LMFDB - Embedded newform 4033.2.a.d.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, 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("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$1$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71802 q^{2} -0.508580 q^{3} +0.951609 q^{4} -0.643447 q^{5} +0.873752 q^{6} +1.62180 q^{7} +1.80116 q^{8} -2.74135 q^{9} +O(q^{10})$	\(q-1.71802 q^{2} -0.508580 q^{3} +0.951609 q^{4} -0.643447 q^{5} +0.873752 q^{6} +1.62180 q^{7} +1.80116 q^{8} -2.74135 q^{9} +1.10546 q^{10} +3.49549 q^{11} -0.483969 q^{12} +2.65198 q^{13} -2.78630 q^{14} +0.327244 q^{15} -4.99766 q^{16} +0.367859 q^{17} +4.70970 q^{18} -1.52337 q^{19} -0.612310 q^{20} -0.824817 q^{21} -6.00533 q^{22} -4.73423 q^{23} -0.916034 q^{24} -4.58598 q^{25} -4.55616 q^{26} +2.91993 q^{27} +1.54332 q^{28} +9.94939 q^{29} -0.562213 q^{30} -5.20688 q^{31} +4.98378 q^{32} -1.77773 q^{33} -0.631991 q^{34} -1.04355 q^{35} -2.60869 q^{36} -1.00000 q^{37} +2.61718 q^{38} -1.34874 q^{39} -1.15895 q^{40} +6.19444 q^{41} +1.41706 q^{42} -5.96711 q^{43} +3.32634 q^{44} +1.76391 q^{45} +8.13353 q^{46} -5.33819 q^{47} +2.54171 q^{48} -4.36975 q^{49} +7.87882 q^{50} -0.187086 q^{51} +2.52365 q^{52} -12.7011 q^{53} -5.01651 q^{54} -2.24916 q^{55} +2.92113 q^{56} +0.774753 q^{57} -17.0933 q^{58} -13.7377 q^{59} +0.311408 q^{60} +7.76954 q^{61} +8.94555 q^{62} -4.44593 q^{63} +1.43306 q^{64} -1.70641 q^{65} +3.05419 q^{66} +9.34196 q^{67} +0.350058 q^{68} +2.40773 q^{69} +1.79284 q^{70} -3.62011 q^{71} -4.93761 q^{72} +15.5161 q^{73} +1.71802 q^{74} +2.33233 q^{75} -1.44965 q^{76} +5.66900 q^{77} +2.31717 q^{78} -12.6202 q^{79} +3.21573 q^{80} +6.73902 q^{81} -10.6422 q^{82} +1.16761 q^{83} -0.784903 q^{84} -0.236698 q^{85} +10.2516 q^{86} -5.06006 q^{87} +6.29593 q^{88} +17.5617 q^{89} -3.03044 q^{90} +4.30099 q^{91} -4.50514 q^{92} +2.64811 q^{93} +9.17114 q^{94} +0.980205 q^{95} -2.53465 q^{96} -11.0033 q^{97} +7.50734 q^{98} -9.58234 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9}+O(q^{10}) $ 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9} - 13 q^{10} - 5 q^{11} - 40 q^{12} - 18 q^{13} - 42 q^{14} - 49 q^{15} + 83 q^{16} - 62 q^{17} - 33 q^{18} - 25 q^{19} - 39 q^{20} - 15 q^{21} - 31 q^{22} - 94 q^{23} - 39 q^{24} + 71 q^{25} - 35 q^{26} - 47 q^{27} - 13 q^{28} - 17 q^{29} + 15 q^{30} - 37 q^{31} - 105 q^{32} - 60 q^{33} + 9 q^{34} - 60 q^{35} + 43 q^{36} - 79 q^{37} - 80 q^{38} - 41 q^{39} - 64 q^{40} - 37 q^{41} - 30 q^{42} - 20 q^{43} - 14 q^{44} - 8 q^{45} + 61 q^{46} - 148 q^{47} - 39 q^{48} + 82 q^{49} - 90 q^{50} - 45 q^{51} - 27 q^{52} - 70 q^{53} - 41 q^{54} - 105 q^{55} - 68 q^{56} - 31 q^{57} - 14 q^{58} - 96 q^{59} - 74 q^{60} - 21 q^{61} + 4 q^{62} - 60 q^{63} + 132 q^{64} - 15 q^{65} + 71 q^{66} - 44 q^{67} - 166 q^{68} - 72 q^{69} - 9 q^{70} - 55 q^{71} - 126 q^{72} - 27 q^{73} + 11 q^{74} - 39 q^{75} - 4 q^{76} - 104 q^{77} - 47 q^{78} - 49 q^{79} - 82 q^{80} + 55 q^{81} + 20 q^{82} - 52 q^{83} - 29 q^{84} + 3 q^{85} - 32 q^{86} - 113 q^{87} + 14 q^{88} - 68 q^{89} - 39 q^{90} - 30 q^{91} - 179 q^{92} - 53 q^{93} - 33 q^{94} - 86 q^{95} + 33 q^{96} - 57 q^{97} - 116 q^{98} + q^{99}+O(q^{100}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9 - 13 * q^10 - 5 * q^11 - 40 * q^12 - 18 * q^13 - 42 * q^14 - 49 * q^15 + 83 * q^16 - 62 * q^17 - 33 * q^18 - 25 * q^19 - 39 * q^20 - 15 * q^21 - 31 * q^22 - 94 * q^23 - 39 * q^24 + 71 * q^25 - 35 * q^26 - 47 * q^27 - 13 * q^28 - 17 * q^29 + 15 * q^30 - 37 * q^31 - 105 * q^32 - 60 * q^33 + 9 * q^34 - 60 * q^35 + 43 * q^36 - 79 * q^37 - 80 * q^38 - 41 * q^39 - 64 * q^40 - 37 * q^41 - 30 * q^42 - 20 * q^43 - 14 * q^44 - 8 * q^45 + 61 * q^46 - 148 * q^47 - 39 * q^48 + 82 * q^49 - 90 * q^50 - 45 * q^51 - 27 * q^52 - 70 * q^53 - 41 * q^54 - 105 * q^55 - 68 * q^56 - 31 * q^57 - 14 * q^58 - 96 * q^59 - 74 * q^60 - 21 * q^61 + 4 * q^62 - 60 * q^63 + 132 * q^64 - 15 * q^65 + 71 * q^66 - 44 * q^67 - 166 * q^68 - 72 * q^69 - 9 * q^70 - 55 * q^71 - 126 * q^72 - 27 * q^73 + 11 * q^74 - 39 * q^75 - 4 * q^76 - 104 * q^77 - 47 * q^78 - 49 * q^79 - 82 * q^80 + 55 * q^81 + 20 * q^82 - 52 * q^83 - 29 * q^84 + 3 * q^85 - 32 * q^86 - 113 * q^87 + 14 * q^88 - 68 * q^89 - 39 * q^90 - 30 * q^91 - 179 * q^92 - 53 * q^93 - 33 * q^94 - 86 * q^95 + 33 * q^96 - 57 * q^97 - 116 * q^98 + 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 $

$2$	−1.71802	−1.21483	−0.607414	−	0.794386i	$-0.707793\pi$
$2$	−1.71802	−1.21483	−0.607414	+	0.794386i	$0.707793\pi$
$3$	−0.508580	−0.293629	−0.146814	−	0.989164i	$-0.546902\pi$
$3$	−0.508580	−0.293629	−0.146814	+	0.989164i	$0.546902\pi$
$4$	0.951609	0.475805
$5$	−0.643447	−0.287758	−0.143879	−	0.989595i	$-0.545958\pi$
$5$	−0.643447	−0.287758	−0.143879	+	0.989595i	$0.545958\pi$
$6$	0.873752	0.356708
$7$	1.62180	0.612985	0.306492	−	0.951873i	$-0.400845\pi$
$7$	1.62180	0.612985	0.306492	+	0.951873i	$0.400845\pi$
$8$	1.80116	0.636807
$9$	−2.74135	−0.913782
$10$	1.10546	0.349576
$11$	3.49549	1.05393	0.526964	−	0.849887i	$-0.323330\pi$
$11$	3.49549	1.05393	0.526964	+	0.849887i	$0.323330\pi$
$12$	−0.483969	−0.139710
$13$	2.65198	0.735526	0.367763	−	0.929920i	$-0.380124\pi$
$13$	2.65198	0.735526	0.367763	+	0.929920i	$0.380124\pi$
$14$	−2.78630	−0.744670
$15$	0.327244	0.0844940
$16$	−4.99766	−1.24941
$17$	0.367859	0.0892190	0.0446095	−	0.999005i	$-0.485796\pi$
$17$	0.367859	0.0892190	0.0446095	+	0.999005i	$0.485796\pi$
$18$	4.70970	1.11009
$19$	−1.52337	−0.349484	−0.174742	−	0.984614i	$-0.555909\pi$
$19$	−1.52337	−0.349484	−0.174742	+	0.984614i	$0.555909\pi$
$20$	−0.612310	−0.136917
$21$	−0.824817	−0.179990
$22$	−6.00533	−1.28034
$23$	−4.73423	−0.987156	−0.493578	−	0.869701i	$-0.664311\pi$
$23$	−4.73423	−0.987156	−0.493578	+	0.869701i	$0.664311\pi$
$24$	−0.916034	−0.186985
$25$	−4.58598	−0.917195
$26$	−4.55616	−0.893537
$27$	2.91993	0.561941
$28$	1.54332	0.291661
$29$	9.94939	1.84756	0.923778	−	0.382928i	$-0.125084\pi$
$29$	9.94939	1.84756	0.923778	+	0.382928i	$0.125084\pi$
$30$	−0.562213	−0.102646
$31$	−5.20688	−0.935184	−0.467592	−	0.883945i	$-0.654878\pi$
