Properties

Label 2520.1.fw.a
Level $2520$
Weight $1$
Character orbit 2520.fw
Analytic conductor $1.258$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -56
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(349,2520)]
 
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, 3, 4, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.349");
 
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.fw (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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{2} q^{2} - \zeta_{24}^{9} q^{3} + \zeta_{24}^{4} q^{4} - \zeta_{24}^{3} q^{5} + \zeta_{24}^{11} q^{6} + \zeta_{24}^{2} q^{7} - \zeta_{24}^{6} q^{8} - \zeta_{24}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{2} q^{2} - \zeta_{24}^{9} q^{3} + \zeta_{24}^{4} q^{4} - \zeta_{24}^{3} q^{5} + \zeta_{24}^{11} q^{6} + \zeta_{24}^{2} q^{7} - \zeta_{24}^{6} q^{8} - \zeta_{24}^{6} q^{9} + \zeta_{24}^{5} q^{10} + \zeta_{24} q^{12} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{13} - \zeta_{24}^{4} q^{14} - q^{15} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{8} q^{18} + (\zeta_{24}^{11} - \zeta_{24}) q^{19} - \zeta_{24}^{7} q^{20} - \zeta_{24}^{11} q^{21} + ( - \zeta_{24}^{8} + 1) q^{23} - \zeta_{24}^{3} q^{24} + \zeta_{24}^{6} q^{25} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{26} - \zeta_{24}^{3} q^{27} + \zeta_{24}^{6} q^{28} + \zeta_{24}^{2} q^{30} - \zeta_{24}^{10} q^{32} - \zeta_{24}^{5} q^{35} - \zeta_{24}^{10} q^{36} + (\zeta_{24}^{3} + \zeta_{24}) q^{38} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{39} + \zeta_{24}^{9} q^{40} - \zeta_{24} q^{42} + \zeta_{24}^{9} q^{45} + (\zeta_{24}^{10} - \zeta_{24}^{2}) q^{46} + \zeta_{24}^{5} q^{48} + \zeta_{24}^{4} q^{49} - \zeta_{24}^{8} q^{50} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{52} + \zeta_{24}^{5} q^{54} - \zeta_{24}^{8} q^{56} + (\zeta_{24}^{10} + \zeta_{24}^{8}) q^{57} + (\zeta_{24}^{7} + \zeta_{24}) q^{59} - \zeta_{24}^{4} q^{60} + ( - \zeta_{24}^{9} - \zeta_{24}^{7}) q^{61} - \zeta_{24}^{8} q^{63} - q^{64} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{65} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{69} + \zeta_{24}^{7} q^{70} + (\zeta_{24}^{10} - \zeta_{24}^{2}) q^{71} - q^{72} + \zeta_{24}^{3} q^{75} + ( - \zeta_{24}^{5} - \zeta_{24}^{3}) q^{76} + (\zeta_{24}^{6} - 1) q^{78} + ( - \zeta_{24}^{10} - \zeta_{24}^{6}) q^{79} - \zeta_{24}^{11} q^{80} - q^{81} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{83} + \zeta_{24}^{3} q^{84} - \zeta_{24}^{11} q^{90} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{91} + (\zeta_{24}^{4} + 1) q^{92} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{95} - \zeta_{24}^{7} q^{96} - \zeta_{24}^{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 4 q^{14} - 8 q^{15} - 4 q^{16} - 4 q^{18} + 12 q^{23} - 4 q^{39} + 4 q^{49} + 4 q^{50} + 4 q^{56} - 4 q^{57} - 4 q^{60} + 4 q^{63} - 8 q^{64} - 4 q^{65} - 8 q^{72} - 8 q^{78} - 8 q^{81} + 12 q^{92} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
