Properties

Label 1232.4.a.d
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, 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("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{3} + 12 q^{5} - 7 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{3} + 12 q^{5} - 7 q^{7} - 11 q^{9} - 11 q^{11} + 38 q^{13} - 48 q^{15} - 48 q^{17} + 70 q^{19} + 28 q^{21} - 12 q^{23} + 19 q^{25} + 152 q^{27} + 126 q^{29} + 70 q^{31} + 44 q^{33} - 84 q^{35} - 358 q^{37} - 152 q^{39} - 216 q^{41} - 344 q^{43} - 132 q^{45} - 390 q^{47} + 49 q^{49} + 192 q^{51} + 438 q^{53} - 132 q^{55} - 280 q^{57} + 552 q^{59} + 830 q^{61} + 77 q^{63} + 456 q^{65} + 196 q^{67} + 48 q^{69} - 648 q^{71} - 16 q^{73} - 76 q^{75} + 77 q^{77} - 1352 q^{79} - 311 q^{81} - 90 q^{83} - 576 q^{85} - 504 q^{87} + 1146 q^{89} - 266 q^{91} - 280 q^{93} + 840 q^{95} - 70 q^{97} + 121 q^{99}+O(q^{100}) \) 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 −4.00000 0 12.0000 0 −7.00000 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(11\) \( +1 \)

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 1232.4.a.d 1
4.b odd 2 1 77.4.a.a 1
12.b even 2 1 693.4.a.b 1
20.d odd 2 1 1925.4.a.a 1
28.d even 2 1 539.4.a.a 1
44.c even 2 1 847.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.a 1 4.b odd 2 1
539.4.a.a 1 28.d even 2 1
693.4.a.b 1 12.b even 2 1
847.4.a.a 1 44.c even 2 1
1232.4.a.d 1 1.a even 1 1 trivial
1925.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3} + 4 \) Copy content Toggle raw display
\( T_{5} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 4 \) Copy content Toggle raw display
$5$ \( T - 12 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 38 \) Copy content Toggle raw display
$17$ \( T + 48 \) Copy content Toggle raw display
$19$ \( T - 70 \) Copy content Toggle raw display
$23$ \( T + 12 \) Copy content Toggle raw display
$29$ \( T - 126 \) Copy content Toggle raw display
$31$ \( T - 70 \) Copy content Toggle raw display
$37$ \( T + 358 \) Copy content Toggle raw display
$41$ \( T + 216 \) Copy content Toggle raw display
$43$ \( T + 344 \) Copy content Toggle raw display
$47$ \( T + 390 \) Copy content Toggle raw display
$53$ \( T - 438 \) Copy content Toggle raw display
$59$ \( T - 552 \) Copy content Toggle raw display
$61$ \( T - 830 \) Copy content Toggle raw display
$67$ \( T - 196 \) Copy content Toggle raw display
$71$ \( T + 648 \) Copy content Toggle raw display
$73$ \( T + 16 \) Copy content Toggle raw display
$79$ \( T + 1352 \) Copy content Toggle raw display
$83$ \( T + 90 \) Copy content Toggle raw display
$89$ \( T - 1146 \) Copy content Toggle raw display
$97$ \( T + 70 \) Copy content Toggle raw display
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