$31$	−5.20688	−0.935184	−0.467592	+	0.883945i	$0.654878\pi$
$32$	4.98378	0.881016
$33$	−1.77773	−0.309464
$34$	−0.631991	−0.108386
$35$	−1.04355	−0.176391
$36$	−2.60869	−0.434782
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	2.61718	0.424563
$39$	−1.34874	−0.215971
$40$	−1.15895	−0.183246
$41$	6.19444	0.967409	0.483705	−	0.875231i	$-0.339291\pi$
$41$	6.19444	0.967409	0.483705	+	0.875231i	$0.339291\pi$
$42$	1.41706	0.218656
$43$	−5.96711	−0.909976	−0.454988	−	0.890497i	$-0.650357\pi$
$43$	−5.96711	−0.909976	−0.454988	+	0.890497i	$0.650357\pi$
$44$	3.32634	0.501464
$45$	1.76391	0.262948
$46$	8.13353	1.19922
$47$	−5.33819	−0.778655	−0.389327	−	0.921099i	$-0.627293\pi$
$47$	−5.33819	−0.778655	−0.389327	+	0.921099i	$0.627293\pi$
$48$	2.54171	0.366864
$49$	−4.36975	−0.624250
$50$	7.87882	1.11423
$51$	−0.187086	−0.0261972
$52$	2.52365	0.349967
$53$	−12.7011	−1.74463	−0.872314	−	0.488946i	$-0.837382\pi$
$53$	−12.7011	−1.74463	−0.872314	+	0.488946i	$0.837382\pi$
$54$	−5.01651	−0.682661
$55$	−2.24916	−0.303277
$56$	2.92113	0.390353
$57$	0.774753	0.102619
$58$	−17.0933	−2.24446
$59$	−13.7377	−1.78849	−0.894247	−	0.447574i	$-0.852288\pi$
$59$	−13.7377	−1.78849	−0.894247	+	0.447574i	$0.852288\pi$
$60$	0.311408	0.0402026
$61$	7.76954	0.994788	0.497394	−	0.867525i	$-0.334290\pi$
$61$	7.76954	0.994788	0.497394	+	0.867525i	$0.334290\pi$
$62$	8.94555	1.13609
$63$	−4.44593	−0.560134
$64$	1.43306	0.179132
$65$	−1.70641	−0.211654
$66$	3.05419	0.375945
$67$	9.34196	1.14130	0.570651	−	0.821193i	$-0.306691\pi$
$67$	9.34196	1.14130	0.570651	+	0.821193i	$0.306691\pi$
$68$	0.350058	0.0424508
$69$	2.40773	0.289857
$70$	1.79284	0.214285
$71$	−3.62011	−0.429628	−0.214814	−	0.976655i	$-0.568914\pi$
$71$	−3.62011	−0.429628	−0.214814	+	0.976655i	$0.568914\pi$
$72$	−4.93761	−0.581903
$73$	15.5161	1.81603	0.908013	−	0.418941i	$-0.137599\pi$
$73$	15.5161	1.81603	0.908013	+	0.418941i	$0.137599\pi$
$74$	1.71802	0.199716
$75$	2.33233	0.269315
$76$	−1.44965	−0.166286
$77$	5.66900	0.646042
$78$	2.31717	0.262368
$79$	−12.6202	−1.41988	−0.709942	−	0.704260i	$-0.751279\pi$
$79$	−12.6202	−1.41988	−0.709942	+	0.704260i	$0.751279\pi$
$80$	3.21573	0.359529
$81$	6.73902	0.748780
$82$	−10.6422	−1.17523
$83$	1.16761	0.128161	0.0640807	−	0.997945i	$-0.479588\pi$
$83$	1.16761	0.128161	0.0640807	+	0.997945i	$0.479588\pi$
$84$	−0.784903	−0.0856400
$85$	−0.236698	−0.0256735
$86$	10.2516	1.10546
$87$	−5.06006	−0.542495
$88$	6.29593	0.671149
$89$	17.5617	1.86154	0.930771	−	0.365603i	$-0.119137\pi$
$89$	17.5617	1.86154	0.930771	+	0.365603i	$0.119137\pi$
$90$	−3.03044	−0.319437
$91$	4.30099	0.450866
$92$	−4.50514	−0.469694
$93$	2.64811	0.274597
$94$	9.17114	0.945931
$95$	0.980205	0.100567
$96$	−2.53465	−0.258691
$97$	−11.0033	−1.11722	−0.558609	−	0.829431i	$-0.688665\pi$
$97$	−11.0033	−1.11722	−0.558609	+	0.829431i	$0.688665\pi$
$98$	7.50734	0.758356
$99$	−9.58234	−0.963061
$100$	−4.36406	−0.436406
$101$	−14.1590	−1.40888	−0.704438	−	0.709766i	$-0.748801\pi$
$101$	−14.1590	−1.40888	−0.704438	+	0.709766i	$0.748801\pi$
$102$	0.321418	0.0318251
$103$	12.8994	1.27101	0.635506	−	0.772096i	$-0.280792\pi$
$103$	12.8994	1.27101	0.635506	+	0.772096i	$0.280792\pi$
$104$	4.77664	0.468388
$105$	0.530726	0.0517935
$106$	21.8208	2.11942
$107$	12.6654	1.22441	0.612205	−	0.790699i	$-0.290283\pi$
$107$	12.6654	1.22441	0.612205	+	0.790699i	$0.290283\pi$
$108$	2.77863	0.267374
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	3.86411	0.368429
$111$	0.508580	0.0482722
$112$	−8.10523	−0.765872
$113$	−2.07287	−0.195000	−0.0974998	−	0.995236i	$-0.531085\pi$
$113$	−2.07287	−0.195000	−0.0974998	+	0.995236i	$0.531085\pi$
$114$	−1.33104	−0.124664
$115$	3.04623	0.284062
$116$	9.46794	0.879076
$117$	−7.26999	−0.672110
$118$	23.6017	2.17271
$119$	0.596596	0.0546899
$120$	0.589419	0.0538063
$121$	1.21842	0.110765
$122$	−13.3483	−1.20850
$123$	−3.15037	−0.284059
$124$	−4.95492	−0.444965
$125$	6.16807	0.551689
$126$	7.63822	0.680466
$127$	3.39828	0.301549	0.150774	−	0.988568i	$-0.451823\pi$
$127$	3.39828	0.301549	0.150774	+	0.988568i	$0.451823\pi$
$128$	−12.4296	−1.09863
$129$	3.03475	0.267195
$130$	2.93165	0.257122
$131$	−7.91588	−0.691614	−0.345807	−	0.938306i	$-0.612395\pi$
$131$	−7.91588	−0.691614	−0.345807	+	0.938306i	$0.612395\pi$
$132$	−1.69171	−0.147244
$133$	−2.47060	−0.214228
$134$	−16.0497	−1.38648
$135$	−1.87882	−0.161703
$136$	0.662574	0.0568152
$137$	−4.81477	−0.411353	−0.205677	−	0.978620i	$-0.565940\pi$
$137$	−4.81477	−0.411353	−0.205677	+	0.978620i	$0.565940\pi$
$138$	−4.13655	−0.352126
$139$	5.40964	0.458839	0.229420	−	0.973328i	$-0.426317\pi$
$139$	5.40964	0.458839	0.229420	+	0.973328i	$0.426317\pi$
$140$	−0.993047	−0.0839278
$141$	2.71489	0.228635
$142$	6.21943	0.521923
$143$	9.26994	0.775192
$144$	13.7003	1.14169
$145$	−6.40191	−0.531649
$146$	−26.6571	−2.20616
$147$	2.22236	0.183298
$148$	−0.951609	−0.0782218
$149$	−22.5291	−1.84566	−0.922828	−	0.385213i	$-0.874128\pi$
$149$	−22.5291	−1.84566	−0.922828	+	0.385213i	$0.874128\pi$
$150$	−4.00701	−0.327171
$151$	3.59487	0.292547	0.146273	−	0.989244i	$-0.453272\pi$
$151$	3.59487	0.292547	0.146273	+	0.989244i	$0.453272\pi$
$152$	−2.74383	−0.222554
$153$	−1.00843	−0.0815267
$154$	−9.73948	−0.784829
$155$	3.35035	0.269107
$156$	−1.28347	−0.102760
$157$	17.6177	1.40604	0.703021	−	0.711169i	$-0.251834\pi$
$157$	17.6177	1.40604	0.703021	+	0.711169i	$0.251834\pi$
$158$	21.6818	1.72491
$159$	6.45951	0.512273
$160$	−3.20680	−0.253520
$161$	−7.67800	−0.605111
$162$	−11.5778	−0.909639
$163$	−14.1851	−1.11106	−0.555530	−	0.831497i	$-0.687484\pi$
$163$	−14.1851	−1.11106	−0.555530	+	0.831497i	$0.687484\pi$
$164$	5.89469	0.460298
$165$	1.14388	0.0890507
$166$	−2.00598	−0.155694
$167$	7.89325	0.610798	0.305399	−	0.952225i	$-0.401210\pi$
$167$	7.89325	0.610798	0.305399	+	0.952225i	$0.401210\pi$
$168$	−1.48563	−0.114619
$169$	−5.96702	−0.459002
$170$	0.406653	0.0311889
$171$	4.17607	0.319352
$172$	−5.67836	−0.432971
$173$	−8.70330	−0.661700	−0.330850	−	0.943683i	$-0.607335\pi$
$173$	−8.70330	−0.661700	−0.330850	+	0.943683i	$0.607335\pi$
$174$	8.69331	0.659038
$175$	−7.43756	−0.562227
