LMFDB - Embedded newform 4031.2.a.c.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.62831 q^{2} +0.935172 q^{3} +4.90799 q^{4} -2.36331 q^{5} -2.45792 q^{6} +0.343928 q^{7} -7.64309 q^{8} -2.12545 q^{9} +O(q^{10})$	\(q-2.62831 q^{2} +0.935172 q^{3} +4.90799 q^{4} -2.36331 q^{5} -2.45792 q^{6} +0.343928 q^{7} -7.64309 q^{8} -2.12545 q^{9} +6.21151 q^{10} +4.42731 q^{11} +4.58982 q^{12} -2.43651 q^{13} -0.903949 q^{14} -2.21010 q^{15} +10.2724 q^{16} +5.05811 q^{17} +5.58634 q^{18} -1.41236 q^{19} -11.5991 q^{20} +0.321632 q^{21} -11.6363 q^{22} +1.96514 q^{23} -7.14761 q^{24} +0.585242 q^{25} +6.40390 q^{26} -4.79318 q^{27} +1.68800 q^{28} -1.00000 q^{29} +5.80883 q^{30} +0.410917 q^{31} -11.7128 q^{32} +4.14030 q^{33} -13.2943 q^{34} -0.812810 q^{35} -10.4317 q^{36} -1.20366 q^{37} +3.71210 q^{38} -2.27856 q^{39} +18.0630 q^{40} -8.28410 q^{41} -0.845348 q^{42} -1.27377 q^{43} +21.7292 q^{44} +5.02311 q^{45} -5.16500 q^{46} -1.89543 q^{47} +9.60647 q^{48} -6.88171 q^{49} -1.53820 q^{50} +4.73020 q^{51} -11.9584 q^{52} +11.0684 q^{53} +12.5979 q^{54} -10.4631 q^{55} -2.62868 q^{56} -1.32080 q^{57} +2.62831 q^{58} +8.43741 q^{59} -10.8472 q^{60} -4.77283 q^{61} -1.08001 q^{62} -0.731004 q^{63} +10.2401 q^{64} +5.75824 q^{65} -10.8820 q^{66} -3.01761 q^{67} +24.8252 q^{68} +1.83775 q^{69} +2.13631 q^{70} +8.62206 q^{71} +16.2450 q^{72} +1.28534 q^{73} +3.16359 q^{74} +0.547302 q^{75} -6.93183 q^{76} +1.52268 q^{77} +5.98875 q^{78} +1.49943 q^{79} -24.2769 q^{80} +1.89391 q^{81} +21.7732 q^{82} +3.80481 q^{83} +1.57857 q^{84} -11.9539 q^{85} +3.34785 q^{86} -0.935172 q^{87} -33.8384 q^{88} +15.3927 q^{89} -13.2023 q^{90} -0.837985 q^{91} +9.64491 q^{92} +0.384278 q^{93} +4.98176 q^{94} +3.33784 q^{95} -10.9535 q^{96} -2.47142 q^{97} +18.0872 q^{98} -9.41005 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.62831	−1.85849	−0.929247	−	0.369460i	$-0.879543\pi$
$2$	−2.62831	−1.85849	−0.929247	+	0.369460i	$0.879543\pi$
$3$	0.935172	0.539922	0.269961	−	0.962871i	$-0.412989\pi$
$3$	0.935172	0.539922	0.269961	+	0.962871i	$0.412989\pi$
$4$	4.90799	2.45400
$5$	−2.36331	−1.05691	−0.528453	−	0.848963i	$-0.677228\pi$
$5$	−2.36331	−1.05691	−0.528453	+	0.848963i	$0.677228\pi$
$6$	−2.45792	−1.00344
$7$	0.343928	0.129993	0.0649964	−	0.997886i	$-0.479296\pi$
$7$	0.343928	0.129993	0.0649964	+	0.997886i	$0.479296\pi$
$8$	−7.64309	−2.70224
$9$	−2.12545	−0.708484
$10$	6.21151	1.96425
$11$	4.42731	1.33489	0.667443	−	0.744661i	$-0.267389\pi$
$11$	4.42731	1.33489	0.667443	+	0.744661i	$0.267389\pi$
$12$	4.58982	1.32497
$13$	−2.43651	−0.675767	−0.337883	−	0.941188i	$-0.609711\pi$
$13$	−2.43651	−0.675767	−0.337883	+	0.941188i	$0.609711\pi$
$14$	−0.903949	−0.241591
$15$	−2.21010	−0.570646
$16$	10.2724	2.56810
$17$	5.05811	1.22677	0.613386	−	0.789783i	$-0.289807\pi$
$17$	5.05811	1.22677	0.613386	+	0.789783i	$0.289807\pi$
$18$	5.58634	1.31671
$19$	−1.41236	−0.324017	−0.162008	−	0.986789i	$-0.551797\pi$
$19$	−1.41236	−0.324017	−0.162008	+	0.986789i	$0.551797\pi$
$20$	−11.5991	−2.59364
$21$	0.321632	0.0701859
$22$	−11.6363	−2.48087
$23$	1.96514	0.409761	0.204880	−	0.978787i	$-0.434319\pi$
$23$	1.96514	0.409761	0.204880	+	0.978787i	$0.434319\pi$
$24$	−7.14761	−1.45900
$25$	0.585242	0.117048
$26$	6.40390	1.25591
$27$	−4.79318	−0.922448
$28$	1.68800	0.319002
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	5.80883	1.06054
$31$	0.410917	0.0738028	0.0369014	−	0.999319i	$-0.488251\pi$
$31$	0.410917	0.0738028	0.0369014	+	0.999319i	$0.488251\pi$
$32$	−11.7128	−2.07056
$33$	4.14030	0.720734
$34$	−13.2943	−2.27995
$35$	−0.812810	−0.137390
$36$	−10.4317	−1.73862
$37$	−1.20366	−0.197880	−0.0989402	−	0.995093i	$-0.531545\pi$
$37$	−1.20366	−0.197880	−0.0989402	+	0.995093i	$0.531545\pi$
$38$	3.71210	0.602183
$39$	−2.27856	−0.364861
$40$	18.0630	2.85601
$41$	−8.28410	−1.29376	−0.646880	−	0.762592i	$-0.723926\pi$
$41$	−8.28410	−1.29376	−0.646880	+	0.762592i	$0.723926\pi$
$42$	−0.845348	−0.130440
$43$	−1.27377	−0.194248	−0.0971240	−	0.995272i	$-0.530964\pi$
$43$	−1.27377	−0.194248	−0.0971240	+	0.995272i	$0.530964\pi$
$44$	21.7292	3.27580
$45$	5.02311	0.748801
$46$	−5.16500	−0.761537
$47$	−1.89543	−0.276476	−0.138238	−	0.990399i	$-0.544144\pi$
$47$	−1.89543	−0.276476	−0.138238	+	0.990399i	$0.544144\pi$
$48$	9.60647	1.38657
$49$	−6.88171	−0.983102
$50$	−1.53820	−0.217534
$51$	4.73020	0.662361
$52$	−11.9584	−1.65833
$53$	11.0684	1.52036	0.760178	−	0.649715i	$-0.225112\pi$
$53$	11.0684	1.52036	0.760178	+	0.649715i	$0.225112\pi$
$54$	12.5979	1.71436
$55$	−10.4631	−1.41085
$56$	−2.62868	−0.351272
$57$	−1.32080	−0.174944
$58$	2.62831	0.345113
$59$	8.43741	1.09846	0.549229	−	0.835672i	$-0.314922\pi$
$59$	8.43741	1.09846	0.549229	+	0.835672i	$0.314922\pi$
$60$	−10.8472	−1.40036
$61$	−4.77283	−0.611098	−0.305549	−	0.952176i	$-0.598840\pi$
$61$	−4.77283	−0.611098	−0.305549	+	0.952176i	$0.598840\pi$
$62$	−1.08001	−0.137162
$63$	−0.731004	−0.0920978
$64$	10.2401	1.28001
$65$	5.75824	0.714221
$66$	−10.8820	−1.33948
$67$	−3.01761	−0.368660	−0.184330	−	0.982864i	$-0.559011\pi$
$67$	−3.01761	−0.368660	−0.184330	+	0.982864i	$0.559011\pi$
$68$	24.8252	3.01049
$69$	1.83775	0.221239
$70$	2.13631	0.255338
$71$	8.62206	1.02325	0.511625	−	0.859209i	$-0.329044\pi$
$71$	8.62206	1.02325	0.511625	+	0.859209i	$0.329044\pi$
$72$	16.2450	1.91450
$73$	1.28534	0.150437	0.0752185	−	0.997167i	$-0.476035\pi$
$73$	1.28534	0.150437	0.0752185	+	0.997167i	$0.476035\pi$
$74$	3.16359	0.367759
$75$	0.547302	0.0631970
