Properties

Label 1120.1.c.a
Level $1120$
Weight $1$
Character orbit 1120.c
Analytic conductor $0.559$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -56
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1120.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{3} q^{5} -\zeta_{8}^{2} q^{7} - q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{3} q^{5} -\zeta_{8}^{2} q^{7} - q^{9} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( 1 + \zeta_{8}^{2} ) q^{15} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{19} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{21} -\zeta_{8}^{2} q^{25} + \zeta_{8} q^{35} -2 q^{39} -\zeta_{8}^{3} q^{45} - q^{49} + 2 \zeta_{8}^{2} q^{57} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{59} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{61} + \zeta_{8}^{2} q^{63} + ( 1 + \zeta_{8}^{2} ) q^{65} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{75} - q^{81} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{91} + ( 1 - \zeta_{8}^{2} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 4q^{15} - 8q^{39} - 4q^{49} + 4q^{65} - 4q^{81} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 1.41421i 0 −0.707107 + 0.707107i 0 1.00000i 0 −1.00000 0
209.2 0 1.41421i 0 0.707107 + 0.707107i 0 1.00000i 0 −1.00000 0
209.3 0 1.41421i 0 −0.707107 0.707107i 0 1.00000i 0 −1.00000 0
209.4 0 1.41421i 0 0.707107 0.707107i 0 1.00000i 0 −1.00000 0
\(n\): 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 1120.1.c.a 4
4.b odd 2 1 280.1.c.a 4
5.b even 2 1 inner 1120.1.c.a 4
7.b odd 2 1 inner 1120.1.c.a 4
8.b even 2 1 inner 1120.1.c.a 4
8.d odd 2 1 280.1.c.a 4
12.b even 2 1 2520.1.h.e 4
20.d odd 2 1 280.1.c.a 4
20.e even 4 1 1400.1.m.b 2
20.e even 4 1 1400.1.m.e 2
24.f even 2 1 2520.1.h.e 4
28.d even 2 1 280.1.c.a 4
28.f even 6 2 1960.1.bk.a 8
28.g odd 6 2 1960.1.bk.a 8
35.c odd 2 1 inner 1120.1.c.a 4
40.e odd 2 1 280.1.c.a 4
40.f even 2 1 inner 1120.1.c.a 4
40.k even 4 1 1400.1.m.b 2
40.k even 4 1 1400.1.m.e 2
56.e even 2 1 280.1.c.a 4
56.h odd 2 1 CM 1120.1.c.a 4
56.k odd 6 2 1960.1.bk.a 8
56.m even 6 2 1960.1.bk.a 8
60.h even 2 1 2520.1.h.e 4
84.h odd 2 1 2520.1.h.e 4
120.m even 2 1 2520.1.h.e 4
140.c even 2 1 280.1.c.a 4
140.j odd 4 1 1400.1.m.b 2
140.j odd 4 1 1400.1.m.e 2
140.p odd 6 2 1960.1.bk.a 8
140.s even 6 2 1960.1.bk.a 8
168.e odd 2 1 2520.1.h.e 4
280.c odd 2 1 inner 1120.1.c.a 4
280.n even 2 1 280.1.c.a 4
280.y odd 4 1 1400.1.m.b 2
280.y odd 4 1 1400.1.m.e 2
280.ba even 6 2 1960.1.bk.a 8
280.bi odd 6 2 1960.1.bk.a 8
420.o odd 2 1 2520.1.h.e 4
840.b odd 2 1 2520.1.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 4.b odd 2 1
280.1.c.a 4 8.d odd 2 1
280.1.c.a 4 20.d odd 2 1
280.1.c.a 4 28.d even 2 1
280.1.c.a 4 40.e odd 2 1
280.1.c.a 4 56.e even 2 1
280.1.c.a 4 140.c even 2 1
280.1.c.a 4 280.n even 2 1
1120.1.c.a 4 1.a even 1 1 trivial
1120.1.c.a 4 5.b even 2 1 inner
1120.1.c.a 4 7.b odd 2 1 inner
1120.1.c.a 4 8.b even 2 1 inner
1120.1.c.a 4 35.c odd 2 1 inner
1120.1.c.a 4 40.f even 2 1 inner
1120.1.c.a 4 56.h odd 2 1 CM
1120.1.c.a 4 280.c odd 2 1 inner
1400.1.m.b 2 20.e even 4 1
1400.1.m.b 2 40.k even 4 1
1400.1.m.b 2 140.j odd 4 1
1400.1.m.b 2 280.y odd 4 1
1400.1.m.e 2 20.e even 4 1
1400.1.m.e 2 40.k even 4 1
1400.1.m.e 2 140.j odd 4 1
1400.1.m.e 2 280.y odd 4 1
1960.1.bk.a 8 28.f even 6 2
1960.1.bk.a 8 28.g odd 6 2
1960.1.bk.a 8 56.k odd 6 2
1960.1.bk.a 8 56.m even 6 2
1960.1.bk.a 8 140.p odd 6 2
1960.1.bk.a 8 140.s even 6 2
1960.1.bk.a 8 280.ba even 6 2
1960.1.bk.a 8 280.bi odd 6 2
2520.1.h.e 4 12.b even 2 1
2520.1.h.e 4 24.f even 2 1
2520.1.h.e 4 60.h even 2 1
2520.1.h.e 4 84.h odd 2 1
2520.1.h.e 4 120.m even 2 1
2520.1.h.e 4 168.e odd 2 1
2520.1.h.e 4 420.o odd 2 1
2520.1.h.e 4 840.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1120, [\chi])\).

Hecke characteristic polynomials

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