$176$	−17.4692	−1.31679
$177$	6.98670	0.525153
$178$	−30.1715	−2.26145
$179$	−12.7067	−0.949740	−0.474870	−	0.880056i	$-0.657505\pi$
$179$	−12.7067	−0.949740	−0.474870	+	0.880056i	$0.657505\pi$
$180$	1.67855	0.125112
$181$	−1.93904	−0.144128	−0.0720638	−	0.997400i	$-0.522959\pi$
$181$	−1.93904	−0.144128	−0.0720638	+	0.997400i	$0.522959\pi$
$182$	−7.38920	−0.547724
$183$	−3.95143	−0.292098
$184$	−8.52712	−0.628628
$185$	0.643447	0.0473072
$186$	−4.54953	−0.333587
$187$	1.28585	0.0940304
$188$	−5.07987	−0.370488
$189$	4.73556	0.344461
$190$	−1.68402	−0.122171
$191$	−12.8361	−0.928791	−0.464396	−	0.885628i	$-0.653728\pi$
$191$	−12.8361	−0.928791	−0.464396	+	0.885628i	$0.653728\pi$
$192$	−0.728825	−0.0525984
$193$	22.2617	1.60243	0.801217	−	0.598374i	$-0.204186\pi$
$193$	22.2617	1.60243	0.801217	+	0.598374i	$0.204186\pi$
$194$	18.9040	1.35723
$195$	0.867843	0.0621475
$196$	−4.15829	−0.297021
$197$	−26.1387	−1.86231	−0.931153	−	0.364628i	$-0.881196\pi$
$197$	−26.1387	−1.86231	−0.931153	+	0.364628i	$0.881196\pi$
$198$	16.4627	1.16995
$199$	17.9359	1.27145	0.635723	−	0.771918i	$-0.280702\pi$
$199$	17.9359	1.27145	0.635723	+	0.771918i	$0.280702\pi$
$200$	−8.26008	−0.584076
$201$	−4.75113	−0.335119
$202$	24.3256	1.71154
$203$	16.1360	1.13252
$204$	−0.178033	−0.0124648
$205$	−3.98579	−0.278380
$206$	−22.1614	−1.54406
$207$	12.9782	0.902046
$208$	−13.2537	−0.918977
$209$	−5.32490	−0.368331
$210$	−0.911800	−0.0629202
$211$	14.5105	0.998941	0.499470	−	0.866331i	$-0.333528\pi$
$211$	14.5105	0.998941	0.499470	+	0.866331i	$0.333528\pi$
$212$	−12.0865	−0.830103
$213$	1.84111	0.126151
$214$	−21.7595	−1.48745
$215$	3.83952	0.261853
$216$	5.25927	0.357848
$217$	−8.44455	−0.573253
$218$	1.71802	0.116359
$219$	−7.89119	−0.533237
$220$	−2.14032	−0.144300
$221$	0.975554	0.0656229
$222$	−0.873752	−0.0586424
$223$	8.96413	0.600283	0.300141	−	0.953895i	$-0.402966\pi$
$223$	8.96413	0.600283	0.300141	+	0.953895i	$0.402966\pi$
$224$	8.08272	0.540049
$225$	12.5718	0.838117
$226$	3.56125	0.236891
$227$	10.4465	0.693362	0.346681	−	0.937983i	$-0.387309\pi$
$227$	10.4465	0.693362	0.346681	+	0.937983i	$0.387309\pi$
$228$	0.737262	0.0488264
$229$	0.407571	0.0269331	0.0134665	−	0.999909i	$-0.495713\pi$
$229$	0.407571	0.0269331	0.0134665	+	0.999909i	$0.495713\pi$
$230$	−5.23350	−0.345086
$231$	−2.88314	−0.189696
$232$	17.9205	1.17654
$233$	−15.4300	−1.01085	−0.505426	−	0.862870i	$-0.668665\pi$
$233$	−15.4300	−1.01085	−0.505426	+	0.862870i	$0.668665\pi$
$234$	12.4900	0.816498
$235$	3.43484	0.224064
$236$	−13.0729	−0.850974
$237$	6.41838	0.416918
$238$	−1.02497	−0.0664387
$239$	21.0684	1.36280	0.681400	−	0.731911i	$-0.261371\pi$
$239$	21.0684	1.36280	0.681400	+	0.731911i	$0.261371\pi$
$240$	−1.63545	−0.105568
$241$	−7.15954	−0.461186	−0.230593	−	0.973050i	$-0.574067\pi$
$241$	−7.15954	−0.461186	−0.230593	+	0.973050i	$0.574067\pi$
$242$	−2.09328	−0.134561
$243$	−12.1871	−0.781804
$244$	7.39357	0.473325
$245$	2.81170	0.179633
$246$	5.41241	0.345082
$247$	−4.03993	−0.257055
$248$	−9.37843	−0.595531
$249$	−0.593821	−0.0376318
$250$	−10.5969	−0.670206
$251$	−6.87939	−0.434223	−0.217112	−	0.976147i	$-0.569664\pi$
$251$	−6.87939	−0.434223	−0.217112	+	0.976147i	$0.569664\pi$
$252$	−4.23079	−0.266515
$253$	−16.5484	−1.04039
$254$	−5.83834	−0.366330
$255$	0.120380	0.00753847
$256$	18.4882	1.15551
$257$	−15.0784	−0.940564	−0.470282	−	0.882516i	$-0.655848\pi$
$257$	−15.0784	−0.940564	−0.470282	+	0.882516i	$0.655848\pi$
$258$	−5.21378	−0.324596
$259$	−1.62180	−0.100774
$260$	−1.62383	−0.100706
$261$	−27.2747	−1.68826
$262$	13.5997	0.840191
$263$	−14.8896	−0.918131	−0.459066	−	0.888402i	$-0.651816\pi$
$263$	−14.8896	−0.918131	−0.459066	+	0.888402i	$0.651816\pi$
$264$	−3.20198	−0.197068
$265$	8.17247	0.502031
$266$	4.24456	0.260250
$267$	−8.93155	−0.546602
$268$	8.88990	0.543037
$269$	−6.25483	−0.381364	−0.190682	−	0.981652i	$-0.561070\pi$
$269$	−6.25483	−0.381364	−0.190682	+	0.981652i	$0.561070\pi$
$270$	3.22786	0.196441
$271$	−4.52848	−0.275086	−0.137543	−	0.990496i	$-0.543920\pi$
$271$	−4.52848	−0.275086	−0.137543	+	0.990496i	$0.543920\pi$
$272$	−1.83843	−0.111471
$273$	−2.18739	−0.132387
$274$	8.27189	0.499723
$275$	−16.0302	−0.966658
$276$	2.29122	0.137915
$277$	−27.7776	−1.66899	−0.834497	−	0.551013i	$-0.814241\pi$
$277$	−27.7776	−1.66899	−0.834497	+	0.551013i	$0.814241\pi$
$278$	−9.29389	−0.557410
$279$	14.2739	0.854554
$280$	−1.87959	−0.112327
$281$	25.3843	1.51430	0.757151	−	0.653240i	$-0.226591\pi$
$281$	25.3843	1.51430	0.757151	+	0.653240i	$0.226591\pi$
$282$	−4.66426	−0.277752
$283$	−14.2057	−0.844441	−0.422220	−	0.906493i	$-0.638749\pi$
$283$	−14.2057	−0.844441	−0.422220	+	0.906493i	$0.638749\pi$
$284$	−3.44493	−0.204419
$285$	−0.498512	−0.0295293
$286$	−15.9260	−0.941724
$287$	10.0462	0.593007
$288$	−13.6623	−0.805057
$289$	−16.8647	−0.992040
$290$	10.9986	0.645862
$291$	5.59607	0.328047
$292$	14.7653	0.864074
$293$	−25.9995	−1.51891	−0.759453	−	0.650562i	$-0.774533\pi$
$293$	−25.9995	−1.51891	−0.759453	+	0.650562i	$0.774533\pi$
$294$	−3.81808	−0.222675
$295$	8.83947	0.514654
$296$	−1.80116	−0.104690
$297$	10.2066	0.592246
$298$	38.7055	2.24215
$299$	−12.5551	−0.726079
$300$	2.21947	0.128141
$301$	−9.67749	−0.557801
$302$	−6.17608	−0.355393
$303$	7.20099	0.413686
$304$	7.61326	0.436651
$305$	−4.99929	−0.286258
$306$	1.73251	0.0990409
$307$	−24.5352	−1.40030	−0.700148	−	0.713998i	$-0.746883\pi$
$307$	−24.5352	−1.40030	−0.700148	+	0.713998i	$0.746883\pi$
$308$	5.39467	0.307390
$309$	−6.56035	−0.373205
$310$	−5.75599	−0.326918
$311$	−10.8012	−0.612478	−0.306239	−	0.951955i	$-0.599071\pi$
$311$	−10.8012	−0.612478	−0.306239	+	0.951955i	$0.599071\pi$
$312$	−2.42930	−0.137532
$313$	21.1134	1.19340	0.596700	−	0.802464i	$-0.296478\pi$
$313$	21.1134	1.19340	0.596700	+	0.802464i	$0.296478\pi$
$314$	−30.2676	−1.70810
$315$	2.86072	0.161183
$316$	−12.0095	−0.675587
$317$	31.3519	1.76090	0.880448	−	0.474143i	$-0.157242\pi$
$317$	31.3519	1.76090	0.880448	+	0.474143i	$0.157242\pi$
$318$	−11.0976	−0.622323
$319$	34.7780	1.94719
$320$	−0.922098	−0.0515468
$321$	−6.44136	−0.359521
$322$	13.1910	0.735106
