LMFDB - Embedded newform 8031.2.a.c.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, 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("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$0$
Dimension:	$121$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11893 q^{2} -1.00000 q^{3} +2.48988 q^{4} +3.42435 q^{5} +2.11893 q^{6} -4.49601 q^{7} -1.03803 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.11893 q^{2} -1.00000 q^{3} +2.48988 q^{4} +3.42435 q^{5} +2.11893 q^{6} -4.49601 q^{7} -1.03803 q^{8} +1.00000 q^{9} -7.25596 q^{10} -5.90928 q^{11} -2.48988 q^{12} +3.86921 q^{13} +9.52675 q^{14} -3.42435 q^{15} -2.78025 q^{16} -0.695427 q^{17} -2.11893 q^{18} -4.06191 q^{19} +8.52621 q^{20} +4.49601 q^{21} +12.5214 q^{22} +6.24148 q^{23} +1.03803 q^{24} +6.72614 q^{25} -8.19860 q^{26} -1.00000 q^{27} -11.1945 q^{28} -0.807465 q^{29} +7.25596 q^{30} +0.0987651 q^{31} +7.96723 q^{32} +5.90928 q^{33} +1.47356 q^{34} -15.3959 q^{35} +2.48988 q^{36} +4.03296 q^{37} +8.60691 q^{38} -3.86921 q^{39} -3.55456 q^{40} -1.08739 q^{41} -9.52675 q^{42} +7.95060 q^{43} -14.7134 q^{44} +3.42435 q^{45} -13.2253 q^{46} -1.33337 q^{47} +2.78025 q^{48} +13.2141 q^{49} -14.2523 q^{50} +0.695427 q^{51} +9.63388 q^{52} -7.73655 q^{53} +2.11893 q^{54} -20.2354 q^{55} +4.66697 q^{56} +4.06191 q^{57} +1.71096 q^{58} -0.270020 q^{59} -8.52621 q^{60} +2.78186 q^{61} -0.209277 q^{62} -4.49601 q^{63} -11.3215 q^{64} +13.2495 q^{65} -12.5214 q^{66} -11.8525 q^{67} -1.73153 q^{68} -6.24148 q^{69} +32.6229 q^{70} -14.6385 q^{71} -1.03803 q^{72} +9.55676 q^{73} -8.54558 q^{74} -6.72614 q^{75} -10.1137 q^{76} +26.5682 q^{77} +8.19860 q^{78} +1.77955 q^{79} -9.52055 q^{80} +1.00000 q^{81} +2.30411 q^{82} -3.82463 q^{83} +11.1945 q^{84} -2.38138 q^{85} -16.8468 q^{86} +0.807465 q^{87} +6.13399 q^{88} +4.22271 q^{89} -7.25596 q^{90} -17.3960 q^{91} +15.5405 q^{92} -0.0987651 q^{93} +2.82532 q^{94} -13.9094 q^{95} -7.96723 q^{96} -17.1152 q^{97} -27.9998 q^{98} -5.90928 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9}+O(q^{10}) $ 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9	\( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9} + 18 q^{10} + 32 q^{11} - 123 q^{12} + 2 q^{13} + 37 q^{14} - 24 q^{15} + 131 q^{16} + 87 q^{17} + 7 q^{18} - 10 q^{19} + 60 q^{20} + 14 q^{21} - 22 q^{22} + 31 q^{23} - 18 q^{24} + 147 q^{25} + 37 q^{26} - 121 q^{27} - 29 q^{28} + 68 q^{29} - 18 q^{30} + 25 q^{31} + 43 q^{32} - 32 q^{33} + 27 q^{34} + 51 q^{35} + 123 q^{36} - 4 q^{37} + 36 q^{38} - 2 q^{39} + 61 q^{40} + 132 q^{41} - 37 q^{42} - 91 q^{43} + 94 q^{44} + 24 q^{45} + 39 q^{47} - 131 q^{48} + 217 q^{49} + 54 q^{50} - 87 q^{51} - 12 q^{52} + 55 q^{53} - 7 q^{54} + 7 q^{55} + 104 q^{56} + 10 q^{57} - 3 q^{58} + 58 q^{59} - 60 q^{60} + 126 q^{61} + 74 q^{62} - 14 q^{63} + 122 q^{64} + 128 q^{65} + 22 q^{66} - 139 q^{67} + 190 q^{68} - 31 q^{69} - 18 q^{70} + 37 q^{71} + 18 q^{72} + 84 q^{73} + 79 q^{74} - 147 q^{75} + 23 q^{76} + 95 q^{77} - 37 q^{78} - 14 q^{79} + 145 q^{80} + 121 q^{81} + 9 q^{82} + 58 q^{83} + 29 q^{84} + 32 q^{85} + 28 q^{86} - 68 q^{87} - 84 q^{88} + 198 q^{89} + 18 q^{90} + 5 q^{91} + 98 q^{92} - 25 q^{93} + 9 q^{94} + 42 q^{95} - 43 q^{96} + 73 q^{97} + 69 q^{98} + 32 q^{99}+O(q^{100}) \) 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9 + 18 * q^10 + 32 * q^11 - 123 * q^12 + 2 * q^13 + 37 * q^14 - 24 * q^15 + 131 * q^16 + 87 * q^17 + 7 * q^18 - 10 * q^19 + 60 * q^20 + 14 * q^21 - 22 * q^22 + 31 * q^23 - 18 * q^24 + 147 * q^25 + 37 * q^26 - 121 * q^27 - 29 * q^28 + 68 * q^29 - 18 * q^30 + 25 * q^31 + 43 * q^32 - 32 * q^33 + 27 * q^34 + 51 * q^35 + 123 * q^36 - 4 * q^37 + 36 * q^38 - 2 * q^39 + 61 * q^40 + 132 * q^41 - 37 * q^42 - 91 * q^43 + 94 * q^44 + 24 * q^45 + 39 * q^47 - 131 * q^48 + 217 * q^49 + 54 * q^50 - 87 * q^51 - 12 * q^52 + 55 * q^53 - 7 * q^54 + 7 * q^55 + 104 * q^56 + 10 * q^57 - 3 * q^58 + 58 * q^59 - 60 * q^60 + 126 * q^61 + 74 * q^62 - 14 * q^63 + 122 * q^64 + 128 * q^65 + 22 * q^66 - 139 * q^67 + 190 * q^68 - 31 * q^69 - 18 * q^70 + 37 * q^71 + 18 * q^72 + 84 * q^73 + 79 * q^74 - 147 * q^75 + 23 * q^76 + 95 * q^77 - 37 * q^78 - 14 * q^79 + 145 * q^80 + 121 * q^81 + 9 * q^82 + 58 * q^83 + 29 * q^84 + 32 * q^85 + 28 * q^86 - 68 * q^87 - 84 * q^88 + 198 * q^89 + 18 * q^90 + 5 * q^91 + 98 * q^92 - 25 * q^93 + 9 * q^94 + 42 * q^95 - 43 * q^96 + 73 * q^97 + 69 * q^98 + 32 * 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 $

$2$	−2.11893	−1.49831	−0.749156	−	0.662393i	$-0.769541\pi$
$2$	−2.11893	−1.49831	−0.749156	+	0.662393i	$0.769541\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.48988	1.24494
$5$	3.42435	1.53141	0.765707	−	0.643190i	$-0.222389\pi$
$5$	3.42435	1.53141	0.765707	+	0.643190i	$0.222389\pi$
$6$	2.11893	0.865051
$7$	−4.49601	−1.69933	−0.849666	−	0.527321i	$-0.823196\pi$
$7$	−4.49601	−1.69933	−0.849666	+	0.527321i	$0.823196\pi$
$8$	−1.03803	−0.366998
$9$	1.00000	0.333333
$10$	−7.25596	−2.29454
$11$	−5.90928	−1.78172	−0.890858	−	0.454282i	$-0.849896\pi$
$11$	−5.90928	−1.78172	−0.890858	+	0.454282i	$0.849896\pi$
$12$	−2.48988	−0.718767
$13$	3.86921	1.07313	0.536563	−	0.843860i	$-0.319722\pi$
$13$	3.86921	1.07313	0.536563	+	0.843860i	$0.319722\pi$
$14$	9.52675	2.54613
$15$	−3.42435	−0.884162
$16$	−2.78025	−0.695064
$17$	−0.695427	−0.168666	−0.0843329	−	0.996438i	$-0.526876\pi$
$17$	−0.695427	−0.168666	−0.0843329	+	0.996438i	$0.526876\pi$
$18$	−2.11893	−0.499438
$19$	−4.06191	−0.931865	−0.465933	−	0.884820i	$-0.654281\pi$
$19$	−4.06191	−0.931865	−0.465933	+	0.884820i	$0.654281\pi$
$20$	8.52621	1.90652
$21$	4.49601	0.981110
$22$	12.5214	2.66957
$23$	6.24148	1.30144	0.650719	−	0.759318i	$-0.274468\pi$
$23$	6.24148	1.30144	0.650719	+	0.759318i	$0.274468\pi$
$24$	1.03803	0.211886
$25$	6.72614	1.34523
$26$	−8.19860	−1.60788
$27$	−1.00000	−0.192450
$28$	−11.1945	−2.11557
$29$	−0.807465	−0.149942	−0.0749712	−	0.997186i	$-0.523886\pi$
$29$	−0.807465	−0.149942	−0.0749712	+	0.997186i	$0.523886\pi$
$30$	7.25596	1.32475
$31$	0.0987651	0.0177387	0.00886936	−	0.999961i	$-0.497177\pi$
