Properties

Label 280.1.c.a
Level $280$
Weight $1$
Character orbit 280.c
Analytic conductor $0.140$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -56
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 280.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.139738203537\)
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: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.11200.2

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
1.00000i 1.41421i −1.00000 0.707107 0.707107i −1.41421 1.00000i 1.00000i −1.00000 −0.707107 0.707107i
69.2 1.00000i 1.41421i −1.00000 −0.707107 + 0.707107i 1.41421 1.00000i 1.00000i −1.00000 0.707107 + 0.707107i
69.3 1.00000i 1.41421i −1.00000 −0.707107 0.707107i 1.41421 1.00000i 1.00000i −1.00000 0.707107 0.707107i
69.4 1.00000i 1.41421i −1.00000 0.707107 + 0.707107i −1.41421 1.00000i 1.00000i −1.00000 −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 280.1.c.a 4
3.b odd 2 1 2520.1.h.e 4
4.b odd 2 1 1120.1.c.a 4
5.b even 2 1 inner 280.1.c.a 4
5.c odd 4 1 1400.1.m.b 2
5.c odd 4 1 1400.1.m.e 2
7.b odd 2 1 inner 280.1.c.a 4
7.c even 3 2 1960.1.bk.a 8
7.d odd 6 2 1960.1.bk.a 8
8.b even 2 1 inner 280.1.c.a 4
8.d odd 2 1 1120.1.c.a 4
15.d odd 2 1 2520.1.h.e 4
20.d odd 2 1 1120.1.c.a 4
21.c even 2 1 2520.1.h.e 4
24.h odd 2 1 2520.1.h.e 4
28.d even 2 1 1120.1.c.a 4
35.c odd 2 1 inner 280.1.c.a 4
35.f even 4 1 1400.1.m.b 2
35.f even 4 1 1400.1.m.e 2
35.i odd 6 2 1960.1.bk.a 8
35.j even 6 2 1960.1.bk.a 8
40.e odd 2 1 1120.1.c.a 4
40.f even 2 1 inner 280.1.c.a 4
40.i odd 4 1 1400.1.m.b 2
40.i odd 4 1 1400.1.m.e 2
56.e even 2 1 1120.1.c.a 4
56.h odd 2 1 CM 280.1.c.a 4
56.j odd 6 2 1960.1.bk.a 8
56.p even 6 2 1960.1.bk.a 8
105.g even 2 1 2520.1.h.e 4
120.i odd 2 1 2520.1.h.e 4
140.c even 2 1 1120.1.c.a 4
168.i even 2 1 2520.1.h.e 4
280.c odd 2 1 inner 280.1.c.a 4
280.n even 2 1 1120.1.c.a 4
280.s even 4 1 1400.1.m.b 2
280.s even 4 1 1400.1.m.e 2
280.bf even 6 2 1960.1.bk.a 8
280.bk odd 6 2 1960.1.bk.a 8
840.u even 2 1 2520.1.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 1.a even 1 1 trivial
280.1.c.a 4 5.b even 2 1 inner
280.1.c.a 4 7.b odd 2 1 inner
280.1.c.a 4 8.b even 2 1 inner
280.1.c.a 4 35.c odd 2 1 inner
280.1.c.a 4 40.f even 2 1 inner
280.1.c.a 4 56.h odd 2 1 CM
280.1.c.a 4 280.c odd 2 1 inner
1120.1.c.a 4 4.b odd 2 1
1120.1.c.a 4 8.d odd 2 1
1120.1.c.a 4 20.d odd 2 1
1120.1.c.a 4 28.d even 2 1
1120.1.c.a 4 40.e odd 2 1
1120.1.c.a 4 56.e even 2 1
1120.1.c.a 4 140.c even 2 1
1120.1.c.a 4 280.n even 2 1
1400.1.m.b 2 5.c odd 4 1
1400.1.m.b 2 35.f even 4 1
1400.1.m.b 2 40.i odd 4 1
1400.1.m.b 2 280.s even 4 1
1400.1.m.e 2 5.c odd 4 1
1400.1.m.e 2 35.f even 4 1
1400.1.m.e 2 40.i odd 4 1
1400.1.m.e 2 280.s even 4 1
1960.1.bk.a 8 7.c even 3 2
1960.1.bk.a 8 7.d odd 6 2
1960.1.bk.a 8 35.i odd 6 2
1960.1.bk.a 8 35.j even 6 2
1960.1.bk.a 8 56.j odd 6 2
1960.1.bk.a 8 56.p even 6 2
1960.1.bk.a 8 280.bf even 6 2
1960.1.bk.a 8 280.bk odd 6 2
2520.1.h.e 4 3.b odd 2 1
2520.1.h.e 4 15.d odd 2 1
2520.1.h.e 4 21.c even 2 1
2520.1.h.e 4 24.h odd 2 1
2520.1.h.e 4 105.g even 2 1
2520.1.h.e 4 120.i odd 2 1
2520.1.h.e 4 168.i even 2 1
2520.1.h.e 4 840.u even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(280, [\chi])\).

Hecke characteristic polynomials

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