$323$	−0.560384	−0.0311806
$324$	6.41292	0.356273
$325$	−12.1619	−0.674621
$326$	24.3703	1.34974
$327$	0.508580	0.0281245
$328$	11.1572	0.616053
$329$	−8.65750	−0.477304
$330$	−1.96521	−0.108181
$331$	15.4219	0.847664	0.423832	−	0.905741i	$-0.360685\pi$
$331$	15.4219	0.847664	0.423832	+	0.905741i	$0.360685\pi$
$332$	1.11111	0.0609798
$333$	2.74135	0.150225
$334$	−13.5608	−0.742013
$335$	−6.01106	−0.328419
$336$	4.12215	0.224882
$337$	20.9198	1.13958	0.569788	−	0.821792i	$-0.307025\pi$
$337$	20.9198	1.13958	0.569788	+	0.821792i	$0.307025\pi$
$338$	10.2515	0.557608
$339$	1.05422	0.0572574
$340$	−0.225244	−0.0122156
$341$	−18.2006	−0.985617
$342$	−7.17460	−0.387958
$343$	−18.4395	−0.995640
$344$	−10.7477	−0.579479
$345$	−1.54925	−0.0834088
$346$	14.9525	0.803850
$347$	−22.2919	−1.19669	−0.598347	−	0.801237i	$-0.704176\pi$
$347$	−22.2919	−1.19669	−0.598347	+	0.801237i	$0.704176\pi$
$348$	−4.81520	−0.258122
$349$	−24.4566	−1.30913	−0.654566	−	0.756005i	$-0.727149\pi$
$349$	−24.4566	−1.30913	−0.654566	+	0.756005i	$0.727149\pi$
$350$	12.7779	0.683008
$351$	7.74359	0.413322
$352$	17.4207	0.928528
$353$	−25.0111	−1.33121	−0.665604	−	0.746305i	$-0.731826\pi$
$353$	−25.0111	−1.33121	−0.665604	+	0.746305i	$0.731826\pi$
$354$	−12.0033	−0.637970
$355$	2.32935	0.123629
$356$	16.7119	0.885730
$357$	−0.303416	−0.0160585
$358$	21.8304	1.15377
$359$	−23.3911	−1.23454	−0.617268	−	0.786753i	$-0.711761\pi$
$359$	−23.3911	−1.23454	−0.617268	+	0.786753i	$0.711761\pi$
$360$	3.17709	0.167447
$361$	−16.6794	−0.877861
$362$	3.33132	0.175090
$363$	−0.619664	−0.0325239
$364$	4.09286	0.214524
$365$	−9.98381	−0.522577
$366$	6.78865	0.354849
$367$	−1.55335	−0.0810840	−0.0405420	−	0.999178i	$-0.512908\pi$
$367$	−1.55335	−0.0810840	−0.0405420	+	0.999178i	$0.512908\pi$
$368$	23.6601	1.23337
$369$	−16.9811	−0.884001
$370$	−1.10546	−0.0574700
$371$	−20.5987	−1.06943
$372$	2.51997	0.130654
$373$	−4.53330	−0.234725	−0.117363	−	0.993089i	$-0.537444\pi$
$373$	−4.53330	−0.234725	−0.117363	+	0.993089i	$0.537444\pi$
$374$	−2.20912	−0.114231
$375$	−3.13695	−0.161992
$376$	−9.61494	−0.495853
$377$	26.3856	1.35893
$378$	−8.13581	−0.418461
$379$	−24.2008	−1.24311	−0.621557	−	0.783369i	$-0.713500\pi$
$379$	−24.2008	−1.24311	−0.621557	+	0.783369i	$0.713500\pi$
$380$	0.932772	0.0478502
$381$	−1.72830	−0.0885434
$382$	22.0528	1.12832
$383$	−18.4088	−0.940644	−0.470322	−	0.882495i	$-0.655862\pi$
$383$	−18.4088	−0.940644	−0.470322	+	0.882495i	$0.655862\pi$
$384$	6.32144	0.322589
$385$	−3.64770	−0.185904
$386$	−38.2462	−1.94668
$387$	16.3579	0.831520
$388$	−10.4709	−0.531578
$389$	0.254458	0.0129015	0.00645077	−	0.999979i	$-0.497947\pi$
$389$	0.254458	0.0129015	0.00645077	+	0.999979i	$0.497947\pi$
$390$	−1.49098	−0.0754985
$391$	−1.74153	−0.0880731
$392$	−7.87062	−0.397526
$393$	4.02585	0.203077
$394$	44.9070	2.26238
$395$	8.12043	0.408583
$396$	−9.11864	−0.458229
$397$	7.10919	0.356800	0.178400	−	0.983958i	$-0.442908\pi$
$397$	7.10919	0.356800	0.178400	+	0.983958i	$0.442908\pi$
$398$	−30.8144	−1.54459
$399$	1.25650	0.0629036
$400$	22.9191	1.14596
$401$	−18.2045	−0.909087	−0.454544	−	0.890724i	$-0.650198\pi$
$401$	−18.2045	−0.909087	−0.454544	+	0.890724i	$0.650198\pi$
$402$	8.16256	0.407112
$403$	−13.8085	−0.687852
$404$	−13.4739	−0.670350
$405$	−4.33620	−0.215468
$406$	−27.7220	−1.37582
$407$	−3.49549	−0.173265
$408$	−0.336971	−0.0166826
$409$	12.5524	0.620678	0.310339	−	0.950626i	$-0.399557\pi$
$409$	12.5524	0.620678	0.310339	+	0.950626i	$0.399557\pi$
$410$	6.84769	0.338183
$411$	2.44869	0.120785
$412$	12.2752	0.604754
$413$	−22.2798	−1.09632
$414$	−22.2968	−1.09583
$415$	−0.751292	−0.0368795
$416$	13.2169	0.648010
$417$	−2.75123	−0.134728
$418$	9.14832	0.447459
$419$	29.2222	1.42760	0.713798	−	0.700351i	$-0.246973\pi$
$419$	29.2222	1.42760	0.713798	+	0.700351i	$0.246973\pi$
$420$	0.505044	0.0246436
$421$	−16.1200	−0.785643	−0.392821	−	0.919615i	$-0.628501\pi$
$421$	−16.1200	−0.785643	−0.392821	+	0.919615i	$0.628501\pi$
$422$	−24.9293	−1.21354
$423$	14.6338	0.711521
$424$	−22.8767	−1.11099
$425$	−1.68699	−0.0818312
$426$	−3.16308	−0.153252
$427$	12.6007	0.609790
$428$	12.0525	0.582580
$429$	−4.71450	−0.227618
$430$	−6.59639	−0.318106
$431$	1.48253	0.0714109	0.0357055	−	0.999362i	$-0.488632\pi$
$431$	1.48253	0.0714109	0.0357055	+	0.999362i	$0.488632\pi$
$432$	−14.5928	−0.702097
$433$	12.0452	0.578857	0.289428	−	0.957200i	$-0.406535\pi$
$433$	12.0452	0.578857	0.289428	+	0.957200i	$0.406535\pi$
$434$	14.5079	0.696403
$435$	3.25588	0.156107
$436$	−0.951609	−0.0455738
$437$	7.21197	0.344995
$438$	13.5573	0.647791
$439$	−31.4645	−1.50172	−0.750860	−	0.660461i	$-0.770361\pi$
$439$	−31.4645	−1.50172	−0.750860	+	0.660461i	$0.770361\pi$
$440$	−4.05110	−0.193129
$441$	11.9790	0.570428
$442$	−1.67603	−0.0797204
$443$	−23.5690	−1.11980	−0.559898	−	0.828561i	$-0.689160\pi$
$443$	−23.5690	−1.11980	−0.559898	+	0.828561i	$0.689160\pi$
$444$	0.483969	0.0229682
$445$	−11.3001	−0.535674
$446$	−15.4006	−0.729240
$447$	11.4578	0.541937
$448$	2.32414	0.109805
$449$	−36.5621	−1.72547	−0.862737	−	0.505653i	$-0.831252\pi$
$449$	−36.5621	−1.72547	−0.862737	+	0.505653i	$0.831252\pi$
$450$	−21.5986	−1.01817
$451$	21.6526	1.01958
$452$	−1.97257	−0.0927817
$453$	−1.82828	−0.0859000
$454$	−17.9474	−0.842315
$455$	−2.76746	−0.129740
$456$	1.39545	0.0653481
$457$	−0.546431	−0.0255610	−0.0127805	−	0.999918i	$-0.504068\pi$
$457$	−0.546431	−0.0255610	−0.0127805	+	0.999918i	$0.504068\pi$
$458$	−0.700218	−0.0327190
$459$	1.07412	0.0501358
$460$	2.89882	0.135158
$461$	34.5992	1.61145	0.805723	−	0.592293i	$-0.201777\pi$
$461$	34.5992	1.61145	0.805723	+	0.592293i	$0.201777\pi$
$462$	4.95330	0.230448
$463$	−22.4655	−1.04406	−0.522030	−	0.852927i	$-0.674825\pi$
$463$	−22.4655	−1.04406	−0.522030	+	0.852927i	$0.674825\pi$
$464$	−49.7237	−2.30836
$465$	−1.70392	−0.0790174
$466$	26.5091	1.22801
$467$	4.84140	0.224033	0.112017	−	0.993706i	$-0.464269\pi$
$467$	4.84140	0.224033	0.112017	+	0.993706i	$0.464269\pi$
$468$	−6.91819	−0.319793
$469$	15.1508	0.699601
$470$	−5.90114	−0.272199
$471$	−8.95998	−0.412854
$472$	−24.7438	−1.13892
$473$	−20.8580	−0.959050
