LMFDB - Embedded newform 8023.2.a.e.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8023,2,Mod(1,8023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8023, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8023 = 71 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8023.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0639775417$
Analytic rank:	$0$
Dimension:	$172$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58451 q^{2} +1.24090 q^{3} +4.67970 q^{4} +4.06279 q^{5} -3.20713 q^{6} +2.63982 q^{7} -6.92572 q^{8} -1.46016 q^{9} +O(q^{10})$	\(q-2.58451 q^{2} +1.24090 q^{3} +4.67970 q^{4} +4.06279 q^{5} -3.20713 q^{6} +2.63982 q^{7} -6.92572 q^{8} -1.46016 q^{9} -10.5003 q^{10} +3.91875 q^{11} +5.80705 q^{12} +5.29409 q^{13} -6.82265 q^{14} +5.04152 q^{15} +8.54020 q^{16} -2.18719 q^{17} +3.77381 q^{18} +0.681271 q^{19} +19.0126 q^{20} +3.27576 q^{21} -10.1280 q^{22} -0.381531 q^{23} -8.59414 q^{24} +11.5063 q^{25} -13.6826 q^{26} -5.53462 q^{27} +12.3536 q^{28} +0.851877 q^{29} -13.0299 q^{30} -5.01249 q^{31} -8.22082 q^{32} +4.86278 q^{33} +5.65281 q^{34} +10.7250 q^{35} -6.83313 q^{36} -0.517767 q^{37} -1.76075 q^{38} +6.56945 q^{39} -28.1377 q^{40} +0.384721 q^{41} -8.46623 q^{42} +0.924969 q^{43} +18.3386 q^{44} -5.93233 q^{45} +0.986072 q^{46} +7.94746 q^{47} +10.5976 q^{48} -0.0313491 q^{49} -29.7380 q^{50} -2.71408 q^{51} +24.7748 q^{52} +5.86149 q^{53} +14.3043 q^{54} +15.9210 q^{55} -18.2827 q^{56} +0.845390 q^{57} -2.20169 q^{58} +6.54799 q^{59} +23.5928 q^{60} +3.20239 q^{61} +12.9548 q^{62} -3.85457 q^{63} +4.16639 q^{64} +21.5088 q^{65} -12.5679 q^{66} -3.67335 q^{67} -10.2354 q^{68} -0.473443 q^{69} -27.7190 q^{70} -1.00000 q^{71} +10.1127 q^{72} -3.35400 q^{73} +1.33818 q^{74} +14.2781 q^{75} +3.18814 q^{76} +10.3448 q^{77} -16.9788 q^{78} -10.7090 q^{79} +34.6970 q^{80} -2.48744 q^{81} -0.994316 q^{82} -15.8903 q^{83} +15.3296 q^{84} -8.88607 q^{85} -2.39059 q^{86} +1.05710 q^{87} -27.1401 q^{88} +7.70794 q^{89} +15.3322 q^{90} +13.9755 q^{91} -1.78545 q^{92} -6.22000 q^{93} -20.5403 q^{94} +2.76786 q^{95} -10.2012 q^{96} +4.69934 q^{97} +0.0810222 q^{98} -5.72201 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9}+O(q^{10}) $ 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9	\( 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9} + 14 q^{10} + 20 q^{11} + 54 q^{12} + 36 q^{13} + 26 q^{14} + 32 q^{15} + 196 q^{16} + 123 q^{17} + 74 q^{18} + 20 q^{19} + 70 q^{20} + 37 q^{21} + 11 q^{22} + 22 q^{23} + 62 q^{24} + 210 q^{25} + 50 q^{26} + 69 q^{27} + 42 q^{28} + 58 q^{29} + 36 q^{30} + 10 q^{31} + 168 q^{32} + 124 q^{33} + 5 q^{34} + 59 q^{35} + 192 q^{36} + 40 q^{37} + 58 q^{38} + 15 q^{39} + 7 q^{40} + 155 q^{41} - 6 q^{42} + 19 q^{43} + 22 q^{44} + 76 q^{45} + q^{46} + 71 q^{47} + 144 q^{48} + 206 q^{49} + 126 q^{50} + 33 q^{51} + 71 q^{52} + 101 q^{53} + 92 q^{54} - 2 q^{55} + 57 q^{56} + 114 q^{57} + 4 q^{58} + 71 q^{59} + 38 q^{60} + 50 q^{61} + 86 q^{62} + 14 q^{63} + 240 q^{64} + 143 q^{65} + 21 q^{66} + 8 q^{67} + 192 q^{68} + 41 q^{69} - 12 q^{70} - 172 q^{71} + 156 q^{72} + 128 q^{73} + 30 q^{74} + 72 q^{75} + 74 q^{76} + 127 q^{77} + 107 q^{78} + 2 q^{79} + 50 q^{80} + 236 q^{81} + 42 q^{82} + 140 q^{83} + 71 q^{84} + 55 q^{85} + 46 q^{86} + 100 q^{87} - 31 q^{88} + 215 q^{89} - 7 q^{90} + 22 q^{91} - 15 q^{92} + 60 q^{93} + 5 q^{94} + 74 q^{95} + 182 q^{96} + 120 q^{97} + 164 q^{98} + 60 q^{99}+O(q^{100}) \) 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9 + 14 * q^10 + 20 * q^11 + 54 * q^12 + 36 * q^13 + 26 * q^14 + 32 * q^15 + 196 * q^16 + 123 * q^17 + 74 * q^18 + 20 * q^19 + 70 * q^20 + 37 * q^21 + 11 * q^22 + 22 * q^23 + 62 * q^24 + 210 * q^25 + 50 * q^26 + 69 * q^27 + 42 * q^28 + 58 * q^29 + 36 * q^30 + 10 * q^31 + 168 * q^32 + 124 * q^33 + 5 * q^34 + 59 * q^35 + 192 * q^36 + 40 * q^37 + 58 * q^38 + 15 * q^39 + 7 * q^40 + 155 * q^41 - 6 * q^42 + 19 * q^43 + 22 * q^44 + 76 * q^45 + q^46 + 71 * q^47 + 144 * q^48 + 206 * q^49 + 126 * q^50 + 33 * q^51 + 71 * q^52 + 101 * q^53 + 92 * q^54 - 2 * q^55 + 57 * q^56 + 114 * q^57 + 4 * q^58 + 71 * q^59 + 38 * q^60 + 50 * q^61 + 86 * q^62 + 14 * q^63 + 240 * q^64 + 143 * q^65 + 21 * q^66 + 8 * q^67 + 192 * q^68 + 41 * q^69 - 12 * q^70 - 172 * q^71 + 156 * q^72 + 128 * q^73 + 30 * q^74 + 72 * q^75 + 74 * q^76 + 127 * q^77 + 107 * q^78 + 2 * q^79 + 50 * q^80 + 236 * q^81 + 42 * q^82 + 140 * q^83 + 71 * q^84 + 55 * q^85 + 46 * q^86 + 100 * q^87 - 31 * q^88 + 215 * q^89 - 7 * q^90 + 22 * q^91 - 15 * q^92 + 60 * q^93 + 5 * q^94 + 74 * q^95 + 182 * q^96 + 120 * q^97 + 164 * q^98 + 60 * 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 $

$2$	−2.58451	−1.82753	−0.913763	−	0.406248i	$-0.866837\pi$
$2$	−2.58451	−1.82753	−0.913763	+	0.406248i	$0.866837\pi$
$3$	1.24090	0.716435	0.358217	−	0.933638i	$-0.383385\pi$
$3$	1.24090	0.716435	0.358217	+	0.933638i	$0.383385\pi$
$4$	4.67970	2.33985
$5$	4.06279	1.81693	0.908467	−	0.417956i	$-0.137253\pi$
$5$	4.06279	1.81693	0.908467	+	0.417956i	$0.137253\pi$
$6$	−3.20713	−1.30930
$7$	2.63982	0.997758	0.498879	−	0.866672i	$-0.333745\pi$
$7$	2.63982	0.997758	0.498879	+	0.866672i	$0.333745\pi$
$8$	−6.92572	−2.44861
$9$	−1.46016	−0.486721
$10$	−10.5003	−3.32049
$11$	3.91875	1.18155	0.590773	−	0.806838i	$-0.298823\pi$
$11$	3.91875	1.18155	0.590773	+	0.806838i	$0.298823\pi$
$12$	5.80705	1.67635
$13$	5.29409	1.46832	0.734159	−	0.678978i	$-0.237577\pi$
$13$	5.29409	1.46832	0.734159	+	0.678978i	$0.237577\pi$
$14$	−6.82265	−1.82343
$15$	5.04152	1.30172
$16$	8.54020	2.13505
$17$	−2.18719	−0.530470	−0.265235	−	0.964184i	$-0.585450\pi$
$17$	−2.18719	−0.530470	−0.265235	+	0.964184i	$0.585450\pi$
$18$	3.77381	0.889495
$19$	0.681271	0.156294	0.0781471	−	0.996942i	$-0.475100\pi$
$19$	0.681271	0.156294	0.0781471	+	0.996942i	$0.475100\pi$
$20$	19.0126	4.25136
$21$	3.27576	0.714829
$22$	−10.1280	−2.15931
$23$	−0.381531	−0.0795547	−0.0397774	−	0.999209i	$-0.512665\pi$
$23$	−0.381531	−0.0795547	−0.0397774	+	0.999209i	$0.512665\pi$
$24$	−8.59414	−1.75427
$25$	11.5063	2.30125
$26$	−13.6826	−2.68339
$27$	−5.53462	−1.06514
$28$	12.3536	2.33461
