Properties

Label 960.2.k.a
Level $960$
Weight $2$
Character orbit 960.k
Analytic conductor $7.666$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{3} - i q^{5} - 4 q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} - i q^{5} - 4 q^{7} - q^{9} - 4 i q^{11} + 2 i q^{13} - q^{15} + 6 i q^{19} + 4 i q^{21} - 2 q^{23} - q^{25} + i q^{27} + 10 i q^{29} - 8 q^{31} - 4 q^{33} + 4 i q^{35} - 6 i q^{37} + 2 q^{39} - 2 q^{41} + i q^{45} - 10 q^{47} + 9 q^{49} + 6 i q^{53} - 4 q^{55} + 6 q^{57} + 4 i q^{61} + 4 q^{63} + 2 q^{65} - 12 i q^{67} + 2 i q^{69} + 12 q^{71} - 14 q^{73} + i q^{75} + 16 i q^{77} - 8 q^{79} + q^{81} - 4 i q^{83} + 10 q^{87} + 18 q^{89} - 8 i q^{91} + 8 i q^{93} + 6 q^{95} - 6 q^{97} + 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} - 4 q^{23} - 2 q^{25} - 16 q^{31} - 8 q^{33} + 4 q^{39} - 4 q^{41} - 20 q^{47} + 18 q^{49} - 8 q^{55} + 12 q^{57} + 8 q^{63} + 4 q^{65} + 24 q^{71} - 28 q^{73} - 16 q^{79} + 2 q^{81} + 20 q^{87} + 36 q^{89} + 12 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
1.00000i
1.00000i
0 1.00000i 0 1.00000i 0 −4.00000 0 −1.00000 0
481.2 0 1.00000i 0 1.00000i 0 −4.00000 0 −1.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.k.a 2
3.b odd 2 1 2880.2.k.a 2
4.b odd 2 1 960.2.k.d yes 2
5.b even 2 1 4800.2.k.h 2
5.c odd 4 1 4800.2.d.c 2
5.c odd 4 1 4800.2.d.f 2
8.b even 2 1 inner 960.2.k.a 2
8.d odd 2 1 960.2.k.d yes 2
12.b even 2 1 2880.2.k.d 2
16.e even 4 1 3840.2.a.n 1
16.e even 4 1 3840.2.a.u 1
16.f odd 4 1 3840.2.a.a 1
16.f odd 4 1 3840.2.a.v 1
20.d odd 2 1 4800.2.k.a 2
20.e even 4 1 4800.2.d.b 2
20.e even 4 1 4800.2.d.g 2
24.f even 2 1 2880.2.k.d 2
24.h odd 2 1 2880.2.k.a 2
40.e odd 2 1 4800.2.k.a 2
40.f even 2 1 4800.2.k.h 2
40.i odd 4 1 4800.2.d.c 2
40.i odd 4 1 4800.2.d.f 2
40.k even 4 1 4800.2.d.b 2
40.k even 4 1 4800.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.k.a 2 1.a even 1 1 trivial
960.2.k.a 2 8.b even 2 1 inner
960.2.k.d yes 2 4.b odd 2 1
960.2.k.d yes 2 8.d odd 2 1
2880.2.k.a 2 3.b odd 2 1
2880.2.k.a 2 24.h odd 2 1
2880.2.k.d 2 12.b even 2 1
2880.2.k.d 2 24.f even 2 1
3840.2.a.a 1 16.f odd 4 1
3840.2.a.n 1 16.e even 4 1
3840.2.a.u 1 16.e even 4 1
3840.2.a.v 1 16.f odd 4 1
4800.2.d.b 2 20.e even 4 1
4800.2.d.b 2 40.k even 4 1
4800.2.d.c 2 5.c odd 4 1
4800.2.d.c 2 40.i odd 4 1
4800.2.d.f 2 5.c odd 4 1
4800.2.d.f 2 40.i odd 4 1
4800.2.d.g 2 20.e even 4 1
4800.2.d.g 2 40.k even 4 1
4800.2.k.a 2 20.d odd 2 1
4800.2.k.a 2 40.e odd 2 1
4800.2.k.h 2 5.b even 2 1
4800.2.k.h 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{23} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 100 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 14)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 18)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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