$474$	−11.0269	−0.506484
$475$	6.98612	0.320545
$476$	0.567726	0.0260217
$477$	34.8181	1.59421
$478$	−36.1960	−1.65557
$479$	−27.4718	−1.25522	−0.627608	−	0.778529i	$-0.715966\pi$
$479$	−27.4718	−1.25522	−0.627608	+	0.778529i	$0.715966\pi$
$480$	1.63091	0.0744406
$481$	−2.65198	−0.120920
$482$	12.3003	0.560262
$483$	3.90488	0.177678
$484$	1.15946	0.0527027
$485$	7.08006	0.321489
$486$	20.9378	0.949757
$487$	33.5210	1.51898	0.759491	−	0.650518i	$-0.225448\pi$
$487$	33.5210	1.51898	0.759491	+	0.650518i	$0.225448\pi$
$488$	13.9942	0.633487
$489$	7.21423	0.326239
$490$	−4.83057	−0.218223
$491$	−5.67780	−0.256235	−0.128118	−	0.991759i	$-0.540894\pi$
$491$	−5.67780	−0.256235	−0.128118	+	0.991759i	$0.540894\pi$
$492$	−2.99792	−0.135157
$493$	3.65998	0.164837
$494$	6.94070	0.312277
$495$	6.16573	0.277129
$496$	26.0222	1.16843
$497$	−5.87111	−0.263355
$498$	1.02020	0.0457162
$499$	−17.7054	−0.792604	−0.396302	−	0.918120i	$-0.629707\pi$
$499$	−17.7054	−0.792604	−0.396302	+	0.918120i	$0.629707\pi$
$500$	5.86959	0.262496
$501$	−4.01434	−0.179348
$502$	11.8190	0.527506
$503$	−15.6261	−0.696733	−0.348366	−	0.937358i	$-0.613264\pi$
$503$	−15.6261	−0.696733	−0.348366	+	0.937358i	$0.613264\pi$
$504$	−8.00783	−0.356697
$505$	9.11058	0.405416
$506$	28.4306	1.26390
$507$	3.03471	0.134776
$508$	3.23384	0.143478
$509$	17.1751	0.761275	0.380637	−	0.924724i	$-0.375705\pi$
$509$	17.1751	0.761275	0.380637	+	0.924724i	$0.375705\pi$
$510$	−0.206815	−0.00915794
$511$	25.1641	1.11320
$512$	−6.90405	−0.305119
$513$	−4.44812	−0.196389
$514$	25.9050	1.14262
$515$	−8.30005	−0.365744
$516$	2.88790	0.127133
$517$	−18.6596	−0.820647
$518$	2.78630	0.122423
$519$	4.42632	0.194294
$520$	−3.07351	−0.134782
$521$	12.6825	0.555629	0.277814	−	0.960635i	$-0.410390\pi$
$521$	12.6825	0.555629	0.277814	+	0.960635i	$0.410390\pi$
$522$	46.8587	2.05095
$523$	−14.1960	−0.620748	−0.310374	−	0.950614i	$-0.600454\pi$
$523$	−14.1960	−0.620748	−0.310374	+	0.950614i	$0.600454\pi$
$524$	−7.53283	−0.329073
$525$	3.78259	0.165086
$526$	25.5807	1.11537
$527$	−1.91540	−0.0834361
$528$	8.88450	0.386648
$529$	−0.587024	−0.0255228
$530$	−14.0405	−0.609881
$531$	37.6597	1.63429
$532$	−2.35105	−0.101931
$533$	16.4275	0.711555
$534$	15.3446	0.664027
$535$	−8.14950	−0.352334
$536$	16.8264	0.726789
$537$	6.46235	0.278871
$538$	10.7460	0.463291
$539$	−15.2744	−0.657915
$540$	−1.78790	−0.0769391
$541$	27.4486	1.18011	0.590053	−	0.807365i	$-0.299107\pi$
$541$	27.4486	1.18011	0.590053	+	0.807365i	$0.299107\pi$
$542$	7.78004	0.334181
$543$	0.986156	0.0423200
$544$	1.83333	0.0786033
$545$	0.643447	0.0275622
$546$	3.75800	0.160827
$547$	−16.4065	−0.701493	−0.350746	−	0.936470i	$-0.614072\pi$
$547$	−16.4065	−0.701493	−0.350746	+	0.936470i	$0.614072\pi$
$548$	−4.58178	−0.195724
$549$	−21.2990	−0.909019
$550$	27.5403	1.17432
$551$	−15.1566	−0.645692
$552$	4.33672	0.184583
$553$	−20.4675	−0.870367
$554$	47.7226	2.02754
$555$	−0.327244	−0.0138907
$556$	5.14786	0.218318
$557$	−21.1307	−0.895335	−0.447668	−	0.894200i	$-0.647745\pi$
$557$	−21.1307	−0.895335	−0.447668	+	0.894200i	$0.647745\pi$
$558$	−24.5229	−1.03814
$559$	−15.8246	−0.669311
$560$	5.21528	0.220386
$561$	−0.653955	−0.0276100
$562$	−43.6109	−1.83962
$563$	−40.9065	−1.72400	−0.862001	−	0.506907i	$-0.830789\pi$
$563$	−40.9065	−1.72400	−0.862001	+	0.506907i	$0.830789\pi$
$564$	2.58352	0.108786
$565$	1.33378	0.0561127
$566$	24.4057	1.02585
$567$	10.9294	0.458991
$568$	−6.52039	−0.273590
$569$	−5.62396	−0.235769	−0.117884	−	0.993027i	$-0.537611\pi$
$569$	−5.62396	−0.235769	−0.117884	+	0.993027i	$0.537611\pi$
$570$	0.856456	0.0358730
$571$	28.4749	1.19164	0.595819	−	0.803119i	$-0.296828\pi$
$571$	28.4749	1.19164	0.595819	+	0.803119i	$0.296828\pi$
$572$	8.82137	0.368840
$573$	6.52820	0.272720
$574$	−17.2596	−0.720401
$575$	21.7111	0.905415
$576$	−3.92851	−0.163688
$577$	−26.7193	−1.11234	−0.556170	−	0.831069i	$-0.687730\pi$
$577$	−26.7193	−1.11234	−0.556170	+	0.831069i	$0.687730\pi$
$578$	28.9739	1.20516
$579$	−11.3219	−0.470520
$580$	−6.09211	−0.252961
$581$	1.89363	0.0785610
$582$	−9.61418	−0.398521
$583$	−44.3965	−1.83871
$584$	27.9471	1.15646
$585$	4.67785	0.193405
$586$	44.6678	1.84521
$587$	5.54626	0.228918	0.114459	−	0.993428i	$-0.463486\pi$
$587$	5.54626	0.228918	0.114459	+	0.993428i	$0.463486\pi$
$588$	2.11482	0.0872139
$589$	7.93199	0.326832
$590$	−15.1864	−0.625215
$591$	13.2936	0.546826
$592$	4.99766	0.205402
$593$	5.35771	0.220015	0.110007	−	0.993931i	$-0.464913\pi$
$593$	5.35771	0.220015	0.110007	+	0.993931i	$0.464913\pi$
$594$	−17.5352	−0.719476
$595$	−0.383878	−0.0157375
$596$	−21.4389	−0.878172
$597$	−9.12185	−0.373333
$598$	21.5699	0.882060
$599$	20.1301	0.822494	0.411247	−	0.911524i	$-0.365093\pi$
$599$	20.1301	0.822494	0.411247	+	0.911524i	$0.365093\pi$
$600$	4.20091	0.171501
$601$	−2.83784	−0.115758	−0.0578789	−	0.998324i	$-0.518434\pi$
$601$	−2.83784	−0.115758	−0.0578789	+	0.998324i	$0.518434\pi$
$602$	16.6262	0.677632
$603$	−25.6096	−1.04290
$604$	3.42091	0.139195
$605$	−0.783989	−0.0318737
$606$	−12.3715	−0.502557
$607$	−21.9266	−0.889974	−0.444987	−	0.895537i	$-0.646792\pi$
$607$	−21.9266	−0.889974	−0.444987	+	0.895537i	$0.646792\pi$
$608$	−7.59212	−0.307901
$609$	−8.20643	−0.332541
$610$	8.58890	0.347754
$611$	−14.1568	−0.572721
$612$	−0.959631	−0.0387908
$613$	23.0658	0.931620	0.465810	−	0.884885i	$-0.345763\pi$
$613$	23.0658	0.931620	0.465810	+	0.884885i	$0.345763\pi$
$614$	42.1520	1.70112
$615$	2.02709	0.0817403
$616$	10.2108	0.411404
$617$	1.76699	0.0711364	0.0355682	−	0.999367i	$-0.488676\pi$
$617$	1.76699	0.0711364	0.0355682	+	0.999367i	$0.488676\pi$
$618$	11.2708	0.453380
$619$	−21.5115	−0.864620	−0.432310	−	0.901725i	$-0.642301\pi$
$619$	−21.5115	−0.864620	−0.432310	+	0.901725i	$0.642301\pi$
$620$	3.18823	0.128042
$621$	−13.8236	−0.554724
$622$	18.5567	0.744055
$623$	28.4817	1.14110
$624$	6.74055	0.269838
$625$	18.9611	0.758442
$626$	−36.2734	−1.44978
$627$	2.70814	0.108153
$628$	16.7651	0.669001
$629$	−0.367859	−0.0146675
$630$	−4.91479	−0.195810
$631$	−35.6404	−1.41882	−0.709411	−	0.704795i	$-0.751039\pi$
$631$	−35.6404	−1.41882	−0.709411	+	0.704795i	$0.751039\pi$