$29$	0.851877	0.158190	0.0790948	−	0.996867i	$-0.474797\pi$
$29$	0.851877	0.158190	0.0790948	+	0.996867i	$0.474797\pi$
$30$	−13.0299	−2.37892
$31$	−5.01249	−0.900269	−0.450135	−	0.892961i	$-0.648624\pi$
$31$	−5.01249	−0.900269	−0.450135	+	0.892961i	$0.648624\pi$
$32$	−8.22082	−1.45325
$33$	4.86278	0.846501
$34$	5.65281	0.969448
$35$	10.7250	1.81286
$36$	−6.83313	−1.13885
$37$	−0.517767	−0.0851204	−0.0425602	−	0.999094i	$-0.513551\pi$
$37$	−0.517767	−0.0851204	−0.0425602	+	0.999094i	$0.513551\pi$
$38$	−1.76075	−0.285632
$39$	6.56945	1.05195
$40$	−28.1377	−4.44897
$41$	0.384721	0.0600833	0.0300417	−	0.999549i	$-0.490436\pi$
$41$	0.384721	0.0600833	0.0300417	+	0.999549i	$0.490436\pi$
$42$	−8.46623	−1.30637
$43$	0.924969	0.141056	0.0705282	−	0.997510i	$-0.477532\pi$
$43$	0.924969	0.141056	0.0705282	+	0.997510i	$0.477532\pi$
$44$	18.3386	2.76464
$45$	−5.93233	−0.884340
$46$	0.986072	0.145388
$47$	7.94746	1.15926	0.579628	−	0.814881i	$-0.303198\pi$
$47$	7.94746	1.15926	0.579628	+	0.814881i	$0.303198\pi$
$48$	10.5976	1.52963
$49$	−0.0313491	−0.00447845
$50$	−29.7380	−4.20559
$51$	−2.71408	−0.380048
$52$	24.7748	3.43564
$53$	5.86149	0.805137	0.402569	−	0.915390i	$-0.368117\pi$
$53$	5.86149	0.805137	0.402569	+	0.915390i	$0.368117\pi$
$54$	14.3043	1.94657
$55$	15.9210	2.14679
$56$	−18.2827	−2.44312
$57$	0.845390	0.111975
$58$	−2.20169	−0.289096
$59$	6.54799	0.852476	0.426238	−	0.904611i	$-0.359839\pi$
$59$	6.54799	0.852476	0.426238	+	0.904611i	$0.359839\pi$
$60$	23.5928	3.04582
$61$	3.20239	0.410024	0.205012	−	0.978759i	$-0.434277\pi$
$61$	3.20239	0.410024	0.205012	+	0.978759i	$0.434277\pi$
$62$	12.9548	1.64526
$63$	−3.85457	−0.485630
$64$	4.16639	0.520799
$65$	21.5088	2.66784
$66$	−12.5679	−1.54700
$67$	−3.67335	−0.448772	−0.224386	−	0.974500i	$-0.572038\pi$
$67$	−3.67335	−0.448772	−0.224386	+	0.974500i	$0.572038\pi$
$68$	−10.2354	−1.24122
$69$	−0.473443	−0.0569958
$70$	−27.7190	−3.31305
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	10.1127	1.19179
$73$	−3.35400	−0.392556	−0.196278	−	0.980548i	$-0.562886\pi$
$73$	−3.35400	−0.392556	−0.196278	+	0.980548i	$0.562886\pi$
$74$	1.33818	0.155560
$75$	14.2781	1.64870
$76$	3.18814	0.365705
$77$	10.3448	1.17890
$78$	−16.9788	−1.92247
$79$	−10.7090	−1.20486	−0.602431	−	0.798171i	$-0.705801\pi$
$79$	−10.7090	−1.20486	−0.602431	+	0.798171i	$0.705801\pi$
$80$	34.6970	3.87925
$81$	−2.48744	−0.276382
$82$	−0.994316	−0.109804
$83$	−15.8903	−1.74419	−0.872094	−	0.489339i	$-0.837238\pi$
$83$	−15.8903	−1.74419	−0.872094	+	0.489339i	$0.837238\pi$
$84$	15.3296	1.67259
$85$	−8.88607	−0.963830
$86$	−2.39059	−0.257784
$87$	1.05710	0.113333
$88$	−27.1401	−2.89315
$89$	7.70794	0.817040	0.408520	−	0.912749i	$-0.366045\pi$
$89$	7.70794	0.817040	0.408520	+	0.912749i	$0.366045\pi$
$90$	15.3322	1.61615
$91$	13.9755	1.46503
$92$	−1.78545	−0.186146
$93$	−6.22000	−0.644984
$94$	−20.5403	−2.11857
$95$	2.76786	0.283976
$96$	−10.2012	−1.04116
$97$	4.69934	0.477146	0.238573	−	0.971125i	$-0.423320\pi$
$97$	4.69934	0.477146	0.238573	+	0.971125i	$0.423320\pi$
$98$	0.0810222	0.00818448
$99$	−5.72201	−0.575084
$100$	53.8458	5.38458
$101$	11.2213	1.11656	0.558281	−	0.829652i	$-0.311461\pi$
$101$	11.2213	1.11656	0.558281	+	0.829652i	$0.311461\pi$
$102$	7.01458	0.694547
$103$	−17.9110	−1.76482	−0.882412	−	0.470477i	$-0.844082\pi$
$103$	−17.9110	−1.76482	−0.882412	+	0.470477i	$0.844082\pi$
$104$	−36.6654	−3.59534
$105$	13.3087	1.29880
$106$	−15.1491	−1.47141
$107$	−4.59266	−0.443989	−0.221995	−	0.975048i	$-0.571257\pi$
$107$	−4.59266	−0.443989	−0.221995	+	0.975048i	$0.571257\pi$
$108$	−25.9004	−2.49227
$109$	−2.92317	−0.279989	−0.139994	−	0.990152i	$-0.544708\pi$
$109$	−2.92317	−0.279989	−0.139994	+	0.990152i	$0.544708\pi$
$110$	−41.1481	−3.92332
$111$	−0.642498	−0.0609833
$112$	22.5446	2.13026
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	−2.18492	−0.204637
$115$	−1.55008	−0.144546
$116$	3.98653	0.370140
$117$	−7.73024	−0.714661
$118$	−16.9234	−1.55792
$119$	−5.77378	−0.529281
$120$	−34.9162	−3.18740
$121$	4.35658	0.396053
$122$	−8.27662	−0.749330
$123$	0.477401	0.0430458
$124$	−23.4569	−2.10650
$125$	26.4335	2.36429
$126$	9.96218	0.887501
$127$	10.2723	0.911520	0.455760	−	0.890103i	$-0.349368\pi$
$127$	10.2723	0.911520	0.455760	+	0.890103i	$0.349368\pi$
$128$	5.67355	0.501475
$129$	1.14780	0.101058
$130$	−55.5897	−4.87554
$131$	−10.5883	−0.925101	−0.462550	−	0.886593i	$-0.653066\pi$
$131$	−10.5883	−0.925101	−0.462550	+	0.886593i	$0.653066\pi$
$132$	22.7564	1.98069
$133$	1.79843	0.155944
$134$	9.49382	0.820142
$135$	−22.4860	−1.93529
$136$	15.1478	1.29892
$137$	1.24552	0.106412	0.0532059	−	0.998584i	$-0.483056\pi$
$137$	1.24552	0.106412	0.0532059	+	0.998584i	$0.483056\pi$
$138$	1.22362	0.104161
$139$	−2.94137	−0.249484	−0.124742	−	0.992189i	$-0.539810\pi$
$139$	−2.94137	−0.249484	−0.124742	+	0.992189i	$0.539810\pi$
$140$	50.1899	4.24182
$141$	9.86201	0.830531
$142$	2.58451	0.216887
$143$	20.7462	1.73489
$144$	−12.4701	−1.03917
$145$	3.46100	0.287420
$146$	8.66846	0.717407
$147$	−0.0389012	−0.00320852
$148$	−2.42300	−0.199169
$149$	9.89793	0.810870	0.405435	−	0.914124i	$-0.367120\pi$
$149$	9.89793	0.810870	0.405435	+	0.914124i	$0.367120\pi$
$150$	−36.9020	−3.01303
$151$	22.5114	1.83195	0.915977	−	0.401230i	$-0.131417\pi$
$151$	22.5114	1.83195	0.915977	+	0.401230i	$0.131417\pi$
$152$	−4.71829	−0.382704
$153$	3.19365	0.258191
$154$	−26.7362	−2.15447
$155$	−20.3647	−1.63573
$156$	30.7431	2.46142
$157$	−1.51591	−0.120983	−0.0604913	−	0.998169i	$-0.519267\pi$
$157$	−1.51591	−0.120983	−0.0604913	+	0.998169i	$0.519267\pi$
$158$	27.6776	2.20191
$159$	7.27353	0.576828
$160$	−33.3994	−2.64046
$161$	−1.00717	−0.0793764
$162$	6.42881	0.505095
$163$	−9.64235	−0.755247	−0.377624	−	0.925959i	$-0.623259\pi$
$163$	−9.64235	−0.755247	−0.377624	+	0.925959i	$0.623259\pi$
$164$	1.80038	0.140586
$165$	19.7564	1.53804
$166$	41.0687	3.18755
$167$	22.5358	1.74388	0.871938	−	0.489616i	$-0.162863\pi$
$167$	22.5358	1.74388	0.871938	+	0.489616i	$0.162863\pi$
$168$	−22.6870	−1.75034
$169$	15.0274	1.15596
$170$	22.9662	1.76142
$171$	−0.994767	−0.0760717
$172$	4.32858	0.330051
