LMFDB - Embedded newform 4030.2.a.l.1.2

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} - 14x^{6} - 6x^{5} + 54x^{4} + 46x^{3} - 32x^{2} - 43x - 11 $ x^8 - 14x^6 - 6x^5 + 54x^4 + 46x^3 - 32x^2 - 43x - 11
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			1.2
Root			$-0.717467$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.80420 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.80420 q^{6} +1.26189 q^{7} -1.00000 q^{8} +0.255155 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.80420 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.80420 q^{6} +1.26189 q^{7} -1.00000 q^{8} +0.255155 q^{9} -1.00000 q^{10} +5.20315 q^{11} -1.80420 q^{12} -1.00000 q^{13} -1.26189 q^{14} -1.80420 q^{15} +1.00000 q^{16} +3.04916 q^{17} -0.255155 q^{18} +2.55020 q^{19} +1.00000 q^{20} -2.27672 q^{21} -5.20315 q^{22} +3.11332 q^{23} +1.80420 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.95226 q^{27} +1.26189 q^{28} -8.20104 q^{29} +1.80420 q^{30} -1.00000 q^{31} -1.00000 q^{32} -9.38756 q^{33} -3.04916 q^{34} +1.26189 q^{35} +0.255155 q^{36} +9.46853 q^{37} -2.55020 q^{38} +1.80420 q^{39} -1.00000 q^{40} +1.94424 q^{41} +2.27672 q^{42} +10.3182 q^{43} +5.20315 q^{44} +0.255155 q^{45} -3.11332 q^{46} +3.22661 q^{47} -1.80420 q^{48} -5.40762 q^{49} -1.00000 q^{50} -5.50131 q^{51} -1.00000 q^{52} +9.82886 q^{53} -4.95226 q^{54} +5.20315 q^{55} -1.26189 q^{56} -4.60109 q^{57} +8.20104 q^{58} -13.2513 q^{59} -1.80420 q^{60} -4.83536 q^{61} +1.00000 q^{62} +0.321979 q^{63} +1.00000 q^{64} -1.00000 q^{65} +9.38756 q^{66} +9.67742 q^{67} +3.04916 q^{68} -5.61707 q^{69} -1.26189 q^{70} -0.325306 q^{71} -0.255155 q^{72} -8.34787 q^{73} -9.46853 q^{74} -1.80420 q^{75} +2.55020 q^{76} +6.56583 q^{77} -1.80420 q^{78} -4.90554 q^{79} +1.00000 q^{80} -9.70036 q^{81} -1.94424 q^{82} +1.56202 q^{83} -2.27672 q^{84} +3.04916 q^{85} -10.3182 q^{86} +14.7964 q^{87} -5.20315 q^{88} +3.26919 q^{89} -0.255155 q^{90} -1.26189 q^{91} +3.11332 q^{92} +1.80420 q^{93} -3.22661 q^{94} +2.55020 q^{95} +1.80420 q^{96} -6.86145 q^{97} +5.40762 q^{98} +1.32761 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9	$ 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9} - 8 q^{10} + q^{12} - 8 q^{13} - 11 q^{14} + q^{15} + 8 q^{16} + 7 q^{17} - 9 q^{18} - 2 q^{19} + 8 q^{20} + q^{21} + 8 q^{23} - q^{24} + 8 q^{25} + 8 q^{26} + 7 q^{27} + 11 q^{28} - q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 14 q^{33} - 7 q^{34} + 11 q^{35} + 9 q^{36} + q^{37} + 2 q^{38} - q^{39} - 8 q^{40} + 16 q^{41} - q^{42} + 3 q^{43} + 9 q^{45} - 8 q^{46} + 29 q^{47} + q^{48} + 11 q^{49} - 8 q^{50} + 11 q^{51} - 8 q^{52} + 22 q^{53} - 7 q^{54} - 11 q^{56} + 33 q^{57} + q^{58} - 8 q^{59} + q^{60} + 4 q^{61} + 8 q^{62} + 38 q^{63} + 8 q^{64} - 8 q^{65} - 14 q^{66} + 28 q^{67} + 7 q^{68} - 42 q^{69} - 11 q^{70} + 4 q^{71} - 9 q^{72} + 39 q^{73} - q^{74} + q^{75} - 2 q^{76} + 11 q^{77} + q^{78} - 16 q^{79} + 8 q^{80} + 32 q^{81} - 16 q^{82} + 25 q^{83} + q^{84} + 7 q^{85} - 3 q^{86} + 13 q^{87} + 21 q^{89} - 9 q^{90} - 11 q^{91} + 8 q^{92} - q^{93} - 29 q^{94} - 2 q^{95} - q^{96} + 28 q^{97} - 11 q^{98} + 23 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9 - 8 * q^10 + q^12 - 8 * q^13 - 11 * q^14 + q^15 + 8 * q^16 + 7 * q^17 - 9 * q^18 - 2 * q^19 + 8 * q^20 + q^21 + 8 * q^23 - q^24 + 8 * q^25 + 8 * q^26 + 7 * q^27 + 11 * q^28 - q^29 - q^30 - 8 * q^31 - 8 * q^32 + 14 * q^33 - 7 * q^34 + 11 * q^35 + 9 * q^36 + q^37 + 2 * q^38 - q^39 - 8 * q^40 + 16 * q^41 - q^42 + 3 * q^43 + 9 * q^45 - 8 * q^46 + 29 * q^47 + q^48 + 11 * q^49 - 8 * q^50 + 11 * q^51 - 8 * q^52 + 22 * q^53 - 7 * q^54 - 11 * q^56 + 33 * q^57 + q^58 - 8 * q^59 + q^60 + 4 * q^61 + 8 * q^62 + 38 * q^63 + 8 * q^64 - 8 * q^65 - 14 * q^66 + 28 * q^67 + 7 * q^68 - 42 * q^69 - 11 * q^70 + 4 * q^71 - 9 * q^72 + 39 * q^73 - q^74 + q^75 - 2 * q^76 + 11 * q^77 + q^78 - 16 * q^79 + 8 * q^80 + 32 * q^81 - 16 * q^82 + 25 * q^83 + q^84 + 7 * q^85 - 3 * q^86 + 13 * q^87 + 21 * q^89 - 9 * q^90 - 11 * q^91 + 8 * q^92 - q^93 - 29 * q^94 - 2 * q^95 - q^96 + 28 * q^97 - 11 * q^98 + 23 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.80420	−1.04166	−0.520829	−	0.853661i	$-0.674377\pi$
$3$	−1.80420	−1.04166	−0.520829	+	0.853661i	$0.674377\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.80420	0.736564
$7$	1.26189	0.476951	0.238476	−	0.971148i	$-0.423352\pi$
$7$	1.26189	0.476951	0.238476	+	0.971148i	$0.423352\pi$
$8$	−1.00000	−0.353553
$9$	0.255155	0.0850518
$10$	−1.00000	−0.316228
$11$	5.20315	1.56881	0.784405	−	0.620249i	$-0.212968\pi$
$11$	5.20315	1.56881	0.784405	+	0.620249i	$0.212968\pi$
$12$	−1.80420	−0.520829
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.26189	−0.337255
$15$	−1.80420	−0.465844
$16$	1.00000	0.250000
$17$	3.04916	0.739530	0.369765	−	0.929125i	$-0.379438\pi$
$17$	3.04916	0.739530	0.369765	+	0.929125i	$0.379438\pi$
$18$	−0.255155	−0.0601407
$19$	2.55020	0.585057	0.292529	−	0.956257i	$-0.405503\pi$
$19$	2.55020	0.585057	0.292529	+	0.956257i	$0.405503\pi$
$20$	1.00000	0.223607
$21$	−2.27672	−0.496820
$22$	−5.20315	−1.10932
$23$	3.11332	0.649172	0.324586	−	0.945856i	$-0.394775\pi$
$23$	3.11332	0.649172	0.324586	+	0.945856i	$0.394775\pi$
$24$	1.80420	0.368282
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	4.95226	0.953063
$28$	1.26189	0.238476
$29$	−8.20104	−1.52289	−0.761447	−	0.648227i	$-0.775511\pi$
$29$	−8.20104	−1.52289	−0.761447	+	0.648227i	$0.775511\pi$
$30$	1.80420	0.329401
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−9.38756	−1.63416
$34$	−3.04916	−0.522927
$35$	1.26189	0.213299
$36$	0.255155	0.0425259
$37$	9.46853	1.55662	0.778308	−	0.627883i	$-0.216078\pi$
$37$	9.46853	1.55662	0.778308	+	0.627883i	$0.216078\pi$
$38$	−2.55020	−0.413698
$39$	1.80420	0.288904
$40$	−1.00000	−0.158114
$41$	1.94424	0.303639	0.151819	−	0.988408i	$-0.451487\pi$
$41$	1.94424	0.303639	0.151819	+	0.988408i	$0.451487\pi$
$42$	2.27672	0.351305
$43$	10.3182	1.57350	0.786752	−	0.617270i	$-0.211761\pi$
$43$	10.3182	1.57350	0.786752	+	0.617270i	$0.211761\pi$
