LMFDB - Embedded newform 231.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,2,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 231 = 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	231.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:	$1.84454428669$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	231.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} +1.46260 q^{6} -1.00000 q^{7} -2.72161 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} +1.46260 q^{6} -1.00000 q^{7} -2.72161 q^{8} +1.00000 q^{9} +3.50761 q^{10} -1.00000 q^{11} +0.139194 q^{12} +5.04502 q^{13} -1.46260 q^{14} +2.39821 q^{15} -4.25901 q^{16} -6.36842 q^{17} +1.46260 q^{18} -5.32340 q^{19} +0.333816 q^{20} -1.00000 q^{21} -1.46260 q^{22} +4.92520 q^{23} -2.72161 q^{24} +0.751399 q^{25} +7.37883 q^{26} +1.00000 q^{27} -0.139194 q^{28} +5.04502 q^{29} +3.50761 q^{30} -7.57201 q^{31} -0.786003 q^{32} -1.00000 q^{33} -9.31444 q^{34} -2.39821 q^{35} +0.139194 q^{36} +4.24860 q^{37} -7.78600 q^{38} +5.04502 q^{39} -6.52699 q^{40} -0.646809 q^{41} -1.46260 q^{42} -10.5180 q^{43} -0.139194 q^{44} +2.39821 q^{45} +7.20359 q^{46} +0.526989 q^{47} -4.25901 q^{48} +1.00000 q^{49} +1.09899 q^{50} -6.36842 q^{51} +0.702237 q^{52} +3.72161 q^{53} +1.46260 q^{54} -2.39821 q^{55} +2.72161 q^{56} -5.32340 q^{57} +7.37883 q^{58} +7.97021 q^{59} +0.333816 q^{60} -2.00000 q^{61} -11.0748 q^{62} -1.00000 q^{63} +7.36842 q^{64} +12.0990 q^{65} -1.46260 q^{66} +8.76663 q^{67} -0.886447 q^{68} +4.92520 q^{69} -3.50761 q^{70} -11.4432 q^{71} -2.72161 q^{72} -13.0450 q^{73} +6.21400 q^{74} +0.751399 q^{75} -0.740987 q^{76} +1.00000 q^{77} +7.37883 q^{78} +11.4432 q^{79} -10.2140 q^{80} +1.00000 q^{81} -0.946021 q^{82} +13.1648 q^{83} -0.139194 q^{84} -15.2728 q^{85} -15.3836 q^{86} +5.04502 q^{87} +2.72161 q^{88} +11.8504 q^{89} +3.50761 q^{90} -5.04502 q^{91} +0.685559 q^{92} -7.57201 q^{93} +0.770774 q^{94} -12.7666 q^{95} -0.786003 q^{96} -1.87122 q^{97} +1.46260 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 + 3 * q^3 + 6 * q^4 + 4 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 11 q^{10} - 3 q^{11} + 6 q^{12} - 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} + 8 q^{17} + 2 q^{18} - 8 q^{19} - 3 q^{20} - 3 q^{21} - 2 q^{22} + 10 q^{23} + 3 q^{24} + 15 q^{25} - q^{26} + 3 q^{27} - 6 q^{28} - 4 q^{29} - 11 q^{30} - 2 q^{31} + 8 q^{32} - 3 q^{33} - 4 q^{34} - 4 q^{35} + 6 q^{36} - 13 q^{38} - 4 q^{39} - 18 q^{40} + 14 q^{41} - 2 q^{42} - 14 q^{43} - 6 q^{44} + 4 q^{45} + 28 q^{46} - 4 q^{48} + 3 q^{49} - 19 q^{50} + 8 q^{51} - 29 q^{52} + 2 q^{54} - 4 q^{55} - 3 q^{56} - 8 q^{57} - q^{58} - 3 q^{60} - 6 q^{61} - 38 q^{62} - 3 q^{63} - 5 q^{64} + 14 q^{65} - 2 q^{66} - 4 q^{67} + 42 q^{68} + 10 q^{69} + 11 q^{70} - 12 q^{71} + 3 q^{72} - 20 q^{73} + 29 q^{74} + 15 q^{75} - 11 q^{76} + 3 q^{77} - q^{78} + 12 q^{79} - 41 q^{80} + 3 q^{81} - 6 q^{82} + 6 q^{83} - 6 q^{84} - 6 q^{85} + 24 q^{86} - 4 q^{87} - 3 q^{88} + 26 q^{89} - 11 q^{90} + 4 q^{91} + 26 q^{92} - 2 q^{93} + 35 q^{94} - 8 q^{95} + 8 q^{96} - 4 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 3 * q^3 + 6 * q^4 + 4 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 - 11 * q^10 - 3 * q^11 + 6 * q^12 - 4 * q^13 - 2 * q^14 + 4 * q^15 - 4 * q^16 + 8 * q^17 + 2 * q^18 - 8 * q^19 - 3 * q^20 - 3 * q^21 - 2 * q^22 + 10 * q^23 + 3 * q^24 + 15 * q^25 - q^26 + 3 * q^27 - 6 * q^28 - 4 * q^29 - 11 * q^30 - 2 * q^31 + 8 * q^32 - 3 * q^33 - 4 * q^34 - 4 * q^35 + 6 * q^36 - 13 * q^38 - 4 * q^39 - 18 * q^40 + 14 * q^41 - 2 * q^42 - 14 * q^43 - 6 * q^44 + 4 * q^45 + 28 * q^46 - 4 * q^48 + 3 * q^49 - 19 * q^50 + 8 * q^51 - 29 * q^52 + 2 * q^54 - 4 * q^55 - 3 * q^56 - 8 * q^57 - q^58 - 3 * q^60 - 6 * q^61 - 38 * q^62 - 3 * q^63 - 5 * q^64 + 14 * q^65 - 2 * q^66 - 4 * q^67 + 42 * q^68 + 10 * q^69 + 11 * q^70 - 12 * q^71 + 3 * q^72 - 20 * q^73 + 29 * q^74 + 15 * q^75 - 11 * q^76 + 3 * q^77 - q^78 + 12 * q^79 - 41 * q^80 + 3 * q^81 - 6 * q^82 + 6 * q^83 - 6 * q^84 - 6 * q^85 + 24 * q^86 - 4 * q^87 - 3 * q^88 + 26 * q^89 - 11 * q^90 + 4 * q^91 + 26 * q^92 - 2 * q^93 + 35 * q^94 - 8 * q^95 + 8 * q^96 - 4 * q^97 + 2 * q^98 - 3 * 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 $

$2$	1.46260	1.03421	0.517107	−	0.855921i	$-0.327009\pi$
$2$	1.46260	1.03421	0.517107	+	0.855921i	$0.327009\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.139194	0.0695971
$5$	2.39821	1.07251	0.536255	−	0.844056i	$-0.319838\pi$
$5$	2.39821	1.07251	0.536255	+	0.844056i	$0.319838\pi$
$6$	1.46260	0.597103
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.72161	−0.962235
$9$	1.00000	0.333333
$10$	3.50761	1.10921
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.139194	0.0401819
$13$	5.04502	1.39924	0.699618	−	0.714517i	$-0.253354\pi$
$13$	5.04502	1.39924	0.699618	+	0.714517i	$0.253354\pi$
$14$	−1.46260	−0.390896
$15$	2.39821	0.619214
$16$	−4.25901	−1.06475
$17$	−6.36842	−1.54457	−0.772284	−	0.635277i	$-0.780886\pi$
$17$	−6.36842	−1.54457	−0.772284	+	0.635277i	$0.780886\pi$
$18$	1.46260	0.344738
$19$	−5.32340	−1.22127	−0.610636	−	0.791911i	$-0.709086\pi$
$19$	−5.32340	−1.22127	−0.610636	+	0.791911i	$0.709086\pi$
$20$	0.333816	0.0746436
$21$	−1.00000	−0.218218
$22$	−1.46260	−0.311827
$23$	4.92520	1.02697	0.513487	−	0.858097i	$-0.328353\pi$
$23$	4.92520	1.02697	0.513487	+	0.858097i	$0.328353\pi$
$24$	−2.72161	−0.555547
$25$	0.751399	0.150280
$26$	7.37883	1.44711
$27$	1.00000	0.192450
$28$	−0.139194	−0.0263052
$29$	5.04502	0.936836	0.468418	−	0.883507i	$-0.344824\pi$
$29$	5.04502	0.936836	0.468418	+	0.883507i	$0.344824\pi$
$30$	3.50761	0.640400
$31$	−7.57201	−1.35997	−0.679986	−	0.733225i	$-0.738014\pi$
$31$	−7.57201	−1.35997	−0.679986	+	0.733225i	$0.738014\pi$
$32$	−0.786003	−0.138947
$33$	−1.00000	−0.174078
$34$	−9.31444	−1.59741
$35$	−2.39821	−0.405371
$36$	0.139194	0.0231990
$37$	4.24860	0.698466	0.349233	−	0.937036i	$-0.386442\pi$
$37$	4.24860	0.698466	0.349233	+	0.937036i	$0.386442\pi$
$38$	−7.78600	−1.26306
$39$	5.04502	0.807849
$40$	−6.52699	−1.03201
$41$	−0.646809	−0.101015	−0.0505073	−	0.998724i	$-0.516084\pi$
