LMFDB - Embedded newform 4026.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.46101901.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 5x^{4} + 12x^{3} + 6x^{2} - 5x - 2 $ x^6 - 3x^5 - 5x^4 + 12x^3 + 6x^2 - 5*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.29072$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.656263 q^{5} -1.00000 q^{6} +4.00423 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.656263 q^{5} -1.00000 q^{6} +4.00423 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.656263 q^{10} -1.00000 q^{11} -1.00000 q^{12} -0.713512 q^{13} +4.00423 q^{14} +0.656263 q^{15} +1.00000 q^{16} -6.79616 q^{17} +1.00000 q^{18} -2.11237 q^{19} -0.656263 q^{20} -4.00423 q^{21} -1.00000 q^{22} -5.33405 q^{23} -1.00000 q^{24} -4.56932 q^{25} -0.713512 q^{26} -1.00000 q^{27} +4.00423 q^{28} -3.73080 q^{29} +0.656263 q^{30} -6.16594 q^{31} +1.00000 q^{32} +1.00000 q^{33} -6.79616 q^{34} -2.62783 q^{35} +1.00000 q^{36} -4.72014 q^{37} -2.11237 q^{38} +0.713512 q^{39} -0.656263 q^{40} +8.71067 q^{41} -4.00423 q^{42} +1.79594 q^{43} -1.00000 q^{44} -0.656263 q^{45} -5.33405 q^{46} -8.19049 q^{47} -1.00000 q^{48} +9.03386 q^{49} -4.56932 q^{50} +6.79616 q^{51} -0.713512 q^{52} +1.24351 q^{53} -1.00000 q^{54} +0.656263 q^{55} +4.00423 q^{56} +2.11237 q^{57} -3.73080 q^{58} -10.0801 q^{59} +0.656263 q^{60} +1.00000 q^{61} -6.16594 q^{62} +4.00423 q^{63} +1.00000 q^{64} +0.468252 q^{65} +1.00000 q^{66} +7.84450 q^{67} -6.79616 q^{68} +5.33405 q^{69} -2.62783 q^{70} -11.3551 q^{71} +1.00000 q^{72} -8.18174 q^{73} -4.72014 q^{74} +4.56932 q^{75} -2.11237 q^{76} -4.00423 q^{77} +0.713512 q^{78} +8.71421 q^{79} -0.656263 q^{80} +1.00000 q^{81} +8.71067 q^{82} +13.1943 q^{83} -4.00423 q^{84} +4.46007 q^{85} +1.79594 q^{86} +3.73080 q^{87} -1.00000 q^{88} +11.7623 q^{89} -0.656263 q^{90} -2.85707 q^{91} -5.33405 q^{92} +6.16594 q^{93} -8.19049 q^{94} +1.38627 q^{95} -1.00000 q^{96} +5.56803 q^{97} +9.03386 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} + q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 + q^7 + 6 * q^8 + 6 * q^9	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} + q^{7} + 6 q^{8} + 6 q^{9} - 6 q^{10} - 6 q^{11} - 6 q^{12} + 2 q^{13} + q^{14} + 6 q^{15} + 6 q^{16} - 13 q^{17} + 6 q^{18} + q^{19} - 6 q^{20} - q^{21} - 6 q^{22} - 11 q^{23} - 6 q^{24} + 8 q^{25} + 2 q^{26} - 6 q^{27} + q^{28} - 14 q^{29} + 6 q^{30} - 5 q^{31} + 6 q^{32} + 6 q^{33} - 13 q^{34} - 13 q^{35} + 6 q^{36} - 6 q^{37} + q^{38} - 2 q^{39} - 6 q^{40} - 25 q^{41} - q^{42} + 19 q^{43} - 6 q^{44} - 6 q^{45} - 11 q^{46} - 10 q^{47} - 6 q^{48} - 5 q^{49} + 8 q^{50} + 13 q^{51} + 2 q^{52} - 17 q^{53} - 6 q^{54} + 6 q^{55} + q^{56} - q^{57} - 14 q^{58} - 14 q^{59} + 6 q^{60} + 6 q^{61} - 5 q^{62} + q^{63} + 6 q^{64} + 6 q^{65} + 6 q^{66} + 12 q^{67} - 13 q^{68} + 11 q^{69} - 13 q^{70} + 6 q^{71} + 6 q^{72} - 29 q^{73} - 6 q^{74} - 8 q^{75} + q^{76} - q^{77} - 2 q^{78} - 24 q^{79} - 6 q^{80} + 6 q^{81} - 25 q^{82} - 9 q^{83} - q^{84} - 22 q^{85} + 19 q^{86} + 14 q^{87} - 6 q^{88} - 4 q^{89} - 6 q^{90} - 29 q^{91} - 11 q^{92} + 5 q^{93} - 10 q^{94} - 27 q^{95} - 6 q^{96} - 5 q^{97} - 5 q^{98} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 + q^7 + 6 * q^8 + 6 * q^9 - 6 * q^10 - 6 * q^11 - 6 * q^12 + 2 * q^13 + q^14 + 6 * q^15 + 6 * q^16 - 13 * q^17 + 6 * q^18 + q^19 - 6 * q^20 - q^21 - 6 * q^22 - 11 * q^23 - 6 * q^24 + 8 * q^25 + 2 * q^26 - 6 * q^27 + q^28 - 14 * q^29 + 6 * q^30 - 5 * q^31 + 6 * q^32 + 6 * q^33 - 13 * q^34 - 13 * q^35 + 6 * q^36 - 6 * q^37 + q^38 - 2 * q^39 - 6 * q^40 - 25 * q^41 - q^42 + 19 * q^43 - 6 * q^44 - 6 * q^45 - 11 * q^46 - 10 * q^47 - 6 * q^48 - 5 * q^49 + 8 * q^50 + 13 * q^51 + 2 * q^52 - 17 * q^53 - 6 * q^54 + 6 * q^55 + q^56 - q^57 - 14 * q^58 - 14 * q^59 + 6 * q^60 + 6 * q^61 - 5 * q^62 + q^63 + 6 * q^64 + 6 * q^65 + 6 * q^66 + 12 * q^67 - 13 * q^68 + 11 * q^69 - 13 * q^70 + 6 * q^71 + 6 * q^72 - 29 * q^73 - 6 * q^74 - 8 * q^75 + q^76 - q^77 - 2 * q^78 - 24 * q^79 - 6 * q^80 + 6 * q^81 - 25 * q^82 - 9 * q^83 - q^84 - 22 * q^85 + 19 * q^86 + 14 * q^87 - 6 * q^88 - 4 * q^89 - 6 * q^90 - 29 * q^91 - 11 * q^92 + 5 * q^93 - 10 * q^94 - 27 * q^95 - 6 * q^96 - 5 * q^97 - 5 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.656263	−0.293490	−0.146745	−	0.989174i	$-0.546880\pi$
$5$	−0.656263	−0.293490	−0.146745	+	0.989174i	$0.546880\pi$
$6$	−1.00000	−0.408248
$7$	4.00423	1.51346	0.756728	−	0.653729i	$-0.226797\pi$
$7$	4.00423	1.51346	0.756728	+	0.653729i	$0.226797\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.656263	−0.207529
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−0.713512	−0.197893	−0.0989463	−	0.995093i	$-0.531547\pi$
$13$	−0.713512	−0.197893	−0.0989463	+	0.995093i	$0.531547\pi$
$14$	4.00423	1.07018
$15$	0.656263	0.169446
$16$	1.00000	0.250000
$17$	−6.79616	−1.64831	−0.824156	−	0.566363i	$-0.808350\pi$
$17$	−6.79616	−1.64831	−0.824156	+	0.566363i	$0.808350\pi$
$18$	1.00000	0.235702
$19$	−2.11237	−0.484611	−0.242305	−	0.970200i	$-0.577904\pi$
$19$	−2.11237	−0.484611	−0.242305	+	0.970200i	$0.577904\pi$
$20$	−0.656263	−0.146745
$21$	−4.00423	−0.873795
$22$	−1.00000	−0.213201
$23$	−5.33405	−1.11223	−0.556113	−	0.831107i	$-0.687708\pi$
$23$	−5.33405	−1.11223	−0.556113	+	0.831107i	$0.687708\pi$
$24$	−1.00000	−0.204124
$25$	−4.56932	−0.913864
$26$	−0.713512	−0.139931
$27$	−1.00000	−0.192450
$28$	4.00423	0.756728
$29$	−3.73080	−0.692793	−0.346396	−	0.938088i	$-0.612595\pi$
$29$	−3.73080	−0.692793	−0.346396	+	0.938088i	$0.612595\pi$
$30$	0.656263	0.119817
$31$	−6.16594	−1.10743	−0.553717	−	0.832705i	$-0.686791\pi$
$31$	−6.16594	−1.10743	−0.553717	+	0.832705i	$0.686791\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−6.79616	−1.16553
$35$	−2.62783	−0.444184
$36$	1.00000	0.166667
$37$	−4.72014	−0.775986	−0.387993	−	0.921662i	$-0.626832\pi$
$37$	−4.72014	−0.775986	−0.387993	+	0.921662i	$0.626832\pi$
$38$	−2.11237	−0.342672
$39$	0.713512	0.114253
$40$	−0.656263	−0.103764
$41$	8.71067	1.36038	0.680189	−	0.733037i	$-0.261898\pi$
