LMFDB - Embedded newform 4031.2.a.c.1.36

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			1.36
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)

$f(q)$	$=$	$q+0.615616 q^{2} -0.601096 q^{3} -1.62102 q^{4} -3.97491 q^{5} -0.370044 q^{6} +2.02870 q^{7} -2.22916 q^{8} -2.63868 q^{9} +O(q^{10})$	\(q+0.615616 q^{2} -0.601096 q^{3} -1.62102 q^{4} -3.97491 q^{5} -0.370044 q^{6} +2.02870 q^{7} -2.22916 q^{8} -2.63868 q^{9} -2.44702 q^{10} -1.11202 q^{11} +0.974386 q^{12} +3.85425 q^{13} +1.24890 q^{14} +2.38930 q^{15} +1.86973 q^{16} +5.07668 q^{17} -1.62442 q^{18} -3.93973 q^{19} +6.44340 q^{20} -1.21944 q^{21} -0.684580 q^{22} +1.65525 q^{23} +1.33994 q^{24} +10.7999 q^{25} +2.37274 q^{26} +3.38939 q^{27} -3.28855 q^{28} -1.00000 q^{29} +1.47089 q^{30} -1.06636 q^{31} +5.60935 q^{32} +0.668432 q^{33} +3.12529 q^{34} -8.06390 q^{35} +4.27735 q^{36} -1.15005 q^{37} -2.42536 q^{38} -2.31677 q^{39} +8.86071 q^{40} -2.93208 q^{41} -0.750708 q^{42} +4.25294 q^{43} +1.80261 q^{44} +10.4885 q^{45} +1.01900 q^{46} +2.70624 q^{47} -1.12388 q^{48} -2.88438 q^{49} +6.64862 q^{50} -3.05157 q^{51} -6.24779 q^{52} +9.71282 q^{53} +2.08656 q^{54} +4.42020 q^{55} -4.52229 q^{56} +2.36816 q^{57} -0.615616 q^{58} +7.35669 q^{59} -3.87310 q^{60} -12.1435 q^{61} -0.656470 q^{62} -5.35309 q^{63} -0.286248 q^{64} -15.3203 q^{65} +0.411498 q^{66} -7.65775 q^{67} -8.22938 q^{68} -0.994962 q^{69} -4.96427 q^{70} -3.84285 q^{71} +5.88204 q^{72} +7.70041 q^{73} -0.707989 q^{74} -6.49180 q^{75} +6.38637 q^{76} -2.25596 q^{77} -1.42624 q^{78} +2.65058 q^{79} -7.43201 q^{80} +5.87871 q^{81} -1.80503 q^{82} +12.8977 q^{83} +1.97674 q^{84} -20.1794 q^{85} +2.61818 q^{86} +0.601096 q^{87} +2.47887 q^{88} -1.51478 q^{89} +6.45692 q^{90} +7.81910 q^{91} -2.68318 q^{92} +0.640986 q^{93} +1.66601 q^{94} +15.6601 q^{95} -3.37175 q^{96} -13.5584 q^{97} -1.77567 q^{98} +2.93428 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.615616	0.435307	0.217653	−	0.976026i	$-0.430160\pi$
$2$	0.615616	0.435307	0.217653	+	0.976026i	$0.430160\pi$
$3$	−0.601096	−0.347043	−0.173521	−	0.984830i	$-0.555515\pi$
$3$	−0.601096	−0.347043	−0.173521	+	0.984830i	$0.555515\pi$
$4$	−1.62102	−0.810508
$5$	−3.97491	−1.77764	−0.888818	−	0.458261i	$-0.848473\pi$
$5$	−3.97491	−1.77764	−0.888818	+	0.458261i	$0.848473\pi$
$6$	−0.370044	−0.151070
$7$	2.02870	0.766776	0.383388	−	0.923587i	$-0.374757\pi$
$7$	2.02870	0.766776	0.383388	+	0.923587i	$0.374757\pi$
$8$	−2.22916	−0.788126
$9$	−2.63868	−0.879561
$10$	−2.44702	−0.773816
$11$	−1.11202	−0.335288	−0.167644	−	0.985848i	$-0.553616\pi$
$11$	−1.11202	−0.335288	−0.167644	+	0.985848i	$0.553616\pi$
$12$	0.974386	0.281281
$13$	3.85425	1.06898	0.534488	−	0.845176i	$-0.320505\pi$
$13$	3.85425	1.06898	0.534488	+	0.845176i	$0.320505\pi$
$14$	1.24890	0.333783
$15$	2.38930	0.616915
$16$	1.86973	0.467432
$17$	5.07668	1.23128	0.615638	−	0.788029i	$-0.288899\pi$
$17$	5.07668	1.23128	0.615638	+	0.788029i	$0.288899\pi$
$18$	−1.62442	−0.382879
$19$	−3.93973	−0.903836	−0.451918	−	0.892059i	$-0.649260\pi$
$19$	−3.93973	−0.903836	−0.451918	+	0.892059i	$0.649260\pi$
$20$	6.44340	1.44079
$21$	−1.21944	−0.266104
$22$	−0.684580	−0.145953
$23$	1.65525	0.345143	0.172571	−	0.984997i	$-0.444792\pi$
$23$	1.65525	0.345143	0.172571	+	0.984997i	$0.444792\pi$
$24$	1.33994	0.273513
$25$	10.7999	2.15999
$26$	2.37274	0.465332
$27$	3.38939	0.652288
$28$	−3.28855	−0.621478
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	1.47089	0.268547
$31$	−1.06636	−0.191524	−0.0957622	−	0.995404i	$-0.530529\pi$
$31$	−1.06636	−0.191524	−0.0957622	+	0.995404i	$0.530529\pi$
$32$	5.60935	0.991602
$33$	0.668432	0.116359
$34$	3.12529	0.535982
$35$	−8.06390	−1.36305
$36$	4.27735	0.712892
$37$	−1.15005	−0.189067	−0.0945334	−	0.995522i	$-0.530136\pi$
$37$	−1.15005	−0.189067	−0.0945334	+	0.995522i	$0.530136\pi$
$38$	−2.42536	−0.393446
$39$	−2.31677	−0.370980
$40$	8.86071	1.40100
$41$	−2.93208	−0.457914	−0.228957	−	0.973437i	$-0.573531\pi$
$41$	−2.93208	−0.457914	−0.228957	+	0.973437i	$0.573531\pi$
$42$	−0.750708	−0.115837
$43$	4.25294	0.648567	0.324284	−	0.945960i	$-0.394877\pi$
$43$	4.25294	0.648567	0.324284	+	0.945960i	$0.394877\pi$
$44$	1.80261	0.271753
$45$	10.4885	1.56354
$46$	1.01900	0.150243
$47$	2.70624	0.394746	0.197373	−	0.980328i	$-0.436759\pi$
$47$	2.70624	0.394746	0.197373	+	0.980328i	$0.436759\pi$
$48$	−1.12388	−0.162219
$49$	−2.88438	−0.412055
$50$	6.64862	0.940257
$51$	−3.05157	−0.427305
$52$	−6.24779	−0.866413
$53$	9.71282	1.33416	0.667079	−	0.744987i	$-0.267544\pi$
$53$	9.71282	1.33416	0.667079	+	0.744987i	$0.267544\pi$
$54$	2.08656	0.283945
$55$	4.42020	0.596019
$56$	−4.52229	−0.604316
$57$	2.36816	0.313670
$58$	−0.615616	−0.0808344
$59$	7.35669	0.957760	0.478880	−	0.877880i	$-0.341043\pi$
$59$	7.35669	0.957760	0.478880	+	0.877880i	$0.341043\pi$
$60$	−3.87310	−0.500015
$61$	−12.1435	−1.55481	−0.777406	−	0.628999i	$-0.783465\pi$
$61$	−12.1435	−1.55481	−0.777406	+	0.628999i	$0.783465\pi$
$62$	−0.656470	−0.0833718
$63$	−5.35309	−0.674427
$64$	−0.286248	−0.0357810
$65$	−15.3203	−1.90025
$66$	0.411498	0.0506519
$67$	−7.65775	−0.935543	−0.467772	−	0.883849i	$-0.654943\pi$
$67$	−7.65775	−0.935543	−0.467772	+	0.883849i	$0.654943\pi$
$68$	−8.22938	−0.997959
$69$	−0.994962	−0.119779
$70$	−4.96427	−0.593344
$71$	−3.84285	−0.456063	−0.228031	−	0.973654i	$-0.573229\pi$
$71$	−3.84285	−0.456063	−0.228031	+	0.973654i	$0.573229\pi$
$72$	5.88204	0.693205
$73$	7.70041	0.901265	0.450633	−	0.892710i	$-0.351198\pi$