$41$	−0.646809	−0.101015	−0.0505073	+	0.998724i	$0.516084\pi$
$42$	−1.46260	−0.225684
$43$	−10.5180	−1.60398	−0.801992	−	0.597335i	$-0.796226\pi$
$43$	−10.5180	−1.60398	−0.801992	+	0.597335i	$0.796226\pi$
$44$	−0.139194	−0.0209843
$45$	2.39821	0.357504
$46$	7.20359	1.06211
$47$	0.526989	0.0768693	0.0384347	−	0.999261i	$-0.487763\pi$
$47$	0.526989	0.0768693	0.0384347	+	0.999261i	$0.487763\pi$
$48$	−4.25901	−0.614736
$49$	1.00000	0.142857
$50$	1.09899	0.155421
$51$	−6.36842	−0.891757
$52$	0.702237	0.0973827
$53$	3.72161	0.511203	0.255601	−	0.966782i	$-0.417727\pi$
$53$	3.72161	0.511203	0.255601	+	0.966782i	$0.417727\pi$
$54$	1.46260	0.199034
$55$	−2.39821	−0.323374
$56$	2.72161	0.363691
$57$	−5.32340	−0.705102
$58$	7.37883	0.968888
$59$	7.97021	1.03763	0.518817	−	0.854886i	$-0.326373\pi$
$59$	7.97021	1.03763	0.518817	+	0.854886i	$0.326373\pi$
$60$	0.333816	0.0430955
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−11.0748	−1.40650
$63$	−1.00000	−0.125988
$64$	7.36842	0.921053
$65$	12.0990	1.50070
$66$	−1.46260	−0.180033
$67$	8.76663	1.07101	0.535507	−	0.844531i	$-0.320121\pi$
$67$	8.76663	1.07101	0.535507	+	0.844531i	$0.320121\pi$
$68$	−0.886447	−0.107497
$69$	4.92520	0.592924
$70$	−3.50761	−0.419240
$71$	−11.4432	−1.35806	−0.679030	−	0.734110i	$-0.737600\pi$
$71$	−11.4432	−1.35806	−0.679030	+	0.734110i	$0.737600\pi$
$72$	−2.72161	−0.320745
$73$	−13.0450	−1.52680	−0.763402	−	0.645924i	$-0.776472\pi$
$73$	−13.0450	−1.52680	−0.763402	+	0.645924i	$0.776472\pi$
$74$	6.21400	0.722363
$75$	0.751399	0.0867641
$76$	−0.740987	−0.0849970
$77$	1.00000	0.113961
$78$	7.37883	0.835488
$79$	11.4432	1.28746	0.643732	−	0.765251i	$-0.277385\pi$
$79$	11.4432	1.28746	0.643732	+	0.765251i	$0.277385\pi$
$80$	−10.2140	−1.14196
$81$	1.00000	0.111111
$82$	−0.946021	−0.104471
$83$	13.1648	1.44503	0.722514	−	0.691356i	$-0.242986\pi$
$83$	13.1648	1.44503	0.722514	+	0.691356i	$0.242986\pi$
$84$	−0.139194	−0.0151873
$85$	−15.2728	−1.65657
$86$	−15.3836	−1.65886
$87$	5.04502	0.540882
$88$	2.72161	0.290125
$89$	11.8504	1.25614	0.628070	−	0.778157i	$-0.283845\pi$
$89$	11.8504	1.25614	0.628070	+	0.778157i	$0.283845\pi$
$90$	3.50761	0.369735
$91$	−5.04502	−0.528861
$92$	0.685559	0.0714744
$93$	−7.57201	−0.785180
$94$	0.770774	0.0794993
$95$	−12.7666	−1.30983
$96$	−0.786003	−0.0802211
$97$	−1.87122	−0.189993	−0.0949967	−	0.995478i	$-0.530284\pi$
$97$	−1.87122	−0.189993	−0.0949967	+	0.995478i	$0.530284\pi$
$98$	1.46260	0.147745
$99$	−1.00000	−0.100504
$100$	0.104590	0.0104590
$101$	4.51803	0.449560	0.224780	−	0.974409i	$-0.427834\pi$
$101$	4.51803	0.449560	0.224780	+	0.974409i	$0.427834\pi$
$102$	−9.31444	−0.922267
$103$	−10.6468	−1.04906	−0.524531	−	0.851392i	$-0.675759\pi$
$103$	−10.6468	−1.04906	−0.524531	+	0.851392i	$0.675759\pi$
$104$	−13.7306	−1.34639
$105$	−2.39821	−0.234041
$106$	5.44322	0.528693
$107$	15.9702	1.54390	0.771949	−	0.635684i	$-0.219282\pi$
$107$	15.9702	1.54390	0.771949	+	0.635684i	$0.219282\pi$
$108$	0.139194	0.0133940
$109$	12.7756	1.22368	0.611840	−	0.790982i	$-0.290430\pi$
$109$	12.7756	1.22368	0.611840	+	0.790982i	$0.290430\pi$
$110$	−3.50761	−0.334438
$111$	4.24860	0.403259
$112$	4.25901	0.402439
$113$	18.7368	1.76261	0.881307	−	0.472544i	$-0.156664\pi$
$113$	18.7368	1.76261	0.881307	+	0.472544i	$0.156664\pi$
$114$	−7.78600	−0.729226
$115$	11.8116	1.10144
$116$	0.702237	0.0652010
$117$	5.04502	0.466412
$118$	11.6572	1.07313
$119$	6.36842	0.583792
$120$	−6.52699	−0.595830
$121$	1.00000	0.0909091
$122$	−2.92520	−0.264835
$123$	−0.646809	−0.0583208
$124$	−1.05398	−0.0946501
$125$	−10.1890	−0.911334
$126$	−1.46260	−0.130299
$127$	−2.27839	−0.202174	−0.101087	−	0.994878i	$-0.532232\pi$
$127$	−2.27839	−0.202174	−0.101087	+	0.994878i	$0.532232\pi$
$128$	12.3490	1.09151
$129$	−10.5180	−0.926061
$130$	17.6960	1.55204
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−0.139194	−0.0121153
$133$	5.32340	0.461598
$134$	12.8221	1.10766
$135$	2.39821	0.206405
$136$	17.3324	1.48624
$137$	−4.77559	−0.408006	−0.204003	−	0.978970i	$-0.565395\pi$
$137$	−4.77559	−0.408006	−0.204003	+	0.978970i	$0.565395\pi$
$138$	7.20359	0.613210
$139$	−15.4432	−1.30988	−0.654939	−	0.755682i	$-0.727306\pi$
$139$	−15.4432	−1.30988	−0.654939	+	0.755682i	$0.727306\pi$
$140$	−0.333816	−0.0282126
$141$	0.526989	0.0443805
$142$	−16.7368	−1.40452
$143$	−5.04502	−0.421885
$144$	−4.25901	−0.354918
$145$	12.0990	1.00477
$146$	−19.0796	−1.57904
$147$	1.00000	0.0824786
$148$	0.591380	0.0486112
$149$	−9.84143	−0.806241	−0.403121	−	0.915147i	$-0.632075\pi$
$149$	−9.84143	−0.806241	−0.403121	+	0.915147i	$0.632075\pi$
$150$	1.09899	0.0897326
$151$	−4.12878	−0.335996	−0.167998	−	0.985787i	$-0.553730\pi$
$151$	−4.12878	−0.335996	−0.167998	+	0.985787i	$0.553730\pi$
$152$	14.4882	1.17515
$153$	−6.36842	−0.514856
$154$	1.46260	0.117860
$155$	−18.1592	−1.45859
$156$	0.702237	0.0562239
$157$	−0.946021	−0.0755007	−0.0377504	−	0.999287i	$-0.512019\pi$
$157$	−0.946021	−0.0755007	−0.0377504	+	0.999287i	$0.512019\pi$
$158$	16.7368	1.33151
$159$	3.72161	0.295143
$160$	−1.88500	−0.149022
$161$	−4.92520	−0.388160
$162$	1.46260	0.114913
$163$	8.76663	0.686655	0.343328	−	0.939216i	$-0.388446\pi$
$163$	8.76663	0.686655	0.343328	+	0.939216i	$0.388446\pi$
$164$	−0.0900320	−0.00703032
$165$	−2.39821	−0.186700
$166$	19.2549	1.49447
$167$	−24.3684	−1.88568	−0.942842	−	0.333239i	$-0.891858\pi$
$167$	−24.3684	−1.88568	−0.942842	+	0.333239i	$0.891858\pi$
$168$	2.72161	0.209977
$169$	12.4522	0.957860
$170$	−22.3380	−1.71324
$171$	−5.32340	−0.407091
$172$	−1.46405	−0.111633
$173$	12.3476	0.938770	0.469385	−	0.882994i	$-0.344476\pi$
$173$	12.3476	0.938770	0.469385	+	0.882994i	$0.344476\pi$
$174$	7.37883	0.559388
$175$	−0.751399	−0.0568004
$176$	4.25901	0.321035
$177$	7.97021	0.599078
$178$	17.3324	1.29912
$179$	−5.59283	−0.418028	−0.209014	−	0.977913i	$-0.567025\pi$
$179$	−5.59283	−0.418028	−0.209014	+	0.977913i	$0.567025\pi$
$180$	0.333816	0.0248812
$181$	13.5720	1.00880	0.504400	−	0.863470i	$-0.331714\pi$
$181$	13.5720	1.00880	0.504400	+	0.863470i	$0.331714\pi$
$182$	−7.37883	−0.546955
$183$	−2.00000	−0.147844
