Properties

Label 6400.2.a.m.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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] = [6400,2,Mod(1,6400)] 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("6400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,0,0,-3,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-4, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6400.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{9} +4.00000 q^{13} +2.00000 q^{17} -4.00000 q^{29} -12.0000 q^{37} -10.0000 q^{41} -7.00000 q^{49} +4.00000 q^{53} +12.0000 q^{61} +6.00000 q^{73} +9.00000 q^{81} +10.0000 q^{89} +18.0000 q^{97} +O(q^{100})\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12.0000 −1.97279 −0.986394 − 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
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 6400.2.a.m.1.1 1
4.3 odd 2 CM 6400.2.a.m.1.1 1
5.4 even 2 256.2.a.b.1.1 1
8.3 odd 2 6400.2.a.l.1.1 1
8.5 even 2 6400.2.a.l.1.1 1
15.14 odd 2 2304.2.a.p.1.1 1
16.3 odd 4 3200.2.d.e.1601.2 2
16.5 even 4 3200.2.d.e.1601.1 2
16.11 odd 4 3200.2.d.e.1601.1 2
16.13 even 4 3200.2.d.e.1601.2 2
20.19 odd 2 256.2.a.b.1.1 1
40.19 odd 2 256.2.a.c.1.1 1
40.29 even 2 256.2.a.c.1.1 1
60.59 even 2 2304.2.a.p.1.1 1
80.3 even 4 3200.2.f.c.449.1 2
80.13 odd 4 3200.2.f.c.449.1 2
80.19 odd 4 128.2.b.b.65.2 yes 2
80.27 even 4 3200.2.f.c.449.2 2
80.29 even 4 128.2.b.b.65.2 yes 2
80.37 odd 4 3200.2.f.c.449.2 2
80.43 even 4 3200.2.f.d.449.1 2
80.53 odd 4 3200.2.f.d.449.1 2
80.59 odd 4 128.2.b.b.65.1 ✓ 2
80.67 even 4 3200.2.f.d.449.2 2
80.69 even 4 128.2.b.b.65.1 ✓ 2
80.77 odd 4 3200.2.f.d.449.2 2
120.29 odd 2 2304.2.a.a.1.1 1
120.59 even 2 2304.2.a.a.1.1 1
160.19 odd 8 1024.2.e.k.257.1 4
160.29 even 8 1024.2.e.k.257.2 4
160.59 odd 8 1024.2.e.k.769.1 4
160.69 even 8 1024.2.e.k.769.1 4
160.99 odd 8 1024.2.e.k.257.2 4
160.109 even 8 1024.2.e.k.257.1 4
160.139 odd 8 1024.2.e.k.769.2 4
160.149 even 8 1024.2.e.k.769.2 4
240.29 odd 4 1152.2.d.d.577.1 2
240.59 even 4 1152.2.d.d.577.2 2
240.149 odd 4 1152.2.d.d.577.2 2
240.179 even 4 1152.2.d.d.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.b.65.1 ✓ 2 80.59 odd 4
128.2.b.b.65.1 ✓ 2 80.69 even 4
128.2.b.b.65.2 yes 2 80.19 odd 4
128.2.b.b.65.2 yes 2 80.29 even 4
256.2.a.b.1.1 1 5.4 even 2
256.2.a.b.1.1 1 20.19 odd 2
256.2.a.c.1.1 1 40.19 odd 2
256.2.a.c.1.1 1 40.29 even 2
1024.2.e.k.257.1 4 160.19 odd 8
1024.2.e.k.257.1 4 160.109 even 8
1024.2.e.k.257.2 4 160.29 even 8
1024.2.e.k.257.2 4 160.99 odd 8
1024.2.e.k.769.1 4 160.59 odd 8
1024.2.e.k.769.1 4 160.69 even 8
1024.2.e.k.769.2 4 160.139 odd 8
1024.2.e.k.769.2 4 160.149 even 8
1152.2.d.d.577.1 2 240.29 odd 4
1152.2.d.d.577.1 2 240.179 even 4
1152.2.d.d.577.2 2 240.59 even 4
1152.2.d.d.577.2 2 240.149 odd 4
2304.2.a.a.1.1 1 120.29 odd 2
2304.2.a.a.1.1 1 120.59 even 2
2304.2.a.p.1.1 1 15.14 odd 2
2304.2.a.p.1.1 1 60.59 even 2
3200.2.d.e.1601.1 2 16.5 even 4
3200.2.d.e.1601.1 2 16.11 odd 4
3200.2.d.e.1601.2 2 16.3 odd 4
3200.2.d.e.1601.2 2 16.13 even 4
3200.2.f.c.449.1 2 80.3 even 4
3200.2.f.c.449.1 2 80.13 odd 4
3200.2.f.c.449.2 2 80.27 even 4
3200.2.f.c.449.2 2 80.37 odd 4
3200.2.f.d.449.1 2 80.43 even 4
3200.2.f.d.449.1 2 80.53 odd 4
3200.2.f.d.449.2 2 80.67 even 4
3200.2.f.d.449.2 2 80.77 odd 4
6400.2.a.l.1.1 1 8.3 odd 2
6400.2.a.l.1.1 1 8.5 even 2
6400.2.a.m.1.1 1 1.1 even 1 trivial
6400.2.a.m.1.1 1 4.3 odd 2 CM