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Properties

Label 2160.3.c.l
Level $2160$
Weight $3$
Character orbit 2160.c
Analytic conductor $58.856$
Analytic rank $0$
Dimension $4$
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] = [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: \(\Q(\sqrt{-11}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 540)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{5} - \beta_{2} q^{7} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{11} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{13} + ( - \beta_{3} - \beta_1 + 1) q^{17} + ( - 3 \beta_{3} - 3 \beta_1 - 3) q^{19} + ( - \beta_{3} - \beta_1 + 7) q^{23} + (5 \beta_{2} + 3 \beta_1 + 3) q^{25} + ( - 3 \beta_{3} + 9 \beta_{2} + 3 \beta_1) q^{29} + ( - 6 \beta_{3} - 6 \beta_1 - 7) q^{31} + (5 \beta_{3} - 10) q^{35} + (6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{37} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{41} + ( - 9 \beta_{3} + \beta_{2} + 9 \beta_1) q^{43} + ( - 4 \beta_{3} - 4 \beta_1 - 14) q^{47} + (3 \beta_{3} + 3 \beta_1 + 21) q^{49} + ( - 6 \beta_{3} - 6 \beta_1 - 51) q^{53} + ( - 15 \beta_{3} + 10 \beta_{2} + \cdots - 14) q^{55}+ \cdots + (3 \beta_{3} - 28 \beta_{2} - 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{5} + 6 q^{17} - 6 q^{19} + 30 q^{23} + 9 q^{25} - 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} - 192 q^{53} - 47 q^{55} + 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} + 288 q^{83} - 100 q^{85} - 84 q^{91} - 318 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 15x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 19\nu + 14 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 19\nu + 14 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 19\beta_{2} - 11\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
− 3.83776i
3.83776i
− 0.521137i
0.521137i
0 0 0 −2.86421 − 4.09833i 0 − 7.15439i 0 0 0
1889.2 0 0 0 −2.86421 + 4.09833i 0 7.15439i 0 0 0
1889.3 0 0 0 4.36421 − 2.44002i 0 2.79549i 0 0 0
1889.4 0 0 0 4.36421 + 2.44002i 0 − 2.79549i 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.l 4
3.b odd 2 1 2160.3.c.h 4
4.b odd 2 1 540.3.b.b yes 4
5.b even 2 1 2160.3.c.h 4
12.b even 2 1 540.3.b.a ✓ 4
15.d odd 2 1 inner 2160.3.c.l 4
20.d odd 2 1 540.3.b.a ✓ 4
20.e even 4 2 2700.3.g.s 8
36.f odd 6 2 1620.3.t.a 8
36.h even 6 2 1620.3.t.d 8
60.h even 2 1 540.3.b.b yes 4
60.l odd 4 2 2700.3.g.s 8
180.n even 6 2 1620.3.t.a 8
180.p odd 6 2 1620.3.t.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.a ✓ 4 12.b even 2 1
540.3.b.a ✓ 4 20.d odd 2 1
540.3.b.b yes 4 4.b odd 2 1
540.3.b.b yes 4 60.h even 2 1
1620.3.t.a 8 36.f odd 6 2
1620.3.t.a 8 180.n even 6 2
1620.3.t.d 8 36.h even 6 2
1620.3.t.d 8 180.p odd 6 2
2160.3.c.h 4 3.b odd 2 1
2160.3.c.h 4 5.b even 2 1
2160.3.c.l 4 1.a even 1 1 trivial
2160.3.c.l 4 15.d odd 2 1 inner
2700.3.g.s 8 20.e even 4 2
2700.3.g.s 8 60.l odd 4 2

Hecke kernels

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

\( T_{7}^{4} + 59T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{17}^{2} - 3T_{17} - 50 \) 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
$5$ \( T^{4} - 3 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 59T^{2} + 400 \) Copy content Toggle raw display
$11$ \( T^{4} + 355T^{2} + 8464 \) Copy content Toggle raw display
$13$ \( T^{4} + 540T^{2} + 5184 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T - 50)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T - 468)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 15 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 1865)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1216)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 6620 T^{2} + 9684544 \) Copy content Toggle raw display
$47$ \( (T^{2} + 24 T - 692)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 96 T + 423)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6880 T^{2} + 4129024 \) Copy content Toggle raw display
$61$ \( (T^{2} - 19 T - 380)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 11372 T^{2} + 541696 \) Copy content Toggle raw display
$71$ \( T^{4} + 16956 T^{2} + 68558400 \) Copy content Toggle raw display
$73$ \( T^{4} + 11795 T^{2} + 6091024 \) Copy content Toggle raw display
$79$ \( (T^{2} - 3 T - 4230)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 144 T - 41)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24700 T^{2} + 19430464 \) Copy content Toggle raw display
$97$ \( T^{4} + 39515 T^{2} + 266603584 \) Copy content Toggle raw display
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