$41$	8.71067	1.36038	0.680189	+	0.733037i	$0.261898\pi$
$42$	−4.00423	−0.617866
$43$	1.79594	0.273878	0.136939	−	0.990579i	$-0.456274\pi$
$43$	1.79594	0.273878	0.136939	+	0.990579i	$0.456274\pi$
$44$	−1.00000	−0.150756
$45$	−0.656263	−0.0978299
$46$	−5.33405	−0.786462
$47$	−8.19049	−1.19471	−0.597353	−	0.801978i	$-0.703781\pi$
$47$	−8.19049	−1.19471	−0.597353	+	0.801978i	$0.703781\pi$
$48$	−1.00000	−0.144338
$49$	9.03386	1.29055
$50$	−4.56932	−0.646199
$51$	6.79616	0.951653
$52$	−0.713512	−0.0989463
$53$	1.24351	0.170809	0.0854047	−	0.996346i	$-0.472782\pi$
$53$	1.24351	0.170809	0.0854047	+	0.996346i	$0.472782\pi$
$54$	−1.00000	−0.136083
$55$	0.656263	0.0884905
$56$	4.00423	0.535088
$57$	2.11237	0.279790
$58$	−3.73080	−0.489878
$59$	−10.0801	−1.31232	−0.656158	−	0.754623i	$-0.727820\pi$
$59$	−10.0801	−1.31232	−0.656158	+	0.754623i	$0.727820\pi$
$60$	0.656263	0.0847232
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−6.16594	−0.783075
$63$	4.00423	0.504486
$64$	1.00000	0.125000
$65$	0.468252	0.0580795
$66$	1.00000	0.123091
$67$	7.84450	0.958358	0.479179	−	0.877717i	$-0.340935\pi$
$67$	7.84450	0.958358	0.479179	+	0.877717i	$0.340935\pi$
$68$	−6.79616	−0.824156
$69$	5.33405	0.642144
$70$	−2.62783	−0.314086
$71$	−11.3551	−1.34761	−0.673804	−	0.738910i	$-0.735341\pi$
$71$	−11.3551	−1.34761	−0.673804	+	0.738910i	$0.735341\pi$
$72$	1.00000	0.117851
$73$	−8.18174	−0.957600	−0.478800	−	0.877924i	$-0.658928\pi$
$73$	−8.18174	−0.957600	−0.478800	+	0.877924i	$0.658928\pi$
$74$	−4.72014	−0.548705
$75$	4.56932	0.527619
$76$	−2.11237	−0.242305
$77$	−4.00423	−0.456324
$78$	0.713512	0.0807893
$79$	8.71421	0.980426	0.490213	−	0.871603i	$-0.336919\pi$
$79$	8.71421	0.980426	0.490213	+	0.871603i	$0.336919\pi$
$80$	−0.656263	−0.0733724
$81$	1.00000	0.111111
$82$	8.71067	0.961933
$83$	13.1943	1.44826	0.724129	−	0.689665i	$-0.242242\pi$
$83$	13.1943	1.44826	0.724129	+	0.689665i	$0.242242\pi$
$84$	−4.00423	−0.436897
$85$	4.46007	0.483762
$86$	1.79594	0.193661
$87$	3.73080	0.399984
$88$	−1.00000	−0.106600
$89$	11.7623	1.24680	0.623399	−	0.781904i	$-0.285751\pi$
$89$	11.7623	1.24680	0.623399	+	0.781904i	$0.285751\pi$
$90$	−0.656263	−0.0691762
$91$	−2.85707	−0.299502
$92$	−5.33405	−0.556113
$93$	6.16594	0.639378
$94$	−8.19049	−0.844785
$95$	1.38627	0.142228
$96$	−1.00000	−0.102062
$97$	5.56803	0.565348	0.282674	−	0.959216i	$-0.408779\pi$
$97$	5.56803	0.565348	0.282674	+	0.959216i	$0.408779\pi$
$98$	9.03386	0.912558
$99$	−1.00000	−0.100504
$100$	−4.56932	−0.456932
$101$	−14.5329	−1.44608	−0.723041	−	0.690805i	$-0.757256\pi$
$101$	−14.5329	−1.44608	−0.723041	+	0.690805i	$0.757256\pi$
$102$	6.79616	0.672920
$103$	−6.19014	−0.609933	−0.304966	−	0.952363i	$-0.598645\pi$
$103$	−6.19014	−0.609933	−0.304966	+	0.952363i	$0.598645\pi$
$104$	−0.713512	−0.0699656
$105$	2.62783	0.256450
$106$	1.24351	0.120780
$107$	−0.908643	−0.0878418	−0.0439209	−	0.999035i	$-0.513985\pi$
$107$	−0.908643	−0.0878418	−0.0439209	+	0.999035i	$0.513985\pi$
$108$	−1.00000	−0.0962250
$109$	15.1500	1.45110	0.725551	−	0.688168i	$-0.241585\pi$
$109$	15.1500	1.45110	0.725551	+	0.688168i	$0.241585\pi$
$110$	0.656263	0.0625722
$111$	4.72014	0.448016
$112$	4.00423	0.378364
$113$	−15.6746	−1.47454	−0.737271	−	0.675597i	$-0.763886\pi$
$113$	−15.6746	−1.47454	−0.737271	+	0.675597i	$0.763886\pi$
$114$	2.11237	0.197841
$115$	3.50054	0.326427
$116$	−3.73080	−0.346396
$117$	−0.713512	−0.0659642
$118$	−10.0801	−0.927948
$119$	−27.2134	−2.49465
$120$	0.656263	0.0599083
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	−8.71067	−0.785415
$124$	−6.16594	−0.553717
$125$	6.27999	0.561699
$126$	4.00423	0.356725
$127$	15.9448	1.41487	0.707436	−	0.706778i	$-0.249852\pi$
$127$	15.9448	1.41487	0.707436	+	0.706778i	$0.249852\pi$
$128$	1.00000	0.0883883
$129$	−1.79594	−0.158124
$130$	0.468252	0.0410684
$131$	−2.67329	−0.233566	−0.116783	−	0.993157i	$-0.537258\pi$
$131$	−2.67329	−0.233566	−0.116783	+	0.993157i	$0.537258\pi$
$132$	1.00000	0.0870388
$133$	−8.45841	−0.733437
$134$	7.84450	0.677661
$135$	0.656263	0.0564821
$136$	−6.79616	−0.582766
$137$	12.1598	1.03889	0.519443	−	0.854505i	$-0.326140\pi$
$137$	12.1598	1.03889	0.519443	+	0.854505i	$0.326140\pi$
$138$	5.33405	0.454064
$139$	−20.3027	−1.72206	−0.861028	−	0.508557i	$-0.830179\pi$
$139$	−20.3027	−1.72206	−0.861028	+	0.508557i	$0.830179\pi$
$140$	−2.62783	−0.222092
$141$	8.19049	0.689764
$142$	−11.3551	−0.952902
$143$	0.713512	0.0596669
$144$	1.00000	0.0833333
$145$	2.44839	0.203328
$146$	−8.18174	−0.677125
$147$	−9.03386	−0.745100
$148$	−4.72014	−0.387993
$149$	−14.0856	−1.15394	−0.576969	−	0.816766i	$-0.695765\pi$
$149$	−14.0856	−1.15394	−0.576969	+	0.816766i	$0.695765\pi$
$150$	4.56932	0.373083
$151$	4.46115	0.363043	0.181522	−	0.983387i	$-0.441898\pi$
$151$	4.46115	0.363043	0.181522	+	0.983387i	$0.441898\pi$
$152$	−2.11237	−0.171336
$153$	−6.79616	−0.549437
$154$	−4.00423	−0.322670
$155$	4.04648	0.325021
$156$	0.713512	0.0571267
$157$	−0.913169	−0.0728788	−0.0364394	−	0.999336i	$-0.511602\pi$
$157$	−0.913169	−0.0728788	−0.0364394	+	0.999336i	$0.511602\pi$
$158$	8.71421	0.693266
$159$	−1.24351	−0.0986168
$160$	−0.656263	−0.0518821
$161$	−21.3588	−1.68331
$162$	1.00000	0.0785674
$163$	0.808887	0.0633569	0.0316784	−	0.999498i	$-0.489915\pi$
$163$	0.808887	0.0633569	0.0316784	+	0.999498i	$0.489915\pi$
$164$	8.71067	0.680189
$165$	−0.656263	−0.0510900
$166$	13.1943	1.02407
$167$	−21.8712	−1.69245	−0.846224	−	0.532828i	$-0.821129\pi$
$167$	−21.8712	−1.69245	−0.846224	+	0.532828i	$0.821129\pi$
$168$	−4.00423	−0.308933
$169$	−12.4909	−0.960838
$170$	4.46007	0.342072
$171$	−2.11237	−0.161537
$172$	1.79594	0.136939
$173$	23.8917	1.81645	0.908226	−	0.418480i	$-0.137437\pi$
$173$	23.8917	1.81645	0.908226	+	0.418480i	$0.137437\pi$
$174$	3.73080	0.282831
$175$	−18.2966	−1.38309
$176$	−1.00000	−0.0753778
$177$	10.0801	0.757666
$178$	11.7623	0.881619
$179$	26.4579	1.97756	0.988778	−	0.149395i	$-0.0477326\pi$
$179$	26.4579	1.97756	0.988778	+	0.149395i	$0.0477326\pi$
$180$	−0.656263	−0.0489150
$181$	−16.2248	−1.20598	−0.602991	−	0.797748i	$-0.706024\pi$
