Properties

Label 50.8.a.h.1.1
Level $50$
Weight $8$
Character 50.1
Self dual yes
Analytic conductor $15.619$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,8,Mod(1,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,8,43] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.6192512742\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 50.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +43.0000 q^{3} +64.0000 q^{4} +344.000 q^{6} +974.000 q^{7} +512.000 q^{8} -338.000 q^{9} +87.0000 q^{11} +2752.00 q^{12} +14828.0 q^{13} +7792.00 q^{14} +4096.00 q^{16} -35571.0 q^{17} -2704.00 q^{18} +20615.0 q^{19} +41882.0 q^{21} +696.000 q^{22} +22218.0 q^{23} +22016.0 q^{24} +118624. q^{26} -108575. q^{27} +62336.0 q^{28} -5760.00 q^{29} +302942. q^{31} +32768.0 q^{32} +3741.00 q^{33} -284568. q^{34} -21632.0 q^{36} -199366. q^{37} +164920. q^{38} +637604. q^{39} -668523. q^{41} +335056. q^{42} -143212. q^{43} +5568.00 q^{44} +177744. q^{46} -338316. q^{47} +176128. q^{48} +125133. q^{49} -1.52955e6 q^{51} +948992. q^{52} -1.09432e6 q^{53} -868600. q^{54} +498688. q^{56} +886445. q^{57} -46080.0 q^{58} -2.13552e6 q^{59} -1.93932e6 q^{61} +2.42354e6 q^{62} -329212. q^{63} +262144. q^{64} +29928.0 q^{66} -3.34875e6 q^{67} -2.27654e6 q^{68} +955374. q^{69} +3.00565e6 q^{71} -173056. q^{72} -3.04840e6 q^{73} -1.59493e6 q^{74} +1.31936e6 q^{76} +84738.0 q^{77} +5.10083e6 q^{78} +5.48513e6 q^{79} -3.92952e6 q^{81} -5.34818e6 q^{82} +5.20593e6 q^{83} +2.68045e6 q^{84} -1.14570e6 q^{86} -247680. q^{87} +44544.0 q^{88} -832665. q^{89} +1.44425e7 q^{91} +1.42195e6 q^{92} +1.30265e7 q^{93} -2.70653e6 q^{94} +1.40902e6 q^{96} +5.31475e6 q^{97} +1.00106e6 q^{98} -29406.0 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 43.0000 0.919484 0.459742 − 0.888053i \(-0.347942\pi\)
0.459742 + 0.888053i \(0.347942\pi\)
\(4\) 64.0000 0.500000
\(5\) 0 0
\(6\) 344.000 0.650173
\(7\) 974.000 1.07329 0.536643 − 0.843809i \(-0.319692\pi\)
0.536643 + 0.843809i \(0.319692\pi\)
\(8\) 512.000 0.353553
\(9\) −338.000 −0.154550
\(10\) 0 0
\(11\) 87.0000 0.0197081 0.00985405 − 0.999951i \(-0.496863\pi\)
0.00985405 + 0.999951i \(0.496863\pi\)
\(12\) 2752.00 0.459742
\(13\) 14828.0 1.87189 0.935946 − 0.352143i \(-0.114547\pi\)
0.935946 + 0.352143i \(0.114547\pi\)
\(14\) 7792.00 0.758928
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) −35571.0 −1.75600 −0.878001 − 0.478659i \(-0.841123\pi\)
−0.878001 + 0.478659i \(0.841123\pi\)
\(18\) −2704.00 −0.109283
\(19\) 20615.0 0.689518 0.344759 − 0.938691i \(-0.387961\pi\)
0.344759 + 0.938691i \(0.387961\pi\)
\(20\) 0 0
\(21\) 41882.0 0.986870
\(22\) 696.000 0.0139357
\(23\) 22218.0 0.380765 0.190383 − 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) 22016.0 0.325087
\(25\) 0 0
\(26\) 118624. 1.32363
\(27\) −108575. −1.06159
\(28\) 62336.0 0.536643
\(29\) −5760.00 −0.0438560 −0.0219280 − 0.999760i \(-0.506980\pi\)
−0.0219280 + 0.999760i \(0.506980\pi\)
\(30\) 0 0
\(31\) 302942. 1.82639 0.913195 − 0.407523i \(-0.133607\pi\)
0.913195 + 0.407523i \(0.133607\pi\)
\(32\) 32768.0 0.176777
\(33\) 3741.00 0.0181213
\(34\) −284568. −1.24168
\(35\) 0 0
\(36\) −21632.0 −0.0772748
\(37\) −199366. −0.647061 −0.323530 − 0.946218i \(-0.604870\pi\)
