Properties

Label 400.8.a.q.1.1
Level $400$
Weight $8$
Character 400.1
Self dual yes
Analytic conductor $124.954$
Analytic rank $1$
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] = [400,8,Mod(1,400)] 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("400.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,57,0,0,0,-1174] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(124.954010194\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-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)
 
\(f(q)\) \(=\) \(q+57.0000 q^{3} -1174.00 q^{7} +1062.00 q^{9} +7563.00 q^{11} -5372.00 q^{13} -24021.0 q^{17} +51235.0 q^{19} -66918.0 q^{21} -57618.0 q^{23} -64125.0 q^{27} +47040.0 q^{29} +192358. q^{31} +431091. q^{33} -197066. q^{37} -306204. q^{39} -237723. q^{41} +653012. q^{43} -826884. q^{47} +554733. q^{49} -1.36920e6 q^{51} -569022. q^{53} +2.92040e6 q^{57} -1.50108e6 q^{59} -2.06892e6 q^{61} -1.24679e6 q^{63} -3.44435e6 q^{67} -3.28423e6 q^{69} -4.12105e6 q^{71} +83653.0 q^{73} -8.87896e6 q^{77} -1.45403e6 q^{79} -5.97772e6 q^{81} +1.62657e6 q^{83} +2.68128e6 q^{87} +6.00434e6 q^{89} +6.30673e6 q^{91} +1.09644e7 q^{93} -3.41175e6 q^{97} +8.03191e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 57.0000 1.21885 0.609425 − 0.792844i \(-0.291400\pi\)
0.609425 + 0.792844i \(0.291400\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1174.00 −1.29367 −0.646837 − 0.762628i \(-0.723909\pi\)
−0.646837 + 0.762628i \(0.723909\pi\)
\(8\) 0 0
\(9\) 1062.00 0.485597
\(10\) 0 0
\(11\) 7563.00 1.71325 0.856623 − 0.515943i \(-0.172558\pi\)
0.856623 + 0.515943i \(0.172558\pi\)
\(12\) 0 0
\(13\) −5372.00 −0.678163 −0.339082 − 0.940757i \(-0.610116\pi\)
−0.339082 + 0.940757i \(0.610116\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −24021.0 −1.18582 −0.592911 − 0.805268i \(-0.702022\pi\)
−0.592911 + 0.805268i \(0.702022\pi\)
\(18\) 0 0
\(19\) 51235.0 1.71368 0.856839 − 0.515584i \(-0.172425\pi\)
0.856839 + 0.515584i \(0.172425\pi\)
\(20\) 0 0
\(21\) −66918.0 −1.57680
\(22\) 0 0
\(23\) −57618.0 −0.987440 −0.493720 − 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −64125.0 −0.626981
\(28\) 0 0
\(29\) 47040.0 0.358158 0.179079 − 0.983835i \(-0.442688\pi\)
0.179079 + 0.983835i \(0.442688\pi\)
\(30\) 0 0
\(31\) 192358. 1.15970 0.579848 − 0.814725i \(-0.303112\pi\)
0.579848 + 0.814725i \(0.303112\pi\)
\(32\) 0 0
\(33\) 431091. 2.08819
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −197066. −0.639596 −0.319798 − 0.947486i \(-0.603615\pi\)
−0.319798 + 0.947486i \(0.603615\pi\)
\(38\) 0 0
\(39\) −306204. −0.826580
\(40\) 0 0
\(41\) −237723. −0.538676 −0.269338 − 0.963046i \(-0.586805\pi\)
−0.269338 + 0.963046i \(0.586805\pi\)
\(42\) 0 0
