LMFDB - Embedded newform 8030.2.a.bb.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, 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("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$1$
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} - 3x^{7} - 8x^{6} + 20x^{5} + 23x^{4} - 32x^{3} - 16x^{2} + 17x - 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 23x^4 - 32x^3 - 16x^2 + 17*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.93960$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.93960 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.93960 q^{6} -0.152382 q^{7} +1.00000 q^{8} +0.762032 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.93960 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.93960 q^{6} -0.152382 q^{7} +1.00000 q^{8} +0.762032 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.93960 q^{12} -3.91609 q^{13} -0.152382 q^{14} -1.93960 q^{15} +1.00000 q^{16} -3.46218 q^{17} +0.762032 q^{18} -3.90418 q^{19} -1.00000 q^{20} -0.295559 q^{21} +1.00000 q^{22} +0.355343 q^{23} +1.93960 q^{24} +1.00000 q^{25} -3.91609 q^{26} -4.34075 q^{27} -0.152382 q^{28} +5.86761 q^{29} -1.93960 q^{30} -6.35862 q^{31} +1.00000 q^{32} +1.93960 q^{33} -3.46218 q^{34} +0.152382 q^{35} +0.762032 q^{36} +4.42115 q^{37} -3.90418 q^{38} -7.59563 q^{39} -1.00000 q^{40} -7.51395 q^{41} -0.295559 q^{42} +7.83646 q^{43} +1.00000 q^{44} -0.762032 q^{45} +0.355343 q^{46} -1.04188 q^{47} +1.93960 q^{48} -6.97678 q^{49} +1.00000 q^{50} -6.71523 q^{51} -3.91609 q^{52} -12.8993 q^{53} -4.34075 q^{54} -1.00000 q^{55} -0.152382 q^{56} -7.57252 q^{57} +5.86761 q^{58} +5.89714 q^{59} -1.93960 q^{60} -6.92085 q^{61} -6.35862 q^{62} -0.116120 q^{63} +1.00000 q^{64} +3.91609 q^{65} +1.93960 q^{66} +0.981624 q^{67} -3.46218 q^{68} +0.689222 q^{69} +0.152382 q^{70} +3.05780 q^{71} +0.762032 q^{72} -1.00000 q^{73} +4.42115 q^{74} +1.93960 q^{75} -3.90418 q^{76} -0.152382 q^{77} -7.59563 q^{78} +6.98610 q^{79} -1.00000 q^{80} -10.7054 q^{81} -7.51395 q^{82} -13.5996 q^{83} -0.295559 q^{84} +3.46218 q^{85} +7.83646 q^{86} +11.3808 q^{87} +1.00000 q^{88} -6.41862 q^{89} -0.762032 q^{90} +0.596741 q^{91} +0.355343 q^{92} -12.3332 q^{93} -1.04188 q^{94} +3.90418 q^{95} +1.93960 q^{96} -12.8595 q^{97} -6.97678 q^{98} +0.762032 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9	\( 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9} - 8 q^{10} + 8 q^{11} - 3 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 8 q^{16} + 2 q^{17} + q^{18} - q^{19} - 8 q^{20} + 7 q^{21} + 8 q^{22} - 16 q^{23} - 3 q^{24} + 8 q^{25} - 7 q^{26} - 21 q^{27} - 4 q^{28} - 5 q^{29} + 3 q^{30} - 22 q^{31} + 8 q^{32} - 3 q^{33} + 2 q^{34} + 4 q^{35} + q^{36} - 9 q^{37} - q^{38} - 18 q^{39} - 8 q^{40} + 6 q^{41} + 7 q^{42} + 15 q^{43} + 8 q^{44} - q^{45} - 16 q^{46} - 7 q^{47} - 3 q^{48} - 6 q^{49} + 8 q^{50} + q^{51} - 7 q^{52} - 11 q^{53} - 21 q^{54} - 8 q^{55} - 4 q^{56} - 17 q^{57} - 5 q^{58} - 11 q^{59} + 3 q^{60} - 22 q^{61} - 22 q^{62} + q^{63} + 8 q^{64} + 7 q^{65} - 3 q^{66} + 21 q^{67} + 2 q^{68} + q^{69} + 4 q^{70} - 28 q^{71} + q^{72} - 8 q^{73} - 9 q^{74} - 3 q^{75} - q^{76} - 4 q^{77} - 18 q^{78} - 28 q^{79} - 8 q^{80} + 12 q^{81} + 6 q^{82} - 5 q^{83} + 7 q^{84} - 2 q^{85} + 15 q^{86} + 36 q^{87} + 8 q^{88} - 17 q^{89} - q^{90} - 29 q^{91} - 16 q^{92} + 42 q^{93} - 7 q^{94} + q^{95} - 3 q^{96} + q^{97} - 6 q^{98} + q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9 - 8 * q^10 + 8 * q^11 - 3 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 8 * q^16 + 2 * q^17 + q^18 - q^19 - 8 * q^20 + 7 * q^21 + 8 * q^22 - 16 * q^23 - 3 * q^24 + 8 * q^25 - 7 * q^26 - 21 * q^27 - 4 * q^28 - 5 * q^29 + 3 * q^30 - 22 * q^31 + 8 * q^32 - 3 * q^33 + 2 * q^34 + 4 * q^35 + q^36 - 9 * q^37 - q^38 - 18 * q^39 - 8 * q^40 + 6 * q^41 + 7 * q^42 + 15 * q^43 + 8 * q^44 - q^45 - 16 * q^46 - 7 * q^47 - 3 * q^48 - 6 * q^49 + 8 * q^50 + q^51 - 7 * q^52 - 11 * q^53 - 21 * q^54 - 8 * q^55 - 4 * q^56 - 17 * q^57 - 5 * q^58 - 11 * q^59 + 3 * q^60 - 22 * q^61 - 22 * q^62 + q^63 + 8 * q^64 + 7 * q^65 - 3 * q^66 + 21 * q^67 + 2 * q^68 + q^69 + 4 * q^70 - 28 * q^71 + q^72 - 8 * q^73 - 9 * q^74 - 3 * q^75 - q^76 - 4 * q^77 - 18 * q^78 - 28 * q^79 - 8 * q^80 + 12 * q^81 + 6 * q^82 - 5 * q^83 + 7 * q^84 - 2 * q^85 + 15 * q^86 + 36 * q^87 + 8 * q^88 - 17 * q^89 - q^90 - 29 * q^91 - 16 * q^92 + 42 * q^93 - 7 * q^94 + q^95 - 3 * q^96 + q^97 - 6 * q^98 + 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
$3$	1.93960	1.11983	0.559913	−	0.828551i	$-0.310835\pi$
$3$	1.93960	1.11983	0.559913	+	0.828551i	$0.310835\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.93960	0.791837
$7$	−0.152382	−0.0575949	−0.0287975	−	0.999585i	$-0.509168\pi$
$7$	−0.152382	−0.0575949	−0.0287975	+	0.999585i	$0.509168\pi$
$8$	1.00000	0.353553
$9$	0.762032	0.254011
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.93960	0.559913
$13$	−3.91609	−1.08613	−0.543064	−	0.839691i	$-0.682736\pi$
$13$	−3.91609	−1.08613	−0.543064	+	0.839691i	$0.682736\pi$
$14$	−0.152382	−0.0407258
$15$	−1.93960	−0.500801
$16$	1.00000	0.250000
$17$	−3.46218	−0.839702	−0.419851	−	0.907593i	$-0.637918\pi$
$17$	−3.46218	−0.839702	−0.419851	+	0.907593i	$0.637918\pi$
$18$	0.762032	0.179613
$19$	−3.90418	−0.895680	−0.447840	−	0.894114i	$-0.647807\pi$
$19$	−3.90418	−0.895680	−0.447840	+	0.894114i	$0.647807\pi$
$20$	−1.00000	−0.223607
$21$	−0.295559	−0.0644963
$22$	1.00000	0.213201
$23$	0.355343	0.0740941	0.0370471	−	0.999314i	$-0.488205\pi$
$23$	0.355343	0.0740941	0.0370471	+	0.999314i	$0.488205\pi$
$24$	1.93960	0.395918
$25$	1.00000	0.200000
$26$	−3.91609	−0.768009
$27$	−4.34075	−0.835378
$28$	−0.152382	−0.0287975
$29$	5.86761	1.08959	0.544794	−	0.838570i	$-0.316608\pi$
$29$	5.86761	1.08959	0.544794	+	0.838570i	$0.316608\pi$
$30$	−1.93960	−0.354120
$31$	−6.35862	−1.14204	−0.571021	−	0.820936i	$-0.693452\pi$
$31$	−6.35862	−1.14204	−0.571021	+	0.820936i	$0.693452\pi$
$32$	1.00000	0.176777
$33$	1.93960	0.337640
$34$	−3.46218	−0.593759
$35$	0.152382	0.0257572
$36$	0.762032	0.127005
$37$	4.42115	0.726833	0.363416	−	0.931627i	$-0.381610\pi$
