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Properties

Label 324.3.f.p
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.207360000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{5} + 1) q^{4} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{5} + ( - 2 \beta_{5} - \beta_1 + 2) q^{7} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{5} + 1) q^{4} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{5} + ( - 2 \beta_{5} - \beta_1 + 2) q^{7} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{8} + (2 \beta_{7} + 2 \beta_{5} + 8 \beta_1 - 2) q^{10} + (\beta_{6} - 3 \beta_{4} + \cdots - \beta_{2}) q^{11}+ \cdots + (8 \beta_{6} - 16 \beta_{4} + \cdots + 8 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 12 q^{7} + 16 q^{10} + 4 q^{13} - 112 q^{16} - 120 q^{22} - 12 q^{25} - 108 q^{28} - 96 q^{31} + 280 q^{34} - 56 q^{37} + 256 q^{40} - 240 q^{43} + 288 q^{46} + 56 q^{49} - 356 q^{52} - 224 q^{58} - 20 q^{61} - 352 q^{64} + 228 q^{67} + 312 q^{70} + 232 q^{73} + 660 q^{79} + 448 q^{82} - 208 q^{85} - 240 q^{88} + 168 q^{94} + 124 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{6} - 16\nu^{4} - 96\nu^{2} - 8 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} - 24\nu^{3} + 24\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} + 32\nu^{5} + 160\nu^{3} + 184\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - 16\nu^{5} - 64\nu^{3} + 264\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} - 48\nu^{4} - 224\nu^{2} - 312 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 16\nu^{5} - 80\nu^{3} + 24\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{6} - 48\nu^{4} - 224\nu^{2} + 120 ) / 64 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{6} - \beta_{4} + 3\beta_{3} - \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + \beta_{5} - 9\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} + 5\beta_{4} - 3\beta_{3} - 7\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 4\beta_{5} + 14\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -22\beta_{6} - 4\beta_{4} - 12\beta_{3} + 56\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -32\beta_{7} + 32\beta_{5} + 216 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 136\beta_{6} - 116\beta_{4} + 156\beta_{3} - 116\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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
0.437016 + 0.756934i
1.14412 + 1.98168i
−1.14412 − 1.98168i
−0.437016 − 0.756934i
1.14412 − 1.98168i
0.437016 − 0.756934i
−0.437016 + 0.756934i
−1.14412 + 1.98168i
−1.58114 − 1.22474i 0 1.00000 + 3.87298i 3.70246 + 6.41285i 0 8.20820 + 4.73901i 3.16228 − 7.34847i 0 2.00000 − 14.6742i
55.2 −1.58114 + 1.22474i 0 1.00000 − 3.87298i −0.540182 − 0.935622i 0 −5.20820 − 3.00696i 3.16228 + 7.34847i 0 2.00000 + 0.817763i
55.3 1.58114 − 1.22474i 0 1.00000 − 3.87298i 0.540182 + 0.935622i 0 −5.20820 − 3.00696i −3.16228 − 7.34847i 0 2.00000 + 0.817763i
55.4 1.58114 + 1.22474i 0 1.00000 + 3.87298i −3.70246 − 6.41285i 0 8.20820 + 4.73901i −3.16228 + 7.34847i 0 2.00000 − 14.6742i
271.1 −1.58114 − 1.22474i 0 1.00000 + 3.87298i −0.540182 + 0.935622i 0 −5.20820 + 3.00696i 3.16228 − 7.34847i 0 2.00000 − 0.817763i
271.2 −1.58114 + 1.22474i 0 1.00000 − 3.87298i 3.70246 − 6.41285i 0 8.20820 − 4.73901i 3.16228 + 7.34847i 0 2.00000 + 14.6742i
271.3 1.58114 − 1.22474i 0 1.00000 − 3.87298i −3.70246 + 6.41285i 0 8.20820 − 4.73901i −3.16228 − 7.34847i 0 2.00000 + 14.6742i
271.4 1.58114 + 1.22474i 0 1.00000 + 3.87298i 0.540182 − 0.935622i 0 −5.20820 + 3.00696i −3.16228 + 7.34847i 0 2.00000 − 0.817763i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
36.f odd 6 1 inner
36.h even 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.p 8
3.b odd 2 1 inner 324.3.f.p 8
4.b odd 2 1 324.3.f.o 8
9.c even 3 1 108.3.d.d ✓ 8
9.c even 3 1 324.3.f.o 8
9.d odd 6 1 108.3.d.d ✓ 8
9.d odd 6 1 324.3.f.o 8
12.b even 2 1 324.3.f.o 8
36.f odd 6 1 108.3.d.d ✓ 8
36.f odd 6 1 inner 324.3.f.p 8
36.h even 6 1 108.3.d.d ✓ 8
36.h even 6 1 inner 324.3.f.p 8
72.j odd 6 1 1728.3.g.l 8
72.l even 6 1 1728.3.g.l 8
72.n even 6 1 1728.3.g.l 8
72.p odd 6 1 1728.3.g.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.d ✓ 8 9.c even 3 1
108.3.d.d ✓ 8 9.d odd 6 1
108.3.d.d ✓ 8 36.f odd 6 1
108.3.d.d ✓ 8 36.h even 6 1
324.3.f.o 8 4.b odd 2 1
324.3.f.o 8 9.c even 3 1
324.3.f.o 8 9.d odd 6 1
324.3.f.o 8 12.b even 2 1
324.3.f.p 8 1.a even 1 1 trivial
324.3.f.p 8 3.b odd 2 1 inner
324.3.f.p 8 36.f odd 6 1 inner
324.3.f.p 8 36.h even 6 1 inner
1728.3.g.l 8 72.j odd 6 1
1728.3.g.l 8 72.l even 6 1
1728.3.g.l 8 72.n even 6 1
1728.3.g.l 8 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 \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{8} + 56T_{5}^{6} + 3072T_{5}^{4} + 3584T_{5}^{2} + 4096 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} - 45T_{7}^{2} + 342T_{7} + 3249 \) Copy content Toggle raw display
\( T_{11}^{8} - 360T_{11}^{6} + 115200T_{11}^{4} - 5184000T_{11}^{2} + 207360000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 56 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{3} + \cdots + 3249)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 360 T^{6} + \cdots + 207360000 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1016 T^{2} + 222784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 306402103296 \) Copy content Toggle raw display
$29$ \( T^{8} + 896 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{3} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T - 131)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 68719476736 \) Copy content Toggle raw display
$43$ \( (T^{4} + 120 T^{3} + \cdots + 921600)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20816369750016 \) Copy content Toggle raw display
$53$ \( (T^{4} - 11744 T^{2} + 33547264)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 34524743602176 \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + \cdots + 24025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 114 T^{3} + \cdots + 7601049)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 19872 T^{2} + 90326016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 58 T + 121)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 330 T^{3} + \cdots + 37638225)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 53084160000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1016 T^{2} + 222784)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 62 T^{3} + \cdots + 111492481)^{2} \) Copy content Toggle raw display
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