Properties

Label 1960.1.bi.b
Level $1960$
Weight $1$
Character orbit 1960.bi
Analytic conductor $0.978$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6} q^{5} - q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{10} - \zeta_{6}^{2} q^{11} - q^{13} - \zeta_{6} q^{16} - \zeta_{6}^{2} q^{18} - \zeta_{6} q^{19} + q^{20} + q^{22} - \zeta_{6} q^{23} + \zeta_{6}^{2} q^{25} - \zeta_{6} q^{26} - \zeta_{6}^{2} q^{32} + q^{36} - \zeta_{6} q^{37} - \zeta_{6}^{2} q^{38} + \zeta_{6} q^{40} + q^{41} + \zeta_{6} q^{44} + \zeta_{6}^{2} q^{45} - \zeta_{6}^{2} q^{46} + \zeta_{6} q^{47} - q^{50} - \zeta_{6}^{2} q^{52} + \zeta_{6}^{2} q^{53} - q^{55} - \zeta_{6}^{2} q^{59} + q^{64} + \zeta_{6} q^{65} + \zeta_{6} q^{72} - \zeta_{6}^{2} q^{74} + q^{76} + \zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} + \zeta_{6}^{2} q^{88} + \zeta_{6} q^{89} - q^{90} + q^{92} + \zeta_{6}^{2} q^{94} + \zeta_{6}^{2} q^{95} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{5} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - q^{5} - 2 q^{8} - q^{9} + q^{10} + q^{11} - 2 q^{13} - q^{16} + q^{18} - q^{19} + 2 q^{20} + 2 q^{22} - q^{23} - q^{25} - q^{26} + q^{32} + 2 q^{36} - q^{37} + q^{38} + q^{40} + 2 q^{41} + q^{44} - q^{45} + q^{46} + q^{47} - 2 q^{50} + q^{52} - q^{53} - 2 q^{55} + 2 q^{59} + 2 q^{64} + q^{65} + q^{72} + q^{74} + 2 q^{76} - q^{80} - q^{81} + q^{82} - q^{88} + 2 q^{89} - 2 q^{90} + 2 q^{92} - q^{94} - q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 0 −1.00000 −0.500000 0.866025i 0.500000 0.866025i
1059.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 0 −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 1960.1.bi.b 2
5.b even 2 1 1960.1.bi.a 2
7.b odd 2 1 280.1.bi.b yes 2
7.c even 3 1 1960.1.i.b 1
7.c even 3 1 inner 1960.1.bi.b 2
7.d odd 6 1 280.1.bi.b yes 2
7.d odd 6 1 1960.1.i.a 1
8.d odd 2 1 1960.1.bi.a 2
21.c even 2 1 2520.1.ef.a 2
21.g even 6 1 2520.1.ef.a 2
28.d even 2 1 1120.1.by.b 2
28.f even 6 1 1120.1.by.b 2
35.c odd 2 1 280.1.bi.a 2
35.f even 4 2 1400.1.ba.a 4
35.i odd 6 1 280.1.bi.a 2
35.i odd 6 1 1960.1.i.d 1
35.j even 6 1 1960.1.i.c 1
35.j even 6 1 1960.1.bi.a 2
35.k even 12 2 1400.1.ba.a 4
40.e odd 2 1 CM 1960.1.bi.b 2
56.e even 2 1 280.1.bi.a 2
56.h odd 2 1 1120.1.by.a 2
56.j odd 6 1 1120.1.by.a 2
56.k odd 6 1 1960.1.i.c 1
56.k odd 6 1 1960.1.bi.a 2
56.m even 6 1 280.1.bi.a 2
56.m even 6 1 1960.1.i.d 1
105.g even 2 1 2520.1.ef.b 2
105.p even 6 1 2520.1.ef.b 2
140.c even 2 1 1120.1.by.a 2
140.s even 6 1 1120.1.by.a 2
168.e odd 2 1 2520.1.ef.b 2
168.be odd 6 1 2520.1.ef.b 2
280.c odd 2 1 1120.1.by.b 2
280.n even 2 1 280.1.bi.b yes 2
280.y odd 4 2 1400.1.ba.a 4
280.ba even 6 1 280.1.bi.b yes 2
280.ba even 6 1 1960.1.i.a 1
280.bi odd 6 1 1960.1.i.b 1
280.bi odd 6 1 inner 1960.1.bi.b 2
280.bk odd 6 1 1120.1.by.b 2
280.bp odd 12 2 1400.1.ba.a 4
840.b odd 2 1 2520.1.ef.a 2
840.ct odd 6 1 2520.1.ef.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 35.c odd 2 1
280.1.bi.a 2 35.i odd 6 1
280.1.bi.a 2 56.e even 2 1
280.1.bi.a 2 56.m even 6 1
280.1.bi.b yes 2 7.b odd 2 1
280.1.bi.b yes 2 7.d odd 6 1
280.1.bi.b yes 2 280.n even 2 1
280.1.bi.b yes 2 280.ba even 6 1
1120.1.by.a 2 56.h odd 2 1
1120.1.by.a 2 56.j odd 6 1
1120.1.by.a 2 140.c even 2 1
1120.1.by.a 2 140.s even 6 1
1120.1.by.b 2 28.d even 2 1
1120.1.by.b 2 28.f even 6 1
1120.1.by.b 2 280.c odd 2 1
1120.1.by.b 2 280.bk odd 6 1
1400.1.ba.a 4 35.f even 4 2
1400.1.ba.a 4 35.k even 12 2
1400.1.ba.a 4 280.y odd 4 2
1400.1.ba.a 4 280.bp odd 12 2
1960.1.i.a 1 7.d odd 6 1
1960.1.i.a 1 280.ba even 6 1
1960.1.i.b 1 7.c even 3 1
1960.1.i.b 1 280.bi odd 6 1
1960.1.i.c 1 35.j even 6 1
1960.1.i.c 1 56.k odd 6 1
1960.1.i.d 1 35.i odd 6 1
1960.1.i.d 1 56.m even 6 1
1960.1.bi.a 2 5.b even 2 1
1960.1.bi.a 2 8.d odd 2 1
1960.1.bi.a 2 35.j even 6 1
1960.1.bi.a 2 56.k odd 6 1
1960.1.bi.b 2 1.a even 1 1 trivial
1960.1.bi.b 2 7.c even 3 1 inner
1960.1.bi.b 2 40.e odd 2 1 CM
1960.1.bi.b 2 280.bi odd 6 1 inner
2520.1.ef.a 2 21.c even 2 1
2520.1.ef.a 2 21.g even 6 1
2520.1.ef.a 2 840.b odd 2 1
2520.1.ef.a 2 840.ct odd 6 1
2520.1.ef.b 2 105.g even 2 1
2520.1.ef.b 2 105.p even 6 1
2520.1.ef.b 2 168.e odd 2 1
2520.1.ef.b 2 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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