LMFDB - Embedded newform 4030.2.a.k.1.5

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} - 18x^{6} + 12x^{5} + 98x^{4} - 18x^{3} - 173x^{2} - 48x + 32 $ x^8 - x^7 - 18x^6 + 12x^5 + 98x^4 - 18x^3 - 173x^2 - 48x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.815721$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +0.815721 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.815721 q^{6} +0.327652 q^{7} -1.00000 q^{8} -2.33460 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.815721 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.815721 q^{6} +0.327652 q^{7} -1.00000 q^{8} -2.33460 q^{9} +1.00000 q^{10} +2.28397 q^{11} +0.815721 q^{12} +1.00000 q^{13} -0.327652 q^{14} -0.815721 q^{15} +1.00000 q^{16} +1.51306 q^{17} +2.33460 q^{18} +2.59453 q^{19} -1.00000 q^{20} +0.267273 q^{21} -2.28397 q^{22} -5.27917 q^{23} -0.815721 q^{24} +1.00000 q^{25} -1.00000 q^{26} -4.35154 q^{27} +0.327652 q^{28} +4.04058 q^{29} +0.815721 q^{30} -1.00000 q^{31} -1.00000 q^{32} +1.86308 q^{33} -1.51306 q^{34} -0.327652 q^{35} -2.33460 q^{36} +0.482797 q^{37} -2.59453 q^{38} +0.815721 q^{39} +1.00000 q^{40} +3.28114 q^{41} -0.267273 q^{42} +6.93730 q^{43} +2.28397 q^{44} +2.33460 q^{45} +5.27917 q^{46} +4.58925 q^{47} +0.815721 q^{48} -6.89264 q^{49} -1.00000 q^{50} +1.23424 q^{51} +1.00000 q^{52} +0.858276 q^{53} +4.35154 q^{54} -2.28397 q^{55} -0.327652 q^{56} +2.11641 q^{57} -4.04058 q^{58} +8.54187 q^{59} -0.815721 q^{60} -4.89971 q^{61} +1.00000 q^{62} -0.764937 q^{63} +1.00000 q^{64} -1.00000 q^{65} -1.86308 q^{66} -8.76914 q^{67} +1.51306 q^{68} -4.30633 q^{69} +0.327652 q^{70} +1.67927 q^{71} +2.33460 q^{72} -9.74767 q^{73} -0.482797 q^{74} +0.815721 q^{75} +2.59453 q^{76} +0.748347 q^{77} -0.815721 q^{78} -12.3833 q^{79} -1.00000 q^{80} +3.45416 q^{81} -3.28114 q^{82} +6.40058 q^{83} +0.267273 q^{84} -1.51306 q^{85} -6.93730 q^{86} +3.29599 q^{87} -2.28397 q^{88} +12.5066 q^{89} -2.33460 q^{90} +0.327652 q^{91} -5.27917 q^{92} -0.815721 q^{93} -4.58925 q^{94} -2.59453 q^{95} -0.815721 q^{96} -3.12072 q^{97} +6.89264 q^{98} -5.33215 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.815721	0.470957	0.235478	−	0.971880i	$-0.424334\pi$
$3$	0.815721	0.470957	0.235478	+	0.971880i	$0.424334\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.815721	−0.333017
$7$	0.327652	0.123841	0.0619205	−	0.998081i	$-0.480277\pi$
$7$	0.327652	0.123841	0.0619205	+	0.998081i	$0.480277\pi$
$8$	−1.00000	−0.353553
$9$	−2.33460	−0.778200
$10$	1.00000	0.316228
$11$	2.28397	0.688642	0.344321	−	0.938852i	$-0.388109\pi$
$11$	2.28397	0.688642	0.344321	+	0.938852i	$0.388109\pi$
$12$	0.815721	0.235478
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.327652	−0.0875688
$15$	−0.815721	−0.210618
$16$	1.00000	0.250000
$17$	1.51306	0.366971	0.183486	−	0.983022i	$-0.441262\pi$
$17$	1.51306	0.366971	0.183486	+	0.983022i	$0.441262\pi$
$18$	2.33460	0.550270
$19$	2.59453	0.595225	0.297612	−	0.954687i	$-0.403810\pi$
$19$	2.59453	0.595225	0.297612	+	0.954687i	$0.403810\pi$
$20$	−1.00000	−0.223607
$21$	0.267273	0.0583237
$22$	−2.28397	−0.486943
$23$	−5.27917	−1.10078	−0.550391	−	0.834907i	$-0.685522\pi$
$23$	−5.27917	−1.10078	−0.550391	+	0.834907i	$0.685522\pi$
$24$	−0.815721	−0.166508
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−4.35154	−0.837455
$28$	0.327652	0.0619205
$29$	4.04058	0.750318	0.375159	−	0.926961i	$-0.377588\pi$
$29$	4.04058	0.750318	0.375159	+	0.926961i	$0.377588\pi$
$30$	0.815721	0.148930
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	1.86308	0.324320
$34$	−1.51306	−0.259488
$35$	−0.327652	−0.0553834
$36$	−2.33460	−0.389100
$37$	0.482797	0.0793714	0.0396857	−	0.999212i	$-0.487364\pi$
$37$	0.482797	0.0793714	0.0396857	+	0.999212i	$0.487364\pi$
$38$	−2.59453	−0.420888
$39$	0.815721	0.130620
$40$	1.00000	0.158114
$41$	3.28114	0.512428	0.256214	−	0.966620i	$-0.417525\pi$
$41$	3.28114	0.512428	0.256214	+	0.966620i	$0.417525\pi$
$42$	−0.267273	−0.0412411
$43$	6.93730	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$43$	6.93730	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$44$	2.28397	0.344321
$45$	2.33460	0.348022
$46$	5.27917	0.778371
$47$	4.58925	0.669412	0.334706	−	0.942323i	$-0.391363\pi$
$47$	4.58925	0.669412	0.334706	+	0.942323i	$0.391363\pi$
$48$	0.815721	0.117739
$49$	−6.89264	−0.984663
$50$	−1.00000	−0.141421
$51$	1.23424	0.172828
$52$	1.00000	0.138675
$53$	0.858276	0.117893	0.0589467	−	0.998261i	$-0.481226\pi$
$53$	0.858276	0.117893	0.0589467	+	0.998261i	$0.481226\pi$
$54$	4.35154	0.592170
$55$	−2.28397	−0.307970
$56$	−0.327652	−0.0437844
$57$	2.11641	0.280325
$58$	−4.04058	−0.530555
$59$	8.54187	1.11206	0.556029	−	0.831163i	$-0.312324\pi$
$59$	8.54187	1.11206	0.556029	+	0.831163i	$0.312324\pi$
$60$	−0.815721	−0.105309
$61$	−4.89971	−0.627343	−0.313672	−	0.949532i	$-0.601559\pi$
$61$	−4.89971	−0.627343	−0.313672	+	0.949532i	$0.601559\pi$
$62$	1.00000	0.127000
$63$	−0.764937	−0.0963730
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−1.86308	−0.229329
$67$	−8.76914	−1.07132	−0.535660	−	0.844434i	$-0.679937\pi$
$67$	−8.76914	−1.07132	−0.535660	+	0.844434i	$0.679937\pi$
$68$	1.51306	0.183486
$69$	−4.30633	−0.518421
$70$	0.327652	0.0391620
$71$	1.67927	0.199292	0.0996462	−	0.995023i	$-0.468229\pi$
$71$	1.67927	0.199292	0.0996462	+	0.995023i	$0.468229\pi$
$72$	2.33460	0.275135
$73$	−9.74767	−1.14088	−0.570439	−	0.821340i	$-0.693227\pi$
$73$	−9.74767	−1.14088	−0.570439	+	0.821340i	$0.693227\pi$
$74$	−0.482797	−0.0561241
$75$	0.815721	0.0941913
$76$	2.59453	0.297612
$77$	0.748347	0.0852821
