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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2816,2,Mod(2815,2816)] 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("2816.2815"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,4,0,-10,0,6,0,0,0,0,0,0,0,16,0,0,0,0,0,-2,0,0,0, 0,0,0,0,-4,0,-4,0,8,0,-4,0,0,0,12,0,32,0,0,0,6,0,12,0,-16,0,-28,0,0,0, 0,0,0,0,-56,0,0,0,0,0,-40,0,0,0,0,0,0,0,8,0,8,0,14,0,-4,0,0,0,-48,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(89)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.4858732092\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.141376000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 45x^{2} + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1408)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{3} q^{5} + ( - \beta_{5} + 1) q^{7} + (\beta_{5} - \beta_{3} - 2) q^{9} + ( - \beta_{4} + \beta_{3} + 1) q^{11} + (\beta_{4} - \beta_{2}) q^{13} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{15}+ \cdots + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{7} - 10 q^{9} + 6 q^{11} + 16 q^{19} - 2 q^{25} - 4 q^{33} - 4 q^{35} + 8 q^{37} - 4 q^{39} + 12 q^{43} + 32 q^{45} + 6 q^{49} + 12 q^{51} - 16 q^{53} - 28 q^{55} - 56 q^{63} - 40 q^{69} + 8 q^{77}+ \cdots - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 45x^{2} + 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 11\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 12\nu^{3} + 21\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 13\nu^{2} + 26 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{5} + 13\beta_{3} + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{4} - 24\beta_{2} + 63\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2815.1
− 3.13686i
− 1.60979i
− 1.25246i
1.25246i
1.60979i
3.13686i
0 − 3.13686i 0 −2.29240 0 3.54751 0 −6.83991 0
2815.2 0 − 1.60979i 0 2.89511 0 1.48654 0 0.408571 0
2815.3 0 − 1.25246i 0 −0.602705 0 −3.03404 0 1.43134 0
2815.4 0 1.25246i 0 −0.602705 0 −3.03404 0 1.43134 0
2815.5 0 1.60979i 0 2.89511 0 1.48654 0 0.408571 0
2815.6 0 3.13686i 0 −2.29240 0 3.54751 0 −6.83991 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2815.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2816.2.e.j 6
4.b odd 2 1 2816.2.e.g 6
8.b even 2 1 2816.2.e.i 6
8.d odd 2 1 2816.2.e.h 6
11.b odd 2 1 2816.2.e.g 6
16.e even 4 2 1408.2.g.f ✓ 12
16.f odd 4 2 1408.2.g.g yes 12
44.c even 2 1 inner 2816.2.e.j 6
88.b odd 2 1 2816.2.e.h 6
88.g even 2 1 2816.2.e.i 6
176.i even 4 2 1408.2.g.f ✓ 12
176.l odd 4 2 1408.2.g.g yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.g.f ✓ 12 16.e even 4 2
1408.2.g.f ✓ 12 176.i even 4 2
1408.2.g.g yes 12 16.f odd 4 2
1408.2.g.g yes 12 176.l odd 4 2
2816.2.e.g 6 4.b odd 2 1
2816.2.e.g 6 11.b odd 2 1
2816.2.e.h 6 8.d odd 2 1
2816.2.e.h 6 88.b odd 2 1
2816.2.e.i 6 8.b even 2 1
2816.2.e.i 6 88.g even 2 1
2816.2.e.j 6 1.a even 1 1 trivial
2816.2.e.j 6 44.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2816, [\chi])\):

\( T_{3}^{6} + 14T_{3}^{4} + 45T_{3}^{2} + 40 \) Copy content Toggle raw display
\( T_{5}^{3} - 7T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 10T_{7} + 16 \) Copy content Toggle raw display
\( T_{13}^{6} + 56T_{13}^{4} + 660T_{13}^{2} + 160 \) Copy content Toggle raw display
\( T_{19}^{3} - 8T_{19}^{2} + 10T_{19} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + \cdots + 40 \) Copy content Toggle raw display
$5$ \( (T^{3} - 7 T - 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 10 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + \cdots + 1331 \) Copy content Toggle raw display
$13$ \( T^{6} + 56 T^{4} + \cdots + 160 \) Copy content Toggle raw display
$17$ \( T^{6} + 64 T^{4} + \cdots + 640 \) Copy content Toggle raw display
$19$ \( (T^{3} - 8 T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 90 T^{4} + \cdots + 1000 \) Copy content Toggle raw display
$29$ \( T^{6} + 104 T^{4} + \cdots + 10240 \) Copy content Toggle raw display
$31$ \( T^{6} + 66 T^{4} + \cdots + 1000 \) Copy content Toggle raw display
$37$ \( (T^{3} - 4 T^{2} + \cdots + 188)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 176 T^{4} + \cdots + 40960 \) Copy content Toggle raw display
$43$ \( (T^{3} - 6 T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 136 T^{4} + \cdots + 40960 \) Copy content Toggle raw display
$53$ \( (T^{3} + 8 T^{2} - 44 T - 80)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 246 T^{4} + \cdots + 1000 \) Copy content Toggle raw display
$61$ \( T^{6} + 104 T^{4} + \cdots + 10240 \) Copy content Toggle raw display
$67$ \( T^{6} + 14 T^{4} + \cdots + 40 \) Copy content Toggle raw display
$71$ \( T^{6} + 186 T^{4} + \cdots + 179560 \) Copy content Toggle raw display
$73$ \( T^{6} + 240 T^{4} + \cdots + 400000 \) Copy content Toggle raw display
$79$ \( (T^{3} - 4 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 2 T^{2} + \cdots - 1136)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 4 T^{2} + \cdots - 1114)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 12 T^{2} + \cdots + 746)^{2} \) Copy content Toggle raw display
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