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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,4,Mod(361,882)] 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("882.361"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,-8,-5,0,0,-32,0,10,67,0,-82] 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: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{1345})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 2 \beta_{2} - \beta_1) q^{5} - 8 q^{8} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 6) q^{10} + (\beta_{3} - 34 \beta_{2} + \beta_1 + 33) q^{11} + ( - \beta_{3} - 20) q^{13}+ \cdots + ( - 35 \beta_{3} - 959) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 5 q^{5} - 32 q^{8} + 10 q^{10} + 67 q^{11} - 82 q^{13} - 32 q^{16} + 92 q^{17} + 43 q^{19} + 40 q^{20} + 268 q^{22} + 148 q^{23} - 435 q^{25} - 82 q^{26} - 154 q^{29} - 520 q^{31}+ \cdots - 3906 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 337\nu^{2} - 337\nu + 112896 ) / 113232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 673 ) / 337 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 336\beta_{2} + \beta _1 - 337 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 337\beta_{3} - 673 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
9.41856 + 16.3134i
−8.91856 − 15.4474i
9.41856 − 16.3134i
−8.91856 + 15.4474i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −10.4186 − 18.0455i 0 0 −8.00000 0 20.8371 − 36.0910i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 7.91856 + 13.7153i 0 0 −8.00000 0 −15.8371 + 27.4307i
667.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −10.4186 + 18.0455i 0 0 −8.00000 0 20.8371 + 36.0910i
667.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 7.91856 − 13.7153i 0 0 −8.00000 0 −15.8371 − 27.4307i
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.bf 4
3.b odd 2 1 294.4.e.l 4
7.b odd 2 1 126.4.g.g 4
7.c even 3 1 882.4.a.z 2
7.c even 3 1 inner 882.4.g.bf 4
7.d odd 6 1 126.4.g.g 4
7.d odd 6 1 882.4.a.v 2
21.c even 2 1 42.4.e.c ✓ 4
21.g even 6 1 42.4.e.c ✓ 4
21.g even 6 1 294.4.a.n 2
21.h odd 6 1 294.4.a.m 2
21.h odd 6 1 294.4.e.l 4
84.h odd 2 1 336.4.q.j 4
84.j odd 6 1 336.4.q.j 4
84.j odd 6 1 2352.4.a.bq 2
84.n even 6 1 2352.4.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c ✓ 4 21.c even 2 1
42.4.e.c ✓ 4 21.g even 6 1
126.4.g.g 4 7.b odd 2 1
126.4.g.g 4 7.d odd 6 1
294.4.a.m 2 21.h odd 6 1
294.4.a.n 2 21.g even 6 1
294.4.e.l 4 3.b odd 2 1
294.4.e.l 4 21.h odd 6 1
336.4.q.j 4 84.h odd 2 1
336.4.q.j 4 84.j odd 6 1
882.4.a.v 2 7.d odd 6 1
882.4.a.z 2 7.c even 3 1
882.4.g.bf 4 1.a even 1 1 trivial
882.4.g.bf 4 7.c even 3 1 inner
2352.4.a.bq 2 84.j odd 6 1
2352.4.a.ca 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 5T_{5}^{3} + 355T_{5}^{2} - 1650T_{5} + 108900 \) Copy content Toggle raw display
\( T_{11}^{4} - 67T_{11}^{3} + 3703T_{11}^{2} - 52662T_{11} + 617796 \) Copy content Toggle raw display
\( T_{13}^{2} + 41T_{13} + 84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + \cdots + 108900 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 67 T^{3} + \cdots + 617796 \) Copy content Toggle raw display
$13$ \( (T^{2} + 41 T + 84)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 92 T^{3} + \cdots + 10653696 \) Copy content Toggle raw display
$19$ \( T^{4} - 43 T^{3} + \cdots + 6574096 \) Copy content Toggle raw display
$23$ \( T^{4} - 148 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( (T^{2} + 77 T - 39204)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 4389725025 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + \cdots + 741146176 \) Copy content Toggle raw display
$41$ \( (T^{2} + 426 T + 33264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 107 T - 72794)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 1191906576 \) Copy content Toggle raw display
$53$ \( T^{4} + 243 T^{3} + \cdots + 137733696 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 44160500736 \) Copy content Toggle raw display
$61$ \( T^{4} - 224 T^{3} + \cdots + 51322896 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 5976217636 \) Copy content Toggle raw display
$71$ \( (T^{2} + 472 T - 78804)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 39938423716 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 2270427201 \) Copy content Toggle raw display
$83$ \( (T^{2} + 221 T - 197946)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 196582277376 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1953 T + 541646)^{2} \) Copy content Toggle raw display
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