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Properties

Label 2646.2.h.l
Level $2646$
Weight $2$
Character orbit 2646.h
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + ( - \beta_{2} + 1) q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_1 - 1) q^{4} + ( - \beta_{2} + 1) q^{5} + q^{8} + (\beta_{3} - 2 \beta_1) q^{10} + ( - \beta_{2} - 2) q^{11} + 2 \beta_1 q^{13} - \beta_1 q^{16} + (\beta_{3} + \beta_1) q^{17} + ( - 5 \beta_1 + 5) q^{19} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{20} + (\beta_{3} + \beta_1) q^{22} + (\beta_{2} + 5) q^{23} + ( - 3 \beta_{2} + 4) q^{25} + ( - 2 \beta_1 + 2) q^{26} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{29}+ \cdots + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 6 q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 6 q^{5} + 4 q^{8} - 3 q^{10} - 6 q^{11} + 4 q^{13} - 2 q^{16} + 3 q^{17} + 10 q^{19} - 3 q^{20} + 3 q^{22} + 18 q^{23} + 22 q^{25} + 4 q^{26} - 6 q^{29} + 4 q^{31} - 2 q^{32} + 3 q^{34} - 4 q^{37} - 20 q^{38} + 6 q^{40} - 15 q^{41} - q^{43} + 3 q^{44} - 9 q^{46} - 11 q^{50} - 8 q^{52} - 6 q^{53} + 24 q^{55} + 12 q^{58} + 3 q^{59} - 11 q^{61} - 8 q^{62} + 4 q^{64} + 6 q^{65} - 13 q^{67} - 6 q^{68} - 6 q^{71} + 7 q^{73} + 8 q^{74} + 10 q^{76} - 7 q^{79} - 3 q^{80} - 15 q^{82} - 12 q^{83} - 12 q^{85} + 2 q^{86} - 6 q^{88} - 18 q^{89} - 9 q^{92} + 15 q^{95} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.37228 0 0 1.00000 0 0.686141 − 1.18843i
361.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 4.37228 0 0 1.00000 0 −2.18614 + 3.78651i
667.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.37228 0 0 1.00000 0 0.686141 + 1.18843i
667.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 4.37228 0 0 1.00000 0 −2.18614 − 3.78651i
\(n\): 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.l 4
3.b odd 2 1 882.2.h.n 4
7.b odd 2 1 2646.2.h.k 4
7.c even 3 1 2646.2.e.m 4
7.c even 3 1 2646.2.f.j 4
7.d odd 6 1 378.2.f.c 4
7.d odd 6 1 2646.2.e.n 4
9.c even 3 1 2646.2.e.m 4
9.d odd 6 1 882.2.e.k 4
21.c even 2 1 882.2.h.m 4
21.g even 6 1 126.2.f.d ✓ 4
21.g even 6 1 882.2.e.l 4
21.h odd 6 1 882.2.e.k 4
21.h odd 6 1 882.2.f.k 4
28.f even 6 1 3024.2.r.f 4
63.g even 3 1 inner 2646.2.h.l 4
63.g even 3 1 7938.2.a.bs 2
63.h even 3 1 2646.2.f.j 4
63.i even 6 1 126.2.f.d ✓ 4
63.j odd 6 1 882.2.f.k 4
63.k odd 6 1 1134.2.a.n 2
63.k odd 6 1 2646.2.h.k 4
63.l odd 6 1 2646.2.e.n 4
63.n odd 6 1 882.2.h.n 4
63.n odd 6 1 7938.2.a.bh 2
63.o even 6 1 882.2.e.l 4
63.s even 6 1 882.2.h.m 4
63.s even 6 1 1134.2.a.k 2
63.t odd 6 1 378.2.f.c 4
84.j odd 6 1 1008.2.r.f 4
252.n even 6 1 9072.2.a.bb 2
252.r odd 6 1 1008.2.r.f 4
252.bj even 6 1 3024.2.r.f 4
252.bn odd 6 1 9072.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d ✓ 4 21.g even 6 1
126.2.f.d ✓ 4 63.i even 6 1
378.2.f.c 4 7.d odd 6 1
378.2.f.c 4 63.t odd 6 1
882.2.e.k 4 9.d odd 6 1
882.2.e.k 4 21.h odd 6 1
882.2.e.l 4 21.g even 6 1
882.2.e.l 4 63.o even 6 1
882.2.f.k 4 21.h odd 6 1
882.2.f.k 4 63.j odd 6 1
882.2.h.m 4 21.c even 2 1
882.2.h.m 4 63.s even 6 1
882.2.h.n 4 3.b odd 2 1
882.2.h.n 4 63.n odd 6 1
1008.2.r.f 4 84.j odd 6 1
1008.2.r.f 4 252.r odd 6 1
1134.2.a.k 2 63.s even 6 1
1134.2.a.n 2 63.k odd 6 1
2646.2.e.m 4 7.c even 3 1
2646.2.e.m 4 9.c even 3 1
2646.2.e.n 4 7.d odd 6 1
2646.2.e.n 4 63.l odd 6 1
2646.2.f.j 4 7.c even 3 1
2646.2.f.j 4 63.h even 3 1
2646.2.h.k 4 7.b odd 2 1
2646.2.h.k 4 63.k odd 6 1
2646.2.h.l 4 1.a even 1 1 trivial
2646.2.h.l 4 63.g even 3 1 inner
3024.2.r.f 4 28.f even 6 1
3024.2.r.f 4 252.bj even 6 1
7938.2.a.bh 2 63.n odd 6 1
7938.2.a.bs 2 63.g even 3 1
9072.2.a.bb 2 252.n even 6 1
9072.2.a.bm 2 252.bn odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 15 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$43$ \( T^{4} + T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{4} + 11 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 7 T^{3} + \cdots + 3844 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} + \cdots + 3844 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$89$ \( T^{4} + 18 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( T^{4} - T^{3} + \cdots + 5476 \) Copy content Toggle raw display
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