LMFDB - Embedded newform 4030.2.a.o.1.4

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} - 3x^{7} - 12x^{6} + 40x^{5} + 24x^{4} - 118x^{3} + 25x^{2} + 30x + 4 $ x^8 - 3x^7 - 12x^6 + 40x^5 + 24x^4 - 118x^3 + 25x^2 + 30*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.194747$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.194747 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.194747 q^{6} +1.49202 q^{7} +1.00000 q^{8} -2.96207 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.194747 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.194747 q^{6} +1.49202 q^{7} +1.00000 q^{8} -2.96207 q^{9} +1.00000 q^{10} +0.169811 q^{11} -0.194747 q^{12} -1.00000 q^{13} +1.49202 q^{14} -0.194747 q^{15} +1.00000 q^{16} +5.11291 q^{17} -2.96207 q^{18} +6.38010 q^{19} +1.00000 q^{20} -0.290567 q^{21} +0.169811 q^{22} -0.653117 q^{23} -0.194747 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.16110 q^{27} +1.49202 q^{28} -1.18690 q^{29} -0.194747 q^{30} +1.00000 q^{31} +1.00000 q^{32} -0.0330701 q^{33} +5.11291 q^{34} +1.49202 q^{35} -2.96207 q^{36} -8.97690 q^{37} +6.38010 q^{38} +0.194747 q^{39} +1.00000 q^{40} +4.93656 q^{41} -0.290567 q^{42} +3.92912 q^{43} +0.169811 q^{44} -2.96207 q^{45} -0.653117 q^{46} -3.31579 q^{47} -0.194747 q^{48} -4.77387 q^{49} +1.00000 q^{50} -0.995725 q^{51} -1.00000 q^{52} +10.2361 q^{53} +1.16110 q^{54} +0.169811 q^{55} +1.49202 q^{56} -1.24251 q^{57} -1.18690 q^{58} -5.54991 q^{59} -0.194747 q^{60} +12.7713 q^{61} +1.00000 q^{62} -4.41948 q^{63} +1.00000 q^{64} -1.00000 q^{65} -0.0330701 q^{66} +1.52326 q^{67} +5.11291 q^{68} +0.127193 q^{69} +1.49202 q^{70} +2.47886 q^{71} -2.96207 q^{72} +15.8319 q^{73} -8.97690 q^{74} -0.194747 q^{75} +6.38010 q^{76} +0.253361 q^{77} +0.194747 q^{78} -13.9494 q^{79} +1.00000 q^{80} +8.66010 q^{81} +4.93656 q^{82} +4.63771 q^{83} -0.290567 q^{84} +5.11291 q^{85} +3.92912 q^{86} +0.231146 q^{87} +0.169811 q^{88} +12.0385 q^{89} -2.96207 q^{90} -1.49202 q^{91} -0.653117 q^{92} -0.194747 q^{93} -3.31579 q^{94} +6.38010 q^{95} -0.194747 q^{96} +4.22780 q^{97} -4.77387 q^{98} -0.502992 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.194747	−0.112437	−0.0562186	−	0.998418i	$-0.517904\pi$
$3$	−0.194747	−0.112437	−0.0562186	+	0.998418i	$0.517904\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.194747	−0.0795051
$7$	1.49202	0.563931	0.281965	−	0.959425i	$-0.409014\pi$
$7$	1.49202	0.563931	0.281965	+	0.959425i	$0.409014\pi$
$8$	1.00000	0.353553
$9$	−2.96207	−0.987358
$10$	1.00000	0.316228
$11$	0.169811	0.0511999	0.0255999	−	0.999672i	$-0.491850\pi$
$11$	0.169811	0.0511999	0.0255999	+	0.999672i	$0.491850\pi$
$12$	−0.194747	−0.0562186
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.49202	0.398759
$15$	−0.194747	−0.0502835
$16$	1.00000	0.250000
$17$	5.11291	1.24006	0.620032	−	0.784577i	$-0.287120\pi$
$17$	5.11291	1.24006	0.620032	+	0.784577i	$0.287120\pi$
$18$	−2.96207	−0.698167
$19$	6.38010	1.46370	0.731848	−	0.681468i	$-0.238658\pi$
$19$	6.38010	1.46370	0.731848	+	0.681468i	$0.238658\pi$
$20$	1.00000	0.223607
$21$	−0.290567	−0.0634068
$22$	0.169811	0.0362038
$23$	−0.653117	−0.136184	−0.0680922	−	0.997679i	$-0.521691\pi$
$23$	−0.653117	−0.136184	−0.0680922	+	0.997679i	$0.521691\pi$
$24$	−0.194747	−0.0397526
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	1.16110	0.223453
$28$	1.49202	0.281965
$29$	−1.18690	−0.220403	−0.110201	−	0.993909i	$-0.535150\pi$
$29$	−1.18690	−0.220403	−0.110201	+	0.993909i	$0.535150\pi$
$30$	−0.194747	−0.0355558
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−0.0330701	−0.00575677
$34$	5.11291	0.876857
$35$	1.49202	0.252198
$36$	−2.96207	−0.493679
$37$	−8.97690	−1.47579	−0.737897	−	0.674913i	$-0.764181\pi$
$37$	−8.97690	−1.47579	−0.737897	+	0.674913i	$0.764181\pi$
$38$	6.38010	1.03499
$39$	0.194747	0.0311845
$40$	1.00000	0.158114
$41$	4.93656	0.770961	0.385480	−	0.922716i	$-0.374036\pi$
$41$	4.93656	0.770961	0.385480	+	0.922716i	$0.374036\pi$
$42$	−0.290567	−0.0448354
$43$	3.92912	0.599185	0.299592	−	0.954067i	$-0.403149\pi$
$43$	3.92912	0.599185	0.299592	+	0.954067i	$0.403149\pi$
$44$	0.169811	0.0255999
$45$	−2.96207	−0.441560
$46$	−0.653117	−0.0962969
$47$	−3.31579	−0.483658	−0.241829	−	0.970319i	$-0.577747\pi$
$47$	−3.31579	−0.483658	−0.241829	+	0.970319i	$0.577747\pi$
$48$	−0.194747	−0.0281093
$49$	−4.77387	−0.681982
$50$	1.00000	0.141421
$51$	−0.995725	−0.139429
$52$	−1.00000	−0.138675
$53$	10.2361	1.40604	0.703018	−	0.711172i	$-0.251835\pi$
$53$	10.2361	1.40604	0.703018	+	0.711172i	$0.251835\pi$
$54$	1.16110	0.158005
$55$	0.169811	0.0228973
$56$	1.49202	0.199380
$57$	−1.24251	−0.164574
$58$	−1.18690	−0.155848
$59$	−5.54991	−0.722537	−0.361269	−	0.932462i	$-0.617656\pi$
$59$	−5.54991	−0.722537	−0.361269	+	0.932462i	$0.617656\pi$
$60$	−0.194747	−0.0251417
$61$	12.7713	1.63520	0.817598	−	0.575789i	$-0.195305\pi$
$61$	12.7713	1.63520	0.817598	+	0.575789i	$0.195305\pi$
$62$	1.00000	0.127000
$63$	−4.41948	−0.556802
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−0.0330701	−0.00407065
$67$	1.52326	0.186096	0.0930480	−	0.995662i	$-0.470339\pi$
$67$	1.52326	0.186096	0.0930480	+	0.995662i	$0.470339\pi$
$68$	5.11291	0.620032
$69$	0.127193	0.0153122
$70$	1.49202	0.178331
$71$	2.47886	0.294187	0.147093	−	0.989123i	$-0.453008\pi$
$71$	2.47886	0.294187	0.147093	+	0.989123i	$0.453008\pi$
$72$	−2.96207	−0.349084
$73$	15.8319	1.85299	0.926494	−	0.376310i	$-0.122807\pi$
$73$	15.8319	1.85299	0.926494	+	0.376310i	$0.122807\pi$
$74$	−8.97690	−1.04354
$75$	−0.194747	−0.0224874
$76$	6.38010	0.731848
$77$	0.253361	0.0288732
