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Properties

Label 2268.2.j.h
Level $2268$
Weight $2$
Character orbit 2268.j
Analytic conductor $18.110$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2268,2,Mod(757,2268)] 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, 2, 0])) 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("2268.757"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.j (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-1,0,0,0,3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{11} + 4 \zeta_{6} q^{13} + 6 q^{17} - 4 q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + (6 \zeta_{6} - 6) q^{29} - 2 \zeta_{6} q^{31} - 7 q^{37} + 12 \zeta_{6} q^{41} + \cdots + ( - 4 \zeta_{6} + 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{7} + 3 q^{11} + 4 q^{13} + 12 q^{17} - 8 q^{19} + 5 q^{25} - 6 q^{29} - 2 q^{31} - 14 q^{37} + 12 q^{41} + 7 q^{43} + 6 q^{47} - q^{49} + 6 q^{53} + 6 q^{59} - 2 q^{61} + 13 q^{67} + 18 q^{71}+ \cdots + 4 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −0.500000 − 0.866025i 0 0 0
1513.1 0 0 0 0 0 −0.500000 + 0.866025i 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
9.c 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.j.h 2
3.b odd 2 1 2268.2.j.g 2
9.c even 3 1 2268.2.a.b ✓ 1
9.c even 3 1 inner 2268.2.j.h 2
9.d odd 6 1 2268.2.a.c yes 1
9.d odd 6 1 2268.2.j.g 2
36.f odd 6 1 9072.2.a.m 1
36.h even 6 1 9072.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.a.b ✓ 1 9.c even 3 1
2268.2.a.c yes 1 9.d odd 6 1
2268.2.j.g 2 3.b odd 2 1
2268.2.j.g 2 9.d odd 6 1
2268.2.j.h 2 1.a even 1 1 trivial
2268.2.j.h 2 9.c even 3 1 inner
9072.2.a.l 1 36.h even 6 1
9072.2.a.m 1 36.f odd 6 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) 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
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$71$ \( (T - 9)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
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