$37$	4.42115	0.726833	0.363416	+	0.931627i	$0.381610\pi$
$38$	−3.90418	−0.633341
$39$	−7.59563	−1.21627
$40$	−1.00000	−0.158114
$41$	−7.51395	−1.17348	−0.586741	−	0.809775i	$-0.699589\pi$
$41$	−7.51395	−1.17348	−0.586741	+	0.809775i	$0.699589\pi$
$42$	−0.295559	−0.0456058
$43$	7.83646	1.19505	0.597524	−	0.801851i	$-0.296151\pi$
$43$	7.83646	1.19505	0.597524	+	0.801851i	$0.296151\pi$
$44$	1.00000	0.150756
$45$	−0.762032	−0.113597
$46$	0.355343	0.0523924
$47$	−1.04188	−0.151974	−0.0759870	−	0.997109i	$-0.524211\pi$
$47$	−1.04188	−0.151974	−0.0759870	+	0.997109i	$0.524211\pi$
$48$	1.93960	0.279957
$49$	−6.97678	−0.996683
$50$	1.00000	0.141421
$51$	−6.71523	−0.940320
$52$	−3.91609	−0.543064
$53$	−12.8993	−1.77185	−0.885925	−	0.463829i	$-0.846475\pi$
$53$	−12.8993	−1.77185	−0.885925	+	0.463829i	$0.846475\pi$
$54$	−4.34075	−0.590702
$55$	−1.00000	−0.134840
$56$	−0.152382	−0.0203629
$57$	−7.57252	−1.00301
$58$	5.86761	0.770455
$59$	5.89714	0.767742	0.383871	−	0.923387i	$-0.374591\pi$
$59$	5.89714	0.767742	0.383871	+	0.923387i	$0.374591\pi$
$60$	−1.93960	−0.250401
$61$	−6.92085	−0.886124	−0.443062	−	0.896491i	$-0.646108\pi$
$61$	−6.92085	−0.886124	−0.443062	+	0.896491i	$0.646108\pi$
$62$	−6.35862	−0.807546
$63$	−0.116120	−0.0146297
$64$	1.00000	0.125000
$65$	3.91609	0.485731
$66$	1.93960	0.238748
$67$	0.981624	0.119924	0.0599622	−	0.998201i	$-0.480902\pi$
$67$	0.981624	0.119924	0.0599622	+	0.998201i	$0.480902\pi$
$68$	−3.46218	−0.419851
$69$	0.689222	0.0829725
$70$	0.152382	0.0182131
$71$	3.05780	0.362894	0.181447	−	0.983401i	$-0.441922\pi$
$71$	3.05780	0.362894	0.181447	+	0.983401i	$0.441922\pi$
$72$	0.762032	0.0898063
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	4.42115	0.513948
$75$	1.93960	0.223965
$76$	−3.90418	−0.447840
$77$	−0.152382	−0.0173655
$78$	−7.59563	−0.860036
$79$	6.98610	0.785998	0.392999	−	0.919539i	$-0.371438\pi$
$79$	6.98610	0.785998	0.392999	+	0.919539i	$0.371438\pi$
$80$	−1.00000	−0.111803
$81$	−10.7054	−1.18949
$82$	−7.51395	−0.829777
$83$	−13.5996	−1.49275	−0.746376	−	0.665525i	$-0.768208\pi$
$83$	−13.5996	−1.49275	−0.746376	+	0.665525i	$0.768208\pi$
$84$	−0.295559	−0.0322482
$85$	3.46218	0.375526
$86$	7.83646	0.845027
$87$	11.3808	1.22015
$88$	1.00000	0.106600
$89$	−6.41862	−0.680372	−0.340186	−	0.940358i	$-0.610490\pi$
$89$	−6.41862	−0.680372	−0.340186	+	0.940358i	$0.610490\pi$
$90$	−0.762032	−0.0803252
$91$	0.596741	0.0625555
$92$	0.355343	0.0370471
$93$	−12.3332	−1.27889
$94$	−1.04188	−0.107462
$95$	3.90418	0.400560
$96$	1.93960	0.197959
$97$	−12.8595	−1.30569	−0.652845	−	0.757492i	$-0.726425\pi$
$97$	−12.8595	−1.30569	−0.652845	+	0.757492i	$0.726425\pi$
$98$	−6.97678	−0.704761
$99$	0.762032	0.0765871
$100$	1.00000	0.100000
$101$	19.1666	1.90715	0.953574	−	0.301157i	$-0.0973729\pi$
$101$	19.1666	1.90715	0.953574	+	0.301157i	$0.0973729\pi$
$102$	−6.71523	−0.664907
$103$	0.262376	0.0258527	0.0129263	−	0.999916i	$-0.495885\pi$
$103$	0.262376	0.0258527	0.0129263	+	0.999916i	$0.495885\pi$
$104$	−3.91609	−0.384004
$105$	0.295559	0.0288436
$106$	−12.8993	−1.25289
$107$	−19.6477	−1.89942	−0.949709	−	0.313135i	$-0.898621\pi$
$107$	−19.6477	−1.89942	−0.949709	+	0.313135i	$0.898621\pi$
$108$	−4.34075	−0.417689
$109$	−5.95630	−0.570510	−0.285255	−	0.958452i	$-0.592078\pi$
$109$	−5.95630	−0.570510	−0.285255	+	0.958452i	$0.592078\pi$
$110$	−1.00000	−0.0953463
$111$	8.57524	0.813926
$112$	−0.152382	−0.0143987
$113$	−1.69194	−0.159164	−0.0795822	−	0.996828i	$-0.525359\pi$
$113$	−1.69194	−0.159164	−0.0795822	+	0.996828i	$0.525359\pi$
$114$	−7.57252	−0.709232
$115$	−0.355343	−0.0331359
$116$	5.86761	0.544794
$117$	−2.98419	−0.275888
$118$	5.89714	0.542875
$119$	0.527573	0.0483626
$120$	−1.93960	−0.177060
$121$	1.00000	0.0909091
$122$	−6.92085	−0.626584
$123$	−14.5740	−1.31410
$124$	−6.35862	−0.571021
$125$	−1.00000	−0.0894427
$126$	−0.116120	−0.0103448
$127$	4.05044	0.359419	0.179709	−	0.983720i	$-0.442484\pi$
$127$	4.05044	0.359419	0.179709	+	0.983720i	$0.442484\pi$
$128$	1.00000	0.0883883
$129$	15.1996	1.33825
$130$	3.91609	0.343464
$131$	−2.76827	−0.241865	−0.120932	−	0.992661i	$-0.538588\pi$
$131$	−2.76827	−0.241865	−0.120932	+	0.992661i	$0.538588\pi$
$132$	1.93960	0.168820
$133$	0.594926	0.0515866
$134$	0.981624	0.0847994
$135$	4.34075	0.373593
$136$	−3.46218	−0.296879
$137$	9.08594	0.776264	0.388132	−	0.921604i	$-0.373120\pi$
$137$	9.08594	0.776264	0.388132	+	0.921604i	$0.373120\pi$
$138$	0.689222	0.0586704
$139$	−3.36050	−0.285034	−0.142517	−	0.989792i	$-0.545520\pi$
$139$	−3.36050	−0.285034	−0.142517	+	0.989792i	$0.545520\pi$
$140$	0.152382	0.0128786
$141$	−2.02083	−0.170185
$142$	3.05780	0.256605
$143$	−3.91609	−0.327480
$144$	0.762032	0.0635027
$145$	−5.86761	−0.487278
$146$	−1.00000	−0.0827606
$147$	−13.5321	−1.11611
$148$	4.42115	0.363416
$149$	−21.8231	−1.78782	−0.893911	−	0.448245i	$-0.852049\pi$
$149$	−21.8231	−1.78782	−0.893911	+	0.448245i	$0.852049\pi$
$150$	1.93960	0.158367
$151$	−14.8734	−1.21038	−0.605191	−	0.796080i	$-0.706903\pi$
$151$	−14.8734	−1.21038	−0.605191	+	0.796080i	$0.706903\pi$
$152$	−3.90418	−0.316671
$153$	−2.63829	−0.213293
$154$	−0.152382	−0.0122793
$155$	6.35862	0.510737
$156$	−7.59563	−0.608137
$157$	22.0169	1.75714	0.878572	−	0.477611i	$-0.158497\pi$
$157$	22.0169	1.75714	0.878572	+	0.477611i	$0.158497\pi$
$158$	6.98610	0.555785
$159$	−25.0194	−1.98416
$160$	−1.00000	−0.0790569
$161$	−0.0541478	−0.00426744
$162$	−10.7054	−0.841096
$163$	−12.6570	−0.991374	−0.495687	−	0.868501i	$-0.665084\pi$
$163$	−12.6570	−0.991374	−0.495687	+	0.868501i	$0.665084\pi$
$164$	−7.51395	−0.586741
$165$	−1.93960	−0.150997
$166$	−13.5996	−1.05553
$167$	1.86255	0.144128	0.0720641	−	0.997400i	$-0.477041\pi$
$167$	1.86255	0.144128	0.0720641	+	0.997400i	$0.477041\pi$
$168$	−0.295559	−0.0228029
$169$	2.33577	0.179675
$170$	3.46218	0.265537
$171$	−2.97511	−0.227512
$172$	7.83646	0.597524
$173$	2.85920	0.217381	0.108690	−	0.994076i	$-0.465334\pi$
$173$	2.85920	0.217381	0.108690	+	0.994076i	$0.465334\pi$
$174$	11.3808	0.862776
$175$	−0.152382	−0.0115190
$176$	1.00000	0.0753778
$177$	11.4381	0.859737
$178$	−6.41862	−0.481096
$179$	4.85245	0.362689	0.181345	−	0.983420i	$-0.441955\pi$