$173$	−11.9132	−0.905744	−0.452872	−	0.891576i	$-0.649601\pi$
$173$	−11.9132	−0.905744	−0.452872	+	0.891576i	$0.649601\pi$
$174$	−2.73208	−0.207118
$175$	30.3744	2.29609
$176$	33.4669	2.52266
$177$	8.12541	0.610743
$178$	−19.9213	−1.49316
$179$	9.96477	0.744802	0.372401	−	0.928072i	$-0.378535\pi$
$179$	9.96477	0.744802	0.372401	+	0.928072i	$0.378535\pi$
$180$	−27.7616	−2.06922
$181$	−0.360991	−0.0268322	−0.0134161	−	0.999910i	$-0.504271\pi$
$181$	−0.360991	−0.0268322	−0.0134161	+	0.999910i	$0.504271\pi$
$182$	−36.1197	−2.67737
$183$	3.97385	0.293756
$184$	2.64238	0.194799
$185$	−2.10358	−0.154658
$186$	16.0757	1.17873
$187$	−8.57103	−0.626776
$188$	37.1917	2.71249
$189$	−14.6104	−1.06275
$190$	−7.15357	−0.518974
$191$	3.15755	0.228472	0.114236	−	0.993454i	$-0.463558\pi$
$191$	3.15755	0.228472	0.114236	+	0.993454i	$0.463558\pi$
$192$	5.17008	0.373119
$193$	19.7382	1.42078	0.710392	−	0.703807i	$-0.248518\pi$
$193$	19.7382	1.42078	0.710392	+	0.703807i	$0.248518\pi$
$194$	−12.1455	−0.871997
$195$	26.6903	1.91133
$196$	−0.146705	−0.0104789
$197$	−16.1940	−1.15378	−0.576889	−	0.816823i	$-0.695733\pi$
$197$	−16.1940	−1.15378	−0.576889	+	0.816823i	$0.695733\pi$
$198$	14.7886	1.05098
$199$	10.0759	0.714262	0.357131	−	0.934054i	$-0.383755\pi$
$199$	10.0759	0.714262	0.357131	+	0.934054i	$0.383755\pi$
$200$	−79.6891	−5.63487
$201$	−4.55827	−0.321516
$202$	−29.0016	−2.04055
$203$	2.24880	0.157835
$204$	−12.7011	−0.889254
$205$	1.56304	0.109167
$206$	46.2912	3.22526
$207$	0.557098	0.0387210
$208$	45.2126	3.13493
$209$	2.66973	0.184669
$210$	−34.3965	−2.37359
$211$	−5.52596	−0.380423	−0.190211	−	0.981743i	$-0.560917\pi$
$211$	−5.52596	−0.380423	−0.190211	+	0.981743i	$0.560917\pi$
$212$	27.4300	1.88390
$213$	−1.24090	−0.0850252
$214$	11.8698	0.811402
$215$	3.75795	0.256290
$216$	38.3313	2.60811
$217$	−13.2321	−0.898251
$218$	7.55496	0.511686
$219$	−4.16199	−0.281241
$220$	74.5057	5.02317
$221$	−11.5792	−0.778899
$222$	1.66054	0.111448
$223$	−0.436576	−0.0292353	−0.0146177	−	0.999893i	$-0.504653\pi$
$223$	−0.436576	−0.0292353	−0.0146177	+	0.999893i	$0.504653\pi$
$224$	−21.7015	−1.44999
$225$	−16.8010	−1.12007
$226$	−2.58451	−0.171919
$227$	−17.3299	−1.15023	−0.575113	−	0.818074i	$-0.695042\pi$
$227$	−17.3299	−1.15023	−0.575113	+	0.818074i	$0.695042\pi$
$228$	3.95617	0.262004
$229$	−17.8265	−1.17801	−0.589005	−	0.808129i	$-0.700480\pi$
$229$	−17.8265	−1.17801	−0.589005	+	0.808129i	$0.700480\pi$
$230$	4.00620	0.264161
$231$	12.8369	0.844604
$232$	−5.89986	−0.387345
$233$	14.1568	0.927445	0.463722	−	0.885981i	$-0.346514\pi$
$233$	14.1568	0.927445	0.463722	+	0.885981i	$0.346514\pi$
$234$	19.9789	1.30606
$235$	32.2888	2.10629
$236$	30.6426	1.99467
$237$	−13.2889	−0.863205
$238$	14.9224	0.967275
$239$	−3.44658	−0.222941	−0.111470	−	0.993768i	$-0.535556\pi$
$239$	−3.44658	−0.222941	−0.111470	+	0.993768i	$0.535556\pi$
$240$	43.0556	2.77923
$241$	14.7103	0.947573	0.473787	−	0.880640i	$-0.342887\pi$
$241$	14.7103	0.947573	0.473787	+	0.880640i	$0.342887\pi$
$242$	−11.2596	−0.723796
$243$	13.5172	0.867129
$244$	14.9862	0.959396
$245$	−0.127365	−0.00813704
$246$	−1.23385	−0.0786673
$247$	3.60671	0.229490
$248$	34.7151	2.20441
$249$	−19.7183	−1.24960
$250$	−68.3178	−4.32079
$251$	18.3587	1.15879	0.579394	−	0.815047i	$-0.303289\pi$
$251$	18.3587	1.15879	0.579394	+	0.815047i	$0.303289\pi$
$252$	−18.0382	−1.13630
$253$	−1.49512	−0.0939976
$254$	−26.5489	−1.66583
$255$	−11.0267	−0.690521
$256$	−22.9961	−1.43726
$257$	8.23942	0.513961	0.256981	−	0.966417i	$-0.417272\pi$
$257$	8.23942	0.513961	0.256981	+	0.966417i	$0.417272\pi$
$258$	−2.96649	−0.184686
$259$	−1.36681	−0.0849296
$260$	100.655	6.24234
$261$	−1.24388	−0.0769942
$262$	27.3655	1.69065
$263$	12.1256	0.747699	0.373849	−	0.927489i	$-0.378038\pi$
$263$	12.1256	0.747699	0.373849	+	0.927489i	$0.378038\pi$
$264$	−33.6783	−2.07275
$265$	23.8140	1.46288
$266$	−4.64807	−0.284992
$267$	9.56480	0.585356
$268$	−17.1902	−1.05006
$269$	19.4539	1.18613	0.593063	−	0.805156i	$-0.297919\pi$
$269$	19.4539	1.18613	0.593063	+	0.805156i	$0.297919\pi$
$270$	58.1154	3.53679
$271$	1.42636	0.0866450	0.0433225	−	0.999061i	$-0.486206\pi$
$271$	1.42636	0.0866450	0.0433225	+	0.999061i	$0.486206\pi$
$272$	−18.6790	−1.13258
$273$	17.3422	1.04960
$274$	−3.21906	−0.194470
$275$	45.0901	2.71903
$276$	−2.21557	−0.133362
$277$	−20.1836	−1.21271	−0.606356	−	0.795193i	$-0.707369\pi$
$277$	−20.1836	−1.21271	−0.606356	+	0.795193i	$0.707369\pi$
$278$	7.60200	0.455938
$279$	7.31905	0.438180
$280$	−74.2786	−4.43899
$281$	−3.42455	−0.204292	−0.102146	−	0.994769i	$-0.532571\pi$
$281$	−3.42455	−0.204292	−0.102146	+	0.994769i	$0.532571\pi$
$282$	−25.4885	−1.51782
$283$	−4.62390	−0.274862	−0.137431	−	0.990511i	$-0.543885\pi$
$283$	−4.62390	−0.274862	−0.137431	+	0.990511i	$0.543885\pi$
$284$	−4.67970	−0.277689
$285$	3.43464	0.203451
$286$	−53.6188	−3.17055
$287$	1.01559	0.0599486
$288$	12.0037	0.707327
$289$	−12.2162	−0.718601
$290$	−8.94498	−0.525268
$291$	5.83142	0.341844
$292$	−15.6957	−0.918523
$293$	2.58143	0.150808	0.0754042	−	0.997153i	$-0.475975\pi$
$293$	2.58143	0.150808	0.0754042	+	0.997153i	$0.475975\pi$
$294$	0.100541	0.00586364
$295$	26.6031	1.54889
$296$	3.58591	0.208427
$297$	−21.6888	−1.25851
$298$	−25.5813	−1.48189
$299$	−2.01986	−0.116812
$300$	66.8174	3.85770
$301$	2.44175	0.140740
$302$	−58.1811	−3.34794
$303$	13.9246	0.799945
$304$	5.81819	0.333696
$305$	13.0106	0.744987
$306$	−8.25402	−0.471851
$307$	24.2204	1.38233	0.691166	−	0.722696i	$-0.257097\pi$
$307$	24.2204	1.38233	0.691166	+	0.722696i	$0.257097\pi$
$308$	48.4105	2.75845
$309$	−22.2258	−1.26438
$310$	52.6327	2.98934
$311$	−29.4913	−1.67230	−0.836148	−	0.548503i	$-0.815198\pi$
$311$	−29.4913	−1.67230	−0.836148	+	0.548503i	$0.815198\pi$
$312$	−45.4982	−2.57583
$313$	−14.8582	−0.839834	−0.419917	−	0.907563i	$-0.637941\pi$
$313$	−14.8582	−0.839834	−0.419917	+	0.907563i	$0.637941\pi$
$314$	3.91788	0.221099
$315$	−15.6603	−0.882358
$316$	−50.1151	−2.81920
$317$	−25.6324	−1.43966	−0.719828	−	0.694152i	$-0.755780\pi$
$317$	−25.6324	−1.43966	−0.719828	+	0.694152i	$0.755780\pi$
$318$	−18.7985	−1.05417
$319$	3.33829	0.186908
