+	0.771238i	$0.719638\pi$
$422$	−12.1262	−0.590297
$423$	−19.1131	−0.929309
$424$	0.196877	0.00956118
$425$	0.593004	0.0287649
$426$	2.84606	0.137892
$427$	30.8746	1.49413
$428$	−7.62707	−0.368668
$429$	0.0839209	0.00405174
$430$	−0.792271	−0.0382067
$431$	−37.0395	−1.78413	−0.892066	−	0.451905i	$-0.850745\pi$
$431$	−37.0395	−1.78413	−0.892066	+	0.451905i	$0.850745\pi$
$432$	4.87946	0.234763
$433$	24.1252	1.15938	0.579691	−	0.814836i	$-0.303173\pi$
$433$	24.1252	1.15938	0.579691	+	0.814836i	$0.303173\pi$
$434$	−35.8005	−1.71848
$435$	−4.85126	−0.232600
$436$	12.0772	0.578393
$437$	−1.74687	−0.0835643
$438$	−1.96528	−0.0939046
$439$	12.7412	0.608106	0.304053	−	0.952655i	$-0.401660\pi$
$439$	12.7412	0.608106	0.304053	+	0.952655i	$0.401660\pi$
$440$	−0.119798	−0.00571113
$441$	−10.7189	−0.510423
$442$	−0.0517020	−0.00245921
$443$	−37.5796	−1.78546	−0.892730	−	0.450593i	$-0.851213\pi$
$443$	−37.5796	−1.78546	−0.892730	+	0.450593i	$0.851213\pi$
$444$	−0.139816	−0.00663536
$445$	−1.30366	−0.0617995
$446$	5.50508	0.260673
$447$	13.2539	0.626887
$448$	−3.48762	−0.164774
$449$	19.1264	0.902633	0.451316	−	0.892364i	$-0.350955\pi$
$449$	19.1264	0.902633	0.451316	+	0.892364i	$0.350955\pi$
$450$	−9.72711	−0.458540
$451$	1.60502	0.0755776
$452$	11.4559	0.538840
$453$	−17.4890	−0.821704
$454$	−3.85998	−0.181158
$455$	0.798758	0.0374463
$456$	−0.286751	−0.0134283
$457$	32.5581	1.52300	0.761502	−	0.648163i	$-0.224462\pi$
$457$	32.5581	1.52300	0.761502	+	0.648163i	$0.224462\pi$
$458$	17.1815	0.802840
$459$	−0.617523	−0.0288235
$460$	3.28304	0.153072
$461$	11.6947	0.544676	0.272338	−	0.962202i	$-0.412203\pi$
$461$	11.6947	0.544676	0.272338	+	0.962202i	$0.412203\pi$
$462$	−0.716430	−0.0333313
$463$	−4.87863	−0.226729	−0.113365	−	0.993553i	$-0.536163\pi$
$463$	−4.87863	−0.226729	−0.113365	+	0.993553i	$0.536163\pi$
$464$	9.00193	0.417904
$465$	5.53197	0.256539
$466$	5.83824	0.270451
$467$	0.808710	0.0374226	0.0187113	−	0.999825i	$-0.494044\pi$
$467$	0.808710	0.0374226	0.0187113	+	0.999825i	$0.494044\pi$
$468$	0.848073	0.0392022
$469$	6.85762	0.316656
$470$	−5.16159	−0.238086
$471$	2.85328	0.131472
$472$	−7.44726	−0.342788
$473$	0.301995	0.0138857
$474$	2.37724	0.109190
$475$	1.39773	0.0641322
$476$	0.441378	0.0202305
$477$	0.408698	0.0187130
$478$	−20.6068	−0.942532
$479$	3.59450	0.164237	0.0821185	−	0.996623i	$-0.473831\pi$
$479$	3.59450	0.164237	0.0821185	+	0.996623i	$0.473831\pi$
$480$	0.538913	0.0245979
$481$	−0.0594188	−0.00270926
$482$	13.2149	0.601924
$483$	19.6337	0.893362
$484$	−10.9543	−0.497924
$485$	−3.68863	−0.167492
$486$	16.1160	0.731036
$487$	31.6714	1.43517	0.717584	−	0.696471i	$-0.245248\pi$
$487$	31.6714	1.43517	0.717584	+	0.696471i	$0.245248\pi$
$488$	8.85264	0.400740
$489$	−4.00570	−0.181144
$490$	−2.89469	−0.130769
$491$	−5.62956	−0.254059	−0.127029	−	0.991899i	$-0.540544\pi$
$491$	−5.62956	−0.254059	−0.127029	+	0.991899i	$0.540544\pi$
$492$	−7.22025	−0.325514
$493$	−1.13925	−0.0513091
$494$	−0.121863	−0.00548288
$495$	−0.248689	−0.0111777
$496$	−10.2650	−0.460914
$497$	−10.3256	−0.463167
$498$	12.4305	0.557025
$499$	4.35579	0.194992	0.0974959	−	0.995236i	$-0.468917\pi$
$499$	4.35579	0.194992	0.0974959	+	0.995236i	$0.468917\pi$
$500$	−5.42991	−0.242833
$501$	11.8854	0.530999
$502$	−14.3087	−0.638627
$503$	−10.3378	−0.460939	−0.230470	−	0.973080i	$-0.574026\pi$
$503$	−10.3378	−0.460939	−0.230470	+	0.973080i	$0.574026\pi$
