LMFDB - Embedded newform 4030.2.a.p.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 16x^{7} + 46x^{6} + 80x^{5} - 212x^{4} - 133x^{3} + 294x^{2} + 52x - 112 $ x^9 - 3x^8 - 16x^7 + 46x^6 + 80x^5 - 212x^4 - 133x^3 + 294x^2 + 52x - 112
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			$2.42233$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.42233 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.42233 q^{6} -2.05608 q^{7} -1.00000 q^{8} +2.86766 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.42233 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.42233 q^{6} -2.05608 q^{7} -1.00000 q^{8} +2.86766 q^{9} +1.00000 q^{10} +5.02015 q^{11} -2.42233 q^{12} -1.00000 q^{13} +2.05608 q^{14} +2.42233 q^{15} +1.00000 q^{16} +2.89533 q^{17} -2.86766 q^{18} +5.54063 q^{19} -1.00000 q^{20} +4.98050 q^{21} -5.02015 q^{22} -7.87732 q^{23} +2.42233 q^{24} +1.00000 q^{25} +1.00000 q^{26} +0.320566 q^{27} -2.05608 q^{28} +10.0342 q^{29} -2.42233 q^{30} +1.00000 q^{31} -1.00000 q^{32} -12.1604 q^{33} -2.89533 q^{34} +2.05608 q^{35} +2.86766 q^{36} -0.769049 q^{37} -5.54063 q^{38} +2.42233 q^{39} +1.00000 q^{40} -2.79038 q^{41} -4.98050 q^{42} +1.72439 q^{43} +5.02015 q^{44} -2.86766 q^{45} +7.87732 q^{46} -10.4197 q^{47} -2.42233 q^{48} -2.77253 q^{49} -1.00000 q^{50} -7.01342 q^{51} -1.00000 q^{52} +5.73411 q^{53} -0.320566 q^{54} -5.02015 q^{55} +2.05608 q^{56} -13.4212 q^{57} -10.0342 q^{58} +4.54182 q^{59} +2.42233 q^{60} -13.7541 q^{61} -1.00000 q^{62} -5.89615 q^{63} +1.00000 q^{64} +1.00000 q^{65} +12.1604 q^{66} +14.8174 q^{67} +2.89533 q^{68} +19.0814 q^{69} -2.05608 q^{70} -11.5182 q^{71} -2.86766 q^{72} +13.1624 q^{73} +0.769049 q^{74} -2.42233 q^{75} +5.54063 q^{76} -10.3218 q^{77} -2.42233 q^{78} -5.12166 q^{79} -1.00000 q^{80} -9.37950 q^{81} +2.79038 q^{82} -6.54133 q^{83} +4.98050 q^{84} -2.89533 q^{85} -1.72439 q^{86} -24.3062 q^{87} -5.02015 q^{88} -6.00252 q^{89} +2.86766 q^{90} +2.05608 q^{91} -7.87732 q^{92} -2.42233 q^{93} +10.4197 q^{94} -5.54063 q^{95} +2.42233 q^{96} +1.81088 q^{97} +2.77253 q^{98} +14.3961 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 - 9 * q^5 + 3 * q^6 - 3 * q^7 - 9 * q^8 + 14 * q^9	\( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9} + 9 q^{10} + 14 q^{11} - 3 q^{12} - 9 q^{13} + 3 q^{14} + 3 q^{15} + 9 q^{16} + q^{17} - 14 q^{18} + 6 q^{19} - 9 q^{20} + q^{21} - 14 q^{22} - 4 q^{23} + 3 q^{24} + 9 q^{25} + 9 q^{26} - 15 q^{27} - 3 q^{28} + 15 q^{29} - 3 q^{30} + 9 q^{31} - 9 q^{32} + 14 q^{33} - q^{34} + 3 q^{35} + 14 q^{36} - 9 q^{37} - 6 q^{38} + 3 q^{39} + 9 q^{40} + 18 q^{41} - q^{42} - 23 q^{43} + 14 q^{44} - 14 q^{45} + 4 q^{46} + 3 q^{47} - 3 q^{48} + 12 q^{49} - 9 q^{50} - 11 q^{51} - 9 q^{52} - 6 q^{53} + 15 q^{54} - 14 q^{55} + 3 q^{56} + 17 q^{57} - 15 q^{58} + 28 q^{59} + 3 q^{60} - 9 q^{62} + 12 q^{63} + 9 q^{64} + 9 q^{65} - 14 q^{66} - 16 q^{67} + q^{68} - 6 q^{69} - 3 q^{70} + 32 q^{71} - 14 q^{72} - 11 q^{73} + 9 q^{74} - 3 q^{75} + 6 q^{76} - 29 q^{77} - 3 q^{78} - 8 q^{79} - 9 q^{80} + 9 q^{81} - 18 q^{82} + 15 q^{83} + q^{84} - q^{85} + 23 q^{86} - 19 q^{87} - 14 q^{88} + 51 q^{89} + 14 q^{90} + 3 q^{91} - 4 q^{92} - 3 q^{93} - 3 q^{94} - 6 q^{95} + 3 q^{96} - 26 q^{97} - 12 q^{98} + 27 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 - 9 * q^5 + 3 * q^6 - 3 * q^7 - 9 * q^8 + 14 * q^9 + 9 * q^10 + 14 * q^11 - 3 * q^12 - 9 * q^13 + 3 * q^14 + 3 * q^15 + 9 * q^16 + q^17 - 14 * q^18 + 6 * q^19 - 9 * q^20 + q^21 - 14 * q^22 - 4 * q^23 + 3 * q^24 + 9 * q^25 + 9 * q^26 - 15 * q^27 - 3 * q^28 + 15 * q^29 - 3 * q^30 + 9 * q^31 - 9 * q^32 + 14 * q^33 - q^34 + 3 * q^35 + 14 * q^36 - 9 * q^37 - 6 * q^38 + 3 * q^39 + 9 * q^40 + 18 * q^41 - q^42 - 23 * q^43 + 14 * q^44 - 14 * q^45 + 4 * q^46 + 3 * q^47 - 3 * q^48 + 12 * q^49 - 9 * q^50 - 11 * q^51 - 9 * q^52 - 6 * q^53 + 15 * q^54 - 14 * q^55 + 3 * q^56 + 17 * q^57 - 15 * q^58 + 28 * q^59 + 3 * q^60 - 9 * q^62 + 12 * q^63 + 9 * q^64 + 9 * q^65 - 14 * q^66 - 16 * q^67 + q^68 - 6 * q^69 - 3 * q^70 + 32 * q^71 - 14 * q^72 - 11 * q^73 + 9 * q^74 - 3 * q^75 + 6 * q^76 - 29 * q^77 - 3 * q^78 - 8 * q^79 - 9 * q^80 + 9 * q^81 - 18 * q^82 + 15 * q^83 + q^84 - q^85 + 23 * q^86 - 19 * q^87 - 14 * q^88 + 51 * q^89 + 14 * q^90 + 3 * q^91 - 4 * q^92 - 3 * q^93 - 3 * q^94 - 6 * q^95 + 3 * q^96 - 26 * q^97 - 12 * q^98 + 27 * 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
$3$	−2.42233	−1.39853	−0.699265	−	0.714862i	$-0.746489\pi$
$3$	−2.42233	−1.39853	−0.699265	+	0.714862i	$0.746489\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.42233	0.988910
$7$	−2.05608	−0.777126	−0.388563	−	0.921422i	$-0.627028\pi$
$7$	−2.05608	−0.777126	−0.388563	+	0.921422i	$0.627028\pi$
$8$	−1.00000	−0.353553
$9$	2.86766	0.955887
$10$	1.00000	0.316228
$11$	5.02015	1.51363	0.756815	−	0.653629i	$-0.226754\pi$
$11$	5.02015	1.51363	0.756815	+	0.653629i	$0.226754\pi$
$12$	−2.42233	−0.699265
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.05608	0.549511
$15$	2.42233	0.625442
$16$	1.00000	0.250000
$17$	2.89533	0.702220	0.351110	−	0.936334i	$-0.385804\pi$
$17$	2.89533	0.702220	0.351110	+	0.936334i	$0.385804\pi$
$18$	−2.86766	−0.675914
$19$	5.54063	1.27111	0.635554	−	0.772056i	$-0.280772\pi$
$19$	5.54063	1.27111	0.635554	+	0.772056i	$0.280772\pi$
$20$	−1.00000	−0.223607
$21$	4.98050	1.08683
$22$	−5.02015	−1.07030
$23$	−7.87732	−1.64254	−0.821268	−	0.570543i	$-0.806733\pi$
$23$	−7.87732	−1.64254	−0.821268	+	0.570543i	$0.806733\pi$
$24$	2.42233	0.494455
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0.320566	0.0616929
$28$	−2.05608	−0.388563
$29$	10.0342	1.86331	0.931655	−	0.363345i	$-0.118365\pi$
$29$	10.0342	1.86331	0.931655	+	0.363345i	$0.118365\pi$
$30$	−2.42233	−0.442254
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−12.1604	−2.11686
$34$	−2.89533	−0.496544
$35$	2.05608	0.347541
$36$	2.86766	0.477944
$37$	−0.769049	−0.126431	−0.0632155	−	0.998000i	$-0.520136\pi$
$37$	−0.769049	−0.126431	−0.0632155	+	0.998000i	$0.520136\pi$
$38$	−5.54063	−0.898809
$39$	2.42233	0.387883
$40$	1.00000	0.158114
