LMFDB - Embedded newform 4030.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 16x^{6} + 18x^{5} + 64x^{4} - 84x^{3} - 19x^{2} + 22x + 4 $ x^8 - x^7 - 16x^6 + 18x^5 + 64x^4 - 84x^3 - 19x^2 + 22x + 4
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			1.1
Root			$3.08271$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.08271 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.08271 q^{6} -2.62527 q^{7} +1.00000 q^{8} +6.50312 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.08271 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.08271 q^{6} -2.62527 q^{7} +1.00000 q^{8} +6.50312 q^{9} -1.00000 q^{10} +0.190183 q^{11} -3.08271 q^{12} -1.00000 q^{13} -2.62527 q^{14} +3.08271 q^{15} +1.00000 q^{16} -6.93705 q^{17} +6.50312 q^{18} -0.237470 q^{19} -1.00000 q^{20} +8.09297 q^{21} +0.190183 q^{22} +4.00840 q^{23} -3.08271 q^{24} +1.00000 q^{25} -1.00000 q^{26} -10.7991 q^{27} -2.62527 q^{28} +0.0461297 q^{29} +3.08271 q^{30} -1.00000 q^{31} +1.00000 q^{32} -0.586281 q^{33} -6.93705 q^{34} +2.62527 q^{35} +6.50312 q^{36} -9.02092 q^{37} -0.237470 q^{38} +3.08271 q^{39} -1.00000 q^{40} -2.50696 q^{41} +8.09297 q^{42} -2.72309 q^{43} +0.190183 q^{44} -6.50312 q^{45} +4.00840 q^{46} -10.0962 q^{47} -3.08271 q^{48} -0.107936 q^{49} +1.00000 q^{50} +21.3849 q^{51} -1.00000 q^{52} -1.99103 q^{53} -10.7991 q^{54} -0.190183 q^{55} -2.62527 q^{56} +0.732053 q^{57} +0.0461297 q^{58} +10.7728 q^{59} +3.08271 q^{60} +0.753431 q^{61} -1.00000 q^{62} -17.0725 q^{63} +1.00000 q^{64} +1.00000 q^{65} -0.586281 q^{66} +13.2211 q^{67} -6.93705 q^{68} -12.3568 q^{69} +2.62527 q^{70} +6.14349 q^{71} +6.50312 q^{72} -5.91955 q^{73} -9.02092 q^{74} -3.08271 q^{75} -0.237470 q^{76} -0.499284 q^{77} +3.08271 q^{78} +0.817179 q^{79} -1.00000 q^{80} +13.7812 q^{81} -2.50696 q^{82} +0.435097 q^{83} +8.09297 q^{84} +6.93705 q^{85} -2.72309 q^{86} -0.142205 q^{87} +0.190183 q^{88} -1.09084 q^{89} -6.50312 q^{90} +2.62527 q^{91} +4.00840 q^{92} +3.08271 q^{93} -10.0962 q^{94} +0.237470 q^{95} -3.08271 q^{96} +8.17526 q^{97} -0.107936 q^{98} +1.23679 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} + q^{14} + q^{15} + 8 q^{16} - 5 q^{17} + 9 q^{18} + 2 q^{19} - 8 q^{20} + 17 q^{21} + 4 q^{22} + 4 q^{23} - q^{24} + 8 q^{25} - 8 q^{26} + 11 q^{27} + q^{28} + 11 q^{29} + q^{30} - 8 q^{31} + 8 q^{32} + 10 q^{33} - 5 q^{34} - q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} + q^{39} - 8 q^{40} + 10 q^{41} + 17 q^{42} + 19 q^{43} + 4 q^{44} - 9 q^{45} + 4 q^{46} + 11 q^{47} - q^{48} + 11 q^{49} + 8 q^{50} + 7 q^{51} - 8 q^{52} + 8 q^{53} + 11 q^{54} - 4 q^{55} + q^{56} - 11 q^{57} + 11 q^{58} + 28 q^{59} + q^{60} - 12 q^{61} - 8 q^{62} + 20 q^{63} + 8 q^{64} + 8 q^{65} + 10 q^{66} + 24 q^{67} - 5 q^{68} + 30 q^{69} - q^{70} + 18 q^{71} + 9 q^{72} - 3 q^{73} + 19 q^{74} - q^{75} + 2 q^{76} - 7 q^{77} + q^{78} + 22 q^{79} - 8 q^{80} + 24 q^{81} + 10 q^{82} + 17 q^{83} + 17 q^{84} + 5 q^{85} + 19 q^{86} + 11 q^{87} + 4 q^{88} + 17 q^{89} - 9 q^{90} - q^{91} + 4 q^{92} + q^{93} + 11 q^{94} - 2 q^{95} - q^{96} - 24 q^{97} + 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9 - 8 * q^10 + 4 * q^11 - q^12 - 8 * q^13 + q^14 + q^15 + 8 * q^16 - 5 * q^17 + 9 * q^18 + 2 * q^19 - 8 * q^20 + 17 * q^21 + 4 * q^22 + 4 * q^23 - q^24 + 8 * q^25 - 8 * q^26 + 11 * q^27 + q^28 + 11 * q^29 + q^30 - 8 * q^31 + 8 * q^32 + 10 * q^33 - 5 * q^34 - q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 + q^39 - 8 * q^40 + 10 * q^41 + 17 * q^42 + 19 * q^43 + 4 * q^44 - 9 * q^45 + 4 * q^46 + 11 * q^47 - q^48 + 11 * q^49 + 8 * q^50 + 7 * q^51 - 8 * q^52 + 8 * q^53 + 11 * q^54 - 4 * q^55 + q^56 - 11 * q^57 + 11 * q^58 + 28 * q^59 + q^60 - 12 * q^61 - 8 * q^62 + 20 * q^63 + 8 * q^64 + 8 * q^65 + 10 * q^66 + 24 * q^67 - 5 * q^68 + 30 * q^69 - q^70 + 18 * q^71 + 9 * q^72 - 3 * q^73 + 19 * q^74 - q^75 + 2 * q^76 - 7 * q^77 + q^78 + 22 * q^79 - 8 * q^80 + 24 * q^81 + 10 * q^82 + 17 * q^83 + 17 * q^84 + 5 * q^85 + 19 * q^86 + 11 * q^87 + 4 * q^88 + 17 * q^89 - 9 * q^90 - q^91 + 4 * q^92 + q^93 + 11 * q^94 - 2 * q^95 - q^96 - 24 * q^97 + 11 * q^98 + 23 * 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
$3$	−3.08271	−1.77980	−0.889902	−	0.456151i	$-0.849228\pi$
$3$	−3.08271	−1.77980	−0.889902	+	0.456151i	$0.849228\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.08271	−1.25851
$7$	−2.62527	−0.992260	−0.496130	−	0.868248i	$-0.665246\pi$
$7$	−2.62527	−0.992260	−0.496130	+	0.868248i	$0.665246\pi$
$8$	1.00000	0.353553
$9$	6.50312	2.16771
$10$	−1.00000	−0.316228
$11$	0.190183	0.0573425	0.0286712	−	0.999589i	$-0.490872\pi$
$11$	0.190183	0.0573425	0.0286712	+	0.999589i	$0.490872\pi$
$12$	−3.08271	−0.889902
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.62527	−0.701634
$15$	3.08271	0.795953
$16$	1.00000	0.250000
$17$	−6.93705	−1.68248	−0.841241	−	0.540660i	$-0.818174\pi$
$17$	−6.93705	−1.68248	−0.841241	+	0.540660i	$0.818174\pi$
$18$	6.50312	1.53280
$19$	−0.237470	−0.0544794	−0.0272397	−	0.999629i	$-0.508672\pi$
$19$	−0.237470	−0.0544794	−0.0272397	+	0.999629i	$0.508672\pi$
$20$	−1.00000	−0.223607
$21$	8.09297	1.76603
$22$	0.190183	0.0405472
$23$	4.00840	0.835810	0.417905	−	0.908491i	$-0.362765\pi$
$23$	4.00840	0.835810	0.417905	+	0.908491i	$0.362765\pi$
$24$	−3.08271	−0.629256
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−10.7991	−2.07829
$28$	−2.62527	−0.496130
$29$	0.0461297	0.00856607	0.00428303	−	0.999991i	$-0.498637\pi$
$29$	0.0461297	0.00856607	0.00428303	+	0.999991i	$0.498637\pi$
$30$	3.08271	0.562824
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	−0.586281	−0.102058
$34$	−6.93705	−1.18969
$35$	2.62527	0.443752
$36$	6.50312	1.08385
$37$	−9.02092	−1.48303	−0.741515	−	0.670936i	$-0.765892\pi$
$37$	−9.02092	−1.48303	−0.741515	+	0.670936i	$0.765892\pi$
$38$	−0.237470	−0.0385228
$39$	3.08271	0.493629
$40$	−1.00000	−0.158114
$41$	−2.50696	−0.391522	−0.195761	−	0.980652i	$-0.562718\pi$
$41$	−2.50696	−0.391522	−0.195761	+	0.980652i	$0.562718\pi$
$42$	8.09297	1.24877
$43$	−2.72309	−0.415267	−0.207633	−	0.978207i	$-0.566576\pi$