$181$	−16.2248	−1.20598	−0.602991	+	0.797748i	$0.706024\pi$
$182$	−2.85707	−0.211780
$183$	−1.00000	−0.0739221
$184$	−5.33405	−0.393231
$185$	3.09765	0.227744
$186$	6.16594	0.452108
$187$	6.79616	0.496985
$188$	−8.19049	−0.597353
$189$	−4.00423	−0.291265
$190$	1.38627	0.100571
$191$	−27.3752	−1.98080	−0.990401	−	0.138224i	$-0.955861\pi$
$191$	−27.3752	−1.98080	−0.990401	+	0.138224i	$0.955861\pi$
$192$	−1.00000	−0.0721688
$193$	13.1566	0.947031	0.473515	−	0.880786i	$-0.342985\pi$
$193$	13.1566	0.947031	0.473515	+	0.880786i	$0.342985\pi$
$194$	5.56803	0.399761
$195$	−0.468252	−0.0335322
$196$	9.03386	0.645276
$197$	−16.5155	−1.17668	−0.588340	−	0.808614i	$-0.700218\pi$
$197$	−16.5155	−1.17668	−0.588340	+	0.808614i	$0.700218\pi$
$198$	−1.00000	−0.0710669
$199$	4.20859	0.298339	0.149169	−	0.988812i	$-0.452340\pi$
$199$	4.20859	0.298339	0.149169	+	0.988812i	$0.452340\pi$
$200$	−4.56932	−0.323100
$201$	−7.84450	−0.553308
$202$	−14.5329	−1.02253
$203$	−14.9390	−1.04851
$204$	6.79616	0.475826
$205$	−5.71649	−0.399257
$206$	−6.19014	−0.431288
$207$	−5.33405	−0.370742
$208$	−0.713512	−0.0494732
$209$	2.11237	0.146116
$210$	2.62783	0.181337
$211$	−1.57738	−0.108591	−0.0542957	−	0.998525i	$-0.517291\pi$
$211$	−1.57738	−0.108591	−0.0542957	+	0.998525i	$0.517291\pi$
$212$	1.24351	0.0854047
$213$	11.3551	0.778042
$214$	−0.908643	−0.0621135
$215$	−1.17861	−0.0803804
$216$	−1.00000	−0.0680414
$217$	−24.6898	−1.67605
$218$	15.1500	1.02608
$219$	8.18174	0.552870
$220$	0.656263	0.0442452
$221$	4.84914	0.326189
$222$	4.72014	0.316795
$223$	−17.4661	−1.16962	−0.584808	−	0.811172i	$-0.698830\pi$
$223$	−17.4661	−1.16962	−0.584808	+	0.811172i	$0.698830\pi$
$224$	4.00423	0.267544
$225$	−4.56932	−0.304621
$226$	−15.6746	−1.04266
$227$	11.9588	0.793731	0.396866	−	0.917877i	$-0.370098\pi$
$227$	11.9588	0.793731	0.396866	+	0.917877i	$0.370098\pi$
$228$	2.11237	0.139895
$229$	14.7092	0.972012	0.486006	−	0.873956i	$-0.338453\pi$
$229$	14.7092	0.972012	0.486006	+	0.873956i	$0.338453\pi$
$230$	3.50054	0.230819
$231$	4.00423	0.263459
$232$	−3.73080	−0.244939
$233$	−11.0711	−0.725295	−0.362647	−	0.931926i	$-0.618127\pi$
$233$	−11.0711	−0.725295	−0.362647	+	0.931926i	$0.618127\pi$
$234$	−0.713512	−0.0466437
$235$	5.37512	0.350634
$236$	−10.0801	−0.656158
$237$	−8.71421	−0.566049
$238$	−27.2134	−1.76398
$239$	−16.7529	−1.08365	−0.541827	−	0.840490i	$-0.682267\pi$
$239$	−16.7529	−1.08365	−0.541827	+	0.840490i	$0.682267\pi$
$240$	0.656263	0.0423616
$241$	10.3576	0.667191	0.333596	−	0.942716i	$-0.391738\pi$
$241$	10.3576	0.667191	0.333596	+	0.942716i	$0.391738\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	1.00000	0.0640184
$245$	−5.92859	−0.378764
$246$	−8.71067	−0.555372
$247$	1.50720	0.0959009
$248$	−6.16594	−0.391537
$249$	−13.1943	−0.836152
$250$	6.27999	0.397181
$251$	−16.4885	−1.04075	−0.520374	−	0.853939i	$-0.674207\pi$
$251$	−16.4885	−1.04075	−0.520374	+	0.853939i	$0.674207\pi$
$252$	4.00423	0.252243
$253$	5.33405	0.335349
$254$	15.9448	1.00046
$255$	−4.46007	−0.279300
$256$	1.00000	0.0625000
$257$	9.19399	0.573506	0.286753	−	0.958005i	$-0.407424\pi$
$257$	9.19399	0.573506	0.286753	+	0.958005i	$0.407424\pi$
$258$	−1.79594	−0.111810
$259$	−18.9005	−1.17442
$260$	0.468252	0.0290397
$261$	−3.73080	−0.230931
$262$	−2.67329	−0.165156
$263$	29.6162	1.82621	0.913105	−	0.407724i	$-0.133677\pi$
$263$	29.6162	1.82621	0.913105	+	0.407724i	$0.133677\pi$
$264$	1.00000	0.0615457
$265$	−0.816070	−0.0501308
$266$	−8.45841	−0.518619
$267$	−11.7623	−0.719839
$268$	7.84450	0.479179
$269$	2.41860	0.147465	0.0737323	−	0.997278i	$-0.476509\pi$
$269$	2.41860	0.147465	0.0737323	+	0.997278i	$0.476509\pi$
$270$	0.656263	0.0399389
$271$	20.1206	1.22224	0.611119	−	0.791539i	$-0.290720\pi$
$271$	20.1206	1.22224	0.611119	+	0.791539i	$0.290720\pi$
$272$	−6.79616	−0.412078
$273$	2.85707	0.172918
$274$	12.1598	0.734603
$275$	4.56932	0.275540
$276$	5.33405	0.321072
$277$	26.1571	1.57163	0.785814	−	0.618462i	$-0.212244\pi$
$277$	26.1571	1.57163	0.785814	+	0.618462i	$0.212244\pi$
$278$	−20.3027	−1.21768
$279$	−6.16594	−0.369145
$280$	−2.62783	−0.157043
$281$	−22.2517	−1.32742	−0.663712	−	0.747989i	$-0.731020\pi$
$281$	−22.2517	−1.32742	−0.663712	+	0.747989i	$0.731020\pi$
$282$	8.19049	0.487737
$283$	19.9109	1.18358	0.591790	−	0.806092i	$-0.298422\pi$
$283$	19.9109	1.18358	0.591790	+	0.806092i	$0.298422\pi$
$284$	−11.3551	−0.673804
$285$	−1.38627	−0.0821155
$286$	0.713512	0.0421909
$287$	34.8795	2.05887
$288$	1.00000	0.0589256
$289$	29.1878	1.71693
$290$	2.44839	0.143774
$291$	−5.56803	−0.326404
$292$	−8.18174	−0.478800
$293$	−6.69120	−0.390904	−0.195452	−	0.980713i	$-0.562617\pi$
$293$	−6.69120	−0.390904	−0.195452	+	0.980713i	$0.562617\pi$
$294$	−9.03386	−0.526865
$295$	6.61519	0.385151
$296$	−4.72014	−0.274353
$297$	1.00000	0.0580259
$298$	−14.0856	−0.815958
$299$	3.80591	0.220101
$300$	4.56932	0.263810
$301$	7.19135	0.414503
$302$	4.46115	0.256710
$303$	14.5329	0.834896
$304$	−2.11237	−0.121153
$305$	−0.656263	−0.0375775
$306$	−6.79616	−0.388511
$307$	−14.1765	−0.809097	−0.404548	−	0.914517i	$-0.632571\pi$
$307$	−14.1765	−0.809097	−0.404548	+	0.914517i	$0.632571\pi$
$308$	−4.00423	−0.228162
$309$	6.19014	0.352145
$310$	4.04648	0.229824
$311$	−24.5928	−1.39453	−0.697266	−	0.716812i	$-0.745600\pi$
$311$	−24.5928	−1.39453	−0.697266	+	0.716812i	$0.745600\pi$
$312$	0.713512	0.0403947
$313$	−14.4907	−0.819061	−0.409531	−	0.912296i	$-0.634307\pi$
$313$	−14.4907	−0.819061	−0.409531	+	0.912296i	$0.634307\pi$
$314$	−0.913169	−0.0515331
$315$	−2.62783	−0.148061
$316$	8.71421	0.490213
$317$	−20.8435	−1.17069	−0.585343	−	0.810786i	$-0.699040\pi$
$317$	−20.8435	−1.17069	−0.585343	+	0.810786i	$0.699040\pi$
$318$	−1.24351	−0.0697326
$319$	3.73080	0.208885
$320$	−0.656263	−0.0366862
$321$	0.908643	0.0507155
$322$	−21.3588	−1.19028
$323$	14.3560	0.798789
$324$	1.00000	0.0555556
$325$	3.26026	0.180847
$326$	0.808887	0.0448001
$327$	−15.1500	−0.837794
$328$	8.71067	0.480966
$329$	−32.7966	−1.80814
$330$	−0.656263	−0.0361261
$331$	35.0585	1.92699	0.963495	−	0.267725i	$-0.0862720\pi$
