Properties

Label 96.8.f.b
Level 96
Weight 8
Character orbit 96.f
Analytic conductor 29.989
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 96.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9889624465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-26})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + ( 9 + 9 \beta_{1} ) q^{3} -5 \beta_{2} q^{5} + 31 \beta_{3} q^{7} + ( -2025 + 162 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 9 + 9 \beta_{1} ) q^{3} -5 \beta_{2} q^{5} + 31 \beta_{3} q^{7} + ( -2025 + 162 \beta_{1} ) q^{9} + 350 \beta_{1} q^{11} -203 \beta_{3} q^{13} + ( -45 \beta_{2} - 90 \beta_{3} ) q^{15} + 5404 \beta_{1} q^{17} -11570 q^{19} + ( -3627 \beta_{2} + 279 \beta_{3} ) q^{21} + 2834 \beta_{2} q^{23} -68525 q^{25} + ( -56133 - 16767 \beta_{1} ) q^{27} + 2795 \beta_{2} q^{29} -1435 \beta_{3} q^{31} + ( -81900 + 3150 \beta_{1} ) q^{33} -29760 \beta_{1} q^{35} -5661 \beta_{3} q^{37} + ( 23751 \beta_{2} - 1827 \beta_{3} ) q^{39} -79160 \beta_{1} q^{41} -495062 q^{43} + ( 10125 \beta_{2} - 1620 \beta_{3} ) q^{45} -57968 \beta_{2} q^{47} -1575113 q^{49} + ( -1264536 + 48636 \beta_{1} ) q^{51} + 29211 \beta_{2} q^{53} -3500 \beta_{3} q^{55} + ( -104130 - 104130 \beta_{1} ) q^{57} + 282506 \beta_{1} q^{59} -35755 \beta_{3} q^{61} + ( -65286 \beta_{2} - 62775 \beta_{3} ) q^{63} + 194880 \beta_{1} q^{65} -1400126 q^{67} + ( 25506 \beta_{2} + 51012 \beta_{3} ) q^{69} + 182154 \beta_{2} q^{71} -2223598 q^{73} + ( -616725 - 616725 \beta_{1} ) q^{75} -141050 \beta_{2} q^{77} + 111901 \beta_{3} q^{79} + ( 3418281 - 656100 \beta_{1} ) q^{81} -590962 \beta_{1} q^{83} -54040 \beta_{3} q^{85} + ( 25155 \beta_{2} + 50310 \beta_{3} ) q^{87} + 1146364 \beta_{1} q^{89} + 15707328 q^{91} + ( 167895 \beta_{2} - 12915 \beta_{3} ) q^{93} + 57850 \beta_{2} q^{95} + 6867926 q^{97} + ( -1474200 - 708750 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 36q^{3} - 8100q^{9} + O(q^{10}) \) \( 4q + 36q^{3} - 8100q^{9} - 46280q^{19} - 274100q^{25} - 224532q^{27} - 327600q^{33} - 1980248q^{43} - 6300452q^{49} - 5058144q^{51} - 416520q^{57} - 5600504q^{67} - 8894392q^{73} - 2466900q^{75} + 13673124q^{81} + 62829312q^{91} + 27471704q^{97} - 5896800q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 10 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 18 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\( -\nu^{3} - 2 \nu \)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} + 40 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 8 \beta_{1}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 40\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{2} - 8 \beta_{1}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.22474 2.54951i
−1.22474 2.54951i
1.22474 + 2.54951i
−1.22474 + 2.54951i
0 9.00000 45.8912i 0 −97.9796 0 1548.76i 0 −2025.00 826.041i 0
47.2 0 9.00000 45.8912i 0 97.9796 0 1548.76i 0 −2025.00 826.041i 0
47.3 0 9.00000 + 45.8912i 0 −97.9796 0 1548.76i 0 −2025.00 + 826.041i 0
47.4 0 9.00000 + 45.8912i 0 97.9796 0 1548.76i 0 −2025.00 + 826.041i 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.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.8.f.b 4
3.b odd 2 1 inner 96.8.f.b 4
4.b odd 2 1 24.8.f.b 4
8.b even 2 1 24.8.f.b 4
8.d odd 2 1 inner 96.8.f.b 4
12.b even 2 1 24.8.f.b 4
24.f even 2 1 inner 96.8.f.b 4
24.h odd 2 1 24.8.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.b 4 4.b odd 2 1
24.8.f.b 4 8.b even 2 1
24.8.f.b 4 12.b even 2 1
24.8.f.b 4 24.h odd 2 1
96.8.f.b 4 1.a even 1 1 trivial
96.8.f.b 4 3.b odd 2 1 inner
96.8.f.b 4 8.d odd 2 1 inner
96.8.f.b 4 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9600 \) acting on \(S_{8}^{\mathrm{new}}(96, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - 18 T + 2187 T^{2} )^{2} \)
$5$ \( ( 1 + 146650 T^{2} + 6103515625 T^{4} )^{2} \)
$7$ \( ( 1 + 751570 T^{2} + 678223072849 T^{4} )^{2} \)
$11$ \( ( 1 - 35789342 T^{2} + 379749833583241 T^{4} )^{2} \)
$13$ \( ( 1 - 22639370 T^{2} + 3937376385699289 T^{4} )^{2} \)
$17$ \( ( 1 - 61393730 T^{2} + 168377826559400929 T^{4} )^{2} \)
$19$ \( ( 1 + 11570 T + 893871739 T^{2} )^{4} \)
$23$ \( ( 1 + 3725533390 T^{2} + 11592836324538749809 T^{4} )^{2} \)
$29$ \( ( 1 + 31499935018 T^{2} + \)\(29\!\cdots\!81\)\( T^{4} )^{2} \)
$31$ \( ( 1 - 49885402622 T^{2} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$37$ \( ( 1 - 109874639450 T^{2} + \)\(90\!\cdots\!89\)\( T^{4} )^{2} \)
$41$ \( ( 1 - 226584602162 T^{2} + \)\(37\!\cdots\!61\)\( T^{4} )^{2} \)
$43$ \( ( 1 + 495062 T + 271818611107 T^{2} )^{4} \)
$47$ \( ( 1 - 277104744290 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} )^{2} \)
$53$ \( ( 1 + 2021761791610 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} )^{2} \)
$59$ \( ( 1 - 2902252328702 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \)
$61$ \( ( 1 - 3094549289642 T^{2} + \)\(98\!\cdots\!41\)\( T^{4} )^{2} \)
$67$ \( ( 1 + 1400126 T + 6060711605323 T^{2} )^{4} \)
$71$ \( ( 1 + 5449089705838 T^{2} + \)\(82\!\cdots\!81\)\( T^{4} )^{2} \)
$73$ \( ( 1 + 2223598 T + 11047398519097 T^{2} )^{4} \)
$79$ \( ( 1 - 7153320805022 T^{2} + \)\(36\!\cdots\!81\)\( T^{4} )^{2} \)
$83$ \( ( 1 - 45191963757710 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} )^{2} \)
$89$ \( ( 1 - 54294758858162 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} )^{2} \)
$97$ \( ( 1 - 6867926 T + 80798284478113 T^{2} )^{4} \)
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