$76$	−6.93183	−0.795136
$77$	1.52268	0.173525
$78$	5.98875	0.678092
$79$	1.49943	0.168699	0.0843493	−	0.996436i	$-0.473119\pi$
$79$	1.49943	0.168699	0.0843493	+	0.996436i	$0.473119\pi$
$80$	−24.2769	−2.71424
$81$	1.89391	0.210435
$82$	21.7732	2.40444
$83$	3.80481	0.417632	0.208816	−	0.977955i	$-0.433039\pi$
$83$	3.80481	0.417632	0.208816	+	0.977955i	$0.433039\pi$
$84$	1.57857	0.172236
$85$	−11.9539	−1.29658
$86$	3.34785	0.361008
$87$	−0.935172	−0.100261
$88$	−33.8384	−3.60718
$89$	15.3927	1.63163	0.815814	−	0.578315i	$-0.196289\pi$
$89$	15.3927	1.63163	0.815814	+	0.578315i	$0.196289\pi$
$90$	−13.2023	−1.39164
$91$	−0.837985	−0.0878447
$92$	9.64491	1.00555
$93$	0.384278	0.0398478
$94$	4.98176	0.513829
$95$	3.33784	0.342455
$96$	−10.9535	−1.11794
$97$	−2.47142	−0.250935	−0.125467	−	0.992098i	$-0.540043\pi$
$97$	−2.47142	−0.250935	−0.125467	+	0.992098i	$0.540043\pi$
$98$	18.0872	1.82709
$99$	−9.41005	−0.945745
$100$	2.87237	0.287237
$101$	−8.65643	−0.861347	−0.430673	−	0.902508i	$-0.641724\pi$
$101$	−8.65643	−0.861347	−0.430673	+	0.902508i	$0.641724\pi$
$102$	−12.4324	−1.23099
$103$	−0.934989	−0.0921272	−0.0460636	−	0.998939i	$-0.514668\pi$
$103$	−0.934989	−0.0921272	−0.0460636	+	0.998939i	$0.514668\pi$
$104$	18.6225	1.82609
$105$	−0.760117	−0.0741798
$106$	−29.0910	−2.82557
$107$	−2.56781	−0.248239	−0.124120	−	0.992267i	$-0.539611\pi$
$107$	−2.56781	−0.248239	−0.124120	+	0.992267i	$0.539611\pi$
$108$	−23.5249	−2.26368
$109$	−9.54608	−0.914349	−0.457174	−	0.889377i	$-0.651138\pi$
$109$	−9.54608	−0.914349	−0.457174	+	0.889377i	$0.651138\pi$
$110$	27.5003	2.62205
$111$	−1.12563	−0.106840
$112$	3.53297	0.333834
$113$	−14.0861	−1.32511	−0.662557	−	0.749012i	$-0.730529\pi$
$113$	−14.0861	−1.32511	−0.662557	+	0.749012i	$0.730529\pi$
$114$	3.47146	0.325132
$115$	−4.64425	−0.433078
$116$	−4.90799	−0.455696
$117$	5.17869	0.478770
$118$	−22.1761	−2.04148
$119$	1.73963	0.159471
$120$	16.8920	1.54202
$121$	8.60110	0.781918
$122$	12.5445	1.13572
$123$	−7.74706	−0.698529
$124$	2.01678	0.181112
$125$	10.4334	0.933196
$126$	1.92130	0.171163
$127$	17.5443	1.55681	0.778405	−	0.627763i	$-0.216029\pi$
$127$	17.5443	1.55681	0.778405	+	0.627763i	$0.216029\pi$
$128$	−3.48847	−0.308341
$129$	−1.19119	−0.104879
$130$	−15.1344	−1.32738
$131$	−15.7016	−1.37185	−0.685926	−	0.727671i	$-0.740603\pi$
$131$	−15.7016	−1.37185	−0.685926	+	0.727671i	$0.740603\pi$
$132$	20.3206	1.76868
$133$	−0.485749	−0.0421198
$134$	7.93121	0.685152
$135$	11.3278	0.974940
$136$	−38.6596	−3.31503
$137$	−13.8369	−1.18217	−0.591084	−	0.806610i	$-0.701300\pi$
$137$	−13.8369	−1.18217	−0.591084	+	0.806610i	$0.701300\pi$
$138$	−4.83016	−0.411171
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−3.98927	−0.337154
$141$	−1.77255	−0.149276
$142$	−22.6614	−1.90170
$143$	−10.7872	−0.902071
$144$	−21.8335	−1.81946
$145$	2.36331	0.196262
$146$	−3.37825	−0.279586
$147$	−6.43559	−0.530798
$148$	−5.90755	−0.485598
$149$	11.0049	0.901558	0.450779	−	0.892636i	$-0.351146\pi$
$149$	11.0049	0.901558	0.450779	+	0.892636i	$0.351146\pi$
$150$	−1.43848	−0.117451
$151$	−10.7938	−0.878390	−0.439195	−	0.898392i	$-0.644736\pi$
$151$	−10.7938	−0.878390	−0.439195	+	0.898392i	$0.644736\pi$
$152$	10.7948	0.875572
$153$	−10.7508	−0.869149
$154$	−4.00207	−0.322496
$155$	−0.971124	−0.0780026
$156$	−11.1831	−0.895368
$157$	6.44625	0.514466	0.257233	−	0.966349i	$-0.417189\pi$
$157$	6.44625	0.514466	0.257233	+	0.966349i	$0.417189\pi$
$158$	−3.94095	−0.313525
$159$	10.3508	0.820873
$160$	27.6811	2.18838
$161$	0.675868	0.0532659
$162$	−4.97778	−0.391091
$163$	−11.8070	−0.924797	−0.462399	−	0.886672i	$-0.653011\pi$
$163$	−11.8070	−0.924797	−0.462399	+	0.886672i	$0.653011\pi$
$164$	−40.6583	−3.17488
$165$	−9.78482	−0.761747
$166$	−10.0002	−0.776167
$167$	−1.37241	−0.106201	−0.0531003	−	0.998589i	$-0.516910\pi$
$167$	−1.37241	−0.106201	−0.0531003	+	0.998589i	$0.516910\pi$
$168$	−2.45827	−0.189659
$169$	−7.06341	−0.543339
$170$	31.4185	2.40969
$171$	3.00190	0.229561
$172$	−6.25165	−0.476684
$173$	−11.4830	−0.873035	−0.436517	−	0.899696i	$-0.643788\pi$
$173$	−11.4830	−0.873035	−0.436517	+	0.899696i	$0.643788\pi$
$174$	2.45792	0.186334
$175$	0.201281	0.0152155
$176$	45.4792	3.42812
$177$	7.89043	0.593081
$178$	−40.4568	−3.03237
$179$	−0.618466	−0.0462263	−0.0231132	−	0.999733i	$-0.507358\pi$
$179$	−0.618466	−0.0462263	−0.0231132	+	0.999733i	$0.507358\pi$
$180$	24.6534	1.83755
$181$	−14.7884	−1.09921	−0.549607	−	0.835423i	$-0.685223\pi$
$181$	−14.7884	−1.09921	−0.549607	+	0.835423i	$0.685223\pi$
$182$	2.20248	0.163259
$183$	−4.46341	−0.329945
$184$	−15.0198	−1.10727
$185$	2.84462	0.209141
$186$	−1.01000	−0.0740568
$187$	22.3938	1.63760
$188$	−9.30274	−0.678472
$189$	−1.64851	−0.119912
$190$	−8.77286	−0.636450
$191$	−22.4065	−1.62128	−0.810638	−	0.585547i	$-0.800880\pi$
$191$	−22.4065	−1.62128	−0.810638	+	0.585547i	$0.800880\pi$
$192$	9.57627	0.691108
$193$	13.6778	0.984550	0.492275	−	0.870440i	$-0.336165\pi$
$193$	13.6778	0.984550	0.492275	+	0.870440i	$0.336165\pi$
$194$	6.49565	0.466360
$195$	5.38494	0.385624
$196$	−33.7754	−2.41253
$197$	9.69504	0.690743	0.345372	−	0.938466i	$-0.387753\pi$
$197$	9.69504	0.690743	0.345372	+	0.938466i	$0.387753\pi$
$198$	24.7325	1.75766
$199$	−7.34376	−0.520585	−0.260293	−	0.965530i	$-0.583819\pi$
$199$	−7.34376	−0.520585	−0.260293	+	0.965530i	$0.583819\pi$
$200$	−4.47306	−0.316293
$201$	−2.82199	−0.199048
$202$	22.7517	1.60081
$203$	−0.343928	−0.0241390
$204$	23.2158	1.62543