$504$	−7.23997	−0.322494
$505$	−7.64323	−0.340119
$506$	−1.25142	−0.0556322
$507$	12.3364	0.547880
$508$	−7.67527	−0.340535
$509$	−18.4839	−0.819285	−0.409643	−	0.912246i	$-0.634347\pi$
$509$	−18.4839	−0.819285	−0.409643	+	0.912246i	$0.634347\pi$
$510$	−0.0682026	−0.00302006
$511$	7.13008	0.315416
$512$	−1.00000	−0.0441942
$513$	−1.45552	−0.0642628
$514$	−22.0433	−0.972287
$515$	2.02409	0.0891920
$516$	−1.35853	−0.0598061
$517$	1.96748	0.0865295
$518$	0.507256	0.0222875
$519$	9.99769	0.438850
$520$	0.229027	0.0100435
$521$	−7.40330	−0.324344	−0.162172	−	0.986762i	$-0.551850\pi$
$521$	−7.40330	−0.324344	−0.162172	+	0.986762i	$0.551850\pi$
$522$	18.6872	0.817915
$523$	−24.4497	−1.06911	−0.534556	−	0.845133i	$-0.679521\pi$
$523$	−24.4497	−1.06911	−0.534556	+	0.845133i	$0.679521\pi$
$524$	−10.9517	−0.478426
$525$	−15.7095	−0.685619
$526$	10.9804	0.478767
$527$	1.29910	0.0565897
$528$	−0.205421	−0.00893980
$529$	11.2949	0.491081
$530$	0.110371	0.00479422
$531$	−15.4598	−0.670899
$532$	1.04034	0.0451045
$533$	−3.06845	−0.132909
$534$	−2.23543	−0.0967367
$535$	−4.27581	−0.184860
$536$	1.96628	0.0849303
$537$	10.3166	0.445192
$538$	5.45407	0.235142
$539$	1.10339	0.0475263
$540$	2.73547	0.117716
$541$	−40.5714	−1.74430	−0.872150	−	0.489238i	$-0.837275\pi$
$541$	−40.5714	−1.74430	−0.872150	+	0.489238i	$0.837275\pi$
$542$	−2.69639	−0.115820
$543$	−16.7615	−0.719307
$544$	0.126556	0.00542603
$545$	6.77060	0.290021
$546$	1.36966	0.0586159
$547$	33.6716	1.43970	0.719848	−	0.694132i	$-0.244212\pi$
$547$	33.6716	1.43970	0.719848	+	0.694132i	$0.244212\pi$
$548$	−8.72351	−0.372650
$549$	18.3773	0.784322
$550$	1.00130	0.0426955
$551$	−2.68524	−0.114395
$552$	5.62953	0.239609
$553$	−8.62471	−0.366760
$554$	20.5534	0.873231
$555$	−0.0783822	−0.00332714
$556$	2.63051	0.111559
$557$	−0.140568	−0.00595607	−0.00297804	−	0.999996i	$-0.500948\pi$
$557$	−0.140568	−0.00595607	−0.00297804	+	0.999996i	$0.500948\pi$
$558$	−21.3093	−0.902093
$559$	−0.577348	−0.0244192
$560$	−1.95519	−0.0826220
$561$	0.0259972	0.00109760
$562$	−5.41389	−0.228371
$563$	13.0901	0.551684	0.275842	−	0.961203i	$-0.411043\pi$
$563$	13.0901	0.551684	0.275842	+	0.961203i	$0.411043\pi$
$564$	−8.85075	−0.372684
$565$	6.42230	0.270188
$566$	4.10829	0.172684
$567$	−5.36085	−0.225134
$568$	−2.96065	−0.124226
$569$	−34.0441	−1.42720	−0.713602	−	0.700551i	$-0.752938\pi$
$569$	−34.0441	−1.42720	−0.713602	+	0.700551i	$0.752938\pi$
$570$	−0.160755	−0.00673331
$571$	−13.2052	−0.552619	−0.276310	−	0.961069i	$-0.589111\pi$
$571$	−13.2052	−0.552619	−0.276310	+	0.961069i	$0.589111\pi$
$572$	−0.0872996	−0.00365018
$573$	9.00426	0.376158
$574$	26.1953	1.09337
$575$	−27.4404	−1.14434
$576$	−2.07591	−0.0864961
$577$	−36.7842	−1.53134	−0.765672	−	0.643231i	$-0.777594\pi$
$577$	−36.7842	−1.53134	−0.765672	+	0.643231i	$0.777594\pi$
$578$	16.9840	0.706441
$579$	5.36838	0.223102
$580$	5.04658	0.209548
$581$	−45.0983	−1.87099
$582$	−6.32501	−0.262180
$583$	−0.0420709	−0.00174240
$584$	2.04440	0.0845979
$585$	0.475438	0.0196570
$586$	−33.2092	−1.37186
$587$	30.8718	1.27422	0.637109	−	0.770774i	$-0.280130\pi$
$587$	30.8718	1.27422	0.637109	+	0.770774i	$0.280130\pi$
$588$	−4.96363	−0.204697
$589$	3.06201	0.126168
$590$	−4.17501	−0.171882
$591$	16.8464	0.692967
$592$	0.145445	0.00597775
$593$	−17.9366	−0.736568	−0.368284	−	0.929713i	$-0.620055\pi$
$593$	−17.9366	−0.736568	−0.368284	+	0.929713i	$0.620055\pi$
