Properties

Label 1232.4.a.f
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{3} + 18 q^{5} - 7 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 18 q^{5} - 7 q^{7} - 23 q^{9} + 11 q^{11} + 56 q^{13} + 36 q^{15} + 36 q^{17} + 28 q^{19} - 14 q^{21} - 180 q^{23} + 199 q^{25} - 100 q^{27} - 54 q^{29} + 334 q^{31} + 22 q^{33} - 126 q^{35} + 386 q^{37} + 112 q^{39} - 444 q^{41} + 316 q^{43} - 414 q^{45} + 402 q^{47} + 49 q^{49} + 72 q^{51} - 486 q^{53} + 198 q^{55} + 56 q^{57} + 282 q^{59} + 380 q^{61} + 161 q^{63} + 1008 q^{65} - 176 q^{67} - 360 q^{69} + 324 q^{71} + 800 q^{73} + 398 q^{75} - 77 q^{77} + 1144 q^{79} + 421 q^{81} - 468 q^{83} + 648 q^{85} - 108 q^{87} - 870 q^{89} - 392 q^{91} + 668 q^{93} + 504 q^{95} - 1330 q^{97} - 253 q^{99}+O(q^{100}) \) 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 18.0000 0 −7.00000 0 −23.0000 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.f 1
4.b odd 2 1 154.4.a.d 1
12.b even 2 1 1386.4.a.a 1
28.d even 2 1 1078.4.a.g 1
44.c even 2 1 1694.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.d 1 4.b odd 2 1
1078.4.a.g 1 28.d even 2 1
1232.4.a.f 1 1.a even 1 1 trivial
1386.4.a.a 1 12.b even 2 1
1694.4.a.c 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} - 18 \) 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
$5$ \( T - 18 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 56 \) Copy content Toggle raw display
$17$ \( T - 36 \) Copy content Toggle raw display
$19$ \( T - 28 \) Copy content Toggle raw display
$23$ \( T + 180 \) Copy content Toggle raw display
$29$ \( T + 54 \) Copy content Toggle raw display
$31$ \( T - 334 \) Copy content Toggle raw display
$37$ \( T - 386 \) Copy content Toggle raw display
$41$ \( T + 444 \) Copy content Toggle raw display
$43$ \( T - 316 \) Copy content Toggle raw display
$47$ \( T - 402 \) Copy content Toggle raw display
$53$ \( T + 486 \) Copy content Toggle raw display
$59$ \( T - 282 \) Copy content Toggle raw display
$61$ \( T - 380 \) Copy content Toggle raw display
$67$ \( T + 176 \) Copy content Toggle raw display
$71$ \( T - 324 \) Copy content Toggle raw display
$73$ \( T - 800 \) Copy content Toggle raw display
$79$ \( T - 1144 \) Copy content Toggle raw display
$83$ \( T + 468 \) Copy content Toggle raw display
$89$ \( T + 870 \) Copy content Toggle raw display
$97$ \( T + 1330 \) Copy content Toggle raw display
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