Properties

Label 960.2.d.e
Level $960$
Weight $2$
Character orbit 960.d
Analytic conductor $7.666$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{4}) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{4}) q^{7} + q^{9} + \beta_1 q^{11} + \beta_{2} q^{13} + \beta_{4} q^{15} - \beta_1 q^{17} - \beta_{3} q^{19} + ( - \beta_{5} + \beta_{4}) q^{21} + \beta_{6} q^{25} - q^{27} + \beta_{7} q^{29} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{31} - \beta_1 q^{33} + (\beta_{6} - 5) q^{35} + (2 \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{37} - \beta_{2} q^{39} + 6 q^{41} + (2 \beta_{6} + \beta_{3} - 2) q^{43} - \beta_{4} q^{45} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{47} + (2 \beta_{6} + \beta_{3} - 3) q^{49} + \beta_1 q^{51} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{53} + (\beta_{7} + \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{55} + \beta_{3} q^{57} + ( - \beta_{3} + \beta_1) q^{59} + (\beta_{7} + 3 \beta_{5} - 3 \beta_{4}) q^{61} + (\beta_{5} - \beta_{4}) q^{63} + ( - \beta_{3} - \beta_1 + 2) q^{65} + (2 \beta_{6} + \beta_{3} + 6) q^{67} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{71} - 3 \beta_{3} q^{73} - \beta_{6} q^{75} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}) q^{77} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{79} + q^{81} + ( - 2 \beta_{6} - \beta_{3} - 2) q^{83} + ( - \beta_{7} - \beta_{5} - \beta_{4} - 3 \beta_{2}) q^{85} - \beta_{7} q^{87} - 2 q^{89} + ( - 2 \beta_{3} - 2 \beta_1) q^{91} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{93} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{95} + 2 \beta_1 q^{97} + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 40 q^{35} + 48 q^{41} - 16 q^{43} - 24 q^{49} + 16 q^{65} + 48 q^{67} + 8 q^{81} - 16 q^{83} - 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - \nu^{6} + 5\nu^{5} + 5\nu^{4} + 15\nu^{3} + 15\nu^{2} + 30\nu + 8 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} + \nu^{6} + 5\nu^{5} - 5\nu^{4} + 15\nu^{3} - 15\nu^{2} + 30\nu - 8 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 10\nu^{6} + 5\nu^{5} + 30\nu^{4} - 5\nu^{3} + 10\nu^{2} + 16\nu + 60 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{6} + 15\nu^{4} + 45\nu^{2} + 96 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 3\beta_{5} + 3\beta_{4} - \beta_{3} - 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 5\beta_{3} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} + 6\beta_{6} - 4\beta_{5} + 4\beta_{4} + 3\beta_{3} - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} - 5\beta_{4} - 3\beta_{3} - 6\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} + 15\beta_{5} - 15\beta_{4} - 36 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20\beta_{5} + 20\beta_{4} - 3\beta_{3} + 6\beta_{2} - 14\beta_1 ) / 8 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.228425 + 1.39564i
−0.228425 1.39564i
−1.09445 0.895644i
−1.09445 + 0.895644i
1.09445 + 0.895644i
1.09445 0.895644i
0.228425 1.39564i
0.228425 + 1.39564i
0 −1.00000 0 −2.18890 0.456850i 0 0.913701i 0 1.00000 0
289.2 0 −1.00000 0 −2.18890 + 0.456850i 0 0.913701i 0 1.00000 0
289.3 0 −1.00000 0 −0.456850 2.18890i 0 4.37780i 0 1.00000 0
289.4 0 −1.00000 0 −0.456850 + 2.18890i 0 4.37780i 0 1.00000 0