$31$	0.0987651	0.0177387	0.00886936	+	0.999961i	$0.497177\pi$
$32$	7.96723	1.40842
$33$	5.90928	1.02867
$34$	1.47356	0.252714
$35$	−15.3959	−2.60238
$36$	2.48988	0.414980
$37$	4.03296	0.663015	0.331507	−	0.943453i	$-0.392443\pi$
$37$	4.03296	0.663015	0.331507	+	0.943453i	$0.392443\pi$
$38$	8.60691	1.39623
$39$	−3.86921	−0.619570
$40$	−3.55456	−0.562025
$41$	−1.08739	−0.169822	−0.0849110	−	0.996389i	$-0.527061\pi$
$41$	−1.08739	−0.169822	−0.0849110	+	0.996389i	$0.527061\pi$
$42$	−9.52675	−1.47001
$43$	7.95060	1.21246	0.606228	−	0.795291i	$-0.292682\pi$
$43$	7.95060	1.21246	0.606228	+	0.795291i	$0.292682\pi$
$44$	−14.7134	−2.21813
$45$	3.42435	0.510471
$46$	−13.2253	−1.94996
$47$	−1.33337	−0.194492	−0.0972458	−	0.995260i	$-0.531003\pi$
$47$	−1.33337	−0.194492	−0.0972458	+	0.995260i	$0.531003\pi$
$48$	2.78025	0.401295
$49$	13.2141	1.88773
$50$	−14.2523	−2.01557
$51$	0.695427	0.0973792
$52$	9.63388	1.33598
$53$	−7.73655	−1.06270	−0.531348	−	0.847154i	$-0.678314\pi$
$53$	−7.73655	−1.06270	−0.531348	+	0.847154i	$0.678314\pi$
$54$	2.11893	0.288350
$55$	−20.2354	−2.72854
$56$	4.66697	0.623651
$57$	4.06191	0.538013
$58$	1.71096	0.224661
$59$	−0.270020	−0.0351536	−0.0175768	−	0.999846i	$-0.505595\pi$
$59$	−0.270020	−0.0351536	−0.0175768	+	0.999846i	$0.505595\pi$
$60$	−8.52621	−1.10073
$61$	2.78186	0.356181	0.178090	−	0.984014i	$-0.443008\pi$
$61$	2.78186	0.356181	0.178090	+	0.984014i	$0.443008\pi$
$62$	−0.209277	−0.0265782
$63$	−4.49601	−0.566444
$64$	−11.3215	−1.41519
$65$	13.2495	1.64340
$66$	−12.5214	−1.54128
$67$	−11.8525	−1.44801	−0.724006	−	0.689794i	$-0.757701\pi$
$67$	−11.8525	−1.44801	−0.724006	+	0.689794i	$0.757701\pi$
$68$	−1.73153	−0.209979
$69$	−6.24148	−0.751386
$70$	32.6229	3.89918
$71$	−14.6385	−1.73727	−0.868636	−	0.495451i	$-0.835003\pi$
$71$	−14.6385	−1.73727	−0.868636	+	0.495451i	$0.835003\pi$
$72$	−1.03803	−0.122333
$73$	9.55676	1.11853	0.559267	−	0.828988i	$-0.311083\pi$
$73$	9.55676	1.11853	0.559267	+	0.828988i	$0.311083\pi$
$74$	−8.54558	−0.993403
$75$	−6.72614	−0.776668
$76$	−10.1137	−1.16012
$77$	26.5682	3.02773
$78$	8.19860	0.928309
$79$	1.77955	0.200215	0.100108	−	0.994977i	$-0.468081\pi$
$79$	1.77955	0.200215	0.100108	+	0.994977i	$0.468081\pi$
$80$	−9.52055	−1.06443
$81$	1.00000	0.111111
$82$	2.30411	0.254446
$83$	−3.82463	−0.419808	−0.209904	−	0.977722i	$-0.567315\pi$
$83$	−3.82463	−0.419808	−0.209904	+	0.977722i	$0.567315\pi$
$84$	11.1945	1.22142
$85$	−2.38138	−0.258297
$86$	−16.8468	−1.81664
$87$	0.807465	0.0865693
$88$	6.13399	0.653885
$89$	4.22271	0.447606	0.223803	−	0.974634i	$-0.428153\pi$
$89$	4.22271	0.447606	0.223803	+	0.974634i	$0.428153\pi$
$90$	−7.25596	−0.764846
$91$	−17.3960	−1.82360
$92$	15.5405	1.62021
$93$	−0.0987651	−0.0102415
$94$	2.82532	0.291409
$95$	−13.9094	−1.42707
$96$	−7.96723	−0.813152
$97$	−17.1152	−1.73779	−0.868894	−	0.494998i	$-0.835169\pi$
$97$	−17.1152	−1.73779	−0.868894	+	0.494998i	$0.835169\pi$
$98$	−27.9998	−2.82841
$99$	−5.90928	−0.593905
$100$	16.7473	1.67473
$101$	−12.7548	−1.26915	−0.634575	−	0.772861i	$-0.718825\pi$
$101$	−12.7548	−1.26915	−0.634575	+	0.772861i	$0.718825\pi$
$102$	−1.47356	−0.145905
$103$	−11.8842	−1.17099	−0.585494	−	0.810677i	$-0.699099\pi$
$103$	−11.8842	−1.17099	−0.585494	+	0.810677i	$0.699099\pi$
$104$	−4.01634	−0.393835
$105$	15.3959	1.50249
$106$	16.3932	1.59225
$107$	18.1256	1.75227	0.876135	−	0.482066i	$-0.160114\pi$
$107$	18.1256	1.75227	0.876135	+	0.482066i	$0.160114\pi$
$108$	−2.48988	−0.239589
$109$	−16.3679	−1.56776	−0.783881	−	0.620911i	$-0.786763\pi$
$109$	−16.3679	−1.56776	−0.783881	+	0.620911i	$0.786763\pi$
$110$	42.8775	4.08821
$111$	−4.03296	−0.382792
$112$	12.5001	1.18114
$113$	−9.64805	−0.907612	−0.453806	−	0.891101i	$-0.649934\pi$
$113$	−9.64805	−0.907612	−0.453806	+	0.891101i	$0.649934\pi$
$114$	−8.60691	−0.806111
$115$	21.3730	1.99304
$116$	−2.01049	−0.186669
$117$	3.86921	0.357709
$118$	0.572154	0.0526710
$119$	3.12665	0.286619
$120$	3.55456	0.324485
$121$	23.9196	2.17451
$122$	−5.89458	−0.533670
$123$	1.08739	0.0980468
$124$	0.245913	0.0220837
$125$	5.91092	0.528688
$126$	9.52675	0.848710
$127$	−15.9469	−1.41506	−0.707531	−	0.706683i	$-0.750191\pi$
$127$	−15.9469	−1.41506	−0.707531	+	0.706683i	$0.750191\pi$
$128$	8.05510	0.711977
$129$	−7.95060	−0.700011
$130$	−28.0749	−2.46233
$131$	9.86981	0.862329	0.431165	−	0.902273i	$-0.358103\pi$
$131$	9.86981	0.862329	0.431165	+	0.902273i	$0.358103\pi$
$132$	14.7134	1.28064
$133$	18.2624	1.58355
$134$	25.1146	2.16958
$135$	−3.42435	−0.294721
$136$	0.721871	0.0618999
$137$	19.2852	1.64765	0.823824	−	0.566845i	$-0.191836\pi$
$137$	19.2852	1.64765	0.823824	+	0.566845i	$0.191836\pi$
$138$	13.2253	1.12581
$139$	9.83479	0.834176	0.417088	−	0.908866i	$-0.363051\pi$
$139$	9.83479	0.834176	0.417088	+	0.908866i	$0.363051\pi$
$140$	−38.3339	−3.23981
$141$	1.33337	0.112290
$142$	31.0180	2.60298
$143$	−22.8643	−1.91201
$144$	−2.78025	−0.231688
$145$	−2.76504	−0.229624
$146$	−20.2501	−1.67591
$147$	−13.2141	−1.08988
$148$	10.0416	0.825414
$149$	15.1498	1.24112	0.620559	−	0.784160i	$-0.286906\pi$
$149$	15.1498	1.24112	0.620559	+	0.784160i	$0.286906\pi$
$150$	14.2523	1.16369
$151$	9.58713	0.780190	0.390095	−	0.920775i	$-0.372442\pi$
$151$	9.58713	0.780190	0.390095	+	0.920775i	$0.372442\pi$
$152$	4.21636	0.341992
$153$	−0.695427	−0.0562219
$154$	−56.2963	−4.53648
$155$	0.338206	0.0271653
$156$	−9.63388	−0.771327
$157$	10.7216	0.855680	0.427840	−	0.903854i	$-0.359275\pi$
$157$	10.7216	0.855680	0.427840	+	0.903854i	$0.359275\pi$
$158$	−3.77075	−0.299985
$159$	7.73655	0.613548
$160$	27.2825	2.15687
$161$	−28.0618	−2.21158
$162$	−2.11893	−0.166479
$163$	−9.53862	−0.747123	−0.373561	−	0.927605i	$-0.621863\pi$
$163$	−9.53862	−0.747123	−0.373561	+	0.927605i	$0.621863\pi$
$164$	−2.70747	−0.211418
$165$	20.2354	1.57533
$166$	8.10414	0.629003
$167$	4.15551	0.321563	0.160782	−	0.986990i	$-0.448599\pi$
$167$	4.15551	0.321563	0.160782	+	0.986990i	$0.448599\pi$
$168$	−4.66697	−0.360065
$169$	1.97080	0.151600
$170$	5.04599	0.387010
$171$	−4.06191	−0.310622
$172$	19.7960	1.50943
$173$	−23.0553	−1.75287	−0.876433	−	0.481524i	$-0.840083\pi$