$205$	19.5779	1.36738
$206$	2.45744	0.171218
$207$	−4.17682	−0.290309
$208$	−25.0288	−1.73544
$209$	−6.25294	−0.432525
$210$	1.99782	0.137863
$211$	−5.79229	−0.398758	−0.199379	−	0.979922i	$-0.563892\pi$
$211$	−5.79229	−0.398758	−0.199379	+	0.979922i	$0.563892\pi$
$212$	54.3234	3.73095
$213$	8.06311	0.552475
$214$	6.74898	0.461351
$215$	3.01031	0.205302
$216$	36.6347	2.49268
$217$	0.141326	0.00959383
$218$	25.0900	1.69931
$219$	1.20201	0.0812243
$220$	−51.3529	−3.46221
$221$	−12.3241	−0.829011
$222$	2.95850	0.198561
$223$	−1.39323	−0.0932978	−0.0466489	−	0.998911i	$-0.514854\pi$
$223$	−1.39323	−0.0932978	−0.0466489	+	0.998911i	$0.514854\pi$
$224$	−4.02838	−0.269157
$225$	−1.24391	−0.0829270
$226$	37.0227	2.46271
$227$	4.47162	0.296792	0.148396	−	0.988928i	$-0.452589\pi$
$227$	4.47162	0.296792	0.148396	+	0.988928i	$0.452589\pi$
$228$	−6.48246	−0.429311
$229$	12.0981	0.799465	0.399732	−	0.916632i	$-0.369103\pi$
$229$	12.0981	0.799465	0.399732	+	0.916632i	$0.369103\pi$
$230$	12.2065	0.804873
$231$	1.42397	0.0936901
$232$	7.64309	0.501794
$233$	13.2664	0.869111	0.434555	−	0.900645i	$-0.356906\pi$
$233$	13.2664	0.869111	0.434555	+	0.900645i	$0.356906\pi$
$234$	−13.6112	−0.889791
$235$	4.47948	0.292209
$236$	41.4108	2.69561
$237$	1.40222	0.0910841
$238$	−4.57227	−0.296376
$239$	−15.6568	−1.01275	−0.506377	−	0.862312i	$-0.669016\pi$
$239$	−15.6568	−1.01275	−0.506377	+	0.862312i	$0.669016\pi$
$240$	−22.7031	−1.46548
$241$	−2.14541	−0.138198	−0.0690990	−	0.997610i	$-0.522012\pi$
$241$	−2.14541	−0.138198	−0.0690990	+	0.997610i	$0.522012\pi$
$242$	−22.6063	−1.45319
$243$	16.1507	1.03607
$244$	−23.4250	−1.49963
$245$	16.2636	1.03905
$246$	20.3616	1.29821
$247$	3.44122	0.218960
$248$	−3.14068	−0.199433
$249$	3.55815	0.225489
$250$	−27.4223	−1.73434
$251$	0.600131	0.0378799	0.0189400	−	0.999821i	$-0.493971\pi$
$251$	0.600131	0.0378799	0.0189400	+	0.999821i	$0.493971\pi$
$252$	−3.58776	−0.226008
$253$	8.70030	0.546983
$254$	−46.1119	−2.89332
$255$	−11.1789	−0.700052
$256$	−11.3114	−0.706966
$257$	−6.48982	−0.404824	−0.202412	−	0.979300i	$-0.564878\pi$
$257$	−6.48982	−0.404824	−0.202412	+	0.979300i	$0.564878\pi$
$258$	3.13082	0.194916
$259$	−0.413973	−0.0257230
$260$	28.2614	1.75270
$261$	2.12545	0.131562
$262$	41.2685	2.54958
$263$	−5.76283	−0.355352	−0.177676	−	0.984089i	$-0.556858\pi$
$263$	−5.76283	−0.355352	−0.177676	+	0.984089i	$0.556858\pi$
$264$	−31.6447	−1.94760
$265$	−26.1580	−1.60687
$266$	1.27670	0.0782794
$267$	14.3949	0.880951
$268$	−14.8104	−0.904690
$269$	−3.31761	−0.202278	−0.101139	−	0.994872i	$-0.532249\pi$
$269$	−3.31761	−0.202278	−0.101139	+	0.994872i	$0.532249\pi$
$270$	−29.7729	−1.81192
$271$	−2.88698	−0.175372	−0.0876859	−	0.996148i	$-0.527947\pi$
$271$	−2.88698	−0.175372	−0.0876859	+	0.996148i	$0.527947\pi$
$272$	51.9590	3.15047
$273$	−0.783661	−0.0474293
$274$	36.3677	2.19705
$275$	2.59105	0.156246
$276$	9.01965	0.542919
$277$	14.3044	0.859469	0.429734	−	0.902955i	$-0.358607\pi$
$277$	14.3044	0.859469	0.429734	+	0.902955i	$0.358607\pi$
$278$	2.62831	0.157635
$279$	−0.873384	−0.0522882
$280$	6.21238	0.371261
$281$	−19.6101	−1.16984	−0.584919	−	0.811092i	$-0.698874\pi$
$281$	−19.6101	−1.16984	−0.584919	+	0.811092i	$0.698874\pi$
$282$	4.65880	0.277428
$283$	−6.04611	−0.359404	−0.179702	−	0.983721i	$-0.557513\pi$
$283$	−6.04611	−0.359404	−0.179702	+	0.983721i	$0.557513\pi$
$284$	42.3170	2.51105
$285$	3.12145	0.184899
$286$	28.3521	1.67649
$287$	−2.84914	−0.168179
$288$	24.8951	1.46696
$289$	8.58446	0.504968
$290$	−6.21151	−0.364752
$291$	−2.31120	−0.135485
$292$	6.30842	0.369172
$293$	7.13013	0.416547	0.208273	−	0.978071i	$-0.433216\pi$
$293$	7.13013	0.416547	0.208273	+	0.978071i	$0.433216\pi$
$294$	16.9147	0.986485
$295$	−19.9402	−1.16097
$296$	9.19968	0.534721
$297$	−21.2209	−1.23136
$298$	−28.9243	−1.67554
$299$	−4.78809	−0.276903
$300$	2.68616	0.155085
$301$	−0.438085	−0.0252508
$302$	28.3695	1.63248
$303$	−8.09525	−0.465060
$304$	−14.5083	−0.832108
$305$	11.2797	0.645873
$306$	28.2563	1.61531
$307$	−21.6699	−1.23677	−0.618384	−	0.785876i	$-0.712212\pi$
$307$	−21.6699	−1.23677	−0.618384	+	0.785876i	$0.712212\pi$
$308$	7.47330	0.425831
$309$	−0.874376	−0.0497415
$310$	2.55241	0.144967
$311$	8.99244	0.509915	0.254957	−	0.966952i	$-0.417939\pi$
$311$	8.99244	0.509915	0.254957	+	0.966952i	$0.417939\pi$
$312$	17.4152	0.985943
$313$	9.51385	0.537755	0.268877	−	0.963174i	$-0.413347\pi$
$313$	9.51385	0.537755	0.268877	+	0.963174i	$0.413347\pi$
$314$	−16.9427	−0.956132
$315$	1.72759	0.0973386
$316$	7.35917	0.413986
$317$	−12.8457	−0.721484	−0.360742	−	0.932666i	$-0.617476\pi$
$317$	−12.8457	−0.721484	−0.360742	+	0.932666i	$0.617476\pi$
$318$	−27.2051	−1.52559
$319$	−4.42731	−0.247882
$320$	−24.2006	−1.35285
$321$	−2.40134	−0.134030
$322$	−1.77639	−0.0989943
$323$	−7.14385	−0.397495
$324$	9.29530	0.516406
$325$	−1.42595	−0.0790975
$326$	31.0325	1.71873
$327$	−8.92723	−0.493677
$328$	63.3162	3.49605
$329$	−0.651891	−0.0359399
$330$	25.7175	1.41570
$331$	−23.5705	−1.29555	−0.647777	−	0.761830i	$-0.724301\pi$
$331$	−23.5705	−1.29555	−0.647777	+	0.761830i	$0.724301\pi$
$332$	18.6740	1.02487
$333$	2.55832	0.140195
$334$	3.60713	0.197373
$335$	7.13156	0.389639
$336$	3.30394	0.180245
$337$	−12.3825	−0.674516	−0.337258	−	0.941412i	$-0.609499\pi$
$337$	−12.3825	−0.674516	−0.337258	+	0.941412i	$0.609499\pi$
$338$	18.5648	1.00979
$339$	−13.1730	−0.715458
$340$	−58.6696	−3.18181
$341$	1.81926	0.0985183
$342$	−7.88990	−0.426637
$343$	−4.77431	−0.257789