$78$	−0.815721	−0.0923622
$79$	−12.3833	−1.39323	−0.696614	−	0.717446i	$-0.745311\pi$
$79$	−12.3833	−1.39323	−0.696614	+	0.717446i	$0.745311\pi$
$80$	−1.00000	−0.111803
$81$	3.45416	0.383795
$82$	−3.28114	−0.362341
$83$	6.40058	0.702555	0.351277	−	0.936271i	$-0.385747\pi$
$83$	6.40058	0.702555	0.351277	+	0.936271i	$0.385747\pi$
$84$	0.267273	0.0291619
$85$	−1.51306	−0.164115
$86$	−6.93730	−0.748068
$87$	3.29599	0.353367
$88$	−2.28397	−0.243472
$89$	12.5066	1.32570	0.662850	−	0.748752i	$-0.269347\pi$
$89$	12.5066	1.32570	0.662850	+	0.748752i	$0.269347\pi$
$90$	−2.33460	−0.246088
$91$	0.327652	0.0343473
$92$	−5.27917	−0.550391
$93$	−0.815721	−0.0845863
$94$	−4.58925	−0.473345
$95$	−2.59453	−0.266193
$96$	−0.815721	−0.0832541
$97$	−3.12072	−0.316862	−0.158431	−	0.987370i	$-0.550643\pi$
$97$	−3.12072	−0.316862	−0.158431	+	0.987370i	$0.550643\pi$
$98$	6.89264	0.696262
$99$	−5.33215	−0.535901
$100$	1.00000	0.100000
$101$	15.0334	1.49588	0.747940	−	0.663766i	$-0.231043\pi$
$101$	15.0334	1.49588	0.747940	+	0.663766i	$0.231043\pi$
$102$	−1.23424	−0.122208
$103$	19.7317	1.94422	0.972110	−	0.234526i	$-0.0753537\pi$
$103$	19.7317	1.94422	0.972110	+	0.234526i	$0.0753537\pi$
$104$	−1.00000	−0.0980581
$105$	−0.267273	−0.0260832
$106$	−0.858276	−0.0833632
$107$	−8.34187	−0.806439	−0.403219	−	0.915103i	$-0.632109\pi$
$107$	−8.34187	−0.806439	−0.403219	+	0.915103i	$0.632109\pi$
$108$	−4.35154	−0.418727
$109$	11.2595	1.07847	0.539233	−	0.842157i	$-0.318714\pi$
$109$	11.2595	1.07847	0.539233	+	0.842157i	$0.318714\pi$
$110$	2.28397	0.217768
$111$	0.393828	0.0373805
$112$	0.327652	0.0309602
$113$	18.9508	1.78274	0.891372	−	0.453273i	$-0.149744\pi$
$113$	18.9508	1.78274	0.891372	+	0.453273i	$0.149744\pi$
$114$	−2.11641	−0.198220
$115$	5.27917	0.492285
$116$	4.04058	0.375159
$117$	−2.33460	−0.215834
$118$	−8.54187	−0.786343
$119$	0.495758	0.0454461
$120$	0.815721	0.0744648
$121$	−5.78350	−0.525772
$122$	4.89971	0.443599
$123$	2.67649	0.241331
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0.764937	0.0681460
$127$	8.58121	0.761459	0.380730	−	0.924686i	$-0.375673\pi$
$127$	8.58121	0.761459	0.380730	+	0.924686i	$0.375673\pi$
$128$	−1.00000	−0.0883883
$129$	5.65890	0.498238
$130$	1.00000	0.0877058
$131$	15.3998	1.34549	0.672744	−	0.739876i	$-0.265116\pi$
$131$	15.3998	1.34549	0.672744	+	0.739876i	$0.265116\pi$
$132$	1.86308	0.162160
$133$	0.850103	0.0737132
$134$	8.76914	0.757538
$135$	4.35154	0.374521
$136$	−1.51306	−0.129744
$137$	−2.38083	−0.203408	−0.101704	−	0.994815i	$-0.532429\pi$
$137$	−2.38083	−0.203408	−0.101704	+	0.994815i	$0.532429\pi$
$138$	4.30633	0.366579
$139$	13.8532	1.17501	0.587506	−	0.809220i	$-0.300110\pi$
$139$	13.8532	1.17501	0.587506	+	0.809220i	$0.300110\pi$
$140$	−0.327652	−0.0276917
$141$	3.74355	0.315264
$142$	−1.67927	−0.140921
$143$	2.28397	0.190995
$144$	−2.33460	−0.194550
$145$	−4.04058	−0.335552
$146$	9.74767	0.806723
$147$	−5.62247	−0.463734
$148$	0.482797	0.0396857
$149$	−11.7895	−0.965832	−0.482916	−	0.875667i	$-0.660422\pi$
$149$	−11.7895	−0.965832	−0.482916	+	0.875667i	$0.660422\pi$
$150$	−0.815721	−0.0666033
$151$	−13.4141	−1.09162	−0.545811	−	0.837909i	$-0.683778\pi$
$151$	−13.4141	−1.09162	−0.545811	+	0.837909i	$0.683778\pi$
$152$	−2.59453	−0.210444
$153$	−3.53239	−0.285577
$154$	−0.748347	−0.0603035
$155$	1.00000	0.0803219
$156$	0.815721	0.0653099
$157$	19.1006	1.52439	0.762197	−	0.647345i	$-0.224121\pi$
$157$	19.1006	1.52439	0.762197	+	0.647345i	$0.224121\pi$
$158$	12.3833	0.985161
$159$	0.700114	0.0555226
$160$	1.00000	0.0790569
$161$	−1.72973	−0.136322
$162$	−3.45416	−0.271384
$163$	−4.10394	−0.321446	−0.160723	−	0.987000i	$-0.551383\pi$
$163$	−4.10394	−0.321446	−0.160723	+	0.987000i	$0.551383\pi$
$164$	3.28114	0.256214
$165$	−1.86308	−0.145040
$166$	−6.40058	−0.496781
$167$	3.61073	0.279406	0.139703	−	0.990193i	$-0.455385\pi$
$167$	3.61073	0.279406	0.139703	+	0.990193i	$0.455385\pi$
$168$	−0.267273	−0.0206205
$169$	1.00000	0.0769231
$170$	1.51306	0.116047
$171$	−6.05718	−0.463204
$172$	6.93730	0.528964
$173$	5.75829	0.437794	0.218897	−	0.975748i	$-0.429754\pi$
$173$	5.75829	0.437794	0.218897	+	0.975748i	$0.429754\pi$
$174$	−3.29599	−0.249868
$175$	0.327652	0.0247682
$176$	2.28397	0.172160
$177$	6.96778	0.523731
$178$	−12.5066	−0.937412
$179$	19.0892	1.42680	0.713399	−	0.700758i	$-0.247155\pi$
$179$	19.0892	1.42680	0.713399	+	0.700758i	$0.247155\pi$
$180$	2.33460	0.174011
$181$	6.87206	0.510796	0.255398	−	0.966836i	$-0.417793\pi$
$181$	6.87206	0.510796	0.255398	+	0.966836i	$0.417793\pi$
$182$	−0.327652	−0.0242872
$183$	−3.99679	−0.295451
$184$	5.27917	0.389186
$185$	−0.482797	−0.0354960
$186$	0.815721	0.0598115
$187$	3.45578	0.252712
$188$	4.58925	0.334706
$189$	−1.42579	−0.103711
$190$	2.59453	0.188227
$191$	5.30946	0.384179	0.192089	−	0.981377i	$-0.438474\pi$
$191$	5.30946	0.384179	0.192089	+	0.981377i	$0.438474\pi$
$192$	0.815721	0.0588696
$193$	−18.6861	−1.34506	−0.672528	−	0.740071i	$-0.734792\pi$
$193$	−18.6861	−1.34506	−0.672528	+	0.740071i	$0.734792\pi$
$194$	3.12072	0.224055
$195$	−0.815721	−0.0584150
$196$	−6.89264	−0.492332
$197$	−15.8659	−1.13040	−0.565201	−	0.824953i	$-0.691201\pi$
$197$	−15.8659	−1.13040	−0.565201	+	0.824953i	$0.691201\pi$
$198$	5.33215	0.378939
$199$	−18.3784	−1.30281	−0.651406	−	0.758730i	$-0.725820\pi$
$199$	−18.3784	−1.30281	−0.651406	+	0.758730i	$0.725820\pi$
$200$	−1.00000	−0.0707107
$201$	−7.15317	−0.504545
$202$	−15.0334	−1.05775
$203$	1.32391	0.0929201
$204$	1.23424	0.0864138
$205$	−3.28114	−0.229165
$206$	−19.7317	−1.37477
$207$	12.3247	0.856629