$78$	0.194747	0.0220508
$79$	−13.9494	−1.56943	−0.784716	−	0.619856i	$-0.787191\pi$
$79$	−13.9494	−1.56943	−0.784716	+	0.619856i	$0.787191\pi$
$80$	1.00000	0.111803
$81$	8.66010	0.962233
$82$	4.93656	0.545151
$83$	4.63771	0.509055	0.254528	−	0.967066i	$-0.418080\pi$
$83$	4.63771	0.509055	0.254528	+	0.967066i	$0.418080\pi$
$84$	−0.290567	−0.0317034
$85$	5.11291	0.554573
$86$	3.92912	0.423688
$87$	0.231146	0.0247815
$88$	0.169811	0.0181019
$89$	12.0385	1.27608	0.638038	−	0.770004i	$-0.279746\pi$
$89$	12.0385	1.27608	0.638038	+	0.770004i	$0.279746\pi$
$90$	−2.96207	−0.312230
$91$	−1.49202	−0.156406
$92$	−0.653117	−0.0680922
$93$	−0.194747	−0.0201943
$94$	−3.31579	−0.341998
$95$	6.38010	0.654585
$96$	−0.194747	−0.0198763
$97$	4.22780	0.429268	0.214634	−	0.976695i	$-0.431144\pi$
$97$	4.22780	0.429268	0.214634	+	0.976695i	$0.431144\pi$
$98$	−4.77387	−0.482234
$99$	−0.502992	−0.0505526
$100$	1.00000	0.100000
$101$	−1.53618	−0.152856	−0.0764278	−	0.997075i	$-0.524351\pi$
$101$	−1.53618	−0.152856	−0.0764278	+	0.997075i	$0.524351\pi$
$102$	−0.995725	−0.0985914
$103$	11.4454	1.12775	0.563873	−	0.825861i	$-0.309311\pi$
$103$	11.4454	1.12775	0.563873	+	0.825861i	$0.309311\pi$
$104$	−1.00000	−0.0980581
$105$	−0.290567	−0.0283564
$106$	10.2361	0.994218
$107$	−13.4241	−1.29775	−0.648876	−	0.760894i	$-0.724761\pi$
$107$	−13.4241	−1.29775	−0.648876	+	0.760894i	$0.724761\pi$
$108$	1.16110	0.111727
$109$	9.87156	0.945524	0.472762	−	0.881190i	$-0.343257\pi$
$109$	9.87156	0.945524	0.472762	+	0.881190i	$0.343257\pi$
$110$	0.169811	0.0161908
$111$	1.74823	0.165934
$112$	1.49202	0.140983
$113$	−13.2317	−1.24473	−0.622367	−	0.782726i	$-0.713829\pi$
$113$	−13.2317	−1.24473	−0.622367	+	0.782726i	$0.713829\pi$
$114$	−1.24251	−0.116371
$115$	−0.653117	−0.0609035
$116$	−1.18690	−0.110201
$117$	2.96207	0.273844
$118$	−5.54991	−0.510911
$119$	7.62858	0.699310
$120$	−0.194747	−0.0177779
$121$	−10.9712	−0.997379
$122$	12.7713	1.15626
$123$	−0.961380	−0.0866847
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	−4.41948	−0.393718
$127$	3.41859	0.303351	0.151676	−	0.988430i	$-0.451533\pi$
$127$	3.41859	0.303351	0.151676	+	0.988430i	$0.451533\pi$
$128$	1.00000	0.0883883
$129$	−0.765184	−0.0673707
$130$	−1.00000	−0.0877058
$131$	16.0579	1.40298	0.701492	−	0.712678i	$-0.252518\pi$
$131$	16.0579	1.40298	0.701492	+	0.712678i	$0.252518\pi$
$132$	−0.0330701	−0.00287839
$133$	9.51925	0.825424
$134$	1.52326	0.131590
$135$	1.16110	0.0999312
$136$	5.11291	0.438429
$137$	−5.33646	−0.455925	−0.227962	−	0.973670i	$-0.573206\pi$
$137$	−5.33646	−0.455925	−0.227962	+	0.973670i	$0.573206\pi$
$138$	0.127193	0.0108274
$139$	−13.2478	−1.12366	−0.561831	−	0.827252i	$-0.689903\pi$
$139$	−13.2478	−1.12366	−0.561831	+	0.827252i	$0.689903\pi$
$140$	1.49202	0.126099
$141$	0.645741	0.0543812
$142$	2.47886	0.208021
$143$	−0.169811	−0.0142003
$144$	−2.96207	−0.246839
$145$	−1.18690	−0.0985671
$146$	15.8319	1.31026
$147$	0.929697	0.0766802
$148$	−8.97690	−0.737897
$149$	8.10018	0.663592	0.331796	−	0.943351i	$-0.392345\pi$
$149$	8.10018	0.663592	0.331796	+	0.943351i	$0.392345\pi$
$150$	−0.194747	−0.0159010
$151$	17.1912	1.39900	0.699499	−	0.714633i	$-0.253406\pi$
$151$	17.1912	1.39900	0.699499	+	0.714633i	$0.253406\pi$
$152$	6.38010	0.517495
$153$	−15.1448	−1.22439
$154$	0.253361	0.0204164
$155$	1.00000	0.0803219
$156$	0.194747	0.0155922
$157$	12.0685	0.963168	0.481584	−	0.876400i	$-0.340062\pi$
$157$	12.0685	0.963168	0.481584	+	0.876400i	$0.340062\pi$
$158$	−13.9494	−1.10976
$159$	−1.99345	−0.158091
$160$	1.00000	0.0790569
$161$	−0.974465	−0.0767986
$162$	8.66010	0.680402
$163$	−22.9349	−1.79640	−0.898200	−	0.439587i	$-0.855125\pi$
$163$	−22.9349	−1.79640	−0.898200	+	0.439587i	$0.855125\pi$
$164$	4.93656	0.385480
$165$	−0.0330701	−0.00257451
$166$	4.63771	0.359956
$167$	−0.721250	−0.0558120	−0.0279060	−	0.999611i	$-0.508884\pi$
$167$	−0.721250	−0.0558120	−0.0279060	+	0.999611i	$0.508884\pi$
$168$	−0.290567	−0.0224177
$169$	1.00000	0.0769231
$170$	5.11291	0.392143
$171$	−18.8983	−1.44519
$172$	3.92912	0.299592
$173$	13.8472	1.05278	0.526390	−	0.850243i	$-0.323545\pi$
$173$	13.8472	1.05278	0.526390	+	0.850243i	$0.323545\pi$
$174$	0.231146	0.0175231
$175$	1.49202	0.112786
$176$	0.169811	0.0128000
$177$	1.08083	0.0812401
$178$	12.0385	0.902323
$179$	−7.13984	−0.533656	−0.266828	−	0.963744i	$-0.585976\pi$
$179$	−7.13984	−0.533656	−0.266828	+	0.963744i	$0.585976\pi$
$180$	−2.96207	−0.220780
$181$	−22.2725	−1.65550	−0.827750	−	0.561096i	$-0.810380\pi$
$181$	−22.2725	−1.65550	−0.827750	+	0.561096i	$0.810380\pi$
$182$	−1.49202	−0.110596
$183$	−2.48717	−0.183857
$184$	−0.653117	−0.0481484
$185$	−8.97690	−0.659995
$186$	−0.194747	−0.0142795
$187$	0.868228	0.0634911
$188$	−3.31579	−0.241829
$189$	1.73238	0.126012
$190$	6.38010	0.462861
$191$	17.7019	1.28086	0.640432	−	0.768015i	$-0.278755\pi$
$191$	17.7019	1.28086	0.640432	+	0.768015i	$0.278755\pi$
$192$	−0.194747	−0.0140547
$193$	−14.7000	−1.05813	−0.529066	−	0.848581i	$-0.677458\pi$
$193$	−14.7000	−1.05813	−0.529066	+	0.848581i	$0.677458\pi$
$194$	4.22780	0.303538
$195$	0.194747	0.0139461
$196$	−4.77387	−0.340991
$197$	−11.7200	−0.835013	−0.417506	−	0.908674i	$-0.637096\pi$
$197$	−11.7200	−0.835013	−0.417506	+	0.908674i	$0.637096\pi$
$198$	−0.502992	−0.0357461
$199$	2.17634	0.154276	0.0771382	−	0.997020i	$-0.475422\pi$
$199$	2.17634	0.154276	0.0771382	+	0.997020i	$0.475422\pi$
$200$	1.00000	0.0707107
$201$	−0.296651	−0.0209241
$202$	−1.53618	−0.108085
$203$	−1.77089	−0.124292
$204$	−0.995725	−0.0697147
$205$	4.93656	0.344784
$206$	11.4454	0.797437
$207$	1.93458	0.134463