$344$	9.73554	0.524905
$345$	−4.34317	−0.233828
$346$	30.1808	1.62253
$347$	−29.4210	−1.57940	−0.789700	−	0.613493i	$-0.789764\pi$
$347$	−29.4210	−1.57940	−0.789700	+	0.613493i	$0.789764\pi$
$348$	−4.58982	−0.246040
$349$	23.9935	1.28434	0.642171	−	0.766561i	$-0.278034\pi$
$349$	23.9935	1.28434	0.642171	+	0.766561i	$0.278034\pi$
$350$	−0.529029	−0.0282778
$351$	11.6786	0.623360
$352$	−51.8564	−2.76396
$353$	10.9643	0.583569	0.291785	−	0.956484i	$-0.405751\pi$
$353$	10.9643	0.583569	0.291785	+	0.956484i	$0.405751\pi$
$354$	−20.7385	−1.10224
$355$	−20.3766	−1.08148
$356$	75.5475	4.00401
$357$	1.62685	0.0861021
$358$	1.62552	0.0859113
$359$	13.2791	0.700845	0.350422	−	0.936592i	$-0.386038\pi$
$359$	13.2791	0.700845	0.350422	+	0.936592i	$0.386038\pi$
$360$	−38.3921	−2.02344
$361$	−17.0052	−0.895013
$362$	38.8685	2.04288
$363$	8.04351	0.422175
$364$	−4.11283	−0.215571
$365$	−3.03765	−0.158998
$366$	11.7312	0.613201
$367$	−13.5100	−0.705214	−0.352607	−	0.935772i	$-0.614705\pi$
$367$	−13.5100	−0.705214	−0.352607	+	0.935772i	$0.614705\pi$
$368$	20.1868	1.05231
$369$	17.6075	0.916608
$370$	−7.47654	−0.388687
$371$	3.80672	0.197635
$372$	1.88603	0.0977863
$373$	1.16663	0.0604059	0.0302030	−	0.999544i	$-0.490385\pi$
$373$	1.16663	0.0604059	0.0302030	+	0.999544i	$0.490385\pi$
$374$	−58.8578	−3.04347
$375$	9.75707	0.503853
$376$	14.4869	0.747106
$377$	2.43651	0.125487
$378$	4.33279	0.222855
$379$	−14.0363	−0.720997	−0.360498	−	0.932760i	$-0.617393\pi$
$379$	−14.0363	−0.720997	−0.360498	+	0.932760i	$0.617393\pi$
$380$	16.3821	0.840383
$381$	16.4070	0.840555
$382$	58.8911	3.01313
$383$	8.90894	0.455226	0.227613	−	0.973752i	$-0.426908\pi$
$383$	8.90894	0.455226	0.227613	+	0.973752i	$0.426908\pi$
$384$	−3.26232	−0.166480
$385$	−3.59856	−0.183400
$386$	−35.9495	−1.82978
$387$	2.70734	0.137622
$388$	−12.1297	−0.615793
$389$	−12.5763	−0.637642	−0.318821	−	0.947815i	$-0.603287\pi$
$389$	−12.5763	−0.637642	−0.318821	+	0.947815i	$0.603287\pi$
$390$	−14.1533	−0.716679
$391$	9.93991	0.502683
$392$	52.5976	2.65658
$393$	−14.6837	−0.740693
$394$	−25.4815	−1.28374
$395$	−3.54361	−0.178298
$396$	−46.1844	−2.32086
$397$	−29.2307	−1.46705	−0.733524	−	0.679664i	$-0.762126\pi$
$397$	−29.2307	−1.46705	−0.733524	+	0.679664i	$0.762126\pi$
$398$	19.3016	0.967504
$399$	−0.454259	−0.0227414
$400$	6.01185	0.300592
$401$	−33.4939	−1.67260	−0.836302	−	0.548269i	$-0.815287\pi$
$401$	−33.4939	−1.67260	−0.836302	+	0.548269i	$0.815287\pi$
$402$	7.41704	0.369928
$403$	−1.00120	−0.0498735
$404$	−42.4857	−2.11374
$405$	−4.47590	−0.222409
$406$	0.903949	0.0448622
$407$	−5.32898	−0.264148
$408$	−36.1534	−1.78986
$409$	−2.18589	−0.108085	−0.0540426	−	0.998539i	$-0.517211\pi$
$409$	−2.18589	−0.108085	−0.0540426	+	0.998539i	$0.517211\pi$
$410$	−51.4568	−2.54127
$411$	−12.9399	−0.638279
$412$	−4.58892	−0.226080
$413$	2.90187	0.142791
$414$	10.9780	0.539537
$415$	−8.99196	−0.441398
$416$	28.5385	1.39921
$417$	−0.935172	−0.0457956
$418$	16.4346	0.803845
$419$	−20.3932	−0.996271	−0.498135	−	0.867099i	$-0.665982\pi$
$419$	−20.3932	−0.996271	−0.498135	+	0.867099i	$0.665982\pi$
$420$	−3.73065	−0.182037
$421$	−7.13279	−0.347631	−0.173816	−	0.984778i	$-0.555610\pi$
$421$	−7.13279	−0.347631	−0.173816	+	0.984778i	$0.555610\pi$
$422$	15.2239	0.741089
$423$	4.02864	0.195879
$424$	−84.5965	−4.10837
$425$	2.96022	0.143592
$426$	−21.1923	−1.02677
$427$	−1.64151	−0.0794383
$428$	−12.6028	−0.609178
$429$	−10.0879	−0.487048
$430$	−7.91202	−0.381552
$431$	−20.0149	−0.964084	−0.482042	−	0.876148i	$-0.660105\pi$
$431$	−20.0149	−0.964084	−0.482042	+	0.876148i	$0.660105\pi$
$432$	−49.2375	−2.36894
$433$	4.12398	0.198186	0.0990928	−	0.995078i	$-0.468406\pi$
$433$	4.12398	0.198186	0.0990928	+	0.995078i	$0.468406\pi$
$434$	−0.371448	−0.0178301
$435$	2.21010	0.105966
$436$	−46.8521	−2.24381
$437$	−2.77548	−0.132769
$438$	−3.15925	−0.150955
$439$	−30.0141	−1.43249	−0.716247	−	0.697847i	$-0.754142\pi$
$439$	−30.0141	−1.43249	−0.716247	+	0.697847i	$0.754142\pi$
$440$	79.9706	3.81245
$441$	14.6268	0.696512
$442$	32.3916	1.54071
$443$	−4.56016	−0.216660	−0.108330	−	0.994115i	$-0.534550\pi$
$443$	−4.56016	−0.216660	−0.108330	+	0.994115i	$0.534550\pi$
$444$	−5.52458	−0.262185
$445$	−36.3778	−1.72448
$446$	3.66184	0.173393
$447$	10.2915	0.486771
$448$	3.52187	0.166393
$449$	8.95557	0.422640	0.211320	−	0.977417i	$-0.432224\pi$
$449$	8.95557	0.422640	0.211320	+	0.977417i	$0.432224\pi$
$450$	3.26936	0.154119
$451$	−36.6763	−1.72702
$452$	−69.1347	−3.25182
$453$	−10.0941	−0.474262
$454$	−11.7528	−0.551586
$455$	1.98042	0.0928436
$456$	10.0950	0.472740
$457$	22.5126	1.05310	0.526549	−	0.850145i	$-0.323486\pi$
$457$	22.5126	1.05310	0.526549	+	0.850145i	$0.323486\pi$
$458$	−31.7975	−1.48580
$459$	−24.2444	−1.13163
$460$	−22.7939	−1.06277
$461$	−3.32266	−0.154752	−0.0773758	−	0.997002i	$-0.524654\pi$
$461$	−3.32266	−0.154752	−0.0773758	+	0.997002i	$0.524654\pi$
$462$	−3.74262	−0.174122
$463$	−8.73764	−0.406073	−0.203036	−	0.979171i	$-0.565081\pi$
$463$	−8.73764	−0.406073	−0.203036	+	0.979171i	$0.565081\pi$
$464$	−10.2724	−0.476885
$465$	−0.908168	−0.0421153
$466$	−34.8682	−1.61524
$467$	−17.8064	−0.823980	−0.411990	−	0.911188i	$-0.635166\pi$
$467$	−17.8064	−0.823980	−0.411990	+	0.911188i	$0.635166\pi$
$468$	25.4170	1.17490
$469$	−1.03784	−0.0479231
$470$	−11.7735	−0.543069
$471$	6.02835	0.277772
$472$	−64.4879	−2.96830
$473$	−5.63937	−0.259299
$474$	−3.68547	−0.169279
$475$	−0.826571	−0.0379257
$476$	8.53808	0.391342
