LMFDB - Embedded newform 4031.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42000 q^{2} -2.29529 q^{3} +3.85639 q^{4} +1.32254 q^{5} +5.55461 q^{6} +0.993943 q^{7} -4.49245 q^{8} +2.26837 q^{9} +O(q^{10})$	\(q-2.42000 q^{2} -2.29529 q^{3} +3.85639 q^{4} +1.32254 q^{5} +5.55461 q^{6} +0.993943 q^{7} -4.49245 q^{8} +2.26837 q^{9} -3.20054 q^{10} +2.10381 q^{11} -8.85154 q^{12} +3.07912 q^{13} -2.40534 q^{14} -3.03561 q^{15} +3.15895 q^{16} +6.60217 q^{17} -5.48946 q^{18} -0.0631268 q^{19} +5.10022 q^{20} -2.28139 q^{21} -5.09121 q^{22} -5.94544 q^{23} +10.3115 q^{24} -3.25089 q^{25} -7.45147 q^{26} +1.67929 q^{27} +3.83303 q^{28} -1.00000 q^{29} +7.34618 q^{30} +5.42341 q^{31} +1.34025 q^{32} -4.82886 q^{33} -15.9772 q^{34} +1.31453 q^{35} +8.74773 q^{36} -5.88877 q^{37} +0.152767 q^{38} -7.06749 q^{39} -5.94144 q^{40} -5.46764 q^{41} +5.52096 q^{42} +0.835709 q^{43} +8.11310 q^{44} +3.00001 q^{45} +14.3880 q^{46} -0.379470 q^{47} -7.25073 q^{48} -6.01208 q^{49} +7.86715 q^{50} -15.1539 q^{51} +11.8743 q^{52} -2.25880 q^{53} -4.06389 q^{54} +2.78237 q^{55} -4.46524 q^{56} +0.144895 q^{57} +2.42000 q^{58} -7.33926 q^{59} -11.7065 q^{60} -4.55335 q^{61} -13.1246 q^{62} +2.25464 q^{63} -9.56131 q^{64} +4.07226 q^{65} +11.6858 q^{66} -8.31257 q^{67} +25.4605 q^{68} +13.6465 q^{69} -3.18115 q^{70} -7.18358 q^{71} -10.1906 q^{72} -15.1404 q^{73} +14.2508 q^{74} +7.46175 q^{75} -0.243442 q^{76} +2.09106 q^{77} +17.1033 q^{78} -1.93701 q^{79} +4.17784 q^{80} -10.6596 q^{81} +13.2317 q^{82} +7.16441 q^{83} -8.79793 q^{84} +8.73162 q^{85} -2.02241 q^{86} +2.29529 q^{87} -9.45126 q^{88} -3.20563 q^{89} -7.26002 q^{90} +3.06047 q^{91} -22.9279 q^{92} -12.4483 q^{93} +0.918316 q^{94} -0.0834877 q^{95} -3.07627 q^{96} -4.74236 q^{97} +14.5492 q^{98} +4.77222 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	−2.42000	−1.71120	−0.855598	−	0.517640i	$-0.826811\pi$
$2$	−2.42000	−1.71120	−0.855598	+	0.517640i	$0.826811\pi$
$3$	−2.29529	−1.32519	−0.662594	−	0.748978i	$-0.730545\pi$
$3$	−2.29529	−1.32519	−0.662594	+	0.748978i	$0.730545\pi$
$4$	3.85639	1.92819
$5$	1.32254	0.591457	0.295729	−	0.955272i	$-0.404438\pi$
$5$	1.32254	0.591457	0.295729	+	0.955272i	$0.404438\pi$
$6$	5.55461	2.26766
$7$	0.993943	0.375675	0.187838	−	0.982200i	$-0.439852\pi$
$7$	0.993943	0.375675	0.187838	+	0.982200i	$0.439852\pi$
$8$	−4.49245	−1.58832
$9$	2.26837	0.756125
$10$	−3.20054	−1.01210
$11$	2.10381	0.634322	0.317161	−	0.948372i	$-0.397270\pi$
$11$	2.10381	0.634322	0.317161	+	0.948372i	$0.397270\pi$
$12$	−8.85154	−2.55522
$13$	3.07912	0.853995	0.426997	−	0.904253i	$-0.359571\pi$
$13$	3.07912	0.853995	0.426997	+	0.904253i	$0.359571\pi$
$14$	−2.40534	−0.642854
$15$	−3.03561	−0.783792
$16$	3.15895	0.789738
$17$	6.60217	1.60126	0.800630	−	0.599159i	$-0.204498\pi$
$17$	6.60217	1.60126	0.800630	+	0.599159i	$0.204498\pi$
$18$	−5.48946	−1.29388
$19$	−0.0631268	−0.0144823	−0.00724114	−	0.999974i	$-0.502305\pi$
$19$	−0.0631268	−0.0144823	−0.00724114	+	0.999974i	$0.502305\pi$
$20$	5.10022	1.14044
$21$	−2.28139	−0.497841
$22$	−5.09121	−1.08545
$23$	−5.94544	−1.23971	−0.619855	−	0.784716i	$-0.712809\pi$
$23$	−5.94544	−1.23971	−0.619855	+	0.784716i	$0.712809\pi$
$24$	10.3115	2.10483
$25$	−3.25089	−0.650178
$26$	−7.45147	−1.46135
$27$	1.67929	0.323180
$28$	3.83303	0.724375
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	7.34618	1.34122
$31$	5.42341	0.974073	0.487036	−	0.873382i	$-0.338078\pi$
$31$	5.42341	0.974073	0.487036	+	0.873382i	$0.338078\pi$
$32$	1.34025	0.236925
$33$	−4.82886	−0.840596
$34$	−15.9772	−2.74007
$35$	1.31453	0.222196
$36$	8.74773	1.45796
$37$	−5.88877	−0.968109	−0.484054	−	0.875038i	$-0.660836\pi$
$37$	−5.88877	−0.968109	−0.484054	+	0.875038i	$0.660836\pi$
$38$	0.152767	0.0247820
$39$	−7.06749	−1.13170
$40$	−5.94144	−0.939425
$41$	−5.46764	−0.853902	−0.426951	−	0.904275i	$-0.640412\pi$
$41$	−5.46764	−0.853902	−0.426951	+	0.904275i	$0.640412\pi$
$42$	5.52096	0.851903
$43$	0.835709	0.127444	0.0637222	−	0.997968i	$-0.479703\pi$
$43$	0.835709	0.127444	0.0637222	+	0.997968i	$0.479703\pi$
$44$	8.11310	1.22310
$45$	3.00001	0.447216
$46$	14.3880	2.12139
$47$	−0.379470	−0.0553514	−0.0276757	−	0.999617i	$-0.508811\pi$
$47$	−0.379470	−0.0553514	−0.0276757	+	0.999617i	$0.508811\pi$
$48$	−7.25073	−1.04655
$49$	−6.01208	−0.858868
$50$	7.86715	1.11258
$51$	−15.1539	−2.12197
$52$	11.8743	1.64667
$53$	−2.25880	−0.310270	−0.155135	−	0.987893i	$-0.549581\pi$
$53$	−2.25880	−0.310270	−0.155135	+	0.987893i	$0.549581\pi$
$54$	−4.06389	−0.553025
$55$	2.78237	0.375174
$56$	−4.46524	−0.596693
$57$	0.144895	0.0191918
$58$	2.42000	0.317761
$59$	−7.33926	−0.955490	−0.477745	−	0.878499i	$-0.658546\pi$
$59$	−7.33926	−0.955490	−0.477745	+	0.878499i	$0.658546\pi$
$60$	−11.7065	−1.51130
$61$	−4.55335	−0.582996	−0.291498	−	0.956571i	$-0.594154\pi$
$61$	−4.55335	−0.582996	−0.291498	+	0.956571i	$0.594154\pi$
$62$	−13.1246	−1.66683
$63$	2.25464	0.284057
$64$	−9.56131	−1.19516
$65$	4.07226	0.505101
$66$	11.6858	1.43842
$67$	−8.31257	−1.01554	−0.507771	−	0.861492i	$-0.669530\pi$
$67$	−8.31257	−1.01554	−0.507771	+	0.861492i	$0.669530\pi$
$68$	25.4605	3.08754
$69$	13.6465	1.64285
$70$	−3.18115	−0.380221
$71$	−7.18358	−0.852535	−0.426267	−	0.904597i	$-0.640172\pi$
$71$	−7.18358	−0.852535	−0.426267	+	0.904597i	$0.640172\pi$
$72$	−10.1906	−1.20097
$73$	−15.1404	−1.77205	−0.886026	−	0.463635i	$-0.846545\pi$
$73$	−15.1404	−1.77205	−0.886026	+	0.463635i	$0.846545\pi$
$74$	14.2508	1.65662
$75$	7.46175	0.861609
