Properties

Label 960.3.j.e
Level $960$
Weight $3$
Character orbit 960.j
Analytic conductor $26.158$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.389136420864.4
Defining polynomial: \(x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{3} ) q^{5} + ( -3 \beta_{1} - \beta_{4} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{3} ) q^{5} + ( -3 \beta_{1} - \beta_{4} ) q^{7} + 3 q^{9} + \beta_{6} q^{11} + ( -\beta_{3} - \beta_{5} ) q^{13} + ( \beta_{2} + \beta_{4} ) q^{15} + \beta_{7} q^{17} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{19} + ( -8 - \beta_{3} + \beta_{5} ) q^{21} + ( 6 \beta_{1} - 2 \beta_{4} ) q^{23} + ( 5 + 2 \beta_{5} - \beta_{7} ) q^{25} + 3 \beta_{1} q^{27} + ( 26 - 3 \beta_{3} + 3 \beta_{5} ) q^{29} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{31} -\beta_{7} q^{33} + ( -10 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{35} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{39} + ( -26 - 6 \beta_{3} + 6 \beta_{5} ) q^{41} + ( -4 \beta_{1} - 8 \beta_{4} ) q^{43} + ( -3 + 3 \beta_{3} ) q^{45} + ( 12 \beta_{1} - 8 \beta_{4} ) q^{47} + ( -9 + 6 \beta_{3} - 6 \beta_{5} ) q^{49} -3 \beta_{6} q^{51} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{53} + ( 5 \beta_{1} + 8 \beta_{2} - 13 \beta_{4} - 2 \beta_{6} ) q^{55} + ( 6 \beta_{3} + 6 \beta_{5} ) q^{57} + ( 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} - \beta_{6} ) q^{59} -38 q^{61} + ( -9 \beta_{1} - 3 \beta_{4} ) q^{63} + ( 20 - 2 \beta_{5} + \beta_{7} ) q^{65} + ( 18 \beta_{1} - 14 \beta_{4} ) q^{67} + ( 20 - 2 \beta_{3} + 2 \beta_{5} ) q^{69} + ( 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} - 6 \beta_{6} ) q^{71} + 2 \beta_{7} q^{73} + ( 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} + 3 \beta_{6} ) q^{75} + ( -8 \beta_{3} - 8 \beta_{5} + 2 \beta_{7} ) q^{77} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{79} + 9 q^{81} + ( 24 \beta_{1} - 4 \beta_{4} ) q^{83} + ( -12 - 3 \beta_{3} - 21 \beta_{5} - 2 \beta_{7} ) q^{85} + ( 23 \beta_{1} - 9 \beta_{4} ) q^{87} + ( 66 + 4 \beta_{3} - 4 \beta_{5} ) q^{89} + ( 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{91} + ( 6 \beta_{3} + 6 \beta_{5} - 2 \beta_{7} ) q^{93} + ( -40 \beta_{1} + 4 \beta_{2} - 8 \beta_{4} + 6 \beta_{6} ) q^{95} + ( -22 \beta_{3} - 22 \beta_{5} - 2 \beta_{7} ) q^{97} + 3 \beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} + 24q^{9} + O(q^{10}) \) \( 8q - 4q^{5} + 24q^{9} - 72q^{21} + 32q^{25} + 184q^{29} - 256q^{41} - 12q^{45} - 24q^{49} - 304q^{61} + 168q^{65} + 144q^{69} + 72q^{81} - 24q^{85} + 560q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + \nu^{3} + 4 \nu \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 5 \nu^{5} - 12 \nu^{4} - 24 \nu^{3} - 16 \nu - 64 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{6} + 5 \nu^{5} + 20 \nu^{4} + 24 \nu^{3} + 32 \nu^{2} + 144 \nu + 192 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 3 \nu^{5} - 22 \nu^{3} - 8 \nu \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} + 5 \nu^{5} - 20 \nu^{4} + 24 \nu^{3} - 32 \nu^{2} + 144 \nu - 192 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{4} + 40 \nu^{2} + 80 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} - 9 \nu^{5} + 24 \nu^{3} - 16 \nu \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{3} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} - \beta_{3} - 4\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 5 \beta_{4} - 3 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} - 4 \beta_{1} - 28\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} - 4 \beta_{5} + \beta_{4} - 4 \beta_{3} + 71 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-3 \beta_{6} - 15 \beta_{5} - 20 \beta_{4} + 15 \beta_{3} + 40 \beta_{2} + 20 \beta_{1} - 20\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-19 \beta_{7} + 4 \beta_{5} - 29 \beta_{4} + 4 \beta_{3} - 139 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