289.5 0 −1.00000 0 0.456850 2.18890i 0 4.37780i 0 1.00000 0
289.6 0 −1.00000 0 0.456850 + 2.18890i 0 4.37780i 0 1.00000 0
289.7 0 −1.00000 0 2.18890 0.456850i 0 0.913701i 0 1.00000 0
289.8 0 −1.00000 0 2.18890 + 0.456850i 0 0.913701i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f 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.d.e 8
3.b odd 2 1 2880.2.d.j 8
4.b odd 2 1 960.2.d.f yes 8
5.b even 2 1 960.2.d.f yes 8
5.c odd 4 1 4800.2.k.q 8
5.c odd 4 1 4800.2.k.r 8
8.b even 2 1 960.2.d.f yes 8
8.d odd 2 1 inner 960.2.d.e 8
12.b even 2 1 2880.2.d.i 8
15.d odd 2 1 2880.2.d.i 8
16.e even 4 1 3840.2.f.i 8
16.e even 4 1 3840.2.f.k 8
16.f odd 4 1 3840.2.f.i 8
16.f odd 4 1 3840.2.f.k 8
20.d odd 2 1 inner 960.2.d.e 8
20.e even 4 1 4800.2.k.q 8
20.e even 4 1 4800.2.k.r 8
24.f even 2 1 2880.2.d.j 8
24.h odd 2 1 2880.2.d.i 8
40.e odd 2 1 960.2.d.f yes 8
40.f even 2 1 inner 960.2.d.e 8
40.i odd 4 1 4800.2.k.q 8
40.i odd 4 1 4800.2.k.r 8
40.k even 4 1 4800.2.k.q 8
40.k even 4 1 4800.2.k.r 8
60.h even 2 1 2880.2.d.j 8
80.k odd 4 1 3840.2.f.i 8
80.k odd 4 1 3840.2.f.k 8
80.q even 4 1 3840.2.f.i 8
80.q even 4 1 3840.2.f.k 8
120.i odd 2 1 2880.2.d.j 8
120.m even 2 1 2880.2.d.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.e 8 1.a even 1 1 trivial
960.2.d.e 8 8.d odd 2 1 inner
960.2.d.e 8 20.d odd 2 1 inner
960.2.d.e 8 40.f even 2 1 inner
960.2.d.f yes 8 4.b odd 2 1
960.2.d.f yes 8 5.b even 2 1
960.2.d.f yes 8 8.b even 2 1
960.2.d.f yes 8 40.e odd 2 1
2880.2.d.i 8 12.b even 2 1
2880.2.d.i 8 15.d odd 2 1
2880.2.d.i 8 24.h odd 2 1
2880.2.d.i 8 120.m even 2 1
2880.2.d.j 8 3.b odd 2 1
2880.2.d.j 8 24.f even 2 1
2880.2.d.j 8 60.h even 2 1
2880.2.d.j 8 120.i odd 2 1
3840.2.f.i 8 16.e even 4 1
3840.2.f.i 8 16.f odd 4 1
3840.2.f.i 8 80.k odd 4 1
3840.2.f.i 8 80.q even 4 1
3840.2.f.k 8 16.e even 4 1
3840.2.f.k 8 16.f odd 4 1
3840.2.f.k 8 80.k odd 4 1
3840.2.f.k 8 80.q even 4 1
4800.2.k.q 8 5.c odd 4 1
4800.2.k.q 8 20.e even 4 1
4800.2.k.q 8 40.i odd 4 1
4800.2.k.q 8 40.k even 4 1
4800.2.k.r 8 5.c odd 4 1
4800.2.k.r 8 20.e even 4 1
4800.2.k.r 8 40.i odd 4 1
4800.2.k.r 8 40.k even 4 1

Hecke kernels

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

\( T_{7}^{4} + 20T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 20T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{31}^{2} - 28 \) Copy content Toggle raw display
\( T_{43}^{2} + 4T_{43} - 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 34T^{4} + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 20 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 44 T^{2} + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 20 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 44 T^{2} + 400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 80)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 60 T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 112)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T - 48)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 80)^{4} \) Copy content Toggle raw display
$89$ \( (T + 2)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 176 T^{2} + 6400)^{2} \) Copy content Toggle raw display
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