LMFDB - Embedded newform 8027.2.a.f.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8027,2,Mod(1,8027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8027, 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("8027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8027 = 23 \cdot 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8027.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:	$64.0959177025$
Analytic rank:	$0$
Dimension:	$176$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45769 q^{2} +0.342548 q^{3} +4.04026 q^{4} -0.529727 q^{5} -0.841878 q^{6} -2.69173 q^{7} -5.01433 q^{8} -2.88266 q^{9} +O(q^{10})$	\(q-2.45769 q^{2} +0.342548 q^{3} +4.04026 q^{4} -0.529727 q^{5} -0.841878 q^{6} -2.69173 q^{7} -5.01433 q^{8} -2.88266 q^{9} +1.30191 q^{10} +3.44506 q^{11} +1.38398 q^{12} -3.97647 q^{13} +6.61545 q^{14} -0.181457 q^{15} +4.24317 q^{16} +4.73579 q^{17} +7.08470 q^{18} +0.229013 q^{19} -2.14024 q^{20} -0.922047 q^{21} -8.46689 q^{22} +1.00000 q^{23} -1.71765 q^{24} -4.71939 q^{25} +9.77295 q^{26} -2.01509 q^{27} -10.8753 q^{28} +1.12006 q^{29} +0.445966 q^{30} -2.98049 q^{31} -0.399747 q^{32} +1.18010 q^{33} -11.6391 q^{34} +1.42588 q^{35} -11.6467 q^{36} +1.41985 q^{37} -0.562845 q^{38} -1.36213 q^{39} +2.65623 q^{40} +4.11935 q^{41} +2.26611 q^{42} -8.35321 q^{43} +13.9189 q^{44} +1.52702 q^{45} -2.45769 q^{46} +4.03088 q^{47} +1.45349 q^{48} +0.245415 q^{49} +11.5988 q^{50} +1.62223 q^{51} -16.0660 q^{52} -2.06114 q^{53} +4.95248 q^{54} -1.82494 q^{55} +13.4972 q^{56} +0.0784480 q^{57} -2.75276 q^{58} -4.70019 q^{59} -0.733133 q^{60} +5.09790 q^{61} +7.32514 q^{62} +7.75935 q^{63} -7.50388 q^{64} +2.10645 q^{65} -2.90032 q^{66} +11.6013 q^{67} +19.1338 q^{68} +0.342548 q^{69} -3.50439 q^{70} -9.99153 q^{71} +14.4546 q^{72} +4.39154 q^{73} -3.48956 q^{74} -1.61662 q^{75} +0.925273 q^{76} -9.27316 q^{77} +3.34770 q^{78} +6.05625 q^{79} -2.24772 q^{80} +7.95772 q^{81} -10.1241 q^{82} -4.61756 q^{83} -3.72531 q^{84} -2.50868 q^{85} +20.5296 q^{86} +0.383673 q^{87} -17.2746 q^{88} -16.5032 q^{89} -3.75296 q^{90} +10.7036 q^{91} +4.04026 q^{92} -1.02096 q^{93} -9.90666 q^{94} -0.121315 q^{95} -0.136932 q^{96} -18.0047 q^{97} -0.603154 q^{98} -9.93093 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) $ 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.45769	−1.73785	−0.868926	−	0.494942i	$-0.835189\pi$
$2$	−2.45769	−1.73785	−0.868926	+	0.494942i	$0.835189\pi$
$3$	0.342548	0.197770	0.0988851	−	0.995099i	$-0.468472\pi$
$3$	0.342548	0.197770	0.0988851	+	0.995099i	$0.468472\pi$
$4$	4.04026	2.02013
$5$	−0.529727	−0.236901	−0.118451	−	0.992960i	$-0.537793\pi$
$5$	−0.529727	−0.236901	−0.118451	+	0.992960i	$0.537793\pi$
$6$	−0.841878	−0.343695
$7$	−2.69173	−1.01738	−0.508689	−	0.860950i	$-0.669870\pi$
$7$	−2.69173	−1.01738	−0.508689	+	0.860950i	$0.669870\pi$
$8$	−5.01433	−1.77283
$9$	−2.88266	−0.960887
$10$	1.30191	0.411699
$11$	3.44506	1.03872	0.519362	−	0.854555i	$-0.326170\pi$
$11$	3.44506	1.03872	0.519362	+	0.854555i	$0.326170\pi$
$12$	1.38398	0.399521
$13$	−3.97647	−1.10287	−0.551437	−	0.834216i	$-0.685920\pi$
$13$	−3.97647	−1.10287	−0.551437	+	0.834216i	$0.685920\pi$
$14$	6.61545	1.76805
$15$	−0.181457	−0.0468520
$16$	4.24317	1.06079
$17$	4.73579	1.14860	0.574299	−	0.818646i	$-0.305275\pi$
$17$	4.73579	1.14860	0.574299	+	0.818646i	$0.305275\pi$
$18$	7.08470	1.66988
$19$	0.229013	0.0525393	0.0262696	−	0.999655i	$-0.491637\pi$
$19$	0.229013	0.0525393	0.0262696	+	0.999655i	$0.491637\pi$
$20$	−2.14024	−0.478571
$21$	−0.922047	−0.201207
$22$	−8.46689	−1.80515
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.71765	−0.350613
$25$	−4.71939	−0.943878
$26$	9.77295	1.91663
$27$	−2.01509	−0.387805
$28$	−10.8753	−2.05524
$29$	1.12006	0.207989	0.103995	−	0.994578i	$-0.466838\pi$
$29$	1.12006	0.207989	0.103995	+	0.994578i	$0.466838\pi$
$30$	0.445966	0.0814218
$31$	−2.98049	−0.535313	−0.267656	−	0.963514i	$-0.586249\pi$
$31$	−2.98049	−0.535313	−0.267656	+	0.963514i	$0.586249\pi$
$32$	−0.399747	−0.0706660
$33$	1.18010	0.205428
$34$	−11.6391	−1.99609
$35$	1.42588	0.241018
$36$	−11.6467	−1.94112
$37$	1.41985	0.233422	0.116711	−	0.993166i	$-0.462765\pi$
$37$	1.41985	0.233422	0.116711	+	0.993166i	$0.462765\pi$
$38$	−0.562845	−0.0913055
$39$	−1.36213	−0.218116
$40$	2.65623	0.419986
$41$	4.11935	0.643334	0.321667	−	0.946853i	$-0.395757\pi$
$41$	4.11935	0.643334	0.321667	+	0.946853i	$0.395757\pi$
$42$	2.26611	0.349668
$43$	−8.35321	−1.27385	−0.636926	−	0.770925i	$-0.719794\pi$
$43$	−8.35321	−1.27385	−0.636926	+	0.770925i	$0.719794\pi$
$44$	13.9189	2.09836
$45$	1.52702	0.227635
$46$	−2.45769	−0.362367
$47$	4.03088	0.587964	0.293982	−	0.955811i	$-0.405019\pi$
$47$	4.03088	0.587964	0.293982	+	0.955811i	$0.405019\pi$
$48$	1.45349	0.209793
$49$	0.245415	0.0350593
$50$	11.5988	1.64032
$51$	1.62223	0.227158
$52$	−16.0660	−2.22795
$53$	−2.06114	−0.283119	−0.141559	−	0.989930i	$-0.545212\pi$
$53$	−2.06114	−0.283119	−0.141559	+	0.989930i	$0.545212\pi$
$54$	4.95248	0.673947
$55$	−1.82494	−0.246075
$56$	13.4972	1.80364
$57$	0.0784480	0.0103907
$58$	−2.75276	−0.361455
$59$	−4.70019	−0.611913	−0.305956	−	0.952046i	$-0.598976\pi$
$59$	−4.70019	−0.611913	−0.305956	+	0.952046i	$0.598976\pi$
$60$	−0.733133	−0.0946471
$61$	5.09790	0.652720	0.326360	−	0.945246i	$-0.394178\pi$
$61$	5.09790	0.652720	0.326360	+	0.945246i	$0.394178\pi$
$62$	7.32514	0.930294
$63$	7.75935	0.977586
$64$	−7.50388	−0.937985
$65$	2.10645	0.261272
$66$	−2.90032	−0.357004
$67$	11.6013	1.41732	0.708660	−	0.705550i	$-0.249300\pi$
