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Properties

Label 324.8.e.e
Level $324$
Weight $8$
Character orbit 324.e
Analytic conductor $101.213$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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 = [2,0,0,0,270,0,-1112] 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 12)
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.

\(f(q)\) \(=\) \( q + 270 \zeta_{6} q^{5} + (1112 \zeta_{6} - 1112) q^{7} + (5724 \zeta_{6} - 5724) q^{11} + 4570 \zeta_{6} q^{13} + 36558 q^{17} + 51740 q^{19} + 22248 \zeta_{6} q^{23} + ( - 5225 \zeta_{6} + 5225) q^{25}+ \cdots + (13719074 \zeta_{6} - 13719074) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 270 q^{5} - 1112 q^{7} - 5724 q^{11} + 4570 q^{13} + 73116 q^{17} + 103480 q^{19} + 22248 q^{23} + 5225 q^{25} - 157194 q^{29} + 103936 q^{31} - 600480 q^{35} - 189668 q^{37} + 659610 q^{41} + 75772 q^{43}+ \cdots - 13719074 q^{97}+O(q^{100}) \) 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 0 135.000 − 233.827i 0 −556.000 − 963.020i 0 0 0
217.1 0 0 0 135.000 + 233.827i 0 −556.000 + 963.020i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.8.e.e 2
3.b odd 2 1 324.8.e.b 2
9.c even 3 1 36.8.a.a 1
9.c even 3 1 inner 324.8.e.e 2
9.d odd 6 1 12.8.a.b ✓ 1
9.d odd 6 1 324.8.e.b 2
36.f odd 6 1 144.8.a.c 1
36.h even 6 1 48.8.a.d 1
45.h odd 6 1 300.8.a.a 1
45.l even 12 2 300.8.d.a 2
63.i even 6 1 588.8.i.g 2
63.j odd 6 1 588.8.i.b 2
63.n odd 6 1 588.8.i.b 2
63.o even 6 1 588.8.a.a 1
63.s even 6 1 588.8.i.g 2
72.j odd 6 1 192.8.a.b 1
72.l even 6 1 192.8.a.j 1
72.n even 6 1 576.8.a.v 1
72.p odd 6 1 576.8.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.8.a.b ✓ 1 9.d odd 6 1
36.8.a.a 1 9.c even 3 1
48.8.a.d 1 36.h even 6 1
144.8.a.c 1 36.f odd 6 1
192.8.a.b 1 72.j odd 6 1
192.8.a.j 1 72.l even 6 1
300.8.a.a 1 45.h odd 6 1
300.8.d.a 2 45.l even 12 2
324.8.e.b 2 3.b odd 2 1
324.8.e.b 2 9.d odd 6 1
324.8.e.e 2 1.a even 1 1 trivial
324.8.e.e 2 9.c even 3 1 inner
576.8.a.u 1 72.p odd 6 1
576.8.a.v 1 72.n even 6 1
588.8.a.a 1 63.o even 6 1
588.8.i.b 2 63.j odd 6 1
588.8.i.b 2 63.n odd 6 1
588.8.i.g 2 63.i even 6 1
588.8.i.g 2 63.s even 6 1

Hecke kernels

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

\( T_{5}^{2} - 270T_{5} + 72900 \) Copy content Toggle raw display
\( T_{7}^{2} + 1112T_{7} + 1236544 \) 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
$5$ \( T^{2} - 270T + 72900 \) Copy content Toggle raw display
$7$ \( T^{2} + 1112 T + 1236544 \) Copy content Toggle raw display
$11$ \( T^{2} + 5724 T + 32764176 \) Copy content Toggle raw display
$13$ \( T^{2} - 4570 T + 20884900 \) Copy content Toggle raw display
$17$ \( (T - 36558)^{2} \) Copy content Toggle raw display
$19$ \( (T - 51740)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 22248 T + 494973504 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 24709953636 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 10802692096 \) Copy content Toggle raw display
$37$ \( (T + 94834)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 435085352100 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 5741395984 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 164550299904 \) Copy content Toggle raw display
$53$ \( (T - 1346274)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1700113485456 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 3362756423524 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1875223494544 \) Copy content Toggle raw display
$71$ \( (T + 2714040)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2868794)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 1276104603904 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 34952075072784 \) Copy content Toggle raw display
$89$ \( (T - 897750)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 188212991417476 \) Copy content Toggle raw display
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