Properties

Label 768.2.c.j
Level $768$
Weight $2$
Character orbit 768.c
Analytic conductor $6.133$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -\beta_{1} q^{5} -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -\beta_{1} q^{5} -\beta_{3} q^{7} + 3 q^{9} -2 \beta_{2} q^{11} + \beta_{3} q^{15} + 3 \beta_{1} q^{21} -3 q^{25} -3 \beta_{2} q^{27} + \beta_{1} q^{29} + \beta_{3} q^{31} + 6 q^{33} -8 \beta_{2} q^{35} -3 \beta_{1} q^{45} -17 q^{49} -5 \beta_{1} q^{53} + 2 \beta_{3} q^{55} -6 \beta_{2} q^{59} -3 \beta_{3} q^{63} -14 q^{73} + 3 \beta_{2} q^{75} + 6 \beta_{1} q^{77} -3 \beta_{3} q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} -\beta_{3} q^{87} -3 \beta_{1} q^{93} + 2 q^{97} -6 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 12q^{25} + 24q^{33} - 68q^{49} - 56q^{73} + 36q^{81} + 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
0.517638i
0.517638i
1.93185i
1.93185i
0 −1.73205 0 2.82843i 0 4.89898i 0 3.00000 0
767.2 0 −1.73205 0 2.82843i 0 4.89898i 0 3.00000 0
767.3 0 1.73205 0 2.82843i 0 4.89898i 0 3.00000 0
767.4 0 1.73205 0 2.82843i 0 4.89898i 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
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 768.2.c.j 4
3.b odd 2 1 inner 768.2.c.j 4
4.b odd 2 1 inner 768.2.c.j 4
8.b even 2 1 inner 768.2.c.j 4
8.d odd 2 1 inner 768.2.c.j 4
12.b even 2 1 inner 768.2.c.j 4
16.e even 4 2 384.2.f.a 4
16.f odd 4 2 384.2.f.a 4
24.f even 2 1 inner 768.2.c.j 4
24.h odd 2 1 CM 768.2.c.j 4
48.i odd 4 2 384.2.f.a 4
48.k even 4 2 384.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.f.a 4 16.e even 4 2
384.2.f.a 4 16.f odd 4 2
384.2.f.a 4 48.i odd 4 2
384.2.f.a 4 48.k even 4 2
768.2.c.j 4 1.a even 1 1 trivial
768.2.c.j 4 3.b odd 2 1 inner
768.2.c.j 4 4.b odd 2 1 inner
768.2.c.j 4 8.b even 2 1 inner
768.2.c.j 4 8.d odd 2 1 inner
768.2.c.j 4 12.b even 2 1 inner
768.2.c.j 4 24.f even 2 1 inner
768.2.c.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 \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 8 \)
\( T_{7}^{2} + 24 \)
\( T_{11}^{2} - 12 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 8 + T^{2} )^{2} \)
$7$ \( ( 24 + T^{2} )^{2} \)
$11$ \( ( -12 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 24 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 200 + T^{2} )^{2} \)
$59$ \( ( -108 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 14 + T )^{4} \)
$79$ \( ( 216 + T^{2} )^{2} \)
$83$ \( ( -300 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -2 + T )^{4} \)
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