$73$	7.70041	0.901265	0.450633	+	0.892710i	$0.351198\pi$
$74$	−0.707989	−0.0823020
$75$	−6.49180	−0.749608
$76$	6.38637	0.732567
$77$	−2.25596	−0.257090
$78$	−1.42624	−0.161490
$79$	2.65058	0.298214	0.149107	−	0.988821i	$-0.452360\pi$
$79$	2.65058	0.298214	0.149107	+	0.988821i	$0.452360\pi$
$80$	−7.43201	−0.830924
$81$	5.87871	0.653190
$82$	−1.80503	−0.199333
$83$	12.8977	1.41571	0.707855	−	0.706358i	$-0.249663\pi$
$83$	12.8977	1.41571	0.707855	+	0.706358i	$0.249663\pi$
$84$	1.97674	0.215679
$85$	−20.1794	−2.18876
$86$	2.61818	0.282326
$87$	0.601096	0.0644442
$88$	2.47887	0.264249
$89$	−1.51478	−0.160566	−0.0802832	−	0.996772i	$-0.525582\pi$
$89$	−1.51478	−0.160566	−0.0802832	+	0.996772i	$0.525582\pi$
$90$	6.45692	0.680619
$91$	7.81910	0.819665
$92$	−2.68318	−0.279741
$93$	0.640986	0.0664671
$94$	1.66601	0.171835
$95$	15.6601	1.60669
$96$	−3.37175	−0.344128
$97$	−13.5584	−1.37665	−0.688325	−	0.725402i	$-0.741654\pi$
$97$	−13.5584	−1.37665	−0.688325	+	0.725402i	$0.741654\pi$
$98$	−1.77567	−0.179370
$99$	2.93428	0.294906
$100$	−17.5069	−1.75069
$101$	8.99135	0.894673	0.447336	−	0.894366i	$-0.352373\pi$
$101$	8.99135	0.894673	0.447336	+	0.894366i	$0.352373\pi$
$102$	−1.87860	−0.186009
$103$	−18.1518	−1.78855	−0.894276	−	0.447517i	$-0.852309\pi$
$103$	−18.1518	−1.78855	−0.894276	+	0.447517i	$0.852309\pi$
$104$	−8.59172	−0.842487
$105$	4.84718	0.473036
$106$	5.97937	0.580768
$107$	−5.50313	−0.532008	−0.266004	−	0.963972i	$-0.585703\pi$
$107$	−5.50313	−0.532008	−0.266004	+	0.963972i	$0.585703\pi$
$108$	−5.49425	−0.528685
$109$	−7.87069	−0.753875	−0.376938	−	0.926239i	$-0.623023\pi$
$109$	−7.87069	−0.753875	−0.376938	+	0.926239i	$0.623023\pi$
$110$	2.72115	0.259451
$111$	0.691289	0.0656143
$112$	3.79311	0.358416
$113$	−7.01168	−0.659603	−0.329802	−	0.944050i	$-0.606982\pi$
$113$	−7.01168	−0.659603	−0.329802	+	0.944050i	$0.606982\pi$
$114$	1.45788	0.136543
$115$	−6.57947	−0.613539
$116$	1.62102	0.150508
$117$	−10.1701	−0.940229
$118$	4.52890	0.416919
$119$	10.2990	0.944112
$120$	−5.32613	−0.486207
$121$	−9.76340	−0.887582
$122$	−7.47572	−0.676820
$123$	1.76246	0.158916
$124$	1.72859	0.155232
$125$	−23.0543	−2.06204
$126$	−3.29545	−0.293582
$127$	−14.5986	−1.29542	−0.647710	−	0.761887i	$-0.724273\pi$
$127$	−14.5986	−1.29542	−0.647710	+	0.761887i	$0.724273\pi$
$128$	−11.3949	−1.00718
$129$	−2.55642	−0.225081
$130$	−9.43142	−0.827191
$131$	−0.715609	−0.0625231	−0.0312615	−	0.999511i	$-0.509952\pi$
$131$	−0.715609	−0.0625231	−0.0312615	+	0.999511i	$0.509952\pi$
$132$	−1.08354	−0.0943100
$133$	−7.99253	−0.693040
$134$	−4.71424	−0.407248
$135$	−13.4725	−1.15953
$136$	−11.3167	−0.970400
$137$	−16.0288	−1.36944	−0.684718	−	0.728808i	$-0.740075\pi$
$137$	−16.0288	−1.36944	−0.684718	+	0.728808i	$0.740075\pi$
$138$	−0.612515	−0.0521407
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	13.0717	1.10476
$141$	−1.62671	−0.136994
$142$	−2.36572	−0.198527
$143$	−4.28601	−0.358414
$144$	−4.93362	−0.411135
$145$	3.97491	0.330099
$146$	4.74050	0.392327
$147$	1.73379	0.143001
$148$	1.86425	0.153240
$149$	6.35112	0.520304	0.260152	−	0.965568i	$-0.416227\pi$
$149$	6.35112	0.520304	0.260152	+	0.965568i	$0.416227\pi$
$150$	−3.99646	−0.326309
$151$	18.4580	1.50209	0.751044	−	0.660253i	$-0.229551\pi$
$151$	18.4580	1.50209	0.751044	+	0.660253i	$0.229551\pi$
$152$	8.78228	0.712337
$153$	−13.3958	−1.08298
$154$	−1.38881	−0.111913
$155$	4.23870	0.340461
$156$	3.75552	0.300682
$157$	−3.16247	−0.252392	−0.126196	−	0.992005i	$-0.540277\pi$
$157$	−3.16247	−0.252392	−0.126196	+	0.992005i	$0.540277\pi$
$158$	1.63174	0.129814
$159$	−5.83834	−0.463010
$160$	−22.2967	−1.76271
$161$	3.35800	0.264647
$162$	3.61903	0.284338
$163$	19.0835	1.49474	0.747369	−	0.664409i	$-0.231317\pi$
$163$	19.0835	1.49474	0.747369	+	0.664409i	$0.231317\pi$
$164$	4.75295	0.371143
$165$	−2.65696	−0.206844
$166$	7.94005	0.616268
$167$	−11.0288	−0.853434	−0.426717	−	0.904385i	$-0.640330\pi$
$167$	−11.0288	−0.853434	−0.426717	+	0.904385i	$0.640330\pi$
$168$	2.71833	0.209723
$169$	1.85521	0.142708
$170$	−12.4227	−0.952781
$171$	10.3957	0.794980
$172$	−6.89409	−0.525669
$173$	−5.76735	−0.438483	−0.219242	−	0.975671i	$-0.570358\pi$
$173$	−5.76735	−0.438483	−0.219242	+	0.975671i	$0.570358\pi$
$174$	0.370044	0.0280530
$175$	21.9098	1.65623
$176$	−2.07918	−0.156724
$177$	−4.42207	−0.332383
$178$	−0.932524	−0.0698956
$179$	−21.4899	−1.60623	−0.803117	−	0.595822i	$-0.796826\pi$
$179$	−21.4899	−1.60623	−0.803117	+	0.595822i	$0.796826\pi$
$180$	−17.0021	−1.26726
$181$	11.9839	0.890756	0.445378	−	0.895343i	$-0.353069\pi$
$181$	11.9839	0.890756	0.445378	+	0.895343i	$0.353069\pi$
$182$	4.81357	0.356805
$183$	7.29939	0.539586
$184$	−3.68981	−0.272016
$185$	4.57135	0.336092
$186$	0.394601	0.0289336
$187$	−5.64538	−0.412831
$188$	−4.38686	−0.319945
$189$	6.87605	0.500159
$190$	9.64061	0.699403
$191$	17.1105	1.23807	0.619036	−	0.785362i	$-0.287523\pi$
$191$	17.1105	1.23807	0.619036	+	0.785362i	$0.287523\pi$
$192$	0.172062	0.0124175
$193$	−20.0620	−1.44409	−0.722046	−	0.691845i	$-0.756798\pi$
$193$	−20.0620	−1.44409	−0.722046	+	0.691845i	$0.756798\pi$
$194$	−8.34679	−0.599265
$195$	9.20896	0.659467
$196$	4.67563	0.333974
$197$	21.6008	1.53900	0.769498	−	0.638650i	$-0.220507\pi$
$197$	21.6008	1.53900	0.769498	+	0.638650i	$0.220507\pi$
$198$	1.80639	0.128374
$199$	−14.8104	−1.04988	−0.524940	−	0.851139i	$-0.675912\pi$
$199$	−14.8104	−1.04988	−0.524940	+	0.851139i	$0.675912\pi$
$200$	−24.0748	−1.70234
$201$	4.60304	0.324673
$202$	5.53522	0.389457
$203$	−2.02870	−0.142387