$184$	−13.4045	−0.988191
$185$	10.1890	0.749112
$186$	−11.0748	−0.812044
$187$	6.36842	0.465705
$188$	0.0733538	0.00534988
$189$	−1.00000	−0.0727393
$190$	−18.6724	−1.35464
$191$	−9.42240	−0.681781	−0.340890	−	0.940103i	$-0.610729\pi$
$191$	−9.42240	−0.681781	−0.340890	+	0.940103i	$0.610729\pi$
$192$	7.36842	0.531770
$193$	−10.1288	−0.729086	−0.364543	−	0.931187i	$-0.618775\pi$
$193$	−10.1288	−0.729086	−0.364543	+	0.931187i	$0.618775\pi$
$194$	−2.73684	−0.196494
$195$	12.0990	0.866427
$196$	0.139194	0.00994244
$197$	−2.25756	−0.160845	−0.0804224	−	0.996761i	$-0.525627\pi$
$197$	−2.25756	−0.160845	−0.0804224	+	0.996761i	$0.525627\pi$
$198$	−1.46260	−0.103942
$199$	3.07480	0.217967	0.108984	−	0.994044i	$-0.465240\pi$
$199$	3.07480	0.217967	0.108984	+	0.994044i	$0.465240\pi$
$200$	−2.04502	−0.144604
$201$	8.76663	0.618350
$202$	6.60806	0.464941
$203$	−5.04502	−0.354091
$204$	−0.886447	−0.0620637
$205$	−1.55118	−0.108339
$206$	−15.5720	−1.08495
$207$	4.92520	0.342325
$208$	−21.4868	−1.48984
$209$	5.32340	0.368228
$210$	−3.50761	−0.242048
$211$	−14.6468	−1.00833	−0.504164	−	0.863608i	$-0.668199\pi$
$211$	−14.6468	−1.00833	−0.504164	+	0.863608i	$0.668199\pi$
$212$	0.518027	0.0355782
$213$	−11.4432	−0.784077
$214$	23.3580	1.59672
$215$	−25.2244	−1.72029
$216$	−2.72161	−0.185182
$217$	7.57201	0.514021
$218$	18.6856	1.26555
$219$	−13.0450	−0.881500
$220$	−0.333816	−0.0225059
$221$	−32.1288	−2.16122
$222$	6.21400	0.417056
$223$	−1.90997	−0.127901	−0.0639505	−	0.997953i	$-0.520370\pi$
$223$	−1.90997	−0.127901	−0.0639505	+	0.997953i	$0.520370\pi$
$224$	0.786003	0.0525170
$225$	0.751399	0.0500933
$226$	27.4045	1.82292
$227$	−3.20359	−0.212629	−0.106315	−	0.994333i	$-0.533905\pi$
$227$	−3.20359	−0.212629	−0.106315	+	0.994333i	$0.533905\pi$
$228$	−0.740987	−0.0490730
$229$	−18.3088	−1.20988	−0.604941	−	0.796270i	$-0.706803\pi$
$229$	−18.3088	−1.20988	−0.604941	+	0.796270i	$0.706803\pi$
$230$	17.2757	1.13913
$231$	1.00000	0.0657952
$232$	−13.7306	−0.901456
$233$	−16.5872	−1.08667	−0.543333	−	0.839517i	$-0.682838\pi$
$233$	−16.5872	−1.08667	−0.543333	+	0.839517i	$0.682838\pi$
$234$	7.37883	0.482369
$235$	1.26383	0.0824432
$236$	1.10941	0.0722162
$237$	11.4432	0.743317
$238$	9.31444	0.603766
$239$	2.91623	0.188635	0.0943177	−	0.995542i	$-0.469933\pi$
$239$	2.91623	0.188635	0.0943177	+	0.995542i	$0.469933\pi$
$240$	−10.2140	−0.659311
$241$	−6.09899	−0.392871	−0.196435	−	0.980517i	$-0.562937\pi$
$241$	−6.09899	−0.392871	−0.196435	+	0.980517i	$0.562937\pi$
$242$	1.46260	0.0940194
$243$	1.00000	0.0641500
$244$	−0.278388	−0.0178220
$245$	2.39821	0.153216
$246$	−0.946021	−0.0603161
$247$	−26.8567	−1.70885
$248$	20.6081	1.30861
$249$	13.1648	0.834288
$250$	−14.9025	−0.942514
$251$	1.62262	0.102419	0.0512093	−	0.998688i	$-0.483692\pi$
$251$	1.62262	0.102419	0.0512093	+	0.998688i	$0.483692\pi$
$252$	−0.139194	−0.00876841
$253$	−4.92520	−0.309644
$254$	−3.33237	−0.209091
$255$	−15.2728	−0.956419
$256$	3.32485	0.207803
$257$	−6.89541	−0.430124	−0.215062	−	0.976600i	$-0.568995\pi$
$257$	−6.89541	−0.430124	−0.215062	+	0.976600i	$0.568995\pi$
$258$	−15.3836	−0.957744
$259$	−4.24860	−0.263995
$260$	1.68411	0.104444
$261$	5.04502	0.312279
$262$	5.85039	0.361439
$263$	5.08377	0.313478	0.156739	−	0.987640i	$-0.449902\pi$
$263$	5.08377	0.313478	0.156739	+	0.987640i	$0.449902\pi$
$264$	2.72161	0.167504
$265$	8.92520	0.548270
$266$	7.78600	0.477390
$267$	11.8504	0.725232
$268$	1.22026	0.0745394
$269$	−0.886447	−0.0540476	−0.0270238	−	0.999635i	$-0.508603\pi$
$269$	−0.886447	−0.0540476	−0.0270238	+	0.999635i	$0.508603\pi$
$270$	3.50761	0.213467
$271$	−25.3234	−1.53829	−0.769144	−	0.639076i	$-0.779317\pi$
$271$	−25.3234	−1.53829	−0.769144	+	0.639076i	$0.779317\pi$
$272$	27.1232	1.64458
$273$	−5.04502	−0.305338
$274$	−6.98477	−0.421965
$275$	−0.751399	−0.0453111
$276$	0.685559	0.0412658
$277$	−24.8269	−1.49170	−0.745851	−	0.666113i	$-0.767957\pi$
$277$	−24.8269	−1.49170	−0.745851	+	0.666113i	$0.767957\pi$
$278$	−22.5872	−1.35469
$279$	−7.57201	−0.453324
$280$	6.52699	0.390062
$281$	1.90101	0.113404	0.0567022	−	0.998391i	$-0.481941\pi$
$281$	1.90101	0.113404	0.0567022	+	0.998391i	$0.481941\pi$
$282$	0.770774	0.0458989
$283$	−22.3178	−1.32666	−0.663328	−	0.748329i	$-0.730857\pi$
$283$	−22.3178	−1.32666	−0.663328	+	0.748329i	$0.730857\pi$
$284$	−1.59283	−0.0945171
$285$	−12.7666	−0.756230
$286$	−7.37883	−0.436320
$287$	0.646809	0.0381799
$288$	−0.786003	−0.0463157
$289$	23.5568	1.38569
$290$	17.6960	1.03914
$291$	−1.87122	−0.109693
$292$	−1.81579	−0.106261
$293$	12.0900	0.706307	0.353154	−	0.935565i	$-0.385109\pi$
$293$	12.0900	0.706307	0.353154	+	0.935565i	$0.385109\pi$
$294$	1.46260	0.0853005
$295$	19.1142	1.11287
$296$	−11.5630	−0.672088
$297$	−1.00000	−0.0580259
$298$	−14.3941	−0.833826
$299$	24.8477	1.43698
$300$	0.104590	0.00603853
$301$	10.5180	0.606249
$302$	−6.03875	−0.347491
$303$	4.51803	0.259554
$304$	22.6724	1.30035
$305$	−4.79641	−0.274642
$306$	−9.31444	−0.532471
$307$	−13.5928	−0.775784	−0.387892	−	0.921705i	$-0.626797\pi$
$307$	−13.5928	−0.775784	−0.387892	+	0.921705i	$0.626797\pi$
$308$	0.139194	0.00793132
$309$	−10.6468	−0.605676
$310$	−26.5597	−1.50849
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−13.7306	−0.777341
$313$	14.9252	0.843622	0.421811	−	0.906684i	$-0.361395\pi$
$313$	14.9252	0.843622	0.421811	+	0.906684i	$0.361395\pi$
$314$	−1.38365	−0.0780838
$315$	−2.39821	−0.135124
$316$	1.59283	0.0896037
$317$	3.97918	0.223493	0.111746	−	0.993737i	$-0.464356\pi$
$317$	3.97918	0.223493	0.111746	+	0.993737i	$0.464356\pi$
$318$	5.44322	0.305241
$319$	−5.04502	−0.282467
$320$	17.6710	0.987839
$321$	15.9702	0.891370
$322$	−7.20359	−0.401440
$323$	33.9017	1.88634
$324$	0.139194	0.00773301
$325$	3.79082	0.210277
$326$	12.8221	0.710148
$327$	12.7756	0.706492
$328$	1.76036	0.0971997
$329$	−0.526989	−0.0290539
$330$	−3.50761	−0.193088
$331$	23.4432	1.28856	0.644278	−	0.764791i	$-0.277158\pi$
$331$	23.4432	1.28856	0.644278	+	0.764791i	$0.277158\pi$
$332$	1.83247	0.100570
$333$	4.24860	0.232822
$334$	−35.6412	−1.95020
$335$	21.0242	1.14867
$336$	4.25901	0.232348