$320$	16.9272	0.946257
$321$	−5.69904	−0.318089
$322$	2.60305	0.145062
$323$	−1.49007	−0.0829095
$324$	−11.6405	−0.646692
$325$	60.9152	3.37897
$326$	24.9208	1.38023
$327$	−3.62736	−0.200594
$328$	−2.66447	−0.147121
$329$	20.9799	1.15666
$330$	−51.0608	−2.81080
$331$	9.58342	0.526752	0.263376	−	0.964693i	$-0.415164\pi$
$331$	9.58342	0.526752	0.263376	+	0.964693i	$0.415164\pi$
$332$	−74.3619	−4.08114
$333$	0.756025	0.0414299
$334$	−58.2442	−3.18698
$335$	−14.9241	−0.815388
$336$	27.9756	1.52620
$337$	24.4909	1.33410	0.667052	−	0.745011i	$-0.267556\pi$
$337$	24.4909	1.33410	0.667052	+	0.745011i	$0.267556\pi$
$338$	−38.8386	−2.11254
$339$	1.24090	0.0673965
$340$	−41.5842	−2.25522
$341$	−19.6427	−1.06371
$342$	2.57099	0.139023
$343$	−18.5615	−1.00223
$344$	−6.40608	−0.345392
$345$	−1.92350	−0.103558
$346$	30.7898	1.65527
$347$	14.5671	0.782002	0.391001	−	0.920390i	$-0.372129\pi$
$347$	14.5671	0.782002	0.391001	+	0.920390i	$0.372129\pi$
$348$	4.94689	0.265181
$349$	−13.2708	−0.710371	−0.355185	−	0.934796i	$-0.615582\pi$
$349$	−13.2708	−0.710371	−0.355185	+	0.934796i	$0.615582\pi$
$350$	−78.5031	−4.19617
$351$	−29.3008	−1.56396
$352$	−32.2153	−1.71708
$353$	−10.7142	−0.570259	−0.285130	−	0.958489i	$-0.592037\pi$
$353$	−10.7142	−0.570259	−0.285130	+	0.958489i	$0.592037\pi$
$354$	−21.0002	−1.11615
$355$	−4.06279	−0.215630
$356$	36.0709	1.91175
$357$	−7.16469	−0.379196
$358$	−25.7541	−1.36114
$359$	9.12228	0.481455	0.240728	−	0.970593i	$-0.422614\pi$
$359$	9.12228	0.481455	0.240728	+	0.970593i	$0.422614\pi$
$360$	41.0857	2.16541
$361$	−18.5359	−0.975572
$362$	0.932985	0.0490366
$363$	5.40609	0.283746
$364$	65.4010	3.42794
$365$	−13.6266	−0.713249
$366$	−10.2705	−0.536846
$367$	−1.35739	−0.0708552	−0.0354276	−	0.999372i	$-0.511279\pi$
$367$	−1.35739	−0.0708552	−0.0354276	+	0.999372i	$0.511279\pi$
$368$	−3.25835	−0.169853
$369$	−0.561755	−0.0292438
$370$	5.43673	0.282642
$371$	15.4733	0.803332
$372$	−29.1078	−1.50917
$373$	23.7346	1.22893	0.614466	−	0.788943i	$-0.289371\pi$
$373$	23.7346	1.22893	0.614466	+	0.788943i	$0.289371\pi$
$374$	22.1519	1.14545
$375$	32.8014	1.69386
$376$	−55.0419	−2.83857
$377$	4.50992	0.232272
$378$	37.7608	1.94221
$379$	−24.9609	−1.28215	−0.641076	−	0.767477i	$-0.721512\pi$
$379$	−24.9609	−1.28215	−0.641076	+	0.767477i	$0.721512\pi$
$380$	12.9528	0.664462
$381$	12.7469	0.653044
$382$	−8.16073	−0.417539
$383$	25.5728	1.30671	0.653355	−	0.757052i	$-0.273361\pi$
$383$	25.5728	1.30671	0.653355	+	0.757052i	$0.273361\pi$
$384$	7.04031	0.359274
$385$	42.0287	2.14198
$386$	−51.0135	−2.59652
$387$	−1.35061	−0.0686551
$388$	21.9915	1.11645
$389$	−18.0152	−0.913406	−0.456703	−	0.889619i	$-0.650970\pi$
$389$	−18.0152	−0.913406	−0.456703	+	0.889619i	$0.650970\pi$
$390$	−68.9814	−3.49301
$391$	0.834479	0.0422014
$392$	0.217115	0.0109660
$393$	−13.1390	−0.662774
$394$	41.8537	2.10856
$395$	−43.5086	−2.18915
$396$	−26.7773	−1.34561
$397$	−11.4885	−0.576589	−0.288294	−	0.957542i	$-0.593088\pi$
$397$	−11.4885	−0.576589	−0.288294	+	0.957542i	$0.593088\pi$
$398$	−26.0413	−1.30533
$399$	2.23168	0.111724
$400$	98.2657	4.91329
$401$	−19.7298	−0.985258	−0.492629	−	0.870239i	$-0.663964\pi$
$401$	−19.7298	−0.985258	−0.492629	+	0.870239i	$0.663964\pi$
$402$	11.7809	0.587578
$403$	−26.5366	−1.32188
$404$	52.5124	2.61259
$405$	−10.1059	−0.502167
$406$	−5.81205	−0.288447
$407$	−2.02900	−0.100574
$408$	18.7970	0.930589
$409$	−34.1830	−1.69024	−0.845121	−	0.534575i	$-0.820472\pi$
$409$	−34.1830	−1.69024	−0.845121	+	0.534575i	$0.820472\pi$
$410$	−4.03970	−0.199506
$411$	1.54557	0.0762371
$412$	−83.8182	−4.12943
$413$	17.2855	0.850565
$414$	−1.43983	−0.0707636
$415$	−64.5589	−3.16907
$416$	−43.5218	−2.13383
$417$	−3.64995	−0.178739
$418$	−6.89995	−0.337487
$419$	−25.9459	−1.26754	−0.633769	−	0.773522i	$-0.718493\pi$
$419$	−25.9459	−1.26754	−0.633769	+	0.773522i	$0.718493\pi$
$420$	62.2808	3.03899
$421$	15.1335	0.737562	0.368781	−	0.929516i	$-0.379775\pi$
$421$	15.1335	0.737562	0.368781	+	0.929516i	$0.379775\pi$
$422$	14.2819	0.695232
$423$	−11.6046	−0.564234
$424$	−40.5950	−1.97147
$425$	−25.1663	−1.22075
$426$	3.20713	0.155386
$427$	8.45374	0.409105
$428$	−21.4923	−1.03887
$429$	25.7440	1.24293
$430$	−9.71247	−0.468377
$431$	−30.0446	−1.44720	−0.723600	−	0.690220i	$-0.757514\pi$
$431$	−30.0446	−1.44720	−0.723600	+	0.690220i	$0.757514\pi$
$432$	−47.2668	−2.27413
$433$	20.8937	1.00409	0.502045	−	0.864842i	$-0.332581\pi$
$433$	20.8937	1.00409	0.502045	+	0.864842i	$0.332581\pi$
$434$	34.1984	1.64158
$435$	4.29476	0.205918
$436$	−13.6796	−0.655132
$437$	−0.259926	−0.0124340
$438$	10.7567	0.513975
$439$	−3.29187	−0.157112	−0.0785562	−	0.996910i	$-0.525031\pi$
$439$	−3.29187	−0.157112	−0.0785562	+	0.996910i	$0.525031\pi$
$440$	−110.265	−5.25666
$441$	0.0457748	0.00217975
$442$	29.9265	1.42346
$443$	34.5340	1.64076	0.820381	−	0.571818i	$-0.193762\pi$
$443$	34.5340	1.64076	0.820381	+	0.571818i	$0.193762\pi$
$444$	−3.00670	−0.142692
$445$	31.3157	1.48451
$446$	1.12834	0.0534283
$447$	12.2824	0.580935
$448$	10.9985	0.519631
$449$	31.5397	1.48845	0.744225	−	0.667929i	$-0.232819\pi$
$449$	31.5397	1.48845	0.744225	+	0.667929i	$0.232819\pi$
$450$	43.4224	2.04695
$451$	1.50762	0.0709913
$452$	4.67970	0.220115
$453$	27.9345	1.31248
$454$	44.7893	2.10207
$455$	56.7793	2.66186
$456$	−5.85494	−0.274183
$457$	−16.7604	−0.784018	−0.392009	−	0.919961i	$-0.628220\pi$
$457$	−16.7604	−0.784018	−0.392009	+	0.919961i	$0.628220\pi$
$458$	46.0729	2.15284
$459$	12.1052	0.565025
$460$	−7.25391	−0.338215
$461$	−24.4907	−1.14065	−0.570324	−	0.821420i	$-0.693182\pi$
$461$	−24.4907	−1.14065	−0.570324	+	0.821420i	$0.693182\pi$
$462$	−33.1770	−1.54354
$463$	−14.8035	−0.687977	−0.343988	−	0.938974i	$-0.611778\pi$
$463$	−14.8035	−0.687977	−0.343988	+	0.938974i	$0.611778\pi$
$464$	7.27520	0.337743
$465$	−25.2706	−1.17189
$466$	−36.5885	−1.69493
$467$	−14.5761	−0.674501	−0.337250	−	0.941415i	$-0.609497\pi$
$467$	−14.5761	−0.674501	−0.337250	+	0.941415i	$0.609497\pi$
$468$	−36.1752	−1.67220
$469$	−9.69699	−0.447765
$470$	−83.4509	−3.84930
$471$	−1.88109	−0.0866762
$472$	−45.3496	−2.08738
$473$	3.62472	0.166665
