LMFDB - Embedded newform 4031.2.a.b.1.11

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:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-1.84055 q^{2} +2.78518 q^{3} +1.38763 q^{4} +3.40108 q^{5} -5.12627 q^{6} -2.70641 q^{7} +1.12710 q^{8} +4.75723 q^{9} +O(q^{10})$	\(q-1.84055 q^{2} +2.78518 q^{3} +1.38763 q^{4} +3.40108 q^{5} -5.12627 q^{6} -2.70641 q^{7} +1.12710 q^{8} +4.75723 q^{9} -6.25987 q^{10} -2.96001 q^{11} +3.86479 q^{12} -3.44469 q^{13} +4.98129 q^{14} +9.47263 q^{15} -4.84975 q^{16} -3.66642 q^{17} -8.75592 q^{18} -4.73149 q^{19} +4.71944 q^{20} -7.53784 q^{21} +5.44805 q^{22} -3.68840 q^{23} +3.13919 q^{24} +6.56738 q^{25} +6.34013 q^{26} +4.89420 q^{27} -3.75549 q^{28} +1.00000 q^{29} -17.4349 q^{30} +3.67132 q^{31} +6.67199 q^{32} -8.24417 q^{33} +6.74824 q^{34} -9.20473 q^{35} +6.60126 q^{36} -1.58125 q^{37} +8.70855 q^{38} -9.59408 q^{39} +3.83338 q^{40} +0.222984 q^{41} +13.8738 q^{42} +2.55774 q^{43} -4.10739 q^{44} +16.1797 q^{45} +6.78868 q^{46} +1.00725 q^{47} -13.5074 q^{48} +0.324657 q^{49} -12.0876 q^{50} -10.2116 q^{51} -4.77994 q^{52} -9.88448 q^{53} -9.00802 q^{54} -10.0673 q^{55} -3.05041 q^{56} -13.1781 q^{57} -1.84055 q^{58} -4.50967 q^{59} +13.1445 q^{60} +3.25908 q^{61} -6.75724 q^{62} -12.8750 q^{63} -2.58065 q^{64} -11.7157 q^{65} +15.1738 q^{66} -1.85857 q^{67} -5.08763 q^{68} -10.2729 q^{69} +16.9418 q^{70} +11.1183 q^{71} +5.36189 q^{72} +3.93510 q^{73} +2.91038 q^{74} +18.2913 q^{75} -6.56555 q^{76} +8.01101 q^{77} +17.6584 q^{78} -8.52439 q^{79} -16.4944 q^{80} -0.640459 q^{81} -0.410414 q^{82} +5.27801 q^{83} -10.4597 q^{84} -12.4698 q^{85} -4.70765 q^{86} +2.78518 q^{87} -3.33624 q^{88} -10.4523 q^{89} -29.7796 q^{90} +9.32274 q^{91} -5.11812 q^{92} +10.2253 q^{93} -1.85390 q^{94} -16.0922 q^{95} +18.5827 q^{96} -0.0866808 q^{97} -0.597548 q^{98} -14.0815 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * 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 $

$2$	−1.84055	−1.30147	−0.650733	−	0.759307i	$-0.725538\pi$
$2$	−1.84055	−1.30147	−0.650733	+	0.759307i	$0.725538\pi$
$3$	2.78518	1.60802	0.804012	−	0.594613i	$-0.202695\pi$
$3$	2.78518	1.60802	0.804012	+	0.594613i	$0.202695\pi$
$4$	1.38763	0.693813
$5$	3.40108	1.52101	0.760506	−	0.649331i	$-0.224951\pi$
$5$	3.40108	1.52101	0.760506	+	0.649331i	$0.224951\pi$
$6$	−5.12627	−2.09279
$7$	−2.70641	−1.02293	−0.511463	−	0.859305i	$-0.670896\pi$
$7$	−2.70641	−1.02293	−0.511463	+	0.859305i	$0.670896\pi$
$8$	1.12710	0.398492
$9$	4.75723	1.58574
$10$	−6.25987	−1.97954
$11$	−2.96001	−0.892477	−0.446239	−	0.894914i	$-0.647237\pi$
$11$	−2.96001	−0.892477	−0.446239	+	0.894914i	$0.647237\pi$
$12$	3.86479	1.11567
$13$	−3.44469	−0.955385	−0.477693	−	0.878527i	$-0.658527\pi$
$13$	−3.44469	−0.955385	−0.477693	+	0.878527i	$0.658527\pi$
$14$	4.98129	1.33130
$15$	9.47263	2.44582
$16$	−4.84975	−1.21244
$17$	−3.66642	−0.889238	−0.444619	−	0.895720i	$-0.646661\pi$
$17$	−3.66642	−0.889238	−0.444619	+	0.895720i	$0.646661\pi$
$18$	−8.75592	−2.06379
$19$	−4.73149	−1.08548	−0.542740	−	0.839901i	$-0.682613\pi$
$19$	−4.73149	−1.08548	−0.542740	+	0.839901i	$0.682613\pi$
$20$	4.71944	1.05530
$21$	−7.53784	−1.64489
$22$	5.44805	1.16153
$23$	−3.68840	−0.769084	−0.384542	−	0.923107i	$-0.625641\pi$
$23$	−3.68840	−0.769084	−0.384542	+	0.923107i	$0.625641\pi$
$24$	3.13919	0.640784
$25$	6.56738	1.31348
$26$	6.34013	1.24340
$27$	4.89420	0.941889
$28$	−3.75549	−0.709720
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−17.4349	−3.18316
$31$	3.67132	0.659388	0.329694	−	0.944088i	$-0.393054\pi$
$31$	3.67132	0.659388	0.329694	+	0.944088i	$0.393054\pi$
$32$	6.67199	1.17945
$33$	−8.24417	−1.43513
$34$	6.74824	1.15731
$35$	−9.20473	−1.55588
$36$	6.60126	1.10021
$37$	−1.58125	−0.259957	−0.129978	−	0.991517i	$-0.541491\pi$
$37$	−1.58125	−0.259957	−0.129978	+	0.991517i	$0.541491\pi$
$38$	8.70855	1.41271
$39$	−9.59408	−1.53628
$40$	3.83338	0.606110
$41$	0.222984	0.0348243	0.0174122	−	0.999848i	$-0.494457\pi$
$41$	0.222984	0.0348243	0.0174122	+	0.999848i	$0.494457\pi$
$42$	13.8738	2.14077
$43$	2.55774	0.390052	0.195026	−	0.980798i	$-0.437521\pi$
$43$	2.55774	0.390052	0.195026	+	0.980798i	$0.437521\pi$
$44$	−4.10739	−0.619213
$45$	16.1797	2.41193
$46$	6.78868	1.00094
$47$	1.00725	0.146923	0.0734614	−	0.997298i	$-0.476595\pi$
$47$	1.00725	0.146923	0.0734614	+	0.997298i	$0.476595\pi$
$48$	−13.5074	−1.94963
$49$	0.324657	0.0463796
$50$	−12.0876	−1.70944
$51$	−10.2116	−1.42992
$52$	−4.77994	−0.662859
$53$	−9.88448	−1.35774	−0.678869	−	0.734260i	$-0.737530\pi$
$53$	−9.88448	−1.35774	−0.678869	+	0.734260i	$0.737530\pi$
$54$	−9.00802	−1.22584
$55$	−10.0673	−1.35747
$56$	−3.05041	−0.407628
$57$	−13.1781	−1.74548
$58$	−1.84055	−0.241676
$59$	−4.50967	−0.587109	−0.293555	−	0.955942i	$-0.594838\pi$
$59$	−4.50967	−0.587109	−0.293555	+	0.955942i	$0.594838\pi$
$60$	13.1445	1.69694
$61$	3.25908	0.417282	0.208641	−	0.977992i	$-0.433096\pi$
$61$	3.25908	0.417282	0.208641	+	0.977992i	$0.433096\pi$
$62$	−6.75724	−0.858170
$63$	−12.8750	−1.62210
$64$	−2.58065	−0.322581
$65$	−11.7157	−1.45315
$66$	15.1738	1.86777
$67$	−1.85857	−0.227061	−0.113530	−	0.993535i	$-0.536216\pi$
$67$	−1.85857	−0.227061	−0.113530	+	0.993535i	$0.536216\pi$
$68$	−5.08763	−0.616965
$69$	−10.2729	−1.23671
$70$	16.9418	2.02493
$71$	11.1183	1.31950	0.659750	−	0.751485i	$-0.270662\pi$
$71$	11.1183	1.31950	0.659750	+	0.751485i	$0.270662\pi$
$72$	5.36189	0.631905
$73$	3.93510	0.460569	0.230284	−	0.973123i	$-0.426034\pi$