$76$	−0.243442	−0.0279247
$77$	2.09106	0.238299
$78$	17.1033	1.93657
$79$	−1.93701	−0.217930	−0.108965	−	0.994046i	$-0.534754\pi$
$79$	−1.93701	−0.217930	−0.108965	+	0.994046i	$0.534754\pi$
$80$	4.17784	0.467096
$81$	−10.6596	−1.18440
$82$	13.2317	1.46119
$83$	7.16441	0.786396	0.393198	−	0.919454i	$-0.371369\pi$
$83$	7.16441	0.786396	0.393198	+	0.919454i	$0.371369\pi$
$84$	−8.79793	−0.959933
$85$	8.73162	0.947077
$86$	−2.02241	−0.218083
$87$	2.29529	0.246081
$88$	−9.45126	−1.00751
$89$	−3.20563	−0.339796	−0.169898	−	0.985462i	$-0.554344\pi$
$89$	−3.20563	−0.339796	−0.169898	+	0.985462i	$0.554344\pi$
$90$	−7.26002	−0.765274
$91$	3.06047	0.320825
$92$	−22.9279	−2.39040
$93$	−12.4483	−1.29083
$94$	0.918316	0.0947171
$95$	−0.0834877	−0.00856565
$96$	−3.07627	−0.313971
$97$	−4.74236	−0.481514	−0.240757	−	0.970585i	$-0.577396\pi$
$97$	−4.74236	−0.481514	−0.240757	+	0.970585i	$0.577396\pi$
$98$	14.5492	1.46969
$99$	4.77222	0.479626
$100$	−12.5367	−1.25367
$101$	−13.7873	−1.37189	−0.685943	−	0.727655i	$-0.740610\pi$
$101$	−13.7873	−1.37189	−0.685943	+	0.727655i	$0.740610\pi$
$102$	36.6724	3.63111
$103$	−7.19163	−0.708613	−0.354306	−	0.935129i	$-0.615283\pi$
$103$	−7.19163	−0.708613	−0.354306	+	0.935129i	$0.615283\pi$
$104$	−13.8328	−1.35642
$105$	−3.01723	−0.294451
$106$	5.46628	0.530932
$107$	9.55918	0.924121	0.462060	−	0.886848i	$-0.347110\pi$
$107$	9.55918	0.924121	0.462060	+	0.886848i	$0.347110\pi$
$108$	6.47601	0.623155
$109$	−7.46758	−0.715264	−0.357632	−	0.933863i	$-0.616416\pi$
$109$	−7.46758	−0.715264	−0.357632	+	0.933863i	$0.616416\pi$
$110$	−6.73332	−0.641997
$111$	13.5165	1.28293
$112$	3.13982	0.296685
$113$	5.21525	0.490610	0.245305	−	0.969446i	$-0.421112\pi$
$113$	5.21525	0.490610	0.245305	+	0.969446i	$0.421112\pi$
$114$	−0.350645	−0.0328409
$115$	−7.86308	−0.733236
$116$	−3.85639	−0.358057
$117$	6.98460	0.645727
$118$	17.7610	1.63503
$119$	6.56218	0.601554
$120$	13.6374	1.24492
$121$	−6.57400	−0.597636
$122$	11.0191	0.997622
$123$	12.5498	1.13158
$124$	20.9148	1.87820
$125$	−10.9121	−0.976010
$126$	−5.45621	−0.486078
$127$	15.9418	1.41460	0.707301	−	0.706913i	$-0.249913\pi$
$127$	15.9418	1.41460	0.707301	+	0.706913i	$0.249913\pi$
$128$	20.4578	1.80824
$129$	−1.91820	−0.168888
$130$	−9.85485	−0.864328
$131$	8.12307	0.709716	0.354858	−	0.934920i	$-0.384529\pi$
$131$	8.12307	0.709716	0.354858	+	0.934920i	$0.384529\pi$
$132$	−18.6219	−1.62083
$133$	−0.0627445	−0.00544064
$134$	20.1164	1.73779
$135$	2.22093	0.191147
$136$	−29.6599	−2.54332
$137$	13.2174	1.12924	0.564621	−	0.825350i	$-0.309022\pi$
$137$	13.2174	1.12924	0.564621	+	0.825350i	$0.309022\pi$
$138$	−33.0246	−2.81124
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	5.06933	0.428437
$141$	0.870995	0.0733510
$142$	17.3843	1.45885
$143$	6.47788	0.541707
$144$	7.16569	0.597141
$145$	−1.32254	−0.109831
$146$	36.6398	3.03233
$147$	13.7995	1.13816
$148$	−22.7094	−1.86670
$149$	−15.3222	−1.25524	−0.627621	−	0.778519i	$-0.715971\pi$
$149$	−15.3222	−1.25524	−0.627621	+	0.778519i	$0.715971\pi$
$150$	−18.0574	−1.47438
$151$	10.3431	0.841711	0.420855	−	0.907128i	$-0.361730\pi$
$151$	10.3431	0.841711	0.420855	+	0.907128i	$0.361730\pi$
$152$	0.283594	0.0230025
$153$	14.9762	1.21075
$154$	−5.06037	−0.407776
$155$	7.17267	0.576122
$156$	−27.2550	−2.18215
$157$	−10.2141	−0.815174	−0.407587	−	0.913166i	$-0.633630\pi$
$157$	−10.2141	−0.815174	−0.407587	+	0.913166i	$0.633630\pi$
$158$	4.68755	0.372922
$159$	5.18460	0.411166
$160$	1.77253	0.140131
$161$	−5.90943	−0.465728
$162$	25.7962	2.02674
$163$	−2.60802	−0.204276	−0.102138	−	0.994770i	$-0.532568\pi$
$163$	−2.60802	−0.204276	−0.102138	+	0.994770i	$0.532568\pi$
$164$	−21.0853	−1.64649
$165$	−6.38635	−0.497176
$166$	−17.3378	−1.34568
$167$	13.1116	1.01461	0.507305	−	0.861767i	$-0.330642\pi$
$167$	13.1116	1.01461	0.507305	+	0.861767i	$0.330642\pi$
$168$	10.2490	0.790731
$169$	−3.51901	−0.270693
$170$	−21.1305	−1.62064
$171$	−0.143195	−0.0109504
$172$	3.22282	0.245738
$173$	−3.00964	−0.228819	−0.114409	−	0.993434i	$-0.536498\pi$
$173$	−3.00964	−0.228819	−0.114409	+	0.993434i	$0.536498\pi$
$174$	−5.55461	−0.421094
$175$	−3.23120	−0.244256
$176$	6.64583	0.500948
$177$	16.8458	1.26620
$178$	7.75763	0.581459
$179$	−11.3054	−0.845002	−0.422501	−	0.906362i	$-0.638848\pi$
$179$	−11.3054	−0.845002	−0.422501	+	0.906362i	$0.638848\pi$
$180$	11.5692	0.862318
$181$	−1.18911	−0.0883860	−0.0441930	−	0.999023i	$-0.514072\pi$
$181$	−1.18911	−0.0883860	−0.0441930	+	0.999023i	$0.514072\pi$
$182$	−7.40634	−0.548994
$183$	10.4513	0.772580
$184$	26.7096	1.96906
$185$	−7.78813	−0.572595
$186$	30.1249	2.20886
$187$	13.8897	1.01571
$188$	−1.46338	−0.106728
$189$	1.66912	0.121411
$190$	0.202040	0.0146575
$191$	10.9855	0.794880	0.397440	−	0.917628i	$-0.369899\pi$
$191$	10.9855	0.794880	0.397440	+	0.917628i	$0.369899\pi$
$192$	21.9460	1.58382
$193$	21.9625	1.58089	0.790446	−	0.612531i	$-0.209849\pi$
$193$	21.9625	1.58089	0.790446	+	0.612531i	$0.209849\pi$
$194$	11.4765	0.823965
$195$	−9.34703	−0.669355
$196$	−23.1849	−1.65606
$197$	18.2380	1.29940	0.649702	−	0.760189i	$-0.274893\pi$
$197$	18.2380	1.29940	0.649702	+	0.760189i	$0.274893\pi$
$198$	−11.5488	−0.820735
$199$	−9.54505	−0.676631	−0.338315	−	0.941033i	$-0.609857\pi$
$199$	−9.54505	−0.676631	−0.338315	+	0.941033i	$0.609857\pi$
$200$	14.6045	1.03269
$201$	19.0798	1.34578
$202$	33.3652	2.34757
$203$	−0.993943	−0.0697611
$204$	−58.4394	−4.09157
$205$	−7.23117	−0.505047
$206$	17.4037	1.21258
$207$	−13.4865	−0.937376