$44$	5.20315	0.784405
$45$	0.255155	0.0380363
$46$	−3.11332	−0.459034
$47$	3.22661	0.470649	0.235325	−	0.971917i	$-0.424385\pi$
$47$	3.22661	0.470649	0.235325	+	0.971917i	$0.424385\pi$
$48$	−1.80420	−0.260415
$49$	−5.40762	−0.772517
$50$	−1.00000	−0.141421
$51$	−5.50131	−0.770338
$52$	−1.00000	−0.138675
$53$	9.82886	1.35010	0.675049	−	0.737773i	$-0.264123\pi$
$53$	9.82886	1.35010	0.675049	+	0.737773i	$0.264123\pi$
$54$	−4.95226	−0.673918
$55$	5.20315	0.701593
$56$	−1.26189	−0.168628
$57$	−4.60109	−0.609430
$58$	8.20104	1.07685
$59$	−13.2513	−1.72518	−0.862588	−	0.505907i	$-0.831158\pi$
$59$	−13.2513	−1.72518	−0.862588	+	0.505907i	$0.831158\pi$
$60$	−1.80420	−0.232922
$61$	−4.83536	−0.619105	−0.309552	−	0.950882i	$-0.600179\pi$
$61$	−4.83536	−0.619105	−0.309552	+	0.950882i	$0.600179\pi$
$62$	1.00000	0.127000
$63$	0.321979	0.0405656
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	9.38756	1.15553
$67$	9.67742	1.18228	0.591142	−	0.806567i	$-0.298677\pi$
$67$	9.67742	1.18228	0.591142	+	0.806567i	$0.298677\pi$
$68$	3.04916	0.369765
$69$	−5.61707	−0.676216
$70$	−1.26189	−0.150825
$71$	−0.325306	−0.0386068	−0.0193034	−	0.999814i	$-0.506145\pi$
$71$	−0.325306	−0.0386068	−0.0193034	+	0.999814i	$0.506145\pi$
$72$	−0.255155	−0.0300704
$73$	−8.34787	−0.977044	−0.488522	−	0.872552i	$-0.662464\pi$
$73$	−8.34787	−0.977044	−0.488522	+	0.872552i	$0.662464\pi$
$74$	−9.46853	−1.10069
$75$	−1.80420	−0.208332
$76$	2.55020	0.292529
$77$	6.56583	0.748246
$78$	−1.80420	−0.204286
$79$	−4.90554	−0.551917	−0.275958	−	0.961170i	$-0.588995\pi$
$79$	−4.90554	−0.551917	−0.275958	+	0.961170i	$0.588995\pi$
$80$	1.00000	0.111803
$81$	−9.70036	−1.07782
$82$	−1.94424	−0.214705
$83$	1.56202	0.171454	0.0857272	−	0.996319i	$-0.472679\pi$
$83$	1.56202	0.171454	0.0857272	+	0.996319i	$0.472679\pi$
$84$	−2.27672	−0.248410
$85$	3.04916	0.330728
$86$	−10.3182	−1.11263
$87$	14.7964	1.58634
$88$	−5.20315	−0.554658
$89$	3.26919	0.346534	0.173267	−	0.984875i	$-0.444568\pi$
$89$	3.26919	0.346534	0.173267	+	0.984875i	$0.444568\pi$
$90$	−0.255155	−0.0268957
$91$	−1.26189	−0.132282
$92$	3.11332	0.324586
$93$	1.80420	0.187087
$94$	−3.22661	−0.332799
$95$	2.55020	0.261645
$96$	1.80420	0.184141
$97$	−6.86145	−0.696674	−0.348337	−	0.937369i	$-0.613254\pi$
$97$	−6.86145	−0.696674	−0.348337	+	0.937369i	$0.613254\pi$
$98$	5.40762	0.546252
$99$	1.32761	0.133430
$100$	1.00000	0.100000
$101$	9.14296	0.909759	0.454879	−	0.890553i	$-0.349682\pi$
$101$	9.14296	0.909759	0.454879	+	0.890553i	$0.349682\pi$
$102$	5.50131	0.544711
$103$	−15.0334	−1.48129	−0.740643	−	0.671899i	$-0.765479\pi$
$103$	−15.0334	−1.48129	−0.740643	+	0.671899i	$0.765479\pi$
$104$	1.00000	0.0980581
$105$	−2.27672	−0.222185
$106$	−9.82886	−0.954663
$107$	−9.54646	−0.922891	−0.461446	−	0.887169i	$-0.652669\pi$
$107$	−9.54646	−0.922891	−0.461446	+	0.887169i	$0.652669\pi$
$108$	4.95226	0.476532
$109$	−1.96907	−0.188603	−0.0943013	−	0.995544i	$-0.530062\pi$
$109$	−1.96907	−0.188603	−0.0943013	+	0.995544i	$0.530062\pi$
$110$	−5.20315	−0.496101
$111$	−17.0832	−1.62146
$112$	1.26189	0.119238
$113$	15.2647	1.43598	0.717990	−	0.696053i	$-0.245062\pi$
$113$	15.2647	1.43598	0.717990	+	0.696053i	$0.245062\pi$
$114$	4.60109	0.430932
$115$	3.11332	0.290319
$116$	−8.20104	−0.761447
$117$	−0.255155	−0.0235891
$118$	13.2513	1.21988
$119$	3.84772	0.352720
$120$	1.80420	0.164701
$121$	16.0728	1.46116
$122$	4.83536	0.437773
$123$	−3.50780	−0.316288
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	−0.321979	−0.0286842
$127$	8.28048	0.734774	0.367387	−	0.930068i	$-0.380253\pi$
$127$	8.28048	0.734774	0.367387	+	0.930068i	$0.380253\pi$
$128$	−1.00000	−0.0883883
$129$	−18.6161	−1.63905
$130$	1.00000	0.0877058
$131$	−2.73114	−0.238621	−0.119310	−	0.992857i	$-0.538068\pi$
$131$	−2.73114	−0.238621	−0.119310	+	0.992857i	$0.538068\pi$
$132$	−9.38756	−0.817082
$133$	3.21809	0.279044
$134$	−9.67742	−0.836001
$135$	4.95226	0.426223
$136$	−3.04916	−0.261463
$137$	1.06266	0.0907889	0.0453944	−	0.998969i	$-0.485546\pi$
$137$	1.06266	0.0907889	0.0453944	+	0.998969i	$0.485546\pi$
$138$	5.61707	0.478157
$139$	4.64474	0.393961	0.196981	−	0.980407i	$-0.436886\pi$
$139$	4.64474	0.393961	0.196981	+	0.980407i	$0.436886\pi$
$140$	1.26189	0.106650
$141$	−5.82146	−0.490255
$142$	0.325306	0.0272991
$143$	−5.20315	−0.435110
$144$	0.255155	0.0212630
$145$	−8.20104	−0.681059
$146$	8.34787	0.690875
$147$	9.75646	0.804699
$148$	9.46853	0.778308
$149$	13.1281	1.07550	0.537748	−	0.843105i	$-0.319275\pi$
$149$	13.1281	1.07550	0.537748	+	0.843105i	$0.319275\pi$
$150$	1.80420	0.147313
$151$	4.73358	0.385213	0.192607	−	0.981276i	$-0.438306\pi$
$151$	4.73358	0.385213	0.192607	+	0.981276i	$0.438306\pi$
$152$	−2.55020	−0.206849
$153$	0.778010	0.0628984
$154$	−6.56583	−0.529090
$155$	−1.00000	−0.0803219
$156$	1.80420	0.144452
$157$	15.7113	1.25390	0.626951	−	0.779059i	$-0.284303\pi$
$157$	15.7113	1.25390	0.626951	+	0.779059i	$0.284303\pi$
$158$	4.90554	0.390264
$159$	−17.7333	−1.40634
$160$	−1.00000	−0.0790569
$161$	3.92868	0.309624
$162$	9.70036	0.762132
$163$	−20.0642	−1.57155	−0.785776	−	0.618512i	$-0.787736\pi$
$163$	−20.0642	−1.57155	−0.785776	+	0.618512i	$0.787736\pi$
$164$	1.94424	0.151819
$165$	−9.38756	−0.730820
$166$	−1.56202	−0.121237
$167$	−17.1371	−1.32611	−0.663056	−	0.748570i	$-0.730741\pi$
$167$	−17.1371	−1.32611	−0.663056	+	0.748570i	$0.730741\pi$
$168$	2.27672	0.175652
$169$	1.00000	0.0769231
$170$	−3.04916	−0.233860
$171$	0.650699	0.0497602
$172$	10.3182	0.786752
$173$	−21.9585	−1.66948	−0.834738	−	0.550647i	$-0.814381\pi$
$173$	−21.9585	−1.66948	−0.834738	+	0.550647i	$0.814381\pi$
$174$	−14.7964	−1.12171
$175$	1.26189	0.0953903
$176$	5.20315	0.392202
$177$	23.9081	1.79704
$178$	−3.26919	−0.245036
$179$	−14.9512	−1.11751	−0.558754	−	0.829333i	$-0.688721\pi$
$179$	−14.9512	−1.11751	−0.558754	+	0.829333i	$0.688721\pi$
$180$	0.255155	0.0190182
$181$	18.1416	1.34846	0.674228	−	0.738524i	$-0.264477\pi$
$181$	18.1416	1.34846	0.674228	+	0.738524i	$0.264477\pi$
$182$	1.26189	0.0935378