$632$	−22.7310	−0.904191
$633$	−7.37972	−0.293318
$634$	−53.8633	−2.13918
$635$	−2.18661	−0.0867732
$636$	6.14693	0.243742
$637$	−11.5885	−0.459152
$638$	−59.7494	−2.36550
$639$	9.92397	0.392586
$640$	7.99778	0.316140
$641$	13.9842	0.552342	0.276171	−	0.961108i	$-0.410934\pi$
$641$	13.9842	0.552342	0.276171	+	0.961108i	$0.410934\pi$
$642$	11.0664	0.436756
$643$	−28.8947	−1.13950	−0.569749	−	0.821819i	$-0.692959\pi$
$643$	−28.8947	−1.13950	−0.569749	+	0.821819i	$0.692959\pi$
$644$	−7.30646	−0.287915
$645$	−1.95270	−0.0768875
$646$	0.962754	0.0378791
$647$	11.0242	0.433405	0.216702	−	0.976238i	$-0.430470\pi$
$647$	11.0242	0.433405	0.216702	+	0.976238i	$0.430470\pi$
$648$	12.1381	0.476828
$649$	−48.0199	−1.88494
$650$	20.8944	0.819548
$651$	4.29472	0.168323
$652$	−13.4986	−0.528647
$653$	−12.7476	−0.498851	−0.249425	−	0.968394i	$-0.580242\pi$
$653$	−12.7476	−0.498851	−0.249425	+	0.968394i	$0.580242\pi$
$654$	−0.873752	−0.0341664
$655$	5.09345	0.199017
$656$	−30.9577	−1.20870
$657$	−42.5351	−1.65945
$658$	14.8738	0.579841
$659$	48.3706	1.88425	0.942126	−	0.335258i	$-0.108823\pi$
$659$	48.3706	1.88425	0.942126	+	0.335258i	$0.108823\pi$
$660$	1.08852	0.0423707
$661$	23.3589	0.908557	0.454278	−	0.890860i	$-0.349897\pi$
$661$	23.3589	0.908557	0.454278	+	0.890860i	$0.349897\pi$
$662$	−26.4952	−1.02976
$663$	−0.496147	−0.0192687
$664$	2.10305	0.0816140
$665$	1.58970	0.0616460
$666$	−4.70970	−0.182497
$667$	−47.1028	−1.82383
$668$	7.51129	0.290620
$669$	−4.55897	−0.176260
$670$	10.3271	0.398972
$671$	27.1583	1.04844
$672$	−4.11070	−0.158574
$673$	−3.54881	−0.136797	−0.0683983	−	0.997658i	$-0.521789\pi$
$673$	−3.54881	−0.136797	−0.0683983	+	0.997658i	$0.521789\pi$
$674$	−35.9408	−1.38439
$675$	−13.3907	−0.515410
$676$	−5.67827	−0.218395
$677$	19.6795	0.756345	0.378172	−	0.925735i	$-0.376553\pi$
$677$	19.6795	0.756345	0.378172	+	0.925735i	$0.376553\pi$
$678$	−1.81118	−0.0695579
$679$	−17.8453	−0.684838
$680$	−0.426331	−0.0163490
$681$	−5.31290	−0.203591
$682$	31.2691	1.19735
$683$	−41.5244	−1.58889	−0.794444	−	0.607337i	$-0.792238\pi$
$683$	−41.5244	−1.58889	−0.794444	+	0.607337i	$0.792238\pi$
$684$	3.97399	0.151949
$685$	3.09805	0.118370
$686$	31.6795	1.20953
$687$	−0.207282	−0.00790832
$688$	29.8216	1.13694
$689$	−33.6830	−1.28322
$690$	2.66165	0.101327
$691$	−49.1434	−1.86950	−0.934751	−	0.355302i	$-0.884378\pi$
$691$	−49.1434	−1.86950	−0.934751	+	0.355302i	$0.884378\pi$
$692$	−8.28214	−0.314840
$693$	−15.5407	−0.590342
$694$	38.2981	1.45378
$695$	−3.48081	−0.132035
$696$	−9.11398	−0.345465
$697$	2.27868	0.0863113
$698$	42.0171	1.59037
$699$	7.84737	0.296815
$700$	−7.07765	−0.267510
$701$	40.8169	1.54163	0.770816	−	0.637058i	$-0.219849\pi$
$701$	40.8169	1.54163	0.770816	+	0.637058i	$0.219849\pi$
$702$	−13.3037	−0.502115
$703$	1.52337	0.0574548
$704$	5.00924	0.188793
$705$	−1.74689	−0.0657917
$706$	42.9697	1.61719
$707$	−22.9632	−0.863619
$708$	6.64861	0.249870
$709$	−16.4624	−0.618260	−0.309130	−	0.951020i	$-0.600038\pi$
$709$	−16.4624	−0.618260	−0.309130	+	0.951020i	$0.600038\pi$
$710$	−4.00187	−0.150188
$711$	34.5964	1.29746
$712$	31.6315	1.18544
$713$	24.6506	0.923172
$714$	0.521277	0.0195083
$715$	−5.96472	−0.223068
$716$	−12.0918	−0.451891
$717$	−10.7149	−0.400157
$718$	40.1865	1.49975
$719$	27.4346	1.02314	0.511569	−	0.859242i	$-0.329065\pi$
$719$	27.4346	1.02314	0.511569	+	0.859242i	$0.329065\pi$
$720$	−8.81542	−0.328531
$721$	20.9202	0.779111
$722$	28.6555	1.06645
$723$	3.64119	0.135417
$724$	−1.84521	−0.0685766
$725$	−45.6277	−1.69457
$726$	1.06460	0.0395109
$727$	−6.02390	−0.223414	−0.111707	−	0.993741i	$-0.535632\pi$
$727$	−6.02390	−0.223414	−0.111707	+	0.993741i	$0.535632\pi$
$728$	7.74677	0.287114
$729$	−14.0189	−0.519220
$730$	17.1524	0.634840
$731$	−2.19506	−0.0811872
$732$	−3.76022	−0.138982
$733$	−5.65901	−0.209020	−0.104510	−	0.994524i	$-0.533327\pi$
$733$	−5.65901	−0.209020	−0.104510	+	0.994524i	$0.533327\pi$
$734$	2.66869	0.0985030
$735$	−1.42997	−0.0527454
$736$	−23.5944	−0.869700
$737$	32.6547	1.20285
$738$	29.1740	1.07391
$739$	13.7609	0.506203	0.253101	−	0.967440i	$-0.418549\pi$
$739$	13.7609	0.506203	0.253101	+	0.967440i	$0.418549\pi$
$740$	0.612310	0.0225090
$741$	2.05463	0.0754786
$742$	35.3891	1.29917
$743$	37.0420	1.35894	0.679470	−	0.733703i	$-0.262210\pi$
$743$	37.0420	1.35894	0.679470	+	0.733703i	$0.262210\pi$
$744$	4.76968	0.174865
$745$	14.4963	0.531102
$746$	7.78831	0.285150
$747$	−3.20081	−0.117112
$748$	1.22362	0.0447401
$749$	20.5408	0.750544
$750$	5.38936	0.196792
$751$	40.9954	1.49594	0.747971	−	0.663731i	$-0.231028\pi$
$751$	40.9954	1.49594	0.747971	+	0.663731i	$0.231028\pi$
$752$	26.6784	0.972863
$753$	3.49872	0.127500
$754$	−45.3310	−1.65086
$755$	−2.31311	−0.0841827
$756$	4.50640	0.163896
$757$	31.8929	1.15917	0.579584	−	0.814912i	$-0.303215\pi$
$757$	31.8929	1.15917	0.579584	+	0.814912i	$0.303215\pi$
$758$	41.5777	1.51017
$759$	8.41620	0.305489
$760$	1.76551	0.0640417
$761$	−40.6663	−1.47415	−0.737075	−	0.675811i	$-0.763794\pi$
$761$	−40.6663	−1.47415	−0.737075	+	0.675811i	$0.763794\pi$
$762$	2.96926	0.107565
$763$	−1.62180	−0.0587133
$764$	−12.2150	−0.441923
$765$	0.648871	0.0234600
$766$	31.6267	1.14272
$767$	−36.4320	−1.31548
$768$	−9.40273	−0.339292
$769$	48.0279	1.73193	0.865965	−	0.500104i	$-0.166705\pi$
$769$	48.0279	1.73193	0.865965	+	0.500104i	$0.166705\pi$
$770$	6.26683	0.225841
$771$	7.66856	0.276176
$772$	21.1845	0.762446
$773$	−43.0957	−1.55005	−0.775023	−	0.631933i	$-0.782262\pi$
$773$	−43.0957	−1.55005	−0.775023	+	0.631933i	$0.782262\pi$
$774$	−28.1033	−1.01015
$775$	23.8786	0.857746
$776$	−19.8188	−0.711452
$777$	0.824817	0.0295901
$778$	−0.437165	−0.0156731
$779$	−9.43640	−0.338094
$780$	0.825848	0.0295701
$781$	−12.6540	−0.452797
$782$	2.99200	0.106994
$783$	29.0516	1.03822
$784$	21.8385	0.779947
$785$	−11.3360	−0.404600
$786$	−6.91652	−0.246704
$787$	30.2001	1.07652	0.538258	−	0.842780i	$-0.319083\pi$
$787$	30.2001	1.07652	0.538258	+	0.842780i	$0.319083\pi$
$788$	−24.8738	−0.886094
$789$	7.57254	0.269589
$790$	−13.9511	−0.496358
$791$	−3.36180	−0.119532
$792$	−17.2593	−0.613284
