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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2736,2,Mod(577,2736)] 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("2736.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,-6,0,0,0,12,0,1] 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1368)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{5} - 3 q^{7} + 6 q^{11} + \zeta_{6} q^{13} + (2 \zeta_{6} - 2) q^{17} + ( - 2 \zeta_{6} + 5) q^{19} + \zeta_{6} q^{25} - 2 \zeta_{6} q^{29} + q^{31} + ( - 6 \zeta_{6} + 6) q^{35} + \cdots + (10 \zeta_{6} - 10) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 6 q^{7} + 12 q^{11} + q^{13} - 2 q^{17} + 8 q^{19} + q^{25} - 2 q^{29} + 2 q^{31} + 6 q^{35} - 14 q^{37} - q^{43} + 4 q^{49} - 4 q^{53} - 12 q^{55} + 8 q^{59} + 11 q^{61} - 4 q^{65} + 15 q^{67}+ \cdots - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 −1.00000 − 1.73205i 0 −3.00000 0 0 0
1873.1 0 0 0 −1.00000 + 1.73205i 0 −3.00000 0 0 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.e 2
3.b odd 2 1 2736.2.s.n 2
4.b odd 2 1 1368.2.s.c ✓ 2
12.b even 2 1 1368.2.s.f yes 2
19.c even 3 1 inner 2736.2.s.e 2
57.h odd 6 1 2736.2.s.n 2
76.g odd 6 1 1368.2.s.c ✓ 2
228.m even 6 1 1368.2.s.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.s.c ✓ 2 4.b odd 2 1
1368.2.s.c ✓ 2 76.g odd 6 1
1368.2.s.f yes 2 12.b even 2 1
1368.2.s.f yes 2 228.m even 6 1
2736.2.s.e 2 1.a even 1 1 trivial
2736.2.s.e 2 19.c even 3 1 inner
2736.2.s.n 2 3.b odd 2 1
2736.2.s.n 2 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$79$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$83$ \( (T - 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
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