Properties

Label 2520.1.hf.a
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(1.25764383184\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
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

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.608761 0.793353i
0.608761 + 0.793353i
0.793353 0.608761i
−0.793353 + 0.608761i
0.130526 0.991445i
−0.130526 + 0.991445i
0.991445 + 0.130526i
−0.991445 0.130526i
0.130526 + 0.991445i
−0.130526 0.991445i
0.991445 0.130526i
−0.991445 + 0.130526i
−0.608761 + 0.793353i
0.608761 0.793353i
0.793353 + 0.608761i
−0.793353 0.608761i
−0.258819 + 0.965926i −0.382683 0.923880i −0.866025 0.500000i −0.923880 0.382683i 0.991445 0.130526i 0.965926 + 0.258819i 0.707107 0.707107i −0.707107 + 0.707107i 0.608761 0.793353i
293.2 −0.258819 + 0.965926i 0.382683 + 0.923880i −0.866025 0.500000i 0.923880 + 0.382683i −0.991445 + 0.130526i 0.965926 + 0.258819i 0.707107 0.707107i −0.707107 + 0.707107i −0.608761 + 0.793353i
293.3 0.258819 0.965926i −0.923880 + 0.382683i −0.866025 0.500000i 0.382683 0.923880i 0.130526 + 0.991445i −0.965926 0.258819i −0.707107 + 0.707107i 0.707107 0.707107i −0.793353 0.608761i
293.4 0.258819 0.965926i 0.923880 0.382683i −0.866025 0.500000i −0.382683 + 0.923880i −0.130526 0.991445i −0.965926 0.258819i −0.707107 + 0.707107i 0.707107 0.707107i 0.793353 + 0.608761i
797.1 −0.965926 0.258819i −0.923880 0.382683i 0.866025 + 0.500000i 0.382683 + 0.923880i 0.793353 + 0.608761i 0.258819 0.965926i −0.707107 0.707107i 0.707107 + 0.707107i −0.130526 0.991445i
797.2 −0.965926 0.258819i 0.923880 + 0.382683i 0.866025 + 0.500000i −0.382683 0.923880i −0.793353 0.608761i 0.258819 0.965926i −0.707107 0.707107i 0.707107 + 0.707107i 0.130526 + 0.991445i
797.3 0.965926 + 0.258819i −0.382683 + 0.923880i 0.866025 + 0.500000i −0.923880 + 0.382683i −0.608761 + 0.793353i −0.258819 + 0.965926i 0.707107 + 0.707107i −0.707107 0.707107i −0.991445 + 0.130526i
797.4 0.965926 + 0.258819i 0.382683 0.923880i 0.866025 + 0.500000i 0.923880 0.382683i 0.608761 0.793353i −0.258819 + 0.965926i 0.707107 + 0.707107i −0.707107 0.707107i 0.991445 0.130526i
1973.1 −0.965926 + 0.258819i −0.923880 + 0.382683i 0.866025 0.500000i 0.382683 0.923880i 0.793353 0.608761i 0.258819 + 0.965926i −0.707107 + 0.707107i 0.707107 0.707107i −0.130526 + 0.991445i
1973.2 −0.965926 + 0.258819i 0.923880 0.382683i 0.866025 0.500000i −0.382683 + 0.923880i −0.793353 + 0.608761i 0.258819 + 0.965926i −0.707107 + 0.707107i 0.707107 0.707107i 0.130526 0.991445i
1973.3 0.965926 0.258819i −0.382683 0.923880i 0.866025 0.500000i −0.923880 0.382683i −0.608761 0.793353i −0.258819 0.965926i 0.707107 0.707107i −0.707107 + 0.707107i −0.991445 0.130526i
1973.4 0.965926 0.258819i 0.382683 + 0.923880i 0.866025 0.500000i 0.923880 + 0.382683i 0.608761 + 0.793353i −0.258819 0.965926i 0.707107 0.707107i −0.707107 + 0.707107i 0.991445 + 0.130526i