$67$	11.6013	1.41732	0.708660	+	0.705550i	$0.249300\pi$
$68$	19.1338	2.32031
$69$	0.342548	0.0412379
$70$	−3.50439	−0.418854
$71$	−9.99153	−1.18578	−0.592888	−	0.805285i	$-0.702012\pi$
$71$	−9.99153	−1.18578	−0.592888	+	0.805285i	$0.702012\pi$
$72$	14.4546	1.70349
$73$	4.39154	0.513991	0.256996	−	0.966413i	$-0.417267\pi$
$73$	4.39154	0.513991	0.256996	+	0.966413i	$0.417267\pi$
$74$	−3.48956	−0.405653
$75$	−1.61662	−0.186671
$76$	0.925273	0.106136
$77$	−9.27316	−1.05677
$78$	3.34770	0.379053
$79$	6.05625	0.681382	0.340691	−	0.940175i	$-0.389339\pi$
$79$	6.05625	0.681382	0.340691	+	0.940175i	$0.389339\pi$
$80$	−2.24772	−0.251303
$81$	7.95772	0.884191
$82$	−10.1241	−1.11802
$83$	−4.61756	−0.506843	−0.253422	−	0.967356i	$-0.581556\pi$
$83$	−4.61756	−0.506843	−0.253422	+	0.967356i	$0.581556\pi$
$84$	−3.72531	−0.406464
$85$	−2.50868	−0.272104
$86$	20.5296	2.21377
$87$	0.383673	0.0411341
$88$	−17.2746	−1.84148
$89$	−16.5032	−1.74933	−0.874666	−	0.484725i	$-0.838920\pi$
$89$	−16.5032	−1.74933	−0.874666	+	0.484725i	$0.838920\pi$
$90$	−3.75296	−0.395597
$91$	10.7036	1.12204
$92$	4.04026	0.421226
$93$	−1.02096	−0.105869
$94$	−9.90666	−1.02179
$95$	−0.121315	−0.0124466
$96$	−0.136932	−0.0139756
$97$	−18.0047	−1.82810	−0.914049	−	0.405603i	$-0.867062\pi$
$97$	−18.0047	−1.82810	−0.914049	+	0.405603i	$0.867062\pi$
$98$	−0.603154	−0.0609278
$99$	−9.93093	−0.998096
$100$	−19.0675	−1.90675
$101$	−12.7298	−1.26667	−0.633334	−	0.773879i	$-0.718314\pi$
$101$	−12.7298	−1.26667	−0.633334	+	0.773879i	$0.718314\pi$
$102$	−3.98696	−0.394767
$103$	−7.48581	−0.737599	−0.368800	−	0.929509i	$-0.620231\pi$
$103$	−7.48581	−0.737599	−0.368800	+	0.929509i	$0.620231\pi$
$104$	19.9393	1.95521
$105$	0.488433	0.0476662
$106$	5.06564	0.492019
$107$	11.1675	1.07960	0.539799	−	0.841794i	$-0.318500\pi$
$107$	11.1675	1.07960	0.539799	+	0.841794i	$0.318500\pi$
$108$	−8.14150	−0.783416
$109$	−15.8363	−1.51685	−0.758423	−	0.651762i	$-0.774030\pi$
$109$	−15.8363	−1.51685	−0.758423	+	0.651762i	$0.774030\pi$
$110$	4.48514	0.427642
$111$	0.486368	0.0461640
$112$	−11.4215	−1.07923
$113$	13.8554	1.30341	0.651705	−	0.758473i	$-0.274054\pi$
$113$	13.8554	1.30341	0.651705	+	0.758473i	$0.274054\pi$
$114$	−0.192801	−0.0180575
$115$	−0.529727	−0.0493973
$116$	4.52532	0.420165
$117$	11.4628	1.05974
$118$	11.5516	1.06341
$119$	−12.7475	−1.16856
$120$	0.909885	0.0830608
$121$	0.868409	0.0789463
$122$	−12.5291	−1.13433
$123$	1.41107	0.127232
$124$	−12.0420	−1.08140
$125$	5.14863	0.460507
$126$	−19.0701	−1.69890
$127$	20.8679	1.85173	0.925865	−	0.377855i	$-0.123338\pi$
$127$	20.8679	1.85173	0.925865	+	0.377855i	$0.123338\pi$
$128$	19.2417	1.70074
$129$	−2.86137	−0.251930
$130$	−5.17700	−0.454053
$131$	−6.97451	−0.609366	−0.304683	−	0.952454i	$-0.598550\pi$
$131$	−6.97451	−0.609366	−0.304683	+	0.952454i	$0.598550\pi$
$132$	4.76789	0.414992
$133$	−0.616442	−0.0534523
$134$	−28.5123	−2.46309
$135$	1.06745	0.0918715
$136$	−23.7468	−2.03627
$137$	−12.2238	−1.04435	−0.522175	−	0.852838i	$-0.674879\pi$
$137$	−12.2238	−1.04435	−0.522175	+	0.852838i	$0.674879\pi$
$138$	−0.841878	−0.0716654
$139$	3.15644	0.267726	0.133863	−	0.991000i	$-0.457262\pi$
$139$	3.15644	0.267726	0.133863	+	0.991000i	$0.457262\pi$
$140$	5.76094	0.486888
$141$	1.38077	0.116282
$142$	24.5561	2.06070
$143$	−13.6992	−1.14558
$144$	−12.2316	−1.01930
$145$	−0.593325	−0.0492729
$146$	−10.7931	−0.893240
$147$	0.0840663	0.00693367
$148$	5.73657	0.471543
$149$	−1.72361	−0.141204	−0.0706020	−	0.997505i	$-0.522492\pi$
$149$	−1.72361	−0.141204	−0.0706020	+	0.997505i	$0.522492\pi$
$150$	3.97315	0.324406
$151$	−9.43083	−0.767470	−0.383735	−	0.923443i	$-0.625362\pi$
$151$	−9.43083	−0.767470	−0.383735	+	0.923443i	$0.625362\pi$
$152$	−1.14835	−0.0931433
$153$	−13.6517	−1.10367
$154$	22.7906	1.83652
$155$	1.57885	0.126816
$156$	−5.50336	−0.440622
$157$	7.00554	0.559103	0.279552	−	0.960131i	$-0.409814\pi$
$157$	7.00554	0.559103	0.279552	+	0.960131i	$0.409814\pi$
$158$	−14.8844	−1.18414
$159$	−0.706038	−0.0559924
$160$	0.211757	0.0167409
$161$	−2.69173	−0.212138
$162$	−19.5576	−1.53659
$163$	−9.05829	−0.709500	−0.354750	−	0.934961i	$-0.615434\pi$
$163$	−9.05829	−0.709500	−0.354750	+	0.934961i	$0.615434\pi$
$164$	16.6432	1.29962
$165$	−0.625130	−0.0486663
$166$	11.3486	0.880819
$167$	4.62025	0.357525	0.178763	−	0.983892i	$-0.442791\pi$
$167$	4.62025	0.357525	0.178763	+	0.983892i	$0.442791\pi$
$168$	4.62344	0.356707
$169$	2.81232	0.216332
$170$	6.16556	0.472877
$171$	−0.660168	−0.0504843
$172$	−33.7491	−2.57335
$173$	−11.0639	−0.841174	−0.420587	−	0.907252i	$-0.638176\pi$
$173$	−11.0639	−0.841174	−0.420587	+	0.907252i	$0.638176\pi$
$174$	−0.942951	−0.0714849
$175$	12.7033	0.960281
$176$	14.6179	1.10187
$177$	−1.61004	−0.121018
$178$	40.5597	3.04008
$179$	−21.3674	−1.59707	−0.798537	−	0.601945i	$-0.794392\pi$
$179$	−21.3674	−1.59707	−0.798537	+	0.601945i	$0.794392\pi$
$180$	6.16957	0.459853
$181$	21.5933	1.60502	0.802510	−	0.596639i	$-0.203497\pi$
$181$	21.5933	1.60502	0.802510	+	0.596639i	$0.203497\pi$
$182$	−26.3061	−1.94994
$183$	1.74628	0.129088
$184$	−5.01433	−0.369661
$185$	−0.752135	−0.0552981
$186$	2.50921	0.183984
$187$	16.3151	1.19308
$188$	16.2858	1.18776
$189$	5.42409	0.394544
$190$	0.298154	0.0216304
$191$	3.09135	0.223683	0.111841	−	0.993726i	$-0.464325\pi$
$191$	3.09135	0.223683	0.111841	+	0.993726i	$0.464325\pi$
$192$	−2.57044	−0.185505
$193$	9.88437	0.711492	0.355746	−	0.934583i	$-0.384227\pi$
$193$	9.88437	0.711492	0.355746	+	0.934583i	$0.384227\pi$
$194$	44.2500	3.17696
$195$	0.721559	0.0516719
$196$	0.991539	0.0708242
