Properties

Label 324.8.e.f
Level $324$
Weight $8$
Character orbit 324.e
Analytic conductor $101.213$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,8,Mod(109,324)]
 
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([0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//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);
 
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

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: \(101.212748257\)
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 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 378 \zeta_{6} q^{5} + ( - 832 \zeta_{6} + 832) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 378 \zeta_{6} q^{5} + ( - 832 \zeta_{6} + 832) q^{7} + ( - 2484 \zeta_{6} + 2484) q^{11} - 14870 \zeta_{6} q^{13} - 22302 q^{17} - 16300 q^{19} + 115128 \zeta_{6} q^{23} + (64759 \zeta_{6} - 64759) q^{25} + (157086 \zeta_{6} - 157086) q^{29} + 16456 \zeta_{6} q^{31} + 314496 q^{35} - 149266 q^{37} + 241110 \zeta_{6} q^{41} + ( - 443188 \zeta_{6} + 443188) q^{43} + (922752 \zeta_{6} - 922752) q^{47} + 131319 \zeta_{6} q^{49} - 697626 q^{53} + 938952 q^{55} - 870156 \zeta_{6} q^{59} + (2067062 \zeta_{6} - 2067062) q^{61} + ( - 5620860 \zeta_{6} + 5620860) q^{65} + 1680748 \zeta_{6} q^{67} - 1070280 q^{71} - 2403334 q^{73} - 2066688 \zeta_{6} q^{77} + (2301512 \zeta_{6} - 2301512) q^{79} + (4708692 \zeta_{6} - 4708692) q^{83} - 8430156 \zeta_{6} q^{85} + 4143690 q^{89} - 12371840 q^{91} - 6161400 \zeta_{6} q^{95} + ( - 1622974 \zeta_{6} + 1622974) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 378 q^{5} + 832 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 378 q^{5} + 832 q^{7} + 2484 q^{11} - 14870 q^{13} - 44604 q^{17} - 32600 q^{19} + 115128 q^{23} - 64759 q^{25} - 157086 q^{29} + 16456 q^{31} + 628992 q^{35} - 298532 q^{37} + 241110 q^{41} + 443188 q^{43} - 922752 q^{47} + 131319 q^{49} - 1395252 q^{53} + 1877904 q^{55} - 870156 q^{59} - 2067062 q^{61} + 5620860 q^{65} + 1680748 q^{67} - 2140560 q^{71} - 4806668 q^{73} - 2066688 q^{77} - 2301512 q^{79} - 4708692 q^{83} - 8430156 q^{85} + 8287380 q^{89} - 24743680 q^{91} - 6161400 q^{95} + 1622974 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 189.000 327.358i 0 416.000 + 720.533i 0 0 0
217.1 0 0 0 189.000 + 327.358i 0 416.000 720.533i 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 324.8.e.f 2
3.b odd 2 1 324.8.e.a 2
9.c even 3 1 12.8.a.a 1
9.c even 3 1 inner 324.8.e.f 2
9.d odd 6 1 36.8.a.c 1
9.d odd 6 1 324.8.e.a 2
36.f odd 6 1 48.8.a.e 1
36.h even 6 1 144.8.a.j 1
45.j even 6 1 300.8.a.g 1
45.k odd 12 2 300.8.d.c 2
63.g even 3 1 588.8.i.h 2
63.h even 3 1 588.8.i.h 2
63.k odd 6 1 588.8.i.a 2
63.l odd 6 1 588.8.a.d 1
63.t odd 6 1 588.8.i.a 2
72.j odd 6 1 576.8.a.d 1
72.l even 6 1 576.8.a.e 1
72.n even 6 1 192.8.a.o 1
72.p odd 6 1 192.8.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.8.a.a 1 9.c even 3 1
36.8.a.c 1 9.d odd 6 1
48.8.a.e 1 36.f odd 6 1
144.8.a.j 1 36.h even 6 1
192.8.a.g 1 72.p odd 6 1
192.8.a.o 1 72.n even 6 1
300.8.a.g 1 45.j even 6 1
300.8.d.c 2 45.k odd 12 2
324.8.e.a 2 3.b odd 2 1
324.8.e.a 2 9.d odd 6 1
324.8.e.f 2 1.a even 1 1 trivial
324.8.e.f 2 9.c even 3 1 inner
576.8.a.d 1 72.j odd 6 1
576.8.a.e 1 72.l even 6 1
588.8.a.d 1 63.l odd 6 1
588.8.i.a 2 63.k odd 6 1
588.8.i.a 2 63.t odd 6 1
588.8.i.h 2 63.g even 3 1
588.8.i.h 2 63.h even 3 1

Hecke kernels

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

\( T_{5}^{2} - 378T_{5} + 142884 \) Copy content Toggle raw display
\( T_{7}^{2} - 832T_{7} + 692224 \) 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
$5$ \( T^{2} - 378T + 142884 \) Copy content Toggle raw display
$7$ \( T^{2} - 832T + 692224 \) Copy content Toggle raw display
$11$ \( T^{2} - 2484 T + 6170256 \) Copy content Toggle raw display
$13$ \( T^{2} + 14870 T + 221116900 \) Copy content Toggle raw display
$17$ \( (T + 22302)^{2} \) Copy content Toggle raw display
$19$ \( (T + 16300)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 115128 T + 13254456384 \) Copy content Toggle raw display
$29$ \( T^{2} + 157086 T + 24676011396 \) Copy content Toggle raw display
$31$ \( T^{2} - 16456 T + 270799936 \) Copy content Toggle raw display
$37$ \( (T + 149266)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 241110 T + 58134032100 \) Copy content Toggle raw display
$43$ \( T^{2} - 443188 T + 196415603344 \) Copy content Toggle raw display
$47$ \( T^{2} + 922752 T + 851471253504 \) Copy content Toggle raw display
$53$ \( (T + 697626)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 870156 T + 757171464336 \) Copy content Toggle raw display
$61$ \( T^{2} + 2067062 T + 4272745311844 \) Copy content Toggle raw display
$67$ \( T^{2} - 1680748 T + 2824913839504 \) Copy content Toggle raw display
$71$ \( (T + 1070280)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2403334)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2301512 T + 5296957486144 \) Copy content Toggle raw display
$83$ \( T^{2} + 4708692 T + 22171780350864 \) Copy content Toggle raw display
$89$ \( (T - 4143690)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1622974 T + 2634044604676 \) Copy content Toggle raw display
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