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Properties

Label 784.6.a.y
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
Analytic rank $1$
Dimension $2$
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] = [784,6,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-118] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{46}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{46}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 7 \beta q^{5} - 59 q^{9} - 476 q^{11} - 71 \beta q^{13} + 1288 q^{15} + 66 \beta q^{17} + 47 \beta q^{19} - 3696 q^{23} + 5891 q^{25} - 302 \beta q^{27} + 1394 q^{29} + 142 \beta q^{31} + \cdots + 28084 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 118 q^{9} - 952 q^{11} + 2576 q^{15} - 7392 q^{23} + 11782 q^{25} + 2788 q^{29} + 24180 q^{37} - 26128 q^{39} - 19448 q^{43} + 24288 q^{51} + 8620 q^{53} + 17296 q^{57} - 182896 q^{65} - 40472 q^{67}+ \cdots + 56168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−6.78233
6.78233
0 −13.5647 0 −94.9526 0 0 0 −59.0000 0
1.2 0 13.5647 0 94.9526 0 0 0 −59.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.a.y 2
4.b odd 2 1 98.6.a.e ✓ 2
7.b odd 2 1 inner 784.6.a.y 2
12.b even 2 1 882.6.a.bo 2
28.d even 2 1 98.6.a.e ✓ 2
28.f even 6 2 98.6.c.g 4
28.g odd 6 2 98.6.c.g 4
84.h odd 2 1 882.6.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.6.a.e ✓ 2 4.b odd 2 1
98.6.a.e ✓ 2 28.d even 2 1
98.6.c.g 4 28.f even 6 2
98.6.c.g 4 28.g odd 6 2
784.6.a.y 2 1.a even 1 1 trivial
784.6.a.y 2 7.b odd 2 1 inner
882.6.a.bo 2 12.b even 2 1
882.6.a.bo 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 184 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 184 \) Copy content Toggle raw display
$5$ \( T^{2} - 9016 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 476)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 927544 \) Copy content Toggle raw display
$17$ \( T^{2} - 801504 \) Copy content Toggle raw display
$19$ \( T^{2} - 406456 \) Copy content Toggle raw display
$23$ \( (T + 3696)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1394)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3710176 \) Copy content Toggle raw display
$37$ \( (T - 12090)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 231634656 \) Copy content Toggle raw display
$43$ \( (T + 9724)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 856881376 \) Copy content Toggle raw display
$53$ \( (T - 4310)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 434675896 \) Copy content Toggle raw display
$61$ \( T^{2} - 86337400 \) Copy content Toggle raw display
$67$ \( (T + 20236)^{2} \) Copy content Toggle raw display
$71$ \( (T + 29792)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 127369216 \) Copy content Toggle raw display
$79$ \( (T - 33176)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 12534264 \) Copy content Toggle raw display
$89$ \( T^{2} - 5006024704 \) Copy content Toggle raw display
$97$ \( T^{2} - 315762400 \) Copy content Toggle raw display
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