$179$	4.85245	0.362689	0.181345	+	0.983420i	$0.441955\pi$
$180$	−0.762032	−0.0567985
$181$	−13.6458	−1.01428	−0.507141	−	0.861863i	$-0.669298\pi$
$181$	−13.6458	−1.01428	−0.507141	+	0.861863i	$0.669298\pi$
$182$	0.596741	0.0442334
$183$	−13.4236	−0.992305
$184$	0.355343	0.0261962
$185$	−4.42115	−0.325049
$186$	−12.3332	−0.904311
$187$	−3.46218	−0.253180
$188$	−1.04188	−0.0759870
$189$	0.661452	0.0481136
$190$	3.90418	0.283239
$191$	−13.1125	−0.948784	−0.474392	−	0.880314i	$-0.657332\pi$
$191$	−13.1125	−0.948784	−0.474392	+	0.880314i	$0.657332\pi$
$192$	1.93960	0.139978
$193$	24.9771	1.79789	0.898945	−	0.438061i	$-0.144335\pi$
$193$	24.9771	1.79789	0.898945	+	0.438061i	$0.144335\pi$
$194$	−12.8595	−0.923262
$195$	7.59563	0.543935
$196$	−6.97678	−0.498341
$197$	5.78658	0.412276	0.206138	−	0.978523i	$-0.433910\pi$
$197$	5.78658	0.412276	0.206138	+	0.978523i	$0.433910\pi$
$198$	0.762032	0.0541553
$199$	−5.32173	−0.377248	−0.188624	−	0.982049i	$-0.560403\pi$
$199$	−5.32173	−0.377248	−0.188624	+	0.982049i	$0.560403\pi$
$200$	1.00000	0.0707107
$201$	1.90395	0.134295
$202$	19.1666	1.34856
$203$	−0.894117	−0.0627547
$204$	−6.71523	−0.470160
$205$	7.51395	0.524797
$206$	0.262376	0.0182806
$207$	0.270783	0.0188207
$208$	−3.91609	−0.271532
$209$	−3.90418	−0.270058
$210$	0.295559	0.0203955
$211$	16.3442	1.12518	0.562592	−	0.826735i	$-0.309804\pi$
$211$	16.3442	1.12518	0.562592	+	0.826735i	$0.309804\pi$
$212$	−12.8993	−0.885925
$213$	5.93089	0.406378
$214$	−19.6477	−1.34309
$215$	−7.83646	−0.534442
$216$	−4.34075	−0.295351
$217$	0.968938	0.0657758
$218$	−5.95630	−0.403411
$219$	−1.93960	−0.131066
$220$	−1.00000	−0.0674200
$221$	13.5582	0.912024
$222$	8.57524	0.575533
$223$	4.49820	0.301221	0.150611	−	0.988593i	$-0.451876\pi$
$223$	4.49820	0.301221	0.150611	+	0.988593i	$0.451876\pi$
$224$	−0.152382	−0.0101814
$225$	0.762032	0.0508021
$226$	−1.69194	−0.112546
$227$	25.1756	1.67096	0.835481	−	0.549519i	$-0.185189\pi$
$227$	25.1756	1.67096	0.835481	+	0.549519i	$0.185189\pi$
$228$	−7.57252	−0.501503
$229$	−3.61393	−0.238815	−0.119407	−	0.992845i	$-0.538099\pi$
$229$	−3.61393	−0.238815	−0.119407	+	0.992845i	$0.538099\pi$
$230$	−0.355343	−0.0234306
$231$	−0.295559	−0.0194464
$232$	5.86761	0.385227
$233$	6.98700	0.457734	0.228867	−	0.973458i	$-0.426498\pi$
$233$	6.98700	0.457734	0.228867	+	0.973458i	$0.426498\pi$
$234$	−2.98419	−0.195082
$235$	1.04188	0.0679649
$236$	5.89714	0.383871
$237$	13.5502	0.880181
$238$	0.527573	0.0341975
$239$	8.26212	0.534432	0.267216	−	0.963637i	$-0.413896\pi$
$239$	8.26212	0.534432	0.267216	+	0.963637i	$0.413896\pi$
$240$	−1.93960	−0.125200
$241$	6.16591	0.397181	0.198590	−	0.980083i	$-0.436364\pi$
$241$	6.16591	0.397181	0.198590	+	0.980083i	$0.436364\pi$
$242$	1.00000	0.0642824
$243$	−7.74189	−0.496643
$244$	−6.92085	−0.443062
$245$	6.97678	0.445730
$246$	−14.5740	−0.929206
$247$	15.2891	0.972823
$248$	−6.35862	−0.403773
$249$	−26.3778	−1.67162
$250$	−1.00000	−0.0632456
$251$	16.3953	1.03486	0.517431	−	0.855725i	$-0.326888\pi$
$251$	16.3953	1.03486	0.517431	+	0.855725i	$0.326888\pi$
$252$	−0.116120	−0.00731486
$253$	0.355343	0.0223402
$254$	4.05044	0.254147
$255$	6.71523	0.420524
$256$	1.00000	0.0625000
$257$	−6.42786	−0.400959	−0.200480	−	0.979698i	$-0.564250\pi$
$257$	−6.42786	−0.400959	−0.200480	+	0.979698i	$0.564250\pi$
$258$	15.1996	0.946283
$259$	−0.673703	−0.0418619
$260$	3.91609	0.242866
$261$	4.47131	0.276767
$262$	−2.76827	−0.171024
$263$	−3.67445	−0.226577	−0.113288	−	0.993562i	$-0.536138\pi$
$263$	−3.67445	−0.226577	−0.113288	+	0.993562i	$0.536138\pi$
$264$	1.93960	0.119374
$265$	12.8993	0.792395
$266$	0.594926	0.0364772
$267$	−12.4495	−0.761899
$268$	0.981624	0.0599622
$269$	−20.4867	−1.24910	−0.624549	−	0.780986i	$-0.714717\pi$
$269$	−20.4867	−1.24910	−0.624549	+	0.780986i	$0.714717\pi$
$270$	4.34075	0.264170
$271$	−0.251425	−0.0152730	−0.00763650	−	0.999971i	$-0.502431\pi$
$271$	−0.251425	−0.0152730	−0.00763650	+	0.999971i	$0.502431\pi$
$272$	−3.46218	−0.209925
$273$	1.15744	0.0700513
$274$	9.08594	0.548901
$275$	1.00000	0.0603023
$276$	0.689222	0.0414863
$277$	4.04919	0.243293	0.121646	−	0.992574i	$-0.461183\pi$
$277$	4.04919	0.243293	0.121646	+	0.992574i	$0.461183\pi$
$278$	−3.36050	−0.201550
$279$	−4.84547	−0.290091
$280$	0.152382	0.00910656
$281$	19.1585	1.14290	0.571452	−	0.820636i	$-0.306381\pi$
$281$	19.1585	1.14290	0.571452	+	0.820636i	$0.306381\pi$
$282$	−2.02083	−0.120339
$283$	0.819438	0.0487105	0.0243553	−	0.999703i	$-0.492247\pi$
$283$	0.819438	0.0487105	0.0243553	+	0.999703i	$0.492247\pi$
$284$	3.05780	0.181447
$285$	7.57252	0.448558
$286$	−3.91609	−0.231563
$287$	1.14499	0.0675866
$288$	0.762032	0.0449032
$289$	−5.01332	−0.294901
$290$	−5.86761	−0.344558
$291$	−24.9423	−1.46214
$292$	−1.00000	−0.0585206
$293$	−5.06398	−0.295841	−0.147921	−	0.988999i	$-0.547258\pi$
$293$	−5.06398	−0.295841	−0.147921	+	0.988999i	$0.547258\pi$
$294$	−13.5321	−0.789210
$295$	−5.89714	−0.343345
$296$	4.42115	0.256974
$297$	−4.34075	−0.251876
$298$	−21.8231	−1.26418
$299$	−1.39156	−0.0804757
$300$	1.93960	0.111983
$301$	−1.19413	−0.0688287
$302$	−14.8734	−0.855869
$303$	37.1755	2.13568
$304$	−3.90418	−0.223920
$305$	6.92085	0.396287
$306$	−2.63829	−0.150821
$307$	6.52870	0.372613	0.186306	−	0.982492i	$-0.440348\pi$
$307$	6.52870	0.372613	0.186306	+	0.982492i	$0.440348\pi$
$308$	−0.152382	−0.00868276
$309$	0.508904	0.0289505
$310$	6.35862	0.361145
$311$	−20.0524	−1.13706	−0.568532	−	0.822661i	$-0.692489\pi$
$311$	−20.0524	−1.13706	−0.568532	+	0.822661i	$0.692489\pi$
$312$	−7.59563	−0.430018
$313$	−17.7529	−1.00345	−0.501726	−	0.865026i	$-0.667302\pi$
$313$	−17.7529	−1.00345	−0.501726	+	0.865026i	$0.667302\pi$
$314$	22.0169	1.24249
$315$	0.116120	0.00654261
$316$	6.98610	0.392999
$317$	10.9494	0.614979	0.307489	−	0.951551i	$-0.400511\pi$
$317$	10.9494	0.614979	0.307489	+	0.951551i	$0.400511\pi$
$318$	−25.0194	−1.40302
$319$	5.86761	0.328523
$320$	−1.00000	−0.0559017
$321$	−38.1086	−2.12702
$322$	−0.0541478	−0.00301754
$323$	13.5170	0.752104
$324$	−10.7054	−0.594745
$325$	−3.91609	−0.217226
$326$	−12.6570	−0.701007
$327$	−11.5528	−0.638872