$43$	−2.72309	−0.415267	−0.207633	+	0.978207i	$0.566576\pi$
$44$	0.190183	0.0286712
$45$	−6.50312	−0.969427
$46$	4.00840	0.591007
$47$	−10.0962	−1.47268	−0.736341	−	0.676611i	$-0.763448\pi$
$47$	−10.0962	−1.47268	−0.736341	+	0.676611i	$0.763448\pi$
$48$	−3.08271	−0.444951
$49$	−0.107936	−0.0154195
$50$	1.00000	0.141421
$51$	21.3849	2.99449
$52$	−1.00000	−0.138675
$53$	−1.99103	−0.273489	−0.136745	−	0.990606i	$-0.543664\pi$
$53$	−1.99103	−0.273489	−0.136745	+	0.990606i	$0.543664\pi$
$54$	−10.7991	−1.46957
$55$	−0.190183	−0.0256443
$56$	−2.62527	−0.350817
$57$	0.732053	0.0969628
$58$	0.0461297	0.00605712
$59$	10.7728	1.40250	0.701248	−	0.712917i	$-0.252627\pi$
$59$	10.7728	1.40250	0.701248	+	0.712917i	$0.252627\pi$
$60$	3.08271	0.397976
$61$	0.753431	0.0964670	0.0482335	−	0.998836i	$-0.484641\pi$
$61$	0.753431	0.0964670	0.0482335	+	0.998836i	$0.484641\pi$
$62$	−1.00000	−0.127000
$63$	−17.0725	−2.15093
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−0.586281	−0.0721662
$67$	13.2211	1.61522	0.807608	−	0.589719i	$-0.200762\pi$
$67$	13.2211	1.61522	0.807608	+	0.589719i	$0.200762\pi$
$68$	−6.93705	−0.841241
$69$	−12.3568	−1.48758
$70$	2.62527	0.313780
$71$	6.14349	0.729098	0.364549	−	0.931184i	$-0.381223\pi$
$71$	6.14349	0.729098	0.364549	+	0.931184i	$0.381223\pi$
$72$	6.50312	0.766400
$73$	−5.91955	−0.692831	−0.346415	−	0.938081i	$-0.612601\pi$
$73$	−5.91955	−0.692831	−0.346415	+	0.938081i	$0.612601\pi$
$74$	−9.02092	−1.04866
$75$	−3.08271	−0.355961
$76$	−0.237470	−0.0272397
$77$	−0.499284	−0.0568986
$78$	3.08271	0.349048
$79$	0.817179	0.0919398	0.0459699	−	0.998943i	$-0.485362\pi$
$79$	0.817179	0.0919398	0.0459699	+	0.998943i	$0.485362\pi$
$80$	−1.00000	−0.111803
$81$	13.7812	1.53124
$82$	−2.50696	−0.276848
$83$	0.435097	0.0477581	0.0238791	−	0.999715i	$-0.492398\pi$
$83$	0.435097	0.0477581	0.0238791	+	0.999715i	$0.492398\pi$
$84$	8.09297	0.883015
$85$	6.93705	0.752429
$86$	−2.72309	−0.293638
$87$	−0.142205	−0.0152459
$88$	0.190183	0.0202736
$89$	−1.09084	−0.115629	−0.0578145	−	0.998327i	$-0.518413\pi$
$89$	−1.09084	−0.115629	−0.0578145	+	0.998327i	$0.518413\pi$
$90$	−6.50312	−0.685489
$91$	2.62527	0.275203
$92$	4.00840	0.417905
$93$	3.08271	0.319662
$94$	−10.0962	−1.04134
$95$	0.237470	0.0243639
$96$	−3.08271	−0.314628
$97$	8.17526	0.830072	0.415036	−	0.909805i	$-0.363769\pi$
$97$	8.17526	0.830072	0.415036	+	0.909805i	$0.363769\pi$
$98$	−0.107936	−0.0109032
$99$	1.23679	0.124302
$100$	1.00000	0.100000
$101$	−5.96499	−0.593538	−0.296769	−	0.954949i	$-0.595909\pi$
$101$	−5.96499	−0.593538	−0.296769	+	0.954949i	$0.595909\pi$
$102$	21.3849	2.11742
$103$	14.3243	1.41141	0.705707	−	0.708504i	$-0.250629\pi$
$103$	14.3243	1.41141	0.705707	+	0.708504i	$0.250629\pi$
$104$	−1.00000	−0.0980581
$105$	−8.09297	−0.789793
$106$	−1.99103	−0.193386
$107$	−1.30962	−0.126606	−0.0633028	−	0.997994i	$-0.520163\pi$
$107$	−1.30962	−0.126606	−0.0633028	+	0.997994i	$0.520163\pi$
$108$	−10.7991	−1.03914
$109$	−0.402902	−0.0385910	−0.0192955	−	0.999814i	$-0.506142\pi$
$109$	−0.402902	−0.0385910	−0.0192955	+	0.999814i	$0.506142\pi$
$110$	−0.190183	−0.0181333
$111$	27.8089	2.63950
$112$	−2.62527	−0.248065
$113$	2.66374	0.250584	0.125292	−	0.992120i	$-0.460013\pi$
$113$	2.66374	0.250584	0.125292	+	0.992120i	$0.460013\pi$
$114$	0.732053	0.0685630
$115$	−4.00840	−0.373786
$116$	0.0461297	0.00428303
$117$	−6.50312	−0.601213
$118$	10.7728	0.991714
$119$	18.2117	1.66946
$120$	3.08271	0.281412
$121$	−10.9638	−0.996712
$122$	0.753431	0.0682124
$123$	7.72824	0.696832
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−17.0725	−1.52094
$127$	10.3607	0.919366	0.459683	−	0.888083i	$-0.347963\pi$
$127$	10.3607	0.919366	0.459683	+	0.888083i	$0.347963\pi$
$128$	1.00000	0.0883883
$129$	8.39449	0.739094
$130$	1.00000	0.0877058
$131$	−10.1846	−0.889836	−0.444918	−	0.895571i	$-0.646767\pi$
$131$	−10.1846	−0.889836	−0.444918	+	0.895571i	$0.646767\pi$
$132$	−0.586281	−0.0510292
$133$	0.623425	0.0540578
$134$	13.2211	1.14213
$135$	10.7991	0.929439
$136$	−6.93705	−0.594847
$137$	6.23374	0.532584	0.266292	−	0.963892i	$-0.414201\pi$
$137$	6.23374	0.532584	0.266292	+	0.963892i	$0.414201\pi$
$138$	−12.3568	−1.05188
$139$	3.87231	0.328445	0.164222	−	0.986423i	$-0.447489\pi$
$139$	3.87231	0.328445	0.164222	+	0.986423i	$0.447489\pi$
$140$	2.62527	0.221876
$141$	31.1237	2.62109
$142$	6.14349	0.515550
$143$	−0.190183	−0.0159039
$144$	6.50312	0.541926
$145$	−0.0461297	−0.00383086
$146$	−5.91955	−0.489905
$147$	0.332736	0.0274436
$148$	−9.02092	−0.741515
$149$	15.1295	1.23946	0.619729	−	0.784816i	$-0.287243\pi$
$149$	15.1295	1.23946	0.619729	+	0.784816i	$0.287243\pi$
$150$	−3.08271	−0.251702
$151$	8.12239	0.660991	0.330495	−	0.943808i	$-0.392784\pi$
$151$	8.12239	0.660991	0.330495	+	0.943808i	$0.392784\pi$
$152$	−0.237470	−0.0192614
$153$	−45.1124	−3.64713
$154$	−0.499284	−0.0402334
$155$	1.00000	0.0803219
$156$	3.08271	0.246815
$157$	19.0833	1.52301	0.761507	−	0.648157i	$-0.224460\pi$
$157$	19.0833	1.52301	0.761507	+	0.648157i	$0.224460\pi$
$158$	0.817179	0.0650113
$159$	6.13778	0.486758
$160$	−1.00000	−0.0790569
$161$	−10.5232	−0.829341
$162$	13.7812	1.08275
$163$	−4.24035	−0.332130	−0.166065	−	0.986115i	$-0.553106\pi$
$163$	−4.24035	−0.332130	−0.166065	+	0.986115i	$0.553106\pi$
$164$	−2.50696	−0.195761
$165$	0.586281	0.0456419
$166$	0.435097	0.0337701
$167$	13.5092	1.04537	0.522687	−	0.852524i	$-0.324929\pi$
$167$	13.5092	1.04537	0.522687	+	0.852524i	$0.324929\pi$
$168$	8.09297	0.624386
$169$	1.00000	0.0769231
$170$	6.93705	0.532047
$171$	−1.54430	−0.118095
$172$	−2.72309	−0.207633
$173$	1.08997	0.0828688	0.0414344	−	0.999141i	$-0.486807\pi$
$173$	1.08997	0.0828688	0.0414344	+	0.999141i	$0.486807\pi$
$174$	−0.142205	−0.0107805
$175$	−2.62527	−0.198452
$176$	0.190183	0.0143356
$177$	−33.2094	−2.49617
$178$	−1.09084	−0.0817620
$179$	12.1337	0.906915	0.453457	−	0.891278i	$-0.350190\pi$
$179$	12.1337	0.906915	0.453457	+	0.891278i	$0.350190\pi$
$180$	−6.50312	−0.484714
$181$	3.25448	0.241904	0.120952	−	0.992658i	$-0.461405\pi$
