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Properties

Label 2160.3.c.n
Level $2160$
Weight $3$
Character orbit 2160.c
Analytic conductor $58.856$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: 8.0.345436422246400.13
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 81x^{6} + 1780x^{4} + 8100x^{2} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{3} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{6} q^{11} - \beta_{3} q^{13} + ( - \beta_{4} + \beta_{2}) q^{17} + ( - \beta_{5} + 5) q^{19} + ( - \beta_{7} + 2 \beta_{4} + 2 \beta_{2}) q^{23} + ( - \beta_{5} + \beta_{3} - \beta_1 + 2) q^{25} + 3 \beta_{6} q^{29} + (\beta_{5} - 6) q^{31} + ( - 3 \beta_{6} + 2 \beta_{4} - \beta_{2}) q^{35} + ( - \beta_{3} - 5 \beta_1) q^{37} + (\beta_{7} + 2 \beta_{6} + \cdots + 5 \beta_{2}) q^{41}+ \cdots + (31 \beta_{3} + 11 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{19} + 12 q^{25} - 44 q^{31} - 108 q^{49} - 40 q^{55} - 116 q^{61} - 312 q^{79} + 160 q^{85} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 81x^{6} + 1780x^{4} + 8100x^{2} + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 76\nu^{5} + 1325\nu^{3} - 3850\nu ) / 1500 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{7} - 15\nu^{6} + 841\nu^{5} - 1165\nu^{4} + 16030\nu^{3} - 24150\nu^{2} + 29600\nu - 77000 ) / 6000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 79\nu^{5} - 1598\nu^{3} - 3920\nu ) / 400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 75\nu^{6} + 841\nu^{5} - 5625\nu^{4} + 16030\nu^{3} - 104550\nu^{2} + 29600\nu - 207000 ) / 6000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 243\nu^{4} - 5040\nu^{2} - 12100 ) / 100 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + 157\nu^{5} + 3205\nu^{3} + 9350\nu ) / 300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\nu^{7} + 75\nu^{6} + 1682\nu^{5} + 5725\nu^{4} + 32060\nu^{3} + 112650\nu^{2} + 59200\nu + 296000 ) / 3000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 5\beta_{6} + \beta_{4} - 10\beta_{3} + 5\beta_{2} - 15\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{7} + 5\beta_{5} + 27\beta_{4} - 75\beta_{2} - 610 ) / 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -41\beta_{7} + 295\beta_{6} - 41\beta_{4} + 610\beta_{3} - 205\beta_{2} + 465\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -204\beta_{7} - 135\beta_{5} - 309\beta_{4} + 1125\beta_{2} + 7570 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1141\beta_{7} - 13895\beta_{6} + 1141\beta_{4} - 33610\beta_{3} + 5705\beta_{2} - 18465\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 29412\beta_{7} + 23405\beta_{5} + 29727\beta_{4} - 147375\beta_{2} - 935710 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -28541\beta_{7} + 645895\beta_{6} - 28541\beta_{4} + 1707610\beta_{3} - 142705\beta_{2} + 774465\beta_1 ) / 30 \) 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
1.47294i
− 1.47294i
1.84653i
− 1.84653i
− 5.41555i
5.41555i
− 6.78915i
6.78915i
0 0 0 −4.98930 − 0.326908i 0 9.26209i 0 0 0
1889.2 0 0 0 −4.98930 + 0.326908i 0 − 9.26209i 0 0 0
1889.3 0 0 0 −1.26762 − 4.83664i 0 6.26209i 0 0 0
1889.4 0 0 0 −1.26762 + 4.83664i 0 − 6.26209i 0 0 0
1889.5 0 0 0 1.26762 − 4.83664i 0 − 6.26209i 0 0 0
1889.6 0 0 0 1.26762 + 4.83664i 0 6.26209i 0 0 0
1889.7 0 0 0 4.98930 − 0.326908i 0 − 9.26209i 0 0 0
1889.8 0 0 0 4.98930 + 0.326908i 0 9.26209i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
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.n 8
3.b odd 2 1 inner 2160.3.c.n 8
4.b odd 2 1 540.3.b.c ✓ 8
5.b even 2 1 inner 2160.3.c.n 8
12.b even 2 1 540.3.b.c ✓ 8
15.d odd 2 1 inner 2160.3.c.n 8
20.d odd 2 1 540.3.b.c ✓ 8
20.e even 4 1 2700.3.g.o 4
20.e even 4 1 2700.3.g.p 4
36.f odd 6 2 1620.3.t.e 16
36.h even 6 2 1620.3.t.e 16
60.h even 2 1 540.3.b.c ✓ 8
60.l odd 4 1 2700.3.g.o 4
60.l odd 4 1 2700.3.g.p 4
180.n even 6 2 1620.3.t.e 16
180.p odd 6 2 1620.3.t.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.c ✓ 8 4.b odd 2 1
540.3.b.c ✓ 8 12.b even 2 1
540.3.b.c ✓ 8 20.d odd 2 1
540.3.b.c ✓ 8 60.h even 2 1
1620.3.t.e 16 36.f odd 6 2
1620.3.t.e 16 36.h even 6 2
1620.3.t.e 16 180.n even 6 2
1620.3.t.e 16 180.p odd 6 2
2160.3.c.n 8 1.a even 1 1 trivial
2160.3.c.n 8 3.b odd 2 1 inner
2160.3.c.n 8 5.b even 2 1 inner
2160.3.c.n 8 15.d odd 2 1 inner
2700.3.g.o 4 20.e even 4 1
2700.3.g.o 4 60.l odd 4 1
2700.3.g.p 4 20.e even 4 1
2700.3.g.p 4 60.l odd 4 1

Hecke kernels

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

\( T_{7}^{4} + 125T_{7}^{2} + 3364 \) Copy content Toggle raw display
\( T_{17}^{4} - 265T_{17}^{2} + 4000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 125 T^{2} + 3364)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 235 T^{2} + 250)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 265 T^{2} + 4000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 9 T - 522)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2635 T^{2} + 1722250)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2115 T^{2} + 20250)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 11 T - 512)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3053 T^{2} + 2208196)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2890 T^{2} + 1600000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1745 T^{2} + 565504)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 6625 T^{2} + 2500000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 5310 T^{2} + 6561000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 15460 T^{2} + 59536000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 29 T - 332)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 3773 T^{2} + 1267876)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 18990 T^{2} + 50625000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12953 T^{2} + 11999296)^{2} \) Copy content Toggle raw display
$79$ \( (T + 39)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10090 T^{2} + 24964000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 15460 T^{2} + 59536000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 26285 T^{2} + 2067844)^{2} \) Copy content Toggle raw display
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