Properties

Label 6400.2.a.bc.1.2
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 1.2
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)
 
\(f(q)\) \(=\) \(q+1.44949 q^{3} -0.898979 q^{9} -0.550510 q^{11} +7.89898 q^{17} -8.34847 q^{19} -5.65153 q^{27} -0.797959 q^{33} -12.7980 q^{41} -10.0000 q^{43} -7.00000 q^{49} +11.4495 q^{51} -12.1010 q^{57} +6.00000 q^{59} +14.3485 q^{67} -13.6969 q^{73} -5.49490 q^{81} -11.4495 q^{83} +13.8990 q^{89} +10.0000 q^{97} +0.494897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 8 q^{9} - 6 q^{11} + 6 q^{17} - 2 q^{19} - 26 q^{27} + 18 q^{33} - 6 q^{41} - 20 q^{43} - 14 q^{49} + 18 q^{51} - 34 q^{57} + 12 q^{59} + 14 q^{67} + 2 q^{73} + 38 q^{81} - 18 q^{83} + 18 q^{89}+ \cdots - 48 q^{99}+O(q^{100}) \) 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
\(3\) 1.44949 0.836863 0.418432 − 0.908248i \(-0.362580\pi\)
0.418432 + 0.908248i \(0.362580\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −0.898979 −0.299660
\(10\) 0 0
\(11\) −0.550510 −0.165985 −0.0829925 − 0.996550i \(-0.526448\pi\)
−0.0829925 + 0.996550i \(0.526448\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.89898 1.91578 0.957892 − 0.287129i \(-0.0927008\pi\)
0.957892 + 0.287129i \(0.0927008\pi\)
\(18\) 0 0
\(19\) −8.34847 −1.91527 −0.957635 − 0.287984i \(-0.907015\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 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
\(27\) −5.65153 −1.08764
\(28\) 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
\(33\) −0.797959 −0.138907
\(34\) 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
\(41\) −12.7980 −1.99871 −0.999353 − 0.0359748i \(-0.988546\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 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
\(51\) 11.4495 1.60325
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.1010 −1.60282
\(58\) 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
\(67\) 14.3485 1.75294 0.876472 − 0.481452i \(-0.159891\pi\)
0.876472 + 0.481452i \(0.159891\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −13.6969 −1.60311 −0.801553 − 0.597924i \(-0.795992\pi\)
−0.801553 + 0.597924i \(0.795992\pi\)
\(74\) 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
\(81\) −5.49490 −0.610544
\(82\) 0 0
\(83\) −11.4495 −1.25674 −0.628372 − 0.777913i \(-0.716279\pi\)
−0.628372 + 0.777913i \(0.716279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.8990 1.47329 0.736644 − 0.676280i \(-0.236409\pi\)
0.736644 + 0.676280i \(0.236409\pi\)
\(90\) 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
\(99\) 0.494897 0.0497391
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.1.2 2
4.3 odd 2 6400.2.a.ci.1.1 2
5.4 even 2 6400.2.a.ch.1.1 2
8.3 odd 2 CM 6400.2.a.bc.1.2 2
8.5 even 2 6400.2.a.ci.1.1 2
16.3 odd 4 1600.2.d.d.801.2 yes 4
16.5 even 4 1600.2.d.d.801.2 yes 4
16.11 odd 4 1600.2.d.d.801.3 yes 4
16.13 even 4 1600.2.d.d.801.3 yes 4
20.19 odd 2 6400.2.a.bd.1.2 2
40.19 odd 2 6400.2.a.ch.1.1 2
40.29 even 2 6400.2.a.bd.1.2 2
80.3 even 4 1600.2.f.f.1249.3 4
80.13 odd 4 1600.2.f.j.1249.2 4
80.19 odd 4 1600.2.d.c.801.3 yes 4
80.27 even 4 1600.2.f.f.1249.4 4
80.29 even 4 1600.2.d.c.801.2 ✓ 4
80.37 odd 4 1600.2.f.j.1249.1 4
80.43 even 4 1600.2.f.j.1249.2 4
80.53 odd 4 1600.2.f.f.1249.3 4
80.59 odd 4 1600.2.d.c.801.2 ✓ 4
80.67 even 4 1600.2.f.j.1249.1 4
80.69 even 4 1600.2.d.c.801.3 yes 4
80.77 odd 4 1600.2.f.f.1249.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.c.801.2 ✓ 4 80.29 even 4
1600.2.d.c.801.2 ✓ 4 80.59 odd 4
1600.2.d.c.801.3 yes 4 80.19 odd 4
1600.2.d.c.801.3 yes 4 80.69 even 4
1600.2.d.d.801.2 yes 4 16.3 odd 4
1600.2.d.d.801.2 yes 4 16.5 even 4
1600.2.d.d.801.3 yes 4 16.11 odd 4
1600.2.d.d.801.3 yes 4 16.13 even 4
1600.2.f.f.1249.3 4 80.3 even 4
1600.2.f.f.1249.3 4 80.53 odd 4
1600.2.f.f.1249.4 4 80.27 even 4
1600.2.f.f.1249.4 4 80.77 odd 4
1600.2.f.j.1249.1 4 80.37 odd 4
1600.2.f.j.1249.1 4 80.67 even 4
1600.2.f.j.1249.2 4 80.13 odd 4
1600.2.f.j.1249.2 4 80.43 even 4
6400.2.a.bc.1.2 2 1.1 even 1 trivial
6400.2.a.bc.1.2 2 8.3 odd 2 CM
6400.2.a.bd.1.2 2 20.19 odd 2
6400.2.a.bd.1.2 2 40.29 even 2
6400.2.a.ch.1.1 2 5.4 even 2
6400.2.a.ch.1.1 2 40.19 odd 2
6400.2.a.ci.1.1 2 4.3 odd 2
6400.2.a.ci.1.1 2 8.5 even 2