$183$	8.72398	0.644895
$184$	−3.11332	−0.229517
$185$	9.46853	0.696140
$186$	−1.80420	−0.132291
$187$	15.8653	1.16018
$188$	3.22661	0.235325
$189$	6.24923	0.454565
$190$	−2.55020	−0.185011
$191$	−2.43217	−0.175985	−0.0879927	−	0.996121i	$-0.528045\pi$
$191$	−2.43217	−0.175985	−0.0879927	+	0.996121i	$0.528045\pi$
$192$	−1.80420	−0.130207
$193$	−6.62320	−0.476748	−0.238374	−	0.971173i	$-0.576614\pi$
$193$	−6.62320	−0.476748	−0.238374	+	0.971173i	$0.576614\pi$
$194$	6.86145	0.492623
$195$	1.80420	0.129202
$196$	−5.40762	−0.386259
$197$	−22.6787	−1.61579	−0.807896	−	0.589325i	$-0.799394\pi$
$197$	−22.6787	−1.61579	−0.807896	+	0.589325i	$0.799394\pi$
$198$	−1.32761	−0.0943494
$199$	−20.6502	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$199$	−20.6502	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$200$	−1.00000	−0.0707107
$201$	−17.4600	−1.23154
$202$	−9.14296	−0.643297
$203$	−10.3488	−0.726346
$204$	−5.50131	−0.385169
$205$	1.94424	0.135791
$206$	15.0334	1.04743
$207$	0.794381	0.0552133
$208$	−1.00000	−0.0693375
$209$	13.2691	0.917843
$210$	2.27672	0.157108
$211$	20.3608	1.40170	0.700849	−	0.713310i	$-0.252805\pi$
$211$	20.3608	1.40170	0.700849	+	0.713310i	$0.252805\pi$
$212$	9.82886	0.675049
$213$	0.586919	0.0402151
$214$	9.54646	0.652582
$215$	10.3182	0.703692
$216$	−4.95226	−0.336959
$217$	−1.26189	−0.0856630
$218$	1.96907	0.133362
$219$	15.0613	1.01775
$220$	5.20315	0.350797
$221$	−3.04916	−0.205109
$222$	17.0832	1.14655
$223$	−3.82178	−0.255926	−0.127963	−	0.991779i	$-0.540844\pi$
$223$	−3.82178	−0.255926	−0.127963	+	0.991779i	$0.540844\pi$
$224$	−1.26189	−0.0843139
$225$	0.255155	0.0170104
$226$	−15.2647	−1.01539
$227$	−12.7995	−0.849534	−0.424767	−	0.905303i	$-0.639644\pi$
$227$	−12.7995	−0.849534	−0.424767	+	0.905303i	$0.639644\pi$
$228$	−4.60109	−0.304715
$229$	−12.9787	−0.857660	−0.428830	−	0.903385i	$-0.641074\pi$
$229$	−12.9787	−0.857660	−0.428830	+	0.903385i	$0.641074\pi$
$230$	−3.11332	−0.205286
$231$	−11.8461	−0.779416
$232$	8.20104	0.538424
$233$	24.8642	1.62891	0.814453	−	0.580229i	$-0.197037\pi$
$233$	24.8642	1.62891	0.814453	+	0.580229i	$0.197037\pi$
$234$	0.255155	0.0166800
$235$	3.22661	0.210481
$236$	−13.2513	−0.862588
$237$	8.85060	0.574908
$238$	−3.84772	−0.249411
$239$	28.9999	1.87585	0.937925	−	0.346838i	$-0.112745\pi$
$239$	28.9999	1.87585	0.937925	+	0.346838i	$0.112745\pi$
$240$	−1.80420	−0.116461
$241$	5.96626	0.384320	0.192160	−	0.981364i	$-0.438451\pi$
$241$	5.96626	0.384320	0.192160	+	0.981364i	$0.438451\pi$
$242$	−16.0728	−1.03320
$243$	2.64466	0.169655
$244$	−4.83536	−0.309552
$245$	−5.40762	−0.345480
$246$	3.50780	0.223649
$247$	−2.55020	−0.162266
$248$	1.00000	0.0635001
$249$	−2.81821	−0.178597
$250$	−1.00000	−0.0632456
$251$	−16.2566	−1.02610	−0.513052	−	0.858357i	$-0.671485\pi$
$251$	−16.2566	−1.02610	−0.513052	+	0.858357i	$0.671485\pi$
$252$	0.321979	0.0202828
$253$	16.1991	1.01843
$254$	−8.28048	−0.519563
$255$	−5.50131	−0.344505
$256$	1.00000	0.0625000
$257$	19.2490	1.20072	0.600360	−	0.799730i	$-0.295024\pi$
$257$	19.2490	1.20072	0.600360	+	0.799730i	$0.295024\pi$
$258$	18.6161	1.15899
$259$	11.9483	0.742430
$260$	−1.00000	−0.0620174
$261$	−2.09254	−0.129525
$262$	2.73114	0.168730
$263$	14.9522	0.921990	0.460995	−	0.887403i	$-0.347493\pi$
$263$	14.9522	0.921990	0.460995	+	0.887403i	$0.347493\pi$
$264$	9.38756	0.577764
$265$	9.82886	0.603782
$266$	−3.21809	−0.197314
$267$	−5.89829	−0.360969
$268$	9.67742	0.591142
$269$	30.0709	1.83346	0.916729	−	0.399509i	$-0.130820\pi$
$269$	30.0709	1.83346	0.916729	+	0.399509i	$0.130820\pi$
$270$	−4.95226	−0.301385
$271$	18.7935	1.14162	0.570812	−	0.821081i	$-0.306628\pi$
$271$	18.7935	1.14162	0.570812	+	0.821081i	$0.306628\pi$
$272$	3.04916	0.184883
$273$	2.27672	0.137793
$274$	−1.06266	−0.0641974
$275$	5.20315	0.313762
$276$	−5.61707	−0.338108
$277$	−6.57335	−0.394954	−0.197477	−	0.980308i	$-0.563275\pi$
$277$	−6.57335	−0.394954	−0.197477	+	0.980308i	$0.563275\pi$
$278$	−4.64474	−0.278573
$279$	−0.255155	−0.0152758
$280$	−1.26189	−0.0754126
$281$	9.87327	0.588990	0.294495	−	0.955653i	$-0.404849\pi$
$281$	9.87327	0.588990	0.294495	+	0.955653i	$0.404849\pi$
$282$	5.82146	0.346663
$283$	24.0156	1.42758	0.713791	−	0.700359i	$-0.246977\pi$
$283$	24.0156	1.42758	0.713791	+	0.700359i	$0.246977\pi$
$284$	−0.325306	−0.0193034
$285$	−4.60109	−0.272545
$286$	5.20315	0.307669
$287$	2.45342	0.144821
$288$	−0.255155	−0.0150352
$289$	−7.70262	−0.453095
$290$	8.20104	0.481581
$291$	12.3795	0.725696
$292$	−8.34787	−0.488522
$293$	15.8910	0.928362	0.464181	−	0.885740i	$-0.346349\pi$
$293$	15.8910	0.928362	0.464181	+	0.885740i	$0.346349\pi$
$294$	−9.75646	−0.569008
$295$	−13.2513	−0.771522
$296$	−9.46853	−0.550347
$297$	25.7674	1.49517
$298$	−13.1281	−0.760491
$299$	−3.11332	−0.180048
$300$	−1.80420	−0.104166
$301$	13.0204	0.750484
$302$	−4.73358	−0.272387
$303$	−16.4958	−0.947658
$304$	2.55020	0.146264
$305$	−4.83536	−0.276872
$306$	−0.778010	−0.0444759
$307$	12.0612	0.688371	0.344186	−	0.938902i	$-0.388155\pi$
$307$	12.0612	0.688371	0.344186	+	0.938902i	$0.388155\pi$
$308$	6.56583	0.374123
$309$	27.1233	1.54299
$310$	1.00000	0.0567962
$311$	21.5639	1.22278	0.611389	−	0.791331i	$-0.290611\pi$
$311$	21.5639	1.22278	0.611389	+	0.791331i	$0.290611\pi$
$312$	−1.80420	−0.102143
$313$	19.7406	1.11580	0.557901	−	0.829907i	$-0.311607\pi$
$313$	19.7406	1.11580	0.557901	+	0.829907i	$0.311607\pi$
$314$	−15.7113	−0.886642
$315$	0.321979	0.0181415
$316$	−4.90554	−0.275958
$317$	21.4862	1.20679	0.603393	−	0.797444i	$-0.293815\pi$
$317$	21.4862	1.20679	0.603393	+	0.797444i	$0.293815\pi$
$318$	17.7333	0.994433
$319$	−42.6713	−2.38913
$320$	1.00000	0.0559017
$321$	17.2238	0.961337
$322$	−3.92868	−0.218937
$323$	7.77599	0.432667
$324$	−9.70036	−0.538909
$325$	−1.00000	−0.0554700
$326$	20.0642	1.11125
$327$	3.55260	0.196459
$328$	−1.94424	−0.107352
$329$	4.07164	0.224477
$330$	9.38756	0.516768
$331$	−4.30482	−0.236614	−0.118307	−	0.992977i	$-0.537747\pi$
$331$	−4.30482	−0.236614	−0.118307	+	0.992977i	$0.537747\pi$