$173$	−23.0553	−1.75287	−0.876433	+	0.481524i	$0.840083\pi$
$174$	−1.71096	−0.129708
$175$	−30.2408	−2.28599
$176$	16.4293	1.23841
$177$	0.270020	0.0202959
$178$	−8.94764	−0.670654
$179$	7.56539	0.565464	0.282732	−	0.959199i	$-0.408759\pi$
$179$	7.56539	0.565464	0.282732	+	0.959199i	$0.408759\pi$
$180$	8.52621	0.635506
$181$	17.9138	1.33153	0.665763	−	0.746164i	$-0.268106\pi$
$181$	17.9138	1.33153	0.665763	+	0.746164i	$0.268106\pi$
$182$	36.8610	2.73232
$183$	−2.78186	−0.205641
$184$	−6.47882	−0.477625
$185$	13.8102	1.01535
$186$	0.209277	0.0153449
$187$	4.10947	0.300515
$188$	−3.31992	−0.242130
$189$	4.49601	0.327037
$190$	29.4730	2.13820
$191$	6.60985	0.478272	0.239136	−	0.970986i	$-0.423136\pi$
$191$	6.60985	0.478272	0.239136	+	0.970986i	$0.423136\pi$
$192$	11.3215	0.817060
$193$	23.4792	1.69007	0.845033	−	0.534714i	$-0.179581\pi$
$193$	23.4792	1.69007	0.845033	+	0.534714i	$0.179581\pi$
$194$	36.2660	2.60375
$195$	−13.2495	−0.948818
$196$	32.9016	2.35011
$197$	2.43913	0.173781	0.0868905	−	0.996218i	$-0.472307\pi$
$197$	2.43913	0.173781	0.0868905	+	0.996218i	$0.472307\pi$
$198$	12.5214	0.889856
$199$	−14.7171	−1.04327	−0.521635	−	0.853169i	$-0.674678\pi$
$199$	−14.7171	−1.04327	−0.521635	+	0.853169i	$0.674678\pi$
$200$	−6.98191	−0.493696
$201$	11.8525	0.836010
$202$	27.0266	1.90158
$203$	3.63037	0.254802
$204$	1.73153	0.121231
$205$	−3.72360	−0.260068
$206$	25.1819	1.75451
$207$	6.24148	0.433813
$208$	−10.7574	−0.745891
$209$	24.0030	1.66032
$210$	−32.6229	−2.25119
$211$	6.31230	0.434557	0.217278	−	0.976110i	$-0.430282\pi$
$211$	6.31230	0.434557	0.217278	+	0.976110i	$0.430282\pi$
$212$	−19.2631	−1.32299
$213$	14.6385	1.00301
$214$	−38.4070	−2.62545
$215$	27.2256	1.85677
$216$	1.03803	0.0706287
$217$	−0.444049	−0.0301440
$218$	34.6825	2.34900
$219$	−9.55676	−0.645786
$220$	−50.3838	−3.39688
$221$	−2.69075	−0.181000
$222$	8.54558	0.573542
$223$	−2.44286	−0.163586	−0.0817931	−	0.996649i	$-0.526065\pi$
$223$	−2.44286	−0.163586	−0.0817931	+	0.996649i	$0.526065\pi$
$224$	−35.8207	−2.39337
$225$	6.72614	0.448410
$226$	20.4436	1.35989
$227$	8.02966	0.532947	0.266474	−	0.963842i	$-0.414141\pi$
$227$	8.02966	0.532947	0.266474	+	0.963842i	$0.414141\pi$
$228$	10.1137	0.669794
$229$	−8.30926	−0.549091	−0.274546	−	0.961574i	$-0.588527\pi$
$229$	−8.30926	−0.549091	−0.274546	+	0.961574i	$0.588527\pi$
$230$	−45.2879	−2.98620
$231$	−26.5682	−1.74806
$232$	0.838169	0.0550285
$233$	6.09757	0.399465	0.199732	−	0.979850i	$-0.435993\pi$
$233$	6.09757	0.399465	0.199732	+	0.979850i	$0.435993\pi$
$234$	−8.19860	−0.535960
$235$	−4.56591	−0.297847
$236$	−0.672317	−0.0437641
$237$	−1.77955	−0.115594
$238$	−6.62516	−0.429445
$239$	−10.4499	−0.675949	−0.337975	−	0.941155i	$-0.609742\pi$
$239$	−10.4499	−0.675949	−0.337975	+	0.941155i	$0.609742\pi$
$240$	9.52055	0.614549
$241$	−13.5698	−0.874108	−0.437054	−	0.899435i	$-0.643978\pi$
$241$	−13.5698	−0.874108	−0.437054	+	0.899435i	$0.643978\pi$
$242$	−50.6841	−3.25810
$243$	−1.00000	−0.0641500
$244$	6.92650	0.443424
$245$	45.2497	2.89090
$246$	−2.30411	−0.146905
$247$	−15.7164	−1.00001
$248$	−0.102521	−0.00651007
$249$	3.82463	0.242376
$250$	−12.5248	−0.792140
$251$	7.72880	0.487838	0.243919	−	0.969796i	$-0.421567\pi$
$251$	7.72880	0.487838	0.243919	+	0.969796i	$0.421567\pi$
$252$	−11.1945	−0.705189
$253$	−36.8827	−2.31879
$254$	33.7905	2.12020
$255$	2.38138	0.149128
$256$	5.57482	0.348426
$257$	3.61870	0.225728	0.112864	−	0.993610i	$-0.463997\pi$
$257$	3.61870	0.225728	0.112864	+	0.993610i	$0.463997\pi$
$258$	16.8468	1.04884
$259$	−18.1322	−1.12668
$260$	32.9897	2.04594
$261$	−0.807465	−0.0499808
$262$	−20.9135	−1.29204
$263$	−12.1724	−0.750581	−0.375290	−	0.926907i	$-0.622457\pi$
$263$	−12.1724	−0.750581	−0.375290	+	0.926907i	$0.622457\pi$
$264$	−6.13399	−0.377521
$265$	−26.4926	−1.62743
$266$	−38.6968	−2.37265
$267$	−4.22271	−0.258426
$268$	−29.5113	−1.80269
$269$	10.1959	0.621653	0.310826	−	0.950467i	$-0.399394\pi$
$269$	10.1959	0.621653	0.310826	+	0.950467i	$0.399394\pi$
$270$	7.25596	0.441584
$271$	28.6688	1.74150	0.870752	−	0.491723i	$-0.163633\pi$
$271$	28.6688	1.74150	0.870752	+	0.491723i	$0.163633\pi$
$272$	1.93346	0.117233
$273$	17.3960	1.05285
$274$	−40.8641	−2.46869
$275$	−39.7467	−2.39682
$276$	−15.5405	−0.935431
$277$	−21.9683	−1.31995	−0.659973	−	0.751290i	$-0.729432\pi$
$277$	−21.9683	−1.31995	−0.659973	+	0.751290i	$0.729432\pi$
$278$	−20.8393	−1.24986
$279$	0.0987651	0.00591291
$280$	15.9813	0.955067
$281$	26.5057	1.58119	0.790597	−	0.612336i	$-0.209770\pi$
$281$	26.5057	1.58119	0.790597	+	0.612336i	$0.209770\pi$
$282$	−2.82532	−0.168245
$283$	2.51307	0.149387	0.0746933	−	0.997207i	$-0.476202\pi$
$283$	2.51307	0.149387	0.0746933	+	0.997207i	$0.476202\pi$
$284$	−36.4481	−2.16280
$285$	13.9094	0.823920
$286$	48.4479	2.86478
$287$	4.88892	0.288584
$288$	7.96723	0.469473
$289$	−16.5164	−0.971552
$290$	5.85894	0.344049
$291$	17.1152	1.00331
$292$	23.7952	1.39251
$293$	−14.7385	−0.861034	−0.430517	−	0.902582i	$-0.641669\pi$
$293$	−14.7385	−0.861034	−0.430517	+	0.902582i	$0.641669\pi$
$294$	27.9998	1.63298
$295$	−0.924641	−0.0538347
$296$	−4.18632	−0.243325
$297$	5.90928	0.342891
$298$	−32.1014	−1.85958
$299$	24.1496	1.39661
$300$	−16.7473	−0.966906
$301$	−35.7460	−2.06036
$302$	−20.3145	−1.16897
$303$	12.7548	0.732744
$304$	11.2931	0.647705
$305$	9.52605	0.545460
$306$	1.47356	0.0842380
$307$	−4.21563	−0.240599	−0.120299	−	0.992738i	$-0.538385\pi$
$307$	−4.21563	−0.240599	−0.120299	+	0.992738i	$0.538385\pi$
$308$	66.1517	3.76934
$309$	11.8842	0.676070
$310$	−0.716636	−0.0407022
$311$	23.5222	1.33382	0.666912	−	0.745137i	$-0.267616\pi$
$311$	23.5222	1.33382	0.666912	+	0.745137i	$0.267616\pi$
$312$	4.01634	0.227381
$313$	6.99878	0.395594	0.197797	−	0.980243i	$-0.436621\pi$
$313$	6.99878	0.395594	0.197797	+	0.980243i	$0.436621\pi$
$314$	−22.7184	−1.28208
$315$	−15.3959	−0.867460
$316$	4.43088	0.249256
$317$	−0.901870	−0.0506541	−0.0253270	−	0.999679i	$-0.508063\pi$
$317$	−0.901870	−0.0506541	−0.0253270	+	0.999679i	$0.508063\pi$
$318$	−16.3932	−0.919287
$319$	4.77154	0.267155
$320$	−38.7688	−2.16724
$321$	−18.1256	−1.01167