$41$	−2.79038	−0.435784	−0.217892	−	0.975973i	$-0.569918\pi$
$41$	−2.79038	−0.435784	−0.217892	+	0.975973i	$0.569918\pi$
$42$	−4.98050	−0.768507
$43$	1.72439	0.262967	0.131483	−	0.991318i	$-0.458026\pi$
$43$	1.72439	0.262967	0.131483	+	0.991318i	$0.458026\pi$
$44$	5.02015	0.756815
$45$	−2.86766	−0.427486
$46$	7.87732	1.16145
$47$	−10.4197	−1.51987	−0.759935	−	0.650000i	$-0.774769\pi$
$47$	−10.4197	−1.51987	−0.759935	+	0.650000i	$0.774769\pi$
$48$	−2.42233	−0.349633
$49$	−2.77253	−0.396076
$50$	−1.00000	−0.141421
$51$	−7.01342	−0.982076
$52$	−1.00000	−0.138675
$53$	5.73411	0.787640	0.393820	−	0.919188i	$-0.371153\pi$
$53$	5.73411	0.787640	0.393820	+	0.919188i	$0.371153\pi$
$54$	−0.320566	−0.0436235
$55$	−5.02015	−0.676916
$56$	2.05608	0.274755
$57$	−13.4212	−1.77768
$58$	−10.0342	−1.31756
$59$	4.54182	0.591295	0.295647	−	0.955297i	$-0.404465\pi$
$59$	4.54182	0.591295	0.295647	+	0.955297i	$0.404465\pi$
$60$	2.42233	0.312721
$61$	−13.7541	−1.76103	−0.880517	−	0.474014i	$-0.842805\pi$
$61$	−13.7541	−1.76103	−0.880517	+	0.474014i	$0.842805\pi$
$62$	−1.00000	−0.127000
$63$	−5.89615	−0.742844
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	12.1604	1.49685
$67$	14.8174	1.81024	0.905119	−	0.425159i	$-0.139782\pi$
$67$	14.8174	1.81024	0.905119	+	0.425159i	$0.139782\pi$
$68$	2.89533	0.351110
$69$	19.0814	2.29714
$70$	−2.05608	−0.245749
$71$	−11.5182	−1.36696	−0.683482	−	0.729967i	$-0.739535\pi$
$71$	−11.5182	−1.36696	−0.683482	+	0.729967i	$0.739535\pi$
$72$	−2.86766	−0.337957
$73$	13.1624	1.54054	0.770270	−	0.637718i	$-0.220121\pi$
$73$	13.1624	1.54054	0.770270	+	0.637718i	$0.220121\pi$
$74$	0.769049	0.0894001
$75$	−2.42233	−0.279706
$76$	5.54063	0.635554
$77$	−10.3218	−1.17628
$78$	−2.42233	−0.274274
$79$	−5.12166	−0.576232	−0.288116	−	0.957596i	$-0.593029\pi$
$79$	−5.12166	−0.576232	−0.288116	+	0.957596i	$0.593029\pi$
$80$	−1.00000	−0.111803
$81$	−9.37950	−1.04217
$82$	2.79038	0.308146
$83$	−6.54133	−0.718005	−0.359002	−	0.933337i	$-0.616883\pi$
$83$	−6.54133	−0.718005	−0.359002	+	0.933337i	$0.616883\pi$
$84$	4.98050	0.543417
$85$	−2.89533	−0.314042
$86$	−1.72439	−0.185945
$87$	−24.3062	−2.60589
$88$	−5.02015	−0.535149
$89$	−6.00252	−0.636265	−0.318133	−	0.948046i	$-0.603056\pi$
$89$	−6.00252	−0.636265	−0.318133	+	0.948046i	$0.603056\pi$
$90$	2.86766	0.302278
$91$	2.05608	0.215536
$92$	−7.87732	−0.821268
$93$	−2.42233	−0.251183
$94$	10.4197	1.07471
$95$	−5.54063	−0.568457
$96$	2.42233	0.247228
$97$	1.81088	0.183867	0.0919335	−	0.995765i	$-0.470695\pi$
$97$	1.81088	0.183867	0.0919335	+	0.995765i	$0.470695\pi$
$98$	2.77253	0.280068
$99$	14.3961	1.44686
$100$	1.00000	0.100000
$101$	12.4218	1.23602	0.618010	−	0.786170i	$-0.287939\pi$
$101$	12.4218	1.23602	0.618010	+	0.786170i	$0.287939\pi$
$102$	7.01342	0.694432
$103$	10.3261	1.01746	0.508732	−	0.860925i	$-0.330115\pi$
$103$	10.3261	1.01746	0.508732	+	0.860925i	$0.330115\pi$
$104$	1.00000	0.0980581
$105$	−4.98050	−0.486047
$106$	−5.73411	−0.556946
$107$	−1.23257	−0.119157	−0.0595786	−	0.998224i	$-0.518976\pi$
$107$	−1.23257	−0.119157	−0.0595786	+	0.998224i	$0.518976\pi$
$108$	0.320566	0.0308465
$109$	−11.7860	−1.12890	−0.564448	−	0.825468i	$-0.690911\pi$
$109$	−11.7860	−1.12890	−0.564448	+	0.825468i	$0.690911\pi$
$110$	5.02015	0.478652
$111$	1.86289	0.176817
$112$	−2.05608	−0.194281
$113$	8.31076	0.781810	0.390905	−	0.920431i	$-0.372162\pi$
$113$	8.31076	0.781810	0.390905	+	0.920431i	$0.372162\pi$
$114$	13.4212	1.25701
$115$	7.87732	0.734564
$116$	10.0342	0.931655
$117$	−2.86766	−0.265115
$118$	−4.54182	−0.418109
$119$	−5.95302	−0.545713
$120$	−2.42233	−0.221127
$121$	14.2019	1.29108
$122$	13.7541	1.24524
$123$	6.75920	0.609457
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	5.89615	0.525270
$127$	−17.2271	−1.52866	−0.764328	−	0.644828i	$-0.776929\pi$
$127$	−17.2271	−1.52866	−0.764328	+	0.644828i	$0.776929\pi$
$128$	−1.00000	−0.0883883
$129$	−4.17703	−0.367767
$130$	−1.00000	−0.0877058
$131$	2.67312	0.233551	0.116776	−	0.993158i	$-0.462744\pi$
$131$	2.67312	0.233551	0.116776	+	0.993158i	$0.462744\pi$
$132$	−12.1604	−1.05843
$133$	−11.3920	−0.987811
$134$	−14.8174	−1.28003
$135$	−0.320566	−0.0275899
$136$	−2.89533	−0.248272
$137$	−2.31346	−0.197652	−0.0988262	−	0.995105i	$-0.531509\pi$
$137$	−2.31346	−0.197652	−0.0988262	+	0.995105i	$0.531509\pi$
$138$	−19.0814	−1.62432
$139$	−15.3723	−1.30386	−0.651932	−	0.758278i	$-0.726041\pi$
$139$	−15.3723	−1.30386	−0.651932	+	0.758278i	$0.726041\pi$
$140$	2.05608	0.173771
$141$	25.2399	2.12558
$142$	11.5182	0.966590
$143$	−5.02015	−0.419806
$144$	2.86766	0.238972
$145$	−10.0342	−0.833297
$146$	−13.1624	−1.08933
$147$	6.71597	0.553924
$148$	−0.769049	−0.0632155
$149$	−2.40555	−0.197071	−0.0985353	−	0.995134i	$-0.531416\pi$
$149$	−2.40555	−0.197071	−0.0985353	+	0.995134i	$0.531416\pi$
$150$	2.42233	0.197782
$151$	−14.7926	−1.20380	−0.601901	−	0.798570i	$-0.705590\pi$
$151$	−14.7926	−1.20380	−0.601901	+	0.798570i	$0.705590\pi$
$152$	−5.54063	−0.449405
$153$	8.30282	0.671243
$154$	10.3218	0.831756
$155$	−1.00000	−0.0803219
$156$	2.42233	0.193941
$157$	7.11202	0.567601	0.283800	−	0.958883i	$-0.408405\pi$
$157$	7.11202	0.567601	0.283800	+	0.958883i	$0.408405\pi$
$158$	5.12166	0.407458
$159$	−13.8899	−1.10154
$160$	1.00000	0.0790569
$161$	16.1964	1.27646
$162$	9.37950	0.736923
$163$	7.12027	0.557702	0.278851	−	0.960334i	$-0.410046\pi$
$163$	7.12027	0.557702	0.278851	+	0.960334i	$0.410046\pi$
$164$	−2.79038	−0.217892
$165$	12.1604	0.946688
$166$	6.54133	0.507706
$167$	21.2215	1.64217	0.821083	−	0.570809i	$-0.193370\pi$
$167$	21.2215	1.64217	0.821083	+	0.570809i	$0.193370\pi$
$168$	−4.98050	−0.384254
$169$	1.00000	0.0769231
$170$	2.89533	0.222061
$171$	15.8887	1.21504
$172$	1.72439	0.131483
$173$	12.1871	0.926568	0.463284	−	0.886210i	$-0.346671\pi$
$173$	12.1871	0.926568	0.463284	+	0.886210i	$0.346671\pi$
$174$	24.3062	1.84265
$175$	−2.05608	−0.155425
$176$	5.02015	0.378408
$177$	−11.0018	−0.826944
$178$	6.00252	0.449908
$179$	5.71537	0.427187	0.213593	−	0.976923i	$-0.431483\pi$
$179$	5.71537	0.427187	0.213593	+	0.976923i	$0.431483\pi$