−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 −0.707107 + 0.707107i 0.500000 + 0.866025i 0.707107 + 0.707107i 0.965926 0.258819i 0.866025 + 0.500000i 1.00000i 1.00000i −0.258819 0.965926i
349.2 −0.866025 0.500000i 0.707107 0.707107i 0.500000 + 0.866025i −0.707107 0.707107i −0.965926 + 0.258819i 0.866025 + 0.500000i 1.00000i 1.00000i 0.258819 + 0.965926i
349.3 0.866025 + 0.500000i −0.707107 0.707107i 0.500000 + 0.866025i 0.707107 0.707107i −0.258819 0.965926i −0.866025 0.500000i 1.00000i 1.00000i 0.965926 0.258819i
349.4 0.866025 + 0.500000i 0.707107 + 0.707107i 0.500000 + 0.866025i −0.707107 + 0.707107i 0.258819 + 0.965926i −0.866025 0.500000i 1.00000i 1.00000i −0.965926 + 0.258819i
2029.1 −0.866025 + 0.500000i −0.707107 0.707107i 0.500000 0.866025i 0.707107 0.707107i 0.965926 + 0.258819i 0.866025 0.500000i 1.00000i 1.00000i −0.258819 + 0.965926i
2029.2 −0.866025 + 0.500000i 0.707107 + 0.707107i 0.500000 0.866025i −0.707107 + 0.707107i −0.965926 0.258819i 0.866025 0.500000i 1.00000i 1.00000i 0.258819 0.965926i
2029.3 0.866025 0.500000i −0.707107 + 0.707107i 0.500000 0.866025i 0.707107 + 0.707107i −0.258819 + 0.965926i −0.866025 + 0.500000i 1.00000i 1.00000i 0.965926 + 0.258819i
2029.4 0.866025 0.500000i 0.707107 0.707107i 0.500000 0.866025i −0.707107 0.707107i 0.258819 0.965926i −0.866025 + 0.500000i 1.00000i 1.00000i −0.965926 0.258819i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
45.j even 6 1 inner
315.bg odd 6 1 inner
360.bk even 6 1 inner
2520.fw 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.fw.a 8
5.b even 2 1 2520.1.fw.b yes 8
7.b odd 2 1 inner 2520.1.fw.a 8
8.b even 2 1 inner 2520.1.fw.a 8
9.c even 3 1 2520.1.fw.b yes 8
35.c odd 2 1 2520.1.fw.b yes 8
40.f even 2 1 2520.1.fw.b yes 8
45.j even 6 1 inner 2520.1.fw.a 8
56.h odd 2 1 CM 2520.1.fw.a 8
63.l odd 6 1 2520.1.fw.b yes 8
72.n even 6 1 2520.1.fw.b yes 8
280.c odd 2 1 2520.1.fw.b yes 8
315.bg odd 6 1 inner 2520.1.fw.a 8
360.bk even 6 1 inner 2520.1.fw.a 8
504.bn odd 6 1 2520.1.fw.b yes 8
2520.fw odd 6 1 inner 2520.1.fw.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.fw.a 8 1.a even 1 1 trivial
2520.1.fw.a 8 7.b odd 2 1 inner
2520.1.fw.a 8 8.b even 2 1 inner
2520.1.fw.a 8 45.j even 6 1 inner
2520.1.fw.a 8 56.h odd 2 1 CM
2520.1.fw.a 8 315.bg odd 6 1 inner
2520.1.fw.a 8 360.bk even 6 1 inner
2520.1.fw.a 8 2520.fw odd 6 1 inner
2520.1.fw.b yes 8 5.b even 2 1
2520.1.fw.b yes 8 9.c even 3 1
2520.1.fw.b yes 8 35.c odd 2 1
2520.1.fw.b yes 8 40.f even 2 1
2520.1.fw.b yes 8 63.l odd 6 1
2520.1.fw.b yes 8 72.n even 6 1
2520.1.fw.b yes 8 280.c odd 2 1
2520.1.fw.b yes 8 504.bn odd 6 1

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
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{2} + 1)^{2} \) 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
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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