$322$	59.4610	3.31363
$323$	2.82476	0.157174
$324$	2.48988	0.138327
$325$	26.0249	1.44360
$326$	20.2117	1.11942
$327$	16.3679	0.905148
$328$	1.12874	0.0623242
$329$	5.99483	0.330506
$330$	−42.8775	−2.36033
$331$	−4.02030	−0.220976	−0.110488	−	0.993877i	$-0.535241\pi$
$331$	−4.02030	−0.220976	−0.110488	+	0.993877i	$0.535241\pi$
$332$	−9.52288	−0.522636
$333$	4.03296	0.221005
$334$	−8.80525	−0.481802
$335$	−40.5870	−2.21751
$336$	−12.5001	−0.681934
$337$	9.75534	0.531407	0.265704	−	0.964055i	$-0.414396\pi$
$337$	9.75534	0.531407	0.265704	+	0.964055i	$0.414396\pi$
$338$	−4.17599	−0.227144
$339$	9.64805	0.524010
$340$	−5.92936	−0.321565
$341$	−0.583631	−0.0316054
$342$	8.60691	0.465408
$343$	−27.9387	−1.50855
$344$	−8.25293	−0.444968
$345$	−21.3730	−1.15068
$346$	48.8528	2.62634
$347$	11.2610	0.604525	0.302262	−	0.953225i	$-0.402258\pi$
$347$	11.2610	0.604525	0.302262	+	0.953225i	$0.402258\pi$
$348$	2.01049	0.107774
$349$	4.57803	0.245056	0.122528	−	0.992465i	$-0.460900\pi$
$349$	4.57803	0.245056	0.122528	+	0.992465i	$0.460900\pi$
$350$	64.0783	3.42513
$351$	−3.86921	−0.206523
$352$	−47.0806	−2.50940
$353$	−6.52775	−0.347437	−0.173718	−	0.984795i	$-0.555578\pi$
$353$	−6.52775	−0.347437	−0.173718	+	0.984795i	$0.555578\pi$
$354$	−0.572154	−0.0304096
$355$	−50.1273	−2.66048
$356$	10.5140	0.557243
$357$	−3.12665	−0.165480
$358$	−16.0306	−0.847241
$359$	17.6646	0.932300	0.466150	−	0.884706i	$-0.345641\pi$
$359$	17.6646	0.932300	0.466150	+	0.884706i	$0.345641\pi$
$360$	−3.55456	−0.187342
$361$	−2.50092	−0.131627
$362$	−37.9582	−1.99504
$363$	−23.9196	−1.25545
$364$	−43.3140	−2.27027
$365$	32.7256	1.71294
$366$	5.89458	0.308114
$367$	−4.73746	−0.247294	−0.123647	−	0.992326i	$-0.539459\pi$
$367$	−4.73746	−0.247294	−0.123647	+	0.992326i	$0.539459\pi$
$368$	−17.3529	−0.904582
$369$	−1.08739	−0.0566073
$370$	−29.2630	−1.52131
$371$	34.7836	1.80587
$372$	−0.245913	−0.0127500
$373$	−28.9427	−1.49860	−0.749298	−	0.662233i	$-0.769609\pi$
$373$	−28.9427	−1.49860	−0.749298	+	0.662233i	$0.769609\pi$
$374$	−8.70770	−0.450265
$375$	−5.91092	−0.305238
$376$	1.38407	0.0713779
$377$	−3.12425	−0.160907
$378$	−9.52675	−0.490003
$379$	−29.8880	−1.53524	−0.767621	−	0.640904i	$-0.778560\pi$
$379$	−29.8880	−1.53524	−0.767621	+	0.640904i	$0.778560\pi$
$380$	−34.6327	−1.77662
$381$	15.9469	0.816986
$382$	−14.0058	−0.716601
$383$	29.0542	1.48460	0.742299	−	0.670068i	$-0.233735\pi$
$383$	29.0542	1.48460	0.742299	+	0.670068i	$0.233735\pi$
$384$	−8.05510	−0.411060
$385$	90.9787	4.63670
$386$	−49.7508	−2.53225
$387$	7.95060	0.404152
$388$	−42.6149	−2.16344
$389$	−0.174742	−0.00885976	−0.00442988	−	0.999990i	$-0.501410\pi$
$389$	−0.174742	−0.00885976	−0.00442988	+	0.999990i	$0.501410\pi$
$390$	28.0749	1.42163
$391$	−4.34049	−0.219508
$392$	−13.7166	−0.692792
$393$	−9.86981	−0.497866
$394$	−5.16836	−0.260378
$395$	6.09380	0.306613
$396$	−14.7134	−0.739377
$397$	−15.4194	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$397$	−15.4194	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$398$	31.1846	1.56314
$399$	−18.2624	−0.914262
$400$	−18.7004	−0.935020
$401$	10.6223	0.530453	0.265226	−	0.964186i	$-0.414553\pi$
$401$	10.6223	0.530453	0.265226	+	0.964186i	$0.414553\pi$
$402$	−25.1146	−1.25260
$403$	0.382143	0.0190359
$404$	−31.7579	−1.58002
$405$	3.42435	0.170157
$406$	−7.69252	−0.381773
$407$	−23.8319	−1.18130
$408$	−0.721871	−0.0357379
$409$	7.79280	0.385329	0.192665	−	0.981265i	$-0.438287\pi$
$409$	7.79280	0.385329	0.192665	+	0.981265i	$0.438287\pi$
$410$	7.89007	0.389663
$411$	−19.2852	−0.951270
$412$	−29.5903	−1.45781
$413$	1.21401	0.0597376
$414$	−13.2253	−0.649987
$415$	−13.0969	−0.642900
$416$	30.8269	1.51141
$417$	−9.83479	−0.481612
$418$	−50.8607	−2.48768
$419$	26.8703	1.31270	0.656350	−	0.754456i	$-0.272099\pi$
$419$	26.8703	1.31270	0.656350	+	0.754456i	$0.272099\pi$
$420$	38.3339	1.87050
$421$	14.0264	0.683603	0.341801	−	0.939772i	$-0.388963\pi$
$421$	14.0264	0.683603	0.341801	+	0.939772i	$0.388963\pi$
$422$	−13.3753	−0.651102
$423$	−1.33337	−0.0648305
$424$	8.03073	0.390007
$425$	−4.67754	−0.226894
$426$	−31.0180	−1.50283
$427$	−12.5073	−0.605269
$428$	45.1306	2.18147
$429$	22.8643	1.10390
$430$	−57.6893	−2.78202
$431$	20.4380	0.984462	0.492231	−	0.870465i	$-0.336182\pi$
$431$	20.4380	0.984462	0.492231	+	0.870465i	$0.336182\pi$
$432$	2.78025	0.133765
$433$	16.7876	0.806763	0.403381	−	0.915032i	$-0.367835\pi$
$433$	16.7876	0.806763	0.403381	+	0.915032i	$0.367835\pi$
$434$	0.940910	0.0451651
$435$	2.76504	0.132573
$436$	−40.7542	−1.95177
$437$	−25.3523	−1.21277
$438$	20.2501	0.967589
$439$	25.7944	1.23110	0.615550	−	0.788098i	$-0.288934\pi$
$439$	25.7944	1.23110	0.615550	+	0.788098i	$0.288934\pi$
$440$	21.0049	1.00137
$441$	13.2141	0.629243
$442$	5.70153	0.271194
$443$	27.1370	1.28932	0.644659	−	0.764470i	$-0.276999\pi$
$443$	27.1370	1.28932	0.644659	+	0.764470i	$0.276999\pi$
$444$	−10.0416	−0.476553
$445$	14.4600	0.685471
$446$	5.17626	0.245103
$447$	−15.1498	−0.716560
$448$	50.9017	2.40488
$449$	−5.91599	−0.279193	−0.139596	−	0.990208i	$-0.544581\pi$
$449$	−5.91599	−0.279193	−0.139596	+	0.990208i	$0.544581\pi$
$450$	−14.2523	−0.671858
$451$	6.42570	0.302575
$452$	−24.0225	−1.12992
$453$	−9.58713	−0.450443
$454$	−17.0143	−0.798521
$455$	−59.5700	−2.79268
$456$	−4.21636	−0.197449
$457$	−25.9518	−1.21397	−0.606987	−	0.794711i	$-0.707622\pi$
$457$	−25.9518	−1.21397	−0.606987	+	0.794711i	$0.707622\pi$
$458$	17.6068	0.822710
$459$	0.695427	0.0324597
$460$	53.2162	2.48122
$461$	−29.2951	−1.36441	−0.682205	−	0.731161i	$-0.738979\pi$
$461$	−29.2951	−1.36441	−0.682205	+	0.731161i	$0.738979\pi$
$462$	56.2963	2.61914
$463$	−18.0275	−0.837807	−0.418904	−	0.908031i	$-0.637585\pi$
$463$	−18.0275	−0.837807	−0.418904	+	0.908031i	$0.637585\pi$
$464$	2.24496	0.104220
$465$	−0.338206	−0.0156839
$466$	−12.9203	−0.598523
$467$	27.1192	1.25493	0.627464	−	0.778645i	$-0.284093\pi$
$467$	27.1192	1.25493	0.627464	+	0.778645i	$0.284093\pi$
$468$	9.63388	0.445326
$469$	53.2889	2.46065
$470$	9.67486	0.446268
$471$	−10.7216	−0.494027
$472$	0.280287	0.0129013
$473$	−46.9823	−2.16025
