Properties

Label 153.2.a.d
Level $153$
Weight $2$
Character orbit 153.a
Self dual yes
Analytic conductor $1.222$
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] = [153,2,Mod(1,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 153.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: \(1.22171115093\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{10} + 3 q^{11} - 5 q^{13} - 4 q^{14} - 4 q^{16} + q^{17} - q^{19} + 2 q^{20} + 6 q^{22} + 7 q^{23} - 4 q^{25} - 10 q^{26} - 4 q^{28} - 6 q^{29} + 4 q^{31} - 8 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 9 q^{41} + q^{43} + 6 q^{44} + 14 q^{46} + 12 q^{47} - 3 q^{49} - 8 q^{50} - 10 q^{52} + 12 q^{53} + 3 q^{55} - 12 q^{58} - 6 q^{59} + 2 q^{61} + 8 q^{62} - 8 q^{64} - 5 q^{65} + 4 q^{67} + 2 q^{68} - 4 q^{70} + 8 q^{71} + 20 q^{74} - 2 q^{76} - 6 q^{77} - 6 q^{79} - 4 q^{80} - 18 q^{82} - 4 q^{83} + q^{85} + 2 q^{86} - 2 q^{89} + 10 q^{91} + 14 q^{92} + 24 q^{94} - q^{95} + 8 q^{97} - 6 q^{98}+O(q^{100}) \) 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 2.00000 1.00000 0 −2.00000 0 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)

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 153.2.a.d yes 1
3.b odd 2 1 153.2.a.a 1
4.b odd 2 1 2448.2.a.m 1
5.b even 2 1 3825.2.a.b 1
7.b odd 2 1 7497.2.a.p 1
8.b even 2 1 9792.2.a.p 1
8.d odd 2 1 9792.2.a.w 1
12.b even 2 1 2448.2.a.g 1
15.d odd 2 1 3825.2.a.o 1
17.b even 2 1 2601.2.a.l 1
21.c even 2 1 7497.2.a.a 1
24.f even 2 1 9792.2.a.bp 1
24.h odd 2 1 9792.2.a.bm 1
51.c odd 2 1 2601.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.a.a 1 3.b odd 2 1
153.2.a.d yes 1 1.a even 1 1 trivial
2448.2.a.g 1 12.b even 2 1
2448.2.a.m 1 4.b odd 2 1
2601.2.a.b 1 51.c odd 2 1
2601.2.a.l 1 17.b even 2 1
3825.2.a.b 1 5.b even 2 1
3825.2.a.o 1 15.d odd 2 1
7497.2.a.a 1 21.c even 2 1
7497.2.a.p 1 7.b odd 2 1
9792.2.a.p 1 8.b even 2 1
9792.2.a.w 1 8.d odd 2 1
9792.2.a.bm 1 24.h odd 2 1
9792.2.a.bp 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(153))\). 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 7 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 6 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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