$208$	9.72680	0.674432
$209$	−0.132807	−0.00918643
$210$	7.30169	0.503864
$211$	−1.12096	−0.0771700	−0.0385850	−	0.999255i	$-0.512285\pi$
$211$	−1.12096	−0.0771700	−0.0385850	+	0.999255i	$0.512285\pi$
$212$	−8.71080	−0.598260
$213$	16.4884	1.12977
$214$	−23.1332	−1.58135
$215$	1.10526	0.0753779
$216$	−7.54416	−0.513315
$217$	5.39056	0.365935
$218$	18.0715	1.22396
$219$	34.7517	2.34830
$220$	10.7299	0.723409
$221$	20.3289	1.36747
$222$	−32.7098	−2.19534
$223$	0.555399	0.0371923	0.0185961	−	0.999827i	$-0.494080\pi$
$223$	0.555399	0.0371923	0.0185961	+	0.999827i	$0.494080\pi$
$224$	1.33213	0.0890069
$225$	−7.37424	−0.491616
$226$	−12.6209	−0.839530
$227$	−9.06931	−0.601951	−0.300975	−	0.953632i	$-0.597312\pi$
$227$	−9.06931	−0.601951	−0.300975	+	0.953632i	$0.597312\pi$
$228$	0.558770	0.0370054
$229$	−6.88442	−0.454936	−0.227468	−	0.973786i	$-0.573045\pi$
$229$	−6.88442	−0.454936	−0.227468	+	0.973786i	$0.573045\pi$
$230$	19.0286	1.25471
$231$	−4.79961	−0.315791
$232$	4.49245	0.294944
$233$	−4.09828	−0.268487	−0.134243	−	0.990948i	$-0.542860\pi$
$233$	−4.09828	−0.268487	−0.134243	+	0.990948i	$0.542860\pi$
$234$	−16.9027	−1.10497
$235$	−0.501864	−0.0327380
$236$	−28.3030	−1.84237
$237$	4.44600	0.288799
$238$	−15.8805	−1.02938
$239$	13.7817	0.891461	0.445731	−	0.895167i	$-0.352944\pi$
$239$	13.7817	0.891461	0.445731	+	0.895167i	$0.352944\pi$
$240$	−9.58936	−0.618991
$241$	14.6615	0.944429	0.472214	−	0.881484i	$-0.343455\pi$
$241$	14.6615	0.944429	0.472214	+	0.881484i	$0.343455\pi$
$242$	15.9091	1.02267
$243$	19.4290	1.24637
$244$	−17.5595	−1.12413
$245$	−7.95120	−0.507984
$246$	−30.3706	−1.93636
$247$	−0.194375	−0.0123678
$248$	−24.3644	−1.54714
$249$	−16.4444	−1.04212
$250$	26.4073	1.67014
$251$	14.5009	0.915292	0.457646	−	0.889135i	$-0.348693\pi$
$251$	14.5009	0.915292	0.457646	+	0.889135i	$0.348693\pi$
$252$	8.69475	0.547718
$253$	−12.5081	−0.786375
$254$	−38.5790	−2.42066
$255$	−20.0416	−1.25506
$256$	−30.3853	−1.89908
$257$	8.45239	0.527246	0.263623	−	0.964626i	$-0.415083\pi$
$257$	8.45239	0.527246	0.263623	+	0.964626i	$0.415083\pi$
$258$	4.64204	0.289001
$259$	−5.85311	−0.363694
$260$	15.7042	0.973933
$261$	−2.26837	−0.140409
$262$	−19.6578	−1.21446
$263$	−0.0694720	−0.00428383	−0.00214191	−	0.999998i	$-0.500682\pi$
$263$	−0.0694720	−0.00428383	−0.00214191	+	0.999998i	$0.500682\pi$
$264$	21.6934	1.33514
$265$	−2.98735	−0.183511
$266$	0.151842	0.00931000
$267$	7.35787	0.450294
$268$	−32.0565	−1.95816
$269$	−20.5705	−1.25421	−0.627103	−	0.778936i	$-0.715760\pi$
$269$	−20.5705	−1.25421	−0.627103	+	0.778936i	$0.715760\pi$
$270$	−5.37465	−0.327091
$271$	−22.6403	−1.37530	−0.687651	−	0.726042i	$-0.741358\pi$
$271$	−22.6403	−1.37530	−0.687651	+	0.726042i	$0.741358\pi$
$272$	20.8559	1.26458
$273$	−7.02468	−0.425153
$274$	−31.9862	−1.93235
$275$	−6.83925	−0.412422
$276$	52.6263	3.16773
$277$	−16.4296	−0.987161	−0.493580	−	0.869700i	$-0.664312\pi$
$277$	−16.4296	−0.987161	−0.493580	+	0.869700i	$0.664312\pi$
$278$	2.42000	0.145142
$279$	12.3023	0.736521
$280$	−5.90546	−0.352919
$281$	1.40158	0.0836115	0.0418057	−	0.999126i	$-0.486689\pi$
$281$	1.40158	0.0836115	0.0418057	+	0.999126i	$0.486689\pi$
$282$	−2.10781	−0.125518
$283$	−29.8875	−1.77663	−0.888313	−	0.459239i	$-0.848122\pi$
$283$	−29.8875	−1.77663	−0.888313	+	0.459239i	$0.848122\pi$
$284$	−27.7027	−1.64385
$285$	0.191629	0.0113511
$286$	−15.6764	−0.926968
$287$	−5.43452	−0.320790
$288$	3.04019	0.179145
$289$	26.5886	1.56404
$290$	3.20054	0.187942
$291$	10.8851	0.638097
$292$	−58.3873	−3.41686
$293$	19.9743	1.16691	0.583456	−	0.812145i	$-0.301700\pi$
$293$	19.9743	1.16691	0.583456	+	0.812145i	$0.301700\pi$
$294$	−33.3947	−1.94762
$295$	−9.70645	−0.565131
$296$	26.4551	1.53767
$297$	3.53291	0.205000
$298$	37.0796	2.14797
$299$	−18.3067	−1.05871
$300$	28.7754	1.66135
$301$	0.830648	0.0478777
$302$	−25.0303	−1.44033
$303$	31.6459	1.81801
$304$	−0.199415	−0.0114372
$305$	−6.02198	−0.344817
$306$	−36.2423	−2.07184
$307$	−1.31117	−0.0748326	−0.0374163	−	0.999300i	$-0.511913\pi$
$307$	−1.31117	−0.0748326	−0.0374163	+	0.999300i	$0.511913\pi$
$308$	8.06396	0.459487
$309$	16.5069	0.939046
$310$	−17.3578	−0.985859
$311$	−8.29311	−0.470259	−0.235129	−	0.971964i	$-0.575551\pi$
$311$	−8.29311	−0.470259	−0.235129	+	0.971964i	$0.575551\pi$
$312$	31.7504	1.79751
$313$	−25.6938	−1.45230	−0.726149	−	0.687537i	$-0.758692\pi$
$313$	−25.6938	−1.45230	−0.726149	+	0.687537i	$0.758692\pi$
$314$	24.7181	1.39492
$315$	2.98184	0.168008
$316$	−7.46985	−0.420212
$317$	−19.1223	−1.07402	−0.537008	−	0.843577i	$-0.680446\pi$
$317$	−19.1223	−1.07402	−0.537008	+	0.843577i	$0.680446\pi$
$318$	−12.5467	−0.703585
$319$	−2.10381	−0.117791
$320$	−12.6452	−0.706888
$321$	−21.9411	−1.22463
$322$	14.3008	0.796953
$323$	−0.416774	−0.0231899
$324$	−41.1076	−2.28375
$325$	−10.0099	−0.555249
$326$	6.31140	0.349556
$327$	17.1403	0.947860
$328$	24.5631	1.35627
$329$	−0.377172	−0.0207941
$330$	15.4549	0.850767
$331$	−11.8325	−0.650371	−0.325185	−	0.945650i	$-0.605427\pi$
$331$	−11.8325	−0.650371	−0.325185	+	0.945650i	$0.605427\pi$
$332$	27.6287	1.51632
$333$	−13.3579	−0.732011
$334$	−31.7302	−1.73620
$335$	−10.9937	−0.600649
$336$	−7.20681	−0.393164
$337$	4.99610	0.272155	0.136078	−	0.990698i	$-0.456550\pi$
$337$	4.99610	0.272155	0.136078	+	0.990698i	$0.456550\pi$
$338$	8.51599	0.463209
$339$	−11.9705	−0.650151
$340$	33.6725	1.82615
$341$	11.4098	0.617875
$342$	0.346532	0.0187383
$343$	−12.9333	−0.698331
$344$	−3.75439	−0.202423