$594$	−1.04270	−0.0427824
$595$	0.247441	0.0101441
$596$	−13.7875	−0.564758
$597$	−7.68445	−0.314504
$598$	2.39243	0.0978339
$599$	15.1851	0.620445	0.310222	−	0.950664i	$-0.399597\pi$
$599$	15.1851	0.620445	0.310222	+	0.950664i	$0.399597\pi$
$600$	−4.50437	−0.183890
$601$	−37.3941	−1.52534	−0.762668	−	0.646790i	$-0.776111\pi$
$601$	−37.3941	−1.52534	−0.762668	+	0.646790i	$0.776111\pi$
$602$	4.92880	0.200883
$603$	4.08181	0.166224
$604$	18.1931	0.740266
$605$	−6.14111	−0.249672
$606$	−13.1061	−0.532399
$607$	−9.84378	−0.399547	−0.199773	−	0.979842i	$-0.564021\pi$
$607$	−9.84378	−0.399547	−0.199773	+	0.979842i	$0.564021\pi$
$608$	0.298296	0.0120975
$609$	30.1802	1.22296
$610$	4.96288	0.200941
$611$	−3.76138	−0.152169
$612$	0.262718	0.0106197
$613$	−43.2354	−1.74626	−0.873132	−	0.487485i	$-0.837915\pi$
$613$	−43.2354	−1.74626	−0.873132	+	0.487485i	$0.837915\pi$
$614$	1.42629	0.0575604
$615$	−4.04775	−0.163221
$616$	0.745274	0.0300279
$617$	−6.69227	−0.269420	−0.134710	−	0.990885i	$-0.543010\pi$
$617$	−6.69227	−0.269420	−0.134710	+	0.990885i	$0.543010\pi$
$618$	3.47077	0.139615
$619$	25.2279	1.01399	0.506997	−	0.861948i	$-0.330755\pi$
$619$	25.2279	1.01399	0.506997	+	0.861948i	$0.330755\pi$
$620$	−5.75469	−0.231114
$621$	28.5750	1.14667
$622$	−11.5613	−0.463566
$623$	8.11022	0.324929
$624$	0.392720	0.0157214
$625$	20.3845	0.815381
$626$	−23.8970	−0.955115
$627$	0.0612762	0.00244713
$628$	−2.96816	−0.118442
$629$	−0.0184069	−0.000733930 0
$630$	−4.05880	−0.161706
$631$	41.6834	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$631$	41.6834	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$632$	−2.47295	−0.0983688
$633$	−11.6569	−0.463322
$634$	−9.21583	−0.366007
$635$	−4.30284	−0.170753
$636$	0.189257	0.00750453
$637$	−2.10944	−0.0835790
$638$	−1.92364	−0.0761575
$639$	−6.14603	−0.243133
$640$	−0.560610	−0.0221601
$641$	22.6799	0.895802	0.447901	−	0.894083i	$-0.352172\pi$
$641$	22.6799	0.895802	0.447901	+	0.894083i	$0.352172\pi$
$642$	−7.33188	−0.289366
$643$	−15.9946	−0.630766	−0.315383	−	0.948964i	$-0.602133\pi$
$643$	−15.9946	−0.630766	−0.315383	+	0.948964i	$0.602133\pi$
$644$	−20.4241	−0.804823
$645$	−0.761608	−0.0299883
$646$	−0.0377510	−0.00148529
$647$	−3.31212	−0.130213	−0.0651064	−	0.997878i	$-0.520739\pi$
$647$	−3.31212	−0.130213	−0.0651064	+	0.997878i	$0.520739\pi$
$648$	−1.53711	−0.0603834
$649$	1.59141	0.0624685
$650$	−1.91426	−0.0750835
$651$	−34.4149	−1.34883
$652$	4.16697	0.163191
$653$	−41.3713	−1.61898	−0.809492	−	0.587131i	$-0.800257\pi$
$653$	−41.3713	−1.61898	−0.809492	+	0.587131i	$0.800257\pi$
$654$	11.6098	0.453978
$655$	−6.13963	−0.239895
$656$	7.51094	0.293253
$657$	4.24399	0.165574
$658$	32.1108	1.25181
$659$	10.5981	0.412845	0.206422	−	0.978463i	$-0.433818\pi$
$659$	10.5981	0.412845	0.206422	+	0.978463i	$0.433818\pi$
$660$	−0.115161	−0.00448264
$661$	−20.8787	−0.812086	−0.406043	−	0.913854i	$-0.633092\pi$
$661$	−20.8787	−0.812086	−0.406043	+	0.913854i	$0.633092\pi$
$662$	29.4587	1.14494
$663$	−0.0497010	−0.00193023
$664$	−12.9310	−0.501819
$665$	0.583226	0.0226165
$666$	0.301930	0.0116995
$667$	52.7170	2.04121
$668$	−12.3639	−0.478373
$669$	5.29202	0.204601
$670$	1.10232	0.0425862
$671$	−1.89173	−0.0730295
$672$	−3.35264	−0.129331
$673$	−42.9437	−1.65536	−0.827680	−	0.561201i	$-0.810340\pi$
$673$	−42.9437	−1.65536	−0.827680	+	0.561201i	$0.810340\pi$
$674$	−10.6837	−0.411519
