Properties

Label 156.4.a.b
Level $156$
Weight $4$
Character orbit 156.a
Self dual yes
Analytic conductor $9.204$
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] = [156,4,Mod(1,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, 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("156.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 156.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: \(9.20429796090\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} - 2 q^{5} - 32 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 2 q^{5} - 32 q^{7} + 9 q^{9} - 68 q^{11} + 13 q^{13} - 6 q^{15} - 14 q^{17} + 4 q^{19} - 96 q^{21} + 72 q^{23} - 121 q^{25} + 27 q^{27} + 102 q^{29} - 136 q^{31} - 204 q^{33} + 64 q^{35} - 386 q^{37} + 39 q^{39} + 250 q^{41} - 140 q^{43} - 18 q^{45} - 296 q^{47} + 681 q^{49} - 42 q^{51} + 526 q^{53} + 136 q^{55} + 12 q^{57} + 332 q^{59} - 410 q^{61} - 288 q^{63} - 26 q^{65} + 596 q^{67} + 216 q^{69} - 880 q^{71} + 506 q^{73} - 363 q^{75} + 2176 q^{77} - 640 q^{79} + 81 q^{81} + 1380 q^{83} + 28 q^{85} + 306 q^{87} + 1450 q^{89} - 416 q^{91} - 408 q^{93} - 8 q^{95} - 446 q^{97} - 612 q^{99}+O(q^{100}) \) 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 −2.00000 0 −32.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

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 156.4.a.b 1
3.b odd 2 1 468.4.a.a 1
4.b odd 2 1 624.4.a.b 1
8.b even 2 1 2496.4.a.d 1
8.d odd 2 1 2496.4.a.m 1
12.b even 2 1 1872.4.a.i 1
13.b even 2 1 2028.4.a.b 1
13.d odd 4 2 2028.4.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.b 1 1.a even 1 1 trivial
468.4.a.a 1 3.b odd 2 1
624.4.a.b 1 4.b odd 2 1
1872.4.a.i 1 12.b even 2 1
2028.4.a.b 1 13.b even 2 1
2028.4.b.d 2 13.d odd 4 2
2496.4.a.d 1 8.b even 2 1
2496.4.a.m 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(156))\). 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
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T + 68 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T + 14 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T - 102 \) Copy content Toggle raw display
$31$ \( T + 136 \) Copy content Toggle raw display
$37$ \( T + 386 \) Copy content Toggle raw display
$41$ \( T - 250 \) Copy content Toggle raw display
$43$ \( T + 140 \) Copy content Toggle raw display
$47$ \( T + 296 \) Copy content Toggle raw display
$53$ \( T - 526 \) Copy content Toggle raw display
$59$ \( T - 332 \) Copy content Toggle raw display
$61$ \( T + 410 \) Copy content Toggle raw display
$67$ \( T - 596 \) Copy content Toggle raw display
$71$ \( T + 880 \) Copy content Toggle raw display
$73$ \( T - 506 \) Copy content Toggle raw display
$79$ \( T + 640 \) Copy content Toggle raw display
$83$ \( T - 1380 \) Copy content Toggle raw display
$89$ \( T - 1450 \) Copy content Toggle raw display
$97$ \( T + 446 \) Copy content Toggle raw display
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