$204$	4.94664	0.346334
$205$	11.6548	0.814003
$206$	−11.1746	−0.778568
$207$	−4.36768	−0.303574
$208$	7.20639	0.499673
$209$	4.38107	0.303045
$210$	2.98400	0.205916
$211$	7.22427	0.497339	0.248670	−	0.968588i	$-0.420007\pi$
$211$	7.22427	0.497339	0.248670	+	0.968588i	$0.420007\pi$
$212$	−15.7446	−1.08135
$213$	2.30992	0.158273
$214$	−3.38782	−0.231587
$215$	−16.9051	−1.15292
$216$	−7.55548	−0.514085
$217$	−2.16333	−0.146856
$218$	−4.84532	−0.328167
$219$	−4.62868	−0.312777
$220$	−7.16521	−0.483078
$221$	19.5668	1.31620
$222$	0.425569	0.0285623
$223$	−21.2526	−1.42318	−0.711591	−	0.702594i	$-0.752025\pi$
$223$	−21.2526	−1.42318	−0.711591	+	0.702594i	$0.752025\pi$
$224$	11.3797	0.760337
$225$	−28.4976	−1.89984
$226$	−4.31651	−0.287130
$227$	10.6364	0.705961	0.352980	−	0.935631i	$-0.385168\pi$
$227$	10.6364	0.705961	0.352980	+	0.935631i	$0.385168\pi$
$228$	−3.83882	−0.254232
$229$	−5.26420	−0.347868	−0.173934	−	0.984757i	$-0.555648\pi$
$229$	−5.26420	−0.347868	−0.173934	+	0.984757i	$0.555648\pi$
$230$	−4.05043	−0.267077
$231$	1.35605	0.0892214
$232$	2.22916	0.146351
$233$	14.1390	0.926278	0.463139	−	0.886286i	$-0.346723\pi$
$233$	14.1390	0.926278	0.463139	+	0.886286i	$0.346723\pi$
$234$	−6.26090	−0.409288
$235$	−10.7571	−0.701714
$236$	−11.9253	−0.776272
$237$	−1.59325	−0.103493
$238$	6.34026	0.410978
$239$	−19.8238	−1.28230	−0.641149	−	0.767417i	$-0.721542\pi$
$239$	−19.8238	−1.28230	−0.641149	+	0.767417i	$0.721542\pi$
$240$	4.46735	0.288366
$241$	27.6751	1.78271	0.891356	−	0.453304i	$-0.149755\pi$
$241$	27.6751	1.78271	0.891356	+	0.453304i	$0.149755\pi$
$242$	−6.01051	−0.386370
$243$	−13.7018	−0.878973
$244$	19.6848	1.26019
$245$	11.4652	0.732483
$246$	1.08500	0.0691770
$247$	−15.1847	−0.966179
$248$	2.37709	0.150945
$249$	−7.75277	−0.491312
$250$	−14.1926	−0.897618
$251$	14.0246	0.885222	0.442611	−	0.896714i	$-0.354052\pi$
$251$	14.0246	0.885222	0.442611	+	0.896714i	$0.354052\pi$
$252$	8.67745	0.546628
$253$	−1.84067	−0.115722
$254$	−8.98716	−0.563905
$255$	12.1297	0.759593
$256$	−6.44240	−0.402650
$257$	−23.3636	−1.45738	−0.728692	−	0.684842i	$-0.759871\pi$
$257$	−23.3636	−1.45738	−0.728692	+	0.684842i	$0.759871\pi$
$258$	−1.57378	−0.0979790
$259$	−2.33310	−0.144972
$260$	24.8344	1.54017
$261$	2.63868	0.163330
$262$	−0.440541	−0.0272167
$263$	2.28010	0.140597	0.0702984	−	0.997526i	$-0.477605\pi$
$263$	2.28010	0.140597	0.0702984	+	0.997526i	$0.477605\pi$
$264$	−1.49004	−0.0917056
$265$	−38.6076	−2.37165
$266$	−4.92033	−0.301685
$267$	0.910528	0.0557234
$268$	12.4133	0.758265
$269$	14.6816	0.895153	0.447577	−	0.894246i	$-0.352287\pi$
$269$	14.6816	0.895153	0.447577	+	0.894246i	$0.352287\pi$
$270$	−8.29391	−0.504751
$271$	−21.5027	−1.30620	−0.653099	−	0.757273i	$-0.726531\pi$
$271$	−21.5027	−1.30620	−0.653099	+	0.757273i	$0.726531\pi$
$272$	9.49200	0.575537
$273$	−4.70003	−0.284459
$274$	−9.86762	−0.596125
$275$	−12.0098	−0.724217
$276$	1.61285	0.0970822
$277$	12.4775	0.749698	0.374849	−	0.927086i	$-0.377695\pi$
$277$	12.4775	0.749698	0.374849	+	0.927086i	$0.377695\pi$
$278$	−0.615616	−0.0369222
$279$	2.81379	0.168457
$280$	17.9757	1.07425
$281$	13.1413	0.783945	0.391973	−	0.919977i	$-0.371793\pi$
$281$	13.1413	0.783945	0.391973	+	0.919977i	$0.371793\pi$
$282$	−1.00143	−0.0596342
$283$	−5.10844	−0.303665	−0.151833	−	0.988406i	$-0.548517\pi$
$283$	−5.10844	−0.303665	−0.151833	+	0.988406i	$0.548517\pi$
$284$	6.22933	0.369642
$285$	−9.41321	−0.557591
$286$	−2.63854	−0.156020
$287$	−5.94830	−0.351117
$288$	−14.8013	−0.872175
$289$	8.77266	0.516039
$290$	2.44702	0.143694
$291$	8.14991	0.477756
$292$	−12.4825	−0.730483
$293$	16.6263	0.971318	0.485659	−	0.874148i	$-0.338580\pi$
$293$	16.6263	0.971318	0.485659	+	0.874148i	$0.338580\pi$
$294$	1.06735	0.0622491
$295$	−29.2422	−1.70255
$296$	2.56364	0.149009
$297$	−3.76908	−0.218704
$298$	3.90985	0.226492
$299$	6.37973	0.368949
$300$	10.5233	0.607564
$301$	8.62793	0.497306
$302$	11.3630	0.653868
$303$	−5.40466	−0.310490
$304$	−7.36622	−0.422482
$305$	48.2693	2.76389
$306$	−8.24664	−0.471429
$307$	12.6283	0.720733	0.360366	−	0.932811i	$-0.382652\pi$
$307$	12.6283	0.720733	0.360366	+	0.932811i	$0.382652\pi$
$308$	3.65695	0.208374
$309$	10.9110	0.620704
$310$	2.60941	0.148205
$311$	14.6237	0.829235	0.414618	−	0.909996i	$-0.363915\pi$
$311$	14.6237	0.829235	0.414618	+	0.909996i	$0.363915\pi$
$312$	5.16444	0.292379
$313$	−23.5069	−1.32869	−0.664346	−	0.747426i	$-0.731290\pi$
$313$	−23.5069	−1.32869	−0.664346	+	0.747426i	$0.731290\pi$
$314$	−1.94687	−0.109868
$315$	21.2781	1.19888
$316$	−4.29664	−0.241705
$317$	30.3158	1.70271	0.851353	−	0.524594i	$-0.175783\pi$
$317$	30.3158	1.70271	0.851353	+	0.524594i	$0.175783\pi$
$318$	−3.59417	−0.201551
$319$	1.11202	0.0622613
$320$	1.13781	0.0636055
$321$	3.30791	0.184629
$322$	2.06724	0.115203
$323$	−20.0007	−1.11287
$324$	−9.52948	−0.529416
$325$	41.6256	2.30897
$326$	11.7481	0.650670
$327$	4.73104	0.261627
$328$	6.53606	0.360894
$329$	5.49015	0.302682
$330$	−1.63567	−0.0900406
$331$	−12.0201	−0.660683	−0.330342	−	0.943861i	$-0.607164\pi$
$331$	−12.0201	−0.660683	−0.330342	+	0.943861i	$0.607164\pi$
$332$	−20.9074	−1.14744
$333$	3.03462	0.166296
$334$	−6.78951	−0.371505
$335$	30.4389	1.66305
$336$	−2.28002	−0.124385
$337$	−4.57770	−0.249363	−0.124682	−	0.992197i	$-0.539791\pi$
$337$	−4.57770	−0.249363	−0.124682	+	0.992197i	$0.539791\pi$
$338$	1.14209	0.0621218
$339$	4.21469	0.228911
$340$	32.7111	1.77401
$341$	1.18582	0.0642157
$342$	6.39977	0.346060
$343$	−20.0524	−1.08273
$344$	−9.48047	−0.511153