$73$	3.93510	0.460569	0.230284	+	0.973123i	$0.426034\pi$
$74$	2.91038	0.338325
$75$	18.2913	2.11210
$76$	−6.56555	−0.753120
$77$	8.01101	0.912939
$78$	17.6584	1.99942
$79$	−8.52439	−0.959069	−0.479534	−	0.877523i	$-0.659194\pi$
$79$	−8.52439	−0.959069	−0.479534	+	0.877523i	$0.659194\pi$
$80$	−16.4944	−1.84413
$81$	−0.640459	−0.0711621
$82$	−0.410414	−0.0453226
$83$	5.27801	0.579337	0.289669	−	0.957127i	$-0.406455\pi$
$83$	5.27801	0.579337	0.289669	+	0.957127i	$0.406455\pi$
$84$	−10.4597	−1.14125
$85$	−12.4698	−1.35254
$86$	−4.70765	−0.507639
$87$	2.78518	0.298603
$88$	−3.33624	−0.355645
$89$	−10.4523	−1.10794	−0.553969	−	0.832537i	$-0.686888\pi$
$89$	−10.4523	−1.10794	−0.553969	+	0.832537i	$0.686888\pi$
$90$	−29.7796	−3.13905
$91$	9.32274	0.977289
$92$	−5.11812	−0.533601
$93$	10.2253	1.06031
$94$	−1.85390	−0.191215
$95$	−16.0922	−1.65103
$96$	18.5827	1.89659
$97$	−0.0866808	−0.00880110	−0.00440055	−	0.999990i	$-0.501401\pi$
$97$	−0.0866808	−0.00880110	−0.00440055	+	0.999990i	$0.501401\pi$
$98$	−0.597548	−0.0603614
$99$	−14.0815	−1.41524
$100$	9.11307	0.911307
$101$	−12.0683	−1.20085	−0.600423	−	0.799683i	$-0.705001\pi$
$101$	−12.0683	−1.20085	−0.600423	+	0.799683i	$0.705001\pi$
$102$	18.7951	1.86099
$103$	8.93146	0.880043	0.440021	−	0.897987i	$-0.354971\pi$
$103$	8.93146	0.880043	0.440021	+	0.897987i	$0.354971\pi$
$104$	−3.88253	−0.380713
$105$	−25.6368	−2.50190
$106$	18.1929	1.76705
$107$	−16.0328	−1.54995	−0.774976	−	0.631991i	$-0.782238\pi$
$107$	−16.0328	−1.54995	−0.774976	+	0.631991i	$0.782238\pi$
$108$	6.79132	0.653495
$109$	−12.0000	−1.14939	−0.574694	−	0.818369i	$-0.694879\pi$
$109$	−12.0000	−1.14939	−0.574694	+	0.818369i	$0.694879\pi$
$110$	18.5293	1.76670
$111$	−4.40408	−0.418017
$112$	13.1254	1.24023
$113$	14.8882	1.40056	0.700281	−	0.713867i	$-0.253058\pi$
$113$	14.8882	1.40056	0.700281	+	0.713867i	$0.253058\pi$
$114$	24.2549	2.27168
$115$	−12.5446	−1.16979
$116$	1.38763	0.128838
$117$	−16.3872	−1.51500
$118$	8.30028	0.764103
$119$	9.92284	0.909626
$120$	10.6766	0.974640
$121$	−2.23832	−0.203484
$122$	−5.99850	−0.543079
$123$	0.621052	0.0559983
$124$	5.09441	0.457492
$125$	5.33078	0.476799
$126$	23.6971	2.11111
$127$	−3.30057	−0.292878	−0.146439	−	0.989220i	$-0.546781\pi$
$127$	−3.30057	−0.292878	−0.146439	+	0.989220i	$0.546781\pi$
$128$	−8.59417	−0.759624
$129$	7.12377	0.627213
$130$	21.5633	1.89123
$131$	−0.579716	−0.0506500	−0.0253250	−	0.999679i	$-0.508062\pi$
$131$	−0.579716	−0.0506500	−0.0253250	+	0.999679i	$0.508062\pi$
$132$	−11.4398	−0.995709
$133$	12.8054	1.11037
$134$	3.42080	0.295512
$135$	16.6456	1.43262
$136$	−4.13244	−0.354354
$137$	−18.5850	−1.58783	−0.793913	−	0.608031i	$-0.791960\pi$
$137$	−18.5850	−1.58783	−0.793913	+	0.608031i	$0.791960\pi$
$138$	18.9077	1.60953
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	−12.7727	−1.07949
$141$	2.80538	0.236255
$142$	−20.4638	−1.71729
$143$	10.1963	0.852660
$144$	−23.0714	−1.92261
$145$	3.40108	0.282445
$146$	−7.24275	−0.599414
$147$	0.904229	0.0745795
$148$	−2.19419	−0.180361
$149$	−23.3636	−1.91402	−0.957010	−	0.290056i	$-0.906326\pi$
$149$	−23.3636	−1.91402	−0.957010	+	0.290056i	$0.906326\pi$
$150$	−33.6661	−2.74883
$151$	−0.477277	−0.0388403	−0.0194201	−	0.999811i	$-0.506182\pi$
$151$	−0.477277	−0.0388403	−0.0194201	+	0.999811i	$0.506182\pi$
$152$	−5.33289	−0.432554
$153$	−17.4420	−1.41010
$154$	−14.7447	−1.18816
$155$	12.4865	1.00294
$156$	−13.3130	−1.06589
$157$	−10.4134	−0.831081	−0.415540	−	0.909575i	$-0.636408\pi$
$157$	−10.4134	−0.831081	−0.415540	+	0.909575i	$0.636408\pi$
$158$	15.6896	1.24819
$159$	−27.5301	−2.18328
$160$	22.6920	1.79396
$161$	9.98232	0.786717
$162$	1.17880	0.0926150
$163$	7.82859	0.613183	0.306591	−	0.951841i	$-0.400811\pi$
$163$	7.82859	0.613183	0.306591	+	0.951841i	$0.400811\pi$
$164$	0.309419	0.0241616
$165$	−28.0391	−2.18284
$166$	−9.71445	−0.753988
$167$	−16.4700	−1.27448	−0.637242	−	0.770664i	$-0.719925\pi$
$167$	−16.4700	−1.27448	−0.637242	+	0.770664i	$0.719925\pi$
$168$	−8.49593	−0.655475
$169$	−1.13411	−0.0872393
$170$	22.9513	1.76029
$171$	−22.5088	−1.72129
$172$	3.54919	0.270623
$173$	−0.167778	−0.0127559	−0.00637796	−	0.999980i	$-0.502030\pi$
$173$	−0.167778	−0.0127559	−0.00637796	+	0.999980i	$0.502030\pi$
$174$	−5.12627	−0.388621
$175$	−17.7740	−1.34359
$176$	14.3553	1.08207
$177$	−12.5602	−0.944086
$178$	19.2379	1.44194
$179$	7.13255	0.533112	0.266556	−	0.963819i	$-0.414114\pi$
$179$	7.13255	0.533112	0.266556	+	0.963819i	$0.414114\pi$
$180$	22.4514	1.67343
$181$	14.6967	1.09240	0.546200	−	0.837655i	$-0.316074\pi$
$181$	14.6967	1.09240	0.546200	+	0.837655i	$0.316074\pi$
$182$	−17.1590	−1.27191
$183$	9.07712	0.671000
$184$	−4.15721	−0.306474
$185$	−5.37798	−0.395397
$186$	−18.8201	−1.37996
$187$	10.8527	0.793625
$188$	1.39769	0.101937
$189$	−13.2457	−0.963484
$190$	29.6185	2.14875
$191$	1.90725	0.138004	0.0690019	−	0.997617i	$-0.478019\pi$
$191$	1.90725	0.138004	0.0690019	+	0.997617i	$0.478019\pi$
$192$	−7.18758	−0.518719
$193$	11.6486	0.838484	0.419242	−	0.907874i	$-0.362296\pi$
$193$	11.6486	0.838484	0.419242	+	0.907874i	$0.362296\pi$
$194$	0.159540	0.0114543
$195$	−32.6303	−2.33670
$196$	0.450503	0.0321788
$197$	−1.92432	−0.137102	−0.0685509	−	0.997648i	$-0.521838\pi$
$197$	−1.92432	−0.137102	−0.0685509	+	0.997648i	$0.521838\pi$
$198$	25.9176	1.84189
$199$	25.0660	1.77688	0.888441	−	0.458992i	$-0.151789\pi$
$199$	25.0660	1.77688	0.888441	+	0.458992i	$0.151789\pi$
$200$	7.40212	0.523409
$201$	−5.17646	−0.365120
$202$	22.2124	1.56286
