Properties

Label 2646.2.h.p
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2646,2,Mod(361,2646)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 1) q^{2} - \beta_{4} q^{4} + (\beta_1 + 2) q^{5} - q^{8} + (\beta_{5} - 2 \beta_{4} + 2) q^{10} + (2 \beta_{3} - \beta_1) q^{11} + (\beta_{5} - \beta_{4} + 1) q^{13} + (\beta_{4} - 1) q^{16}+ \cdots + (10 \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + 10 q^{5} - 6 q^{8} + 5 q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{16} - 4 q^{17} + 3 q^{19} - 5 q^{20} - q^{22} - 14 q^{23} + 4 q^{25} - 2 q^{26} + 5 q^{29} + 14 q^{31} + 3 q^{32} + 4 q^{34}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(-\beta_{4}\)

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.

comment: embeddings in the coefficient field
 
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 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
0.500000 0.866025i 0 −0.500000 0.866025i −0.460505 0 0 −1.00000 0 −0.230252 + 0.398809i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.76088 0 0 −1.00000 0 0.880438 1.52496i
361.3 0.500000 0.866025i 0 −0.500000 0.866025i 3.69963 0 0 −1.00000 0 1.84981 3.20397i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.460505 0 0 −1.00000 0 −0.230252 0.398809i
667.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.76088 0 0 −1.00000 0 0.880438 + 1.52496i
667.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.69963 0 0 −1.00000 0 1.84981 + 3.20397i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
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.p 6
3.b odd 2 1 882.2.h.o 6
7.b odd 2 1 378.2.h.d 6
7.c even 3 1 2646.2.e.o 6
7.c even 3 1 2646.2.f.n 6
7.d odd 6 1 378.2.e.c 6
7.d odd 6 1 2646.2.f.o 6
9.c even 3 1 2646.2.e.o 6
9.d odd 6 1 882.2.e.p 6
21.c even 2 1 126.2.h.c yes 6
21.g even 6 1 126.2.e.d 6
21.g even 6 1 882.2.f.l 6
21.h odd 6 1 882.2.e.p 6
21.h odd 6 1 882.2.f.m 6
28.d even 2 1 3024.2.t.g 6
28.f even 6 1 3024.2.q.h 6
63.g even 3 1 inner 2646.2.h.p 6
63.g even 3 1 7938.2.a.bx 3
63.h even 3 1 2646.2.f.n 6
63.i even 6 1 882.2.f.l 6
63.i even 6 1 1134.2.g.k 6
63.j odd 6 1 882.2.f.m 6
63.k odd 6 1 378.2.h.d 6
63.k odd 6 1 7938.2.a.bu 3
63.l odd 6 1 378.2.e.c 6
63.l odd 6 1 1134.2.g.n 6
63.n odd 6 1 882.2.h.o 6
63.n odd 6 1 7938.2.a.by 3
63.o even 6 1 126.2.e.d 6
63.o even 6 1 1134.2.g.k 6
63.s even 6 1 126.2.h.c yes 6
63.s even 6 1 7938.2.a.cb 3
63.t odd 6 1 1134.2.g.n 6
63.t odd 6 1 2646.2.f.o 6
84.h odd 2 1 1008.2.t.g 6
84.j odd 6 1 1008.2.q.h 6
252.n even 6 1 3024.2.t.g 6
252.s odd 6 1 1008.2.q.h 6
252.bi even 6 1 3024.2.q.h 6
252.bn odd 6 1 1008.2.t.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 21.g even 6 1
126.2.e.d 6 63.o even 6 1
126.2.h.c yes 6 21.c even 2 1
126.2.h.c yes 6 63.s even 6 1
378.2.e.c 6 7.d odd 6 1
378.2.e.c 6 63.l odd 6 1
378.2.h.d 6 7.b odd 2 1
378.2.h.d 6 63.k odd 6 1
882.2.e.p 6 9.d odd 6 1
882.2.e.p 6 21.h odd 6 1
882.2.f.l 6 21.g even 6 1
882.2.f.l 6 63.i even 6 1
882.2.f.m 6 21.h odd 6 1
882.2.f.m 6 63.j odd 6 1
882.2.h.o 6 3.b odd 2 1
882.2.h.o 6 63.n odd 6 1
1008.2.q.h 6 84.j odd 6 1
1008.2.q.h 6 252.s odd 6 1
1008.2.t.g 6 84.h odd 2 1
1008.2.t.g 6 252.bn odd 6 1
1134.2.g.k 6 63.i even 6 1
1134.2.g.k 6 63.o even 6 1
1134.2.g.n 6 63.l odd 6 1
1134.2.g.n 6 63.t odd 6 1
2646.2.e.o 6 7.c even 3 1
2646.2.e.o 6 9.c even 3 1
2646.2.f.n 6 7.c even 3 1
2646.2.f.n 6 63.h even 3 1
2646.2.f.o 6 7.d odd 6 1
2646.2.f.o 6 63.t odd 6 1
2646.2.h.p 6 1.a even 1 1 trivial
2646.2.h.p 6 63.g even 3 1 inner
3024.2.q.h 6 28.f even 6 1
3024.2.q.h 6 252.bi even 6 1
3024.2.t.g 6 28.d even 2 1
3024.2.t.g 6 252.n even 6 1
7938.2.a.bu 3 63.k odd 6 1
7938.2.a.bx 3 63.g even 3 1
7938.2.a.by 3 63.n odd 6 1
7938.2.a.cb 3 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 \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 26T_{11} + 33 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} + 7T_{13}^{4} + 15T_{13}^{2} - 9T_{13} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 5 T^{2} + 4 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 26 T + 33)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 28224 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( (T^{3} + 7 T^{2} + 4 T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + \cdots + 1089 \) Copy content Toggle raw display
$31$ \( T^{6} - 14 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( T^{6} - 18 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( T^{6} + 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} - 4 T^{5} + \cdots + 31329 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 514089 \) Copy content Toggle raw display
$67$ \( T^{6} - 5 T^{5} + \cdots + 22201 \) Copy content Toggle raw display
$71$ \( (T^{3} + 7 T^{2} - 50 T - 99)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 25 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{6} - 7 T^{5} + \cdots + 594441 \) Copy content Toggle raw display
$83$ \( T^{6} - 8 T^{5} + \cdots + 8649 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + \cdots + 287296 \) Copy content Toggle raw display
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