Properties

Label 297.2.a.d
Level $297$
Weight $2$
Character orbit 297.a
Self dual yes
Analytic conductor $2.372$
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] = [297,2,Mod(1,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, 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("297.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.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: \(2.37155694003\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 4 q^{10} - q^{11} - 5 q^{13} + 2 q^{14} - 4 q^{16} + 2 q^{17} + 3 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} - q^{25} - 10 q^{26} + 2 q^{28} + 6 q^{29} - 8 q^{31} - 8 q^{32} + 4 q^{34} + 2 q^{35} - 9 q^{37} + 6 q^{38} - 4 q^{41} - 2 q^{44} + 8 q^{46} + 10 q^{47} - 6 q^{49} - 2 q^{50} - 10 q^{52} - 6 q^{53} - 2 q^{55} + 12 q^{58} - 14 q^{59} + 9 q^{61} - 16 q^{62} - 8 q^{64} - 10 q^{65} + 5 q^{67} + 4 q^{68} + 4 q^{70} + 12 q^{71} + 7 q^{73} - 18 q^{74} + 6 q^{76} - q^{77} + 11 q^{79} - 8 q^{80} - 8 q^{82} + 12 q^{83} + 4 q^{85} + 6 q^{89} - 5 q^{91} + 8 q^{92} + 20 q^{94} + 6 q^{95} - 7 q^{97} - 12 q^{98}+O(q^{100}) \) 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 2.00000 0 1.00000 0 0 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

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 297.2.a.d yes 1
3.b odd 2 1 297.2.a.a 1
4.b odd 2 1 4752.2.a.o 1
5.b even 2 1 7425.2.a.b 1
9.c even 3 2 891.2.e.a 2
9.d odd 6 2 891.2.e.l 2
11.b odd 2 1 3267.2.a.a 1
12.b even 2 1 4752.2.a.c 1
15.d odd 2 1 7425.2.a.u 1
33.d even 2 1 3267.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.a.a 1 3.b odd 2 1
297.2.a.d yes 1 1.a even 1 1 trivial
891.2.e.a 2 9.c even 3 2
891.2.e.l 2 9.d odd 6 2
3267.2.a.a 1 11.b odd 2 1
3267.2.a.k 1 33.d even 2 1
4752.2.a.c 1 12.b even 2 1
4752.2.a.o 1 4.b odd 2 1
7425.2.a.b 1 5.b even 2 1
7425.2.a.u 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(297))\). 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 3 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 9 \) Copy content Toggle raw display
$41$ \( T + 4 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 10 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 14 \) Copy content Toggle raw display
$61$ \( T - 9 \) Copy content Toggle raw display
$67$ \( T - 5 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 7 \) Copy content Toggle raw display
$79$ \( T - 11 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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