$208$	1.00000	0.0693375
$209$	5.92581	0.409897
$210$	0.267273	0.0184436
$211$	22.4710	1.54697	0.773484	−	0.633816i	$-0.218512\pi$
$211$	22.4710	1.54697	0.773484	+	0.633816i	$0.218512\pi$
$212$	0.858276	0.0589467
$213$	1.36981	0.0938581
$214$	8.34187	0.570238
$215$	−6.93730	−0.473120
$216$	4.35154	0.296085
$217$	−0.327652	−0.0222425
$218$	−11.2595	−0.762590
$219$	−7.95138	−0.537304
$220$	−2.28397	−0.153985
$221$	1.51306	0.101780
$222$	−0.393828	−0.0264320
$223$	15.2612	1.02197	0.510984	−	0.859590i	$-0.329281\pi$
$223$	15.2612	1.02197	0.510984	+	0.859590i	$0.329281\pi$
$224$	−0.327652	−0.0218922
$225$	−2.33460	−0.155640
$226$	−18.9508	−1.26059
$227$	−4.81665	−0.319693	−0.159846	−	0.987142i	$-0.551100\pi$
$227$	−4.81665	−0.319693	−0.159846	+	0.987142i	$0.551100\pi$
$228$	2.11641	0.140163
$229$	6.45040	0.426254	0.213127	−	0.977024i	$-0.431635\pi$
$229$	6.45040	0.426254	0.213127	+	0.977024i	$0.431635\pi$
$230$	−5.27917	−0.348098
$231$	0.610442	0.0401642
$232$	−4.04058	−0.265277
$233$	21.2675	1.39328	0.696641	−	0.717420i	$-0.254677\pi$
$233$	21.2675	1.39328	0.696641	+	0.717420i	$0.254677\pi$
$234$	2.33460	0.152618
$235$	−4.58925	−0.299370
$236$	8.54187	0.556029
$237$	−10.1013	−0.656150
$238$	−0.495758	−0.0321352
$239$	−11.0755	−0.716413	−0.358206	−	0.933642i	$-0.616612\pi$
$239$	−11.0755	−0.716413	−0.358206	+	0.933642i	$0.616612\pi$
$240$	−0.815721	−0.0526545
$241$	22.3080	1.43698	0.718491	−	0.695536i	$-0.244833\pi$
$241$	22.3080	1.43698	0.718491	+	0.695536i	$0.244833\pi$
$242$	5.78350	0.371777
$243$	15.8723	1.01821
$244$	−4.89971	−0.313672
$245$	6.89264	0.440355
$246$	−2.67649	−0.170647
$247$	2.59453	0.165086
$248$	1.00000	0.0635001
$249$	5.22108	0.330873
$250$	1.00000	0.0632456
$251$	23.5624	1.48725	0.743624	−	0.668598i	$-0.233105\pi$
$251$	23.5624	1.48725	0.743624	+	0.668598i	$0.233105\pi$
$252$	−0.764937	−0.0481865
$253$	−12.0574	−0.758045
$254$	−8.58121	−0.538433
$255$	−1.23424	−0.0772909
$256$	1.00000	0.0625000
$257$	−24.4870	−1.52746	−0.763729	−	0.645537i	$-0.776634\pi$
$257$	−24.4870	−1.52746	−0.763729	+	0.645537i	$0.776634\pi$
$258$	−5.65890	−0.352308
$259$	0.158190	0.00982943
$260$	−1.00000	−0.0620174
$261$	−9.43315	−0.583897
$262$	−15.3998	−0.951403
$263$	−10.5438	−0.650158	−0.325079	−	0.945687i	$-0.605391\pi$
$263$	−10.5438	−0.650158	−0.325079	+	0.945687i	$0.605391\pi$
$264$	−1.86308	−0.114665
$265$	−0.858276	−0.0527235
$266$	−0.850103	−0.0521231
$267$	10.2019	0.624347
$268$	−8.76914	−0.535660
$269$	−24.0774	−1.46802	−0.734012	−	0.679137i	$-0.762354\pi$
$269$	−24.0774	−1.46802	−0.734012	+	0.679137i	$0.762354\pi$
$270$	−4.35154	−0.264826
$271$	0.878650	0.0533742	0.0266871	−	0.999644i	$-0.491504\pi$
$271$	0.878650	0.0533742	0.0266871	+	0.999644i	$0.491504\pi$
$272$	1.51306	0.0917429
$273$	0.267273	0.0161761
$274$	2.38083	0.143831
$275$	2.28397	0.137728
$276$	−4.30633	−0.259210
$277$	14.1390	0.849529	0.424764	−	0.905304i	$-0.360357\pi$
$277$	14.1390	0.849529	0.424764	+	0.905304i	$0.360357\pi$
$278$	−13.8532	−0.830858
$279$	2.33460	0.139769
$280$	0.327652	0.0195810
$281$	27.3535	1.63177	0.815886	−	0.578213i	$-0.196250\pi$
$281$	27.3535	1.63177	0.815886	+	0.578213i	$0.196250\pi$
$282$	−3.74355	−0.222925
$283$	25.7309	1.52954	0.764772	−	0.644301i	$-0.222852\pi$
$283$	25.7309	1.52954	0.764772	+	0.644301i	$0.222852\pi$
$284$	1.67927	0.0996462
$285$	−2.11641	−0.125365
$286$	−2.28397	−0.135054
$287$	1.07507	0.0634596
$288$	2.33460	0.137568
$289$	−14.7106	−0.865332
$290$	4.04058	0.237271
$291$	−2.54564	−0.149228
$292$	−9.74767	−0.570439
$293$	−9.98660	−0.583423	−0.291712	−	0.956506i	$-0.594225\pi$
$293$	−9.98660	−0.583423	−0.291712	+	0.956506i	$0.594225\pi$
$294$	5.62247	0.327909
$295$	−8.54187	−0.497327
$296$	−0.482797	−0.0280620
$297$	−9.93878	−0.576706
$298$	11.7895	0.682946
$299$	−5.27917	−0.305302
$300$	0.815721	0.0470957
$301$	2.27302	0.131015
$302$	13.4141	0.771893
$303$	12.2631	0.704495
$304$	2.59453	0.148806
$305$	4.89971	0.280556
$306$	3.53239	0.201934
$307$	−11.4212	−0.651845	−0.325922	−	0.945397i	$-0.605675\pi$
$307$	−11.4212	−0.651845	−0.325922	+	0.945397i	$0.605675\pi$
$308$	0.748347	0.0426410
$309$	16.0955	0.915643
$310$	−1.00000	−0.0567962
$311$	24.5954	1.39468	0.697339	−	0.716742i	$-0.254367\pi$
$311$	24.5954	1.39468	0.697339	+	0.716742i	$0.254367\pi$
$312$	−0.815721	−0.0461811
$313$	−15.6764	−0.886083	−0.443041	−	0.896501i	$-0.646101\pi$
$313$	−15.6764	−0.886083	−0.443041	+	0.896501i	$0.646101\pi$
$314$	−19.1006	−1.07791
$315$	0.764937	0.0430993
$316$	−12.3833	−0.696614
$317$	−15.3745	−0.863519	−0.431759	−	0.901989i	$-0.642107\pi$
$317$	−15.3745	−0.863519	−0.431759	+	0.901989i	$0.642107\pi$
$318$	−0.700114	−0.0392604
$319$	9.22856	0.516700
$320$	−1.00000	−0.0559017
$321$	−6.80463	−0.379798
$322$	1.72973	0.0963942
$323$	3.92568	0.218431
$324$	3.45416	0.191898
$325$	1.00000	0.0554700
$326$	4.10394	0.227296
$327$	9.18461	0.507910
$328$	−3.28114	−0.181171
$329$	1.50368	0.0829006
$330$	1.86308	0.102559
$331$	31.0143	1.70470	0.852350	−	0.522971i	$-0.175176\pi$
$331$	31.0143	1.70470	0.852350	+	0.522971i	$0.175176\pi$
$332$	6.40058	0.351277
$333$	−1.12714	−0.0617668
$334$	−3.61073	−0.197570
$335$	8.76914	0.479109
$336$	0.267273	0.0145809
$337$	−11.4807	−0.625394	−0.312697	−	0.949853i	$-0.601232\pi$
$337$	−11.4807	−0.625394	−0.312697	+	0.949853i	$0.601232\pi$
$338$	−1.00000	−0.0543928
$339$	15.4586	0.839595
$340$	−1.51306	−0.0820573
$341$	−2.28397	−0.123684
$342$	6.05718	0.327535
$343$	−4.55196	−0.245783
$344$	−6.93730	−0.374034
$345$	4.30633	0.231845
$346$	−5.75829	−0.309567