$675$	−22.8638	−0.880026
$676$	−12.8331	−0.493581
$677$	−25.9354	−0.996779	−0.498390	−	0.866953i	$-0.666075\pi$
$677$	−25.9354	−0.996779	−0.498390	+	0.866953i	$0.666075\pi$
$678$	11.0125	0.422934
$679$	22.9473	0.880638
$680$	0.0709485	0.00272075
$681$	−3.71059	−0.142190
$682$	2.19355	0.0839954
$683$	23.4063	0.895619	0.447809	−	0.894129i	$-0.352204\pi$
$683$	23.4063	0.895619	0.447809	+	0.894129i	$0.352204\pi$
$684$	0.619234	0.0236770
$685$	−4.89049	−0.186856
$686$	−6.40512	−0.244549
$687$	16.5166	0.630146
$688$	1.41323	0.0538789
$689$	0.0804303	0.00306415
$690$	3.15598	0.120146
$691$	−21.4015	−0.814153	−0.407076	−	0.913394i	$-0.633452\pi$
$691$	−21.4015	−0.814153	−0.407076	+	0.913394i	$0.633452\pi$
$692$	−10.4002	−0.395356
$693$	1.54712	0.0587702
$694$	−10.7497	−0.408054
$695$	1.47469	0.0559383
$696$	8.65354	0.328011
$697$	−0.950552	−0.0360047
$698$	25.2992	0.957590
$699$	5.61229	0.212276
$700$	16.3420	0.617669
$701$	−20.0351	−0.756714	−0.378357	−	0.925660i	$-0.623511\pi$
$701$	−20.0351	−0.756714	−0.378357	+	0.925660i	$0.623511\pi$
$702$	1.99341	0.0752365
$703$	−0.0433855	−0.00163632
$704$	0.213691	0.00805380
$705$	−4.96182	−0.186873
$706$	−16.6217	−0.625565
$707$	47.5494	1.78828
$708$	−7.15903	−0.269053
$709$	−10.3148	−0.387379	−0.193689	−	0.981063i	$-0.562045\pi$
$709$	−10.3148	−0.387379	−0.193689	+	0.981063i	$0.562045\pi$
$710$	−1.65977	−0.0622901
$711$	−5.13362	−0.192526
$712$	2.32543	0.0871493
$713$	−60.1139	−2.25128
$714$	0.424296	0.0158789
$715$	−0.0489411	−0.00183029
$716$	−10.7319	−0.401070
$717$	−19.8093	−0.739790
$718$	22.5144	0.840229
$719$	−19.3036	−0.719904	−0.359952	−	0.932971i	$-0.617207\pi$
$719$	−19.3036	−0.719904	−0.359952	+	0.932971i	$0.617207\pi$
$720$	−1.16377	−0.0433713
$721$	−12.5921	−0.468953
$722$	18.9110	0.703795
$723$	12.7035	0.472448
$724$	17.4364	0.648018
$725$	−42.1805	−1.56654
$726$	−10.5304	−0.390819
$727$	−0.959621	−0.0355904	−0.0177952	−	0.999842i	$-0.505665\pi$
$727$	−0.959621	−0.0355904	−0.0177952	+	0.999842i	$0.505665\pi$
$728$	−1.42480	−0.0528066
$729$	10.8809	0.402998
$730$	1.14611	0.0424195
$731$	−0.178852	−0.00661509
$732$	8.51002	0.314539
$733$	−51.3082	−1.89511	−0.947556	−	0.319589i	$-0.896455\pi$
$733$	−51.3082	−1.89511	−0.947556	+	0.319589i	$0.896455\pi$
$734$	20.5883	0.759929
$735$	−2.78266	−0.102640
$736$	−5.85618	−0.215862
$737$	−0.420177	−0.0154774
$738$	15.5920	0.573950
$739$	−19.8923	−0.731750	−0.365875	−	0.930664i	$-0.619230\pi$
$739$	−19.8923	−0.731750	−0.365875	+	0.930664i	$0.619230\pi$
$740$	0.0815379	0.00299739
$741$	−0.117147	−0.00430349
$742$	−0.686630	−0.0252070
$743$	9.29792	0.341108	0.170554	−	0.985348i	$-0.445444\pi$
$743$	9.29792	0.341108	0.170554	+	0.985348i	$0.445444\pi$
$744$	−9.86776	−0.361769
$745$	−7.72941	−0.283184
$746$	−31.2699	−1.14487
$747$	−26.8435	−0.982152
$748$	−0.0270439	−0.000988822 0
$749$	26.6003	0.971954
$750$	−5.21976	−0.190599
$751$	−46.5011	−1.69685	−0.848424	−	0.529318i	$-0.822448\pi$
$751$	−46.5011	−1.69685	−0.848424	+	0.529318i	$0.822448\pi$
$752$	9.20709	0.335748
$753$	−13.7549	−0.501256
$754$	3.67757	0.133929
$755$	10.1992	0.371188
$756$	−17.0177	−0.618927
$757$	−8.36608	−0.304070	−0.152035	−	0.988375i	$-0.548583\pi$
$757$	−8.36608	−0.304070	−0.152035	+	0.988375i	$0.548583\pi$
$758$	36.3973	1.32201
$759$	−1.20298	−0.0436655
$760$	0.167228	0.00606598
$761$	0.400562	0.0145204	0.00726018	−	0.999974i	$-0.497689\pi$