$181$	3.25448	0.241904	0.120952	+	0.992658i	$0.461405\pi$
$182$	2.62527	0.194598
$183$	−2.32261	−0.171692
$184$	4.00840	0.295503
$185$	9.02092	0.663231
$186$	3.08271	0.226035
$187$	−1.31931	−0.0964776
$188$	−10.0962	−0.736341
$189$	28.3506	2.06220
$190$	0.237470	0.0172279
$191$	−10.6539	−0.770892	−0.385446	−	0.922730i	$-0.625952\pi$
$191$	−10.6539	−0.770892	−0.385446	+	0.922730i	$0.625952\pi$
$192$	−3.08271	−0.222476
$193$	−12.5653	−0.904472	−0.452236	−	0.891898i	$-0.649374\pi$
$193$	−12.5653	−0.904472	−0.452236	+	0.891898i	$0.649374\pi$
$194$	8.17526	0.586950
$195$	−3.08271	−0.220758
$196$	−0.107936	−0.00770973
$197$	9.06106	0.645574	0.322787	−	0.946472i	$-0.395380\pi$
$197$	9.06106	0.645574	0.322787	+	0.946472i	$0.395380\pi$
$198$	1.23679	0.0878945
$199$	25.5763	1.81305	0.906527	−	0.422148i	$-0.138724\pi$
$199$	25.5763	1.81305	0.906527	+	0.422148i	$0.138724\pi$
$200$	1.00000	0.0707107
$201$	−40.7569	−2.87477
$202$	−5.96499	−0.419695
$203$	−0.121103	−0.00849977
$204$	21.3849	1.49724
$205$	2.50696	0.175094
$206$	14.3243	0.998021
$207$	26.0671	1.81179
$208$	−1.00000	−0.0693375
$209$	−0.0451629	−0.00312399
$210$	−8.09297	−0.558468
$211$	8.79692	0.605605	0.302802	−	0.953053i	$-0.402078\pi$
$211$	8.79692	0.605605	0.302802	+	0.953053i	$0.402078\pi$
$212$	−1.99103	−0.136745
$213$	−18.9386	−1.29765
$214$	−1.30962	−0.0895237
$215$	2.72309	0.185713
$216$	−10.7991	−0.734786
$217$	2.62527	0.178215
$218$	−0.402902	−0.0272880
$219$	18.2483	1.23310
$220$	−0.190183	−0.0128222
$221$	6.93705	0.466637
$222$	27.8089	1.86641
$223$	−26.7938	−1.79425	−0.897124	−	0.441779i	$-0.854347\pi$
$223$	−26.7938	−1.79425	−0.897124	+	0.441779i	$0.854347\pi$
$224$	−2.62527	−0.175409
$225$	6.50312	0.433541
$226$	2.66374	0.177189
$227$	10.2446	0.679958	0.339979	−	0.940433i	$-0.389580\pi$
$227$	10.2446	0.679958	0.339979	+	0.940433i	$0.389580\pi$
$228$	0.732053	0.0484814
$229$	−11.9399	−0.789010	−0.394505	−	0.918894i	$-0.629084\pi$
$229$	−11.9399	−0.789010	−0.394505	+	0.918894i	$0.629084\pi$
$230$	−4.00840	−0.264306
$231$	1.53915	0.101268
$232$	0.0461297	0.00302856
$233$	25.8498	1.69348	0.846739	−	0.532008i	$-0.178562\pi$
$233$	25.8498	1.69348	0.846739	+	0.532008i	$0.178562\pi$
$234$	−6.50312	−0.425122
$235$	10.0962	0.658603
$236$	10.7728	0.701248
$237$	−2.51913	−0.163635
$238$	18.2117	1.18049
$239$	−4.74305	−0.306802	−0.153401	−	0.988164i	$-0.549023\pi$
$239$	−4.74305	−0.306802	−0.153401	+	0.988164i	$0.549023\pi$
$240$	3.08271	0.198988
$241$	−5.76720	−0.371498	−0.185749	−	0.982597i	$-0.559471\pi$
$241$	−5.76720	−0.371498	−0.185749	+	0.982597i	$0.559471\pi$
$242$	−10.9638	−0.704782
$243$	−10.0861	−0.647024
$244$	0.753431	0.0482335
$245$	0.107936	0.00689579
$246$	7.72824	0.492735
$247$	0.237470	0.0151099
$248$	−1.00000	−0.0635001
$249$	−1.34128	−0.0850001
$250$	−1.00000	−0.0632456
$251$	21.6019	1.36350	0.681751	−	0.731584i	$-0.261219\pi$
$251$	21.6019	1.36350	0.681751	+	0.731584i	$0.261219\pi$
$252$	−17.0725	−1.07546
$253$	0.762332	0.0479274
$254$	10.3607	0.650090
$255$	−21.3849	−1.33918
$256$	1.00000	0.0625000
$257$	15.1584	0.945556	0.472778	−	0.881182i	$-0.343251\pi$
$257$	15.1584	0.945556	0.472778	+	0.881182i	$0.343251\pi$
$258$	8.39449	0.522618
$259$	23.6824	1.47155
$260$	1.00000	0.0620174
$261$	0.299987	0.0185687
$262$	−10.1846	−0.629209
$263$	1.44308	0.0889843	0.0444921	−	0.999010i	$-0.485833\pi$
$263$	1.44308	0.0889843	0.0444921	+	0.999010i	$0.485833\pi$
$264$	−0.586281	−0.0360831
$265$	1.99103	0.122308
$266$	0.623425	0.0382246
$267$	3.36275	0.205797
$268$	13.2211	0.807608
$269$	−9.94188	−0.606167	−0.303084	−	0.952964i	$-0.598016\pi$
$269$	−9.94188	−0.606167	−0.303084	+	0.952964i	$0.598016\pi$
$270$	10.7991	0.657212
$271$	23.9828	1.45685	0.728427	−	0.685124i	$-0.240252\pi$
$271$	23.9828	1.45685	0.728427	+	0.685124i	$0.240252\pi$
$272$	−6.93705	−0.420620
$273$	−8.09297	−0.489809
$274$	6.23374	0.376594
$275$	0.190183	0.0114685
$276$	−12.3568	−0.743789
$277$	−14.2200	−0.854394	−0.427197	−	0.904158i	$-0.640499\pi$
$277$	−14.2200	−0.854394	−0.427197	+	0.904158i	$0.640499\pi$
$278$	3.87231	0.232246
$279$	−6.50312	−0.389331
$280$	2.62527	0.156890
$281$	15.7629	0.940339	0.470169	−	0.882576i	$-0.344193\pi$
$281$	15.7629	0.940339	0.470169	+	0.882576i	$0.344193\pi$
$282$	31.1237	1.85339
$283$	−18.2060	−1.08223	−0.541117	−	0.840947i	$-0.681999\pi$
$283$	−18.2060	−1.08223	−0.541117	+	0.840947i	$0.681999\pi$
$284$	6.14349	0.364549
$285$	−0.732053	−0.0433631
$286$	−0.190183	−0.0112458
$287$	6.58146	0.388492
$288$	6.50312	0.383200
$289$	31.1227	1.83075
$290$	−0.0461297	−0.00270883
$291$	−25.2020	−1.47737
$292$	−5.91955	−0.346415
$293$	13.6870	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$293$	13.6870	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$294$	0.332736	0.0194056
$295$	−10.7728	−0.627215
$296$	−9.02092	−0.524330
$297$	−2.05381	−0.119174
$298$	15.1295	0.876429
$299$	−4.00840	−0.231812
$300$	−3.08271	−0.177980
$301$	7.14884	0.412053
$302$	8.12239	0.467391
$303$	18.3883	1.05638
$304$	−0.237470	−0.0136199
$305$	−0.753431	−0.0431413
$306$	−45.1124	−2.57891
$307$	20.5185	1.17105	0.585527	−	0.810653i	$-0.300888\pi$
$307$	20.5185	1.17105	0.585527	+	0.810653i	$0.300888\pi$
$308$	−0.499284	−0.0284493
$309$	−44.1577	−2.51204
$310$	1.00000	0.0567962
$311$	−18.9628	−1.07528	−0.537642	−	0.843173i	$-0.680685\pi$
$311$	−18.9628	−1.07528	−0.537642	+	0.843173i	$0.680685\pi$
$312$	3.08271	0.174524
$313$	−15.8179	−0.894080	−0.447040	−	0.894514i	$-0.647522\pi$
$313$	−15.8179	−0.894080	−0.447040	+	0.894514i	$0.647522\pi$
$314$	19.0833	1.07693
$315$	17.0725	0.961924
$316$	0.817179	0.0459699
$317$	31.6064	1.77519	0.887595	−	0.460624i	$-0.152374\pi$
$317$	31.6064	1.77519	0.887595	+	0.460624i	$0.152374\pi$
$318$	6.13778	0.344190
$319$	0.00877310	0.000491199 0
$320$	−1.00000	−0.0559017
$321$	4.03718	0.225333
$322$	−10.5232	−0.586433
$323$	1.64734	0.0916607
$324$	13.7812	0.765621
$325$	−1.00000	−0.0554700
$326$	−4.24035	−0.234851
$327$	1.24203	0.0686845
$328$	−2.50696	−0.138424
$329$	26.5053	1.46128
$330$	0.586281	0.0322737
$331$	16.2940	0.895600	0.447800	−	0.894134i	$-0.352208\pi$