$474$	34.3452	1.57753
$475$	7.83888	0.359672
$476$	−27.0195	−1.23844
$477$	−8.55873	−0.391877
$478$	8.90772	0.407430
$479$	33.9816	1.55266	0.776329	−	0.630328i	$-0.217079\pi$
$479$	33.9816	1.55266	0.776329	+	0.630328i	$0.217079\pi$
$480$	−41.4454	−1.89172
$481$	−2.74111	−0.124984
$482$	−38.0189	−1.73172
$483$	−1.24980	−0.0568680
$484$	20.3875	0.926704
$485$	19.0924	0.866943
$486$	−34.9354	−1.58470
$487$	−20.6941	−0.937740	−0.468870	−	0.883267i	$-0.655339\pi$
$487$	−20.6941	−0.937740	−0.468870	+	0.883267i	$0.655339\pi$
$488$	−22.1789	−1.00399
$489$	−11.9652	−0.541085
$490$	0.329176	0.0148707
$491$	−20.2508	−0.913906	−0.456953	−	0.889491i	$-0.651059\pi$
$491$	−20.2508	−0.913906	−0.456953	+	0.889491i	$0.651059\pi$
$492$	2.23409	0.100721
$493$	−1.86321	−0.0839149
$494$	−9.32159	−0.419398
$495$	−23.2473	−1.04489
$496$	−42.8077	−1.92212
$497$	−2.63982	−0.118412
$498$	50.9622	2.28367
$499$	−39.4584	−1.76640	−0.883200	−	0.468997i	$-0.844616\pi$
$499$	−39.4584	−1.76640	−0.883200	+	0.468997i	$0.844616\pi$
$500$	123.701	5.53208
$501$	27.9648	1.24937
$502$	−47.4482	−2.11772
$503$	−16.9558	−0.756022	−0.378011	−	0.925801i	$-0.623392\pi$
$503$	−16.9558	−0.756022	−0.378011	+	0.925801i	$0.623392\pi$
$504$	26.6957	1.18912
$505$	45.5898	2.02872
$506$	3.86417	0.171783
$507$	18.6476	0.828167
$508$	48.0713	2.13282
$509$	7.27280	0.322361	0.161181	−	0.986925i	$-0.448470\pi$
$509$	7.27280	0.322361	0.161181	+	0.986925i	$0.448470\pi$
$510$	28.4987	1.26195
$511$	−8.85396	−0.391676
$512$	48.0867	2.12515
$513$	−3.77058	−0.166475
$514$	−21.2949	−0.939277
$515$	−72.7687	−3.20657
$516$	5.37134	0.236460
$517$	31.1441	1.36972
$518$	3.53254	0.155211
$519$	−14.7831	−0.648907
$520$	−148.964	−6.53250
$521$	−5.96137	−0.261172	−0.130586	−	0.991437i	$-0.541686\pi$
$521$	−5.96137	−0.261172	−0.130586	+	0.991437i	$0.541686\pi$
$522$	3.21482	0.140709
$523$	−32.7377	−1.43152	−0.715760	−	0.698347i	$-0.753919\pi$
$523$	−32.7377	−1.43152	−0.715760	+	0.698347i	$0.753919\pi$
$524$	−49.5499	−2.16460
$525$	37.6917	1.64500
$526$	−31.3389	−1.36644
$527$	10.9632	0.477566
$528$	41.5291	1.80732
$529$	−22.8544	−0.993671
$530$	−61.5475	−2.67345
$531$	−9.56113	−0.414918
$532$	8.41613	0.364885
$533$	2.03675	0.0882214
$534$	−24.7203	−1.06975
$535$	−18.6590	−0.806699
$536$	25.4406	1.09887
$537$	12.3653	0.533602
$538$	−50.2789	−2.16768
$539$	−0.122849	−0.00529149
$540$	−105.228	−4.52828
$541$	11.0928	0.476917	0.238458	−	0.971153i	$-0.423358\pi$
$541$	11.0928	0.476917	0.238458	+	0.971153i	$0.423358\pi$
$542$	−3.68644	−0.158346
$543$	−0.447954	−0.0192236
$544$	17.9805	0.770905
$545$	−11.8762	−0.508721
$546$	−44.8210	−1.91816
$547$	−42.6823	−1.82496	−0.912481	−	0.409118i	$-0.865836\pi$
$547$	−42.6823	−1.82496	−0.912481	+	0.409118i	$0.865836\pi$
$548$	5.82865	0.248988
$549$	−4.67601	−0.199567
$550$	−116.536	−4.96911
$551$	0.580359	0.0247241
$552$	3.27893	0.139561
$553$	−28.2699	−1.20216
$554$	52.1646	2.21626
$555$	−2.61034	−0.110803
$556$	−13.7647	−0.583754
$557$	3.64811	0.154576	0.0772878	−	0.997009i	$-0.475374\pi$
$557$	3.64811	0.154576	0.0772878	+	0.997009i	$0.475374\pi$
$558$	−18.9162	−0.800785
$559$	4.89687	0.207116
$560$	91.5940	3.87055
$561$	−10.6358	−0.449044
$562$	8.85079	0.373348
$563$	33.5722	1.41490	0.707450	−	0.706764i	$-0.249846\pi$
$563$	33.5722	1.41490	0.707450	+	0.706764i	$0.249846\pi$
$564$	46.1513	1.94332
$565$	4.06279	0.170923
$566$	11.9505	0.502318
$567$	−6.56638	−0.275762
$568$	6.92572	0.290597
$569$	24.1069	1.01061	0.505307	−	0.862940i	$-0.331379\pi$
$569$	24.1069	1.01061	0.505307	+	0.862940i	$0.331379\pi$
$570$	−8.87687	−0.371811
$571$	35.5532	1.48786	0.743928	−	0.668260i	$-0.232960\pi$
$571$	35.5532	1.48786	0.743928	+	0.668260i	$0.232960\pi$
$572$	97.0861	4.05937
$573$	3.91821	0.163686
$574$	−2.62482	−0.109558
$575$	−4.38999	−0.183075
$576$	−6.08361	−0.253484
$577$	6.70355	0.279073	0.139536	−	0.990217i	$-0.455439\pi$
$577$	6.70355	0.279073	0.139536	+	0.990217i	$0.455439\pi$
$578$	31.5730	1.31326
$579$	24.4931	1.01790
$580$	16.1964	0.672520
$581$	−41.9475	−1.74028
$582$	−15.0714	−0.624729
$583$	22.9697	0.951307
$584$	23.2289	0.961218
$585$	−31.4063	−1.29849
$586$	−6.67172	−0.275606
$587$	0.906374	0.0374101	0.0187050	−	0.999825i	$-0.494046\pi$
$587$	0.906374	0.0374101	0.0187050	+	0.999825i	$0.494046\pi$
$588$	−0.182046	−0.00750745
$589$	−3.41486	−0.140707
$590$	−68.7560	−2.83064
$591$	−20.0952	−0.826606
$592$	−4.42184	−0.181736
$593$	−19.0742	−0.783282	−0.391641	−	0.920118i	$-0.628092\pi$
$593$	−19.0742	−0.783282	−0.391641	+	0.920118i	$0.628092\pi$
$594$	56.0549	2.29996
$595$	−23.4576	−0.961669
$596$	46.3193	1.89731
$597$	12.5032	0.511722
$598$	5.22036	0.213476
$599$	21.4935	0.878201	0.439101	−	0.898438i	$-0.355297\pi$
$599$	21.4935	0.878201	0.439101	+	0.898438i	$0.355297\pi$
$600$	−98.8863	−4.03702
$601$	−12.6366	−0.515457	−0.257729	−	0.966217i	$-0.582974\pi$
$601$	−12.6366	−0.515457	−0.257729	+	0.966217i	$0.582974\pi$
$602$	−6.31074	−0.257206
$603$	5.36369	0.218427
$604$	105.347	4.28650
$605$	17.6999	0.719602
$606$	−35.9882	−1.46192
$607$	16.3855	0.665068	0.332534	−	0.943091i	$-0.392096\pi$
$607$	16.3855	0.665068	0.332534	+	0.943091i	$0.392096\pi$
$608$	−5.60060	−0.227134
$609$	2.79054	0.113078
$610$	−33.6262	−1.36148
$611$	42.0746	1.70216
$612$	14.9453	0.604129
$613$	11.6926	0.472259	0.236130	−	0.971722i	$-0.424121\pi$
$613$	11.6926	0.472259	0.236130	+	0.971722i	$0.424121\pi$
$614$	−62.5979	−2.52625
$615$	1.93958	0.0782114
$616$	−71.6451	−2.88666
$617$	−10.4201	−0.419496	−0.209748	−	0.977755i	$-0.567264\pi$
$617$	−10.4201	−0.419496	−0.209748	+	0.977755i	$0.567264\pi$
$618$	57.4429	2.31069
$619$	−42.7698	−1.71906	−0.859532	−	0.511081i	$-0.829245\pi$
$619$	−42.7698	−1.71906	−0.859532	+	0.511081i	$0.829245\pi$
$620$	−95.3006	−3.82736
$621$	2.11163	0.0847368
$622$	76.2206	3.05617
$623$	20.3476	0.815208
$624$	56.1044	2.24598
$625$	49.8626	1.99450
$626$	38.4011	1.53482
$627$	3.31287	0.132303
$628$	−7.09400	−0.283081
$629$	1.13245	0.0451539
$630$	40.4742	1.61253
$631$	8.74945	0.348310	0.174155	−	0.984718i	$-0.444281\pi$
$631$	8.74945	0.348310	0.174155	+	0.984718i	$0.444281\pi$
$632$	74.1678	2.95024
$633$	−6.85717	−0.272548