$347$	−19.1045	−1.02559	−0.512793	−	0.858513i	$-0.671389\pi$
$347$	−19.1045	−1.02559	−0.512793	+	0.858513i	$0.671389\pi$
$348$	3.29599	0.176684
$349$	2.91600	0.156090	0.0780451	−	0.996950i	$-0.475132\pi$
$349$	2.91600	0.156090	0.0780451	+	0.996950i	$0.475132\pi$
$350$	−0.327652	−0.0175138
$351$	−4.35154	−0.232268
$352$	−2.28397	−0.121736
$353$	−2.02524	−0.107793	−0.0538963	−	0.998547i	$-0.517164\pi$
$353$	−2.02524	−0.107793	−0.0538963	+	0.998547i	$0.517164\pi$
$354$	−6.96778	−0.370334
$355$	−1.67927	−0.0891263
$356$	12.5066	0.662850
$357$	0.404400	0.0214031
$358$	−19.0892	−1.00890
$359$	−29.2326	−1.54284	−0.771418	−	0.636329i	$-0.780452\pi$
$359$	−29.2326	−1.54284	−0.771418	+	0.636329i	$0.780452\pi$
$360$	−2.33460	−0.123044
$361$	−12.2684	−0.645707
$362$	−6.87206	−0.361187
$363$	−4.71772	−0.247616
$364$	0.327652	0.0171737
$365$	9.74767	0.510216
$366$	3.99679	0.208916
$367$	24.0465	1.25522	0.627609	−	0.778529i	$-0.284034\pi$
$367$	24.0465	1.25522	0.627609	+	0.778529i	$0.284034\pi$
$368$	−5.27917	−0.275196
$369$	−7.66015	−0.398771
$370$	0.482797	0.0250994
$371$	0.281216	0.0146000
$372$	−0.815721	−0.0422931
$373$	5.21643	0.270097	0.135048	−	0.990839i	$-0.456881\pi$
$373$	5.21643	0.270097	0.135048	+	0.990839i	$0.456881\pi$
$374$	−3.45578	−0.178694
$375$	−0.815721	−0.0421236
$376$	−4.58925	−0.236673
$377$	4.04058	0.208101
$378$	1.42579	0.0733349
$379$	26.3452	1.35326	0.676632	−	0.736321i	$-0.263439\pi$
$379$	26.3452	1.35326	0.676632	+	0.736321i	$0.263439\pi$
$380$	−2.59453	−0.133096
$381$	6.99987	0.358614
$382$	−5.30946	−0.271656
$383$	3.06950	0.156844	0.0784222	−	0.996920i	$-0.475012\pi$
$383$	3.06950	0.156844	0.0784222	+	0.996920i	$0.475012\pi$
$384$	−0.815721	−0.0416271
$385$	−0.748347	−0.0381393
$386$	18.6861	0.951099
$387$	−16.1958	−0.823280
$388$	−3.12072	−0.158431
$389$	32.4778	1.64669	0.823345	−	0.567541i	$-0.192105\pi$
$389$	32.4778	1.64669	0.823345	+	0.567541i	$0.192105\pi$
$390$	0.815721	0.0413056
$391$	−7.98771	−0.403956
$392$	6.89264	0.348131
$393$	12.5619	0.633666
$394$	15.8659	0.799315
$395$	12.3833	0.623070
$396$	−5.33215	−0.267951
$397$	−19.1885	−0.963043	−0.481521	−	0.876434i	$-0.659916\pi$
$397$	−19.1885	−0.963043	−0.481521	+	0.876434i	$0.659916\pi$
$398$	18.3784	0.921227
$399$	0.693446	0.0347157
$400$	1.00000	0.0500000
$401$	17.5058	0.874199	0.437100	−	0.899413i	$-0.356006\pi$
$401$	17.5058	0.874199	0.437100	+	0.899413i	$0.356006\pi$
$402$	7.15317	0.356767
$403$	−1.00000	−0.0498135
$404$	15.0334	0.747940
$405$	−3.45416	−0.171638
$406$	−1.32391	−0.0657044
$407$	1.10269	0.0546585
$408$	−1.23424	−0.0611038
$409$	−8.48455	−0.419534	−0.209767	−	0.977751i	$-0.567271\pi$
$409$	−8.48455	−0.419534	−0.209767	+	0.977751i	$0.567271\pi$
$410$	3.28114	0.162044
$411$	−1.94209	−0.0957962
$412$	19.7317	0.972110
$413$	2.79877	0.137718
$414$	−12.3247	−0.605728
$415$	−6.40058	−0.314192
$416$	−1.00000	−0.0490290
$417$	11.3003	0.553379
$418$	−5.92581	−0.289841
$419$	−9.61612	−0.469778	−0.234889	−	0.972022i	$-0.575473\pi$
$419$	−9.61612	−0.469778	−0.234889	+	0.972022i	$0.575473\pi$
$420$	−0.267273	−0.0130416
$421$	31.8513	1.55234	0.776169	−	0.630525i	$-0.217160\pi$
$421$	31.8513	1.55234	0.776169	+	0.630525i	$0.217160\pi$
$422$	−22.4710	−1.09387
$423$	−10.7141	−0.520936
$424$	−0.858276	−0.0416816
$425$	1.51306	0.0733943
$426$	−1.36981	−0.0663677
$427$	−1.60540	−0.0776908
$428$	−8.34187	−0.403219
$429$	1.86308	0.0899503
$430$	6.93730	0.334546
$431$	15.7673	0.759486	0.379743	−	0.925092i	$-0.376012\pi$
$431$	15.7673	0.759486	0.379743	+	0.925092i	$0.376012\pi$
$432$	−4.35154	−0.209364
$433$	−16.2330	−0.780110	−0.390055	−	0.920792i	$-0.627544\pi$
$433$	−16.2330	−0.780110	−0.390055	+	0.920792i	$0.627544\pi$
$434$	0.327652	0.0157278
$435$	−3.29599	−0.158031
$436$	11.2595	0.539233
$437$	−13.6969	−0.655213
$438$	7.95138	0.379932
$439$	−12.4922	−0.596219	−0.298109	−	0.954532i	$-0.596356\pi$
$439$	−12.4922	−0.596219	−0.298109	+	0.954532i	$0.596356\pi$
$440$	2.28397	0.108884
$441$	16.0916	0.766265
$442$	−1.51306	−0.0719690
$443$	6.89601	0.327639	0.163820	−	0.986490i	$-0.447618\pi$
$443$	6.89601	0.327639	0.163820	+	0.986490i	$0.447618\pi$
$444$	0.393828	0.0186902
$445$	−12.5066	−0.592871
$446$	−15.2612	−0.722641
$447$	−9.61692	−0.454865
$448$	0.327652	0.0154801
$449$	34.7650	1.64066	0.820331	−	0.571889i	$-0.193789\pi$
$449$	34.7650	1.64066	0.820331	+	0.571889i	$0.193789\pi$
$450$	2.33460	0.110054
$451$	7.49401	0.352879
$452$	18.9508	0.891372
$453$	−10.9421	−0.514106
$454$	4.81665	0.226057
$455$	−0.327652	−0.0153606
$456$	−2.11641	−0.0991099
$457$	−2.18101	−0.102023	−0.0510117	−	0.998698i	$-0.516245\pi$
$457$	−2.18101	−0.102023	−0.0510117	+	0.998698i	$0.516245\pi$
$458$	−6.45040	−0.301407
$459$	−6.58415	−0.307322
$460$	5.27917	0.246143
$461$	2.28704	0.106518	0.0532591	−	0.998581i	$-0.483039\pi$
$461$	2.28704	0.106518	0.0532591	+	0.998581i	$0.483039\pi$
$462$	−0.610442	−0.0284003
$463$	−29.1690	−1.35560	−0.677799	−	0.735248i	$-0.737066\pi$
$463$	−29.1690	−1.35560	−0.677799	+	0.735248i	$0.737066\pi$
$464$	4.04058	0.187579
$465$	0.815721	0.0378281
$466$	−21.2675	−0.985199
$467$	29.4271	1.36172	0.680861	−	0.732413i	$-0.261606\pi$
$467$	29.4271	1.36172	0.680861	+	0.732413i	$0.261606\pi$
$468$	−2.33460	−0.107917
$469$	−2.87323	−0.132673
$470$	4.58925	0.211687
$471$	15.5808	0.717924
$472$	−8.54187	−0.393172
$473$	15.8446	0.728534
$474$	10.1013	0.463968
$475$	2.59453	0.119045
$476$	0.495758	0.0227231
$477$	−2.00373	−0.0917446
$478$	11.0755	0.506580
$479$	0.338575	0.0154699	0.00773495	−	0.999970i	$-0.497538\pi$
$479$	0.338575	0.0154699	0.00773495	+	0.999970i	$0.497538\pi$