$203$	−2.70641	−0.189953
$204$	−14.1700	−0.992095
$205$	0.758389	0.0529682
$206$	−16.4388	−1.14535
$207$	−17.5466	−1.21957
$208$	16.7059	1.15834
$209$	14.0053	0.968766
$210$	47.1859	3.25614
$211$	9.08897	0.625710	0.312855	−	0.949801i	$-0.398715\pi$
$211$	9.08897	0.625710	0.312855	+	0.949801i	$0.398715\pi$
$212$	−13.7160	−0.942016
$213$	30.9665	2.12179
$214$	29.5092	2.01721
$215$	8.69909	0.593273
$216$	5.51627	0.375335
$217$	−9.93609	−0.674505
$218$	22.0865	1.49589
$219$	10.9600	0.740606
$220$	−13.9696	−0.941829
$221$	12.6297	0.849565
$222$	8.10593	0.544034
$223$	11.1602	0.747340	0.373670	−	0.927562i	$-0.378099\pi$
$223$	11.1602	0.747340	0.373670	+	0.927562i	$0.378099\pi$
$224$	−18.0572	−1.20649
$225$	31.2425	2.08283
$226$	−27.4024	−1.82278
$227$	10.5347	0.699214	0.349607	−	0.936897i	$-0.386315\pi$
$227$	10.5347	0.699214	0.349607	+	0.936897i	$0.386315\pi$
$228$	−18.2862	−1.21104
$229$	−18.7136	−1.23663	−0.618315	−	0.785930i	$-0.712185\pi$
$229$	−18.7136	−1.23663	−0.618315	+	0.785930i	$0.712185\pi$
$230$	23.0889	1.52244
$231$	22.3121	1.46803
$232$	1.12710	0.0739980
$233$	13.6999	0.897511	0.448756	−	0.893654i	$-0.351867\pi$
$233$	13.6999	0.897511	0.448756	+	0.893654i	$0.351867\pi$
$234$	30.1614	1.97171
$235$	3.42575	0.223471
$236$	−6.25774	−0.407344
$237$	−23.7420	−1.54221
$238$	−18.2635	−1.18385
$239$	−27.2806	−1.76464	−0.882318	−	0.470654i	$-0.844018\pi$
$239$	−27.2806	−1.76464	−0.882318	+	0.470654i	$0.844018\pi$
$240$	−45.9399	−2.96541
$241$	−1.93701	−0.124774	−0.0623869	−	0.998052i	$-0.519871\pi$
$241$	−1.93701	−0.124774	−0.0623869	+	0.998052i	$0.519871\pi$
$242$	4.11975	0.264828
$243$	−16.4664	−1.05632
$244$	4.52238	0.289516
$245$	1.10419	0.0705439
$246$	−1.14308	−0.0728799
$247$	16.2985	1.03705
$248$	4.13796	0.262760
$249$	14.7002	0.931588
$250$	−9.81157	−0.620538
$251$	15.0206	0.948090	0.474045	−	0.880501i	$-0.342793\pi$
$251$	15.0206	0.948090	0.474045	+	0.880501i	$0.342793\pi$
$252$	−17.8657	−1.12543
$253$	10.9177	0.686390
$254$	6.07487	0.381171
$255$	−34.7307	−2.17492
$256$	20.9793	1.31121
$257$	16.0181	0.999180	0.499590	−	0.866262i	$-0.333484\pi$
$257$	16.0181	0.999180	0.499590	+	0.866262i	$0.333484\pi$
$258$	−13.1117	−0.816296
$259$	4.27952	0.265917
$260$	−16.2570	−1.00822
$261$	4.75723	0.294465
$262$	1.06700	0.0659193
$263$	14.4206	0.889214	0.444607	−	0.895726i	$-0.353343\pi$
$263$	14.4206	0.889214	0.444607	+	0.895726i	$0.353343\pi$
$264$	−9.29204	−0.571885
$265$	−33.6179	−2.06513
$266$	−23.5689	−1.44510
$267$	−29.1115	−1.78159
$268$	−2.57901	−0.157538
$269$	−4.47452	−0.272816	−0.136408	−	0.990653i	$-0.543556\pi$
$269$	−4.47452	−0.272816	−0.136408	+	0.990653i	$0.543556\pi$
$270$	−30.6370	−1.86451
$271$	−14.3858	−0.873876	−0.436938	−	0.899492i	$-0.643937\pi$
$271$	−14.3858	−0.873876	−0.436938	+	0.899492i	$0.643937\pi$
$272$	17.7812	1.07814
$273$	25.9655	1.57151
$274$	34.2067	2.06650
$275$	−19.4395	−1.17225
$276$	−14.2549	−0.858043
$277$	15.6985	0.943233	0.471617	−	0.881804i	$-0.343671\pi$
$277$	15.6985	0.943233	0.471617	+	0.881804i	$0.343671\pi$
$278$	−1.84055	−0.110389
$279$	17.4653	1.04562
$280$	−10.3747	−0.620006
$281$	27.4686	1.63864	0.819320	−	0.573336i	$-0.194351\pi$
$281$	27.4686	1.63864	0.819320	+	0.573336i	$0.194351\pi$
$282$	−5.16344	−0.307478
$283$	−14.5043	−0.862191	−0.431095	−	0.902306i	$-0.641873\pi$
$283$	−14.5043	−0.862191	−0.431095	+	0.902306i	$0.641873\pi$
$284$	15.4281	0.915487
$285$	−44.8197	−2.65489
$286$	−18.7669	−1.10971
$287$	−0.603487	−0.0356227
$288$	31.7402	1.87031
$289$	−3.55734	−0.209256
$290$	−6.25987	−0.367592
$291$	−0.241422	−0.0141524
$292$	5.46045	0.319549
$293$	−10.6283	−0.620912	−0.310456	−	0.950588i	$-0.600482\pi$
$293$	−10.6283	−0.620912	−0.310456	+	0.950588i	$0.600482\pi$
$294$	−1.66428	−0.0970627
$295$	−15.3378	−0.893000
$296$	−1.78224	−0.103591
$297$	−14.4869	−0.840615
$298$	43.0019	2.49103
$299$	12.7054	0.734771
$300$	25.3815	1.46540
$301$	−6.92230	−0.398995
$302$	0.878453	0.0505493
$303$	−33.6125	−1.93099
$304$	22.9465	1.31607
$305$	11.0844	0.634691
$306$	32.1029	1.83520
$307$	−7.28751	−0.415920	−0.207960	−	0.978137i	$-0.566682\pi$
$307$	−7.28751	−0.415920	−0.207960	+	0.978137i	$0.566682\pi$
$308$	11.1163	0.633409
$309$	24.8757	1.41513
$310$	−22.9819	−1.30529
$311$	12.7492	0.722941	0.361470	−	0.932384i	$-0.382275\pi$
$311$	12.7492	0.722941	0.361470	+	0.932384i	$0.382275\pi$
$312$	−10.8135	−0.612196
$313$	−13.8306	−0.781753	−0.390877	−	0.920443i	$-0.627828\pi$
$313$	−13.8306	−0.781753	−0.390877	+	0.920443i	$0.627828\pi$
$314$	19.1664	1.08162
$315$	−43.7890	−2.46723
$316$	−11.8287	−0.665414
$317$	11.5486	0.648635	0.324318	−	0.945948i	$-0.394865\pi$
$317$	11.5486	0.648635	0.324318	+	0.945948i	$0.394865\pi$
$318$	50.6705	2.84146
$319$	−2.96001	−0.165729
$320$	−8.77701	−0.490650
$321$	−44.6543	−2.49236
$322$	−18.3730	−1.02389
$323$	17.3477	0.965249
$324$	−0.888717	−0.0493732
$325$	−22.6226	−1.25487
$326$	−14.4089	−0.798036
$327$	−33.4220	−1.84824
$328$	0.251327	0.0138772
$329$	−2.72604	−0.150291
$330$	51.6074	2.84089
$331$	16.7129	0.918625	0.459312	−	0.888275i	$-0.348096\pi$
$331$	16.7129	0.918625	0.459312	+	0.888275i	$0.348096\pi$
$332$	7.32391	0.401952
$333$	−7.52239	−0.412224
$334$	30.3138	1.65870
$335$	−6.32117	−0.345362
$336$	36.5566	1.99433
$337$	−0.318569	−0.0173536	−0.00867678	−	0.999962i	$-0.502762\pi$
$337$	−0.318569	−0.0173536	−0.00867678	+	0.999962i	$0.502762\pi$
$338$	2.08739	0.113539
$339$	41.4662	2.25214
$340$	−17.3034	−0.938411
$341$	−10.8671	−0.588489
$342$	41.4286	2.24020