$328$	−7.51395	−0.414889
$329$	0.158764	0.00875294
$330$	−1.93960	−0.106771
$331$	28.4222	1.56223	0.781113	−	0.624390i	$-0.214652\pi$
$331$	28.4222	1.56223	0.781113	+	0.624390i	$0.214652\pi$
$332$	−13.5996	−0.746376
$333$	3.36906	0.184623
$334$	1.86255	0.101914
$335$	−0.981624	−0.0536318
$336$	−0.295559	−0.0161241
$337$	9.36026	0.509886	0.254943	−	0.966956i	$-0.417943\pi$
$337$	9.36026	0.509886	0.254943	+	0.966956i	$0.417943\pi$
$338$	2.33577	0.127049
$339$	−3.28168	−0.178236
$340$	3.46218	0.187763
$341$	−6.35862	−0.344339
$342$	−2.97511	−0.160875
$343$	2.12981	0.114999
$344$	7.83646	0.422513
$345$	−0.689222	−0.0371064
$346$	2.85920	0.153711
$347$	−5.35456	−0.287448	−0.143724	−	0.989618i	$-0.545908\pi$
$347$	−5.35456	−0.287448	−0.143724	+	0.989618i	$0.545908\pi$
$348$	11.3808	0.610074
$349$	7.78539	0.416742	0.208371	−	0.978050i	$-0.433184\pi$
$349$	7.78539	0.416742	0.208371	+	0.978050i	$0.433184\pi$
$350$	−0.152382	−0.00814515
$351$	16.9988	0.907328
$352$	1.00000	0.0533002
$353$	−17.7713	−0.945869	−0.472935	−	0.881098i	$-0.656805\pi$
$353$	−17.7713	−0.945869	−0.472935	+	0.881098i	$0.656805\pi$
$354$	11.4381	0.607926
$355$	−3.05780	−0.162291
$356$	−6.41862	−0.340186
$357$	1.02328	0.0541577
$358$	4.85245	0.256460
$359$	17.8063	0.939780	0.469890	−	0.882725i	$-0.344294\pi$
$359$	17.8063	0.939780	0.469890	+	0.882725i	$0.344294\pi$
$360$	−0.762032	−0.0401626
$361$	−3.75740	−0.197758
$362$	−13.6458	−0.717206
$363$	1.93960	0.101802
$364$	0.596741	0.0312777
$365$	1.00000	0.0523424
$366$	−13.4236	−0.701665
$367$	−1.15129	−0.0600967	−0.0300483	−	0.999548i	$-0.509566\pi$
$367$	−1.15129	−0.0600967	−0.0300483	+	0.999548i	$0.509566\pi$
$368$	0.355343	0.0185235
$369$	−5.72587	−0.298077
$370$	−4.42115	−0.229845
$371$	1.96561	0.102050
$372$	−12.3332	−0.639444
$373$	−10.3389	−0.535327	−0.267663	−	0.963512i	$-0.586252\pi$
$373$	−10.3389	−0.535327	−0.267663	+	0.963512i	$0.586252\pi$
$374$	−3.46218	−0.179025
$375$	−1.93960	−0.100160
$376$	−1.04188	−0.0537310
$377$	−22.9781	−1.18343
$378$	0.661452	0.0340214
$379$	28.6565	1.47198	0.735992	−	0.676991i	$-0.236716\pi$
$379$	28.6565	1.47198	0.735992	+	0.676991i	$0.236716\pi$
$380$	3.90418	0.200280
$381$	7.85622	0.402486
$382$	−13.1125	−0.670891
$383$	6.88281	0.351695	0.175848	−	0.984417i	$-0.443733\pi$
$383$	6.88281	0.351695	0.175848	+	0.984417i	$0.443733\pi$
$384$	1.93960	0.0989796
$385$	0.152382	0.00776610
$386$	24.9771	1.27130
$387$	5.97163	0.303555
$388$	−12.8595	−0.652845
$389$	−26.1772	−1.32723	−0.663617	−	0.748072i	$-0.730980\pi$
$389$	−26.1772	−1.32723	−0.663617	+	0.748072i	$0.730980\pi$
$390$	7.59563	0.384620
$391$	−1.23026	−0.0622170
$392$	−6.97678	−0.352381
$393$	−5.36932	−0.270847
$394$	5.78658	0.291523
$395$	−6.98610	−0.351509
$396$	0.762032	0.0382935
$397$	7.67051	0.384972	0.192486	−	0.981300i	$-0.438345\pi$
$397$	7.67051	0.384972	0.192486	+	0.981300i	$0.438345\pi$
$398$	−5.32173	−0.266754
$399$	1.15392	0.0577680
$400$	1.00000	0.0500000
$401$	18.7920	0.938427	0.469213	−	0.883085i	$-0.344538\pi$
$401$	18.7920	0.938427	0.469213	+	0.883085i	$0.344538\pi$
$402$	1.90395	0.0949606
$403$	24.9009	1.24040
$404$	19.1666	0.953574
$405$	10.7054	0.531956
$406$	−0.894117	−0.0443743
$407$	4.42115	0.219148
$408$	−6.71523	−0.332453
$409$	−3.23088	−0.159757	−0.0798783	−	0.996805i	$-0.525453\pi$
$409$	−3.23088	−0.159757	−0.0798783	+	0.996805i	$0.525453\pi$
$410$	7.51395	0.371088
$411$	17.6230	0.869281
$412$	0.262376	0.0129263
$413$	−0.898617	−0.0442180
$414$	0.270783	0.0133082
$415$	13.5996	0.667579
$416$	−3.91609	−0.192002
$417$	−6.51802	−0.319189
$418$	−3.90418	−0.190960
$419$	33.3188	1.62773	0.813865	−	0.581053i	$-0.197359\pi$
$419$	33.3188	1.62773	0.813865	+	0.581053i	$0.197359\pi$
$420$	0.295559	0.0144218
$421$	0.897233	0.0437285	0.0218642	−	0.999761i	$-0.493040\pi$
$421$	0.897233	0.0437285	0.0218642	+	0.999761i	$0.493040\pi$
$422$	16.3442	0.795625
$423$	−0.793947	−0.0386030
$424$	−12.8993	−0.626444
$425$	−3.46218	−0.167940
$426$	5.93089	0.287353
$427$	1.05461	0.0510362
$428$	−19.6477	−0.949709
$429$	−7.59563	−0.366721
$430$	−7.83646	−0.377908
$431$	−30.4221	−1.46538	−0.732690	−	0.680563i	$-0.761735\pi$
$431$	−30.4221	−1.46538	−0.732690	+	0.680563i	$0.761735\pi$
$432$	−4.34075	−0.208845
$433$	−13.9244	−0.669162	−0.334581	−	0.942367i	$-0.608595\pi$
$433$	−13.9244	−0.669162	−0.334581	+	0.942367i	$0.608595\pi$
$434$	0.968938	0.0465105
$435$	−11.3808	−0.545667
$436$	−5.95630	−0.285255
$437$	−1.38732	−0.0663646
$438$	−1.93960	−0.0926775
$439$	−28.4284	−1.35681	−0.678407	−	0.734687i	$-0.737329\pi$
$439$	−28.4284	−1.35681	−0.678407	+	0.734687i	$0.737329\pi$
$440$	−1.00000	−0.0476731
$441$	−5.31653	−0.253168
$442$	13.5582	0.644898
$443$	2.33741	0.111054	0.0555268	−	0.998457i	$-0.482316\pi$
$443$	2.33741	0.111054	0.0555268	+	0.998457i	$0.482316\pi$
$444$	8.57524	0.406963
$445$	6.41862	0.304272
$446$	4.49820	0.212996
$447$	−42.3281	−2.00205
$448$	−0.152382	−0.00719937
$449$	−36.0560	−1.70159	−0.850793	−	0.525501i	$-0.823878\pi$
$449$	−36.0560	−1.70159	−0.850793	+	0.525501i	$0.823878\pi$
$450$	0.762032	0.0359225
$451$	−7.51395	−0.353818
$452$	−1.69194	−0.0795822
$453$	−28.8484	−1.35542
$454$	25.1756	1.18155
$455$	−0.596741	−0.0279757
$456$	−7.57252	−0.354616
$457$	30.4983	1.42665	0.713324	−	0.700834i	$-0.247189\pi$
$457$	30.4983	1.42665	0.713324	+	0.700834i	$0.247189\pi$
$458$	−3.61393	−0.168868
$459$	15.0285	0.701469
$460$	−0.355343	−0.0165679
$461$	−32.0471	−1.49258	−0.746292	−	0.665619i	$-0.768168\pi$
$461$	−32.0471	−1.49258	−0.746292	+	0.665619i	$0.768168\pi$
$462$	−0.295559	−0.0137507
$463$	−11.0735	−0.514631	−0.257316	−	0.966327i	$-0.582838\pi$
$463$	−11.0735	−0.514631	−0.257316	+	0.966327i	$0.582838\pi$
$464$	5.86761	0.272397
$465$	12.3332	0.571936
$466$	6.98700	0.323667
$467$	6.90127	0.319353	0.159676	−	0.987169i	$-0.448955\pi$
$467$	6.90127	0.319353	0.159676	+	0.987169i	$0.448955\pi$
$468$	−2.98419	−0.137944
$469$	−0.149582	−0.00690704
$470$	1.04188	0.0480584
$471$	42.7040	1.96770
$472$	5.89714	0.271438
$473$	7.83646	0.360321
$474$	13.5502	0.622382
$475$	−3.90418	−0.179136
$476$	0.527573	0.0241813
$477$	−9.82965	−0.450069
$478$	8.26212	0.377901