$180$	−2.86766	−0.213743
$181$	−2.23387	−0.166042	−0.0830211	−	0.996548i	$-0.526457\pi$
$181$	−2.23387	−0.166042	−0.0830211	+	0.996548i	$0.526457\pi$
$182$	−2.05608	−0.152407
$183$	33.3170	2.46286
$184$	7.87732	0.580724
$185$	0.769049	0.0565416
$186$	2.42233	0.177614
$187$	14.5350	1.06290
$188$	−10.4197	−0.759935
$189$	−0.659109	−0.0479431
$190$	5.54063	0.401960
$191$	0.324835	0.0235043	0.0117521	−	0.999931i	$-0.496259\pi$
$191$	0.324835	0.0235043	0.0117521	+	0.999931i	$0.496259\pi$
$192$	−2.42233	−0.174816
$193$	22.9217	1.64994	0.824971	−	0.565175i	$-0.191191\pi$
$193$	22.9217	1.64994	0.824971	+	0.565175i	$0.191191\pi$
$194$	−1.81088	−0.130014
$195$	−2.42233	−0.173466
$196$	−2.77253	−0.198038
$197$	2.96938	0.211559	0.105780	−	0.994390i	$-0.466266\pi$
$197$	2.96938	0.211559	0.105780	+	0.994390i	$0.466266\pi$
$198$	−14.3961	−1.02308
$199$	10.2069	0.723546	0.361773	−	0.932266i	$-0.382171\pi$
$199$	10.2069	0.723546	0.361773	+	0.932266i	$0.382171\pi$
$200$	−1.00000	−0.0707107
$201$	−35.8927	−2.53167
$202$	−12.4218	−0.873998
$203$	−20.6312	−1.44803
$204$	−7.01342	−0.491038
$205$	2.79038	0.194888
$206$	−10.3261	−0.719455
$207$	−22.5895	−1.57008
$208$	−1.00000	−0.0693375
$209$	27.8148	1.92399
$210$	4.98050	0.343687
$211$	8.45075	0.581774	0.290887	−	0.956757i	$-0.406050\pi$
$211$	8.45075	0.581774	0.290887	+	0.956757i	$0.406050\pi$
$212$	5.73411	0.393820
$213$	27.9010	1.91174
$214$	1.23257	0.0842568
$215$	−1.72439	−0.117602
$216$	−0.320566	−0.0218117
$217$	−2.05608	−0.139576
$218$	11.7860	0.798251
$219$	−31.8836	−2.15449
$220$	−5.02015	−0.338458
$221$	−2.89533	−0.194761
$222$	−1.86289	−0.125029
$223$	−19.1113	−1.27979	−0.639893	−	0.768464i	$-0.721021\pi$
$223$	−19.1113	−1.27979	−0.639893	+	0.768464i	$0.721021\pi$
$224$	2.05608	0.137378
$225$	2.86766	0.191177
$226$	−8.31076	−0.552823
$227$	−22.5761	−1.49843	−0.749216	−	0.662326i	$-0.769569\pi$
$227$	−22.5761	−1.49843	−0.749216	+	0.662326i	$0.769569\pi$
$228$	−13.4212	−0.888842
$229$	−1.49592	−0.0988529	−0.0494264	−	0.998778i	$-0.515739\pi$
$229$	−1.49592	−0.0988529	−0.0494264	+	0.998778i	$0.515739\pi$
$230$	−7.87732	−0.519415
$231$	25.0028	1.64507
$232$	−10.0342	−0.658779
$233$	1.60166	0.104928	0.0524642	−	0.998623i	$-0.483292\pi$
$233$	1.60166	0.104928	0.0524642	+	0.998623i	$0.483292\pi$
$234$	2.86766	0.187465
$235$	10.4197	0.679706
$236$	4.54182	0.295647
$237$	12.4063	0.805878
$238$	5.95302	0.385877
$239$	22.7408	1.47098	0.735491	−	0.677534i	$-0.236951\pi$
$239$	22.7408	1.47098	0.735491	+	0.677534i	$0.236951\pi$
$240$	2.42233	0.156360
$241$	−0.00217295	−0.000139972 0	−6.99860e−5	−	1.00000i	$-0.500022\pi$
$241$	−0.00217295	−0.000139972 0	−6.99860e−5		1.00000i	$0.500022\pi$
$242$	−14.2019	−0.912930
$243$	21.7585	1.39581
$244$	−13.7541	−0.880517
$245$	2.77253	0.177131
$246$	−6.75920	−0.430951
$247$	−5.54063	−0.352542
$248$	−1.00000	−0.0635001
$249$	15.8452	1.00415
$250$	1.00000	0.0632456
$251$	−14.9776	−0.945380	−0.472690	−	0.881229i	$-0.656717\pi$
$251$	−14.9776	−0.945380	−0.472690	+	0.881229i	$0.656717\pi$
$252$	−5.89615	−0.371422
$253$	−39.5453	−2.48619
$254$	17.2271	1.08092
$255$	7.01342	0.439198
$256$	1.00000	0.0625000
$257$	−11.6719	−0.728072	−0.364036	−	0.931385i	$-0.618601\pi$
$257$	−11.6719	−0.728072	−0.364036	+	0.931385i	$0.618601\pi$
$258$	4.17703	0.260050
$259$	1.58123	0.0982527
$260$	1.00000	0.0620174
$261$	28.7748	1.78111
$262$	−2.67312	−0.165146
$263$	22.8152	1.40685	0.703424	−	0.710771i	$-0.251654\pi$
$263$	22.8152	1.40685	0.703424	+	0.710771i	$0.251654\pi$
$264$	12.1604	0.748423
$265$	−5.73411	−0.352243
$266$	11.3920	0.698488
$267$	14.5400	0.889836
$268$	14.8174	0.905119
$269$	28.6660	1.74780	0.873900	−	0.486106i	$-0.161583\pi$
$269$	28.6660	1.74780	0.873900	+	0.486106i	$0.161583\pi$
$270$	0.320566	0.0195090
$271$	−13.6926	−0.831767	−0.415883	−	0.909418i	$-0.636528\pi$
$271$	−13.6926	−0.831767	−0.415883	+	0.909418i	$0.636528\pi$
$272$	2.89533	0.175555
$273$	−4.98050	−0.301433
$274$	2.31346	0.139761
$275$	5.02015	0.302726
$276$	19.0814	1.14857
$277$	7.89954	0.474638	0.237319	−	0.971432i	$-0.423731\pi$
$277$	7.89954	0.474638	0.237319	+	0.971432i	$0.423731\pi$
$278$	15.3723	0.921971
$279$	2.86766	0.171682
$280$	−2.05608	−0.122874
$281$	31.1656	1.85919	0.929593	−	0.368588i	$-0.120159\pi$
$281$	31.1656	1.85919	0.929593	+	0.368588i	$0.120159\pi$
$282$	−25.2399	−1.50301
$283$	−18.0929	−1.07551	−0.537756	−	0.843101i	$-0.680728\pi$
$283$	−18.0929	−1.07551	−0.537756	+	0.843101i	$0.680728\pi$
$284$	−11.5182	−0.683482
$285$	13.4212	0.795004
$286$	5.02015	0.296847
$287$	5.73724	0.338659
$288$	−2.86766	−0.168979
$289$	−8.61709	−0.506887
$290$	10.0342	0.589230
$291$	−4.38654	−0.257144
$292$	13.1624	0.770270
$293$	5.40439	0.315728	0.157864	−	0.987461i	$-0.449539\pi$
$293$	5.40439	0.315728	0.157864	+	0.987461i	$0.449539\pi$
$294$	−6.71597	−0.391684
$295$	−4.54182	−0.264435
$296$	0.769049	0.0447001
$297$	1.60929	0.0933803
$298$	2.40555	0.139350
$299$	7.87732	0.455557
$300$	−2.42233	−0.139853
$301$	−3.54548	−0.204358
$302$	14.7926	0.851217
$303$	−30.0898	−1.72861
$304$	5.54063	0.317777
$305$	13.7541	0.787559
$306$	−8.30282	−0.474640
$307$	2.30129	0.131342	0.0656709	−	0.997841i	$-0.479081\pi$
$307$	2.30129	0.131342	0.0656709	+	0.997841i	$0.479081\pi$
$308$	−10.3218	−0.588141
$309$	−25.0132	−1.42295
$310$	1.00000	0.0567962
$311$	9.59681	0.544185	0.272093	−	0.962271i	$-0.412284\pi$
$311$	9.59681	0.544185	0.272093	+	0.962271i	$0.412284\pi$
$312$	−2.42233	−0.137137
$313$	5.54079	0.313184	0.156592	−	0.987663i	$-0.449949\pi$
$313$	5.54079	0.313184	0.156592	+	0.987663i	$0.449949\pi$
$314$	−7.11202	−0.401354
$315$	5.89615	0.332210
$316$	−5.12166	−0.288116
$317$	0.799728	0.0449172	0.0224586	−	0.999748i	$-0.492851\pi$
$317$	0.799728	0.0449172	0.0224586	+	0.999748i	$0.492851\pi$
$318$	13.8899	0.778906
$319$	50.3733	2.82036
$320$	−1.00000	−0.0559017
$321$	2.98569	0.166645
$322$	−16.1964	−0.902591
$323$	16.0419	0.892597
$324$	−9.37950	−0.521083
$325$	−1.00000	−0.0554700
$326$	−7.12027	−0.394355
$327$	28.5496	1.57880
$328$	2.79038	0.154073
$329$	21.4237	1.18113
$330$	−12.1604	−0.669410