$331$	35.0585	1.92699	0.963495	+	0.267725i	$0.0862720\pi$
$332$	13.1943	0.724129
$333$	−4.72014	−0.258662
$334$	−21.8712	−1.19674
$335$	−5.14805	−0.281268
$336$	−4.00423	−0.218449
$337$	−6.44860	−0.351278	−0.175639	−	0.984455i	$-0.556199\pi$
$337$	−6.44860	−0.351278	−0.175639	+	0.984455i	$0.556199\pi$
$338$	−12.4909	−0.679415
$339$	15.6746	0.851327
$340$	4.46007	0.241881
$341$	6.16594	0.333904
$342$	−2.11237	−0.114224
$343$	8.14404	0.439737
$344$	1.79594	0.0968305
$345$	−3.50054	−0.188463
$346$	23.8917	1.28443
$347$	−28.7772	−1.54484	−0.772422	−	0.635110i	$-0.780955\pi$
$347$	−28.7772	−1.54484	−0.772422	+	0.635110i	$0.780955\pi$
$348$	3.73080	0.199992
$349$	−25.8936	−1.38605	−0.693026	−	0.720912i	$-0.743723\pi$
$349$	−25.8936	−1.38605	−0.693026	+	0.720912i	$0.743723\pi$
$350$	−18.2966	−0.977995
$351$	0.713512	0.0380845
$352$	−1.00000	−0.0533002
$353$	7.13670	0.379848	0.189924	−	0.981799i	$-0.439176\pi$
$353$	7.13670	0.379848	0.189924	+	0.981799i	$0.439176\pi$
$354$	10.0801	0.535751
$355$	7.45196	0.395509
$356$	11.7623	0.623399
$357$	27.2134	1.44029
$358$	26.4579	1.39834
$359$	26.2850	1.38727	0.693635	−	0.720327i	$-0.256008\pi$
$359$	26.2850	1.38727	0.693635	+	0.720327i	$0.256008\pi$
$360$	−0.656263	−0.0345881
$361$	−14.5379	−0.765152
$362$	−16.2248	−0.852757
$363$	−1.00000	−0.0524864
$364$	−2.85707	−0.149751
$365$	5.36937	0.281046
$366$	−1.00000	−0.0522708
$367$	−10.3643	−0.541011	−0.270505	−	0.962718i	$-0.587191\pi$
$367$	−10.3643	−0.541011	−0.270505	+	0.962718i	$0.587191\pi$
$368$	−5.33405	−0.278056
$369$	8.71067	0.453459
$370$	3.09765	0.161039
$371$	4.97930	0.258513
$372$	6.16594	0.319689
$373$	−23.5014	−1.21686	−0.608428	−	0.793609i	$-0.708200\pi$
$373$	−23.5014	−1.21686	−0.608428	+	0.793609i	$0.708200\pi$
$374$	6.79616	0.351421
$375$	−6.27999	−0.324297
$376$	−8.19049	−0.422392
$377$	2.66197	0.137099
$378$	−4.00423	−0.205955
$379$	−11.8214	−0.607225	−0.303613	−	0.952796i	$-0.598193\pi$
$379$	−11.8214	−0.607225	−0.303613	+	0.952796i	$0.598193\pi$
$380$	1.38627	0.0711141
$381$	−15.9448	−0.816876
$382$	−27.3752	−1.40064
$383$	12.8517	0.656692	0.328346	−	0.944558i	$-0.393509\pi$
$383$	12.8517	0.656692	0.328346	+	0.944558i	$0.393509\pi$
$384$	−1.00000	−0.0510310
$385$	2.62783	0.133927
$386$	13.1566	0.669652
$387$	1.79594	0.0912927
$388$	5.56803	0.282674
$389$	−2.87690	−0.145865	−0.0729323	−	0.997337i	$-0.523236\pi$
$389$	−2.87690	−0.145865	−0.0729323	+	0.997337i	$0.523236\pi$
$390$	−0.468252	−0.0237108
$391$	36.2510	1.83329
$392$	9.03386	0.456279
$393$	2.67329	0.134849
$394$	−16.5155	−0.832038
$395$	−5.71882	−0.287745
$396$	−1.00000	−0.0502519
$397$	−13.9395	−0.699603	−0.349801	−	0.936824i	$-0.613751\pi$
$397$	−13.9395	−0.699603	−0.349801	+	0.936824i	$0.613751\pi$
$398$	4.20859	0.210957
$399$	8.45841	0.423450
$400$	−4.56932	−0.228466
$401$	33.5896	1.67739	0.838694	−	0.544604i	$-0.183320\pi$
$401$	33.5896	1.67739	0.838694	+	0.544604i	$0.183320\pi$
$402$	−7.84450	−0.391248
$403$	4.39947	0.219153
$404$	−14.5329	−0.723041
$405$	−0.656263	−0.0326100
$406$	−14.9390	−0.741410
$407$	4.72014	0.233969
$408$	6.79616	0.336460
$409$	−30.7280	−1.51940	−0.759700	−	0.650273i	$-0.774654\pi$
$409$	−30.7280	−1.51940	−0.759700	+	0.650273i	$0.774654\pi$
$410$	−5.71649	−0.282317
$411$	−12.1598	−0.599801
$412$	−6.19014	−0.304966
$413$	−40.3630	−1.98613
$414$	−5.33405	−0.262154
$415$	−8.65890	−0.425049
$416$	−0.713512	−0.0349828
$417$	20.3027	0.994230
$418$	2.11237	0.103319
$419$	29.4540	1.43892	0.719461	−	0.694533i	$-0.244389\pi$
$419$	29.4540	1.43892	0.719461	+	0.694533i	$0.244389\pi$
$420$	2.62783	0.128225
$421$	7.89348	0.384705	0.192352	−	0.981326i	$-0.438388\pi$
$421$	7.89348	0.384705	0.192352	+	0.981326i	$0.438388\pi$
$422$	−1.57738	−0.0767857
$423$	−8.19049	−0.398235
$424$	1.24351	0.0603902
$425$	31.0538	1.50633
$426$	11.3551	0.550158
$427$	4.00423	0.193778
$428$	−0.908643	−0.0439209
$429$	−0.713512	−0.0344487
$430$	−1.17861	−0.0568375
$431$	37.7385	1.81780	0.908899	−	0.417016i	$-0.136924\pi$
$431$	37.7385	1.81780	0.908899	+	0.417016i	$0.136924\pi$
$432$	−1.00000	−0.0481125
$433$	−39.2942	−1.88836	−0.944179	−	0.329433i	$-0.893142\pi$
$433$	−39.2942	−1.88836	−0.944179	+	0.329433i	$0.893142\pi$
$434$	−24.6898	−1.18515
$435$	−2.44839	−0.117391
$436$	15.1500	0.725551
$437$	11.2675	0.538996
$438$	8.18174	0.390938
$439$	14.6566	0.699520	0.349760	−	0.936839i	$-0.386263\pi$
$439$	14.6566	0.699520	0.349760	+	0.936839i	$0.386263\pi$
$440$	0.656263	0.0312861
$441$	9.03386	0.430184
$442$	4.84914	0.230650
$443$	18.2295	0.866110	0.433055	−	0.901368i	$-0.357436\pi$
$443$	18.2295	0.866110	0.433055	+	0.901368i	$0.357436\pi$
$444$	4.72014	0.224008
$445$	−7.71914	−0.365922
$446$	−17.4661	−0.827044
$447$	14.0856	0.666227
$448$	4.00423	0.189182
$449$	−24.5681	−1.15944	−0.579720	−	0.814816i	$-0.696838\pi$
$449$	−24.5681	−1.15944	−0.579720	+	0.814816i	$0.696838\pi$
$450$	−4.56932	−0.215400
$451$	−8.71067	−0.410169
$452$	−15.6746	−0.737271
$453$	−4.46115	−0.209603
$454$	11.9588	0.561253
$455$	1.87499	0.0879008
$456$	2.11237	0.0989207
$457$	16.0312	0.749906	0.374953	−	0.927044i	$-0.377659\pi$
$457$	16.0312	0.749906	0.374953	+	0.927044i	$0.377659\pi$
$458$	14.7092	0.687316
$459$	6.79616	0.317218
$460$	3.50054	0.163213
$461$	38.5537	1.79562	0.897812	−	0.440379i	$-0.145156\pi$
$461$	38.5537	1.79562	0.897812	+	0.440379i	$0.145156\pi$
$462$	4.00423	0.186294
$463$	4.84718	0.225267	0.112634	−	0.993637i	$-0.464071\pi$
$463$	4.84718	0.225267	0.112634	+	0.993637i	$0.464071\pi$
$464$	−3.73080	−0.173198
$465$	−4.04648	−0.187651
$466$	−11.0711	−0.512861
$467$	24.5887	1.13783	0.568916	−	0.822396i	$-0.307363\pi$
$467$	24.5887	1.13783	0.568916	+	0.822396i	$0.307363\pi$
$468$	−0.713512	−0.0329821
$469$	31.4112	1.45043
$470$	5.37512	0.247936
$471$	0.913169	0.0420766
$472$	−10.0801	−0.463974
$473$	−1.79594	−0.0825774
$474$	−8.71421	−0.400257
$475$	9.65209	0.442868
$476$	−27.2134	−1.24732
$477$	1.24351	0.0569364
$478$	−16.7529	−0.766259
$479$	−32.8682	−1.50179	−0.750894	−	0.660423i	$-0.770377\pi$
