Properties

Label 324.3.f.l
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{4} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{5} + (4 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{7} + (\beta_{3} - 9 \beta_{2} - 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{4} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{5} + (4 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{7} + (\beta_{3} - 9 \beta_{2} - 5) q^{8} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{10} + (7 \beta_{2} - 7) q^{11} - 16 \beta_{2} q^{13} + ( - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 10) q^{14} + ( - 10 \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 11) q^{16} + (8 \beta_{3} - 4) q^{17} + ( - 12 \beta_{3} + 12 \beta_{2} - 24 \beta_1 + 18) q^{19} + (4 \beta_{2} + 5 \beta_1 - 9) q^{20} + (7 \beta_{3} - 7 \beta_{2} - 7 \beta_1) q^{22} + ( - 10 \beta_{2} - 20) q^{23} + (12 \beta_{2} + 12) q^{25} + ( - 16 \beta_{3} + 16 \beta_{2} + 16) q^{26} + ( - 5 \beta_{3} + 5 \beta_{2} - 10 \beta_1 + 27) q^{28} + (14 \beta_{2} - 28 \beta_1 + 14) q^{29} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{31} + (11 \beta_{3} + 9 \beta_{2} + 11 \beta_1 + 18) q^{32} + ( - 8 \beta_{3} - 24 \beta_{2} - 4 \beta_1 - 20) q^{34} + (26 \beta_{2} + 13) q^{35} + 26 q^{37} + ( - 48 \beta_{2} + 18 \beta_1 + 30) q^{38} + (9 \beta_{3} + 11 \beta_{2} - 9 \beta_1) q^{40} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{41} + ( - 8 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{43} + ( - 21 \beta_{3} - 35 \beta_{2} - 7) q^{44} + ( - 10 \beta_{3} + 10 \beta_{2} - 20 \beta_1 + 30) q^{46} + ( - 2 \beta_{2} + 2) q^{47} + 10 \beta_{2} q^{49} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 - 24) q^{50} + (32 \beta_{3} + 32 \beta_{2} + 16 \beta_1 + 16) q^{52} + (38 \beta_{3} - 19) q^{53} + (14 \beta_{3} - 14 \beta_{2} + 28 \beta_1 - 21) q^{55} + ( - 20 \beta_{2} + 27 \beta_1 - 7) q^{56} + ( - 14 \beta_{3} - 98 \beta_{2} + 14 \beta_1) q^{58} + (44 \beta_{2} + 88) q^{59} + ( - 8 \beta_{2} - 8) q^{61} + ( - 3 \beta_{3} - 13 \beta_{2} - 5) q^{62} + (9 \beta_{3} - 9 \beta_{2} + 18 \beta_1 - 71) q^{64} + (16 \beta_{2} - 32 \beta_1 + 16) q^{65} + (20 \beta_{3} + 10 \beta_{2} - 20 \beta_1) q^{67} + ( - 20 \beta_{3} + 36 \beta_{2} - 20 \beta_1 + 72) q^{68} + (26 \beta_{3} - 26 \beta_{2} + 13 \beta_1 - 39) q^{70} + (72 \beta_{2} + 36) q^{71} - 19 q^{73} + (26 \beta_1 - 26) q^{74} + ( - 30 \beta_{3} + 102 \beta_{2} + 30 \beta_1) q^{76} + ( - 42 \beta_{3} + 21 \beta_{2} - 42 \beta_1 + 42) q^{77} + ( - 32 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 24) q^{79} + ( - 7 \beta_{3} - 65 \beta_{2} - 29) q^{80} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 10) q^{82} + (67 \beta_{2} - 67) q^{83} + 52 \beta_{2} q^{85} + (6 \beta_{3} + 10 \beta_{2} + 6 \beta_1 + 20) q^{86} + ( - 14 \beta_{3} + 98 \beta_{2} - 7 \beta_1 + 105) q^{88} + (44 \beta_{3} - 22) q^{89} + (32 \beta_{3} - 32 \beta_{2} + 64 \beta_1 - 48) q^{91} + ( - 40 \beta_{2} + 30 \beta_1 + 10) q^{92} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{94} + ( - 78 \beta_{2} - 156) q^{95} + ( - 119 \beta_{2} - 119) q^{97} + (10 \beta_{3} - 10 \beta_{2} - 10) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 5 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 5 q^{4} + 26 q^{10} - 42 q^{11} + 32 q^{13} - 39 q^{14} + 7 q^{16} - 39 q^{20} + 21 q^{22} - 60 q^{23} + 24 q^{25} + 78 q^{28} + 87 q^{32} - 52 q^{34} + 104 q^{37} + 234 q^{38} - 13 q^{40} + 60 q^{46} + 12 q^{47} - 20 q^{49} - 36 q^{50} + 80 q^{52} + 39 q^{56} + 182 q^{58} + 264 q^{59} - 16 q^{61} - 230 q^{64} + 156 q^{68} - 39 q^{70} - 76 q^{73} - 78 q^{74} - 234 q^{76} - 52 q^{82} - 402 q^{83} - 104 q^{85} + 78 q^{86} + 189 q^{88} + 150 q^{92} - 6 q^{94} - 468 q^{95} - 238 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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
−0.651388 + 1.12824i
1.15139 1.99426i
−0.651388 1.12824i
1.15139 + 1.99426i
−1.65139 + 1.12824i 0 1.45416 3.72631i −1.80278 3.12250i 0 5.40833 + 3.12250i 1.80278 + 7.79423i 0 6.50000 + 3.12250i
55.2 0.151388 1.99426i 0 −3.95416 0.603814i 1.80278 + 3.12250i 0 −5.40833 3.12250i −1.80278 + 7.79423i 0 6.50000 3.12250i
271.1 −1.65139 1.12824i 0 1.45416 + 3.72631i −1.80278 + 3.12250i 0 5.40833 3.12250i 1.80278 7.79423i 0 6.50000 3.12250i
271.2 0.151388 + 1.99426i 0 −3.95416 + 0.603814i 1.80278 3.12250i 0 −5.40833 + 3.12250i −1.80278 7.79423i 0 6.50000 + 3.12250i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

\( T_{5}^{4} + 13T_{5}^{2} + 169 \) Copy content Toggle raw display
\( T_{7}^{4} - 39T_{7}^{2} + 1521 \) Copy content Toggle raw display
\( T_{11}^{2} + 21T_{11} + 147 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + 7 T^{2} + 12 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} - 39T^{2} + 1521 \) Copy content Toggle raw display
$11$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 16 T + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 208)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1404)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 30 T + 300)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2548 T^{2} + \cdots + 6492304 \) Copy content Toggle raw display
$31$ \( T^{4} - 39T^{2} + 1521 \) Copy content Toggle raw display
$37$ \( (T - 26)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$43$ \( T^{4} - 156 T^{2} + 24336 \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4693)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 132 T + 5808)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 3900 T^{2} + \cdots + 15210000 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3888)^{2} \) Copy content Toggle raw display
$73$ \( (T + 19)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 2496 T^{2} + \cdots + 6230016 \) Copy content Toggle raw display
$83$ \( (T^{2} + 201 T + 13467)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6292)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 119 T + 14161)^{2} \) Copy content Toggle raw display
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