$761$	0.400562	0.0145204	0.00726018	+	0.999974i	$0.497689\pi$
$762$	−7.37822	−0.267285
$763$	−42.1206	−1.52487
$764$	−9.36677	−0.338878
$765$	0.147282	0.00532501
$766$	−14.3222	−0.517483
$767$	−3.04244	−0.109856
$768$	−0.961298	−0.0346878
$769$	6.35491	0.229164	0.114582	−	0.993414i	$-0.463447\pi$
$769$	6.35491	0.229164	0.114582	+	0.993414i	$0.463447\pi$
$770$	0.417808	0.0150568
$771$	−21.1901	−0.763144
$772$	−5.58452	−0.200991
$773$	−32.5075	−1.16921	−0.584607	−	0.811317i	$-0.698751\pi$
$773$	−32.5075	−1.16921	−0.584607	+	0.811317i	$0.698751\pi$
$774$	2.93373	0.105451
$775$	48.0990	1.72777
$776$	6.57966	0.236196
$777$	0.487624	0.0174934
$778$	−16.3354	−0.585651
$779$	−2.24048	−0.0802735
$780$	0.220163	0.00788310
$781$	0.632665	0.0226385
$782$	0.741134	0.0265029
$783$	43.9245	1.56974
$784$	5.16347	0.184410
$785$	−1.66398	−0.0593899
$786$	−10.5278	−0.375515
$787$	18.1748	0.647860	0.323930	−	0.946081i	$-0.394996\pi$
$787$	18.1748	0.647860	0.323930	+	0.946081i	$0.394996\pi$
$788$	−17.5246	−0.624288
$789$	10.5554	0.375782
$790$	−1.38636	−0.0493246
$791$	−39.9538	−1.42059
$792$	0.443603	0.0157628
$793$	3.61658	0.128429
$794$	24.5577	0.871519
$795$	0.106100	0.00376296
$796$	7.99383	0.283334
$797$	−19.2018	−0.680161	−0.340081	−	0.940396i	$-0.610454\pi$
$797$	−19.2018	−0.680161	−0.340081	+	0.940396i	$0.610454\pi$
$798$	1.00008	0.0354023
$799$	−1.16521	−0.0412222
$800$	4.68572	0.165665
$801$	4.82738	0.170567
$802$	37.1183	1.31069
$803$	−0.436871	−0.0154168
$804$	1.89018	0.0666615
$805$	−11.4500	−0.403558
$806$	−4.19359	−0.147713
$807$	5.24299	0.184562
$808$	13.6338	0.479634
$809$	12.3159	0.433003	0.216502	−	0.976282i	$-0.430535\pi$
$809$	12.3159	0.433003	0.216502	+	0.976282i	$0.430535\pi$
$810$	−0.861720	−0.0302777
$811$	23.6203	0.829422	0.414711	−	0.909953i	$-0.363883\pi$
$811$	23.6203	0.829422	0.414711	+	0.909953i	$0.363883\pi$
$812$	−31.3953	−1.10176
$813$	−2.59203	−0.0909065
$814$	−0.0310803	−0.00108936
$815$	2.33605	0.0818282
$816$	0.121658	0.00425887
$817$	−0.421560	−0.0147485
$818$	9.74723	0.340804
$819$	−2.95775	−0.103352
$820$	4.21071	0.147044
$821$	51.9055	1.81151	0.905757	−	0.423797i	$-0.139303\pi$
$821$	51.9055	1.81151	0.905757	+	0.423797i	$0.139303\pi$
$822$	−8.38589	−0.292491
$823$	8.72505	0.304136	0.152068	−	0.988370i	$-0.451407\pi$
$823$	8.72505	0.304136	0.152068	+	0.988370i	$0.451407\pi$
$824$	−3.61051	−0.125778
$825$	0.962545	0.0335115
$826$	25.9732	0.903722
$827$	30.0612	1.04533	0.522665	−	0.852538i	$-0.324938\pi$
$827$	30.0612	1.04533	0.522665	+	0.852538i	$0.324938\pi$
$828$	−12.1569	−0.422481
$829$	−39.8950	−1.38561	−0.692806	−	0.721124i	$-0.743626\pi$
$829$	−39.8950	−1.38561	−0.692806	+	0.721124i	$0.743626\pi$
$830$	−7.24924	−0.251625
$831$	19.7580	0.685396
$832$	−0.408531	−0.0141633
$833$	−0.653467	−0.0226413
$834$	2.52871	0.0875619
$835$	−6.93132	−0.239868
$836$	−0.0637432	−0.00220460
$837$	−50.0878	−1.73129
$838$	27.1709	0.938605
$839$	−35.5902	−1.22871	−0.614355	−	0.789029i	$-0.710584\pi$
$839$	−35.5902	−1.22871	−0.614355	+	0.789029i	$0.710584\pi$
$840$	−1.87952	−0.0648497
$841$	52.0348	1.79430
$842$	26.1217	0.900213
$843$	−5.20436	−0.179248
$844$	12.1262	0.417403
$845$	−7.19437	−0.247494
$846$	19.1131	0.657120
$847$	38.2045	1.31272
$848$	−0.196877	−0.00676077
$849$	3.94929	0.135539
$850$	−0.593004	−0.0203399
$851$	0.851751	0.0291977
$852$	−2.84606	−0.0975045
$853$	36.3884	1.24592	0.622958	−	0.782256i	$-0.285931\pi$
