Properties

Label 280.1.bi.b
Level $280$
Weight $1$
Character orbit 280.bi
Analytic conductor $0.140$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 280.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.139738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} - q^{7} - q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} - q^{7} - q^{8} + \zeta_{6}^{2} q^{9} -\zeta_{6} q^{10} + \zeta_{6} q^{11} + q^{13} + \zeta_{6}^{2} q^{14} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} -\zeta_{6}^{2} q^{19} - q^{20} + q^{22} + \zeta_{6}^{2} q^{23} -\zeta_{6} q^{25} -\zeta_{6}^{2} q^{26} + \zeta_{6} q^{28} + \zeta_{6} q^{32} + \zeta_{6}^{2} q^{35} + q^{36} + \zeta_{6}^{2} q^{37} -\zeta_{6} q^{38} + \zeta_{6}^{2} q^{40} - q^{41} -\zeta_{6}^{2} q^{44} + \zeta_{6} q^{45} + \zeta_{6} q^{46} + \zeta_{6}^{2} q^{47} + q^{49} - q^{50} -\zeta_{6} q^{52} -\zeta_{6} q^{53} + q^{55} + q^{56} -2 \zeta_{6} q^{59} -\zeta_{6}^{2} q^{63} + q^{64} -\zeta_{6}^{2} q^{65} + \zeta_{6} q^{70} -\zeta_{6}^{2} q^{72} + \zeta_{6} q^{74} - q^{76} -\zeta_{6} q^{77} + \zeta_{6} q^{80} -\zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} -\zeta_{6} q^{88} + 2 \zeta_{6}^{2} q^{89} + q^{90} - q^{91} + q^{92} + \zeta_{6} q^{94} -\zeta_{6} q^{95} -\zeta_{6}^{2} q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + q^{5} - 2q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + q^{5} - 2q^{7} - 2q^{8} - q^{9} - q^{10} + q^{11} + 2q^{13} - q^{14} - q^{16} + q^{18} + q^{19} - 2q^{20} + 2q^{22} - q^{23} - q^{25} + q^{26} + q^{28} + q^{32} - q^{35} + 2q^{36} - q^{37} - q^{38} - q^{40} - 2q^{41} + q^{44} + q^{45} + q^{46} - q^{47} + 2q^{49} - 2q^{50} - q^{52} - q^{53} + 2q^{55} + 2q^{56} - 2q^{59} + q^{63} + 2q^{64} + q^{65} + q^{70} + q^{72} + q^{74} - 2q^{76} - q^{77} + q^{80} - q^{81} - q^{82} - q^{88} - 2q^{89} + 2q^{90} - 2q^{91} + 2q^{92} + q^{94} - q^{95} + q^{98} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.00000 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
219.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −1.00000 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): 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 280.1.bi.b yes 2
3.b odd 2 1 2520.1.ef.a 2
4.b odd 2 1 1120.1.by.b 2
5.b even 2 1 280.1.bi.a 2
5.c odd 4 2 1400.1.ba.a 4
7.b odd 2 1 1960.1.bi.b 2
7.c even 3 1 inner 280.1.bi.b yes 2
7.c even 3 1 1960.1.i.a 1
7.d odd 6 1 1960.1.i.b 1
7.d odd 6 1 1960.1.bi.b 2
8.b even 2 1 1120.1.by.a 2
8.d odd 2 1 280.1.bi.a 2
15.d odd 2 1 2520.1.ef.b 2
20.d odd 2 1 1120.1.by.a 2
21.h odd 6 1 2520.1.ef.a 2
24.f even 2 1 2520.1.ef.b 2
28.g odd 6 1 1120.1.by.b 2
35.c odd 2 1 1960.1.bi.a 2
35.i odd 6 1 1960.1.i.c 1
35.i odd 6 1 1960.1.bi.a 2
35.j even 6 1 280.1.bi.a 2
35.j even 6 1 1960.1.i.d 1
35.l odd 12 2 1400.1.ba.a 4
40.e odd 2 1 CM 280.1.bi.b yes 2
40.f even 2 1 1120.1.by.b 2
40.k even 4 2 1400.1.ba.a 4
56.e even 2 1 1960.1.bi.a 2
56.k odd 6 1 280.1.bi.a 2
56.k odd 6 1 1960.1.i.d 1
56.m even 6 1 1960.1.i.c 1
56.m even 6 1 1960.1.bi.a 2
56.p even 6 1 1120.1.by.a 2
105.o odd 6 1 2520.1.ef.b 2
120.m even 2 1 2520.1.ef.a 2
140.p odd 6 1 1120.1.by.a 2
168.v even 6 1 2520.1.ef.b 2
280.n even 2 1 1960.1.bi.b 2
280.ba even 6 1 1960.1.i.b 1
280.ba even 6 1 1960.1.bi.b 2
280.bf even 6 1 1120.1.by.b 2
280.bi odd 6 1 inner 280.1.bi.b yes 2
280.bi odd 6 1 1960.1.i.a 1
280.br even 12 2 1400.1.ba.a 4
840.cv even 6 1 2520.1.ef.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 5.b even 2 1
280.1.bi.a 2 8.d odd 2 1
280.1.bi.a 2 35.j even 6 1
280.1.bi.a 2 56.k odd 6 1
280.1.bi.b yes 2 1.a even 1 1 trivial
280.1.bi.b yes 2 7.c even 3 1 inner
280.1.bi.b yes 2 40.e odd 2 1 CM
280.1.bi.b yes 2 280.bi odd 6 1 inner
1120.1.by.a 2 8.b even 2 1
1120.1.by.a 2 20.d odd 2 1
1120.1.by.a 2 56.p even 6 1
1120.1.by.a 2 140.p odd 6 1
1120.1.by.b 2 4.b odd 2 1
1120.1.by.b 2 28.g odd 6 1
1120.1.by.b 2 40.f even 2 1
1120.1.by.b 2 280.bf even 6 1
1400.1.ba.a 4 5.c odd 4 2
1400.1.ba.a 4 35.l odd 12 2
1400.1.ba.a 4 40.k even 4 2
1400.1.ba.a 4 280.br even 12 2
1960.1.i.a 1 7.c even 3 1
1960.1.i.a 1 280.bi odd 6 1
1960.1.i.b 1 7.d odd 6 1
1960.1.i.b 1 280.ba even 6 1
1960.1.i.c 1 35.i odd 6 1
1960.1.i.c 1 56.m even 6 1
1960.1.i.d 1 35.j even 6 1
1960.1.i.d 1 56.k odd 6 1
1960.1.bi.a 2 35.c odd 2 1
1960.1.bi.a 2 35.i odd 6 1
1960.1.bi.a 2 56.e even 2 1
1960.1.bi.a 2 56.m even 6 1
1960.1.bi.b 2 7.b odd 2 1
1960.1.bi.b 2 7.d odd 6 1
1960.1.bi.b 2 280.n even 2 1
1960.1.bi.b 2 280.ba even 6 1
2520.1.ef.a 2 3.b odd 2 1
2520.1.ef.a 2 21.h odd 6 1
2520.1.ef.a 2 120.m even 2 1
2520.1.ef.a 2 840.cv even 6 1
2520.1.ef.b 2 15.d odd 2 1
2520.1.ef.b 2 24.f even 2 1
2520.1.ef.b 2 105.o odd 6 1
2520.1.ef.b 2 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(280, [\chi])\).

Hecke characteristic polynomials

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