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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,8,Mod(109,324)] 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("324.109"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,508] 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: \(101.212748257\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 508 \zeta_{6} + 508) q^{7} + 14614 \zeta_{6} q^{13} - 57448 q^{19} + ( - 78125 \zeta_{6} + 78125) q^{25} - 178916 \zeta_{6} q^{31} + 279710 q^{37} + (1035224 \zeta_{6} - 1035224) q^{43} + 565479 \zeta_{6} q^{49} + \cdots + (12245198 \zeta_{6} - 12245198) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 508 q^{7} + 14614 q^{13} - 114896 q^{19} + 78125 q^{25} - 178916 q^{31} + 559420 q^{37} - 1035224 q^{43} + 565479 q^{49} + 3535546 q^{61} + 385072 q^{67} - 12549620 q^{73} - 8763044 q^{79} + 14847824 q^{91}+ \cdots - 12245198 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 254.000 + 439.941i 0 0 0
217.1 0 0 0 0 0 254.000 − 439.941i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.8.e.c 2
3.b odd 2 1 CM 324.8.e.c 2
9.c even 3 1 36.8.a.b ✓ 1
9.c even 3 1 inner 324.8.e.c 2
9.d odd 6 1 36.8.a.b ✓ 1
9.d odd 6 1 inner 324.8.e.c 2
36.f odd 6 1 144.8.a.e 1
36.h even 6 1 144.8.a.e 1
72.j odd 6 1 576.8.a.q 1
72.l even 6 1 576.8.a.r 1
72.n even 6 1 576.8.a.q 1
72.p odd 6 1 576.8.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.8.a.b ✓ 1 9.c even 3 1
36.8.a.b ✓ 1 9.d odd 6 1
144.8.a.e 1 36.f odd 6 1
144.8.a.e 1 36.h even 6 1
324.8.e.c 2 1.a even 1 1 trivial
324.8.e.c 2 3.b odd 2 1 CM
324.8.e.c 2 9.c even 3 1 inner
324.8.e.c 2 9.d odd 6 1 inner
576.8.a.q 1 72.j odd 6 1
576.8.a.q 1 72.n even 6 1
576.8.a.r 1 72.l even 6 1
576.8.a.r 1 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(324, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 508T_{7} + 258064 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 508T + 258064 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 14614 T + 213568996 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 57448)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 32010935056 \) Copy content Toggle raw display
$37$ \( (T - 279710)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 1071688730176 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 12500085518116 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 148280445184 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 6274810)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 76790940145936 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 149944874059204 \) Copy content Toggle raw display
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