LMFDB - Embedded newform 4134.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4134,2,Mod(1,4134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4134, 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("4134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4134 = 2 \cdot 3 \cdot 13 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4134.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:	$33.0101561956$
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} - 2x^{7} - 21x^{6} + 43x^{5} + 96x^{4} - 235x^{3} + 136x^{2} - 8x - 8 $ x^8 - 2x^7 - 21x^6 + 43x^5 + 96x^4 - 235x^3 + 136x^2 - 8*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.188218$ of defining polynomial
Character	$\chi$	$=$	4134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.188218 q^{5} +1.00000 q^{6} -4.10516 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.188218 q^{5} +1.00000 q^{6} -4.10516 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.188218 q^{10} +1.03292 q^{11} +1.00000 q^{12} -1.00000 q^{13} -4.10516 q^{14} -0.188218 q^{15} +1.00000 q^{16} +5.92314 q^{17} +1.00000 q^{18} +0.127490 q^{19} -0.188218 q^{20} -4.10516 q^{21} +1.03292 q^{22} +0.590525 q^{23} +1.00000 q^{24} -4.96457 q^{25} -1.00000 q^{26} +1.00000 q^{27} -4.10516 q^{28} -0.323003 q^{29} -0.188218 q^{30} +3.65842 q^{31} +1.00000 q^{32} +1.03292 q^{33} +5.92314 q^{34} +0.772664 q^{35} +1.00000 q^{36} +8.84615 q^{37} +0.127490 q^{38} -1.00000 q^{39} -0.188218 q^{40} +5.78835 q^{41} -4.10516 q^{42} +3.21602 q^{43} +1.03292 q^{44} -0.188218 q^{45} +0.590525 q^{46} +9.12097 q^{47} +1.00000 q^{48} +9.85235 q^{49} -4.96457 q^{50} +5.92314 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -0.194414 q^{55} -4.10516 q^{56} +0.127490 q^{57} -0.323003 q^{58} +11.1774 q^{59} -0.188218 q^{60} -10.5396 q^{61} +3.65842 q^{62} -4.10516 q^{63} +1.00000 q^{64} +0.188218 q^{65} +1.03292 q^{66} -3.83464 q^{67} +5.92314 q^{68} +0.590525 q^{69} +0.772664 q^{70} +8.79070 q^{71} +1.00000 q^{72} +5.34461 q^{73} +8.84615 q^{74} -4.96457 q^{75} +0.127490 q^{76} -4.24031 q^{77} -1.00000 q^{78} -1.04846 q^{79} -0.188218 q^{80} +1.00000 q^{81} +5.78835 q^{82} +1.10767 q^{83} -4.10516 q^{84} -1.11484 q^{85} +3.21602 q^{86} -0.323003 q^{87} +1.03292 q^{88} -10.8453 q^{89} -0.188218 q^{90} +4.10516 q^{91} +0.590525 q^{92} +3.65842 q^{93} +9.12097 q^{94} -0.0239959 q^{95} +1.00000 q^{96} -13.4044 q^{97} +9.85235 q^{98} +1.03292 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 7 q^{11} + 8 q^{12} - 8 q^{13} + 7 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + 11 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 5 q^{23} + 8 q^{24} + 6 q^{25} - 8 q^{26} + 8 q^{27} + 7 q^{28} + 13 q^{29} + 2 q^{30} + 12 q^{31} + 8 q^{32} + 7 q^{33} + 8 q^{34} + 14 q^{35} + 8 q^{36} + 10 q^{37} + 11 q^{38} - 8 q^{39} + 2 q^{40} + 19 q^{41} + 7 q^{42} + 10 q^{43} + 7 q^{44} + 2 q^{45} + 5 q^{46} + 6 q^{47} + 8 q^{48} + 15 q^{49} + 6 q^{50} + 8 q^{51} - 8 q^{52} + 8 q^{53} + 8 q^{54} + 5 q^{55} + 7 q^{56} + 11 q^{57} + 13 q^{58} + 11 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} + 5 q^{67} + 8 q^{68} + 5 q^{69} + 14 q^{70} - 6 q^{71} + 8 q^{72} + 18 q^{73} + 10 q^{74} + 6 q^{75} + 11 q^{76} - 8 q^{77} - 8 q^{78} + 17 q^{79} + 2 q^{80} + 8 q^{81} + 19 q^{82} + 16 q^{83} + 7 q^{84} + 25 q^{85} + 10 q^{86} + 13 q^{87} + 7 q^{88} - 8 q^{89} + 2 q^{90} - 7 q^{91} + 5 q^{92} + 12 q^{93} + 6 q^{94} - 30 q^{95} + 8 q^{96} + 11 q^{97} + 15 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 7 * q^11 + 8 * q^12 - 8 * q^13 + 7 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + 11 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 5 * q^23 + 8 * q^24 + 6 * q^25 - 8 * q^26 + 8 * q^27 + 7 * q^28 + 13 * q^29 + 2 * q^30 + 12 * q^31 + 8 * q^32 + 7 * q^33 + 8 * q^34 + 14 * q^35 + 8 * q^36 + 10 * q^37 + 11 * q^38 - 8 * q^39 + 2 * q^40 + 19 * q^41 + 7 * q^42 + 10 * q^43 + 7 * q^44 + 2 * q^45 + 5 * q^46 + 6 * q^47 + 8 * q^48 + 15 * q^49 + 6 * q^50 + 8 * q^51 - 8 * q^52 + 8 * q^53 + 8 * q^54 + 5 * q^55 + 7 * q^56 + 11 * q^57 + 13 * q^58 + 11 * q^59 + 2 * q^60 + 6 * q^61 + 12 * q^62 + 7 * q^63 + 8 * q^64 - 2 * q^65 + 7 * q^66 + 5 * q^67 + 8 * q^68 + 5 * q^69 + 14 * q^70 - 6 * q^71 + 8 * q^72 + 18 * q^73 + 10 * q^74 + 6 * q^75 + 11 * q^76 - 8 * q^77 - 8 * q^78 + 17 * q^79 + 2 * q^80 + 8 * q^81 + 19 * q^82 + 16 * q^83 + 7 * q^84 + 25 * q^85 + 10 * q^86 + 13 * q^87 + 7 * q^88 - 8 * q^89 + 2 * q^90 - 7 * q^91 + 5 * q^92 + 12 * q^93 + 6 * q^94 - 30 * q^95 + 8 * q^96 + 11 * q^97 + 15 * q^98 + 7 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−0.188218	−0.0841735	−0.0420868	−	0.999114i	$-0.513401\pi$
$5$	−0.188218	−0.0841735	−0.0420868	+	0.999114i	$0.513401\pi$
$6$	1.00000	0.408248
$7$	−4.10516	−1.55161	−0.775803	−	0.630976i	$-0.782655\pi$
$7$	−4.10516	−1.55161	−0.775803	+	0.630976i	$0.782655\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.188218	−0.0595197
$11$	1.03292	0.311438	0.155719	−	0.987801i	$-0.450231\pi$
$11$	1.03292	0.311438	0.155719	+	0.987801i	$0.450231\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.10516	−1.09715
$15$	−0.188218	−0.0485976
$16$	1.00000	0.250000
$17$	5.92314	1.43657	0.718286	−	0.695748i	$-0.244927\pi$
$17$	5.92314	1.43657	0.718286	+	0.695748i	$0.244927\pi$
$18$	1.00000	0.235702
$19$	0.127490	0.0292483	0.0146241	−	0.999893i	$-0.495345\pi$
$19$	0.127490	0.0292483	0.0146241	+	0.999893i	$0.495345\pi$
$20$	−0.188218	−0.0420868
$21$	−4.10516	−0.895820
$22$	1.03292	0.220220
$23$	0.590525	0.123133	0.0615665	−	0.998103i	$-0.480390\pi$
$23$	0.590525	0.123133	0.0615665	+	0.998103i	$0.480390\pi$
$24$	1.00000	0.204124
$25$	−4.96457	−0.992915
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−4.10516	−0.775803
$29$	−0.323003	−0.0599802	−0.0299901	−	0.999550i	$-0.509548\pi$
$29$	−0.323003	−0.0599802	−0.0299901	+	0.999550i	$0.509548\pi$
$30$	−0.188218	−0.0343637
$31$	3.65842	0.657072	0.328536	−	0.944492i	$-0.393445\pi$
$31$	3.65842	0.657072	0.328536	+	0.944492i	$0.393445\pi$
$32$	1.00000	0.176777
$33$	1.03292	0.179809
$34$	5.92314	1.01581
$35$	0.772664	0.130604
$36$	1.00000	0.166667
$37$	8.84615	1.45430	0.727149	−	0.686480i	$-0.240845\pi$