$479$	19.4416	0.888309	0.444154	−	0.895950i	$-0.353504\pi$
$479$	19.4416	0.888309	0.444154	+	0.895950i	$0.353504\pi$
$480$	−1.93960	−0.0885300
$481$	−17.3136	−0.789433
$482$	6.16591	0.280849
$483$	−0.105025	−0.00477880
$484$	1.00000	0.0454545
$485$	12.8595	0.583922
$486$	−7.74189	−0.351179
$487$	18.6819	0.846559	0.423279	−	0.905999i	$-0.360879\pi$
$487$	18.6819	0.846559	0.423279	+	0.905999i	$0.360879\pi$
$488$	−6.92085	−0.313292
$489$	−24.5495	−1.11017
$490$	6.97678	0.315179
$491$	15.5225	0.700522	0.350261	−	0.936652i	$-0.386093\pi$
$491$	15.5225	0.700522	0.350261	+	0.936652i	$0.386093\pi$
$492$	−14.5740	−0.657048
$493$	−20.3147	−0.914929
$494$	15.2891	0.687890
$495$	−0.762032	−0.0342508
$496$	−6.35862	−0.285510
$497$	−0.465953	−0.0209008
$498$	−26.3778	−1.18202
$499$	28.7986	1.28920	0.644601	−	0.764519i	$-0.277023\pi$
$499$	28.7986	1.28920	0.644601	+	0.764519i	$0.277023\pi$
$500$	−1.00000	−0.0447214
$501$	3.61259	0.161398
$502$	16.3953	0.731758
$503$	−20.7787	−0.926476	−0.463238	−	0.886234i	$-0.653312\pi$
$503$	−20.7787	−0.926476	−0.463238	+	0.886234i	$0.653312\pi$
$504$	−0.116120	−0.00517239
$505$	−19.1666	−0.852903
$506$	0.355343	0.0157969
$507$	4.53045	0.201204
$508$	4.05044	0.179709
$509$	−19.5182	−0.865127	−0.432564	−	0.901603i	$-0.642391\pi$
$509$	−19.5182	−0.865127	−0.432564	+	0.901603i	$0.642391\pi$
$510$	6.71523	0.297355
$511$	0.152382	0.00674098
$512$	1.00000	0.0441942
$513$	16.9471	0.748231
$514$	−6.42786	−0.283521
$515$	−0.262376	−0.0115617
$516$	15.1996	0.669123
$517$	−1.04188	−0.0458219
$518$	−0.673703	−0.0296008
$519$	5.54569	0.243429
$520$	3.91609	0.171732
$521$	3.15235	0.138107	0.0690535	−	0.997613i	$-0.478002\pi$
$521$	3.15235	0.138107	0.0690535	+	0.997613i	$0.478002\pi$
$522$	4.47131	0.195704
$523$	−28.7069	−1.25527	−0.627633	−	0.778509i	$-0.715976\pi$
$523$	−28.7069	−1.25527	−0.627633	+	0.778509i	$0.715976\pi$
$524$	−2.76827	−0.120932
$525$	−0.295559	−0.0128993
$526$	−3.67445	−0.160214
$527$	22.0147	0.958975
$528$	1.93960	0.0844101
$529$	−22.8737	−0.994510
$530$	12.8993	0.560308
$531$	4.49381	0.195015
$532$	0.594926	0.0257933
$533$	29.4253	1.27455
$534$	−12.4495	−0.538744
$535$	19.6477	0.849445
$536$	0.981624	0.0423997
$537$	9.41180	0.406149
$538$	−20.4867	−0.883245
$539$	−6.97678	−0.300511
$540$	4.34075	0.186796
$541$	−26.4046	−1.13522	−0.567612	−	0.823296i	$-0.692133\pi$
$541$	−26.4046	−1.13522	−0.567612	+	0.823296i	$0.692133\pi$
$542$	−0.251425	−0.0107996
$543$	−26.4673	−1.13582
$544$	−3.46218	−0.148440
$545$	5.95630	0.255140
$546$	1.15744	0.0495337
$547$	−6.30074	−0.269400	−0.134700	−	0.990886i	$-0.543007\pi$
$547$	−6.30074	−0.269400	−0.134700	+	0.990886i	$0.543007\pi$
$548$	9.08594	0.388132
$549$	−5.27391	−0.225085
$550$	1.00000	0.0426401
$551$	−22.9082	−0.975922
$552$	0.689222	0.0293352
$553$	−1.06456	−0.0452695
$554$	4.04919	0.172034
$555$	−8.57524	−0.363999
$556$	−3.36050	−0.142517
$557$	29.2771	1.24051	0.620255	−	0.784400i	$-0.287029\pi$
$557$	29.2771	1.24051	0.620255	+	0.784400i	$0.287029\pi$
$558$	−4.84547	−0.205125
$559$	−30.6883	−1.29798
$560$	0.152382	0.00643931
$561$	−6.71523	−0.283517
$562$	19.1585	0.808154
$563$	17.5177	0.738281	0.369140	−	0.929374i	$-0.379652\pi$
$563$	17.5177	0.738281	0.369140	+	0.929374i	$0.379652\pi$
$564$	−2.02083	−0.0850923
$565$	1.69194	0.0711804
$566$	0.819438	0.0344435
$567$	1.63131	0.0685085
$568$	3.05780	0.128302
$569$	2.23776	0.0938118	0.0469059	−	0.998899i	$-0.485064\pi$
$569$	2.23776	0.0938118	0.0469059	+	0.998899i	$0.485064\pi$
$570$	7.57252	0.317178
$571$	42.3046	1.77039	0.885195	−	0.465220i	$-0.154025\pi$
$571$	42.3046	1.77039	0.885195	+	0.465220i	$0.154025\pi$
$572$	−3.91609	−0.163740
$573$	−25.4329	−1.06247
$574$	1.14499	0.0477910
$575$	0.355343	0.0148188
$576$	0.762032	0.0317513
$577$	−5.27174	−0.219465	−0.109733	−	0.993961i	$-0.534999\pi$
$577$	−5.27174	−0.219465	−0.109733	+	0.993961i	$0.534999\pi$
$578$	−5.01332	−0.208526
$579$	48.4455	2.01332
$580$	−5.86761	−0.243639
$581$	2.07233	0.0859749
$582$	−24.9423	−1.03389
$583$	−12.8993	−0.534233
$584$	−1.00000	−0.0413803
$585$	2.98419	0.123381
$586$	−5.06398	−0.209191
$587$	−10.4262	−0.430334	−0.215167	−	0.976577i	$-0.569030\pi$
$587$	−10.4262	−0.430334	−0.215167	+	0.976577i	$0.569030\pi$
$588$	−13.5321	−0.558056
$589$	24.8252	1.02290
$590$	−5.89714	−0.242781
$591$	11.2236	0.461678
$592$	4.42115	0.181708
$593$	39.9109	1.63894	0.819472	−	0.573119i	$-0.194267\pi$
$593$	39.9109	1.63894	0.819472	+	0.573119i	$0.194267\pi$
$594$	−4.34075	−0.178103
$595$	−0.527573	−0.0216284
$596$	−21.8231	−0.893911
$597$	−10.3220	−0.422452
$598$	−1.39156	−0.0569049
$599$	4.91754	0.200925	0.100463	−	0.994941i	$-0.467968\pi$
$599$	4.91754	0.200925	0.100463	+	0.994941i	$0.467968\pi$
$600$	1.93960	0.0791837
$601$	21.3873	0.872408	0.436204	−	0.899848i	$-0.356323\pi$
$601$	21.3873	0.872408	0.436204	+	0.899848i	$0.356323\pi$
$602$	−1.19413	−0.0486693
$603$	0.748029	0.0304621
$604$	−14.8734	−0.605191
$605$	−1.00000	−0.0406558
$606$	37.1755	1.51015
$607$	−22.5098	−0.913646	−0.456823	−	0.889558i	$-0.651013\pi$
$607$	−22.5098	−0.913646	−0.456823	+	0.889558i	$0.651013\pi$
$608$	−3.90418	−0.158335
$609$	−1.73423	−0.0702744
$610$	6.92085	0.280217
$611$	4.08010	0.165063
$612$	−2.63829	−0.106647
$613$	−30.5424	−1.23360	−0.616798	−	0.787122i	$-0.711570\pi$
$613$	−30.5424	−1.23360	−0.616798	+	0.787122i	$0.711570\pi$
$614$	6.52870	0.263477
$615$	14.5740	0.587682
$616$	−0.152382	−0.00613964
$617$	−23.2224	−0.934900	−0.467450	−	0.884020i	$-0.654827\pi$
$617$	−23.2224	−0.934900	−0.467450	+	0.884020i	$0.654827\pi$
$618$	0.508904	0.0204711
$619$	−17.6012	−0.707451	−0.353726	−	0.935349i	$-0.615085\pi$
$619$	−17.6012	−0.707451	−0.353726	+	0.935349i	$0.615085\pi$
$620$	6.35862	0.255368
$621$	−1.54246	−0.0618966
$622$	−20.0524	−0.804026
$623$	0.978081	0.0391860
$624$	−7.59563	−0.304069
$625$	1.00000	0.0400000
$626$	−17.7529	−0.709548
$627$	−7.57252	−0.302418
$628$	22.0169	0.878572
$629$	−15.3068	−0.610323
$630$	0.116120	0.00462632
$631$	−11.6038	−0.461939	−0.230969	−	0.972961i	$-0.574190\pi$
$631$	−11.6038	−0.461939	−0.230969	+	0.972961i	$0.574190\pi$
$632$	6.98610	0.277892