$332$	1.56202	0.0857272
$333$	2.41595	0.132393
$334$	17.1371	0.937703
$335$	9.67742	0.528734
$336$	−2.27672	−0.124205
$337$	−0.461400	−0.0251340	−0.0125670	−	0.999921i	$-0.504000\pi$
$337$	−0.461400	−0.0251340	−0.0125670	+	0.999921i	$0.504000\pi$
$338$	−1.00000	−0.0543928
$339$	−27.5406	−1.49580
$340$	3.04916	0.165364
$341$	−5.20315	−0.281767
$342$	−0.650699	−0.0351858
$343$	−15.6571	−0.845404
$344$	−10.3182	−0.556317
$345$	−5.61707	−0.302413
$346$	21.9585	1.18050
$347$	29.4631	1.58166	0.790831	−	0.612035i	$-0.209649\pi$
$347$	29.4631	1.58166	0.790831	+	0.612035i	$0.209649\pi$
$348$	14.7964	0.793168
$349$	−10.5639	−0.565474	−0.282737	−	0.959197i	$-0.591242\pi$
$349$	−10.5639	−0.565474	−0.282737	+	0.959197i	$0.591242\pi$
$350$	−1.26189	−0.0674511
$351$	−4.95226	−0.264332
$352$	−5.20315	−0.277329
$353$	36.6754	1.95203	0.976016	−	0.217697i	$-0.0698546\pi$
$353$	36.6754	1.95203	0.976016	+	0.217697i	$0.0698546\pi$
$354$	−23.9081	−1.27070
$355$	−0.325306	−0.0172655
$356$	3.26919	0.173267
$357$	−6.94207	−0.367414
$358$	14.9512	0.790198
$359$	34.0967	1.79956	0.899778	−	0.436347i	$-0.143728\pi$
$359$	34.0967	1.79956	0.899778	+	0.436347i	$0.143728\pi$
$360$	−0.255155	−0.0134479
$361$	−12.4965	−0.657708
$362$	−18.1416	−0.953502
$363$	−28.9986	−1.52203
$364$	−1.26189	−0.0661412
$365$	−8.34787	−0.436947
$366$	−8.72398	−0.456010
$367$	−27.4453	−1.43263	−0.716316	−	0.697776i	$-0.754173\pi$
$367$	−27.4453	−1.43263	−0.716316	+	0.697776i	$0.754173\pi$
$368$	3.11332	0.162293
$369$	0.496082	0.0258250
$370$	−9.46853	−0.492245
$371$	12.4030	0.643931
$372$	1.80420	0.0935437
$373$	3.94698	0.204367	0.102183	−	0.994766i	$-0.467417\pi$
$373$	3.94698	0.204367	0.102183	+	0.994766i	$0.467417\pi$
$374$	−15.8653	−0.820373
$375$	−1.80420	−0.0931687
$376$	−3.22661	−0.166400
$377$	8.20104	0.422375
$378$	−6.24923	−0.321426
$379$	−22.0697	−1.13365	−0.566823	−	0.823839i	$-0.691828\pi$
$379$	−22.0697	−1.13365	−0.566823	+	0.823839i	$0.691828\pi$
$380$	2.55020	0.130823
$381$	−14.9397	−0.765383
$382$	2.43217	0.124440
$383$	−21.0227	−1.07421	−0.537106	−	0.843515i	$-0.680483\pi$
$383$	−21.0227	−1.07421	−0.537106	+	0.843515i	$0.680483\pi$
$384$	1.80420	0.0920704
$385$	6.56583	0.334626
$386$	6.62320	0.337112
$387$	2.63273	0.133829
$388$	−6.86145	−0.348337
$389$	−1.12256	−0.0569160	−0.0284580	−	0.999595i	$-0.509060\pi$
$389$	−1.12256	−0.0569160	−0.0284580	+	0.999595i	$0.509060\pi$
$390$	−1.80420	−0.0913595
$391$	9.49302	0.480083
$392$	5.40762	0.273126
$393$	4.92754	0.248561
$394$	22.6787	1.14254
$395$	−4.90554	−0.246825
$396$	1.32761	0.0667151
$397$	−16.1935	−0.812730	−0.406365	−	0.913711i	$-0.633204\pi$
$397$	−16.1935	−0.812730	−0.406365	+	0.913711i	$0.633204\pi$
$398$	20.6502	1.03510
$399$	−5.80609	−0.290668
$400$	1.00000	0.0500000
$401$	−2.24768	−0.112244	−0.0561220	−	0.998424i	$-0.517874\pi$
$401$	−2.24768	−0.112244	−0.0561220	+	0.998424i	$0.517874\pi$
$402$	17.4600	0.870828
$403$	1.00000	0.0498135
$404$	9.14296	0.454879
$405$	−9.70036	−0.482015
$406$	10.3488	0.513604
$407$	49.2662	2.44203
$408$	5.50131	0.272356
$409$	10.5232	0.520339	0.260170	−	0.965563i	$-0.416221\pi$
$409$	10.5232	0.520339	0.260170	+	0.965563i	$0.416221\pi$
$410$	−1.94424	−0.0960190
$411$	−1.91725	−0.0945710
$412$	−15.0334	−0.740643
$413$	−16.7218	−0.822825
$414$	−0.794381	−0.0390417
$415$	1.56202	0.0766767
$416$	1.00000	0.0490290
$417$	−8.38006	−0.410373
$418$	−13.2691	−0.649013
$419$	3.01900	0.147488	0.0737439	−	0.997277i	$-0.476505\pi$
$419$	3.01900	0.147488	0.0737439	+	0.997277i	$0.476505\pi$
$420$	−2.27672	−0.111092
$421$	−6.09983	−0.297287	−0.148644	−	0.988891i	$-0.547491\pi$
$421$	−6.09983	−0.297287	−0.148644	+	0.988891i	$0.547491\pi$
$422$	−20.3608	−0.991150
$423$	0.823287	0.0400296
$424$	−9.82886	−0.477332
$425$	3.04916	0.147906
$426$	−0.586919	−0.0284363
$427$	−6.10172	−0.295283
$428$	−9.54646	−0.461446
$429$	9.38756	0.453235
$430$	−10.3182	−0.497585
$431$	−14.0806	−0.678240	−0.339120	−	0.940743i	$-0.610129\pi$
$431$	−14.0806	−0.678240	−0.339120	+	0.940743i	$0.610129\pi$
$432$	4.95226	0.238266
$433$	26.0066	1.24980	0.624899	−	0.780705i	$-0.285140\pi$
$433$	26.0066	1.24980	0.624899	+	0.780705i	$0.285140\pi$
$434$	1.26189	0.0605729
$435$	14.7964	0.709431
$436$	−1.96907	−0.0943013
$437$	7.93961	0.379803
$438$	−15.0613	−0.719655
$439$	5.74031	0.273970	0.136985	−	0.990573i	$-0.456259\pi$
$439$	5.74031	0.273970	0.136985	+	0.990573i	$0.456259\pi$
$440$	−5.20315	−0.248051
$441$	−1.37978	−0.0657040
$442$	3.04916	0.145034
$443$	17.4703	0.830039	0.415019	−	0.909813i	$-0.363775\pi$
$443$	17.4703	0.830039	0.415019	+	0.909813i	$0.363775\pi$
$444$	−17.0832	−0.810731
$445$	3.26919	0.154975
$446$	3.82178	0.180967
$447$	−23.6858	−1.12030
$448$	1.26189	0.0596189
$449$	19.4289	0.916906	0.458453	−	0.888719i	$-0.348404\pi$
$449$	19.4289	0.916906	0.458453	+	0.888719i	$0.348404\pi$
$450$	−0.255155	−0.0120281
$451$	10.1162	0.476351
$452$	15.2647	0.717990
$453$	−8.54035	−0.401261
$454$	12.7995	0.600711
$455$	−1.26189	−0.0591585
$456$	4.60109	0.215466
$457$	35.7071	1.67031	0.835154	−	0.550017i	$-0.185379\pi$
$457$	35.7071	1.67031	0.835154	+	0.550017i	$0.185379\pi$
$458$	12.9787	0.606457
$459$	15.1002	0.704819
$460$	3.11332	0.145159
$461$	32.8642	1.53064	0.765318	−	0.643652i	$-0.222582\pi$
$461$	32.8642	1.53064	0.765318	+	0.643652i	$0.222582\pi$
$462$	11.8461	0.551131
$463$	−22.5716	−1.04899	−0.524495	−	0.851414i	$-0.675746\pi$
$463$	−22.5716	−1.04899	−0.524495	+	0.851414i	$0.675746\pi$
$464$	−8.20104	−0.380724
$465$	1.80420	0.0836680
$466$	−24.8642	−1.15181
$467$	36.4635	1.68733	0.843665	−	0.536870i	$-0.180394\pi$
$467$	36.4635	1.68733	0.843665	+	0.536870i	$0.180394\pi$
$468$	−0.255155	−0.0117946
$469$	12.2119	0.563892
$470$	−3.22661	−0.148832
$471$	−28.3465	−1.30614
$472$	13.2513	0.609942
$473$	53.6869	2.46853
$474$	−8.85060	−0.406522
$475$	2.55020	0.117011
$476$	3.84772	0.176360
$477$	2.50789	0.114828
$478$	−28.9999	−1.32643
$479$	6.99030	0.319395	0.159697	−	0.987166i	$-0.448948\pi$
$479$	6.99030	0.319395	0.159697	+	0.987166i	$0.448948\pi$
