Properties

Label 2960.1.o.a
Level $2960$
Weight $1$
Character orbit 2960.o
Self dual yes
Analytic conductor $1.477$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -740
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.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.1620896000.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} - q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} - q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{2} ) q^{9} + \beta_{2} q^{13} + \beta_{1} q^{15} -\beta_{2} q^{17} -\beta_{1} q^{19} -\beta_{2} q^{21} + q^{25} + ( -\beta_{1} - \beta_{3} ) q^{27} + \beta_{3} q^{31} -\beta_{3} q^{35} + q^{37} + ( -\beta_{1} - \beta_{3} ) q^{39} + ( -1 - \beta_{2} ) q^{45} -\beta_{3} q^{47} + ( 1 - \beta_{2} ) q^{49} + ( \beta_{1} + \beta_{3} ) q^{51} + ( 2 + \beta_{2} ) q^{57} + \beta_{3} q^{59} + \beta_{1} q^{63} -\beta_{2} q^{65} + \beta_{1} q^{67} -\beta_{1} q^{75} -\beta_{3} q^{79} + ( 1 + \beta_{2} ) q^{81} + \beta_{1} q^{83} + \beta_{2} q^{85} + ( \beta_{1} - \beta_{3} ) q^{91} -\beta_{2} q^{93} + \beta_{1} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{5} + 4q^{9} + 4q^{25} + 4q^{37} - 4q^{45} + 4q^{49} + 8q^{57} + 4q^{81} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2959.1
1.84776
0.765367
−0.765367
−1.84776
0 −1.84776 0 −1.00000 0 0.765367 0 2.41421 0
2959.2 0 −0.765367 0 −1.00000 0 −1.84776 0 −0.414214 0
2959.3 0 0.765367 0 −1.00000 0 1.84776 0 −0.414214 0
2959.4 0 1.84776 0 −1.00000 0 −0.765367 0 2.41421 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
740.g odd 2 1 CM by \(\Q(\sqrt{-185}) \)
4.b odd 2 1 inner
185.d even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{193} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2960, [\chi])\).

Hecke characteristic polynomials

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