$37$	8.84615	1.45430	0.727149	+	0.686480i	$0.240845\pi$
$38$	0.127490	0.0206816
$39$	−1.00000	−0.160128
$40$	−0.188218	−0.0297598
$41$	5.78835	0.903989	0.451995	−	0.892021i	$-0.350713\pi$
$41$	5.78835	0.903989	0.451995	+	0.892021i	$0.350713\pi$
$42$	−4.10516	−0.633440
$43$	3.21602	0.490439	0.245219	−	0.969468i	$-0.421140\pi$
$43$	3.21602	0.490439	0.245219	+	0.969468i	$0.421140\pi$
$44$	1.03292	0.155719
$45$	−0.188218	−0.0280578
$46$	0.590525	0.0870682
$47$	9.12097	1.33043	0.665215	−	0.746652i	$-0.268340\pi$
$47$	9.12097	1.33043	0.665215	+	0.746652i	$0.268340\pi$
$48$	1.00000	0.144338
$49$	9.85235	1.40748
$50$	−4.96457	−0.702097
$51$	5.92314	0.829405
$52$	−1.00000	−0.138675
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	−0.194414	−0.0262148
$56$	−4.10516	−0.548575
$57$	0.127490	0.0168865
$58$	−0.323003	−0.0424124
$59$	11.1774	1.45517	0.727587	−	0.686016i	$-0.240642\pi$
$59$	11.1774	1.45517	0.727587	+	0.686016i	$0.240642\pi$
$60$	−0.188218	−0.0242988
$61$	−10.5396	−1.34946	−0.674731	−	0.738063i	$-0.735740\pi$
$61$	−10.5396	−1.34946	−0.674731	+	0.738063i	$0.735740\pi$
$62$	3.65842	0.464620
$63$	−4.10516	−0.517202
$64$	1.00000	0.125000
$65$	0.188218	0.0233455
$66$	1.03292	0.127144
$67$	−3.83464	−0.468476	−0.234238	−	0.972179i	$-0.575259\pi$
$67$	−3.83464	−0.468476	−0.234238	+	0.972179i	$0.575259\pi$
$68$	5.92314	0.718286
$69$	0.590525	0.0710909
$70$	0.772664	0.0923510
$71$	8.79070	1.04326	0.521632	−	0.853170i	$-0.325323\pi$
$71$	8.79070	1.04326	0.521632	+	0.853170i	$0.325323\pi$
$72$	1.00000	0.117851
$73$	5.34461	0.625540	0.312770	−	0.949829i	$-0.398743\pi$
$73$	5.34461	0.625540	0.312770	+	0.949829i	$0.398743\pi$
$74$	8.84615	1.02834
$75$	−4.96457	−0.573260
$76$	0.127490	0.0146241
$77$	−4.24031	−0.483228
$78$	−1.00000	−0.113228
$79$	−1.04846	−0.117961	−0.0589807	−	0.998259i	$-0.518785\pi$
$79$	−1.04846	−0.117961	−0.0589807	+	0.998259i	$0.518785\pi$
$80$	−0.188218	−0.0210434
$81$	1.00000	0.111111
$82$	5.78835	0.639217
$83$	1.10767	0.121582	0.0607910	−	0.998151i	$-0.480638\pi$
$83$	1.10767	0.121582	0.0607910	+	0.998151i	$0.480638\pi$
$84$	−4.10516	−0.447910
$85$	−1.11484	−0.120921
$86$	3.21602	0.346793
$87$	−0.323003	−0.0346296
$88$	1.03292	0.110110
$89$	−10.8453	−1.14960	−0.574802	−	0.818292i	$-0.694921\pi$
$89$	−10.8453	−1.14960	−0.574802	+	0.818292i	$0.694921\pi$
$90$	−0.188218	−0.0198399
$91$	4.10516	0.430338
$92$	0.590525	0.0615665
$93$	3.65842	0.379360
$94$	9.12097	0.940756
$95$	−0.0239959	−0.00246193
$96$	1.00000	0.102062
$97$	−13.4044	−1.36101	−0.680504	−	0.732745i	$-0.738239\pi$
$97$	−13.4044	−1.36101	−0.680504	+	0.732745i	$0.738239\pi$
$98$	9.85235	0.995237
$99$	1.03292	0.103813
$100$	−4.96457	−0.496457
$101$	14.1838	1.41134	0.705669	−	0.708542i	$-0.250647\pi$
$101$	14.1838	1.41134	0.705669	+	0.708542i	$0.250647\pi$
$102$	5.92314	0.586478
$103$	−9.37572	−0.923817	−0.461908	−	0.886928i	$-0.652835\pi$
$103$	−9.37572	−0.923817	−0.461908	+	0.886928i	$0.652835\pi$
$104$	−1.00000	−0.0980581
$105$	0.772664	0.0754043
$106$	1.00000	0.0971286
$107$	−7.79394	−0.753468	−0.376734	−	0.926321i	$-0.622953\pi$
$107$	−7.79394	−0.753468	−0.376734	+	0.926321i	$0.622953\pi$
$108$	1.00000	0.0962250
$109$	1.39870	0.133972	0.0669858	−	0.997754i	$-0.478662\pi$
$109$	1.39870	0.133972	0.0669858	+	0.997754i	$0.478662\pi$
$110$	−0.194414	−0.0185367
$111$	8.84615	0.839640
$112$	−4.10516	−0.387901
$113$	18.0360	1.69668	0.848340	−	0.529451i	$-0.177602\pi$
$113$	18.0360	1.69668	0.848340	+	0.529451i	$0.177602\pi$
$114$	0.127490	0.0119405
$115$	−0.111147	−0.0103645
$116$	−0.323003	−0.0299901
$117$	−1.00000	−0.0924500
$118$	11.1774	1.02896
$119$	−24.3154	−2.22899
$120$	−0.188218	−0.0171818
$121$	−9.93307	−0.903007
$122$	−10.5396	−0.954214
$123$	5.78835	0.521918
$124$	3.65842	0.328536
$125$	1.87551	0.167751
$126$	−4.10516	−0.365717
$127$	17.3568	1.54017	0.770083	−	0.637944i	$-0.220215\pi$
$127$	17.3568	1.54017	0.770083	+	0.637944i	$0.220215\pi$
$128$	1.00000	0.0883883
$129$	3.21602	0.283155
$130$	0.188218	0.0165078
$131$	13.4860	1.17827	0.589137	−	0.808033i	$-0.299468\pi$
$131$	13.4860	1.17827	0.589137	+	0.808033i	$0.299468\pi$
$132$	1.03292	0.0899043
$133$	−0.523368	−0.0453817
$134$	−3.83464	−0.331262
$135$	−0.188218	−0.0161992
$136$	5.92314	0.507905
$137$	−6.15911	−0.526208	−0.263104	−	0.964767i	$-0.584746\pi$
$137$	−6.15911	−0.526208	−0.263104	+	0.964767i	$0.584746\pi$
$138$	0.590525	0.0502689
$139$	20.1563	1.70964	0.854818	−	0.518927i	$-0.173668\pi$
$139$	20.1563	1.70964	0.854818	+	0.518927i	$0.173668\pi$
$140$	0.772664	0.0653020
$141$	9.12097	0.768124
$142$	8.79070	0.737699
$143$	−1.03292	−0.0863773
$144$	1.00000	0.0833333
$145$	0.0607949	0.00504874
$146$	5.34461	0.442323
$147$	9.85235	0.812608
$148$	8.84615	0.727149
$149$	15.9012	1.30268	0.651338	−	0.758788i	$-0.274208\pi$
$149$	15.9012	1.30268	0.651338	+	0.758788i	$0.274208\pi$
$150$	−4.96457	−0.405356
$151$	−3.82856	−0.311564	−0.155782	−	0.987791i	$-0.549790\pi$
$151$	−3.82856	−0.311564	−0.155782	+	0.987791i	$0.549790\pi$
$152$	0.127490	0.0103408
$153$	5.92314	0.478857
$154$	−4.24031	−0.341694
$155$	−0.688579	−0.0553080
$156$	−1.00000	−0.0800641
$157$	−7.96248	−0.635475	−0.317737	−	0.948179i	$-0.602923\pi$
$157$	−7.96248	−0.635475	−0.317737	+	0.948179i	$0.602923\pi$
$158$	−1.04846	−0.0834112
$159$	1.00000	0.0793052
$160$	−0.188218	−0.0148799
$161$	−2.42420	−0.191054
$162$	1.00000	0.0785674
$163$	11.7366	0.919284	0.459642	−	0.888104i	$-0.347978\pi$
$163$	11.7366	0.919284	0.459642	+	0.888104i	$0.347978\pi$
$164$	5.78835	0.451995
$165$	−0.194414	−0.0151351
$166$	1.10767	0.0859715
$167$	0.758697	0.0587097	0.0293549	−	0.999569i	$-0.490655\pi$
$167$	0.758697	0.0587097	0.0293549	+	0.999569i	$0.490655\pi$
$168$	−4.10516	−0.316720
$169$	1.00000	0.0769231
$170$	−1.11484	−0.0855043
$171$	0.127490	0.00974942
$172$	3.21602	0.245219
$173$	−22.9403	−1.74412	−0.872058	−	0.489402i	$-0.837215\pi$
$173$	−22.9403	−1.74412	−0.872058	+	0.489402i	$0.837215\pi$
$174$	−0.323003	−0.0244868
$175$	20.3804	1.54061
$176$	1.03292	0.0778594
$177$	11.1774	0.840145
$178$	−10.8453	−0.812893
$179$	−1.94655	−0.145492	−0.0727460	−	0.997351i	$-0.523176\pi$
