Properties

Label 960.3.l.f
Level $960$
Weight $3$
Character orbit 960.l
Analytic conductor $26.158$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 - 5 \beta_{1} + \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 - 5 \beta_{1} + \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{11} + 10 q^{13} + ( -5 - \beta_{2} + \beta_{3} ) q^{15} + ( -4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 8 - 10 \beta_{1} + 2 \beta_{3} ) q^{19} + ( 13 + 2 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{21} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{23} -5 q^{25} + ( 7 + 10 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{27} -12 \beta_{2} q^{29} -8 q^{31} + ( 6 + 12 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{35} + ( 22 + 20 \beta_{1} - 4 \beta_{3} ) q^{37} + ( 10 - 10 \beta_{1} - 10 \beta_{2} ) q^{39} + ( 8 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} ) q^{41} + ( 14 + 15 \beta_{1} - 3 \beta_{3} ) q^{43} + ( -5 + 10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{45} + ( 5 \beta_{1} + 30 \beta_{2} + 5 \beta_{3} ) q^{47} + ( 45 + 20 \beta_{1} - 4 \beta_{3} ) q^{49} + ( 18 - 24 \beta_{1} + 18 \beta_{2} - 6 \beta_{3} ) q^{51} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 38 - 8 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} ) q^{57} + ( -12 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{59} + ( 16 - 20 \beta_{1} + 4 \beta_{3} ) q^{61} + ( 64 + 19 \beta_{1} - 22 \beta_{2} - \beta_{3} ) q^{63} -10 \beta_{2} q^{65} + ( -82 + 15 \beta_{1} - 3 \beta_{3} ) q^{67} + ( 33 + 6 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{69} + ( -10 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} ) q^{71} + ( 50 - 20 \beta_{1} + 4 \beta_{3} ) q^{73} + ( -5 + 5 \beta_{1} + 5 \beta_{2} ) q^{75} + ( -4 \beta_{1} - 36 \beta_{2} - 4 \beta_{3} ) q^{77} + ( 28 - 10 \beta_{1} + 2 \beta_{3} ) q^{79} + ( 7 + 28 \beta_{1} - 28 \beta_{2} - 4 \beta_{3} ) q^{81} + ( 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{83} + ( 30 - 20 \beta_{1} + 4 \beta_{3} ) q^{85} + ( -60 - 12 \beta_{2} + 12 \beta_{3} ) q^{87} + ( 8 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} ) q^{89} + ( -20 - 50 \beta_{1} + 10 \beta_{3} ) q^{91} + ( -8 + 8 \beta_{1} + 8 \beta_{2} ) q^{93} + ( 10 \beta_{1} - 8 \beta_{2} + 10 \beta_{3} ) q^{95} + ( 74 + 20 \beta_{1} - 4 \beta_{3} ) q^{97} + ( -48 + 6 \beta_{1} - 24 \beta_{2} - 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 8q^{7} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 8q^{7} - 8q^{9} + 40q^{13} - 20q^{15} + 32q^{19} + 52q^{21} - 20q^{25} + 28q^{27} - 32q^{31} + 24q^{33} + 88q^{37} + 40q^{39} + 56q^{43} - 20q^{45} + 180q^{49} + 72q^{51} + 152q^{57} + 64q^{61} + 256q^{63} - 328q^{67} + 132q^{69} + 200q^{73} - 20q^{75} + 112q^{79} + 28q^{81} + 120q^{85} - 240q^{87} - 80q^{91} - 32q^{93} + 296q^{97} - 192q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
−1.58114 + 0.707107i
0 −0.581139 2.94317i 0 2.23607i 0 −11.4868 0 −8.32456 + 3.42079i 0
641.2 0 −0.581139 + 2.94317i 0 2.23607i 0 −11.4868 0 −8.32456 3.42079i 0
641.3 0 2.58114 1.52896i 0 2.23607i 0 7.48683 0 4.32456 7.89292i 0
641.4 0 2.58114 + 1.52896i 0 2.23607i 0 7.48683 0 4.32456 + 7.89292i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.l.f 4
3.b odd 2 1 inner 960.3.l.f 4
4.b odd 2 1 960.3.l.e 4
8.b even 2 1 240.3.l.c 4
8.d odd 2 1 30.3.d.a 4
12.b even 2 1 960.3.l.e 4
24.f even 2 1 30.3.d.a 4
24.h odd 2 1 240.3.l.c 4
40.e odd 2 1 150.3.d.c 4
40.f even 2 1 1200.3.l.u 4
40.i odd 4 2 1200.3.c.k 8
40.k even 4 2 150.3.b.b 8
72.l even 6 2 810.3.h.a 8
72.p odd 6 2 810.3.h.a 8
120.i odd 2 1 1200.3.l.u 4
120.m even 2 1 150.3.d.c 4
120.q odd 4 2 150.3.b.b 8
120.w even 4 2 1200.3.c.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 8.d odd 2 1
30.3.d.a 4 24.f even 2 1
150.3.b.b 8 40.k even 4 2
150.3.b.b 8 120.q odd 4 2
150.3.d.c 4 40.e odd 2 1
150.3.d.c 4 120.m even 2 1
240.3.l.c 4 8.b even 2 1
240.3.l.c 4 24.h odd 2 1
810.3.h.a 8 72.l even 6 2
810.3.h.a 8 72.p odd 6 2
960.3.l.e 4 4.b odd 2 1
960.3.l.e 4 12.b even 2 1
960.3.l.f 4 1.a even 1 1 trivial
960.3.l.f 4 3.b odd 2 1 inner
1200.3.c.k 8 40.i odd 4 2
1200.3.c.k 8 120.w even 4 2
1200.3.l.u 4 40.f even 2 1
1200.3.l.u 4 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 T_{7} - 86 \) acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 36 T + 12 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( -86 + 4 T + T^{2} )^{2} \)
$11$ \( ( 72 + T^{2} )^{2} \)
$13$ \( ( -10 + T )^{4} \)
$17$ \( 11664 + 936 T^{2} + T^{4} \)
$19$ \( ( -296 - 16 T + T^{2} )^{2} \)
$23$ \( 26244 + 396 T^{2} + T^{4} \)
$29$ \( ( 720 + T^{2} )^{2} \)
$31$ \( ( 8 + T )^{4} \)
$37$ \( ( -956 - 44 T + T^{2} )^{2} \)
$41$ \( 944784 + 2664 T^{2} + T^{4} \)
$43$ \( ( -614 - 28 T + T^{2} )^{2} \)
$47$ \( 16402500 + 9900 T^{2} + T^{4} \)
$53$ \( 11664 + 936 T^{2} + T^{4} \)
$59$ \( 3504384 + 6624 T^{2} + T^{4} \)
$61$ \( ( -1184 - 32 T + T^{2} )^{2} \)
$67$ \( ( 5914 + 164 T + T^{2} )^{2} \)
$71$ \( 1166400 + 5040 T^{2} + T^{4} \)
$73$ \( ( 1060 - 100 T + T^{2} )^{2} \)
$79$ \( ( 424 - 56 T + T^{2} )^{2} \)
$83$ \( 324 + 684 T^{2} + T^{4} \)
$89$ \( 186624 + 3744 T^{2} + T^{4} \)
$97$ \( ( 4036 - 148 T + T^{2} )^{2} \)
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