$480$	1.80420	0.0823503
$481$	−9.46853	−0.431728
$482$	−5.96626	−0.271756
$483$	−7.08815	−0.322522
$484$	16.0728	0.730582
$485$	−6.86145	−0.311562
$486$	−2.64466	−0.119964
$487$	−2.34010	−0.106040	−0.0530199	−	0.998593i	$-0.516885\pi$
$487$	−2.34010	−0.106040	−0.0530199	+	0.998593i	$0.516885\pi$
$488$	4.83536	0.218887
$489$	36.2000	1.63702
$490$	5.40762	0.244291
$491$	14.9917	0.676564	0.338282	−	0.941045i	$-0.390154\pi$
$491$	14.9917	0.676564	0.338282	+	0.941045i	$0.390154\pi$
$492$	−3.50780	−0.158144
$493$	−25.0063	−1.12623
$494$	2.55020	0.114739
$495$	1.32761	0.0596718
$496$	−1.00000	−0.0449013
$497$	−0.410502	−0.0184135
$498$	2.81821	0.126287
$499$	27.6968	1.23988	0.619940	−	0.784649i	$-0.287157\pi$
$499$	27.6968	1.23988	0.619940	+	0.784649i	$0.287157\pi$
$500$	1.00000	0.0447214
$501$	30.9189	1.38136
$502$	16.2566	0.725565
$503$	0.889597	0.0396652	0.0198326	−	0.999803i	$-0.493687\pi$
$503$	0.889597	0.0396652	0.0198326	+	0.999803i	$0.493687\pi$
$504$	−0.321979	−0.0143421
$505$	9.14296	0.406857
$506$	−16.1991	−0.720137
$507$	−1.80420	−0.0801276
$508$	8.28048	0.367387
$509$	−17.4512	−0.773512	−0.386756	−	0.922182i	$-0.626405\pi$
$509$	−17.4512	−0.773512	−0.386756	+	0.922182i	$0.626405\pi$
$510$	5.50131	0.243602
$511$	−10.5341	−0.466002
$512$	−1.00000	−0.0441942
$513$	12.6293	0.557596
$514$	−19.2490	−0.849038
$515$	−15.0334	−0.662451
$516$	−18.6161	−0.819526
$517$	16.7885	0.738359
$518$	−11.9483	−0.524977
$519$	39.6177	1.73902
$520$	1.00000	0.0438529
$521$	−18.9804	−0.831546	−0.415773	−	0.909468i	$-0.636489\pi$
$521$	−18.9804	−0.831546	−0.415773	+	0.909468i	$0.636489\pi$
$522$	2.09254	0.0915880
$523$	−19.3652	−0.846783	−0.423392	−	0.905947i	$-0.639161\pi$
$523$	−19.3652	−0.846783	−0.423392	+	0.905947i	$0.639161\pi$
$524$	−2.73114	−0.119310
$525$	−2.27672	−0.0993640
$526$	−14.9522	−0.651945
$527$	−3.04916	−0.132824
$528$	−9.38756	−0.408541
$529$	−13.3072	−0.578575
$530$	−9.82886	−0.426938
$531$	−3.38115	−0.146729
$532$	3.21809	0.139522
$533$	−1.94424	−0.0842142
$534$	5.89829	0.255244
$535$	−9.54646	−0.412729
$536$	−9.67742	−0.418001
$537$	26.9751	1.16406
$538$	−30.0709	−1.29645
$539$	−28.1367	−1.21193
$540$	4.95226	0.213111
$541$	−23.6836	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$541$	−23.6836	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$542$	−18.7935	−0.807250
$543$	−32.7312	−1.40463
$544$	−3.04916	−0.130732
$545$	−1.96907	−0.0843456
$546$	−2.27672	−0.0974345
$547$	−8.38543	−0.358535	−0.179268	−	0.983800i	$-0.557373\pi$
$547$	−8.38543	−0.358535	−0.179268	+	0.983800i	$0.557373\pi$
$548$	1.06266	0.0453944
$549$	−1.23377	−0.0526560
$550$	−5.20315	−0.221863
$551$	−20.9143	−0.890980
$552$	5.61707	0.239078
$553$	−6.19028	−0.263237
$554$	6.57335	0.279275
$555$	−17.0832	−0.725140
$556$	4.64474	0.196981
$557$	−33.9605	−1.43895	−0.719475	−	0.694518i	$-0.755618\pi$
$557$	−33.9605	−1.43895	−0.719475	+	0.694518i	$0.755618\pi$
$558$	0.255155	0.0108016
$559$	−10.3182	−0.436411
$560$	1.26189	0.0533248
$561$	−28.6242	−1.20851
$562$	−9.87327	−0.416479
$563$	28.3258	1.19379	0.596896	−	0.802319i	$-0.296400\pi$
$563$	28.3258	1.19379	0.596896	+	0.802319i	$0.296400\pi$
$564$	−5.82146	−0.245128
$565$	15.2647	0.642190
$566$	−24.0156	−1.00945
$567$	−12.2408	−0.514067
$568$	0.325306	0.0136496
$569$	−4.01139	−0.168166	−0.0840832	−	0.996459i	$-0.526796\pi$
$569$	−4.01139	−0.168166	−0.0840832	+	0.996459i	$0.526796\pi$
$570$	4.60109	0.192719
$571$	−33.8135	−1.41505	−0.707525	−	0.706688i	$-0.750189\pi$
$571$	−33.8135	−1.41505	−0.707525	+	0.706688i	$0.750189\pi$
$572$	−5.20315	−0.217555
$573$	4.38813	0.183317
$574$	−2.45342	−0.102404
$575$	3.11332	0.129834
$576$	0.255155	0.0106315
$577$	12.2579	0.510302	0.255151	−	0.966901i	$-0.417875\pi$
$577$	12.2579	0.510302	0.255151	+	0.966901i	$0.417875\pi$
$578$	7.70262	0.320387
$579$	11.9496	0.496609
$580$	−8.20104	−0.340530
$581$	1.97111	0.0817754
$582$	−12.3795	−0.513145
$583$	51.1411	2.11805
$584$	8.34787	0.345437
$585$	−0.255155	−0.0105494
$586$	−15.8910	−0.656451
$587$	−22.5883	−0.932319	−0.466159	−	0.884701i	$-0.654363\pi$
$587$	−22.5883	−0.932319	−0.466159	+	0.884701i	$0.654363\pi$
$588$	9.75646	0.402350
$589$	−2.55020	−0.105079
$590$	13.2513	0.545549
$591$	40.9171	1.68310
$592$	9.46853	0.389154
$593$	−35.6148	−1.46252	−0.731262	−	0.682097i	$-0.761068\pi$
$593$	−35.6148	−1.46252	−0.731262	+	0.682097i	$0.761068\pi$
$594$	−25.7674	−1.05725
$595$	3.84772	0.157741
$596$	13.1281	0.537748
$597$	37.2572	1.52484
$598$	3.11332	0.127313
$599$	36.8552	1.50586	0.752931	−	0.658099i	$-0.228639\pi$
$599$	36.8552	1.50586	0.752931	+	0.658099i	$0.228639\pi$
$600$	1.80420	0.0736564
$601$	0.243213	0.00992085	0.00496042	−	0.999988i	$-0.498421\pi$
$601$	0.243213	0.00992085	0.00496042	+	0.999988i	$0.498421\pi$
$602$	−13.0204	−0.530673
$603$	2.46925	0.100555
$604$	4.73358	0.192607
$605$	16.0728	0.653452
$606$	16.4958	0.670095
$607$	−7.42121	−0.301218	−0.150609	−	0.988593i	$-0.548123\pi$
$607$	−7.42121	−0.301218	−0.150609	+	0.988593i	$0.548123\pi$
$608$	−2.55020	−0.103424
$609$	18.6714	0.756605
$610$	4.83536	0.195778
$611$	−3.22661	−0.130535
$612$	0.778010	0.0314492
$613$	−18.8248	−0.760326	−0.380163	−	0.924920i	$-0.624132\pi$
$613$	−18.8248	−0.760326	−0.380163	+	0.924920i	$0.624132\pi$
$614$	−12.0612	−0.486752
$615$	−3.50780	−0.141448
$616$	−6.56583	−0.264545
$617$	46.3996	1.86798	0.933989	−	0.357302i	$-0.116303\pi$
$617$	46.3996	1.86798	0.933989	+	0.357302i	$0.116303\pi$
$618$	−27.1233	−1.09106
$619$	43.7210	1.75730	0.878648	−	0.477470i	$-0.158446\pi$
$619$	43.7210	1.75730	0.878648	+	0.477470i	$0.158446\pi$
$620$	−1.00000	−0.0401610
$621$	15.4180	0.618702
$622$	−21.5639	−0.864634
$623$	4.12537	0.165280
$624$	1.80420	0.0722260
$625$	1.00000	0.0400000
$626$	−19.7406	−0.788992
$627$	−23.9402	−0.956079
$628$	15.7113	0.626951
$629$	28.8711	1.15116
$630$	−0.321979	−0.0128280
$631$	−32.3008	−1.28587	−0.642937	−	0.765919i	$-0.722284\pi$
$631$	−32.3008	−1.28587	−0.642937	+	0.765919i	$0.722284\pi$
$632$	4.90554	0.195132
$633$	−36.7351	−1.46009
$634$	−21.4862	−0.853327