$331$	16.2940	0.895600	0.447800	+	0.894134i	$0.352208\pi$
$332$	0.435097	0.0238791
$333$	−58.6641	−3.21477
$334$	13.5092	0.739192
$335$	−13.2211	−0.722347
$336$	8.09297	0.441507
$337$	15.4158	0.839753	0.419876	−	0.907581i	$-0.362073\pi$
$337$	15.4158	0.839753	0.419876	+	0.907581i	$0.362073\pi$
$338$	1.00000	0.0543928
$339$	−8.21155	−0.445990
$340$	6.93705	0.376214
$341$	−0.190183	−0.0102990
$342$	−1.54430	−0.0835061
$343$	18.6603	1.00756
$344$	−2.72309	−0.146819
$345$	12.3568	0.665265
$346$	1.08997	0.0585971
$347$	7.68442	0.412521	0.206261	−	0.978497i	$-0.433871\pi$
$347$	7.68442	0.412521	0.206261	+	0.978497i	$0.433871\pi$
$348$	−0.142205	−0.00762296
$349$	16.5847	0.887760	0.443880	−	0.896086i	$-0.353602\pi$
$349$	16.5847	0.887760	0.443880	+	0.896086i	$0.353602\pi$
$350$	−2.62527	−0.140327
$351$	10.7991	0.576413
$352$	0.190183	0.0101368
$353$	−33.2411	−1.76925	−0.884623	−	0.466307i	$-0.845584\pi$
$353$	−33.2411	−1.76925	−0.884623	+	0.466307i	$0.845584\pi$
$354$	−33.2094	−1.76506
$355$	−6.14349	−0.326063
$356$	−1.09084	−0.0578145
$357$	−56.1413	−2.97131
$358$	12.1337	0.641286
$359$	21.8698	1.15424	0.577121	−	0.816658i	$-0.304176\pi$
$359$	21.8698	1.15424	0.577121	+	0.816658i	$0.304176\pi$
$360$	−6.50312	−0.342744
$361$	−18.9436	−0.997032
$362$	3.25448	0.171052
$363$	33.7983	1.77395
$364$	2.62527	0.137602
$365$	5.91955	0.309843
$366$	−2.32261	−0.121405
$367$	−33.8742	−1.76822	−0.884108	−	0.467282i	$-0.845233\pi$
$367$	−33.8742	−1.76822	−0.884108	+	0.467282i	$0.845233\pi$
$368$	4.00840	0.208952
$369$	−16.3031	−0.848704
$370$	9.02092	0.468975
$371$	5.22701	0.271373
$372$	3.08271	0.159831
$373$	−15.7122	−0.813549	−0.406775	−	0.913529i	$-0.633347\pi$
$373$	−15.7122	−0.813549	−0.406775	+	0.913529i	$0.633347\pi$
$374$	−1.31931	−0.0682200
$375$	3.08271	0.159191
$376$	−10.0962	−0.520672
$377$	−0.0461297	−0.00237580
$378$	28.3506	1.45820
$379$	4.04448	0.207751	0.103875	−	0.994590i	$-0.466876\pi$
$379$	4.04448	0.207751	0.103875	+	0.994590i	$0.466876\pi$
$380$	0.237470	0.0121820
$381$	−31.9391	−1.63629
$382$	−10.6539	−0.545103
$383$	2.70970	0.138459	0.0692296	−	0.997601i	$-0.477946\pi$
$383$	2.70970	0.138459	0.0692296	+	0.997601i	$0.477946\pi$
$384$	−3.08271	−0.157314
$385$	0.499284	0.0254458
$386$	−12.5653	−0.639559
$387$	−17.7085	−0.900176
$388$	8.17526	0.415036
$389$	14.2920	0.724632	0.362316	−	0.932055i	$-0.381986\pi$
$389$	14.2920	0.724632	0.362316	+	0.932055i	$0.381986\pi$
$390$	−3.08271	−0.156099
$391$	−27.8065	−1.40624
$392$	−0.107936	−0.00545160
$393$	31.3963	1.58374
$394$	9.06106	0.456489
$395$	−0.817179	−0.0411167
$396$	1.23679	0.0621508
$397$	−3.48803	−0.175059	−0.0875295	−	0.996162i	$-0.527897\pi$
$397$	−3.48803	−0.175059	−0.0875295	+	0.996162i	$0.527897\pi$
$398$	25.5763	1.28202
$399$	−1.92184	−0.0962123
$400$	1.00000	0.0500000
$401$	−20.2318	−1.01033	−0.505165	−	0.863023i	$-0.668568\pi$
$401$	−20.2318	−1.01033	−0.505165	+	0.863023i	$0.668568\pi$
$402$	−40.7569	−2.03277
$403$	1.00000	0.0498135
$404$	−5.96499	−0.296769
$405$	−13.7812	−0.684792
$406$	−0.121103	−0.00601024
$407$	−1.71563	−0.0850406
$408$	21.3849	1.05871
$409$	26.7867	1.32452	0.662259	−	0.749275i	$-0.269598\pi$
$409$	26.7867	1.32452	0.662259	+	0.749275i	$0.269598\pi$
$410$	2.50696	0.123810
$411$	−19.2168	−0.947896
$412$	14.3243	0.705707
$413$	−28.2815	−1.39164
$414$	26.0671	1.28113
$415$	−0.435097	−0.0213581
$416$	−1.00000	−0.0490290
$417$	−11.9372	−0.584568
$418$	−0.0451629	−0.00220899
$419$	−37.0735	−1.81116	−0.905579	−	0.424177i	$-0.860563\pi$
$419$	−37.0735	−1.81116	−0.905579	+	0.424177i	$0.860563\pi$
$420$	−8.09297	−0.394896
$421$	8.02881	0.391300	0.195650	−	0.980674i	$-0.437318\pi$
$421$	8.02881	0.391300	0.195650	+	0.980674i	$0.437318\pi$
$422$	8.79692	0.428227
$423$	−65.6567	−3.19234
$424$	−1.99103	−0.0966931
$425$	−6.93705	−0.336496
$426$	−18.9386	−0.917579
$427$	−1.97796	−0.0957203
$428$	−1.30962	−0.0633028
$429$	0.586281	0.0283059
$430$	2.72309	0.131319
$431$	−25.3589	−1.22149	−0.610746	−	0.791826i	$-0.709131\pi$
$431$	−25.3589	−1.22149	−0.610746	+	0.791826i	$0.709131\pi$
$432$	−10.7991	−0.519572
$433$	−28.3882	−1.36425	−0.682125	−	0.731236i	$-0.738944\pi$
$433$	−28.3882	−1.36425	−0.682125	+	0.731236i	$0.738944\pi$
$434$	2.62527	0.126017
$435$	0.142205	0.00681819
$436$	−0.402902	−0.0192955
$437$	−0.951877	−0.0455345
$438$	18.2483	0.871936
$439$	−18.2165	−0.869427	−0.434713	−	0.900569i	$-0.643150\pi$
$439$	−18.2165	−0.869427	−0.434713	+	0.900569i	$0.643150\pi$
$440$	−0.190183	−0.00906664
$441$	−0.701922	−0.0334248
$442$	6.93705	0.329962
$443$	−19.5676	−0.929683	−0.464842	−	0.885394i	$-0.653889\pi$
$443$	−19.5676	−0.929683	−0.464842	+	0.885394i	$0.653889\pi$
$444$	27.8089	1.31975
$445$	1.09084	0.0517108
$446$	−26.7938	−1.26872
$447$	−46.6399	−2.20599
$448$	−2.62527	−0.124033
$449$	−8.23469	−0.388619	−0.194310	−	0.980940i	$-0.562247\pi$
$449$	−8.23469	−0.388619	−0.194310	+	0.980940i	$0.562247\pi$
$450$	6.50312	0.306560
$451$	−0.476783	−0.0224508
$452$	2.66374	0.125292
$453$	−25.0390	−1.17643
$454$	10.2446	0.480803
$455$	−2.62527	−0.123075
$456$	0.732053	0.0342815
$457$	−24.4634	−1.14435	−0.572175	−	0.820131i	$-0.693900\pi$
$457$	−24.4634	−1.14435	−0.572175	+	0.820131i	$0.693900\pi$
$458$	−11.9399	−0.557914
$459$	74.9139	3.49668
$460$	−4.00840	−0.186893
$461$	−8.11358	−0.377887	−0.188944	−	0.981988i	$-0.560506\pi$
$461$	−8.11358	−0.377887	−0.188944	+	0.981988i	$0.560506\pi$
$462$	1.53915	0.0716076
$463$	10.0863	0.468749	0.234375	−	0.972146i	$-0.424696\pi$
$463$	10.0863	0.468749	0.234375	+	0.972146i	$0.424696\pi$
$464$	0.0461297	0.00214152
$465$	−3.08271	−0.142957
$466$	25.8498	1.19747
$467$	25.0000	1.15686	0.578432	−	0.815731i	$-0.303665\pi$
$467$	25.0000	1.15686	0.578432	+	0.815731i	$0.303665\pi$
$468$	−6.50312	−0.300607
$469$	−34.7091	−1.60272
$470$	10.0962	0.465703
$471$	−58.8283	−2.71067
$472$	10.7728	0.495857
$473$	−0.517886	−0.0238124
$474$	−2.51913	−0.115707
$475$	−0.237470	−0.0108959
$476$	18.2117	0.834730
$477$	−12.9479	−0.592844
$478$	−4.74305	−0.216942
