Properties

Label 2520.1.hf.b
Level $2520$
Weight $1$
Character orbit 2520.hf
Analytic conductor $1.258$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
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(293,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 10, 9, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.293");
 
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.hf (of order \(12\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{48}^{14} q^{2} + \zeta_{48}^{15} q^{3} - \zeta_{48}^{4} q^{4} + \zeta_{48}^{23} q^{5} - \zeta_{48}^{5} q^{6} - \zeta_{48}^{2} q^{7} - \zeta_{48}^{18} q^{8} - \zeta_{48}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{48}^{14} q^{2} + \zeta_{48}^{15} q^{3} - \zeta_{48}^{4} q^{4} + \zeta_{48}^{23} q^{5} - \zeta_{48}^{5} q^{6} - \zeta_{48}^{2} q^{7} - \zeta_{48}^{18} q^{8} - \zeta_{48}^{6} q^{9} - \zeta_{48}^{13} q^{10} - \zeta_{48}^{19} q^{12} + ( - \zeta_{48}^{19} + \zeta_{48}) q^{13} - \zeta_{48}^{16} q^{14} - \zeta_{48}^{14} q^{15} + \zeta_{48}^{8} q^{16} - \zeta_{48}^{20} q^{18} + (\zeta_{48}^{19} + \zeta_{48}^{5}) q^{19} + \zeta_{48}^{3} q^{20} - \zeta_{48}^{17} q^{21} + ( - \zeta_{48}^{12} - \zeta_{48}^{8}) q^{23} + \zeta_{48}^{9} q^{24} - \zeta_{48}^{22} q^{25} + (\zeta_{48}^{15} + \zeta_{48}^{9}) q^{26} - \zeta_{48}^{21} q^{27} + \zeta_{48}^{6} q^{28} + \zeta_{48}^{4} q^{30} + \zeta_{48}^{22} q^{32} + \zeta_{48} q^{35} + \zeta_{48}^{10} q^{36} + (\zeta_{48}^{19} - \zeta_{48}^{9}) q^{38} + (\zeta_{48}^{16} + \zeta_{48}^{10}) q^{39} + \zeta_{48}^{17} q^{40} + \zeta_{48}^{7} q^{42} + \zeta_{48}^{5} q^{45} + ( - \zeta_{48}^{22} + \zeta_{48}^{2}) q^{46} + \zeta_{48}^{23} q^{48} + \zeta_{48}^{4} q^{49} + \zeta_{48}^{12} q^{50} + (\zeta_{48}^{23} - \zeta_{48}^{5}) q^{52} + \zeta_{48}^{11} q^{54} + \zeta_{48}^{20} q^{56} + (\zeta_{48}^{20} - \zeta_{48}^{10}) q^{57} + (\zeta_{48}^{7} - \zeta_{48}) q^{59} + \zeta_{48}^{18} q^{60} + (\zeta_{48}^{13} + \zeta_{48}^{3}) q^{61} + \zeta_{48}^{8} q^{63} - \zeta_{48}^{12} q^{64} + (\zeta_{48}^{18} - 1) q^{65} + ( - \zeta_{48}^{23} + \zeta_{48}^{3}) q^{69} + \zeta_{48}^{15} q^{70} + (\zeta_{48}^{22} + \zeta_{48}^{2}) q^{71} - q^{72} + \zeta_{48}^{13} q^{75} + ( - \zeta_{48}^{23} - \zeta_{48}^{9}) q^{76} + ( - \zeta_{48}^{6} - 1) q^{78} + ( - \zeta_{48}^{10} + \zeta_{48}^{6}) q^{79} - \zeta_{48}^{7} q^{80} + \zeta_{48}^{12} q^{81} + ( - \zeta_{48}^{17} + \zeta_{48}^{11}) q^{83} + \zeta_{48}^{21} q^{84} + \zeta_{48}^{19} q^{90} + (\zeta_{48}^{21} - \zeta_{48}^{3}) q^{91} + (\zeta_{48}^{16} + \zeta_{48}^{12}) q^{92} + ( - \zeta_{48}^{18} - \zeta_{48}^{4}) q^{95} - \zeta_{48}^{13} q^{96} + \zeta_{48}^{18} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{14} + 8 q^{16} - 8 q^{23} - 8 q^{39} + 8 q^{63} - 16 q^{65} - 16 q^{72} - 16 q^{78} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(-\zeta_{48}^{16}\) \(1\) \(-1\) \(-1\) \(-\zeta_{48}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.991445 + 0.130526i
−0.991445 0.130526i
0.130526 0.991445i
−0.130526 + 0.991445i
0.793353 0.608761i
−0.793353 + 0.608761i
−0.608761 0.793353i
0.608761 + 0.793353i
0.793353 + 0.608761i
−0.793353 0.608761i
−0.608761 + 0.793353i
0.608761 0.793353i
0.991445 0.130526i
−0.991445 + 0.130526i
0.130526 + 0.991445i
−0.130526 0.991445i
−0.258819 + 0.965926i −0.382683 + 0.923880i −0.866025 0.500000i −0.991445 + 0.130526i −0.793353 0.608761i −0.965926 0.258819i 0.707107 0.707107i −0.707107 0.707107i 0.130526 0.991445i
293.2 −0.258819 + 0.965926i 0.382683 0.923880i −0.866025 0.500000i 0.991445 0.130526i 0.793353 + 0.608761i −0.965926 0.258819i 0.707107 0.707107i −0.707107 0.707107i −0.130526 + 0.991445i
293.3 0.258819 0.965926i −0.923880 0.382683i −0.866025 0.500000i −0.130526 0.991445i −0.608761 + 0.793353i 0.965926 + 0.258819i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.991445 0.130526i
293.4 0.258819 0.965926i 0.923880 + 0.382683i −0.866025 0.500000i 0.130526 + 0.991445i 0.608761 0.793353i 0.965926 + 0.258819i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.991445 + 0.130526i
797.1 −0.965926 0.258819i −0.923880 + 0.382683i 0.866025 + 0.500000i −0.793353 0.608761i 0.991445 0.130526i −0.258819 + 0.965926i −0.707107 0.707107i 0.707107 0.707107i 0.608761 + 0.793353i
