Properties

Label 1120.1.by.a
Level $1120$
Weight $1$
Character orbit 1120.by
Analytic conductor $0.559$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1120.by (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1960.1
Artin image $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{5} - q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{5} - q^{7} + \zeta_{6}^{2} q^{9} -\zeta_{6} q^{11} - q^{13} + \zeta_{6}^{2} q^{19} + \zeta_{6}^{2} q^{23} -\zeta_{6} q^{25} -\zeta_{6}^{2} q^{35} -\zeta_{6}^{2} q^{37} - q^{41} -\zeta_{6} q^{45} + \zeta_{6}^{2} q^{47} + q^{49} + \zeta_{6} q^{53} + q^{55} + 2 \zeta_{6} q^{59} -\zeta_{6}^{2} q^{63} -\zeta_{6}^{2} q^{65} + \zeta_{6} q^{77} -\zeta_{6} q^{81} + 2 \zeta_{6}^{2} q^{89} + q^{91} -\zeta_{6} q^{95} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} - 2q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{5} - 2q^{7} - q^{9} - q^{11} - 2q^{13} - q^{19} - q^{23} - q^{25} + q^{35} + q^{37} - 2q^{41} - q^{45} - q^{47} + 2q^{49} + q^{53} + 2q^{55} + 2q^{59} + q^{63} + q^{65} + q^{77} - q^{81} - 2q^{89} + 2q^{91} - q^{95} + 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0
879.1 0 0 0 −0.500000 0.866025i 0 −1.00000 0 −0.500000 0.866025i 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
7.c even 3 1 inner
280.bi odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.1.by.a 2
4.b odd 2 1 280.1.bi.a 2
5.b even 2 1 1120.1.by.b 2
7.c even 3 1 inner 1120.1.by.a 2
8.b even 2 1 280.1.bi.b yes 2
8.d odd 2 1 1120.1.by.b 2
12.b even 2 1 2520.1.ef.b 2
20.d odd 2 1 280.1.bi.b yes 2
20.e even 4 2 1400.1.ba.a 4
24.h odd 2 1 2520.1.ef.a 2
28.d even 2 1 1960.1.bi.a 2
28.f even 6 1 1960.1.i.c 1
28.f even 6 1 1960.1.bi.a 2
28.g odd 6 1 280.1.bi.a 2
28.g odd 6 1 1960.1.i.d 1
35.j even 6 1 1120.1.by.b 2
40.e odd 2 1 CM 1120.1.by.a 2
40.f even 2 1 280.1.bi.a 2
40.i odd 4 2 1400.1.ba.a 4
56.h odd 2 1 1960.1.bi.b 2
56.j odd 6 1 1960.1.i.b 1
56.j odd 6 1 1960.1.bi.b 2
56.k odd 6 1 1120.1.by.b 2
56.p even 6 1 280.1.bi.b yes 2
56.p even 6 1 1960.1.i.a 1
60.h even 2 1 2520.1.ef.a 2
84.n even 6 1 2520.1.ef.b 2
120.i odd 2 1 2520.1.ef.b 2
140.c even 2 1 1960.1.bi.b 2
140.p odd 6 1 280.1.bi.b yes 2
140.p odd 6 1 1960.1.i.a 1
140.s even 6 1 1960.1.i.b 1
140.s even 6 1 1960.1.bi.b 2
140.w even 12 2 1400.1.ba.a 4
168.s odd 6 1 2520.1.ef.a 2
280.c odd 2 1 1960.1.bi.a 2
280.bf even 6 1 280.1.bi.a 2
280.bf even 6 1 1960.1.i.d 1
280.bi odd 6 1 inner 1120.1.by.a 2
280.bk odd 6 1 1960.1.i.c 1
280.bk odd 6 1 1960.1.bi.a 2
280.bt odd 12 2 1400.1.ba.a 4
420.ba even 6 1 2520.1.ef.a 2
840.cg odd 6 1 2520.1.ef.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 4.b odd 2 1
280.1.bi.a 2 28.g odd 6 1
280.1.bi.a 2 40.f even 2 1
280.1.bi.a 2 280.bf even 6 1
280.1.bi.b yes 2 8.b even 2 1
280.1.bi.b yes 2 20.d odd 2 1
280.1.bi.b yes 2 56.p even 6 1
280.1.bi.b yes 2 140.p odd 6 1
1120.1.by.a 2 1.a even 1 1 trivial
1120.1.by.a 2 7.c even 3 1 inner
1120.1.by.a 2 40.e odd 2 1 CM
1120.1.by.a 2 280.bi odd 6 1 inner
1120.1.by.b 2 5.b even 2 1
1120.1.by.b 2 8.d odd 2 1
1120.1.by.b 2 35.j even 6 1
1120.1.by.b 2 56.k odd 6 1
1400.1.ba.a 4 20.e even 4 2
1400.1.ba.a 4 40.i odd 4 2
1400.1.ba.a 4 140.w even 12 2
1400.1.ba.a 4 280.bt odd 12 2
1960.1.i.a 1 56.p even 6 1
1960.1.i.a 1 140.p odd 6 1
1960.1.i.b 1 56.j odd 6 1
1960.1.i.b 1 140.s even 6 1
1960.1.i.c 1 28.f even 6 1
1960.1.i.c 1 280.bk odd 6 1
1960.1.i.d 1 28.g odd 6 1
1960.1.i.d 1 280.bf even 6 1
1960.1.bi.a 2 28.d even 2 1
1960.1.bi.a 2 28.f even 6 1
1960.1.bi.a 2 280.c odd 2 1
1960.1.bi.a 2 280.bk odd 6 1
1960.1.bi.b 2 56.h odd 2 1
1960.1.bi.b 2 56.j odd 6 1
1960.1.bi.b 2 140.c even 2 1
1960.1.bi.b 2 140.s even 6 1
2520.1.ef.a 2 24.h odd 2 1
2520.1.ef.a 2 60.h even 2 1
2520.1.ef.a 2 168.s odd 6 1
2520.1.ef.a 2 420.ba even 6 1
2520.1.ef.b 2 12.b even 2 1
2520.1.ef.b 2 84.n even 6 1
2520.1.ef.b 2 120.i odd 2 1
2520.1.ef.b 2 840.cg odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1120, [\chi])\).

Hecke characteristic polynomials

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