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Properties

Label 2646.2.h.j
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
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] = [2646,2,Mod(361,2646)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) 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("2646.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,-1,2,0,0,-2,0,1,4,0,2] 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: \(21.1284163748\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 882)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + q^{5} - q^{8} + ( - \zeta_{6} + 1) q^{10} + 2 q^{11} + ( - 2 \zeta_{6} + 2) q^{13} + (\zeta_{6} - 1) q^{16} - 7 \zeta_{6} q^{19} - \zeta_{6} q^{20} + ( - 2 \zeta_{6} + 2) q^{22} + \cdots - 2 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{16} - 7 q^{19} - q^{20} + 2 q^{22} - 6 q^{23} - 8 q^{25} - 2 q^{26} - 8 q^{29} + 4 q^{31} + q^{32} + 6 q^{37} - 14 q^{38}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.00000 0 0 −1.00000 0 0.500000 − 0.866025i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 0 −1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
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 2646.2.h.j 2
3.b odd 2 1 882.2.h.a 2
7.b odd 2 1 2646.2.h.g 2
7.c even 3 1 2646.2.e.a 2
7.c even 3 1 2646.2.f.f 2
7.d odd 6 1 2646.2.e.d 2
7.d odd 6 1 2646.2.f.h 2
9.c even 3 1 2646.2.e.a 2
9.d odd 6 1 882.2.e.j 2
21.c even 2 1 882.2.h.d 2
21.g even 6 1 882.2.e.f 2
21.g even 6 1 882.2.f.b ✓ 2
21.h odd 6 1 882.2.e.j 2
21.h odd 6 1 882.2.f.c yes 2
63.g even 3 1 inner 2646.2.h.j 2
63.g even 3 1 7938.2.a.k 1
63.h even 3 1 2646.2.f.f 2
63.i even 6 1 882.2.f.b ✓ 2
63.j odd 6 1 882.2.f.c yes 2
63.k odd 6 1 2646.2.h.g 2
63.k odd 6 1 7938.2.a.f 1
63.l odd 6 1 2646.2.e.d 2
63.n odd 6 1 882.2.h.a 2
63.n odd 6 1 7938.2.a.v 1
63.o even 6 1 882.2.e.f 2
63.s even 6 1 882.2.h.d 2
63.s even 6 1 7938.2.a.ba 1
63.t odd 6 1 2646.2.f.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.f 2 21.g even 6 1
882.2.e.f 2 63.o even 6 1
882.2.e.j 2 9.d odd 6 1
882.2.e.j 2 21.h odd 6 1
882.2.f.b ✓ 2 21.g even 6 1
882.2.f.b ✓ 2 63.i even 6 1
882.2.f.c yes 2 21.h odd 6 1
882.2.f.c yes 2 63.j odd 6 1
882.2.h.a 2 3.b odd 2 1
882.2.h.a 2 63.n odd 6 1
882.2.h.d 2 21.c even 2 1
882.2.h.d 2 63.s even 6 1
2646.2.e.a 2 7.c even 3 1
2646.2.e.a 2 9.c even 3 1
2646.2.e.d 2 7.d odd 6 1
2646.2.e.d 2 63.l odd 6 1
2646.2.f.f 2 7.c even 3 1
2646.2.f.f 2 63.h even 3 1
2646.2.f.h 2 7.d odd 6 1
2646.2.f.h 2 63.t odd 6 1
2646.2.h.g 2 7.b odd 2 1
2646.2.h.g 2 63.k odd 6 1
2646.2.h.j 2 1.a even 1 1 trivial
2646.2.h.j 2 63.g even 3 1 inner
7938.2.a.f 1 63.k odd 6 1
7938.2.a.k 1 63.g even 3 1
7938.2.a.v 1 63.n odd 6 1
7938.2.a.ba 1 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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