Properties

Label 768.3.e.j
Level $768$
Weight $3$
Character orbit 768.e
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{3} q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{3} q^{7} -9 q^{9} -10 \beta_{1} q^{11} -3 \beta_{3} q^{15} -3 \beta_{2} q^{21} -71 q^{25} -27 \beta_{1} q^{27} + 3 \beta_{2} q^{29} + 5 \beta_{3} q^{31} + 30 q^{33} -96 \beta_{1} q^{35} -9 \beta_{2} q^{45} + 47 q^{49} + 5 \beta_{2} q^{53} + 10 \beta_{3} q^{55} + 10 \beta_{1} q^{59} + 9 \beta_{3} q^{63} + 50 q^{73} -213 \beta_{1} q^{75} + 10 \beta_{2} q^{77} -15 \beta_{3} q^{79} + 81 q^{81} -134 \beta_{1} q^{83} -9 \beta_{3} q^{87} + 15 \beta_{2} q^{93} -190 q^{97} + 90 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 36q^{9} + O(q^{10}) \) \( 4q - 36q^{9} - 284q^{25} + 120q^{33} + 188q^{49} + 200q^{73} + 324q^{81} - 760q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.22474 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 3.00000i 0 9.79796i 0 −9.79796 0 −9.00000 0
257.2 0 3.00000i 0 9.79796i 0 9.79796 0 −9.00000 0
257.3 0 3.00000i 0 9.79796i 0 9.79796 0 −9.00000 0
257.4 0 3.00000i 0 9.79796i 0 −9.79796 0 −9.00000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 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 768.3.e.j 4
3.b odd 2 1 inner 768.3.e.j 4
4.b odd 2 1 inner 768.3.e.j 4
8.b even 2 1 inner 768.3.e.j 4
8.d odd 2 1 inner 768.3.e.j 4
12.b even 2 1 inner 768.3.e.j 4
16.e even 4 1 384.3.h.a 2
16.e even 4 1 384.3.h.d yes 2
16.f odd 4 1 384.3.h.a 2
16.f odd 4 1 384.3.h.d yes 2
24.f even 2 1 inner 768.3.e.j 4
24.h odd 2 1 CM 768.3.e.j 4
48.i odd 4 1 384.3.h.a 2
48.i odd 4 1 384.3.h.d yes 2
48.k even 4 1 384.3.h.a 2
48.k even 4 1 384.3.h.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.a 2 16.e even 4 1
384.3.h.a 2 16.f odd 4 1
384.3.h.a 2 48.i odd 4 1
384.3.h.a 2 48.k even 4 1
384.3.h.d yes 2 16.e even 4 1
384.3.h.d yes 2 16.f odd 4 1
384.3.h.d yes 2 48.i odd 4 1
384.3.h.d yes 2 48.k even 4 1
768.3.e.j 4 1.a even 1 1 trivial
768.3.e.j 4 3.b odd 2 1 inner
768.3.e.j 4 4.b odd 2 1 inner
768.3.e.j 4 8.b even 2 1 inner
768.3.e.j 4 8.d odd 2 1 inner
768.3.e.j 4 12.b even 2 1 inner
768.3.e.j 4 24.f even 2 1 inner
768.3.e.j 4 24.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 96 \)
\( T_{7}^{2} - 96 \)
\( T_{11}^{2} + 100 \)
\( T_{19} \)
\( T_{37} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( ( 96 + T^{2} )^{2} \)
$7$ \( ( -96 + T^{2} )^{2} \)
$11$ \( ( 100 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 864 + T^{2} )^{2} \)
$31$ \( ( -2400 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 2400 + T^{2} )^{2} \)
$59$ \( ( 100 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -50 + T )^{4} \)
$79$ \( ( -21600 + T^{2} )^{2} \)
$83$ \( ( 17956 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 190 + T )^{4} \)
show more
show less