$337$	11.1648	0.608187	0.304094	−	0.952642i	$-0.401646\pi$
$337$	11.1648	0.608187	0.304094	+	0.952642i	$0.401646\pi$
$338$	18.2125	0.990632
$339$	18.7368	1.01765
$340$	−2.12588	−0.115292
$341$	7.57201	0.410047
$342$	−7.78600	−0.421019
$343$	−1.00000	−0.0539949
$344$	28.6260	1.54341
$345$	11.8116	0.635918
$346$	18.0596	0.970889
$347$	−22.5872	−1.21255	−0.606273	−	0.795256i	$-0.707336\pi$
$347$	−22.5872	−1.21255	−0.606273	+	0.795256i	$0.707336\pi$
$348$	0.702237	0.0376438
$349$	27.9315	1.49514	0.747568	−	0.664185i	$-0.231221\pi$
$349$	27.9315	1.49514	0.747568	+	0.664185i	$0.231221\pi$
$350$	−1.09899	−0.0587437
$351$	5.04502	0.269283
$352$	0.786003	0.0418941
$353$	−16.5478	−0.880751	−0.440376	−	0.897814i	$-0.645155\pi$
$353$	−16.5478	−0.880751	−0.440376	+	0.897814i	$0.645155\pi$
$354$	11.6572	0.619574
$355$	−27.4432	−1.45654
$356$	1.64951	0.0874236
$357$	6.36842	0.337053
$358$	−8.18006	−0.432330
$359$	22.0305	1.16272	0.581362	−	0.813645i	$-0.302520\pi$
$359$	22.0305	1.16272	0.581362	+	0.813645i	$0.302520\pi$
$360$	−6.52699	−0.344003
$361$	9.33863	0.491507
$362$	19.8504	1.04331
$363$	1.00000	0.0524864
$364$	−0.702237	−0.0368072
$365$	−31.2847	−1.63751
$366$	−2.92520	−0.152902
$367$	−19.3836	−1.01182	−0.505909	−	0.862587i	$-0.668843\pi$
$367$	−19.3836	−1.01182	−0.505909	+	0.862587i	$0.668843\pi$
$368$	−20.9765	−1.09347
$369$	−0.646809	−0.0336715
$370$	14.9025	0.774742
$371$	−3.72161	−0.193216
$372$	−1.05398	−0.0546463
$373$	29.2549	1.51476	0.757380	−	0.652975i	$-0.226479\pi$
$373$	29.2549	1.51476	0.757380	+	0.652975i	$0.226479\pi$
$374$	9.31444	0.481638
$375$	−10.1890	−0.526159
$376$	−1.43426	−0.0739663
$377$	25.4522	1.31085
$378$	−1.46260	−0.0752279
$379$	12.5270	0.643468	0.321734	−	0.946830i	$-0.395734\pi$
$379$	12.5270	0.643468	0.321734	+	0.946830i	$0.395734\pi$
$380$	−1.77704	−0.0911602
$381$	−2.27839	−0.116725
$382$	−13.7812	−0.705107
$383$	−17.5928	−0.898952	−0.449476	−	0.893293i	$-0.648389\pi$
$383$	−17.5928	−0.898952	−0.449476	+	0.893293i	$0.648389\pi$
$384$	12.3490	0.630185
$385$	2.39821	0.122224
$386$	−14.8143	−0.754030
$387$	−10.5180	−0.534661
$388$	−0.260463	−0.0132230
$389$	20.0900	1.01861	0.509303	−	0.860588i	$-0.329903\pi$
$389$	20.0900	1.01861	0.509303	+	0.860588i	$0.329903\pi$
$390$	17.6960	0.896070
$391$	−31.3657	−1.58623
$392$	−2.72161	−0.137462
$393$	4.00000	0.201773
$394$	−3.30191	−0.166348
$395$	27.4432	1.38082
$396$	−0.139194	−0.00699477
$397$	−35.1053	−1.76188	−0.880941	−	0.473226i	$-0.843090\pi$
$397$	−35.1053	−1.76188	−0.880941	+	0.473226i	$0.843090\pi$
$398$	4.49720	0.225424
$399$	5.32340	0.266504
$400$	−3.20022	−0.160011
$401$	9.57201	0.478003	0.239002	−	0.971019i	$-0.423180\pi$
$401$	9.57201	0.478003	0.239002	+	0.971019i	$0.423180\pi$
$402$	12.8221	0.639506
$403$	−38.2009	−1.90292
$404$	0.628883	0.0312881
$405$	2.39821	0.119168
$406$	−7.37883	−0.366205
$407$	−4.24860	−0.210595
$408$	17.3324	0.858080
$409$	38.1801	1.88788	0.943941	−	0.330113i	$-0.107087\pi$
$409$	38.1801	1.88788	0.943941	+	0.330113i	$0.107087\pi$
$410$	−2.26875	−0.112046
$411$	−4.77559	−0.235563
$412$	−1.48197	−0.0730116
$413$	−7.97021	−0.392189
$414$	7.20359	0.354037
$415$	31.5720	1.54981
$416$	−3.96540	−0.194420
$417$	−15.4432	−0.756258
$418$	7.78600	0.380826
$419$	7.17380	0.350463	0.175231	−	0.984527i	$-0.443933\pi$
$419$	7.17380	0.350463	0.175231	+	0.984527i	$0.443933\pi$
$420$	−0.333816	−0.0162886
$421$	15.1530	0.738511	0.369255	−	0.929328i	$-0.379613\pi$
$421$	15.1530	0.738511	0.369255	+	0.929328i	$0.379613\pi$
$422$	−21.4224	−1.04283
$423$	0.526989	0.0256231
$424$	−10.1288	−0.491897
$425$	−4.78522	−0.232117
$426$	−16.7368	−0.810903
$427$	2.00000	0.0967868
$428$	2.22296	0.107451
$429$	−5.04502	−0.243576
$430$	−36.8932	−1.77915
$431$	5.56304	0.267962	0.133981	−	0.990984i	$-0.457224\pi$
$431$	5.56304	0.267962	0.133981	+	0.990984i	$0.457224\pi$
$432$	−4.25901	−0.204912
$433$	−25.6412	−1.23224	−0.616119	−	0.787653i	$-0.711296\pi$
$433$	−25.6412	−1.23224	−0.616119	+	0.787653i	$0.711296\pi$
$434$	11.0748	0.531608
$435$	12.0990	0.580102
$436$	1.77829	0.0851645
$437$	−26.2188	−1.25422
$438$	−19.0796	−0.911659
$439$	23.6710	1.12976	0.564878	−	0.825175i	$-0.308923\pi$
$439$	23.6710	1.12976	0.564878	+	0.825175i	$0.308923\pi$
$440$	6.52699	0.311162
$441$	1.00000	0.0476190
$442$	−46.9915	−2.23516
$443$	−18.0305	−0.856653	−0.428326	−	0.903624i	$-0.640897\pi$
$443$	−18.0305	−0.856653	−0.428326	+	0.903624i	$0.640897\pi$
$444$	0.591380	0.0280657
$445$	28.4197	1.34722
$446$	−2.79352	−0.132277
$447$	−9.84143	−0.465484
$448$	−7.36842	−0.348125
$449$	−34.9765	−1.65064	−0.825321	−	0.564664i	$-0.809006\pi$
$449$	−34.9765	−1.65064	−0.825321	+	0.564664i	$0.809006\pi$
$450$	1.09899	0.0518071
$451$	0.646809	0.0304570
$452$	2.60806	0.122673
$453$	−4.12878	−0.193987
$454$	−4.68556	−0.219904
$455$	−12.0990	−0.567210
$456$	14.4882	0.678474
$457$	4.53595	0.212183	0.106091	−	0.994356i	$-0.466166\pi$
$457$	4.53595	0.212183	0.106091	+	0.994356i	$0.466166\pi$
$458$	−26.7785	−1.25128
$459$	−6.36842	−0.297252
$460$	1.64411	0.0766571
$461$	−2.79641	−0.130242	−0.0651210	−	0.997877i	$-0.520743\pi$
$461$	−2.79641	−0.130242	−0.0651210	+	0.997877i	$0.520743\pi$
$462$	1.46260	0.0680462
$463$	38.3595	1.78272	0.891358	−	0.453301i	$-0.149754\pi$
$463$	38.3595	1.78272	0.891358	+	0.453301i	$0.149754\pi$
$464$	−21.4868	−0.997499
$465$	−18.1592	−0.842115
$466$	−24.2605	−1.12384
$467$	−20.4674	−0.947119	−0.473560	−	0.880762i	$-0.657031\pi$
$467$	−20.4674	−0.947119	−0.473560	+	0.880762i	$0.657031\pi$
$468$	0.702237	0.0324609
$469$	−8.76663	−0.404805
$470$	1.84848	0.0852638
$471$	−0.946021	−0.0435904
$472$	−21.6918	−0.998447
$473$	10.5180	0.483619
$474$	16.7368	0.768749
$475$	−4.00000	−0.183533
$476$	0.886447	0.0406302
$477$	3.72161	0.170401
$478$	4.26528	0.195089
$479$	−11.6137	−0.530641	−0.265321	−	0.964160i	$-0.585478\pi$
$479$	−11.6137	−0.530641	−0.265321	+	0.964160i	$0.585478\pi$
$480$	−1.88500	−0.0860380
$481$	21.4343	0.977318
$482$	−8.92038	−0.406312
$483$	−4.92520	−0.224104
$484$	0.139194	0.00632701
$485$	−4.48757	−0.203770
$486$	1.46260	0.0663448
$487$	32.4793	1.47178	0.735888	−	0.677103i	$-0.236765\pi$