$179$	−1.94655	−0.145492	−0.0727460	+	0.997351i	$0.523176\pi$
$180$	−0.188218	−0.0140289
$181$	−24.4229	−1.81534	−0.907671	−	0.419682i	$-0.862142\pi$
$181$	−24.4229	−1.81534	−0.907671	+	0.419682i	$0.862142\pi$
$182$	4.10516	0.304295
$183$	−10.5396	−0.779113
$184$	0.590525	0.0435341
$185$	−1.66500	−0.122413
$186$	3.65842	0.268248
$187$	6.11814	0.447403
$188$	9.12097	0.665215
$189$	−4.10516	−0.298607
$190$	−0.0239959	−0.00174085
$191$	15.5159	1.12269	0.561347	−	0.827580i	$-0.310283\pi$
$191$	15.5159	1.12269	0.561347	+	0.827580i	$0.310283\pi$
$192$	1.00000	0.0721688
$193$	15.1730	1.09217	0.546087	−	0.837729i	$-0.316117\pi$
$193$	15.1730	1.09217	0.546087	+	0.837729i	$0.316117\pi$
$194$	−13.4044	−0.962378
$195$	0.188218	0.0134785
$196$	9.85235	0.703739
$197$	−16.2878	−1.16046	−0.580229	−	0.814453i	$-0.697037\pi$
$197$	−16.2878	−1.16046	−0.580229	+	0.814453i	$0.697037\pi$
$198$	1.03292	0.0734066
$199$	−4.76296	−0.337637	−0.168819	−	0.985647i	$-0.553995\pi$
$199$	−4.76296	−0.337637	−0.168819	+	0.985647i	$0.553995\pi$
$200$	−4.96457	−0.351048
$201$	−3.83464	−0.270475
$202$	14.1838	0.997966
$203$	1.32598	0.0930656
$204$	5.92314	0.414703
$205$	−1.08947	−0.0760919
$206$	−9.37572	−0.653237
$207$	0.590525	0.0410444
$208$	−1.00000	−0.0693375
$209$	0.131687	0.00910901
$210$	0.772664	0.0533189
$211$	−12.7735	−0.879362	−0.439681	−	0.898154i	$-0.644908\pi$
$211$	−12.7735	−0.879362	−0.439681	+	0.898154i	$0.644908\pi$
$212$	1.00000	0.0686803
$213$	8.79070	0.602329
$214$	−7.79394	−0.532783
$215$	−0.605313	−0.0412820
$216$	1.00000	0.0680414
$217$	−15.0184	−1.01952
$218$	1.39870	0.0947322
$219$	5.34461	0.361155
$220$	−0.194414	−0.0131074
$221$	−5.92314	−0.398434
$222$	8.84615	0.593715
$223$	13.5206	0.905404	0.452702	−	0.891662i	$-0.350460\pi$
$223$	13.5206	0.905404	0.452702	+	0.891662i	$0.350460\pi$
$224$	−4.10516	−0.274288
$225$	−4.96457	−0.330972
$226$	18.0360	1.19973
$227$	−10.7311	−0.712246	−0.356123	−	0.934439i	$-0.615902\pi$
$227$	−10.7311	−0.712246	−0.356123	+	0.934439i	$0.615902\pi$
$228$	0.127490	0.00844324
$229$	−22.4811	−1.48559	−0.742797	−	0.669517i	$-0.766501\pi$
$229$	−22.4811	−1.48559	−0.742797	+	0.669517i	$0.766501\pi$
$230$	−0.111147	−0.00732884
$231$	−4.24031	−0.278992
$232$	−0.323003	−0.0212062
$233$	−18.8158	−1.23267	−0.616333	−	0.787485i	$-0.711383\pi$
$233$	−18.8158	−1.23267	−0.616333	+	0.787485i	$0.711383\pi$
$234$	−1.00000	−0.0653720
$235$	−1.71673	−0.111987
$236$	11.1774	0.727587
$237$	−1.04846	−0.0681050
$238$	−24.3154	−1.57614
$239$	6.68189	0.432216	0.216108	−	0.976369i	$-0.430664\pi$
$239$	6.68189	0.432216	0.216108	+	0.976369i	$0.430664\pi$
$240$	−0.188218	−0.0121494
$241$	21.0808	1.35794	0.678968	−	0.734168i	$-0.262428\pi$
$241$	21.0808	1.35794	0.678968	+	0.734168i	$0.262428\pi$
$242$	−9.93307	−0.638522
$243$	1.00000	0.0641500
$244$	−10.5396	−0.674731
$245$	−1.85439	−0.118472
$246$	5.78835	0.369052
$247$	−0.127490	−0.00811201
$248$	3.65842	0.232310
$249$	1.10767	0.0701954
$250$	1.87551	0.118618
$251$	−9.07825	−0.573014	−0.286507	−	0.958078i	$-0.592494\pi$
$251$	−9.07825	−0.573014	−0.286507	+	0.958078i	$0.592494\pi$
$252$	−4.10516	−0.258601
$253$	0.609967	0.0383483
$254$	17.3568	1.08906
$255$	−1.11484	−0.0698140
$256$	1.00000	0.0625000
$257$	11.2159	0.699630	0.349815	−	0.936819i	$-0.386245\pi$
$257$	11.2159	0.699630	0.349815	+	0.936819i	$0.386245\pi$
$258$	3.21602	0.200221
$259$	−36.3149	−2.25650
$260$	0.188218	0.0116728
$261$	−0.323003	−0.0199934
$262$	13.4860	0.833165
$263$	−29.0370	−1.79050	−0.895250	−	0.445564i	$-0.853003\pi$
$263$	−29.0370	−1.79050	−0.895250	+	0.445564i	$0.853003\pi$
$264$	1.03292	0.0635719
$265$	−0.188218	−0.0115621
$266$	−0.523368	−0.0320897
$267$	−10.8453	−0.663725
$268$	−3.83464	−0.234238
$269$	−27.5271	−1.67836	−0.839179	−	0.543855i	$-0.816964\pi$
$269$	−27.5271	−1.67836	−0.839179	+	0.543855i	$0.816964\pi$
$270$	−0.188218	−0.0114546
$271$	−12.7424	−0.774047	−0.387023	−	0.922070i	$-0.626497\pi$
$271$	−12.7424	−0.774047	−0.387023	+	0.922070i	$0.626497\pi$
$272$	5.92314	0.359143
$273$	4.10516	0.248456
$274$	−6.15911	−0.372085
$275$	−5.12802	−0.309231
$276$	0.590525	0.0355455
$277$	19.3027	1.15979	0.579894	−	0.814692i	$-0.303094\pi$
$277$	19.3027	1.15979	0.579894	+	0.814692i	$0.303094\pi$
$278$	20.1563	1.20890
$279$	3.65842	0.219024
$280$	0.772664	0.0461755
$281$	18.9805	1.13228	0.566140	−	0.824309i	$-0.308436\pi$
$281$	18.9805	1.13228	0.566140	+	0.824309i	$0.308436\pi$
$282$	9.12097	0.543146
$283$	−13.3878	−0.795822	−0.397911	−	0.917424i	$-0.630265\pi$
$283$	−13.3878	−0.795822	−0.397911	+	0.917424i	$0.630265\pi$
$284$	8.79070	0.521632
$285$	−0.0239959	−0.00142140
$286$	−1.03292	−0.0610779
$287$	−23.7621	−1.40263
$288$	1.00000	0.0589256
$289$	18.0836	1.06374
$290$	0.0607949	0.00357000
$291$	−13.4044	−0.785778
$292$	5.34461	0.312770
$293$	1.79309	0.104753	0.0523767	−	0.998627i	$-0.483320\pi$
$293$	1.79309	0.104753	0.0523767	+	0.998627i	$0.483320\pi$
$294$	9.85235	0.574601
$295$	−2.10378	−0.122487
$296$	8.84615	0.514172
$297$	1.03292	0.0599362
$298$	15.9012	0.921130
$299$	−0.590525	−0.0341510
$300$	−4.96457	−0.286630
$301$	−13.2023	−0.760968
$302$	−3.82856	−0.220309
$303$	14.1838	0.814836
$304$	0.127490	0.00731206
$305$	1.98375	0.113589
$306$	5.92314	0.338603
$307$	21.7172	1.23946	0.619732	−	0.784813i	$-0.287241\pi$
$307$	21.7172	1.23946	0.619732	+	0.784813i	$0.287241\pi$
$308$	−4.24031	−0.241614
$309$	−9.37572	−0.533366
$310$	−0.688579	−0.0391087
$311$	8.93712	0.506778	0.253389	−	0.967365i	$-0.418455\pi$
$311$	8.93712	0.506778	0.253389	+	0.967365i	$0.418455\pi$
$312$	−1.00000	−0.0566139
$313$	12.1911	0.689082	0.344541	−	0.938771i	$-0.388035\pi$
$313$	12.1911	0.689082	0.344541	+	0.938771i	$0.388035\pi$
$314$	−7.96248	−0.449349
$315$	0.772664	0.0435347
$316$	−1.04846	−0.0589807
$317$	−27.8254	−1.56283	−0.781415	−	0.624012i	$-0.785502\pi$
$317$	−27.8254	−1.56283	−0.781415	+	0.624012i	$0.785502\pi$
$318$	1.00000	0.0560772
$319$	−0.333637	−0.0186801
$320$	−0.188218	−0.0105217
$321$	−7.79394	−0.435015
$322$	−2.42420	−0.135095
$323$	0.755142	0.0420172
$324$	1.00000	0.0555556
$325$	4.96457	0.275385
$326$	11.7366	0.650032