$345$	18.0481	0.971676
$346$	7.28333	0.391554
$347$	15.5398	0.834218	0.417109	−	0.908856i	$-0.363043\pi$
$347$	15.5398	0.834218	0.417109	+	0.908856i	$0.363043\pi$
$348$	8.85154	0.474493
$349$	−10.6995	−0.572732	−0.286366	−	0.958120i	$-0.592447\pi$
$349$	−10.6995	−0.572732	−0.286366	+	0.958120i	$0.592447\pi$
$350$	7.81950	0.417970
$351$	5.17075	0.275994
$352$	2.81963	0.150287
$353$	−22.3963	−1.19203	−0.596017	−	0.802972i	$-0.703251\pi$
$353$	−22.3963	−1.19203	−0.596017	+	0.802972i	$0.703251\pi$
$354$	−40.7667	−2.16672
$355$	−9.50057	−0.504238
$356$	−12.3622	−0.655194
$357$	−15.0621	−0.797172
$358$	27.3589	1.44597
$359$	16.4105	0.866111	0.433056	−	0.901367i	$-0.357435\pi$
$359$	16.4105	0.866111	0.433056	+	0.901367i	$0.357435\pi$
$360$	−13.4774	−0.710323
$361$	−18.9960	−0.999790
$362$	2.87765	0.151246
$363$	15.0893	0.791980
$364$	11.8024	0.618612
$365$	−20.0238	−1.04809
$366$	−25.2921	−1.32204
$367$	−32.5064	−1.69682	−0.848411	−	0.529338i	$-0.822441\pi$
$367$	−32.5064	−1.69682	−0.848411	+	0.529338i	$0.822441\pi$
$368$	−18.7814	−0.979047
$369$	−12.4027	−0.645657
$370$	18.8473	0.979822
$371$	−2.24512	−0.116561
$372$	−48.0055	−2.48897
$373$	18.2762	0.946304	0.473152	−	0.880981i	$-0.343116\pi$
$373$	18.2762	0.946304	0.473152	+	0.880981i	$0.343116\pi$
$374$	−33.6130	−1.73809
$375$	25.0465	1.29340
$376$	1.70475	0.0879158
$377$	−3.07912	−0.158583
$378$	−4.03928	−0.207758
$379$	−5.15184	−0.264632	−0.132316	−	0.991208i	$-0.542241\pi$
$379$	−5.15184	−0.264632	−0.132316	+	0.991208i	$0.542241\pi$
$380$	−0.321961	−0.0165162
$381$	−36.5910	−1.87461
$382$	−26.5848	−1.36020
$383$	−24.5106	−1.25244	−0.626218	−	0.779648i	$-0.715398\pi$
$383$	−24.5106	−1.25244	−0.626218	+	0.779648i	$0.715398\pi$
$384$	−46.9568	−2.39625
$385$	2.76551	0.140944
$386$	−53.1491	−2.70522
$387$	1.89570	0.0963639
$388$	−18.2884	−0.928452
$389$	4.22748	0.214342	0.107171	−	0.994241i	$-0.465821\pi$
$389$	4.22748	0.214342	0.107171	+	0.994241i	$0.465821\pi$
$390$	22.6198	1.14540
$391$	−39.2528	−1.98510
$392$	27.0090	1.36416
$393$	−18.6448	−0.940508
$394$	−44.1360	−2.22354
$395$	−2.56177	−0.128897
$396$	18.4035	0.924813
$397$	−0.130869	−0.00656810	−0.00328405	−	0.999995i	$-0.501045\pi$
$397$	−0.130869	−0.00656810	−0.00328405	+	0.999995i	$0.501045\pi$
$398$	23.0990	1.15785
$399$	0.144017	0.00720987
$400$	−10.2694	−0.513471
$401$	1.40604	0.0702141	0.0351071	−	0.999384i	$-0.488823\pi$
$401$	1.40604	0.0702141	0.0351071	+	0.999384i	$0.488823\pi$
$402$	−46.1730	−2.30290
$403$	16.6993	0.831853
$404$	−53.1691	−2.64526
$405$	−14.0977	−0.700522
$406$	2.40534	0.119375
$407$	−12.3888	−0.614092
$408$	68.0783	3.37038
$409$	1.30490	0.0645232	0.0322616	−	0.999479i	$-0.489729\pi$
$409$	1.30490	0.0645232	0.0322616	+	0.999479i	$0.489729\pi$
$410$	17.4994	0.864234
$411$	−30.3379	−1.49646
$412$	−27.7337	−1.36634
$413$	−7.29481	−0.358954
$414$	32.6373	1.60403
$415$	9.47520	0.465119
$416$	4.12680	0.202333
$417$	2.29529	0.112401
$418$	0.321392	0.0157198
$419$	32.6894	1.59698	0.798491	−	0.602006i	$-0.205632\pi$
$419$	32.6894	1.59698	0.798491	+	0.602006i	$0.205632\pi$
$420$	−11.6356	−0.567759
$421$	11.3690	0.554092	0.277046	−	0.960857i	$-0.410645\pi$
$421$	11.3690	0.554092	0.277046	+	0.960857i	$0.410645\pi$
$422$	2.71272	0.132053
$423$	−0.860780	−0.0418526
$424$	10.1475	0.492808
$425$	−21.4629	−1.04110
$426$	−39.9020	−1.93326
$427$	−4.52577	−0.219017
$428$	36.8639	1.78188
$429$	−14.8686	−0.717864
$430$	−2.67472	−0.128986
$431$	17.6376	0.849574	0.424787	−	0.905293i	$-0.360349\pi$
$431$	17.6376	0.849574	0.424787	+	0.905293i	$0.360349\pi$
$432$	5.30481	0.255228
$433$	0.471773	0.0226720	0.0113360	−	0.999936i	$-0.496392\pi$
$433$	0.471773	0.0226720	0.0113360	+	0.999936i	$0.496392\pi$
$434$	−13.0451	−0.626187
$435$	3.03561	0.145547
$436$	−28.7979	−1.37917
$437$	0.375317	0.0179538
$438$	−84.0991	−4.01841
$439$	11.0842	0.529019	0.264510	−	0.964383i	$-0.414790\pi$
$439$	11.0842	0.529019	0.264510	+	0.964383i	$0.414790\pi$
$440$	−12.4997	−0.595898
$441$	−13.6376	−0.649412
$442$	−49.1958	−2.34001
$443$	−38.7939	−1.84316	−0.921578	−	0.388194i	$-0.873099\pi$
$443$	−38.7939	−1.84316	−0.921578	+	0.388194i	$0.873099\pi$
$444$	52.1248	2.47373
$445$	−4.23957	−0.200975
$446$	−1.34406	−0.0636433
$447$	35.1689	1.66343
$448$	−9.50340	−0.448993
$449$	−21.3325	−1.00674	−0.503371	−	0.864070i	$-0.667907\pi$
$449$	−21.3325	−1.00674	−0.503371	+	0.864070i	$0.667907\pi$
$450$	17.8456	0.841252
$451$	−11.5029	−0.541649
$452$	20.1120	0.945991
$453$	−23.7405	−1.11543
$454$	21.9477	1.03006
$455$	4.04759	0.189754
$456$	−0.650933	−0.0304827
$457$	−1.35756	−0.0635042	−0.0317521	−	0.999496i	$-0.510109\pi$
$457$	−1.35756	−0.0635042	−0.0317521	+	0.999496i	$0.510109\pi$
$458$	16.6603	0.778484
$459$	11.0870	0.517496
$460$	−30.3231	−1.41382
$461$	20.1214	0.937149	0.468574	−	0.883424i	$-0.344768\pi$
$461$	20.1214	0.937149	0.468574	+	0.883424i	$0.344768\pi$
$462$	11.6150	0.540381
$463$	−23.3357	−1.08450	−0.542252	−	0.840216i	$-0.682428\pi$
$463$	−23.3357	−1.08450	−0.542252	+	0.840216i	$0.682428\pi$
$464$	−3.15895	−0.146651
$465$	−16.4634	−0.763471
$466$	9.91782	0.459434
$467$	3.22537	0.149252	0.0746262	−	0.997212i	$-0.476224\pi$
$467$	3.22537	0.149252	0.0746262	+	0.997212i	$0.476224\pi$
$468$	26.9353	1.24509
$469$	−8.26222	−0.381514
$470$	1.21451	0.0560211
$471$	23.4444	1.08026
$472$	32.9713	1.51763
$473$	1.75817	0.0808408
$474$	−10.7593	−0.494192
$475$	0.205219	0.00941607
$476$	25.3063	1.15991
$477$	−5.12380	−0.234603
$478$	−33.3516	−1.52547