$208$	−1.00000	−0.0693375
$209$	1.08341	0.0749411
$210$	−0.290567	−0.0200510
$211$	10.7733	0.741666	0.370833	−	0.928700i	$-0.379072\pi$
$211$	10.7733	0.741666	0.370833	+	0.928700i	$0.379072\pi$
$212$	10.2361	0.703018
$213$	−0.482751	−0.0330776
$214$	−13.4241	−0.917649
$215$	3.92912	0.267964
$216$	1.16110	0.0790026
$217$	1.49202	0.101285
$218$	9.87156	0.668586
$219$	−3.08322	−0.208345
$220$	0.169811	0.0114486
$221$	−5.11291	−0.343932
$222$	1.74823	0.117333
$223$	13.9307	0.932868	0.466434	−	0.884556i	$-0.345539\pi$
$223$	13.9307	0.932868	0.466434	+	0.884556i	$0.345539\pi$
$224$	1.49202	0.0996899
$225$	−2.96207	−0.197472
$226$	−13.2317	−0.880159
$227$	1.95721	0.129905	0.0649523	−	0.997888i	$-0.479310\pi$
$227$	1.95721	0.129905	0.0649523	+	0.997888i	$0.479310\pi$
$228$	−1.24251	−0.0822870
$229$	−19.5925	−1.29471	−0.647356	−	0.762188i	$-0.724125\pi$
$229$	−19.5925	−1.29471	−0.647356	+	0.762188i	$0.724125\pi$
$230$	−0.653117	−0.0430653
$231$	−0.0493413	−0.00324642
$232$	−1.18690	−0.0779241
$233$	−21.9759	−1.43969	−0.719845	−	0.694135i	$-0.755787\pi$
$233$	−21.9759	−1.43969	−0.719845	+	0.694135i	$0.755787\pi$
$234$	2.96207	0.193637
$235$	−3.31579	−0.216299
$236$	−5.54991	−0.361269
$237$	2.71661	0.176463
$238$	7.62858	0.494487
$239$	14.2844	0.923983	0.461992	−	0.886884i	$-0.347135\pi$
$239$	14.2844	0.923983	0.461992	+	0.886884i	$0.347135\pi$
$240$	−0.194747	−0.0125709
$241$	−19.5000	−1.25611	−0.628054	−	0.778170i	$-0.716148\pi$
$241$	−19.5000	−1.25611	−0.628054	+	0.778170i	$0.716148\pi$
$242$	−10.9712	−0.705253
$243$	−5.16982	−0.331644
$244$	12.7713	0.817598
$245$	−4.77387	−0.304992
$246$	−0.961380	−0.0612953
$247$	−6.38010	−0.405956
$248$	1.00000	0.0635001
$249$	−0.903181	−0.0572368
$250$	1.00000	0.0632456
$251$	12.2609	0.773900	0.386950	−	0.922101i	$-0.373529\pi$
$251$	12.2609	0.773900	0.386950	+	0.922101i	$0.373529\pi$
$252$	−4.41948	−0.278401
$253$	−0.110906	−0.00697262
$254$	3.41859	0.214502
$255$	−0.995725	−0.0623547
$256$	1.00000	0.0625000
$257$	−4.41274	−0.275259	−0.137629	−	0.990484i	$-0.543948\pi$
$257$	−4.41274	−0.275259	−0.137629	+	0.990484i	$0.543948\pi$
$258$	−0.765184	−0.0476383
$259$	−13.3937	−0.832246
$260$	−1.00000	−0.0620174
$261$	3.51570	0.217616
$262$	16.0579	0.992059
$263$	7.98926	0.492639	0.246319	−	0.969189i	$-0.420779\pi$
$263$	7.98926	0.492639	0.246319	+	0.969189i	$0.420779\pi$
$264$	−0.0330701	−0.00203533
$265$	10.2361	0.628799
$266$	9.51925	0.583663
$267$	−2.34446	−0.143479
$268$	1.52326	0.0930480
$269$	20.0487	1.22239	0.611196	−	0.791479i	$-0.290689\pi$
$269$	20.0487	1.22239	0.611196	+	0.791479i	$0.290689\pi$
$270$	1.16110	0.0706621
$271$	−11.6529	−0.707863	−0.353932	−	0.935271i	$-0.615155\pi$
$271$	−11.6529	−0.707863	−0.353932	+	0.935271i	$0.615155\pi$
$272$	5.11291	0.310016
$273$	0.290567	0.0175859
$274$	−5.33646	−0.322388
$275$	0.169811	0.0102400
$276$	0.127193	0.00765609
$277$	4.32904	0.260107	0.130053	−	0.991507i	$-0.458485\pi$
$277$	4.32904	0.260107	0.130053	+	0.991507i	$0.458485\pi$
$278$	−13.2478	−0.794550
$279$	−2.96207	−0.177335
$280$	1.49202	0.0891653
$281$	−23.9282	−1.42743	−0.713717	−	0.700434i	$-0.752990\pi$
$281$	−23.9282	−1.42743	−0.713717	+	0.700434i	$0.752990\pi$
$282$	0.645741	0.0384533
$283$	19.5407	1.16158	0.580788	−	0.814055i	$-0.302745\pi$
$283$	19.5407	1.16158	0.580788	+	0.814055i	$0.302745\pi$
$284$	2.47886	0.147093
$285$	−1.24251	−0.0735997
$286$	−0.169811	−0.0100411
$287$	7.36545	0.434769
$288$	−2.96207	−0.174542
$289$	9.14189	0.537758
$290$	−1.18690	−0.0696974
$291$	−0.823351	−0.0482657
$292$	15.8319	0.926494
$293$	−30.4667	−1.77988	−0.889942	−	0.456075i	$-0.849255\pi$
$293$	−30.4667	−1.77988	−0.889942	+	0.456075i	$0.849255\pi$
$294$	0.929697	0.0542211
$295$	−5.54991	−0.323129
$296$	−8.97690	−0.521772
$297$	0.197167	0.0114408
$298$	8.10018	0.469231
$299$	0.653117	0.0377707
$300$	−0.194747	−0.0112437
$301$	5.86232	0.337899
$302$	17.1912	0.989241
$303$	0.299167	0.0171867
$304$	6.38010	0.365924
$305$	12.7713	0.731282
$306$	−15.1448	−0.865772
$307$	−9.75516	−0.556756	−0.278378	−	0.960472i	$-0.589797\pi$
$307$	−9.75516	−0.556756	−0.278378	+	0.960472i	$0.589797\pi$
$308$	0.253361	0.0144366
$309$	−2.22895	−0.126801
$310$	1.00000	0.0567962
$311$	14.1872	0.804481	0.402241	−	0.915534i	$-0.368232\pi$
$311$	14.1872	0.804481	0.402241	+	0.915534i	$0.368232\pi$
$312$	0.194747	0.0110254
$313$	19.3328	1.09276	0.546378	−	0.837539i	$-0.316006\pi$
$313$	19.3328	1.09276	0.546378	+	0.837539i	$0.316006\pi$
$314$	12.0685	0.681063
$315$	−4.41948	−0.249009
$316$	−13.9494	−0.784716
$317$	−28.0254	−1.57406	−0.787031	−	0.616913i	$-0.788383\pi$
$317$	−28.0254	−1.57406	−0.787031	+	0.616913i	$0.788383\pi$
$318$	−1.99345	−0.111787
$319$	−0.201549	−0.0112846
$320$	1.00000	0.0559017
$321$	2.61429	0.145916
$322$	−0.974465	−0.0543048
$323$	32.6209	1.81508
$324$	8.66010	0.481117
$325$	−1.00000	−0.0554700
$326$	−22.9349	−1.27025
$327$	−1.92246	−0.106312
$328$	4.93656	0.272576
$329$	−4.94724	−0.272750
$330$	−0.0330701	−0.00182045
$331$	28.4136	1.56175	0.780876	−	0.624686i	$-0.214773\pi$
$331$	28.4136	1.56175	0.780876	+	0.624686i	$0.214773\pi$
$332$	4.63771	0.254528
$333$	26.5903	1.45714
$334$	−0.721250	−0.0394650
$335$	1.52326	0.0832246
$336$	−0.290567	−0.0158517
$337$	−3.49649	−0.190466	−0.0952329	−	0.995455i	$-0.530360\pi$
$337$	−3.49649	−0.190466	−0.0952329	+	0.995455i	$0.530360\pi$
$338$	1.00000	0.0543928
$339$	2.57683	0.139954
$340$	5.11291	0.277287
$341$	0.169811	0.00919577
$342$	−18.8983	−1.02190
$343$	−17.5669	−0.948522
$344$	3.92912	0.211844
$345$	0.127193	0.00684782
$346$	13.8472	0.744428
$347$	28.8624	1.54941	0.774707	−	0.632320i	$-0.217897\pi$