$487$	32.4793	1.47178	0.735888	+	0.677103i	$0.236765\pi$
$488$	5.44322	0.246403
$489$	8.76663	0.396441
$490$	3.50761	0.158458
$491$	26.6766	1.20390	0.601949	−	0.798535i	$-0.294391\pi$
$491$	26.6766	1.20390	0.601949	+	0.798535i	$0.294391\pi$
$492$	−0.0900320	−0.00405895
$493$	−32.1288	−1.44701
$494$	−39.2805	−1.76731
$495$	−2.39821	−0.107791
$496$	32.2493	1.44804
$497$	11.4432	0.513299
$498$	19.2549	0.862831
$499$	−41.2459	−1.84642	−0.923210	−	0.384296i	$-0.874444\pi$
$499$	−41.2459	−1.84642	−0.923210	+	0.384296i	$0.874444\pi$
$500$	−1.41825	−0.0634262
$501$	−24.3684	−1.08870
$502$	2.37324	0.105923
$503$	30.5180	1.36073	0.680366	−	0.732873i	$-0.261821\pi$
$503$	30.5180	1.36073	0.680366	+	0.732873i	$0.261821\pi$
$504$	2.72161	0.121230
$505$	10.8352	0.482159
$506$	−7.20359	−0.320238
$507$	12.4522	0.553021
$508$	−0.317138	−0.0140707
$509$	−18.9944	−0.841912	−0.420956	−	0.907081i	$-0.638305\pi$
$509$	−18.9944	−0.841912	−0.420956	+	0.907081i	$0.638305\pi$
$510$	−22.3380	−0.989142
$511$	13.0450	0.577078
$512$	−19.8352	−0.876599
$513$	−5.32340	−0.235034
$514$	−10.0852	−0.444840
$515$	−25.5333	−1.12513
$516$	−1.46405	−0.0644511
$517$	−0.526989	−0.0231770
$518$	−6.21400	−0.273027
$519$	12.3476	0.541999
$520$	−32.9288	−1.44402
$521$	25.2430	1.10592	0.552958	−	0.833209i	$-0.313499\pi$
$521$	25.2430	1.10592	0.552958	+	0.833209i	$0.313499\pi$
$522$	7.37883	0.322963
$523$	−2.93416	−0.128302	−0.0641509	−	0.997940i	$-0.520434\pi$
$523$	−2.93416	−0.128302	−0.0641509	+	0.997940i	$0.520434\pi$
$524$	0.556777	0.0243229
$525$	−0.751399	−0.0327937
$526$	7.43551	0.324204
$527$	48.2217	2.10057
$528$	4.25901	0.185350
$529$	1.25756	0.0546767
$530$	13.0540	0.567029
$531$	7.97021	0.345878
$532$	0.740987	0.0321258
$533$	−3.26316	−0.141343
$534$	17.3324	0.750045
$535$	38.2999	1.65585
$536$	−23.8594	−1.03057
$537$	−5.59283	−0.241348
$538$	−1.29652	−0.0558968
$539$	−1.00000	−0.0430730
$540$	0.333816	0.0143652
$541$	8.90437	0.382829	0.191414	−	0.981509i	$-0.438693\pi$
$541$	8.90437	0.382829	0.191414	+	0.981509i	$0.438693\pi$
$542$	−37.0380	−1.59092
$543$	13.5720	0.582430
$544$	5.00560	0.214613
$545$	30.6385	1.31241
$546$	−7.37883	−0.315785
$547$	−29.4737	−1.26020	−0.630102	−	0.776513i	$-0.716987\pi$
$547$	−29.4737	−1.26020	−0.630102	+	0.776513i	$0.716987\pi$
$548$	−0.664734	−0.0283960
$549$	−2.00000	−0.0853579
$550$	−1.09899	−0.0468613
$551$	−26.8567	−1.14413
$552$	−13.4045	−0.570532
$553$	−11.4432	−0.486615
$554$	−36.3117	−1.54274
$555$	10.1890	0.432500
$556$	−2.14961	−0.0911636
$557$	14.8954	0.631139	0.315569	−	0.948903i	$-0.397804\pi$
$557$	14.8954	0.631139	0.315569	+	0.948903i	$0.397804\pi$
$558$	−11.0748	−0.468834
$559$	−53.0636	−2.24435
$560$	10.2140	0.431620
$561$	6.36842	0.268875
$562$	2.78041	0.117284
$563$	−7.81164	−0.329222	−0.164611	−	0.986359i	$-0.552637\pi$
$563$	−7.81164	−0.329222	−0.164611	+	0.986359i	$0.552637\pi$
$564$	0.0733538	0.00308875
$565$	44.9348	1.89042
$566$	−32.6420	−1.37205
$567$	−1.00000	−0.0419961
$568$	31.1440	1.30677
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	−18.6724	−0.782103
$571$	25.5512	1.06928	0.534642	−	0.845079i	$-0.320447\pi$
$571$	25.5512	1.06928	0.534642	+	0.845079i	$0.320447\pi$
$572$	−0.702237	−0.0293620
$573$	−9.42240	−0.393626
$574$	0.946021	0.0394862
$575$	3.70079	0.154334
$576$	7.36842	0.307018
$577$	−29.5124	−1.22862	−0.614309	−	0.789065i	$-0.710565\pi$
$577$	−29.5124	−1.22862	−0.614309	+	0.789065i	$0.710565\pi$
$578$	34.4541	1.43310
$579$	−10.1288	−0.420938
$580$	1.68411	0.0699288
$581$	−13.1648	−0.546169
$582$	−2.73684	−0.113446
$583$	−3.72161	−0.154133
$584$	35.5035	1.46914
$585$	12.0990	0.500232
$586$	17.6829	0.730472
$587$	31.3955	1.29583	0.647916	−	0.761712i	$-0.275641\pi$
$587$	31.3955	1.29583	0.647916	+	0.761712i	$0.275641\pi$
$588$	0.139194	0.00574027
$589$	40.3088	1.66090
$590$	27.9564	1.15095
$591$	−2.25756	−0.0928638
$592$	−18.0948	−0.743694
$593$	−7.90997	−0.324823	−0.162412	−	0.986723i	$-0.551927\pi$
$593$	−7.90997	−0.324823	−0.162412	+	0.986723i	$0.551927\pi$
$594$	−1.46260	−0.0600111
$595$	15.2728	0.626123
$596$	−1.36987	−0.0561120
$597$	3.07480	0.125843
$598$	36.3422	1.48614
$599$	27.4432	1.12130	0.560650	−	0.828053i	$-0.310551\pi$
$599$	27.4432	1.12130	0.560650	+	0.828053i	$0.310551\pi$
$600$	−2.04502	−0.0834874
$601$	31.9910	1.30494	0.652471	−	0.757814i	$-0.273732\pi$
$601$	31.9910	1.30494	0.652471	+	0.757814i	$0.273732\pi$
$602$	15.3836	0.626991
$603$	8.76663	0.357005
$604$	−0.574702	−0.0233843
$605$	2.39821	0.0975010
$606$	6.60806	0.268434
$607$	7.41344	0.300902	0.150451	−	0.988617i	$-0.451927\pi$
$607$	7.41344	0.300902	0.150451	+	0.988617i	$0.451927\pi$
$608$	4.18421	0.169692
$609$	−5.04502	−0.204434
$610$	−7.01523	−0.284038
$611$	2.65867	0.107558
$612$	−0.886447	−0.0358325
$613$	−33.9917	−1.37291	−0.686456	−	0.727171i	$-0.740835\pi$
$613$	−33.9917	−1.37291	−0.686456	+	0.727171i	$0.740835\pi$
$614$	−19.8809	−0.802326
$615$	−1.55118	−0.0625497
$616$	−2.72161	−0.109657
$617$	−44.0305	−1.77260	−0.886300	−	0.463112i	$-0.846733\pi$
$617$	−44.0305	−1.77260	−0.886300	+	0.463112i	$0.846733\pi$
$618$	−15.5720	−0.626398
$619$	40.0096	1.60812	0.804061	−	0.594546i	$-0.202668\pi$
$619$	40.0096	1.60812	0.804061	+	0.594546i	$0.202668\pi$
$620$	−2.52766	−0.101513
$621$	4.92520	0.197641
$622$	11.7008	0.469159
$623$	−11.8504	−0.474776
$624$	−21.4868	−0.860160
$625$	−28.1924	−1.12770
$626$	21.8296	0.872485
$627$	5.32340	0.212596
$628$	−0.131681	−0.00525463
$629$	−27.0569	−1.07883
$630$	−3.50761	−0.139747
$631$	28.5568	1.13683	0.568414	−	0.822743i	$-0.307557\pi$
$631$	28.5568	1.13683	0.568414	+	0.822743i	$0.307557\pi$
$632$	−31.1440	−1.23884
$633$	−14.6468	−0.582158
$634$	5.81994	0.231139
$635$	−5.46405	−0.216834
$636$	0.518027	0.0205411
$637$	5.04502	0.199891
$638$	−7.37883	−0.292131
$639$	−11.4432	−0.452687
$640$	29.6156	1.17066
$641$	−31.1053	−1.22858	−0.614292	−	0.789079i	$-0.710558\pi$
$641$	−31.1053	−1.22858	−0.614292	+	0.789079i	$0.710558\pi$
$642$	23.3580	0.921867
$643$	−5.48197	−0.216188	−0.108094	−	0.994141i	$-0.534475\pi$
$643$	−5.48197	−0.216188	−0.108094	+	0.994141i	$0.534475\pi$