$480$	0.815721	0.0372324
$481$	0.482797	0.0220137
$482$	−22.3080	−1.01610
$483$	−1.41098	−0.0642018
$484$	−5.78350	−0.262886
$485$	3.12072	0.141705
$486$	−15.8723	−0.719980
$487$	−21.2802	−0.964299	−0.482149	−	0.876089i	$-0.660144\pi$
$487$	−21.2802	−0.964299	−0.482149	+	0.876089i	$0.660144\pi$
$488$	4.89971	0.221799
$489$	−3.34767	−0.151387
$490$	−6.89264	−0.311378
$491$	37.4826	1.69157	0.845784	−	0.533526i	$-0.179133\pi$
$491$	37.4826	1.69157	0.845784	+	0.533526i	$0.179133\pi$
$492$	2.67649	0.120666
$493$	6.11365	0.275345
$494$	−2.59453	−0.116733
$495$	5.33215	0.239662
$496$	−1.00000	−0.0449013
$497$	0.550216	0.0246806
$498$	−5.22108	−0.233962
$499$	−26.7258	−1.19641	−0.598205	−	0.801343i	$-0.704119\pi$
$499$	−26.7258	−1.19641	−0.598205	+	0.801343i	$0.704119\pi$
$500$	−1.00000	−0.0447214
$501$	2.94534	0.131588
$502$	−23.5624	−1.05164
$503$	−15.1597	−0.675935	−0.337968	−	0.941158i	$-0.609739\pi$
$503$	−15.1597	−0.675935	−0.337968	+	0.941158i	$0.609739\pi$
$504$	0.764937	0.0340730
$505$	−15.0334	−0.668978
$506$	12.0574	0.536019
$507$	0.815721	0.0362274
$508$	8.58121	0.380730
$509$	23.6854	1.04984	0.524919	−	0.851152i	$-0.324096\pi$
$509$	23.6854	1.04984	0.524919	+	0.851152i	$0.324096\pi$
$510$	1.23424	0.0546529
$511$	−3.19385	−0.141288
$512$	−1.00000	−0.0441942
$513$	−11.2902	−0.498474
$514$	24.4870	1.08008
$515$	−19.7317	−0.869482
$516$	5.65890	0.249119
$517$	10.4817	0.460985
$518$	−0.158190	−0.00695046
$519$	4.69715	0.206182
$520$	1.00000	0.0438529
$521$	8.18313	0.358510	0.179255	−	0.983803i	$-0.442631\pi$
$521$	8.18313	0.358510	0.179255	+	0.983803i	$0.442631\pi$
$522$	9.43315	0.412878
$523$	−27.1939	−1.18910	−0.594552	−	0.804057i	$-0.702671\pi$
$523$	−27.1939	−1.18910	−0.594552	+	0.804057i	$0.702671\pi$
$524$	15.3998	0.672744
$525$	0.267273	0.0116647
$526$	10.5438	0.459731
$527$	−1.51306	−0.0659100
$528$	1.86308	0.0810801
$529$	4.86963	0.211723
$530$	0.858276	0.0372811
$531$	−19.9419	−0.865403
$532$	0.850103	0.0368566
$533$	3.28114	0.142122
$534$	−10.2019	−0.441480
$535$	8.34187	0.360650
$536$	8.76914	0.378769
$537$	15.5715	0.671960
$538$	24.0774	1.03805
$539$	−15.7426	−0.678080
$540$	4.35154	0.187261
$541$	25.9843	1.11715	0.558576	−	0.829453i	$-0.311348\pi$
$541$	25.9843	1.11715	0.558576	+	0.829453i	$0.311348\pi$
$542$	−0.878650	−0.0377413
$543$	5.60568	0.240563
$544$	−1.51306	−0.0648720
$545$	−11.2595	−0.482304
$546$	−0.267273	−0.0114382
$547$	−0.538154	−0.0230098	−0.0115049	−	0.999934i	$-0.503662\pi$
$547$	−0.538154	−0.0230098	−0.0115049	+	0.999934i	$0.503662\pi$
$548$	−2.38083	−0.101704
$549$	11.4389	0.488198
$550$	−2.28397	−0.0973887
$551$	10.4834	0.446608
$552$	4.30633	0.183289
$553$	−4.05741	−0.172539
$554$	−14.1390	−0.600708
$555$	−0.393828	−0.0167171
$556$	13.8532	0.587506
$557$	5.26005	0.222875	0.111438	−	0.993771i	$-0.464454\pi$
$557$	5.26005	0.222875	0.111438	+	0.993771i	$0.464454\pi$
$558$	−2.33460	−0.0988315
$559$	6.93730	0.293417
$560$	−0.327652	−0.0138458
$561$	2.81895	0.119016
$562$	−27.3535	−1.15384
$563$	2.60164	0.109646	0.0548231	−	0.998496i	$-0.482541\pi$
$563$	2.60164	0.109646	0.0548231	+	0.998496i	$0.482541\pi$
$564$	3.74355	0.157632
$565$	−18.9508	−0.797267
$566$	−25.7309	−1.08155
$567$	1.13176	0.0475296
$568$	−1.67927	−0.0704605
$569$	12.5022	0.524119	0.262059	−	0.965052i	$-0.415598\pi$
$569$	12.5022	0.524119	0.262059	+	0.965052i	$0.415598\pi$
$570$	2.11641	0.0886466
$571$	−46.1871	−1.93287	−0.966435	−	0.256911i	$-0.917295\pi$
$571$	−46.1871	−1.93287	−0.966435	+	0.256911i	$0.917295\pi$
$572$	2.28397	0.0954974
$573$	4.33104	0.180932
$574$	−1.07507	−0.0448727
$575$	−5.27917	−0.220157
$576$	−2.33460	−0.0972750
$577$	−14.4105	−0.599918	−0.299959	−	0.953952i	$-0.596973\pi$
$577$	−14.4105	−0.599918	−0.299959	+	0.953952i	$0.596973\pi$
$578$	14.7106	0.611882
$579$	−15.2427	−0.633463
$580$	−4.04058	−0.167776
$581$	2.09716	0.0870051
$582$	2.54564	0.105520
$583$	1.96027	0.0811863
$584$	9.74767	0.403362
$585$	2.33460	0.0965238
$586$	9.98660	0.412543
$587$	−21.0548	−0.869023	−0.434512	−	0.900666i	$-0.643079\pi$
$587$	−21.0548	−0.869023	−0.434512	+	0.900666i	$0.643079\pi$
$588$	−5.62247	−0.231867
$589$	−2.59453	−0.106906
$590$	8.54187	0.351663
$591$	−12.9422	−0.532370
$592$	0.482797	0.0198429
$593$	−15.3675	−0.631068	−0.315534	−	0.948914i	$-0.602184\pi$
$593$	−15.3675	−0.631068	−0.315534	+	0.948914i	$0.602184\pi$
$594$	9.93878	0.407793
$595$	−0.495758	−0.0203241
$596$	−11.7895	−0.482916
$597$	−14.9916	−0.613567
$598$	5.27917	0.215881
$599$	−11.6587	−0.476362	−0.238181	−	0.971221i	$-0.576551\pi$
$599$	−11.6587	−0.476362	−0.238181	+	0.971221i	$0.576551\pi$
$600$	−0.815721	−0.0333017
$601$	19.9222	0.812644	0.406322	−	0.913730i	$-0.366811\pi$
$601$	19.9222	0.812644	0.406322	+	0.913730i	$0.366811\pi$
$602$	−2.27302	−0.0926415
$603$	20.4724	0.833702
$604$	−13.4141	−0.545811
$605$	5.78350	0.235133
$606$	−12.2631	−0.498153
$607$	−22.3407	−0.906783	−0.453391	−	0.891312i	$-0.649786\pi$
$607$	−22.3407	−0.906783	−0.453391	+	0.891312i	$0.649786\pi$
$608$	−2.59453	−0.105222
$609$	1.07994	0.0437613
$610$	−4.89971	−0.198383
$611$	4.58925	0.185661
$612$	−3.53239	−0.142789
$613$	28.1828	1.13829	0.569146	−	0.822237i	$-0.307274\pi$
$613$	28.1828	1.13829	0.569146	+	0.822237i	$0.307274\pi$
$614$	11.4212	0.460924
$615$	−2.67649	−0.107927
$616$	−0.748347	−0.0301518
$617$	16.6884	0.671848	0.335924	−	0.941889i	$-0.390951\pi$
$617$	16.6884	0.671848	0.335924	+	0.941889i	$0.390951\pi$
$618$	−16.0955	−0.647457
$619$	−33.5443	−1.34826	−0.674130	−	0.738613i	$-0.735481\pi$
$619$	−33.5443	−1.34826	−0.674130	+	0.738613i	$0.735481\pi$