$331$	−6.59439	−0.362460	−0.181230	−	0.983441i	$-0.558008\pi$
$331$	−6.59439	−0.362460	−0.181230	+	0.983441i	$0.558008\pi$
$332$	−6.54133	−0.359002
$333$	−2.20537	−0.120854
$334$	−21.2215	−1.16119
$335$	−14.8174	−0.809563
$336$	4.98050	0.271708
$337$	23.8880	1.30126	0.650631	−	0.759394i	$-0.274504\pi$
$337$	23.8880	1.30126	0.650631	+	0.759394i	$0.274504\pi$
$338$	−1.00000	−0.0543928
$339$	−20.1314	−1.09339
$340$	−2.89533	−0.157021
$341$	5.02015	0.271856
$342$	−15.8887	−0.859160
$343$	20.0931	1.08493
$344$	−1.72439	−0.0929727
$345$	−19.0814	−1.02731
$346$	−12.1871	−0.655183
$347$	−9.59538	−0.515107	−0.257553	−	0.966264i	$-0.582916\pi$
$347$	−9.59538	−0.515107	−0.257553	+	0.966264i	$0.582916\pi$
$348$	−24.3062	−1.30295
$349$	18.9612	1.01497	0.507485	−	0.861660i	$-0.330575\pi$
$349$	18.9612	1.01497	0.507485	+	0.861660i	$0.330575\pi$
$350$	2.05608	0.109902
$351$	−0.320566	−0.0171105
$352$	−5.02015	−0.267575
$353$	4.57335	0.243415	0.121708	−	0.992566i	$-0.461163\pi$
$353$	4.57335	0.243415	0.121708	+	0.992566i	$0.461163\pi$
$354$	11.0018	0.584738
$355$	11.5182	0.611325
$356$	−6.00252	−0.318133
$357$	14.4202	0.763196
$358$	−5.71537	−0.302067
$359$	12.1852	0.643113	0.321556	−	0.946890i	$-0.395794\pi$
$359$	12.1852	0.643113	0.321556	+	0.946890i	$0.395794\pi$
$360$	2.86766	0.151139
$361$	11.6986	0.615716
$362$	2.23387	0.117410
$363$	−34.4015	−1.80561
$364$	2.05608	0.107768
$365$	−13.1624	−0.688951
$366$	−33.3170	−1.74151
$367$	−23.2062	−1.21135	−0.605676	−	0.795711i	$-0.707097\pi$
$367$	−23.2062	−1.21135	−0.605676	+	0.795711i	$0.707097\pi$
$368$	−7.87732	−0.410634
$369$	−8.00186	−0.416560
$370$	−0.769049	−0.0399810
$371$	−11.7898	−0.612095
$372$	−2.42233	−0.125592
$373$	−10.4209	−0.539573	−0.269786	−	0.962920i	$-0.586953\pi$
$373$	−10.4209	−0.539573	−0.269786	+	0.962920i	$0.586953\pi$
$374$	−14.5350	−0.751585
$375$	2.42233	0.125088
$376$	10.4197	0.537355
$377$	−10.0342	−0.516789
$378$	0.659109	0.0339009
$379$	26.1796	1.34476	0.672378	−	0.740208i	$-0.265273\pi$
$379$	26.1796	1.34476	0.672378	+	0.740208i	$0.265273\pi$
$380$	−5.54063	−0.284228
$381$	41.7296	2.13787
$382$	−0.324835	−0.0166200
$383$	12.5008	0.638761	0.319380	−	0.947627i	$-0.396525\pi$
$383$	12.5008	0.638761	0.319380	+	0.947627i	$0.396525\pi$
$384$	2.42233	0.123614
$385$	10.3218	0.526049
$386$	−22.9217	−1.16669
$387$	4.94496	0.251366
$388$	1.81088	0.0919335
$389$	12.3493	0.626135	0.313068	−	0.949731i	$-0.398643\pi$
$389$	12.3493	0.626135	0.313068	+	0.949731i	$0.398643\pi$
$390$	2.42233	0.122659
$391$	−22.8074	−1.15342
$392$	2.77253	0.140034
$393$	−6.47516	−0.326628
$394$	−2.96938	−0.149595
$395$	5.12166	0.257699
$396$	14.3961	0.723430
$397$	−25.9935	−1.30457	−0.652287	−	0.757972i	$-0.726190\pi$
$397$	−25.9935	−1.30457	−0.652287	+	0.757972i	$0.726190\pi$
$398$	−10.2069	−0.511625
$399$	27.5951	1.38148
$400$	1.00000	0.0500000
$401$	23.7366	1.18535	0.592675	−	0.805442i	$-0.298072\pi$
$401$	23.7366	1.18535	0.592675	+	0.805442i	$0.298072\pi$
$402$	35.8927	1.79016
$403$	−1.00000	−0.0498135
$404$	12.4218	0.618010
$405$	9.37950	0.466071
$406$	20.6312	1.02391
$407$	−3.86074	−0.191370
$408$	7.01342	0.347216
$409$	30.6514	1.51561	0.757807	−	0.652478i	$-0.226271\pi$
$409$	30.6514	1.51561	0.757807	+	0.652478i	$0.226271\pi$
$410$	−2.79038	−0.137807
$411$	5.60396	0.276423
$412$	10.3261	0.508732
$413$	−9.33835	−0.459510
$414$	22.5895	1.11021
$415$	6.54133	0.321101
$416$	1.00000	0.0490290
$417$	37.2368	1.82349
$418$	−27.8148	−1.36047
$419$	13.5505	0.661985	0.330993	−	0.943633i	$-0.392616\pi$
$419$	13.5505	0.661985	0.330993	+	0.943633i	$0.392616\pi$
$420$	−4.98050	−0.243023
$421$	2.05345	0.100079	0.0500394	−	0.998747i	$-0.484065\pi$
$421$	2.05345	0.100079	0.0500394	+	0.998747i	$0.484065\pi$
$422$	−8.45075	−0.411376
$423$	−29.8802	−1.45282
$424$	−5.73411	−0.278473
$425$	2.89533	0.140444
$426$	−27.9010	−1.35181
$427$	28.2796	1.36855
$428$	−1.23257	−0.0595786
$429$	12.1604	0.587111
$430$	1.72439	0.0831573
$431$	−5.84842	−0.281709	−0.140854	−	0.990030i	$-0.544985\pi$
$431$	−5.84842	−0.281709	−0.140854	+	0.990030i	$0.544985\pi$
$432$	0.320566	0.0154232
$433$	−6.23028	−0.299408	−0.149704	−	0.988731i	$-0.547832\pi$
$433$	−6.23028	−0.299408	−0.149704	+	0.988731i	$0.547832\pi$
$434$	2.05608	0.0986950
$435$	24.3062	1.16539
$436$	−11.7860	−0.564448
$437$	−43.6453	−2.08784
$438$	31.8836	1.52346
$439$	−24.1985	−1.15493	−0.577466	−	0.816415i	$-0.695958\pi$
$439$	−24.1985	−1.15493	−0.577466	+	0.816415i	$0.695958\pi$
$440$	5.02015	0.239326
$441$	−7.95068	−0.378604
$442$	2.89533	0.137717
$443$	39.1603	1.86056	0.930282	−	0.366845i	$-0.119562\pi$
$443$	39.1603	1.86056	0.930282	+	0.366845i	$0.119562\pi$
$444$	1.86289	0.0884087
$445$	6.00252	0.284547
$446$	19.1113	0.904945
$447$	5.82704	0.275609
$448$	−2.05608	−0.0971407
$449$	−33.5736	−1.58443	−0.792217	−	0.610239i	$-0.791073\pi$
$449$	−33.5736	−1.58443	−0.792217	+	0.610239i	$0.791073\pi$
$450$	−2.86766	−0.135183
$451$	−14.0081	−0.659616
$452$	8.31076	0.390905
$453$	35.8324	1.68355
$454$	22.5761	1.05955
$455$	−2.05608	−0.0963906
$456$	13.4212	0.628506
$457$	18.6702	0.873355	0.436678	−	0.899618i	$-0.356155\pi$
$457$	18.6702	0.873355	0.436678	+	0.899618i	$0.356155\pi$
$458$	1.49592	0.0698995
$459$	0.928143	0.0433220
$460$	7.87732	0.367282
$461$	19.4343	0.905146	0.452573	−	0.891727i	$-0.350506\pi$
$461$	19.4343	0.905146	0.452573	+	0.891727i	$0.350506\pi$
$462$	−25.0028	−1.16324
$463$	33.5442	1.55893	0.779466	−	0.626445i	$-0.215491\pi$
$463$	33.5442	1.55893	0.779466	+	0.626445i	$0.215491\pi$
$464$	10.0342	0.465827
$465$	2.42233	0.112333
$466$	−1.60166	−0.0741955
$467$	1.54920	0.0716884	0.0358442	−	0.999357i	$-0.488588\pi$
$467$	1.54920	0.0716884	0.0358442	+	0.999357i	$0.488588\pi$
$468$	−2.86766	−0.132558
$469$	−30.4658	−1.40678
$470$	−10.4197	−0.480625
$471$	−17.2276	−0.793807
$472$	−4.54182	−0.209054
$473$	8.65667	0.398034
$474$	−12.4063	−0.569842
$475$	5.54063	0.254222
$476$	−5.95302	−0.272856
$477$	16.4435	0.752895
$478$	−22.7408	−1.04014
$479$	−15.2803	−0.698173	−0.349087	−	0.937090i	$-0.613508\pi$
$479$	−15.2803	−0.698173	−0.349087	+	0.937090i	$0.613508\pi$