$634$	66.2471	2.63101
$635$	41.7342	1.65617
$636$	34.0380	1.34969
$637$	−0.165965	−0.00657578
$638$	−8.62785	−0.341580
$639$	1.46016	0.0577632
$640$	23.0504	0.911148
$641$	−25.9523	−1.02506	−0.512528	−	0.858671i	$-0.671291\pi$
$641$	−25.9523	−1.02506	−0.512528	+	0.858671i	$0.671291\pi$
$642$	14.7292	0.581317
$643$	25.6262	1.01060	0.505300	−	0.862944i	$-0.331382\pi$
$643$	25.6262	1.01060	0.505300	+	0.862944i	$0.331382\pi$
$644$	−4.71327	−0.185729
$645$	4.66325	0.183615
$646$	3.85109	0.151519
$647$	−16.5570	−0.650922	−0.325461	−	0.945556i	$-0.605519\pi$
$647$	−16.5570	−0.650922	−0.325461	+	0.945556i	$0.605519\pi$
$648$	17.2273	0.676752
$649$	25.6599	1.00724
$650$	−157.436	−6.17515
$651$	−16.4197	−0.643538
$652$	−45.1233	−1.76717
$653$	10.9660	0.429131	0.214565	−	0.976710i	$-0.431166\pi$
$653$	10.9660	0.429131	0.214565	+	0.976710i	$0.431166\pi$
$654$	9.37496	0.366590
$655$	−43.0179	−1.68085
$656$	3.28560	0.128281
$657$	4.89739	0.191065
$658$	−54.2227	−2.11382
$659$	1.67043	0.0650708	0.0325354	−	0.999471i	$-0.489642\pi$
$659$	1.67043	0.0650708	0.0325354	+	0.999471i	$0.489642\pi$
$660$	92.4543	3.59878
$661$	0.0916292	0.00356396	0.00178198	−	0.999998i	$-0.499433\pi$
$661$	0.0916292	0.00356396	0.00178198	+	0.999998i	$0.499433\pi$
$662$	−24.7685	−0.962653
$663$	−14.3686	−0.558030
$664$	110.052	4.27084
$665$	7.30665	0.283340
$666$	−1.95395	−0.0757142
$667$	−0.325018	−0.0125847
$668$	105.461	4.08041
$669$	−0.541748	−0.0209452
$670$	38.5714	1.49014
$671$	12.5494	0.484463
$672$	−26.9294	−1.03882
$673$	23.2845	0.897550	0.448775	−	0.893645i	$-0.351860\pi$
$673$	23.2845	0.897550	0.448775	+	0.893645i	$0.351860\pi$
$674$	−63.2970	−2.43811
$675$	−63.6828	−2.45115
$676$	70.3239	2.70476
$677$	−33.6378	−1.29280	−0.646402	−	0.762997i	$-0.723727\pi$
$677$	−33.6378	−1.29280	−0.646402	+	0.762997i	$0.723727\pi$
$678$	−3.20713	−0.123169
$679$	12.4054	0.476076
$680$	61.5425	2.36005
$681$	−21.5047	−0.824062
$682$	50.7667	1.94396
$683$	−46.2845	−1.77103	−0.885513	−	0.464614i	$-0.846193\pi$
$683$	−46.2845	−1.77103	−0.885513	+	0.464614i	$0.846193\pi$
$684$	−4.65521	−0.177996
$685$	5.06028	0.193343
$686$	47.9724	1.83160
$687$	−22.1210	−0.843968
$688$	7.89942	0.301163
$689$	31.0313	1.18220
$690$	4.97130	0.189254
$691$	−20.2936	−0.772003	−0.386002	−	0.922498i	$-0.626144\pi$
$691$	−20.2936	−0.772003	−0.386002	+	0.922498i	$0.626144\pi$
$692$	−55.7502	−2.11931
$693$	−15.1051	−0.573794
$694$	−37.6488	−1.42913
$695$	−11.9502	−0.453295
$696$	−7.32115	−0.277507
$697$	−0.841456	−0.0318724
$698$	34.2986	1.29822
$699$	17.5672	0.664454
$700$	142.143	5.37251
$701$	30.7612	1.16184	0.580918	−	0.813962i	$-0.302694\pi$
$701$	30.7612	1.16184	0.580918	+	0.813962i	$0.302694\pi$
$702$	75.7283	2.85818
$703$	−0.352740	−0.0133038
$704$	16.3270	0.615348
$705$	40.0673	1.50902
$706$	27.6910	1.04216
$707$	29.6223	1.11406
$708$	38.0245	1.42905
$709$	−1.17787	−0.0442358	−0.0221179	−	0.999755i	$-0.507041\pi$
$709$	−1.17787	−0.0442358	−0.0221179	+	0.999755i	$0.507041\pi$
$710$	10.5003	0.394070
$711$	15.6369	0.586431
$712$	−53.3830	−2.00061
$713$	1.91242	0.0716207
$714$	18.5172	0.692990
$715$	84.2875	3.15217
$716$	46.6321	1.74272
$717$	−4.27686	−0.159722
$718$	−23.5766	−0.879872
$719$	−38.6727	−1.44225	−0.721125	−	0.692805i	$-0.756374\pi$
$719$	−38.6727	−1.44225	−0.721125	+	0.692805i	$0.756374\pi$
$720$	−50.6633	−1.88811
$721$	−47.2819	−1.76087
$722$	47.9062	1.78288
$723$	18.2540	0.678875
$724$	−1.68933	−0.0627834
$725$	9.80191	0.364034
$726$	−13.9721	−0.518553
$727$	26.3117	0.975849	0.487924	−	0.872886i	$-0.337754\pi$
$727$	26.3117	0.975849	0.487924	+	0.872886i	$0.337754\pi$
$728$	−96.7901	−3.58728
$729$	24.2358	0.897623
$730$	35.2181	1.30348
$731$	−2.02308	−0.0748263
$732$	18.5964	0.687344
$733$	−24.4610	−0.903488	−0.451744	−	0.892148i	$-0.649198\pi$
$733$	−24.4610	−0.903488	−0.451744	+	0.892148i	$0.649198\pi$
$734$	3.50819	0.129490
$735$	−0.158047	−0.00582966
$736$	3.13650	0.115613
$737$	−14.3949	−0.530244
$738$	1.45186	0.0534438
$739$	−33.5267	−1.23330	−0.616649	−	0.787238i	$-0.711510\pi$
$739$	−33.5267	−1.23330	−0.616649	+	0.787238i	$0.711510\pi$
$740$	−9.84412	−0.361877
$741$	4.47558	0.164414
$742$	−39.9909	−1.46811
$743$	12.5083	0.458883	0.229442	−	0.973322i	$-0.426310\pi$
$743$	12.5083	0.458883	0.229442	+	0.973322i	$0.426310\pi$
$744$	43.0780	1.57932
$745$	40.2132	1.47330
$746$	−61.3424	−2.24591
$747$	23.2024	0.848933
$748$	−40.1098	−1.46656
$749$	−12.1238	−0.442994
$750$	−84.7756	−3.09557
$751$	21.3683	0.779739	0.389870	−	0.920870i	$-0.372520\pi$
$751$	21.3683	0.779739	0.389870	+	0.920870i	$0.372520\pi$
$752$	67.8729	2.47507
$753$	22.7813	0.830197
$754$	−11.6559	−0.424484
$755$	91.4592	3.32854
$756$	−68.3724	−2.48668
$757$	5.83375	0.212031	0.106016	−	0.994364i	$-0.466191\pi$
$757$	5.83375	0.212031	0.106016	+	0.994364i	$0.466191\pi$
$758$	64.5116	2.34317
$759$	−1.85530	−0.0673432
$760$	−19.1694	−0.695348
$761$	0.764555	0.0277151	0.0138575	−	0.999904i	$-0.495589\pi$
$761$	0.764555	0.0277151	0.0138575	+	0.999904i	$0.495589\pi$
$762$	−32.9446	−1.19346
$763$	−7.71664	−0.279361
$764$	14.7764	0.534591
$765$	12.9751	0.469116
$766$	−66.0932	−2.38805
$767$	34.6657	1.25170
$768$	−28.5359	−1.02970
$769$	−40.6294	−1.46513	−0.732567	−	0.680695i	$-0.761678\pi$
$769$	−40.6294	−1.46513	−0.732567	+	0.680695i	$0.761678\pi$
$770$	−108.624	−3.91452
$771$	10.2243	0.368220
$772$	92.3687	3.32442
$773$	9.53455	0.342934	0.171467	−	0.985190i	$-0.445149\pi$
$773$	9.53455	0.342934	0.171467	+	0.985190i	$0.445149\pi$
$774$	3.49066	0.125469
$775$	−57.6749	−2.07174
$776$	−32.5463	−1.16835
$777$	−1.69608	−0.0608465
$778$	46.5605	1.66927
$779$	0.262099	0.00939068
$780$	124.903	4.47223
$781$	−3.91875	−0.140224
$782$	−2.15672	−0.0771242
$783$	−4.71482	−0.168494
$784$	−0.267728	−0.00956171
$785$	−6.15881	−0.219817
$786$	33.9579	1.21124
$787$	−9.89660	−0.352776	−0.176388	−	0.984321i	$-0.556441\pi$
$787$	−9.89660	−0.352776	−0.176388	+	0.984321i	$0.556441\pi$
$788$	−75.7832	−2.69967
$789$	15.0467	0.535678
$790$	112.448	4.00073
$791$	2.63982	0.0938612
$792$	39.6290	1.40816
$793$	16.9538	0.602046
$794$	29.6920	1.05373
$795$	29.5508	1.04806
$796$	47.1522	1.67127