$620$	1.00000	0.0401610
$621$	22.9725	0.921856
$622$	−24.5954	−0.986186
$623$	4.09783	0.164176
$624$	0.815721	0.0326550
$625$	1.00000	0.0400000
$626$	15.6764	0.626555
$627$	4.83381	0.193044
$628$	19.1006	0.762197
$629$	0.730502	0.0291270
$630$	−0.764937	−0.0304758
$631$	18.2250	0.725524	0.362762	−	0.931882i	$-0.381834\pi$
$631$	18.2250	0.725524	0.362762	+	0.931882i	$0.381834\pi$
$632$	12.3833	0.492580
$633$	18.3301	0.728554
$634$	15.3745	0.610600
$635$	−8.58121	−0.340535
$636$	0.700114	0.0277613
$637$	−6.89264	−0.273096
$638$	−9.22856	−0.365362
$639$	−3.92042	−0.155089
$640$	1.00000	0.0395285
$641$	0.569286	0.0224854	0.0112427	−	0.999937i	$-0.496421\pi$
$641$	0.569286	0.0224854	0.0112427	+	0.999937i	$0.496421\pi$
$642$	6.80463	0.268558
$643$	33.5904	1.32467	0.662337	−	0.749206i	$-0.269565\pi$
$643$	33.5904	1.32467	0.662337	+	0.749206i	$0.269565\pi$
$644$	−1.72973	−0.0681610
$645$	−5.65890	−0.222819
$646$	−3.92568	−0.154454
$647$	14.7072	0.578199	0.289100	−	0.957299i	$-0.406644\pi$
$647$	14.7072	0.578199	0.289100	+	0.957299i	$0.406644\pi$
$648$	−3.45416	−0.135692
$649$	19.5094	0.765809
$650$	−1.00000	−0.0392232
$651$	−0.267273	−0.0104752
$652$	−4.10394	−0.160723
$653$	39.8067	1.55776	0.778878	−	0.627176i	$-0.215789\pi$
$653$	39.8067	1.55776	0.778878	+	0.627176i	$0.215789\pi$
$654$	−9.18461	−0.359147
$655$	−15.3998	−0.601720
$656$	3.28114	0.128107
$657$	22.7569	0.887832
$658$	−1.50368	−0.0586196
$659$	29.2316	1.13870	0.569352	−	0.822094i	$-0.307194\pi$
$659$	29.2316	1.13870	0.569352	+	0.822094i	$0.307194\pi$
$660$	−1.86308	−0.0725202
$661$	−1.68087	−0.0653782	−0.0326891	−	0.999466i	$-0.510407\pi$
$661$	−1.68087	−0.0653782	−0.0326891	+	0.999466i	$0.510407\pi$
$662$	−31.0143	−1.20541
$663$	1.23424	0.0479338
$664$	−6.40058	−0.248391
$665$	−0.850103	−0.0329656
$666$	1.12714	0.0436757
$667$	−21.3309	−0.825937
$668$	3.61073	0.139703
$669$	12.4489	0.481303
$670$	−8.76914	−0.338781
$671$	−11.1908	−0.432015
$672$	−0.267273	−0.0103103
$673$	23.0297	0.887729	0.443864	−	0.896094i	$-0.353607\pi$
$673$	23.0297	0.887729	0.443864	+	0.896094i	$0.353607\pi$
$674$	11.4807	0.442220
$675$	−4.35154	−0.167491
$676$	1.00000	0.0384615
$677$	−7.54206	−0.289865	−0.144932	−	0.989442i	$-0.546297\pi$
$677$	−7.54206	−0.289865	−0.144932	+	0.989442i	$0.546297\pi$
$678$	−15.4586	−0.593683
$679$	−1.02251	−0.0392404
$680$	1.51306	0.0580233
$681$	−3.92904	−0.150561
$682$	2.28397	0.0874576
$683$	−1.44071	−0.0551271	−0.0275635	−	0.999620i	$-0.508775\pi$
$683$	−1.44071	−0.0551271	−0.0275635	+	0.999620i	$0.508775\pi$
$684$	−6.05718	−0.231602
$685$	2.38083	0.0909667
$686$	4.55196	0.173795
$687$	5.26172	0.200747
$688$	6.93730	0.264482
$689$	0.858276	0.0326977
$690$	−4.30633	−0.163939
$691$	−9.91399	−0.377146	−0.188573	−	0.982059i	$-0.560386\pi$
$691$	−9.91399	−0.377146	−0.188573	+	0.982059i	$0.560386\pi$
$692$	5.75829	0.218897
$693$	−1.74709	−0.0663665
$694$	19.1045	0.725198
$695$	−13.8532	−0.525481
$696$	−3.29599	−0.124934
$697$	4.96457	0.188046
$698$	−2.91600	−0.110372
$699$	17.3484	0.656175
$700$	0.327652	0.0123841
$701$	−23.9356	−0.904033	−0.452017	−	0.892010i	$-0.649295\pi$
$701$	−23.9356	−0.904033	−0.452017	+	0.892010i	$0.649295\pi$
$702$	4.35154	0.164238
$703$	1.25263	0.0472438
$704$	2.28397	0.0860802
$705$	−3.74355	−0.140990
$706$	2.02524	0.0762208
$707$	4.92574	0.185251
$708$	6.96778	0.261865
$709$	−41.5093	−1.55892	−0.779458	−	0.626455i	$-0.784505\pi$
$709$	−41.5093	−1.55892	−0.779458	+	0.626455i	$0.784505\pi$
$710$	1.67927	0.0630218
$711$	28.9100	1.08421
$712$	−12.5066	−0.468706
$713$	5.27917	0.197706
$714$	−0.404400	−0.0151343
$715$	−2.28397	−0.0854155
$716$	19.0892	0.713399
$717$	−9.03449	−0.337399
$718$	29.2326	1.09095
$719$	14.8439	0.553582	0.276791	−	0.960930i	$-0.410729\pi$
$719$	14.8439	0.553582	0.276791	+	0.960930i	$0.410729\pi$
$720$	2.33460	0.0870054
$721$	6.46513	0.240774
$722$	12.2684	0.456584
$723$	18.1971	0.676756
$724$	6.87206	0.255398
$725$	4.04058	0.150064
$726$	4.71772	0.175091
$727$	−23.8899	−0.886029	−0.443014	−	0.896515i	$-0.646091\pi$
$727$	−23.8899	−0.886029	−0.443014	+	0.896515i	$0.646091\pi$
$728$	−0.327652	−0.0121436
$729$	2.58486	0.0957355
$730$	−9.74767	−0.360778
$731$	10.4966	0.388229
$732$	−3.99679	−0.147726
$733$	−1.52696	−0.0563996	−0.0281998	−	0.999602i	$-0.508977\pi$
$733$	−1.52696	−0.0563996	−0.0281998	+	0.999602i	$0.508977\pi$
$734$	−24.0465	−0.887573
$735$	5.62247	0.207388
$736$	5.27917	0.194593
$737$	−20.0284	−0.737756
$738$	7.66015	0.281974
$739$	−15.9500	−0.586730	−0.293365	−	0.956001i	$-0.594775\pi$
$739$	−15.9500	−0.586730	−0.293365	+	0.956001i	$0.594775\pi$
$740$	−0.482797	−0.0177480
$741$	2.11641	0.0777482
$742$	−0.281216	−0.0103238
$743$	−12.9085	−0.473568	−0.236784	−	0.971562i	$-0.576093\pi$
$743$	−12.9085	−0.473568	−0.236784	+	0.971562i	$0.576093\pi$
$744$	0.815721	0.0299058
$745$	11.7895	0.431933
$746$	−5.21643	−0.190987
$747$	−14.9428	−0.546728
$748$	3.45578	0.126356
$749$	−2.73323	−0.0998702
$750$	0.815721	0.0297859
$751$	−48.4530	−1.76808	−0.884038	−	0.467415i	$-0.845185\pi$
$751$	−48.4530	−1.76808	−0.884038	+	0.467415i	$0.845185\pi$
$752$	4.58925	0.167353
$753$	19.2204	0.700429
$754$	−4.04058	−0.147149
$755$	13.4141	0.488188
$756$	−1.42579	−0.0518556
$757$	−50.4663	−1.83423	−0.917114	−	0.398626i	$-0.869487\pi$
$757$	−50.4663	−1.83423	−0.917114	+	0.398626i	$0.869487\pi$
$758$	−26.3452	−0.956902
$759$	−9.83551	−0.357006
$760$	2.59453	0.0941133
$761$	−48.9744	−1.77532	−0.887661	−	0.460498i	$-0.847671\pi$
$761$	−48.9744	−1.77532	−0.887661	+	0.460498i	$0.847671\pi$