$474$	3.77075	0.173197
$475$	−27.3210	−1.25357
$476$	7.78498	0.356824
$477$	−7.73655	−0.354232
$478$	22.1427	1.01278
$479$	−20.9066	−0.955249	−0.477624	−	0.878564i	$-0.658502\pi$
$479$	−20.9066	−0.955249	−0.477624	+	0.878564i	$0.658502\pi$
$480$	−27.2825	−1.24527
$481$	15.6044	0.711498
$482$	28.7535	1.30969
$483$	28.0618	1.27685
$484$	59.5570	2.70714
$485$	−58.6084	−2.66127
$486$	2.11893	0.0961168
$487$	31.7780	1.44000	0.720000	−	0.693974i	$-0.244142\pi$
$487$	31.7780	1.44000	0.720000	+	0.693974i	$0.244142\pi$
$488$	−2.88764	−0.130717
$489$	9.53862	0.431351
$490$	−95.8811	−4.33147
$491$	31.1534	1.40593	0.702967	−	0.711223i	$-0.251858\pi$
$491$	31.1534	1.40593	0.702967	+	0.711223i	$0.251858\pi$
$492$	2.70747	0.122062
$493$	0.561533	0.0252902
$494$	33.3020	1.49833
$495$	−20.2354	−0.909515
$496$	−0.274592	−0.0123295
$497$	65.8149	2.95220
$498$	−8.10414	−0.363155
$499$	−20.4679	−0.916268	−0.458134	−	0.888883i	$-0.651482\pi$
$499$	−20.4679	−0.916268	−0.458134	+	0.888883i	$0.651482\pi$
$500$	14.7175	0.658186
$501$	−4.15551	−0.185655
$502$	−16.3768	−0.730934
$503$	−9.44204	−0.421000	−0.210500	−	0.977594i	$-0.567509\pi$
$503$	−9.44204	−0.421000	−0.210500	+	0.977594i	$0.567509\pi$
$504$	4.66697	0.207884
$505$	−43.6768	−1.94359
$506$	78.1519	3.47428
$507$	−1.97080	−0.0875262
$508$	−39.7060	−1.76167
$509$	−33.9825	−1.50625	−0.753124	−	0.657879i	$-0.771454\pi$
$509$	−33.9825	−1.50625	−0.753124	+	0.657879i	$0.771454\pi$
$510$	−5.04599	−0.223440
$511$	−42.9673	−1.90076
$512$	−27.9229	−1.23403
$513$	4.06191	0.179338
$514$	−7.66779	−0.338212
$515$	−40.6957	−1.79327
$516$	−19.7960	−0.871472
$517$	7.87924	0.346529
$518$	38.4210	1.68812
$519$	23.0553	1.01202
$520$	−13.7533	−0.603124
$521$	43.4103	1.90184	0.950919	−	0.309439i	$-0.100141\pi$
$521$	43.4103	1.90184	0.950919	+	0.309439i	$0.100141\pi$
$522$	1.71096	0.0748869
$523$	−7.96954	−0.348484	−0.174242	−	0.984703i	$-0.555747\pi$
$523$	−7.96954	−0.348484	−0.174242	+	0.984703i	$0.555747\pi$
$524$	24.5746	1.07355
$525$	30.2408	1.31982
$526$	25.7925	1.12460
$527$	−0.0686839	−0.00299192
$528$	−16.4293	−0.714994
$529$	15.9561	0.693742
$530$	56.1361	2.43840
$531$	−0.270020	−0.0117179
$532$	45.4711	1.97142
$533$	−4.20735	−0.182240
$534$	8.94764	0.387202
$535$	62.0684	2.68345
$536$	12.3032	0.531417
$537$	−7.56539	−0.326471
$538$	−21.6044	−0.931430
$539$	−78.0859	−3.36340
$540$	−8.52621	−0.366910
$541$	17.9027	0.769699	0.384849	−	0.922979i	$-0.374253\pi$
$541$	17.9027	0.769699	0.384849	+	0.922979i	$0.374253\pi$
$542$	−60.7472	−2.60932
$543$	−17.9138	−0.768756
$544$	−5.54062	−0.237552
$545$	−56.0494	−2.40089
$546$	−36.8610	−1.57751
$547$	10.6068	0.453516	0.226758	−	0.973951i	$-0.427187\pi$
$547$	10.6068	0.453516	0.226758	+	0.973951i	$0.427187\pi$
$548$	48.0179	2.05122
$549$	2.78186	0.118727
$550$	84.2206	3.59118
$551$	3.27985	0.139726
$552$	6.47882	0.275757
$553$	−8.00089	−0.340232
$554$	46.5493	1.97769
$555$	−13.8102	−0.586212
$556$	24.4875	1.03850
$557$	21.9765	0.931175	0.465587	−	0.885002i	$-0.345843\pi$
$557$	21.9765	0.931175	0.465587	+	0.885002i	$0.345843\pi$
$558$	−0.209277	−0.00885939
$559$	30.7626	1.30112
$560$	42.8045	1.80882
$561$	−4.10947	−0.173502
$562$	−56.1637	−2.36912
$563$	22.8189	0.961702	0.480851	−	0.876802i	$-0.340328\pi$
$563$	22.8189	0.961702	0.480851	+	0.876802i	$0.340328\pi$
$564$	3.31992	0.139794
$565$	−33.0382	−1.38993
$566$	−5.32503	−0.223828
$567$	−4.49601	−0.188815
$568$	15.1952	0.637574
$569$	11.4447	0.479789	0.239894	−	0.970799i	$-0.422887\pi$
$569$	11.4447	0.479789	0.239894	+	0.970799i	$0.422887\pi$
$570$	−29.4730	−1.23449
$571$	38.8677	1.62656	0.813281	−	0.581872i	$-0.197679\pi$
$571$	38.8677	1.62656	0.813281	+	0.581872i	$0.197679\pi$
$572$	−56.9293	−2.38033
$573$	−6.60985	−0.276131
$574$	−10.3593	−0.432389
$575$	41.9811	1.75073
$576$	−11.3215	−0.471730
$577$	11.8427	0.493019	0.246509	−	0.969140i	$-0.420716\pi$
$577$	11.8427	0.493019	0.246509	+	0.969140i	$0.420716\pi$
$578$	34.9971	1.45569
$579$	−23.4792	−0.975760
$580$	−6.88462	−0.285868
$581$	17.1956	0.713393
$582$	−36.2660	−1.50328
$583$	45.7174	1.89342
$584$	−9.92016	−0.410499
$585$	13.2495	0.547800
$586$	31.2300	1.29010
$587$	42.2251	1.74282	0.871409	−	0.490557i	$-0.163207\pi$
$587$	42.2251	1.74282	0.871409	+	0.490557i	$0.163207\pi$
$588$	−32.9016	−1.35684
$589$	−0.401174	−0.0165301
$590$	1.95925	0.0806611
$591$	−2.43913	−0.100332
$592$	−11.2127	−0.460837
$593$	−0.771600	−0.0316858	−0.0158429	−	0.999874i	$-0.505043\pi$
$593$	−0.771600	−0.0316858	−0.0158429	+	0.999874i	$0.505043\pi$
$594$	−12.5214	−0.513758
$595$	10.7067	0.438933
$596$	37.7211	1.54512
$597$	14.7171	0.602332
$598$	−51.1714	−2.09255
$599$	33.6694	1.37569	0.687847	−	0.725856i	$-0.258556\pi$
$599$	33.6694	1.37569	0.687847	+	0.725856i	$0.258556\pi$
$600$	6.98191	0.285035
$601$	−28.8844	−1.17822	−0.589110	−	0.808053i	$-0.700521\pi$
$601$	−28.8844	−1.17822	−0.589110	+	0.808053i	$0.700521\pi$
$602$	75.7434	3.08707
$603$	−11.8525	−0.482671
$604$	23.8708	0.971290
$605$	81.9091	3.33008
$606$	−27.0266	−1.09788
$607$	7.33859	0.297864	0.148932	−	0.988847i	$-0.452416\pi$
$607$	7.33859	0.297864	0.148932	+	0.988847i	$0.452416\pi$
$608$	−32.3621	−1.31246
$609$	−3.63037	−0.147110
$610$	−20.1851	−0.817270
$611$	−5.15908	−0.208714
$612$	−1.73153	−0.0699930
$613$	46.0417	1.85961	0.929803	−	0.368058i	$-0.119977\pi$
$613$	46.0417	1.85961	0.929803	+	0.368058i	$0.119977\pi$
$614$	8.93265	0.360492
$615$	3.72360	0.150150
$616$	−27.5785	−1.11117
$617$	4.36847	0.175868	0.0879340	−	0.996126i	$-0.471974\pi$
$617$	4.36847	0.175868	0.0879340	+	0.996126i	$0.471974\pi$
$618$	−25.1819	−1.01296
$619$	30.9802	1.24520	0.622599	−	0.782541i	$-0.286077\pi$
$619$	30.9802	1.24520	0.622599	+	0.782541i	$0.286077\pi$
$620$	0.842092	0.0338192
$621$	−6.24148	−0.250462
$622$	−49.8421	−1.99848
$623$	−18.9853	−0.760632
$624$	10.7574	0.430640
$625$	−13.3897	−0.535588
$626$	−14.8299	−0.592724
$627$	−24.0030	−0.958586
$628$	26.6956	1.06527
$629$	−2.80463	−0.111828
$630$	32.6229	1.29973
$631$	1.33119	0.0529938	0.0264969	−	0.999649i	$-0.491565\pi$
$631$	1.33119	0.0529938	0.0264969	+	0.999649i	$0.491565\pi$
$632$	−1.84722	−0.0734785