$479$	−26.5870	−1.21479	−0.607395	−	0.794400i	$-0.707785\pi$
$479$	−26.5870	−1.21479	−0.607395	+	0.794400i	$0.707785\pi$
$480$	3.08271	0.140706
$481$	9.02092	0.411319
$482$	−5.76720	−0.262689
$483$	32.4399	1.47607
$484$	−10.9638	−0.498356
$485$	−8.17526	−0.371220
$486$	−10.0861	−0.457515
$487$	−6.50732	−0.294875	−0.147437	−	0.989071i	$-0.547103\pi$
$487$	−6.50732	−0.294875	−0.147437	+	0.989071i	$0.547103\pi$
$488$	0.753431	0.0341062
$489$	13.0718	0.591126
$490$	0.107936	0.00487606
$491$	−30.5048	−1.37666	−0.688331	−	0.725397i	$-0.741656\pi$
$491$	−30.5048	−1.37666	−0.688331	+	0.725397i	$0.741656\pi$
$492$	7.72824	0.348416
$493$	−0.320004	−0.0144123
$494$	0.237470	0.0106843
$495$	−1.23679	−0.0555894
$496$	−1.00000	−0.0449013
$497$	−16.1284	−0.723455
$498$	−1.34128	−0.0601042
$499$	12.5736	0.562873	0.281437	−	0.959580i	$-0.409189\pi$
$499$	12.5736	0.562873	0.281437	+	0.959580i	$0.409189\pi$
$500$	−1.00000	−0.0447214
$501$	−41.6450	−1.86056
$502$	21.6019	0.964142
$503$	10.9320	0.487436	0.243718	−	0.969846i	$-0.421633\pi$
$503$	10.9320	0.487436	0.243718	+	0.969846i	$0.421633\pi$
$504$	−17.0725	−0.760468
$505$	5.96499	0.265438
$506$	0.762332	0.0338898
$507$	−3.08271	−0.136908
$508$	10.3607	0.459683
$509$	21.9412	0.972525	0.486263	−	0.873813i	$-0.338360\pi$
$509$	21.9412	0.972525	0.486263	+	0.873813i	$0.338360\pi$
$510$	−21.3849	−0.946941
$511$	15.5404	0.687468
$512$	1.00000	0.0441942
$513$	2.56447	0.113224
$514$	15.1584	0.668609
$515$	−14.3243	−0.631204
$516$	8.39449	0.369547
$517$	−1.92013	−0.0844472
$518$	23.6824	1.04054
$519$	−3.36006	−0.147490
$520$	1.00000	0.0438529
$521$	10.7473	0.470849	0.235424	−	0.971893i	$-0.424352\pi$
$521$	10.7473	0.470849	0.235424	+	0.971893i	$0.424352\pi$
$522$	0.299987	0.0131301
$523$	12.7032	0.555472	0.277736	−	0.960658i	$-0.410416\pi$
$523$	12.7032	0.555472	0.277736	+	0.960658i	$0.410416\pi$
$524$	−10.1846	−0.444918
$525$	8.09297	0.353206
$526$	1.44308	0.0629214
$527$	6.93705	0.302183
$528$	−0.586281	−0.0255146
$529$	−6.93270	−0.301422
$530$	1.99103	0.0864849
$531$	70.0566	3.04020
$532$	0.623425	0.0270289
$533$	2.50696	0.108589
$534$	3.36275	0.145520
$535$	1.30962	0.0566198
$536$	13.2211	0.571065
$537$	−37.4047	−1.61413
$538$	−9.94188	−0.428625
$539$	−0.0205277	−0.000884189 0
$540$	10.7991	0.464719
$541$	4.21317	0.181138	0.0905692	−	0.995890i	$-0.471131\pi$
$541$	4.21317	0.181138	0.0905692	+	0.995890i	$0.471131\pi$
$542$	23.9828	1.03015
$543$	−10.0326	−0.430541
$544$	−6.93705	−0.297424
$545$	0.402902	0.0172584
$546$	−8.09297	−0.346347
$547$	34.7736	1.48681	0.743406	−	0.668841i	$-0.233209\pi$
$547$	34.7736	1.48681	0.743406	+	0.668841i	$0.233209\pi$
$548$	6.23374	0.266292
$549$	4.89965	0.209112
$550$	0.190183	0.00810945
$551$	−0.0109544	−0.000466674 0
$552$	−12.3568	−0.525938
$553$	−2.14532	−0.0912282
$554$	−14.2200	−0.604148
$555$	−27.8089	−1.18042
$556$	3.87231	0.164222
$557$	−28.6834	−1.21535	−0.607677	−	0.794184i	$-0.707899\pi$
$557$	−28.6834	−1.21535	−0.607677	+	0.794184i	$0.707899\pi$
$558$	−6.50312	−0.275299
$559$	2.72309	0.115174
$560$	2.62527	0.110938
$561$	4.06706	0.171711
$562$	15.7629	0.664920
$563$	38.6802	1.63018	0.815089	−	0.579336i	$-0.196688\pi$
$563$	38.6802	1.63018	0.815089	+	0.579336i	$0.196688\pi$
$564$	31.1237	1.31054
$565$	−2.66374	−0.112064
$566$	−18.2060	−0.765256
$567$	−36.1794	−1.51939
$568$	6.14349	0.257775
$569$	−32.2884	−1.35360	−0.676801	−	0.736166i	$-0.736634\pi$
$569$	−32.2884	−1.35360	−0.676801	+	0.736166i	$0.736634\pi$
$570$	−0.732053	−0.0306623
$571$	23.4429	0.981056	0.490528	−	0.871425i	$-0.336804\pi$
$571$	23.4429	0.981056	0.490528	+	0.871425i	$0.336804\pi$
$572$	−0.190183	−0.00795197
$573$	32.8430	1.37204
$574$	6.58146	0.274705
$575$	4.00840	0.167162
$576$	6.50312	0.270963
$577$	−41.4785	−1.72677	−0.863387	−	0.504543i	$-0.831661\pi$
$577$	−41.4785	−1.72677	−0.863387	+	0.504543i	$0.831661\pi$
$578$	31.1227	1.29453
$579$	38.7353	1.60978
$580$	−0.0461297	−0.00191543
$581$	−1.14225	−0.0473885
$582$	−25.2020	−1.04466
$583$	−0.378661	−0.0156825
$584$	−5.91955	−0.244953
$585$	6.50312	0.268871
$586$	13.6870	0.565405
$587$	11.7320	0.484232	0.242116	−	0.970247i	$-0.422159\pi$
$587$	11.7320	0.484232	0.242116	+	0.970247i	$0.422159\pi$
$588$	0.332736	0.0137218
$589$	0.237470	0.00978480
$590$	−10.7728	−0.443508
$591$	−27.9326	−1.14900
$592$	−9.02092	−0.370758
$593$	14.8281	0.608918	0.304459	−	0.952525i	$-0.401524\pi$
$593$	14.8281	0.608918	0.304459	+	0.952525i	$0.401524\pi$
$594$	−2.05381	−0.0842689
$595$	−18.2117	−0.746605
$596$	15.1295	0.619729
$597$	−78.8443	−3.22688
$598$	−4.00840	−0.163916
$599$	−8.04349	−0.328648	−0.164324	−	0.986406i	$-0.552544\pi$
$599$	−8.04349	−0.328648	−0.164324	+	0.986406i	$0.552544\pi$
$600$	−3.08271	−0.125851
$601$	−10.8367	−0.442040	−0.221020	−	0.975269i	$-0.570939\pi$
$601$	−10.8367	−0.442040	−0.221020	+	0.975269i	$0.570939\pi$
$602$	7.14884	0.291365
$603$	85.9785	3.50131
$604$	8.12239	0.330495
$605$	10.9638	0.445743
$606$	18.3883	0.746975
$607$	38.2829	1.55386	0.776928	−	0.629589i	$-0.216777\pi$
$607$	38.2829	1.55386	0.776928	+	0.629589i	$0.216777\pi$
$608$	−0.237470	−0.00963070
$609$	0.373326	0.0151279
$610$	−0.753431	−0.0305055
$611$	10.0962	0.408448
$612$	−45.1124	−1.82356
$613$	6.03017	0.243556	0.121778	−	0.992557i	$-0.461140\pi$
$613$	6.03017	0.243556	0.121778	+	0.992557i	$0.461140\pi$
$614$	20.5185	0.828061
$615$	−7.72824	−0.311633
$616$	−0.499284	−0.0201167
$617$	1.89394	0.0762470	0.0381235	−	0.999273i	$-0.487862\pi$
$617$	1.89394	0.0762470	0.0381235	+	0.999273i	$0.487862\pi$
$618$	−44.1577	−1.77628
$619$	−24.3574	−0.979005	−0.489503	−	0.872002i	$-0.662822\pi$
$619$	−24.3574	−0.979005	−0.489503	+	0.872002i	$0.662822\pi$
$620$	1.00000	0.0401610
$621$	−43.2872	−1.73705
$622$	−18.9628	−0.760340
$623$	2.86376	0.114734
$624$	3.08271	0.123407
$625$	1.00000	0.0400000
$626$	−15.8179	−0.632210
$627$	0.139224	0.00556008
$628$	19.0833	0.761507
$629$	62.5786	2.49517
$630$	17.0725	0.680183
$631$	−6.27936	−0.249977	−0.124989	−	0.992158i	$-0.539889\pi$
$631$	−6.27936	−0.249977	−0.124989	+	0.992158i	$0.539889\pi$
$632$	0.817179	0.0325056