$477$	−23.5253	−1.07715
$478$	41.1509	1.88220
$479$	−41.4867	−1.89558	−0.947788	−	0.318900i	$-0.896687\pi$
$479$	−41.4867	−1.89558	−0.947788	+	0.318900i	$0.896687\pi$
$480$	25.8866	1.18156
$481$	2.93273	0.133721
$482$	5.63879	0.256840
$483$	0.632053	0.0287594
$484$	42.2141	1.91882
$485$	5.84074	0.265214
$486$	−42.4489	−1.92552
$487$	−16.1437	−0.731539	−0.365770	−	0.930705i	$-0.619194\pi$
$487$	−16.1437	−0.731539	−0.365770	+	0.930705i	$0.619194\pi$
$488$	36.4792	1.65133
$489$	−11.0416	−0.499318
$490$	−42.7458	−1.93106
$491$	−10.5206	−0.474790	−0.237395	−	0.971413i	$-0.576294\pi$
$491$	−10.5206	−0.474790	−0.237395	+	0.971413i	$0.576294\pi$
$492$	−38.0225	−1.71419
$493$	−5.05811	−0.227806
$494$	−9.04459	−0.406935
$495$	22.2389	0.999563
$496$	4.22110	0.189533
$497$	2.96537	0.133015
$498$	−9.35192	−0.419069
$499$	14.0505	0.628986	0.314493	−	0.949260i	$-0.398165\pi$
$499$	14.0505	0.628986	0.314493	+	0.949260i	$0.398165\pi$
$500$	51.2073	2.29006
$501$	−1.28344	−0.0573401
$502$	−1.57733	−0.0703995
$503$	34.6649	1.54563	0.772815	−	0.634632i	$-0.218848\pi$
$503$	34.6649	1.54563	0.772815	+	0.634632i	$0.218848\pi$
$504$	5.58713	0.248871
$505$	20.4578	0.910362
$506$	−22.8671	−1.01656
$507$	−6.60550	−0.293361
$508$	86.1075	3.82040
$509$	−7.33670	−0.325194	−0.162597	−	0.986693i	$-0.551987\pi$
$509$	−7.33670	−0.325194	−0.162597	+	0.986693i	$0.551987\pi$
$510$	29.3817	1.30104
$511$	0.442063	0.0195557
$512$	36.7069	1.62223
$513$	6.76968	0.298889
$514$	17.0572	0.752362
$515$	2.20967	0.0973697
$516$	−5.84637	−0.257372
$517$	−8.39165	−0.369064
$518$	1.08805	0.0478060
$519$	−10.7386	−0.471370
$520$	−44.0107	−1.93000
$521$	40.0828	1.75606	0.878029	−	0.478608i	$-0.158858\pi$
$521$	40.0828	1.75606	0.878029	+	0.478608i	$0.158858\pi$
$522$	−5.58634	−0.244508
$523$	−14.3247	−0.626374	−0.313187	−	0.949691i	$-0.601397\pi$
$523$	−14.3247	−0.626374	−0.313187	+	0.949691i	$0.601397\pi$
$524$	−77.0632	−3.36652
$525$	0.188233	0.00821515
$526$	15.1465	0.660419
$527$	2.07846	0.0905392
$528$	42.5308	1.85092
$529$	−19.1382	−0.832096
$530$	68.7511	2.98636
$531$	−17.9333	−0.778240
$532$	−2.38405	−0.103362
$533$	20.1843	0.874279
$534$	−37.8341	−1.63724
$535$	6.06853	0.262365
$536$	23.0639	0.996208
$537$	−0.578372	−0.0249586
$538$	8.71968	0.375932
$539$	−30.4675	−1.31233
$540$	55.5967	2.39250
$541$	29.7731	1.28004	0.640022	−	0.768356i	$-0.278925\pi$
$541$	29.7731	1.28004	0.640022	+	0.768356i	$0.278925\pi$
$542$	7.58787	0.325927
$543$	−13.8297	−0.593490
$544$	−59.2448	−2.54010
$545$	22.5604	0.966380
$546$	2.05970	0.0881470
$547$	7.00405	0.299472	0.149736	−	0.988726i	$-0.452158\pi$
$547$	7.00405	0.299472	0.149736	+	0.988726i	$0.452158\pi$
$548$	−67.9115	−2.90104
$549$	10.1444	0.432953
$550$	−6.81008	−0.290383
$551$	1.41236	0.0601684
$552$	−14.0461	−0.597841
$553$	0.515695	0.0219296
$554$	−37.5964	−1.59732
$555$	2.66021	0.112920
$556$	−4.90799	−0.208145
$557$	−17.8046	−0.754407	−0.377204	−	0.926130i	$-0.623114\pi$
$557$	−17.8046	−0.754407	−0.377204	+	0.926130i	$0.623114\pi$
$558$	2.29552	0.0971772
$559$	3.10355	0.131266
$560$	−8.34951	−0.352831
$561$	20.9421	0.884175
$562$	51.5413	2.17414
$563$	−1.26512	−0.0533185	−0.0266592	−	0.999645i	$-0.508487\pi$
$563$	−1.26512	−0.0533185	−0.0266592	+	0.999645i	$0.508487\pi$
$564$	−8.69966	−0.366322
$565$	33.2900	1.40052
$566$	15.8910	0.667950
$567$	0.651370	0.0273550
$568$	−65.8993	−2.76507
$569$	−42.2205	−1.76998	−0.884988	−	0.465614i	$-0.845834\pi$
$569$	−42.2205	−1.76998	−0.884988	+	0.465614i	$0.845834\pi$
$570$	−8.20413	−0.343633
$571$	−16.3627	−0.684757	−0.342379	−	0.939562i	$-0.611233\pi$
$571$	−16.3627	−0.684757	−0.342379	+	0.939562i	$0.611233\pi$
$572$	−52.9435	−2.21368
$573$	−20.9539	−0.875362
$574$	7.48841	0.312560
$575$	1.15009	0.0479619
$576$	−21.7649	−0.906870
$577$	−1.54929	−0.0644980	−0.0322490	−	0.999480i	$-0.510267\pi$
$577$	−1.54929	−0.0644980	−0.0322490	+	0.999480i	$0.510267\pi$
$578$	−22.5626	−0.938480
$579$	12.7911	0.531580
$580$	11.5991	0.481627
$581$	1.30858	0.0542892
$582$	6.07455	0.251798
$583$	49.0031	2.02950
$584$	−9.82394	−0.406517
$585$	−12.2389	−0.506015
$586$	−18.7402	−0.774149
$587$	24.9553	1.03002	0.515009	−	0.857185i	$-0.327789\pi$
$587$	24.9553	1.03002	0.515009	+	0.857185i	$0.327789\pi$
$588$	−31.5858	−1.30258
$589$	−0.580361	−0.0239134
$590$	52.4090	2.15765
$591$	9.06653	0.372947
$592$	−12.3645	−0.508177
$593$	−19.8936	−0.816932	−0.408466	−	0.912773i	$-0.633936\pi$
$593$	−19.8936	−0.816932	−0.408466	+	0.912773i	$0.633936\pi$
$594$	55.7750	2.28848
$595$	−4.11128	−0.168546
$596$	54.0120	2.21242
$597$	−6.86768	−0.281075
$598$	12.5846	0.514622
$599$	10.1041	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$599$	10.1041	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$600$	−4.18308	−0.170774
$601$	−48.6642	−1.98505	−0.992527	−	0.122029i	$-0.961060\pi$
$601$	−48.6642	−1.98505	−0.992527	+	0.122029i	$0.961060\pi$
$602$	1.15142	0.0469285
$603$	6.41379	0.261190
$604$	−52.9761	−2.15557
$605$	−20.3271	−0.826413
$606$	21.2768	0.864311
$607$	9.53920	0.387184	0.193592	−	0.981082i	$-0.437986\pi$
$607$	9.53920	0.387184	0.193592	+	0.981082i	$0.437986\pi$
$608$	16.5427	0.670895
$609$	−0.321632	−0.0130332
$610$	−29.6464	−1.20035
$611$	4.61823	0.186834
$612$	−52.7647	−2.13289
$613$	−20.2305	−0.817104	−0.408552	−	0.912735i	$-0.633966\pi$
$613$	−20.2305	−0.817104	−0.408552	+	0.912735i	$0.633966\pi$
$614$	56.9552	2.29853
$615$	18.3087	0.738279
$616$	−11.6380	−0.468907
$617$	48.8716	1.96750	0.983749	−	0.179550i	$-0.0574640\pi$