$633$	31.7012	1.26001
$634$	10.9494	0.434856
$635$	−4.05044	−0.160737
$636$	−25.0194	−0.992082
$637$	27.3217	1.08253
$638$	5.86761	0.232301
$639$	2.33014	0.0921789
$640$	−1.00000	−0.0395285
$641$	−40.4063	−1.59595	−0.797977	−	0.602688i	$-0.794096\pi$
$641$	−40.4063	−1.59595	−0.797977	+	0.602688i	$0.794096\pi$
$642$	−38.1086	−1.50403
$643$	8.20712	0.323657	0.161829	−	0.986819i	$-0.448261\pi$
$643$	8.20712	0.323657	0.161829	+	0.986819i	$0.448261\pi$
$644$	−0.0541478	−0.00213372
$645$	−15.1996	−0.598482
$646$	13.5170	0.531818
$647$	−19.4145	−0.763262	−0.381631	−	0.924315i	$-0.624637\pi$
$647$	−19.4145	−0.763262	−0.381631	+	0.924315i	$0.624637\pi$
$648$	−10.7054	−0.420548
$649$	5.89714	0.231483
$650$	−3.91609	−0.153602
$651$	1.87935	0.0736575
$652$	−12.6570	−0.495687
$653$	−21.0566	−0.824007	−0.412004	−	0.911182i	$-0.635171\pi$
$653$	−21.0566	−0.824007	−0.412004	+	0.911182i	$0.635171\pi$
$654$	−11.5528	−0.451750
$655$	2.76827	0.108165
$656$	−7.51395	−0.293371
$657$	−0.762032	−0.0297297
$658$	0.158764	0.00618926
$659$	32.0887	1.25000	0.624999	−	0.780625i	$-0.285099\pi$
$659$	32.0887	1.25000	0.624999	+	0.780625i	$0.285099\pi$
$660$	−1.93960	−0.0754987
$661$	−18.9903	−0.738635	−0.369318	−	0.929303i	$-0.620409\pi$
$661$	−18.9903	−0.738635	−0.369318	+	0.929303i	$0.620409\pi$
$662$	28.4222	1.10466
$663$	26.2974	1.02131
$664$	−13.5996	−0.527767
$665$	−0.594926	−0.0230702
$666$	3.36906	0.130548
$667$	2.08501	0.0807320
$668$	1.86255	0.0720641
$669$	8.72468	0.337316
$670$	−0.981624	−0.0379234
$671$	−6.92085	−0.267176
$672$	−0.295559	−0.0114014
$673$	−6.62061	−0.255206	−0.127603	−	0.991825i	$-0.540728\pi$
$673$	−6.62061	−0.255206	−0.127603	+	0.991825i	$0.540728\pi$
$674$	9.36026	0.360544
$675$	−4.34075	−0.167076
$676$	2.33577	0.0898374
$677$	7.33258	0.281814	0.140907	−	0.990023i	$-0.454998\pi$
$677$	7.33258	0.281814	0.140907	+	0.990023i	$0.454998\pi$
$678$	−3.28168	−0.126032
$679$	1.95956	0.0752011
$680$	3.46218	0.132769
$681$	48.8305	1.87119
$682$	−6.35862	−0.243484
$683$	36.9396	1.41346	0.706728	−	0.707486i	$-0.250171\pi$
$683$	36.9396	1.41346	0.706728	+	0.707486i	$0.250171\pi$
$684$	−2.97511	−0.113756
$685$	−9.08594	−0.347156
$686$	2.12981	0.0813164
$687$	−7.00956	−0.267431
$688$	7.83646	0.298762
$689$	50.5147	1.92446
$690$	−0.689222	−0.0262382
$691$	11.1188	0.422979	0.211489	−	0.977380i	$-0.432169\pi$
$691$	11.1188	0.422979	0.211489	+	0.977380i	$0.432169\pi$
$692$	2.85920	0.108690
$693$	−0.116120	−0.00441103
$694$	−5.35456	−0.203256
$695$	3.36050	0.127471
$696$	11.3808	0.431388
$697$	26.0146	0.985375
$698$	7.78539	0.294681
$699$	13.5520	0.512582
$700$	−0.152382	−0.00575949
$701$	34.1990	1.29168	0.645839	−	0.763474i	$-0.276508\pi$
$701$	34.1990	1.29168	0.645839	+	0.763474i	$0.276508\pi$
$702$	16.9988	0.641578
$703$	−17.2609	−0.651009
$704$	1.00000	0.0376889
$705$	2.02083	0.0761089
$706$	−17.7713	−0.668830
$707$	−2.92064	−0.109842
$708$	11.4381	0.429869
$709$	8.17702	0.307095	0.153547	−	0.988141i	$-0.450930\pi$
$709$	8.17702	0.307095	0.153547	+	0.988141i	$0.450930\pi$
$710$	−3.05780	−0.114757
$711$	5.32363	0.199652
$712$	−6.41862	−0.240548
$713$	−2.25949	−0.0846186
$714$	1.02328	0.0382952
$715$	3.91609	0.146454
$716$	4.85245	0.181345
$717$	16.0252	0.598471
$718$	17.8063	0.664525
$719$	7.46405	0.278362	0.139181	−	0.990267i	$-0.455553\pi$
$719$	7.46405	0.278362	0.139181	+	0.990267i	$0.455553\pi$
$720$	−0.762032	−0.0283993
$721$	−0.0399814	−0.00148898
$722$	−3.75740	−0.139836
$723$	11.9594	0.444774
$724$	−13.6458	−0.507141
$725$	5.86761	0.217918
$726$	1.93960	0.0719852
$727$	34.1286	1.26576	0.632880	−	0.774250i	$-0.281873\pi$
$727$	34.1286	1.26576	0.632880	+	0.774250i	$0.281873\pi$
$728$	0.596741	0.0221167
$729$	17.1001	0.633336
$730$	1.00000	0.0370117
$731$	−27.1312	−1.00348
$732$	−13.4236	−0.496152
$733$	−33.2529	−1.22822	−0.614112	−	0.789219i	$-0.710486\pi$
$733$	−33.2529	−1.22822	−0.614112	+	0.789219i	$0.710486\pi$
$734$	−1.15129	−0.0424948
$735$	13.5321	0.499140
$736$	0.355343	0.0130981
$737$	0.981624	0.0361586
$738$	−5.72587	−0.210772
$739$	26.2944	0.967256	0.483628	−	0.875274i	$-0.339319\pi$
$739$	26.2944	0.967256	0.483628	+	0.875274i	$0.339319\pi$
$740$	−4.42115	−0.162525
$741$	29.6547	1.08939
$742$	1.96561	0.0721599
$743$	−2.43739	−0.0894191	−0.0447095	−	0.999000i	$-0.514236\pi$
$743$	−2.43739	−0.0894191	−0.0447095	+	0.999000i	$0.514236\pi$
$744$	−12.3332	−0.452155
$745$	21.8231	0.799538
$746$	−10.3389	−0.378533
$747$	−10.3633	−0.379175
$748$	−3.46218	−0.126590
$749$	2.99396	0.109397
$750$	−1.93960	−0.0708240
$751$	45.7240	1.66849	0.834246	−	0.551393i	$-0.185903\pi$
$751$	45.7240	1.66849	0.834246	+	0.551393i	$0.185903\pi$
$752$	−1.04188	−0.0379935
$753$	31.8003	1.15887
$754$	−22.9781	−0.836813
$755$	14.8734	0.541299
$756$	0.661452	0.0240568
$757$	10.2421	0.372256	0.186128	−	0.982525i	$-0.440406\pi$
$757$	10.2421	0.372256	0.186128	+	0.982525i	$0.440406\pi$
$758$	28.6565	1.04085
$759$	0.689222	0.0250172
$760$	3.90418	0.141619
$761$	35.2490	1.27777	0.638887	−	0.769300i	$-0.279395\pi$
$761$	35.2490	1.27777	0.638887	+	0.769300i	$0.279395\pi$
$762$	7.85622	0.284601
$763$	0.907631	0.0328585
$764$	−13.1125	−0.474392
$765$	2.63829	0.0953876
$766$	6.88281	0.248686
$767$	−23.0937	−0.833866
$768$	1.93960	0.0699891
$769$	−24.1953	−0.872503	−0.436252	−	0.899825i	$-0.643694\pi$
$769$	−24.1953	−0.872503	−0.436252	+	0.899825i	$0.643694\pi$
$770$	0.152382	0.00549146
$771$	−12.4675	−0.449005
$772$	24.9771	0.898945
$773$	2.69279	0.0968531	0.0484265	−	0.998827i	$-0.484579\pi$
$773$	2.69279	0.0968531	0.0484265	+	0.998827i	$0.484579\pi$
$774$	5.97163	0.214646
$775$	−6.35862	−0.228408
$776$	−12.8595	−0.461631
$777$	−1.30671	−0.0468780
$778$	−26.1772	−0.938497
$779$	29.3358	1.05106
$780$	7.59563	0.271967
$781$	3.05780	0.109417
$782$	−1.23026	−0.0439940
$783$	−25.4698	−0.910218
$784$	−6.97678	−0.249171
$785$	−22.0169	−0.785818
$786$	−5.36932	−0.191517
$787$	−17.5472	−0.625492	−0.312746	−	0.949837i	$-0.601249\pi$
$787$	−17.5472	−0.625492	−0.312746	+	0.949837i	$0.601249\pi$
$788$	5.78658	0.206138
$789$	−7.12696	−0.253726
$790$	−6.98610	−0.248554
$791$	0.257821	0.00916706
$792$	0.762032	0.0270776
$793$	27.1027	0.962444