$635$	8.28048	0.328601
$636$	−17.7333	−0.703170
$637$	5.40762	0.214258
$638$	42.6713	1.68937
$639$	−0.0830037	−0.00328358
$640$	−1.00000	−0.0395285
$641$	−43.8580	−1.73229	−0.866144	−	0.499794i	$-0.833409\pi$
$641$	−43.8580	−1.73229	−0.866144	+	0.499794i	$0.833409\pi$
$642$	−17.2238	−0.679768
$643$	−50.0487	−1.97373	−0.986865	−	0.161548i	$-0.948351\pi$
$643$	−50.0487	−1.97373	−0.986865	+	0.161548i	$0.948351\pi$
$644$	3.92868	0.154812
$645$	−18.6161	−0.733007
$646$	−7.77599	−0.305942
$647$	3.80337	0.149526	0.0747630	−	0.997201i	$-0.476180\pi$
$647$	3.80337	0.149526	0.0747630	+	0.997201i	$0.476180\pi$
$648$	9.70036	0.381066
$649$	−68.9487	−2.70647
$650$	1.00000	0.0392232
$651$	2.27672	0.0892315
$652$	−20.0642	−0.785776
$653$	−9.17212	−0.358933	−0.179466	−	0.983764i	$-0.557437\pi$
$653$	−9.17212	−0.358933	−0.179466	+	0.983764i	$0.557437\pi$
$654$	−3.55260	−0.138918
$655$	−2.73114	−0.106714
$656$	1.94424	0.0759097
$657$	−2.13000	−0.0830994
$658$	−4.07164	−0.158729
$659$	−37.3070	−1.45327	−0.726636	−	0.687022i	$-0.758917\pi$
$659$	−37.3070	−1.45327	−0.726636	+	0.687022i	$0.758917\pi$
$660$	−9.38756	−0.365410
$661$	12.0426	0.468405	0.234202	−	0.972188i	$-0.424752\pi$
$661$	12.0426	0.468405	0.234202	+	0.972188i	$0.424752\pi$
$662$	4.30482	0.167311
$663$	5.50131	0.213653
$664$	−1.56202	−0.0606183
$665$	3.21809	0.124792
$666$	−2.41595	−0.0936160
$667$	−25.5325	−0.988621
$668$	−17.1371	−0.663056
$669$	6.89528	0.266587
$670$	−9.67742	−0.373871
$671$	−25.1591	−0.971257
$672$	2.27672	0.0878262
$673$	−38.0035	−1.46493	−0.732464	−	0.680805i	$-0.761630\pi$
$673$	−38.0035	−1.46493	−0.732464	+	0.680805i	$0.761630\pi$
$674$	0.461400	0.0177724
$675$	4.95226	0.190613
$676$	1.00000	0.0384615
$677$	−10.3291	−0.396980	−0.198490	−	0.980103i	$-0.563604\pi$
$677$	−10.3291	−0.396980	−0.198490	+	0.980103i	$0.563604\pi$
$678$	27.5406	1.05769
$679$	−8.65842	−0.332280
$680$	−3.04916	−0.116930
$681$	23.0930	0.884924
$682$	5.20315	0.199239
$683$	28.7012	1.09822	0.549111	−	0.835750i	$-0.314967\pi$
$683$	28.7012	1.09822	0.549111	+	0.835750i	$0.314967\pi$
$684$	0.650699	0.0248801
$685$	1.06266	0.0406020
$686$	15.6571	0.597791
$687$	23.4163	0.893388
$688$	10.3182	0.393376
$689$	−9.82886	−0.374450
$690$	5.61707	0.213838
$691$	29.3589	1.11687	0.558433	−	0.829549i	$-0.311403\pi$
$691$	29.3589	1.11687	0.558433	+	0.829549i	$0.311403\pi$
$692$	−21.9585	−0.834738
$693$	1.67531	0.0636397
$694$	−29.4631	−1.11840
$695$	4.64474	0.176185
$696$	−14.7964	−0.560854
$697$	5.92829	0.224550
$698$	10.5639	0.399851
$699$	−44.8601	−1.69676
$700$	1.26189	0.0476951
$701$	8.89904	0.336112	0.168056	−	0.985777i	$-0.446251\pi$
$701$	8.89904	0.336112	0.168056	+	0.985777i	$0.446251\pi$
$702$	4.95226	0.186911
$703$	24.1467	0.910709
$704$	5.20315	0.196101
$705$	−5.82146	−0.219249
$706$	−36.6754	−1.38030
$707$	11.5375	0.433911
$708$	23.9081	0.898522
$709$	−17.9207	−0.673026	−0.336513	−	0.941679i	$-0.609248\pi$
$709$	−17.9207	−0.673026	−0.336513	+	0.941679i	$0.609248\pi$
$710$	0.325306	0.0122085
$711$	−1.25168	−0.0469415
$712$	−3.26919	−0.122518
$713$	−3.11332	−0.116595
$714$	6.94207	0.259801
$715$	−5.20315	−0.194587
$716$	−14.9512	−0.558754
$717$	−52.3218	−1.95399
$718$	−34.0967	−1.27248
$719$	−33.4437	−1.24724	−0.623619	−	0.781728i	$-0.714338\pi$
$719$	−33.4437	−1.24724	−0.623619	+	0.781728i	$0.714338\pi$
$720$	0.255155	0.00950908
$721$	−18.9706	−0.706501
$722$	12.4965	0.465070
$723$	−10.7644	−0.400331
$724$	18.1416	0.674228
$725$	−8.20104	−0.304579
$726$	28.9986	1.07624
$727$	11.3288	0.420160	0.210080	−	0.977684i	$-0.432627\pi$
$727$	11.3288	0.420160	0.210080	+	0.977684i	$0.432627\pi$
$728$	1.26189	0.0467689
$729$	24.3296	0.901096
$730$	8.34787	0.308968
$731$	31.4617	1.16365
$732$	8.72398	0.322448
$733$	21.0295	0.776741	0.388370	−	0.921503i	$-0.373038\pi$
$733$	21.0295	0.776741	0.388370	+	0.921503i	$0.373038\pi$
$734$	27.4453	1.01302
$735$	9.75646	0.359872
$736$	−3.11332	−0.114759
$737$	50.3531	1.85478
$738$	−0.496082	−0.0182610
$739$	−4.94727	−0.181988	−0.0909941	−	0.995851i	$-0.529004\pi$
$739$	−4.94727	−0.181988	−0.0909941	+	0.995851i	$0.529004\pi$
$740$	9.46853	0.348070
$741$	4.60109	0.169025
$742$	−12.4030	−0.455328
$743$	−22.1597	−0.812959	−0.406480	−	0.913660i	$-0.633244\pi$
$743$	−22.1597	−0.812959	−0.406480	+	0.913660i	$0.633244\pi$
$744$	−1.80420	−0.0661454
$745$	13.1281	0.480977
$746$	−3.94698	−0.144509
$747$	0.398559	0.0145825
$748$	15.8653	0.580091
$749$	−12.0466	−0.440174
$750$	1.80420	0.0658802
$751$	−37.0788	−1.35302	−0.676512	−	0.736431i	$-0.736509\pi$
$751$	−37.0788	−1.35302	−0.676512	+	0.736431i	$0.736509\pi$
$752$	3.22661	0.117662
$753$	29.3302	1.06885
$754$	−8.20104	−0.298664
$755$	4.73358	0.172273
$756$	6.24923	0.227282
$757$	38.2614	1.39063	0.695316	−	0.718704i	$-0.255264\pi$
$757$	38.2614	1.39063	0.695316	+	0.718704i	$0.255264\pi$
$758$	22.0697	0.801609
$759$	−29.2265	−1.06085
$760$	−2.55020	−0.0925057
$761$	5.23166	0.189647	0.0948237	−	0.995494i	$-0.469771\pi$
$761$	5.23166	0.189647	0.0948237	+	0.995494i	$0.469771\pi$
$762$	14.9397	0.541207
$763$	−2.48476	−0.0899542
$764$	−2.43217	−0.0879927
$765$	0.778010	0.0281290
$766$	21.0227	0.759583
$767$	13.2513	0.478478
$768$	−1.80420	−0.0651036
$769$	−19.9557	−0.719621	−0.359811	−	0.933025i	$-0.617159\pi$
$769$	−19.9557	−0.719621	−0.359811	+	0.933025i	$0.617159\pi$
$770$	−6.56583	−0.236616
$771$	−34.7292	−1.25074
$772$	−6.62320	−0.238374
$773$	−33.5819	−1.20786	−0.603929	−	0.797038i	$-0.706399\pi$
$773$	−33.5819	−1.20786	−0.603929	+	0.797038i	$0.706399\pi$
$774$	−2.63273	−0.0946316
$775$	−1.00000	−0.0359211
$776$	6.86145	0.246312
$777$	−21.5571	−0.773358
$778$	1.12256	0.0402457
$779$	4.95820	0.177646
$780$	1.80420	0.0646009
$781$	−1.69262	−0.0605667
$782$	−9.49302	−0.339470
$783$	−40.6137	−1.45141
$784$	−5.40762	−0.193129
$785$	15.7113	0.560762
$786$	−4.92754	−0.175759
$787$	−5.90563	−0.210513	−0.105257	−	0.994445i	$-0.533566\pi$
$787$	−5.90563	−0.210513	−0.105257	+	0.994445i	$0.533566\pi$
$788$	−22.6787	−0.807896
$789$	−26.9768	−0.960399
$790$	4.90554	0.174531
