Properties

Label 234.4.a.c
Level $234$
Weight $4$
Character orbit 234.a
Self dual yes
Analytic conductor $13.806$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 8 q^{8} + 8 q^{10} - 2 q^{11} - 13 q^{13} - 8 q^{14} + 16 q^{16} + 6 q^{17} - 36 q^{19} - 16 q^{20} + 4 q^{22} + 20 q^{23} - 109 q^{25} + 26 q^{26} + 16 q^{28} + 14 q^{29} - 152 q^{31} - 32 q^{32} - 12 q^{34} - 16 q^{35} - 258 q^{37} + 72 q^{38} + 32 q^{40} - 84 q^{41} - 188 q^{43} - 8 q^{44} - 40 q^{46} - 254 q^{47} - 327 q^{49} + 218 q^{50} - 52 q^{52} - 366 q^{53} + 8 q^{55} - 32 q^{56} - 28 q^{58} - 550 q^{59} - 14 q^{61} + 304 q^{62} + 64 q^{64} + 52 q^{65} + 448 q^{67} + 24 q^{68} + 32 q^{70} - 926 q^{71} + 254 q^{73} + 516 q^{74} - 144 q^{76} - 8 q^{77} + 1328 q^{79} - 64 q^{80} + 168 q^{82} - 186 q^{83} - 24 q^{85} + 376 q^{86} + 16 q^{88} + 336 q^{89} - 52 q^{91} + 80 q^{92} + 508 q^{94} + 144 q^{95} + 614 q^{97} + 654 q^{98}+O(q^{100}) \) 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 −4.00000 0 4.00000 −8.00000 0 8.00000
\(n\): 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.c 1
3.b odd 2 1 78.4.a.f 1
4.b odd 2 1 1872.4.a.f 1
12.b even 2 1 624.4.a.c 1
15.d odd 2 1 1950.4.a.a 1
24.f even 2 1 2496.4.a.l 1
24.h odd 2 1 2496.4.a.c 1
39.d odd 2 1 1014.4.a.e 1
39.f even 4 2 1014.4.b.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.f 1 3.b odd 2 1
234.4.a.c 1 1.a even 1 1 trivial
624.4.a.c 1 12.b even 2 1
1014.4.a.e 1 39.d odd 2 1
1014.4.b.g 2 39.f even 4 2
1872.4.a.f 1 4.b odd 2 1
1950.4.a.a 1 15.d odd 2 1
2496.4.a.c 1 24.h odd 2 1
2496.4.a.l 1 24.f even 2 1

Hecke kernels

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

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} - 4 \) 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
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T + 13 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 36 \) Copy content Toggle raw display
$23$ \( T - 20 \) Copy content Toggle raw display
$29$ \( T - 14 \) Copy content Toggle raw display
$31$ \( T + 152 \) Copy content Toggle raw display
$37$ \( T + 258 \) Copy content Toggle raw display
$41$ \( T + 84 \) Copy content Toggle raw display
$43$ \( T + 188 \) Copy content Toggle raw display
$47$ \( T + 254 \) Copy content Toggle raw display
$53$ \( T + 366 \) Copy content Toggle raw display
$59$ \( T + 550 \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T - 448 \) Copy content Toggle raw display
$71$ \( T + 926 \) Copy content Toggle raw display
$73$ \( T - 254 \) Copy content Toggle raw display
$79$ \( T - 1328 \) Copy content Toggle raw display
$83$ \( T + 186 \) Copy content Toggle raw display
$89$ \( T - 336 \) Copy content Toggle raw display
$97$ \( T - 614 \) Copy content Toggle raw display
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