$347$	28.8624	1.54941	0.774707	+	0.632320i	$0.217897\pi$
$348$	0.231146	0.0123907
$349$	12.2321	0.654771	0.327386	−	0.944891i	$-0.393832\pi$
$349$	12.2321	0.654771	0.327386	+	0.944891i	$0.393832\pi$
$350$	1.49202	0.0797519
$351$	−1.16110	−0.0619747
$352$	0.169811	0.00905094
$353$	−3.78914	−0.201675	−0.100838	−	0.994903i	$-0.532152\pi$
$353$	−3.78914	−0.201675	−0.100838	+	0.994903i	$0.532152\pi$
$354$	1.08083	0.0574454
$355$	2.47886	0.131564
$356$	12.0385	0.638038
$357$	−1.48564	−0.0786285
$358$	−7.13984	−0.377352
$359$	−18.8937	−0.997173	−0.498587	−	0.866840i	$-0.666147\pi$
$359$	−18.8937	−0.997173	−0.498587	+	0.866840i	$0.666147\pi$
$360$	−2.96207	−0.156115
$361$	21.7057	1.14241
$362$	−22.2725	−1.17062
$363$	2.13660	0.112142
$364$	−1.49202	−0.0782032
$365$	15.8319	0.828681
$366$	−2.48717	−0.130007
$367$	7.50655	0.391839	0.195919	−	0.980620i	$-0.437231\pi$
$367$	7.50655	0.391839	0.195919	+	0.980620i	$0.437231\pi$
$368$	−0.653117	−0.0340461
$369$	−14.6224	−0.761214
$370$	−8.97690	−0.466687
$371$	15.2725	0.792908
$372$	−0.194747	−0.0100972
$373$	−15.7555	−0.815789	−0.407894	−	0.913029i	$-0.633737\pi$
$373$	−15.7555	−0.815789	−0.407894	+	0.913029i	$0.633737\pi$
$374$	0.868228	0.0448950
$375$	−0.194747	−0.0100567
$376$	−3.31579	−0.170999
$377$	1.18690	0.0611287
$378$	1.73238	0.0891040
$379$	−28.4986	−1.46387	−0.731937	−	0.681372i	$-0.761384\pi$
$379$	−28.4986	−1.46387	−0.731937	+	0.681372i	$0.761384\pi$
$380$	6.38010	0.327292
$381$	−0.665761	−0.0341080
$382$	17.7019	0.905708
$383$	9.08199	0.464068	0.232034	−	0.972708i	$-0.425462\pi$
$383$	9.08199	0.464068	0.232034	+	0.972708i	$0.425462\pi$
$384$	−0.194747	−0.00993814
$385$	0.253361	0.0129125
$386$	−14.7000	−0.748212
$387$	−11.6383	−0.591610
$388$	4.22780	0.214634
$389$	0.311576	0.0157975	0.00789877	−	0.999969i	$-0.497486\pi$
$389$	0.311576	0.0157975	0.00789877	+	0.999969i	$0.497486\pi$
$390$	0.194747	0.00986140
$391$	−3.33933	−0.168877
$392$	−4.77387	−0.241117
$393$	−3.12722	−0.157748
$394$	−11.7200	−0.590443
$395$	−13.9494	−0.701871
$396$	−0.502992	−0.0252763
$397$	17.0526	0.855845	0.427922	−	0.903816i	$-0.359246\pi$
$397$	17.0526	0.855845	0.427922	+	0.903816i	$0.359246\pi$
$398$	2.17634	0.109090
$399$	−1.85385	−0.0928083
$400$	1.00000	0.0500000
$401$	33.5173	1.67377	0.836886	−	0.547377i	$-0.184374\pi$
$401$	33.5173	1.67377	0.836886	+	0.547377i	$0.184374\pi$
$402$	−0.296651	−0.0147956
$403$	−1.00000	−0.0498135
$404$	−1.53618	−0.0764278
$405$	8.66010	0.430324
$406$	−1.77089	−0.0878876
$407$	−1.52438	−0.0755605
$408$	−0.995725	−0.0492957
$409$	−22.0484	−1.09022	−0.545112	−	0.838363i	$-0.683513\pi$
$409$	−22.0484	−1.09022	−0.545112	+	0.838363i	$0.683513\pi$
$410$	4.93656	0.243799
$411$	1.03926	0.0512629
$412$	11.4454	0.563873
$413$	−8.28059	−0.407461
$414$	1.93458	0.0950795
$415$	4.63771	0.227656
$416$	−1.00000	−0.0490290
$417$	2.57997	0.126342
$418$	1.08341	0.0529913
$419$	−19.6789	−0.961376	−0.480688	−	0.876892i	$-0.659613\pi$
$419$	−19.6789	−0.961376	−0.480688	+	0.876892i	$0.659613\pi$
$420$	−0.290567	−0.0141782
$421$	−2.47790	−0.120766	−0.0603828	−	0.998175i	$-0.519232\pi$
$421$	−2.47790	−0.120766	−0.0603828	+	0.998175i	$0.519232\pi$
$422$	10.7733	0.524437
$423$	9.82163	0.477544
$424$	10.2361	0.497109
$425$	5.11291	0.248013
$426$	−0.482751	−0.0233894
$427$	19.0550	0.922138
$428$	−13.4241	−0.648876
$429$	0.0330701	0.00159664
$430$	3.92912	0.189479
$431$	−37.5191	−1.80723	−0.903617	−	0.428342i	$-0.859098\pi$
$431$	−37.5191	−1.80723	−0.903617	+	0.428342i	$0.859098\pi$
$432$	1.16110	0.0558633
$433$	−8.62732	−0.414602	−0.207301	−	0.978277i	$-0.566468\pi$
$433$	−8.62732	−0.414602	−0.207301	+	0.978277i	$0.566468\pi$
$434$	1.49202	0.0716193
$435$	0.231146	0.0110826
$436$	9.87156	0.472762
$437$	−4.16695	−0.199332
$438$	−3.08322	−0.147322
$439$	−37.9525	−1.81138	−0.905688	−	0.423944i	$-0.860645\pi$
$439$	−37.9525	−1.81138	−0.905688	+	0.423944i	$0.860645\pi$
$440$	0.169811	0.00809541
$441$	14.1406	0.673360
$442$	−5.11291	−0.243197
$443$	8.46371	0.402123	0.201061	−	0.979579i	$-0.435561\pi$
$443$	8.46371	0.402123	0.201061	+	0.979579i	$0.435561\pi$
$444$	1.74823	0.0829671
$445$	12.0385	0.570679
$446$	13.9307	0.659637
$447$	−1.57748	−0.0746125
$448$	1.49202	0.0704914
$449$	−1.54173	−0.0727589	−0.0363794	−	0.999338i	$-0.511582\pi$
$449$	−1.54173	−0.0727589	−0.0363794	+	0.999338i	$0.511582\pi$
$450$	−2.96207	−0.139633
$451$	0.838280	0.0394731
$452$	−13.2317	−0.622367
$453$	−3.34793	−0.157299
$454$	1.95721	0.0918564
$455$	−1.49202	−0.0699470
$456$	−1.24251	−0.0581857
$457$	28.4183	1.32935	0.664676	−	0.747132i	$-0.268570\pi$
$457$	28.4183	1.32935	0.664676	+	0.747132i	$0.268570\pi$
$458$	−19.5925	−0.915499
$459$	5.93658	0.277096
$460$	−0.653117	−0.0304517
$461$	−4.82082	−0.224528	−0.112264	−	0.993678i	$-0.535810\pi$
$461$	−4.82082	−0.224528	−0.112264	+	0.993678i	$0.535810\pi$
$462$	−0.0493413	−0.00229557
$463$	−15.7801	−0.733364	−0.366682	−	0.930346i	$-0.619506\pi$
$463$	−15.7801	−0.733364	−0.366682	+	0.930346i	$0.619506\pi$
$464$	−1.18690	−0.0551007
$465$	−0.194747	−0.00903118
$466$	−21.9759	−1.01801
$467$	−13.4171	−0.620868	−0.310434	−	0.950595i	$-0.600474\pi$
$467$	−13.4171	−0.620868	−0.310434	+	0.950595i	$0.600474\pi$
$468$	2.96207	0.136922
$469$	2.27274	0.104945
$470$	−3.31579	−0.152946
$471$	−2.35030	−0.108296
$472$	−5.54991	−0.255456
$473$	0.667206	0.0306782
$474$	2.71661	0.124778
$475$	6.38010	0.292739
$476$	7.62858	0.349655
$477$	−30.3201	−1.38826
$478$	14.2844	0.653355
$479$	−0.539588	−0.0246544	−0.0123272	−	0.999924i	$-0.503924\pi$
$479$	−0.539588	−0.0246544	−0.0123272	+	0.999924i	$0.503924\pi$
