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Properties

Label 2646.2.e.s
Level $2646$
Weight $2$
Character orbit 2646.e
Analytic conductor $21.128$
Analytic rank $0$
Dimension $8$
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] = [2646,2,Mod(1549,2646)] 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("2646.1549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8,0,8,0,0,0,8,0,0,2,0,0] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.31116960000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_{4} q^{5} + q^{8} - \beta_{4} q^{10} + \beta_{6} q^{11} + ( - 2 \beta_{7} + 2 \beta_1) q^{13} + q^{16} + ( - 2 \beta_{4} - \beta_1) q^{17} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{19}+ \cdots + 5 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 2 q^{11} + 8 q^{16} + 2 q^{22} + 6 q^{23} - 2 q^{25} - 16 q^{29} + 8 q^{32} - 24 q^{37} - 10 q^{43} + 2 q^{44} + 6 q^{46} - 2 q^{50} - 16 q^{53} - 16 q^{58} + 8 q^{64}+ \cdots + 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 8\nu^{5} - 64\nu^{3} + 135\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 8\nu^{4} + 8\nu^{2} - 81 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 8\nu^{5} + 44\nu^{3} + 27\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 4\nu^{5} + 4\nu^{3} - 15\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{4} - 17\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 8\nu^{2} + 17 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} + 2\nu^{3} - 15\nu ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{4} + 2\beta_{3} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{2} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{4} + 5\beta_{3} - 4\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} + 2\beta_{5} + 25\beta_{2} + 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 14\beta_{4} + 5\beta_{3} + 14\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16\beta_{6} + 8\beta_{5} - 8\beta_{2} - 43 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -31\beta_{7} + 41\beta_{4} - 10\beta_{3} - 10\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
−1.62968 + 0.586627i
−0.306808 − 1.70466i
0.306808 + 1.70466i
1.62968 − 0.586627i
−1.62968 − 0.586627i
−0.306808 + 1.70466i
0.306808 − 1.70466i
1.62968 + 0.586627i
1.00000 0 1.00000 −1.62968 − 2.82269i 0 0 1.00000 0 −1.62968 − 2.82269i
1549.2 1.00000 0 1.00000 −0.306808 − 0.531407i 0 0 1.00000 0 −0.306808 − 0.531407i
1549.3 1.00000 0 1.00000 0.306808 + 0.531407i 0 0 1.00000 0 0.306808 + 0.531407i
1549.4 1.00000 0 1.00000 1.62968 + 2.82269i 0 0 1.00000 0 1.62968 + 2.82269i
2125.1 1.00000 0 1.00000 −1.62968 + 2.82269i 0 0 1.00000 0 −1.62968 + 2.82269i
2125.2 1.00000 0 1.00000 −0.306808 + 0.531407i 0 0 1.00000 0 −0.306808 + 0.531407i
2125.3 1.00000 0 1.00000 0.306808 − 0.531407i 0 0 1.00000 0 0.306808 − 0.531407i
2125.4 1.00000 0 1.00000 1.62968 − 2.82269i 0 0 1.00000 0 1.62968 − 2.82269i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.h even 3 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 2646.2.e.s 8
3.b odd 2 1 882.2.e.r 8
7.b odd 2 1 inner 2646.2.e.s 8
7.c even 3 1 2646.2.f.p 8
7.c even 3 1 2646.2.h.r 8
7.d odd 6 1 2646.2.f.p 8
7.d odd 6 1 2646.2.h.r 8
9.c even 3 1 2646.2.h.r 8
9.d odd 6 1 882.2.h.s 8
21.c even 2 1 882.2.e.r 8
21.g even 6 1 882.2.f.r ✓ 8
21.g even 6 1 882.2.h.s 8
21.h odd 6 1 882.2.f.r ✓ 8
21.h odd 6 1 882.2.h.s 8
63.g even 3 1 2646.2.f.p 8
63.h even 3 1 inner 2646.2.e.s 8
63.h even 3 1 7938.2.a.cq 4
63.i even 6 1 882.2.e.r 8
63.i even 6 1 7938.2.a.ch 4
63.j odd 6 1 882.2.e.r 8
63.j odd 6 1 7938.2.a.ch 4
63.k odd 6 1 2646.2.f.p 8
63.l odd 6 1 2646.2.h.r 8
63.n odd 6 1 882.2.f.r ✓ 8
63.o even 6 1 882.2.h.s 8
63.s even 6 1 882.2.f.r ✓ 8
63.t odd 6 1 inner 2646.2.e.s 8
63.t odd 6 1 7938.2.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.r 8 3.b odd 2 1
882.2.e.r 8 21.c even 2 1
882.2.e.r 8 63.i even 6 1
882.2.e.r 8 63.j odd 6 1
882.2.f.r ✓ 8 21.g even 6 1
882.2.f.r ✓ 8 21.h odd 6 1
882.2.f.r ✓ 8 63.n odd 6 1
882.2.f.r ✓ 8 63.s even 6 1
882.2.h.s 8 9.d odd 6 1
882.2.h.s 8 21.g even 6 1
882.2.h.s 8 21.h odd 6 1
882.2.h.s 8 63.o even 6 1
2646.2.e.s 8 1.a even 1 1 trivial
2646.2.e.s 8 7.b odd 2 1 inner
2646.2.e.s 8 63.h even 3 1 inner
2646.2.e.s 8 63.t odd 6 1 inner
2646.2.f.p 8 7.c even 3 1
2646.2.f.p 8 7.d odd 6 1
2646.2.f.p 8 63.g even 3 1
2646.2.f.p 8 63.k odd 6 1
2646.2.h.r 8 7.c even 3 1
2646.2.h.r 8 7.d odd 6 1
2646.2.h.r 8 9.c even 3 1
2646.2.h.r 8 63.l odd 6 1
7938.2.a.ch 4 63.i even 6 1
7938.2.a.ch 4 63.j odd 6 1
7938.2.a.cq 4 63.h even 3 1
7938.2.a.cq 4 63.t odd 6 1

Hecke kernels

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

\( T_{5}^{8} + 11T_{5}^{6} + 117T_{5}^{4} + 44T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} + 27T_{11}^{2} + 26T_{11} + 676 \) Copy content Toggle raw display
\( T_{13}^{8} + 44T_{13}^{6} + 1872T_{13}^{4} + 2816T_{13}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 11 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + 27 T^{2} + \cdots + 676)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} + 39 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$19$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 3 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 44 T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 39 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 112)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 371 T^{2} + 33124)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 71 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 26)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 11 T + 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 179 T^{6} + \cdots + 45212176 \) Copy content Toggle raw display
$79$ \( (T^{2} - 17 T + 46)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 60 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 284 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$97$ \( T^{8} + 275 T^{6} + \cdots + 6250000 \) Copy content Toggle raw display
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