$327$	1.39870	0.0773486
$328$	5.78835	0.319608
$329$	−37.4430	−2.06430
$330$	−0.194414	−0.0107021
$331$	13.0333	0.716374	0.358187	−	0.933650i	$-0.383395\pi$
$331$	13.0333	0.716374	0.358187	+	0.933650i	$0.383395\pi$
$332$	1.10767	0.0607910
$333$	8.84615	0.484766
$334$	0.758697	0.0415141
$335$	0.721747	0.0394333
$336$	−4.10516	−0.223955
$337$	−21.1141	−1.15016	−0.575079	−	0.818098i	$-0.695029\pi$
$337$	−21.1141	−1.15016	−0.575079	+	0.818098i	$0.695029\pi$
$338$	1.00000	0.0543928
$339$	18.0360	0.979579
$340$	−1.11484	−0.0604607
$341$	3.77886	0.204637
$342$	0.127490	0.00689388
$343$	−11.7093	−0.632245
$344$	3.21602	0.173396
$345$	−0.111147	−0.00598397
$346$	−22.9403	−1.23328
$347$	12.3063	0.660636	0.330318	−	0.943870i	$-0.392844\pi$
$347$	12.3063	0.660636	0.330318	+	0.943870i	$0.392844\pi$
$348$	−0.323003	−0.0173148
$349$	−33.2371	−1.77914	−0.889570	−	0.456799i	$-0.848996\pi$
$349$	−33.2371	−1.77914	−0.889570	+	0.456799i	$0.848996\pi$
$350$	20.3804	1.08938
$351$	−1.00000	−0.0533761
$352$	1.03292	0.0550549
$353$	15.4101	0.820197	0.410098	−	0.912041i	$-0.365494\pi$
$353$	15.4101	0.820197	0.410098	+	0.912041i	$0.365494\pi$
$354$	11.1774	0.594072
$355$	−1.65457	−0.0878152
$356$	−10.8453	−0.574802
$357$	−24.3154	−1.28691
$358$	−1.94655	−0.102878
$359$	23.4226	1.23620	0.618100	−	0.786100i	$-0.287903\pi$
$359$	23.4226	1.23620	0.618100	+	0.786100i	$0.287903\pi$
$360$	−0.188218	−0.00991994
$361$	−18.9837	−0.999145
$362$	−24.4229	−1.28364
$363$	−9.93307	−0.521351
$364$	4.10516	0.215169
$365$	−1.00595	−0.0526539
$366$	−10.5396	−0.550916
$367$	−12.5560	−0.655416	−0.327708	−	0.944779i	$-0.606276\pi$
$367$	−12.5560	−0.655416	−0.327708	+	0.944779i	$0.606276\pi$
$368$	0.590525	0.0307833
$369$	5.78835	0.301330
$370$	−1.66500	−0.0865593
$371$	−4.10516	−0.213129
$372$	3.65842	0.189680
$373$	16.4371	0.851083	0.425541	−	0.904939i	$-0.360084\pi$
$373$	16.4371	0.851083	0.425541	+	0.904939i	$0.360084\pi$
$374$	6.11814	0.316362
$375$	1.87551	0.0968509
$376$	9.12097	0.470378
$377$	0.323003	0.0166355
$378$	−4.10516	−0.211147
$379$	26.3140	1.35166	0.675829	−	0.737059i	$-0.263786\pi$
$379$	26.3140	1.35166	0.675829	+	0.737059i	$0.263786\pi$
$380$	−0.0239959	−0.00123096
$381$	17.3568	0.889215
$382$	15.5159	0.793865
$383$	−15.0742	−0.770256	−0.385128	−	0.922863i	$-0.625843\pi$
$383$	−15.0742	−0.770256	−0.385128	+	0.922863i	$0.625843\pi$
$384$	1.00000	0.0510310
$385$	0.798102	0.0406750
$386$	15.1730	0.772283
$387$	3.21602	0.163480
$388$	−13.4044	−0.680504
$389$	−32.0639	−1.62570	−0.812852	−	0.582470i	$-0.802086\pi$
$389$	−32.0639	−1.62570	−0.812852	+	0.582470i	$0.802086\pi$
$390$	0.188218	0.00953077
$391$	3.49776	0.176890
$392$	9.85235	0.497619
$393$	13.4860	0.680277
$394$	−16.2878	−0.820568
$395$	0.197339	0.00992922
$396$	1.03292	0.0519063
$397$	31.3372	1.57277	0.786385	−	0.617737i	$-0.211950\pi$
$397$	31.3372	1.57277	0.786385	+	0.617737i	$0.211950\pi$
$398$	−4.76296	−0.238746
$399$	−0.523368	−0.0262012
$400$	−4.96457	−0.248229
$401$	−10.5063	−0.524660	−0.262330	−	0.964978i	$-0.584491\pi$
$401$	−10.5063	−0.524660	−0.262330	+	0.964978i	$0.584491\pi$
$402$	−3.83464	−0.191254
$403$	−3.65842	−0.182239
$404$	14.1838	0.705669
$405$	−0.188218	−0.00935261
$406$	1.32598	0.0658073
$407$	9.13738	0.452923
$408$	5.92314	0.293239
$409$	9.07531	0.448745	0.224373	−	0.974503i	$-0.427967\pi$
$409$	9.07531	0.448745	0.224373	+	0.974503i	$0.427967\pi$
$410$	−1.08947	−0.0538051
$411$	−6.15911	−0.303806
$412$	−9.37572	−0.461908
$413$	−45.8850	−2.25785
$414$	0.590525	0.0290227
$415$	−0.208482	−0.0102340
$416$	−1.00000	−0.0490290
$417$	20.1563	0.987059
$418$	0.131687	0.00644104
$419$	14.7591	0.721027	0.360514	−	0.932754i	$-0.382601\pi$
$419$	14.7591	0.721027	0.360514	+	0.932754i	$0.382601\pi$
$420$	0.772664	0.0377021
$421$	19.5681	0.953690	0.476845	−	0.878987i	$-0.341780\pi$
$421$	19.5681	0.953690	0.476845	+	0.878987i	$0.341780\pi$
$422$	−12.7735	−0.621803
$423$	9.12097	0.443477
$424$	1.00000	0.0485643
$425$	−29.4059	−1.42639
$426$	8.79070	0.425911
$427$	43.2669	2.09383
$428$	−7.79394	−0.376734
$429$	−1.03292	−0.0498699
$430$	−0.605313	−0.0291908
$431$	−40.4941	−1.95053	−0.975266	−	0.221032i	$-0.929057\pi$
$431$	−40.4941	−1.95053	−0.975266	+	0.221032i	$0.929057\pi$
$432$	1.00000	0.0481125
$433$	−19.0788	−0.916869	−0.458434	−	0.888728i	$-0.651590\pi$
$433$	−19.0788	−0.916869	−0.458434	+	0.888728i	$0.651590\pi$
$434$	−15.0184	−0.720906
$435$	0.0607949	0.00291489
$436$	1.39870	0.0669858
$437$	0.0752862	0.00360143
$438$	5.34461	0.255375
$439$	−0.302671	−0.0144457	−0.00722284	−	0.999974i	$-0.502299\pi$
$439$	−0.302671	−0.0144457	−0.00722284	+	0.999974i	$0.502299\pi$
$440$	−0.194414	−0.00926833
$441$	9.85235	0.469159
$442$	−5.92314	−0.281735
$443$	−9.48578	−0.450683	−0.225342	−	0.974280i	$-0.572350\pi$
$443$	−9.48578	−0.450683	−0.225342	+	0.974280i	$0.572350\pi$
$444$	8.84615	0.419820
$445$	2.04129	0.0967663
$446$	13.5206	0.640217
$447$	15.9012	0.752100
$448$	−4.10516	−0.193951
$449$	−10.5161	−0.496286	−0.248143	−	0.968723i	$-0.579820\pi$
$449$	−10.5161	−0.496286	−0.248143	+	0.968723i	$0.579820\pi$
$450$	−4.96457	−0.234032
$451$	5.97892	0.281536
$452$	18.0360	0.848340
$453$	−3.82856	−0.179881
$454$	−10.7311	−0.503634
$455$	−0.772664	−0.0362230
$456$	0.127490	0.00597027
$457$	14.5722	0.681659	0.340829	−	0.940125i	$-0.389292\pi$
$457$	14.5722	0.681659	0.340829	+	0.940125i	$0.389292\pi$
$458$	−22.4811	−1.05047
$459$	5.92314	0.276468
$460$	−0.111147	−0.00518227
$461$	−36.7350	−1.71092	−0.855459	−	0.517871i	$-0.826725\pi$
$461$	−36.7350	−1.71092	−0.855459	+	0.517871i	$0.826725\pi$
$462$	−4.24031	−0.197277
$463$	−27.1685	−1.26263	−0.631314	−	0.775528i	$-0.717484\pi$
$463$	−27.1685	−1.26263	−0.631314	+	0.775528i	$0.717484\pi$
$464$	−0.323003	−0.0149950
$465$	−0.688579	−0.0319321
$466$	−18.8158	−0.871627
$467$	−12.2266	−0.565779	−0.282890	−	0.959152i	$-0.591293\pi$
$467$	−12.2266	−0.565779	−0.282890	+	0.959152i	$0.591293\pi$
$468$	−1.00000	−0.0462250
$469$	15.7418	0.726889
$470$	−1.71673	−0.0791867
$471$	−7.96248	−0.366892
$472$	11.1774	0.514482
$473$	3.32190	0.152741
$474$	−1.04846	−0.0481575
$475$	−0.632934	−0.0290410
$476$	−24.3154	−1.11450