$197$	11.9903	0.854271	0.427135	−	0.904188i	$-0.359523\pi$
$197$	11.9903	0.854271	0.427135	+	0.904188i	$0.359523\pi$
$198$	24.4072	1.73454
$199$	−7.49757	−0.531488	−0.265744	−	0.964044i	$-0.585618\pi$
$199$	−7.49757	−0.531488	−0.265744	+	0.964044i	$0.585618\pi$
$200$	23.6646	1.67334
$201$	3.97399	0.280304
$202$	31.2861	2.20128
$203$	−3.01489	−0.211604
$204$	6.55425	0.458889
$205$	−2.18213	−0.152407
$206$	18.3978	1.28184
$207$	−2.88266	−0.200359
$208$	−16.8728	−1.16992
$209$	0.788964	0.0545738
$210$	−1.20042	−0.0828368
$211$	−22.4754	−1.54727	−0.773634	−	0.633633i	$-0.781563\pi$
$211$	−22.4754	−1.54727	−0.773634	+	0.633633i	$0.781563\pi$
$212$	−8.32752	−0.571936
$213$	−3.42258	−0.234511
$214$	−27.4462	−1.87618
$215$	4.42493	0.301777
$216$	10.1043	0.687513
$217$	8.02269	0.544615
$218$	38.9209	2.63605
$219$	1.50431	0.101652
$220$	−7.37323	−0.497103
$221$	−18.8317	−1.26676
$222$	−1.19534	−0.0802261
$223$	24.4105	1.63465	0.817325	−	0.576176i	$-0.195456\pi$
$223$	24.4105	1.63465	0.817325	+	0.576176i	$0.195456\pi$
$224$	1.07601	0.0718940
$225$	13.6044	0.906960
$226$	−34.0524	−2.26513
$227$	25.1855	1.67162	0.835810	−	0.549019i	$-0.184999\pi$
$227$	25.1855	1.67162	0.835810	+	0.549019i	$0.184999\pi$
$228$	0.316950	0.0209905
$229$	27.0410	1.78692	0.893458	−	0.449146i	$-0.148272\pi$
$229$	27.0410	1.78692	0.893458	+	0.449146i	$0.148272\pi$
$230$	1.30191	0.0858453
$231$	−3.17650	−0.208999
$232$	−5.61633	−0.368730
$233$	−3.00162	−0.196643	−0.0983213	−	0.995155i	$-0.531347\pi$
$233$	−3.00162	−0.196643	−0.0983213	+	0.995155i	$0.531347\pi$
$234$	−28.1721	−1.84167
$235$	−2.13527	−0.139289
$236$	−18.9900	−1.23614
$237$	2.07456	0.134757
$238$	31.3294	2.03078
$239$	24.4826	1.58364	0.791822	−	0.610751i	$-0.209132\pi$
$239$	24.4826	1.58364	0.791822	+	0.610751i	$0.209132\pi$
$240$	−0.769952	−0.0497002
$241$	3.93065	0.253196	0.126598	−	0.991954i	$-0.459594\pi$
$241$	3.93065	0.253196	0.126598	+	0.991954i	$0.459594\pi$
$242$	−2.13428	−0.137197
$243$	8.77118	0.562671
$244$	20.5968	1.31858
$245$	−0.130003	−0.00830558
$246$	−3.46799	−0.221111
$247$	−0.910665	−0.0579442
$248$	14.9452	0.949020
$249$	−1.58174	−0.100238
$250$	−12.6537	−0.800293
$251$	8.37786	0.528806	0.264403	−	0.964412i	$-0.414825\pi$
$251$	8.37786	0.528806	0.264403	+	0.964412i	$0.414825\pi$
$252$	31.3498	1.97485
$253$	3.44506	0.216589
$254$	−51.2870	−3.21803
$255$	−0.859342	−0.0538141
$256$	−32.2825	−2.01766
$257$	−1.23759	−0.0771989	−0.0385995	−	0.999255i	$-0.512290\pi$
$257$	−1.23759	−0.0771989	−0.0385995	+	0.999255i	$0.512290\pi$
$258$	7.03238	0.437817
$259$	−3.82186	−0.237479
$260$	8.51058	0.527804
$261$	−3.22874	−0.199854
$262$	17.1412	1.05899
$263$	−18.4230	−1.13601	−0.568005	−	0.823025i	$-0.692284\pi$
$263$	−18.4230	−1.13601	−0.568005	+	0.823025i	$0.692284\pi$
$264$	−5.91739	−0.364190
$265$	1.09184	0.0670712
$266$	1.51503	0.0928922
$267$	−5.65313	−0.345966
$268$	46.8721	2.86317
$269$	−10.0610	−0.613428	−0.306714	−	0.951802i	$-0.599230\pi$
$269$	−10.0610	−0.613428	−0.306714	+	0.951802i	$0.599230\pi$
$270$	−2.62347	−0.159659
$271$	−26.6162	−1.61682	−0.808410	−	0.588620i	$-0.799672\pi$
$271$	−26.6162	−1.61682	−0.808410	+	0.588620i	$0.799672\pi$
$272$	20.0947	1.21842
$273$	3.66649	0.221906
$274$	30.0424	1.81493
$275$	−16.2586	−0.980428
$276$	1.38398	0.0833059
$277$	2.68365	0.161245	0.0806223	−	0.996745i	$-0.474309\pi$
$277$	2.68365	0.161245	0.0806223	+	0.996745i	$0.474309\pi$
$278$	−7.75756	−0.465268
$279$	8.59175	0.514375
$280$	−7.14985	−0.427285
$281$	11.6101	0.692598	0.346299	−	0.938124i	$-0.387438\pi$
$281$	11.6101	0.692598	0.346299	+	0.938124i	$0.387438\pi$
$282$	−3.39350	−0.202080
$283$	20.7572	1.23389	0.616945	−	0.787006i	$-0.288370\pi$
$283$	20.7572	1.23389	0.616945	+	0.787006i	$0.288370\pi$
$284$	−40.3684	−2.39542
$285$	−0.0415561	−0.00246157
$286$	33.6683	1.99085
$287$	−11.0882	−0.654515
$288$	1.15234	0.0679020
$289$	5.42769	0.319276
$290$	1.45821	0.0856291
$291$	−6.16747	−0.361543
$292$	17.7430	1.03833
$293$	3.31042	0.193397	0.0966985	−	0.995314i	$-0.469172\pi$
$293$	3.31042	0.193397	0.0966985	+	0.995314i	$0.469172\pi$
$294$	−0.206609	−0.0120497
$295$	2.48982	0.144963
$296$	−7.11961	−0.413819
$297$	−6.94211	−0.402822
$298$	4.23612	0.245392
$299$	−3.97647	−0.229965
$300$	−6.53155	−0.377099
$301$	22.4846	1.29599
$302$	23.1781	1.33375
$303$	−4.36058	−0.250509
$304$	0.971742	0.0557332
$305$	−2.70050	−0.154630
$306$	33.5516	1.91802
$307$	−32.2315	−1.83955	−0.919774	−	0.392448i	$-0.871628\pi$
$307$	−32.2315	−1.83955	−0.919774	+	0.392448i	$0.871628\pi$
$308$	−37.4660	−2.13482
$309$	−2.56425	−0.145875
$310$	−3.88033	−0.220388
$311$	5.42662	0.307715	0.153858	−	0.988093i	$-0.450830\pi$
$311$	5.42662	0.307715	0.153858	+	0.988093i	$0.450830\pi$
$312$	6.83018	0.386683
$313$	21.1179	1.19365	0.596826	−	0.802371i	$-0.296428\pi$
$313$	21.1179	1.19365	0.596826	+	0.802371i	$0.296428\pi$
$314$	−17.2175	−0.971638
$315$	−4.11034	−0.231591
$316$	24.4688	1.37648
$317$	10.1993	0.572852	0.286426	−	0.958102i	$-0.407533\pi$
$317$	10.1993	0.572852	0.286426	+	0.958102i	$0.407533\pi$
$318$	1.73522	0.0973066
$319$	3.85866	0.216043
$320$	3.97501	0.222210
$321$	3.82539	0.213512
$322$	6.61545	0.368665
$323$	1.08456	0.0603465
$324$	32.1512	1.78618
$325$	18.7665	1.04098
$326$	22.2625	1.23301
$327$	−5.42471	−0.299987
$328$	−20.6558	−1.14052
$329$	−10.8500	−0.598182
$330$	1.53638	0.0845748
$331$	21.6720	1.19120	0.595601	−	0.803281i	$-0.296914\pi$
$331$	21.6720	1.19120	0.595601	+	0.803281i	$0.296914\pi$
$332$	−18.6561	−1.02389
$333$	−4.09295	−0.224293
$334$	−11.3552	−0.621326
$335$	−6.14551	−0.335765
