Properties

Label 960.4.a.l
Level $960$
Weight $4$
Character orbit 960.a
Self dual yes
Analytic conductor $56.642$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} + 5 q^{5} - 20 q^{7} + 9 q^{9} + O(q^{10}) \) \( q - 3 q^{3} + 5 q^{5} - 20 q^{7} + 9 q^{9} - 24 q^{11} - 74 q^{13} - 15 q^{15} + 54 q^{17} - 124 q^{19} + 60 q^{21} + 120 q^{23} + 25 q^{25} - 27 q^{27} + 78 q^{29} - 200 q^{31} + 72 q^{33} - 100 q^{35} + 70 q^{37} + 222 q^{39} + 330 q^{41} + 92 q^{43} + 45 q^{45} + 24 q^{47} + 57 q^{49} - 162 q^{51} - 450 q^{53} - 120 q^{55} + 372 q^{57} + 24 q^{59} + 322 q^{61} - 180 q^{63} - 370 q^{65} - 196 q^{67} - 360 q^{69} + 288 q^{71} - 430 q^{73} - 75 q^{75} + 480 q^{77} + 520 q^{79} + 81 q^{81} + 156 q^{83} + 270 q^{85} - 234 q^{87} + 1026 q^{89} + 1480 q^{91} + 600 q^{93} - 620 q^{95} - 286 q^{97} - 216 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3.00000 0 5.00000 0 −20.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.4.a.l 1
4.b odd 2 1 960.4.a.bi 1
8.b even 2 1 240.4.a.f 1
8.d odd 2 1 15.4.a.b 1
24.f even 2 1 45.4.a.b 1
24.h odd 2 1 720.4.a.r 1
40.e odd 2 1 75.4.a.a 1
40.f even 2 1 1200.4.a.o 1
40.i odd 4 2 1200.4.f.m 2
40.k even 4 2 75.4.b.a 2
56.e even 2 1 735.4.a.i 1
72.l even 6 2 405.4.e.k 2
72.p odd 6 2 405.4.e.d 2
88.g even 2 1 1815.4.a.a 1
120.m even 2 1 225.4.a.g 1
120.q odd 4 2 225.4.b.d 2
168.e odd 2 1 2205.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 8.d odd 2 1
45.4.a.b 1 24.f even 2 1
75.4.a.a 1 40.e odd 2 1
75.4.b.a 2 40.k even 4 2
225.4.a.g 1 120.m even 2 1
225.4.b.d 2 120.q odd 4 2
240.4.a.f 1 8.b even 2 1
405.4.e.d 2 72.p odd 6 2
405.4.e.k 2 72.l even 6 2
720.4.a.r 1 24.h odd 2 1
735.4.a.i 1 56.e even 2 1
960.4.a.l 1 1.a even 1 1 trivial
960.4.a.bi 1 4.b odd 2 1
1200.4.a.o 1 40.f even 2 1
1200.4.f.m 2 40.i odd 4 2
1815.4.a.a 1 88.g even 2 1
2205.4.a.c 1 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(960))\):

\( T_{7} + 20 \)
\( T_{11} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -5 + T \)
$7$ \( 20 + T \)
$11$ \( 24 + T \)
$13$ \( 74 + T \)
$17$ \( -54 + T \)
$19$ \( 124 + T \)
$23$ \( -120 + T \)
$29$ \( -78 + T \)
$31$ \( 200 + T \)
$37$ \( -70 + T \)
$41$ \( -330 + T \)
$43$ \( -92 + T \)
$47$ \( -24 + T \)
$53$ \( 450 + T \)
$59$ \( -24 + T \)
$61$ \( -322 + T \)
$67$ \( 196 + T \)
$71$ \( -288 + T \)
$73$ \( 430 + T \)
$79$ \( -520 + T \)
$83$ \( -156 + T \)
$89$ \( -1026 + T \)
$97$ \( 286 + T \)
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