Properties

Label 960.2.h.g
Level $960$
Weight $2$
Character orbit 960.h
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.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
Defining polynomial: \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} -\beta_{1} q^{5} + ( \beta_{3} + \beta_{4} ) q^{7} + \beta_{6} q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} -\beta_{1} q^{5} + ( \beta_{3} + \beta_{4} ) q^{7} + \beta_{6} q^{9} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} + ( 2 - \beta_{1} + \beta_{6} - \beta_{7} ) q^{13} -\beta_{3} q^{15} -2 \beta_{1} q^{17} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + ( -1 + 3 \beta_{1} + \beta_{7} ) q^{21} + ( -\beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{23} - q^{25} + ( \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{27} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{29} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( -2 + 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{33} + ( \beta_{2} + \beta_{5} ) q^{35} + ( -2 + \beta_{1} - \beta_{6} + \beta_{7} ) q^{37} + ( 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{39} + ( -3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{43} + ( 1 - \beta_{7} ) q^{45} + ( \beta_{2} + \beta_{5} ) q^{47} + ( 1 + \beta_{1} - \beta_{6} + \beta_{7} ) q^{49} -2 \beta_{3} q^{51} + ( -4 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{55} + ( 2 + 3 \beta_{1} - \beta_{6} + \beta_{7} ) q^{57} + ( 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{59} + ( \beta_{1} - \beta_{6} + \beta_{7} ) q^{61} + ( -3 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{63} + ( -\beta_{1} - \beta_{6} - \beta_{7} ) q^{65} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( -1 + 6 \beta_{1} - \beta_{6} - 2 \beta_{7} ) q^{69} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} -\beta_{5} q^{75} -4 \beta_{1} q^{77} + ( 3 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{79} + ( -1 + 9 \beta_{1} - \beta_{6} + \beta_{7} ) q^{81} + ( -5 \beta_{2} - 5 \beta_{5} ) q^{83} -2 q^{85} + ( -3 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{87} + ( -2 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{89} + ( -2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -4 + 3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{93} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{95} -6 q^{97} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{9} + 8q^{13} - 4q^{21} - 8q^{25} - 16q^{33} - 8q^{37} + 4q^{45} + 16q^{49} + 24q^{57} + 8q^{61} - 12q^{69} - 16q^{85} - 32q^{93} - 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - 3 \nu^{5} - 5 \nu^{4} - 2 \nu^{3} + 2 \nu^{2} - 8 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - \nu^{5} + 3 \nu^{4} + 6 \nu^{3} + 10 \nu^{2} - 8 \nu + 8 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} - 3 \nu^{5} + 5 \nu^{4} - 2 \nu^{3} - 2 \nu^{2} + 8 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} + 3 \nu^{5} - 4 \nu^{4} + 10 \nu^{3} + 24 \nu - 16 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + \nu^{5} + 2 \nu^{4} + 4 \nu^{3} + 12 \nu + 8 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 7 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} - \beta_{6} + 5 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_{1} - 8\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} + 9 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(7 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} - \beta_{4} - \beta_{3} + 5 \beta_{2} + 7 \beta_{1} - 8\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{7} + \beta_{6} - 7 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} - 41 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−1.17915 + 0.780776i