$336$	−3.91240	−0.213439
$337$	36.2570	1.97505	0.987523	−	0.157472i	$-0.0503344\pi$
$337$	36.2570	1.97505	0.987523	+	0.157472i	$0.0503344\pi$
$338$	−6.91182	−0.375953
$339$	4.74615	0.257775
$340$	−10.1357	−0.549686
$341$	−10.2680	−0.556042
$342$	1.62249	0.0877342
$343$	18.1815	0.981710
$344$	41.8857	2.25833
$345$	−0.181457	−0.00976932
$346$	27.1917	1.46184
$347$	−21.0934	−1.13235	−0.566175	−	0.824285i	$-0.691578\pi$
$347$	−21.0934	−1.13235	−0.566175	+	0.824285i	$0.691578\pi$
$348$	1.55014	0.0830961
$349$	−1.00000	−0.0535288
$349$	−1.00000	−0.0535288
$350$	−31.2209	−1.66883
$351$	8.01296	0.427700
$352$	−1.37715	−0.0734024
$353$	−13.0137	−0.692650	−0.346325	−	0.938115i	$-0.612571\pi$
$353$	−13.0137	−0.692650	−0.346325	+	0.938115i	$0.612571\pi$
$354$	3.95699	0.210311
$355$	5.29279	0.280912
$356$	−66.6771	−3.53388
$357$	−4.36662	−0.231106
$358$	52.5145	2.77548
$359$	20.0172	1.05647	0.528234	−	0.849099i	$-0.322855\pi$
$359$	20.0172	1.05647	0.528234	+	0.849099i	$0.322855\pi$
$360$	−7.65700	−0.403559
$361$	−18.9476	−0.997240
$362$	−53.0698	−2.78929
$363$	0.297472	0.0156132
$364$	43.2453	2.26667
$365$	−2.32632	−0.121765
$366$	−4.29181	−0.224337
$367$	10.2313	0.534067	0.267034	−	0.963687i	$-0.413957\pi$
$367$	10.2313	0.534067	0.267034	+	0.963687i	$0.413957\pi$
$368$	4.24317	0.221190
$369$	−11.8747	−0.618172
$370$	1.84852	0.0960998
$371$	5.54802	0.288039
$372$	−4.12495	−0.213869
$373$	−24.1064	−1.24818	−0.624090	−	0.781352i	$-0.714530\pi$
$373$	−24.1064	−1.24818	−0.624090	+	0.781352i	$0.714530\pi$
$374$	−40.0974	−2.07339
$375$	1.76365	0.0910746
$376$	−20.2121	−1.04236
$377$	−4.45387	−0.229386
$378$	−13.3307	−0.685660
$379$	25.5176	1.31075	0.655376	−	0.755303i	$-0.272510\pi$
$379$	25.5176	1.31075	0.655376	+	0.755303i	$0.272510\pi$
$380$	−0.490142	−0.0251438
$381$	7.14827	0.366217
$382$	−7.59760	−0.388727
$383$	−11.2623	−0.575478	−0.287739	−	0.957709i	$-0.592904\pi$
$383$	−11.2623	−0.575478	−0.287739	+	0.957709i	$0.592904\pi$
$384$	6.59121	0.336356
$385$	4.91225	0.250351
$386$	−24.2927	−1.23647
$387$	24.0795	1.22403
$388$	−72.7436	−3.69300
$389$	31.5865	1.60150	0.800751	−	0.598998i	$-0.204434\pi$
$389$	31.5865	1.60150	0.800751	+	0.598998i	$0.204434\pi$
$390$	−1.77337	−0.0897981
$391$	4.73579	0.239499
$392$	−1.23059	−0.0621542
$393$	−2.38910	−0.120514
$394$	−29.4684	−1.48460
$395$	−3.20816	−0.161420
$396$	−40.1235	−2.01628
$397$	17.2659	0.866549	0.433274	−	0.901262i	$-0.357358\pi$
$397$	17.2659	0.866549	0.433274	+	0.901262i	$0.357358\pi$
$398$	18.4267	0.923648
$399$	−0.211161	−0.0105713
$400$	−20.0252	−1.00126
$401$	−9.22126	−0.460488	−0.230244	−	0.973133i	$-0.573952\pi$
$401$	−9.22126	−0.460488	−0.230244	+	0.973133i	$0.573952\pi$
$402$	−9.76685	−0.487126
$403$	11.8518	0.590383
$404$	−51.4319	−2.55883
$405$	−4.21542	−0.209466
$406$	7.40968	0.367736
$407$	4.89147	0.242461
$408$	−8.13442	−0.402714
$409$	−33.5080	−1.65686	−0.828432	−	0.560090i	$-0.810766\pi$
$409$	−33.5080	−1.65686	−0.828432	+	0.560090i	$0.810766\pi$
$410$	5.36301	0.264860
$411$	−4.18724	−0.206541
$412$	−30.2446	−1.49005
$413$	12.6517	0.622547
$414$	7.08470	0.348194
$415$	2.44605	0.120072
$416$	1.58958	0.0779357
$417$	1.08123	0.0529482
$418$	−1.93903	−0.0948411
$419$	−23.7481	−1.16017	−0.580086	−	0.814555i	$-0.696981\pi$
$419$	−23.7481	−1.16017	−0.580086	+	0.814555i	$0.696981\pi$
$420$	1.97340	0.0962919
$421$	7.08343	0.345225	0.172613	−	0.984990i	$-0.444779\pi$
$421$	7.08343	0.345225	0.172613	+	0.984990i	$0.444779\pi$
$422$	55.2376	2.68892
$423$	−11.6196	−0.564967
$424$	10.3352	0.501922
$425$	−22.3500	−1.08414
$426$	8.41165	0.407546
$427$	−13.7222	−0.664063
$428$	45.1194	2.18093
$429$	−4.69262	−0.226562
$430$	−10.8751	−0.524444
$431$	28.5327	1.37437	0.687187	−	0.726481i	$-0.258845\pi$
$431$	28.5327	1.37437	0.687187	+	0.726481i	$0.258845\pi$
$432$	−8.55038	−0.411380
$433$	−3.77190	−0.181266	−0.0906330	−	0.995884i	$-0.528889\pi$
$433$	−3.77190	−0.181266	−0.0906330	+	0.995884i	$0.528889\pi$
$434$	−19.7173	−0.946461
$435$	−0.203242	−0.00974471
$436$	−63.9829	−3.06423
$437$	0.229013	0.0109552
$438$	−3.69714	−0.176656
$439$	2.63964	0.125983	0.0629916	−	0.998014i	$-0.479936\pi$
$439$	2.63964	0.125983	0.0629916	+	0.998014i	$0.479936\pi$
$440$	9.15085	0.436250
$441$	−0.707448	−0.0336880
$442$	46.2826	2.20144
$443$	5.94252	0.282338	0.141169	−	0.989986i	$-0.454914\pi$
$443$	5.94252	0.282338	0.141169	+	0.989986i	$0.454914\pi$
$444$	1.96505	0.0932572
$445$	8.74218	0.414419
$446$	−59.9936	−2.84078
$447$	−0.590420	−0.0279259
$448$	20.1984	0.954286
$449$	−13.8175	−0.652090	−0.326045	−	0.945354i	$-0.605716\pi$
$449$	−13.8175	−0.652090	−0.326045	+	0.945354i	$0.605716\pi$
$450$	−33.4354	−1.57616
$451$	14.1914	0.668246
$452$	55.9795	2.63306
$453$	−3.23051	−0.151783
$454$	−61.8982	−2.90503
$455$	−5.66998	−0.265813
$456$	−0.393364	−0.0184210
$457$	23.9348	1.11962	0.559810	−	0.828621i	$-0.310874\pi$
$457$	23.9348	1.11962	0.559810	+	0.828621i	$0.310874\pi$
$458$	−66.4584	−3.10540
$459$	−9.54306	−0.445432
$460$	−2.14024	−0.0997890
$461$	−13.6400	−0.635279	−0.317640	−	0.948211i	$-0.602890\pi$
$461$	−13.6400	−0.635279	−0.317640	+	0.948211i	$0.602890\pi$
$462$	7.80687	0.363208
$463$	−8.63044	−0.401091	−0.200545	−	0.979684i	$-0.564271\pi$
$463$	−8.63044	−0.401091	−0.200545	+	0.979684i	$0.564271\pi$
$464$	4.75259	0.220633
$465$	0.540832	0.0250805
$466$	7.37706	0.341736
$467$	1.62345	0.0751242	0.0375621	−	0.999294i	$-0.488041\pi$
$467$	1.62345	0.0751242	0.0375621	+	0.999294i	$0.488041\pi$
$468$	46.3127	2.14081
$469$	−31.2275	−1.44195
$470$	5.24783	0.242064
$471$	2.39973	0.110574
$472$	23.5683	1.08482