$633$	−6.31230	−0.250891
$634$	1.91100	0.0758957
$635$	−54.6078	−2.16704
$636$	19.2631	0.763831
$637$	51.1282	2.02577
$638$	−10.1106	−0.400282
$639$	−14.6385	−0.579090
$640$	27.5834	1.09033
$641$	−10.8685	−0.429279	−0.214639	−	0.976693i	$-0.568858\pi$
$641$	−10.8685	−0.429279	−0.214639	+	0.976693i	$0.568858\pi$
$642$	38.4070	1.51580
$643$	−36.2662	−1.43020	−0.715099	−	0.699024i	$-0.753618\pi$
$643$	−36.2662	−1.43020	−0.715099	+	0.699024i	$0.753618\pi$
$644$	−69.8704	−2.75328
$645$	−27.2256	−1.07201
$646$	−5.98548	−0.235495
$647$	−19.0435	−0.748676	−0.374338	−	0.927292i	$-0.622130\pi$
$647$	−19.0435	−0.748676	−0.374338	+	0.927292i	$0.622130\pi$
$648$	−1.03803	−0.0407775
$649$	1.59562	0.0626337
$650$	−55.1450	−2.16296
$651$	0.444049	0.0174036
$652$	−23.7500	−0.930123
$653$	−8.71939	−0.341216	−0.170608	−	0.985339i	$-0.554573\pi$
$653$	−8.71939	−0.341216	−0.170608	+	0.985339i	$0.554573\pi$
$654$	−34.6825	−1.35619
$655$	33.7976	1.32058
$656$	3.02322	0.118037
$657$	9.55676	0.372845
$658$	−12.7026	−0.495201
$659$	−7.89491	−0.307542	−0.153771	−	0.988106i	$-0.549142\pi$
$659$	−7.89491	−0.307542	−0.153771	+	0.988106i	$0.549142\pi$
$660$	50.3838	1.96119
$661$	−29.5912	−1.15096	−0.575482	−	0.817814i	$-0.695186\pi$
$661$	−29.5912	−1.15096	−0.575482	+	0.817814i	$0.695186\pi$
$662$	8.51875	0.331091
$663$	2.69075	0.104500
$664$	3.97007	0.154068
$665$	62.5367	2.42507
$666$	−8.54558	−0.331134
$667$	−5.03978	−0.195141
$668$	10.3467	0.400327
$669$	2.44286	0.0944465
$670$	86.0012	3.32252
$671$	−16.4388	−0.634613
$672$	35.8207	1.38181
$673$	−43.5446	−1.67852	−0.839261	−	0.543729i	$-0.817012\pi$
$673$	−43.5446	−1.67852	−0.839261	+	0.543729i	$0.817012\pi$
$674$	−20.6709	−0.796214
$675$	−6.72614	−0.258889
$676$	4.90705	0.188733
$677$	−3.33678	−0.128243	−0.0641213	−	0.997942i	$-0.520424\pi$
$677$	−3.33678	−0.128243	−0.0641213	+	0.997942i	$0.520424\pi$
$678$	−20.4436	−0.785131
$679$	76.9502	2.95308
$680$	2.47194	0.0947944
$681$	−8.02966	−0.307697
$682$	1.23667	0.0473547
$683$	17.0804	0.653564	0.326782	−	0.945100i	$-0.394036\pi$
$683$	17.0804	0.653564	0.326782	+	0.945100i	$0.394036\pi$
$684$	−10.1137	−0.386706
$685$	66.0393	2.52323
$686$	59.2002	2.26028
$687$	8.30926	0.317018
$688$	−22.1047	−0.842733
$689$	−29.9343	−1.14041
$690$	45.2879	1.72408
$691$	20.6884	0.787025	0.393512	−	0.919319i	$-0.371260\pi$
$691$	20.6884	0.787025	0.393512	+	0.919319i	$0.371260\pi$
$692$	−57.4051	−2.18221
$693$	26.5682	1.00924
$694$	−23.8614	−0.905767
$695$	33.6777	1.27747
$696$	−0.838169	−0.0317707
$697$	0.756201	0.0286432
$698$	−9.70053	−0.367171
$699$	−6.09757	−0.230631
$700$	−75.2960	−2.84592
$701$	34.8993	1.31813	0.659064	−	0.752087i	$-0.270953\pi$
$701$	34.8993	1.31813	0.659064	+	0.752087i	$0.270953\pi$
$702$	8.19860	0.309436
$703$	−16.3815	−0.617840
$704$	66.9021	2.52147
$705$	4.56591	0.171962
$706$	13.8319	0.520569
$707$	57.3457	2.15671
$708$	0.672317	0.0252672
$709$	−46.7280	−1.75491	−0.877453	−	0.479662i	$-0.840759\pi$
$709$	−46.7280	−1.75491	−0.877453	+	0.479662i	$0.840759\pi$
$710$	106.216	3.98623
$711$	1.77955	0.0667384
$712$	−4.38328	−0.164270
$713$	0.616440	0.0230859
$714$	6.62516	0.247940
$715$	−78.2952	−2.92807
$716$	18.8369	0.703969
$717$	10.4499	0.390259
$718$	−37.4300	−1.39688
$719$	37.6127	1.40272	0.701359	−	0.712808i	$-0.252577\pi$
$719$	37.6127	1.40272	0.701359	+	0.712808i	$0.252577\pi$
$720$	−9.52055	−0.354810
$721$	53.4316	1.98990
$722$	5.29928	0.197219
$723$	13.5698	0.504666
$724$	44.6033	1.65767
$725$	−5.43113	−0.201707
$726$	50.6841	1.88106
$727$	19.6732	0.729637	0.364818	−	0.931079i	$-0.381131\pi$
$727$	19.6732	0.729637	0.364818	+	0.931079i	$0.381131\pi$
$728$	18.0575	0.669256
$729$	1.00000	0.0370370
$730$	−69.3435	−2.56652
$731$	−5.52906	−0.204500
$732$	−6.92650	−0.256011
$733$	28.8142	1.06428	0.532138	−	0.846658i	$-0.321389\pi$
$733$	28.8142	1.06428	0.532138	+	0.846658i	$0.321389\pi$
$734$	10.0384	0.370523
$735$	−45.2497	−1.66906
$736$	49.7273	1.83297
$737$	70.0397	2.57995
$738$	2.30411	0.0848155
$739$	26.0052	0.956616	0.478308	−	0.878192i	$-0.341250\pi$
$739$	26.0052	0.956616	0.478308	+	0.878192i	$0.341250\pi$
$740$	34.3859	1.26405
$741$	15.7164	0.577355
$742$	−73.7041	−2.70576
$743$	−12.4520	−0.456821	−0.228411	−	0.973565i	$-0.573353\pi$
$743$	−12.4520	−0.456821	−0.228411	+	0.973565i	$0.573353\pi$
$744$	0.102521	0.00375859
$745$	51.8781	1.90067
$746$	61.3277	2.24537
$747$	−3.82463	−0.139936
$748$	10.2321	0.374123
$749$	−81.4930	−2.97769
$750$	12.5248	0.457343
$751$	−12.8825	−0.470088	−0.235044	−	0.971985i	$-0.575523\pi$
$751$	−12.8825	−0.470088	−0.235044	+	0.971985i	$0.575523\pi$
$752$	3.70710	0.135184
$753$	−7.72880	−0.281653
$754$	6.62009	0.241089
$755$	32.8297	1.19479
$756$	11.1945	0.407141
$757$	12.8236	0.466082	0.233041	−	0.972467i	$-0.425132\pi$
$757$	12.8236	0.466082	0.233041	+	0.972467i	$0.425132\pi$
$758$	63.3307	2.30027
$759$	36.8827	1.33876
$760$	14.4383	0.523732
$761$	34.1128	1.23659	0.618294	−	0.785947i	$-0.287824\pi$
$761$	34.1128	1.23659	0.618294	+	0.785947i	$0.287824\pi$
$762$	−33.7905	−1.22410
$763$	73.5903	2.66415
$764$	16.4577	0.595420
$765$	−2.38138	−0.0860991
$766$	−61.5639	−2.22439
$767$	−1.04476	−0.0377242
$768$	−5.57482	−0.201164
$769$	41.1376	1.48346	0.741729	−	0.670699i	$-0.234006\pi$
$769$	41.1376	1.48346	0.741729	+	0.670699i	$0.234006\pi$
$770$	−192.778	−6.94723
$771$	−3.61870	−0.130324
$772$	58.4603	2.10403
$773$	−33.9937	−1.22267	−0.611334	−	0.791373i	$-0.709367\pi$
$773$	−33.9937	−1.22267	−0.611334	+	0.791373i	$0.709367\pi$
$774$	−16.8468	−0.605546
$775$	0.664308	0.0238626
$776$	17.7660	0.637764
$777$	18.1322	0.650490
$778$	0.370266	0.0132747
$779$	4.41688	0.158251
$780$	−32.9897	−1.18122
$781$	86.5031	3.09532
$782$	9.19722	0.328892
$783$	0.807465	0.0288564
$784$	−36.7386	−1.31209
$785$	36.7146	1.31040
$786$	20.9135	0.745959
$787$	−20.9320	−0.746146	−0.373073	−	0.927802i	$-0.621696\pi$
$787$	−20.9320	−0.746146	−0.373073	+	0.927802i	$0.621696\pi$
$788$	6.07315	0.216347
$789$	12.1724	0.433348
$790$	−12.9124	−0.459401
$791$	43.3777	1.54233
$792$	6.13399	0.217962
$793$	10.7636	0.382227
$794$	32.6728	1.15951
$795$	26.4926	0.939596