$479$	9.40374	0.429668	0.214834	−	0.976651i	$-0.431079\pi$
$479$	9.40374	0.429668	0.214834	+	0.976651i	$0.431079\pi$
$480$	−4.06849	−0.185700
$481$	−18.1323	−0.826760
$482$	−35.4807	−1.61610
$483$	13.5639	0.617178
$484$	−25.3519	−1.15236
$485$	−6.27195	−0.284795
$486$	−47.0182	−2.13279
$487$	25.1250	1.13852	0.569262	−	0.822156i	$-0.307229\pi$
$487$	25.1250	1.13852	0.569262	+	0.822156i	$0.307229\pi$
$488$	20.4557	0.925986
$489$	5.98617	0.270704
$490$	19.2419	0.869260
$491$	26.0291	1.17468	0.587339	−	0.809341i	$-0.300176\pi$
$491$	26.0291	1.17468	0.587339	+	0.809341i	$0.300176\pi$
$492$	48.3971	2.18191
$493$	−6.60217	−0.297347
$494$	0.470388	0.0211637
$495$	6.31145	0.283679
$496$	17.1323	0.769262
$497$	−7.14007	−0.320276
$498$	39.7955	1.78328
$499$	−37.8778	−1.69564	−0.847821	−	0.530282i	$-0.822086\pi$
$499$	−37.8778	−1.69564	−0.847821	+	0.530282i	$0.822086\pi$
$500$	−42.0814	−1.88194
$501$	−30.0951	−1.34455
$502$	−35.0923	−1.56624
$503$	32.1149	1.43193	0.715966	−	0.698135i	$-0.245986\pi$
$503$	32.1149	1.43193	0.715966	+	0.698135i	$0.245986\pi$
$504$	−10.1288	−0.451175
$505$	−18.2342	−0.811412
$506$	30.2695	1.34564
$507$	8.07716	0.358719
$508$	61.4776	2.72763
$509$	−21.7665	−0.964783	−0.482392	−	0.875956i	$-0.660232\pi$
$509$	−21.7665	−0.964783	−0.482392	+	0.875956i	$0.660232\pi$
$510$	48.5007	2.14765
$511$	−15.0487	−0.665716
$512$	32.6167	1.44147
$513$	−0.106009	−0.00468039
$514$	−20.4548	−0.902221
$515$	−9.51121	−0.419114
$516$	−7.39732	−0.325649
$517$	−0.798331	−0.0351106
$518$	14.1645	0.622353
$519$	6.90801	0.303228
$520$	−18.2944	−0.802264
$521$	−10.1407	−0.444274	−0.222137	−	0.975015i	$-0.571303\pi$
$521$	−10.1407	−0.444274	−0.222137	+	0.975015i	$0.571303\pi$
$522$	5.48946	0.240267
$523$	−3.77378	−0.165016	−0.0825080	−	0.996590i	$-0.526293\pi$
$523$	−3.77378	−0.165016	−0.0825080	+	0.996590i	$0.526293\pi$
$524$	31.3257	1.36847
$525$	7.41656	0.323685
$526$	0.168122	0.00733048
$527$	35.8062	1.55974
$528$	−15.2541	−0.663851
$529$	12.3483	0.536882
$530$	7.22937	0.314024
$531$	−16.6482	−0.722470
$532$	−0.241967	−0.0104906
$533$	−16.8355	−0.729228
$534$	−17.8060	−0.770542
$535$	12.6424	0.546578
$536$	37.3438	1.61301
$537$	25.9491	1.11979
$538$	49.7806	2.14619
$539$	−12.6482	−0.544799
$540$	8.56478	0.368569
$541$	−32.0384	−1.37744	−0.688719	−	0.725028i	$-0.741827\pi$
$541$	−32.0384	−1.37744	−0.688719	+	0.725028i	$0.741827\pi$
$542$	54.7895	2.35341
$543$	2.72936	0.117128
$544$	8.84856	0.379379
$545$	−9.87616	−0.423048
$546$	16.9997	0.727521
$547$	32.9681	1.40962	0.704808	−	0.709398i	$-0.251033\pi$
$547$	32.9681	1.40962	0.704808	+	0.709398i	$0.251033\pi$
$548$	50.9716	2.17740
$549$	−10.3287	−0.440818
$550$	16.5510	0.705736
$551$	0.0631268	0.00268929
$552$	−61.3065	−2.60938
$553$	−1.92528	−0.0818711
$554$	39.7597	1.68923
$555$	17.8761	0.758796
$556$	−3.85639	−0.163547
$557$	−2.28887	−0.0969827	−0.0484914	−	0.998824i	$-0.515441\pi$
$557$	−2.28887	−0.0969827	−0.0484914	+	0.998824i	$0.515441\pi$
$558$	−29.7716	−1.26033
$559$	2.57325	0.108837
$560$	4.15253	0.175477
$561$	−31.8809	−1.34601
$562$	−3.39183	−0.143076
$563$	25.6742	1.08204	0.541020	−	0.841010i	$-0.318038\pi$
$563$	25.6742	1.08204	0.541020	+	0.841010i	$0.318038\pi$
$564$	3.35889	0.141435
$565$	6.89738	0.290175
$566$	72.3276	3.04016
$567$	−10.5950	−0.444950
$568$	32.2719	1.35410
$569$	16.8490	0.706348	0.353174	−	0.935558i	$-0.385102\pi$
$569$	16.8490	0.706348	0.353174	+	0.935558i	$0.385102\pi$
$570$	−0.463741	−0.0194240
$571$	−31.7691	−1.32950	−0.664748	−	0.747068i	$-0.731461\pi$
$571$	−31.7691	−1.32950	−0.664748	+	0.747068i	$0.731461\pi$
$572$	24.9812	1.04452
$573$	−25.2149	−1.05337
$574$	13.1515	0.548935
$575$	19.3280	0.806033
$576$	−21.6886	−0.903693
$577$	−6.81810	−0.283841	−0.141921	−	0.989878i	$-0.545328\pi$
$577$	−6.81810	−0.283841	−0.141921	+	0.989878i	$0.545328\pi$
$578$	−64.3443	−2.67637
$579$	−50.4103	−2.09498
$580$	−5.10022	−0.211775
$581$	7.12101	0.295429
$582$	−26.3419	−1.09191
$583$	−4.75207	−0.196811
$584$	68.0177	2.81459
$585$	9.23741	0.381920
$586$	−48.3378	−1.99682
$587$	13.7641	0.568106	0.284053	−	0.958809i	$-0.408321\pi$
$587$	13.7641	0.568106	0.284053	+	0.958809i	$0.408321\pi$
$588$	53.2162	2.19460
$589$	−0.342363	−0.0141068
$590$	23.4896	0.967051
$591$	−41.8616	−1.72196
$592$	−18.6024	−0.764552
$593$	−10.7738	−0.442427	−0.221213	−	0.975225i	$-0.571002\pi$
$593$	−10.7738	−0.442427	−0.221213	+	0.975225i	$0.571002\pi$
$594$	−8.54964	−0.350796
$595$	8.67873	0.355793
$596$	−59.0883	−2.42035
$597$	21.9087	0.896663
$598$	44.3023	1.81165
$599$	−17.9555	−0.733642	−0.366821	−	0.930291i	$-0.619554\pi$
$599$	−17.9555	−0.733642	−0.366821	+	0.930291i	$0.619554\pi$
$600$	−33.5216	−1.36851
$601$	−19.2342	−0.784579	−0.392290	−	0.919842i	$-0.628317\pi$
$601$	−19.2342	−0.784579	−0.392290	+	0.919842i	$0.628317\pi$
$602$	−2.01016	−0.0819282
$603$	−18.8560	−0.767876
$604$	39.8871	1.62298
$605$	−8.69436	−0.353476
$606$	−76.5829	−3.11097
$607$	26.8813	1.09108	0.545539	−	0.838085i	$-0.316325\pi$
$607$	26.8813	1.09108	0.545539	+	0.838085i	$0.316325\pi$
$608$	−0.0846058	−0.00343122
$609$	2.28139	0.0924467
$610$	14.5732	0.590050
$611$	−1.16843	−0.0472698
$612$	57.7540	2.33457
$613$	37.0311	1.49567	0.747836	−	0.663884i	$-0.231093\pi$
$613$	37.0311	1.49567	0.747836	+	0.663884i	$0.231093\pi$
$614$	3.17304	0.128053
$615$	16.5977	0.669282
$616$	−9.39401	−0.378496
$617$	−42.1368	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$617$	−42.1368	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$618$	−39.9467	−1.60689