$473$	−28.7773	−1.32318
$474$	−5.09863	−0.234188
$475$	−1.08080	−0.0495906
$476$	−51.5031	−2.36064
$477$	5.94156	0.272045
$478$	−60.1706	−2.75214
$479$	−2.92359	−0.133582	−0.0667911	−	0.997767i	$-0.521276\pi$
$479$	−2.92359	−0.133582	−0.0667911	+	0.997767i	$0.521276\pi$
$480$	0.0725369	0.00331084
$481$	−5.64600	−0.257436
$482$	−9.66034	−0.440017
$483$	−0.922047	−0.0419546
$484$	3.50860	0.159482
$485$	9.53758	0.433079
$486$	−21.5569	−0.977839
$487$	−6.32402	−0.286569	−0.143284	−	0.989682i	$-0.545766\pi$
$487$	−6.32402	−0.286569	−0.143284	+	0.989682i	$0.545766\pi$
$488$	−25.5626	−1.15716
$489$	−3.10290	−0.140318
$490$	0.319507	0.0144339
$491$	3.48555	0.157301	0.0786504	−	0.996902i	$-0.474939\pi$
$491$	3.48555	0.157301	0.0786504	+	0.996902i	$0.474939\pi$
$492$	5.70110	0.257026
$493$	5.30435	0.238896
$494$	2.23814	0.100698
$495$	5.26068	0.236450
$496$	−12.6467	−0.567855
$497$	26.8945	1.20638
$498$	3.88742	0.174200
$499$	24.7377	1.10741	0.553706	−	0.832712i	$-0.313213\pi$
$499$	24.7377	1.10741	0.553706	+	0.832712i	$0.313213\pi$
$500$	20.8018	0.930284
$501$	1.58266	0.0707079
$502$	−20.5902	−0.918987
$503$	−6.94866	−0.309825	−0.154913	−	0.987928i	$-0.549510\pi$
$503$	−6.94866	−0.309825	−0.154913	+	0.987928i	$0.549510\pi$
$504$	−38.9079	−1.73310
$505$	6.74335	0.300075
$506$	−8.46689	−0.376399
$507$	0.963354	0.0427841
$508$	84.3118	3.74073
$509$	28.6540	1.27007	0.635033	−	0.772485i	$-0.280987\pi$
$509$	28.6540	1.27007	0.635033	+	0.772485i	$0.280987\pi$
$510$	2.11200	0.0935209
$511$	−11.8208	−0.522924
$512$	40.8571	1.80564
$513$	−0.461483	−0.0203750
$514$	3.04162	0.134160
$515$	3.96544	0.174738
$516$	−11.5607	−0.508931
$517$	13.8866	0.610732
$518$	9.39297	0.412703
$519$	−3.78992	−0.166359
$520$	−10.5624	−0.463192
$521$	−6.04848	−0.264989	−0.132494	−	0.991184i	$-0.542299\pi$
$521$	−6.04848	−0.264989	−0.132494	+	0.991184i	$0.542299\pi$
$522$	7.93526	0.347317
$523$	−39.7852	−1.73968	−0.869842	−	0.493331i	$-0.835779\pi$
$523$	−39.7852	−1.73968	−0.869842	+	0.493331i	$0.835779\pi$
$524$	−28.1788	−1.23100
$525$	4.35150	0.189915
$526$	45.2780	1.97422
$527$	−14.1150	−0.614859
$528$	5.00735	0.217917
$529$	1.00000	0.0434783
$530$	−2.68341	−0.116560
$531$	13.5491	0.587979
$532$	−2.49059	−0.107981
$533$	−16.3805	−0.709517
$534$	13.8937	0.601237
$535$	−5.91571	−0.255758
$536$	−58.1725	−2.51267
$537$	−7.31936	−0.315854
$538$	24.7268	1.06605
$539$	0.845468	0.0364169
$540$	4.31277	0.185592
$541$	−37.5995	−1.61653	−0.808265	−	0.588819i	$-0.799593\pi$
$541$	−37.5995	−1.61653	−0.808265	+	0.588819i	$0.799593\pi$
$542$	65.4145	2.80979
$543$	7.39675	0.317425
$544$	−1.89312	−0.0811667
$545$	8.38895	0.359343
$546$	−9.01111	−0.385640
$547$	15.1906	0.649503	0.324751	−	0.945799i	$-0.394719\pi$
$547$	15.1906	0.649503	0.324751	+	0.945799i	$0.394719\pi$
$548$	−49.3873	−2.10972
$549$	−14.6955	−0.627190
$550$	39.9586	1.70384
$551$	0.256508	0.0109276
$552$	−1.71765	−0.0731079
$553$	−16.3018	−0.693223
$554$	−6.59558	−0.280219
$555$	−0.257642	−0.0109363
$556$	12.7528	0.540841
$557$	−37.4321	−1.58605	−0.793024	−	0.609191i	$-0.791494\pi$
$557$	−37.4321	−1.58605	−0.793024	+	0.609191i	$0.791494\pi$
$558$	−21.1159	−0.893907
$559$	33.2163	1.40490
$560$	6.05026	0.255670
$561$	5.58869	0.235955
$562$	−28.5340	−1.20363
$563$	15.3323	0.646180	0.323090	−	0.946368i	$-0.395278\pi$
$563$	15.3323	0.646180	0.323090	+	0.946368i	$0.395278\pi$
$564$	5.57866	0.234904
$565$	−7.33960	−0.308779
$566$	−51.0149	−2.14432
$567$	−21.4200	−0.899557
$568$	50.1008	2.10218
$569$	15.8440	0.664214	0.332107	−	0.943242i	$-0.392240\pi$
$569$	15.8440	0.664214	0.332107	+	0.943242i	$0.392240\pi$
$570$	0.102132	0.00427784
$571$	25.1086	1.05076	0.525380	−	0.850867i	$-0.323923\pi$
$571$	25.1086	1.05076	0.525380	+	0.850867i	$0.323923\pi$
$572$	−55.3481	−2.31422
$573$	1.05894	0.0442377
$574$	27.2514	1.13745
$575$	−4.71939	−0.196812
$576$	21.6311	0.901297
$577$	12.1029	0.503849	0.251925	−	0.967747i	$-0.418936\pi$
$577$	12.1029	0.503849	0.251925	+	0.967747i	$0.418936\pi$
$578$	−13.3396	−0.554855
$579$	3.38587	0.140712
$580$	−2.39718	−0.0995377
$581$	12.4292	0.515652
$582$	15.1577	0.628309
$583$	−7.10073	−0.294082
$584$	−22.0206	−0.911220
$585$	−6.07217	−0.251053
$586$	−8.13601	−0.336095
$587$	39.2732	1.62098	0.810490	−	0.585753i	$-0.199201\pi$
$587$	39.2732	1.62098	0.810490	+	0.585753i	$0.199201\pi$
$588$	0.339650	0.0140069
$589$	−0.682573	−0.0281249
$590$	−6.11922	−0.251924
$591$	4.10724	0.168949
$592$	6.02467	0.247613
$593$	13.0513	0.535953	0.267976	−	0.963425i	$-0.413645\pi$
$593$	13.0513	0.535953	0.267976	+	0.963425i	$0.413645\pi$
$594$	17.0616	0.700045
$595$	6.75268	0.276833
$596$	−6.96384	−0.285250
$597$	−2.56828	−0.105113
$598$	9.77295	0.399646
$599$	36.4842	1.49070	0.745352	−	0.666671i	$-0.232282\pi$
$599$	36.4842	1.49070	0.745352	+	0.666671i	$0.232282\pi$
$600$	8.10625	0.330936
$601$	−4.04469	−0.164986	−0.0824932	−	0.996592i	$-0.526288\pi$
$601$	−4.04469	−0.164986	−0.0824932	+	0.996592i	$0.526288\pi$
$602$	−55.2602	−2.25224
$603$	−33.4425	−1.36188
$604$	−38.1030	−1.55039
$605$	−0.460020	−0.0187025
$606$	10.7170	0.435347
$607$	21.2322	0.861789	0.430895	−	0.902402i	$-0.358198\pi$
$607$	21.2322	0.861789	0.430895	+	0.902402i	$0.358198\pi$
$608$	−0.0915474	−0.00371274
$609$	−1.03274	−0.0418489
$610$	6.63700	0.268724
$611$	−16.0287	−0.648450
$612$	−55.1563	−2.22956
$613$	−13.7027	−0.553447	−0.276723	−	0.960950i	$-0.589249\pi$
$613$	−13.7027	−0.553447	−0.276723	+	0.960950i	$0.589249\pi$
$614$	79.2151	3.19686
$615$	−0.747485	−0.0301415
$616$	46.4987	1.87349
$617$	−3.77209	−0.151859	−0.0759293	−	0.997113i	$-0.524192\pi$