$633$	−27.1184	−1.07786
$634$	31.6064	1.25525
$635$	−10.3607	−0.411153
$636$	6.13778	0.243379
$637$	0.107936	0.00427659
$638$	0.00877310	0.000347330 0
$639$	39.9519	1.58047
$640$	−1.00000	−0.0395285
$641$	−42.2402	−1.66839	−0.834193	−	0.551472i	$-0.814066\pi$
$641$	−42.2402	−1.66839	−0.834193	+	0.551472i	$0.814066\pi$
$642$	4.03718	0.159335
$643$	27.9359	1.10168	0.550841	−	0.834610i	$-0.314307\pi$
$643$	27.9359	1.10168	0.550841	+	0.834610i	$0.314307\pi$
$644$	−10.5232	−0.414671
$645$	−8.39449	−0.330533
$646$	1.64734	0.0648139
$647$	13.8574	0.544791	0.272395	−	0.962185i	$-0.412184\pi$
$647$	13.8574	0.544791	0.272395	+	0.962185i	$0.412184\pi$
$648$	13.7812	0.541376
$649$	2.04880	0.0804226
$650$	−1.00000	−0.0392232
$651$	−8.09297	−0.317188
$652$	−4.24035	−0.166065
$653$	9.24792	0.361899	0.180950	−	0.983492i	$-0.442083\pi$
$653$	9.24792	0.361899	0.180950	+	0.983492i	$0.442083\pi$
$654$	1.24203	0.0485673
$655$	10.1846	0.397947
$656$	−2.50696	−0.0978804
$657$	−38.4955	−1.50185
$658$	26.5053	1.03328
$659$	28.2732	1.10137	0.550685	−	0.834713i	$-0.314367\pi$
$659$	28.2732	1.10137	0.550685	+	0.834713i	$0.314367\pi$
$660$	0.586281	0.0228210
$661$	31.4799	1.22443	0.612213	−	0.790693i	$-0.290280\pi$
$661$	31.4799	1.22443	0.612213	+	0.790693i	$0.290280\pi$
$662$	16.2940	0.633285
$663$	−21.3849	−0.830522
$664$	0.435097	0.0168850
$665$	−0.623425	−0.0241754
$666$	−58.6641	−2.27319
$667$	0.184906	0.00715960
$668$	13.5092	0.522687
$669$	82.5977	3.19341
$670$	−13.2211	−0.510776
$671$	0.143290	0.00553165
$672$	8.09297	0.312193
$673$	−9.55452	−0.368300	−0.184150	−	0.982898i	$-0.558953\pi$
$673$	−9.55452	−0.368300	−0.184150	+	0.982898i	$0.558953\pi$
$674$	15.4158	0.593795
$675$	−10.7991	−0.415658
$676$	1.00000	0.0384615
$677$	−32.0117	−1.23031	−0.615155	−	0.788406i	$-0.710907\pi$
$677$	−32.0117	−1.23031	−0.615155	+	0.788406i	$0.710907\pi$
$678$	−8.21155	−0.315363
$679$	−21.4623	−0.823648
$680$	6.93705	0.266024
$681$	−31.5812	−1.21019
$682$	−0.190183	−0.00728250
$683$	−19.9713	−0.764182	−0.382091	−	0.924125i	$-0.624796\pi$
$683$	−19.9713	−0.764182	−0.382091	+	0.924125i	$0.624796\pi$
$684$	−1.54430	−0.0590477
$685$	−6.23374	−0.238179
$686$	18.6603	0.712453
$687$	36.8072	1.40428
$688$	−2.72309	−0.103817
$689$	1.99103	0.0758523
$690$	12.3568	0.470414
$691$	−20.2688	−0.771062	−0.385531	−	0.922695i	$-0.625982\pi$
$691$	−20.2688	−0.771062	−0.385531	+	0.922695i	$0.625982\pi$
$692$	1.08997	0.0414344
$693$	−3.24690	−0.123340
$694$	7.68442	0.291696
$695$	−3.87231	−0.146885
$696$	−0.142205	−0.00539025
$697$	17.3909	0.658728
$698$	16.5847	0.627741
$699$	−79.6876	−3.01406
$700$	−2.62527	−0.0992260
$701$	37.6355	1.42147	0.710736	−	0.703458i	$-0.248362\pi$
$701$	37.6355	1.42147	0.710736	+	0.703458i	$0.248362\pi$
$702$	10.7991	0.407586
$703$	2.14220	0.0807947
$704$	0.190183	0.00716781
$705$	−31.1237	−1.17219
$706$	−33.2411	−1.25105
$707$	15.6597	0.588944
$708$	−33.2094	−1.24808
$709$	24.3657	0.915074	0.457537	−	0.889191i	$-0.348732\pi$
$709$	24.3657	0.915074	0.457537	+	0.889191i	$0.348732\pi$
$710$	−6.14349	−0.230561
$711$	5.31421	0.199298
$712$	−1.09084	−0.0408810
$713$	−4.00840	−0.150116
$714$	−56.1413	−2.10104
$715$	0.190183	0.00711246
$716$	12.1337	0.453457
$717$	14.6215	0.546048
$718$	21.8698	0.816173
$719$	−18.8095	−0.701475	−0.350738	−	0.936474i	$-0.614069\pi$
$719$	−18.8095	−0.701475	−0.350738	+	0.936474i	$0.614069\pi$
$720$	−6.50312	−0.242357
$721$	−37.6052	−1.40049
$722$	−18.9436	−0.705008
$723$	17.7786	0.661194
$724$	3.25448	0.120952
$725$	0.0461297	0.00171321
$726$	33.7983	1.25437
$727$	−24.4092	−0.905286	−0.452643	−	0.891692i	$-0.649519\pi$
$727$	−24.4092	−0.905286	−0.452643	+	0.891692i	$0.649519\pi$
$728$	2.62527	0.0972991
$729$	−10.2510	−0.379666
$730$	5.91955	0.219092
$731$	18.8902	0.698678
$732$	−2.32261	−0.0858462
$733$	−1.05303	−0.0388946	−0.0194473	−	0.999811i	$-0.506191\pi$
$733$	−1.05303	−0.0388946	−0.0194473	+	0.999811i	$0.506191\pi$
$734$	−33.8742	−1.25032
$735$	−0.332736	−0.0122732
$736$	4.00840	0.147752
$737$	2.51444	0.0926205
$738$	−16.3031	−0.600124
$739$	−1.40499	−0.0516834	−0.0258417	−	0.999666i	$-0.508227\pi$
$739$	−1.40499	−0.0516834	−0.0258417	+	0.999666i	$0.508227\pi$
$740$	9.02092	0.331616
$741$	−0.732053	−0.0268926
$742$	5.22701	0.191889
$743$	−9.58396	−0.351602	−0.175801	−	0.984426i	$-0.556251\pi$
$743$	−9.58396	−0.351602	−0.175801	+	0.984426i	$0.556251\pi$
$744$	3.08271	0.113018
$745$	−15.1295	−0.554302
$746$	−15.7122	−0.575266
$747$	2.82949	0.103526
$748$	−1.31931	−0.0482388
$749$	3.43811	0.125626
$750$	3.08271	0.112565
$751$	−9.05783	−0.330525	−0.165263	−	0.986250i	$-0.552847\pi$
$751$	−9.05783	−0.330525	−0.165263	+	0.986250i	$0.552847\pi$
$752$	−10.0962	−0.368170
$753$	−66.5926	−2.42677
$754$	−0.0461297	−0.00167994
$755$	−8.12239	−0.295604
$756$	28.3506	1.03110
$757$	38.3596	1.39420	0.697101	−	0.716973i	$-0.254473\pi$
$757$	38.3596	1.39420	0.697101	+	0.716973i	$0.254473\pi$
$758$	4.04448	0.146902
$759$	−2.35005	−0.0853014
$760$	0.237470	0.00861396
$761$	24.8063	0.899229	0.449614	−	0.893223i	$-0.351561\pi$
$761$	24.8063	0.899229	0.449614	+	0.893223i	$0.351561\pi$
$762$	−31.9391	−1.15703
$763$	1.05773	0.0382923
$764$	−10.6539	−0.385446
$765$	45.1124	1.63104
$766$	2.70970	0.0979055
$767$	−10.7728	−0.388982
$768$	−3.08271	−0.111238
$769$	8.41317	0.303387	0.151693	−	0.988428i	$-0.451527\pi$
$769$	8.41317	0.303387	0.151693	+	0.988428i	$0.451527\pi$
$770$	0.499284	0.0179929
$771$	−46.7290	−1.68290
$772$	−12.5653	−0.452236
$773$	31.5147	1.13350	0.566752	−	0.823888i	$-0.308200\pi$
$773$	31.5147	1.13350	0.566752	+	0.823888i	$0.308200\pi$
$774$	−17.7085	−0.636520
$775$	−1.00000	−0.0359211
$776$	8.17526	0.293475
$777$	−73.0060	−2.61908
$778$	14.2920	0.512392
$779$	0.595329	0.0213299
$780$	−3.08271	−0.110379
$781$	1.16839	0.0418083
$782$	−27.8065	−0.994358
$783$	−0.498159	−0.0178028
$784$	−0.107936	−0.00385486
$785$	−19.0833	−0.681112
$786$	31.3963	1.11987
$787$	−35.7119	−1.27299	−0.636495	−	0.771281i	$-0.719616\pi$
$787$	−35.7119	−1.27299	−0.636495	+	0.771281i	$0.719616\pi$
$788$	9.06106	0.322787
$789$	−4.44861	−0.158375