$480$	−2.42233	−0.110564
$481$	0.769049	0.0350656
$482$	0.00217295	9.89752e−5 0
$483$	−39.2330	−1.78516
$484$	14.2019	0.645539
$485$	−1.81088	−0.0822278
$486$	−21.7585	−0.986986
$487$	−15.7402	−0.713256	−0.356628	−	0.934246i	$-0.616074\pi$
$487$	−15.7402	−0.713256	−0.356628	+	0.934246i	$0.616074\pi$
$488$	13.7541	0.622620
$489$	−17.2476	−0.779964
$490$	−2.77253	−0.125250
$491$	4.68458	0.211412	0.105706	−	0.994397i	$-0.466290\pi$
$491$	4.68458	0.211412	0.105706	+	0.994397i	$0.466290\pi$
$492$	6.75920	0.304728
$493$	29.0524	1.30845
$494$	5.54063	0.249285
$495$	−14.3961	−0.647056
$496$	1.00000	0.0449013
$497$	23.6825	1.06230
$498$	−15.8452	−0.710042
$499$	31.6582	1.41722	0.708608	−	0.705603i	$-0.249324\pi$
$499$	31.6582	1.41722	0.708608	+	0.705603i	$0.249324\pi$
$500$	−1.00000	−0.0447214
$501$	−51.4053	−2.29662
$502$	14.9776	0.668485
$503$	−25.5374	−1.13866	−0.569328	−	0.822111i	$-0.692797\pi$
$503$	−25.5374	−1.13866	−0.569328	+	0.822111i	$0.692797\pi$
$504$	5.89615	0.262635
$505$	−12.4218	−0.552765
$506$	39.5453	1.75800
$507$	−2.42233	−0.107579
$508$	−17.2271	−0.764328
$509$	12.4705	0.552745	0.276373	−	0.961051i	$-0.410868\pi$
$509$	12.4705	0.552745	0.276373	+	0.961051i	$0.410868\pi$
$510$	−7.01342	−0.310560
$511$	−27.0629	−1.19719
$512$	−1.00000	−0.0441942
$513$	1.77614	0.0784184
$514$	11.6719	0.514824
$515$	−10.3261	−0.455023
$516$	−4.17703	−0.183883
$517$	−52.3084	−2.30052
$518$	−1.58123	−0.0694751
$519$	−29.5211	−1.29583
$520$	−1.00000	−0.0438529
$521$	19.0823	0.836010	0.418005	−	0.908445i	$-0.362729\pi$
$521$	19.0823	0.836010	0.418005	+	0.908445i	$0.362729\pi$
$522$	−28.7748	−1.25944
$523$	−37.6016	−1.64420	−0.822102	−	0.569341i	$-0.807199\pi$
$523$	−37.6016	−1.64420	−0.822102	+	0.569341i	$0.807199\pi$
$524$	2.67312	0.116776
$525$	4.98050	0.217367
$526$	−22.8152	−0.994792
$527$	2.89533	0.126122
$528$	−12.1604	−0.529215
$529$	39.0522	1.69792
$530$	5.73411	0.249074
$531$	13.0244	0.565211
$532$	−11.3920	−0.493905
$533$	2.79038	0.120865
$534$	−14.5400	−0.629209
$535$	1.23257	0.0532887
$536$	−14.8174	−0.640016
$537$	−13.8445	−0.597434
$538$	−28.6660	−1.23588
$539$	−13.9185	−0.599513
$540$	−0.320566	−0.0137950
$541$	20.0627	0.862562	0.431281	−	0.902218i	$-0.358062\pi$
$541$	20.0627	0.862562	0.431281	+	0.902218i	$0.358062\pi$
$542$	13.6926	0.588148
$543$	5.41116	0.232215
$544$	−2.89533	−0.124136
$545$	11.7860	0.504858
$546$	4.98050	0.213146
$547$	−14.2004	−0.607165	−0.303582	−	0.952805i	$-0.598183\pi$
$547$	−14.2004	−0.607165	−0.303582	+	0.952805i	$0.598183\pi$
$548$	−2.31346	−0.0988262
$549$	−39.4422	−1.68335
$550$	−5.02015	−0.214060
$551$	55.5960	2.36847
$552$	−19.0814	−0.812160
$553$	10.5306	0.447805
$554$	−7.89954	−0.335619
$555$	−1.86289	−0.0790752
$556$	−15.3723	−0.651932
$557$	16.3961	0.694726	0.347363	−	0.937731i	$-0.387077\pi$
$557$	16.3961	0.694726	0.347363	+	0.937731i	$0.387077\pi$
$558$	−2.86766	−0.121398
$559$	−1.72439	−0.0729338
$560$	2.05608	0.0868853
$561$	−35.2084	−1.48650
$562$	−31.1656	−1.31464
$563$	9.45054	0.398293	0.199146	−	0.979970i	$-0.436183\pi$
$563$	9.45054	0.398293	0.199146	+	0.979970i	$0.436183\pi$
$564$	25.2399	1.06279
$565$	−8.31076	−0.349636
$566$	18.0929	0.760501
$567$	19.2850	0.809894
$568$	11.5182	0.483295
$569$	28.3360	1.18791	0.593954	−	0.804499i	$-0.297566\pi$
$569$	28.3360	1.18791	0.593954	+	0.804499i	$0.297566\pi$
$570$	−13.4212	−0.562153
$571$	−0.0453818	−0.00189917	−0.000949585	−	1.00000i	$-0.500302\pi$
$571$	−0.0453818	−0.00189917	−0.000949585		1.00000i	$0.500302\pi$
$572$	−5.02015	−0.209903
$573$	−0.786857	−0.0328714
$574$	−5.73724	−0.239468
$575$	−7.87732	−0.328507
$576$	2.86766	0.119486
$577$	26.4025	1.09915	0.549576	−	0.835444i	$-0.314789\pi$
$577$	26.4025	1.09915	0.549576	+	0.835444i	$0.314789\pi$
$578$	8.61709	0.358424
$579$	−55.5239	−2.30749
$580$	−10.0342	−0.416649
$581$	13.4495	0.557980
$582$	4.38654	0.181828
$583$	28.7861	1.19220
$584$	−13.1624	−0.544663
$585$	2.86766	0.118563
$586$	−5.40439	−0.223253
$587$	27.9726	1.15455	0.577277	−	0.816548i	$-0.304115\pi$
$587$	27.9726	1.15455	0.577277	+	0.816548i	$0.304115\pi$
$588$	6.71597	0.276962
$589$	5.54063	0.228298
$590$	4.54182	0.186984
$591$	−7.19280	−0.295872
$592$	−0.769049	−0.0316077
$593$	25.6946	1.05515	0.527576	−	0.849508i	$-0.323101\pi$
$593$	25.6946	1.05515	0.527576	+	0.849508i	$0.323101\pi$
$594$	−1.60929	−0.0660299
$595$	5.95302	0.244050
$596$	−2.40555	−0.0985353
$597$	−24.7244	−1.01190
$598$	−7.87732	−0.322128
$599$	8.20592	0.335285	0.167642	−	0.985848i	$-0.446385\pi$
$599$	8.20592	0.335285	0.167642	+	0.985848i	$0.446385\pi$
$600$	2.42233	0.0988910
$601$	8.52717	0.347831	0.173915	−	0.984761i	$-0.444358\pi$
$601$	8.52717	0.347831	0.173915	+	0.984761i	$0.444358\pi$
$602$	3.54548	0.144503
$603$	42.4914	1.73038
$604$	−14.7926	−0.601901
$605$	−14.2019	−0.577388
$606$	30.0898	1.22231
$607$	−4.05029	−0.164396	−0.0821982	−	0.996616i	$-0.526194\pi$
$607$	−4.05029	−0.164396	−0.0821982	+	0.996616i	$0.526194\pi$
$608$	−5.54063	−0.224702
$609$	49.9754	2.02511
$610$	−13.7541	−0.556888
$611$	10.4197	0.421536
$612$	8.30282	0.335621
$613$	−37.5403	−1.51624	−0.758120	−	0.652116i	$-0.773882\pi$
$613$	−37.5403	−1.51624	−0.758120	+	0.652116i	$0.773882\pi$
$614$	−2.30129	−0.0928726
$615$	−6.75920	−0.272557
$616$	10.3218	0.415878
$617$	26.3326	1.06011	0.530056	−	0.847963i	$-0.322171\pi$
$617$	26.3326	1.06011	0.530056	+	0.847963i	$0.322171\pi$
$618$	25.0132	1.00618
$619$	−16.8705	−0.678082	−0.339041	−	0.940772i	$-0.610102\pi$
$619$	−16.8705	−0.678082	−0.339041	+	0.940772i	$0.610102\pi$
$620$	−1.00000	−0.0401610
$621$	−2.52520	−0.101333
$622$	−9.59681	−0.384797
$623$	12.3417	0.494458
$624$	2.42233	0.0969706
$625$	1.00000	0.0400000
$626$	−5.54079	−0.221454
$627$	−67.3765	−2.69076
$628$	7.11202	0.283800
$629$	−2.22665	−0.0887823
$630$	−5.89615	−0.234908
$631$	1.44255	0.0574268	0.0287134	−	0.999588i	$-0.490859\pi$
$631$	1.44255	0.0574268	0.0287134	+	0.999588i	$0.490859\pi$
$632$	5.12166	0.203729
$633$	−20.4705	−0.813629
$634$	−0.799728	−0.0317613
$635$	17.2271	0.683636
$636$	−13.8899	−0.550769
