Properties

Label 784.4.a.r
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
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] = [784,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
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)
 
\(f(q)\) \(=\) \( q + 7 q^{3} - 7 q^{5} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 7 q^{3} - 7 q^{5} + 22 q^{9} + 5 q^{11} + 14 q^{13} - 49 q^{15} + 21 q^{17} + 49 q^{19} + 159 q^{23} - 76 q^{25} - 35 q^{27} + 58 q^{29} + 147 q^{31} + 35 q^{33} + 219 q^{37} + 98 q^{39} - 350 q^{41} + 124 q^{43} - 154 q^{45} + 525 q^{47} + 147 q^{51} + 303 q^{53} - 35 q^{55} + 343 q^{57} - 105 q^{59} + 413 q^{61} - 98 q^{65} - 415 q^{67} + 1113 q^{69} + 432 q^{71} + 1113 q^{73} - 532 q^{75} + 103 q^{79} - 839 q^{81} + 1092 q^{83} - 147 q^{85} + 406 q^{87} + 329 q^{89} + 1029 q^{93} - 343 q^{95} + 882 q^{97} + 110 q^{99}+O(q^{100}) \) 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 7.00000 0 −7.00000 0 0 0 22.0000 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 784.4.a.r 1
4.b odd 2 1 49.4.a.c 1
7.b odd 2 1 784.4.a.b 1
7.d odd 6 2 112.4.i.c 2
12.b even 2 1 441.4.a.e 1
20.d odd 2 1 1225.4.a.d 1
28.d even 2 1 49.4.a.d 1
28.f even 6 2 7.4.c.a 2
28.g odd 6 2 49.4.c.a 2
56.j odd 6 2 448.4.i.a 2
56.m even 6 2 448.4.i.f 2
84.h odd 2 1 441.4.a.d 1
84.j odd 6 2 63.4.e.b 2
84.n even 6 2 441.4.e.k 2
140.c even 2 1 1225.4.a.c 1
140.s even 6 2 175.4.e.a 2
140.x odd 12 4 175.4.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 28.f even 6 2
49.4.a.c 1 4.b odd 2 1
49.4.a.d 1 28.d even 2 1
49.4.c.a 2 28.g odd 6 2
63.4.e.b 2 84.j odd 6 2
112.4.i.c 2 7.d odd 6 2
175.4.e.a 2 140.s even 6 2
175.4.k.a 4 140.x odd 12 4
441.4.a.d 1 84.h odd 2 1
441.4.a.e 1 12.b even 2 1
441.4.e.k 2 84.n even 6 2
448.4.i.a 2 56.j odd 6 2
448.4.i.f 2 56.m even 6 2
784.4.a.b 1 7.b odd 2 1
784.4.a.r 1 1.a even 1 1 trivial
1225.4.a.c 1 140.c even 2 1
1225.4.a.d 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3} - 7 \) Copy content Toggle raw display
\( T_{5} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 7 \) Copy content Toggle raw display
$5$ \( T + 7 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 5 \) Copy content Toggle raw display
$13$ \( T - 14 \) Copy content Toggle raw display
$17$ \( T - 21 \) Copy content Toggle raw display
$19$ \( T - 49 \) Copy content Toggle raw display
$23$ \( T - 159 \) Copy content Toggle raw display
$29$ \( T - 58 \) Copy content Toggle raw display
$31$ \( T - 147 \) Copy content Toggle raw display
$37$ \( T - 219 \) Copy content Toggle raw display
$41$ \( T + 350 \) Copy content Toggle raw display
$43$ \( T - 124 \) Copy content Toggle raw display
$47$ \( T - 525 \) Copy content Toggle raw display
$53$ \( T - 303 \) Copy content Toggle raw display
$59$ \( T + 105 \) Copy content Toggle raw display
$61$ \( T - 413 \) Copy content Toggle raw display
$67$ \( T + 415 \) Copy content Toggle raw display
$71$ \( T - 432 \) Copy content Toggle raw display
$73$ \( T - 1113 \) Copy content Toggle raw display
$79$ \( T - 103 \) Copy content Toggle raw display
$83$ \( T - 1092 \) Copy content Toggle raw display
$89$ \( T - 329 \) Copy content Toggle raw display
$97$ \( T - 882 \) Copy content Toggle raw display
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