$619$	31.4965	1.26595	0.632975	−	0.774172i	$-0.281834\pi$
$619$	31.4965	1.26595	0.632975	+	0.774172i	$0.281834\pi$
$620$	27.6606	1.11088
$621$	−9.98415	−0.400650
$622$	20.0693	0.804706
$623$	−3.18622	−0.127653
$624$	−22.3259	−0.893750
$625$	1.82276	0.0729103
$626$	62.1789	2.48517
$627$	0.304830	0.0121738
$628$	−39.3895	−1.57181
$629$	−38.8787	−1.55019
$630$	−7.21605	−0.287494
$631$	−38.5253	−1.53367	−0.766833	−	0.641846i	$-0.778169\pi$
$631$	−38.5253	−1.53367	−0.766833	+	0.641846i	$0.778169\pi$
$632$	8.70192	0.346144
$633$	2.57293	0.102265
$634$	46.2760	1.83785
$635$	21.0836	0.836676
$636$	19.9938	0.792807
$637$	−18.5119	−0.733469
$638$	5.09121	0.201563
$639$	−16.2951	−0.644623
$640$	27.0563	1.06949
$641$	44.8125	1.76999	0.884994	−	0.465602i	$-0.154162\pi$
$641$	44.8125	1.76999	0.884994	+	0.465602i	$0.154162\pi$
$642$	53.0975	2.09559
$643$	1.57148	0.0619733	0.0309866	−	0.999520i	$-0.490135\pi$
$643$	1.57148	0.0619733	0.0309866	+	0.999520i	$0.490135\pi$
$644$	−22.7891	−0.898015
$645$	−2.53689	−0.0998900
$646$	1.00859	0.0396825
$647$	32.1307	1.26319	0.631593	−	0.775300i	$-0.282401\pi$
$647$	32.1307	1.26319	0.631593	+	0.775300i	$0.282401\pi$
$648$	47.8878	1.88121
$649$	−15.4404	−0.606088
$650$	24.2239	0.950140
$651$	−12.3729	−0.484933
$652$	−10.0575	−0.393883
$653$	17.1292	0.670319	0.335159	−	0.942161i	$-0.391210\pi$
$653$	17.1292	0.670319	0.335159	+	0.942161i	$0.391210\pi$
$654$	−41.4795	−1.62198
$655$	10.7431	0.419767
$656$	−17.2720	−0.674359
$657$	−34.3442	−1.33989
$658$	0.912754	0.0355829
$659$	−34.5549	−1.34607	−0.673035	−	0.739611i	$-0.735010\pi$
$659$	−34.5549	−1.34607	−0.673035	+	0.739611i	$0.735010\pi$
$660$	−24.6282	−0.958653
$661$	35.5387	1.38230	0.691148	−	0.722714i	$-0.257105\pi$
$661$	35.5387	1.38230	0.691148	+	0.722714i	$0.257105\pi$
$662$	28.6345	1.11291
$663$	−46.6607	−1.81215
$664$	−32.1858	−1.24905
$665$	−0.0829820	−0.00321790
$666$	32.3262	1.25261
$667$	5.94544	0.230208
$668$	50.5636	1.95636
$669$	−1.27480	−0.0492867
$670$	26.6047	1.02783
$671$	−9.57936	−0.369807
$672$	−3.05764	−0.117951
$673$	13.7134	0.528612	0.264306	−	0.964439i	$-0.414857\pi$
$673$	13.7134	0.528612	0.264306	+	0.964439i	$0.414857\pi$
$674$	−12.0906	−0.465711
$675$	−5.45921	−0.210125
$676$	−13.5707	−0.521948
$677$	31.6369	1.21590	0.607952	−	0.793973i	$-0.291991\pi$
$677$	31.6369	1.21590	0.607952	+	0.793973i	$0.291991\pi$
$678$	28.9687	1.11254
$679$	−4.71364	−0.180893
$680$	−39.2264	−1.50426
$681$	20.8167	0.797699
$682$	−27.6117	−1.05731
$683$	−29.1284	−1.11457	−0.557284	−	0.830322i	$-0.688157\pi$
$683$	−29.1284	−1.11457	−0.557284	+	0.830322i	$0.688157\pi$
$684$	−0.552217	−0.0211145
$685$	17.4806	0.667898
$686$	31.2985	1.19498
$687$	15.8018	0.602875
$688$	2.63997	0.100648
$689$	−6.95511	−0.264969
$690$	−43.6763	−1.66273
$691$	19.3505	0.736129	0.368065	−	0.929800i	$-0.380021\pi$
$691$	19.3505	0.736129	0.368065	+	0.929800i	$0.380021\pi$
$692$	−11.6063	−0.441207
$693$	4.74332	0.180184
$694$	−37.6062	−1.42751
$695$	−1.32254	−0.0501667
$696$	−10.3115	−0.390857
$697$	−36.0983	−1.36732
$698$	25.8928	0.980057
$699$	9.40675	0.355796
$700$	−12.4608	−0.470973
$701$	−6.91265	−0.261087	−0.130544	−	0.991443i	$-0.541672\pi$
$701$	−6.91265	−0.261087	−0.130544	+	0.991443i	$0.541672\pi$
$702$	−12.5132	−0.472281
$703$	0.371740	0.0140204
$704$	−20.1152	−0.758118
$705$	1.15192	0.0433840
$706$	54.1989	2.03980
$707$	−13.7038	−0.515384
$708$	64.9638	2.44149
$709$	44.1605	1.65848	0.829241	−	0.558891i	$-0.188773\pi$
$709$	44.1605	1.65848	0.829241	+	0.558891i	$0.188773\pi$
$710$	22.9913	0.862850
$711$	−4.39386	−0.164783
$712$	14.4012	0.539706
$713$	−32.2446	−1.20757
$714$	36.4503	1.36412
$715$	8.56724	0.320397
$716$	−43.5979	−1.62933
$717$	−31.6330	−1.18135
$718$	−39.7133	−1.48209
$719$	−36.3925	−1.35721	−0.678605	−	0.734503i	$-0.737415\pi$
$719$	−36.3925	−1.35721	−0.678605	+	0.734503i	$0.737415\pi$
$720$	9.47690	0.353183
$721$	−7.14808	−0.266208
$722$	45.9703	1.71084
$723$	−33.6524	−1.25155
$724$	−4.58568	−0.170425
$725$	3.25089	0.120735
$726$	−36.5160	−1.35523
$727$	−2.63607	−0.0977664	−0.0488832	−	0.998805i	$-0.515566\pi$
$727$	−2.63607	−0.0977664	−0.0488832	+	0.998805i	$0.515566\pi$
$728$	−13.7490	−0.509573
$729$	−12.6165	−0.467279
$730$	48.4575	1.79349
$731$	5.51749	0.204072
$732$	40.3042	1.48968
$733$	−22.0550	−0.814620	−0.407310	−	0.913290i	$-0.633533\pi$
$733$	−22.0550	−0.814620	−0.407310	+	0.913290i	$0.633533\pi$
$734$	78.6655	2.90360
$735$	18.2503	0.673174
$736$	−7.96839	−0.293719
$737$	−17.4880	−0.644180
$738$	30.0144	1.10485
$739$	27.7896	1.02226	0.511129	−	0.859504i	$-0.329227\pi$
$739$	27.7896	1.02226	0.511129	+	0.859504i	$0.329227\pi$
$740$	−30.0341	−1.10407
$741$	0.446148	0.0163897
$742$	5.43317	0.199458
$743$	−43.2896	−1.58814	−0.794071	−	0.607825i	$-0.792042\pi$
$743$	−43.2896	−1.58814	−0.794071	+	0.607825i	$0.792042\pi$
$744$	55.9235	2.05025
$745$	−20.2642	−0.742422
$746$	−44.2283	−1.61931
$747$	16.2516	0.594613
$748$	53.5640	1.95849
$749$	9.50128	0.347169
$750$	−60.6125	−2.21326
$751$	−28.8839	−1.05399	−0.526995	−	0.849868i	$-0.676681\pi$
$751$	−28.8839	−1.05399	−0.526995	+	0.849868i	$0.676681\pi$
$752$	−1.19873	−0.0437131
$753$	−33.2839	−1.21293
$754$	7.45147	0.271366
$755$	13.6792	0.497836
$756$	6.43679	0.234104
$757$	14.6830	0.533662	0.266831	−	0.963743i	$-0.414023\pi$
$757$	14.6830	0.533662	0.266831	+	0.963743i	$0.414023\pi$
$758$	12.4674	0.452838
$759$	28.7097	1.04210
$760$	0.375065	0.0136050
$761$	−40.2124	−1.45770	−0.728850	−	0.684674i	$-0.759945\pi$