$617$	48.8716	1.96750	0.983749	+	0.179550i	$0.0574640\pi$
$618$	2.29813	0.0924442
$619$	−13.2726	−0.533472	−0.266736	−	0.963770i	$-0.585945\pi$
$619$	−13.2726	−0.533472	−0.266736	+	0.963770i	$0.585945\pi$
$620$	−4.76627	−0.191418
$621$	−9.41929	−0.377983
$622$	−23.6349	−0.947673
$623$	5.29400	0.212100
$624$	−23.4063	−0.937001
$625$	−27.5837	−1.10335
$626$	−25.0053	−0.999413
$627$	−5.84758	−0.233530
$628$	31.6381	1.26250
$629$	−6.08824	−0.242754
$630$	−4.54063	−0.180903
$631$	−35.5430	−1.41494	−0.707472	−	0.706742i	$-0.750164\pi$
$631$	−35.5430	−1.41494	−0.707472	+	0.706742i	$0.750164\pi$
$632$	−11.4603	−0.455865
$633$	−5.41679	−0.215298
$634$	33.7623	1.34087
$635$	−41.4628	−1.64540
$636$	50.8017	2.01442
$637$	16.7674	0.664348
$638$	11.6363	0.460687
$639$	−18.3258	−0.724957
$640$	8.24435	0.325887
$641$	35.3938	1.39797	0.698986	−	0.715135i	$-0.253635\pi$
$641$	35.3938	1.39797	0.698986	+	0.715135i	$0.253635\pi$
$642$	6.31146	0.249093
$643$	−15.6360	−0.616625	−0.308312	−	0.951285i	$-0.599764\pi$
$643$	−15.6360	−0.616625	−0.308312	+	0.951285i	$0.599764\pi$
$644$	3.31716	0.130714
$645$	2.81516	0.110847
$646$	18.7762	0.738741
$647$	1.57843	0.0620546	0.0310273	−	0.999519i	$-0.490122\pi$
$647$	1.57843	0.0620546	0.0310273	+	0.999519i	$0.490122\pi$
$648$	−14.4753	−0.568645
$649$	37.3551	1.46631
$650$	3.74783	0.147002
$651$	0.132164	0.00517992
$652$	−57.9488	−2.26945
$653$	34.7245	1.35887	0.679437	−	0.733734i	$-0.262224\pi$
$653$	34.7245	1.35887	0.679437	+	0.733734i	$0.262224\pi$
$654$	23.4635	0.917495
$655$	37.1077	1.44992
$656$	−85.0977	−3.32251
$657$	−2.73192	−0.106582
$658$	1.71337	0.0667941
$659$	−43.3835	−1.68998	−0.844991	−	0.534781i	$-0.820394\pi$
$659$	−43.3835	−1.68998	−0.844991	+	0.534781i	$0.820394\pi$
$660$	−48.0238	−1.86932
$661$	19.3222	0.751548	0.375774	−	0.926711i	$-0.377377\pi$
$661$	19.3222	0.751548	0.375774	+	0.926711i	$0.377377\pi$
$662$	61.9506	2.40778
$663$	−11.5252	−0.447601
$664$	−29.0805	−1.12854
$665$	1.14798	0.0445166
$666$	−6.72405	−0.260552
$667$	−1.96514	−0.0760906
$668$	−6.73580	−0.260616
$669$	−1.30291	−0.0503735
$670$	−18.7439	−0.724140
$671$	−21.1308	−0.815746
$672$	−3.76723	−0.145324
$673$	−35.9844	−1.38710	−0.693549	−	0.720410i	$-0.743954\pi$
$673$	−35.9844	−1.38710	−0.693549	+	0.720410i	$0.743954\pi$
$674$	32.5449	1.25358
$675$	−2.80517	−0.107971
$676$	−34.6672	−1.33335
$677$	25.4825	0.979372	0.489686	−	0.871899i	$-0.337111\pi$
$677$	25.4825	0.979372	0.489686	+	0.871899i	$0.337111\pi$
$678$	34.6226	1.32967
$679$	−0.849991	−0.0326197
$680$	91.3647	3.50368
$681$	4.18174	0.160244
$682$	−4.78156	−0.183096
$683$	−6.35577	−0.243197	−0.121598	−	0.992579i	$-0.538802\pi$
$683$	−6.35577	−0.243197	−0.121598	+	0.992579i	$0.538802\pi$
$684$	14.7333	0.563341
$685$	32.7010	1.24944
$686$	12.5484	0.479099
$687$	11.3138	0.431648
$688$	−13.0847	−0.498848
$689$	−26.9682	−1.02741
$690$	11.4152	0.434568
$691$	7.41721	0.282164	0.141082	−	0.989998i	$-0.454942\pi$
$691$	7.41721	0.282164	0.141082	+	0.989998i	$0.454942\pi$
$692$	−56.3584	−2.14242
$693$	−3.23638	−0.122940
$694$	77.3273	2.93530
$695$	2.36331	0.0896455
$696$	7.14761	0.270929
$697$	−41.9019	−1.58715
$698$	−63.0622	−2.38694
$699$	12.4064	0.469252
$700$	0.987888	0.0373387
$701$	−50.4375	−1.90500	−0.952499	−	0.304543i	$-0.901496\pi$
$701$	−50.4375	−1.90500	−0.952499	+	0.304543i	$0.901496\pi$
$702$	−30.6950	−1.15851
$703$	1.70000	0.0641166
$704$	45.3362	1.70867
$705$	4.18909	0.157770
$706$	−28.8175	−1.08456
$707$	−2.97719	−0.111969
$708$	38.7262	1.45542
$709$	−8.75843	−0.328930	−0.164465	−	0.986383i	$-0.552590\pi$
$709$	−8.75843	−0.328930	−0.164465	+	0.986383i	$0.552590\pi$
$710$	53.5560	2.00992
$711$	−3.18696	−0.119520
$712$	−117.648	−4.40905
$713$	0.807510	0.0302415
$714$	−4.27586	−0.160020
$715$	25.4935	0.953403
$716$	−3.03543	−0.113439
$717$	−14.6418	−0.546808
$718$	−34.9016	−1.30251
$719$	−22.9135	−0.854528	−0.427264	−	0.904127i	$-0.640523\pi$
$719$	−22.9135	−0.854528	−0.427264	+	0.904127i	$0.640523\pi$
$720$	51.5994	1.92300
$721$	−0.321569	−0.0119759
$722$	44.6950	1.66338
$723$	−2.00633	−0.0746161
$724$	−72.5814	−2.69747
$725$	−0.585242	−0.0217354
$726$	−21.1408	−0.784609
$727$	−36.8514	−1.36674	−0.683371	−	0.730071i	$-0.739487\pi$
$727$	−36.8514	−1.36674	−0.683371	+	0.730071i	$0.739487\pi$
$728$	6.40480	0.237378
$729$	9.42192	0.348960
$730$	7.98387	0.295496
$731$	−6.44286	−0.238298
$732$	−21.9064	−0.809684
$733$	18.6553	0.689048	0.344524	−	0.938778i	$-0.388040\pi$
$733$	18.6553	0.689048	0.344524	+	0.938778i	$0.388040\pi$
$734$	35.5083	1.31064
$735$	15.2093	0.561003
$736$	−23.0174	−0.848433
$737$	−13.3599	−0.492119
$738$	−46.2778	−1.70351
$739$	−23.8037	−0.875632	−0.437816	−	0.899065i	$-0.644248\pi$
$739$	−23.8037	−0.875632	−0.437816	+	0.899065i	$0.644248\pi$
$740$	13.9614	0.513231
$741$	3.21813	0.118221
$742$	−10.0052	−0.367303
$743$	23.7484	0.871244	0.435622	−	0.900130i	$-0.356528\pi$
$743$	23.7484	0.871244	0.435622	+	0.900130i	$0.356528\pi$
$744$	−2.93707	−0.107678
$745$	−26.0080	−0.952861
$746$	−3.06627	−0.112264
$747$	−8.08695	−0.295886
$748$	109.909	4.01866
$749$	−0.883141	−0.0322693
$750$	−25.6446	−0.936407
$751$	25.7950	0.941274	0.470637	−	0.882327i	$-0.344024\pi$
$751$	25.7950	0.941274	0.470637	+	0.882327i	$0.344024\pi$
$752$	−19.4706	−0.710019
$753$	0.561225	0.0204522
$754$	−6.40390	−0.233216
$755$	25.5092	0.928375
$756$	−8.09088	−0.294262
$757$	−20.3783	−0.740662	−0.370331	−	0.928900i	$-0.620756\pi$
$757$	−20.3783	−0.740662	−0.370331	+	0.928900i	$0.620756\pi$