$479$	−32.8682	−1.50179	−0.750894	+	0.660423i	$0.770377\pi$
$480$	0.656263	0.0299542
$481$	3.36788	0.153562
$482$	10.3576	0.471776
$483$	21.3588	0.971857
$484$	1.00000	0.0454545
$485$	−3.65409	−0.165924
$486$	−1.00000	−0.0453609
$487$	−11.9114	−0.539759	−0.269880	−	0.962894i	$-0.586984\pi$
$487$	−11.9114	−0.539759	−0.269880	+	0.962894i	$0.586984\pi$
$488$	1.00000	0.0452679
$489$	−0.808887	−0.0365791
$490$	−5.92859	−0.267826
$491$	17.9442	0.809811	0.404906	−	0.914358i	$-0.367304\pi$
$491$	17.9442	0.809811	0.404906	+	0.914358i	$0.367304\pi$
$492$	−8.71067	−0.392707
$493$	25.3551	1.14194
$494$	1.50720	0.0678122
$495$	0.656263	0.0294968
$496$	−6.16594	−0.276859
$497$	−45.4686	−2.03955
$498$	−13.1943	−0.591249
$499$	−6.90193	−0.308973	−0.154486	−	0.987995i	$-0.549372\pi$
$499$	−6.90193	−0.308973	−0.154486	+	0.987995i	$0.549372\pi$
$500$	6.27999	0.280850
$501$	21.8712	0.977135
$502$	−16.4885	−0.735920
$503$	32.9799	1.47050	0.735252	−	0.677794i	$-0.237064\pi$
$503$	32.9799	1.47050	0.735252	+	0.677794i	$0.237064\pi$
$504$	4.00423	0.178363
$505$	9.53743	0.424410
$506$	5.33405	0.237127
$507$	12.4909	0.554740
$508$	15.9448	0.707436
$509$	18.9890	0.841672	0.420836	−	0.907137i	$-0.361737\pi$
$509$	18.9890	0.841672	0.420836	+	0.907137i	$0.361737\pi$
$510$	−4.46007	−0.197495
$511$	−32.7616	−1.44929
$512$	1.00000	0.0441942
$513$	2.11237	0.0932634
$514$	9.19399	0.405530
$515$	4.06236	0.179009
$516$	−1.79594	−0.0790618
$517$	8.19049	0.360217
$518$	−18.9005	−0.830441
$519$	−23.8917	−1.04873
$520$	0.468252	0.0205342
$521$	−42.9313	−1.88086	−0.940428	−	0.339993i	$-0.889575\pi$
$521$	−42.9313	−1.88086	−0.940428	+	0.339993i	$0.889575\pi$
$522$	−3.73080	−0.163293
$523$	13.2149	0.577849	0.288925	−	0.957352i	$-0.406702\pi$
$523$	13.2149	0.577849	0.288925	+	0.957352i	$0.406702\pi$
$524$	−2.67329	−0.116783
$525$	18.2966	0.798529
$526$	29.6162	1.29133
$527$	41.9047	1.82540
$528$	1.00000	0.0435194
$529$	5.45205	0.237046
$530$	−0.816070	−0.0354478
$531$	−10.0801	−0.437439
$532$	−8.45841	−0.366719
$533$	−6.21517	−0.269209
$534$	−11.7623	−0.509003
$535$	0.596309	0.0257807
$536$	7.84450	0.338831
$537$	−26.4579	−1.14174
$538$	2.41860	0.104273
$539$	−9.03386	−0.389116
$540$	0.656263	0.0282411
$541$	34.2625	1.47306	0.736529	−	0.676406i	$-0.236463\pi$
$541$	34.2625	1.47306	0.736529	+	0.676406i	$0.236463\pi$
$542$	20.1206	0.864253
$543$	16.2248	0.696274
$544$	−6.79616	−0.291383
$545$	−9.94235	−0.425884
$546$	2.85707	0.122271
$547$	3.91188	0.167260	0.0836300	−	0.996497i	$-0.473349\pi$
$547$	3.91188	0.167260	0.0836300	+	0.996497i	$0.473349\pi$
$548$	12.1598	0.519443
$549$	1.00000	0.0426790
$550$	4.56932	0.194836
$551$	7.88083	0.335735
$552$	5.33405	0.227032
$553$	34.8937	1.48383
$554$	26.1571	1.11131
$555$	−3.09765	−0.131488
$556$	−20.3027	−0.861028
$557$	−7.56003	−0.320329	−0.160164	−	0.987090i	$-0.551202\pi$
$557$	−7.56003	−0.320329	−0.160164	+	0.987090i	$0.551202\pi$
$558$	−6.16594	−0.261025
$559$	−1.28142	−0.0541985
$560$	−2.62783	−0.111046
$561$	−6.79616	−0.286934
$562$	−22.2517	−0.938630
$563$	33.2684	1.40210	0.701048	−	0.713114i	$-0.252716\pi$
$563$	33.2684	1.40210	0.701048	+	0.713114i	$0.252716\pi$
$564$	8.19049	0.344882
$565$	10.2867	0.432763
$566$	19.9109	0.836917
$567$	4.00423	0.168162
$568$	−11.3551	−0.476451
$569$	8.42763	0.353305	0.176652	−	0.984273i	$-0.443473\pi$
$569$	8.42763	0.353305	0.176652	+	0.984273i	$0.443473\pi$
$570$	−1.38627	−0.0580644
$571$	29.4530	1.23257	0.616284	−	0.787524i	$-0.288637\pi$
$571$	29.4530	1.23257	0.616284	+	0.787524i	$0.288637\pi$
$572$	0.713512	0.0298334
$573$	27.3752	1.14362
$574$	34.8795	1.45584
$575$	24.3730	1.01642
$576$	1.00000	0.0416667
$577$	−25.8575	−1.07646	−0.538231	−	0.842798i	$-0.680907\pi$
$577$	−25.8575	−1.07646	−0.538231	+	0.842798i	$0.680907\pi$
$578$	29.1878	1.21405
$579$	−13.1566	−0.546768
$580$	2.44839	0.101664
$581$	52.8328	2.19187
$582$	−5.56803	−0.230802
$583$	−1.24351	−0.0515010
$584$	−8.18174	−0.338563
$585$	0.468252	0.0193598
$586$	−6.69120	−0.276411
$587$	−41.9111	−1.72986	−0.864928	−	0.501897i	$-0.832636\pi$
$587$	−41.9111	−1.72986	−0.864928	+	0.501897i	$0.832636\pi$
$588$	−9.03386	−0.372550
$589$	13.0247	0.536675
$590$	6.61519	0.272343
$591$	16.5155	0.679356
$592$	−4.72014	−0.193997
$593$	−11.2006	−0.459955	−0.229978	−	0.973196i	$-0.573865\pi$
$593$	−11.2006	−0.459955	−0.229978	+	0.973196i	$0.573865\pi$
$594$	1.00000	0.0410305
$595$	17.8591	0.732153
$596$	−14.0856	−0.576969
$597$	−4.20859	−0.172246
$598$	3.80591	0.155635
$599$	−24.1860	−0.988214	−0.494107	−	0.869401i	$-0.664505\pi$
$599$	−24.1860	−0.988214	−0.494107	+	0.869401i	$0.664505\pi$
$600$	4.56932	0.186542
$601$	23.1866	0.945799	0.472900	−	0.881116i	$-0.343207\pi$
$601$	23.1866	0.945799	0.472900	+	0.881116i	$0.343207\pi$
$602$	7.19135	0.293098
$603$	7.84450	0.319453
$604$	4.46115	0.181522
$605$	−0.656263	−0.0266809
$606$	14.5329	0.590360
$607$	−18.6172	−0.755647	−0.377824	−	0.925878i	$-0.623327\pi$
$607$	−18.6172	−0.755647	−0.377824	+	0.925878i	$0.623327\pi$
$608$	−2.11237	−0.0856679
$609$	14.9390	0.605359
$610$	−0.656263	−0.0265713
$611$	5.84402	0.236424
$612$	−6.79616	−0.274719
$613$	14.5905	0.589304	0.294652	−	0.955605i	$-0.404796\pi$
$613$	14.5905	0.589304	0.294652	+	0.955605i	$0.404796\pi$
$614$	−14.1765	−0.572118
$615$	5.71649	0.230511
$616$	−4.00423	−0.161335
$617$	−33.4220	−1.34552	−0.672760	−	0.739861i	$-0.734891\pi$
$617$	−33.4220	−1.34552	−0.672760	+	0.739861i	$0.734891\pi$
$618$	6.19014	0.249004
$619$	15.4993	0.622968	0.311484	−	0.950251i	$-0.399174\pi$
$619$	15.4993	0.622968	0.311484	+	0.950251i	$0.399174\pi$
$620$	4.04648	0.162510
$621$	5.33405	0.214048
$622$	−24.5928	−0.986083
$623$	47.0988	1.88697
$624$	0.713512	0.0285633
$625$	18.7253	0.749011
$626$	−14.4907	−0.579164
$627$	−2.11237	−0.0843599
$628$	−0.913169	−0.0364394
$629$	32.0788	1.27907
$630$	−2.62783	−0.104695
$631$	30.6553	1.22037	0.610184	−	0.792260i	$-0.291096\pi$
$631$	30.6553	1.22037	0.610184	+	0.792260i	$0.291096\pi$
$632$	8.71421	0.346633
$633$	1.57738	0.0626952
$634$	−20.8435	−0.827800
$635$	−10.4640	−0.415250
$636$	−1.24351	−0.0493084