$343$	18.0662	0.975484
$344$	2.88284	0.155432
$345$	−34.9388	−1.88104
$346$	0.308804	0.0166014
$347$	21.4536	1.15169	0.575845	−	0.817559i	$-0.304673\pi$
$347$	21.4536	1.15169	0.575845	+	0.817559i	$0.304673\pi$
$348$	3.86479	0.207174
$349$	33.8069	1.80964	0.904820	−	0.425794i	$-0.140005\pi$
$349$	33.8069	1.80964	0.904820	+	0.425794i	$0.140005\pi$
$350$	32.7140	1.74864
$351$	−16.8590	−0.899867
$352$	−19.7492	−1.05264
$353$	19.0214	1.01241	0.506203	−	0.862414i	$-0.331049\pi$
$353$	19.0214	1.01241	0.506203	+	0.862414i	$0.331049\pi$
$354$	23.1178	1.22870
$355$	37.8143	2.00698
$356$	−14.5039	−0.768703
$357$	27.6369	1.46270
$358$	−13.1278	−0.693827
$359$	2.77159	0.146279	0.0731395	−	0.997322i	$-0.476698\pi$
$359$	2.77159	0.146279	0.0731395	+	0.997322i	$0.476698\pi$
$360$	18.2363	0.961135
$361$	3.38703	0.178265
$362$	−27.0501	−1.42172
$363$	−6.23414	−0.327207
$364$	12.9365	0.678056
$365$	13.3836	0.700530
$366$	−16.7069	−0.873284
$367$	5.36547	0.280075	0.140038	−	0.990146i	$-0.455278\pi$
$367$	5.36547	0.280075	0.140038	+	0.990146i	$0.455278\pi$
$368$	17.8878	0.932466
$369$	1.06079	0.0552224
$370$	9.89844	0.514596
$371$	26.7515	1.38887
$372$	14.1889	0.735658
$373$	−23.7474	−1.22959	−0.614796	−	0.788686i	$-0.710762\pi$
$373$	−23.7474	−1.22959	−0.614796	+	0.788686i	$0.710762\pi$
$374$	−19.9749	−1.03288
$375$	14.8472	0.766705
$376$	1.13528	0.0585475
$377$	−3.44469	−0.177411
$378$	24.3794	1.25394
$379$	−21.1052	−1.08410	−0.542051	−	0.840346i	$-0.682352\pi$
$379$	−21.1052	−1.08410	−0.542051	+	0.840346i	$0.682352\pi$
$380$	−22.3300	−1.14550
$381$	−9.19269	−0.470956
$382$	−3.51039	−0.179607
$383$	1.65467	0.0845497	0.0422749	−	0.999106i	$-0.486539\pi$
$383$	1.65467	0.0845497	0.0422749	+	0.999106i	$0.486539\pi$
$384$	−23.9363	−1.22149
$385$	27.2461	1.38859
$386$	−21.4398	−1.09126
$387$	12.1678	0.618522
$388$	−0.120281	−0.00610632
$389$	−3.61813	−0.183447	−0.0917233	−	0.995785i	$-0.529238\pi$
$389$	−3.61813	−0.183447	−0.0917233	+	0.995785i	$0.529238\pi$
$390$	60.0577	3.04114
$391$	13.5232	0.683899
$392$	0.365922	0.0184819
$393$	−1.61461	−0.0814465
$394$	3.54180	0.178433
$395$	−28.9922	−1.45875
$396$	−19.5398	−0.981912
$397$	0.151402	0.00759866	0.00379933	−	0.999993i	$-0.498791\pi$
$397$	0.151402	0.00759866	0.00379933	+	0.999993i	$0.498791\pi$
$398$	−46.1352	−2.31255
$399$	35.6652	1.78550
$400$	−31.8501	−1.59251
$401$	−0.182981	−0.00913763	−0.00456881	−	0.999990i	$-0.501454\pi$
$401$	−0.182981	−0.00913763	−0.00456881	+	0.999990i	$0.501454\pi$
$402$	9.52754	0.475191
$403$	−12.6465	−0.629969
$404$	−16.7464	−0.833162
$405$	−2.17825	−0.108238
$406$	4.98129	0.247217
$407$	4.68053	0.232005
$408$	−11.5096	−0.569810
$409$	−22.2120	−1.09831	−0.549156	−	0.835720i	$-0.685051\pi$
$409$	−22.2120	−1.09831	−0.549156	+	0.835720i	$0.685051\pi$
$410$	−1.39585	−0.0689363
$411$	−51.7627	−2.55326
$412$	12.3935	0.610585
$413$	12.2050	0.600570
$414$	32.2953	1.58723
$415$	17.9510	0.881178
$416$	−22.9829	−1.12683
$417$	2.78518	0.136391
$418$	−25.7774	−1.26082
$419$	−17.9762	−0.878197	−0.439099	−	0.898439i	$-0.644702\pi$
$419$	−17.9762	−0.878197	−0.439099	+	0.898439i	$0.644702\pi$
$420$	−35.5743	−1.73585
$421$	−11.7359	−0.571972	−0.285986	−	0.958234i	$-0.592321\pi$
$421$	−11.7359	−0.571972	−0.285986	+	0.958234i	$0.592321\pi$
$422$	−16.7287	−0.814341
$423$	4.79173	0.232982
$424$	−11.1408	−0.541047
$425$	−24.0788	−1.16799
$426$	−56.9954	−2.76144
$427$	−8.82041	−0.426849
$428$	−22.2476	−1.07538
$429$	28.3986	1.37110
$430$	−16.0111	−0.772125
$431$	36.1482	1.74120	0.870600	−	0.491992i	$-0.163731\pi$
$431$	36.1482	1.74120	0.870600	+	0.491992i	$0.163731\pi$
$432$	−23.7356	−1.14198
$433$	−21.4714	−1.03185	−0.515924	−	0.856634i	$-0.672551\pi$
$433$	−21.4714	−1.03185	−0.515924	+	0.856634i	$0.672551\pi$
$434$	18.2879	0.877846
$435$	9.47263	0.454178
$436$	−16.6515	−0.797460
$437$	17.4516	0.834825
$438$	−20.1724	−0.963873
$439$	−32.8396	−1.56735	−0.783675	−	0.621172i	$-0.786657\pi$
$439$	−32.8396	−1.56735	−0.783675	+	0.621172i	$0.786657\pi$
$440$	−11.3468	−0.540940
$441$	1.54447	0.0735461
$442$	−23.2456	−1.10568
$443$	37.2372	1.76919	0.884595	−	0.466359i	$-0.154435\pi$
$443$	37.2372	1.76919	0.884595	+	0.466359i	$0.154435\pi$
$444$	−6.11122	−0.290026
$445$	−35.5491	−1.68519
$446$	−20.5409	−0.972637
$447$	−65.0718	−3.07779
$448$	6.98430	0.329977
$449$	12.3581	0.583213	0.291607	−	0.956538i	$-0.405810\pi$
$449$	12.3581	0.583213	0.291607	+	0.956538i	$0.405810\pi$
$450$	−57.5034	−2.71074
$451$	−0.660037	−0.0310799
$452$	20.6592	0.971728
$453$	−1.32930	−0.0624561
$454$	−19.3897	−0.910003
$455$	31.7074	1.48647
$456$	−14.8531	−0.695558
$457$	−29.7471	−1.39151	−0.695756	−	0.718278i	$-0.744930\pi$
$457$	−29.7471	−1.39151	−0.695756	+	0.718278i	$0.744930\pi$
$458$	34.4434	1.60943
$459$	−17.9442	−0.837564
$460$	−17.4072	−0.811613
$461$	5.95535	0.277368	0.138684	−	0.990337i	$-0.455713\pi$
$461$	5.95535	0.277368	0.138684	+	0.990337i	$0.455713\pi$
$462$	−41.0666	−1.91059
$463$	14.2754	0.663432	0.331716	−	0.943379i	$-0.392372\pi$
$463$	14.2754	0.663432	0.331716	+	0.943379i	$0.392372\pi$
$464$	−4.84975	−0.225144
$465$	34.7770	1.61275
$466$	−25.2154	−1.16808
$467$	2.55459	0.118212	0.0591062	−	0.998252i	$-0.481175\pi$
$467$	2.55459	0.118212	0.0591062	+	0.998252i	$0.481175\pi$
$468$	−22.7393	−1.05112
$469$	5.03006	0.232267
$470$	−6.30526	−0.290840
$471$	−29.0032	−1.33640
$472$	−5.08287	−0.233958
$473$	−7.57095	−0.348113
$474$	43.6983	2.00713
$475$	−31.0735	−1.42575
$476$	13.7692	0.631110
$477$	−47.0227	−2.15302
$478$	50.2113	2.29661
