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Properties

Label 2520.1.eq.b
Level $2520$
Weight $1$
Character orbit 2520.eq
Analytic conductor $1.258$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2520,1,Mod(899,2520)]
 
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([3, 3, 3, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.899");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2520.eq (of order \(6\), 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: \(1.25764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{24}^{10} q^{2} - \zeta_{24}^{8} q^{4} + \zeta_{24}^{4} q^{5} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{6} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{10} q^{2} - \zeta_{24}^{8} q^{4} + \zeta_{24}^{4} q^{5} + \zeta_{24}^{9} q^{7} + \zeta_{24}^{6} q^{8} - \zeta_{24}^{2} q^{10} + (\zeta_{24}^{3} + \zeta_{24}) q^{11} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{13} - \zeta_{24}^{7} q^{14} - \zeta_{24}^{4} q^{16} - \zeta_{24}^{10} q^{19} + q^{20} + (\zeta_{24}^{11} - \zeta_{24}) q^{22} + ( - \zeta_{24}^{8} + 1) q^{23} + \zeta_{24}^{8} q^{25} + ( - \zeta_{24}^{11} + \zeta_{24}^{9}) q^{26} + \zeta_{24}^{5} q^{28} + \zeta_{24}^{2} q^{32} - \zeta_{24} q^{35} + (\zeta_{24}^{11} - \zeta_{24}^{9}) q^{37} + \zeta_{24}^{8} q^{38} + \zeta_{24}^{10} q^{40} + (\zeta_{24}^{11} - \zeta_{24}) q^{41} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{44} + (\zeta_{24}^{10} + \zeta_{24}^{6}) q^{46} + (\zeta_{24}^{6} + \zeta_{24}^{2}) q^{47} - \zeta_{24}^{6} q^{49} - \zeta_{24}^{6} q^{50} + (\zeta_{24}^{9} - \zeta_{24}^{7}) q^{52} - \zeta_{24}^{2} q^{53} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{55} - \zeta_{24}^{3} q^{56} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{59} - q^{64} + ( - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{65} - \zeta_{24}^{11} q^{70} + ( - \zeta_{24}^{9} + \zeta_{24}^{7}) q^{74} - \zeta_{24}^{6} q^{76} + (\zeta_{24}^{10} - 1) q^{77} - \zeta_{24}^{8} q^{80} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{82} + (\zeta_{24}^{9} + \zeta_{24}^{7}) q^{88} + (\zeta_{24}^{7} + \zeta_{24}) q^{89} + ( - \zeta_{24}^{10} + \zeta_{24}^{8}) q^{91} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{92} + ( - \zeta_{24}^{4} - 1) q^{94} + \zeta_{24}^{2} q^{95} + \zeta_{24}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{5} - 4 q^{16} + 8 q^{20} + 12 q^{23} - 4 q^{25} - 4 q^{38} - 8 q^{64} - 8 q^{77} + 4 q^{80} - 4 q^{91} - 12 q^{94} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{24}^{4}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0 −0.866025 − 0.500000i
899.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 0.707107 − 0.707107i 1.00000i 0 −0.866025 − 0.500000i
899.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −0.707107 − 0.707107i − 1.00000i 0 0.866025 + 0.500000i
899.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 0.707107 + 0.707107i − 1.00000i 0 0.866025 + 0.500000i
1979.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −0.707107 − 0.707107i − 1.00000i 0 −0.866025 + 0.500000i
1979.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 0.707107 + 0.707107i − 1.00000i 0 −0.866025 + 0.500000i
1979.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −0.707107 + 0.707107i 1.00000i 0 0.866025 − 0.500000i
1979.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 0.707107 − 0.707107i 1.00000i 0 0.866025 − 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
15.d odd 2 1 inner
21.g even 6 1 inner
24.f even 2 1 inner
35.i odd 6 1 inner
56.m even 6 1 inner
840.ct odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.eq.b yes 8
3.b odd 2 1 2520.1.eq.a ✓ 8
5.b even 2 1 2520.1.eq.a ✓ 8
7.d odd 6 1 2520.1.eq.a ✓ 8
8.d odd 2 1 2520.1.eq.a ✓ 8
15.d odd 2 1 inner 2520.1.eq.b yes 8
21.g even 6 1 inner 2520.1.eq.b yes 8
24.f even 2 1 inner 2520.1.eq.b yes 8
35.i odd 6 1 inner 2520.1.eq.b yes 8
40.e odd 2 1 CM 2520.1.eq.b yes 8
56.m even 6 1 inner 2520.1.eq.b yes 8
105.p even 6 1 2520.1.eq.a ✓ 8
120.m even 2 1 2520.1.eq.a ✓ 8
168.be odd 6 1 2520.1.eq.a ✓ 8
280.ba even 6 1 2520.1.eq.a ✓ 8
840.ct odd 6 1 inner 2520.1.eq.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.eq.a ✓ 8 3.b odd 2 1
2520.1.eq.a ✓ 8 5.b even 2 1
2520.1.eq.a ✓ 8 7.d odd 6 1
2520.1.eq.a ✓ 8 8.d odd 2 1
2520.1.eq.a ✓ 8 105.p even 6 1
2520.1.eq.a ✓ 8 120.m even 2 1
2520.1.eq.a ✓ 8 168.be odd 6 1
2520.1.eq.a ✓ 8 280.ba even 6 1
2520.1.eq.b yes 8 1.a even 1 1 trivial
2520.1.eq.b yes 8 15.d odd 2 1 inner
2520.1.eq.b yes 8 21.g even 6 1 inner
2520.1.eq.b yes 8 24.f even 2 1 inner
2520.1.eq.b yes 8 35.i odd 6 1 inner
2520.1.eq.b yes 8 40.e odd 2 1 CM
2520.1.eq.b yes 8 56.m even 6 1 inner
2520.1.eq.b yes 8 840.ct odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} - 3T_{23} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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