$477$	1.00000	0.0457869
$478$	6.68189	0.305623
$479$	25.2574	1.15404	0.577020	−	0.816730i	$-0.304215\pi$
$479$	25.2574	1.15404	0.577020	+	0.816730i	$0.304215\pi$
$480$	−0.188218	−0.00859092
$481$	−8.84615	−0.403350
$482$	21.0808	0.960206
$483$	−2.42420	−0.110305
$484$	−9.93307	−0.451503
$485$	2.52294	0.114561
$486$	1.00000	0.0453609
$487$	−16.4053	−0.743396	−0.371698	−	0.928354i	$-0.621224\pi$
$487$	−16.4053	−0.743396	−0.371698	+	0.928354i	$0.621224\pi$
$488$	−10.5396	−0.477107
$489$	11.7366	0.530749
$490$	−1.85439	−0.0837726
$491$	−28.2185	−1.27348	−0.636741	−	0.771078i	$-0.719718\pi$
$491$	−28.2185	−1.27348	−0.636741	+	0.771078i	$0.719718\pi$
$492$	5.78835	0.260959
$493$	−1.91319	−0.0861659
$494$	−0.127490	−0.00573605
$495$	−0.194414	−0.00873827
$496$	3.65842	0.164268
$497$	−36.0872	−1.61873
$498$	1.10767	0.0496357
$499$	21.9202	0.981282	0.490641	−	0.871362i	$-0.336763\pi$
$499$	21.9202	0.981282	0.490641	+	0.871362i	$0.336763\pi$
$500$	1.87551	0.0838753
$501$	0.758697	0.0338961
$502$	−9.07825	−0.405182
$503$	−38.9042	−1.73465	−0.867326	−	0.497741i	$-0.834163\pi$
$503$	−38.9042	−1.73465	−0.867326	+	0.497741i	$0.834163\pi$
$504$	−4.10516	−0.182858
$505$	−2.66964	−0.118797
$506$	0.609967	0.0271163
$507$	1.00000	0.0444116
$508$	17.3568	0.770083
$509$	2.70946	0.120095	0.0600473	−	0.998196i	$-0.480875\pi$
$509$	2.70946	0.120095	0.0600473	+	0.998196i	$0.480875\pi$
$510$	−1.11484	−0.0493659
$511$	−21.9405	−0.970590
$512$	1.00000	0.0441942
$513$	0.127490	0.00562883
$514$	11.2159	0.494713
$515$	1.76468	0.0777609
$516$	3.21602	0.141578
$517$	9.42125	0.414346
$518$	−36.3149	−1.59558
$519$	−22.9403	−1.00697
$520$	0.188218	0.00825389
$521$	25.9928	1.13877	0.569383	−	0.822072i	$-0.307182\pi$
$521$	25.9928	1.13877	0.569383	+	0.822072i	$0.307182\pi$
$522$	−0.323003	−0.0141375
$523$	2.07385	0.0906833	0.0453417	−	0.998972i	$-0.485562\pi$
$523$	2.07385	0.0906833	0.0453417	+	0.998972i	$0.485562\pi$
$524$	13.4860	0.589137
$525$	20.3804	0.889473
$526$	−29.0370	−1.26607
$527$	21.6693	0.943931
$528$	1.03292	0.0449522
$529$	−22.6513	−0.984838
$530$	−0.188218	−0.00817565
$531$	11.1774	0.485058
$532$	−0.523368	−0.0226909
$533$	−5.78835	−0.250721
$534$	−10.8453	−0.469324
$535$	1.46696	0.0634221
$536$	−3.83464	−0.165631
$537$	−1.94655	−0.0839998
$538$	−27.5271	−1.18678
$539$	10.1767	0.438342
$540$	−0.188218	−0.00809960
$541$	13.0457	0.560879	0.280440	−	0.959872i	$-0.409520\pi$
$541$	13.0457	0.560879	0.280440	+	0.959872i	$0.409520\pi$
$542$	−12.7424	−0.547334
$543$	−24.4229	−1.04809
$544$	5.92314	0.253953
$545$	−0.263261	−0.0112769
$546$	4.10516	0.175685
$547$	34.1950	1.46207	0.731037	−	0.682338i	$-0.239037\pi$
$547$	34.1950	1.46207	0.731037	+	0.682338i	$0.239037\pi$
$548$	−6.15911	−0.263104
$549$	−10.5396	−0.449821
$550$	−5.12802	−0.218659
$551$	−0.0411797	−0.00175432
$552$	0.590525	0.0251344
$553$	4.30411	0.183029
$554$	19.3027	0.820095
$555$	−1.66500	−0.0706754
$556$	20.1563	0.854818
$557$	28.9573	1.22696	0.613481	−	0.789709i	$-0.289769\pi$
$557$	28.9573	1.22696	0.613481	+	0.789709i	$0.289769\pi$
$558$	3.65842	0.154873
$559$	−3.21602	−0.136023
$560$	0.772664	0.0326510
$561$	6.11814	0.258308
$562$	18.9805	0.800643
$563$	30.5453	1.28733	0.643666	−	0.765306i	$-0.277413\pi$
$563$	30.5453	1.28733	0.643666	+	0.765306i	$0.277413\pi$
$564$	9.12097	0.384062
$565$	−3.39469	−0.142816
$566$	−13.3878	−0.562731
$567$	−4.10516	−0.172401
$568$	8.79070	0.368850
$569$	12.1781	0.510530	0.255265	−	0.966871i	$-0.417837\pi$
$569$	12.1781	0.510530	0.255265	+	0.966871i	$0.417837\pi$
$570$	−0.0239959	−0.00100508
$571$	17.1227	0.716564	0.358282	−	0.933613i	$-0.383363\pi$
$571$	17.1227	0.716564	0.358282	+	0.933613i	$0.383363\pi$
$572$	−1.03292	−0.0431886
$573$	15.5159	0.648188
$574$	−23.7621	−0.991812
$575$	−2.93171	−0.122261
$576$	1.00000	0.0416667
$577$	−12.7986	−0.532812	−0.266406	−	0.963861i	$-0.585836\pi$
$577$	−12.7986	−0.532812	−0.266406	+	0.963861i	$0.585836\pi$
$578$	18.0836	0.752178
$579$	15.1730	0.630567
$580$	0.0607949	0.00252437
$581$	−4.54714	−0.188647
$582$	−13.4044	−0.555629
$583$	1.03292	0.0427793
$584$	5.34461	0.221162
$585$	0.188218	0.00778184
$586$	1.79309	0.0740718
$587$	−15.7183	−0.648763	−0.324382	−	0.945926i	$-0.605156\pi$
$587$	−15.7183	−0.648763	−0.324382	+	0.945926i	$0.605156\pi$
$588$	9.85235	0.406304
$589$	0.466413	0.0192182
$590$	−2.10378	−0.0866114
$591$	−16.2878	−0.669991
$592$	8.84615	0.363575
$593$	−7.57488	−0.311063	−0.155532	−	0.987831i	$-0.549709\pi$
$593$	−7.57488	−0.311063	−0.155532	+	0.987831i	$0.549709\pi$
$594$	1.03292	0.0423813
$595$	4.57660	0.187622
$596$	15.9012	0.651338
$597$	−4.76296	−0.194935
$598$	−0.590525	−0.0241484
$599$	−10.6004	−0.433121	−0.216560	−	0.976269i	$-0.569484\pi$
$599$	−10.6004	−0.433121	−0.216560	+	0.976269i	$0.569484\pi$
$600$	−4.96457	−0.202678
$601$	−32.8666	−1.34066	−0.670329	−	0.742064i	$-0.733847\pi$
$601$	−32.8666	−1.34066	−0.670329	+	0.742064i	$0.733847\pi$
$602$	−13.2023	−0.538085
$603$	−3.83464	−0.156159
$604$	−3.82856	−0.155782
$605$	1.86958	0.0760092
$606$	14.1838	0.576176
$607$	25.8733	1.05016	0.525082	−	0.851051i	$-0.324035\pi$
$607$	25.8733	1.05016	0.525082	+	0.851051i	$0.324035\pi$
$608$	0.127490	0.00517041
$609$	1.32598	0.0537314
$610$	1.98375	0.0803196
$611$	−9.12097	−0.368995
$612$	5.92314	0.239429
$613$	−35.6965	−1.44177	−0.720883	−	0.693057i	$-0.756264\pi$
$613$	−35.6965	−1.44177	−0.720883	+	0.693057i	$0.756264\pi$
$614$	21.7172	0.876434
$615$	−1.08947	−0.0439317
$616$	−4.24031	−0.170847
$617$	43.2561	1.74142	0.870712	−	0.491793i	$-0.163658\pi$
$617$	43.2561	1.74142	0.870712	+	0.491793i	$0.163658\pi$
$618$	−9.37572	−0.377147
$619$	10.8782	0.437233	0.218616	−	0.975811i	$-0.429846\pi$
$619$	10.8782	0.437233	0.218616	+	0.975811i	$0.429846\pi$
$620$	−0.688579	−0.0276540
$621$	0.590525	0.0236970
$622$	8.93712	0.358346
$623$	44.5219	1.78373
$624$	−1.00000	−0.0400320
$625$	24.4699	0.978795
$626$	12.1911	0.487254
$627$	0.131687	0.00525909
$628$	−7.96248	−0.317737
$629$	52.3970	2.08920
$630$	0.772664	0.0307837
$631$	4.02536	0.160247	0.0801236	−	0.996785i	$-0.474469\pi$
$631$	4.02536	0.160247	0.0801236	+	0.996785i	$0.474469\pi$
$632$	−1.04846	−0.0417056