$637$	2.77253	0.109852
$638$	−50.3733	−1.99430
$639$	−33.0304	−1.30666
$640$	1.00000	0.0395285
$641$	19.9390	0.787544	0.393772	−	0.919208i	$-0.371170\pi$
$641$	19.9390	0.787544	0.393772	+	0.919208i	$0.371170\pi$
$642$	−2.98569	−0.117836
$643$	15.6332	0.616512	0.308256	−	0.951303i	$-0.400255\pi$
$643$	15.6332	0.616512	0.308256	+	0.951303i	$0.400255\pi$
$644$	16.1964	0.638228
$645$	4.17703	0.164470
$646$	−16.0419	−0.631162
$647$	−38.2492	−1.50373	−0.751865	−	0.659317i	$-0.770846\pi$
$647$	−38.2492	−1.50373	−0.751865	+	0.659317i	$0.770846\pi$
$648$	9.37950	0.368462
$649$	22.8006	0.895002
$650$	1.00000	0.0392232
$651$	4.98050	0.195201
$652$	7.12027	0.278851
$653$	45.8497	1.79424	0.897118	−	0.441790i	$-0.145656\pi$
$653$	45.8497	1.79424	0.897118	+	0.441790i	$0.145656\pi$
$654$	−28.5496	−1.11638
$655$	−2.67312	−0.104447
$656$	−2.79038	−0.108946
$657$	37.7453	1.47258
$658$	−21.4237	−0.835184
$659$	−40.2686	−1.56864	−0.784321	−	0.620355i	$-0.786988\pi$
$659$	−40.2686	−1.56864	−0.784321	+	0.620355i	$0.786988\pi$
$660$	12.1604	0.473344
$661$	−49.4063	−1.92168	−0.960841	−	0.277099i	$-0.910627\pi$
$661$	−49.4063	−1.92168	−0.960841	+	0.277099i	$0.910627\pi$
$662$	6.59439	0.256298
$663$	7.01342	0.272379
$664$	6.54133	0.253853
$665$	11.3920	0.441762
$666$	2.20537	0.0854565
$667$	−79.0428	−3.06055
$668$	21.2215	0.821083
$669$	46.2937	1.78982
$670$	14.8174	0.572447
$671$	−69.0477	−2.66556
$672$	−4.98050	−0.192127
$673$	6.04759	0.233117	0.116559	−	0.993184i	$-0.462814\pi$
$673$	6.04759	0.233117	0.116559	+	0.993184i	$0.462814\pi$
$674$	−23.8880	−0.920131
$675$	0.320566	0.0123386
$676$	1.00000	0.0384615
$677$	1.26118	0.0484712	0.0242356	−	0.999706i	$-0.492285\pi$
$677$	1.26118	0.0484712	0.0242356	+	0.999706i	$0.492285\pi$
$678$	20.1314	0.773140
$679$	−3.72331	−0.142888
$680$	2.89533	0.111031
$681$	54.6868	2.09560
$682$	−5.02015	−0.192231
$683$	−10.8814	−0.416364	−0.208182	−	0.978090i	$-0.566755\pi$
$683$	−10.8814	−0.416364	−0.208182	+	0.978090i	$0.566755\pi$
$684$	15.8887	0.607518
$685$	2.31346	0.0883928
$686$	−20.0931	−0.767159
$687$	3.62359	0.138249
$688$	1.72439	0.0657416
$689$	−5.73411	−0.218452
$690$	19.0814	0.726418
$691$	−35.3775	−1.34582	−0.672911	−	0.739723i	$-0.734956\pi$
$691$	−35.3775	−1.34582	−0.672911	+	0.739723i	$0.734956\pi$
$692$	12.1871	0.463284
$693$	−29.5995	−1.12439
$694$	9.59538	0.364236
$695$	15.3723	0.583106
$696$	24.3062	0.921323
$697$	−8.07905	−0.306016
$698$	−18.9612	−0.717693
$699$	−3.87974	−0.146745
$700$	−2.05608	−0.0777126
$701$	25.5141	0.963655	0.481827	−	0.876266i	$-0.339973\pi$
$701$	25.5141	0.963655	0.481827	+	0.876266i	$0.339973\pi$
$702$	0.320566	0.0120990
$703$	−4.26102	−0.160707
$704$	5.02015	0.189204
$705$	−25.2399	−0.950590
$706$	−4.57335	−0.172120
$707$	−25.5403	−0.960543
$708$	−11.0018	−0.413472
$709$	19.6059	0.736316	0.368158	−	0.929763i	$-0.379989\pi$
$709$	19.6059	0.736316	0.368158	+	0.929763i	$0.379989\pi$
$710$	−11.5182	−0.432272
$711$	−14.6872	−0.550813
$712$	6.00252	0.224954
$713$	−7.87732	−0.295008
$714$	−14.4202	−0.539661
$715$	5.02015	0.187743
$716$	5.71537	0.213593
$717$	−55.0857	−2.05721
$718$	−12.1852	−0.454749
$719$	−35.0913	−1.30869	−0.654343	−	0.756198i	$-0.727055\pi$
$719$	−35.0913	−1.30869	−0.654343	+	0.756198i	$0.727055\pi$
$720$	−2.86766	−0.106871
$721$	−21.2313	−0.790696
$722$	−11.6986	−0.435377
$723$	0.00526359	0.000195755 0
$724$	−2.23387	−0.0830211
$725$	10.0342	0.372662
$726$	34.4015	1.27676
$727$	−27.1684	−1.00762	−0.503811	−	0.863814i	$-0.668069\pi$
$727$	−27.1684	−1.00762	−0.503811	+	0.863814i	$0.668069\pi$
$728$	−2.05608	−0.0762034
$729$	−24.5677	−0.909915
$730$	13.1624	0.487162
$731$	4.99266	0.184660
$732$	33.3170	1.23143
$733$	−31.2673	−1.15488	−0.577442	−	0.816432i	$-0.695949\pi$
$733$	−31.2673	−1.15488	−0.577442	+	0.816432i	$0.695949\pi$
$734$	23.2062	0.856555
$735$	−6.71597	−0.247722
$736$	7.87732	0.290362
$737$	74.3857	2.74003
$738$	8.00186	0.294553
$739$	−11.3416	−0.417209	−0.208604	−	0.978000i	$-0.566892\pi$
$739$	−11.3416	−0.417209	−0.208604	+	0.978000i	$0.566892\pi$
$740$	0.769049	0.0282708
$741$	13.4212	0.493041
$742$	11.7898	0.432817
$743$	1.81498	0.0665851	0.0332926	−	0.999446i	$-0.489401\pi$
$743$	1.81498	0.0665851	0.0332926	+	0.999446i	$0.489401\pi$
$744$	2.42233	0.0888068
$745$	2.40555	0.0881327
$746$	10.4209	0.381536
$747$	−18.7583	−0.686331
$748$	14.5350	0.531451
$749$	2.53427	0.0926001
$750$	−2.42233	−0.0884508
$751$	−11.0356	−0.402695	−0.201348	−	0.979520i	$-0.564532\pi$
$751$	−11.0356	−0.402695	−0.201348	+	0.979520i	$0.564532\pi$
$752$	−10.4197	−0.379967
$753$	36.2807	1.32214
$754$	10.0342	0.365425
$755$	14.7926	0.538357
$756$	−0.659109	−0.0239716
$757$	−29.9252	−1.08765	−0.543826	−	0.839198i	$-0.683025\pi$
$757$	−29.9252	−1.08765	−0.543826	+	0.839198i	$0.683025\pi$
$758$	−26.1796	−0.950886
$759$	95.7916	3.47702
$760$	5.54063	0.200980
$761$	−29.8875	−1.08342	−0.541710	−	0.840565i	$-0.682223\pi$
$761$	−29.8875	−1.08342	−0.541710	+	0.840565i	$0.682223\pi$
$762$	−41.7296	−1.51170
$763$	24.2330	0.877295
$764$	0.324835	0.0117521
$765$	−8.30282	−0.300189
$766$	−12.5008	−0.451672
$767$	−4.54182	−0.163996
$768$	−2.42233	−0.0874082
$769$	−14.3329	−0.516857	−0.258428	−	0.966030i	$-0.583205\pi$
$769$	−14.3329	−0.516857	−0.258428	+	0.966030i	$0.583205\pi$
$770$	−10.3218	−0.371973
$771$	28.2731	1.01823
$772$	22.9217	0.824971
$773$	54.5805	1.96312	0.981562	−	0.191142i	$-0.0612190\pi$
$773$	54.5805	1.96312	0.981562	+	0.191142i	$0.0612190\pi$
$774$	−4.94496	−0.177743
$775$	1.00000	0.0359211
$776$	−1.81088	−0.0650068
$777$	−3.83025	−0.137409
$778$	−12.3493	−0.442745
$779$	−15.4605	−0.553928
$780$	−2.42233	−0.0867332
$781$	−57.8233	−2.06908
$782$	22.8074	0.815592
$783$	3.21663	0.114953
$784$	−2.77253	−0.0990190
$785$	−7.11202	−0.253839
$786$	6.47516	0.230961
$787$	−10.1824	−0.362963	−0.181481	−	0.983394i	$-0.558089\pi$
$787$	−10.1824	−0.362963	−0.181481	+	0.983394i	$0.558089\pi$
$788$	2.96938	0.105780
$789$	−55.2659	−1.96752
$790$	−5.12166	−0.182221
$791$	−17.0876	−0.607565
$792$	−14.3961	−0.511542
$793$	13.7541	0.488423
$794$	25.9935	0.922473
