Properties

Label 768.5.g.c
Level $768$
Weight $5$
Character orbit 768.g
Analytic conductor $79.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.3881316484\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -8 - \beta_{3} ) q^{5} + ( \beta_{1} + 5 \beta_{2} ) q^{7} -27 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -8 - \beta_{3} ) q^{5} + ( \beta_{1} + 5 \beta_{2} ) q^{7} -27 q^{9} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{11} + ( -120 - 2 \beta_{3} ) q^{13} + ( 9 \beta_{1} + 5 \beta_{2} ) q^{15} + ( 174 + 8 \beta_{3} ) q^{17} + ( 24 \beta_{1} + 36 \beta_{2} ) q^{19} + ( 144 + 3 \beta_{3} ) q^{21} + ( 30 \beta_{1} + 54 \beta_{2} ) q^{23} + ( 447 + 16 \beta_{3} ) q^{25} + 27 \beta_{2} q^{27} + ( -712 - 3 \beta_{3} ) q^{29} + ( 41 \beta_{1} - 83 \beta_{2} ) q^{31} + ( -180 - 24 \beta_{3} ) q^{33} + ( -56 \beta_{1} - 136 \beta_{2} ) q^{35} + ( 168 - 20 \beta_{3} ) q^{37} + ( 18 \beta_{1} + 114 \beta_{2} ) q^{39} + ( 126 - 72 \beta_{3} ) q^{41} + ( 72 \beta_{1} - 180 \beta_{2} ) q^{43} + ( 216 + 27 \beta_{3} ) q^{45} + ( 42 \beta_{1} - 558 \beta_{2} ) q^{47} + ( 1297 - 32 \beta_{3} ) q^{49} + ( -72 \beta_{1} - 150 \beta_{2} ) q^{51} + ( -40 + 25 \beta_{3} ) q^{53} + ( 124 \beta_{1} + 908 \beta_{2} ) q^{55} + ( 1188 + 72 \beta_{3} ) q^{57} -252 \beta_{2} q^{59} + ( 1992 - 88 \beta_{3} ) q^{61} + ( -27 \beta_{1} - 135 \beta_{2} ) q^{63} + ( 2976 + 136 \beta_{3} ) q^{65} + ( 96 \beta_{1} - 468 \beta_{2} ) q^{67} + ( 1728 + 90 \beta_{3} ) q^{69} + ( -318 \beta_{1} + 810 \beta_{2} ) q^{71} + ( 1282 + 32 \beta_{3} ) q^{73} + ( -144 \beta_{1} - 399 \beta_{2} ) q^{75} + ( 3648 + 148 \beta_{3} ) q^{77} + ( -135 \beta_{1} - 1539 \beta_{2} ) q^{79} + 729 q^{81} + ( 280 \beta_{1} - 508 \beta_{2} ) q^{83} + ( -9456 - 238 \beta_{3} ) q^{85} + ( 27 \beta_{1} + 703 \beta_{2} ) q^{87} + ( 3954 + 80 \beta_{3} ) q^{89} + ( -216 \beta_{1} - 792 \beta_{2} ) q^{91} + ( -1872 + 123 \beta_{3} ) q^{93} + ( -588 \beta_{1} - 2844 \beta_{2} ) q^{95} + ( -5650 - 80 \beta_{3} ) q^{97} + ( 216 \beta_{1} + 108 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{5} - 108 q^{9} + O(q^{10}) \) \( 4 q - 32 q^{5} - 108 q^{9} - 480 q^{13} + 696 q^{17} + 576 q^{21} + 1788 q^{25} - 2848 q^{29} - 720 q^{33} + 672 q^{37} + 504 q^{41} + 864 q^{45} + 5188 q^{49} - 160 q^{53} + 4752 q^{57} + 7968 q^{61} + 11904 q^{65} + 6912 q^{69} + 5128 q^{73} + 14592 q^{77} + 2916 q^{81} - 37824 q^{85} + 15816 q^{89} - 7488 q^{93} - 22600 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{3} + 2 \nu^{2} + 56 \nu + 7 \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( 6 \nu^{2} + 21 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( -12 \nu^{3} \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + 3 \beta_{1}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - 21\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{3}\)\()/12\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
0 5.19615i 0 −39.7490 0 46.0431i 0 −27.0000 0
511.2 0 5.19615i 0 23.7490 0 9.38251i 0 −27.0000 0
511.3 0 5.19615i 0 −39.7490 0 46.0431i 0 −27.0000 0
511.4 0 5.19615i 0 23.7490 0 9.38251i 0 −27.0000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.5.g.c 4
4.b odd 2 1 inner 768.5.g.c 4
8.b even 2 1 768.5.g.g 4
8.d odd 2 1 768.5.g.g 4
16.e even 4 2 384.5.b.c 8
16.f odd 4 2 384.5.b.c 8
48.i odd 4 2 1152.5.b.k 8
48.k even 4 2 1152.5.b.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.c 8 16.e even 4 2
384.5.b.c 8 16.f odd 4 2
768.5.g.c 4 1.a even 1 1 trivial
768.5.g.c 4 4.b odd 2 1 inner
768.5.g.g 4 8.b even 2 1
768.5.g.g 4 8.d odd 2 1
1152.5.b.k 8 48.i odd 4 2
1152.5.b.k 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 16 T_{5} - 944 \) acting on \(S_{5}^{\mathrm{new}}(768, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 27 + T^{2} )^{2} \)
$5$ \( ( -944 + 16 T + T^{2} )^{2} \)
$7$ \( 186624 + 2208 T^{2} + T^{4} \)
$11$ \( 412252416 + 45408 T^{2} + T^{4} \)
$13$ \( ( 10368 + 240 T + T^{2} )^{2} \)
$17$ \( ( -34236 - 348 T + T^{2} )^{2} \)
$19$ \( 19955517696 + 491616 T^{2} + T^{4} \)
$23$ \( 36790308864 + 825984 T^{2} + T^{4} \)
$29$ \( ( 497872 + 1424 T + T^{2} )^{2} \)
$31$ \( 189245880576 + 1389216 T^{2} + T^{4} \)
$37$ \( ( -374976 - 336 T + T^{2} )^{2} \)
$41$ \( ( -5209596 - 252 T + T^{2} )^{2} \)
$43$ \( 1176686901504 + 4797792 T^{2} + T^{4} \)
$47$ \( 54724012314624 + 17165952 T^{2} + T^{4} \)
$53$ \( ( -628400 + 80 T + T^{2} )^{2} \)
$59$ \( ( 1714608 + T^{2} )^{2} \)
$61$ \( ( -3837888 - 3984 T + T^{2} )^{2} \)
$67$ \( 4145361152256 + 16458336 T^{2} + T^{4} \)
$71$ \( 424196534145024 + 94718592 T^{2} + T^{4} \)
$73$ \( ( 611332 - 2564 T + T^{2} )^{2} \)
$79$ \( 3797136795711744 + 147736224 T^{2} + T^{4} \)
$83$ \( 470880972843264 + 61970016 T^{2} + T^{4} \)
$89$ \( ( 9182916 - 7908 T + T^{2} )^{2} \)
$97$ \( ( 25471300 + 11300 T + T^{2} )^{2} \)
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