$644$	−0.685559	−0.0270148
$645$	−25.2244	−0.993210
$646$	49.5845	1.95088
$647$	−9.26383	−0.364199	−0.182099	−	0.983280i	$-0.558289\pi$
$647$	−9.26383	−0.364199	−0.182099	+	0.983280i	$0.558289\pi$
$648$	−2.72161	−0.106915
$649$	−7.97021	−0.312858
$650$	5.54445	0.217471
$651$	7.57201	0.296770
$652$	1.22026	0.0477892
$653$	−29.9821	−1.17329	−0.586645	−	0.809844i	$-0.699551\pi$
$653$	−29.9821	−1.17329	−0.586645	+	0.809844i	$0.699551\pi$
$654$	18.6856	0.730663
$655$	9.59283	0.374823
$656$	2.75477	0.107556
$657$	−13.0450	−0.508935
$658$	−0.770774	−0.0300479
$659$	23.9702	0.933747	0.466873	−	0.884324i	$-0.345380\pi$
$659$	23.9702	0.933747	0.466873	+	0.884324i	$0.345380\pi$
$660$	−0.333816	−0.0129938
$661$	−40.4585	−1.57365	−0.786826	−	0.617175i	$-0.788277\pi$
$661$	−40.4585	−1.57365	−0.786826	+	0.617175i	$0.788277\pi$
$662$	34.2880	1.33264
$663$	−32.1288	−1.24778
$664$	−35.8296	−1.39046
$665$	12.7666	0.495069
$666$	6.21400	0.240788
$667$	24.8477	0.962107
$668$	−3.39194	−0.131238
$669$	−1.90997	−0.0738436
$670$	30.7499	1.18797
$671$	2.00000	0.0772091
$672$	0.786003	0.0303207
$673$	21.8712	0.843073	0.421537	−	0.906811i	$-0.361491\pi$
$673$	21.8712	0.843073	0.421537	+	0.906811i	$0.361491\pi$
$674$	16.3297	0.628995
$675$	0.751399	0.0289214
$676$	1.73327	0.0666643
$677$	−1.26316	−0.0485472	−0.0242736	−	0.999705i	$-0.507727\pi$
$677$	−1.26316	−0.0485472	−0.0242736	+	0.999705i	$0.507727\pi$
$678$	27.4045	1.05246
$679$	1.87122	0.0718108
$680$	41.5666	1.59401
$681$	−3.20359	−0.122762
$682$	11.0748	0.424076
$683$	37.6441	1.44041	0.720206	−	0.693760i	$-0.244047\pi$
$683$	37.6441	1.44041	0.720206	+	0.693760i	$0.244047\pi$
$684$	−0.740987	−0.0283323
$685$	−11.4529	−0.437591
$686$	−1.46260	−0.0558423
$687$	−18.3088	−0.698526
$688$	44.7964	1.70785
$689$	18.7756	0.715293
$690$	17.2757	0.657674
$691$	14.3892	0.547393	0.273696	−	0.961816i	$-0.411754\pi$
$691$	14.3892	0.547393	0.273696	+	0.961816i	$0.411754\pi$
$692$	1.71871	0.0653357
$693$	1.00000	0.0379869
$694$	−33.0361	−1.25403
$695$	−37.0361	−1.40486
$696$	−13.7306	−0.520456
$697$	4.11915	0.156024
$698$	40.8525	1.54629
$699$	−16.5872	−0.627387
$700$	−0.104590	−0.00395314
$701$	−39.2936	−1.48410	−0.742050	−	0.670345i	$-0.766146\pi$
$701$	−39.2936	−1.48410	−0.742050	+	0.670345i	$0.766146\pi$
$702$	7.37883	0.278496
$703$	−22.6170	−0.853017
$704$	−7.36842	−0.277708
$705$	1.26383	0.0475986
$706$	−24.2028	−0.910885
$707$	−4.51803	−0.169918
$708$	1.10941	0.0416941
$709$	−49.2430	−1.84936	−0.924680	−	0.380745i	$-0.875667\pi$
$709$	−49.2430	−1.84936	−0.924680	+	0.380745i	$0.875667\pi$
$710$	−40.1384	−1.50637
$711$	11.4432	0.429154
$712$	−32.2522	−1.20870
$713$	−37.2936	−1.39666
$714$	9.31444	0.348584
$715$	−12.0990	−0.452477
$716$	−0.778489	−0.0290935
$717$	2.91623	0.108909
$718$	32.2217	1.20250
$719$	−7.41344	−0.276475	−0.138237	−	0.990399i	$-0.544144\pi$
$719$	−7.41344	−0.276475	−0.138237	+	0.990399i	$0.544144\pi$
$720$	−10.2140	−0.380653
$721$	10.6468	0.396508
$722$	13.6587	0.508323
$723$	−6.09899	−0.226824
$724$	1.88914	0.0702095
$725$	3.79082	0.140787
$726$	1.46260	0.0542821
$727$	18.9557	0.703026	0.351513	−	0.936183i	$-0.385667\pi$
$727$	18.9557	0.703026	0.351513	+	0.936183i	$0.385667\pi$
$728$	13.7306	0.508889
$729$	1.00000	0.0370370
$730$	−45.7569	−1.69354
$731$	66.9832	2.47746
$732$	−0.278388	−0.0102895
$733$	−3.59283	−0.132704	−0.0663521	−	0.997796i	$-0.521136\pi$
$733$	−3.59283	−0.132704	−0.0663521	+	0.997796i	$0.521136\pi$
$734$	−28.3505	−1.04644
$735$	2.39821	0.0884592
$736$	−3.87122	−0.142695
$737$	−8.76663	−0.322923
$738$	−0.946021	−0.0348235
$739$	−26.7756	−0.984956	−0.492478	−	0.870325i	$-0.663909\pi$
$739$	−26.7756	−0.984956	−0.492478	+	0.870325i	$0.663909\pi$
$740$	1.41825	0.0521360
$741$	−26.8567	−0.986604
$742$	−5.44322	−0.199827
$743$	−33.8027	−1.24010	−0.620050	−	0.784562i	$-0.712888\pi$
$743$	−33.8027	−1.24010	−0.620050	+	0.784562i	$0.712888\pi$
$744$	20.6081	0.755528
$745$	−23.6018	−0.864703
$746$	42.7881	1.56658
$747$	13.1648	0.481676
$748$	0.886447	0.0324117
$749$	−15.9702	−0.583539
$750$	−14.9025	−0.544161
$751$	35.3955	1.29160	0.645800	−	0.763506i	$-0.276524\pi$
$751$	35.3955	1.29160	0.645800	+	0.763506i	$0.276524\pi$
$752$	−2.24445	−0.0818468
$753$	1.62262	0.0591314
$754$	37.2263	1.35570
$755$	−9.90168	−0.360359
$756$	−0.139194	−0.00506244
$757$	−29.3442	−1.06653	−0.533267	−	0.845947i	$-0.679036\pi$
$757$	−29.3442	−1.06653	−0.533267	+	0.845947i	$0.679036\pi$
$758$	18.3220	0.665483
$759$	−4.92520	−0.178773
$760$	34.7458	1.26036
$761$	12.9044	0.467783	0.233892	−	0.972263i	$-0.424854\pi$
$761$	12.9044	0.467783	0.233892	+	0.972263i	$0.424854\pi$
$762$	−3.33237	−0.120719
$763$	−12.7756	−0.462507
$764$	−1.31154	−0.0474500
$765$	−15.2728	−0.552189
$766$	−25.7312	−0.929708
$767$	40.2099	1.45189
$768$	3.32485	0.119975
$769$	9.78186	0.352743	0.176371	−	0.984324i	$-0.443564\pi$
$769$	9.78186	0.352743	0.176371	+	0.984324i	$0.443564\pi$
$770$	3.50761	0.126406
$771$	−6.89541	−0.248332
$772$	−1.40987	−0.0507422
$773$	29.7223	1.06904	0.534518	−	0.845157i	$-0.320493\pi$
$773$	29.7223	1.06904	0.534518	+	0.845157i	$0.320493\pi$
$774$	−15.3836	−0.552954
$775$	−5.68960	−0.204376
$776$	5.09273	0.182818
$777$	−4.24860	−0.152418
$778$	29.3836	1.05345
$779$	3.44322	0.123366
$780$	1.68411	0.0603008
$781$	11.4432	0.409471
$782$	−45.8755	−1.64050
$783$	5.04502	0.180294
$784$	−4.25901	−0.152108
$785$	−2.26875	−0.0809753
$786$	5.85039	0.208677
$787$	17.0242	0.606847	0.303423	−	0.952856i	$-0.401870\pi$
$787$	17.0242	0.606847	0.303423	+	0.952856i	$0.401870\pi$
$788$	−0.314240	−0.0111943
$789$	5.08377	0.180987
$790$	40.1384	1.42806
$791$	−18.7368	−0.666205
$792$	2.72161	0.0967083
$793$	−10.0900	−0.358308
$794$	−51.3449	−1.82216
$795$	8.92520	0.316544
$796$	0.427995	0.0151699
$797$	−36.4287	−1.29037	−0.645185	−	0.764027i	$-0.723220\pi$
$797$	−36.4287	−1.29037	−0.645185	+	0.764027i	$0.723220\pi$
$798$	7.78600	0.275622
$799$	−3.35609	−0.118730
$800$	−0.590602	−0.0208809
$801$	11.8504	0.418713
$802$	14.0000	0.494357
$803$	13.0450	0.460349
$804$	1.22026	0.0430354
$805$	−11.8116	−0.416306
$806$	−55.8726	−1.96803