1.52274 1.29664i
1.52274 + 1.29664i
−0.656712 1.88911i
−0.656712 + 1.88911i
−1.52274 1.29664i
−1.52274 + 1.29664i
0.656712 1.88911i
0.656712 + 1.88911i
0 −1.73205 0 −4.27492 2.59328i 0 0.837253 0 3.00000 0
319.2 0 −1.73205 0 −4.27492 + 2.59328i 0 0.837253 0 3.00000 0
319.3 0 −1.73205 0 3.27492 3.77822i 0 9.55505 0 3.00000 0
319.4 0 −1.73205 0 3.27492 + 3.77822i 0 9.55505 0 3.00000 0
319.5 0 1.73205 0 −4.27492 2.59328i 0 −0.837253 0 3.00000 0
319.6 0 1.73205 0 −4.27492 + 2.59328i 0 −0.837253 0 3.00000 0
319.7 0 1.73205 0 3.27492 3.77822i 0 −9.55505 0 3.00000 0
319.8 0 1.73205 0 3.27492 + 3.77822i 0 −9.55505 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d 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.j.e 8
4.b odd 2 1 inner 960.3.j.e 8
5.b even 2 1 inner 960.3.j.e 8
8.b even 2 1 60.3.f.b 8
8.d odd 2 1 60.3.f.b 8
20.d odd 2 1 inner 960.3.j.e 8
24.f even 2 1 180.3.f.h 8
24.h odd 2 1 180.3.f.h 8
40.e odd 2 1 60.3.f.b 8
40.f even 2 1 60.3.f.b 8
40.i odd 4 2 300.3.c.f 8
40.k even 4 2 300.3.c.f 8
120.i odd 2 1 180.3.f.h 8
120.m even 2 1 180.3.f.h 8
120.q odd 4 2 900.3.c.r 8
120.w even 4 2 900.3.c.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b 8 8.b even 2 1
60.3.f.b 8 8.d odd 2 1
60.3.f.b 8 40.e odd 2 1
60.3.f.b 8 40.f even 2 1
180.3.f.h 8 24.f even 2 1
180.3.f.h 8 24.h odd 2 1
180.3.f.h 8 120.i odd 2 1
180.3.f.h 8 120.m even 2 1
300.3.c.f 8 40.i odd 4 2
300.3.c.f 8 40.k even 4 2
900.3.c.r 8 120.q odd 4 2
900.3.c.r 8 120.w even 4 2
960.3.j.e 8 1.a even 1 1 trivial
960.3.j.e 8 4.b odd 2 1 inner
960.3.j.e 8 5.b even 2 1 inner
960.3.j.e 8 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{4} - 92 T_{7}^{2} + 64 \)
\( T_{11}^{4} + 348 T_{11}^{2} + 24576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -3 + T^{2} )^{4} \)
$5$ \( ( 625 + 50 T - 6 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$7$ \( ( 64 - 92 T^{2} + T^{4} )^{2} \)
$11$ \( ( 24576 + 348 T^{2} + T^{4} )^{2} \)
$13$ \( ( 1536 + 84 T^{2} + T^{4} )^{2} \)
$17$ \( ( 221184 + 1044 T^{2} + T^{4} )^{2} \)
$19$ \( ( 221184 + 1008 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1024 - 368 T^{2} + T^{4} )^{2} \)
$29$ \( ( 16 - 46 T + T^{2} )^{4} \)
$31$ \( ( 393216 + 2304 T^{2} + T^{4} )^{2} \)
$37$ \( ( 124416 + 756 T^{2} + T^{4} )^{2} \)
$41$ \( ( -1028 + 64 T + T^{2} )^{4} \)
$43$ \( ( 1364224 - 2528 T^{2} + T^{4} )^{2} \)
$47$ \( ( 614656 - 3296 T^{2} + T^{4} )^{2} \)
$53$ \( ( 124416 + 756 T^{2} + T^{4} )^{2} \)
$59$ \( ( 69033984 + 16668 T^{2} + T^{4} )^{2} \)
$61$ \( ( 38 + T )^{8} \)
$67$ \( ( 7573504 - 9392 T^{2} + T^{4} )^{2} \)
$71$ \( ( 884736 + 17136 T^{2} + T^{4} )^{2} \)
$73$ \( ( 3538944 + 4176 T^{2} + T^{4} )^{2} \)
$79$ \( ( 393216 + 2304 T^{2} + T^{4} )^{2} \)
$83$ \( ( 2027776 - 4064 T^{2} + T^{4} )^{2} \)
$89$ \( ( 3988 - 140 T + T^{2} )^{4} \)
$97$ \( ( 495550464 + 45888 T^{2} + T^{4} )^{2} \)
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