$795$	13.8899	0.492623
$796$	10.2069	0.361773
$797$	5.11703	0.181255	0.0906273	−	0.995885i	$-0.471113\pi$
$797$	5.11703	0.181255	0.0906273	+	0.995885i	$0.471113\pi$
$798$	−27.5951	−0.976856
$799$	−30.1684	−1.06728
$800$	−1.00000	−0.0353553
$801$	−17.2132	−0.608198
$802$	−23.7366	−0.838169
$803$	66.0771	2.33181
$804$	−35.8927	−1.26584
$805$	−16.1964	−0.570849
$806$	1.00000	0.0352235
$807$	−69.4385	−2.44435
$808$	−12.4218	−0.436999
$809$	−37.5576	−1.32046	−0.660228	−	0.751065i	$-0.729540\pi$
$809$	−37.5576	−1.32046	−0.660228	+	0.751065i	$0.729540\pi$
$810$	−9.37950	−0.329562
$811$	−36.8842	−1.29518	−0.647590	−	0.761989i	$-0.724223\pi$
$811$	−36.8842	−1.29518	−0.647590	+	0.761989i	$0.724223\pi$
$812$	−20.6312	−0.724013
$813$	33.1680	1.16325
$814$	3.86074	0.135319
$815$	−7.12027	−0.249412
$816$	−7.01342	−0.245519
$817$	9.55419	0.334259
$818$	−30.6514	−1.07170
$819$	5.89615	0.206028
$820$	2.79038	0.0974442
$821$	−12.9717	−0.452716	−0.226358	−	0.974044i	$-0.572682\pi$
$821$	−12.9717	−0.452716	−0.226358	+	0.974044i	$0.572682\pi$
$822$	−5.60396	−0.195460
$823$	−21.2729	−0.741526	−0.370763	−	0.928727i	$-0.620904\pi$
$823$	−21.2729	−0.741526	−0.370763	+	0.928727i	$0.620904\pi$
$824$	−10.3261	−0.359727
$825$	−12.1604	−0.423372
$826$	9.33835	0.324923
$827$	32.8461	1.14217	0.571086	−	0.820891i	$-0.306522\pi$
$827$	32.8461	1.14217	0.571086	+	0.820891i	$0.306522\pi$
$828$	−22.5895	−0.785039
$829$	−3.98495	−0.138403	−0.0692016	−	0.997603i	$-0.522045\pi$
$829$	−3.98495	−0.138403	−0.0692016	+	0.997603i	$0.522045\pi$
$830$	−6.54133	−0.227053
$831$	−19.1353	−0.663795
$832$	−1.00000	−0.0346688
$833$	−8.02738	−0.278132
$834$	−37.2368	−1.28940
$835$	−21.2215	−0.734399
$836$	27.8148	0.961994
$837$	0.320566	0.0110804
$838$	−13.5505	−0.468094
$839$	−3.96195	−0.136782	−0.0683909	−	0.997659i	$-0.521787\pi$
$839$	−3.96195	−0.136782	−0.0683909	+	0.997659i	$0.521787\pi$
$840$	4.98050	0.171843
$841$	71.6857	2.47192
$842$	−2.05345	−0.0707664
$843$	−75.4933	−2.60013
$844$	8.45075	0.290887
$845$	−1.00000	−0.0344010
$846$	29.8802	1.02730
$847$	−29.2002	−1.00333
$848$	5.73411	0.196910
$849$	43.8269	1.50414
$850$	−2.89533	−0.0993089
$851$	6.05805	0.207667
$852$	27.9010	0.955871
$853$	25.3428	0.867720	0.433860	−	0.900980i	$-0.357151\pi$
$853$	25.3428	0.867720	0.433860	+	0.900980i	$0.357151\pi$
$854$	−28.2796	−0.967707
$855$	−15.8887	−0.543381
$856$	1.23257	0.0421284
$857$	−29.1043	−0.994183	−0.497092	−	0.867698i	$-0.665599\pi$
$857$	−29.1043	−0.994183	−0.497092	+	0.867698i	$0.665599\pi$
$858$	−12.1604	−0.415150
$859$	−18.9110	−0.645236	−0.322618	−	0.946529i	$-0.604563\pi$
$859$	−18.9110	−0.645236	−0.322618	+	0.946529i	$0.604563\pi$
$860$	−1.72439	−0.0588011
$861$	−13.8975	−0.473624
$862$	5.84842	0.199198
$863$	−18.8811	−0.642722	−0.321361	−	0.946957i	$-0.604140\pi$
$863$	−18.8811	−0.642722	−0.321361	+	0.946957i	$0.604140\pi$
$864$	−0.320566	−0.0109059
$865$	−12.1871	−0.414374
$866$	6.23028	0.211713
$867$	20.8734	0.708898
$868$	−2.05608	−0.0697879
$869$	−25.7115	−0.872202
$870$	−24.3062	−0.824056
$871$	−14.8174	−0.502070
$872$	11.7860	0.399125
$873$	5.19299	0.175756
$874$	43.6453	1.47633
$875$	2.05608	0.0695082
$876$	−31.8836	−1.07725
$877$	−23.8624	−0.805776	−0.402888	−	0.915249i	$-0.631994\pi$
$877$	−23.8624	−0.805776	−0.402888	+	0.915249i	$0.631994\pi$
$878$	24.1985	0.816660
$879$	−13.0912	−0.441555
$880$	−5.02015	−0.169229
$881$	−58.3416	−1.96558	−0.982789	−	0.184730i	$-0.940859\pi$
$881$	−58.3416	−1.96558	−0.982789	+	0.184730i	$0.940859\pi$
$882$	7.95068	0.267713
$883$	5.80921	0.195495	0.0977477	−	0.995211i	$-0.468836\pi$
$883$	5.80921	0.195495	0.0977477	+	0.995211i	$0.468836\pi$
$884$	−2.89533	−0.0973804
$885$	11.0018	0.369821
$886$	−39.1603	−1.31562
$887$	5.85744	0.196674	0.0983368	−	0.995153i	$-0.468648\pi$
$887$	5.85744	0.196674	0.0983368	+	0.995153i	$0.468648\pi$
$888$	−1.86289	−0.0625144
$889$	35.4203	1.18796
$890$	−6.00252	−0.201205
$891$	−47.0865	−1.57746
$892$	−19.1113	−0.639893
$893$	−57.7317	−1.93192
$894$	−5.82704	−0.194885
$895$	−5.71537	−0.191044
$896$	2.05608	0.0686888
$897$	−19.0814	−0.637111
$898$	33.5736	1.12036
$899$	10.0342	0.334660
$900$	2.86766	0.0955887
$901$	16.6021	0.553096
$902$	14.0081	0.466419
$903$	8.58831	0.285801
$904$	−8.31076	−0.276412
$905$	2.23387	0.0742563
$906$	−35.8324	−1.19045
$907$	21.9926	0.730253	0.365127	−	0.930958i	$-0.381026\pi$
$907$	21.9926	0.730253	0.365127	+	0.930958i	$0.381026\pi$
$908$	−22.5761	−0.749216
$909$	35.6217	1.18150
$910$	2.05608	0.0681584
$911$	46.0315	1.52509	0.762546	−	0.646934i	$-0.223949\pi$
$911$	46.0315	1.52509	0.762546	+	0.646934i	$0.223949\pi$
$912$	−13.4212	−0.444421
$913$	−32.8384	−1.08679
$914$	−18.6702	−0.617556
$915$	−33.3170	−1.10142
$916$	−1.49592	−0.0494264
$917$	−5.49614	−0.181499
$918$	−0.928143	−0.0306333
$919$	−20.9424	−0.690826	−0.345413	−	0.938451i	$-0.612261\pi$
$919$	−20.9424	−0.690826	−0.345413	+	0.938451i	$0.612261\pi$
$920$	−7.87732	−0.259708
$921$	−5.57448	−0.183685
$922$	−19.4343	−0.640035
$923$	11.5182	0.379128
$924$	25.0028	0.822533
$925$	−0.769049	−0.0252862
$926$	−33.5442	−1.10233
$927$	29.6118	0.972580
$928$	−10.0342	−0.329390
$929$	33.9688	1.11448	0.557239	−	0.830352i	$-0.311861\pi$
$929$	33.9688	1.11448	0.557239	+	0.830352i	$0.311861\pi$
$930$	−2.42233	−0.0794312
$931$	−15.3616	−0.503455
$932$	1.60166	0.0524642
$933$	−23.2466	−0.761060
$934$	−1.54920	−0.0506913
$935$	−14.5350	−0.475344
$936$	2.86766	0.0937325
$937$	−53.3863	−1.74405	−0.872027	−	0.489457i	$-0.837195\pi$
$937$	−53.3863	−1.74405	−0.872027	+	0.489457i	$0.837195\pi$
$938$	30.4658	0.994745
$939$	−13.4216	−0.437997
$940$	10.4197	0.339853
$941$	−39.0745	−1.27379	−0.636896	−	0.770950i	$-0.719782\pi$
$941$	−39.0745	−1.27379	−0.636896	+	0.770950i	$0.719782\pi$
$942$	17.2276	0.561306
$943$	21.9807	0.715790
$944$	4.54182	0.147824
$945$	0.659109	0.0214408
$946$	−8.65667	−0.281453
$947$	39.1736	1.27297	0.636486	−	0.771288i	$-0.280387\pi$
$947$	39.1736	1.27297	0.636486	+	0.771288i	$0.280387\pi$
$948$	12.4063	0.402939
$949$	−13.1624	−0.427269
$950$	−5.54063	−0.179762
$951$	−1.93720	−0.0628181