$796$	−36.6439	−1.29881
$797$	6.96570	0.246738	0.123369	−	0.992361i	$-0.460630\pi$
$797$	6.96570	0.246738	0.123369	+	0.992361i	$0.460630\pi$
$798$	38.6968	1.36985
$799$	0.927259	0.0328041
$800$	53.5887	1.89465
$801$	4.22271	0.149202
$802$	−22.5080	−0.794784
$803$	−56.4736	−1.99291
$804$	29.5113	1.04078
$805$	−96.0931	−3.38684
$806$	−0.809736	−0.0285217
$807$	−10.1959	−0.358911
$808$	13.2398	0.465775
$809$	15.6162	0.549038	0.274519	−	0.961582i	$-0.411481\pi$
$809$	15.6162	0.549038	0.274519	+	0.961582i	$0.411481\pi$
$810$	−7.25596	−0.254949
$811$	−20.6120	−0.723785	−0.361892	−	0.932220i	$-0.617869\pi$
$811$	−20.6120	−0.723785	−0.361892	+	0.932220i	$0.617869\pi$
$812$	9.03919	0.317213
$813$	−28.6688	−1.00546
$814$	50.4982	1.76996
$815$	−32.6635	−1.14415
$816$	−1.93346	−0.0676848
$817$	−32.2946	−1.12984
$818$	−16.5124	−0.577344
$819$	−17.3960	−0.607866
$820$	−9.27133	−0.323769
$821$	−36.3879	−1.26995	−0.634973	−	0.772535i	$-0.718989\pi$
$821$	−36.3879	−1.26995	−0.634973	+	0.772535i	$0.718989\pi$
$822$	40.8641	1.42530
$823$	−18.5413	−0.646308	−0.323154	−	0.946346i	$-0.604743\pi$
$823$	−18.5413	−0.646308	−0.323154	+	0.946346i	$0.604743\pi$
$824$	12.3361	0.429750
$825$	39.7467	1.38380
$826$	−2.57241	−0.0895056
$827$	−6.10385	−0.212252	−0.106126	−	0.994353i	$-0.533845\pi$
$827$	−6.10385	−0.212252	−0.106126	+	0.994353i	$0.533845\pi$
$828$	15.5405	0.540071
$829$	8.26142	0.286931	0.143466	−	0.989655i	$-0.454175\pi$
$829$	8.26142	0.286931	0.143466	+	0.989655i	$0.454175\pi$
$830$	27.7514	0.963265
$831$	21.9683	0.762071
$832$	−43.8054	−1.51868
$833$	−9.18944	−0.318395
$834$	20.8393	0.721605
$835$	14.2299	0.492446
$836$	59.7645	2.06700
$837$	−0.0987651	−0.00341382
$838$	−56.9364	−1.96684
$839$	35.1326	1.21291	0.606455	−	0.795118i	$-0.292591\pi$
$839$	35.1326	1.21291	0.606455	+	0.795118i	$0.292591\pi$
$840$	−15.9813	−0.551408
$841$	−28.3480	−0.977517
$842$	−29.7209	−1.02425
$843$	−26.5057	−0.912903
$844$	15.7169	0.540997
$845$	6.74870	0.232162
$846$	2.82532	0.0971364
$847$	−107.543	−3.69522
$848$	21.5096	0.738641
$849$	−2.51307	−0.0862484
$850$	9.91140	0.339958
$851$	25.1716	0.862873
$852$	36.4481	1.24869
$853$	29.1728	0.998858	0.499429	−	0.866355i	$-0.333543\pi$
$853$	29.1728	0.998858	0.499429	+	0.866355i	$0.333543\pi$
$854$	26.5021	0.906882
$855$	−13.9094	−0.475690
$856$	−18.8149	−0.643079
$857$	53.9446	1.84271	0.921356	−	0.388721i	$-0.127083\pi$
$857$	53.9446	1.84271	0.921356	+	0.388721i	$0.127083\pi$
$858$	−48.4479	−1.65398
$859$	20.3741	0.695156	0.347578	−	0.937651i	$-0.387004\pi$
$859$	20.3741	0.695156	0.347578	+	0.937651i	$0.387004\pi$
$860$	67.7885	2.31157
$861$	−4.88892	−0.166614
$862$	−43.3067	−1.47503
$863$	21.9144	0.745974	0.372987	−	0.927837i	$-0.378334\pi$
$863$	21.9144	0.745974	0.372987	+	0.927837i	$0.378334\pi$
$864$	−7.96723	−0.271051
$865$	−78.9495	−2.68436
$866$	−35.5719	−1.20878
$867$	16.5164	0.560926
$868$	−1.10563	−0.0375275
$869$	−10.5159	−0.356727
$870$	−5.85894	−0.198637
$871$	−45.8598	−1.55390
$872$	16.9903	0.575365
$873$	−17.1152	−0.579263
$874$	53.7199	1.81710
$875$	−26.5755	−0.898417
$876$	−23.7952	−0.803965
$877$	58.5370	1.97665	0.988327	−	0.152350i	$-0.0486842\pi$
$877$	58.5370	1.97665	0.988327	+	0.152350i	$0.0486842\pi$
$878$	−54.6567	−1.84457
$879$	14.7385	0.497118
$880$	56.2596	1.89651
$881$	15.3846	0.518321	0.259161	−	0.965834i	$-0.416554\pi$
$881$	15.3846	0.518321	0.259161	+	0.965834i	$0.416554\pi$
$882$	−27.9998	−0.942803
$883$	−51.0231	−1.71707	−0.858533	−	0.512759i	$-0.828623\pi$
$883$	−51.0231	−1.71707	−0.858533	+	0.512759i	$0.828623\pi$
$884$	−6.69966	−0.225334
$885$	0.924641	0.0310815
$886$	−57.5015	−1.93180
$887$	−16.1845	−0.543422	−0.271711	−	0.962379i	$-0.587590\pi$
$887$	−16.1845	−0.543422	−0.271711	+	0.962379i	$0.587590\pi$
$888$	4.18632	0.140484
$889$	71.6976	2.40466
$890$	−30.6398	−1.02705
$891$	−5.90928	−0.197968
$892$	−6.08244	−0.203655
$893$	5.41601	0.181240
$894$	32.1014	1.07363
$895$	25.9065	0.865959
$896$	−36.2158	−1.20988
$897$	−24.1496	−0.806332
$898$	12.5356	0.418318
$899$	−0.0797493	−0.00265979
$900$	16.7473	0.558243
$901$	5.38020	0.179240
$902$	−13.6156	−0.453351
$903$	35.7460	1.18955
$904$	10.0149	0.333091
$905$	61.3432	2.03912
$906$	20.3145	0.674904
$907$	46.5361	1.54521	0.772603	−	0.634890i	$-0.218954\pi$
$907$	46.5361	1.54521	0.772603	+	0.634890i	$0.218954\pi$
$908$	19.9929	0.663488
$909$	−12.7548	−0.423050
$910$	126.225	4.18431
$911$	−16.0795	−0.532737	−0.266368	−	0.963871i	$-0.585824\pi$
$911$	−16.0795	−0.532737	−0.266368	+	0.963871i	$0.585824\pi$
$912$	−11.2931	−0.373953
$913$	22.6008	0.747978
$914$	54.9902	1.81891
$915$	−9.52605	−0.314922
$916$	−20.6891	−0.683586
$917$	−44.3748	−1.46538
$918$	−1.47356	−0.0486348
$919$	−1.61284	−0.0532027	−0.0266013	−	0.999646i	$-0.508468\pi$
$919$	−1.61284	−0.0532027	−0.0266013	+	0.999646i	$0.508468\pi$
$920$	−22.1857	−0.731441
$921$	4.21563	0.138910
$922$	62.0744	2.04431
$923$	−56.6395	−1.86431
$924$	−66.1517	−2.17623
$925$	27.1263	0.891906
$926$	38.1990	1.25530
$927$	−11.8842	−0.390329
$928$	−6.43326	−0.211182
$929$	−6.18680	−0.202982	−0.101491	−	0.994836i	$-0.532361\pi$
$929$	−6.18680	−0.202982	−0.101491	+	0.994836i	$0.532361\pi$
$930$	0.716636	0.0234994
$931$	−53.6745	−1.75911
$932$	15.1822	0.497310
$933$	−23.5222	−0.770083
$934$	−57.4639	−1.88028
$935$	14.0723	0.460212
$936$	−4.01634	−0.131278
$937$	25.2326	0.824314	0.412157	−	0.911113i	$-0.364776\pi$
$937$	25.2326	0.824314	0.412157	+	0.911113i	$0.364776\pi$
$938$	−112.916	−3.68683
$939$	−6.99878	−0.228396
$940$	−11.3686	−0.370802
$941$	3.80101	0.123910	0.0619548	−	0.998079i	$-0.480267\pi$
$941$	3.80101	0.123910	0.0619548	+	0.998079i	$0.480267\pi$
$942$	22.7184	0.740207
$943$	−6.78693	−0.221013
$944$	0.750723	0.0244340
$945$	15.3959	0.500828
$946$	99.5525	3.23673
$947$	−22.0743	−0.717319	−0.358659	−	0.933469i	$-0.616766\pi$
$947$	−22.0743	−0.717319	−0.358659	+	0.933469i	$0.616766\pi$
$948$	−4.43088	−0.143908
$949$	36.9771	1.20033
$950$	57.8913	1.87824
$951$	0.901870	0.0292451
$952$	−3.24554	−0.105189
$953$	20.2584	0.656233	0.328117	−	0.944637i	$-0.393586\pi$
$953$	20.2584	0.656233	0.328117	+	0.944637i	$0.393586\pi$