797.2 −0.965926 0.258819i 0.923880 0.382683i 0.866025 + 0.500000i 0.793353 + 0.608761i −0.991445 + 0.130526i −0.258819 + 0.965926i −0.707107 0.707107i 0.707107 0.707107i −0.608761 0.793353i
797.3 0.965926 + 0.258819i −0.382683 0.923880i 0.866025 + 0.500000i 0.608761 0.793353i −0.130526 0.991445i 0.258819 0.965926i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.793353 0.608761i
797.4 0.965926 + 0.258819i 0.382683 + 0.923880i 0.866025 + 0.500000i −0.608761 + 0.793353i 0.130526 + 0.991445i 0.258819 0.965926i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.793353 + 0.608761i
1973.1 −0.965926 + 0.258819i −0.923880 0.382683i 0.866025 0.500000i −0.793353 + 0.608761i 0.991445 + 0.130526i −0.258819 0.965926i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.608761 0.793353i
1973.2 −0.965926 + 0.258819i 0.923880 + 0.382683i 0.866025 0.500000i 0.793353 0.608761i −0.991445 0.130526i −0.258819 0.965926i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.608761 + 0.793353i
1973.3 0.965926 0.258819i −0.382683 + 0.923880i 0.866025 0.500000i 0.608761 + 0.793353i −0.130526 + 0.991445i 0.258819 + 0.965926i 0.707107 0.707107i −0.707107 0.707107i 0.793353 + 0.608761i
1973.4 0.965926 0.258819i 0.382683 0.923880i 0.866025 0.500000i −0.608761 0.793353i 0.130526 0.991445i 0.258819 + 0.965926i 0.707107 0.707107i −0.707107 0.707107i −0.793353 0.608761i
2477.1 −0.258819 0.965926i −0.382683 0.923880i −0.866025 + 0.500000i −0.991445 0.130526i −0.793353 + 0.608761i −0.965926 + 0.258819i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.130526 + 0.991445i
2477.2 −0.258819 0.965926i 0.382683 + 0.923880i −0.866025 + 0.500000i 0.991445 + 0.130526i 0.793353 0.608761i −0.965926 + 0.258819i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.130526 0.991445i
2477.3 0.258819 + 0.965926i −0.923880 + 0.382683i −0.866025 + 0.500000i −0.130526 + 0.991445i −0.608761 0.793353i 0.965926 0.258819i −0.707107 0.707107i 0.707107 0.707107i −0.991445 + 0.130526i
2477.4 0.258819 + 0.965926i 0.923880 0.382683i −0.866025 + 0.500000i 0.130526 0.991445i 0.608761 + 0.793353i 0.965926 0.258819i −0.707107 0.707107i 0.707107 0.707107i 0.991445 0.130526i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.4
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.l even 12 1 inner
315.cf odd 12 1 inner
360.br even 12 1 inner
2520.hf odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.hf.b yes 16
5.c odd 4 1 2520.1.hf.a 16
7.b odd 2 1 inner 2520.1.hf.b yes 16
8.b even 2 1 inner 2520.1.hf.b yes 16
9.d odd 6 1 2520.1.hf.a 16
35.f even 4 1 2520.1.hf.a 16
40.i odd 4 1 2520.1.hf.a 16
45.l even 12 1 inner 2520.1.hf.b yes 16
56.h odd 2 1 CM 2520.1.hf.b yes 16
63.o even 6 1 2520.1.hf.a 16
72.j odd 6 1 2520.1.hf.a 16
280.s even 4 1 2520.1.hf.a 16
315.cf odd 12 1 inner 2520.1.hf.b yes 16
360.br even 12 1 inner 2520.1.hf.b yes 16
504.cc even 6 1 2520.1.hf.a 16
2520.hf odd 12 1 inner 2520.1.hf.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.hf.a 16 5.c odd 4 1
2520.1.hf.a 16 9.d odd 6 1
2520.1.hf.a 16 35.f even 4 1
2520.1.hf.a 16 40.i odd 4 1
2520.1.hf.a 16 63.o even 6 1
2520.1.hf.a 16 72.j odd 6 1
2520.1.hf.a 16 280.s even 4 1
2520.1.hf.a 16 504.cc even 6 1
2520.1.hf.b yes 16 1.a even 1 1 trivial
2520.1.hf.b yes 16 7.b odd 2 1 inner
2520.1.hf.b yes 16 8.b even 2 1 inner
2520.1.hf.b yes 16 45.l even 12 1 inner
2520.1.hf.b yes 16 56.h odd 2 1 CM
2520.1.hf.b yes 16 315.cf odd 12 1 inner
2520.1.hf.b yes 16 360.br even 12 1 inner
2520.1.hf.b yes 16 2520.hf odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{4} + 2T_{23}^{3} + 5T_{23}^{2} + 4T_{23} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$7$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 12 T^{12} + 140 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{6} + 20 T^{4} + 16 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 8 T^{14} + 44 T^{12} + 128 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{2} + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 12 T^{12} + 140 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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