$761$	−40.2124	−1.45770	−0.728850	+	0.684674i	$0.759945\pi$
$762$	88.5501	3.20783
$763$	−7.42235	−0.268707
$764$	42.3642	1.53268
$765$	19.8066	0.716109
$766$	59.3157	2.14316
$767$	−22.5985	−0.815983
$768$	69.7432	2.51664
$769$	−47.9126	−1.72777	−0.863887	−	0.503686i	$-0.831977\pi$
$769$	−47.9126	−1.72777	−0.863887	+	0.503686i	$0.831977\pi$
$770$	−6.69254	−0.241182
$771$	−19.4007	−0.698700
$772$	84.6958	3.04827
$773$	50.4137	1.81326	0.906628	−	0.421932i	$-0.138648\pi$
$773$	50.4137	1.81326	0.906628	+	0.421932i	$0.138648\pi$
$774$	−4.58759	−0.164898
$775$	−17.6309	−0.633321
$776$	21.3048	0.764799
$777$	13.4346	0.481964
$778$	−10.2305	−0.366781
$779$	0.345155	0.0123665
$780$	−36.0458	−1.29065
$781$	−15.1129	−0.540781
$782$	94.9917	3.39690
$783$	−1.67929	−0.0600131
$784$	−18.9919	−0.678281
$785$	−13.5085	−0.482141
$786$	45.1205	1.60939
$787$	30.9280	1.10246	0.551232	−	0.834352i	$-0.314158\pi$
$787$	30.9280	1.10246	0.551232	+	0.834352i	$0.314158\pi$
$788$	70.3329	2.50550
$789$	0.159459	0.00567688
$790$	6.19947	0.220567
$791$	5.18367	0.184310
$792$	−21.4390	−0.761801
$793$	−14.0203	−0.497876
$794$	0.316702	0.0112393
$795$	6.85684	0.243187
$796$	−36.8094	−1.30468
$797$	−1.30169	−0.0461083	−0.0230541	−	0.999734i	$-0.507339\pi$
$797$	−1.30169	−0.0461083	−0.0230541	+	0.999734i	$0.507339\pi$
$798$	−0.348521	−0.0123375
$799$	−2.50532	−0.0886320
$800$	−4.35701	−0.154044
$801$	−7.27158	−0.256929
$802$	−3.40260	−0.120150
$803$	−31.8525	−1.12405
$804$	73.5791	2.59493
$805$	−7.81545	−0.275458
$806$	−40.4123	−1.42346
$807$	47.2154	1.66206
$808$	61.9387	2.17900
$809$	−11.2088	−0.394080	−0.197040	−	0.980395i	$-0.563133\pi$
$809$	−11.2088	−0.394080	−0.197040	+	0.980395i	$0.563133\pi$
$810$	34.1165	1.19873
$811$	−48.3721	−1.69857	−0.849287	−	0.527931i	$-0.822968\pi$
$811$	−48.3721	−1.69857	−0.849287	+	0.527931i	$0.822968\pi$
$812$	−3.83303	−0.134513
$813$	51.9662	1.82253
$814$	29.9810	1.05083
$815$	−3.44921	−0.120820
$816$	−47.8705	−1.67580
$817$	−0.0527557	−0.00184569
$818$	−3.15786	−0.110412
$819$	6.94230	0.242584
$820$	−27.8862	−0.973828
$821$	21.9970	0.767701	0.383851	−	0.923395i	$-0.374598\pi$
$821$	21.9970	0.767701	0.383851	+	0.923395i	$0.374598\pi$
$822$	73.4176	2.56073
$823$	43.5196	1.51700	0.758500	−	0.651674i	$-0.225933\pi$
$823$	43.5196	1.51700	0.758500	+	0.651674i	$0.225933\pi$
$824$	32.3081	1.12551
$825$	15.6981	0.546537
$826$	17.6534	0.614241
$827$	39.8202	1.38468	0.692341	−	0.721570i	$-0.256579\pi$
$827$	39.8202	1.38468	0.692341	+	0.721570i	$0.256579\pi$
$828$	−52.0091	−1.80744
$829$	38.3913	1.33339	0.666693	−	0.745333i	$-0.267709\pi$
$829$	38.3913	1.33339	0.666693	+	0.745333i	$0.267709\pi$
$830$	−22.9300	−0.795911
$831$	37.7108	1.30817
$832$	−29.4404	−1.02066
$833$	−39.6927	−1.37527
$834$	−5.55461	−0.192340
$835$	17.3407	0.600098
$836$	−0.512154	−0.0177132
$837$	9.10750	0.314801
$838$	−79.1083	−2.73275
$839$	−43.9201	−1.51629	−0.758146	−	0.652085i	$-0.773895\pi$
$839$	−43.9201	−1.51629	−0.758146	+	0.652085i	$0.773895\pi$
$840$	13.5548	0.467684
$841$	1.00000	0.0344828
$842$	−27.5130	−0.948161
$843$	−3.21705	−0.110801
$844$	−4.32285	−0.148799
$845$	−4.65402	−0.160103
$846$	2.08309	0.0716180
$847$	−6.53418	−0.224517
$848$	−7.13543	−0.245032
$849$	68.6005	2.35436
$850$	51.9402	1.78154
$851$	35.0114	1.20017
$852$	63.5858	2.17841
$853$	26.1199	0.894328	0.447164	−	0.894452i	$-0.352434\pi$
$853$	26.1199	0.894328	0.447164	+	0.894452i	$0.352434\pi$
$854$	10.9523	0.374782
$855$	−0.189381	−0.00647671
$856$	−42.9442	−1.46780
$857$	−41.6365	−1.42228	−0.711138	−	0.703053i	$-0.751820\pi$
$857$	−41.6365	−1.42228	−0.711138	+	0.703053i	$0.751820\pi$
$858$	35.9821	1.22841
$859$	−16.8512	−0.574954	−0.287477	−	0.957788i	$-0.592816\pi$
$859$	−16.8512	−0.574954	−0.287477	+	0.957788i	$0.592816\pi$
$860$	4.26230	0.145343
$861$	12.4738	0.425107
$862$	−42.6830	−1.45379
$863$	35.9569	1.22399	0.611993	−	0.790863i	$-0.290368\pi$
$863$	35.9569	1.22399	0.611993	+	0.790863i	$0.290368\pi$
$864$	2.25068	0.0765696
$865$	−3.98037	−0.135337
$866$	−1.14169	−0.0387962
$867$	−61.0287	−2.07264
$868$	20.7881	0.705594
$869$	−4.07509	−0.138238
$870$	−7.34618	−0.249059
$871$	−25.5954	−0.867267
$872$	33.5478	1.13607
$873$	−10.7575	−0.364085
$874$	−0.908266	−0.0307226
$875$	−10.8460	−0.366663
$876$	134.016	4.52799
$877$	−11.6984	−0.395027	−0.197514	−	0.980300i	$-0.563287\pi$
$877$	−11.6984	−0.395027	−0.197514	+	0.980300i	$0.563287\pi$
$878$	−26.8237	−0.905256
$879$	−45.8469	−1.54638
$880$	8.78936	0.296289
$881$	−21.5111	−0.724728	−0.362364	−	0.932037i	$-0.618030\pi$
$881$	−21.5111	−0.724728	−0.362364	+	0.932037i	$0.618030\pi$
$882$	33.0031	1.11127
$883$	−2.67443	−0.0900019	−0.0450009	−	0.998987i	$-0.514329\pi$
$883$	−2.67443	−0.0900019	−0.0450009	+	0.998987i	$0.514329\pi$
$884$	78.3960	2.63674
$885$	22.2792	0.748906
$886$	93.8812	3.15400
$887$	−30.1820	−1.01341	−0.506707	−	0.862118i	$-0.669137\pi$
$887$	−30.1820	−1.01341	−0.506707	+	0.862118i	$0.669137\pi$
$888$	−60.7221	−2.03770
$889$	15.8452	0.531431
$890$	10.2598	0.343908
$891$	−22.4257	−0.751291
$892$	2.14183	0.0717139
$893$	0.0239547	0.000801615 0
$894$	−85.1087	−2.84646
$895$	−14.9518	−0.499783
$896$	20.3339	0.679309
$897$	42.0194	1.40299
$898$	51.6246	1.72273
$899$	−5.42341	−0.180881
$900$	−28.4379	−0.947931
$901$	−14.9130	−0.496822
$902$	27.8369	0.926867
$903$	−1.90658	−0.0634470
$904$	−23.4293	−0.779247
$905$	−1.57265	−0.0522765
$906$	57.4519	1.90871
$907$	−16.1485	−0.536203	−0.268102	−	0.963391i	$-0.586396\pi$
$907$	−16.1485	−0.536203	−0.268102	+	0.963391i	$0.586396\pi$