$794$	7.67051	0.272216
$795$	25.0194	0.887345
$796$	−5.32173	−0.188624
$797$	−29.2606	−1.03646	−0.518232	−	0.855240i	$-0.673410\pi$
$797$	−29.2606	−1.03646	−0.518232	+	0.855240i	$0.673410\pi$
$798$	1.15392	0.0408482
$799$	3.60718	0.127613
$800$	1.00000	0.0353553
$801$	−4.89119	−0.172822
$802$	18.7920	0.663568
$803$	−1.00000	−0.0352892
$804$	1.90395	0.0671473
$805$	0.0541478	0.00190846
$806$	24.9009	0.877098
$807$	−39.7360	−1.39877
$808$	19.1666	0.674279
$809$	−10.8290	−0.380727	−0.190364	−	0.981714i	$-0.560967\pi$
$809$	−10.8290	−0.380727	−0.190364	+	0.981714i	$0.560967\pi$
$810$	10.7054	0.376150
$811$	−22.6824	−0.796488	−0.398244	−	0.917279i	$-0.630380\pi$
$811$	−22.6824	−0.796488	−0.398244	+	0.917279i	$0.630380\pi$
$812$	−0.894117	−0.0313774
$813$	−0.487663	−0.0171031
$814$	4.42115	0.154961
$815$	12.6570	0.443356
$816$	−6.71523	−0.235080
$817$	−30.5949	−1.07038
$818$	−3.23088	−0.112965
$819$	0.454736	0.0158898
$820$	7.51395	0.262399
$821$	−4.18767	−0.146151	−0.0730754	−	0.997326i	$-0.523281\pi$
$821$	−4.18767	−0.146151	−0.0730754	+	0.997326i	$0.523281\pi$
$822$	17.6230	0.614674
$823$	−31.0815	−1.08343	−0.541716	−	0.840562i	$-0.682225\pi$
$823$	−31.0815	−1.08343	−0.541716	+	0.840562i	$0.682225\pi$
$824$	0.262376	0.00914031
$825$	1.93960	0.0675281
$826$	−0.898617	−0.0312669
$827$	40.1442	1.39595	0.697976	−	0.716121i	$-0.254084\pi$
$827$	40.1442	1.39595	0.697976	+	0.716121i	$0.254084\pi$
$828$	0.270783	0.00941035
$829$	−28.8985	−1.00369	−0.501844	−	0.864958i	$-0.667345\pi$
$829$	−28.8985	−1.00369	−0.501844	+	0.864958i	$0.667345\pi$
$830$	13.5996	0.472050
$831$	7.85380	0.272445
$832$	−3.91609	−0.135766
$833$	24.1549	0.836916
$834$	−6.51802	−0.225701
$835$	−1.86255	−0.0644561
$836$	−3.90418	−0.135029
$837$	27.6012	0.954037
$838$	33.3188	1.15098
$839$	−1.93709	−0.0668759	−0.0334379	−	0.999441i	$-0.510646\pi$
$839$	−1.93709	−0.0668759	−0.0334379	+	0.999441i	$0.510646\pi$
$840$	0.295559	0.0101978
$841$	5.42885	0.187202
$842$	0.897233	0.0309207
$843$	37.1598	1.27985
$844$	16.3442	0.562592
$845$	−2.33577	−0.0803530
$846$	−0.793947	−0.0272965
$847$	−0.152382	−0.00523590
$848$	−12.8993	−0.442962
$849$	1.58938	0.0545473
$850$	−3.46218	−0.118752
$851$	1.57102	0.0538540
$852$	5.93089	0.203189
$853$	11.7482	0.402252	0.201126	−	0.979565i	$-0.435540\pi$
$853$	11.7482	0.402252	0.201126	+	0.979565i	$0.435540\pi$
$854$	1.05461	0.0360881
$855$	2.97511	0.101747
$856$	−19.6477	−0.671545
$857$	10.2552	0.350310	0.175155	−	0.984541i	$-0.443957\pi$
$857$	10.2552	0.350310	0.175155	+	0.984541i	$0.443957\pi$
$858$	−7.59563	−0.259311
$859$	29.5636	1.00870	0.504349	−	0.863500i	$-0.331733\pi$
$859$	29.5636	1.00870	0.504349	+	0.863500i	$0.331733\pi$
$860$	−7.83646	−0.267221
$861$	2.22082	0.0756853
$862$	−30.4221	−1.03618
$863$	−21.8590	−0.744088	−0.372044	−	0.928215i	$-0.621343\pi$
$863$	−21.8590	−0.744088	−0.372044	+	0.928215i	$0.621343\pi$
$864$	−4.34075	−0.147675
$865$	−2.85920	−0.0972156
$866$	−13.9244	−0.473169
$867$	−9.72381	−0.330238
$868$	0.968938	0.0328879
$869$	6.98610	0.236987
$870$	−11.3808	−0.385845
$871$	−3.84413	−0.130253
$872$	−5.95630	−0.201706
$873$	−9.79938	−0.331659
$874$	−1.38732	−0.0469268
$875$	0.152382	0.00515145
$876$	−1.93960	−0.0655329
$877$	−42.0389	−1.41955	−0.709776	−	0.704427i	$-0.751204\pi$
$877$	−42.0389	−1.41955	−0.709776	+	0.704427i	$0.751204\pi$
$878$	−28.4284	−0.959412
$879$	−9.82208	−0.331291
$880$	−1.00000	−0.0337100
$881$	−27.5638	−0.928650	−0.464325	−	0.885665i	$-0.653703\pi$
$881$	−27.5638	−0.928650	−0.464325	+	0.885665i	$0.653703\pi$
$882$	−5.31653	−0.179017
$883$	−11.4232	−0.384421	−0.192211	−	0.981354i	$-0.561566\pi$
$883$	−11.4232	−0.384421	−0.192211	+	0.981354i	$0.561566\pi$
$884$	13.5582	0.456012
$885$	−11.4381	−0.384486
$886$	2.33741	0.0785267
$887$	41.6476	1.39839	0.699195	−	0.714931i	$-0.253542\pi$
$887$	41.6476	1.39839	0.699195	+	0.714931i	$0.253542\pi$
$888$	8.57524	0.287766
$889$	−0.617214	−0.0207007
$890$	6.41862	0.215153
$891$	−10.7054	−0.358644
$892$	4.49820	0.150611
$893$	4.06769	0.136120
$894$	−42.3281	−1.41566
$895$	−4.85245	−0.162200
$896$	−0.152382	−0.00509072
$897$	−2.69905	−0.0901188
$898$	−36.0560	−1.20320
$899$	−37.3099	−1.24435
$900$	0.762032	0.0254011
$901$	44.6596	1.48783
$902$	−7.51395	−0.250187
$903$	−2.31614	−0.0770762
$904$	−1.69194	−0.0562731
$905$	13.6458	0.453601
$906$	−28.8484	−0.958425
$907$	−24.9763	−0.829324	−0.414662	−	0.909975i	$-0.636100\pi$
$907$	−24.9763	−0.829324	−0.414662	+	0.909975i	$0.636100\pi$
$908$	25.1756	0.835481
$909$	14.6056	0.484436
$910$	−0.596741	−0.0197818
$911$	−55.4253	−1.83632	−0.918161	−	0.396208i	$-0.870326\pi$
$911$	−55.4253	−1.83632	−0.918161	+	0.396208i	$0.870326\pi$
$912$	−7.57252	−0.250751
$913$	−13.5996	−0.450082
$914$	30.4983	1.00879
$915$	13.4236	0.443772
$916$	−3.61393	−0.119407
$917$	0.421834	0.0139302
$918$	15.0285	0.496013
$919$	3.32906	0.109816	0.0549078	−	0.998491i	$-0.482513\pi$
$919$	3.32906	0.109816	0.0549078	+	0.998491i	$0.482513\pi$
$920$	−0.355343	−0.0117153
$921$	12.6630	0.417262
$922$	−32.0471	−1.05542
$923$	−11.9746	−0.394149
$924$	−0.295559	−0.00972318
$925$	4.42115	0.145367
$926$	−11.0735	−0.363899
$927$	0.199939	0.00656686
$928$	5.86761	0.192614
$929$	43.0124	1.41119	0.705596	−	0.708614i	$-0.250679\pi$
$929$	43.0124	1.41119	0.705596	+	0.708614i	$0.250679\pi$
$930$	12.3332	0.404420
$931$	27.2386	0.892708
$932$	6.98700	0.228867
$933$	−38.8935	−1.27331
$934$	6.90127	0.225817
$935$	3.46218	0.113225
$936$	−2.98419	−0.0975412
$937$	8.25096	0.269547	0.134774	−	0.990876i	$-0.456969\pi$
$937$	8.25096	0.269547	0.134774	+	0.990876i	$0.456969\pi$
$938$	−0.149582	−0.00488401
$939$	−34.4334	−1.12369
$940$	1.04188	0.0339824
$941$	−24.8469	−0.809987	−0.404993	−	0.914320i	$-0.632726\pi$
$941$	−24.8469	−0.809987	−0.404993	+	0.914320i	$0.632726\pi$
$942$	42.7040	1.39137
$943$	−2.67003	−0.0869481
$944$	5.89714	0.191935
$945$	−0.661452	−0.0215170
$946$	7.83646	0.254785
$947$	7.93758	0.257937	0.128968	−	0.991649i	$-0.458833\pi$
$947$	7.93758	0.257937	0.128968	+	0.991649i	$0.458833\pi$
$948$	13.5502	0.440091
$949$	3.91609	0.127122
$950$	−3.90418	−0.126668
$951$	21.2374	0.688670