$633$	−12.7735	−0.507700
$634$	−27.8254	−1.10509
$635$	−3.26685	−0.129641
$636$	1.00000	0.0396526
$637$	−9.85235	−0.390364
$638$	−0.333637	−0.0132088
$639$	8.79070	0.347755
$640$	−0.188218	−0.00743996
$641$	7.86871	0.310795	0.155398	−	0.987852i	$-0.450334\pi$
$641$	7.86871	0.310795	0.155398	+	0.987852i	$0.450334\pi$
$642$	−7.79394	−0.307602
$643$	43.2881	1.70712	0.853558	−	0.520998i	$-0.174440\pi$
$643$	43.2881	1.70712	0.853558	+	0.520998i	$0.174440\pi$
$644$	−2.42420	−0.0955269
$645$	−0.605313	−0.0238342
$646$	0.755142	0.0297107
$647$	17.7330	0.697155	0.348577	−	0.937280i	$-0.386665\pi$
$647$	17.7330	0.697155	0.348577	+	0.937280i	$0.386665\pi$
$648$	1.00000	0.0392837
$649$	11.5454	0.453196
$650$	4.96457	0.194727
$651$	−15.0184	−0.588618
$652$	11.7366	0.459642
$653$	−21.8537	−0.855200	−0.427600	−	0.903968i	$-0.640641\pi$
$653$	−21.8537	−0.855200	−0.427600	+	0.903968i	$0.640641\pi$
$654$	1.39870	0.0546937
$655$	−2.53830	−0.0991795
$656$	5.78835	0.225997
$657$	5.34461	0.208513
$658$	−37.4430	−1.45968
$659$	−3.78597	−0.147481	−0.0737403	−	0.997277i	$-0.523494\pi$
$659$	−3.78597	−0.147481	−0.0737403	+	0.997277i	$0.523494\pi$
$660$	−0.194414	−0.00756756
$661$	7.42093	0.288641	0.144320	−	0.989531i	$-0.453900\pi$
$661$	7.42093	0.288641	0.144320	+	0.989531i	$0.453900\pi$
$662$	13.0333	0.506553
$663$	−5.92314	−0.230036
$664$	1.10767	0.0429857
$665$	0.0985071	0.00381994
$666$	8.84615	0.342781
$667$	−0.190742	−0.00738555
$668$	0.758697	0.0293549
$669$	13.5206	0.522735
$670$	0.721747	0.0278835
$671$	−10.8866	−0.420273
$672$	−4.10516	−0.158360
$673$	−35.3442	−1.36242	−0.681209	−	0.732089i	$-0.738545\pi$
$673$	−35.3442	−1.36242	−0.681209	+	0.732089i	$0.738545\pi$
$674$	−21.1141	−0.813285
$675$	−4.96457	−0.191087
$676$	1.00000	0.0384615
$677$	30.0634	1.15543	0.577716	−	0.816238i	$-0.303944\pi$
$677$	30.0634	1.15543	0.577716	+	0.816238i	$0.303944\pi$
$678$	18.0360	0.692667
$679$	55.0271	2.11175
$680$	−1.11484	−0.0427522
$681$	−10.7311	−0.411216
$682$	3.77886	0.144700
$683$	−30.1438	−1.15342	−0.576710	−	0.816949i	$-0.695664\pi$
$683$	−30.1438	−1.15342	−0.576710	+	0.816949i	$0.695664\pi$
$684$	0.127490	0.00487471
$685$	1.15925	0.0442928
$686$	−11.7093	−0.447065
$687$	−22.4811	−0.857708
$688$	3.21602	0.122610
$689$	−1.00000	−0.0380970
$690$	−0.111147	−0.00423131
$691$	13.0459	0.496289	0.248145	−	0.968723i	$-0.420179\pi$
$691$	13.0459	0.496289	0.248145	+	0.968723i	$0.420179\pi$
$692$	−22.9403	−0.872058
$693$	−4.24031	−0.161076
$694$	12.3063	0.467140
$695$	−3.79378	−0.143906
$696$	−0.323003	−0.0122434
$697$	34.2852	1.29865
$698$	−33.2371	−1.25804
$699$	−18.8158	−0.711680
$700$	20.3804	0.770306
$701$	41.8506	1.58068	0.790338	−	0.612670i	$-0.209905\pi$
$701$	41.8506	1.58068	0.790338	+	0.612670i	$0.209905\pi$
$702$	−1.00000	−0.0377426
$703$	1.12780	0.0425357
$704$	1.03292	0.0389297
$705$	−1.71673	−0.0646557
$706$	15.4101	0.579967
$707$	−58.2266	−2.18984
$708$	11.1774	0.420072
$709$	−39.5908	−1.48686	−0.743432	−	0.668812i	$-0.766803\pi$
$709$	−39.5908	−1.48686	−0.743432	+	0.668812i	$0.766803\pi$
$710$	−1.65457	−0.0620947
$711$	−1.04846	−0.0393204
$712$	−10.8453	−0.406447
$713$	2.16039	0.0809072
$714$	−24.3154	−0.909983
$715$	0.194414	0.00727068
$716$	−1.94655	−0.0727460
$717$	6.68189	0.249540
$718$	23.4226	0.874125
$719$	−38.0075	−1.41744	−0.708720	−	0.705490i	$-0.750727\pi$
$719$	−38.0075	−1.41744	−0.708720	+	0.705490i	$0.750727\pi$
$720$	−0.188218	−0.00701446
$721$	38.4888	1.43340
$722$	−18.9837	−0.706502
$723$	21.0808	0.784005
$724$	−24.4229	−0.907671
$725$	1.60357	0.0595552
$726$	−9.93307	−0.368651
$727$	−20.4870	−0.759822	−0.379911	−	0.925023i	$-0.624045\pi$
$727$	−20.4870	−0.759822	−0.379911	+	0.925023i	$0.624045\pi$
$728$	4.10516	0.152147
$729$	1.00000	0.0370370
$730$	−1.00595	−0.0372319
$731$	19.0490	0.704551
$732$	−10.5396	−0.389556
$733$	2.17983	0.0805139	0.0402569	−	0.999189i	$-0.487182\pi$
$733$	2.17983	0.0805139	0.0402569	+	0.999189i	$0.487182\pi$
$734$	−12.5560	−0.463449
$735$	−1.85439	−0.0684001
$736$	0.590525	0.0217671
$737$	−3.96088	−0.145901
$738$	5.78835	0.213072
$739$	−15.3292	−0.563895	−0.281947	−	0.959430i	$-0.590980\pi$
$739$	−15.3292	−0.563895	−0.281947	+	0.959430i	$0.590980\pi$
$740$	−1.66500	−0.0612067
$741$	−0.127490	−0.00468347
$742$	−4.10516	−0.150705
$743$	−35.5487	−1.30415	−0.652077	−	0.758152i	$-0.726102\pi$
$743$	−35.5487	−1.30415	−0.652077	+	0.758152i	$0.726102\pi$
$744$	3.65842	0.134124
$745$	−2.99288	−0.109651
$746$	16.4371	0.601807
$747$	1.10767	0.0405273
$748$	6.11814	0.223701
$749$	31.9954	1.16909
$750$	1.87551	0.0684839
$751$	−42.6720	−1.55713	−0.778563	−	0.627567i	$-0.784051\pi$
$751$	−42.6720	−1.55713	−0.778563	+	0.627567i	$0.784051\pi$
$752$	9.12097	0.332608
$753$	−9.07825	−0.330830
$754$	0.323003	0.0117631
$755$	0.720603	0.0262254
$756$	−4.10516	−0.149303
$757$	−10.1725	−0.369728	−0.184864	−	0.982764i	$-0.559184\pi$
$757$	−10.1725	−0.369728	−0.184864	+	0.982764i	$0.559184\pi$
$758$	26.3140	0.955766
$759$	0.609967	0.0221404
$760$	−0.0239959	−0.000870423 0
$761$	29.1729	1.05752	0.528759	−	0.848772i	$-0.322657\pi$
$761$	29.1729	1.05752	0.528759	+	0.848772i	$0.322657\pi$
$762$	17.3568	0.628770
$763$	−5.74191	−0.207871
$764$	15.5159	0.561347
$765$	−1.11484	−0.0403071
$766$	−15.0742	−0.544653
$767$	−11.1774	−0.403593
$768$	1.00000	0.0360844
$769$	4.44147	0.160163	0.0800817	−	0.996788i	$-0.474482\pi$
$769$	4.44147	0.160163	0.0800817	+	0.996788i	$0.474482\pi$
$770$	0.798102	0.0287616
$771$	11.2159	0.403931
$772$	15.1730	0.546087
$773$	−25.9079	−0.931841	−0.465921	−	0.884827i	$-0.654277\pi$
$773$	−25.9079	−0.931841	−0.465921	+	0.884827i	$0.654277\pi$
$774$	3.21602	0.115598
$775$	−18.1625	−0.652416
$776$	−13.4044	−0.481189
$777$	−36.3149	−1.30279
$778$	−32.0639	−1.14955
$779$	0.737958	0.0264401
$780$	0.188218	0.00673927
$781$	9.08011	0.324912
$782$	3.49776	0.125080
$783$	−0.323003	−0.0115432
$784$	9.85235	0.351870
$785$	1.49868	0.0534901
$786$	13.4860	0.481028
$787$	−48.2308	−1.71924	−0.859620	−	0.510933i	$-0.829300\pi$
$787$	−48.2308	−1.71924	−0.859620	+	0.510933i	$0.829300\pi$
$788$	−16.2878	−0.580229
$789$	−29.0370	−1.03375
$790$	0.197339	0.00702102
