Properties

Label 784.6.a.f
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 28)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} + 96 q^{5} - 239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + 96 q^{5} - 239 q^{9} + 720 q^{11} - 572 q^{13} - 192 q^{15} - 1254 q^{17} - 94 q^{19} - 96 q^{23} + 6091 q^{25} + 964 q^{27} - 4374 q^{29} - 6244 q^{31} - 1440 q^{33} - 10798 q^{37} + 1144 q^{39} - 12006 q^{41} + 9160 q^{43} - 22944 q^{45} - 25836 q^{47} + 2508 q^{51} + 1014 q^{53} + 69120 q^{55} + 188 q^{57} + 1242 q^{59} - 7592 q^{61} - 54912 q^{65} - 41132 q^{67} + 192 q^{69} + 37632 q^{71} + 13438 q^{73} - 12182 q^{75} - 6248 q^{79} + 56149 q^{81} - 25254 q^{83} - 120384 q^{85} + 8748 q^{87} + 45126 q^{89} + 12488 q^{93} - 9024 q^{95} - 107222 q^{97} - 172080 q^{99}+O(q^{100}) \) 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 −2.00000 0 96.0000 0 0 0 −239.000 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.f 1
4.b odd 2 1 196.6.a.d 1
7.b odd 2 1 112.6.a.e 1
21.c even 2 1 1008.6.a.bb 1
28.d even 2 1 28.6.a.a 1
28.f even 6 2 196.6.e.f 2
28.g odd 6 2 196.6.e.e 2
56.e even 2 1 448.6.a.i 1
56.h odd 2 1 448.6.a.h 1
84.h odd 2 1 252.6.a.d 1
140.c even 2 1 700.6.a.d 1
140.j odd 4 2 700.6.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.a 1 28.d even 2 1
112.6.a.e 1 7.b odd 2 1
196.6.a.d 1 4.b odd 2 1
196.6.e.e 2 28.g odd 6 2
196.6.e.f 2 28.f even 6 2
252.6.a.d 1 84.h odd 2 1
448.6.a.h 1 56.h odd 2 1
448.6.a.i 1 56.e even 2 1
700.6.a.d 1 140.c even 2 1
700.6.e.d 2 140.j odd 4 2
784.6.a.f 1 1.a even 1 1 trivial
1008.6.a.bb 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) 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
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 96 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 720 \) Copy content Toggle raw display
$13$ \( T + 572 \) Copy content Toggle raw display
$17$ \( T + 1254 \) Copy content Toggle raw display
$19$ \( T + 94 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T + 4374 \) Copy content Toggle raw display
$31$ \( T + 6244 \) Copy content Toggle raw display
$37$ \( T + 10798 \) Copy content Toggle raw display
$41$ \( T + 12006 \) Copy content Toggle raw display
$43$ \( T - 9160 \) Copy content Toggle raw display
$47$ \( T + 25836 \) Copy content Toggle raw display
$53$ \( T - 1014 \) Copy content Toggle raw display
$59$ \( T - 1242 \) Copy content Toggle raw display
$61$ \( T + 7592 \) Copy content Toggle raw display
$67$ \( T + 41132 \) Copy content Toggle raw display
$71$ \( T - 37632 \) Copy content Toggle raw display
$73$ \( T - 13438 \) Copy content Toggle raw display
$79$ \( T + 6248 \) Copy content Toggle raw display
$83$ \( T + 25254 \) Copy content Toggle raw display
$89$ \( T - 45126 \) Copy content Toggle raw display
$97$ \( T + 107222 \) Copy content Toggle raw display
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