$479$	−0.672963	−0.0307485	−0.0153742	−	0.999882i	$-0.504894\pi$
$479$	−0.672963	−0.0307485	−0.0153742	+	0.999882i	$0.504894\pi$
$480$	63.2013	2.88473
$481$	5.44693	0.248359
$482$	3.56517	0.162389
$483$	27.8026	1.26506
$484$	−3.10596	−0.141180
$485$	−0.294809	−0.0133866
$486$	30.3072	1.37476
$487$	−0.885746	−0.0401370	−0.0200685	−	0.999799i	$-0.506388\pi$
$487$	−0.885746	−0.0401370	−0.0200685	+	0.999799i	$0.506388\pi$
$488$	3.67332	0.166283
$489$	21.8040	0.986013
$490$	−2.03231	−0.0918104
$491$	18.9753	0.856342	0.428171	−	0.903698i	$-0.359158\pi$
$491$	18.9753	0.856342	0.428171	+	0.903698i	$0.359158\pi$
$492$	0.861788	0.0388524
$493$	−3.66642	−0.165127
$494$	−29.9983	−1.34969
$495$	−47.8922	−2.15260
$496$	−17.8049	−0.799466
$497$	−30.0907	−1.34975
$498$	−27.0565	−1.21243
$499$	11.4773	0.513795	0.256897	−	0.966439i	$-0.417300\pi$
$499$	11.4773	0.513795	0.256897	+	0.966439i	$0.417300\pi$
$500$	7.39713	0.330810
$501$	−45.8718	−2.04940
$502$	−27.6461	−1.23391
$503$	16.2657	0.725250	0.362625	−	0.931935i	$-0.381881\pi$
$503$	16.2657	0.725250	0.362625	+	0.931935i	$0.381881\pi$
$504$	−14.5115	−0.646393
$505$	−41.0455	−1.82650
$506$	−20.0946	−0.893313
$507$	−3.15870	−0.140283
$508$	−4.57996	−0.203203
$509$	10.3471	0.458627	0.229314	−	0.973353i	$-0.426352\pi$
$509$	10.3471	0.458627	0.229314	+	0.973353i	$0.426352\pi$
$510$	63.9236	2.83058
$511$	−10.6500	−0.471128
$512$	−21.4251	−0.946866
$513$	−23.1569	−1.02240
$514$	−29.4821	−1.30040
$515$	30.3766	1.33856
$516$	9.88513	0.435169
$517$	−2.98148	−0.131125
$518$	−7.87668	−0.346081
$519$	−0.467292	−0.0205118
$520$	−13.2048	−0.579069
$521$	−6.87336	−0.301128	−0.150564	−	0.988600i	$-0.548109\pi$
$521$	−6.87336	−0.301128	−0.150564	+	0.988600i	$0.548109\pi$
$522$	−8.75592	−0.383236
$523$	−19.7599	−0.864040	−0.432020	−	0.901864i	$-0.642199\pi$
$523$	−19.7599	−0.864040	−0.432020	+	0.901864i	$0.642199\pi$
$524$	−0.804429	−0.0351417
$525$	−49.5038	−2.16052
$526$	−26.5419	−1.15728
$527$	−13.4606	−0.586353
$528$	39.9821	1.74000
$529$	−9.39572	−0.408510
$530$	61.8755	2.68770
$531$	−21.4535	−0.931004
$532$	17.7691	0.770387
$533$	−0.768112	−0.0332706
$534$	53.5811	2.31868
$535$	−54.5290	−2.35749
$536$	−2.09481	−0.0904819
$537$	19.8654	0.857257
$538$	8.23558	0.355061
$539$	−0.960989	−0.0413927
$540$	23.0979	0.993974
$541$	−4.00955	−0.172384	−0.0861920	−	0.996279i	$-0.527470\pi$
$541$	−4.00955	−0.172384	−0.0861920	+	0.996279i	$0.527470\pi$
$542$	26.4778	1.13732
$543$	40.9330	1.75661
$544$	−24.4623	−1.04881
$545$	−40.8129	−1.74823
$546$	−47.7909	−2.04526
$547$	−5.61017	−0.239874	−0.119937	−	0.992782i	$-0.538269\pi$
$547$	−5.61017	−0.239874	−0.119937	+	0.992782i	$0.538269\pi$
$548$	−25.7891	−1.10166
$549$	15.5042	0.661703
$550$	35.7794	1.52564
$551$	−4.73149	−0.201568
$552$	−11.5786	−0.492817
$553$	23.0705	0.981057
$554$	−28.8939	−1.22759
$555$	−14.9786	−0.635808
$556$	1.38763	0.0588485
$557$	18.3772	0.778666	0.389333	−	0.921097i	$-0.372705\pi$
$557$	18.3772	0.778666	0.389333	+	0.921097i	$0.372705\pi$
$558$	−32.1457	−1.36084
$559$	−8.81063	−0.372650
$560$	44.6406	1.88641
$561$	30.2266	1.27617
$562$	−50.5574	−2.13263
$563$	−47.2960	−1.99329	−0.996645	−	0.0818493i	$-0.973917\pi$
$563$	−47.2960	−1.99329	−0.996645	+	0.0818493i	$0.973917\pi$
$564$	3.89281	0.163917
$565$	50.6359	2.13027
$566$	26.6959	1.12211
$567$	1.73334	0.0727936
$568$	12.5315	0.525810
$569$	−6.22180	−0.260831	−0.130416	−	0.991459i	$-0.541631\pi$
$569$	−6.22180	−0.260831	−0.130416	+	0.991459i	$0.541631\pi$
$570$	82.4929	3.45525
$571$	−21.9710	−0.919457	−0.459728	−	0.888060i	$-0.652053\pi$
$571$	−21.9710	−0.919457	−0.459728	+	0.888060i	$0.652053\pi$
$572$	14.1487	0.591587
$573$	5.31204	0.221914
$574$	1.11075	0.0463618
$575$	−24.2231	−1.01017
$576$	−12.2767	−0.511531
$577$	−4.85802	−0.202242	−0.101121	−	0.994874i	$-0.532243\pi$
$577$	−4.85802	−0.202242	−0.101121	+	0.994874i	$0.532243\pi$
$578$	6.54747	0.272339
$579$	32.4434	1.34830
$580$	4.71944	0.195964
$581$	−14.2845	−0.592620
$582$	0.444349	0.0184189
$583$	29.2582	1.21175
$584$	4.43527	0.183533
$585$	−55.7342	−2.30432
$586$	19.5619	0.808095
$587$	37.6497	1.55397	0.776986	−	0.629518i	$-0.216748\pi$
$587$	37.6497	1.55397	0.776986	+	0.629518i	$0.216748\pi$
$588$	1.25473	0.0517443
$589$	−17.3708	−0.715751
$590$	28.2299	1.16221
$591$	−5.35957	−0.220463
$592$	7.66868	0.315181
$593$	−20.3741	−0.836666	−0.418333	−	0.908294i	$-0.637385\pi$
$593$	−20.3741	−0.836666	−0.418333	+	0.908294i	$0.637385\pi$
$594$	26.6639	1.09403
$595$	33.7484	1.38355
$596$	−32.4199	−1.32797
$597$	69.8133	2.85727
$598$	−23.3849	−0.956280
$599$	−6.78582	−0.277261	−0.138631	−	0.990344i	$-0.544270\pi$
$599$	−6.78582	−0.277261	−0.138631	+	0.990344i	$0.544270\pi$
$600$	20.6162	0.841654
$601$	−39.5883	−1.61484	−0.807419	−	0.589978i	$-0.799136\pi$
$601$	−39.5883	−1.61484	−0.807419	+	0.589978i	$0.799136\pi$
$602$	12.7408	0.519278
$603$	−8.84166	−0.360060
$604$	−0.662283	−0.0269479
$605$	−7.61273	−0.309502
$606$	61.8655	2.51312
$607$	12.7991	0.519498	0.259749	−	0.965676i	$-0.416360\pi$
$607$	12.7991	0.519498	0.259749	+	0.965676i	$0.416360\pi$
$608$	−31.5685	−1.28027
$609$	−7.53784	−0.305449
$610$	−20.4014	−0.826029
$611$	−3.46967	−0.140368
$612$	−24.2030	−0.978348
$613$	−29.6331	−1.19687	−0.598435	−	0.801171i	$-0.704211\pi$
$613$	−29.6331	−1.19687	−0.598435	+	0.801171i	$0.704211\pi$
$614$	13.4130	0.541306
$615$	2.11225	0.0851741
$616$	9.02924	0.363799
$617$	−12.1608	−0.489576	−0.244788	−	0.969577i	$-0.578718\pi$
$617$	−12.1608	−0.489576	−0.244788	+	0.969577i	$0.578718\pi$
