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Properties

Label 882.2.e.q
Level $882$
Weight $2$
Character orbit 882.e
Analytic conductor $7.043$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,2,Mod(373,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.373");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.e (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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{7} + \beta_{3}) q^{3} + q^{4} - \beta_{7} q^{5} + ( - \beta_{7} - \beta_{3}) q^{6} - q^{8} + (2 \beta_{4} - \beta_{2}) q^{9} + \beta_{7} q^{10} + ( - 2 \beta_{2} - 2 \beta_1) q^{11}+ \cdots + ( - 2 \beta_{4} - 2 \beta_{2} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} - 8 q^{11} + 8 q^{16} + 8 q^{22} - 4 q^{23} + 12 q^{25} - 4 q^{29} - 8 q^{32} + 32 q^{37} + 24 q^{39} + 28 q^{43} - 8 q^{44} + 4 q^{46} - 12 q^{50} - 12 q^{51} - 20 q^{53}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -2\beta_{7} + \beta_{6} + 3\beta_{5} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\beta_{1}\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−1.00000 −1.67303 − 0.448288i 1.00000 0.965926 + 1.67303i 1.67303 + 0.448288i 0 −1.00000 2.59808 + 1.50000i −0.965926 − 1.67303i
373.2 −1.00000 −0.448288 + 1.67303i 1.00000 −0.258819 − 0.448288i 0.448288 − 1.67303i 0 −1.00000 −2.59808 − 1.50000i 0.258819 + 0.448288i
373.3 −1.00000 0.448288 − 1.67303i 1.00000 0.258819 + 0.448288i −0.448288 + 1.67303i 0 −1.00000 −2.59808 − 1.50000i −0.258819 − 0.448288i
373.4 −1.00000 1.67303 + 0.448288i 1.00000 −0.965926 − 1.67303i −1.67303 − 0.448288i 0 −1.00000 2.59808 + 1.50000i 0.965926 + 1.67303i
655.1 −1.00000 −1.67303 + 0.448288i 1.00000 0.965926 − 1.67303i 1.67303 − 0.448288i 0 −1.00000 2.59808 − 1.50000i −0.965926 + 1.67303i
655.2 −1.00000 −0.448288 − 1.67303i 1.00000 −0.258819 + 0.448288i 0.448288 + 1.67303i 0 −1.00000 −2.59808 + 1.50000i 0.258819 − 0.448288i
655.3 −1.00000 0.448288 + 1.67303i 1.00000 0.258819 − 0.448288i −0.448288 − 1.67303i 0 −1.00000 −2.59808 + 1.50000i −0.258819 + 0.448288i
655.4 −1.00000 1.67303 − 0.448288i 1.00000 −0.965926 + 1.67303i −1.67303 + 0.448288i 0 −1.00000 2.59808 − 1.50000i 0.965926 − 1.67303i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.4
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 882.2.e.q 8
3.b odd 2 1 2646.2.e.t 8
7.b odd 2 1 inner 882.2.e.q 8
7.c even 3 1 882.2.f.s ✓ 8
7.c even 3 1 882.2.h.t 8
7.d odd 6 1 882.2.f.s ✓ 8
7.d odd 6 1 882.2.h.t 8
9.c even 3 1 882.2.h.t 8
9.d odd 6 1 2646.2.h.q 8
21.c even 2 1 2646.2.e.t 8
21.g even 6 1 2646.2.f.q 8
21.g even 6 1 2646.2.h.q 8
21.h odd 6 1 2646.2.f.q 8
21.h odd 6 1 2646.2.h.q 8
63.g even 3 1 882.2.f.s ✓ 8
63.h even 3 1 inner 882.2.e.q 8
63.h even 3 1 7938.2.a.cj 4
63.i even 6 1 2646.2.e.t 8
63.i even 6 1 7938.2.a.co 4
63.j odd 6 1 2646.2.e.t 8
63.j odd 6 1 7938.2.a.co 4
63.k odd 6 1 882.2.f.s ✓ 8
63.l odd 6 1 882.2.h.t 8
63.n odd 6 1 2646.2.f.q 8
63.o even 6 1 2646.2.h.q 8
63.s even 6 1 2646.2.f.q 8
63.t odd 6 1 inner 882.2.e.q 8
63.t odd 6 1 7938.2.a.cj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.q 8 1.a even 1 1 trivial
882.2.e.q 8 7.b odd 2 1 inner
882.2.e.q 8 63.h even 3 1 inner
882.2.e.q 8 63.t odd 6 1 inner
882.2.f.s ✓ 8 7.c even 3 1
882.2.f.s ✓ 8 7.d odd 6 1
882.2.f.s ✓ 8 63.g even 3 1
882.2.f.s ✓ 8 63.k odd 6 1
882.2.h.t 8 7.c even 3 1
882.2.h.t 8 7.d odd 6 1
882.2.h.t 8 9.c even 3 1
882.2.h.t 8 63.l odd 6 1
2646.2.e.t 8 3.b odd 2 1
2646.2.e.t 8 21.c even 2 1
2646.2.e.t 8 63.i even 6 1
2646.2.e.t 8 63.j odd 6 1
2646.2.f.q 8 21.g even 6 1
2646.2.f.q 8 21.h odd 6 1
2646.2.f.q 8 63.n odd 6 1
2646.2.f.q 8 63.s even 6 1
2646.2.h.q 8 9.d odd 6 1
2646.2.h.q 8 21.g even 6 1
2646.2.h.q 8 21.h odd 6 1
2646.2.h.q 8 63.o even 6 1
7938.2.a.cj 4 63.h even 3 1
7938.2.a.cj 4 63.t odd 6 1
7938.2.a.co 4 63.i even 6 1
7938.2.a.co 4 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} + 4T_{5}^{6} + 15T_{5}^{4} + 4T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} + 24T_{11}^{2} - 32T_{11} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + 24 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 52 T^{6} + \cdots + 234256 \) Copy content Toggle raw display
$19$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + 6 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 10 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 244 T^{2} + 5476)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{2} + 169)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 18 T + 54)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 9)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 112 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 143)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 98 T^{2} + 9604)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 2097273616 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{6} + \cdots + 256 \) Copy content Toggle raw display
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