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Properties

Label 324.4.e.a
Level $324$
Weight $4$
Character orbit 324.e
Analytic conductor $19.117$
Analytic rank $1$
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] = [324,4,Mod(109,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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, 4, names="a")
 
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, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
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,-18,0,-8] 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: \(19.1166188419\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q - 18 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + ( - 36 \zeta_{6} + 36) q^{11} + 10 \zeta_{6} q^{13} - 18 q^{17} - 100 q^{19} + 72 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} + (234 \zeta_{6} - 234) q^{29} + \cdots + ( - 1054 \zeta_{6} + 1054) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{5} - 8 q^{7} + 36 q^{11} + 10 q^{13} - 36 q^{17} - 200 q^{19} + 72 q^{23} - 199 q^{25} - 234 q^{29} + 16 q^{31} + 288 q^{35} - 452 q^{37} + 90 q^{41} - 452 q^{43} + 432 q^{47} + 279 q^{49} - 828 q^{53}+ \cdots + 1054 q^{97}+O(q^{100}) \) 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 −9.00000 + 15.5885i 0 −4.00000 − 6.92820i 0 0 0
217.1 0 0 0 −9.00000 − 15.5885i 0 −4.00000 + 6.92820i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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 324.4.e.a 2
3.b odd 2 1 324.4.e.h 2
9.c even 3 1 36.4.a.a 1
9.c even 3 1 inner 324.4.e.a 2
9.d odd 6 1 12.4.a.a ✓ 1
9.d odd 6 1 324.4.e.h 2
36.f odd 6 1 144.4.a.g 1
36.h even 6 1 48.4.a.a 1
45.h odd 6 1 300.4.a.b 1
45.j even 6 1 900.4.a.g 1
45.k odd 12 2 900.4.d.c 2
45.l even 12 2 300.4.d.e 2
63.g even 3 1 1764.4.k.b 2
63.h even 3 1 1764.4.k.b 2
63.i even 6 1 588.4.i.e 2
63.j odd 6 1 588.4.i.d 2
63.k odd 6 1 1764.4.k.o 2
63.l odd 6 1 1764.4.a.b 1
63.n odd 6 1 588.4.i.d 2
63.o even 6 1 588.4.a.c 1
63.s even 6 1 588.4.i.e 2
63.t odd 6 1 1764.4.k.o 2
72.j odd 6 1 192.4.a.f 1
72.l even 6 1 192.4.a.l 1
72.n even 6 1 576.4.a.b 1
72.p odd 6 1 576.4.a.a 1
99.g even 6 1 1452.4.a.d 1
117.n odd 6 1 2028.4.a.c 1
117.z even 12 2 2028.4.b.c 2
144.u even 12 2 768.4.d.j 2
144.w odd 12 2 768.4.d.g 2
180.n even 6 1 1200.4.a.be 1
180.v odd 12 2 1200.4.f.d 2
252.s odd 6 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a ✓ 1 9.d odd 6 1
36.4.a.a 1 9.c even 3 1
48.4.a.a 1 36.h even 6 1
144.4.a.g 1 36.f odd 6 1
192.4.a.f 1 72.j odd 6 1
192.4.a.l 1 72.l even 6 1
300.4.a.b 1 45.h odd 6 1
300.4.d.e 2 45.l even 12 2
324.4.e.a 2 1.a even 1 1 trivial
324.4.e.a 2 9.c even 3 1 inner
324.4.e.h 2 3.b odd 2 1
324.4.e.h 2 9.d odd 6 1
576.4.a.a 1 72.p odd 6 1
576.4.a.b 1 72.n even 6 1
588.4.a.c 1 63.o even 6 1
588.4.i.d 2 63.j odd 6 1
588.4.i.d 2 63.n odd 6 1
588.4.i.e 2 63.i even 6 1
588.4.i.e 2 63.s even 6 1
768.4.d.g 2 144.w odd 12 2
768.4.d.j 2 144.u even 12 2
900.4.a.g 1 45.j even 6 1
900.4.d.c 2 45.k odd 12 2
1200.4.a.be 1 180.n even 6 1
1200.4.f.d 2 180.v odd 12 2
1452.4.a.d 1 99.g even 6 1
1764.4.a.b 1 63.l odd 6 1
1764.4.k.b 2 63.g even 3 1
1764.4.k.b 2 63.h even 3 1
1764.4.k.o 2 63.k odd 6 1
1764.4.k.o 2 63.t odd 6 1
2028.4.a.c 1 117.n odd 6 1
2028.4.b.c 2 117.z even 12 2
2352.4.a.bk 1 252.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 18T_{5} + 324 \) Copy content Toggle raw display
\( T_{7}^{2} + 8T_{7} + 64 \) 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
$5$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$11$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$13$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$17$ \( (T + 18)^{2} \) Copy content Toggle raw display
$19$ \( (T + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$29$ \( T^{2} + 234T + 54756 \) Copy content Toggle raw display
$31$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$37$ \( (T + 226)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 90T + 8100 \) Copy content Toggle raw display
$43$ \( T^{2} + 452T + 204304 \) Copy content Toggle raw display
$47$ \( T^{2} - 432T + 186624 \) Copy content Toggle raw display
$53$ \( (T + 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 684T + 467856 \) Copy content Toggle raw display
$61$ \( T^{2} + 422T + 178084 \) Copy content Toggle raw display
$67$ \( T^{2} + 332T + 110224 \) Copy content Toggle raw display
$71$ \( (T - 360)^{2} \) Copy content Toggle raw display
$73$ \( (T - 26)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 512T + 262144 \) Copy content Toggle raw display
$83$ \( T^{2} + 1188 T + 1411344 \) Copy content Toggle raw display
$89$ \( (T - 630)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1054 T + 1110916 \) Copy content Toggle raw display
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