$618$	−45.7850	−1.84174
$619$	8.67709	0.348762	0.174381	−	0.984678i	$-0.444208\pi$
$619$	8.67709	0.348762	0.174381	+	0.984678i	$0.444208\pi$
$620$	17.3265	0.695850
$621$	−18.0518	−0.724392
$622$	−23.4655	−0.940883
$623$	28.2881	1.13334
$624$	46.5289	1.86265
$625$	−14.7065	−0.588258
$626$	25.4560	1.01743
$627$	39.0072	1.55780
$628$	−14.4499	−0.576615
$629$	5.79755	0.231163
$630$	80.5959	3.21102
$631$	33.3579	1.32796	0.663979	−	0.747751i	$-0.268866\pi$
$631$	33.3579	1.32796	0.663979	+	0.747751i	$0.268866\pi$
$632$	−9.60788	−0.382181
$633$	25.3144	1.00616
$634$	−21.2558	−0.844177
$635$	−11.2255	−0.445471
$636$	−38.2014	−1.51479
$637$	−1.11834	−0.0443104
$638$	5.44805	0.215690
$639$	52.8924	2.09239
$640$	−29.2295	−1.15540
$641$	−40.0813	−1.58312	−0.791558	−	0.611094i	$-0.790730\pi$
$641$	−40.0813	−1.58312	−0.791558	+	0.611094i	$0.790730\pi$
$642$	82.1885	3.24372
$643$	−37.4155	−1.47552	−0.737762	−	0.675060i	$-0.764118\pi$
$643$	−37.4155	−1.47552	−0.737762	+	0.675060i	$0.764118\pi$
$644$	13.8517	0.545835
$645$	24.2285	0.953998
$646$	−31.9292	−1.25624
$647$	−7.48976	−0.294453	−0.147226	−	0.989103i	$-0.547035\pi$
$647$	−7.48976	−0.294453	−0.147226	+	0.989103i	$0.547035\pi$
$648$	−0.721864	−0.0283575
$649$	13.3487	0.523982
$650$	41.6380	1.63318
$651$	−27.6738	−1.08462
$652$	10.8632	0.425434
$653$	−5.95236	−0.232934	−0.116467	−	0.993195i	$-0.537157\pi$
$653$	−5.95236	−0.232934	−0.116467	+	0.993195i	$0.537157\pi$
$654$	61.5150	2.40542
$655$	−1.97166	−0.0770392
$656$	−1.08142	−0.0422223
$657$	18.7202	0.730343
$658$	5.01741	0.195599
$659$	−2.99657	−0.116730	−0.0583649	−	0.998295i	$-0.518589\pi$
$659$	−2.99657	−0.116730	−0.0583649	+	0.998295i	$0.518589\pi$
$660$	−38.9078	−1.51448
$661$	−19.5666	−0.761052	−0.380526	−	0.924770i	$-0.624257\pi$
$661$	−19.5666	−0.761052	−0.380526	+	0.924770i	$0.624257\pi$
$662$	−30.7610	−1.19556
$663$	35.1760	1.36612
$664$	5.94887	0.230861
$665$	43.5521	1.68888
$666$	13.8453	0.536496
$667$	−3.68840	−0.142815
$668$	−22.8542	−0.884254
$669$	31.0831	1.20174
$670$	11.6344	0.449477
$671$	−9.64692	−0.372415
$672$	−50.2924	−1.94007
$673$	2.63692	0.101646	0.0508229	−	0.998708i	$-0.483816\pi$
$673$	2.63692	0.101646	0.0508229	+	0.998708i	$0.483816\pi$
$674$	0.586343	0.0225851
$675$	32.1421	1.23715
$676$	−1.57372	−0.0605278
$677$	34.8840	1.34070	0.670351	−	0.742044i	$-0.266144\pi$
$677$	34.8840	1.34070	0.670351	+	0.742044i	$0.266144\pi$
$678$	−76.3207	−2.93108
$679$	0.234594	0.00900289
$680$	−14.0548	−0.538976
$681$	29.3411	1.12435
$682$	20.0015	0.765898
$683$	31.6342	1.21045	0.605225	−	0.796054i	$-0.293083\pi$
$683$	31.6342	1.21045	0.605225	+	0.796054i	$0.293083\pi$
$684$	−31.2338	−1.19425
$685$	−63.2093	−2.41510
$686$	−33.2518	−1.26956
$687$	−52.1208	−1.98853
$688$	−12.4044	−0.472913
$689$	34.0490	1.29716
$690$	64.3067	2.44811
$691$	44.8742	1.70710	0.853548	−	0.521014i	$-0.174446\pi$
$691$	44.8742	1.70710	0.853548	+	0.521014i	$0.174446\pi$
$692$	−0.232813	−0.00885022
$693$	38.1102	1.44769
$694$	−39.4865	−1.49889
$695$	3.40108	0.129010
$696$	3.13919	0.118991
$697$	−0.817555	−0.0309671
$698$	−62.2233	−2.35519
$699$	38.1567	1.44322
$700$	−24.6637	−0.932200
$701$	16.8960	0.638154	0.319077	−	0.947729i	$-0.396627\pi$
$701$	16.8960	0.638154	0.319077	+	0.947729i	$0.396627\pi$
$702$	31.0298	1.17115
$703$	7.48170	0.282177
$704$	7.63876	0.287897
$705$	9.54132	0.359347
$706$	−35.0098	−1.31761
$707$	32.6619	1.22838
$708$	−17.4289	−0.655019
$709$	40.6367	1.52614	0.763071	−	0.646315i	$-0.223691\pi$
$709$	40.6367	1.52614	0.763071	+	0.646315i	$0.223691\pi$
$710$	−69.5992	−2.61201
$711$	−40.5525	−1.52084
$712$	−11.7808	−0.441504
$713$	−13.5413	−0.507125
$714$	−50.8671	−1.90365
$715$	34.6786	1.29690
$716$	9.89731	0.369880
$717$	−75.9814	−2.83758
$718$	−5.10126	−0.190377
$719$	−15.4146	−0.574866	−0.287433	−	0.957801i	$-0.592802\pi$
$719$	−15.4146	−0.574866	−0.287433	+	0.957801i	$0.592802\pi$
$720$	−78.4676	−2.92432
$721$	−24.1722	−0.900220
$722$	−6.23400	−0.232006
$723$	−5.39492	−0.200639
$724$	20.3936	0.757921
$725$	6.56738	0.243906
$726$	11.4742	0.425849
$727$	4.68807	0.173871	0.0869355	−	0.996214i	$-0.472293\pi$
$727$	4.68807	0.173871	0.0869355	+	0.996214i	$0.472293\pi$
$728$	10.5077	0.389442
$729$	−43.9405	−1.62743
$730$	−24.6332	−0.911716
$731$	−9.37776	−0.346849
$732$	12.5957	0.465549
$733$	9.83339	0.363204	0.181602	−	0.983372i	$-0.441872\pi$
$733$	9.83339	0.363204	0.181602	+	0.983372i	$0.441872\pi$
$734$	−9.87541	−0.364508
$735$	3.07536	0.113436
$736$	−24.6090	−0.907098
$737$	5.50140	0.202647
$738$	−1.95243	−0.0718701
$739$	31.8386	1.17120	0.585601	−	0.810599i	$-0.300859\pi$
$739$	31.8386	1.17120	0.585601	+	0.810599i	$0.300859\pi$
$740$	−7.46263	−0.274332
$741$	45.3943	1.66760
$742$	−49.2374	−1.80756
$743$	42.2323	1.54935	0.774676	−	0.632358i	$-0.217913\pi$
$743$	42.2323	1.54935	0.774676	+	0.632358i	$0.217913\pi$
$744$	11.5250	0.422525
$745$	−79.4615	−2.91124
$746$	43.7082	1.60027
$747$	25.1087	0.918680
$748$	15.0594	0.550628
$749$	43.3914	1.58549
$750$	−27.3270	−0.997840
$751$	35.2810	1.28742	0.643711	−	0.765269i	$-0.277394\pi$
$751$	35.2810	1.28742	0.643711	+	0.765269i	$0.277394\pi$
$752$	−4.88491	−0.178134
$753$	41.8350	1.52455
$754$	6.34013	0.230894
$755$	−1.62326	−0.0590765
$756$	−18.3801	−0.668478
$757$	−37.0562	−1.34683	−0.673415	−	0.739265i	$-0.735173\pi$
$757$	−37.0562	−1.34683	−0.673415	+	0.739265i	$0.735173\pi$
$758$	38.8452	1.41092
$759$	30.4078	1.10373
$760$	−18.1376	−0.657920
