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Properties

Label 324.3.f.e
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,3,Mod(55,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
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: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + ( - 8 \zeta_{6} + 8) q^{5} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + ( - 8 \zeta_{6} + 8) q^{5} + 8 q^{8} - 16 q^{10} + ( - 10 \zeta_{6} + 10) q^{13} - 16 \zeta_{6} q^{16} + 16 q^{17} + 32 \zeta_{6} q^{20} - 39 \zeta_{6} q^{25} - 20 q^{26} - 40 \zeta_{6} q^{29} + (32 \zeta_{6} - 32) q^{32} - 32 \zeta_{6} q^{34} - 70 q^{37} + ( - 64 \zeta_{6} + 64) q^{40} + ( - 80 \zeta_{6} + 80) q^{41} + (49 \zeta_{6} - 49) q^{49} + (78 \zeta_{6} - 78) q^{50} + 40 \zeta_{6} q^{52} - 56 q^{53} + (80 \zeta_{6} - 80) q^{58} + 22 \zeta_{6} q^{61} + 64 q^{64} - 80 \zeta_{6} q^{65} + (64 \zeta_{6} - 64) q^{68} + 110 q^{73} + 140 \zeta_{6} q^{74} - 128 q^{80} - 160 q^{82} + ( - 128 \zeta_{6} + 128) q^{85} + 160 q^{89} + 130 \zeta_{6} q^{97} + 98 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 8 q^{5} + 16 q^{8} - 32 q^{10} + 10 q^{13} - 16 q^{16} + 32 q^{17} + 32 q^{20} - 39 q^{25} - 40 q^{26} - 40 q^{29} - 32 q^{32} - 32 q^{34} - 140 q^{37} + 64 q^{40} + 80 q^{41} - 49 q^{49} - 78 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 80 q^{65} - 64 q^{68} + 220 q^{73} + 140 q^{74} - 256 q^{80} - 320 q^{82} + 128 q^{85} + 320 q^{89} + 130 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−1.00000 + 1.73205i 0 −2.00000 − 3.46410i 4.00000 + 6.92820i 0 0 8.00000 0 −16.0000
271.1 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 4.00000 − 6.92820i 0 0 8.00000 0 −16.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.e 2
3.b odd 2 1 324.3.f.f 2
4.b odd 2 1 CM 324.3.f.e 2
9.c even 3 1 36.3.d.b yes 1
9.c even 3 1 inner 324.3.f.e 2
9.d odd 6 1 36.3.d.a ✓ 1
9.d odd 6 1 324.3.f.f 2
12.b even 2 1 324.3.f.f 2
36.f odd 6 1 36.3.d.b yes 1
36.f odd 6 1 inner 324.3.f.e 2
36.h even 6 1 36.3.d.a ✓ 1
36.h even 6 1 324.3.f.f 2
45.h odd 6 1 900.3.c.c 1
45.j even 6 1 900.3.c.b 1
45.k odd 12 2 900.3.f.a 2
45.l even 12 2 900.3.f.b 2
72.j odd 6 1 576.3.g.a 1
72.l even 6 1 576.3.g.a 1
72.n even 6 1 576.3.g.c 1
72.p odd 6 1 576.3.g.c 1
144.u even 12 2 2304.3.b.d 2
144.v odd 12 2 2304.3.b.e 2
144.w odd 12 2 2304.3.b.d 2
144.x even 12 2 2304.3.b.e 2
180.n even 6 1 900.3.c.c 1
180.p odd 6 1 900.3.c.b 1
180.v odd 12 2 900.3.f.b 2
180.x even 12 2 900.3.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a ✓ 1 9.d odd 6 1
36.3.d.a ✓ 1 36.h even 6 1
36.3.d.b yes 1 9.c even 3 1
36.3.d.b yes 1 36.f odd 6 1
324.3.f.e 2 1.a even 1 1 trivial
324.3.f.e 2 4.b odd 2 1 CM
324.3.f.e 2 9.c even 3 1 inner
324.3.f.e 2 36.f odd 6 1 inner
324.3.f.f 2 3.b odd 2 1
324.3.f.f 2 9.d odd 6 1
324.3.f.f 2 12.b even 2 1
324.3.f.f 2 36.h even 6 1
576.3.g.a 1 72.j odd 6 1
576.3.g.a 1 72.l even 6 1
576.3.g.c 1 72.n even 6 1
576.3.g.c 1 72.p odd 6 1
900.3.c.b 1 45.j even 6 1
900.3.c.b 1 180.p odd 6 1
900.3.c.c 1 45.h odd 6 1
900.3.c.c 1 180.n even 6 1
900.3.f.a 2 45.k odd 12 2
900.3.f.a 2 180.x even 12 2
900.3.f.b 2 45.l even 12 2
900.3.f.b 2 180.v odd 12 2
2304.3.b.d 2 144.u even 12 2
2304.3.b.d 2 144.w odd 12 2
2304.3.b.e 2 144.v odd 12 2
2304.3.b.e 2 144.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} - 8T_{5} + 64 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$17$ \( (T - 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 70)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 80T + 6400 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 56)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 110)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 160)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 130T + 16900 \) Copy content Toggle raw display
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