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Properties

Label 882.2.h.k
Level $882$
Weight $2$
Character orbit 882.h
Analytic conductor $7.043$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(67,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, 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("882.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,-2,-2,4,-2,0,4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} - \beta_{2} q^{4} + (\beta_{3} - 2 \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2} - \beta_1) q^{6} + q^{8} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{9}+ \cdots + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{8} + 2 q^{9} - 2 q^{10} + 8 q^{11} + 4 q^{12} - 14 q^{15} - 2 q^{16} - 4 q^{17} - 4 q^{18} - 10 q^{19} - 2 q^{20} - 4 q^{22} - 4 q^{23} - 2 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\beta_{2}\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
−0.500000 + 0.866025i −1.72474 + 0.158919i −0.500000 − 0.866025i 3.44949 0.724745 − 1.57313i 0 1.00000 2.94949 − 0.548188i −1.72474 + 2.98735i
67.2 −0.500000 + 0.866025i 0.724745 + 1.57313i −0.500000 − 0.866025i −1.44949 −1.72474 − 0.158919i 0 1.00000 −1.94949 + 2.28024i 0.724745 − 1.25529i
79.1 −0.500000 − 0.866025i −1.72474 − 0.158919i −0.500000 + 0.866025i 3.44949 0.724745 + 1.57313i 0 1.00000 2.94949 + 0.548188i −1.72474 − 2.98735i
79.2 −0.500000 − 0.866025i 0.724745 − 1.57313i −0.500000 + 0.866025i −1.44949 −1.72474 + 0.158919i 0 1.00000 −1.94949 − 2.28024i 0.724745 + 1.25529i
\(n\): 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 882.2.h.k 4
3.b odd 2 1 2646.2.h.m 4
7.b odd 2 1 882.2.h.l 4
7.c even 3 1 126.2.f.c ✓ 4
7.c even 3 1 882.2.e.m 4
7.d odd 6 1 882.2.e.n 4
7.d odd 6 1 882.2.f.j 4
9.c even 3 1 882.2.e.m 4
9.d odd 6 1 2646.2.e.l 4
21.c even 2 1 2646.2.h.n 4
21.g even 6 1 2646.2.e.k 4
21.g even 6 1 2646.2.f.k 4
21.h odd 6 1 378.2.f.d 4
21.h odd 6 1 2646.2.e.l 4
28.g odd 6 1 1008.2.r.e 4
63.g even 3 1 inner 882.2.h.k 4
63.g even 3 1 1134.2.a.p 2
63.h even 3 1 126.2.f.c ✓ 4
63.i even 6 1 2646.2.f.k 4
63.j odd 6 1 378.2.f.d 4
63.k odd 6 1 882.2.h.l 4
63.k odd 6 1 7938.2.a.bn 2
63.l odd 6 1 882.2.e.n 4
63.n odd 6 1 1134.2.a.i 2
63.n odd 6 1 2646.2.h.m 4
63.o even 6 1 2646.2.e.k 4
63.s even 6 1 2646.2.h.n 4
63.s even 6 1 7938.2.a.bm 2
63.t odd 6 1 882.2.f.j 4
84.n even 6 1 3024.2.r.e 4
252.o even 6 1 9072.2.a.bd 2
252.u odd 6 1 1008.2.r.e 4
252.bb even 6 1 3024.2.r.e 4
252.bl odd 6 1 9072.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c ✓ 4 7.c even 3 1
126.2.f.c ✓ 4 63.h even 3 1
378.2.f.d 4 21.h odd 6 1
378.2.f.d 4 63.j odd 6 1
882.2.e.m 4 7.c even 3 1
882.2.e.m 4 9.c even 3 1
882.2.e.n 4 7.d odd 6 1
882.2.e.n 4 63.l odd 6 1
882.2.f.j 4 7.d odd 6 1
882.2.f.j 4 63.t odd 6 1
882.2.h.k 4 1.a even 1 1 trivial
882.2.h.k 4 63.g even 3 1 inner
882.2.h.l 4 7.b odd 2 1
882.2.h.l 4 63.k odd 6 1
1008.2.r.e 4 28.g odd 6 1
1008.2.r.e 4 252.u odd 6 1
1134.2.a.i 2 63.n odd 6 1
1134.2.a.p 2 63.g even 3 1
2646.2.e.k 4 21.g even 6 1
2646.2.e.k 4 63.o even 6 1
2646.2.e.l 4 9.d odd 6 1
2646.2.e.l 4 21.h odd 6 1
2646.2.f.k 4 21.g even 6 1
2646.2.f.k 4 63.i even 6 1
2646.2.h.m 4 3.b odd 2 1
2646.2.h.m 4 63.n odd 6 1
2646.2.h.n 4 21.c even 2 1
2646.2.h.n 4 63.s even 6 1
3024.2.r.e 4 84.n even 6 1
3024.2.r.e 4 252.bb even 6 1
7938.2.a.bm 2 63.s even 6 1
7938.2.a.bn 2 63.k odd 6 1
9072.2.a.bd 2 252.o even 6 1
9072.2.a.bk 2 252.bl odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 2T_{5} - 5 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 24T_{13}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$41$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$47$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + \cdots + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} + \cdots + 1600 \) Copy content Toggle raw display
$71$ \( (T^{2} - 10 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + \cdots + 225 \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + \cdots + 14400 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 400 \) Copy content Toggle raw display
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