Properties

Label 6400.2.a.bc.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $2$
CM discriminant -8
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 = [2,0,-2,0,0,0,0,0,8,0,-6,0,0,0,0,0,6,0,-2,0,0,0,0,0,0,0,-26,0, 0,0,0] 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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1600)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-3.44949 q^{3} +8.89898 q^{9} -5.44949 q^{11} -1.89898 q^{17} +6.34847 q^{19} -20.3485 q^{27} +18.7980 q^{33} +6.79796 q^{41} -10.0000 q^{43} -7.00000 q^{49} +6.55051 q^{51} -21.8990 q^{57} +6.00000 q^{59} -0.348469 q^{67} +15.6969 q^{73} +43.4949 q^{81} -6.55051 q^{83} +4.10102 q^{89} +10.0000 q^{97} -48.4949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 8 q^{9} - 6 q^{11} + 6 q^{17} - 2 q^{19} - 26 q^{27} + 18 q^{33} - 6 q^{41} - 20 q^{43} - 14 q^{49} + 18 q^{51} - 34 q^{57} + 12 q^{59} + 14 q^{67} + 2 q^{73} + 38 q^{81} - 18 q^{83} + 18 q^{89}+ \cdots - 48 q^{99}+O(q^{100}) \) 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
\(3\) −3.44949 −1.99156 −0.995782 − 0.0917517i \(-0.970753\pi\)
−0.995782 + 0.0917517i \(0.970753\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 8.89898 2.96633
\(10\) 0 0
\(11\) −5.44949 −1.64308 −0.821541 − 0.570149i \(-0.806886\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.89898 −0.460570 −0.230285 − 0.973123i \(-0.573966\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) 6.34847 1.45644 0.728219 − 0.685344i \(-0.240348\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 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
\(27\) −20.3485 −3.91606
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 18.7980 3.27230
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.79796 1.06166 0.530831 − 0.847477i \(-0.321880\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 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
\(51\) 6.55051 0.917255
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −21.8990 −2.90059
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.348469 −0.0425723 −0.0212861 − 0.999773i \(-0.506776\pi\)
−0.0212861 + 0.999773i \(0.506776\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 15.6969 1.83719 0.918594 − 0.395203i \(-0.129326\pi\)
0.918594 + 0.395203i \(0.129326\pi\)
\(74\) 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
\(81\) 43.4949 4.83277
\(82\) 0 0
\(83\) −6.55051 −0.719012 −0.359506 − 0.933143i \(-0.617055\pi\)
−0.359506 + 0.933143i \(0.617055\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.10102 0.434707 0.217354 − 0.976093i \(-0.430258\pi\)
0.217354 + 0.976093i \(0.430258\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −48.4949 −4.87392
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.bc.1.1 2
4.3 odd 2 6400.2.a.ci.1.2 2
5.4 even 2 6400.2.a.ch.1.2 2
8.3 odd 2 CM 6400.2.a.bc.1.1 2
8.5 even 2 6400.2.a.ci.1.2 2
16.3 odd 4 1600.2.d.d.801.4 yes 4
16.5 even 4 1600.2.d.d.801.4 yes 4
16.11 odd 4 1600.2.d.d.801.1 yes 4
16.13 even 4 1600.2.d.d.801.1 yes 4
20.19 odd 2 6400.2.a.bd.1.1 2
40.19 odd 2 6400.2.a.ch.1.2 2
40.29 even 2 6400.2.a.bd.1.1 2
80.3 even 4 1600.2.f.f.1249.1 4
80.13 odd 4 1600.2.f.j.1249.4 4
80.19 odd 4 1600.2.d.c.801.1 ✓ 4
80.27 even 4 1600.2.f.f.1249.2 4
80.29 even 4 1600.2.d.c.801.4 yes 4
80.37 odd 4 1600.2.f.j.1249.3 4
80.43 even 4 1600.2.f.j.1249.4 4
80.53 odd 4 1600.2.f.f.1249.1 4
80.59 odd 4 1600.2.d.c.801.4 yes 4
80.67 even 4 1600.2.f.j.1249.3 4
80.69 even 4 1600.2.d.c.801.1 ✓ 4
80.77 odd 4 1600.2.f.f.1249.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.c.801.1 ✓ 4 80.19 odd 4
1600.2.d.c.801.1 ✓ 4 80.69 even 4
1600.2.d.c.801.4 yes 4 80.29 even 4
1600.2.d.c.801.4 yes 4 80.59 odd 4
1600.2.d.d.801.1 yes 4 16.11 odd 4
1600.2.d.d.801.1 yes 4 16.13 even 4
1600.2.d.d.801.4 yes 4 16.3 odd 4
1600.2.d.d.801.4 yes 4 16.5 even 4
1600.2.f.f.1249.1 4 80.3 even 4
1600.2.f.f.1249.1 4 80.53 odd 4
1600.2.f.f.1249.2 4 80.27 even 4
1600.2.f.f.1249.2 4 80.77 odd 4
1600.2.f.j.1249.3 4 80.37 odd 4
1600.2.f.j.1249.3 4 80.67 even 4
1600.2.f.j.1249.4 4 80.13 odd 4
1600.2.f.j.1249.4 4 80.43 even 4
6400.2.a.bc.1.1 2 1.1 even 1 trivial
6400.2.a.bc.1.1 2 8.3 odd 2 CM
6400.2.a.bd.1.1 2 20.19 odd 2
6400.2.a.bd.1.1 2 40.29 even 2
6400.2.a.ch.1.2 2 5.4 even 2
6400.2.a.ch.1.2 2 40.19 odd 2
6400.2.a.ci.1.2 2 4.3 odd 2
6400.2.a.ci.1.2 2 8.5 even 2