$480$	−0.194747	−0.00888894
$481$	8.97690	0.409312
$482$	−19.5000	−0.888203
$483$	0.189774	0.00863502
$484$	−10.9712	−0.498689
$485$	4.22780	0.191974
$486$	−5.16982	−0.234508
$487$	−2.58596	−0.117181	−0.0585905	−	0.998282i	$-0.518661\pi$
$487$	−2.58596	−0.117181	−0.0585905	+	0.998282i	$0.518661\pi$
$488$	12.7713	0.578129
$489$	4.46650	0.201982
$490$	−4.77387	−0.215662
$491$	−15.1829	−0.685195	−0.342597	−	0.939482i	$-0.611307\pi$
$491$	−15.1829	−0.685195	−0.342597	+	0.939482i	$0.611307\pi$
$492$	−0.961380	−0.0433423
$493$	−6.06854	−0.273313
$494$	−6.38010	−0.287054
$495$	−0.502992	−0.0226078
$496$	1.00000	0.0449013
$497$	3.69851	0.165901
$498$	−0.903181	−0.0404725
$499$	−27.5643	−1.23395	−0.616974	−	0.786983i	$-0.711642\pi$
$499$	−27.5643	−1.23395	−0.616974	+	0.786983i	$0.711642\pi$
$500$	1.00000	0.0447214
$501$	0.140461	0.00627535
$502$	12.2609	0.547230
$503$	17.1848	0.766230	0.383115	−	0.923701i	$-0.374851\pi$
$503$	17.1848	0.766230	0.383115	+	0.923701i	$0.374851\pi$
$504$	−4.41948	−0.196859
$505$	−1.53618	−0.0683591
$506$	−0.110906	−0.00493039
$507$	−0.194747	−0.00864902
$508$	3.41859	0.151676
$509$	−16.4136	−0.727521	−0.363760	−	0.931493i	$-0.618507\pi$
$509$	−16.4136	−0.727521	−0.363760	+	0.931493i	$0.618507\pi$
$510$	−0.995725	−0.0440914
$511$	23.6216	1.04496
$512$	1.00000	0.0441942
$513$	7.40791	0.327067
$514$	−4.41274	−0.194637
$515$	11.4454	0.504344
$516$	−0.765184	−0.0336853
$517$	−0.563058	−0.0247632
$518$	−13.3937	−0.588487
$519$	−2.69669	−0.118372
$520$	−1.00000	−0.0438529
$521$	−30.6560	−1.34306	−0.671531	−	0.740976i	$-0.734363\pi$
$521$	−30.6560	−1.34306	−0.671531	+	0.740976i	$0.734363\pi$
$522$	3.51570	0.153878
$523$	16.1886	0.707879	0.353939	−	0.935268i	$-0.384842\pi$
$523$	16.1886	0.707879	0.353939	+	0.935268i	$0.384842\pi$
$524$	16.0579	0.701492
$525$	−0.290567	−0.0126814
$526$	7.98926	0.348348
$527$	5.11291	0.222722
$528$	−0.0330701	−0.00143919
$529$	−22.5734	−0.981454
$530$	10.2361	0.444628
$531$	16.4393	0.713403
$532$	9.51925	0.412712
$533$	−4.93656	−0.213826
$534$	−2.34446	−0.101455
$535$	−13.4241	−0.580372
$536$	1.52326	0.0657949
$537$	1.39046	0.0600028
$538$	20.0487	0.864362
$539$	−0.810655	−0.0349174
$540$	1.16110	0.0499656
$541$	−0.213063	−0.00916029	−0.00458014	−	0.999990i	$-0.501458\pi$
$541$	−0.213063	−0.00916029	−0.00458014	+	0.999990i	$0.501458\pi$
$542$	−11.6529	−0.500535
$543$	4.33750	0.186140
$544$	5.11291	0.219214
$545$	9.87156	0.422851
$546$	0.290567	0.0124351
$547$	−32.9939	−1.41072	−0.705358	−	0.708851i	$-0.749214\pi$
$547$	−32.9939	−1.41072	−0.705358	+	0.708851i	$0.749214\pi$
$548$	−5.33646	−0.227962
$549$	−37.8295	−1.61452
$550$	0.169811	0.00724076
$551$	−7.57258	−0.322603
$552$	0.127193	0.00541368
$553$	−20.8128	−0.885051
$554$	4.32904	0.183923
$555$	1.74823	0.0742080
$556$	−13.2478	−0.561831
$557$	−13.9757	−0.592171	−0.296085	−	0.955162i	$-0.595681\pi$
$557$	−13.9757	−0.592171	−0.296085	+	0.955162i	$0.595681\pi$
$558$	−2.96207	−0.125395
$559$	−3.92912	−0.166184
$560$	1.49202	0.0630494
$561$	−0.169085	−0.00713876
$562$	−23.9282	−1.00935
$563$	32.8418	1.38412	0.692058	−	0.721842i	$-0.256704\pi$
$563$	32.8418	1.38412	0.692058	+	0.721842i	$0.256704\pi$
$564$	0.645741	0.0271906
$565$	−13.2317	−0.556662
$566$	19.5407	0.821358
$567$	12.9211	0.542633
$568$	2.47886	0.104011
$569$	−29.5496	−1.23878	−0.619392	−	0.785082i	$-0.712621\pi$
$569$	−29.5496	−1.23878	−0.619392	+	0.785082i	$0.712621\pi$
$570$	−1.24251	−0.0520429
$571$	−5.29499	−0.221588	−0.110794	−	0.993843i	$-0.535339\pi$
$571$	−5.29499	−0.221588	−0.110794	+	0.993843i	$0.535339\pi$
$572$	−0.169811	−0.00710015
$573$	−3.44739	−0.144017
$574$	7.36545	0.307428
$575$	−0.653117	−0.0272369
$576$	−2.96207	−0.123420
$577$	−17.0474	−0.709694	−0.354847	−	0.934924i	$-0.615467\pi$
$577$	−17.0474	−0.709694	−0.354847	+	0.934924i	$0.615467\pi$
$578$	9.14189	0.380252
$579$	2.86279	0.118973
$580$	−1.18690	−0.0492835
$581$	6.91957	0.287072
$582$	−0.823351	−0.0341290
$583$	1.73820	0.0719889
$584$	15.8319	0.655130
$585$	2.96207	0.122467
$586$	−30.4667	−1.25857
$587$	9.91726	0.409329	0.204664	−	0.978832i	$-0.434390\pi$
$587$	9.91726	0.409329	0.204664	+	0.978832i	$0.434390\pi$
$588$	0.929697	0.0383401
$589$	6.38010	0.262888
$590$	−5.54991	−0.228486
$591$	2.28243	0.0938865
$592$	−8.97690	−0.368948
$593$	2.42835	0.0997204	0.0498602	−	0.998756i	$-0.484122\pi$
$593$	2.42835	0.0997204	0.0498602	+	0.998756i	$0.484122\pi$
$594$	0.197167	0.00808984
$595$	7.62858	0.312741
$596$	8.10018	0.331796
$597$	−0.423835	−0.0173464
$598$	0.653117	0.0267079
$599$	−22.3970	−0.915118	−0.457559	−	0.889179i	$-0.651276\pi$
$599$	−22.3970	−0.915118	−0.457559	+	0.889179i	$0.651276\pi$
$600$	−0.194747	−0.00795051
$601$	−42.4882	−1.73313	−0.866565	−	0.499065i	$-0.833677\pi$
$601$	−42.4882	−1.73313	−0.866565	+	0.499065i	$0.833677\pi$
$602$	5.86232	0.238931
$603$	−4.51201	−0.183743
$604$	17.1912	0.699499
$605$	−10.9712	−0.446041
$606$	0.299167	0.0121528
$607$	30.4429	1.23564	0.617819	−	0.786320i	$-0.288016\pi$
$607$	30.4429	1.23564	0.617819	+	0.786320i	$0.288016\pi$
$608$	6.38010	0.258747
$609$	0.344875	0.0139750
$610$	12.7713	0.517095
$611$	3.31579	0.134143
$612$	−15.1448	−0.612193
$613$	−3.35342	−0.135443	−0.0677217	−	0.997704i	$-0.521573\pi$
$613$	−3.35342	−0.135443	−0.0677217	+	0.997704i	$0.521573\pi$
$614$	−9.75516	−0.393686
$615$	−0.961380	−0.0387666
$616$	0.253361	0.0102082
$617$	31.5169	1.26882	0.634411	−	0.772996i	$-0.281243\pi$
$617$	31.5169	1.26882	0.634411	+	0.772996i	$0.281243\pi$
$618$	−2.22895	−0.0896616
$619$	−22.1363	−0.889735	−0.444867	−	0.895596i	$-0.646749\pi$