$797$	3.40771	0.120707	0.0603537	−	0.998177i	$-0.480777\pi$
$797$	3.40771	0.120707	0.0603537	+	0.998177i	$0.480777\pi$
$798$	−5.76780	−0.204178
$799$	−17.3826	−0.614951
$800$	−94.5908	−3.34429
$801$	−11.2548	−0.397670
$802$	50.9919	1.80059
$803$	−13.1435	−0.463824
$804$	−21.3313	−0.752299
$805$	−4.09193	−0.144222
$806$	68.5841	2.41577
$807$	24.1404	0.849782
$808$	−77.7157	−2.73403
$809$	38.6154	1.35764	0.678822	−	0.734303i	$-0.262491\pi$
$809$	38.6154	1.35764	0.678822	+	0.734303i	$0.262491\pi$
$810$	26.1189	0.917724
$811$	8.42857	0.295967	0.147984	−	0.988990i	$-0.452722\pi$
$811$	8.42857	0.295967	0.147984	+	0.988990i	$0.452722\pi$
$812$	10.5237	0.369310
$813$	1.76997	0.0620755
$814$	5.24397	0.183801
$815$	−39.1748	−1.37223
$816$	−23.1788	−0.811421
$817$	0.630154	0.0220463
$818$	88.3465	3.08896
$819$	−20.4064	−0.713059
$820$	7.31456	0.255436
$821$	4.64617	0.162153	0.0810763	−	0.996708i	$-0.474164\pi$
$821$	4.64617	0.162153	0.0810763	+	0.996708i	$0.474164\pi$
$822$	−3.99453	−0.139325
$823$	2.23620	0.0779490	0.0389745	−	0.999240i	$-0.487591\pi$
$823$	2.23620	0.0779490	0.0389745	+	0.999240i	$0.487591\pi$
$824$	124.047	4.32137
$825$	55.9524	1.94801
$826$	−44.6746	−1.55443
$827$	43.6829	1.51900	0.759502	−	0.650505i	$-0.225443\pi$
$827$	43.6829	1.51900	0.759502	+	0.650505i	$0.225443\pi$
$828$	2.60705	0.0906013
$829$	−28.3873	−0.985931	−0.492966	−	0.870049i	$-0.664087\pi$
$829$	−28.3873	−0.985931	−0.492966	+	0.870049i	$0.664087\pi$
$830$	166.853	5.79156
$831$	−25.0458	−0.868830
$832$	22.0573	0.764698
$833$	0.0685663	0.00237568
$834$	9.43334	0.326650
$835$	91.5584	3.16851
$836$	12.4935	0.432098
$837$	27.7422	0.958912
$838$	67.0574	2.31646
$839$	−36.3591	−1.25526	−0.627629	−	0.778513i	$-0.715974\pi$
$839$	−36.3591	−1.25526	−0.627629	+	0.778513i	$0.715974\pi$
$840$	−92.1724	−3.18025
$841$	−28.2743	−0.974976
$842$	−39.1127	−1.34791
$843$	−4.24953	−0.146362
$844$	−25.8598	−0.890132
$845$	61.0533	2.10030
$846$	29.9922	1.03115
$847$	11.5006	0.395165
$848$	50.0583	1.71901
$849$	−5.73780	−0.196921
$850$	65.0426	2.23094
$851$	0.197544	0.00677173
$852$	−5.80705	−0.198946
$853$	37.5913	1.28710	0.643551	−	0.765403i	$-0.277460\pi$
$853$	37.5913	1.28710	0.643551	+	0.765403i	$0.277460\pi$
$854$	−21.8488	−0.747650
$855$	−4.04153	−0.138217
$856$	31.8075	1.08716
$857$	37.1746	1.26986	0.634930	−	0.772570i	$-0.281029\pi$
$857$	37.1746	1.26986	0.634930	+	0.772570i	$0.281029\pi$
$858$	−66.5357	−2.27149
$859$	−52.3507	−1.78618	−0.893092	−	0.449874i	$-0.851469\pi$
$859$	−52.3507	−1.78618	−0.893092	+	0.449874i	$0.851469\pi$
$860$	17.5861	0.599681
$861$	1.26025	0.0429493
$862$	77.6507	2.64479
$863$	12.0153	0.409004	0.204502	−	0.978866i	$-0.434442\pi$
$863$	12.0153	0.409004	0.204502	+	0.978866i	$0.434442\pi$
$864$	45.4991	1.54791
$865$	−48.4008	−1.64568
$866$	−54.0001	−1.83500
$867$	−15.1591	−0.514831
$868$	−61.9221	−2.10177
$869$	−41.9660	−1.42360
$870$	−11.0998	−0.376320
$871$	−19.4471	−0.658939
$872$	20.2450	0.685584
$873$	−6.86181	−0.232237
$874$	0.671782	0.0227234
$875$	69.7798	2.35899
$876$	−19.4769	−0.658062
$877$	−53.9180	−1.82068	−0.910341	−	0.413859i	$-0.864181\pi$
$877$	−53.9180	−1.82068	−0.910341	+	0.413859i	$0.864181\pi$
$878$	8.50788	0.287127
$879$	3.20330	0.108044
$880$	135.969	4.58351
$881$	7.82763	0.263719	0.131860	−	0.991268i	$-0.457905\pi$
$881$	7.82763	0.263719	0.131860	+	0.991268i	$0.457905\pi$
$882$	−0.118306	−0.00398356
$883$	−20.2295	−0.680778	−0.340389	−	0.940285i	$-0.610559\pi$
$883$	−20.2295	−0.680778	−0.340389	+	0.940285i	$0.610559\pi$
$884$	−54.1870	−1.82251
$885$	33.0118	1.10968
$886$	−89.2536	−2.99853
$887$	11.4131	0.383215	0.191607	−	0.981472i	$-0.438630\pi$
$887$	11.4131	0.383215	0.191607	+	0.981472i	$0.438630\pi$
$888$	4.44976	0.149324
$889$	27.1170	0.909476
$890$	−80.9359	−2.71298
$891$	−9.74763	−0.326558
$892$	−2.04305	−0.0684062
$893$	5.41437	0.181185
$894$	−31.7439	−1.06167
$895$	40.4847	1.35326
$896$	14.9771	0.500351
$897$	−2.50645	−0.0836879
$898$	−81.5147	−2.72018
$899$	−4.27002	−0.142413
$900$	−78.6237	−2.62079
$901$	−12.8202	−0.427101
$902$	−3.89647	−0.129738
$903$	3.02997	0.100831
$904$	−6.92572	−0.230346
$905$	−1.46663	−0.0487524
$906$	−72.1970	−2.39858
$907$	3.42706	0.113794	0.0568968	−	0.998380i	$-0.481879\pi$
$907$	3.42706	0.113794	0.0568968	+	0.998380i	$0.481879\pi$
$908$	−81.0988	−2.69136
$909$	−16.3850	−0.543455
$910$	−146.747	−4.86461
$911$	47.0474	1.55875	0.779376	−	0.626557i	$-0.215536\pi$
$911$	47.0474	1.55875	0.779376	+	0.626557i	$0.215536\pi$
$912$	7.21981	0.239072
$913$	−62.2701	−2.06084
$914$	43.3174	1.43281
$915$	16.1449	0.533735
$916$	−83.4229	−2.75637
$917$	−27.9511	−0.923027
$918$	−31.2862	−1.03260
$919$	23.8982	0.788330	0.394165	−	0.919040i	$-0.371034\pi$
$919$	23.8982	0.788330	0.394165	+	0.919040i	$0.371034\pi$
$920$	10.7354	0.353936
$921$	30.0551	0.990351
$922$	63.2966	2.08456
$923$	−5.29409	−0.174257
$924$	60.0727	1.97625
$925$	−5.95756	−0.195883
$926$	38.2598	1.25730
$927$	26.1530	0.858977
$928$	−7.00312	−0.229889
$929$	−22.2401	−0.729675	−0.364837	−	0.931071i	$-0.618875\pi$
$929$	−22.2401	−0.729675	−0.364837	+	0.931071i	$0.618875\pi$
$930$	65.3120	2.14167
$931$	−0.0213572	−0.000699955 0
$932$	66.2497	2.17008
$933$	−36.5958	−1.19809
$934$	37.6721	1.23267
$935$	−34.8223	−1.13881
$936$	53.5375	1.74993
$937$	17.6728	0.577346	0.288673	−	0.957428i	$-0.406786\pi$
$937$	17.6728	0.577346	0.288673	+	0.957428i	$0.406786\pi$
$938$	25.0620	0.818303
$939$	−18.4375	−0.601686
$940$	151.102	4.92841
$941$	17.6564	0.575584	0.287792	−	0.957693i	$-0.407079\pi$
$941$	17.6564	0.575584	0.287792	+	0.957693i	$0.407079\pi$
$942$	4.86171	0.158403
$943$	−0.146783	−0.00477991
$944$	55.9212	1.82008
$945$	−59.3590	−1.93095
$946$	−9.36813	−0.304584
$947$	−28.1138	−0.913575	−0.456788	−	0.889576i	$-0.651000\pi$
$947$	−28.1138	−0.913575	−0.456788	+	0.889576i	$0.651000\pi$
$948$	−62.1879	−2.01977
$949$	−17.7564	−0.576397
$950$	−20.2597	−0.657310
$951$	−31.8072	−1.03142
$952$	39.9876	1.29600
$953$	18.8602	0.610943	0.305471	−	0.952201i	$-0.401186\pi$
$953$	18.8602	0.610943	0.305471	+	0.952201i	$0.401186\pi$
$954$	22.1201	0.716166
$955$	12.8285	0.415119