$345$	3.95489	0.212924
$346$	−3.55047	−0.190875
$347$	−13.5257	−0.726100	−0.363050	−	0.931770i	$-0.618265\pi$
$347$	−13.5257	−0.726100	−0.363050	+	0.931770i	$0.618265\pi$
$348$	−0.974386	−0.0522326
$349$	−36.2039	−1.93795	−0.968975	−	0.247160i	$-0.920503\pi$
$349$	−36.2039	−1.93795	−0.968975	+	0.247160i	$0.920503\pi$
$350$	13.4880	0.720967
$351$	13.0635	0.697280
$352$	−6.23773	−0.332472
$353$	−26.9850	−1.43627	−0.718135	−	0.695904i	$-0.755004\pi$
$353$	−26.9850	−1.43627	−0.718135	+	0.695904i	$0.755004\pi$
$354$	−2.72230	−0.144689
$355$	15.2750	0.810713
$356$	2.45548	0.130140
$357$	−6.19071	−0.327647
$358$	−13.2296	−0.699204
$359$	−1.11856	−0.0590353	−0.0295176	−	0.999564i	$-0.509397\pi$
$359$	−1.11856	−0.0590353	−0.0295176	+	0.999564i	$0.509397\pi$
$360$	−23.3806	−1.23227
$361$	−3.47852	−0.183080
$362$	7.37748	0.387752
$363$	5.86874	0.308029
$364$	−12.6749	−0.664345
$365$	−30.6085	−1.60212
$366$	4.49362	0.234885
$367$	8.60130	0.448984	0.224492	−	0.974476i	$-0.427928\pi$
$367$	8.60130	0.448984	0.224492	+	0.974476i	$0.427928\pi$
$368$	3.09486	0.161331
$369$	7.73683	0.402763
$370$	2.81420	0.146303
$371$	19.7044	1.02300
$372$	−1.03905	−0.0538722
$373$	−15.1345	−0.783634	−0.391817	−	0.920043i	$-0.628153\pi$
$373$	−15.1345	−0.783634	−0.391817	+	0.920043i	$0.628153\pi$
$374$	−3.47539	−0.179708
$375$	13.8578	0.715615
$376$	−6.03263	−0.311109
$377$	−3.85425	−0.198504
$378$	4.23301	0.217722
$379$	−37.1079	−1.90611	−0.953053	−	0.302804i	$-0.902077\pi$
$379$	−37.1079	−1.90611	−0.953053	+	0.302804i	$0.902077\pi$
$380$	−25.3853	−1.30224
$381$	8.77518	0.449566
$382$	10.5335	0.538941
$383$	−28.1644	−1.43913	−0.719566	−	0.694424i	$-0.755659\pi$
$383$	−28.1644	−1.43913	−0.719566	+	0.694424i	$0.755659\pi$
$384$	6.84943	0.349534
$385$	8.96725	0.457013
$386$	−12.3505	−0.628623
$387$	−11.2222	−0.570455
$388$	21.9784	1.11579
$389$	−17.7910	−0.902042	−0.451021	−	0.892513i	$-0.648940\pi$
$389$	−17.7910	−0.902042	−0.451021	+	0.892513i	$0.648940\pi$
$390$	5.66919	0.287070
$391$	8.40316	0.424966
$392$	6.42974	0.324751
$393$	0.430150	0.0216982
$394$	13.2978	0.669935
$395$	−10.5358	−0.530116
$396$	−4.75651	−0.239024
$397$	7.57846	0.380352	0.190176	−	0.981750i	$-0.439094\pi$
$397$	7.57846	0.380352	0.190176	+	0.981750i	$0.439094\pi$
$398$	−9.11751	−0.457019
$399$	4.80427	0.240514
$400$	20.1930	1.00965
$401$	15.8085	0.789437	0.394718	−	0.918802i	$-0.370842\pi$
$401$	15.8085	0.789437	0.394718	+	0.918802i	$0.370842\pi$
$402$	2.83371	0.141332
$403$	−4.11002	−0.204735
$404$	−14.5751	−0.725140
$405$	−23.3674	−1.16113
$406$	−1.24890	−0.0619819
$407$	1.27888	0.0633918
$408$	6.80243	0.336770
$409$	−10.3518	−0.511863	−0.255932	−	0.966695i	$-0.582382\pi$
$409$	−10.3518	−0.511863	−0.255932	+	0.966695i	$0.582382\pi$
$410$	7.17486	0.354341
$411$	9.63487	0.475253
$412$	29.4244	1.44964
$413$	14.9245	0.734387
$414$	−2.68881	−0.132148
$415$	−51.2674	−2.51662
$416$	21.6198	1.06000
$417$	0.601096	0.0294358
$418$	2.69706	0.131918
$419$	27.5104	1.34397	0.671986	−	0.740564i	$-0.265442\pi$
$419$	27.5104	1.34397	0.671986	+	0.740564i	$0.265442\pi$
$420$	−7.85735	−0.383400
$421$	19.2900	0.940139	0.470070	−	0.882629i	$-0.344229\pi$
$421$	19.2900	0.940139	0.470070	+	0.882629i	$0.344229\pi$
$422$	4.44738	0.216495
$423$	−7.14091	−0.347203
$424$	−21.6514	−1.05149
$425$	54.8278	2.65954
$426$	1.42203	0.0688974
$427$	−24.6354	−1.19219
$428$	8.92067	0.431197
$429$	2.57630	0.124385
$430$	−10.4070	−0.501872
$431$	−25.2094	−1.21430	−0.607148	−	0.794589i	$-0.707687\pi$
$431$	−25.2094	−1.21430	−0.607148	+	0.794589i	$0.707687\pi$
$432$	6.33723	0.304900
$433$	14.0514	0.675266	0.337633	−	0.941278i	$-0.390374\pi$
$433$	14.0514	0.675266	0.337633	+	0.941278i	$0.390374\pi$
$434$	−1.33178	−0.0639275
$435$	−2.38930	−0.114558
$436$	12.7585	0.611022
$437$	−6.52123	−0.311953
$438$	−2.84949	−0.136154
$439$	−19.5953	−0.935233	−0.467616	−	0.883932i	$-0.654887\pi$
$439$	−19.5953	−0.935233	−0.467616	+	0.883932i	$0.654887\pi$
$440$	−9.85331	−0.469738
$441$	7.61097	0.362427
$442$	12.0456	0.572952
$443$	27.3631	1.30006	0.650031	−	0.759908i	$-0.274756\pi$
$443$	27.3631	1.30006	0.650031	+	0.759908i	$0.274756\pi$
$444$	−1.12059	−0.0531809
$445$	6.02112	0.285429
$446$	−13.0835	−0.619520
$447$	−3.81763	−0.180568
$448$	−0.580711	−0.0274360
$449$	1.72837	0.0815668	0.0407834	−	0.999168i	$-0.487015\pi$
$449$	1.72837	0.0815668	0.0407834	+	0.999168i	$0.487015\pi$
$450$	−17.5436	−0.827014
$451$	3.26054	0.153533
$452$	11.3660	0.534614
$453$	−11.0950	−0.521288
$454$	6.54793	0.307309
$455$	−31.0803	−1.45706
$456$	−5.27899	−0.247211
$457$	−13.7529	−0.643333	−0.321666	−	0.946853i	$-0.604243\pi$
$457$	−13.7529	−0.643333	−0.321666	+	0.946853i	$0.604243\pi$
$458$	−3.24073	−0.151429
$459$	17.2068	0.803146
$460$	10.6654	0.497278
$461$	0.0328874	0.00153172	0.000765860	−	1.00000i	$-0.499756\pi$
$461$	0.0328874	0.00153172	0.000765860		1.00000i	$0.499756\pi$
$462$	0.834805	0.0388386
$463$	27.5517	1.28044	0.640219	−	0.768192i	$-0.278844\pi$
$463$	27.5517	1.28044	0.640219	+	0.768192i	$0.278844\pi$
$464$	−1.86973	−0.0867999
$465$	−2.54786	−0.118154
$466$	8.70421	0.403215
$467$	37.1987	1.72135	0.860676	−	0.509154i	$-0.170041\pi$
$467$	37.1987	1.72135	0.860676	+	0.509154i	$0.170041\pi$
$468$	16.4860	0.762064
$469$	−15.5353	−0.717352
$470$	−6.62223	−0.305461
$471$	1.90095	0.0875909
$472$	−16.3992	−0.754835
$473$	−4.72937	−0.217457
$474$	−0.980833	−0.0450512
$475$	−42.5489	−1.95228
$476$	−16.6949	−0.765211
$477$	−25.6291	−1.17347
$478$	−12.2039	−0.558192
$479$	−6.60073	−0.301595	−0.150798	−	0.988565i	$-0.548184\pi$
