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Properties

Label 5120.2.a.v.1.3
Level $5120$
Weight $2$
Character 5120.1
Self dual yes
Analytic conductor $40.883$
Analytic rank $0$
Dimension $8$
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] = [5120,2,Mod(1,5120)] 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("5120.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5120 = 2^{10} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5120.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,4,0,8,0,4,0,8,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.8834058349\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 80)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.56993\) of defining polynomial
Character \(\chi\) \(=\) 5120.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.296378 q^{3} +1.00000 q^{5} -1.73696 q^{7} -2.91216 q^{9} -0.714786 q^{11} -2.66933 q^{13} -0.296378 q^{15} -4.53524 q^{17} -4.55407 q^{19} +0.514799 q^{21} +8.85045 q^{23} +1.00000 q^{25} +1.75224 q^{27} +3.45151 q^{29} -5.70401 q^{31} +0.211847 q^{33} -1.73696 q^{35} -7.57552 q^{37} +0.791130 q^{39} +10.0343 q^{41} +2.97782 q^{43} -2.91216 q^{45} -4.32303 q^{47} -3.98295 q^{49} +1.34415 q^{51} +1.94396 q^{53} -0.714786 q^{55} +1.34973 q^{57} +9.39236 q^{59} -7.44171 q^{61} +5.05832 q^{63} -2.66933 q^{65} +14.9309 q^{67} -2.62308 q^{69} +14.0437 q^{71} -6.63830 q^{73} -0.296378 q^{75} +1.24156 q^{77} -4.27297 q^{79} +8.21715 q^{81} +12.9469 q^{83} -4.53524 q^{85} -1.02295 q^{87} -3.23826 q^{89} +4.63652 q^{91} +1.69055 q^{93} -4.55407 q^{95} +1.94129 q^{97} +2.08157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 8 q^{5} + 4 q^{7} + 8 q^{9} + 8 q^{11} + 4 q^{15} + 16 q^{19} + 12 q^{23} + 8 q^{25} + 16 q^{27} + 4 q^{35} + 28 q^{43} + 8 q^{45} + 20 q^{47} + 8 q^{49} + 24 q^{51} + 8 q^{55} + 16 q^{59}+ \cdots + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.296378 −0.171114 −0.0855571 − 0.996333i \(-0.527267\pi\)
−0.0855571 + 0.996333i \(0.527267\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.73696 −0.656511 −0.328255 − 0.944589i \(-0.606461\pi\)
−0.328255 + 0.944589i \(0.606461\pi\)
\(8\) 0 0
\(9\) −2.91216 −0.970720
\(10\) 0 0
\(11\) −0.714786 −0.215516 −0.107758 − 0.994177i \(-0.534367\pi\)
−0.107758 + 0.994177i \(0.534367\pi\)
\(12\) 0 0
\(13\) −2.66933 −0.740338 −0.370169 − 0.928964i \(-0.620700\pi\)
−0.370169 + 0.928964i \(0.620700\pi\)
\(14\) 0 0
\(15\) −0.296378 −0.0765246
\(16\) 0 0
\(17\) −4.53524 −1.09996 −0.549979 − 0.835178i \(-0.685364\pi\)
−0.549979 + 0.835178i \(0.685364\pi\)
\(18\) 0 0
\(19\) −4.55407 −1.04478 −0.522388 − 0.852708i \(-0.674959\pi\)
−0.522388 + 0.852708i \(0.674959\pi\)
\(20\) 0 0
\(21\) 0.514799 0.112338
\(22\) 0 0
\(23\) 8.85045 1.84545 0.922723 − 0.385463i \(-0.125958\pi\)
0.922723 + 0.385463i \(0.125958\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.75224 0.337218
\(28\) 0 0
\(29\) 3.45151 0.640929 0.320465 − 0.947260i \(-0.396161\pi\)
0.320465 + 0.947260i \(0.396161\pi\)
\(30\) 0 0
\(31\) −5.70401 −1.02447 −0.512235 − 0.858845i \(-0.671182\pi\)
−0.512235 + 0.858845i \(0.671182\pi\)
\(32\) 0 0
\(33\) 0.211847 0.0368779
\(34\) 0 0
\(35\) −1.73696 −0.293601
\(36\) 0 0
\(37\) −7.57552 −1.24541 −0.622704 − 0.782458i \(-0.713966\pi\)
−0.622704 + 0.782458i \(0.713966\pi\)
\(38\) 0 0
\(39\) 0.791130 0.126682
\(40\) 0 0
\(41\) 10.0343 1.56709 0.783545 − 0.621335i \(-0.213409\pi\)
0.783545 + 0.621335i \(0.213409\pi\)
\(42\) 0 0
\(43\) 2.97782 0.454113 0.227057 − 0.973882i \(-0.427090\pi\)
0.227057 + 0.973882i \(0.427090\pi\)
\(44\) 0 0
\(45\) −2.91216 −0.434119
\(46\) 0 0
\(47\) −4.32303 −0.630578 −0.315289 − 0.948996i \(-0.602101\pi\)