$908$	−34.9748	−1.16068
$909$	−31.2747	−1.03732
$910$	−9.79516	−0.324707
$911$	−19.0676	−0.631739	−0.315869	−	0.948803i	$-0.602296\pi$
$911$	−19.0676	−0.631739	−0.315869	+	0.948803i	$0.602296\pi$
$912$	0.457715	0.0151565
$913$	15.0725	0.498828
$914$	3.28530	0.108668
$915$	13.8222	0.456948
$916$	−26.5490	−0.877204
$917$	8.07387	0.266623
$918$	−26.8305	−0.885538
$919$	−48.5107	−1.60022	−0.800110	−	0.599854i	$-0.795226\pi$
$919$	−48.5107	−1.60022	−0.800110	+	0.599854i	$0.795226\pi$
$920$	35.3245	1.16461
$921$	3.00953	0.0991673
$922$	−48.6938	−1.60365
$923$	−22.1191	−0.728060
$924$	−18.5092	−0.608906
$925$	19.1438	0.629443
$926$	56.4724	1.85580
$927$	−16.3133	−0.535800
$928$	−1.34025	−0.0439959
$929$	−5.06627	−0.166219	−0.0831095	−	0.996540i	$-0.526485\pi$
$929$	−5.06627	−0.166219	−0.0831095	+	0.996540i	$0.526485\pi$
$930$	39.8413	1.30645
$931$	0.379523	0.0124384
$932$	−15.8045	−0.517695
$933$	19.0351	0.623182
$934$	−7.80539	−0.255400
$935$	18.3696	0.600752
$936$	−31.3780	−1.02562
$937$	48.6089	1.58798	0.793992	−	0.607929i	$-0.207999\pi$
$937$	48.6089	1.58798	0.793992	+	0.607929i	$0.207999\pi$
$938$	19.9945	0.652845
$939$	58.9748	1.92457
$940$	−1.93538	−0.0631252
$941$	−10.5540	−0.344051	−0.172025	−	0.985093i	$-0.555031\pi$
$941$	−10.5540	−0.344051	−0.172025	+	0.985093i	$0.555031\pi$
$942$	−56.7353	−1.84854
$943$	32.5075	1.05859
$944$	−23.1844	−0.754587
$945$	2.20748	0.0718093
$946$	−4.25477	−0.138334
$947$	30.1059	0.978310	0.489155	−	0.872197i	$-0.337305\pi$
$947$	30.1059	0.978310	0.489155	+	0.872197i	$0.337305\pi$
$948$	17.1455	0.556860
$949$	−46.6192	−1.51332
$950$	−0.496628	−0.0161128
$951$	43.8914	1.42327
$952$	−29.4803	−0.955462
$953$	24.3969	0.790294	0.395147	−	0.918618i	$-0.370694\pi$
$953$	24.3969	0.790294	0.395147	+	0.918618i	$0.370694\pi$
$954$	12.3996	0.401451
$955$	14.5287	0.470138
$956$	53.1474	1.71891
$957$	4.82886	0.156095
$958$	−22.7570	−0.735246
$959$	13.1374	0.424228
$960$	29.0245	0.936760
$961$	−1.58665	−0.0511823
$962$	43.8800	1.41475
$963$	21.6838	0.698751
$964$	56.5404	1.82104
$965$	29.0462	0.935030
$966$	−32.8246	−1.05611
$967$	−17.7460	−0.570674	−0.285337	−	0.958427i	$-0.592106\pi$
$967$	−17.7460	−0.570674	−0.285337	+	0.958427i	$0.592106\pi$
$968$	29.5334	0.949239
$969$	0.956619	0.0307310
$970$	15.1781	0.487340
$971$	22.4187	0.719452	0.359726	−	0.933058i	$-0.382870\pi$
$971$	22.4187	0.719452	0.359726	+	0.933058i	$0.382870\pi$
$972$	74.9259	2.40325
$973$	−0.993943	−0.0318644
$974$	−60.8025	−1.94824
$975$	22.9756	0.735810
$976$	−14.3838	−0.460415
$977$	−51.0740	−1.63400	−0.817001	−	0.576637i	$-0.804365\pi$
$977$	−51.0740	−1.63400	−0.817001	+	0.576637i	$0.804365\pi$
$978$	−14.4865	−0.463228
$979$	−6.74403	−0.215540
$980$	−30.6629	−0.979491
$981$	−16.9393	−0.540829
$982$	−62.9904	−2.01010
$983$	5.71547	0.182295	0.0911476	−	0.995837i	$-0.470946\pi$
$983$	5.71547	0.182295	0.0911476	+	0.995837i	$0.470946\pi$
$984$	−56.3796	−1.79732
$985$	24.1205	0.768542
$986$	15.9772	0.508819
$987$	0.865720	0.0275562
$988$	−0.749586	−0.0238475
$989$	−4.96866	−0.157994
$990$	−15.2737	−0.485430
$991$	19.9731	0.634465	0.317233	−	0.948348i	$-0.397246\pi$
$991$	19.9731	0.634465	0.317233	+	0.948348i	$0.397246\pi$
$992$	7.26873	0.230782
$993$	27.1590	0.861864
$994$	17.2790	0.548055
$995$	−12.6237	−0.400198
$996$	−63.4161	−2.00941
$997$	31.6745	1.00314	0.501571	−	0.865116i	$-0.332755\pi$
$997$	31.6745	1.00314	0.501571	+	0.865116i	$0.332755\pi$
$998$	91.6641	2.90158
$999$	−9.88899	−0.312874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.5	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.5	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42000 q^{2} -2.29529 q^{3} +3.85639 q^{4} +1.32254 q^{5} +5.55461 q^{6} +0.993943 q^{7} -4.49245 q^{8} +2.26837 q^{9} +O(q^{10})\)	\(q-2.42000 q^{2} -2.29529 q^{3} +3.85639 q^{4} +1.32254 q^{5} +5.55461 q^{6} +0.993943 q^{7} -4.49245 q^{8} +2.26837 q^{9} -3.20054 q^{10} +2.10381 q^{11} -8.85154 q^{12} +3.07912 q^{13} -2.40534 q^{14} -3.03561 q^{15} +3.15895 q^{16} +6.60217 q^{17} -5.48946 q^{18} -0.0631268 q^{19} +5.10022 q^{20} -2.28139 q^{21} -5.09121 q^{22} -5.94544 q^{23} +10.3115 q^{24} -3.25089 q^{25} -7.45147 q^{26} +1.67929 q^{27} +3.83303 q^{28} -1.00000 q^{29} +7.34618 q^{30} +5.42341 q^{31} +1.34025 q^{32} -4.82886 q^{33} -15.9772 q^{34} +1.31453 q^{35} +8.74773 q^{36} -5.88877 q^{37} +0.152767 q^{38} -7.06749 q^{39} -5.94144 q^{40} -5.46764 q^{41} +5.52096 q^{42} +0.835709 q^{43} +8.11310 q^{44} +3.00001 q^{45} +14.3880 q^{46} -0.379470 q^{47} -7.25073 q^{48} -6.01208 q^{49} +7.86715 q^{50} -15.1539 q^{51} +11.8743 q^{52} -2.25880 q^{53} -4.06389 q^{54} +2.78237 q^{55} -4.46524 q^{56} +0.144895 q^{57} +2.42000 q^{58} -7.33926 q^{59} -11.7065 q^{60} -4.55335 q^{61} -13.1246 q^{62} +2.25464 q^{63} -9.56131 q^{64} +4.07226 q^{65} +11.6858 q^{66} -8.31257 q^{67} +25.4605 q^{68} +13.6465 q^{69} -3.18115 q^{70} -7.18358 q^{71} -10.1906 q^{72} -15.1404 q^{73} +14.2508 q^{74} +7.46175 q^{75} -0.243442 q^{76} +2.09106 q^{77} +17.1033 q^{78} -1.93701 q^{79} +4.17784 q^{80} -10.6596 q^{81} +13.2317 q^{82} +7.16441 q^{83} -8.79793 q^{84} +8.73162 q^{85} -2.02241 q^{86} +2.29529 q^{87} -9.45126 q^{88} -3.20563 q^{89} -7.26002 q^{90} +3.06047 q^{91} -22.9279 q^{92} -12.4483 q^{93} +0.918316 q^{94} -0.0834877 q^{95} -3.07627 q^{96} -4.74236 q^{97} +14.5492 q^{98} +4.77222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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