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Properties

Label 2268.2.x.f
Level $2268$
Weight $2$
Character orbit 2268.x
Analytic conductor $18.110$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2268,2,Mod(377,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.377"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,4,0,0,0,0,0,-3,0,0,0,0,0,0,0,0,0,0,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 3) q^{7} + (\zeta_{6} - 2) q^{13} + ( - 10 \zeta_{6} + 5) q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + (6 \zeta_{6} - 12) q^{31} + q^{37} + ( - 8 \zeta_{6} + 8) q^{43} + ( - 8 \zeta_{6} + 5) q^{49}+ \cdots + ( - 11 \zeta_{6} - 11) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7} - 3 q^{13} + 5 q^{25} - 18 q^{31} + 2 q^{37} + 8 q^{43} + 2 q^{49} + 15 q^{61} - 11 q^{67} + 13 q^{79} - 3 q^{91} - 33 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{6}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 2.00000 + 1.73205i 0 0 0
1889.1 0 0 0 0 0 2.00000 − 1.73205i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.l odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.x.f 2
3.b odd 2 1 CM 2268.2.x.f 2
7.b odd 2 1 2268.2.x.d 2
9.c even 3 1 756.2.f.b ✓ 2
9.c even 3 1 2268.2.x.d 2
9.d odd 6 1 756.2.f.b ✓ 2
9.d odd 6 1 2268.2.x.d 2
21.c even 2 1 2268.2.x.d 2
36.f odd 6 1 3024.2.k.c 2
36.h even 6 1 3024.2.k.c 2
63.l odd 6 1 756.2.f.b ✓ 2
63.l odd 6 1 inner 2268.2.x.f 2
63.o even 6 1 756.2.f.b ✓ 2
63.o even 6 1 inner 2268.2.x.f 2
252.s odd 6 1 3024.2.k.c 2
252.bi even 6 1 3024.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.b ✓ 2 9.c even 3 1
756.2.f.b ✓ 2 9.d odd 6 1
756.2.f.b ✓ 2 63.l odd 6 1
756.2.f.b ✓ 2 63.o even 6 1
2268.2.x.d 2 7.b odd 2 1
2268.2.x.d 2 9.c even 3 1
2268.2.x.d 2 9.d odd 6 1
2268.2.x.d 2 21.c even 2 1
2268.2.x.f 2 1.a even 1 1 trivial
2268.2.x.f 2 3.b odd 2 1 CM
2268.2.x.f 2 63.l odd 6 1 inner
2268.2.x.f 2 63.o even 6 1 inner
3024.2.k.c 2 36.f odd 6 1
3024.2.k.c 2 36.h even 6 1
3024.2.k.c 2 252.s odd 6 1
3024.2.k.c 2 252.bi even 6 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 75 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 33T + 363 \) Copy content Toggle raw display
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