$853$	36.3884	1.24592	0.622958	+	0.782256i	$0.285931\pi$
$854$	−30.8746	−1.05651
$855$	0.347149	0.0118722
$856$	7.62707	0.260688
$857$	34.2816	1.17104	0.585518	−	0.810660i	$-0.300891\pi$
$857$	34.2816	1.17104	0.585518	+	0.810660i	$0.300891\pi$
$858$	−0.0839209	−0.00286501
$859$	29.8689	1.01911	0.509557	−	0.860437i	$-0.329809\pi$
$859$	29.8689	1.01911	0.509557	+	0.860437i	$0.329809\pi$
$860$	0.792271	0.0270162
$861$	25.1814	0.858181
$862$	37.0395	1.26157
$863$	−3.82499	−0.130204	−0.0651022	−	0.997879i	$-0.520737\pi$
$863$	−3.82499	−0.130204	−0.0651022	+	0.997879i	$0.520737\pi$
$864$	−4.87946	−0.166003
$865$	−5.83046	−0.198242
$866$	−24.1252	−0.819807
$867$	16.3267	0.554483
$868$	35.8005	1.21515
$869$	0.528449	0.0179264
$870$	4.85126	0.164473
$871$	0.803286	0.0272183
$872$	−12.0772	−0.408985
$873$	13.6588	0.462279
$874$	1.74687	0.0590889
$875$	18.9375	0.640203
$876$	1.96528	0.0664006
$877$	−7.59472	−0.256455	−0.128228	−	0.991745i	$-0.540929\pi$
$877$	−7.59472	−0.256455	−0.128228	+	0.991745i	$0.540929\pi$
$878$	−12.7412	−0.429996
$879$	−31.9239	−1.07677
$880$	0.119798	0.00403838
$881$	−21.4140	−0.721457	−0.360728	−	0.932671i	$-0.617472\pi$
$881$	−21.4140	−0.721457	−0.360728	+	0.932671i	$0.617472\pi$
$882$	10.7189	0.360923
$883$	−28.4517	−0.957477	−0.478738	−	0.877958i	$-0.658906\pi$
$883$	−28.4517	−0.957477	−0.478738	+	0.877958i	$0.658906\pi$
$884$	0.0517020	0.00173893
$885$	−4.01343	−0.134910
$886$	37.5796	1.26251
$887$	41.0840	1.37946	0.689732	−	0.724064i	$-0.257728\pi$
$887$	41.0840	1.37946	0.689732	+	0.724064i	$0.257728\pi$
$888$	0.139816	0.00469191
$889$	26.7684	0.897784
$890$	1.30366	0.0436989
$891$	0.328467	0.0110041
$892$	−5.50508	−0.184324
$893$	−2.74643	−0.0919059
$894$	−13.2539	−0.443276
$895$	−6.01642	−0.201107
$896$	3.48762	0.116513
$897$	2.29984	0.0767894
$898$	−19.1264	−0.638258
$899$	−92.4052	−3.08188
$900$	9.72711	0.324237
$901$	0.0249159	0.000830069 0
$902$	−1.60502	−0.0534414
$903$	4.73805	0.157672
$904$	−11.4559	−0.381018
$905$	9.77501	0.324932
$906$	17.4890	0.581032
$907$	38.7404	1.28635	0.643177	−	0.765718i	$-0.277616\pi$
$907$	38.7404	1.28635	0.643177	+	0.765718i	$0.277616\pi$
$908$	3.85998	0.128098
$909$	28.3024	0.938732
$910$	−0.798758	−0.0264786
$911$	−40.5749	−1.34431	−0.672153	−	0.740412i	$-0.734630\pi$
$911$	−40.5749	−1.34431	−0.672153	+	0.740412i	$0.734630\pi$
$912$	0.286751	0.00949527
$913$	2.76324	0.0914498
$914$	−32.5581	−1.07693
$915$	4.77081	0.157718
$916$	−17.1815	−0.567694
$917$	38.1953	1.26132
$918$	0.617523	0.0203813
$919$	−6.02219	−0.198654	−0.0993269	−	0.995055i	$-0.531669\pi$
$919$	−6.02219	−0.198654	−0.0993269	+	0.995055i	$0.531669\pi$
$920$	−3.28304	−0.108239
$921$	1.37109	0.0451789
$922$	−11.6947	−0.385144
$923$	−1.20952	−0.0398117
$924$	0.716430	0.0235688
$925$	−0.681513	−0.0224080
$926$	4.87863	0.160322
$927$	−7.49508	−0.246171
$928$	−9.00193	−0.295503
$929$	0.854031	0.0280199	0.0140099	−	0.999902i	$-0.495540\pi$
$929$	0.854031	0.0280199	0.0140099	+	0.999902i	$0.495540\pi$
$930$	−5.53197	−0.181400
$931$	−1.54024	−0.0504793
$932$	−5.83824	−0.191238
$933$	−11.1139	−0.363851
$934$	−0.808710	−0.0264618
$935$	−0.0151611	−0.000495820 0
$936$	−0.848073	−0.0277201
$937$	−1.21695	−0.0397560	−0.0198780	−	0.999802i	$-0.506328\pi$
$937$	−1.21695	−0.0397560	−0.0198780	+	0.999802i	$0.506328\pi$
$938$	−6.85762	−0.223909
$939$	−22.9721	−0.749666
$940$	5.16159	0.168352
$941$	−18.1861	−0.592851	−0.296426	−	0.955056i	$-0.595795\pi$