$619$	−22.1363	−0.889735	−0.444867	+	0.895596i	$0.646749\pi$
$620$	1.00000	0.0401610
$621$	−0.758332	−0.0304308
$622$	14.1872	0.568854
$623$	17.9617	0.719619
$624$	0.194747	0.00779612
$625$	1.00000	0.0400000
$626$	19.3328	0.772695
$627$	−0.210991	−0.00842617
$628$	12.0685	0.481584
$629$	−45.8981	−1.83008
$630$	−4.41948	−0.176076
$631$	−38.0136	−1.51330	−0.756650	−	0.653821i	$-0.773165\pi$
$631$	−38.0136	−1.51330	−0.756650	+	0.653821i	$0.773165\pi$
$632$	−13.9494	−0.554878
$633$	−2.09807	−0.0833909
$634$	−28.0254	−1.11303
$635$	3.41859	0.135663
$636$	−1.99345	−0.0790454
$637$	4.77387	0.189148
$638$	−0.201549	−0.00797941
$639$	−7.34257	−0.290468
$640$	1.00000	0.0395285
$641$	−21.2786	−0.840456	−0.420228	−	0.907419i	$-0.638050\pi$
$641$	−21.2786	−0.840456	−0.420228	+	0.907419i	$0.638050\pi$
$642$	2.61429	0.103178
$643$	19.9931	0.788450	0.394225	−	0.919014i	$-0.371013\pi$
$643$	19.9931	0.788450	0.394225	+	0.919014i	$0.371013\pi$
$644$	−0.974465	−0.0383993
$645$	−0.765184	−0.0301291
$646$	32.6209	1.28345
$647$	15.8676	0.623819	0.311910	−	0.950112i	$-0.399031\pi$
$647$	15.8676	0.623819	0.311910	+	0.950112i	$0.399031\pi$
$648$	8.66010	0.340201
$649$	−0.942435	−0.0369938
$650$	−1.00000	−0.0392232
$651$	−0.290567	−0.0113882
$652$	−22.9349	−0.898200
$653$	−20.2410	−0.792092	−0.396046	−	0.918231i	$-0.629618\pi$
$653$	−20.2410	−0.792092	−0.396046	+	0.918231i	$0.629618\pi$
$654$	−1.92246	−0.0751740
$655$	16.0579	0.627433
$656$	4.93656	0.192740
$657$	−46.8954	−1.82956
$658$	−4.94724	−0.192863
$659$	23.3477	0.909497	0.454748	−	0.890620i	$-0.349729\pi$
$659$	23.3477	0.909497	0.454748	+	0.890620i	$0.349729\pi$
$660$	−0.0330701	−0.00128725
$661$	−9.97987	−0.388172	−0.194086	−	0.980985i	$-0.562174\pi$
$661$	−9.97987	−0.388172	−0.194086	+	0.980985i	$0.562174\pi$
$662$	28.4136	1.10433
$663$	0.995725	0.0386707
$664$	4.63771	0.179978
$665$	9.51925	0.369141
$666$	26.5903	1.03035
$667$	0.775188	0.0300154
$668$	−0.721250	−0.0279060
$669$	−2.71296	−0.104889
$670$	1.52326	0.0588487
$671$	2.16870	0.0837219
$672$	−0.290567	−0.0112089
$673$	−19.6645	−0.758010	−0.379005	−	0.925395i	$-0.623734\pi$
$673$	−19.6645	−0.758010	−0.379005	+	0.925395i	$0.623734\pi$
$674$	−3.49649	−0.134680
$675$	1.16110	0.0446906
$676$	1.00000	0.0384615
$677$	−2.51582	−0.0966910	−0.0483455	−	0.998831i	$-0.515395\pi$
$677$	−2.51582	−0.0966910	−0.0483455	+	0.998831i	$0.515395\pi$
$678$	2.57683	0.0989627
$679$	6.30797	0.242077
$680$	5.11291	0.196071
$681$	−0.381161	−0.0146061
$682$	0.169811	0.00650239
$683$	−28.3481	−1.08471	−0.542355	−	0.840149i	$-0.682467\pi$
$683$	−28.3481	−1.08471	−0.542355	+	0.840149i	$0.682467\pi$
$684$	−18.8983	−0.722596
$685$	−5.33646	−0.203896
$686$	−17.5669	−0.670706
$687$	3.81559	0.145574
$688$	3.92912	0.149796
$689$	−10.2361	−0.389964
$690$	0.127193	0.00484214
$691$	−27.5027	−1.04625	−0.523126	−	0.852255i	$-0.675234\pi$
$691$	−27.5027	−1.04625	−0.523126	+	0.852255i	$0.675234\pi$
$692$	13.8472	0.526390
$693$	−0.750475	−0.0285082
$694$	28.8624	1.09560
$695$	−13.2478	−0.502517
$696$	0.231146	0.00876157
$697$	25.2402	0.956040
$698$	12.2321	0.462993
$699$	4.27974	0.161875
$700$	1.49202	0.0563931
$701$	15.8865	0.600024	0.300012	−	0.953935i	$-0.403009\pi$
$701$	15.8865	0.600024	0.300012	+	0.953935i	$0.403009\pi$
$702$	−1.16110	−0.0438227
$703$	−57.2736	−2.16011
$704$	0.169811	0.00639998
$705$	0.645741	0.0243200
$706$	−3.78914	−0.142606
$707$	−2.29201	−0.0862000
$708$	1.08083	0.0406200
$709$	22.7579	0.854690	0.427345	−	0.904089i	$-0.359449\pi$
$709$	22.7579	0.854690	0.427345	+	0.904089i	$0.359449\pi$
$710$	2.47886	0.0930300
$711$	41.3192	1.54959
$712$	12.0385	0.451161
$713$	−0.653117	−0.0244594
$714$	−1.48564	−0.0555988
$715$	−0.169811	−0.00635056
$716$	−7.13984	−0.266828
$717$	−2.78185	−0.103890
$718$	−18.8937	−0.705108
$719$	−2.56575	−0.0956862	−0.0478431	−	0.998855i	$-0.515235\pi$
$719$	−2.56575	−0.0956862	−0.0478431	+	0.998855i	$0.515235\pi$
$720$	−2.96207	−0.110390
$721$	17.0767	0.635971
$722$	21.7057	0.807803
$723$	3.79757	0.141233
$724$	−22.2725	−0.827750
$725$	−1.18690	−0.0440805
$726$	2.13660	0.0792967
$727$	28.4607	1.05555	0.527774	−	0.849385i	$-0.323027\pi$
$727$	28.4607	1.05555	0.527774	+	0.849385i	$0.323027\pi$
$728$	−1.49202	−0.0552980
$729$	−24.9735	−0.924944
$730$	15.8319	0.585966
$731$	20.0892	0.743027
$732$	−2.48717	−0.0919285
$733$	−20.1563	−0.744490	−0.372245	−	0.928134i	$-0.621412\pi$
$733$	−20.1563	−0.744490	−0.372245	+	0.928134i	$0.621412\pi$
$734$	7.50655	0.277072
$735$	0.929697	0.0342924
$736$	−0.653117	−0.0240742
$737$	0.258666	0.00952809
$738$	−14.6224	−0.538260
$739$	−40.5491	−1.49162	−0.745811	−	0.666158i	$-0.767938\pi$
$739$	−40.5491	−1.49162	−0.745811	+	0.666158i	$0.767938\pi$
$740$	−8.97690	−0.329998
$741$	1.24251	0.0456446
$742$	15.2725	0.560670
$743$	7.51602	0.275736	0.137868	−	0.990451i	$-0.455975\pi$
$743$	7.51602	0.275736	0.137868	+	0.990451i	$0.455975\pi$
$744$	−0.194747	−0.00713977
$745$	8.10018	0.296767
$746$	−15.7555	−0.576850
$747$	−13.7372	−0.502620
$748$	0.868228	0.0317456
$749$	−20.0290	−0.731843
$750$	−0.194747	−0.00711116
$751$	−25.7989	−0.941416	−0.470708	−	0.882289i	$-0.656001\pi$
$751$	−25.7989	−0.941416	−0.470708	+	0.882289i	$0.656001\pi$
$752$	−3.31579	−0.120915
$753$	−2.38777	−0.0870152
$754$	1.18690	0.0432245
$755$	17.1912	0.625651
$756$	1.73238	0.0630060
$757$	1.55633	0.0565658	0.0282829	−	0.999600i	$-0.490996\pi$
$757$	1.55633	0.0565658	0.0282829	+	0.999600i	$0.490996\pi$
$758$	−28.4986	−1.03512
$759$	0.0215987	0.000783982 0
$760$	6.38010	0.231431
$761$	−17.4160	−0.631330	−0.315665	−	0.948871i	$-0.602228\pi$