$791$	19.2624	0.684892
$792$	−1.32761	−0.0471747
$793$	4.83536	0.171709
$794$	16.1935	0.574687
$795$	−17.7333	−0.628935
$796$	−20.6502	−0.731928
$797$	21.3835	0.757443	0.378721	−	0.925511i	$-0.376364\pi$
$797$	21.3835	0.757443	0.378721	+	0.925511i	$0.376364\pi$
$798$	5.80609	0.205533
$799$	9.83845	0.348059
$800$	−1.00000	−0.0353553
$801$	0.834152	0.0294733
$802$	2.24768	0.0793685
$803$	−43.4352	−1.53280
$804$	−17.4600	−0.615768
$805$	3.92868	0.138468
$806$	−1.00000	−0.0352235
$807$	−54.2542	−1.90984
$808$	−9.14296	−0.321648
$809$	5.72745	0.201366	0.100683	−	0.994919i	$-0.467897\pi$
$809$	5.72745	0.201366	0.100683	+	0.994919i	$0.467897\pi$
$810$	9.70036	0.340836
$811$	4.31201	0.151415	0.0757076	−	0.997130i	$-0.475878\pi$
$811$	4.31201	0.151415	0.0757076	+	0.997130i	$0.475878\pi$
$812$	−10.3488	−0.363173
$813$	−33.9073	−1.18918
$814$	−49.2662	−1.72678
$815$	−20.0642	−0.702819
$816$	−5.50131	−0.192584
$817$	26.3134	0.920589
$818$	−10.5232	−0.367936
$819$	−0.321979	−0.0112509
$820$	1.94424	0.0678957
$821$	−36.1961	−1.26325	−0.631627	−	0.775272i	$-0.717613\pi$
$821$	−36.1961	−1.26325	−0.631627	+	0.775272i	$0.717613\pi$
$822$	1.91725	0.0668718
$823$	−7.01388	−0.244489	−0.122244	−	0.992500i	$-0.539009\pi$
$823$	−7.01388	−0.244489	−0.122244	+	0.992500i	$0.539009\pi$
$824$	15.0334	0.523714
$825$	−9.38756	−0.326833
$826$	16.7218	0.581825
$827$	3.47620	0.120879	0.0604396	−	0.998172i	$-0.480750\pi$
$827$	3.47620	0.120879	0.0604396	+	0.998172i	$0.480750\pi$
$828$	0.794381	0.0276067
$829$	−49.6039	−1.72281	−0.861407	−	0.507915i	$-0.830416\pi$
$829$	−49.6039	−1.72281	−0.861407	+	0.507915i	$0.830416\pi$
$830$	−1.56202	−0.0542186
$831$	11.8597	0.411407
$832$	−1.00000	−0.0346688
$833$	−16.4887	−0.571300
$834$	8.38006	0.290178
$835$	−17.1371	−0.593055
$836$	13.2691	0.458922
$837$	−4.95226	−0.171175
$838$	−3.01900	−0.104290
$839$	0.819336	0.0282866	0.0141433	−	0.999900i	$-0.495498\pi$
$839$	0.819336	0.0282866	0.0141433	+	0.999900i	$0.495498\pi$
$840$	2.27672	0.0785542
$841$	38.2570	1.31921
$842$	6.09983	0.210214
$843$	−17.8134	−0.613526
$844$	20.3608	0.700849
$845$	1.00000	0.0344010
$846$	−0.823287	−0.0283052
$847$	20.2822	0.696904
$848$	9.82886	0.337524
$849$	−43.3291	−1.48705
$850$	−3.04916	−0.104585
$851$	29.4786	1.01051
$852$	0.586919	0.0201075
$853$	−7.95088	−0.272233	−0.136116	−	0.990693i	$-0.543462\pi$
$853$	−7.95088	−0.272233	−0.136116	+	0.990693i	$0.543462\pi$
$854$	6.10172	0.208796
$855$	0.650699	0.0222534
$856$	9.54646	0.326291
$857$	−3.00926	−0.102794	−0.0513971	−	0.998678i	$-0.516367\pi$
$857$	−3.00926	−0.102794	−0.0513971	+	0.998678i	$0.516367\pi$
$858$	−9.38756	−0.320486
$859$	−39.7065	−1.35477	−0.677384	−	0.735629i	$-0.736887\pi$
$859$	−39.7065	−1.35477	−0.677384	+	0.735629i	$0.736887\pi$
$860$	10.3182	0.351846
$861$	−4.42647	−0.150854
$862$	14.0806	0.479588
$863$	30.2095	1.02834	0.514171	−	0.857688i	$-0.328100\pi$
$863$	30.2095	1.02834	0.514171	+	0.857688i	$0.328100\pi$
$864$	−4.95226	−0.168479
$865$	−21.9585	−0.746613
$866$	−26.0066	−0.883741
$867$	13.8971	0.471970
$868$	−1.26189	−0.0428315
$869$	−25.5243	−0.865852
$870$	−14.7964	−0.501643
$871$	−9.67742	−0.327907
$872$	1.96907	0.0666811
$873$	−1.75074	−0.0592534
$874$	−7.93961	−0.268561
$875$	1.26189	0.0426598
$876$	15.0613	0.508873
$877$	33.5058	1.13141	0.565706	−	0.824607i	$-0.308604\pi$
$877$	33.5058	1.13141	0.565706	+	0.824607i	$0.308604\pi$
$878$	−5.74031	−0.193726
$879$	−28.6706	−0.967036
$880$	5.20315	0.175398
$881$	36.7846	1.23930	0.619652	−	0.784877i	$-0.287274\pi$
$881$	36.7846	1.23930	0.619652	+	0.784877i	$0.287274\pi$
$882$	1.37978	0.0464598
$883$	−51.6431	−1.73793	−0.868964	−	0.494874i	$-0.835214\pi$
$883$	−51.6431	−1.73793	−0.868964	+	0.494874i	$0.835214\pi$
$884$	−3.04916	−0.102554
$885$	23.9081	0.803663
$886$	−17.4703	−0.586926
$887$	−23.2628	−0.781087	−0.390544	−	0.920584i	$-0.627713\pi$
$887$	−23.2628	−0.781087	−0.390544	+	0.920584i	$0.627713\pi$
$888$	17.0832	0.573273
$889$	10.4491	0.350451
$890$	−3.26919	−0.109584
$891$	−50.4725	−1.69089
$892$	−3.82178	−0.127963
$893$	8.22851	0.275357
$894$	23.6858	0.792172
$895$	−14.9512	−0.499765
$896$	−1.26189	−0.0421569
$897$	5.61707	0.187549
$898$	−19.4289	−0.648350
$899$	8.20104	0.273520
$900$	0.255155	0.00850518
$901$	29.9698	0.998438
$902$	−10.1162	−0.336831
$903$	−23.4915	−0.781748
$904$	−15.2647	−0.507696
$905$	18.1416	0.603047
$906$	8.54035	0.283734
$907$	−55.5497	−1.84450	−0.922248	−	0.386599i	$-0.873650\pi$
$907$	−55.5497	−1.84450	−0.922248	+	0.386599i	$0.873650\pi$
$908$	−12.7995	−0.424767
$909$	2.33288	0.0773767
$910$	1.26189	0.0418314
$911$	−19.2770	−0.638675	−0.319337	−	0.947641i	$-0.603460\pi$
$911$	−19.2770	−0.638675	−0.319337	+	0.947641i	$0.603460\pi$
$912$	−4.60109	−0.152357
$913$	8.12745	0.268979
$914$	−35.7071	−1.18109
$915$	8.72398	0.288406
$916$	−12.9787	−0.428830
$917$	−3.44641	−0.113810
$918$	−15.1002	−0.498382
$919$	−26.3476	−0.869127	−0.434564	−	0.900641i	$-0.643097\pi$
$919$	−26.3476	−0.869127	−0.434564	+	0.900641i	$0.643097\pi$
$920$	−3.11332	−0.102643
$921$	−21.7610	−0.717048
$922$	−32.8642	−1.08232
$923$	0.325306	0.0107076
$924$	−11.8461	−0.389708
$925$	9.46853	0.311323
$926$	22.5716	0.741748
$927$	−3.83586	−0.125986
$928$	8.20104	0.269212
$929$	−44.9155	−1.47363	−0.736814	−	0.676095i	$-0.763671\pi$
$929$	−44.9155	−1.47363	−0.736814	+	0.676095i	$0.763671\pi$
$930$	−1.80420	−0.0591622
$931$	−13.7905	−0.451967
$932$	24.8642	0.814453
$933$	−38.9057	−1.27372
$934$	−36.4635	−1.19312
$935$	15.8653	0.518849
$936$	0.255155	0.00834002
$937$	34.2952	1.12038	0.560188	−	0.828366i	$-0.310729\pi$
$937$	34.2952	1.12038	0.560188	+	0.828366i	$0.310729\pi$
$938$	−12.2119	−0.398732
$939$	−35.6160	−1.16229
$940$	3.22661	0.105240
$941$	28.2008	0.919319	0.459659	−	0.888095i	$-0.347971\pi$
$941$	28.2008	0.919319	0.459659	+	0.888095i	$0.347971\pi$
$942$	28.3465	0.923578
$943$	6.05303	0.197114
$944$	−13.2513	−0.431294
$945$	6.24923	0.203288
$946$	−53.6869	−1.74551
$947$	−32.1668	−1.04528	−0.522640	−	0.852553i	$-0.675053\pi$
$947$	−32.1668	−1.04528	−0.522640	+	0.852553i	$0.675053\pi$