$637$	−6.44577	−0.255391
$638$	3.73080	0.147704
$639$	−11.3551	−0.449202
$640$	−0.656263	−0.0259411
$641$	23.4422	0.925912	0.462956	−	0.886381i	$-0.346789\pi$
$641$	23.4422	0.925912	0.462956	+	0.886381i	$0.346789\pi$
$642$	0.908643	0.0358613
$643$	4.07099	0.160544	0.0802720	−	0.996773i	$-0.474421\pi$
$643$	4.07099	0.160544	0.0802720	+	0.996773i	$0.474421\pi$
$644$	−21.3588	−0.841653
$645$	1.17861	0.0464077
$646$	14.3560	0.564829
$647$	−25.7634	−1.01286	−0.506431	−	0.862280i	$-0.669036\pi$
$647$	−25.7634	−1.01286	−0.506431	+	0.862280i	$0.669036\pi$
$648$	1.00000	0.0392837
$649$	10.0801	0.395678
$650$	3.26026	0.127878
$651$	24.6898	0.967671
$652$	0.808887	0.0316784
$653$	−32.6267	−1.27678	−0.638392	−	0.769712i	$-0.720400\pi$
$653$	−32.6267	−1.27678	−0.638392	+	0.769712i	$0.720400\pi$
$654$	−15.1500	−0.592410
$655$	1.75438	0.0685492
$656$	8.71067	0.340095
$657$	−8.18174	−0.319200
$658$	−32.7966	−1.27855
$659$	−4.20676	−0.163872	−0.0819361	−	0.996638i	$-0.526110\pi$
$659$	−4.20676	−0.163872	−0.0819361	+	0.996638i	$0.526110\pi$
$660$	−0.656263	−0.0255450
$661$	−33.0339	−1.28487	−0.642435	−	0.766340i	$-0.722076\pi$
$661$	−33.0339	−1.28487	−0.642435	+	0.766340i	$0.722076\pi$
$662$	35.0585	1.36259
$663$	−4.84914	−0.188325
$664$	13.1943	0.512036
$665$	5.55094	0.215256
$666$	−4.72014	−0.182902
$667$	19.9003	0.770542
$668$	−21.8712	−0.846224
$669$	17.4661	0.675278
$670$	−5.14805	−0.198887
$671$	−1.00000	−0.0386046
$672$	−4.00423	−0.154467
$673$	−17.1183	−0.659863	−0.329931	−	0.944005i	$-0.607026\pi$
$673$	−17.1183	−0.659863	−0.329931	+	0.944005i	$0.607026\pi$
$674$	−6.44860	−0.248391
$675$	4.56932	0.175873
$676$	−12.4909	−0.480419
$677$	17.2033	0.661176	0.330588	−	0.943775i	$-0.392753\pi$
$677$	17.2033	0.661176	0.330588	+	0.943775i	$0.392753\pi$
$678$	15.6746	0.601979
$679$	22.2957	0.855630
$680$	4.46007	0.171036
$681$	−11.9588	−0.458261
$682$	6.16594	0.236106
$683$	−13.3029	−0.509020	−0.254510	−	0.967070i	$-0.581914\pi$
$683$	−13.3029	−0.509020	−0.254510	+	0.967070i	$0.581914\pi$
$684$	−2.11237	−0.0807685
$685$	−7.98005	−0.304902
$686$	8.14404	0.310941
$687$	−14.7092	−0.561191
$688$	1.79594	0.0684695
$689$	−0.887260	−0.0338019
$690$	−3.50054	−0.133263
$691$	−37.8605	−1.44028	−0.720140	−	0.693828i	$-0.755923\pi$
$691$	−37.8605	−1.44028	−0.720140	+	0.693828i	$0.755923\pi$
$692$	23.8917	0.908226
$693$	−4.00423	−0.152108
$694$	−28.7772	−1.09237
$695$	13.3239	0.505406
$696$	3.73080	0.141416
$697$	−59.1991	−2.24233
$698$	−25.8936	−0.980087
$699$	11.0711	0.418749
$700$	−18.2966	−0.691547
$701$	−30.0112	−1.13351	−0.566753	−	0.823888i	$-0.691801\pi$
$701$	−30.0112	−1.13351	−0.566753	+	0.823888i	$0.691801\pi$
$702$	0.713512	0.0269298
$703$	9.97068	0.376051
$704$	−1.00000	−0.0376889
$705$	−5.37512	−0.202439
$706$	7.13670	0.268593
$707$	−58.1933	−2.18858
$708$	10.0801	0.378833
$709$	23.1309	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$709$	23.1309	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$710$	7.45196	0.279667
$711$	8.71421	0.326809
$712$	11.7623	0.440810
$713$	32.8894	1.23172
$714$	27.2134	1.01844
$715$	−0.468252	−0.0175116
$716$	26.4579	0.988778
$717$	16.7529	0.625648
$718$	26.2850	0.980948
$719$	−42.9017	−1.59996	−0.799981	−	0.600026i	$-0.795157\pi$
$719$	−42.9017	−1.59996	−0.799981	+	0.600026i	$0.795157\pi$
$720$	−0.656263	−0.0244575
$721$	−24.7867	−0.923107
$722$	−14.5379	−0.541044
$723$	−10.3576	−0.385203
$724$	−16.2248	−0.602991
$725$	17.0472	0.633118
$726$	−1.00000	−0.0371135
$727$	46.5226	1.72543	0.862713	−	0.505694i	$-0.168763\pi$
$727$	46.5226	1.72543	0.862713	+	0.505694i	$0.168763\pi$
$728$	−2.85707	−0.105890
$729$	1.00000	0.0370370
$730$	5.36937	0.198729
$731$	−12.2055	−0.451436
$732$	−1.00000	−0.0369611
$733$	9.09635	0.335981	0.167991	−	0.985789i	$-0.446272\pi$
$733$	9.09635	0.335981	0.167991	+	0.985789i	$0.446272\pi$
$734$	−10.3643	−0.382552
$735$	5.92859	0.218679
$736$	−5.33405	−0.196616
$737$	−7.84450	−0.288956
$738$	8.71067	0.320644
$739$	−44.4480	−1.63505	−0.817524	−	0.575895i	$-0.804654\pi$
$739$	−44.4480	−1.63505	−0.817524	+	0.575895i	$0.804654\pi$
$740$	3.09765	0.113872
$741$	−1.50720	−0.0553684
$742$	4.97930	0.182796
$743$	36.3720	1.33436	0.667180	−	0.744897i	$-0.267501\pi$
$743$	36.3720	1.33436	0.667180	+	0.744897i	$0.267501\pi$
$744$	6.16594	0.226054
$745$	9.24387	0.338669
$746$	−23.5014	−0.860448
$747$	13.1943	0.482752
$748$	6.79616	0.248492
$749$	−3.63842	−0.132945
$750$	−6.27999	−0.229313
$751$	23.9323	0.873302	0.436651	−	0.899631i	$-0.356165\pi$
$751$	23.9323	0.873302	0.436651	+	0.899631i	$0.356165\pi$
$752$	−8.19049	−0.298677
$753$	16.4885	0.600876
$754$	2.66197	0.0969433
$755$	−2.92769	−0.106550
$756$	−4.00423	−0.145632
$757$	−6.23508	−0.226618	−0.113309	−	0.993560i	$-0.536145\pi$
$757$	−6.23508	−0.226618	−0.113309	+	0.993560i	$0.536145\pi$
$758$	−11.8214	−0.429373
$759$	−5.33405	−0.193614
$760$	1.38627	0.0502853
$761$	−23.1909	−0.840669	−0.420335	−	0.907369i	$-0.638087\pi$
$761$	−23.1909	−0.840669	−0.420335	+	0.907369i	$0.638087\pi$
$762$	−15.9448	−0.577619
$763$	60.6639	2.19618
$764$	−27.3752	−0.990401
$765$	4.46007	0.161254
$766$	12.8517	0.464351
$767$	7.19227	0.259698
$768$	−1.00000	−0.0360844
$769$	−35.7353	−1.28865	−0.644324	−	0.764753i	$-0.722861\pi$
$769$	−35.7353	−1.28865	−0.644324	+	0.764753i	$0.722861\pi$
$770$	2.62783	0.0947004
$771$	−9.19399	−0.331114
$772$	13.1566	0.473515
$773$	24.3193	0.874705	0.437353	−	0.899290i	$-0.355916\pi$
$773$	24.3193	0.874705	0.437353	+	0.899290i	$0.355916\pi$
$774$	1.79594	0.0645537
$775$	28.1741	1.01204
$776$	5.56803	0.199881
$777$	18.9005	0.678053
$778$	−2.87690	−0.103142
$779$	−18.4002	−0.659254
$780$	−0.468252	−0.0167661
$781$	11.3551	0.406319
$782$	36.2510	1.29633
$783$	3.73080	0.133328
$784$	9.03386	0.322638
$785$	0.599279	0.0213892
$786$	2.67329	0.0953529
$787$	19.5836	0.698080	0.349040	−	0.937108i	$-0.386508\pi$
$787$	19.5836	0.698080	0.349040	+	0.937108i	$0.386508\pi$
$788$	−16.5155	−0.588340
$789$	−29.6162	−1.05436
$790$	−5.71882	−0.203466
$791$	−62.7647	−2.23165
$792$	−1.00000	−0.0355335
$793$	−0.713512	−0.0253376