$479$	−6.60073	−0.301595	−0.150798	+	0.988565i	$0.548184\pi$
$480$	13.4024	0.611735
$481$	−4.43257	−0.202108
$482$	17.0373	0.776026
$483$	−2.01848	−0.0918439
$484$	15.8266	0.719393
$485$	53.8936	2.44718
$486$	−8.43507	−0.382623
$487$	9.97276	0.451909	0.225954	−	0.974138i	$-0.427450\pi$
$487$	9.97276	0.451909	0.225954	+	0.974138i	$0.427450\pi$
$488$	27.0697	1.22539
$489$	−11.4710	−0.518738
$490$	7.05815	0.318855
$491$	−34.0992	−1.53887	−0.769437	−	0.638722i	$-0.779463\pi$
$491$	−34.0992	−1.53887	−0.769437	+	0.638722i	$0.779463\pi$
$492$	−2.85697	−0.128802
$493$	−5.07668	−0.228642
$494$	−9.34794	−0.420584
$495$	−11.6635	−0.524235
$496$	−1.99381	−0.0895246
$497$	−7.79599	−0.349698
$498$	−4.77273	−0.213871
$499$	−2.80683	−0.125651	−0.0628254	−	0.998025i	$-0.520011\pi$
$499$	−2.80683	−0.125651	−0.0628254	+	0.998025i	$0.520011\pi$
$500$	37.3714	1.67130
$501$	6.62936	0.296178
$502$	8.63374	0.385343
$503$	−30.9422	−1.37964	−0.689821	−	0.723980i	$-0.742311\pi$
$503$	−30.9422	−1.37964	−0.689821	+	0.723980i	$0.742311\pi$
$504$	11.9329	0.531533
$505$	−35.7399	−1.59040
$506$	−1.13315	−0.0503746
$507$	−1.11516	−0.0495258
$508$	23.6646	1.04995
$509$	4.18478	0.185487	0.0927436	−	0.995690i	$-0.470436\pi$
$509$	4.18478	0.185487	0.0927436	+	0.995690i	$0.470436\pi$
$510$	7.46726	0.330656
$511$	15.6218	0.691068
$512$	18.8238	0.831902
$513$	−13.3533	−0.589562
$514$	−14.3830	−0.634408
$515$	72.1519	3.17939
$516$	4.14400	0.182430
$517$	−3.00940	−0.132353
$518$	−1.43630	−0.0631072
$519$	3.46673	0.152172
$520$	34.1513	1.49764
$521$	20.2995	0.889336	0.444668	−	0.895696i	$-0.353322\pi$
$521$	20.2995	0.889336	0.444668	+	0.895696i	$0.353322\pi$
$522$	1.62442	0.0710988
$523$	2.31065	0.101038	0.0505188	−	0.998723i	$-0.483913\pi$
$523$	2.31065	0.101038	0.0505188	+	0.998723i	$0.483913\pi$
$524$	1.16001	0.0506755
$525$	−13.1699	−0.574782
$526$	1.40366	0.0612027
$527$	−5.41358	−0.235819
$528$	1.24979	0.0543900
$529$	−20.2602	−0.880876
$530$	−23.7675	−1.03239
$531$	−19.4120	−0.842408
$532$	12.9560	0.561715
$533$	−11.3009	−0.489498
$534$	0.560536	0.0242568
$535$	21.8745	0.945716
$536$	17.0703	0.737326
$537$	12.9175	0.557432
$538$	9.03824	0.389666
$539$	3.20750	0.138157
$540$	21.8392	0.939809
$541$	−10.2859	−0.442225	−0.221112	−	0.975248i	$-0.570969\pi$
$541$	−10.2859	−0.442225	−0.221112	+	0.975248i	$0.570969\pi$
$542$	−13.2374	−0.568596
$543$	−7.20346	−0.309130
$544$	28.4769	1.22094
$545$	31.2853	1.34012
$546$	−2.89341	−0.123827
$547$	−31.8539	−1.36198	−0.680988	−	0.732295i	$-0.738449\pi$
$547$	−31.8539	−1.36198	−0.680988	+	0.732295i	$0.738449\pi$
$548$	25.9830	1.10994
$549$	32.0428	1.36755
$550$	−7.39342	−0.315257
$551$	3.93973	0.167838
$552$	2.21793	0.0944012
$553$	5.37724	0.228663
$554$	7.68133	0.326348
$555$	−2.74782	−0.116638
$556$	1.62102	0.0687464
$557$	−14.6446	−0.620513	−0.310256	−	0.950653i	$-0.600415\pi$
$557$	−14.6446	−0.620513	−0.310256	+	0.950653i	$0.600415\pi$
$558$	1.73222	0.0733306
$559$	16.3919	0.693302
$560$	−15.0773	−0.637132
$561$	3.39341	0.143270
$562$	8.09001	0.341256
$563$	−42.4379	−1.78855	−0.894273	−	0.447522i	$-0.852307\pi$
$563$	−42.4379	−1.78855	−0.894273	+	0.447522i	$0.852307\pi$
$564$	2.63692	0.111034
$565$	27.8708	1.17253
$566$	−3.14484	−0.132188
$567$	11.9261	0.500850
$568$	8.56632	0.359435
$569$	3.05541	0.128089	0.0640446	−	0.997947i	$-0.479600\pi$
$569$	3.05541	0.128089	0.0640446	+	0.997947i	$0.479600\pi$
$570$	−5.79493	−0.242723
$571$	37.7646	1.58040	0.790199	−	0.612850i	$-0.209977\pi$
$571$	37.7646	1.58040	0.790199	+	0.612850i	$0.209977\pi$
$572$	6.94769	0.290498
$573$	−10.2850	−0.429664
$574$	−3.66187	−0.152844
$575$	17.8766	0.745505
$576$	0.755318	0.0314716
$577$	38.5479	1.60477	0.802384	−	0.596808i	$-0.203565\pi$
$577$	38.5479	1.60477	0.802384	+	0.596808i	$0.203565\pi$
$578$	5.40059	0.224635
$579$	12.0592	0.501162
$580$	−6.44340	−0.267548
$581$	26.1656	1.08553
$582$	5.01722	0.207970
$583$	−10.8009	−0.447327
$584$	−17.1654	−0.710311
$585$	40.4254	1.67139
$586$	10.2354	0.422821
$587$	−31.2128	−1.28829	−0.644146	−	0.764903i	$-0.722787\pi$
$587$	−31.2128	−1.28829	−0.644146	+	0.764903i	$0.722787\pi$
$588$	−2.81050	−0.115903
$589$	4.20118	0.173107
$590$	−18.0020	−0.741130
$591$	−12.9842	−0.534097
$592$	−2.15028	−0.0883759
$593$	37.0761	1.52253	0.761267	−	0.648439i	$-0.224578\pi$
$593$	37.0761	1.52253	0.761267	+	0.648439i	$0.224578\pi$
$594$	−2.32031	−0.0952033
$595$	−40.9378	−1.67829
$596$	−10.2953	−0.421710
$597$	8.90245	0.364353
$598$	3.92747	0.160606
$599$	5.32743	0.217673	0.108836	−	0.994060i	$-0.465288\pi$
$599$	5.32743	0.217673	0.108836	+	0.994060i	$0.465288\pi$
$600$	14.4712	0.590786
$601$	−16.1212	−0.657599	−0.328800	−	0.944400i	$-0.606644\pi$
$601$	−16.1212	−0.657599	−0.328800	+	0.944400i	$0.606644\pi$
$602$	5.31150	0.216480
$603$	20.2064	0.822868
$604$	−29.9207	−1.21745
$605$	38.8087	1.57780
$606$	−3.32720	−0.135158
$607$	−9.46080	−0.384002	−0.192001	−	0.981395i	$-0.561498\pi$
$607$	−9.46080	−0.384002	−0.192001	+	0.981395i	$0.561498\pi$
$608$	−22.0993	−0.896246
$609$	1.21944	0.0494143
$610$	29.7153	1.20314
$611$	10.4305	0.421973
$612$	21.7147	0.877766
$613$	−6.21076	−0.250850	−0.125425	−	0.992103i	$-0.540029\pi$
$613$	−6.21076	−0.250850	−0.125425	+	0.992103i	$0.540029\pi$
$614$	7.77416	0.313740
$615$	−7.00562	−0.282494
$616$	5.02889	0.202620
$617$	−10.8486	−0.436749	−0.218374	−	0.975865i	$-0.570075\pi$
$617$	−10.8486	−0.436749	−0.218374	+	0.975865i	$0.570075\pi$
$618$	6.71697	0.270196
$619$	−10.6805	−0.429286	−0.214643	−	0.976693i	$-0.568859\pi$
