Properties

Label 960.2.o.c
Level $960$
Weight $2$
Character orbit 960.o
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM discriminant -20
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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( 2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( 2 + \beta_{3} ) q^{9} + ( -\beta_{1} + 3 \beta_{2} ) q^{15} + ( 5 + \beta_{3} ) q^{21} + \beta_{2} q^{23} -5 q^{25} + ( \beta_{1} + 3 \beta_{2} ) q^{27} -4 \beta_{3} q^{29} + 5 \beta_{2} q^{35} -2 \beta_{3} q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + ( -5 + 2 \beta_{3} ) q^{45} + 7 \beta_{2} q^{47} + 3 q^{49} -8 q^{61} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{63} + ( 10 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -1 + \beta_{3} ) q^{69} -5 \beta_{1} q^{75} + ( -1 + 4 \beta_{3} ) q^{81} + 11 \beta_{2} q^{83} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{9} + 20q^{21} - 20q^{25} - 20q^{45} + 12q^{49} - 32q^{61} - 4q^{69} - 4q^{81} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
959.1
−1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 0.707107i
1.58114 + 0.707107i
0 −1.58114 0.707107i 0 2.23607i 0 −3.16228 0 2.00000 + 2.23607i 0
959.2 0 −1.58114 + 0.707107i 0 2.23607i 0 −3.16228 0 2.00000 2.23607i 0
959.3 0 1.58114 0.707107i 0 2.23607i 0 3.16228 0 2.00000 2.23607i 0
959.4 0 1.58114 + 0.707107i 0 2.23607i 0 3.16228 0 2.00000 + 2.23607i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.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.c 4
3.b odd 2 1 inner 960.2.o.c 4
4.b odd 2 1 inner 960.2.o.c 4
5.b even 2 1 inner 960.2.o.c 4
8.b even 2 1 60.2.h.a 4
8.d odd 2 1 60.2.h.a 4
12.b even 2 1 inner 960.2.o.c 4
15.d odd 2 1 inner 960.2.o.c 4
20.d odd 2 1 CM 960.2.o.c 4
24.f even 2 1 60.2.h.a 4
24.h odd 2 1 60.2.h.a 4
40.e odd 2 1 60.2.h.a 4
40.f even 2 1 60.2.h.a 4
40.i odd 4 2 300.2.e.b 4
40.k even 4 2 300.2.e.b 4
60.h even 2 1 inner 960.2.o.c 4
120.i odd 2 1 60.2.h.a 4
120.m even 2 1 60.2.h.a 4
120.q odd 4 2 300.2.e.b 4
120.w even 4 2 300.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.a 4 8.b even 2 1
60.2.h.a 4 8.d odd 2 1
60.2.h.a 4 24.f even 2 1
60.2.h.a 4 24.h odd 2 1
60.2.h.a 4 40.e odd 2 1
60.2.h.a 4 40.f even 2 1
60.2.h.a 4 120.i odd 2 1
60.2.h.a 4 120.m even 2 1
300.2.e.b 4 40.i odd 4 2
300.2.e.b 4 40.k even 4 2
300.2.e.b 4 120.q odd 4 2
300.2.e.b 4 120.w even 4 2
960.2.o.c 4 1.a even 1 1 trivial
960.2.o.c 4 3.b odd 2 1 inner
960.2.o.c 4 4.b odd 2 1 inner
960.2.o.c 4 5.b even 2 1 inner
960.2.o.c 4 12.b even 2 1 inner
960.2.o.c 4 15.d odd 2 1 inner
960.2.o.c 4 20.d odd 2 1 CM
960.2.o.c 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 \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{2} - 10 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 4 T^{2} + T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( -10 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 2 + T^{2} )^{2} \)
$29$ \( ( 80 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 20 + T^{2} )^{2} \)
$43$ \( ( -10 + T^{2} )^{2} \)
$47$ \( ( 98 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 8 + T )^{4} \)
$67$ \( ( -250 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 242 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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