Properties

Label 96.3.h.c
Level 96
Weight 3
Character orbit 96.h
Analytic conductor 2.616
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.61581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 16\)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} ) q^{3} -2 \beta_{1} q^{5} -4 q^{7} + ( -5 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} ) q^{3} -2 \beta_{1} q^{5} -4 q^{7} + ( -5 + \beta_{3} ) q^{9} -3 \beta_{1} q^{11} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{13} + ( 8 + 2 \beta_{3} ) q^{15} -2 \beta_{3} q^{17} + ( \beta_{1} + 2 \beta_{2} ) q^{19} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{21} -4 \beta_{3} q^{23} + 7 q^{25} + ( 10 \beta_{1} + \beta_{2} ) q^{27} + 6 \beta_{1} q^{29} + 4 q^{31} + ( 12 + 3 \beta_{3} ) q^{33} + 8 \beta_{1} q^{35} + ( 10 \beta_{1} + 20 \beta_{2} ) q^{37} + ( -28 + 2 \beta_{3} ) q^{39} -4 \beta_{3} q^{41} + ( -\beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{45} -33 q^{49} + ( -10 \beta_{1} + 8 \beta_{2} ) q^{51} -18 \beta_{1} q^{53} + 48 q^{55} + ( 14 - \beta_{3} ) q^{57} -17 \beta_{1} q^{59} + ( -18 \beta_{1} - 36 \beta_{2} ) q^{61} + ( 20 - 4 \beta_{3} ) q^{63} + 8 \beta_{3} q^{65} + ( -9 \beta_{1} - 18 \beta_{2} ) q^{67} + ( -20 \beta_{1} + 16 \beta_{2} ) q^{69} + 12 \beta_{3} q^{71} -6 q^{73} + ( -7 \beta_{1} - 7 \beta_{2} ) q^{75} + 12 \beta_{1} q^{77} -124 q^{79} + ( -31 - 10 \beta_{3} ) q^{81} + \beta_{1} q^{83} + ( 16 \beta_{1} + 32 \beta_{2} ) q^{85} + ( -24 - 6 \beta_{3} ) q^{87} + 14 \beta_{3} q^{89} + ( 8 \beta_{1} + 16 \beta_{2} ) q^{91} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{93} -4 \beta_{3} q^{95} + 118 q^{97} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{7} - 20q^{9} + O(q^{10}) \) \( 4q - 16q^{7} - 20q^{9} + 32q^{15} + 28q^{25} + 16q^{31} + 48q^{33} - 112q^{39} - 132q^{49} + 192q^{55} + 56q^{57} + 80q^{63} - 24q^{73} - 496q^{79} - 124q^{81} - 96q^{87} + 472q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 2 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} + 2 \nu + 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 10 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + \beta_{1} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} - 5 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.707107 + 1.87083i
0.707107 1.87083i
−0.707107 1.87083i
−0.707107 + 1.87083i
0 −1.41421 2.64575i 0 −5.65685 0 −4.00000 0 −5.00000 + 7.48331i 0
17.2 0 −1.41421 + 2.64575i 0 −5.65685 0 −4.00000 0 −5.00000 7.48331i 0
17.3 0 1.41421 2.64575i 0 5.65685 0 −4.00000 0 −5.00000 7.48331i 0
17.4 0 1.41421 + 2.64575i 0 5.65685 0 −4.00000 0 −5.00000 + 7.48331i 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.h.c 4
3.b odd 2 1 inner 96.3.h.c 4
4.b odd 2 1 24.3.h.c 4
8.b even 2 1 inner 96.3.h.c 4
8.d odd 2 1 24.3.h.c 4
12.b even 2 1 24.3.h.c 4
16.e even 4 2 768.3.e.l 4
16.f odd 4 2 768.3.e.i 4
24.f even 2 1 24.3.h.c 4
24.h odd 2 1 inner 96.3.h.c 4
48.i odd 4 2 768.3.e.l 4
48.k even 4 2 768.3.e.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.c 4 4.b odd 2 1
24.3.h.c 4 8.d odd 2 1
24.3.h.c 4 12.b even 2 1
24.3.h.c 4 24.f even 2 1
96.3.h.c 4 1.a even 1 1 trivial
96.3.h.c 4 3.b odd 2 1 inner
96.3.h.c 4 8.b even 2 1 inner
96.3.h.c 4 24.h odd 2 1 inner
768.3.e.i 4 16.f odd 4 2
768.3.e.i 4 48.k even 4 2
768.3.e.l 4 16.e even 4 2
768.3.e.l 4 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 32 \) acting on \(S_{3}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 10 T^{2} + 81 T^{4} \)
$5$ \( ( 1 + 18 T^{2} + 625 T^{4} )^{2} \)
$7$ \( ( 1 + 4 T + 49 T^{2} )^{4} \)
$11$ \( ( 1 + 170 T^{2} + 14641 T^{4} )^{2} \)
$13$ \( ( 1 - 226 T^{2} + 28561 T^{4} )^{2} \)
$17$ \( ( 1 - 354 T^{2} + 83521 T^{4} )^{2} \)
$19$ \( ( 1 - 694 T^{2} + 130321 T^{4} )^{2} \)
$23$ \( ( 1 - 162 T^{2} + 279841 T^{4} )^{2} \)
$29$ \( ( 1 + 1394 T^{2} + 707281 T^{4} )^{2} \)
$31$ \( ( 1 - 4 T + 961 T^{2} )^{4} \)
$37$ \( ( 1 + 62 T^{2} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 - 2466 T^{2} + 2825761 T^{4} )^{2} \)
$43$ \( ( 1 - 3670 T^{2} + 3418801 T^{4} )^{2} \)
$47$ \( ( 1 - 47 T )^{4}( 1 + 47 T )^{4} \)
$53$ \( ( 1 + 3026 T^{2} + 7890481 T^{4} )^{2} \)
$59$ \( ( 1 + 4650 T^{2} + 12117361 T^{4} )^{2} \)
$61$ \( ( 1 + 1630 T^{2} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 - 6710 T^{2} + 20151121 T^{4} )^{2} \)
$71$ \( ( 1 - 110 T + 5041 T^{2} )^{2}( 1 + 110 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 + 6 T + 5329 T^{2} )^{4} \)
$79$ \( ( 1 + 124 T + 6241 T^{2} )^{4} \)
$83$ \( ( 1 + 13770 T^{2} + 47458321 T^{4} )^{2} \)
$89$ \( ( 1 - 4866 T^{2} + 62742241 T^{4} )^{2} \)
$97$ \( ( 1 - 118 T + 9409 T^{2} )^{4} \)
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