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Properties

Label 2268.2.l.j
Level $2268$
Weight $2$
Character orbit 2268.l
Analytic conductor $18.110$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(109,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.l (of order \(3\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{5} - \beta_{2} - 2) q^{11} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots - 1) q^{17}+ \cdots + (4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 4 q^{7} - 10 q^{11} + 2 q^{13} - 4 q^{17} - 3 q^{19} - 28 q^{23} + 20 q^{25} - 5 q^{29} + 2 q^{31} + 26 q^{35} + 12 q^{41} + 9 q^{43} - 9 q^{47} - 12 q^{49} + 6 q^{53} + 16 q^{55} - 5 q^{59} - 7 q^{61} + 24 q^{65} + 16 q^{67} - 22 q^{71} + q^{73} + 13 q^{77} + 8 q^{79} + 17 q^{83} - 5 q^{85} - 3 q^{89} + 5 q^{91} + 32 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} - \nu^{2} + 3\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 21\nu + 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 11\nu^{3} + 20\nu^{2} - 15\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 4\nu^{4} + 14\nu^{3} - 20\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 5\beta_{4} - 2\beta_{3} - 5\beta _1 - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 11\beta_{4} - 9\beta_{3} - 7\beta_{2} - 7\beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -17\beta_{5} - 16\beta_{4} + 2\beta_{3} - 8\beta_{2} + 12\beta _1 + 31 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 − 1.41036i
0.500000 − 0.224437i
0.500000 − 2.05195i
0.500000 + 1.41036i
0 0 0 −4.28799 0 −2.64400 − 0.0963576i 0 0 0
109.2 0 0 0 0.866926 0 −0.0665372 + 2.64491i 0 0 0
109.3 0 0 0 2.42107 0 0.710533 − 2.54856i 0 0 0
541.1 0 0 0 −4.28799 0 −2.64400 + 0.0963576i 0 0 0
541.2 0 0 0 0.866926 0 −0.0665372 − 2.64491i 0 0 0
541.3 0 0 0 2.42107 0 0.710533 + 2.54856i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.l.j 6
3.b odd 2 1 2268.2.l.k 6
7.c even 3 1 2268.2.i.k 6
9.c even 3 1 756.2.k.f yes 6
9.c even 3 1 2268.2.i.k 6
9.d odd 6 1 756.2.k.e ✓ 6
9.d odd 6 1 2268.2.i.j 6
21.h odd 6 1 2268.2.i.j 6
63.g even 3 1 inner 2268.2.l.j 6
63.g even 3 1 5292.2.a.u 3
63.h even 3 1 756.2.k.f yes 6
63.j odd 6 1 756.2.k.e ✓ 6
63.k odd 6 1 5292.2.a.w 3
63.n odd 6 1 2268.2.l.k 6
63.n odd 6 1 5292.2.a.x 3
63.s even 6 1 5292.2.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.e ✓ 6 9.d odd 6 1
756.2.k.e ✓ 6 63.j odd 6 1
756.2.k.f yes 6 9.c even 3 1
756.2.k.f yes 6 63.h even 3 1
2268.2.i.j 6 9.d odd 6 1
2268.2.i.j 6 21.h odd 6 1
2268.2.i.k 6 7.c even 3 1
2268.2.i.k 6 9.c even 3 1
2268.2.l.j 6 1.a even 1 1 trivial
2268.2.l.j 6 63.g even 3 1 inner
2268.2.l.k 6 3.b odd 2 1
2268.2.l.k 6 63.n odd 6 1
5292.2.a.u 3 63.g even 3 1
5292.2.a.v 3 63.s even 6 1
5292.2.a.w 3 63.k odd 6 1
5292.2.a.x 3 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{3} + T_{5}^{2} - 12T_{5} + 9 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} + 15T_{13}^{4} - 20T_{13}^{3} + 163T_{13}^{2} - 231T_{13} + 441 \) Copy content Toggle raw display
\( T_{19}^{6} + 3T_{19}^{5} + 45T_{19}^{4} - 10T_{19}^{3} + 1443T_{19}^{2} + 1764T_{19} + 2401 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + T^{2} - 12 T + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} - 12 T - 63)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots + 441 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{3} + 14 T^{2} + \cdots - 63)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 5 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$31$ \( T^{6} - 2 T^{5} + \cdots + 441 \) Copy content Toggle raw display
$37$ \( T^{6} + 57 T^{4} + \cdots + 18769 \) Copy content Toggle raw display
$41$ \( T^{6} - 12 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} + \cdots + 78961 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + \cdots + 59049 \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + \cdots + 321489 \) Copy content Toggle raw display
$59$ \( T^{6} + 5 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{6} + 7 T^{5} + \cdots + 321489 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$71$ \( (T^{3} + 11 T^{2} + \cdots - 189)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - T^{5} + \cdots + 25921 \) Copy content Toggle raw display
$79$ \( T^{6} - 8 T^{5} + \cdots + 762129 \) Copy content Toggle raw display
$83$ \( T^{6} - 17 T^{5} + \cdots + 9801 \) Copy content Toggle raw display
$89$ \( T^{6} + 3 T^{5} + \cdots + 1750329 \) Copy content Toggle raw display
$97$ \( T^{6} + 14 T^{5} + \cdots + 3136 \) Copy content Toggle raw display
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