$617$	−3.77209	−0.151859	−0.0759293	+	0.997113i	$0.524192\pi$
$618$	6.30214	0.253509
$619$	−46.0495	−1.85089	−0.925443	−	0.378887i	$-0.876307\pi$
$619$	−46.0495	−1.85089	−0.925443	+	0.378887i	$0.876307\pi$
$620$	6.37896	0.256185
$621$	−2.01509	−0.0808629
$622$	−13.3370	−0.534763
$623$	44.4221	1.77973
$624$	−5.77975	−0.231375
$625$	20.8696	0.834783
$626$	−51.9012	−2.07439
$627$	0.270258	0.0107931
$628$	28.3042	1.12946
$629$	6.72412	0.268108
$630$	10.1020	0.402471
$631$	44.3267	1.76462	0.882308	−	0.470672i	$-0.155988\pi$
$631$	44.3267	1.76462	0.882308	+	0.470672i	$0.155988\pi$
$632$	−30.3680	−1.20798
$633$	−7.69889	−0.306003
$634$	−25.0669	−0.995532
$635$	−11.0543	−0.438677
$636$	−2.85257	−0.113112
$637$	−0.975885	−0.0386660
$638$	−9.48340	−0.375451
$639$	28.8022	1.13940
$640$	−10.1929	−0.402909
$641$	−10.3322	−0.408098	−0.204049	−	0.978961i	$-0.565410\pi$
$641$	−10.3322	−0.408098	−0.204049	+	0.978961i	$0.565410\pi$
$642$	−9.40163	−0.371053
$643$	36.5123	1.43991	0.719953	−	0.694023i	$-0.244163\pi$
$643$	36.5123	1.43991	0.719953	+	0.694023i	$0.244163\pi$
$644$	−10.8753	−0.428546
$645$	1.51575	0.0596825
$646$	−2.66551	−0.104873
$647$	−28.2373	−1.11012	−0.555062	−	0.831809i	$-0.687306\pi$
$647$	−28.2373	−1.11012	−0.555062	+	0.831809i	$0.687306\pi$
$648$	−39.9026	−1.56752
$649$	−16.1924	−0.635608
$650$	−46.1223	−1.80907
$651$	2.74815	0.107709
$652$	−36.5978	−1.43328
$653$	29.1424	1.14043	0.570215	−	0.821496i	$-0.306860\pi$
$653$	29.1424	1.14043	0.570215	+	0.821496i	$0.306860\pi$
$654$	13.3323	0.521333
$655$	3.69459	0.144359
$656$	17.4791	0.682444
$657$	−12.6593	−0.493887
$658$	26.6661	1.03955
$659$	−25.8166	−1.00567	−0.502836	−	0.864382i	$-0.667710\pi$
$659$	−25.8166	−1.00567	−0.502836	+	0.864382i	$0.667710\pi$
$660$	−2.52568	−0.0983121
$661$	9.69006	0.376900	0.188450	−	0.982083i	$-0.439654\pi$
$661$	9.69006	0.376900	0.188450	+	0.982083i	$0.439654\pi$
$662$	−53.2632	−2.07013
$663$	−6.45077	−0.250527
$664$	23.1540	0.898549
$665$	0.326546	0.0126629
$666$	10.0592	0.389787
$667$	1.12006	0.0433688
$668$	18.6670	0.722248
$669$	8.36178	0.323285
$670$	15.1038	0.583510
$671$	17.5626	0.677995
$672$	0.368585	0.0142185
$673$	−21.8461	−0.842107	−0.421053	−	0.907036i	$-0.638340\pi$
$673$	−21.8461	−0.842107	−0.421053	+	0.907036i	$0.638340\pi$
$674$	−89.1087	−3.43234
$675$	9.51001	0.366040
$676$	11.3625	0.437019
$677$	−11.7608	−0.452003	−0.226002	−	0.974127i	$-0.572565\pi$
$677$	−11.7608	−0.452003	−0.226002	+	0.974127i	$0.572565\pi$
$678$	−11.6646	−0.447976
$679$	48.4638	1.85987
$680$	12.5793	0.482395
$681$	8.62724	0.330596
$682$	25.2355	0.966318
$683$	−6.44571	−0.246638	−0.123319	−	0.992367i	$-0.539354\pi$
$683$	−6.44571	−0.246638	−0.123319	+	0.992367i	$0.539354\pi$
$684$	−2.66725	−0.101985
$685$	6.47528	0.247408
$686$	−44.6846	−1.70607
$687$	9.26282	0.353399
$688$	−35.4441	−1.35129
$689$	8.19605	0.312245
$690$	0.445966	0.0169776
$691$	−10.2801	−0.391075	−0.195538	−	0.980696i	$-0.562645\pi$
$691$	−10.2801	−0.391075	−0.195538	+	0.980696i	$0.562645\pi$
$692$	−44.7011	−1.69928
$693$	26.7314	1.01544
$694$	51.8410	1.96786
$695$	−1.67205	−0.0634246
$696$	−1.92386	−0.0729238
$697$	19.5084	0.738932
$698$	2.45769	0.0930251
$699$	−1.02820	−0.0388900
$700$	51.3247	1.93989
$701$	43.4816	1.64228	0.821138	−	0.570729i	$-0.193339\pi$
$701$	43.4816	1.64228	0.821138	+	0.570729i	$0.193339\pi$
$702$	−19.6934	−0.743279
$703$	0.325165	0.0122638
$704$	−25.8513	−0.974307
$705$	−0.731431	−0.0275473
$706$	31.9837	1.20372
$707$	34.2653	1.28868
$708$	−6.50498	−0.244472
$709$	39.0848	1.46786	0.733930	−	0.679225i	$-0.237684\pi$
$709$	39.0848	1.46786	0.733930	+	0.679225i	$0.237684\pi$
$710$	−13.0081	−0.488183
$711$	−17.4581	−0.654731
$712$	82.7523	3.10127
$713$	−2.98049	−0.111620
$714$	10.7318	0.401628
$715$	7.25682	0.271390
$716$	−86.3298	−3.22630
$717$	8.38645	0.313198
$718$	−49.1961	−1.83598
$719$	9.07212	0.338333	0.169166	−	0.985588i	$-0.445892\pi$
$719$	9.07212	0.338333	0.169166	+	0.985588i	$0.445892\pi$
$720$	6.47942	0.241474
$721$	20.1498	0.750418
$722$	46.5673	1.73305
$723$	1.34644	0.0500745
$724$	87.2426	3.24235
$725$	−5.28598	−0.196316
$726$	−0.731095	−0.0271335
$727$	30.1795	1.11930	0.559649	−	0.828730i	$-0.310936\pi$
$727$	30.1795	1.11930	0.559649	+	0.828730i	$0.310936\pi$
$728$	−53.6713	−1.98919
$729$	−20.8686	−0.772911
$730$	5.71738	0.211610
$731$	−39.5590	−1.46314
$732$	7.05541	0.260775
$733$	7.85867	0.290267	0.145133	−	0.989412i	$-0.453639\pi$
$733$	7.85867	0.290267	0.145133	+	0.989412i	$0.453639\pi$
$734$	−25.1453	−0.928129
$735$	−0.0445322	−0.00164260
$736$	−0.399747	−0.0147349
$737$	39.9670	1.47220
$738$	29.1843	1.07429
$739$	24.0500	0.884694	0.442347	−	0.896844i	$-0.354146\pi$
$739$	24.0500	0.884694	0.442347	+	0.896844i	$0.354146\pi$
$740$	−3.03882	−0.111709
$741$	−0.311946	−0.0114596
$742$	−13.6353	−0.500569
$743$	6.46266	0.237092	0.118546	−	0.992949i	$-0.462177\pi$
$743$	6.46266	0.237092	0.118546	+	0.992949i	$0.462177\pi$
$744$	5.11944	0.187688
$745$	0.913046	0.0334514
$746$	59.2460	2.16915
$747$	13.3109	0.487019
$748$	65.9170	2.41017
$749$	−30.0598	−1.09836
$750$	−4.33451	−0.158274
$751$	22.5664	0.823458	0.411729	−	0.911306i	$-0.364925\pi$
$751$	22.5664	0.823458	0.411729	+	0.911306i	$0.364925\pi$
$752$	17.1037	0.623707
$753$	2.86982	0.104582
$754$	10.9463	0.398639
$755$	4.99577	0.181815
$756$	21.9147	0.797030
$757$	−43.7231	−1.58914	−0.794572	−	0.607171i	$-0.792304\pi$
$757$	−43.7231	−1.58914	−0.794572	+	0.607171i	$0.792304\pi$
$758$	−62.7145	−2.27789
$759$	1.18010	0.0428348
$760$	0.608311	0.0220658
