Properties

Label 324.4.a.c
Level $324$
Weight $4$
Character orbit 324.a
Self dual yes
Analytic conductor $19.117$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,4,Mod(1,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

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: yes
Analytic conductor: \(19.1166188419\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1509.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 2) q^{5} + (\beta_{2} - \beta_1 + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 2) q^{5} + (\beta_{2} - \beta_1 + 2) q^{7} + (2 \beta_{2} + 3 \beta_1 - 17) q^{11} + (5 \beta_{2} + 4 \beta_1 - 4) q^{13} + (\beta_{2} - 6 \beta_1 - 37) q^{17} + ( - 7 \beta_{2} - 2 \beta_1 + 5) q^{19} + ( - 5 \beta_{2} - 3 \beta_1 - 70) q^{23} + ( - 3 \beta_{2} - 6 \beta_1 + 1) q^{25} + (5 \beta_{2} - 152) q^{29} + ( - 15 \beta_{2} - 3 \beta_1 - 16) q^{31} + (\beta_{2} + 15 \beta_1 - 184) q^{35} + (10 \beta_{2} + 8 \beta_1 - 16) q^{37} + (14 \beta_{2} + 12 \beta_1 - 299) q^{41} + (27 \beta_1 - 43) q^{43} + ( - 39 \beta_{2} - 21 \beta_1 - 174) q^{47} + (13 \beta_{2} - 22 \beta_1 + 75) q^{49} + ( - 22 \beta_{2} - 368) q^{53} + (33 \beta_{2} - 15 \beta_1 - 36) q^{55} + (34 \beta_{2} - 15 \beta_1 - 151) q^{59} + (3 \beta_{2} - 12 \beta_1 + 134) q^{61} + (37 \beta_{2} - 6 \beta_1 - 370) q^{65} + ( - 24 \beta_{2} - 3 \beta_1 + 71) q^{67} + (52 \beta_{2} + 12 \beta_1 + 20) q^{71} + ( - 21 \beta_{2} + 30 \beta_1 + 125) q^{73} + ( - 65 \beta_{2} + 12 \beta_1 - 376) q^{77} + (23 \beta_{2} - 5 \beta_1 - 184) q^{79} + (15 \beta_{2} + 33 \beta_1 + 204) q^{83} + (30 \beta_{2} + 60 \beta_1 - 396) q^{85} + ( - 44 \beta_{2} - 60 \beta_1 - 154) q^{89} + ( - 75 \beta_{2} - 33 \beta_1 - 44) q^{91} + ( - 44 \beta_{2} - 24 \beta_1 + 728) q^{95} + ( - 70 \beta_{2} - 56 \beta_1 - 31) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53} - 108 q^{55} - 453 q^{59} + 402 q^{61} - 1110 q^{65} + 213 q^{67} + 60 q^{71} + 375 q^{73} - 1128 q^{77} - 552 q^{79} + 612 q^{83} - 1188 q^{85} - 462 q^{89} - 132 q^{91} + 2184 q^{95} - 93 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{2} - 3\nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 30 ) / 6 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.47735
2.92542
0.551929
0 0 0 −13.8439 0 30.7080 0 0 0
1.2 0 0 0 −4.89803 0 −10.6545 0 0 0
1.3 0 0 0 12.7419 0 −14.0535 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.a.c 3
3.b odd 2 1 324.4.a.d 3
4.b odd 2 1 1296.4.a.v 3
9.c even 3 2 36.4.e.a 6
9.d odd 6 2 108.4.e.a 6
12.b even 2 1 1296.4.a.w 3
36.f odd 6 2 144.4.i.d 6
36.h even 6 2 432.4.i.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a 6 9.c even 3 2
108.4.e.a 6 9.d odd 6 2
144.4.i.d 6 36.f odd 6 2
324.4.a.c 3 1.a even 1 1 trivial
324.4.a.d 3 3.b odd 2 1
432.4.i.d 6 36.h even 6 2
1296.4.a.v 3 4.b odd 2 1
1296.4.a.w 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 6T_{5}^{2} - 171T_{5} - 864 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(324))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} + \cdots - 864 \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} + \cdots - 4598 \) Copy content Toggle raw display
$11$ \( T^{3} + 51 T^{2} + \cdots - 66231 \) Copy content Toggle raw display
$13$ \( T^{3} + 12 T^{2} + \cdots - 64466 \) Copy content Toggle raw display
$17$ \( T^{3} + 111 T^{2} + \cdots - 577476 \) Copy content Toggle raw display
$19$ \( T^{3} - 15 T^{2} + \cdots - 216368 \) Copy content Toggle raw display
$23$ \( T^{3} + 210 T^{2} + \cdots + 2322 \) Copy content Toggle raw display
$29$ \( T^{3} + 456 T^{2} + \cdots + 2879658 \) Copy content Toggle raw display
$31$ \( T^{3} + 48 T^{2} + \cdots - 3054788 \) Copy content Toggle raw display
$37$ \( T^{3} + 48 T^{2} + \cdots - 682352 \) Copy content Toggle raw display
$41$ \( T^{3} + 897 T^{2} + \cdots + 11796543 \) Copy content Toggle raw display
$43$ \( T^{3} + 129 T^{2} + \cdots - 1425149 \) Copy content Toggle raw display
$47$ \( T^{3} + 522 T^{2} + \cdots - 64558782 \) Copy content Toggle raw display
$53$ \( T^{3} + 1104 T^{2} + \cdots + 11853648 \) Copy content Toggle raw display
$59$ \( T^{3} + 453 T^{2} + \cdots - 96892713 \) Copy content Toggle raw display
$61$ \( T^{3} - 402 T^{2} + \cdots + 1209736 \) Copy content Toggle raw display
$67$ \( T^{3} - 213 T^{2} + \cdots - 3095063 \) Copy content Toggle raw display
$71$ \( T^{3} - 60 T^{2} + \cdots + 113211648 \) Copy content Toggle raw display
$73$ \( T^{3} - 375 T^{2} + \cdots + 158369284 \) Copy content Toggle raw display
$79$ \( T^{3} + 552 T^{2} + \cdots - 17848772 \) Copy content Toggle raw display
$83$ \( T^{3} - 612 T^{2} + \cdots + 3478788 \) Copy content Toggle raw display
$89$ \( T^{3} + 462 T^{2} + \cdots + 170122248 \) Copy content Toggle raw display
$97$ \( T^{3} + 93 T^{2} + \cdots + 86400523 \) Copy content Toggle raw display
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