$807$	−0.886447	−0.0312044
$808$	−12.2963	−0.432583
$809$	44.4882	1.56412	0.782062	−	0.623201i	$-0.214168\pi$
$809$	44.4882	1.56412	0.782062	+	0.623201i	$0.214168\pi$
$810$	3.50761	0.123245
$811$	7.65307	0.268736	0.134368	−	0.990932i	$-0.457100\pi$
$811$	7.65307	0.268736	0.134368	+	0.990932i	$0.457100\pi$
$812$	−0.702237	−0.0246437
$813$	−25.3234	−0.888131
$814$	−6.21400	−0.217800
$815$	21.0242	0.736445
$816$	27.1232	0.949501
$817$	55.9917	1.95890
$818$	55.8421	1.95247
$819$	−5.04502	−0.176287
$820$	−0.215915	−0.00754009
$821$	−44.3691	−1.54849	−0.774246	−	0.632885i	$-0.781871\pi$
$821$	−44.3691	−1.54849	−0.774246	+	0.632885i	$0.781871\pi$
$822$	−6.98477	−0.243622
$823$	6.61702	0.230655	0.115327	−	0.993328i	$-0.463208\pi$
$823$	6.61702	0.230655	0.115327	+	0.993328i	$0.463208\pi$
$824$	28.9765	1.00944
$825$	−0.751399	−0.0261604
$826$	−11.6572	−0.405607
$827$	39.7126	1.38094	0.690472	−	0.723359i	$-0.257403\pi$
$827$	39.7126	1.38094	0.690472	+	0.723359i	$0.257403\pi$
$828$	0.685559	0.0238248
$829$	−3.90997	−0.135799	−0.0678994	−	0.997692i	$-0.521630\pi$
$829$	−3.90997	−0.135799	−0.0678994	+	0.997692i	$0.521630\pi$
$830$	46.1772	1.60283
$831$	−24.8269	−0.861235
$832$	37.1738	1.28877
$833$	−6.36842	−0.220653
$834$	−22.5872	−0.782132
$835$	−58.4405	−2.02242
$836$	0.740987	0.0256276
$837$	−7.57201	−0.261727
$838$	10.4924	0.362453
$839$	−9.58097	−0.330772	−0.165386	−	0.986229i	$-0.552887\pi$
$839$	−9.58097	−0.330772	−0.165386	+	0.986229i	$0.552887\pi$
$840$	6.52699	0.225203
$841$	−3.54781	−0.122338
$842$	22.1627	0.763778
$843$	1.90101	0.0654741
$844$	−2.03875	−0.0701767
$845$	29.8629	1.02732
$846$	0.770774	0.0264998
$847$	−1.00000	−0.0343604
$848$	−15.8504	−0.544305
$849$	−22.3178	−0.765945
$850$	−6.99886	−0.240059
$851$	20.9252	0.717307
$852$	−1.59283	−0.0545694
$853$	−14.5568	−0.498415	−0.249207	−	0.968450i	$-0.580170\pi$
$853$	−14.5568	−0.498415	−0.249207	+	0.968450i	$0.580170\pi$
$854$	2.92520	0.100098
$855$	−12.7666	−0.436609
$856$	−43.4647	−1.48559
$857$	10.4793	0.357965	0.178983	−	0.983852i	$-0.442719\pi$
$857$	10.4793	0.357965	0.178983	+	0.983852i	$0.442719\pi$
$858$	−7.37883	−0.251909
$859$	2.88645	0.0984843	0.0492421	−	0.998787i	$-0.484319\pi$
$859$	2.88645	0.0984843	0.0492421	+	0.998787i	$0.484319\pi$
$860$	−3.51109	−0.119727
$861$	0.646809	0.0220432
$862$	8.13650	0.277130
$863$	−20.5485	−0.699479	−0.349739	−	0.936847i	$-0.613730\pi$
$863$	−20.5485	−0.699479	−0.349739	+	0.936847i	$0.613730\pi$
$864$	−0.786003	−0.0267404
$865$	29.6121	1.00684
$866$	−37.5028	−1.27440
$867$	23.5568	0.800030
$868$	1.05398	0.0357744
$869$	−11.4432	−0.388185
$870$	17.6960	0.599950
$871$	44.2278	1.49860
$872$	−34.7702	−1.17747
$873$	−1.87122	−0.0633311
$874$	−38.3476	−1.29713
$875$	10.1890	0.344452
$876$	−1.81579	−0.0613499
$877$	59.1149	1.99617	0.998084	−	0.0618724i	$-0.0197072\pi$
$877$	59.1149	1.99617	0.998084	+	0.0618724i	$0.0197072\pi$
$878$	34.6212	1.16841
$879$	12.0900	0.407787
$880$	10.2140	0.344314
$881$	30.3982	1.02414	0.512071	−	0.858943i	$-0.328879\pi$
$881$	30.3982	1.02414	0.512071	+	0.858943i	$0.328879\pi$
$882$	1.46260	0.0492483
$883$	−35.6114	−1.19842	−0.599210	−	0.800592i	$-0.704519\pi$
$883$	−35.6114	−1.19842	−0.599210	+	0.800592i	$0.704519\pi$
$884$	−4.47214	−0.150414
$885$	19.1142	0.642518
$886$	−26.3713	−0.885962
$887$	22.9736	0.771377	0.385689	−	0.922629i	$-0.373964\pi$
$887$	22.9736	0.771377	0.385689	+	0.922629i	$0.373964\pi$
$888$	−11.5630	−0.388030
$889$	2.27839	0.0764147
$890$	41.5666	1.39332
$891$	−1.00000	−0.0335013
$892$	−0.265856	−0.00890153
$893$	−2.80538	−0.0938784
$894$	−14.3941	−0.481409
$895$	−13.4128	−0.448339
$896$	−12.3490	−0.412553
$897$	24.8477	0.829640
$898$	−51.1565	−1.70712
$899$	−38.2009	−1.27407
$900$	0.104590	0.00348634
$901$	−23.7008	−0.789588
$902$	0.946021	0.0314991
$903$	10.5180	0.350018
$904$	−50.9944	−1.69605
$905$	32.5485	1.08195
$906$	−6.03875	−0.200624
$907$	−57.1745	−1.89845	−0.949224	−	0.314602i	$-0.898129\pi$
$907$	−57.1745	−1.89845	−0.949224	+	0.314602i	$0.898129\pi$
$908$	−0.445920	−0.0147984
$909$	4.51803	0.149853
$910$	−17.6960	−0.586616
$911$	6.82687	0.226184	0.113092	−	0.993584i	$-0.463924\pi$
$911$	6.82687	0.226184	0.113092	+	0.993584i	$0.463924\pi$
$912$	22.6724	0.750760
$913$	−13.1648	−0.435692
$914$	6.63428	0.219442
$915$	−4.79641	−0.158565
$916$	−2.54848	−0.0842043
$917$	−4.00000	−0.132092
$918$	−9.31444	−0.307422
$919$	12.0692	0.398126	0.199063	−	0.979987i	$-0.436210\pi$
$919$	12.0692	0.398126	0.199063	+	0.979987i	$0.436210\pi$
$920$	−32.1467	−1.05985
$921$	−13.5928	−0.447899
$922$	−4.09003	−0.134698
$923$	−57.7312	−1.90025
$924$	0.139194	0.00457915
$925$	3.19239	0.104965
$926$	56.1045	1.84371
$927$	−10.6468	−0.349687
$928$	−3.96540	−0.130171
$929$	26.8954	0.882410	0.441205	−	0.897406i	$-0.354551\pi$
$929$	26.8954	0.882410	0.441205	+	0.897406i	$0.354551\pi$
$930$	−26.5597	−0.870926
$931$	−5.32340	−0.174468
$932$	−2.30885	−0.0756288
$933$	8.00000	0.261908
$934$	−29.9356	−0.979523
$935$	15.2728	0.499474
$936$	−13.7306	−0.448798
$937$	−14.9944	−0.489846	−0.244923	−	0.969543i	$-0.578763\pi$
$937$	−14.9944	−0.489846	−0.244923	+	0.969543i	$0.578763\pi$
$938$	−12.8221	−0.418655
$939$	14.9252	0.487065
$940$	0.175918	0.00573780
$941$	−30.1205	−0.981900	−0.490950	−	0.871188i	$-0.663350\pi$
$941$	−30.1205	−0.981900	−0.490950	+	0.871188i	$0.663350\pi$
$942$	−1.38365	−0.0450817
$943$	−3.18566	−0.103739
$944$	−33.9452	−1.10482
$945$	−2.39821	−0.0780137
$946$	15.3836	0.500166
$947$	−17.3532	−0.563903	−0.281951	−	0.959429i	$-0.590982\pi$
$947$	−17.3532	−0.563903	−0.281951	+	0.959429i	$0.590982\pi$
$948$	1.59283	0.0517327
$949$	−65.8123	−2.13636
$950$	−5.85039	−0.189812
$951$	3.97918	0.129034
$952$	−17.3324	−0.561745
$953$	−2.14064	−0.0693422	−0.0346711	−	0.999399i	$-0.511038\pi$
$953$	−2.14064	−0.0693422	−0.0346711	+	0.999399i	$0.511038\pi$
$954$	5.44322	0.176231
$955$	−22.5969	−0.731217
$956$	0.405923	0.0131285
$957$	−5.04502	−0.163082
$958$	−16.9861	−0.548796
$959$	4.77559	0.154212
$960$	17.6710	0.570329
$961$	26.3353	0.849525
$962$	31.3497	1.01076
$963$	15.9702	0.514633
$964$	−0.848944	−0.0273427