$952$	5.95302	0.192939
$953$	−5.61349	−0.181839	−0.0909194	−	0.995858i	$-0.528981\pi$
$953$	−5.61349	−0.181839	−0.0909194	+	0.995858i	$0.528981\pi$
$954$	−16.4435	−0.532377
$955$	−0.324835	−0.0105114
$956$	22.7408	0.735491
$957$	−122.020	−3.94436
$958$	15.2803	0.493683
$959$	4.75666	0.153601
$960$	2.42233	0.0781802
$961$	1.00000	0.0322581
$962$	−0.769049	−0.0247951
$963$	−3.53460	−0.113901
$964$	−0.00217295	−6.99860e−5 0
$965$	−22.9217	−0.737876
$966$	39.2330	1.26230
$967$	49.2231	1.58291	0.791454	−	0.611228i	$-0.209324\pi$
$967$	49.2231	1.58291	0.791454	+	0.611228i	$0.209324\pi$
$968$	−14.2019	−0.456465
$969$	−38.8588	−1.24832
$970$	1.81088	0.0581438
$971$	22.6812	0.727874	0.363937	−	0.931424i	$-0.381432\pi$
$971$	22.6812	0.727874	0.363937	+	0.931424i	$0.381432\pi$
$972$	21.7585	0.697904
$973$	31.6067	1.01327
$974$	15.7402	0.504348
$975$	2.42233	0.0775765
$976$	−13.7541	−0.440259
$977$	−53.8692	−1.72343	−0.861714	−	0.507394i	$-0.830609\pi$
$977$	−53.8692	−1.72343	−0.861714	+	0.507394i	$0.830609\pi$
$978$	17.2476	0.551518
$979$	−30.1335	−0.963071
$980$	2.77253	0.0885653
$981$	−33.7984	−1.07910
$982$	−4.68458	−0.149491
$983$	9.26528	0.295517	0.147758	−	0.989023i	$-0.452794\pi$
$983$	9.26528	0.295517	0.147758	+	0.989023i	$0.452794\pi$
$984$	−6.75920	−0.215476
$985$	−2.96938	−0.0946123
$986$	−29.0524	−0.925216
$987$	−51.8953	−1.65184
$988$	−5.54063	−0.176271
$989$	−13.5836	−0.431932
$990$	14.3961	0.457537
$991$	−40.6439	−1.29110	−0.645549	−	0.763719i	$-0.723371\pi$
$991$	−40.6439	−1.29110	−0.645549	+	0.763719i	$0.723371\pi$
$992$	−1.00000	−0.0317500
$993$	15.9738	0.506912
$994$	−23.6825	−0.751162
$995$	−10.2069	−0.323580
$996$	15.8452	0.502076
$997$	42.8314	1.35648	0.678241	−	0.734839i	$-0.262742\pi$
$997$	42.8314	1.35648	0.678241	+	0.734839i	$0.262742\pi$
$998$	−31.6582	−1.00212
$999$	−0.246531	−0.00779989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.p.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.p.1.3	✓	9	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.42233 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.42233 q^{6} -2.05608 q^{7} -1.00000 q^{8} +2.86766 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.42233 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.42233 q^{6} -2.05608 q^{7} -1.00000 q^{8} +2.86766 q^{9} +1.00000 q^{10} +5.02015 q^{11} -2.42233 q^{12} -1.00000 q^{13} +2.05608 q^{14} +2.42233 q^{15} +1.00000 q^{16} +2.89533 q^{17} -2.86766 q^{18} +5.54063 q^{19} -1.00000 q^{20} +4.98050 q^{21} -5.02015 q^{22} -7.87732 q^{23} +2.42233 q^{24} +1.00000 q^{25} +1.00000 q^{26} +0.320566 q^{27} -2.05608 q^{28} +10.0342 q^{29} -2.42233 q^{30} +1.00000 q^{31} -1.00000 q^{32} -12.1604 q^{33} -2.89533 q^{34} +2.05608 q^{35} +2.86766 q^{36} -0.769049 q^{37} -5.54063 q^{38} +2.42233 q^{39} +1.00000 q^{40} -2.79038 q^{41} -4.98050 q^{42} +1.72439 q^{43} +5.02015 q^{44} -2.86766 q^{45} +7.87732 q^{46} -10.4197 q^{47} -2.42233 q^{48} -2.77253 q^{49} -1.00000 q^{50} -7.01342 q^{51} -1.00000 q^{52} +5.73411 q^{53} -0.320566 q^{54} -5.02015 q^{55} +2.05608 q^{56} -13.4212 q^{57} -10.0342 q^{58} +4.54182 q^{59} +2.42233 q^{60} -13.7541 q^{61} -1.00000 q^{62} -5.89615 q^{63} +1.00000 q^{64} +1.00000 q^{65} +12.1604 q^{66} +14.8174 q^{67} +2.89533 q^{68} +19.0814 q^{69} -2.05608 q^{70} -11.5182 q^{71} -2.86766 q^{72} +13.1624 q^{73} +0.769049 q^{74} -2.42233 q^{75} +5.54063 q^{76} -10.3218 q^{77} -2.42233 q^{78} -5.12166 q^{79} -1.00000 q^{80} -9.37950 q^{81} +2.79038 q^{82} -6.54133 q^{83} +4.98050 q^{84} -2.89533 q^{85} -1.72439 q^{86} -24.3062 q^{87} -5.02015 q^{88} -6.00252 q^{89} +2.86766 q^{90} +2.05608 q^{91} -7.87732 q^{92} -2.42233 q^{93} +10.4197 q^{94} -5.54063 q^{95} +2.42233 q^{96} +1.81088 q^{97} +2.77253 q^{98} +14.3961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 - 9 * q^5 + 3 * q^6 - 3 * q^7 - 9 * q^8 + 14 * q^9	\( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9} + 9 q^{10} + 14 q^{11} - 3 q^{12} - 9 q^{13} + 3 q^{14} + 3 q^{15} + 9 q^{16} + q^{17} - 14 q^{18} + 6 q^{19} - 9 q^{20} + q^{21} - 14 q^{22} - 4 q^{23} + 3 q^{24} + 9 q^{25} + 9 q^{26} - 15 q^{27} - 3 q^{28} + 15 q^{29} - 3 q^{30} + 9 q^{31} - 9 q^{32} + 14 q^{33} - q^{34} + 3 q^{35} + 14 q^{36} - 9 q^{37} - 6 q^{38} + 3 q^{39} + 9 q^{40} + 18 q^{41} - q^{42} - 23 q^{43} + 14 q^{44} - 14 q^{45} + 4 q^{46} + 3 q^{47} - 3 q^{48} + 12 q^{49} - 9 q^{50} - 11 q^{51} - 9 q^{52} - 6 q^{53} + 15 q^{54} - 14 q^{55} + 3 q^{56} + 17 q^{57} - 15 q^{58} + 28 q^{59} + 3 q^{60} - 9 q^{62} + 12 q^{63} + 9 q^{64} + 9 q^{65} - 14 q^{66} - 16 q^{67} + q^{68} - 6 q^{69} - 3 q^{70} + 32 q^{71} - 14 q^{72} - 11 q^{73} + 9 q^{74} - 3 q^{75} + 6 q^{76} - 29 q^{77} - 3 q^{78} - 8 q^{79} - 9 q^{80} + 9 q^{81} - 18 q^{82} + 15 q^{83} + q^{84} - q^{85} + 23 q^{86} - 19 q^{87} - 14 q^{88} + 51 q^{89} + 14 q^{90} + 3 q^{91} - 4 q^{92} - 3 q^{93} - 3 q^{94} - 6 q^{95} + 3 q^{96} - 26 q^{97} - 12 q^{98} + 27 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 - 9 * q^5 + 3 * q^6 - 3 * q^7 - 9 * q^8 + 14 * q^9 + 9 * q^10 + 14 * q^11 - 3 * q^12 - 9 * q^13 + 3 * q^14 + 3 * q^15 + 9 * q^16 + q^17 - 14 * q^18 + 6 * q^19 - 9 * q^20 + q^21 - 14 * q^22 - 4 * q^23 + 3 * q^24 + 9 * q^25 + 9 * q^26 - 15 * q^27 - 3 * q^28 + 15 * q^29 - 3 * q^30 + 9 * q^31 - 9 * q^32 + 14 * q^33 - q^34 + 3 * q^35 + 14 * q^36 - 9 * q^37 - 6 * q^38 + 3 * q^39 + 9 * q^40 + 18 * q^41 - q^42 - 23 * q^43 + 14 * q^44 - 14 * q^45 + 4 * q^46 + 3 * q^47 - 3 * q^48 + 12 * q^49 - 9 * q^50 - 11 * q^51 - 9 * q^52 - 6 * q^53 + 15 * q^54 - 14 * q^55 + 3 * q^56 + 17 * q^57 - 15 * q^58 + 28 * q^59 + 3 * q^60 - 9 * q^62 + 12 * q^63 + 9 * q^64 + 9 * q^65 - 14 * q^66 - 16 * q^67 + q^68 - 6 * q^69 - 3 * q^70 + 32 * q^71 - 14 * q^72 - 11 * q^73 + 9 * q^74 - 3 * q^75 + 6 * q^76 - 29 * q^77 - 3 * q^78 - 8 * q^79 - 9 * q^80 + 9 * q^81 - 18 * q^82 + 15 * q^83 + q^84 - q^85 + 23 * q^86 - 19 * q^87 - 14 * q^88 + 51 * q^89 + 14 * q^90 + 3 * q^91 - 4 * q^92 - 3 * q^93 - 3 * q^94 - 6 * q^95 + 3 * q^96 - 26 * q^97 - 12 * q^98 + 27 * q^99

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