Properties

Label 1960.1.bk.a
Level $1960$
Weight $1$
Character orbit 1960.bk
Analytic conductor $0.978$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -56
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1960.bk (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
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.

\(f(q)\) \(=\) \( q -\zeta_{24}^{2} q^{2} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{3} + \zeta_{24}^{4} q^{4} -\zeta_{24}^{11} q^{5} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{6} -\zeta_{24}^{6} q^{8} -\zeta_{24}^{8} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{2} q^{2} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{3} + \zeta_{24}^{4} q^{4} -\zeta_{24}^{11} q^{5} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{6} -\zeta_{24}^{6} q^{8} -\zeta_{24}^{8} q^{9} -\zeta_{24} q^{10} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{12} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{13} + ( -1 + \zeta_{24}^{6} ) q^{15} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{10} q^{18} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{19} + \zeta_{24}^{3} q^{20} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{24} -\zeta_{24}^{10} q^{25} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{26} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{30} -\zeta_{24}^{10} q^{32} + q^{36} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{38} -2 \zeta_{24}^{4} q^{39} -\zeta_{24}^{5} q^{40} -\zeta_{24}^{7} q^{45} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{48} - q^{50} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{52} -2 \zeta_{24}^{6} q^{57} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{59} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{60} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{61} - q^{64} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{65} -\zeta_{24}^{2} q^{72} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{75} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{76} + 2 \zeta_{24}^{6} q^{78} + \zeta_{24}^{7} q^{80} + \zeta_{24}^{4} q^{81} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{83} + \zeta_{24}^{9} q^{90} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{95} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 4q^{9} - 8q^{15} - 4q^{16} + 4q^{30} + 8q^{36} - 8q^{39} - 8q^{50} - 4q^{60} - 8q^{64} - 4q^{65} + 4q^{81} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{8}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
509.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.866025 0.500000i −1.22474 + 0.707107i 0.500000 + 0.866025i 0.965926 0.258819i 1.41421 0 1.00000i 0.500000 0.866025i −0.965926 0.258819i
509.2 −0.866025 0.500000i 1.22474 0.707107i 0.500000 + 0.866025i −0.965926 + 0.258819i −1.41421 0 1.00000i 0.500000 0.866025i 0.965926 + 0.258819i
509.3 0.866025 + 0.500000i −1.22474 + 0.707107i 0.500000 + 0.866025i 0.258819 + 0.965926i −1.41421 0 1.00000i 0.500000 0.866025i −0.258819 + 0.965926i
509.4 0.866025 + 0.500000i 1.22474 0.707107i 0.500000 + 0.866025i −0.258819 0.965926i 1.41421 0 1.00000i 0.500000 0.866025i 0.258819 0.965926i
1109.1 −0.866025 + 0.500000i −1.22474 0.707107i 0.500000 0.866025i 0.965926 + 0.258819i 1.41421 0 1.00000i 0.500000 + 0.866025i −0.965926 + 0.258819i
1109.2 −0.866025 + 0.500000i 1.22474 + 0.707107i 0.500000 0.866025i −0.965926 0.258819i −1.41421 0 1.00000i 0.500000 + 0.866025i 0.965926 0.258819i
1109.3 0.866025 0.500000i −1.22474 0.707107i 0.500000 0.866025i 0.258819 0.965926i −1.41421 0 1.00000i 0.500000 + 0.866025i −0.258819 0.965926i
