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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [936,2,Mod(217,936)] 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("936.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.t (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,4,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.27870912.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - \beta_{5} + \beta_{3}) q^{11} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{2} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{5} - q^{7} - q^{13} + 2 q^{17} - 8 q^{23} - 2 q^{25} + 2 q^{29} - 10 q^{31} + 12 q^{35} - 6 q^{41} - 5 q^{43} + 8 q^{47} + 8 q^{49} + 28 q^{53} - 4 q^{55} - 4 q^{59} - 5 q^{61} - 22 q^{65}+ \cdots - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 38\nu^{2} + 12\nu - 84 ) / 254 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 49\nu^{4} - 89\nu^{3} + 266\nu^{2} - 84\nu + 1096 ) / 254 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{5} - 20\nu^{4} + 140\nu^{3} + 91\nu^{2} + 760\nu + 14 ) / 254 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -85\nu^{5} + 87\nu^{4} - 609\nu^{3} - 72\nu^{2} - 3306\nu + 1044 ) / 254 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{4} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 7\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 26\beta_{4} + 7\beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} - 30\beta_{4} - 51\beta_{2} - 51\beta _1 + 30 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
1.42789 − 2.47317i
0.160819 − 0.278546i
−1.08870 + 1.88569i
1.42789 + 2.47317i
0.160819 + 0.278546i
−1.08870 − 1.88569i
0 0 0 −1.85577 0 −1.42789 − 2.47317i 0 0 0
217.2 0 0 0 0.678363 0 −0.160819 − 0.278546i 0 0 0
217.3 0 0 0 3.17741 0 1.08870 + 1.88569i 0 0 0
289.1 0 0 0 −1.85577 0 −1.42789 + 2.47317i 0 0 0
289.2 0 0 0 0.678363 0 −0.160819 + 0.278546i 0 0 0
289.3 0 0 0 3.17741 0 1.08870 − 1.88569i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 217.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.t.i yes 6
3.b odd 2 1 936.2.t.g ✓ 6
4.b odd 2 1 1872.2.t.v 6
12.b even 2 1 1872.2.t.t 6
13.c even 3 1 inner 936.2.t.i yes 6
39.i odd 6 1 936.2.t.g ✓ 6
52.j odd 6 1 1872.2.t.v 6
156.p even 6 1 1872.2.t.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.t.g ✓ 6 3.b odd 2 1
936.2.t.g ✓ 6 39.i odd 6 1
936.2.t.i yes 6 1.a even 1 1 trivial
936.2.t.i yes 6 13.c even 3 1 inner
1872.2.t.t 6 12.b even 2 1
1872.2.t.t 6 156.p even 6 1
1872.2.t.v 6 4.b odd 2 1
1872.2.t.v 6 52.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 \(S_{2}^{\mathrm{new}}(936, [\chi])\):

\( T_{5}^{3} - 2T_{5}^{2} - 5T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + T_{7}^{5} + 7T_{7}^{4} - 2T_{7}^{3} + 38T_{7}^{2} + 12T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 2 T^{2} - 5 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + 7 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{6} + 14 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{6} + 14 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$31$ \( (T^{3} + 5 T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 57 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( T^{6} + 5 T^{5} + \cdots + 13924 \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T^{2} - 26 T + 96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 14 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} + \cdots + 1557504 \) Copy content Toggle raw display
$61$ \( T^{6} + 5 T^{5} + \cdots + 257049 \) Copy content Toggle raw display
$67$ \( T^{6} + 13 T^{5} + \cdots + 2377764 \) Copy content Toggle raw display
$71$ \( T^{6} + 16 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$73$ \( (T^{3} - 7 T^{2} - 81 T - 9)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - T^{2} - 152 T + 624)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 20 T^{2} + \cdots + 712)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T + 16)^{3} \) Copy content Toggle raw display
$97$ \( T^{6} + 5 T^{5} + \cdots + 256 \) Copy content Toggle raw display
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