\(43\) 653012. 1.25251 0.626256 − 0.779618i \(-0.284587\pi\)
0.626256 + 0.779618i \(0.284587\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −826884. −1.16172 −0.580861 − 0.814003i \(-0.697284\pi\)
−0.580861 + 0.814003i \(0.697284\pi\)
\(48\) 0 0
\(49\) 554733. 0.673593
\(50\) 0 0
\(51\) −1.36920e6 −1.44534
\(52\) 0 0
\(53\) −569022. −0.525005 −0.262503 − 0.964931i \(-0.584548\pi\)
−0.262503 + 0.964931i \(0.584548\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.92040e6 2.08872
\(58\) 0 0
\(59\) −1.50108e6 −0.951528 −0.475764 − 0.879573i \(-0.657828\pi\)
−0.475764 + 0.879573i \(0.657828\pi\)
\(60\) 0 0
\(61\) −2.06892e6 −1.16705 −0.583524 − 0.812096i \(-0.698327\pi\)
−0.583524 + 0.812096i \(0.698327\pi\)
\(62\) 0 0
\(63\) −1.24679e6 −0.628204
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.44435e6 −1.39909 −0.699544 − 0.714589i \(-0.746614\pi\)
−0.699544 + 0.714589i \(0.746614\pi\)
\(68\) 0 0
\(69\) −3.28423e6 −1.20354
\(70\) 0 0
\(71\) −4.12105e6 −1.36648 −0.683241 − 0.730193i \(-0.739430\pi\)
−0.683241 + 0.730193i \(0.739430\pi\)
\(72\) 0 0
\(73\) 83653.0 0.0251682 0.0125841 − 0.999921i \(-0.495994\pi\)
0.0125841 + 0.999921i \(0.495994\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.87896e6 −2.21638
\(78\) 0 0
\(79\) −1.45403e6 −0.331802 −0.165901 − 0.986142i \(-0.553053\pi\)
−0.165901 + 0.986142i \(0.553053\pi\)
\(80\) 0 0
\(81\) −5.97772e6 −1.24979
\(82\) 0 0
\(83\) 1.62657e6 0.312247 0.156124 − 0.987738i \(-0.450100\pi\)
0.156124 + 0.987738i \(0.450100\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.68128e6 0.436541
\(88\) 0 0
\(89\) 6.00434e6 0.902817 0.451409 − 0.892317i \(-0.350922\pi\)
0.451409 + 0.892317i \(0.350922\pi\)
\(90\) 0 0
\(91\) 6.30673e6 0.877322
\(92\) 0 0
\(93\) 1.09644e7 1.41350
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.41175e6 −0.379556 −0.189778 − 0.981827i \(-0.560777\pi\)
−0.189778 + 0.981827i \(0.560777\pi\)
\(98\) 0 0
\(99\) 8.03191e6 0.831947
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 400.8.a.q.1.1 1
4.3 odd 2 50.8.a.f.1.1 yes 1
5.2 odd 4 400.8.c.d.49.1 2
5.3 odd 4 400.8.c.d.49.2 2
5.4 even 2 400.8.a.d.1.1 1
12.11 even 2 450.8.a.l.1.1 1
20.3 even 4 50.8.b.b.49.1 2
20.7 even 4 50.8.b.b.49.2 2
20.19 odd 2 50.8.a.c.1.1 ✓ 1
60.23 odd 4 450.8.c.q.199.2 2
60.47 odd 4 450.8.c.q.199.1 2
60.59 even 2 450.8.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.c.1.1 ✓ 1 20.19 odd 2
50.8.a.f.1.1 yes 1 4.3 odd 2
50.8.b.b.49.1 2 20.3 even 4
50.8.b.b.49.2 2 20.7 even 4
400.8.a.d.1.1 1 5.4 even 2
400.8.a.q.1.1 1 1.1 even 1 trivial
400.8.c.d.49.1 2 5.2 odd 4
400.8.c.d.49.2 2 5.3 odd 4
450.8.a.l.1.1 1 12.11 even 2
450.8.a.p.1.1 1 60.59 even 2
450.8.c.q.199.1 2 60.47 odd 4
450.8.c.q.199.2 2 60.23 odd 4