$965$	−24.2909	−0.781952
$966$	−7.20359	−0.231772
$967$	−1.53326	−0.0493062	−0.0246531	−	0.999696i	$-0.507848\pi$
$967$	−1.53326	−0.0493062	−0.0246531	+	0.999696i	$0.507848\pi$
$968$	−2.72161	−0.0874759
$969$	33.9017	1.08908
$970$	−6.56351	−0.210742
$971$	−26.5574	−0.852269	−0.426135	−	0.904660i	$-0.640125\pi$
$971$	−26.5574	−0.852269	−0.426135	+	0.904660i	$0.640125\pi$
$972$	0.139194	0.00446465
$973$	15.4432	0.495087
$974$	47.5041	1.52213
$975$	3.79082	0.121403
$976$	8.51803	0.272655
$977$	−55.9017	−1.78845	−0.894227	−	0.447615i	$-0.852274\pi$
$977$	−55.9017	−1.78845	−0.894227	+	0.447615i	$0.852274\pi$
$978$	12.8221	0.410004
$979$	−11.8504	−0.378740
$980$	0.333816	0.0106634
$981$	12.7756	0.407893
$982$	39.0171	1.24509
$983$	−53.0361	−1.69159	−0.845794	−	0.533510i	$-0.820873\pi$
$983$	−53.0361	−1.69159	−0.845794	+	0.533510i	$0.820873\pi$
$984$	1.76036	0.0561183
$985$	−5.41411	−0.172508
$986$	−46.9915	−1.49651
$987$	−0.526989	−0.0167743
$988$	−3.73829	−0.118931
$989$	−51.8034	−1.64725
$990$	−3.50761	−0.111479
$991$	14.7362	0.468110	0.234055	−	0.972223i	$-0.424800\pi$
$991$	14.7362	0.468110	0.234055	+	0.972223i	$0.424800\pi$
$992$	5.95162	0.188964
$993$	23.4432	0.743948
$994$	16.7368	0.530860
$995$	7.37402	0.233772
$996$	1.83247	0.0580640
$997$	−45.0665	−1.42727	−0.713635	−	0.700517i	$-0.752953\pi$
$997$	−45.0665	−1.42727	−0.713635	+	0.700517i	$0.752953\pi$
$998$	−60.3262	−1.90959
$999$	4.24860	0.134420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.a.e.1.2	✓	3
3.2	odd	2		693.2.a.l.1.2		3
4.3	odd	2		3696.2.a.bo.1.2		3
5.4	even	2		5775.2.a.bp.1.2		3
7.6	odd	2		1617.2.a.t.1.2		3
11.10	odd	2		2541.2.a.bg.1.2		3
21.20	even	2		4851.2.a.bi.1.2		3
33.32	even	2		7623.2.a.cd.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.2	✓	3	1.1	even	1	trivial
693.2.a.l.1.2		3	3.2	odd	2
1617.2.a.t.1.2		3	7.6	odd	2
2541.2.a.bg.1.2		3	11.10	odd	2
3696.2.a.bo.1.2		3	4.3	odd	2
4851.2.a.bi.1.2		3	21.20	even	2
5775.2.a.bp.1.2		3	5.4	even	2
7623.2.a.cd.1.2		3	33.32	even	2

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:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} +1.46260 q^{6} -1.00000 q^{7} -2.72161 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.46260 q^{2} +1.00000 q^{3} +0.139194 q^{4} +2.39821 q^{5} +1.46260 q^{6} -1.00000 q^{7} -2.72161 q^{8} +1.00000 q^{9} +3.50761 q^{10} -1.00000 q^{11} +0.139194 q^{12} +5.04502 q^{13} -1.46260 q^{14} +2.39821 q^{15} -4.25901 q^{16} -6.36842 q^{17} +1.46260 q^{18} -5.32340 q^{19} +0.333816 q^{20} -1.00000 q^{21} -1.46260 q^{22} +4.92520 q^{23} -2.72161 q^{24} +0.751399 q^{25} +7.37883 q^{26} +1.00000 q^{27} -0.139194 q^{28} +5.04502 q^{29} +3.50761 q^{30} -7.57201 q^{31} -0.786003 q^{32} -1.00000 q^{33} -9.31444 q^{34} -2.39821 q^{35} +0.139194 q^{36} +4.24860 q^{37} -7.78600 q^{38} +5.04502 q^{39} -6.52699 q^{40} -0.646809 q^{41} -1.46260 q^{42} -10.5180 q^{43} -0.139194 q^{44} +2.39821 q^{45} +7.20359 q^{46} +0.526989 q^{47} -4.25901 q^{48} +1.00000 q^{49} +1.09899 q^{50} -6.36842 q^{51} +0.702237 q^{52} +3.72161 q^{53} +1.46260 q^{54} -2.39821 q^{55} +2.72161 q^{56} -5.32340 q^{57} +7.37883 q^{58} +7.97021 q^{59} +0.333816 q^{60} -2.00000 q^{61} -11.0748 q^{62} -1.00000 q^{63} +7.36842 q^{64} +12.0990 q^{65} -1.46260 q^{66} +8.76663 q^{67} -0.886447 q^{68} +4.92520 q^{69} -3.50761 q^{70} -11.4432 q^{71} -2.72161 q^{72} -13.0450 q^{73} +6.21400 q^{74} +0.751399 q^{75} -0.740987 q^{76} +1.00000 q^{77} +7.37883 q^{78} +11.4432 q^{79} -10.2140 q^{80} +1.00000 q^{81} -0.946021 q^{82} +13.1648 q^{83} -0.139194 q^{84} -15.2728 q^{85} -15.3836 q^{86} +5.04502 q^{87} +2.72161 q^{88} +11.8504 q^{89} +3.50761 q^{90} -5.04502 q^{91} +0.685559 q^{92} -7.57201 q^{93} +0.770774 q^{94} -12.7666 q^{95} -0.786003 q^{96} -1.87122 q^{97} +1.46260 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 + 3 * q^3 + 6 * q^4 + 4 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 2 q^{2} + 3 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 11 q^{10} - 3 q^{11} + 6 q^{12} - 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} + 8 q^{17} + 2 q^{18} - 8 q^{19} - 3 q^{20} - 3 q^{21} - 2 q^{22} + 10 q^{23} + 3 q^{24} + 15 q^{25} - q^{26} + 3 q^{27} - 6 q^{28} - 4 q^{29} - 11 q^{30} - 2 q^{31} + 8 q^{32} - 3 q^{33} - 4 q^{34} - 4 q^{35} + 6 q^{36} - 13 q^{38} - 4 q^{39} - 18 q^{40} + 14 q^{41} - 2 q^{42} - 14 q^{43} - 6 q^{44} + 4 q^{45} + 28 q^{46} - 4 q^{48} + 3 q^{49} - 19 q^{50} + 8 q^{51} - 29 q^{52} + 2 q^{54} - 4 q^{55} - 3 q^{56} - 8 q^{57} - q^{58} - 3 q^{60} - 6 q^{61} - 38 q^{62} - 3 q^{63} - 5 q^{64} + 14 q^{65} - 2 q^{66} - 4 q^{67} + 42 q^{68} + 10 q^{69} + 11 q^{70} - 12 q^{71} + 3 q^{72} - 20 q^{73} + 29 q^{74} + 15 q^{75} - 11 q^{76} + 3 q^{77} - q^{78} + 12 q^{79} - 41 q^{80} + 3 q^{81} - 6 q^{82} + 6 q^{83} - 6 q^{84} - 6 q^{85} + 24 q^{86} - 4 q^{87} - 3 q^{88} + 26 q^{89} - 11 q^{90} + 4 q^{91} + 26 q^{92} - 2 q^{93} + 35 q^{94} - 8 q^{95} + 8 q^{96} - 4 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 3 * q^3 + 6 * q^4 + 4 * q^5 + 2 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 - 11 * q^10 - 3 * q^11 + 6 * q^12 - 4 * q^13 - 2 * q^14 + 4 * q^15 - 4 * q^16 + 8 * q^17 + 2 * q^18 - 8 * q^19 - 3 * q^20 - 3 * q^21 - 2 * q^22 + 10 * q^23 + 3 * q^24 + 15 * q^25 - q^26 + 3 * q^27 - 6 * q^28 - 4 * q^29 - 11 * q^30 - 2 * q^31 + 8 * q^32 - 3 * q^33 - 4 * q^34 - 4 * q^35 + 6 * q^36 - 13 * q^38 - 4 * q^39 - 18 * q^40 + 14 * q^41 - 2 * q^42 - 14 * q^43 - 6 * q^44 + 4 * q^45 + 28 * q^46 - 4 * q^48 + 3 * q^49 - 19 * q^50 + 8 * q^51 - 29 * q^52 + 2 * q^54 - 4 * q^55 - 3 * q^56 - 8 * q^57 - q^58 - 3 * q^60 - 6 * q^61 - 38 * q^62 - 3 * q^63 - 5 * q^64 + 14 * q^65 - 2 * q^66 - 4 * q^67 + 42 * q^68 + 10 * q^69 + 11 * q^70 - 12 * q^71 + 3 * q^72 - 20 * q^73 + 29 * q^74 + 15 * q^75 - 11 * q^76 + 3 * q^77 - q^78 + 12 * q^79 - 41 * q^80 + 3 * q^81 - 6 * q^82 + 6 * q^83 - 6 * q^84 - 6 * q^85 + 24 * q^86 - 4 * q^87 - 3 * q^88 + 26 * q^89 - 11 * q^90 + 4 * q^91 + 26 * q^92 - 2 * q^93 + 35 * q^94 - 8 * q^95 + 8 * q^96 - 4 * q^97 + 2 * q^98 - 3 * q^99

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