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Properties

Label 297.3.h.a
Level $297$
Weight $3$
Character orbit 297.h
Analytic conductor $8.093$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,3,Mod(10,297)]
 
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([2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.h (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: \(8.09266385150\)
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: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q + ( - 4 \beta_1 - 4) q^{4} + ( - \beta_{3} + \beta_1 + 1) q^{5} + 11 \beta_1 q^{11} + 16 \beta_1 q^{16} + (4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{20} + (35 \beta_1 + 35) q^{23} + ( - \beta_{3} + \beta_{2} + 50 \beta_1 + 1) q^{25}+ \cdots + (17 \beta_{3} - 17 \beta_{2} + \cdots - 17) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + q^{5} - 22 q^{11} - 32 q^{16} + 4 q^{20} + 70 q^{23} - 99 q^{25} - 37 q^{31} + 50 q^{37} + 176 q^{44} - 50 q^{47} - 98 q^{49} - 140 q^{53} - 22 q^{55} - 107 q^{59} + 256 q^{64} + 35 q^{67}+ \cdots + 95 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{3} + 4\nu^{2} + 14\nu - 51 ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 23\beta _1 + 24 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 33 ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
0 0 −2.00000 + 3.46410i −4.05842 + 7.02939i 0 0 0 0 0
10.2 0 0 −2.00000 + 3.46410i 4.55842 − 7.89542i 0 0 0 0 0
208.1 0 0 −2.00000 − 3.46410i −4.05842 − 7.02939i 0 0 0 0 0
208.2 0 0 −2.00000 − 3.46410i 4.55842 + 7.89542i 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.c even 3 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.h.a 4
3.b odd 2 1 99.3.h.a ✓ 4
9.c even 3 1 inner 297.3.h.a 4
9.c even 3 1 891.3.c.a 2
9.d odd 6 1 99.3.h.a ✓ 4
9.d odd 6 1 891.3.c.b 2
11.b odd 2 1 CM 297.3.h.a 4
33.d even 2 1 99.3.h.a ✓ 4
99.g even 6 1 99.3.h.a ✓ 4
99.g even 6 1 891.3.c.b 2
99.h odd 6 1 inner 297.3.h.a 4
99.h odd 6 1 891.3.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.h.a ✓ 4 3.b odd 2 1
99.3.h.a ✓ 4 9.d odd 6 1
99.3.h.a ✓ 4 33.d even 2 1
99.3.h.a ✓ 4 99.g even 6 1
297.3.h.a 4 1.a even 1 1 trivial
297.3.h.a 4 9.c even 3 1 inner
297.3.h.a 4 11.b odd 2 1 CM
297.3.h.a 4 99.h odd 6 1 inner
891.3.c.a 2 9.c even 3 1
891.3.c.a 2 99.h odd 6 1
891.3.c.b 2 9.d odd 6 1
891.3.c.b 2 99.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). 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
$5$ \( T^{4} - T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11 T + 121)^{2} \) 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
$23$ \( (T^{2} - 35 T + 1225)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 37 T^{3} + \cdots + 2292196 \) Copy content Toggle raw display
$37$ \( (T^{2} - 25 T - 3482)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 50 T^{3} + \cdots + 17032129 \) Copy content Toggle raw display
$53$ \( (T^{2} + 70 T - 3527)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 107 T^{3} + \cdots + 1012036 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 35 T^{3} + \cdots + 149866564 \) Copy content Toggle raw display
$71$ \( (T^{2} + 133 T + 2566)^{2} \) 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
$89$ \( (T - 97)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 95 T^{3} + \cdots + 368716804 \) Copy content Toggle raw display
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