$794$	−13.9395	−0.494694
$795$	0.816070	0.0289430
$796$	4.20859	0.149169
$797$	−42.2855	−1.49783	−0.748915	−	0.662666i	$-0.769425\pi$
$797$	−42.2855	−1.49783	−0.748915	+	0.662666i	$0.769425\pi$
$798$	8.45841	0.299425
$799$	55.6639	1.96925
$800$	−4.56932	−0.161550
$801$	11.7623	0.415599
$802$	33.5896	1.18609
$803$	8.18174	0.288727
$804$	−7.84450	−0.276654
$805$	14.0170	0.494033
$806$	4.39947	0.154965
$807$	−2.41860	−0.0851388
$808$	−14.5329	−0.511267
$809$	−17.5917	−0.618490	−0.309245	−	0.950982i	$-0.600076\pi$
$809$	−17.5917	−0.618490	−0.309245	+	0.950982i	$0.600076\pi$
$810$	−0.656263	−0.0230587
$811$	−43.7900	−1.53767	−0.768837	−	0.639445i	$-0.779164\pi$
$811$	−43.7900	−1.53767	−0.768837	+	0.639445i	$0.779164\pi$
$812$	−14.9390	−0.524256
$813$	−20.1206	−0.705660
$814$	4.72014	0.165441
$815$	−0.530842	−0.0185946
$816$	6.79616	0.237913
$817$	−3.79369	−0.132724
$818$	−30.7280	−1.07438
$819$	−2.85707	−0.0998340
$820$	−5.71649	−0.199629
$821$	27.5098	0.960099	0.480050	−	0.877241i	$-0.340619\pi$
$821$	27.5098	0.960099	0.480050	+	0.877241i	$0.340619\pi$
$822$	−12.1598	−0.424123
$823$	−1.80423	−0.0628914	−0.0314457	−	0.999505i	$-0.510011\pi$
$823$	−1.80423	−0.0628914	−0.0314457	+	0.999505i	$0.510011\pi$
$824$	−6.19014	−0.215644
$825$	−4.56932	−0.159083
$826$	−40.3630	−1.40441
$827$	−27.5193	−0.956939	−0.478469	−	0.878104i	$-0.658808\pi$
$827$	−27.5193	−0.956939	−0.478469	+	0.878104i	$0.658808\pi$
$828$	−5.33405	−0.185371
$829$	18.5211	0.643264	0.321632	−	0.946865i	$-0.395769\pi$
$829$	18.5211	0.643264	0.321632	+	0.946865i	$0.395769\pi$
$830$	−8.65890	−0.300555
$831$	−26.1571	−0.907380
$832$	−0.713512	−0.0247366
$833$	−61.3956	−2.12723
$834$	20.3027	0.703026
$835$	14.3533	0.496716
$836$	2.11237	0.0730578
$837$	6.16594	0.213126
$838$	29.4540	1.01747
$839$	−27.0130	−0.932593	−0.466297	−	0.884628i	$-0.654412\pi$
$839$	−27.0130	−0.932593	−0.466297	+	0.884628i	$0.654412\pi$
$840$	2.62783	0.0906687
$841$	−15.0811	−0.520038
$842$	7.89348	0.272027
$843$	22.2517	0.766388
$844$	−1.57738	−0.0542957
$845$	8.19732	0.281996
$846$	−8.19049	−0.281595
$847$	4.00423	0.137587
$848$	1.24351	0.0427023
$849$	−19.9109	−0.683340
$850$	31.0538	1.06514
$851$	25.1774	0.863072
$852$	11.3551	0.389021
$853$	−27.1416	−0.929311	−0.464656	−	0.885491i	$-0.653822\pi$
$853$	−27.1416	−0.929311	−0.464656	+	0.885491i	$0.653822\pi$
$854$	4.00423	0.137022
$855$	1.38627	0.0474094
$856$	−0.908643	−0.0310568
$857$	32.7647	1.11922	0.559610	−	0.828756i	$-0.310951\pi$
$857$	32.7647	1.11922	0.559610	+	0.828756i	$0.310951\pi$
$858$	−0.713512	−0.0243589
$859$	−4.05463	−0.138342	−0.0691712	−	0.997605i	$-0.522035\pi$
$859$	−4.05463	−0.138342	−0.0691712	+	0.997605i	$0.522035\pi$
$860$	−1.17861	−0.0401902
$861$	−34.8795	−1.18869
$862$	37.7385	1.28538
$863$	−17.1944	−0.585305	−0.292652	−	0.956219i	$-0.594538\pi$
$863$	−17.1944	−0.585305	−0.292652	+	0.956219i	$0.594538\pi$
$864$	−1.00000	−0.0340207
$865$	−15.6792	−0.533110
$866$	−39.2942	−1.33527
$867$	−29.1878	−0.991270
$868$	−24.6898	−0.838027
$869$	−8.71421	−0.295609
$870$	−2.44839	−0.0830081
$871$	−5.59714	−0.189652
$872$	15.1500	0.513042
$873$	5.56803	0.188449
$874$	11.2675	0.381128
$875$	25.1465	0.850108
$876$	8.18174	0.276435
$877$	−19.1211	−0.645673	−0.322836	−	0.946455i	$-0.604636\pi$
$877$	−19.1211	−0.645673	−0.322836	+	0.946455i	$0.604636\pi$
$878$	14.6566	0.494635
$879$	6.69120	0.225688
$880$	0.656263	0.0221226
$881$	−51.7050	−1.74198	−0.870992	−	0.491297i	$-0.836523\pi$
$881$	−51.7050	−1.74198	−0.870992	+	0.491297i	$0.836523\pi$
$882$	9.03386	0.304186
$883$	41.2063	1.38670	0.693352	−	0.720599i	$-0.256133\pi$
$883$	41.2063	1.38670	0.693352	+	0.720599i	$0.256133\pi$
$884$	4.84914	0.163094
$885$	−6.61519	−0.222367
$886$	18.2295	0.612432
$887$	3.18978	0.107102	0.0535511	−	0.998565i	$-0.482946\pi$
$887$	3.18978	0.107102	0.0535511	+	0.998565i	$0.482946\pi$
$888$	4.72014	0.158397
$889$	63.8466	2.14135
$890$	−7.71914	−0.258746
$891$	−1.00000	−0.0335013
$892$	−17.4661	−0.584808
$893$	17.3013	0.578967
$894$	14.0856	0.471094
$895$	−17.3633	−0.580392
$896$	4.00423	0.133772
$897$	−3.80591	−0.127076
$898$	−24.5681	−0.819848
$899$	23.0039	0.767223
$900$	−4.56932	−0.152311
$901$	−8.45110	−0.281547
$902$	−8.71067	−0.290034
$903$	−7.19135	−0.239313
$904$	−15.6746	−0.521329
$905$	10.6477	0.353943
$906$	−4.46115	−0.148212
$907$	36.0306	1.19638	0.598188	−	0.801356i	$-0.295888\pi$
$907$	36.0306	1.19638	0.598188	+	0.801356i	$0.295888\pi$
$908$	11.9588	0.396866
$909$	−14.5329	−0.482027
$910$	1.87499	0.0621552
$911$	48.4572	1.60546	0.802729	−	0.596344i	$-0.203381\pi$
$911$	48.4572	1.60546	0.802729	+	0.596344i	$0.203381\pi$
$912$	2.11237	0.0699475
$913$	−13.1943	−0.436666
$914$	16.0312	0.530263
$915$	0.656263	0.0216954
$916$	14.7092	0.486006
$917$	−10.7045	−0.353492
$918$	6.79616	0.224307
$919$	4.98299	0.164374	0.0821868	−	0.996617i	$-0.473810\pi$
$919$	4.98299	0.164374	0.0821868	+	0.996617i	$0.473810\pi$
$920$	3.50054	0.115409
$921$	14.1765	0.467132
$922$	38.5537	1.26970
$923$	8.10203	0.266682
$924$	4.00423	0.131730
$925$	21.5678	0.709146
$926$	4.84718	0.159288
$927$	−6.19014	−0.203311
$928$	−3.73080	−0.122470
$929$	−40.5376	−1.32999	−0.664997	−	0.746846i	$-0.731567\pi$
$929$	−40.5376	−1.32999	−0.664997	+	0.746846i	$0.731567\pi$
$930$	−4.04648	−0.132689
$931$	−19.0828	−0.625415
$932$	−11.0711	−0.362647
$933$	24.5928	0.805133
$934$	24.5887	0.804568
$935$	−4.46007	−0.145860
$936$	−0.713512	−0.0233219
$937$	31.6107	1.03268	0.516338	−	0.856385i	$-0.327295\pi$
$937$	31.6107	1.03268	0.516338	+	0.856385i	$0.327295\pi$
$938$	31.4112	1.02561
$939$	14.4907	0.472885
$940$	5.37512	0.175317
$941$	51.7156	1.68588	0.842941	−	0.538007i	$-0.180823\pi$
$941$	51.7156	1.68588	0.842941	+	0.538007i	$0.180823\pi$
$942$	0.913169	0.0297526
$943$	−46.4631	−1.51305
$944$	−10.0801	−0.328079
$945$	2.62783	0.0854833
$946$	−1.79594	−0.0583910
$947$	49.3114	1.60241	0.801203	−	0.598393i	$-0.204194\pi$
$947$	49.3114	1.60241	0.801203	+	0.598393i	$0.204194\pi$
$948$	−8.71421	−0.283025
$949$	5.83777	0.189502
$950$	9.65209	0.313155
$951$	20.8435	0.675896