$941$	−18.1861	−0.592851	−0.296426	+	0.955056i	$0.595795\pi$
$942$	−2.85328	−0.0929649
$943$	43.9854	1.43236
$944$	7.44726	0.242388
$945$	−9.54028	−0.310345
$946$	−0.301995	−0.00981870
$947$	30.2672	0.983552	0.491776	−	0.870722i	$-0.336348\pi$
$947$	30.2672	0.983552	0.491776	+	0.870722i	$0.336348\pi$
$948$	−2.37724	−0.0772093
$949$	0.835202	0.0271118
$950$	−1.39773	−0.0453483
$951$	−8.85915	−0.287278
$952$	−0.441378	−0.0143051
$953$	−14.8656	−0.481543	−0.240772	−	0.970582i	$-0.577400\pi$
$953$	−14.8656	−0.481543	−0.240772	+	0.970582i	$0.577400\pi$
$954$	−0.408698	−0.0132321
$955$	−5.25111	−0.169922
$956$	20.6068	0.666471
$957$	−1.84919	−0.0597757
$958$	−3.59450	−0.116133
$959$	30.4242	0.982450
$960$	−0.538913	−0.0173934
$961$	74.3710	2.39906
$962$	0.0594188	0.00191574
$963$	15.8331	0.510214
$964$	−13.2149	−0.425624
$965$	−3.13074	−0.100782
$966$	−19.6337	−0.631702
$967$	−33.8138	−1.08738	−0.543689	−	0.839287i	$-0.682973\pi$
$967$	−33.8138	−1.08738	−0.543689	+	0.839287i	$0.682973\pi$
$968$	10.9543	0.352086
$969$	−0.0362900	−0.00116580
$970$	3.68863	0.118435
$971$	−55.7114	−1.78786	−0.893932	−	0.448202i	$-0.852065\pi$
$971$	−55.7114	−1.78786	−0.893932	+	0.448202i	$0.852065\pi$
$972$	−16.1160	−0.516921
$973$	−9.17422	−0.294112
$974$	−31.6714	−1.01482
$975$	−1.84018	−0.0589328
$976$	−8.85264	−0.283366
$977$	19.6357	0.628203	0.314102	−	0.949389i	$-0.398297\pi$
$977$	19.6357	0.628203	0.314102	+	0.949389i	$0.398297\pi$
$978$	4.00570	0.128088
$979$	−0.496925	−0.0158818
$980$	2.89469	0.0924676
$981$	−25.0711	−0.800459
$982$	5.62956	0.179647
$983$	26.1874	0.835247	0.417624	−	0.908620i	$-0.362863\pi$
$983$	26.1874	0.835247	0.417624	+	0.908620i	$0.362863\pi$
$984$	7.22025	0.230173
$985$	−9.82447	−0.313034
$986$	1.13925	0.0362810
$987$	30.8680	0.982540
$988$	0.121863	0.00387698
$989$	8.27613	0.263166
$990$	0.248689	0.00790384
$991$	20.0604	0.637239	0.318619	−	0.947883i	$-0.396781\pi$
$991$	20.0604	0.637239	0.318619	+	0.947883i	$0.396781\pi$
$992$	10.2650	0.325915
$993$	28.3185	0.898662
$994$	10.3256	0.327508
$995$	4.48143	0.142071
$996$	−12.4305	−0.393876
$997$	56.9055	1.80221	0.901107	−	0.433596i	$-0.142755\pi$
$997$	56.9055	1.80221	0.901107	+	0.433596i	$0.142755\pi$
$998$	−4.35579	−0.137880
$999$	0.709692	0.0224537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.d.1.19	✓	55

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.d.1.19	✓	55	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.961298 q^{3} +1.00000 q^{4} +0.560610 q^{5} +0.961298 q^{6} -3.48762 q^{7} -1.00000 q^{8} -2.07591 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.961298 q^{3} +1.00000 q^{4} +0.560610 q^{5} +0.961298 q^{6} -3.48762 q^{7} -1.00000 q^{8} -2.07591 q^{9} -0.560610 q^{10} +0.213691 q^{11} -0.961298 q^{12} -0.408531 q^{13} +3.48762 q^{14} -0.538913 q^{15} +1.00000 q^{16} -0.126556 q^{17} +2.07591 q^{18} -0.298296 q^{19} +0.560610 q^{20} +3.35264 q^{21} -0.213691 q^{22} +5.85618 q^{23} +0.961298 q^{24} -4.68572 q^{25} +0.408531 q^{26} +4.87946 q^{27} -3.48762 q^{28} +9.00193 q^{29} +0.538913 q^{30} -10.2650 q^{31} -1.00000 q^{32} -0.205421 q^{33} +0.126556 q^{34} -1.95519 q^{35} -2.07591 q^{36} +0.145445 q^{37} +0.298296 q^{38} +0.392720 q^{39} -0.560610 q^{40} +7.51094 q^{41} -3.35264 q^{42} +1.41323 q^{43} +0.213691 q^{44} -1.16377 q^{45} -5.85618 q^{46} +9.20709 q^{47} -0.961298 q^{48} +5.16347 q^{49} +4.68572 