$762$	−6.99987	−0.253579
$763$	3.68920	0.133558
$764$	5.30946	0.192089
$765$	3.53239	0.127714
$766$	−3.06950	−0.110906
$767$	8.54187	0.308429
$768$	0.815721	0.0294348
$769$	43.0159	1.55119	0.775597	−	0.631228i	$-0.217449\pi$
$769$	43.0159	1.55119	0.775597	+	0.631228i	$0.217449\pi$
$770$	0.748347	0.0269686
$771$	−19.9746	−0.719367
$772$	−18.6861	−0.672528
$773$	−2.33542	−0.0839990	−0.0419995	−	0.999118i	$-0.513373\pi$
$773$	−2.33542	−0.0839990	−0.0419995	+	0.999118i	$0.513373\pi$
$774$	16.1958	0.582147
$775$	−1.00000	−0.0359211
$776$	3.12072	0.112027
$777$	0.129039	0.00462924
$778$	−32.4778	−1.16439
$779$	8.51300	0.305010
$780$	−0.815721	−0.0292075
$781$	3.83539	0.137241
$782$	7.98771	0.285640
$783$	−17.5828	−0.628357
$784$	−6.89264	−0.246166
$785$	−19.1006	−0.681730
$786$	−12.5619	−0.448070
$787$	−33.2187	−1.18412	−0.592059	−	0.805895i	$-0.701685\pi$
$787$	−33.2187	−1.18412	−0.592059	+	0.805895i	$0.701685\pi$
$788$	−15.8659	−0.565201
$789$	−8.60079	−0.306196
$790$	−12.3833	−0.440577
$791$	6.20928	0.220777
$792$	5.33215	0.189470
$793$	−4.89971	−0.173994
$794$	19.1885	0.680974
$795$	−0.700114	−0.0248305
$796$	−18.3784	−0.651406
$797$	0.837881	0.0296793	0.0148396	−	0.999890i	$-0.495276\pi$
$797$	0.837881	0.0296793	0.0148396	+	0.999890i	$0.495276\pi$
$798$	−0.693446	−0.0245477
$799$	6.94383	0.245655
$800$	−1.00000	−0.0353553
$801$	−29.1980	−1.03166
$802$	−17.5058	−0.618152
$803$	−22.2634	−0.785657
$804$	−7.15317	−0.252273
$805$	1.72973	0.0609651
$806$	1.00000	0.0352235
$807$	−19.6404	−0.691375
$808$	−15.0334	−0.528874
$809$	−11.5325	−0.405461	−0.202730	−	0.979235i	$-0.564981\pi$
$809$	−11.5325	−0.405461	−0.202730	+	0.979235i	$0.564981\pi$
$810$	3.45416	0.121367
$811$	−45.7837	−1.60768	−0.803842	−	0.594843i	$-0.797214\pi$
$811$	−45.7837	−1.60768	−0.803842	+	0.594843i	$0.797214\pi$
$812$	1.32391	0.0464600
$813$	0.716733	0.0251369
$814$	−1.10269	−0.0386494
$815$	4.10394	0.143755
$816$	1.23424	0.0432069
$817$	17.9990	0.629705
$818$	8.48455	0.296655
$819$	−0.764937	−0.0267291
$820$	−3.28114	−0.114582
$821$	−16.2285	−0.566377	−0.283189	−	0.959064i	$-0.591392\pi$
$821$	−16.2285	−0.566377	−0.283189	+	0.959064i	$0.591392\pi$
$822$	1.94209	0.0677382
$823$	26.1925	0.913014	0.456507	−	0.889720i	$-0.349100\pi$
$823$	26.1925	0.913014	0.456507	+	0.889720i	$0.349100\pi$
$824$	−19.7317	−0.687386
$825$	1.86308	0.0648641
$826$	−2.79877	−0.0973815
$827$	−30.8252	−1.07190	−0.535949	−	0.844250i	$-0.680046\pi$
$827$	−30.8252	−1.07190	−0.535949	+	0.844250i	$0.680046\pi$
$828$	12.3247	0.428315
$829$	−6.83640	−0.237438	−0.118719	−	0.992928i	$-0.537879\pi$
$829$	−6.83640	−0.237438	−0.118719	+	0.992928i	$0.537879\pi$
$830$	6.40058	0.222167
$831$	11.5335	0.400091
$832$	1.00000	0.0346688
$833$	−10.4290	−0.361343
$834$	−11.3003	−0.391298
$835$	−3.61073	−0.124954
$836$	5.92581	0.204948
$837$	4.35154	0.150411
$838$	9.61612	0.332183
$839$	−4.16142	−0.143668	−0.0718341	−	0.997417i	$-0.522885\pi$
$839$	−4.16142	−0.143668	−0.0718341	+	0.997417i	$0.522885\pi$
$840$	0.267273	0.00922179
$841$	−12.6737	−0.437023
$842$	−31.8513	−1.09767
$843$	22.3128	0.768494
$844$	22.4710	0.773484
$845$	−1.00000	−0.0344010
$846$	10.7141	0.368357
$847$	−1.89498	−0.0651122
$848$	0.858276	0.0294733
$849$	20.9892	0.720348
$850$	−1.51306	−0.0518976
$851$	−2.54877	−0.0873707
$852$	1.36981	0.0469290
$853$	−21.8626	−0.748560	−0.374280	−	0.927316i	$-0.622110\pi$
$853$	−21.8626	−0.748560	−0.374280	+	0.927316i	$0.622110\pi$
$854$	1.60540	0.0549357
$855$	6.05718	0.207151
$856$	8.34187	0.285119
$857$	20.8140	0.710992	0.355496	−	0.934678i	$-0.384312\pi$
$857$	20.8140	0.710992	0.355496	+	0.934678i	$0.384312\pi$
$858$	−1.86308	−0.0636045
$859$	21.6686	0.739322	0.369661	−	0.929167i	$-0.379474\pi$
$859$	21.6686	0.739322	0.369661	+	0.929167i	$0.379474\pi$
$860$	−6.93730	−0.236560
$861$	0.876960	0.0298867
$862$	−15.7673	−0.537038
$863$	−35.6643	−1.21403	−0.607013	−	0.794692i	$-0.707633\pi$
$863$	−35.6643	−1.21403	−0.607013	+	0.794692i	$0.707633\pi$
$864$	4.35154	0.148043
$865$	−5.75829	−0.195788
$866$	16.2330	0.551621
$867$	−11.9998	−0.407534
$868$	−0.327652	−0.0111212
$869$	−28.2830	−0.959435
$870$	3.29599	0.111744
$871$	−8.76914	−0.297131
$872$	−11.2595	−0.381295
$873$	7.28564	0.246582
$874$	13.6969	0.463306
$875$	−0.327652	−0.0110767
$876$	−7.95138	−0.268652
$877$	−32.8662	−1.10981	−0.554906	−	0.831913i	$-0.687246\pi$
$877$	−32.8662	−1.10981	−0.554906	+	0.831913i	$0.687246\pi$
$878$	12.4922	0.421590
$879$	−8.14627	−0.274767
$880$	−2.28397	−0.0769925
$881$	−3.65040	−0.122985	−0.0614926	−	0.998108i	$-0.519586\pi$
$881$	−3.65040	−0.122985	−0.0614926	+	0.998108i	$0.519586\pi$
$882$	−16.0916	−0.541831
$883$	7.76606	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$883$	7.76606	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$884$	1.51306	0.0508898
$885$	−6.96778	−0.234219
$886$	−6.89601	−0.231676
$887$	−26.0176	−0.873586	−0.436793	−	0.899562i	$-0.643886\pi$
$887$	−26.0176	−0.873586	−0.436793	+	0.899562i	$0.643886\pi$
$888$	−0.393828	−0.0132160
$889$	2.81165	0.0942999
$890$	12.5066	0.419223
$891$	7.88918	0.264297
$892$	15.2612	0.510984
$893$	11.9069	0.398451
$894$	9.61692	0.321638
$895$	−19.0892	−0.638083
$896$	−0.327652	−0.0109461
$897$	−4.30633	−0.143784
$898$	−34.7650	−1.16012
$899$	−4.04058	−0.134761
$900$	−2.33460	−0.0778200
$901$	1.29863	0.0432635
$902$	−7.49401	−0.249523
$903$	1.85415	0.0617023
$904$	−18.9508	−0.630295
$905$	−6.87206	−0.228435
$906$	10.9421	0.363528
$907$	−20.0420	−0.665482	−0.332741	−	0.943018i	$-0.607974\pi$
$907$	−20.0420	−0.665482	−0.332741	+	0.943018i	$0.607974\pi$