$619$	−10.6805	−0.429286	−0.214643	+	0.976693i	$0.568859\pi$
$620$	−6.87100	−0.275946
$621$	5.61028	0.225133
$622$	9.00260	0.360972
$623$	−3.07303	−0.123118
$624$	−4.33173	−0.173408
$625$	37.6391	1.50556
$626$	−14.4713	−0.578388
$627$	−2.63344	−0.105170
$628$	5.12641	0.204566
$629$	−5.83843	−0.232793
$630$	13.0991	0.521882
$631$	−4.98594	−0.198487	−0.0992436	−	0.995063i	$-0.531642\pi$
$631$	−4.98594	−0.198487	−0.0992436	+	0.995063i	$0.531642\pi$
$632$	−5.90857	−0.235030
$633$	−4.34248	−0.172598
$634$	18.6629	0.741199
$635$	58.0284	2.30279
$636$	9.46404	0.375274
$637$	−11.1171	−0.440476
$638$	0.684580	0.0271028
$639$	10.1401	0.401135
$640$	45.2938	1.79040
$641$	0.148411	0.00586187	0.00293094	−	0.999996i	$-0.499067\pi$
$641$	0.148411	0.00586187	0.00293094	+	0.999996i	$0.499067\pi$
$642$	2.03640	0.0803704
$643$	8.24633	0.325203	0.162602	−	0.986692i	$-0.448011\pi$
$643$	8.24633	0.325203	0.162602	+	0.986692i	$0.448011\pi$
$644$	−5.44337	−0.214499
$645$	10.1616	0.400111
$646$	−12.3128	−0.484440
$647$	7.51637	0.295499	0.147750	−	0.989025i	$-0.452797\pi$
$647$	7.51637	0.295499	0.147750	+	0.989025i	$0.452797\pi$
$648$	−13.1046	−0.514796
$649$	−8.18081	−0.321125
$650$	25.6254	1.00511
$651$	1.30037	0.0509654
$652$	−30.9347	−1.21150
$653$	−22.7328	−0.889604	−0.444802	−	0.895629i	$-0.646726\pi$
$653$	−22.7328	−0.889604	−0.444802	+	0.895629i	$0.646726\pi$
$654$	2.91250	0.113888
$655$	2.84449	0.111143
$656$	−5.48219	−0.214043
$657$	−20.3190	−0.792718
$658$	3.37982	0.131759
$659$	2.90476	0.113153	0.0565766	−	0.998398i	$-0.481981\pi$
$659$	2.90476	0.113153	0.0565766	+	0.998398i	$0.481981\pi$
$660$	4.30698	0.167649
$661$	−22.2795	−0.866571	−0.433286	−	0.901257i	$-0.642646\pi$
$661$	−22.2795	−0.866571	−0.433286	+	0.901257i	$0.642646\pi$
$662$	−7.39976	−0.287600
$663$	−11.7615	−0.456779
$664$	−28.7511	−1.11576
$665$	31.7696	1.23197
$666$	1.86816	0.0723897
$667$	−1.65525	−0.0640914
$668$	17.8779	0.691715
$669$	12.7749	0.493905
$670$	18.7387	0.723939
$671$	13.5038	0.521309
$672$	−6.84027	−0.263869
$673$	2.12558	0.0819353	0.0409676	−	0.999160i	$-0.486956\pi$
$673$	2.12558	0.0819353	0.0409676	+	0.999160i	$0.486956\pi$
$674$	−2.81811	−0.108549
$675$	36.6052	1.40893
$676$	−3.00732	−0.115666
$677$	10.6579	0.409617	0.204809	−	0.978802i	$-0.434343\pi$
$677$	10.6579	0.409617	0.204809	+	0.978802i	$0.434343\pi$
$678$	2.59463	0.0996462
$679$	−27.5060	−1.05558
$680$	44.9830	1.72502
$681$	−6.39348	−0.244999
$682$	0.730010	0.0279535
$683$	−20.9959	−0.803386	−0.401693	−	0.915774i	$-0.631578\pi$
$683$	−20.9959	−0.803386	−0.401693	+	0.915774i	$0.631578\pi$
$684$	−16.8516	−0.644337
$685$	63.7133	2.43436
$686$	−12.3446	−0.471319
$687$	3.16429	0.120725
$688$	7.95184	0.303161
$689$	37.4356	1.42618
$690$	2.43469	0.0926872
$691$	35.1483	1.33711	0.668553	−	0.743664i	$-0.266914\pi$
$691$	35.1483	1.33711	0.668553	+	0.743664i	$0.266914\pi$
$692$	9.34897	0.355394
$693$	5.95276	0.226127
$694$	−8.32667	−0.316076
$695$	3.97491	0.150777
$696$	−1.33994	−0.0507902
$697$	−14.8852	−0.563818
$698$	−22.2877	−0.843602
$699$	−8.49890	−0.321458
$700$	−35.5162	−1.34239
$701$	−40.7904	−1.54063	−0.770316	−	0.637663i	$-0.779901\pi$
$701$	−40.7904	−1.54063	−0.770316	+	0.637663i	$0.779901\pi$
$702$	8.04212	0.303530
$703$	4.53088	0.170886
$704$	0.318314	0.0119969
$705$	6.46603	0.243525
$706$	−16.6124	−0.625217
$707$	18.2407	0.686014
$708$	7.16826	0.269400
$709$	44.6748	1.67780	0.838898	−	0.544289i	$-0.183200\pi$
$709$	44.6748	1.67780	0.838898	+	0.544289i	$0.183200\pi$
$710$	9.40354	0.352909
$711$	−6.99405	−0.262297
$712$	3.37668	0.126547
$713$	−1.76509	−0.0661033
$714$	−3.81110	−0.142627
$715$	17.0365	0.637130
$716$	34.8355	1.30187
$717$	11.9160	0.445012
$718$	−0.688603	−0.0256984
$719$	−35.3292	−1.31756	−0.658778	−	0.752338i	$-0.728926\pi$
$719$	−35.3292	−1.31756	−0.658778	+	0.752338i	$0.728926\pi$
$720$	19.6107	0.730848
$721$	−36.8246	−1.37142
$722$	−2.14143	−0.0796958
$723$	−16.6354	−0.618677
$724$	−19.4261	−0.721965
$725$	−10.7999	−0.401100
$726$	3.61289	0.134087
$727$	−19.6178	−0.727585	−0.363793	−	0.931480i	$-0.618518\pi$
$727$	−19.6178	−0.727585	−0.363793	+	0.931480i	$0.618518\pi$
$728$	−17.4300	−0.645999
$729$	−9.40001	−0.348149
$730$	−18.8431	−0.697414
$731$	21.5908	0.798565
$732$	−11.8324	−0.437339
$733$	−29.6250	−1.09422	−0.547112	−	0.837059i	$-0.684273\pi$
$733$	−29.6250	−1.09422	−0.547112	+	0.837059i	$0.684273\pi$
$734$	5.29510	0.195446
$735$	−6.89167	−0.254203
$736$	9.28486	0.342245
$737$	8.51560	0.313676
$738$	4.76292	0.175325
$739$	32.8693	1.20912	0.604558	−	0.796561i	$-0.293350\pi$
$739$	32.8693	1.20912	0.604558	+	0.796561i	$0.293350\pi$
$740$	−7.41023	−0.272405
$741$	9.12745	0.335305
$742$	12.1303	0.445319
$743$	−33.8805	−1.24296	−0.621478	−	0.783432i	$-0.713467\pi$
$743$	−33.8805	−1.24296	−0.621478	+	0.783432i	$0.713467\pi$
$744$	−1.42886	−0.0523845
$745$	−25.2451	−0.924911
$746$	−9.31704	−0.341121
$747$	−34.0330	−1.24520
$748$	9.15126	0.334603
$749$	−11.1642	−0.407931
$750$	8.53110	0.311512
$751$	30.8934	1.12732	0.563658	−	0.826008i	$-0.309394\pi$
$751$	30.8934	1.12732	0.563658	+	0.826008i	$0.309394\pi$
$752$	5.05993	0.184517
$753$	−8.43010	−0.307210
$754$	−2.37274	−0.0864100
$755$	−73.3688	−2.67016
$756$	−11.1462	−0.405383
$757$	−49.7854	−1.80948	−0.904741	−	0.425961i	$-0.859936\pi$
$757$	−49.7854	−1.80948	−0.904741	+	0.425961i	$0.859936\pi$
$758$	−22.8442	−0.829740
$759$	1.10642	0.0401605
$760$	−34.9088	−1.26628
$761$	−15.5545	−0.563849	−0.281924	−	0.959437i	$-0.590973\pi$
$761$	−15.5545	−0.563849	−0.281924	+	0.959437i	$0.590973\pi$
