Properties

Label 1176.4.a.g
Level $1176$
Weight $4$
Character orbit 1176.a
Self dual yes
Analytic conductor $69.386$
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] = [1176,4,Mod(1,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1176.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.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: \(69.3862461668\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} + 10 q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 10 q^{5} + 9 q^{9} - 52 q^{11} + 10 q^{13} - 30 q^{15} + 54 q^{17} + 52 q^{19} + 48 q^{23} - 25 q^{25} - 27 q^{27} - 186 q^{29} - 224 q^{31} + 156 q^{33} + 94 q^{37} - 30 q^{39} + 478 q^{41} - 316 q^{43} + 90 q^{45} - 256 q^{47} - 162 q^{51} - 66 q^{53} - 520 q^{55} - 156 q^{57} - 420 q^{59} - 342 q^{61} + 100 q^{65} + 668 q^{67} - 144 q^{69} - 272 q^{71} + 86 q^{73} + 75 q^{75} + 1360 q^{79} + 81 q^{81} - 188 q^{83} + 540 q^{85} + 558 q^{87} + 366 q^{89} + 672 q^{93} + 520 q^{95} - 1554 q^{97} - 468 q^{99}+O(q^{100}) \) 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 −3.00000 0 10.0000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

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 1176.4.a.g 1
4.b odd 2 1 2352.4.a.bh 1
7.b odd 2 1 168.4.a.e 1
21.c even 2 1 504.4.a.e 1
28.d even 2 1 336.4.a.b 1
56.e even 2 1 1344.4.a.x 1
56.h odd 2 1 1344.4.a.k 1
84.h odd 2 1 1008.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.e 1 7.b odd 2 1
336.4.a.b 1 28.d even 2 1
504.4.a.e 1 21.c even 2 1
1008.4.a.q 1 84.h odd 2 1
1176.4.a.g 1 1.a even 1 1 trivial
1344.4.a.k 1 56.h odd 2 1
1344.4.a.x 1 56.e even 2 1
2352.4.a.bh 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1176))\):

\( T_{5} - 10 \) Copy content Toggle raw display
\( T_{11} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 10 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 52 \) Copy content Toggle raw display
$13$ \( T - 10 \) Copy content Toggle raw display
$17$ \( T - 54 \) Copy content Toggle raw display
$19$ \( T - 52 \) Copy content Toggle raw display
$23$ \( T - 48 \) Copy content Toggle raw display
$29$ \( T + 186 \) Copy content Toggle raw display
$31$ \( T + 224 \) Copy content Toggle raw display
$37$ \( T - 94 \) Copy content Toggle raw display
$41$ \( T - 478 \) Copy content Toggle raw display
$43$ \( T + 316 \) Copy content Toggle raw display
$47$ \( T + 256 \) Copy content Toggle raw display
$53$ \( T + 66 \) Copy content Toggle raw display
$59$ \( T + 420 \) Copy content Toggle raw display
$61$ \( T + 342 \) Copy content Toggle raw display
$67$ \( T - 668 \) Copy content Toggle raw display
$71$ \( T + 272 \) Copy content Toggle raw display
$73$ \( T - 86 \) Copy content Toggle raw display
$79$ \( T - 1360 \) Copy content Toggle raw display
$83$ \( T + 188 \) Copy content Toggle raw display
$89$ \( T - 366 \) Copy content Toggle raw display
$97$ \( T + 1554 \) Copy content Toggle raw display
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