$948$	8.85060	0.287454
$949$	8.34787	0.270983
$950$	−2.55020	−0.0827396
$951$	−38.7656	−1.25706
$952$	−3.84772	−0.124705
$953$	21.6994	0.702911	0.351455	−	0.936205i	$-0.385687\pi$
$953$	21.6994	0.702911	0.351455	+	0.936205i	$0.385687\pi$
$954$	−2.50789	−0.0811959
$955$	−2.43217	−0.0787031
$956$	28.9999	0.937925
$957$	76.9877	2.48866
$958$	−6.99030	−0.225846
$959$	1.34096	0.0433019
$960$	−1.80420	−0.0582305
$961$	1.00000	0.0322581
$962$	9.46853	0.305278
$963$	−2.43583	−0.0784936
$964$	5.96626	0.192160
$965$	−6.62320	−0.213208
$966$	7.08815	0.228057
$967$	12.7041	0.408535	0.204268	−	0.978915i	$-0.434519\pi$
$967$	12.7041	0.408535	0.204268	+	0.978915i	$0.434519\pi$
$968$	−16.0728	−0.516600
$969$	−14.0295	−0.450692
$970$	6.86145	0.220308
$971$	3.12869	0.100404	0.0502022	−	0.998739i	$-0.484013\pi$
$971$	3.12869	0.100404	0.0502022	+	0.998739i	$0.484013\pi$
$972$	2.64466	0.0848274
$973$	5.86117	0.187900
$974$	2.34010	0.0749815
$975$	1.80420	0.0577808
$976$	−4.83536	−0.154776
$977$	40.6057	1.29909	0.649546	−	0.760322i	$-0.274959\pi$
$977$	40.6057	1.29909	0.649546	+	0.760322i	$0.274959\pi$
$978$	−36.2000	−1.15755
$979$	17.0101	0.543645
$980$	−5.40762	−0.172740
$981$	−0.502419	−0.0160410
$982$	−14.9917	−0.478403
$983$	−21.5806	−0.688314	−0.344157	−	0.938912i	$-0.611835\pi$
$983$	−21.5806	−0.688314	−0.344157	+	0.938912i	$0.611835\pi$
$984$	3.50780	0.111825
$985$	−22.6787	−0.722604
$986$	25.0063	0.796362
$987$	−7.34607	−0.233828
$988$	−2.55020	−0.0811328
$989$	32.1237	1.02148
$990$	−1.32761	−0.0421943
$991$	−6.49838	−0.206428	−0.103214	−	0.994659i	$-0.532913\pi$
$991$	−6.49838	−0.206428	−0.103214	+	0.994659i	$0.532913\pi$
$992$	1.00000	0.0317500
$993$	7.76677	0.246471
$994$	0.410502	0.0130203
$995$	−20.6502	−0.654656
$996$	−2.81821	−0.0892984
$997$	26.4930	0.839040	0.419520	−	0.907746i	$-0.362198\pi$
$997$	26.4930	0.839040	0.419520	+	0.907746i	$0.362198\pi$
$998$	−27.6968	−0.876728
$999$	46.8906	1.48355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.l.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.l.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.80420 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.80420 q^{6} +1.26189 q^{7} -1.00000 q^{8} +0.255155 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.80420 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.80420 q^{6} +1.26189 q^{7} -1.00000 q^{8} +0.255155 q^{9} -1.00000 q^{10} +5.20315 q^{11} -1.80420 q^{12} -1.00000 q^{13} -1.26189 q^{14} -1.80420 q^{15} +1.00000 q^{16} +3.04916 q^{17} -0.255155 q^{18} +2.55020 q^{19} +1.00000 q^{20} -2.27672 q^{21} -5.20315 q^{22} +3.11332 q^{23} +1.80420 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.95226 q^{27} +1.26189 q^{28} -8.20104 q^{29} +1.80420 q^{30} -1.00000 q^{31} -1.00000 q^{32} -9.38756 q^{33} -3.04916 q^{34} +1.26189 q^{35} +0.255155 q^{36} +9.46853 q^{37} -2.55020 q^{38} +1.80420 q^{39} -1.00000 q^{40} +1.94424 q^{41} +2.27672 q^{42} +10.3182 q^{43} +5.20315 q^{44} +0.255155 q^{45} -3.11332 q^{46} +3.22661 q^{47} -1.80420 q^{48} -5.40762 q^{49} -1.00000 q^{50} -5.50131 q^{51} -1.00000 q^{52} +9.82886 q^{53} -4.95226 q^{54} +5.20315 q^{55} -1.26189 q^{56} -4.60109 q^{57} +8.20104 q^{58} -13.2513 q^{59} -1.80420 q^{60} -4.83536 q^{61} +1.00000 q^{62} +0.321979 q^{63} +1.00000 q^{64} -1.00000 q^{65} +9.38756 q^{66} +9.67742 q^{67} +3.04916 q^{68} -5.61707 q^{69} -1.26189 q^{70} -0.325306 q^{71} -0.255155 q^{72} -8.34787 q^{73} -9.46853 q^{74} -1.80420 q^{75} +2.55020 q^{76} +6.56583 q^{77} -1.80420 q^{78} -4.90554 q^{79} +1.00000 q^{80} -9.70036 q^{81} -1.94424 q^{82} +1.56202 q^{83} -2.27672 q^{84} +3.04916 q^{85} -10.3182 q^{86} +14.7964 q^{87} -5.20315 q^{88} +3.26919 q^{89} -0.255155 q^{90} -1.26189 q^{91} +3.11332 q^{92} +1.80420 q^{93} -3.22661 q^{94} +2.55020 q^{95} +1.80420 q^{96} -6.86145 q^{97} +5.40762 q^{98} +1.32761 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9	\( 8 q - 8 q^{2} + q^{3} + 8 q^{4} + 8 q^{5} - q^{6} + 11 q^{7} - 8 q^{8} + 9 q^{9} - 8 q^{10} + q^{12} - 8 q^{13} - 11 q^{14} + q^{15} + 8 q^{16} + 7 q^{17} - 9 q^{18} - 2 q^{19} + 8 q^{20} + q^{21} + 8 q^{23} - q^{24} + 8 q^{25} + 8 q^{26} + 7 q^{27} + 11 q^{28} - q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 14 q^{33} - 7 q^{34} + 11 q^{35} + 9 q^{36} + q^{37} + 2 q^{38} - q^{39} - 8 q^{40} + 16 q^{41} - q^{42} + 3 q^{43} + 9 q^{45} - 8 q^{46} + 29 q^{47} + q^{48} + 11 q^{49} - 8 q^{50} + 11 q^{51} - 8 q^{52} + 22 q^{53} - 7 q^{54} - 11 q^{56} + 33 q^{57} + q^{58} - 8 q^{59} + q^{60} + 4 q^{61} + 8 q^{62} + 38 q^{63} + 8 q^{64} - 8 q^{65} - 14 q^{66} + 28 q^{67} + 7 q^{68} - 42 q^{69} - 11 q^{70} + 4 q^{71} - 9 q^{72} + 39 q^{73} - q^{74} + q^{75} - 2 q^{76} + 11 q^{77} + q^{78} - 16 q^{79} + 8 q^{80} + 32 q^{81} - 16 q^{82} + 25 q^{83} + q^{84} + 7 q^{85} - 3 q^{86} + 13 q^{87} + 21 q^{89} - 9 q^{90} - 11 q^{91} + 8 q^{92} - q^{93} - 29 q^{94} - 2 q^{95} - q^{96} + 28 q^{97} - 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + q^3 + 8 * q^4 + 8 * q^5 - q^6 + 11 * q^7 - 8 * q^8 + 9 * q^9 - 8 * q^10 + q^12 - 8 * q^13 - 11 * q^14 + q^15 + 8 * q^16 + 7 * q^17 - 9 * q^18 - 2 * q^19 + 8 * q^20 + q^21 + 8 * q^23 - q^24 + 8 * q^25 + 8 * q^26 + 7 * q^27 + 11 * q^28 - q^29 - q^30 - 8 * q^31 - 8 * q^32 + 14 * q^33 - 7 * q^34 + 11 * q^35 + 9 * q^36 + q^37 + 2 * q^38 - q^39 - 8 * q^40 + 16 * q^41 - q^42 + 3 * q^43 + 9 * q^45 - 8 * q^46 + 29 * q^47 + q^48 + 11 * q^49 - 8 * q^50 + 11 * q^51 - 8 * q^52 + 22 * q^53 - 7 * q^54 - 11 * q^56 + 33 * q^57 + q^58 - 8 * q^59 + q^60 + 4 * q^61 + 8 * q^62 + 38 * q^63 + 8 * q^64 - 8 * q^65 - 14 * q^66 + 28 * q^67 + 7 * q^68 - 42 * q^69 - 11 * q^70 + 4 * q^71 - 9 * q^72 + 39 * q^73 - q^74 + q^75 - 2 * q^76 + 11 * q^77 + q^78 - 16 * q^79 + 8 * q^80 + 32 * q^81 - 16 * q^82 + 25 * q^83 + q^84 + 7 * q^85 - 3 * q^86 + 13 * q^87 + 21 * q^89 - 9 * q^90 - 11 * q^91 + 8 * q^92 - q^93 - 29 * q^94 - 2 * q^95 - q^96 + 28 * q^97 - 11 * q^98 + 23 * q^99

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