$762$	5.40214	0.195699
$763$	−15.9673	−0.578053
$764$	−27.7364	−1.00347
$765$	53.2470	1.92515
$766$	−17.3384	−0.626464
$767$	28.3545	1.02382
$768$	3.87250	0.139737
$769$	−26.8281	−0.967447	−0.483724	−	0.875221i	$-0.660716\pi$
$769$	−26.8281	−0.967447	−0.483724	+	0.875221i	$0.660716\pi$
$770$	5.52038	0.198941
$771$	14.0438	0.505774
$772$	32.5208	1.17045
$773$	−31.8644	−1.14608	−0.573042	−	0.819526i	$-0.694237\pi$
$773$	−31.8644	−1.14608	−0.573042	+	0.819526i	$0.694237\pi$
$774$	−6.90855	−0.248323
$775$	−11.5167	−0.413690
$776$	30.2239	1.08497
$777$	1.40242	0.0503115
$778$	−10.9525	−0.392665
$779$	11.5516	0.413879
$780$	−14.9279	−0.534504
$781$	4.27334	0.152912
$782$	5.17312	0.184990
$783$	−3.38939	−0.121127
$784$	−5.39301	−0.192607
$785$	12.5705	0.448662
$786$	0.264807	0.00944536
$787$	7.31493	0.260749	0.130374	−	0.991465i	$-0.458382\pi$
$787$	7.31493	0.260749	0.130374	+	0.991465i	$0.458382\pi$
$788$	−35.0153	−1.24737
$789$	−1.37056	−0.0487931
$790$	−6.48604	−0.230763
$791$	−14.2246	−0.505768
$792$	−6.54097	−0.232423
$793$	−46.8039	−1.66206
$794$	4.66542	0.165570
$795$	23.2069	0.823063
$796$	24.0079	0.850936
$797$	−50.0545	−1.77302	−0.886511	−	0.462707i	$-0.846878\pi$
$797$	−50.0545	−1.77302	−0.886511	+	0.462707i	$0.846878\pi$
$798$	2.95759	0.104698
$799$	13.7387	0.486041
$800$	60.5807	2.14185
$801$	3.99703	0.141228
$802$	9.73195	0.343647
$803$	−8.56304	−0.302183
$804$	−7.46160	−0.263150
$805$	−13.3478	−0.470447
$806$	−2.53020	−0.0891224
$807$	−8.82505	−0.310656
$808$	−20.0431	−0.705115
$809$	−6.08205	−0.213833	−0.106917	−	0.994268i	$-0.534098\pi$
$809$	−6.08205	−0.213833	−0.106917	+	0.994268i	$0.534098\pi$
$810$	−14.3853	−0.505449
$811$	15.0138	0.527206	0.263603	−	0.964631i	$-0.415089\pi$
$811$	15.0138	0.527206	0.263603	+	0.964631i	$0.415089\pi$
$812$	3.28855	0.115406
$813$	12.9252	0.453306
$814$	0.787300	0.0275949
$815$	−75.8555	−2.65710
$816$	−5.70560	−0.199736
$817$	−16.7554	−0.586199
$818$	−6.37273	−0.222817
$819$	−20.6321	−0.720945
$820$	−18.8926	−0.659756
$821$	−5.33220	−0.186095	−0.0930475	−	0.995662i	$-0.529661\pi$
$821$	−5.33220	−0.186095	−0.0930475	+	0.995662i	$0.529661\pi$
$822$	5.93138	0.206881
$823$	38.9628	1.35816	0.679080	−	0.734065i	$-0.262379\pi$
$823$	38.9628	1.35816	0.679080	+	0.734065i	$0.262379\pi$
$824$	40.4632	1.40960
$825$	7.21903	0.251334
$826$	9.18777	0.319683
$827$	23.3030	0.810326	0.405163	−	0.914245i	$-0.367215\pi$
$827$	23.3030	0.810326	0.405163	+	0.914245i	$0.367215\pi$
$828$	7.08007	0.246050
$829$	−5.21237	−0.181033	−0.0905165	−	0.995895i	$-0.528852\pi$
$829$	−5.21237	−0.181033	−0.0905165	+	0.995895i	$0.528852\pi$
$830$	−31.5610	−1.09550
$831$	−7.50015	−0.260177
$832$	−1.10327	−0.0382490
$833$	−14.6431	−0.507353
$834$	0.370044	0.0128136
$835$	43.8385	1.51709
$836$	−7.10179	−0.245621
$837$	−3.61432	−0.124929
$838$	16.9359	0.585040
$839$	46.8673	1.61804	0.809019	−	0.587782i	$-0.199999\pi$
$839$	46.8673	1.61804	0.809019	+	0.587782i	$0.199999\pi$
$840$	−10.8051	−0.372812
$841$	1.00000	0.0344828
$842$	11.8753	0.409249
$843$	−7.89919	−0.272062
$844$	−11.7107	−0.403097
$845$	−7.37428	−0.253683
$846$	−4.39606	−0.151140
$847$	−19.8070	−0.680577
$848$	18.1603	0.623628
$849$	3.07066	0.105385
$850$	33.7529	1.15772
$851$	−1.90362	−0.0652551
$852$	−3.74442	−0.128282
$853$	37.2671	1.27600	0.638000	−	0.770036i	$-0.279762\pi$
$853$	37.2671	1.27600	0.638000	+	0.770036i	$0.279762\pi$
$854$	−15.1660	−0.518969
$855$	−41.3220	−1.41318
$856$	12.2673	0.419289
$857$	29.4433	1.00576	0.502882	−	0.864355i	$-0.332273\pi$
$857$	29.4433	1.00576	0.502882	+	0.864355i	$0.332273\pi$
$858$	1.58601	0.0541456
$859$	−26.6031	−0.907687	−0.453844	−	0.891081i	$-0.649948\pi$
$859$	−26.6031	−0.907687	−0.453844	+	0.891081i	$0.649948\pi$
$860$	27.4034	0.934448
$861$	3.57550	0.121853
$862$	−15.5193	−0.528591
$863$	−4.62917	−0.157579	−0.0787893	−	0.996891i	$-0.525105\pi$
$863$	−4.62917	−0.157579	−0.0787893	+	0.996891i	$0.525105\pi$
$864$	19.0123	0.646810
$865$	22.9247	0.779464
$866$	8.65026	0.293948
$867$	−5.27321	−0.179088
$868$	3.50679	0.119028
$869$	−2.94751	−0.0999874
$870$	−1.47089	−0.0498680
$871$	−29.5148	−1.00007
$872$	17.5450	0.594149
$873$	35.7764	1.21085
$874$	−4.01458	−0.135795
$875$	−46.7702	−1.58112
$876$	7.50317	0.253509
$877$	−37.9500	−1.28148	−0.640741	−	0.767757i	$-0.721373\pi$
$877$	−37.9500	−1.28148	−0.640741	+	0.767757i	$0.721373\pi$
$878$	−12.0632	−0.407113
$879$	−9.99398	−0.337089
$880$	8.26456	0.278598
$881$	−36.9384	−1.24449	−0.622243	−	0.782824i	$-0.713779\pi$
$881$	−36.9384	−1.24449	−0.622243	+	0.782824i	$0.713779\pi$
$882$	4.68544	0.157767
$883$	6.62091	0.222811	0.111406	−	0.993775i	$-0.464465\pi$
$883$	6.62091	0.222811	0.111406	+	0.993775i	$0.464465\pi$
$884$	−31.7180	−1.06679
$885$	17.5774	0.590857
$886$	16.8452	0.565925
$887$	−31.5031	−1.05777	−0.528886	−	0.848693i	$-0.677390\pi$
$887$	−31.5031	−1.05777	−0.528886	+	0.848693i	$0.677390\pi$
$888$	−1.54099	−0.0517123
$889$	−29.6162	−0.993297
$890$	3.70670	0.124249
$891$	−6.53726	−0.219006
$892$	34.4509	1.15350
$893$	−10.6619	−0.356786
$894$	−2.35019	−0.0786023
$895$	85.4207	2.85530
$896$	−23.1168	−0.772280
$897$	−3.83483	−0.128041
$898$	1.06401	0.0355066
$899$	1.06636	0.0355652
$900$	46.1951	1.53984
$901$	49.3089	1.64272
$902$	2.00724	0.0668338
$903$	−5.18621	−0.172586
$904$	15.6301	0.519851
$905$	−47.6349	−1.58344
$906$	−6.83026	−0.226920
$907$	−35.3168	−1.17267	−0.586337	−	0.810067i	$-0.699431\pi$
$907$	−35.3168	−1.17267	−0.586337	+	0.810067i	$0.699431\pi$
$908$	−17.2417	−0.572187