2477.1 −0.258819 0.965926i −0.382683 + 0.923880i −0.866025 + 0.500000i −0.923880 + 0.382683i 0.991445 + 0.130526i 0.965926 0.258819i 0.707107 + 0.707107i −0.707107 0.707107i 0.608761 + 0.793353i
2477.2 −0.258819 0.965926i 0.382683 0.923880i −0.866025 + 0.500000i 0.923880 0.382683i −0.991445 0.130526i 0.965926 0.258819i 0.707107 + 0.707107i −0.707107 0.707107i −0.608761 0.793353i
2477.3 0.258819 + 0.965926i −0.923880 0.382683i −0.866025 + 0.500000i 0.382683 + 0.923880i 0.130526 0.991445i −0.965926 + 0.258819i −0.707107 0.707107i 0.707107 + 0.707107i −0.793353 + 0.608761i
2477.4 0.258819 + 0.965926i 0.923880 + 0.382683i −0.866025 + 0.500000i −0.382683 0.923880i −0.130526 + 0.991445i −0.965926 + 0.258819i −0.707107 0.707107i 0.707107 + 0.707107i 0.793353 0.608761i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2477.4
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.a 16
5.c odd 4 1 2520.1.hf.b yes 16
7.b odd 2 1 inner 2520.1.hf.a 16
8.b even 2 1 inner 2520.1.hf.a 16
9.d odd 6 1 2520.1.hf.b yes 16
35.f even 4 1 2520.1.hf.b yes 16
40.i odd 4 1 2520.1.hf.b yes 16
45.l even 12 1 inner 2520.1.hf.a 16
56.h odd 2 1 CM 2520.1.hf.a 16
63.o even 6 1 2520.1.hf.b yes 16
72.j odd 6 1 2520.1.hf.b yes 16
280.s even 4 1 2520.1.hf.b yes 16
315.cf odd 12 1 inner 2520.1.hf.a 16
360.br even 12 1 inner 2520.1.hf.a 16
504.cc even 6 1 2520.1.hf.b yes 16
2520.hf odd 12 1 inner 2520.1.hf.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.1.hf.a 16 1.a even 1 1 trivial
2520.1.hf.a 16 7.b odd 2 1 inner
2520.1.hf.a 16 8.b even 2 1 inner
2520.1.hf.a 16 45.l even 12 1 inner
2520.1.hf.a 16 56.h odd 2 1 CM
2520.1.hf.a 16 315.cf odd 12 1 inner
2520.1.hf.a 16 360.br even 12 1 inner
2520.1.hf.a 16 2520.hf odd 12 1 inner
2520.1.hf.b yes 16 5.c odd 4 1
2520.1.hf.b yes 16 9.d odd 6 1
2520.1.hf.b yes 16 35.f even 4 1
2520.1.hf.b yes 16 40.i odd 4 1
2520.1.hf.b yes 16 63.o even 6 1
2520.1.hf.b yes 16 72.j odd 6 1
2520.1.hf.b yes 16 280.s even 4 1
2520.1.hf.b yes 16 504.cc even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( ( 1 + T^{8} )^{2} \)
$5$ \( ( 1 + T^{8} )^{2} \)
$7$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$11$ \( T^{16} \)
$13$ \( 16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16} \)
$17$ \( T^{16} \)
$19$ \( ( 1 + 16 T^{2} + 20 T^{4} + 8 T^{6} + T^{8} )^{2} \)
$23$ \( ( 1 + 2 T + 5 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( ( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$61$ \( 1 + 16 T^{2} + 236 T^{4} + 304 T^{6} + 271 T^{8} + 128 T^{10} + 44 T^{12} + 8 T^{14} + T^{16} \)
$67$ \( T^{16} \)
$71$ \( ( 1 + 4 T^{2} + T^{4} )^{4} \)
$73$ \( T^{16} \)
$79$ \( ( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} )^{2} \)
$83$ \( 16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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