$761$	−32.3972	−1.17440	−0.587199	−	0.809443i	$-0.699769\pi$
$761$	−32.3972	−1.17440	−0.587199	+	0.809443i	$0.699769\pi$
$762$	−17.5682	−0.636431
$763$	42.6272	1.54321
$764$	12.4899	0.451868
$765$	7.23167	0.261461
$766$	27.6794	1.00010
$767$	18.6902	0.674863
$768$	−11.0583	−0.399032
$769$	25.2783	0.911560	0.455780	−	0.890092i	$-0.349360\pi$
$769$	25.2783	0.911560	0.455780	+	0.890092i	$0.349360\pi$
$770$	−12.0728	−0.435074
$771$	−0.423935	−0.0152676
$772$	39.9354	1.43731
$773$	45.2638	1.62803	0.814013	−	0.580846i	$-0.197278\pi$
$773$	45.2638	1.62803	0.814013	+	0.580846i	$0.197278\pi$
$774$	−59.1800	−2.12718
$775$	14.0661	0.505270
$776$	90.2814	3.24091
$777$	−1.30917	−0.0469662
$778$	−77.6301	−2.78317
$779$	0.943386	0.0338003
$780$	2.91528	0.104384
$781$	−34.4214	−1.23169
$782$	−11.6391	−0.416214
$783$	−2.25702	−0.0806592
$784$	1.04134	0.0371906
$785$	−3.71103	−0.132452
$786$	5.87168	0.209436
$787$	−17.0656	−0.608321	−0.304161	−	0.952621i	$-0.598376\pi$
$787$	−17.0656	−0.608321	−0.304161	+	0.952621i	$0.598376\pi$
$788$	48.4437	1.72574
$789$	−6.31075	−0.224669
$790$	7.88469	0.280525
$791$	−37.2951	−1.32606
$792$	49.7969	1.76946
$793$	−20.2717	−0.719868
$794$	−42.4342	−1.50593
$795$	0.374008	0.0132647
$796$	−30.2921	−1.07367
$797$	26.8771	0.952037	0.476019	−	0.879435i	$-0.342079\pi$
$797$	26.8771	0.952037	0.476019	+	0.879435i	$0.342079\pi$
$798$	0.518969	0.0183713
$799$	19.0894	0.675333
$800$	1.88656	0.0667000
$801$	47.5731	1.68091
$802$	22.6630	0.800260
$803$	15.1291	0.533895
$804$	16.0559	0.566249
$805$	1.42588	0.0502558
$806$	−29.1282	−1.02600
$807$	−3.44637	−0.121318
$808$	63.8316	2.24559
$809$	44.2709	1.55648	0.778241	−	0.627966i	$-0.216112\pi$
$809$	44.2709	1.55648	0.778241	+	0.627966i	$0.216112\pi$
$810$	10.3602	0.364021
$811$	−35.7084	−1.25389	−0.626947	−	0.779062i	$-0.715696\pi$
$811$	−35.7084	−1.25389	−0.626947	+	0.779062i	$0.715696\pi$
$812$	−12.1809	−0.427467
$813$	−9.11733	−0.319759
$814$	−12.0217	−0.421362
$815$	4.79842	0.168081
$816$	6.88341	0.240968
$817$	−1.91300	−0.0669273
$818$	82.3523	2.87938
$819$	−30.8548	−1.07815
$820$	−8.81638	−0.307881
$821$	−52.9856	−1.84921	−0.924605	−	0.380928i	$-0.875605\pi$
$821$	−52.9856	−1.84921	−0.924605	+	0.380928i	$0.875605\pi$
$822$	10.2909	0.358938
$823$	−25.6366	−0.893634	−0.446817	−	0.894625i	$-0.647443\pi$
$823$	−25.6366	−0.893634	−0.446817	+	0.894625i	$0.647443\pi$
$824$	37.5363	1.30764
$825$	−5.56933	−0.193899
$826$	−31.0939	−1.08189
$827$	47.9151	1.66617	0.833085	−	0.553144i	$-0.186572\pi$
$827$	47.9151	1.66617	0.833085	+	0.553144i	$0.186572\pi$
$828$	−11.6467	−0.404751
$829$	7.03956	0.244494	0.122247	−	0.992500i	$-0.460990\pi$
$829$	7.03956	0.244494	0.122247	+	0.992500i	$0.460990\pi$
$830$	−6.01164	−0.208667
$831$	0.919277	0.0318894
$832$	29.8389	1.03448
$833$	1.16223	0.0402690
$834$	−2.65734	−0.0920161
$835$	−2.44747	−0.0846983
$836$	3.18762	0.110246
$837$	6.00597	0.207597
$838$	58.3656	2.01621
$839$	−5.24467	−0.181066	−0.0905330	−	0.995893i	$-0.528857\pi$
$839$	−5.24467	−0.181066	−0.0905330	+	0.995893i	$0.528857\pi$
$840$	−2.44917	−0.0845042
$841$	−27.7455	−0.956740
$842$	−17.4089	−0.599950
$843$	3.97700	0.136975
$844$	−90.8063	−3.12568
$845$	−1.48976	−0.0512494
$846$	28.5575	0.981828
$847$	−2.33752	−0.0803183
$848$	−8.74574	−0.300330
$849$	7.11035	0.244026
$850$	54.9295	1.88407
$851$	1.41985	0.0486719
$852$	−13.8281	−0.473743
$853$	35.6976	1.22226	0.611131	−	0.791530i	$-0.290715\pi$
$853$	35.6976	1.22226	0.611131	+	0.791530i	$0.290715\pi$
$854$	33.7249	1.15404
$855$	0.349709	0.0119598
$856$	−55.9973	−1.91395
$857$	−10.3044	−0.351993	−0.175997	−	0.984391i	$-0.556315\pi$
$857$	−10.3044	−0.351993	−0.175997	+	0.984391i	$0.556315\pi$
$858$	11.5330	0.393731
$859$	45.9304	1.56712	0.783562	−	0.621313i	$-0.213401\pi$
$859$	45.9304	1.56712	0.783562	+	0.621313i	$0.213401\pi$
$860$	17.8778	0.609629
$861$	−3.79823	−0.129443
$862$	−70.1247	−2.38846
$863$	−8.63962	−0.294096	−0.147048	−	0.989129i	$-0.546977\pi$
$863$	−8.63962	−0.294096	−0.147048	+	0.989129i	$0.546977\pi$
$864$	0.805527	0.0274046
$865$	5.86086	0.199275
$866$	9.27018	0.315014
$867$	1.85925	0.0631433
$868$	32.4137	1.10019
$869$	20.8641	0.707767
$870$	0.499507	0.0169349
$871$	−46.1321	−1.56313
$872$	79.4086	2.68912
$873$	51.9014	1.75660
$874$	−0.562845	−0.0190385
$875$	−13.8587	−0.468510
$876$	6.07781	0.205350
$877$	35.0843	1.18471	0.592356	−	0.805677i	$-0.298198\pi$
$877$	35.0843	1.18471	0.592356	+	0.805677i	$0.298198\pi$
$878$	−6.48743	−0.218940
$879$	1.13398	0.0382481
$880$	−7.74353	−0.261034
$881$	9.11522	0.307099	0.153550	−	0.988141i	$-0.450929\pi$
$881$	9.11522	0.307099	0.153550	+	0.988141i	$0.450929\pi$
$882$	1.73869	0.0585447
$883$	49.5173	1.66639	0.833196	−	0.552978i	$-0.186509\pi$
$883$	49.5173	1.66639	0.833196	+	0.552978i	$0.186509\pi$
$884$	−76.0850	−2.55902
$885$	0.852883	0.0286693
$886$	−14.6049	−0.490661
$887$	41.0102	1.37699	0.688493	−	0.725243i	$-0.258273\pi$
$887$	41.0102	1.37699	0.688493	+	0.725243i	$0.258273\pi$
$888$	−2.43881	−0.0818410
$889$	−56.1709	−1.88391
$890$	−21.4856	−0.720199
$891$	27.4148	0.918430
$892$	98.6249	3.30220
$893$	0.923124	0.0308912
$894$	1.45107	0.0485311
$895$	11.3189	0.378349
$896$	−51.7935	−1.73030
$897$	−1.36213	−0.0454803
$898$	33.9593	1.13324
$899$	−3.33832	−0.111339
$900$	54.9653	1.83218
$901$	−9.76111	−0.325190
$902$	−34.8781	−1.16131
$903$	7.70205	0.256308
$904$	−69.4757	−2.31073
$905$	−11.4386	−0.380231
$906$	7.93961	0.263776
$907$	7.35913	0.244356	0.122178	−	0.992508i	$-0.461012\pi$
$907$	7.35913	0.244356	0.122178	+	0.992508i	$0.461012\pi$
$908$	101.756	3.37689
$909$	36.6958	1.21712