$954$	16.3932	0.530750
$955$	22.6344	0.732433
$956$	−26.0191	−0.841517
$957$	−4.77154	−0.154242
$958$	44.2998	1.43126
$959$	−86.7066	−2.79990
$960$	38.7688	1.25126
$961$	−30.9902	−0.999685
$962$	−33.0646	−1.06605
$963$	18.1256	0.584090
$964$	−33.7872	−1.08821
$965$	80.4007	2.58819
$966$	−59.4610	−1.91313
$967$	−37.8598	−1.21749	−0.608744	−	0.793367i	$-0.708326\pi$
$967$	−37.8598	−1.21749	−0.608744	+	0.793367i	$0.708326\pi$
$968$	−24.8292	−0.798040
$969$	−2.82476	−0.0907443
$970$	124.187	3.98742
$971$	−44.2119	−1.41883	−0.709413	−	0.704793i	$-0.751040\pi$
$971$	−44.2119	−1.41883	−0.709413	+	0.704793i	$0.751040\pi$
$972$	−2.48988	−0.0798630
$973$	−44.2173	−1.41754
$974$	−67.3355	−2.15757
$975$	−26.0249	−0.833463
$976$	−7.73428	−0.247568
$977$	39.1907	1.25382	0.626910	−	0.779092i	$-0.284319\pi$
$977$	39.1907	1.25382	0.626910	+	0.779092i	$0.284319\pi$
$978$	−20.2117	−0.646299
$979$	−24.9532	−0.797507
$980$	112.666	3.59899
$981$	−16.3679	−0.522587
$982$	−66.0120	−2.10653
$983$	60.3884	1.92609	0.963046	−	0.269338i	$-0.0868049\pi$
$983$	60.3884	1.92609	0.963046	+	0.269338i	$0.0868049\pi$
$984$	−1.12874	−0.0359829
$985$	8.35243	0.266131
$986$	−1.18985	−0.0378926
$987$	−5.99483	−0.190818
$988$	−39.1319	−1.24495
$989$	49.6235	1.57794
$990$	42.8775	1.36274
$991$	59.3710	1.88598	0.942992	−	0.332816i	$-0.107999\pi$
$991$	59.3710	1.88598	0.942992	+	0.332816i	$0.107999\pi$
$992$	0.786884	0.0249836
$993$	4.02030	0.127580
$994$	−139.457	−4.42332
$995$	−50.3965	−1.59768
$996$	9.52288	0.301744
$997$	−18.6203	−0.589711	−0.294856	−	0.955542i	$-0.595272\pi$
$997$	−18.6203	−0.589711	−0.294856	+	0.955542i	$0.595272\pi$
$998$	43.3701	1.37286
$999$	−4.03296	−0.127597

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.c.1.19	✓	121

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.c.1.19	✓	121	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11893 q^{2} -1.00000 q^{3} +2.48988 q^{4} +3.42435 q^{5} +2.11893 q^{6} -4.49601 q^{7} -1.03803 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.11893 q^{2} -1.00000 q^{3} +2.48988 q^{4} +3.42435 q^{5} +2.11893 q^{6} -4.49601 q^{7} -1.03803 q^{8} +1.00000 q^{9} -7.25596 q^{10} -5.90928 q^{11} -2.48988 q^{12} +3.86921 q^{13} +9.52675 q^{14} -3.42435 q^{15} -2.78025 q^{16} -0.695427 q^{17} -2.11893 q^{18} -4.06191 q^{19} +8.52621 q^{20} +4.49601 q^{21} +12.5214 q^{22} +6.24148 q^{23} +1.03803 q^{24} +6.72614 q^{25} -8.19860 q^{26} -1.00000 q^{27} -11.1945 q^{28} -0.807465 q^{29} +7.25596 q^{30} +0.0987651 q^{31} +7.96723 q^{32} +5.90928 q^{33} +1.47356 q^{34} -15.3959 q^{35} +2.48988 q^{36} +4.03296 q^{37} +8.60691 q^{38} -3.86921 q^{39} -3.55456 q^{40} -1.08739 q^{41} -9.52675 q^{42} +7.95060 q^{43} -14.7134 q^{44} +3.42435 q^{45} -13.2253 q^{46} -1.33337 q^{47} +2.78025 q^{48} +13.2141 q^{49} -14.2523 q^{50} +0.695427 q^{51} +9.63388 q^{52} -7.73655 q^{53} +2.11893 q^{54} -20.2354 q^{55} +4.66697 q^{56} +4.06191 q^{57} +1.71096 q^{58} -0.270020 q^{59} -8.52621 q^{60} +2.78186 q^{61} -0.209277 q^{62} -4.49601 q^{63} -11.3215 q^{64} +13.2495 q^{65} -12.5214 q^{66} -11.8525 q^{67} -1.73153 q^{68} -6.24148 q^{69} +32.6229 q^{70} -14.6385 q^{71} -1.03803 q^{72} +9.55676 q^{73} -8.54558 q^{74} -6.72614 q^{75} -10.1137 q^{76} +26.5682 q^{77} +8.19860 q^{78} +1.77955 q^{79} -9.52055 q^{80} +1.00000 q^{81} +2.30411 q^{82} -3.82463 q^{83} +11.1945 q^{84} -2.38138 q^{85} -16.8468 q^{86} +0.807465 q^{87} +6.13399 q^{88} +4.22271 q^{89} -7.25596 q^{90} -17.3960 q^{91} +15.5405 q^{92} -0.0987651 q^{93} +2.82532 q^{94} -13.9094 q^{95} -7.96723 q^{96} -17.1152 q^{97} -27.9998 q^{98} -5.90928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9}+O(q^{10}) \) 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9	\( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9} + 18 q^{10} + 32 q^{11} - 123 q^{12} + 2 q^{13} + 37 q^{14} - 24 q^{15} + 131 q^{16} + 87 q^{17} + 7 q^{18} - 10 q^{19} + 60 q^{20} + 14 q^{21} - 22 q^{22} + 31 q^{23} - 18 q^{24} + 147 q^{25} + 37 q^{26} - 121 q^{27} - 29 q^{28} + 68 q^{29} - 18 q^{30} + 25 q^{31} + 43 q^{32} - 32 q^{33} + 27 q^{34} + 51 q^{35} + 123 q^{36} - 4 q^{37} + 36 q^{38} - 2 q^{39} + 61 q^{40} + 132 q^{41} - 37 q^{42} - 91 q^{43} + 94 q^{44} + 24 q^{45} + 39 q^{47} - 131 q^{48} + 217 q^{49} + 54 q^{50} - 87 q^{51} - 12 q^{52} + 55 q^{53} - 7 q^{54} + 7 q^{55} + 104 q^{56} + 10 q^{57} - 3 q^{58} + 58 q^{59} - 60 q^{60} + 126 q^{61} + 74 q^{62} - 14 q^{63} + 122 q^{64} + 128 q^{65} + 22 q^{66} - 139 q^{67} + 190 q^{68} - 31 q^{69} - 18 q^{70} + 37 q^{71} + 18 q^{72} + 84 q^{73} + 79 q^{74} - 147 q^{75} + 23 q^{76} + 95 q^{77} - 37 q^{78} - 14 q^{79} + 145 q^{80} + 121 q^{81} + 9 q^{82} + 58 q^{83} + 29 q^{84} + 32 q^{85} + 28 q^{86} - 68 q^{87} - 84 q^{88} + 198 q^{89} + 18 q^{90} + 5 q^{91} + 98 q^{92} - 25 q^{93} + 9 q^{94} + 42 q^{95} - 43 q^{96} + 73 q^{97} + 69 q^{98} + 32 q^{99}+O(q^{100}) \) 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9 + 18 * q^10 + 32 * q^11 - 123 * q^12 + 2 * q^13 + 37 * q^14 - 24 * q^15 + 131 * q^16 + 87 * q^17 + 7 * q^18 - 10 * q^19 + 60 * q^20 + 14 * q^21 - 22 * q^22 + 31 * q^23 - 18 * q^24 + 147 * q^25 + 37 * q^26 - 121 * q^27 - 29 * q^28 + 68 * q^29 - 18 * q^30 + 25 * q^31 + 43 * q^32 - 32 * q^33 + 27 * q^34 + 51 * q^35 + 123 * q^36 - 4 * q^37 + 36 * q^38 - 2 * q^39 + 61 * q^40 + 132 * q^41 - 37 * q^42 - 91 * q^43 + 94 * q^44 + 24 * q^45 + 39 * q^47 - 131 * q^48 + 217 * q^49 + 54 * q^50 - 87 * q^51 - 12 * q^52 + 55 * q^53 - 7 * q^54 + 7 * q^55 + 104 * q^56 + 10 * q^57 - 3 * q^58 + 58 * q^59 - 60 * q^60 + 126 * q^61 + 74 * q^62 - 14 * q^63 + 122 * q^64 + 128 * q^65 + 22 * q^66 - 139 * q^67 + 190 * q^68 - 31 * q^69 - 18 * q^70 + 37 * q^71 + 18 * q^72 + 84 * q^73 + 79 * q^74 - 147 * q^75 + 23 * q^76 + 95 * q^77 - 37 * q^78 - 14 * q^79 + 145 * q^80 + 121 * q^81 + 9 * q^82 + 58 * q^83 + 29 * q^84 + 32 * q^85 + 28 * q^86 - 68 * q^87 - 84 * q^88 + 198 * q^89 + 18 * q^90 + 5 * q^91 + 98 * q^92 - 25 * q^93 + 9 * q^94 + 42 * q^95 - 43 * q^96 + 73 * q^97 + 69 * q^98 + 32 * q^99

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