$952$	−27.2134	−0.881991
$953$	26.7318	0.865929	0.432965	−	0.901411i	$-0.357468\pi$
$953$	26.7318	0.865929	0.432965	+	0.901411i	$0.357468\pi$
$954$	1.24351	0.0402601
$955$	17.9653	0.581345
$956$	−16.7529	−0.541827
$957$	−3.73080	−0.120600
$958$	−32.8682	−1.06192
$959$	48.6908	1.57231
$960$	0.656263	0.0211808
$961$	7.01877	0.226412
$962$	3.36788	0.108585
$963$	−0.908643	−0.0292806
$964$	10.3576	0.333596
$965$	−8.63417	−0.277944
$966$	21.3588	0.687207
$967$	−32.5116	−1.04550	−0.522751	−	0.852486i	$-0.675094\pi$
$967$	−32.5116	−1.04550	−0.522751	+	0.852486i	$0.675094\pi$
$968$	1.00000	0.0321412
$969$	−14.3560	−0.461181
$970$	−3.65409	−0.117326
$971$	10.7436	0.344778	0.172389	−	0.985029i	$-0.444851\pi$
$971$	10.7436	0.344778	0.172389	+	0.985029i	$0.444851\pi$
$972$	−1.00000	−0.0320750
$973$	−81.2969	−2.60626
$974$	−11.9114	−0.381667
$975$	−3.26026	−0.104412
$976$	1.00000	0.0320092
$977$	1.34665	0.0430832	0.0215416	−	0.999768i	$-0.493143\pi$
$977$	1.34665	0.0430832	0.0215416	+	0.999768i	$0.493143\pi$
$978$	−0.808887	−0.0258653
$979$	−11.7623	−0.375924
$980$	−5.92859	−0.189382
$981$	15.1500	0.483701
$982$	17.9442	0.572623
$983$	−43.7168	−1.39435	−0.697175	−	0.716901i	$-0.745560\pi$
$983$	−43.7168	−1.39435	−0.697175	+	0.716901i	$0.745560\pi$
$984$	−8.71067	−0.277686
$985$	10.8385	0.345343
$986$	25.3551	0.807472
$987$	32.7966	1.04393
$988$	1.50720	0.0479505
$989$	−9.57962	−0.304614
$990$	0.656263	0.0208574
$991$	35.7439	1.13544	0.567721	−	0.823221i	$-0.307825\pi$
$991$	35.7439	1.13544	0.567721	+	0.823221i	$0.307825\pi$
$992$	−6.16594	−0.195769
$993$	−35.0585	−1.11255
$994$	−45.4686	−1.44218
$995$	−2.76194	−0.0875594
$996$	−13.1943	−0.418076
$997$	−51.8627	−1.64251	−0.821254	−	0.570562i	$-0.806725\pi$
$997$	−51.8627	−1.64251	−0.821254	+	0.570562i	$0.806725\pi$
$998$	−6.90193	−0.218477
$999$	4.72014	0.149339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.x.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.x.1.3	✓	6	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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.656263 q^{5} -1.00000 q^{6} +4.00423 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.656263 q^{5} -1.00000 q^{6} +4.00423 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.656263 q^{10} -1.00000 q^{11} -1.00000 q^{12} -0.713512 q^{13} +4.00423 q^{14} +0.656263 q^{15} +1.00000 q^{16} -6.79616 q^{17} +1.00000 q^{18} -2.11237 q^{19} -0.656263 q^{20} -4.00423 q^{21} -1.00000 q^{22} -5.33405 q^{23} -1.00000 q^{24} -4.56932 q^{25} -0.713512 q^{26} -1.00000 q^{27} +4.00423 q^{28} -3.73080 q^{29} +0.656263 q^{30} -6.16594 q^{31} +1.00000 q^{32} +1.00000 q^{33} -6.79616 q^{34} -2.62783 q^{35} +1.00000 q^{36} -4.72014 q^{37} -2.11237 q^{38} +0.713512 q^{39} -0.656263 q^{40} +8.71067 q^{41} -4.00423 q^{42} +1.79594 q^{43} -1.00000 q^{44} -0.656263 q^{45} -5.33405 q^{46} -8.19049 q^{47} -1.00000 q^{48} +9.03386 q^{49} -4.56932 q^{50} +6.79616 q^{51} -0.713512 q^{52} +1.24351 q^{53} -1.00000 q^{54} +0.656263 q^{55} +4.00423 q^{56} +2.11237 q^{57} -3.73080 q^{58} -10.0801 q^{59} +0.656263 q^{60} +1.00000 q^{61} -6.16594 q^{62} +4.00423 q^{63} +1.00000 q^{64} +0.468252 q^{65} +1.00000 q^{66} +7.84450 q^{67} -6.79616 q^{68} +5.33405 q^{69} -2.62783 q^{70} -11.3551 q^{71} +1.00000 q^{72} -8.18174 q^{73} -4.72014 q^{74} +4.56932 q^{75} -2.11237 q^{76} -4.00423 q^{77} +0.713512 q^{78} +8.71421 q^{79} -0.656263 q^{80} +1.00000 q^{81} +8.71067 q^{82} +13.1943 q^{83} -4.00423 q^{84} +4.46007 q^{85} +1.79594 q^{86} +3.73080 q^{87} -1.00000 q^{88} +11.7623 q^{89} -0.656263 q^{90} -2.85707 q^{91} -5.33405 q^{92} +6.16594 q^{93} -8.19049 q^{94} +1.38627 q^{95} -1.00000 q^{96} +5.56803 q^{97} +9.03386 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} + q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 + q^7 + 6 * q^8 + 6 * q^9	\( 6 q + 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} - 6 q^{6} + q^{7} + 6 q^{8} + 6 q^{9} - 6 q^{10} - 6 q^{11} - 6 q^{12} + 2 q^{13} + q^{14} + 6 q^{15} + 6 q^{16} - 13 q^{17} + 6 q^{18} + q^{19} - 6 q^{20} - q^{21} - 6 q^{22} - 11 q^{23} - 6 q^{24} + 8 q^{25} + 2 q^{26} - 6 q^{27} + q^{28} - 14 q^{29} + 6 q^{30} - 5 q^{31} + 6 q^{32} + 6 q^{33} - 13 q^{34} - 13 q^{35} + 6 q^{36} - 6 q^{37} + q^{38} - 2 q^{39} - 6 q^{40} - 25 q^{41} - q^{42} + 19 q^{43} - 6 q^{44} - 6 q^{45} - 11 q^{46} - 10 q^{47} - 6 q^{48} - 5 q^{49} + 8 q^{50} + 13 q^{51} + 2 q^{52} - 17 q^{53} - 6 q^{54} + 6 q^{55} + q^{56} - q^{57} - 14 q^{58} - 14 q^{59} + 6 q^{60} + 6 q^{61} - 5 q^{62} + q^{63} + 6 q^{64} + 6 q^{65} + 6 q^{66} + 12 q^{67} - 13 q^{68} + 11 q^{69} - 13 q^{70} + 6 q^{71} + 6 q^{72} - 29 q^{73} - 6 q^{74} - 8 q^{75} + q^{76} - q^{77} - 2 q^{78} - 24 q^{79} - 6 q^{80} + 6 q^{81} - 25 q^{82} - 9 q^{83} - q^{84} - 22 q^{85} + 19 q^{86} + 14 q^{87} - 6 q^{88} - 4 q^{89} - 6 q^{90} - 29 q^{91} - 11 q^{92} + 5 q^{93} - 10 q^{94} - 27 q^{95} - 6 q^{96} - 5 q^{97} - 5 q^{98} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 - 6 * q^6 + q^7 + 6 * q^8 + 6 * q^9 - 6 * q^10 - 6 * q^11 - 6 * q^12 + 2 * q^13 + q^14 + 6 * q^15 + 6 * q^16 - 13 * q^17 + 6 * q^18 + q^19 - 6 * q^20 - q^21 - 6 * q^22 - 11 * q^23 - 6 * q^24 + 8 * q^25 + 2 * q^26 - 6 * q^27 + q^28 - 14 * q^29 + 6 * q^30 - 5 * q^31 + 6 * q^32 + 6 * q^33 - 13 * q^34 - 13 * q^35 + 6 * q^36 - 6 * q^37 + q^38 - 2 * q^39 - 6 * q^40 - 25 * q^41 - q^42 + 19 * q^43 - 6 * q^44 - 6 * q^45 - 11 * q^46 - 10 * q^47 - 6 * q^48 - 5 * q^49 + 8 * q^50 + 13 * q^51 + 2 * q^52 - 17 * q^53 - 6 * q^54 + 6 * q^55 + q^56 - q^57 - 14 * q^58 - 14 * q^59 + 6 * q^60 + 6 * q^61 - 5 * q^62 + q^63 + 6 * q^64 + 6 * q^65 + 6 * q^66 + 12 * q^67 - 13 * q^68 + 11 * q^69 - 13 * q^70 + 6 * q^71 + 6 * q^72 - 29 * q^73 - 6 * q^74 - 8 * q^75 + q^76 - q^77 - 2 * q^78 - 24 * q^79 - 6 * q^80 + 6 * q^81 - 25 * q^82 - 9 * q^83 - q^84 - 22 * q^85 + 19 * q^86 + 14 * q^87 - 6 * q^88 - 4 * q^89 - 6 * q^90 - 29 * q^91 - 11 * q^92 + 5 * q^93 - 10 * q^94 - 27 * q^95 - 6 * q^96 - 5 * q^97 - 5 * q^98 - 6 * q^99

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