−1.17915 0.780776i
0.599676 + 1.28078i
0.599676 1.28078i
−0.599676 1.28078i
−0.599676 + 1.28078i
1.17915 0.780776i
1.17915 + 0.780776i
0 −1.51022 0.848071i 0 1.00000i 0 3.02045i 0 1.56155 + 2.56155i 0
191.2 0 −1.51022 + 0.848071i 0 1.00000i 0 3.02045i 0 1.56155 2.56155i 0
191.3 0 −0.468213 1.66757i 0 1.00000i 0 0.936426i 0 −2.56155 + 1.56155i 0
191.4 0 −0.468213 + 1.66757i 0 1.00000i 0 0.936426i 0 −2.56155 1.56155i 0
191.5 0 0.468213 1.66757i 0 1.00000i 0 0.936426i 0 −2.56155 1.56155i 0
191.6 0 0.468213 + 1.66757i 0 1.00000i 0 0.936426i 0 −2.56155 + 1.56155i 0
191.7 0 1.51022 0.848071i 0 1.00000i 0 3.02045i 0 1.56155 2.56155i 0
191.8 0 1.51022 + 0.848071i 0 1.00000i 0 3.02045i 0 1.56155 + 2.56155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.h.g 8
3.b odd 2 1 inner 960.2.h.g 8
4.b odd 2 1 inner 960.2.h.g 8
8.b even 2 1 60.2.e.a 8
8.d odd 2 1 60.2.e.a 8
12.b even 2 1 inner 960.2.h.g 8
24.f even 2 1 60.2.e.a 8
24.h odd 2 1 60.2.e.a 8
40.e odd 2 1 300.2.e.c 8
40.f even 2 1 300.2.e.c 8
40.i odd 4 1 300.2.h.a 8
40.i odd 4 1 300.2.h.b 8
40.k even 4 1 300.2.h.a 8
40.k even 4 1 300.2.h.b 8
120.i odd 2 1 300.2.e.c 8
120.m even 2 1 300.2.e.c 8
120.q odd 4 1 300.2.h.a 8
120.q odd 4 1 300.2.h.b 8
120.w even 4 1 300.2.h.a 8
120.w even 4 1 300.2.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.e.a 8 8.b even 2 1
60.2.e.a 8 8.d odd 2 1
60.2.e.a 8 24.f even 2 1
60.2.e.a 8 24.h odd 2 1
300.2.e.c 8 40.e odd 2 1
300.2.e.c 8 40.f even 2 1
300.2.e.c 8 120.i odd 2 1
300.2.e.c 8 120.m even 2 1
300.2.h.a 8 40.i odd 4 1
300.2.h.a 8 40.k even 4 1
300.2.h.a 8 120.q odd 4 1
300.2.h.a 8 120.w even 4 1
300.2.h.b 8 40.i odd 4 1
300.2.h.b 8 40.k even 4 1
300.2.h.b 8 120.q odd 4 1
300.2.h.b 8 120.w even 4 1
960.2.h.g 8 1.a even 1 1 trivial
960.2.h.g 8 3.b odd 2 1 inner
960.2.h.g 8 4.b odd 2 1 inner
960.2.h.g 8 12.b 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 \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{4} + 10 T_{7}^{2} + 8 \)
\( T_{11}^{4} - 20 T_{11}^{2} + 32 \)
\( T_{23}^{4} - 58 T_{23}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 81 + 18 T^{2} + 2 T^{4} + 2 T^{6} + T^{8} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( ( 8 + 10 T^{2} + T^{4} )^{2} \)
$11$ \( ( 32 - 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( -16 - 2 T + T^{2} )^{4} \)
$17$ \( ( 4 + T^{2} )^{4} \)
$19$ \( ( 32 + 20 T^{2} + T^{4} )^{2} \)
$23$ \( ( 8 - 58 T^{2} + T^{4} )^{2} \)
$29$ \( ( 256 + 36 T^{2} + T^{4} )^{2} \)
$31$ \( ( 128 + 28 T^{2} + T^{4} )^{2} \)
$37$ \( ( -16 + 2 T + T^{2} )^{4} \)
$41$ \( ( 64 + 52 T^{2} + T^{4} )^{2} \)
$43$ \( ( 128 + 62 T^{2} + T^{4} )^{2} \)
$47$ \( ( 8 - 10 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2704 + 168 T^{2} + T^{4} )^{2} \)
$59$ \( ( 10368 - 252 T^{2} + T^{4} )^{2} \)
$61$ \( ( -16 - 2 T + T^{2} )^{4} \)
$67$ \( ( 512 + 46 T^{2} + T^{4} )^{2} \)
$71$ \( ( 512 - 56 T^{2} + T^{4} )^{2} \)
$73$ \( ( -68 + T^{2} )^{4} \)
$79$ \( ( 5408 + 148 T^{2} + T^{4} )^{2} \)
$83$ \( ( 5000 - 250 T^{2} + T^{4} )^{2} \)
$89$ \( ( 4096 + 144 T^{2} + T^{4} )^{2} \)
$97$ \( ( 6 + T )^{8} \)
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