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Properties

Label 2268.2.w.j
Level $2268$
Weight $2$
Character orbit 2268.w
Analytic conductor $18.110$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2268,2,Mod(269,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,4,0,0,0,0,0,12] 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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{7} + 12 q^{13} - 16 q^{25} - 4 q^{37} - 4 q^{43} + 20 q^{49} - 8 q^{67} - 36 q^{73} - 56 q^{79} + 12 q^{85} - 36 q^{91} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 0 0 0 −2.11958 + 3.67122i 0 1.81145 − 1.92838i 0 0 0
269.2 0 0 0 −1.76994 + 3.06563i 0 −2.36878 + 1.17852i 0 0 0
269.3 0 0 0 −1.43229 + 2.48080i 0 2.49882 + 0.869438i 0 0 0
269.4 0 0 0 −1.00175 + 1.73509i 0 1.26572 − 2.32335i 0 0 0
269.5 0 0 0 −0.896692 + 1.55312i 0 2.33265 + 1.24850i 0 0 0
269.6 0 0 0 −0.566658 + 0.981481i 0 −0.814380 + 2.51730i 0 0 0
269.7 0 0 0 −0.440135 + 0.762336i 0 −2.46165 − 0.969691i 0 0 0
269.8 0 0 0 −0.0292047 + 0.0505839i 0 −1.26383 − 2.32438i 0 0 0
269.9 0 0 0 0.0292047 − 0.0505839i 0 −1.26383 − 2.32438i 0 0 0
269.10 0 0 0 0.440135 − 0.762336i 0 −2.46165 − 0.969691i 0 0 0
269.11 0 0 0 0.566658 − 0.981481i 0 −0.814380 + 2.51730i 0 0 0
269.12 0 0 0 0.896692 − 1.55312i 0 2.33265 + 1.24850i 0 0 0
269.13 0 0 0 1.00175 − 1.73509i 0 1.26572 − 2.32335i 0 0 0
269.14 0 0 0 1.43229 − 2.48080i 0 2.49882 + 0.869438i 0 0 0
269.15 0 0 0 1.76994 − 3.06563i 0 −2.36878 + 1.17852i 0 0 0
269.16 0 0 0 2.11958 − 3.67122i 0 1.81145 − 1.92838i 0 0 0
1349.1 0 0 0 −2.11958 − 3.67122i 0 1.81145 + 1.92838i 0 0 0
1349.2 0 0 0 −1.76994 − 3.06563i 0 −2.36878 − 1.17852i 0 0 0
1349.3 0 0 0 −1.43229 − 2.48080i 0 2.49882 − 0.869438i 0 0 0
1349.4 0 0 0 −1.00175 − 1.73509i 0 1.26572 + 2.32335i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.i even 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.w.j 32
3.b odd 2 1 inner 2268.2.w.j 32
7.d odd 6 1 2268.2.bm.j 32
9.c even 3 1 2268.2.t.c ✓ 32
9.c even 3 1 2268.2.bm.j 32
9.d odd 6 1 2268.2.t.c ✓ 32
9.d odd 6 1 2268.2.bm.j 32
21.g even 6 1 2268.2.bm.j 32
63.i even 6 1 inner 2268.2.w.j 32
63.k odd 6 1 2268.2.t.c ✓ 32
63.s even 6 1 2268.2.t.c ✓ 32
63.t odd 6 1 inner 2268.2.w.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.t.c ✓ 32 9.c even 3 1
2268.2.t.c ✓ 32 9.d odd 6 1
2268.2.t.c ✓ 32 63.k odd 6 1
2268.2.t.c ✓ 32 63.s even 6 1
2268.2.w.j 32 1.a even 1 1 trivial
2268.2.w.j 32 3.b odd 2 1 inner
2268.2.w.j 32 63.i even 6 1 inner
2268.2.w.j 32 63.t odd 6 1 inner
2268.2.bm.j 32 7.d odd 6 1
2268.2.bm.j 32 9.c even 3 1
2268.2.bm.j 32 9.d odd 6 1
2268.2.bm.j 32 21.g even 6 1

Hecke kernels

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

\( T_{5}^{32} + 48 T_{5}^{30} + 1440 T_{5}^{28} + 26640 T_{5}^{26} + 358326 T_{5}^{24} + 3385908 T_{5}^{22} + \cdots + 6561 \) Copy content Toggle raw display
\( T_{13}^{16} - 6 T_{13}^{15} - 45 T_{13}^{14} + 342 T_{13}^{13} + 1845 T_{13}^{12} - 8928 T_{13}^{11} + \cdots + 50936769 \) Copy content Toggle raw display