$790$	−0.817179	−0.0290739
$791$	−6.99305	−0.248644
$792$	1.23679	0.0439472
$793$	−0.753431	−0.0267551
$794$	−3.48803	−0.123785
$795$	−6.13778	−0.217685
$796$	25.5763	0.906527
$797$	20.0161	0.709006	0.354503	−	0.935055i	$-0.384650\pi$
$797$	20.0161	0.709006	0.354503	+	0.935055i	$0.384650\pi$
$798$	−1.92184	−0.0680324
$799$	70.0378	2.47776
$800$	1.00000	0.0353553
$801$	−7.09387	−0.250650
$802$	−20.2318	−0.714411
$803$	−1.12580	−0.0397286
$804$	−40.7569	−1.43739
$805$	10.5232	0.370893
$806$	1.00000	0.0352235
$807$	30.6480	1.07886
$808$	−5.96499	−0.209847
$809$	−14.1815	−0.498595	−0.249298	−	0.968427i	$-0.580200\pi$
$809$	−14.1815	−0.498595	−0.249298	+	0.968427i	$0.580200\pi$
$810$	−13.7812	−0.484221
$811$	12.3169	0.432505	0.216253	−	0.976337i	$-0.430617\pi$
$811$	12.3169	0.432505	0.216253	+	0.976337i	$0.430617\pi$
$812$	−0.121103	−0.00424988
$813$	−73.9322	−2.59291
$814$	−1.71563	−0.0601328
$815$	4.24035	0.148533
$816$	21.3849	0.748622
$817$	0.646652	0.0226235
$818$	26.7867	0.936576
$819$	17.0725	0.596560
$820$	2.50696	0.0875469
$821$	32.4098	1.13111	0.565554	−	0.824711i	$-0.308662\pi$
$821$	32.4098	1.13111	0.565554	+	0.824711i	$0.308662\pi$
$822$	−19.2168	−0.670264
$823$	−24.0884	−0.839670	−0.419835	−	0.907600i	$-0.637912\pi$
$823$	−24.0884	−0.839670	−0.419835	+	0.907600i	$0.637912\pi$
$824$	14.3243	0.499010
$825$	−0.586281	−0.0204117
$826$	−28.2815	−0.984039
$827$	−12.6164	−0.438716	−0.219358	−	0.975644i	$-0.570396\pi$
$827$	−12.6164	−0.438716	−0.219358	+	0.975644i	$0.570396\pi$
$828$	26.0671	0.905895
$829$	−39.0359	−1.35577	−0.677887	−	0.735166i	$-0.737104\pi$
$829$	−39.0359	−1.35577	−0.677887	+	0.735166i	$0.737104\pi$
$830$	−0.435097	−0.0151024
$831$	43.8360	1.52066
$832$	−1.00000	−0.0346688
$833$	0.748759	0.0259430
$834$	−11.9372	−0.413352
$835$	−13.5092	−0.467506
$836$	−0.0451629	−0.00156199
$837$	10.7991	0.373272
$838$	−37.0735	−1.28068
$839$	38.9273	1.34392	0.671959	−	0.740588i	$-0.265453\pi$
$839$	38.9273	1.34392	0.671959	+	0.740588i	$0.265453\pi$
$840$	−8.09297	−0.279234
$841$	−28.9979	−0.999927
$842$	8.02881	0.276691
$843$	−48.5926	−1.67362
$844$	8.79692	0.302802
$845$	−1.00000	−0.0344010
$846$	−65.6567	−2.25733
$847$	28.7831	0.988998
$848$	−1.99103	−0.0683723
$849$	56.1239	1.92617
$850$	−6.93705	−0.237939
$851$	−36.1595	−1.23953
$852$	−18.9386	−0.648827
$853$	−15.4680	−0.529615	−0.264808	−	0.964301i	$-0.585308\pi$
$853$	−15.4680	−0.529615	−0.264808	+	0.964301i	$0.585308\pi$
$854$	−1.97796	−0.0676845
$855$	1.54430	0.0528139
$856$	−1.30962	−0.0447618
$857$	23.9701	0.818804	0.409402	−	0.912354i	$-0.365737\pi$
$857$	23.9701	0.818804	0.409402	+	0.912354i	$0.365737\pi$
$858$	0.586281	0.0200153
$859$	10.2235	0.348823	0.174411	−	0.984673i	$-0.444198\pi$
$859$	10.2235	0.348823	0.174411	+	0.984673i	$0.444198\pi$
$860$	2.72309	0.0928564
$861$	−20.2888	−0.691439
$862$	−25.3589	−0.863726
$863$	21.6389	0.736598	0.368299	−	0.929707i	$-0.379940\pi$
$863$	21.6389	0.736598	0.368299	+	0.929707i	$0.379940\pi$
$864$	−10.7991	−0.367393
$865$	−1.08997	−0.0370600
$866$	−28.3882	−0.964670
$867$	−95.9422	−3.25837
$868$	2.62527	0.0891076
$869$	0.155414	0.00527206
$870$	0.142205	0.00482119
$871$	−13.2211	−0.447980
$872$	−0.402902	−0.0136440
$873$	53.1647	1.79935
$874$	−0.951877	−0.0321977
$875$	2.62527	0.0887505
$876$	18.2483	0.616552
$877$	44.3224	1.49666	0.748330	−	0.663327i	$-0.230856\pi$
$877$	44.3224	1.49666	0.748330	+	0.663327i	$0.230856\pi$
$878$	−18.2165	−0.614778
$879$	−42.1931	−1.42314
$880$	−0.190183	−0.00641108
$881$	43.6937	1.47208	0.736040	−	0.676938i	$-0.236694\pi$
$881$	43.6937	1.47208	0.736040	+	0.676938i	$0.236694\pi$
$882$	−0.701922	−0.0236349
$883$	23.6293	0.795189	0.397595	−	0.917561i	$-0.369845\pi$
$883$	23.6293	0.795189	0.397595	+	0.917561i	$0.369845\pi$
$884$	6.93705	0.233318
$885$	33.2094	1.11632
$886$	−19.5676	−0.657385
$887$	−26.3809	−0.885783	−0.442891	−	0.896575i	$-0.646047\pi$
$887$	−26.3809	−0.885783	−0.442891	+	0.896575i	$0.646047\pi$
$888$	27.8089	0.933206
$889$	−27.1997	−0.912250
$890$	1.09084	0.0365651
$891$	2.62095	0.0878052
$892$	−26.7938	−0.897124
$893$	2.39755	0.0802309
$894$	−46.6399	−1.55987
$895$	−12.1337	−0.405585
$896$	−2.62527	−0.0877043
$897$	12.3568	0.412580
$898$	−8.23469	−0.274795
$899$	−0.0461297	−0.00153851
$900$	6.50312	0.216771
$901$	13.8119	0.460141
$902$	−0.476783	−0.0158751
$903$	−22.0378	−0.733373
$904$	2.66374	0.0885947
$905$	−3.25448	−0.108183
$906$	−25.0390	−0.831865
$907$	−15.9224	−0.528696	−0.264348	−	0.964427i	$-0.585157\pi$
$907$	−15.9224	−0.528696	−0.264348	+	0.964427i	$0.585157\pi$
$908$	10.2446	0.339979
$909$	−38.7910	−1.28662
$910$	−2.62527	−0.0870270
$911$	−5.84421	−0.193627	−0.0968136	−	0.995303i	$-0.530865\pi$
$911$	−5.84421	−0.193627	−0.0968136	+	0.995303i	$0.530865\pi$
$912$	0.732053	0.0242407
$913$	0.0827483	0.00273857
$914$	−24.4634	−0.809178
$915$	2.32261	0.0767832
$916$	−11.9399	−0.394505
$917$	26.7375	0.882949
$918$	74.9139	2.47253
$919$	−39.4296	−1.30066	−0.650331	−	0.759651i	$-0.725370\pi$
$919$	−39.4296	−1.30066	−0.650331	+	0.759651i	$0.725370\pi$
$920$	−4.00840	−0.132153
$921$	−63.2527	−2.08425
$922$	−8.11358	−0.267207
$923$	−6.14349	−0.202216
$924$	1.53915	0.0506342
$925$	−9.02092	−0.296606
$926$	10.0863	0.331456
$927$	93.1526	3.05953
$928$	0.0461297	0.00151428
$929$	40.9322	1.34294	0.671471	−	0.741031i	$-0.265663\pi$
$929$	40.9322	1.34294	0.671471	+	0.741031i	$0.265663\pi$
$930$	−3.08271	−0.101086
$931$	0.0256316	0.000840043 0
$932$	25.8498	0.846739
$933$	58.4570	1.91379
$934$	25.0000	0.818026
$935$	1.31931	0.0431461
$936$	−6.50312	−0.212561
$937$	44.8535	1.46530	0.732651	−	0.680605i	$-0.238283\pi$
$937$	44.8535	1.46530	0.732651	+	0.680605i	$0.238283\pi$
$938$	−34.7091	−1.13329
$939$	48.7620	1.59129
$940$	10.0962	0.329302
$941$	2.31410	0.0754376	0.0377188	−	0.999288i	$-0.487991\pi$
$941$	2.31410	0.0754376	0.0377188	+	0.999288i	$0.487991\pi$
$942$	−58.8283	−1.91673
$943$	−10.0489	−0.327238
$944$	10.7728	0.350624
$945$	−28.3506	−0.922245
$946$	−0.517886	−0.0168379
$947$	9.66596	0.314102	0.157051	−	0.987591i	$-0.449801\pi$