$791$	−74.0405	−2.63258
$792$	1.03292	0.0367033
$793$	10.5396	0.374274
$794$	31.3372	1.11212
$795$	−0.188218	−0.00667539
$796$	−4.76296	−0.168819
$797$	43.4692	1.53976	0.769879	−	0.638190i	$-0.220317\pi$
$797$	43.4692	1.53976	0.769879	+	0.638190i	$0.220317\pi$
$798$	−0.523368	−0.0185270
$799$	54.0248	1.91126
$800$	−4.96457	−0.175524
$801$	−10.8453	−0.383202
$802$	−10.5063	−0.370991
$803$	5.52057	0.194817
$804$	−3.83464	−0.135237
$805$	0.456278	0.0160817
$806$	−3.65842	−0.128862
$807$	−27.5271	−0.969001
$808$	14.1838	0.498983
$809$	−2.83360	−0.0996241	−0.0498120	−	0.998759i	$-0.515862\pi$
$809$	−2.83360	−0.0996241	−0.0498120	+	0.998759i	$0.515862\pi$
$810$	−0.188218	−0.00661330
$811$	52.8762	1.85673	0.928367	−	0.371665i	$-0.121213\pi$
$811$	52.8762	1.85673	0.928367	+	0.371665i	$0.121213\pi$
$812$	1.32598	0.0465328
$813$	−12.7424	−0.446896
$814$	9.13738	0.320265
$815$	−2.20904	−0.0773794
$816$	5.92314	0.207351
$817$	0.410011	0.0143445
$818$	9.07531	0.317311
$819$	4.10516	0.143446
$820$	−1.08947	−0.0380460
$821$	−36.6199	−1.27804	−0.639021	−	0.769189i	$-0.720660\pi$
$821$	−36.6199	−1.27804	−0.639021	+	0.769189i	$0.720660\pi$
$822$	−6.15911	−0.214823
$823$	−30.3981	−1.05961	−0.529805	−	0.848120i	$-0.677735\pi$
$823$	−30.3981	−1.05961	−0.529805	+	0.848120i	$0.677735\pi$
$824$	−9.37572	−0.326619
$825$	−5.12802	−0.178535
$826$	−45.8850	−1.59654
$827$	−28.4155	−0.988103	−0.494051	−	0.869433i	$-0.664485\pi$
$827$	−28.4155	−0.988103	−0.494051	+	0.869433i	$0.664485\pi$
$828$	0.590525	0.0205222
$829$	−0.539355	−0.0187326	−0.00936628	−	0.999956i	$-0.502981\pi$
$829$	−0.539355	−0.0187326	−0.00936628	+	0.999956i	$0.502981\pi$
$830$	−0.208482	−0.00723652
$831$	19.3027	0.669604
$832$	−1.00000	−0.0346688
$833$	58.3568	2.02194
$834$	20.1563	0.697956
$835$	−0.142800	−0.00494181
$836$	0.131687	0.00455450
$837$	3.65842	0.126453
$838$	14.7591	0.509843
$839$	37.0514	1.27916	0.639579	−	0.768726i	$-0.279109\pi$
$839$	37.0514	1.27916	0.639579	+	0.768726i	$0.279109\pi$
$840$	0.772664	0.0266594
$841$	−28.8957	−0.996402
$842$	19.5681	0.674361
$843$	18.9805	0.653722
$844$	−12.7735	−0.439681
$845$	−0.188218	−0.00647489
$846$	9.12097	0.313585
$847$	40.7769	1.40111
$848$	1.00000	0.0343401
$849$	−13.3878	−0.459468
$850$	−29.4059	−1.00861
$851$	5.22388	0.179072
$852$	8.79070	0.301164
$853$	−13.5986	−0.465607	−0.232803	−	0.972524i	$-0.574790\pi$
$853$	−13.5986	−0.465607	−0.232803	+	0.972524i	$0.574790\pi$
$854$	43.2669	1.48056
$855$	−0.0239959	−0.000820643 0
$856$	−7.79394	−0.266391
$857$	−26.5324	−0.906330	−0.453165	−	0.891427i	$-0.649705\pi$
$857$	−26.5324	−0.906330	−0.453165	+	0.891427i	$0.649705\pi$
$858$	−1.03292	−0.0352634
$859$	−33.0679	−1.12826	−0.564132	−	0.825685i	$-0.690789\pi$
$859$	−33.0679	−1.12826	−0.564132	+	0.825685i	$0.690789\pi$
$860$	−0.605313	−0.0206410
$861$	−23.7621	−0.809811
$862$	−40.4941	−1.37924
$863$	−11.7890	−0.401302	−0.200651	−	0.979663i	$-0.564306\pi$
$863$	−11.7890	−0.401302	−0.200651	+	0.979663i	$0.564306\pi$
$864$	1.00000	0.0340207
$865$	4.31777	0.146808
$866$	−19.0788	−0.648324
$867$	18.0836	0.614151
$868$	−15.0184	−0.509758
$869$	−1.08298	−0.0367376
$870$	0.0607949	0.00206114
$871$	3.83464	0.129932
$872$	1.39870	0.0473661
$873$	−13.4044	−0.453669
$874$	0.0752862	0.00254659
$875$	−7.69927	−0.260283
$876$	5.34461	0.180578
$877$	−4.51207	−0.152362	−0.0761808	−	0.997094i	$-0.524273\pi$
$877$	−4.51207	−0.152362	−0.0761808	+	0.997094i	$0.524273\pi$
$878$	−0.302671	−0.0102146
$879$	1.79309	0.0604794
$880$	−0.194414	−0.00655370
$881$	−1.40722	−0.0474105	−0.0237052	−	0.999719i	$-0.507546\pi$
$881$	−1.40722	−0.0474105	−0.0237052	+	0.999719i	$0.507546\pi$
$882$	9.85235	0.331746
$883$	−21.0372	−0.707959	−0.353979	−	0.935253i	$-0.615172\pi$
$883$	−21.0372	−0.707959	−0.353979	+	0.935253i	$0.615172\pi$
$884$	−5.92314	−0.199217
$885$	−2.10378	−0.0707179
$886$	−9.48578	−0.318681
$887$	2.15846	0.0724741	0.0362371	−	0.999343i	$-0.488463\pi$
$887$	2.15846	0.0724741	0.0362371	+	0.999343i	$0.488463\pi$
$888$	8.84615	0.296857
$889$	−71.2524	−2.38973
$890$	2.04129	0.0684241
$891$	1.03292	0.0346042
$892$	13.5206	0.452702
$893$	1.16283	0.0389128
$894$	15.9012	0.531815
$895$	0.366375	0.0122466
$896$	−4.10516	−0.137144
$897$	−0.590525	−0.0197171
$898$	−10.5161	−0.350927
$899$	−1.18168	−0.0394113
$900$	−4.96457	−0.165486
$901$	5.92314	0.197328
$902$	5.97892	0.199076
$903$	−13.2023	−0.439345
$904$	18.0360	0.599867
$905$	4.59683	0.152804
$906$	−3.82856	−0.127195
$907$	18.6240	0.618400	0.309200	−	0.950997i	$-0.399939\pi$
$907$	18.6240	0.618400	0.309200	+	0.950997i	$0.399939\pi$
$908$	−10.7311	−0.356123
$909$	14.1838	0.470446
$910$	−0.772664	−0.0256136
$911$	−9.18621	−0.304353	−0.152176	−	0.988353i	$-0.548628\pi$
$911$	−9.18621	−0.304353	−0.152176	+	0.988353i	$0.548628\pi$
$912$	0.127490	0.00422162
$913$	1.14413	0.0378652
$914$	14.5722	0.482006
$915$	1.98375	0.0655806
$916$	−22.4811	−0.742797
$917$	−55.3620	−1.82822
$918$	5.92314	0.195493
$919$	6.92909	0.228570	0.114285	−	0.993448i	$-0.463542\pi$
$919$	6.92909	0.228570	0.114285	+	0.993448i	$0.463542\pi$
$920$	−0.111147	−0.00366442
$921$	21.7172	0.715605
$922$	−36.7350	−1.20980
$923$	−8.79070	−0.289349
$924$	−4.24031	−0.139496
$925$	−43.9174	−1.44399
$926$	−27.1685	−0.892812
$927$	−9.37572	−0.307939
$928$	−0.323003	−0.0106031
$929$	20.3785	0.668598	0.334299	−	0.942467i	$-0.391500\pi$
$929$	20.3785	0.668598	0.334299	+	0.942467i	$0.391500\pi$
$930$	−0.688579	−0.0225794
$931$	1.25608	0.0411663
$932$	−18.8158	−0.616333
$933$	8.93712	0.292588
$934$	−12.2266	−0.400066
$935$	−1.15154	−0.0376595
$936$	−1.00000	−0.0326860
$937$	28.5267	0.931926	0.465963	−	0.884804i	$-0.345708\pi$
$937$	28.5267	0.931926	0.465963	+	0.884804i	$0.345708\pi$
$938$	15.7418	0.513988
$939$	12.1911	0.397842
$940$	−1.71673	−0.0559935
$941$	−6.41610	−0.209159	−0.104579	−	0.994517i	$-0.533350\pi$
$941$	−6.41610	−0.209159	−0.104579	+	0.994517i	$0.533350\pi$
$942$	−7.96248	−0.259431
$943$	3.41817	0.111311
$944$	11.1774	0.363793
$945$	0.772664	0.0251348
$946$	3.32190	0.108004
$947$	−48.7647	−1.58464	−0.792319	−	0.610107i	$-0.791127\pi$
$947$	−48.7647	−1.58464	−0.792319	+	0.610107i	$0.791127\pi$