$952$	0.527573	0.0170987
$953$	51.4015	1.66506	0.832529	−	0.553982i	$-0.186892\pi$
$953$	51.4015	1.66506	0.832529	+	0.553982i	$0.186892\pi$
$954$	−9.82965	−0.318247
$955$	13.1125	0.424309
$956$	8.26212	0.267216
$957$	11.3808	0.367889
$958$	19.4416	0.628129
$959$	−1.38453	−0.0447089
$960$	−1.93960	−0.0626002
$961$	9.43205	0.304260
$962$	−17.3136	−0.558214
$963$	−14.9722	−0.482472
$964$	6.16591	0.198590
$965$	−24.9771	−0.804041
$966$	−0.105025	−0.00337912
$967$	41.0729	1.32082	0.660408	−	0.750907i	$-0.270384\pi$
$967$	41.0729	1.32082	0.660408	+	0.750907i	$0.270384\pi$
$968$	1.00000	0.0321412
$969$	26.2174	0.842225
$970$	12.8595	0.412895
$971$	31.4383	1.00890	0.504451	−	0.863440i	$-0.331695\pi$
$971$	31.4383	1.00890	0.504451	+	0.863440i	$0.331695\pi$
$972$	−7.74189	−0.248321
$973$	0.512080	0.0164165
$974$	18.6819	0.598608
$975$	−7.59563	−0.243255
$976$	−6.92085	−0.221531
$977$	23.2979	0.745364	0.372682	−	0.927959i	$-0.378438\pi$
$977$	23.2979	0.745364	0.372682	+	0.927959i	$0.378438\pi$
$978$	−24.5495	−0.785006
$979$	−6.41862	−0.205140
$980$	6.97678	0.222865
$981$	−4.53889	−0.144916
$982$	15.5225	0.495344
$983$	−9.06195	−0.289031	−0.144516	−	0.989503i	$-0.546162\pi$
$983$	−9.06195	−0.289031	−0.144516	+	0.989503i	$0.546162\pi$
$984$	−14.5740	−0.464603
$985$	−5.78658	−0.184376
$986$	−20.3147	−0.646952
$987$	0.307938	0.00980177
$988$	15.2891	0.486411
$989$	2.78463	0.0885460
$990$	−0.762032	−0.0242190
$991$	−58.7906	−1.86754	−0.933772	−	0.357869i	$-0.883503\pi$
$991$	−58.7906	−1.86754	−0.933772	+	0.357869i	$0.883503\pi$
$992$	−6.35862	−0.201886
$993$	55.1276	1.74942
$994$	−0.465953	−0.0147791
$995$	5.32173	0.168710
$996$	−26.3778	−0.835811
$997$	−36.3880	−1.15242	−0.576210	−	0.817302i	$-0.695469\pi$
$997$	−36.3880	−1.15242	−0.576210	+	0.817302i	$0.695469\pi$
$998$	28.7986	0.911604
$999$	−19.1911	−0.607180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.bb.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.bb.1.8	✓	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:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.93960 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.93960 q^{6} -0.152382 q^{7} +1.00000 q^{8} +0.762032 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.93960 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.93960 q^{6} -0.152382 q^{7} +1.00000 q^{8} +0.762032 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.93960 q^{12} -3.91609 q^{13} -0.152382 q^{14} -1.93960 q^{15} +1.00000 q^{16} -3.46218 q^{17} +0.762032 q^{18} -3.90418 q^{19} -1.00000 q^{20} -0.295559 q^{21} +1.00000 q^{22} +0.355343 q^{23} +1.93960 q^{24} +1.00000 q^{25} -3.91609 q^{26} -4.34075 q^{27} -0.152382 q^{28} +5.86761 q^{29} -1.93960 q^{30} -6.35862 q^{31} +1.00000 q^{32} +1.93960 q^{33} -3.46218 q^{34} +0.152382 q^{35} +0.762032 q^{36} +4.42115 q^{37} -3.90418 q^{38} -7.59563 q^{39} -1.00000 q^{40} -7.51395 q^{41} -0.295559 q^{42} +7.83646 q^{43} +1.00000 q^{44} -0.762032 q^{45} +0.355343 q^{46} -1.04188 q^{47} +1.93960 q^{48} -6.97678 q^{49} +1.00000 q^{50} -6.71523 q^{51} -3.91609 q^{52} -12.8993 q^{53} -4.34075 q^{54} -1.00000 q^{55} -0.152382 q^{56} -7.57252 q^{57} +5.86761 q^{58} +5.89714 q^{59} -1.93960 q^{60} -6.92085 q^{61} -6.35862 q^{62} -0.116120 q^{63} +1.00000 q^{64} +3.91609 q^{65} +1.93960 q^{66} +0.981624 q^{67} -3.46218 q^{68} +0.689222 q^{69} +0.152382 q^{70} +3.05780 q^{71} +0.762032 q^{72} -1.00000 q^{73} +4.42115 q^{74} +1.93960 q^{75} -3.90418 q^{76} -0.152382 q^{77} -7.59563 q^{78} +6.98610 q^{79} -1.00000 q^{80} -10.7054 q^{81} -7.51395 q^{82} -13.5996 q^{83} -0.295559 q^{84} +3.46218 q^{85} +7.83646 q^{86} +11.3808 q^{87} +1.00000 q^{88} -6.41862 q^{89} -0.762032 q^{90} +0.596741 q^{91} +0.355343 q^{92} -12.3332 q^{93} -1.04188 q^{94} +3.90418 q^{95} +1.93960 q^{96} -12.8595 q^{97} -6.97678 q^{98} +0.762032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9	\( 8 q + 8 q^{2} - 3 q^{3} + 8 q^{4} - 8 q^{5} - 3 q^{6} - 4 q^{7} + 8 q^{8} + q^{9} - 8 q^{10} + 8 q^{11} - 3 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 8 q^{16} + 2 q^{17} + q^{18} - q^{19} - 8 q^{20} + 7 q^{21} + 8 q^{22} - 16 q^{23} - 3 q^{24} + 8 q^{25} - 7 q^{26} - 21 q^{27} - 4 q^{28} - 5 q^{29} + 3 q^{30} - 22 q^{31} + 8 q^{32} - 3 q^{33} + 2 q^{34} + 4 q^{35} + q^{36} - 9 q^{37} - q^{38} - 18 q^{39} - 8 q^{40} + 6 q^{41} + 7 q^{42} + 15 q^{43} + 8 q^{44} - q^{45} - 16 q^{46} - 7 q^{47} - 3 q^{48} - 6 q^{49} + 8 q^{50} + q^{51} - 7 q^{52} - 11 q^{53} - 21 q^{54} - 8 q^{55} - 4 q^{56} - 17 q^{57} - 5 q^{58} - 11 q^{59} + 3 q^{60} - 22 q^{61} - 22 q^{62} + q^{63} + 8 q^{64} + 7 q^{65} - 3 q^{66} + 21 q^{67} + 2 q^{68} + q^{69} + 4 q^{70} - 28 q^{71} + q^{72} - 8 q^{73} - 9 q^{74} - 3 q^{75} - q^{76} - 4 q^{77} - 18 q^{78} - 28 q^{79} - 8 q^{80} + 12 q^{81} + 6 q^{82} - 5 q^{83} + 7 q^{84} - 2 q^{85} + 15 q^{86} + 36 q^{87} + 8 q^{88} - 17 q^{89} - q^{90} - 29 q^{91} - 16 q^{92} + 42 q^{93} - 7 q^{94} + q^{95} - 3 q^{96} + q^{97} - 6 q^{98} + q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 3 * q^3 + 8 * q^4 - 8 * q^5 - 3 * q^6 - 4 * q^7 + 8 * q^8 + q^9 - 8 * q^10 + 8 * q^11 - 3 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 8 * q^16 + 2 * q^17 + q^18 - q^19 - 8 * q^20 + 7 * q^21 + 8 * q^22 - 16 * q^23 - 3 * q^24 + 8 * q^25 - 7 * q^26 - 21 * q^27 - 4 * q^28 - 5 * q^29 + 3 * q^30 - 22 * q^31 + 8 * q^32 - 3 * q^33 + 2 * q^34 + 4 * q^35 + q^36 - 9 * q^37 - q^38 - 18 * q^39 - 8 * q^40 + 6 * q^41 + 7 * q^42 + 15 * q^43 + 8 * q^44 - q^45 - 16 * q^46 - 7 * q^47 - 3 * q^48 - 6 * q^49 + 8 * q^50 + q^51 - 7 * q^52 - 11 * q^53 - 21 * q^54 - 8 * q^55 - 4 * q^56 - 17 * q^57 - 5 * q^58 - 11 * q^59 + 3 * q^60 - 22 * q^61 - 22 * q^62 + q^63 + 8 * q^64 + 7 * q^65 - 3 * q^66 + 21 * q^67 + 2 * q^68 + q^69 + 4 * q^70 - 28 * q^71 + q^72 - 8 * q^73 - 9 * q^74 - 3 * q^75 - q^76 - 4 * q^77 - 18 * q^78 - 28 * q^79 - 8 * q^80 + 12 * q^81 + 6 * q^82 - 5 * q^83 + 7 * q^84 - 2 * q^85 + 15 * q^86 + 36 * q^87 + 8 * q^88 - 17 * q^89 - q^90 - 29 * q^91 - 16 * q^92 + 42 * q^93 - 7 * q^94 + q^95 - 3 * q^96 + q^97 - 6 * q^98 + q^99

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