Properties

Label 297.2.g.a
Level $297$
Weight $2$
Character orbit 297.g
Analytic conductor $2.372$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,2,Mod(98,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.98");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.g (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1 + 2) q^{5} + (2 \beta_{3} + \beta_{2}) q^{11} - 4 \beta_{2} q^{16} + ( - 2 \beta_{3} - 4 \beta_{2} + 6) q^{20} + (\beta_{2} + 2 \beta_1 - 1) q^{23}+ \cdots + (3 \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 9 q^{5} - 8 q^{16} + 18 q^{20} + 9 q^{25} + 5 q^{31} + 14 q^{37} - 36 q^{47} - 14 q^{49} - 22 q^{55} - 45 q^{59} - 32 q^{64} - 13 q^{67} + 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1
1.68614 + 0.396143i
−1.18614 1.26217i
1.68614 0.396143i
−1.18614 + 1.26217i
0 0 1.00000 1.73205i 0.813859 + 0.469882i 0 0 0 0 0
98.2 0 0 1.00000 1.73205i 3.68614 + 2.12819i 0 0 0 0 0
197.1 0 0 1.00000 + 1.73205i 0.813859 0.469882i 0 0 0 0 0
197.2 0 0 1.00000 + 1.73205i 3.68614 2.12819i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.d odd 6 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.g.a 4
3.b odd 2 1 99.2.g.a 4
9.c even 3 1 99.2.g.a 4
9.c even 3 1 891.2.d.a 4
9.d odd 6 1 inner 297.2.g.a 4
9.d odd 6 1 891.2.d.a 4
11.b odd 2 1 CM 297.2.g.a 4
33.d even 2 1 99.2.g.a 4
99.g even 6 1 inner 297.2.g.a 4
99.g even 6 1 891.2.d.a 4
99.h odd 6 1 99.2.g.a 4
99.h odd 6 1 891.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.g.a 4 3.b odd 2 1
99.2.g.a 4 9.c even 3 1
99.2.g.a 4 33.d even 2 1
99.2.g.a 4 99.h odd 6 1
297.2.g.a 4 1.a even 1 1 trivial
297.2.g.a 4 9.d odd 6 1 inner
297.2.g.a 4 11.b odd 2 1 CM
297.2.g.a 4 99.g even 6 1 inner
891.2.d.a 4 9.c even 3 1
891.2.d.a 4 9.d odd 6 1
891.2.d.a 4 99.g even 6 1
891.2.d.a 4 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 9 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 5 T^{3} + \cdots + 4624 \) Copy content Toggle raw display
$37$ \( (T^{2} - 7 T - 62)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 36 T^{3} + \cdots + 9409 \) Copy content Toggle raw display
$53$ \( T^{4} + 142T^{2} + 289 \) Copy content Toggle raw display
$59$ \( T^{4} + 45 T^{3} + \cdots + 27556 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{4} + 151T^{2} + 3844 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 275)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 17 T^{3} + \cdots + 4 \) Copy content Toggle raw display
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