$908$	−4.81665	−0.159846
$909$	−35.0970	−1.16409
$910$	0.327652	0.0108616
$911$	−36.1401	−1.19737	−0.598687	−	0.800983i	$-0.704311\pi$
$911$	−36.1401	−1.19737	−0.598687	+	0.800983i	$0.704311\pi$
$912$	2.11641	0.0700813
$913$	14.6187	0.483809
$914$	2.18101	0.0721415
$915$	3.99679	0.132130
$916$	6.45040	0.213127
$917$	5.04579	0.166627
$918$	6.58415	0.217309
$919$	−34.6188	−1.14197	−0.570985	−	0.820961i	$-0.693438\pi$
$919$	−34.6188	−1.14197	−0.570985	+	0.820961i	$0.693438\pi$
$920$	−5.27917	−0.174049
$921$	−9.31654	−0.306991
$922$	−2.28704	−0.0753197
$923$	1.67927	0.0552738
$924$	0.610442	0.0200821
$925$	0.482797	0.0158743
$926$	29.1690	0.958552
$927$	−46.0656	−1.51299
$928$	−4.04058	−0.132639
$929$	−32.3563	−1.06157	−0.530787	−	0.847505i	$-0.678104\pi$
$929$	−32.3563	−1.06157	−0.530787	+	0.847505i	$0.678104\pi$
$930$	−0.815721	−0.0267485
$931$	−17.8831	−0.586096
$932$	21.2675	0.696641
$933$	20.0630	0.656832
$934$	−29.4271	−0.962883
$935$	−3.45578	−0.113016
$936$	2.33460	0.0763088
$937$	27.3741	0.894274	0.447137	−	0.894466i	$-0.352444\pi$
$937$	27.3741	0.894274	0.447137	+	0.894466i	$0.352444\pi$
$938$	2.87323	0.0938143
$939$	−12.7876	−0.417306
$940$	−4.58925	−0.149685
$941$	48.4099	1.57812	0.789059	−	0.614317i	$-0.210568\pi$
$941$	48.4099	1.57812	0.789059	+	0.614317i	$0.210568\pi$
$942$	−15.5808	−0.507649
$943$	−17.3217	−0.564072
$944$	8.54187	0.278014
$945$	1.42579	0.0463811
$946$	−15.8446	−0.515151
$947$	−2.77264	−0.0900988	−0.0450494	−	0.998985i	$-0.514345\pi$
$947$	−2.77264	−0.0900988	−0.0450494	+	0.998985i	$0.514345\pi$
$948$	−10.1013	−0.328075
$949$	−9.74767	−0.316423
$950$	−2.59453	−0.0841775
$951$	−12.5413	−0.406680
$952$	−0.495758	−0.0160676
$953$	8.26422	0.267704	0.133852	−	0.991001i	$-0.457265\pi$
$953$	8.26422	0.267704	0.133852	+	0.991001i	$0.457265\pi$
$954$	2.00373	0.0648732
$955$	−5.30946	−0.171810
$956$	−11.0755	−0.358206
$957$	7.52793	0.243343
$958$	−0.338575	−0.0109389
$959$	−0.780084	−0.0251902
$960$	−0.815721	−0.0263273
$961$	1.00000	0.0322581
$962$	−0.482797	−0.0155660
$963$	19.4749	0.627571
$964$	22.3080	0.718491
$965$	18.6861	0.601528
$966$	1.41098	0.0453975
$967$	34.4207	1.10690	0.553448	−	0.832884i	$-0.313312\pi$
$967$	34.4207	1.10690	0.553448	+	0.832884i	$0.313312\pi$
$968$	5.78350	0.185889
$969$	3.20226	0.102871
$970$	−3.12072	−0.100200
$971$	16.0220	0.514170	0.257085	−	0.966389i	$-0.417238\pi$
$971$	16.0220	0.514170	0.257085	+	0.966389i	$0.417238\pi$
$972$	15.8723	0.509103
$973$	4.53903	0.145515
$974$	21.2802	0.681862
$975$	0.815721	0.0261240
$976$	−4.89971	−0.156836
$977$	56.5527	1.80928	0.904640	−	0.426176i	$-0.140140\pi$
$977$	56.5527	1.80928	0.904640	+	0.426176i	$0.140140\pi$
$978$	3.34767	0.107047
$979$	28.5647	0.912933
$980$	6.89264	0.220177
$981$	−26.2864	−0.839262
$982$	−37.4826	−1.19612
$983$	−17.9243	−0.571697	−0.285848	−	0.958275i	$-0.592275\pi$
$983$	−17.9243	−0.571697	−0.285848	+	0.958275i	$0.592275\pi$
$984$	−2.67649	−0.0853235
$985$	15.8659	0.505531
$986$	−6.11365	−0.194698
$987$	1.22658	0.0390426
$988$	2.59453	0.0825429
$989$	−36.6232	−1.16455
$990$	−5.33215	−0.169467
$991$	38.6946	1.22917	0.614587	−	0.788849i	$-0.289323\pi$
$991$	38.6946	1.22917	0.614587	+	0.788849i	$0.289323\pi$
$992$	1.00000	0.0317500
$993$	25.2990	0.802840
$994$	−0.550216	−0.0174518
$995$	18.3784	0.582635
$996$	5.22108	0.165436
$997$	14.4341	0.457133	0.228566	−	0.973528i	$-0.426596\pi$
$997$	14.4341	0.457133	0.228566	+	0.973528i	$0.426596\pi$
$998$	26.7258	0.845990
$999$	−2.10091	−0.0664700

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.k.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.k.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.815721 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.815721 q^{6} +0.327652 q^{7} -1.00000 q^{8} -2.33460 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.815721 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.815721 q^{6} +0.327652 q^{7} -1.00000 q^{8} -2.33460 q^{9} +1.00000 q^{10} +2.28397 q^{11} +0.815721 q^{12} +1.00000 q^{13} -0.327652 q^{14} -0.815721 q^{15} +1.00000 q^{16} +1.51306 q^{17} +2.33460 q^{18} +2.59453 q^{19} -1.00000 q^{20} +0.267273 q^{21} -2.28397 q^{22} -5.27917 q^{23} -0.815721 q^{24} +1.00000 q^{25} -1.00000 q^{26} -4.35154 q^{27} +0.327652 q^{28} +4.04058 q^{29} +0.815721 q^{30} -1.00000 q^{31} -1.00000 q^{32} +1.86308 q^{33} -1.51306 q^{34} -0.327652 q^{35} -2.33460 q^{36} +0.482797 q^{37} -2.59453 q^{38} +0.815721 q^{39} +1.00000 q^{40} +3.28114 q^{41} -0.267273 q^{42} +6.93730 q^{43} +2.28397 q^{44} +2.33460 q^{45} +5.27917 q^{46} +4.58925 q^{47} +0.815721 q^{48} -6.89264 q^{49} -1.00000 q^{50} +1.23424 q^{51} +1.00000 q^{52} +0.858276 q^{53} +4.35154 q^{54} -2.28397 q^{55} -0.327652 q^{56} +2.11641 q^{57} -4.04058 q^{58} +8.54187 q^{59} -0.815721 q^{60} -4.89971 q^{61} +1.00000 q^{62} -0.764937 q^{63} +1.00000 q^{64} -1.00000 q^{65} -1.86308 q^{66} -8.76914 q^{67} +1.51306 q^{68} -4.30633 q^{69} +0.327652 q^{70} +1.67927 q^{71} +2.33460 q^{72} -9.74767 q^{73} -0.482797 q^{74} +0.815721 q^{75} +2.59453 q^{76} +0.748347 q^{77} -0.815721 q^{78} -12.3833 q^{79} -1.00000 q^{80} +3.45416 q^{81} -3.28114 q^{82} +6.40058 q^{83} +0.267273 q^{84} -1.51306 q^{85} -6.93730 q^{86} +3.29599 q^{87} -2.28397 q^{88} +12.5066 q^{89} -2.33460 q^{90} +0.327652 q^{91} -5.27917 q^{92} -0.815721 q^{93} -4.58925 q^{94} -2.59453 q^{95} -0.815721 q^{96} -3.12072 q^{97} +6.89264 q^{98} -5.33215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * q^99

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