$758$	36.8917	1.33997
$759$	8.13628	0.295328
$760$	−25.5114	−0.925396
$761$	−5.83281	−0.211439	−0.105720	−	0.994396i	$-0.533715\pi$
$761$	−5.83281	−0.211439	−0.105720	+	0.994396i	$0.533715\pi$
$762$	−43.1226	−1.56217
$763$	−3.28317	−0.118859
$764$	−109.971	−3.97861
$765$	25.4074	0.918608
$766$	−23.4154	−0.846033
$767$	−20.5579	−0.742301
$768$	−10.5782	−0.381706
$769$	36.2613	1.30762	0.653808	−	0.756660i	$-0.273170\pi$
$769$	36.2613	1.30762	0.653808	+	0.756660i	$0.273170\pi$
$770$	9.45813	0.340847
$771$	−6.06910	−0.218573
$772$	67.1306	2.41608
$773$	40.8478	1.46919	0.734597	−	0.678504i	$-0.237371\pi$
$773$	40.8478	1.46919	0.734597	+	0.678504i	$0.237371\pi$
$774$	−7.11571	−0.255769
$775$	0.240486	0.00863851
$776$	18.8893	0.678086
$777$	−0.387136	−0.0138884
$778$	33.0543	1.18505
$779$	11.7001	0.419200
$780$	26.4293	0.946319
$781$	38.1726	1.36592
$782$	−26.1251	−0.934232
$783$	4.79318	0.171294
$784$	−70.6918	−2.52471
$785$	−15.2345	−0.543742
$786$	38.5932	1.37657
$787$	38.3941	1.36860	0.684301	−	0.729200i	$-0.260108\pi$
$787$	38.3941	1.36860	0.684301	+	0.729200i	$0.260108\pi$
$788$	47.5832	1.69508
$789$	−5.38924	−0.191862
$790$	9.31370	0.331366
$791$	−4.84463	−0.172255
$792$	71.9219	2.55563
$793$	11.6290	0.412960
$794$	76.8272	2.72650
$795$	−24.4622	−0.867585
$796$	−36.0431	−1.27751
$797$	−15.1324	−0.536017	−0.268009	−	0.963417i	$-0.586366\pi$
$797$	−15.1324	−0.536017	−0.268009	+	0.963417i	$0.586366\pi$
$798$	1.19393	0.0422647
$799$	−9.58727	−0.339173
$800$	−6.85485	−0.242356
$801$	−32.7166	−1.15598
$802$	88.0321	3.10852
$803$	5.69058	0.200816
$804$	−13.8503	−0.488462
$805$	−1.59729	−0.0562970
$806$	2.63147	0.0926895
$807$	−3.10253	−0.109214
$808$	66.1619	2.32757
$809$	22.5955	0.794415	0.397207	−	0.917729i	$-0.369979\pi$
$809$	22.5955	0.794415	0.397207	+	0.917729i	$0.369979\pi$
$810$	11.7640	0.413346
$811$	45.2936	1.59047	0.795237	−	0.606299i	$-0.207347\pi$
$811$	45.2936	1.59047	0.795237	+	0.606299i	$0.207347\pi$
$812$	−1.68800	−0.0592371
$813$	−2.69983	−0.0946870
$814$	14.0062	0.490917
$815$	27.9037	0.977423
$816$	48.5906	1.70101
$817$	1.79902	0.0629396
$818$	5.74519	0.200876
$819$	1.78110	0.0622366
$820$	96.0883	3.35555
$821$	31.2674	1.09124	0.545620	−	0.838033i	$-0.316294\pi$
$821$	31.2674	1.09124	0.545620	+	0.838033i	$0.316294\pi$
$822$	34.0100	1.18624
$823$	−24.5296	−0.855050	−0.427525	−	0.904004i	$-0.640614\pi$
$823$	−24.5296	−0.855050	−0.427525	+	0.904004i	$0.640614\pi$
$824$	7.14621	0.248950
$825$	2.42308	0.0843608
$826$	−7.62699	−0.265377
$827$	−36.6656	−1.27499	−0.637493	−	0.770456i	$-0.720029\pi$
$827$	−36.6656	−1.27499	−0.637493	+	0.770456i	$0.720029\pi$
$828$	−20.4998	−0.712417
$829$	38.4943	1.33696	0.668482	−	0.743729i	$-0.266944\pi$
$829$	38.4943	1.33696	0.668482	+	0.743729i	$0.266944\pi$
$830$	23.6336	0.820335
$831$	13.3771	0.464046
$832$	−24.9502	−0.864991
$833$	−34.8085	−1.20604
$834$	2.45792	0.0851108
$835$	3.24344	0.112244
$836$	−30.6894	−1.06142
$837$	−1.96960	−0.0680793
$838$	53.5994	1.85156
$839$	27.3836	0.945388	0.472694	−	0.881227i	$-0.343282\pi$
$839$	27.3836	0.945388	0.472694	+	0.881227i	$0.343282\pi$
$840$	5.80965	0.200452
$841$	1.00000	0.0344828
$842$	18.7472	0.646070
$843$	−18.3388	−0.631621
$844$	−28.4285	−0.978551
$845$	16.6930	0.574258
$846$	−10.5885	−0.364040
$847$	2.95816	0.101644
$848$	113.699	3.90443
$849$	−5.65415	−0.194050
$850$	−7.78036	−0.266864
$851$	−2.36536	−0.0810836
$852$	39.5737	1.35577
$853$	−4.82861	−0.165329	−0.0826643	−	0.996577i	$-0.526343\pi$
$853$	−4.82861	−0.165329	−0.0826643	+	0.996577i	$0.526343\pi$
$854$	4.31439	0.147635
$855$	−7.09442	−0.242624
$856$	19.6260	0.670802
$857$	−10.9199	−0.373018	−0.186509	−	0.982453i	$-0.559717\pi$
$857$	−10.9199	−0.373018	−0.186509	+	0.982453i	$0.559717\pi$
$858$	26.5141	0.905175
$859$	34.0506	1.16179	0.580896	−	0.813978i	$-0.302702\pi$
$859$	34.0506	1.16179	0.580896	+	0.813978i	$0.302702\pi$
$860$	14.7746	0.503809
$861$	−2.66443	−0.0908037
$862$	52.6053	1.79174
$863$	0.410447	0.0139718	0.00698589	−	0.999976i	$-0.497776\pi$
$863$	0.410447	0.0139718	0.00698589	+	0.999976i	$0.497776\pi$
$864$	56.1418	1.90998
$865$	27.1379	0.922715
$866$	−10.8391	−0.368327
$867$	8.02795	0.272643
$868$	0.693627	0.0235432
$869$	6.63843	0.225193
$870$	−5.80883	−0.196938
$871$	7.35244	0.249128
$872$	72.9616	2.47079
$873$	5.25289	0.177783
$874$	7.29482	0.246751
$875$	3.58836	0.121309
$876$	5.89945	0.199324
$877$	36.6213	1.23661	0.618307	−	0.785936i	$-0.287819\pi$
$877$	36.6213	1.23661	0.618307	+	0.785936i	$0.287819\pi$
$878$	78.8862	2.66228
$879$	6.66790	0.224903
$880$	−107.481	−3.62320
$881$	−13.7395	−0.462897	−0.231448	−	0.972847i	$-0.574346\pi$
$881$	−13.7395	−0.462897	−0.231448	+	0.972847i	$0.574346\pi$
$882$	−38.4436	−1.29446
$883$	−14.6612	−0.493389	−0.246694	−	0.969093i	$-0.579344\pi$
$883$	−14.6612	−0.493389	−0.246694	+	0.969093i	$0.579344\pi$
$884$	−60.4868	−2.03439
$885$	−18.6475	−0.626831
$886$	11.9855	0.402661
$887$	−41.9019	−1.40693	−0.703465	−	0.710730i	$-0.748365\pi$
$887$	−41.9019	−1.40693	−0.703465	+	0.710730i	$0.748365\pi$
$888$	8.60329	0.288707
$889$	6.03400	0.202374
$890$	95.6121	3.20493
$891$	8.38494	0.280906
$892$	−6.83798	−0.228952
$893$	2.67702	0.0895830
$894$	−27.0492	−0.904660
$895$	1.46163	0.0488568
$896$	−1.19979	−0.0400820
$897$	−4.47769	−0.149506
$898$	−23.5380	−0.785473
$899$	−0.410917	−0.0137048
$900$	−6.10508	−0.203503
$901$	55.9849	1.86513
$902$	96.3966	3.20965
$903$	−0.409685	−0.0136335
$904$	107.662	3.58078