$761$	−21.5368	−0.780707	−0.390354	−	0.920665i	$-0.627647\pi$
$761$	−21.5368	−0.780707	−0.390354	+	0.920665i	$0.627647\pi$
$762$	16.9196	0.612933
$763$	32.4768	1.17574
$764$	2.64655	0.0957489
$765$	−59.3218	−2.14478
$766$	−3.04551	−0.110039
$767$	15.5344	0.560915
$768$	58.4311	2.10845
$769$	−9.21475	−0.332292	−0.166146	−	0.986101i	$-0.553132\pi$
$769$	−9.21475	−0.332292	−0.166146	+	0.986101i	$0.553132\pi$
$770$	−50.1479	−1.80720
$771$	44.6132	1.60671
$772$	16.1639	0.581752
$773$	27.1412	0.976200	0.488100	−	0.872788i	$-0.337690\pi$
$773$	27.1412	0.976200	0.488100	+	0.872788i	$0.337690\pi$
$774$	−22.3954	−0.804985
$775$	24.1109	0.866089
$776$	−0.0976983	−0.00350717
$777$	11.9192	0.427601
$778$	6.65936	0.238749
$779$	−1.05505	−0.0378011
$780$	−45.2786	−1.62124
$781$	−32.9103	−1.17762
$782$	−24.8902	−0.890071
$783$	4.89420	0.174904
$784$	−1.57450	−0.0562323
$785$	−35.4169	−1.26408
$786$	2.97178	0.106000
$787$	18.7317	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$787$	18.7317	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$788$	−2.67023	−0.0951231
$789$	40.1640	1.42988
$790$	53.3615	1.89852
$791$	−40.2935	−1.43267
$792$	−15.8713	−0.563961
$793$	−11.2265	−0.398665
$794$	−0.278664	−0.00988940
$795$	−93.6320	−3.32079
$796$	34.7822	1.23282
$797$	−26.2257	−0.928960	−0.464480	−	0.885583i	$-0.653759\pi$
$797$	−26.2257	−0.928960	−0.464480	+	0.885583i	$0.653759\pi$
$798$	−65.6437	−2.32376
$799$	−3.69301	−0.130649
$800$	43.8175	1.54918
$801$	−49.7239	−1.75691
$802$	0.336786	0.0118923
$803$	−11.6479	−0.411047
$804$	−7.18300	−0.253325
$805$	33.9507	1.19661
$806$	23.2766	0.819883
$807$	−12.4623	−0.438695
$808$	−13.6023	−0.478527
$809$	−47.6950	−1.67687	−0.838434	−	0.545004i	$-0.816528\pi$
$809$	−47.6950	−1.67687	−0.838434	+	0.545004i	$0.816528\pi$
$810$	4.00919	0.140868
$811$	18.4945	0.649429	0.324715	−	0.945812i	$-0.394732\pi$
$811$	18.4945	0.649429	0.324715	+	0.945812i	$0.394732\pi$
$812$	−3.75549	−0.131792
$813$	−40.0671	−1.40521
$814$	−8.61476	−0.301947
$815$	26.6257	0.932658
$816$	49.5239	1.73368
$817$	−12.1019	−0.423393
$818$	40.8823	1.42942
$819$	44.3504	1.54973
$820$	1.05236	0.0367500
$821$	35.2808	1.23131	0.615654	−	0.788017i	$-0.288892\pi$
$821$	35.2808	1.23131	0.615654	+	0.788017i	$0.288892\pi$
$822$	95.2718	3.32299
$823$	−6.34027	−0.221008	−0.110504	−	0.993876i	$-0.535246\pi$
$823$	−6.34027	−0.221008	−0.110504	+	0.993876i	$0.535246\pi$
$824$	10.0667	0.350690
$825$	−54.1426	−1.88500
$826$	−22.4640	−0.781621
$827$	7.67011	0.266716	0.133358	−	0.991068i	$-0.457424\pi$
$827$	7.67011	0.266716	0.133358	+	0.991068i	$0.457424\pi$
$828$	−24.3481	−0.846154
$829$	47.3853	1.64576	0.822880	−	0.568215i	$-0.192366\pi$
$829$	47.3853	1.64576	0.822880	+	0.568215i	$0.192366\pi$
$830$	−33.0397	−1.14682
$831$	43.7232	1.51674
$832$	8.88954	0.308189
$833$	−1.19033	−0.0412425
$834$	−5.12627	−0.177508
$835$	−56.0158	−1.93851
$836$	19.4341	0.672142
$837$	17.9681	0.621070
$838$	33.0862	1.14294
$839$	−17.3748	−0.599846	−0.299923	−	0.953963i	$-0.596961\pi$
$839$	−17.3748	−0.599846	−0.299923	+	0.953963i	$0.596961\pi$
$840$	−28.8954	−0.996986
$841$	1.00000	0.0344828
$842$	21.6005	0.744402
$843$	76.5051	2.63497
$844$	12.6121	0.434126
$845$	−3.85721	−0.132692
$846$	−8.81941	−0.303218
$847$	6.05783	0.208149
$848$	47.9372	1.64617
$849$	−40.3971	−1.38642
$850$	44.3182	1.52010
$851$	5.83230	0.199929
$852$	42.9699	1.47213
$853$	−3.93506	−0.134734	−0.0673669	−	0.997728i	$-0.521460\pi$
$853$	−3.93506	−0.134734	−0.0673669	+	0.997728i	$0.521460\pi$
$854$	16.2344	0.555530
$855$	−76.5543	−2.61810
$856$	−18.0707	−0.617643
$857$	54.9806	1.87810	0.939051	−	0.343778i	$-0.111707\pi$
$857$	54.9806	1.87810	0.939051	+	0.343778i	$0.111707\pi$
$858$	−52.2691	−1.78444
$859$	−7.74884	−0.264387	−0.132194	−	0.991224i	$-0.542202\pi$
$859$	−7.74884	−0.264387	−0.132194	+	0.991224i	$0.542202\pi$
$860$	12.0711	0.411621
$861$	−1.68082	−0.0572822
$862$	−66.5327	−2.26611
$863$	−55.9313	−1.90392	−0.951962	−	0.306215i	$-0.900937\pi$
$863$	−55.9313	−1.90392	−0.951962	+	0.306215i	$0.900937\pi$
$864$	32.6541	1.11091
$865$	−0.570627	−0.0194019
$866$	39.5191	1.34292
$867$	−9.90785	−0.336488
$868$	−13.7876	−0.467981
$869$	25.2323	0.855947
$870$	−17.4349	−0.591097
$871$	6.40221	0.216931
$872$	−13.5252	−0.458021
$873$	−0.412360	−0.0139563
$874$	−32.1206	−1.08650
$875$	−14.4273	−0.487731
$876$	15.2083	0.513842
$877$	29.6730	1.00199	0.500993	−	0.865451i	$-0.332968\pi$
$877$	29.6730	1.00199	0.500993	+	0.865451i	$0.332968\pi$
$878$	60.4430	2.03985
$879$	−29.6017	−0.998441
$880$	48.8236	1.64584
$881$	−46.3980	−1.56319	−0.781594	−	0.623787i	$-0.785593\pi$
$881$	−46.3980	−1.56319	−0.781594	+	0.623787i	$0.785593\pi$
$882$	−2.84267	−0.0957177
$883$	−13.8658	−0.466622	−0.233311	−	0.972402i	$-0.574956\pi$
$883$	−13.8658	−0.466622	−0.233311	+	0.972402i	$0.574956\pi$
$884$	17.5253	0.589439
$885$	−42.7185	−1.43597
$886$	−68.5369	−2.30254
$887$	−35.3421	−1.18667	−0.593336	−	0.804955i	$-0.702190\pi$
$887$	−35.3421	−1.18667	−0.593336	+	0.804955i	$0.702190\pi$
$888$	−4.96386	−0.166576
$889$	8.93270	0.299593
$890$	65.4299	2.19321
$891$	1.89577	0.0635105
$892$	15.4861	0.518514
$893$	−4.76580	−0.159482
$894$	119.768	4.00564
$895$	24.2584	0.810869
$896$	23.2593	0.777040
$897$	35.3868	1.18153
$898$	−22.7456	−0.759032
$899$	3.67132	0.122445
$900$	43.3529	1.44510
$901$	36.2407	1.20735
$902$	1.21483	0.0404494
$903$	−19.2798	−0.641593
$904$	16.7805	0.558112
$905$	49.9848	1.66155
$906$	2.44665	0.0812845
$907$	−32.4248	−1.07665	−0.538325	−	0.842738i	$-0.680943\pi$
