Properties

Label 784.6.a.h
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,6,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 8 q^{3} - 10 q^{5} - 179 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{3} - 10 q^{5} - 179 q^{9} + 340 q^{11} + 294 q^{13} - 80 q^{15} - 1226 q^{17} + 2432 q^{19} - 2000 q^{23} - 3025 q^{25} - 3376 q^{27} - 6746 q^{29} + 8856 q^{31} + 2720 q^{33} + 9182 q^{37} + 2352 q^{39} + 14574 q^{41} - 8108 q^{43} + 1790 q^{45} - 312 q^{47} - 9808 q^{51} - 14634 q^{53} - 3400 q^{55} + 19456 q^{57} - 27656 q^{59} - 34338 q^{61} - 2940 q^{65} - 12316 q^{67} - 16000 q^{69} - 36920 q^{71} + 61718 q^{73} - 24200 q^{75} + 64752 q^{79} + 16489 q^{81} - 77056 q^{83} + 12260 q^{85} - 53968 q^{87} + 8166 q^{89} + 70848 q^{93} - 24320 q^{95} - 20650 q^{97} - 60860 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
0
0 8.00000 0 −10.0000 0 0 0 −179.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.a.h 1
4.b odd 2 1 98.6.a.b 1
7.b odd 2 1 112.6.a.d 1
12.b even 2 1 882.6.a.g 1
21.c even 2 1 1008.6.a.n 1
28.d even 2 1 14.6.a.b 1
28.f even 6 2 98.6.c.a 2
28.g odd 6 2 98.6.c.b 2
56.e even 2 1 448.6.a.f 1
56.h odd 2 1 448.6.a.k 1
84.h odd 2 1 126.6.a.c 1
140.c even 2 1 350.6.a.b 1
140.j odd 4 2 350.6.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 28.d even 2 1
98.6.a.b 1 4.b odd 2 1
98.6.c.a 2 28.f even 6 2
98.6.c.b 2 28.g odd 6 2
112.6.a.d 1 7.b odd 2 1
126.6.a.c 1 84.h odd 2 1
350.6.a.b 1 140.c even 2 1
350.6.c.f 2 140.j odd 4 2
448.6.a.f 1 56.e even 2 1
448.6.a.k 1 56.h odd 2 1
784.6.a.h 1 1.a even 1 1 trivial
882.6.a.g 1 12.b even 2 1
1008.6.a.n 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 8 \) Copy content Toggle raw display
$5$ \( T + 10 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 340 \) Copy content Toggle raw display
$13$ \( T - 294 \) Copy content Toggle raw display
$17$ \( T + 1226 \) Copy content Toggle raw display
$19$ \( T - 2432 \) Copy content Toggle raw display
$23$ \( T + 2000 \) Copy content Toggle raw display
$29$ \( T + 6746 \) Copy content Toggle raw display
$31$ \( T - 8856 \) Copy content Toggle raw display
$37$ \( T - 9182 \) Copy content Toggle raw display
$41$ \( T - 14574 \) Copy content Toggle raw display
$43$ \( T + 8108 \) Copy content Toggle raw display
$47$ \( T + 312 \) Copy content Toggle raw display
$53$ \( T + 14634 \) Copy content Toggle raw display
$59$ \( T + 27656 \) Copy content Toggle raw display
$61$ \( T + 34338 \) Copy content Toggle raw display
$67$ \( T + 12316 \) Copy content Toggle raw display
$71$ \( T + 36920 \) Copy content Toggle raw display
$73$ \( T - 61718 \) Copy content Toggle raw display
$79$ \( T - 64752 \) Copy content Toggle raw display
$83$ \( T + 77056 \) Copy content Toggle raw display
$89$ \( T - 8166 \) Copy content Toggle raw display
$97$ \( T + 20650 \) Copy content Toggle raw display
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