$905$	34.9496	1.16176
$906$	26.5304	0.881413
$907$	39.4420	1.30965	0.654824	−	0.755781i	$-0.272743\pi$
$907$	39.4420	1.30965	0.654824	+	0.755781i	$0.272743\pi$
$908$	21.9467	0.728326
$909$	18.3988	0.610251
$910$	−5.20515	−0.172549
$911$	−8.13213	−0.269430	−0.134715	−	0.990884i	$-0.543012\pi$
$911$	−8.13213	−0.269430	−0.134715	+	0.990884i	$0.543012\pi$
$912$	−13.5678	−0.449273
$913$	16.8451	0.557491
$914$	−59.1701	−1.95717
$915$	10.5484	0.348721
$916$	59.3774	1.96188
$917$	−5.40022	−0.178331
$918$	63.7218	2.10313
$919$	16.2567	0.536261	0.268130	−	0.963383i	$-0.413594\pi$
$919$	16.2567	0.536261	0.268130	+	0.963383i	$0.413594\pi$
$920$	35.4964	1.17028
$921$	−20.2651	−0.667758
$922$	8.73296	0.287605
$923$	−21.0078	−0.691479
$924$	6.98882	0.229915
$925$	−0.704433	−0.0231616
$926$	22.9652	0.754683
$927$	1.98728	0.0652707
$928$	11.7128	0.384493
$929$	−3.14680	−0.103243	−0.0516215	−	0.998667i	$-0.516439\pi$
$929$	−3.14680	−0.103243	−0.0516215	+	0.998667i	$0.516439\pi$
$930$	2.38694	0.0782710
$931$	9.71943	0.318541
$932$	65.1114	2.13279
$933$	8.40948	0.275314
$934$	46.8006	1.53136
$935$	−52.9236	−1.73079
$936$	−39.5812	−1.29375
$937$	−44.5215	−1.45445	−0.727226	−	0.686398i	$-0.759191\pi$
$937$	−44.5215	−1.45445	−0.727226	+	0.686398i	$0.759191\pi$
$938$	2.72777	0.0890647
$939$	8.89709	0.290346
$940$	21.9853	0.717081
$941$	−5.28426	−0.172262	−0.0861309	−	0.996284i	$-0.527450\pi$
$941$	−5.28426	−0.172262	−0.0861309	+	0.996284i	$0.527450\pi$
$942$	−15.8443	−0.516237
$943$	−16.2794	−0.530132
$944$	86.6725	2.82095
$945$	3.89594	0.126735
$946$	14.8220	0.481905
$947$	−4.63559	−0.150636	−0.0753182	−	0.997160i	$-0.523997\pi$
$947$	−4.63559	−0.150636	−0.0753182	+	0.997160i	$0.523997\pi$
$948$	6.88209	0.223520
$949$	−3.13173	−0.101660
$950$	2.17248	0.0704846
$951$	−12.0129	−0.389545
$952$	−13.2961	−0.430930
$953$	−14.9103	−0.482992	−0.241496	−	0.970402i	$-0.577638\pi$
$953$	−14.9103	−0.482992	−0.241496	+	0.970402i	$0.577638\pi$
$954$	61.8316	2.00187
$955$	52.9535	1.71354
$956$	−76.8434	−2.48529
$957$	−4.14030	−0.133837
$958$	109.040	3.52292
$959$	−4.75891	−0.153673
$960$	−22.6317	−0.730435
$961$	−30.8311	−0.994553
$962$	−7.70811	−0.248520
$963$	5.45775	0.175874
$964$	−10.5297	−0.339137
$965$	−32.3249	−1.04058
$966$	−1.66123	−0.0534492
$967$	40.2853	1.29549	0.647744	−	0.761858i	$-0.275713\pi$
$967$	40.2853	1.29549	0.647744	+	0.761858i	$0.275713\pi$
$968$	−65.7390	−2.11293
$969$	−6.68073	−0.214616
$970$	−15.3512	−0.492899
$971$	12.9114	0.414346	0.207173	−	0.978304i	$-0.433574\pi$
$971$	12.9114	0.414346	0.207173	+	0.978304i	$0.433574\pi$
$972$	79.2674	2.54250
$973$	−0.343928	−0.0110258
$974$	42.4305	1.35956
$975$	−1.33351	−0.0427065
$976$	−49.0284	−1.56936
$977$	−1.55018	−0.0495947	−0.0247974	−	0.999692i	$-0.507894\pi$
$977$	−1.55018	−0.0495947	−0.0247974	+	0.999692i	$0.507894\pi$
$978$	29.0207	0.927979
$979$	68.1485	2.17804
$980$	79.8218	2.54981
$981$	20.2897	0.647802
$982$	27.6515	0.882393
$983$	52.7948	1.68389	0.841946	−	0.539561i	$-0.181410\pi$
$983$	52.7948	1.68389	0.841946	+	0.539561i	$0.181410\pi$
$984$	59.2115	1.88759
$985$	−22.9124	−0.730050
$986$	13.2943	0.423375
$987$	−0.609630	−0.0194047
$988$	16.8895	0.537326
$989$	−2.50314	−0.0795952
$990$	−58.4506	−1.85768
$991$	20.7288	0.658472	0.329236	−	0.944248i	$-0.393209\pi$
$991$	20.7288	0.658472	0.329236	+	0.944248i	$0.393209\pi$
$992$	−4.81300	−0.152813
$993$	−22.0425	−0.699498
$994$	−7.79391	−0.247208
$995$	17.3556	0.550209
$996$	17.4634	0.553349
$997$	−12.4077	−0.392955	−0.196477	−	0.980508i	$-0.562950\pi$
$997$	−12.4077	−0.392955	−0.196477	+	0.980508i	$0.562950\pi$
$998$	−36.9290	−1.16897
$999$	5.76936	0.182534

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.62831 q^{2} +0.935172 q^{3} +4.90799 q^{4} -2.36331 q^{5} -2.45792 q^{6} +0.343928 q^{7} -7.64309 q^{8} -2.12545 q^{9} +O(q^{10})\)	\(q-2.62831 q^{2} +0.935172 q^{3} +4.90799 q^{4} -2.36331 q^{5} -2.45792 q^{6} +0.343928 q^{7} -7.64309 q^{8} -2.12545 q^{9} +6.21151 q^{10} +4.42731 q^{11} +4.58982 q^{12} -2.43651 q^{13} -0.903949 q^{14} -2.21010 q^{15} +10.2724 q^{16} +5.05811 q^{17} +5.58634 q^{18} -1.41236 q^{19} -11.5991 q^{20} +0.321632 q^{21} -11.6363 q^{22} +1.96514 q^{23} -7.14761 q^{24} +0.585242 q^{25} +6.40390 q^{26} -4.79318 q^{27} +1.68800 q^{28} -1.00000 q^{29} +5.80883 q^{30} +0.410917 q^{31} -11.7128 q^{32} +4.14030 q^{33} -13.2943 q^{34} -0.812810 q^{35} -10.4317 q^{36} -1.20366 q^{37} +3.71210 q^{38} -2.27856 q^{39} +18.0630 q^{40} -8.28410 q^{41} -0.845348 q^{42} -1.27377 q^{43} +21.7292 q^{44} +5.02311 q^{45} -5.16500 q^{46} -1.89543 q^{47} +9.60647 q^{48} -6.88171 q^{49} -1.53820 q^{50} +4.73020 q^{51} -11.9584 q^{52} +11.0684 q^{53} +12.5979 q^{54} -10.4631 q^{55} -2.62868 q^{56} -1.32080 q^{57} +2.62831 q^{58} +8.43741 q^{59} -10.8472 q^{60} -4.77283 q^{61} -1.08001 q^{62} -0.731004 q^{63} +10.2401 q^{64} +5.75824 q^{65} -10.8820 q^{66} -3.01761 q^{67} +24.8252 q^{68} +1.83775 q^{69} +2.13631 q^{70} +8.62206 q^{71} +16.2450 q^{72} +1.28534 q^{73} +3.16359 q^{74} +0.547302 q^{75} -6.93183 q^{76} +1.52268 q^{77} +5.98875 q^{78} +1.49943 q^{79} -24.2769 q^{80} +1.89391 q^{81} +21.7732 q^{82} +3.80481 q^{83} +1.57857 q^{84} -11.9539 q^{85} +3.34785 q^{86} -0.935172 q^{87} -33.8384 q^{88} +15.3927 q^{89} -13.2023 q^{90} -0.837985 q^{91} +9.64491 q^{92} +0.384278 q^{93} +4.98176 q^{94} +3.33784 q^{95} -10.9535 q^{96} -2.47142 q^{97} +18.0872 q^{98} -9.41005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more