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Properties

Label 6400.2.a.bj.1.2
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $2$
CM discriminant -40
Inner twists $4$

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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 = [2,0,0,0,0,0,0,0,-6,0,4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,0, 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+4.47214 q^{7} -3.00000 q^{9} +2.00000 q^{11} +4.47214 q^{13} -6.00000 q^{19} -4.47214 q^{23} +4.47214 q^{37} -2.00000 q^{41} +13.4164 q^{47} +13.0000 q^{49} +13.4164 q^{53} +14.0000 q^{59} -13.4164 q^{63} +8.94427 q^{77} +9.00000 q^{81} -14.0000 q^{89} +20.0000 q^{91} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} + 4 q^{11} - 12 q^{19} - 4 q^{41} + 26 q^{49} + 28 q^{59} + 18 q^{81} - 28 q^{89} + 40 q^{91} - 12 q^{99}+O(q^{100}) \) 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 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.47214 1.69031 0.845154 − 0.534522i \(-0.179509\pi\)
0.845154 + 0.534522i \(0.179509\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.47214 −0.932505 −0.466252 − 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 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 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 − 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.4164 1.95698 0.978492 − 0.206284i \(-0.0661372\pi\)
0.978492 + 0.206284i \(0.0661372\pi\)
\(48\) 0 0
\(49\) 13.0000 1.85714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.4164 1.84289 0.921443 − 0.388514i \(-0.127012\pi\)
0.921443 + 0.388514i \(0.127012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.0000 1.82264 0.911322 − 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −13.4164 −1.69031
\(64\) 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
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.94427 1.01929
\(78\) 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
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 20.0000 2.09657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
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.bj.1.2 2
4.3 odd 2 6400.2.a.bi.1.1 2
5.2 odd 4 1280.2.c.c.769.2 2
5.3 odd 4 1280.2.c.c.769.1 2
5.4 even 2 inner 6400.2.a.bj.1.1 2
8.3 odd 2 inner 6400.2.a.bj.1.1 2
8.5 even 2 6400.2.a.bi.1.2 2
16.3 odd 4 1600.2.d.g.801.4 4
16.5 even 4 1600.2.d.g.801.2 4
16.11 odd 4 1600.2.d.g.801.3 4
16.13 even 4 1600.2.d.g.801.1 4
20.3 even 4 1280.2.c.b.769.1 2
20.7 even 4 1280.2.c.b.769.2 2
20.19 odd 2 6400.2.a.bi.1.2 2
40.3 even 4 1280.2.c.c.769.2 2
40.13 odd 4 1280.2.c.b.769.2 2
40.19 odd 2 CM 6400.2.a.bj.1.2 2
40.27 even 4 1280.2.c.c.769.1 2
40.29 even 2 6400.2.a.bi.1.1 2
40.37 odd 4 1280.2.c.b.769.1 2
80.3 even 4 320.2.f.a.289.1 ✓ 4
80.13 odd 4 320.2.f.a.289.2 yes 4
80.19 odd 4 1600.2.d.g.801.2 4
80.27 even 4 320.2.f.a.289.2 yes 4
80.29 even 4 1600.2.d.g.801.3 4
80.37 odd 4 320.2.f.a.289.1 ✓ 4
80.43 even 4 320.2.f.a.289.3 yes 4
80.53 odd 4 320.2.f.a.289.4 yes 4
80.59 odd 4 1600.2.d.g.801.1 4
80.67 even 4 320.2.f.a.289.4 yes 4
80.69 even 4 1600.2.d.g.801.4 4
80.77 odd 4 320.2.f.a.289.3 yes 4
240.53 even 4 2880.2.d.e.289.2 4
240.77 even 4 2880.2.d.e.289.1 4
240.83 odd 4 2880.2.d.e.289.3 4
240.107 odd 4 2880.2.d.e.289.4 4
240.173 even 4 2880.2.d.e.289.4 4
240.197 even 4 2880.2.d.e.289.3 4
240.203 odd 4 2880.2.d.e.289.1 4
240.227 odd 4 2880.2.d.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.a.289.1 ✓ 4 80.3 even 4
320.2.f.a.289.1 ✓ 4 80.37 odd 4
320.2.f.a.289.2 yes 4 80.13 odd 4
320.2.f.a.289.2 yes 4 80.27 even 4
320.2.f.a.289.3 yes 4 80.43 even 4
320.2.f.a.289.3 yes 4 80.77 odd 4
320.2.f.a.289.4 yes 4 80.53 odd 4
320.2.f.a.289.4 yes 4 80.67 even 4
1280.2.c.b.769.1 2 20.3 even 4
1280.2.c.b.769.1 2 40.37 odd 4
1280.2.c.b.769.2 2 20.7 even 4
1280.2.c.b.769.2 2 40.13 odd 4
1280.2.c.c.769.1 2 5.3 odd 4
1280.2.c.c.769.1 2 40.27 even 4
1280.2.c.c.769.2 2 5.2 odd 4
1280.2.c.c.769.2 2 40.3 even 4
1600.2.d.g.801.1 4 16.13 even 4
1600.2.d.g.801.1 4 80.59 odd 4
1600.2.d.g.801.2 4 16.5 even 4
1600.2.d.g.801.2 4 80.19 odd 4
1600.2.d.g.801.3 4 16.11 odd 4
1600.2.d.g.801.3 4 80.29 even 4
1600.2.d.g.801.4 4 16.3 odd 4
1600.2.d.g.801.4 4 80.69 even 4
2880.2.d.e.289.1 4 240.77 even 4
2880.2.d.e.289.1 4 240.203 odd 4
2880.2.d.e.289.2 4 240.53 even 4
2880.2.d.e.289.2 4 240.227 odd 4
2880.2.d.e.289.3 4 240.83 odd 4
2880.2.d.e.289.3 4 240.197 even 4
2880.2.d.e.289.4 4 240.107 odd 4
2880.2.d.e.289.4 4 240.173 even 4
6400.2.a.bi.1.1 2 4.3 odd 2
6400.2.a.bi.1.1 2 40.29 even 2
6400.2.a.bi.1.2 2 8.5 even 2
6400.2.a.bi.1.2 2 20.19 odd 2
6400.2.a.bj.1.1 2 5.4 even 2 inner
6400.2.a.bj.1.1 2 8.3 odd 2 inner
6400.2.a.bj.1.2 2 1.1 even 1 trivial
6400.2.a.bj.1.2 2 40.19 odd 2 CM