$761$	−17.4160	−0.631330	−0.315665	+	0.948871i	$0.602228\pi$
$762$	−0.665761	−0.0241180
$763$	14.7286	0.533210
$764$	17.7019	0.640432
$765$	−15.1448	−0.547562
$766$	9.08199	0.328146
$767$	5.54991	0.200396
$768$	−0.194747	−0.00702733
$769$	14.5244	0.523763	0.261881	−	0.965100i	$-0.415657\pi$
$769$	14.5244	0.523763	0.261881	+	0.965100i	$0.415657\pi$
$770$	0.253361	0.00913051
$771$	0.859367	0.0309494
$772$	−14.7000	−0.529066
$773$	13.2982	0.478303	0.239152	−	0.970982i	$-0.423131\pi$
$773$	13.2982	0.478303	0.239152	+	0.970982i	$0.423131\pi$
$774$	−11.6383	−0.418331
$775$	1.00000	0.0359211
$776$	4.22780	0.151769
$777$	2.60839	0.0935754
$778$	0.311576	0.0111706
$779$	31.4957	1.12845
$780$	0.194747	0.00697306
$781$	0.420937	0.0150623
$782$	−3.33933	−0.119414
$783$	−1.37811	−0.0492496
$784$	−4.77387	−0.170495
$785$	12.0685	0.430742
$786$	−3.12722	−0.111544
$787$	−17.5775	−0.626570	−0.313285	−	0.949659i	$-0.601429\pi$
$787$	−17.5775	−0.626570	−0.313285	+	0.949659i	$0.601429\pi$
$788$	−11.7200	−0.417506
$789$	−1.55588	−0.0553909
$790$	−13.9494	−0.496298
$791$	−19.7420	−0.701944
$792$	−0.502992	−0.0178730
$793$	−12.7713	−0.453522
$794$	17.0526	0.605174
$795$	−1.99345	−0.0707004
$796$	2.17634	0.0771382
$797$	6.88561	0.243901	0.121950	−	0.992536i	$-0.461085\pi$
$797$	6.88561	0.243901	0.121950	+	0.992536i	$0.461085\pi$
$798$	−1.85385	−0.0656254
$799$	−16.9534	−0.599767
$800$	1.00000	0.0353553
$801$	−35.6589	−1.25994
$802$	33.5173	1.18354
$803$	2.68843	0.0948727
$804$	−0.296651	−0.0104621
$805$	−0.974465	−0.0343454
$806$	−1.00000	−0.0352235
$807$	−3.90443	−0.137442
$808$	−1.53618	−0.0540426
$809$	−45.9168	−1.61435	−0.807174	−	0.590314i	$-0.799004\pi$
$809$	−45.9168	−1.61435	−0.807174	+	0.590314i	$0.799004\pi$
$810$	8.66010	0.304285
$811$	−8.68702	−0.305043	−0.152521	−	0.988300i	$-0.548739\pi$
$811$	−8.68702	−0.305043	−0.152521	+	0.988300i	$0.548739\pi$
$812$	−1.77089	−0.0621460
$813$	2.26937	0.0795902
$814$	−1.52438	−0.0534293
$815$	−22.9349	−0.803375
$816$	−0.995725	−0.0348573
$817$	25.0682	0.877024
$818$	−22.0484	−0.770905
$819$	4.41948	0.154429
$820$	4.93656	0.172392
$821$	13.0173	0.454308	0.227154	−	0.973859i	$-0.427058\pi$
$821$	13.0173	0.454308	0.227154	+	0.973859i	$0.427058\pi$
$822$	1.03926	0.0362484
$823$	17.3297	0.604077	0.302038	−	0.953296i	$-0.402333\pi$
$823$	17.3297	0.604077	0.302038	+	0.953296i	$0.402333\pi$
$824$	11.4454	0.398719
$825$	−0.0330701	−0.00115135
$826$	−8.28059	−0.288119
$827$	16.3136	0.567280	0.283640	−	0.958931i	$-0.408458\pi$
$827$	16.3136	0.567280	0.283640	+	0.958931i	$0.408458\pi$
$828$	1.93458	0.0672313
$829$	−30.7007	−1.06628	−0.533141	−	0.846027i	$-0.678988\pi$
$829$	−30.7007	−1.06628	−0.533141	+	0.846027i	$0.678988\pi$
$830$	4.63771	0.160977
$831$	−0.843067	−0.0292457
$832$	−1.00000	−0.0346688
$833$	−24.4084	−0.845701
$834$	2.57997	0.0893370
$835$	−0.721250	−0.0249599
$836$	1.08341	0.0374705
$837$	1.16110	0.0401333
$838$	−19.6789	−0.679796
$839$	56.7435	1.95900	0.979502	−	0.201433i	$-0.0645598\pi$
$839$	56.7435	1.95900	0.979502	+	0.201433i	$0.0645598\pi$
$840$	−0.290567	−0.0100255
$841$	−27.5913	−0.951423
$842$	−2.47790	−0.0853941
$843$	4.65994	0.160497
$844$	10.7733	0.370833
$845$	1.00000	0.0344010
$846$	9.82163	0.337675
$847$	−16.3692	−0.562453
$848$	10.2361	0.351509
$849$	−3.80550	−0.130604
$850$	5.11291	0.175371
$851$	5.86297	0.200980
$852$	−0.482751	−0.0165388
$853$	−2.43205	−0.0832719	−0.0416360	−	0.999133i	$-0.513257\pi$
$853$	−2.43205	−0.0832719	−0.0416360	+	0.999133i	$0.513257\pi$
$854$	19.0550	0.652050
$855$	−18.8983	−0.646309
$856$	−13.4241	−0.458825
$857$	31.2999	1.06918	0.534591	−	0.845111i	$-0.320466\pi$
$857$	31.2999	1.06918	0.534591	+	0.845111i	$0.320466\pi$
$858$	0.0330701	0.00112900
$859$	8.94269	0.305121	0.152560	−	0.988294i	$-0.451248\pi$
$859$	8.94269	0.305121	0.152560	+	0.988294i	$0.451248\pi$
$860$	3.92912	0.133982
$861$	−1.43440	−0.0488842
$862$	−37.5191	−1.27791
$863$	9.51114	0.323763	0.161881	−	0.986810i	$-0.448244\pi$
$863$	9.51114	0.323763	0.161881	+	0.986810i	$0.448244\pi$
$864$	1.16110	0.0395013
$865$	13.8472	0.470817
$866$	−8.62732	−0.293168
$867$	−1.78035	−0.0604640
$868$	1.49202	0.0506425
$869$	−2.36876	−0.0803547
$870$	0.231146	0.00783659
$871$	−1.52326	−0.0516137
$872$	9.87156	0.334293
$873$	−12.5231	−0.423841
$874$	−4.16695	−0.140949
$875$	1.49202	0.0504395
$876$	−3.08322	−0.104172
$877$	−51.8963	−1.75241	−0.876207	−	0.481935i	$-0.839934\pi$
$877$	−51.8963	−1.75241	−0.876207	+	0.481935i	$0.839934\pi$
$878$	−37.9525	−1.28084
$879$	5.93330	0.200125
$880$	0.169811	0.00572432
$881$	−18.4275	−0.620838	−0.310419	−	0.950600i	$-0.600469\pi$
$881$	−18.4275	−0.620838	−0.310419	+	0.950600i	$0.600469\pi$
$882$	14.1406	0.476138
$883$	−7.94957	−0.267524	−0.133762	−	0.991013i	$-0.542706\pi$
$883$	−7.94957	−0.267524	−0.133762	+	0.991013i	$0.542706\pi$
$884$	−5.11291	−0.171966
$885$	1.08083	0.0363317
$886$	8.46371	0.284344
$887$	−31.7867	−1.06729	−0.533647	−	0.845708i	$-0.679179\pi$
$887$	−31.7867	−1.06729	−0.533647	+	0.845708i	$0.679179\pi$
$888$	1.74823	0.0586666
$889$	5.10061	0.171069
$890$	12.0385	0.403531
$891$	1.47058	0.0492662
$892$	13.9307	0.466434
$893$	−21.1551	−0.707929
$894$	−1.57748	−0.0527590
$895$	−7.13984	−0.238658
$896$	1.49202	0.0498449
$897$	−0.127193	−0.00424684
$898$	−1.54173	−0.0514483
$899$	−1.18690	−0.0395855
$900$	−2.96207	−0.0987358
$901$	52.3363	1.74358
$902$	0.838280	0.0279117
$903$	−1.14167	−0.0379924
$904$	−13.2317	−0.440080
$905$	−22.2725	−0.740363
$906$	−3.34793	−0.111228
$907$	42.8434	1.42259	0.711296	−	0.702892i	$-0.248108\pi$
$907$	42.8434	1.42259	0.711296	+	0.702892i	$0.248108\pi$