$956$	−16.1290	−0.521648
$957$	4.14249	0.133908
$958$	−87.8258	−2.83752
$959$	3.28794	0.106173
$960$	21.0050	0.677932
$961$	−5.87499	−0.189516
$962$	7.08443	0.228411
$963$	6.70603	0.216099
$964$	68.8398	2.21718
$965$	80.1919	2.58147
$966$	3.23013	0.103928
$967$	−38.5097	−1.23839	−0.619195	−	0.785237i	$-0.712541\pi$
$967$	−38.5097	−1.23839	−0.619195	+	0.785237i	$0.712541\pi$
$968$	−30.1724	−0.969779
$969$	−1.84903	−0.0593993
$970$	−49.3446	−1.58436
$971$	−1.01746	−0.0326519	−0.0163260	−	0.999867i	$-0.505197\pi$
$971$	−1.01746	−0.0326519	−0.0163260	+	0.999867i	$0.505197\pi$
$972$	63.2565	2.02895
$973$	−7.76468	−0.248924
$974$	53.4842	1.71374
$975$	75.5897	2.42081
$976$	27.3491	0.875423
$977$	−31.8426	−1.01873	−0.509367	−	0.860549i	$-0.670121\pi$
$977$	−31.8426	−1.01873	−0.509367	+	0.860549i	$0.670121\pi$
$978$	30.9242	0.988848
$979$	30.2055	0.965371
$980$	−0.596030	−0.0190395
$981$	4.26830	0.136276
$982$	52.3384	1.67019
$983$	33.7639	1.07690	0.538450	−	0.842657i	$-0.319010\pi$
$983$	33.7639	1.07690	0.538450	+	0.842657i	$0.319010\pi$
$984$	−3.30635	−0.105402
$985$	−65.7929	−2.09634
$986$	4.81550	0.153357
$987$	26.0339	0.828670
$988$	16.8783	0.536971
$989$	−0.352904	−0.0112217
$990$	60.0830	1.90956
$991$	36.0814	1.14616	0.573082	−	0.819498i	$-0.305748\pi$
$991$	36.0814	1.14616	0.573082	+	0.819498i	$0.305748\pi$
$992$	41.2067	1.30831
$993$	11.8921	0.377384
$994$	6.82265	0.216401
$995$	40.9363	1.29777
$996$	−92.2758	−2.92387
$997$	−19.1232	−0.605637	−0.302819	−	0.953048i	$-0.597928\pi$
$997$	−19.1232	−0.605637	−0.302819	+	0.953048i	$0.597928\pi$
$998$	101.981	3.22814
$999$	2.86565	0.0906651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8023.2.a.e.1.8	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8023.2.a.e.1.8	✓	172	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:	\( N \)	\(=\)	\( 8023 = 71 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58451 q^{2} +1.24090 q^{3} +4.67970 q^{4} +4.06279 q^{5} -3.20713 q^{6} +2.63982 q^{7} -6.92572 q^{8} -1.46016 q^{9} +O(q^{10})\)	\(q-2.58451 q^{2} +1.24090 q^{3} +4.67970 q^{4} +4.06279 q^{5} -3.20713 q^{6} +2.63982 q^{7} -6.92572 q^{8} -1.46016 q^{9} -10.5003 q^{10} +3.91875 q^{11} +5.80705 q^{12} +5.29409 q^{13} -6.82265 q^{14} +5.04152 q^{15} +8.54020 q^{16} -2.18719 q^{17} +3.77381 q^{18} +0.681271 q^{19} +19.0126 q^{20} +3.27576 q^{21} -10.1280 q^{22} -0.381531 q^{23} -8.59414 q^{24} +11.5063 q^{25} -13.6826 q^{26} -5.53462 q^{27} +12.3536 q^{28} +0.851877 q^{29} -13.0299 q^{30} -5.01249 q^{31} -8.22082 q^{32} +4.86278 q^{33} +5.65281 q^{34} +10.7250 q^{35} -6.83313 q^{36} -0.517767 q^{37} -1.76075 q^{38} +6.56945 q^{39} -28.1377 q^{40} +0.384721 q^{41} -8.46623 q^{42} +0.924969 q^{43} +18.3386 q^{44} -5.93233 q^{45} +0.986072 q^{46} +7.94746 q^{47} +10.5976 q^{48} -0.0313491 q^{49} -29.7380 q^{50} -2.71408 q^{51} +24.7748 q^{52} +5.86149 q^{53} +14.3043 q^{54} +15.9210 q^{55} -18.2827 q^{56} +0.845390 q^{57} -2.20169 q^{58} +6.54799 q^{59} +23.5928 q^{60} +3.20239 q^{61} +12.9548 q^{62} -3.85457 q^{63} +4.16639 q^{64} +21.5088 q^{65} -12.5679 q^{66} -3.67335 q^{67} -10.2354 q^{68} -0.473443 q^{69} -27.7190 q^{70} -1.00000 q^{71} +10.1127 q^{72} -3.35400 q^{73} +1.33818 q^{74} +14.2781 q^{75} +3.18814 q^{76} +10.3448 q^{77} -16.9788 q^{78} -10.7090 q^{79} +34.6970 q^{80} -2.48744 q^{81} -0.994316 q^{82} -15.8903 q^{83} +15.3296 q^{84} -8.88607 q^{85} -2.39059 q^{86} +1.05710 q^{87} -27.1401 q^{88} +7.70794 q^{89} +15.3322 q^{90} +13.9755 q^{91} -1.78545 q^{92} -6.22000 q^{93} -20.5403 q^{94} +2.76786 q^{95} -10.2012 q^{96} +4.69934 q^{97} +0.0810222 q^{98} -5.72201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9}+O(q^{10}) \) 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9	\( 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9} + 14 q^{10} + 20 q^{11} + 54 q^{12} + 36 q^{13} + 26 q^{14} + 32 q^{15} + 196 q^{16} + 123 q^{17} + 74 q^{18} + 20 q^{19} + 70 q^{20} + 37 q^{21} + 11 q^{22} + 22 q^{23} + 62 q^{24} + 210 q^{25} + 50 q^{26} + 69 q^{27} + 42 q^{28} + 58 q^{29} + 36 q^{30} + 10 q^{31} + 168 q^{32} + 124 q^{33} + 5 q^{34} + 59 q^{35} + 192 q^{36} + 40 q^{37} + 58 q^{38} + 15 q^{39} + 7 q^{40} + 155 q^{41} - 6 q^{42} + 19 q^{43} + 22 q^{44} + 76 q^{45} + q^{46} + 71 q^{47} + 144 q^{48} + 206 q^{49} + 126 q^{50} + 33 q^{51} + 71 q^{52} + 101 q^{53} + 92 q^{54} - 2 q^{55} + 57 q^{56} + 114 q^{57} + 4 q^{58} + 71 q^{59} + 38 q^{60} + 50 q^{61} + 86 q^{62} + 14 q^{63} + 240 q^{64} + 143 q^{65} + 21 q^{66} + 8 q^{67} + 192 q^{68} + 41 q^{69} - 12 q^{70} - 172 q^{71} + 156 q^{72} + 128 q^{73} + 30 q^{74} + 72 q^{75} + 74 q^{76} + 127 q^{77} + 107 q^{78} + 2 q^{79} + 50 q^{80} + 236 q^{81} + 42 q^{82} + 140 q^{83} + 71 q^{84} + 55 q^{85} + 46 q^{86} + 100 q^{87} - 31 q^{88} + 215 q^{89} - 7 q^{90} + 22 q^{91} - 15 q^{92} + 60 q^{93} + 5 q^{94} + 74 q^{95} + 182 q^{96} + 120 q^{97} + 164 q^{98} + 60 q^{99}+O(q^{100}) \) 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9 + 14 * q^10 + 20 * q^11 + 54 * q^12 + 36 * q^13 + 26 * q^14 + 32 * q^15 + 196 * q^16 + 123 * q^17 + 74 * q^18 + 20 * q^19 + 70 * q^20 + 37 * q^21 + 11 * q^22 + 22 * q^23 + 62 * q^24 + 210 * q^25 + 50 * q^26 + 69 * q^27 + 42 * q^28 + 58 * q^29 + 36 * q^30 + 10 * q^31 + 168 * q^32 + 124 * q^33 + 5 * q^34 + 59 * q^35 + 192 * q^36 + 40 * q^37 + 58 * q^38 + 15 * q^39 + 7 * q^40 + 155 * q^41 - 6 * q^42 + 19 * q^43 + 22 * q^44 + 76 * q^45 + q^46 + 71 * q^47 + 144 * q^48 + 206 * q^49 + 126 * q^50 + 33 * q^51 + 71 * q^52 + 101 * q^53 + 92 * q^54 - 2 * q^55 + 57 * q^56 + 114 * q^57 + 4 * q^58 + 71 * q^59 + 38 * q^60 + 50 * q^61 + 86 * q^62 + 14 * q^63 + 240 * q^64 + 143 * q^65 + 21 * q^66 + 8 * q^67 + 192 * q^68 + 41 * q^69 - 12 * q^70 - 172 * q^71 + 156 * q^72 + 128 * q^73 + 30 * q^74 + 72 * q^75 + 74 * q^76 + 127 * q^77 + 107 * q^78 + 2 * q^79 + 50 * q^80 + 236 * q^81 + 42 * q^82 + 140 * q^83 + 71 * q^84 + 55 * q^85 + 46 * q^86 + 100 * q^87 - 31 * q^88 + 215 * q^89 - 7 * q^90 + 22 * q^91 - 15 * q^92 + 60 * q^93 + 5 * q^94 + 74 * q^95 + 182 * q^96 + 120 * q^97 + 164 * q^98 + 60 * q^99

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