q^{50} +0.121658 q^{51} -0.408531 q^{52} -0.196877 q^{53} -4.87946 q^{54} +0.119798 q^{55} +3.48762 q^{56} +0.286751 q^{57} -9.00193 q^{58} +7.44726 q^{59} -0.538913 q^{60} -8.85264 q^{61} +10.2650 q^{62} +7.23997 q^{63} +1.00000 q^{64} -0.229027 q^{65} +0.205421 q^{66} -1.96628 q^{67} -0.126556 q^{68} -5.62953 q^{69} +1.95519 q^{70} +2.96065 q^{71} +2.07591 q^{72} -2.04440 q^{73} -0.145445 q^{74} +4.50437 q^{75} -0.298296 q^{76} -0.745274 q^{77} -0.392720 q^{78} +2.47295 q^{79} +0.560610 q^{80} +1.53711 q^{81} -7.51094 q^{82} +12.9310 q^{83} +3.35264 q^{84} -0.0709485 q^{85} -1.41323 q^{86} -8.65354 q^{87} -0.213691 q^{88} -2.32543 q^{89} +1.16377 q^{90} +1.42480 q^{91} +5.85618 q^{92} +9.86776 q^{93} -9.20709 q^{94} -0.167228 q^{95} +0.961298 q^{96} -6.57966 q^{97} -5.16347 q^{98} -0.443603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 55 q - 55 q^{2} - 4 q^{3} + 55 q^{4} - 7 q^{5} + 4 q^{6} + 17 q^{7} - 55 q^{8} + 29 q^{9}+O(q^{10}) \) 55 * q - 55 * q^2 - 4 * q^3 + 55 * q^4 - 7 * q^5 + 4 * q^6 + 17 * q^7 - 55 * q^8 + 29 * q^9	\( 55 q - 55 q^{2} - 4 q^{3} + 55 q^{4} - 7 q^{5} + 4 q^{6} + 17 q^{7} - 55 q^{8} + 29 q^{9} + 7 q^{10} - 28 q^{11} - 4 q^{12} + q^{13} - 17 q^{14} - 8 q^{15} + 55 q^{16} - 32 q^{17} - 29 q^{18} - 3 q^{19} - 7 q^{20} - 25 q^{21} + 28 q^{22} - 27 q^{23} + 4 q^{24} + 30 q^{25} - q^{26} - q^{27} + 17 q^{28} - 69 q^{29} + 8 q^{30} - 13 q^{31} - 55 q^{32} - 18 q^{33} + 32 q^{34} - 23 q^{35} + 29 q^{36} + 3 q^{37} + 3 q^{38} - 28 q^{39} + 7 q^{40} - 51 q^{41} + 25 q^{42} + 23 q^{43} - 28 q^{44} - 28 q^{45} + 27 q^{46} - 27 q^{47} - 4 q^{48} + 8 q^{49} - 30 q^{50} - 42 q^{51} + q^{52} - 61 q^{53} + q^{54} + 5 q^{55} - 17 q^{56} - 52 q^{57} + 69 q^{58} - 71 q^{59} - 8 q^{60} - 16 q^{61} + 13 q^{62} + 14 q^{63} + 55 q^{64} - 82 q^{65} + 18 q^{66} + 32 q^{67} - 32 q^{68} - 44 q^{69} + 23 q^{70} - 84 q^{71} - 29 q^{72} - 43 q^{73} - 3 q^{74} - 37 q^{75} - 3 q^{76} - 47 q^{77} + 28 q^{78} - 20 q^{79} - 7 q^{80} - 33 q^{81} + 51 q^{82} + 17 q^{83} - 25 q^{84} + 10 q^{85} - 23 q^{86} - q^{87} + 28 q^{88} - 92 q^{89} + 28 q^{90} - 34 q^{91} - 27 q^{92} - 13 q^{93} + 27 q^{94} - 60 q^{95} + 4 q^{96} - 45 q^{97} - 8 q^{98} - 73 q^{99}+O(q^{100}) \) 55 * q - 55 * q^2 - 4 * q^3 + 55 * q^4 - 7 * q^5 + 4 * q^6 + 17 * q^7 - 55 * q^8 + 29 * q^9 + 7 * q^10 - 28 * q^11 - 4 * q^12 + q^13 - 17 * q^14 - 8 * q^15 + 55 * q^16 - 32 * q^17 - 29 * q^18 - 3 * q^19 - 7 * q^20 - 25 * q^21 + 28 * q^22 - 27 * q^23 + 4 * q^24 + 30 * q^25 - q^26 - q^27 + 17 * q^28 - 69 * q^29 + 8 * q^30 - 13 * q^31 - 55 * q^32 - 18 * q^33 + 32 * q^34 - 23 * q^35 + 29 * q^36 + 3 * q^37 + 3 * q^38 - 28 * q^39 + 7 * q^40 - 51 * q^41 + 25 * q^42 + 23 * q^43 - 28 * q^44 - 28 * q^45 + 27 * q^46 - 27 * q^47 - 4 * q^48 + 8 * q^49 - 30 * q^50 - 42 * q^51 + q^52 - 61 * q^53 + q^54 + 5 * q^55 - 17 * q^56 - 52 * q^57 + 69 * q^58 - 71 * q^59 - 8 * q^60 - 16 * q^61 + 13 * q^62 + 14 * q^63 + 55 * q^64 - 82 * q^65 + 18 * q^66 + 32 * q^67 - 32 * q^68 - 44 * q^69 + 23 * q^70 - 84 * q^71 - 29 * q^72 - 43 * q^73 - 3 * q^74 - 37 * q^75 - 3 * q^76 - 47 * q^77 + 28 * q^78 - 20 * q^79 - 7 * q^80 - 33 * q^81 + 51 * q^82 + 17 * q^83 - 25 * q^84 + 10 * q^85 - 23 * q^86 - q^87 + 28 * q^88 - 92 * q^89 + 28 * q^90 - 34 * q^91 - 27 * q^92 - 13 * q^93 + 27 * q^94 - 60 * q^95 + 4 * q^96 - 45 * q^97 - 8 * q^98 - 73 * q^99

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