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Properties

Label 2520.2.bi.e
Level $2520$
Weight $2$
Character orbit 2520.bi
Analytic conductor $20.122$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2520,2,Mod(361,2520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-1,0,4,0,0,0,-1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + (\zeta_{6} - 1) q^{11} - 3 q^{13} + (2 \zeta_{6} - 2) q^{17} + 5 \zeta_{6} q^{19} + 7 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} + \cdots + 6 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 4 q^{7} - q^{11} - 6 q^{13} - 2 q^{17} + 5 q^{19} + 7 q^{23} - q^{25} + 12 q^{29} - 4 q^{31} - 5 q^{35} + 5 q^{37} + 10 q^{41} + 12 q^{43} - 9 q^{47} + 2 q^{49} + 11 q^{53} + 2 q^{55} + 8 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 −0.500000 − 0.866025i 0 2.00000 − 1.73205i 0 0 0
1801.1 0 0 0 −0.500000 + 0.866025i 0 2.00000 + 1.73205i 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.e 2
3.b odd 2 1 280.2.q.c ✓ 2
7.c even 3 1 inner 2520.2.bi.e 2
12.b even 2 1 560.2.q.c 2
15.d odd 2 1 1400.2.q.a 2
15.e even 4 2 1400.2.bh.e 4
21.c even 2 1 1960.2.q.c 2
21.g even 6 1 1960.2.a.m 1
21.g even 6 1 1960.2.q.c 2
21.h odd 6 1 280.2.q.c ✓ 2
21.h odd 6 1 1960.2.a.a 1
84.j odd 6 1 3920.2.a.i 1
84.n even 6 1 560.2.q.c 2
84.n even 6 1 3920.2.a.bf 1
105.o odd 6 1 1400.2.q.a 2
105.o odd 6 1 9800.2.a.bi 1
105.p even 6 1 9800.2.a.g 1
105.x even 12 2 1400.2.bh.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.c ✓ 2 3.b odd 2 1
280.2.q.c ✓ 2 21.h odd 6 1
560.2.q.c 2 12.b even 2 1
560.2.q.c 2 84.n even 6 1
1400.2.q.a 2 15.d odd 2 1
1400.2.q.a 2 105.o odd 6 1
1400.2.bh.e 4 15.e even 4 2
1400.2.bh.e 4 105.x even 12 2
1960.2.a.a 1 21.h odd 6 1
1960.2.a.m 1 21.g even 6 1
1960.2.q.c 2 21.c even 2 1
1960.2.q.c 2 21.g even 6 1
2520.2.bi.e 2 1.a even 1 1 trivial
2520.2.bi.e 2 7.c even 3 1 inner
3920.2.a.i 1 84.j odd 6 1
3920.2.a.bf 1 84.n even 6 1
9800.2.a.g 1 105.p even 6 1
9800.2.a.bi 1 105.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 \(S_{2}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11}^{2} + T_{11} + 1 \) Copy content Toggle raw display
\( T_{13} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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