$793$	20.6046	0.731692
$794$	−12.2138	−0.433450
$795$	−4.15635	−0.147411
$796$	17.0680	0.604960
$797$	−42.0775	−1.49046	−0.745231	−	0.666806i	$-0.767661\pi$
$797$	−42.0775	−1.49046	−0.745231	+	0.666806i	$0.767661\pi$
$798$	−2.15869	−0.0764169
$799$	−1.96370	−0.0694708
$800$	−22.8555	−0.808064
$801$	−48.1428	−1.70104
$802$	31.2757	1.10438
$803$	54.2364	1.91396
$804$	−4.52122	−0.159451
$805$	4.94039	0.174126
$806$	23.7234	0.835621
$807$	3.18108	0.111979
$808$	−25.5027	−0.897181
$809$	25.2271	0.886937	0.443469	−	0.896290i	$-0.353748\pi$
$809$	25.2271	0.886937	0.443469	+	0.896290i	$0.353748\pi$
$810$	7.44970	0.261756
$811$	−15.1035	−0.530356	−0.265178	−	0.964199i	$-0.585431\pi$
$811$	−15.1035	−0.530356	−0.265178	+	0.964199i	$0.585431\pi$
$812$	15.3551	0.538860
$813$	2.30309	0.0807730
$814$	6.00533	0.210487
$815$	9.12733	0.319716
$816$	0.934990	0.0327312
$817$	9.09010	0.318022
$818$	−21.5654	−0.754017
$819$	−11.7905	−0.411993
$820$	−3.79292	−0.132454
$821$	−15.7451	−0.549508	−0.274754	−	0.961515i	$-0.588596\pi$
$821$	−15.7451	−0.549508	−0.274754	+	0.961515i	$0.588596\pi$
$822$	−4.20692	−0.146733
$823$	−41.2400	−1.43754	−0.718768	−	0.695250i	$-0.755294\pi$
$823$	−41.2400	−1.43754	−0.718768	+	0.695250i	$0.755294\pi$
$824$	23.2338	0.809389
$825$	8.15264	0.283838
$826$	38.2773	1.33184
$827$	−35.5330	−1.23560	−0.617801	−	0.786334i	$-0.711976\pi$
$827$	−35.5330	−1.23560	−0.617801	+	0.786334i	$0.711976\pi$
$828$	12.3502	0.429198
$829$	24.5052	0.851102	0.425551	−	0.904934i	$-0.360080\pi$
$829$	24.5052	0.851102	0.425551	+	0.904934i	$0.360080\pi$
$830$	1.29074	0.0448022
$831$	14.1271	0.490064
$832$	3.80044	0.131757
$833$	−1.60745	−0.0556949
$834$	4.72668	0.163672
$835$	−5.07888	−0.175762
$836$	−5.06723	−0.175254
$837$	−15.2037	−0.525518
$838$	−50.2044	−1.73428
$839$	44.3256	1.53029	0.765145	−	0.643858i	$-0.222667\pi$
$839$	44.3256	1.53029	0.765145	+	0.643858i	$0.222667\pi$
$840$	0.955922	0.0329825
$841$	69.9905	2.41346
$842$	27.6946	0.954420
$843$	−12.9100	−0.444642
$844$	13.8083	0.475301
$845$	3.83946	0.132081
$846$	−25.1413	−0.864375
$847$	1.97604	0.0678975
$848$	63.4757	2.17976
$849$	7.22472	0.247952
$850$	2.89830	0.0994108
$851$	4.73423	0.162287
$852$	1.75202	0.0600232
$853$	17.1542	0.587347	0.293674	−	0.955906i	$-0.405122\pi$
$853$	17.1542	0.587347	0.293674	+	0.955906i	$0.405122\pi$
$854$	−21.6483	−0.740789
$855$	−2.68708	−0.0918963
$856$	22.8124	0.779712
$857$	−41.3447	−1.41231	−0.706154	−	0.708059i	$-0.749571\pi$
$857$	−41.3447	−1.41231	−0.706154	+	0.708059i	$0.749571\pi$
$858$	8.09964	0.276517
$859$	52.2064	1.78126	0.890630	−	0.454728i	$-0.150264\pi$
$859$	52.2064	1.78126	0.890630	+	0.454728i	$0.150264\pi$
$860$	3.65372	0.124591
$861$	−5.10928	−0.174124
$862$	−2.54702	−0.0867519
$863$	−15.5586	−0.529622	−0.264811	−	0.964300i	$-0.585310\pi$
$863$	−15.5586	−0.529622	−0.264811	+	0.964300i	$0.585310\pi$
$864$	14.5523	0.495079
$865$	5.60011	0.190409
$866$	−20.6940	−0.703211
$867$	8.57703	0.291291
$868$	−8.03591	−0.272757
$869$	−44.1137	−1.49646
$870$	−5.59368	−0.189644
$871$	24.7747	0.839457
$872$	−1.80116	−0.0609950
$873$	30.1639	1.02089
$874$	−12.3903	−0.419110
$875$	10.0034	0.338177
$876$	−7.50933	−0.253717
$877$	−5.27712	−0.178196	−0.0890978	−	0.996023i	$-0.528398\pi$
$877$	−5.27712	−0.178196	−0.0890978	+	0.996023i	$0.528398\pi$
$878$	54.0568	1.82433
$879$	13.2228	0.445994
$880$	11.2405	0.378918
$881$	34.9960	1.17905	0.589523	−	0.807752i	$-0.299316\pi$
$881$	34.9960	1.17905	0.589523	+	0.807752i	$0.299316\pi$
$882$	−20.5802	−0.692972
$883$	25.2831	0.850845	0.425423	−	0.904995i	$-0.360125\pi$
$883$	25.2831	0.850845	0.425423	+	0.904995i	$0.360125\pi$
$884$	0.928346	0.0312237
$885$	−4.49557	−0.151117
$886$	40.4921	1.36036
$887$	−2.94007	−0.0987179	−0.0493590	−	0.998781i	$-0.515718\pi$
$887$	−2.94007	−0.0987179	−0.0493590	+	0.998781i	$0.515718\pi$
$888$	0.916034	0.0307401
$889$	5.51135	0.184845
$890$	19.4138	0.650751
$891$	23.5562	0.789161
$892$	8.53035	0.285617
$893$	8.13202	0.272128
$894$	−19.6848	−0.658360
$895$	8.17606	0.273296
$896$	−20.1584	−0.673444
$897$	6.38526	0.213197
$898$	62.8147	2.09615
$899$	−51.8053	−1.72780
$900$	11.9634	0.398780
$901$	−4.67221	−0.155654
$902$	−37.1997	−1.23861
$903$	4.92178	0.163786
$904$	−3.73358	−0.124177
$905$	1.24767	0.0414739
$906$	3.14103	0.104354
$907$	28.0051	0.929893	0.464947	−	0.885339i	$-0.346073\pi$
$907$	28.0051	0.929893	0.464947	+	0.885339i	$0.346073\pi$
$908$	9.94103	0.329905
$909$	38.8148	1.28741
$910$	4.75456	0.157612
$911$	−28.3305	−0.938632	−0.469316	−	0.883030i	$-0.655499\pi$
$911$	−28.3305	−0.938632	−0.469316	+	0.883030i	$0.655499\pi$
$912$	−3.87195	−0.128213
$913$	4.08135	0.135073
$914$	0.938782	0.0310521
$915$	2.54253	0.0840536
$916$	0.387849	0.0128149
$917$	−12.8380	−0.423948
$918$	−1.84537	−0.0609063
$919$	11.9543	0.394334	0.197167	−	0.980370i	$-0.436826\pi$
$919$	11.9543	0.394334	0.197167	+	0.980370i	$0.436826\pi$
$920$	5.48675	0.180893
$921$	12.4781	0.411167
$922$	−59.4423	−1.95763
$923$	−9.60044	−0.316002
$924$	−2.74362	−0.0902584
$925$	4.58598	0.150786
$926$	38.5963	1.26835
$927$	−35.3616	−1.16143
$928$	49.5856	1.62773
$929$	−19.0344	−0.624499	−0.312250	−	0.950000i	$-0.601083\pi$
$929$	−19.0344	−0.624499	−0.312250	+	0.950000i	$0.601083\pi$
$930$	2.92738	0.0959925
$931$	6.65673	0.218165
$932$	−14.6833	−0.480968
$933$	5.49325	0.179841
$934$	−8.31764	−0.272162
$935$	−0.827374	−0.0270580
$936$	−13.0944	−0.428004
$937$	−45.0855	−1.47288	−0.736440	−	0.676503i	$-0.763495\pi$
$937$	−45.0855	−1.47288	−0.736440	+	0.676503i	$0.763495\pi$
$938$	−26.0295	−0.849894
$939$	−10.7379	−0.350417
$940$	3.26863	0.106611
$941$	6.38129	0.208024	0.104012	−	0.994576i	$-0.466832\pi$
$941$	6.38129	0.208024	0.104012	+	0.994576i	$0.466832\pi$
$942$	15.3935	0.501546
$943$	−29.3259	−0.954984
$944$	68.6562	2.23457
$945$	−3.04708	−0.0991215
$946$	35.8345	1.16508
$947$	−26.3782	−0.857177	−0.428589	−	0.903500i	$-0.640989\pi$
$947$	−26.3782	−0.857177	−0.428589	+	0.903500i	$0.640989\pi$
$948$	6.10779	0.198372
$949$	41.1484	1.33573
$950$	−12.0023	−0.389407
$951$	−15.9449	−0.517049
$952$	1.07457	0.0348269