1109.4 0.866025 0.500000i 1.22474 + 0.707107i 0.500000 0.866025i −0.258819 + 0.965926i 1.41421 0 1.00000i 0.500000 + 0.866025i 0.258819 + 0.965926i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1109.4
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
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
40.f even 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner
280.c odd 2 1 inner
280.bf even 6 1 inner
280.bk odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.1.bk.a 8
5.b even 2 1 inner 1960.1.bk.a 8
7.b odd 2 1 inner 1960.1.bk.a 8
7.c even 3 1 280.1.c.a 4
7.c even 3 1 inner 1960.1.bk.a 8
7.d odd 6 1 280.1.c.a 4
7.d odd 6 1 inner 1960.1.bk.a 8
8.b even 2 1 inner 1960.1.bk.a 8
21.g even 6 1 2520.1.h.e 4
21.h odd 6 1 2520.1.h.e 4
28.f even 6 1 1120.1.c.a 4
28.g odd 6 1 1120.1.c.a 4
35.c odd 2 1 inner 1960.1.bk.a 8
35.i odd 6 1 280.1.c.a 4
35.i odd 6 1 inner 1960.1.bk.a 8
35.j even 6 1 280.1.c.a 4
35.j even 6 1 inner 1960.1.bk.a 8
35.k even 12 1 1400.1.m.b 2
35.k even 12 1 1400.1.m.e 2
35.l odd 12 1 1400.1.m.b 2
35.l odd 12 1 1400.1.m.e 2
40.f even 2 1 inner 1960.1.bk.a 8
56.h odd 2 1 CM 1960.1.bk.a 8
56.j odd 6 1 280.1.c.a 4
56.j odd 6 1 inner 1960.1.bk.a 8
56.k odd 6 1 1120.1.c.a 4
56.m even 6 1 1120.1.c.a 4
56.p even 6 1 280.1.c.a 4
56.p even 6 1 inner 1960.1.bk.a 8
105.o odd 6 1 2520.1.h.e 4
105.p even 6 1 2520.1.h.e 4
140.p odd 6 1 1120.1.c.a 4
140.s even 6 1 1120.1.c.a 4
168.s odd 6 1 2520.1.h.e 4
168.ba even 6 1 2520.1.h.e 4
280.c odd 2 1 inner 1960.1.bk.a 8
280.ba even 6 1 1120.1.c.a 4
280.bf even 6 1 280.1.c.a 4
280.bf even 6 1 inner 1960.1.bk.a 8
280.bi odd 6 1 1120.1.c.a 4
280.bk odd 6 1 280.1.c.a 4
280.bk odd 6 1 inner 1960.1.bk.a 8
280.bt odd 12 1 1400.1.m.b 2
280.bt odd 12 1 1400.1.m.e 2
280.bv even 12 1 1400.1.m.b 2
280.bv even 12 1 1400.1.m.e 2
840.cb even 6 1 2520.1.h.e 4
840.cg odd 6 1 2520.1.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 7.c even 3 1
280.1.c.a 4 7.d odd 6 1
280.1.c.a 4 35.i odd 6 1
280.1.c.a 4 35.j even 6 1
280.1.c.a 4 56.j odd 6 1
280.1.c.a 4 56.p even 6 1
280.1.c.a 4 280.bf even 6 1
280.1.c.a 4 280.bk odd 6 1
1120.1.c.a 4 28.f even 6 1
1120.1.c.a 4 28.g odd 6 1
1120.1.c.a 4 56.k odd 6 1
1120.1.c.a 4 56.m even 6 1
1120.1.c.a 4 140.p odd 6 1
1120.1.c.a 4 140.s even 6 1
1120.1.c.a 4 280.ba even 6 1
1120.1.c.a 4 280.bi odd 6 1
1400.1.m.b 2 35.k even 12 1
1400.1.m.b 2 35.l odd 12 1
1400.1.m.b 2 280.bt odd 12 1
1400.1.m.b 2 280.bv even 12 1
1400.1.m.e 2 35.k even 12 1
1400.1.m.e 2 35.l odd 12 1
1400.1.m.e 2 280.bt odd 12 1
1400.1.m.e 2 280.bv even 12 1
1960.1.bk.a 8 1.a even 1 1 trivial
1960.1.bk.a 8 5.b even 2 1 inner
1960.1.bk.a 8 7.b odd 2 1 inner
1960.1.bk.a 8 7.c even 3 1 inner
1960.1.bk.a 8 7.d odd 6 1 inner
1960.1.bk.a 8 8.b even 2 1 inner
1960.1.bk.a 8 35.c odd 2 1 inner
1960.1.bk.a 8 35.i odd 6 1 inner
1960.1.bk.a 8 35.j even 6 1 inner
1960.1.bk.a 8 40.f even 2 1 inner
1960.1.bk.a 8 56.h odd 2 1 CM
1960.1.bk.a 8 56.j odd 6 1 inner
1960.1.bk.a 8 56.p even 6 1 inner
1960.1.bk.a 8 280.c odd 2 1 inner
1960.1.bk.a 8 280.bf even 6 1 inner
1960.1.bk.a 8 280.bk odd 6 1 inner
2520.1.h.e 4 21.g even 6 1
2520.1.h.e 4 21.h odd 6 1
2520.1.h.e 4 105.o odd 6 1
2520.1.h.e 4 105.p even 6 1
2520.1.h.e 4 168.s odd 6 1
2520.1.h.e 4 168.ba even 6 1
2520.1.h.e 4 840.cb even 6 1
2520.1.h.e 4 840.cg odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$5$ \( 1 - T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 2 + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 2 + T^{2} )^{4} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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