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Properties

Label 324.10.e.a
Level $324$
Weight $10$
Character orbit 324.e
Analytic conductor $166.872$
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] = [324,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
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,-990,0,-8576] 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: \(166.871610917\)
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.

\(f(q)\) \(=\) \( q - 990 \zeta_{6} q^{5} + (8576 \zeta_{6} - 8576) q^{7} + (70596 \zeta_{6} - 70596) q^{11} + 2530 \zeta_{6} q^{13} - 200574 q^{17} - 695620 q^{19} - 2472696 \zeta_{6} q^{23} + ( - 973025 \zeta_{6} + 973025) q^{25} + \cdots + ( - 872463358 \zeta_{6} + 872463358) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 990 q^{5} - 8576 q^{7} - 70596 q^{11} + 2530 q^{13} - 401148 q^{17} - 1391240 q^{19} - 2472696 q^{23} + 973025 q^{25} - 5474214 q^{29} - 3732104 q^{31} + 16980480 q^{35} - 43797044 q^{37} + 23818950 q^{41}+ \cdots + 872463358 q^{97}+O(q^{100}) \) 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 −495.000 + 857.365i 0 −4288.00 − 7427.03i 0 0 0
217.1 0 0 0 −495.000 − 857.365i 0 −4288.00 + 7427.03i 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 324.10.e.a 2
3.b odd 2 1 324.10.e.f 2
9.c even 3 1 12.10.a.a ✓ 1
9.c even 3 1 inner 324.10.e.a 2
9.d odd 6 1 36.10.a.a 1
9.d odd 6 1 324.10.e.f 2
36.f odd 6 1 48.10.a.g 1
36.h even 6 1 144.10.a.c 1
45.j even 6 1 300.10.a.a 1
45.k odd 12 2 300.10.d.c 2
72.n even 6 1 192.10.a.i 1
72.p odd 6 1 192.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.10.a.a ✓ 1 9.c even 3 1
36.10.a.a 1 9.d odd 6 1
48.10.a.g 1 36.f odd 6 1
144.10.a.c 1 36.h even 6 1
192.10.a.b 1 72.p odd 6 1
192.10.a.i 1 72.n even 6 1
300.10.a.a 1 45.j even 6 1
300.10.d.c 2 45.k odd 12 2
324.10.e.a 2 1.a even 1 1 trivial
324.10.e.a 2 9.c even 3 1 inner
324.10.e.f 2 3.b odd 2 1
324.10.e.f 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 990T_{5} + 980100 \) Copy content Toggle raw display
\( T_{7}^{2} + 8576T_{7} + 73547776 \) 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
$5$ \( T^{2} + 990T + 980100 \) Copy content Toggle raw display
$7$ \( T^{2} + 8576 T + 73547776 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 4983795216 \) Copy content Toggle raw display
$13$ \( T^{2} - 2530 T + 6400900 \) Copy content Toggle raw display
$17$ \( (T + 200574)^{2} \) Copy content Toggle raw display
$19$ \( (T + 695620)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 6114225508416 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 29967018917796 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 13928600266816 \) Copy content Toggle raw display
$37$ \( (T + 21898522)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 567342379102500 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 112628891880976 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 5752629559296 \) Copy content Toggle raw display
$53$ \( (T + 8994978)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 392208634930564 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T + 194801400)^{2} \) Copy content Toggle raw display
$73$ \( (T - 148729418)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 908067116759104 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T - 909502650)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
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