Properties

Label 1960.1.i.a
Level $1960$
Weight $1$
Character orbit 1960.i
Self dual yes
Analytic conductor $0.978$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -40
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1960.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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 $D_6$
Artin field Galois closure of 6.2.19208000.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{5} - q^{8} + q^{9} + q^{10} - q^{11} + q^{13} + q^{16} - q^{18} - q^{19} - q^{20} + q^{22} + q^{23} + q^{25} - q^{26} - q^{32} + q^{36} + q^{37} + q^{38} + q^{40} - q^{41} - q^{44} - q^{45} - q^{46} + q^{47} - q^{50} + q^{52} + q^{53} + q^{55} + 2q^{59} + q^{64} - q^{65} - q^{72} - q^{74} - q^{76} - q^{80} + q^{81} + q^{82} + q^{88} + 2q^{89} + q^{90} + q^{92} - q^{94} + q^{95} - q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 1.00000 1.00000
\(n\): 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 \(\Q(\sqrt{-10}) \)

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1960, [\chi])\):

\( T_{13} - 1 \)
\( T_{19} + 1 \)

Hecke characteristic polynomials

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