$908$	1.95721	0.0649523
$909$	4.55028	0.150923
$910$	−1.49202	−0.0494600
$911$	49.3153	1.63389	0.816944	−	0.576717i	$-0.195666\pi$
$911$	49.3153	1.63389	0.816944	+	0.576717i	$0.195666\pi$
$912$	−1.24251	−0.0411435
$913$	0.787534	0.0260636
$914$	28.4183	0.939993
$915$	−2.48717	−0.0822234
$916$	−19.5925	−0.647356
$917$	23.9587	0.791186
$918$	5.93658	0.195936
$919$	−30.1494	−0.994536	−0.497268	−	0.867597i	$-0.665663\pi$
$919$	−30.1494	−0.994536	−0.497268	+	0.867597i	$0.665663\pi$
$920$	−0.653117	−0.0215326
$921$	1.89979	0.0626001
$922$	−4.82082	−0.158765
$923$	−2.47886	−0.0815927
$924$	−0.0493413	−0.00162321
$925$	−8.97690	−0.295159
$926$	−15.7801	−0.518567
$927$	−33.9020	−1.11349
$928$	−1.18690	−0.0389621
$929$	−26.6805	−0.875359	−0.437679	−	0.899131i	$-0.644200\pi$
$929$	−26.6805	−0.875359	−0.437679	+	0.899131i	$0.644200\pi$
$930$	−0.194747	−0.00638601
$931$	−30.4578	−0.998214
$932$	−21.9759	−0.719845
$933$	−2.76291	−0.0904536
$934$	−13.4171	−0.439020
$935$	0.868228	0.0283941
$936$	2.96207	0.0968184
$937$	−30.7355	−1.00409	−0.502043	−	0.864843i	$-0.667418\pi$
$937$	−30.7355	−1.00409	−0.502043	+	0.864843i	$0.667418\pi$
$938$	2.27274	0.0742075
$939$	−3.76501	−0.122866
$940$	−3.31579	−0.108149
$941$	−23.4496	−0.764435	−0.382218	−	0.924072i	$-0.624840\pi$
$941$	−23.4496	−0.764435	−0.382218	+	0.924072i	$0.624840\pi$
$942$	−2.35030	−0.0765768
$943$	−3.22415	−0.104993
$944$	−5.54991	−0.180634
$945$	1.73238	0.0563543
$946$	0.667206	0.0216927
$947$	12.4516	0.404623	0.202312	−	0.979321i	$-0.435155\pi$
$947$	12.4516	0.404623	0.202312	+	0.979321i	$0.435155\pi$
$948$	2.71661	0.0882313
$949$	−15.8319	−0.513926
$950$	6.38010	0.206998
$951$	5.45786	0.176983
$952$	7.62858	0.247244
$953$	3.98365	0.129043	0.0645215	−	0.997916i	$-0.479448\pi$
$953$	3.98365	0.129043	0.0645215	+	0.997916i	$0.479448\pi$
$954$	−30.3201	−0.981649
$955$	17.7019	0.572820
$956$	14.2844	0.461992
$957$	0.0392511	0.00126881
$958$	−0.539588	−0.0174333
$959$	−7.96212	−0.257110
$960$	−0.194747	−0.00628543
$961$	1.00000	0.0322581
$962$	8.97690	0.289427
$963$	39.7630	1.28135
$964$	−19.5000	−0.628054
$965$	−14.7000	−0.473211
$966$	0.189774	0.00610588
$967$	50.9201	1.63748	0.818740	−	0.574164i	$-0.194673\pi$
$967$	50.9201	1.63748	0.818740	+	0.574164i	$0.194673\pi$
$968$	−10.9712	−0.352627
$969$	−6.35283	−0.204082
$970$	4.22780	0.135746
$971$	23.6878	0.760178	0.380089	−	0.924950i	$-0.375893\pi$
$971$	23.6878	0.760178	0.380089	+	0.924950i	$0.375893\pi$
$972$	−5.16982	−0.165822
$973$	−19.7660	−0.633668
$974$	−2.58596	−0.0828595
$975$	0.194747	0.00623690
$976$	12.7713	0.408799
$977$	47.1531	1.50856	0.754280	−	0.656552i	$-0.227986\pi$
$977$	47.1531	1.50856	0.754280	+	0.656552i	$0.227986\pi$
$978$	4.46650	0.142823
$979$	2.04426	0.0653350
$980$	−4.77387	−0.152496
$981$	−29.2403	−0.933571
$982$	−15.1829	−0.484506
$983$	48.2987	1.54049	0.770244	−	0.637749i	$-0.220134\pi$
$983$	48.2987	1.54049	0.770244	+	0.637749i	$0.220134\pi$
$984$	−0.961380	−0.0306477
$985$	−11.7200	−0.373429
$986$	−6.06854	−0.193262
$987$	0.963459	0.0306672
$988$	−6.38010	−0.202978
$989$	−2.56617	−0.0815996
$990$	−0.502992	−0.0159861
$991$	5.62714	0.178752	0.0893760	−	0.995998i	$-0.471513\pi$
$991$	5.62714	0.178752	0.0893760	+	0.995998i	$0.471513\pi$
$992$	1.00000	0.0317500
$993$	−5.53346	−0.175599
$994$	3.69851	0.117310
$995$	2.17634	0.0689945
$996$	−0.903181	−0.0286184
$997$	−17.4684	−0.553229	−0.276615	−	0.960981i	$-0.589213\pi$
$997$	−17.4684	−0.553229	−0.276615	+	0.960981i	$0.589213\pi$
$998$	−27.5643	−0.872534
$999$	−10.4230	−0.329771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.o.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.o.1.4	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.194747 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.194747 q^{6} +1.49202 q^{7} +1.00000 q^{8} -2.96207 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.194747 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.194747 q^{6} +1.49202 q^{7} +1.00000 q^{8} -2.96207 q^{9} +1.00000 q^{10} +0.169811 q^{11} -0.194747 q^{12} -1.00000 q^{13} +1.49202 q^{14} -0.194747 q^{15} +1.00000 q^{16} +5.11291 q^{17} -2.96207 q^{18} +6.38010 q^{19} +1.00000 q^{20} -0.290567 q^{21} +0.169811 q^{22} -0.653117 q^{23} -0.194747 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.16110 q^{27} +1.49202 q^{28} -1.18690 q^{29} -0.194747 q^{30} +1.00000 q^{31} +1.00000 q^{32} -0.0330701 q^{33} +5.11291 q^{34} +1.49202 q^{35} -2.96207 q^{36} -8.97690 q^{37} +6.38010 q^{38} +0.194747 q^{39} +1.00000 q^{40} +4.93656 q^{41} -0.290567 q^{42} +3.92912 q^{43} +0.169811 q^{44} -2.96207 q^{45} -0.653117 q^{46} -3.31579 q^{47} -0.194747 q^{48} -4.77387 q^{49} +1.00000 q^{50} -0.995725 q^{51} -1.00000 q^{52} +10.2361 q^{53} +1.16110 q^{54} +0.169811 q^{55} +1.49202 q^{56} -1.24251 q^{57} -1.18690 q^{58} -5.54991 q^{59} -0.194747 q^{60} +12.7713 q^{61} +1.00000 q^{62} -4.41948 q^{63} +1.00000 q^{64} -1.00000 q^{65} -0.0330701 q^{66} +1.52326 q^{67} +5.11291 q^{68} +0.127193 q^{69} +1.49202 q^{70} +2.47886 q^{71} -2.96207 q^{72} +15.8319 q^{73} -8.97690 q^{74} -0.194747 q^{75} +6.38010 q^{76} +0.253361 q^{77} +0.194747 q^{78} -13.9494 q^{79} +1.00000 q^{80} +8.66010 q^{81} +4.93656 q^{82} +4.63771 q^{83} -0.290567 q^{84} +5.11291 q^{85} +3.92912 q^{86} +0.231146 q^{87} +0.169811 q^{88} +12.0385 q^{89} -2.96207 q^{90} -1.49202 q^{91} -0.653117 q^{92} -0.194747 q^{93} -3.31579 q^{94} +6.38010 q^{95} -0.194747 q^{96} +4.22780 q^{97} -4.77387 q^{98} -0.502992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * q^99

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