$947$	9.66596	0.314102	0.157051	+	0.987591i	$0.449801\pi$
$948$	−2.51913	−0.0818175
$949$	5.91955	0.192157
$950$	−0.237470	−0.00770456
$951$	−97.4334	−3.15949
$952$	18.2117	0.590243
$953$	−37.1792	−1.20435	−0.602177	−	0.798363i	$-0.705700\pi$
$953$	−37.1792	−1.20435	−0.602177	+	0.798363i	$0.705700\pi$
$954$	−12.9479	−0.419204
$955$	10.6539	0.344753
$956$	−4.74305	−0.153401
$957$	−0.0270449	−0.000874239 0
$958$	−26.5870	−0.858986
$959$	−16.3653	−0.528462
$960$	3.08271	0.0994941
$961$	1.00000	0.0322581
$962$	9.02092	0.290846
$963$	−8.51660	−0.274444
$964$	−5.76720	−0.185749
$965$	12.5653	0.404492
$966$	32.4399	1.04374
$967$	−21.0756	−0.677744	−0.338872	−	0.940832i	$-0.610045\pi$
$967$	−21.0756	−0.677744	−0.338872	+	0.940832i	$0.610045\pi$
$968$	−10.9638	−0.352391
$969$	−5.07829	−0.163138
$970$	−8.17526	−0.262492
$971$	−23.1383	−0.742544	−0.371272	−	0.928524i	$-0.621078\pi$
$971$	−23.1383	−0.742544	−0.371272	+	0.928524i	$0.621078\pi$
$972$	−10.0861	−0.323512
$973$	−10.1659	−0.325903
$974$	−6.50732	−0.208508
$975$	3.08271	0.0987258
$976$	0.753431	0.0241167
$977$	57.3418	1.83453	0.917264	−	0.398280i	$-0.130393\pi$
$977$	57.3418	1.83453	0.917264	+	0.398280i	$0.130393\pi$
$978$	13.0718	0.417989
$979$	−0.207460	−0.00663045
$980$	0.107936	0.00344789
$981$	−2.62012	−0.0836540
$982$	−30.5048	−0.973447
$983$	30.1178	0.960610	0.480305	−	0.877102i	$-0.340526\pi$
$983$	30.1178	0.960610	0.480305	+	0.877102i	$0.340526\pi$
$984$	7.72824	0.246367
$985$	−9.06106	−0.288709
$986$	−0.320004	−0.0101910
$987$	−81.7082	−2.60080
$988$	0.237470	0.00755494
$989$	−10.9152	−0.347084
$990$	−1.23679	−0.0393076
$991$	30.2927	0.962280	0.481140	−	0.876644i	$-0.340223\pi$
$991$	30.2927	0.962280	0.481140	+	0.876644i	$0.340223\pi$
$992$	−1.00000	−0.0317500
$993$	−50.2298	−1.59399
$994$	−16.1284	−0.511560
$995$	−25.5763	−0.810822
$996$	−1.34128	−0.0425001
$997$	−26.1923	−0.829518	−0.414759	−	0.909931i	$-0.636134\pi$
$997$	−26.1923	−0.829518	−0.414759	+	0.909931i	$0.636134\pi$
$998$	12.5736	0.398012
$999$	97.4178	3.08216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.n.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.n.1.1	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.08271 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.08271 q^{6} -2.62527 q^{7} +1.00000 q^{8} +6.50312 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.08271 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.08271 q^{6} -2.62527 q^{7} +1.00000 q^{8} +6.50312 q^{9} -1.00000 q^{10} +0.190183 q^{11} -3.08271 q^{12} -1.00000 q^{13} -2.62527 q^{14} +3.08271 q^{15} +1.00000 q^{16} -6.93705 q^{17} +6.50312 q^{18} -0.237470 q^{19} -1.00000 q^{20} +8.09297 q^{21} +0.190183 q^{22} +4.00840 q^{23} -3.08271 q^{24} +1.00000 q^{25} -1.00000 q^{26} -10.7991 q^{27} -2.62527 q^{28} +0.0461297 q^{29} +3.08271 q^{30} -1.00000 q^{31} +1.00000 q^{32} -0.586281 q^{33} -6.93705 q^{34} +2.62527 q^{35} +6.50312 q^{36} -9.02092 q^{37} -0.237470 q^{38} +3.08271 q^{39} -1.00000 q^{40} -2.50696 q^{41} +8.09297 q^{42} -2.72309 q^{43} +0.190183 q^{44} -6.50312 q^{45} +4.00840 q^{46} -10.0962 q^{47} -3.08271 q^{48} -0.107936 q^{49} +1.00000 q^{50} +21.3849 q^{51} -1.00000 q^{52} -1.99103 q^{53} -10.7991 q^{54} -0.190183 q^{55} -2.62527 q^{56} +0.732053 q^{57} +0.0461297 q^{58} +10.7728 q^{59} +3.08271 q^{60} +0.753431 q^{61} -1.00000 q^{62} -17.0725 q^{63} +1.00000 q^{64} +1.00000 q^{65} -0.586281 q^{66} +13.2211 q^{67} -6.93705 q^{68} -12.3568 q^{69} +2.62527 q^{70} +6.14349 q^{71} +6.50312 q^{72} -5.91955 q^{73} -9.02092 q^{74} -3.08271 q^{75} -0.237470 q^{76} -0.499284 q^{77} +3.08271 q^{78} +0.817179 q^{79} -1.00000 q^{80} +13.7812 q^{81} -2.50696 q^{82} +0.435097 q^{83} +8.09297 q^{84} +6.93705 q^{85} -2.72309 q^{86} -0.142205 q^{87} +0.190183 q^{88} -1.09084 q^{89} -6.50312 q^{90} +2.62527 q^{91} +4.00840 q^{92} +3.08271 q^{93} -10.0962 q^{94} +0.237470 q^{95} -3.08271 q^{96} +8.17526 q^{97} -0.107936 q^{98} +1.23679 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} + q^{14} + q^{15} + 8 q^{16} - 5 q^{17} + 9 q^{18} + 2 q^{19} - 8 q^{20} + 17 q^{21} + 4 q^{22} + 4 q^{23} - q^{24} + 8 q^{25} - 8 q^{26} + 11 q^{27} + q^{28} + 11 q^{29} + q^{30} - 8 q^{31} + 8 q^{32} + 10 q^{33} - 5 q^{34} - q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} + q^{39} - 8 q^{40} + 10 q^{41} + 17 q^{42} + 19 q^{43} + 4 q^{44} - 9 q^{45} + 4 q^{46} + 11 q^{47} - q^{48} + 11 q^{49} + 8 q^{50} + 7 q^{51} - 8 q^{52} + 8 q^{53} + 11 q^{54} - 4 q^{55} + q^{56} - 11 q^{57} + 11 q^{58} + 28 q^{59} + q^{60} - 12 q^{61} - 8 q^{62} + 20 q^{63} + 8 q^{64} + 8 q^{65} + 10 q^{66} + 24 q^{67} - 5 q^{68} + 30 q^{69} - q^{70} + 18 q^{71} + 9 q^{72} - 3 q^{73} + 19 q^{74} - q^{75} + 2 q^{76} - 7 q^{77} + q^{78} + 22 q^{79} - 8 q^{80} + 24 q^{81} + 10 q^{82} + 17 q^{83} + 17 q^{84} + 5 q^{85} + 19 q^{86} + 11 q^{87} + 4 q^{88} + 17 q^{89} - 9 q^{90} - q^{91} + 4 q^{92} + q^{93} + 11 q^{94} - 2 q^{95} - q^{96} - 24 q^{97} + 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9 - 8 * q^10 + 4 * q^11 - q^12 - 8 * q^13 + q^14 + q^15 + 8 * q^16 - 5 * q^17 + 9 * q^18 + 2 * q^19 - 8 * q^20 + 17 * q^21 + 4 * q^22 + 4 * q^23 - q^24 + 8 * q^25 - 8 * q^26 + 11 * q^27 + q^28 + 11 * q^29 + q^30 - 8 * q^31 + 8 * q^32 + 10 * q^33 - 5 * q^34 - q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 + q^39 - 8 * q^40 + 10 * q^41 + 17 * q^42 + 19 * q^43 + 4 * q^44 - 9 * q^45 + 4 * q^46 + 11 * q^47 - q^48 + 11 * q^49 + 8 * q^50 + 7 * q^51 - 8 * q^52 + 8 * q^53 + 11 * q^54 - 4 * q^55 + q^56 - 11 * q^57 + 11 * q^58 + 28 * q^59 + q^60 - 12 * q^61 - 8 * q^62 + 20 * q^63 + 8 * q^64 + 8 * q^65 + 10 * q^66 + 24 * q^67 - 5 * q^68 + 30 * q^69 - q^70 + 18 * q^71 + 9 * q^72 - 3 * q^73 + 19 * q^74 - q^75 + 2 * q^76 - 7 * q^77 + q^78 + 22 * q^79 - 8 * q^80 + 24 * q^81 + 10 * q^82 + 17 * q^83 + 17 * q^84 + 5 * q^85 + 19 * q^86 + 11 * q^87 + 4 * q^88 + 17 * q^89 - 9 * q^90 - q^91 + 4 * q^92 + q^93 + 11 * q^94 - 2 * q^95 - q^96 - 24 * q^97 + 11 * q^98 + 23 * q^99

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