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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 = [2,0,0,0,0,0,0,0,-10,0,6,0,0,0,0,0,0,0,-4,0,0,0,0,0,-10,0,0,0, 0,0,0,0,8,0,0,0,0,0,0,0,0,0,-20,0,0,0,0,0,-14,0,-32,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,-36,0,0,0,0,0,36] 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: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1408)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} - 5 q^{9} + ( - \beta + 3) q^{11} + 4 \beta q^{17} - 2 q^{19} - 5 q^{25} - 4 \beta q^{27} + (6 \beta + 4) q^{33} + 8 \beta q^{41} - 10 q^{43} - 7 q^{49} - 16 q^{51} - 4 \beta q^{57} - 10 \beta q^{59} + \cdots + (5 \beta - 15) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{9} + 6 q^{11} - 4 q^{19} - 10 q^{25} + 8 q^{33} - 20 q^{43} - 14 q^{49} - 32 q^{51} + 2 q^{81} - 36 q^{83} + 36 q^{89} - 20 q^{97} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
− 1.41421i
1.41421i
0 − 2.82843i 0 0 0 0 0 −5.00000 0
2815.2 0 2.82843i 0 0 0 0 0 −5.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
44.c even 2 1 inner
88.b odd 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.b 2
4.b odd 2 1 2816.2.e.a 2
8.b even 2 1 2816.2.e.a 2
8.d odd 2 1 CM 2816.2.e.b 2
11.b odd 2 1 2816.2.e.a 2
16.e even 4 2 1408.2.g.c ✓ 4
16.f odd 4 2 1408.2.g.c ✓ 4
44.c even 2 1 inner 2816.2.e.b 2
88.b odd 2 1 inner 2816.2.e.b 2
88.g even 2 1 2816.2.e.a 2
176.i even 4 2 1408.2.g.c ✓ 4
176.l odd 4 2 1408.2.g.c ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.g.c ✓ 4 16.e even 4 2
1408.2.g.c ✓ 4 16.f odd 4 2
1408.2.g.c ✓ 4 176.i even 4 2
1408.2.g.c ✓ 4 176.l odd 4 2
2816.2.e.a 2 4.b odd 2 1
2816.2.e.a 2 8.b even 2 1
2816.2.e.a 2 11.b odd 2 1
2816.2.e.a 2 88.g even 2 1
2816.2.e.b 2 1.a even 1 1 trivial
2816.2.e.b 2 8.d odd 2 1 CM
2816.2.e.b 2 44.c even 2 1 inner
2816.2.e.b 2 88.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 11 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 32 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 128 \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 200 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 72 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 288 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 18)^{2} \) Copy content Toggle raw display
$89$ \( (T - 18)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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