Properties

Label 2520.1.h.e
Level $2520$
Weight $1$
Character orbit 2520.h
Analytic conductor $1.258$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.25764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.11200.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} - q^{4} + \zeta_{8}^{3} q^{5} + \zeta_{8}^{2} q^{7} -\zeta_{8}^{2} q^{8} +O(q^{10})\) \( q + \zeta_{8}^{2} q^{2} - q^{4} + \zeta_{8}^{3} q^{5} + \zeta_{8}^{2} q^{7} -\zeta_{8}^{2} q^{8} -\zeta_{8} q^{10} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{13} - q^{14} + q^{16} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{19} -\zeta_{8}^{3} q^{20} -\zeta_{8}^{2} q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{26} -\zeta_{8}^{2} q^{28} + \zeta_{8}^{2} q^{32} -\zeta_{8} q^{35} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{38} + \zeta_{8} q^{40} - q^{49} + q^{50} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{52} + q^{56} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{61} - q^{64} + ( -1 - \zeta_{8}^{2} ) q^{65} -\zeta_{8}^{3} q^{70} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{76} + \zeta_{8}^{3} q^{80} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{83} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{91} + ( 1 - \zeta_{8}^{2} ) q^{95} -\zeta_{8}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{14} + 4q^{16} - 4q^{49} + 4q^{50} + 4q^{56} - 4q^{64} - 4q^{65} + 4q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
1.00000i 0 −1.00000 −0.707107 0.707107i 0 1.00000i 1.00000i 0 −0.707107 + 0.707107i
1189.2 1.00000i 0 −1.00000 0.707107 + 0.707107i 0 1.00000i 1.00000i 0 0.707107 0.707107i
1189.3 1.00000i 0 −1.00000 −0.707107 + 0.707107i 0 1.00000i 1.00000i 0 −0.707107 0.707107i
1189.4 1.00000i 0 −1.00000 0.707107 0.707107i 0 1.00000i 1.00000i 0 0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
280.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.1.h.e 4
3.b odd 2 1 280.1.c.a 4
5.b even 2 1 inner 2520.1.h.e 4
7.b odd 2 1 inner 2520.1.h.e 4
8.b even 2 1 inner 2520.1.h.e 4
12.b even 2 1 1120.1.c.a 4
15.d odd 2 1 280.1.c.a 4
15.e even 4 1 1400.1.m.b 2
15.e even 4 1 1400.1.m.e 2
21.c even 2 1 280.1.c.a 4
21.g even 6 2 1960.1.bk.a 8
21.h odd 6 2 1960.1.bk.a 8
24.f even 2 1 1120.1.c.a 4
24.h odd 2 1 280.1.c.a 4
35.c odd 2 1 inner 2520.1.h.e 4
40.f even 2 1 inner 2520.1.h.e 4
56.h odd 2 1 CM 2520.1.h.e 4
60.h even 2 1 1120.1.c.a 4
84.h odd 2 1 1120.1.c.a 4
105.g even 2 1 280.1.c.a 4
105.k odd 4 1 1400.1.m.b 2
105.k odd 4 1 1400.1.m.e 2
105.o odd 6 2 1960.1.bk.a 8
105.p even 6 2 1960.1.bk.a 8
120.i odd 2 1 280.1.c.a 4
120.m even 2 1 1120.1.c.a 4
120.w even 4 1 1400.1.m.b 2
120.w even 4 1 1400.1.m.e 2
168.e odd 2 1 1120.1.c.a 4
168.i even 2 1 280.1.c.a 4
168.s odd 6 2 1960.1.bk.a 8
168.ba even 6 2 1960.1.bk.a 8
280.c odd 2 1 inner 2520.1.h.e 4
420.o odd 2 1 1120.1.c.a 4
840.b odd 2 1 1120.1.c.a 4
840.u even 2 1 280.1.c.a 4
840.bp odd 4 1 1400.1.m.b 2
840.bp odd 4 1 1400.1.m.e 2
840.cb even 6 2 1960.1.bk.a 8
840.cg odd 6 2 1960.1.bk.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 3.b odd 2 1
280.1.c.a 4 15.d odd 2 1
280.1.c.a 4 21.c even 2 1
280.1.c.a 4 24.h odd 2 1
280.1.c.a 4 105.g even 2 1
280.1.c.a 4 120.i odd 2 1
280.1.c.a 4 168.i even 2 1
280.1.c.a 4 840.u even 2 1
1120.1.c.a 4 12.b even 2 1
1120.1.c.a 4 24.f even 2 1
1120.1.c.a 4 60.h even 2 1
1120.1.c.a 4 84.h odd 2 1
1120.1.c.a 4 120.m even 2 1
1120.1.c.a 4 168.e odd 2 1
1120.1.c.a 4 420.o odd 2 1
1120.1.c.a 4 840.b odd 2 1
1400.1.m.b 2 15.e even 4 1
1400.1.m.b 2 105.k odd 4 1
1400.1.m.b 2 120.w even 4 1
1400.1.m.b 2 840.bp odd 4 1
1400.1.m.e 2 15.e even 4 1
1400.1.m.e 2 105.k odd 4 1
1400.1.m.e 2 120.w even 4 1
1400.1.m.e 2 840.bp odd 4 1
1960.1.bk.a 8 21.g even 6 2
1960.1.bk.a 8 21.h odd 6 2
1960.1.bk.a 8 105.o odd 6 2
1960.1.bk.a 8 105.p even 6 2
1960.1.bk.a 8 168.s odd 6 2
1960.1.bk.a 8 168.ba even 6 2
1960.1.bk.a 8 840.cb even 6 2
1960.1.bk.a 8 840.cg odd 6 2
2520.1.h.e 4 1.a even 1 1 trivial
2520.1.h.e 4 5.b even 2 1 inner
2520.1.h.e 4 7.b odd 2 1 inner
2520.1.h.e 4 8.b even 2 1 inner
2520.1.h.e 4 35.c odd 2 1 inner
2520.1.h.e 4 40.f even 2 1 inner
2520.1.h.e 4 56.h odd 2 1 CM
2520.1.h.e 4 280.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2520, [\chi])\):

\( T_{13}^{2} + 2 \)
\( T_{53} \)
\( T_{59}^{2} - 2 \)

Hecke characteristic polynomials

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