$907$	−32.4248	−1.07665	−0.538325	+	0.842738i	$0.680943\pi$
$908$	14.6183	0.485124
$909$	−57.4119	−1.90423
$910$	−58.3592	−1.93459
$911$	25.5411	0.846215	0.423108	−	0.906079i	$-0.360939\pi$
$911$	25.5411	0.846215	0.423108	+	0.906079i	$0.360939\pi$
$912$	63.9103	2.11628
$913$	−15.6230	−0.517045
$914$	54.7511	1.81100
$915$	30.8721	1.02060
$916$	−25.9675	−0.857991
$917$	1.56895	0.0518113
$918$	33.0272	1.09006
$919$	−19.2004	−0.633362	−0.316681	−	0.948532i	$-0.602568\pi$
$919$	−19.2004	−0.633362	−0.316681	+	0.948532i	$0.602568\pi$
$920$	−14.1390	−0.466150
$921$	−20.2970	−0.668810
$922$	−10.9611	−0.360985
$923$	−38.2991	−1.26063
$924$	30.9609	1.01854
$925$	−10.3847	−0.341447
$926$	−26.2745	−0.863434
$927$	42.4890	1.39552
$928$	6.67199	0.219019
$929$	−22.7117	−0.745146	−0.372573	−	0.928003i	$-0.621524\pi$
$929$	−22.7117	−0.745146	−0.372573	+	0.928003i	$0.621524\pi$
$930$	−64.0089	−2.09893
$931$	−1.53611	−0.0503441
$932$	19.0104	0.622705
$933$	35.5088	1.16251
$934$	−4.70186	−0.153849
$935$	36.9108	1.20711
$936$	−18.4701	−0.603713
$937$	39.3272	1.28476	0.642381	−	0.766385i	$-0.277947\pi$
$937$	39.3272	1.28476	0.642381	+	0.766385i	$0.277947\pi$
$938$	−9.25809	−0.302287
$939$	−38.5208	−1.25708
$940$	4.75366	0.155047
$941$	−27.7029	−0.903089	−0.451545	−	0.892249i	$-0.649127\pi$
$941$	−27.7029	−0.903089	−0.451545	+	0.892249i	$0.649127\pi$
$942$	53.3819	1.73928
$943$	−0.822455	−0.0267828
$944$	21.8708	0.711833
$945$	−45.0498	−1.46547
$946$	13.9347	0.453057
$947$	3.78318	0.122937	0.0614685	−	0.998109i	$-0.480422\pi$
$947$	3.78318	0.122937	0.0614685	+	0.998109i	$0.480422\pi$
$948$	−32.9450	−1.07000
$949$	−13.5552	−0.440020
$950$	57.1923	1.85556
$951$	32.1650	1.04302
$952$	11.1841	0.362478
$953$	−16.4474	−0.532783	−0.266392	−	0.963865i	$-0.585831\pi$
$953$	−16.4474	−0.532783	−0.266392	+	0.963865i	$0.585831\pi$
$954$	86.5477	2.80209
$955$	6.48672	0.209905
$956$	−37.8553	−1.22433
$957$	−8.24417	−0.266496
$958$	1.23862	0.0400181
$959$	50.2987	1.62423
$960$	−24.4456	−0.788977
$961$	−17.5214	−0.565208
$962$	−10.0254	−0.323230
$963$	−76.2718	−2.45782
$964$	−2.68785	−0.0865697
$965$	39.6179	1.27534
$966$	−51.1720	−1.64643
$967$	2.60570	0.0837938	0.0418969	−	0.999122i	$-0.486660\pi$
$967$	2.60570	0.0837938	0.0418969	+	0.999122i	$0.486660\pi$
$968$	−2.52283	−0.0810867
$969$	48.3163	1.55214
$970$	0.542610	0.0174222
$971$	56.6950	1.81943	0.909715	−	0.415233i	$-0.136300\pi$
$971$	56.6950	1.81943	0.909715	+	0.415233i	$0.136300\pi$
$972$	−22.8492	−0.732889
$973$	−2.70641	−0.0867635
$974$	1.63026	0.0522369
$975$	−63.0079	−2.01787
$976$	−15.8057	−0.505928
$977$	−56.4096	−1.80470	−0.902352	−	0.431000i	$-0.858161\pi$
$977$	−56.4096	−1.80470	−0.902352	+	0.431000i	$0.858161\pi$
$978$	−40.1314	−1.28326
$979$	30.9389	0.988810
$980$	1.53220	0.0489443
$981$	−57.0865	−1.82263
$982$	−34.9250	−1.11450
$983$	24.3816	0.777652	0.388826	−	0.921311i	$-0.372881\pi$
$983$	24.3816	0.777652	0.388826	+	0.921311i	$0.372881\pi$
$984$	0.699990	0.0223149
$985$	−6.54476	−0.208533
$986$	6.74824	0.214908
$987$	−7.59250	−0.241672
$988$	22.6163	0.719519
$989$	−9.43397	−0.299983
$990$	88.1481	2.80153
$991$	−35.8330	−1.13827	−0.569136	−	0.822243i	$-0.692722\pi$
$991$	−35.8330	−1.13827	−0.569136	+	0.822243i	$0.692722\pi$
$992$	24.4950	0.777717
$993$	46.5485	1.47717
$994$	55.3835	1.75666
$995$	85.2516	2.70266
$996$	20.3984	0.646348
$997$	−48.8932	−1.54846	−0.774231	−	0.632903i	$-0.781863\pi$
$997$	−48.8932	−1.54846	−0.774231	+	0.632903i	$0.781863\pi$
$998$	−21.1246	−0.668686
$999$	−7.73898	−0.244850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.11	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.11	✓	59	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)

\(f(q)\)	\(=\)	\(q-1.84055 q^{2} +2.78518 q^{3} +1.38763 q^{4} +3.40108 q^{5} -5.12627 q^{6} -2.70641 q^{7} +1.12710 q^{8} +4.75723 q^{9} +O(q^{10})\)	\(q-1.84055 q^{2} +2.78518 q^{3} +1.38763 q^{4} +3.40108 q^{5} -5.12627 q^{6} -2.70641 q^{7} +1.12710 q^{8} +4.75723 q^{9} -6.25987 q^{10} -2.96001 q^{11} +3.86479 q^{12} -3.44469 q^{13} +4.98129 q^{14} +9.47263 q^{15} -4.84975 q^{16} -3.66642 q^{17} -8.75592 q^{18} -4.73149 q^{19} +4.71944 q^{20} -7.53784 q^{21} +5.44805 q^{22} -3.68840 q^{23} +3.13919 q^{24} +6.56738 q^{25} +6.34013 q^{26} +4.89420 q^{27} -3.75549 q^{28} +1.00000 q^{29} -17.4349 q^{30} +3.67132 q^{31} +6.67199 q^{32} -8.24417 q^{33} +6.74824 q^{34} -9.20473 q^{35} +6.60126 q^{36} -1.58125 q^{37} +8.70855 q^{38} -9.59408 q^{39} +3.83338 q^{40} +0.222984 q^{41} +13.8738 q^{42} +2.55774 q^{43} -4.10739 q^{44} +16.1797 q^{45} +6.78868 q^{46} +1.00725 q^{47} -13.5074 q^{48} +0.324657 q^{49} -12.0876 q^{50} -10.2116 q^{51} -4.77994 q^{52} -9.88448 q^{53} -9.00802 q^{54} -10.0673 q^{55} -3.05041 q^{56} -13.1781 q^{57} -1.84055 q^{58} -4.50967 q^{59} +13.1445 q^{60} +3.25908 q^{61} -6.75724 q^{62} -12.8750 q^{63} -2.58065 q^{64} -11.7157 q^{65} +15.1738 q^{66} -1.85857 q^{67} -5.08763 q^{68} -10.2729 q^{69} +16.9418 q^{70} +11.1183 q^{71} +5.36189 q^{72} +3.93510 q^{73} +2.91038 q^{74} +18.2913 q^{75} -6.56555 q^{76} +8.01101 q^{77} +17.6584 q^{78} -8.52439 q^{79} -16.4944 q^{80} -0.640459 q^{81} -0.410414 q^{82} +5.27801 q^{83} -10.4597 q^{84} -12.4698 q^{85} -4.70765 q^{86} +2.78518 q^{87} -3.33624 q^{88} -10.4523 q^{89} -29.7796 q^{90} +9.32274 q^{91} -5.11812 q^{92} +10.2253 q^{93} -1.85390 q^{94} -16.0922 q^{95} +18.5827 q^{96} -0.0866808 q^{97} -0.597548 q^{98} -14.0815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more