−0.315289 + 0.948996i \(0.602101\pi\)
\(48\) 0 0
\(49\) −3.98295 −0.568993
\(50\) 0 0
\(51\) 1.34415 0.188218
\(52\) 0 0
\(53\) 1.94396 0.267023 0.133511 − 0.991047i \(-0.457375\pi\)
0.133511 + 0.991047i \(0.457375\pi\)
\(54\) 0 0
\(55\) −0.714786 −0.0963817
\(56\) 0 0
\(57\) 1.34973 0.178776
\(58\) 0 0
\(59\) 9.39236 1.22278 0.611390 − 0.791329i \(-0.290611\pi\)
0.611390 + 0.791329i \(0.290611\pi\)
\(60\) 0 0
\(61\) −7.44171 −0.952813 −0.476407 − 0.879225i \(-0.658061\pi\)
−0.476407 + 0.879225i \(0.658061\pi\)
\(62\) 0 0
\(63\) 5.05832 0.637288
\(64\) 0 0
\(65\) −2.66933 −0.331089
\(66\) 0 0
\(67\) 14.9309 1.82411 0.912053 − 0.410073i \(-0.134497\pi\)
0.912053 + 0.410073i \(0.134497\pi\)
\(68\) 0 0
\(69\) −2.62308 −0.315782
\(70\) 0 0
\(71\) 14.0437 1.66668 0.833338 − 0.552764i \(-0.186427\pi\)
0.833338 + 0.552764i \(0.186427\pi\)
\(72\) 0 0
\(73\) −6.63830 −0.776954 −0.388477 − 0.921458i \(-0.626999\pi\)
−0.388477 + 0.921458i \(0.626999\pi\)
\(74\) 0 0
\(75\) −0.296378 −0.0342228
\(76\) 0 0
\(77\) 1.24156 0.141489
\(78\) 0 0
\(79\) −4.27297 −0.480746 −0.240373 − 0.970681i \(-0.577270\pi\)
−0.240373 + 0.970681i \(0.577270\pi\)
\(80\) 0 0
\(81\) 8.21715 0.913017
\(82\) 0 0
\(83\) 12.9469 1.42111 0.710553 − 0.703644i \(-0.248445\pi\)
0.710553 + 0.703644i \(0.248445\pi\)
\(84\) 0 0
\(85\) −4.53524 −0.491916
\(86\) 0 0
\(87\) −1.02295 −0.109672
\(88\) 0 0
\(89\) −3.23826 −0.343255 −0.171627 − 0.985162i \(-0.554903\pi\)
−0.171627 + 0.985162i \(0.554903\pi\)
\(90\) 0 0
\(91\) 4.63652 0.486040
\(92\) 0 0
\(93\) 1.69055 0.175301
\(94\) 0 0
\(95\) −4.55407 −0.467238
\(96\) 0 0
\(97\) 1.94129 0.197108 0.0985541 − 0.995132i \(-0.468578\pi\)
0.0985541 + 0.995132i \(0.468578\pi\)
\(98\) 0 0
\(99\) 2.08157 0.209206
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 5120.2.a.v.1.3 8
4.3 odd 2 5120.2.a.t.1.6 8
8.3 odd 2 5120.2.a.u.1.3 8
8.5 even 2 5120.2.a.s.1.6 8
32.3 odd 8 640.2.l.a.161.5 16
32.5 even 8 80.2.l.a.21.6 ✓ 16
32.11 odd 8 640.2.l.a.481.5 16
32.13 even 8 80.2.l.a.61.6 yes 16
32.19 odd 8 320.2.l.a.81.4 16
32.21 even 8 640.2.l.b.481.4 16
32.27 odd 8 320.2.l.a.241.4 16
32.29 even 8 640.2.l.b.161.4 16
96.5 odd 8 720.2.t.c.181.3 16
96.59 even 8 2880.2.t.c.2161.4 16
96.77 odd 8 720.2.t.c.541.3 16
96.83 even 8 2880.2.t.c.721.1 16
160.13 odd 8 400.2.q.g.349.7 16
160.19 odd 8 1600.2.l.i.401.5 16
160.27 even 8 1600.2.q.h.49.5 16
160.37 odd 8 400.2.q.g.149.7 16
160.59 odd 8 1600.2.l.i.1201.5 16
160.69 even 8 400.2.l.h.101.3 16
160.77 odd 8 400.2.q.h.349.2 16
160.83 even 8 1600.2.q.h.849.5 16
160.109 even 8 400.2.l.h.301.3 16
160.123 even 8 1600.2.q.g.49.4 16
160.133 odd 8 400.2.q.h.149.2 16
160.147 even 8 1600.2.q.g.849.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.6 ✓ 16 32.5 even 8
80.2.l.a.61.6 yes 16 32.13 even 8
320.2.l.a.81.4 16 32.19 odd 8
320.2.l.a.241.4 16 32.27 odd 8
400.2.l.h.101.3 16 160.69 even 8
400.2.l.h.301.3 16 160.109 even 8
400.2.q.g.149.7 16 160.37 odd 8
400.2.q.g.349.7 16 160.13 odd 8
400.2.q.h.149.2 16 160.133 odd 8
400.2.q.h.349.2 16 160.77 odd 8
640.2.l.a.161.5 16 32.3 odd 8
640.2.l.a.481.5 16 32.11 odd 8
640.2.l.b.161.4 16 32.29 even 8
640.2.l.b.481.4 16 32.21 even 8
720.2.t.c.181.3 16 96.5 odd 8
720.2.t.c.541.3 16 96.77 odd 8
1600.2.l.i.401.5 16 160.19 odd 8
1600.2.l.i.1201.5 16 160.59 odd 8
1600.2.q.g.49.4 16 160.123 even 8
1600.2.q.g.849.4 16 160.147 even 8
1600.2.q.h.49.5 16 160.27 even 8
1600.2.q.h.849.5 16 160.83 even 8
2880.2.t.c.721.1 16 96.83 even 8
2880.2.t.c.2161.4 16 96.59 even 8
5120.2.a.s.1.6 8 8.5 even 2
5120.2.a.t.1.6 8 4.3 odd 2
5120.2.a.u.1.3 8 8.3 odd 2
5120.2.a.v.1.3 8 1.1 even 1 trivial