$953$	23.3669	0.756929	0.378465	−	0.925616i	$-0.376452\pi$
$953$	23.3669	0.756929	0.378465	+	0.925616i	$0.376452\pi$
$954$	−59.8183	−1.93669
$955$	8.25938	0.267267
$956$	20.0489	0.648427
$957$	−17.6874	−0.571751
$958$	47.1972	1.52487
$959$	−7.80862	−0.252153
$960$	0.468960	0.0151356
$961$	−3.88838	−0.125432
$962$	4.55616	0.146897
$963$	−34.7202	−1.11884
$964$	−6.81308	−0.219435
$965$	−14.3242	−0.461114
$966$	−6.70867	−0.215848
$967$	−4.75130	−0.152792	−0.0763958	−	0.997078i	$-0.524341\pi$
$967$	−4.75130	−0.152792	−0.0763958	+	0.997078i	$0.524341\pi$
$968$	2.19457	0.0705362
$969$	0.285000	0.00915552
$970$	−12.1637	−0.390553
$971$	41.4763	1.33104	0.665519	−	0.746381i	$-0.268210\pi$
$971$	41.4763	1.33104	0.665519	+	0.746381i	$0.268210\pi$
$972$	−11.5974	−0.371986
$973$	8.77337	0.281261
$974$	−57.5899	−1.84530
$975$	6.18529	0.198088
$976$	−38.8295	−1.24290
$977$	28.2002	0.902203	0.451102	−	0.892473i	$-0.351031\pi$
$977$	28.2002	0.902203	0.451102	+	0.892473i	$0.351031\pi$
$978$	−12.3942	−0.396324
$979$	61.3868	1.96193
$980$	2.67564	0.0854702
$981$	2.74135	0.0875245
$982$	9.75460	0.311282
$983$	−9.33801	−0.297836	−0.148918	−	0.988850i	$-0.547579\pi$
$983$	−9.33801	−0.297836	−0.148918	+	0.988850i	$0.547579\pi$
$984$	−5.67432	−0.180891
$985$	16.8189	0.535894
$986$	−6.28793	−0.200249
$987$	4.40303	0.140150
$988$	−3.84444	−0.122308
$989$	28.2497	0.898289
$990$	−10.5929	−0.336664
$991$	4.59385	0.145929	0.0729643	−	0.997335i	$-0.476754\pi$
$991$	4.59385	0.145929	0.0729643	+	0.997335i	$0.476754\pi$
$992$	−25.9500	−0.823912
$993$	−7.84326	−0.248898
$994$	10.0867	0.319931
$995$	−11.5408	−0.365869
$996$	−0.565085	−0.0179054
$997$	9.44762	0.299209	0.149605	−	0.988746i	$-0.452200\pi$
$997$	9.44762	0.299209	0.149605	+	0.988746i	$0.452200\pi$
$998$	30.4184	0.962877
$999$	−2.91993	−0.0923825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.d.1.21	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.d.1.21	✓	79	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-1.71802 q^{2} -0.508580 q^{3} +0.951609 q^{4} -0.643447 q^{5} +0.873752 q^{6} +1.62180 q^{7} +1.80116 q^{8} -2.74135 q^{9} +O(q^{10})\)	\(q-1.71802 q^{2} -0.508580 q^{3} +0.951609 q^{4} -0.643447 q^{5} +0.873752 q^{6} +1.62180 q^{7} +1.80116 q^{8} -2.74135 q^{9} +1.10546 q^{10} +3.49549 q^{11} -0.483969 q^{12} +2.65198 q^{13} -2.78630 q^{14} +0.327244 q^{15} -4.99766 q^{16} +0.367859 q^{17} +4.70970 q^{18} -1.52337 q^{19} -0.612310 q^{20} -0.824817 q^{21} -6.00533 q^{22} -4.73423 q^{23} -0.916034 q^{24} -4.58598 q^{25} -4.55616 q^{26} +2.91993 q^{27} +1.54332 q^{28} +9.94939 q^{29} -0.562213 q^{30} -5.20688 q^{31} +4.98378 q^{32} -1.77773 q^{33} -0.631991 q^{34} -1.04355 q^{35} -2.60869 q^{36} -1.00000 q^{37} +2.61718 q^{38} -1.34874 q^{39} -1.15895 q^{40} +6.19444 q^{41} +1.41706 q^{42} -5.96711 q^{43} +3.32634 q^{44} +1.76391 q^{45} +8.13353 q^{46} -5.33819 q^{47} +2.54171 q^{48} -4.36975 q^{49} +7.87882 q^{50} -0.187086 q^{51} +2.52365 q^{52} -12.7011 q^{53} -5.01651 q^{54} -2.24916 q^{55} +2.92113 q^{56} +0.774753 q^{57} -17.0933 q^{58} -13.7377 q^{59} +0.311408 q^{60} +7.76954 q^{61} +8.94555 q^{62} -4.44593 q^{63} +1.43306 q^{64} -1.70641 q^{65} +3.05419 q^{66} +9.34196 q^{67} +0.350058 q^{68} +2.40773 q^{69} +1.79284 q^{70} -3.62011 q^{71} -4.93761 q^{72} +15.5161 q^{73} +1.71802 q^{74} +2.33233 q^{75} -1.44965 q^{76} +5.66900 q^{77} +2.31717 q^{78} -12.6202 q^{79} +3.21573 q^{80} +6.73902 q^{81} -10.6422 q^{82} +1.16761 q^{83} -0.784903 q^{84} -0.236698 q^{85} +10.2516 q^{86} -5.06006 q^{87} +6.29593 q^{88} +17.5617 q^{89} -3.03044 q^{90} +4.30099 q^{91} -4.50514 q^{92} +2.64811 q^{93} +9.17114 q^{94} +0.980205 q^{95} -2.53465 q^{96} -11.0033 q^{97} +7.50734 q^{98} -9.58234 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9}+O(q^{10}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9} - 13 q^{10} - 5 q^{11} - 40 q^{12} - 18 q^{13} - 42 q^{14} - 49 q^{15} + 83 q^{16} - 62 q^{17} - 33 q^{18} - 25 q^{19} - 39 q^{20} - 15 q^{21} - 31 q^{22} - 94 q^{23} - 39 q^{24} + 71 q^{25} - 35 q^{26} - 47 q^{27} - 13 q^{28} - 17 q^{29} + 15 q^{30} - 37 q^{31} - 105 q^{32} - 60 q^{33} + 9 q^{34} - 60 q^{35} + 43 q^{36} - 79 q^{37} - 80 q^{38} - 41 q^{39} - 64 q^{40} - 37 q^{41} - 30 q^{42} - 20 q^{43} - 14 q^{44} - 8 q^{45} + 61 q^{46} - 148 q^{47} - 39 q^{48} + 82 q^{49} - 90 q^{50} - 45 q^{51} - 27 q^{52} - 70 q^{53} - 41 q^{54} - 105 q^{55} - 68 q^{56} - 31 q^{57} - 14 q^{58} - 96 q^{59} - 74 q^{60} - 21 q^{61} + 4 q^{62} - 60 q^{63} + 132 q^{64} - 15 q^{65} + 71 q^{66} - 44 q^{67} - 166 q^{68} - 72 q^{69} - 9 q^{70} - 55 q^{71} - 126 q^{72} - 27 q^{73} + 11 q^{74} - 39 q^{75} - 4 q^{76} - 104 q^{77} - 47 q^{78} - 49 q^{79} - 82 q^{80} + 55 q^{81} + 20 q^{82} - 52 q^{83} - 29 q^{84} + 3 q^{85} - 32 q^{86} - 113 q^{87} + 14 q^{88} - 68 q^{89} - 39 q^{90} - 30 q^{91} - 179 q^{92} - 53 q^{93} - 33 q^{94} - 86 q^{95} + 33 q^{96} - 57 q^{97} - 116 q^{98} + q^{99}+O(q^{100}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9 - 13 * q^10 - 5 * q^11 - 40 * q^12 - 18 * q^13 - 42 * q^14 - 49 * q^15 + 83 * q^16 - 62 * q^17 - 33 * q^18 - 25 * q^19 - 39 * q^20 - 15 * q^21 - 31 * q^22 - 94 * q^23 - 39 * q^24 + 71 * q^25 - 35 * q^26 - 47 * q^27 - 13 * q^28 - 17 * q^29 + 15 * q^30 - 37 * q^31 - 105 * q^32 - 60 * q^33 + 9 * q^34 - 60 * q^35 + 43 * q^36 - 79 * q^37 - 80 * q^38 - 41 * q^39 - 64 * q^40 - 37 * q^41 - 30 * q^42 - 20 * q^43 - 14 * q^44 - 8 * q^45 + 61 * q^46 - 148 * q^47 - 39 * q^48 + 82 * q^49 - 90 * q^50 - 45 * q^51 - 27 * q^52 - 70 * q^53 - 41 * q^54 - 105 * q^55 - 68 * q^56 - 31 * q^57 - 14 * q^58 - 96 * q^59 - 74 * q^60 - 21 * q^61 + 4 * q^62 - 60 * q^63 + 132 * q^64 - 15 * q^65 + 71 * q^66 - 44 * q^67 - 166 * q^68 - 72 * q^69 - 9 * q^70 - 55 * q^71 - 126 * q^72 - 27 * q^73 + 11 * q^74 - 39 * q^75 - 4 * q^76 - 104 * q^77 - 47 * q^78 - 49 * q^79 - 82 * q^80 + 55 * q^81 + 20 * q^82 - 52 * q^83 - 29 * q^84 + 3 * q^85 - 32 * q^86 - 113 * q^87 + 14 * q^88 - 68 * q^89 - 39 * q^90 - 30 * q^91 - 179 * q^92 - 53 * q^93 - 33 * q^94 - 86 * q^95 + 33 * q^96 - 57 * q^97 - 116 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more