$909$	−23.7253	−0.786920
$910$	−19.1335	−0.634270
$911$	−41.4212	−1.37235	−0.686173	−	0.727439i	$-0.740711\pi$
$911$	−41.4212	−1.37235	−0.686173	+	0.727439i	$0.740711\pi$
$912$	4.42780	0.146619
$913$	−14.3426	−0.474670
$914$	−8.46650	−0.280047
$915$	−29.0144	−0.959188
$916$	8.53335	0.281950
$917$	−1.45176	−0.0479412
$918$	10.5928	0.349615
$919$	−26.9177	−0.887934	−0.443967	−	0.896043i	$-0.646429\pi$
$919$	−26.9177	−0.887934	−0.443967	+	0.896043i	$0.646429\pi$
$920$	14.6667	0.483546
$921$	−7.59079	−0.250125
$922$	0.0202460	0.000666767 0
$923$	−14.8113	−0.487520
$924$	−2.19818	−0.0723146
$925$	−12.4205	−0.408382
$926$	16.9613	0.557383
$927$	47.8969	1.57314
$928$	−5.60935	−0.184136
$929$	−12.8312	−0.420977	−0.210488	−	0.977596i	$-0.567505\pi$
$929$	−12.8312	−0.420977	−0.210488	+	0.977596i	$0.567505\pi$
$930$	−1.56851	−0.0514334
$931$	11.3637	0.372430
$932$	−22.9196	−0.750756
$933$	−8.79026	−0.287780
$934$	22.9001	0.749315
$935$	22.4399	0.733864
$936$	22.6708	0.741019
$937$	−15.6452	−0.511106	−0.255553	−	0.966795i	$-0.582258\pi$
$937$	−15.6452	−0.511106	−0.255553	+	0.966795i	$0.582258\pi$
$938$	−9.56377	−0.312268
$939$	14.1299	0.461113
$940$	17.4374	0.568745
$941$	−21.1585	−0.689748	−0.344874	−	0.938649i	$-0.612078\pi$
$941$	−21.1585	−0.689748	−0.344874	+	0.938649i	$0.612078\pi$
$942$	1.17025	0.0381289
$943$	−4.85331	−0.158046
$944$	13.7550	0.447687
$945$	−27.3317	−0.889100
$946$	−2.91148	−0.0946603
$947$	−3.97387	−0.129133	−0.0645666	−	0.997913i	$-0.520567\pi$
$947$	−3.97387	−0.129133	−0.0645666	+	0.997913i	$0.520567\pi$
$948$	2.58269	0.0838819
$949$	29.6793	0.963430
$950$	−26.1938	−0.849839
$951$	−18.2227	−0.590911
$952$	−22.9582	−0.744079
$953$	6.48506	0.210072	0.105036	−	0.994468i	$-0.466504\pi$
$953$	6.48506	0.210072	0.105036	+	0.994468i	$0.466504\pi$
$954$	−15.7777	−0.510821
$955$	−68.0128	−2.20084
$956$	32.1348	1.03931
$957$	−0.668432	−0.0216073
$958$	−4.06352	−0.131286
$959$	−32.5177	−1.05005
$960$	−0.683933	−0.0220738
$961$	−29.8629	−0.963318
$962$	−2.72876	−0.0879789
$963$	14.5210	0.467934
$964$	−44.8619	−1.44490
$965$	79.7446	2.56707
$966$	−1.24261	−0.0399803
$967$	−8.77847	−0.282297	−0.141148	−	0.989988i	$-0.545079\pi$
$967$	−8.77847	−0.282297	−0.141148	+	0.989988i	$0.545079\pi$
$968$	21.7642	0.699527
$969$	12.0224	0.386214
$970$	33.1778	1.06527
$971$	−32.5946	−1.04601	−0.523006	−	0.852329i	$-0.675189\pi$
$971$	−32.5946	−1.04601	−0.523006	+	0.852329i	$0.675189\pi$
$972$	22.2109	0.712415
$973$	−2.02870	−0.0650371
$974$	6.13939	0.196719
$975$	−25.0210	−0.801313
$976$	−22.7050	−0.726769
$977$	−39.0187	−1.24832	−0.624159	−	0.781297i	$-0.714558\pi$
$977$	−39.0187	−1.24832	−0.624159	+	0.781297i	$0.714558\pi$
$978$	−7.06176	−0.225810
$979$	1.68447	0.0538359
$980$	−18.5852	−0.593684
$981$	20.7683	0.663079
$982$	−20.9920	−0.669882
$983$	25.7395	0.820962	0.410481	−	0.911869i	$-0.365361\pi$
$983$	25.7395	0.820962	0.410481	+	0.911869i	$0.365361\pi$
$984$	−3.92880	−0.125245
$985$	−85.8615	−2.73577
$986$	−3.12529	−0.0995294
$987$	−3.30010	−0.105043
$988$	24.6146	0.783096
$989$	7.03967	0.223848
$990$	−7.18024	−0.228203
$991$	−13.8592	−0.440251	−0.220125	−	0.975472i	$-0.570647\pi$
$991$	−13.8592	−0.440251	−0.220125	+	0.975472i	$0.570647\pi$
$992$	−5.98160	−0.189916
$993$	7.22522	0.229285
$994$	−4.79934	−0.152226
$995$	58.8700	1.86630
$996$	12.5674	0.398212
$997$	35.1566	1.11342	0.556710	−	0.830707i	$-0.312063\pi$
$997$	35.1566	1.11342	0.556710	+	0.830707i	$0.312063\pi$
$998$	−1.72793	−0.0546966
$999$	−3.89796	−0.123326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.36	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.36	✓	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)

\(f(q)\)	\(=\)	\(q+0.615616 q^{2} -0.601096 q^{3} -1.62102 q^{4} -3.97491 q^{5} -0.370044 q^{6} +2.02870 q^{7} -2.22916 q^{8} -2.63868 q^{9} +O(q^{10})\)	\(q+0.615616 q^{2} -0.601096 q^{3} -1.62102 q^{4} -3.97491 q^{5} -0.370044 q^{6} +2.02870 q^{7} -2.22916 q^{8} -2.63868 q^{9} -2.44702 q^{10} -1.11202 q^{11} +0.974386 q^{12} +3.85425 q^{13} +1.24890 q^{14} +2.38930 q^{15} +1.86973 q^{16} +5.07668 q^{17} -1.62442 q^{18} -3.93973 q^{19} +6.44340 q^{20} -1.21944 q^{21} -0.684580 q^{22} +1.65525 q^{23} +1.33994 q^{24} +10.7999 q^{25} +2.37274 q^{26} +3.38939 q^{27} -3.28855 q^{28} -1.00000 q^{29} +1.47089 q^{30} -1.06636 q^{31} +5.60935 q^{32} +0.668432 q^{33} +3.12529 q^{34} -8.06390 q^{35} +4.27735 q^{36} -1.15005 q^{37} -2.42536 q^{38} -2.31677 q^{39} +8.86071 q^{40} -2.93208 q^{41} -0.750708 q^{42} +4.25294 q^{43} +1.80261 q^{44} +10.4885 q^{45} +1.01900 q^{46} +2.70624 q^{47} -1.12388 q^{48} -2.88438 q^{49} +6.64862 q^{50} -3.05157 q^{51} -6.24779 q^{52} +9.71282 q^{53} +2.08656 q^{54} +4.42020 q^{55} -4.52229 q^{56} +2.36816 q^{57} -0.615616 q^{58} +7.35669 q^{59} -3.87310 q^{60} -12.1435 q^{61} -0.656470 q^{62} -5.35309 q^{63} -0.286248 q^{64} -15.3203 q^{65} +0.411498 q^{66} -7.65775 q^{67} -8.22938 q^{68} -0.994962 q^{69} -4.96427 q^{70} -3.84285 q^{71} +5.88204 q^{72} +7.70041 q^{73} -0.707989 q^{74} -6.49180 q^{75} +6.38637 q^{76} -2.25596 q^{77} -1.42624 q^{78} +2.65058 q^{79} -7.43201 q^{80} +5.87871 q^{81} -1.80503 q^{82} +12.8977 q^{83} +1.97674 q^{84} -20.1794 q^{85} +2.61818 q^{86} +0.601096 q^{87} +2.47887 q^{88} -1.51478 q^{89} +6.45692 q^{90} +7.81910 q^{91} -2.68318 q^{92} +0.640986 q^{93} +1.66601 q^{94} +15.6601 q^{95} -3.37175 q^{96} -13.5584 q^{97} -1.77567 q^{98} +2.93428 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 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