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Properties

Label 2520.2.bi.k
Level $2520$
Weight $2$
Character orbit 2520.bi
Analytic conductor $20.122$
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] = [2520,2,Mod(361,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (of order \(3\), degree \(2\), 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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{11} + 2 q^{13} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{17} + ( - 4 \beta_{3} - 4 \beta_1) q^{19} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + ( - 2 \beta_{3} + 5) q^{29} + (2 \beta_{2} - 2 \beta_1 + 2) q^{31} + ( - \beta_{3} - 2 \beta_1 - 1) q^{35} + (4 \beta_{3} + 4 \beta_1) q^{37} + (2 \beta_{3} - 3) q^{41} + (7 \beta_{3} + 3) q^{43} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{47} + (2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{49} - 4 \beta_1 q^{53} + ( - 2 \beta_{3} - 2) q^{55} + ( - 4 \beta_{2} - 4) q^{59} + (4 \beta_{3} + \beta_{2} + 4 \beta_1) q^{61} - 2 \beta_{2} q^{65} + (3 \beta_{2} + 7 \beta_1 + 3) q^{67} + 12 q^{71} + (2 \beta_{2} - 4 \beta_1 + 2) q^{73} + (6 \beta_{2} - 4 \beta_1 + 8) q^{77} - 4 \beta_{2} q^{79} + (3 \beta_{3} + 9) q^{83} + (4 \beta_{3} - 2) q^{85} + ( - 4 \beta_{3} + 11 \beta_{2} - 4 \beta_1) q^{89} + ( - 4 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{91}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 2 q^{7} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 14 q^{23} - 2 q^{25} + 20 q^{29} + 4 q^{31} - 4 q^{35} - 12 q^{41} + 12 q^{43} + 12 q^{47} + 10 q^{49} - 8 q^{55} - 8 q^{59} - 2 q^{61} + 4 q^{65} + 6 q^{67} + 48 q^{71} + 4 q^{73} + 20 q^{77} + 8 q^{79} + 36 q^{83} - 8 q^{85} - 22 q^{89} - 4 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
0 0 0 0.500000 + 0.866025i 0 −2.62132 − 0.358719i 0 0 0
361.2 0 0 0 0.500000 + 0.866025i 0 1.62132 + 2.09077i 0 0 0
1801.1 0 0 0 0.500000 − 0.866025i 0 −2.62132 + 0.358719i 0 0 0
1801.2 0 0 0 0.500000 − 0.866025i 0 1.62132 − 2.09077i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.2.bi.k 4
3.b odd 2 1 280.2.q.d ✓ 4
7.c even 3 1 inner 2520.2.bi.k 4
12.b even 2 1 560.2.q.j 4
15.d odd 2 1 1400.2.q.h 4
15.e even 4 2 1400.2.bh.g 8
21.c even 2 1 1960.2.q.q 4
21.g even 6 1 1960.2.a.t 2
21.g even 6 1 1960.2.q.q 4
21.h odd 6 1 280.2.q.d ✓ 4
21.h odd 6 1 1960.2.a.p 2
84.j odd 6 1 3920.2.a.bp 2
84.n even 6 1 560.2.q.j 4
84.n even 6 1 3920.2.a.bz 2
105.o odd 6 1 1400.2.q.h 4
105.o odd 6 1 9800.2.a.bz 2
105.p even 6 1 9800.2.a.br 2
105.x even 12 2 1400.2.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d ✓ 4 3.b odd 2 1
280.2.q.d ✓ 4 21.h odd 6 1
560.2.q.j 4 12.b even 2 1
560.2.q.j 4 84.n even 6 1
1400.2.q.h 4 15.d odd 2 1
1400.2.q.h 4 105.o odd 6 1
1400.2.bh.g 8 15.e even 4 2
1400.2.bh.g 8 105.x even 12 2
1960.2.a.p 2 21.h odd 6 1
1960.2.a.t 2 21.g even 6 1
1960.2.q.q 4 21.c even 2 1
1960.2.q.q 4 21.g even 6 1
2520.2.bi.k 4 1.a even 1 1 trivial
2520.2.bi.k 4 7.c even 3 1 inner
3920.2.a.bp 2 84.j odd 6 1
3920.2.a.bz 2 84.n even 6 1
9800.2.a.br 2 105.p even 6 1
9800.2.a.bz 2 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11}^{4} + 4T_{11}^{3} + 20T_{11}^{2} - 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} - 2 \) 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
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$19$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$23$ \( T^{4} + 14 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 17)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 89)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} + \cdots + 7921 \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 18 T + 63)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 22 T^{3} + \cdots + 7921 \) Copy content Toggle raw display
$97$ \( (T - 6)^{4} \) Copy content Toggle raw display
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