$948$	−1.04846	−0.0340525
$949$	−5.34461	−0.173493
$950$	−0.632934	−0.0205351
$951$	−27.8254	−0.902300
$952$	−24.3154	−0.788068
$953$	12.6982	0.411335	0.205668	−	0.978622i	$-0.434063\pi$
$953$	12.6982	0.411335	0.205668	+	0.978622i	$0.434063\pi$
$954$	1.00000	0.0323762
$955$	−2.92037	−0.0945011
$956$	6.68189	0.216108
$957$	−0.333637	−0.0107850
$958$	25.2574	0.816030
$959$	25.2841	0.816467
$960$	−0.188218	−0.00607470
$961$	−17.6160	−0.568257
$962$	−8.84615	−0.285211
$963$	−7.79394	−0.251156
$964$	21.0808	0.678968
$965$	−2.85582	−0.0919321
$966$	−2.42420	−0.0779974
$967$	34.9759	1.12475	0.562374	−	0.826883i	$-0.309888\pi$
$967$	34.9759	1.12475	0.562374	+	0.826883i	$0.309888\pi$
$968$	−9.93307	−0.319261
$969$	0.755142	0.0242587
$970$	2.52294	0.0810067
$971$	42.0330	1.34890	0.674452	−	0.738319i	$-0.264380\pi$
$971$	42.0330	1.34890	0.674452	+	0.738319i	$0.264380\pi$
$972$	1.00000	0.0320750
$973$	−82.7449	−2.65268
$974$	−16.4053	−0.525660
$975$	4.96457	0.158994
$976$	−10.5396	−0.337366
$977$	8.99755	0.287857	0.143929	−	0.989588i	$-0.454026\pi$
$977$	8.99755	0.287857	0.143929	+	0.989588i	$0.454026\pi$
$978$	11.7366	0.375296
$979$	−11.2024	−0.358030
$980$	−1.85439	−0.0592362
$981$	1.39870	0.0446572
$982$	−28.2185	−0.900488
$983$	−2.52399	−0.0805029	−0.0402514	−	0.999190i	$-0.512816\pi$
$983$	−2.52399	−0.0805029	−0.0402514	+	0.999190i	$0.512816\pi$
$984$	5.78835	0.184526
$985$	3.06565	0.0976798
$986$	−1.91319	−0.0609285
$987$	−37.4430	−1.19183
$988$	−0.127490	−0.00405600
$989$	1.89914	0.0603893
$990$	−0.194414	−0.00617889
$991$	60.8901	1.93424	0.967120	−	0.254322i	$-0.0818523\pi$
$991$	60.8901	1.93424	0.967120	+	0.254322i	$0.0818523\pi$
$992$	3.65842	0.116155
$993$	13.0333	0.413599
$994$	−36.0872	−1.14462
$995$	0.896474	0.0284201
$996$	1.10767	0.0350977
$997$	1.76250	0.0558189	0.0279095	−	0.999610i	$-0.491115\pi$
$997$	1.76250	0.0558189	0.0279095	+	0.999610i	$0.491115\pi$
$998$	21.9202	0.693871
$999$	8.84615	0.279880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4134.2.a.x.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4134.2.a.x.1.3	✓	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:	\( N \)	\(=\)	\( 4134 = 2 \cdot 3 \cdot 13 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4134.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.188218 q^{5} +1.00000 q^{6} -4.10516 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.188218 q^{5} +1.00000 q^{6} -4.10516 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.188218 q^{10} +1.03292 q^{11} +1.00000 q^{12} -1.00000 q^{13} -4.10516 q^{14} -0.188218 q^{15} +1.00000 q^{16} +5.92314 q^{17} +1.00000 q^{18} +0.127490 q^{19} -0.188218 q^{20} -4.10516 q^{21} +1.03292 q^{22} +0.590525 q^{23} +1.00000 q^{24} -4.96457 q^{25} -1.00000 q^{26} +1.00000 q^{27} -4.10516 q^{28} -0.323003 q^{29} -0.188218 q^{30} +3.65842 q^{31} +1.00000 q^{32} +1.03292 q^{33} +5.92314 q^{34} +0.772664 q^{35} +1.00000 q^{36} +8.84615 q^{37} +0.127490 q^{38} -1.00000 q^{39} -0.188218 q^{40} +5.78835 q^{41} -4.10516 q^{42} +3.21602 q^{43} +1.03292 q^{44} -0.188218 q^{45} +0.590525 q^{46} +9.12097 q^{47} +1.00000 q^{48} +9.85235 q^{49} -4.96457 q^{50} +5.92314 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -0.194414 q^{55} -4.10516 q^{56} +0.127490 q^{57} -0.323003 q^{58} +11.1774 q^{59} -0.188218 q^{60} -10.5396 q^{61} +3.65842 q^{62} -4.10516 q^{63} +1.00000 q^{64} +0.188218 q^{65} +1.03292 q^{66} -3.83464 q^{67} +5.92314 q^{68} +0.590525 q^{69} +0.772664 q^{70} +8.79070 q^{71} +1.00000 q^{72} +5.34461 q^{73} +8.84615 q^{74} -4.96457 q^{75} +0.127490 q^{76} -4.24031 q^{77} -1.00000 q^{78} -1.04846 q^{79} -0.188218 q^{80} +1.00000 q^{81} +5.78835 q^{82} +1.10767 q^{83} -4.10516 q^{84} -1.11484 q^{85} +3.21602 q^{86} -0.323003 q^{87} +1.03292 q^{88} -10.8453 q^{89} -0.188218 q^{90} +4.10516 q^{91} +0.590525 q^{92} +3.65842 q^{93} +9.12097 q^{94} -0.0239959 q^{95} +1.00000 q^{96} -13.4044 q^{97} +9.85235 q^{98} +1.03292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 7 q^{11} + 8 q^{12} - 8 q^{13} + 7 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + 11 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 5 q^{23} + 8 q^{24} + 6 q^{25} - 8 q^{26} + 8 q^{27} + 7 q^{28} + 13 q^{29} + 2 q^{30} + 12 q^{31} + 8 q^{32} + 7 q^{33} + 8 q^{34} + 14 q^{35} + 8 q^{36} + 10 q^{37} + 11 q^{38} - 8 q^{39} + 2 q^{40} + 19 q^{41} + 7 q^{42} + 10 q^{43} + 7 q^{44} + 2 q^{45} + 5 q^{46} + 6 q^{47} + 8 q^{48} + 15 q^{49} + 6 q^{50} + 8 q^{51} - 8 q^{52} + 8 q^{53} + 8 q^{54} + 5 q^{55} + 7 q^{56} + 11 q^{57} + 13 q^{58} + 11 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} + 5 q^{67} + 8 q^{68} + 5 q^{69} + 14 q^{70} - 6 q^{71} + 8 q^{72} + 18 q^{73} + 10 q^{74} + 6 q^{75} + 11 q^{76} - 8 q^{77} - 8 q^{78} + 17 q^{79} + 2 q^{80} + 8 q^{81} + 19 q^{82} + 16 q^{83} + 7 q^{84} + 25 q^{85} + 10 q^{86} + 13 q^{87} + 7 q^{88} - 8 q^{89} + 2 q^{90} - 7 q^{91} + 5 q^{92} + 12 q^{93} + 6 q^{94} - 30 q^{95} + 8 q^{96} + 11 q^{97} + 15 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 7 * q^11 + 8 * q^12 - 8 * q^13 + 7 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + 11 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 5 * q^23 + 8 * q^24 + 6 * q^25 - 8 * q^26 + 8 * q^27 + 7 * q^28 + 13 * q^29 + 2 * q^30 + 12 * q^31 + 8 * q^32 + 7 * q^33 + 8 * q^34 + 14 * q^35 + 8 * q^36 + 10 * q^37 + 11 * q^38 - 8 * q^39 + 2 * q^40 + 19 * q^41 + 7 * q^42 + 10 * q^43 + 7 * q^44 + 2 * q^45 + 5 * q^46 + 6 * q^47 + 8 * q^48 + 15 * q^49 + 6 * q^50 + 8 * q^51 - 8 * q^52 + 8 * q^53 + 8 * q^54 + 5 * q^55 + 7 * q^56 + 11 * q^57 + 13 * q^58 + 11 * q^59 + 2 * q^60 + 6 * q^61 + 12 * q^62 + 7 * q^63 + 8 * q^64 - 2 * q^65 + 7 * q^66 + 5 * q^67 + 8 * q^68 + 5 * q^69 + 14 * q^70 - 6 * q^71 + 8 * q^72 + 18 * q^73 + 10 * q^74 + 6 * q^75 + 11 * q^76 - 8 * q^77 - 8 * q^78 + 17 * q^79 + 2 * q^80 + 8 * q^81 + 19 * q^82 + 16 * q^83 + 7 * q^84 + 25 * q^85 + 10 * q^86 + 13 * q^87 + 7 * q^88 - 8 * q^89 + 2 * q^90 - 7 * q^91 + 5 * q^92 + 12 * q^93 + 6 * q^94 - 30 * q^95 + 8 * q^96 + 11 * q^97 + 15 * q^98 + 7 * q^99

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