Properties

Label 96.3.b.a
Level $96$
Weight $3$
Character orbit 96.b
Analytic conductor $2.616$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 96.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.61581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4752.1
Defining polynomial: \(x^{4} + 3 x^{2} - 6 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{3} ) q^{7} + 3 q^{9} + 8 q^{11} -2 \beta_{3} q^{13} + ( -\beta_{2} + \beta_{3} ) q^{15} + ( -2 - 8 \beta_{1} ) q^{17} + ( -8 + 4 \beta_{1} ) q^{19} + ( \beta_{2} + 2 \beta_{3} ) q^{21} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -11 + 16 \beta_{1} ) q^{25} -3 \beta_{1} q^{27} + ( 3 \beta_{2} + 4 \beta_{3} ) q^{29} + ( 5 \beta_{2} - 3 \beta_{3} ) q^{31} -8 \beta_{1} q^{33} + ( -24 - 4 \beta_{1} ) q^{35} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{37} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 10 + 24 \beta_{1} ) q^{41} + ( -8 + 12 \beta_{1} ) q^{43} -3 \beta_{2} q^{45} + ( -6 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -11 - 32 \beta_{1} ) q^{49} + ( 24 + 2 \beta_{1} ) q^{51} + ( \beta_{2} - 4 \beta_{3} ) q^{53} -8 \beta_{2} q^{55} + ( -12 + 8 \beta_{1} ) q^{57} + ( 32 - 12 \beta_{1} ) q^{59} + ( -6 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{63} + ( 24 - 40 \beta_{1} ) q^{65} + ( 64 - 12 \beta_{1} ) q^{67} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{69} + ( -2 \beta_{2} - 10 \beta_{3} ) q^{71} + ( 50 + 32 \beta_{1} ) q^{73} + ( -48 + 11 \beta_{1} ) q^{75} + ( -8 \beta_{2} - 8 \beta_{3} ) q^{77} + ( \beta_{2} + 9 \beta_{3} ) q^{79} + 9 q^{81} + ( -40 + 16 \beta_{1} ) q^{83} + ( -6 \beta_{2} + 8 \beta_{3} ) q^{85} + ( -5 \beta_{2} - 7 \beta_{3} ) q^{87} + ( -50 + 48 \beta_{1} ) q^{89} + ( -72 - 56 \beta_{1} ) q^{91} + ( 11 \beta_{2} - 2 \beta_{3} ) q^{93} + ( 12 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 14 - 48 \beta_{1} ) q^{97} + 24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 32q^{11} - 8q^{17} - 32q^{19} - 44q^{25} - 96q^{35} + 40q^{41} - 32q^{43} - 44q^{49} + 96q^{51} - 48q^{57} + 128q^{59} + 96q^{65} + 256q^{67} + 200q^{73} - 192q^{75} + 36q^{81} - 160q^{83} - 200q^{89} - 288q^{91} + 56q^{97} + 96q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} - 6 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 6 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} + 8 \nu - 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + 5 \nu^{2} + 8 \nu - 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - \beta_{2} + 4 \beta_{1} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - \beta_{2} - 12 \beta_{1} + 18\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.866025 + 0.719687i
0.866025 0.719687i
−0.866025 + 1.99551i
−0.866025 1.99551i
0 −1.73205 0 2.87875i 0 10.7436i 0 3.00000 0
79.2 0 −1.73205 0 2.87875i 0 10.7436i 0 3.00000 0
79.3 0 1.73205 0 7.98203i 0 2.13878i 0 3.00000 0
79.4 0 1.73205 0 7.98203i 0 2.13878i 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
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.3.b.a 4
3.b odd 2 1 288.3.b.b 4
4.b odd 2 1 24.3.b.a 4
5.b even 2 1 2400.3.g.a 4
5.c odd 4 2 2400.3.p.a 8
8.b even 2 1 24.3.b.a 4
8.d odd 2 1 inner 96.3.b.a 4
12.b even 2 1 72.3.b.b 4
16.e even 4 2 768.3.g.h 8
16.f odd 4 2 768.3.g.h 8
20.d odd 2 1 600.3.g.a 4
20.e even 4 2 600.3.p.a 8
24.f even 2 1 288.3.b.b 4
24.h odd 2 1 72.3.b.b 4
40.e odd 2 1 2400.3.g.a 4
40.f even 2 1 600.3.g.a 4
40.i odd 4 2 600.3.p.a 8
40.k even 4 2 2400.3.p.a 8
48.i odd 4 2 2304.3.g.z 8
48.k even 4 2 2304.3.g.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a 4 4.b odd 2 1
24.3.b.a 4 8.b even 2 1
72.3.b.b 4 12.b even 2 1
72.3.b.b 4 24.h odd 2 1
96.3.b.a 4 1.a even 1 1 trivial
96.3.b.a 4 8.d odd 2 1 inner
288.3.b.b 4 3.b odd 2 1
288.3.b.b 4 24.f even 2 1
600.3.g.a 4 20.d odd 2 1
600.3.g.a 4 40.f even 2 1
600.3.p.a 8 20.e even 4 2
600.3.p.a 8 40.i odd 4 2
768.3.g.h 8 16.e even 4 2
768.3.g.h 8 16.f odd 4 2
2304.3.g.z 8 48.i odd 4 2
2304.3.g.z 8 48.k even 4 2
2400.3.g.a 4 5.b even 2 1
2400.3.g.a 4 40.e odd 2 1
2400.3.p.a 8 5.c odd 4 2
2400.3.p.a 8 40.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( 528 + 72 T^{2} + T^{4} \)
$7$ \( 528 + 120 T^{2} + T^{4} \)
$11$ \( ( -8 + T )^{4} \)
$13$ \( 33792 + 384 T^{2} + T^{4} \)
$17$ \( ( -188 + 4 T + T^{2} )^{2} \)
$19$ \( ( 16 + 16 T + T^{2} )^{2} \)
$23$ \( 8448 + 480 T^{2} + T^{4} \)
$29$ \( 528 + 1608 T^{2} + T^{4} \)
$31$ \( 279312 + 3384 T^{2} + T^{4} \)
$37$ \( 76032 + 864 T^{2} + T^{4} \)
$41$ \( ( -1628 - 20 T + T^{2} )^{2} \)
$43$ \( ( -368 + 16 T + T^{2} )^{2} \)
$47$ \( 8448 + 3552 T^{2} + T^{4} \)
$53$ \( 803088 + 1800 T^{2} + T^{4} \)
$59$ \( ( 592 - 64 T + T^{2} )^{2} \)
$61$ \( 8448 + 3552 T^{2} + T^{4} \)
$67$ \( ( 3664 - 128 T + T^{2} )^{2} \)
$71$ \( 12849408 + 8928 T^{2} + T^{4} \)
$73$ \( ( -572 - 100 T + T^{2} )^{2} \)
$79$ \( 10797072 + 7416 T^{2} + T^{4} \)
$83$ \( ( 832 + 80 T + T^{2} )^{2} \)
$89$ \( ( -4412 + 100 T + T^{2} )^{2} \)
$97$ \( ( -6716 - 28 T + T^{2} )^{2} \)
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