$910$	13.9351	0.461944
$911$	−18.4435	−0.611059	−0.305530	−	0.952183i	$-0.598833\pi$
$911$	−18.4435	−0.611059	−0.305530	+	0.952183i	$0.598833\pi$
$912$	0.332868	0.0110224
$913$	−15.9078	−0.526470
$914$	−58.8243	−1.94573
$915$	−0.925050	−0.0305812
$916$	109.252	3.60980
$917$	18.7735	0.619955
$918$	23.4539	0.774094
$919$	34.4076	1.13500	0.567501	−	0.823373i	$-0.307910\pi$
$919$	34.4076	1.13500	0.567501	+	0.823373i	$0.307910\pi$
$920$	2.65623	0.0875732
$921$	−11.0408	−0.363808
$922$	33.5230	1.10402
$923$	39.7310	1.30776
$924$	−12.8339	−0.422204
$925$	−6.70084	−0.220322
$926$	21.2110	0.697036
$927$	21.5791	0.708749
$928$	−0.447739	−0.0146978
$929$	32.6774	1.07211	0.536056	−	0.844183i	$-0.319914\pi$
$929$	32.6774	1.07211	0.536056	+	0.844183i	$0.319914\pi$
$930$	−1.32920	−0.0435861
$931$	0.0562033	0.00184199
$932$	−12.1273	−0.397243
$933$	1.85888	0.0608569
$934$	−3.98994	−0.130555
$935$	−8.64253	−0.282641
$936$	−57.4783	−1.87874
$937$	−21.0126	−0.686452	−0.343226	−	0.939253i	$-0.611520\pi$
$937$	−21.0126	−0.686452	−0.343226	+	0.939253i	$0.611520\pi$
$938$	76.7476	2.50590
$939$	7.23388	0.236069
$940$	−8.62702	−0.281382
$941$	3.79551	0.123730	0.0618650	−	0.998085i	$-0.480295\pi$
$941$	3.79551	0.123730	0.0618650	+	0.998085i	$0.480295\pi$
$942$	−5.89781	−0.192161
$943$	4.11935	0.134144
$944$	−19.9437	−0.649112
$945$	−2.87329	−0.0934681
$946$	70.7257	2.29949
$947$	2.07168	0.0673206	0.0336603	−	0.999433i	$-0.489284\pi$
$947$	2.07168	0.0673206	0.0336603	+	0.999433i	$0.489284\pi$
$948$	8.38175	0.272226
$949$	−17.4628	−0.566868
$950$	2.65628	0.0861812
$951$	3.49376	0.113293
$952$	63.9200	2.07166
$953$	11.3256	0.366872	0.183436	−	0.983032i	$-0.441278\pi$
$953$	11.3256	0.366872	0.183436	+	0.983032i	$0.441278\pi$
$954$	−14.6025	−0.472774
$955$	−1.63757	−0.0529907
$956$	98.9158	3.19917
$957$	1.32177	0.0427269
$958$	7.18529	0.232146
$959$	32.9032	1.06250
$960$	1.36163	0.0439465
$961$	−22.1167	−0.713441
$962$	13.8761	0.447385
$963$	−32.1920	−1.03737
$964$	15.8809	0.511488
$965$	−5.23602	−0.168553
$966$	2.26611	0.0729108
$967$	24.7639	0.796352	0.398176	−	0.917309i	$-0.369643\pi$
$967$	24.7639	0.796352	0.398176	+	0.917309i	$0.369643\pi$
$968$	−4.35449	−0.139959
$969$	0.371513	0.0119347
$970$	−23.4404	−0.752627
$971$	51.9814	1.66816	0.834082	−	0.551641i	$-0.185998\pi$
$971$	51.9814	1.66816	0.834082	+	0.551641i	$0.185998\pi$
$972$	35.4378	1.13667
$973$	−8.49629	−0.272379
$974$	15.5425	0.498014
$975$	6.42843	0.205874
$976$	21.6313	0.692400
$977$	20.0648	0.641930	0.320965	−	0.947091i	$-0.395993\pi$
$977$	20.0648	0.641930	0.320965	+	0.947091i	$0.395993\pi$
$978$	7.62597	0.243852
$979$	−56.8543	−1.81707
$980$	−0.525245	−0.0167783
$981$	45.6508	1.45752
$982$	−8.56642	−0.273365
$983$	54.2228	1.72944	0.864720	−	0.502255i	$-0.167496\pi$
$983$	54.2228	1.72944	0.864720	+	0.502255i	$0.167496\pi$
$984$	−7.07559	−0.225562
$985$	−6.35157	−0.202378
$986$	−13.0365	−0.415166
$987$	−3.71666	−0.118302
$988$	−3.67932	−0.117055
$989$	−8.35321	−0.265617
$990$	−12.9292	−0.410915
$991$	−20.6690	−0.656572	−0.328286	−	0.944578i	$-0.606471\pi$
$991$	−20.6690	−0.656572	−0.328286	+	0.944578i	$0.606471\pi$
$992$	1.19144	0.0378284
$993$	7.42370	0.235584
$994$	−66.0985	−2.09652
$995$	3.97167	0.125910
$996$	−6.39062	−0.202495
$997$	−20.0372	−0.634585	−0.317293	−	0.948328i	$-0.602774\pi$
$997$	−20.0372	−0.634585	−0.317293	+	0.948328i	$0.602774\pi$
$998$	−60.7978	−1.92452
$999$	−2.86114	−0.0905223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8027.2.a.f.1.14	✓	176

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8027.2.a.f.1.14	✓	176	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:	\( N \)	\(=\)	\( 8027 = 23 \cdot 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45769 q^{2} +0.342548 q^{3} +4.04026 q^{4} -0.529727 q^{5} -0.841878 q^{6} -2.69173 q^{7} -5.01433 q^{8} -2.88266 q^{9} +O(q^{10})\)	\(q-2.45769 q^{2} +0.342548 q^{3} +4.04026 q^{4} -0.529727 q^{5} -0.841878 q^{6} -2.69173 q^{7} -5.01433 q^{8} -2.88266 q^{9} +1.30191 q^{10} +3.44506 q^{11} +1.38398 q^{12} -3.97647 q^{13} +6.61545 q^{14} -0.181457 q^{15} +4.24317 q^{16} +4.73579 q^{17} +7.08470 q^{18} +0.229013 q^{19} -2.14024 q^{20} -0.922047 q^{21} -8.46689 q^{22} +1.00000 q^{23} -1.71765 q^{24} -4.71939 q^{25} +9.77295 q^{26} -2.01509 q^{27} -10.8753 q^{28} +1.12006 q^{29} +0.445966 q^{30} -2.98049 q^{31} -0.399747 q^{32} +1.18010 q^{33} -11.6391 q^{34} +1.42588 q^{35} -11.6467 q^{36} +1.41985 q^{37} -0.562845 q^{38} -1.36213 q^{39} +2.65623 q^{40} +4.11935 q^{41} +2.26611 q^{42} -8.35321 q^{43} +13.9189 q^{44} +1.52702 q^{45} -2.45769 q^{46} +4.03088 q^{47} +1.45349 q^{48} +0.245415 q^{49} +11.5988 q^{50} +1.62223 q^{51} -16.0660 q^{52} -2.06114 q^{53} +4.95248 q^{54} -1.82494 q^{55} +13.4972 q^{56} +0.0784480 q^{57} -2.75276 q^{58} -4.70019 q^{59} -0.733133 q^{60} +5.09790 q^{61} +7.32514 q^{62} +7.75935 q^{63} -7.50388 q^{64} +2.10645 q^{65} -2.90032 q^{66} +11.6013 q^{67} +19.1338 q^{68} +0.342548 q^{69} -3.50439 q^{70} -9.99153 q^{71} +14.4546 q^{72} +4.39154 q^{73} -3.48956 q^{74} -1.61662 q^{75} +0.925273 q^{76} -9.27316 q^{77} +3.34770 q^{78} +6.05625 q^{79} -2.24772 q^{80} +7.95772 q^{81} -10.1241 q^{82} -4.61756 q^{83} -3.72531 q^{84} -2.50868 q^{85} +20.5296 q^{86} +0.383673 q^{87} -17.2746 q^{88} -16.5032 q^{89} -3.75296 q^{90} +10.7036 q^{91} +4.04026 q^{92} -1.02096 q^{93} -9.90666 q^{94} -0.121315 q^{95} -0.136932 q^{96} -18.0047 q^{97} -0.603154 q^{98} -9.93093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

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