Properties

Label 2960.1.bu.a
Level $2960$
Weight $1$
Character orbit 2960.bu
Analytic conductor $1.477$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2960.bu (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.25326500.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{5} -i q^{9} +O(q^{10})\) \( q + q^{5} -i q^{9} + 2 i q^{13} + q^{25} + ( 1 + i ) q^{29} + q^{37} -i q^{45} -i q^{49} + ( -1 - i ) q^{53} + ( 1 - i ) q^{61} + 2 i q^{65} + ( -1 + i ) q^{73} - q^{81} + ( -1 - i ) q^{89} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{25} + 2q^{29} + 2q^{37} - 2q^{53} + 2q^{61} - 2q^{73} - 2q^{81} - 2q^{89} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(-i\) \(-i\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
1.00000i
1.00000i
0 0 0 1.00000 0 0 0 1.00000i 0
1807.1 0 0 0 1.00000 0 0 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.f even 4 1 inner
740.p odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.bu.a 2
4.b odd 2 1 CM 2960.1.bu.a 2
5.c odd 4 1 2960.1.bz.a yes 2
20.e even 4 1 2960.1.bz.a yes 2
37.d odd 4 1 2960.1.bz.a yes 2
148.g even 4 1 2960.1.bz.a yes 2
185.f even 4 1 inner 2960.1.bu.a 2
740.p odd 4 1 inner 2960.1.bu.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.bu.a 2 1.a even 1 1 trivial
2960.1.bu.a 2 4.b odd 2 1 CM
2960.1.bu.a 2 185.f even 4 1 inner
2960.1.bu.a 2 740.p odd 4 1 inner
2960.1.bz.a yes 2 5.c odd 4 1
2960.1.bz.a yes 2 20.e even 4 1
2960.1.bz.a yes 2 37.d odd 4 1
2960.1.bz.a yes 2 148.g even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2960, [\chi])\).

Hecke characteristic polynomials

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