Properties

Label 960.2.o.a
Level $960$
Weight $2$
Character orbit 960.o
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{1} q^{5} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{1} q^{5} -3 q^{9} + \beta_{3} q^{15} -2 \beta_{1} q^{17} + 2 \beta_{3} q^{19} + 2 \beta_{2} q^{23} + 5 q^{25} -3 \beta_{2} q^{27} + 2 \beta_{3} q^{31} + 3 \beta_{1} q^{45} -6 \beta_{2} q^{47} -7 q^{49} + 2 \beta_{3} q^{51} -2 \beta_{1} q^{53} + 6 \beta_{1} q^{57} + 2 q^{61} -6 q^{69} + 5 \beta_{2} q^{75} + 2 \beta_{3} q^{79} + 9 q^{81} + 2 \beta_{2} q^{83} + 10 q^{85} + 6 \beta_{1} q^{93} + 10 \beta_{2} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} + 20q^{25} - 28q^{49} + 8q^{61} - 24q^{69} + 36q^{81} + 40q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 2 \nu^{2} + 6 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + \beta_{1} - 3\)\()/4\)
\(\nu^{3}\)\(=\)\(\beta_{1} - 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
959.1
−0.309017 + 0.535233i
0.809017 1.40126i
−0.309017 0.535233i
0.809017 + 1.40126i
0 1.73205i 0 −2.23607 0 0 0 −3.00000 0
959.2 0 1.73205i 0 2.23607 0 0 0 −3.00000 0
959.3 0 1.73205i 0 −2.23607 0 0 0 −3.00000 0
959.4 0 1.73205i 0 2.23607 0 0 0 −3.00000 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h 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.o.a 4
3.b odd 2 1 inner 960.2.o.a 4
4.b odd 2 1 inner 960.2.o.a 4
5.b even 2 1 inner 960.2.o.a 4
8.b even 2 1 60.2.h.b 4
8.d odd 2 1 60.2.h.b 4
12.b even 2 1 inner 960.2.o.a 4
15.d odd 2 1 CM 960.2.o.a 4
20.d odd 2 1 inner 960.2.o.a 4
24.f even 2 1 60.2.h.b 4
24.h odd 2 1 60.2.h.b 4
40.e odd 2 1 60.2.h.b 4
40.f even 2 1 60.2.h.b 4
40.i odd 4 2 300.2.e.a 4
40.k even 4 2 300.2.e.a 4
60.h even 2 1 inner 960.2.o.a 4
120.i odd 2 1 60.2.h.b 4
120.m even 2 1 60.2.h.b 4
120.q odd 4 2 300.2.e.a 4
120.w even 4 2 300.2.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.b 4 8.b even 2 1
60.2.h.b 4 8.d odd 2 1
60.2.h.b 4 24.f even 2 1
60.2.h.b 4 24.h odd 2 1
60.2.h.b 4 40.e odd 2 1
60.2.h.b 4 40.f even 2 1
60.2.h.b 4 120.i odd 2 1
60.2.h.b 4 120.m even 2 1
300.2.e.a 4 40.i odd 4 2
300.2.e.a 4 40.k even 4 2
300.2.e.a 4 120.q odd 4 2
300.2.e.a 4 120.w even 4 2
960.2.o.a 4 1.a even 1 1 trivial
960.2.o.a 4 3.b odd 2 1 inner
960.2.o.a 4 4.b odd 2 1 inner
960.2.o.a 4 5.b even 2 1 inner
960.2.o.a 4 12.b even 2 1 inner
960.2.o.a 4 15.d odd 2 1 CM
960.2.o.a 4 20.d odd 2 1 inner
960.2.o.a 4 60.h even 2 1 inner

Hecke kernels

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

\( T_{7} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -20 + T^{2} )^{2} \)
$19$ \( ( 60 + T^{2} )^{2} \)
$23$ \( ( 12 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 60 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 108 + T^{2} )^{2} \)
$53$ \( ( -20 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 60 + T^{2} )^{2} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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