−0.323530 + 0.946218i \(0.604870\pi\)
\(38\) 164920. 0.487563
\(39\) 637604. 1.72117
\(40\) 0 0
\(41\) −668523. −1.51486 −0.757431 − 0.652916i \(-0.773546\pi\)
−0.757431 + 0.652916i \(0.773546\pi\)
\(42\) 335056. 0.697822
\(43\) −143212. −0.274688 −0.137344 − 0.990523i \(-0.543857\pi\)
−0.137344 + 0.990523i \(0.543857\pi\)
\(44\) 5568.00 0.00985405
\(45\) 0 0
\(46\) 177744. 0.269242
\(47\) −338316. −0.475313 −0.237657 − 0.971349i \(-0.576379\pi\)
−0.237657 + 0.971349i \(0.576379\pi\)
\(48\) 176128. 0.229871
\(49\) 125133. 0.151945
\(50\) 0 0
\(51\) −1.52955e6 −1.61461
\(52\) 948992. 0.935946
\(53\) −1.09432e6 −1.00967 −0.504835 − 0.863216i \(-0.668447\pi\)
−0.504835 + 0.863216i \(0.668447\pi\)
\(54\) −868600. −0.750657
\(55\) 0 0
\(56\) 498688. 0.379464
\(57\) 886445. 0.634001
\(58\) −46080.0 −0.0310109
\(59\) −2.13552e6 −1.35370 −0.676849 − 0.736122i \(-0.736655\pi\)
−0.676849 + 0.736122i \(0.736655\pi\)
\(60\) 0 0
\(61\) −1.93932e6 −1.09394 −0.546971 − 0.837151i \(-0.684219\pi\)
−0.546971 + 0.837151i \(0.684219\pi\)
\(62\) 2.42354e6 1.29145
\(63\) −329212. −0.165876
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) 29928.0 0.0128137
\(67\) −3.34875e6 −1.36026 −0.680129 − 0.733093i \(-0.738076\pi\)
−0.680129 + 0.733093i \(0.738076\pi\)
\(68\) −2.27654e6 −0.878001
\(69\) 955374. 0.350108
\(70\) 0 0
\(71\) 3.00565e6 0.996631 0.498316 − 0.866996i \(-0.333952\pi\)
0.498316 + 0.866996i \(0.333952\pi\)
\(72\) −173056. −0.0546415
\(73\) −3.04840e6 −0.917152 −0.458576 − 0.888655i \(-0.651640\pi\)
−0.458576 + 0.888655i \(0.651640\pi\)
\(74\) −1.59493e6 −0.457541
\(75\) 0 0
\(76\) 1.31936e6 0.344759
\(77\) 84738.0 0.0211524
\(78\) 5.10083e6 1.21705
\(79\) 5.48513e6 1.25168 0.625838 − 0.779953i \(-0.284757\pi\)
0.625838 + 0.779953i \(0.284757\pi\)
\(80\) 0 0
\(81\) −3.92952e6 −0.821565
\(82\) −5.34818e6 −1.07117
\(83\) 5.20593e6 0.999368 0.499684 − 0.866208i \(-0.333450\pi\)
0.499684 + 0.866208i \(0.333450\pi\)
\(84\) 2.68045e6 0.493435
\(85\) 0 0
\(86\) −1.14570e6 −0.194234
\(87\) −247680. −0.0403249
\(88\) 44544.0 0.00696787
\(89\) −832665. −0.125200 −0.0626001 − 0.998039i \(-0.519939\pi\)
−0.0626001 + 0.998039i \(0.519939\pi\)
\(90\) 0 0
\(91\) 1.44425e7 2.00908
\(92\) 1.42195e6 0.190383
\(93\) 1.30265e7 1.67934
\(94\) −2.70653e6 −0.336097
\(95\) 0 0
\(96\) 1.40902e6 0.162543
\(97\) 5.31475e6 0.591265 0.295632 − 0.955302i \(-0.404470\pi\)
0.295632 + 0.955302i \(0.404470\pi\)
\(98\) 1.00106e6 0.107441
\(99\) −29406.0 −0.00304588
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 50.8.a.h.1.1 yes 1
3.2 odd 2 450.8.a.j.1.1 1
4.3 odd 2 400.8.a.f.1.1 1
5.2 odd 4 50.8.b.e.49.2 2
5.3 odd 4 50.8.b.e.49.1 2
5.4 even 2 50.8.a.a.1.1 ✓ 1
15.2 even 4 450.8.c.i.199.1 2
15.8 even 4 450.8.c.i.199.2 2
15.14 odd 2 450.8.a.q.1.1 1
20.3 even 4 400.8.c.g.49.1 2
20.7 even 4 400.8.c.g.49.2 2
20.19 odd 2 400.8.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.a.1.1 ✓ 1 5.4 even 2
50.8.a.h.1.1 yes 1 1.1 even 1 trivial
50.8.b.e.49.1 2 5.3 odd 4
50.8.b.e.49.2 2 5.2 odd 4
400.8.a.f.1.1 1 4.3 odd 2
400.8.a.o.1.1 1 20.19 odd 2
400.8.c.g.49.1 2 20.3 even 4
400.8.c.g.49.2 2 20.7 even 4
450.8.a.j.1.1 1 3.2 odd 2
450.8.a.q.1.1 1 15.14 odd 2
450.8.c.i.199.1 2 15.2 even 4
450.8.c.i.199.2 2 15.8 even 4