Properties

Label 234.4.a.b
Level $234$
Weight $4$
Character orbit 234.a
Self dual yes
Analytic conductor $13.806$
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] = [234,4,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("234.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(13.8064469413\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 20 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} - 6 q^{5} + 20 q^{7} - 8 q^{8} + 12 q^{10} - 24 q^{11} + 13 q^{13} - 40 q^{14} + 16 q^{16} + 30 q^{17} - 16 q^{19} - 24 q^{20} + 48 q^{22} + 72 q^{23} - 89 q^{25} - 26 q^{26} + 80 q^{28} + 282 q^{29} + 164 q^{31} - 32 q^{32} - 60 q^{34} - 120 q^{35} + 110 q^{37} + 32 q^{38} + 48 q^{40} + 126 q^{41} + 164 q^{43} - 96 q^{44} - 144 q^{46} + 204 q^{47} + 57 q^{49} + 178 q^{50} + 52 q^{52} + 738 q^{53} + 144 q^{55} - 160 q^{56} - 564 q^{58} - 120 q^{59} + 614 q^{61} - 328 q^{62} + 64 q^{64} - 78 q^{65} + 848 q^{67} + 120 q^{68} + 240 q^{70} - 132 q^{71} + 218 q^{73} - 220 q^{74} - 64 q^{76} - 480 q^{77} - 1096 q^{79} - 96 q^{80} - 252 q^{82} - 552 q^{83} - 180 q^{85} - 328 q^{86} + 192 q^{88} - 210 q^{89} + 260 q^{91} + 288 q^{92} - 408 q^{94} + 96 q^{95} - 1726 q^{97} - 114 q^{98}+O(q^{100}) \) 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
−2.00000 0 4.00000 −6.00000 0 20.0000 −8.00000 0 12.0000
\(n\): 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 234.4.a.b 1
3.b odd 2 1 78.4.a.e 1
4.b odd 2 1 1872.4.a.e 1
12.b even 2 1 624.4.a.i 1
15.d odd 2 1 1950.4.a.c 1
24.f even 2 1 2496.4.a.b 1
24.h odd 2 1 2496.4.a.k 1
39.d odd 2 1 1014.4.a.b 1
39.f even 4 2 1014.4.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.e 1 3.b odd 2 1
234.4.a.b 1 1.a even 1 1 trivial
624.4.a.i 1 12.b even 2 1
1014.4.a.b 1 39.d odd 2 1
1014.4.b.c 2 39.f even 4 2
1872.4.a.e 1 4.b odd 2 1
1950.4.a.c 1 15.d odd 2 1
2496.4.a.b 1 24.f even 2 1
2496.4.a.k 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(234))\):

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{7} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T - 20 \) Copy content Toggle raw display
$11$ \( T + 24 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T + 16 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T - 282 \) Copy content Toggle raw display
$31$ \( T - 164 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T - 126 \) Copy content Toggle raw display
$43$ \( T - 164 \) Copy content Toggle raw display
$47$ \( T - 204 \) Copy content Toggle raw display
$53$ \( T - 738 \) Copy content Toggle raw display
$59$ \( T + 120 \) Copy content Toggle raw display
$61$ \( T - 614 \) Copy content Toggle raw display
$67$ \( T - 848 \) Copy content Toggle raw display
$71$ \( T + 132 \) Copy content Toggle raw display
$73$ \( T - 218 \) Copy content Toggle raw display
$79$ \( T + 1096 \) Copy content Toggle raw display
$83$ \( T + 552 \) Copy content Toggle raw display
$89$ \( T + 210 \) Copy content Toggle raw display
$97$ \( T + 1726 \) Copy content Toggle raw display
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