Properties

Label 2160.3.c.m
Level $2160$
Weight $3$
Character orbit 2160.c
Analytic conductor $58.856$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,3,Mod(1889,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.1889");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2160.c (of order \(2\), degree \(1\), 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: \(58.8557371018\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.31744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 14x^{2} + 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 3) q^{5} + (2 \beta_{3} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 3) q^{5} + (2 \beta_{3} + \beta_1) q^{7} + (3 \beta_{3} - 2 \beta_1) q^{11} + 3 \beta_{3} q^{13} + (7 \beta_{2} - 3) q^{17} + (3 \beta_{2} + 3) q^{19} + ( - 14 \beta_{2} - 15) q^{23} + (2 \beta_{3} + 12 \beta_{2} + 4 \beta_1 - 3) q^{25} + 9 \beta_1 q^{29} + (15 \beta_{2} + 17) q^{31} + (9 \beta_{3} - 10 \beta_{2} + \cdots - 18) q^{35}+ \cdots + ( - 22 \beta_{3} + 10 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{5} - 12 q^{17} + 12 q^{19} - 60 q^{23} - 12 q^{25} + 68 q^{31} - 72 q^{35} + 240 q^{47} - 180 q^{49} + 204 q^{53} + 88 q^{55} - 196 q^{61} - 24 q^{65} - 312 q^{77} - 180 q^{79} - 108 q^{83} + 20 q^{85} - 456 q^{91} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 14x^{2} + 31 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 13\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{3} + 13\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
3.35300i
3.35300i
1.66053i
1.66053i
0 0 0 1.58579 4.74186i 0 0.813575i 0 0 0
1889.2 0 0 0 1.58579 + 4.74186i 0 0.813575i 0 0 0
1889.3 0 0 0 4.41421 2.34834i 0 13.6872i 0 0 0
1889.4 0 0 0 4.41421 + 2.34834i 0 13.6872i 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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.3.c.m 4
3.b odd 2 1 2160.3.c.g 4
4.b odd 2 1 270.3.b.d yes 4
5.b even 2 1 2160.3.c.g 4
12.b even 2 1 270.3.b.a 4
15.d odd 2 1 inner 2160.3.c.m 4
20.d odd 2 1 270.3.b.a 4
20.e even 4 2 1350.3.d.o 8
36.f odd 6 2 810.3.j.a 8
36.h even 6 2 810.3.j.f 8
60.h even 2 1 270.3.b.d yes 4
60.l odd 4 2 1350.3.d.o 8
180.n even 6 2 810.3.j.a 8
180.p odd 6 2 810.3.j.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.a 4 12.b even 2 1
270.3.b.a 4 20.d odd 2 1
270.3.b.d yes 4 4.b odd 2 1
270.3.b.d yes 4 60.h even 2 1
810.3.j.a 8 36.f odd 6 2
810.3.j.a 8 180.n even 6 2
810.3.j.f 8 36.h even 6 2
810.3.j.f 8 180.p odd 6 2
1350.3.d.o 8 20.e even 4 2
1350.3.d.o 8 60.l odd 4 2
2160.3.c.g 4 3.b odd 2 1
2160.3.c.g 4 5.b even 2 1
2160.3.c.m 4 1.a even 1 1 trivial
2160.3.c.m 4 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{4} + 188T_{7}^{2} + 124 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 89 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 12 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 188T^{2} + 124 \) Copy content Toggle raw display
$11$ \( T^{4} + 388 T^{2} + 35836 \) Copy content Toggle raw display
$13$ \( T^{4} + 324 T^{2} + 10044 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 89)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 9)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 30 T - 167)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2268 T^{2} + 813564 \) Copy content Toggle raw display
$31$ \( (T^{2} - 34 T - 161)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 5408 T^{2} + 2293504 \) Copy content Toggle raw display
$41$ \( T^{4} + 1372 T^{2} + 65596 \) Copy content Toggle raw display
$43$ \( T^{4} + 1724 T^{2} + 65596 \) Copy content Toggle raw display
$47$ \( (T^{2} - 120 T + 3208)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 102 T + 2439)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3076 T^{2} + 2327356 \) Copy content Toggle raw display
$61$ \( (T^{2} + 98 T + 1249)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 24464 T^{2} + 131041216 \) Copy content Toggle raw display
$71$ \( T^{4} + 15876 T^{2} + 53524476 \) Copy content Toggle raw display
$73$ \( T^{4} + 8036 T^{2} + 35836 \) Copy content Toggle raw display
$79$ \( (T^{2} + 90 T - 5913)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 54 T - 5999)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 23548 T^{2} + 87702844 \) Copy content Toggle raw display
$97$ \( T^{4} + 18464 T^{2} + 84193024 \) Copy content Toggle raw display
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