Properties

Label 768.3.g.g
Level $768$
Weight $3$
Character orbit 768.g
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( 4 + \beta_{3} ) q^{5} -\beta_{1} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( 4 + \beta_{3} ) q^{5} -\beta_{1} q^{7} -3 q^{9} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{11} + ( 4 - 2 \beta_{3} ) q^{13} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{15} + ( -2 - 4 \beta_{3} ) q^{17} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{19} + \beta_{3} q^{21} + ( 2 \beta_{1} + 16 \beta_{2} ) q^{23} + ( 15 + 8 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} + ( 20 - 5 \beta_{3} ) q^{29} + ( -9 \beta_{1} + 16 \beta_{2} ) q^{31} + ( -12 - 4 \beta_{3} ) q^{33} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{35} + ( 4 + 8 \beta_{3} ) q^{37} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{39} + ( -18 + 4 \beta_{3} ) q^{41} + ( -20 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -12 - 3 \beta_{3} ) q^{45} + ( 6 \beta_{1} + 32 \beta_{2} ) q^{47} + 41 q^{49} + ( -12 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 36 + 7 \beta_{3} ) q^{53} + ( 28 \beta_{1} + 48 \beta_{2} ) q^{55} + ( -12 - 4 \beta_{3} ) q^{57} -20 \beta_{2} q^{59} + ( -44 - 4 \beta_{3} ) q^{61} + 3 \beta_{1} q^{63} + ( -32 - 4 \beta_{3} ) q^{65} + ( 16 \beta_{1} - 28 \beta_{2} ) q^{67} + ( -48 - 2 \beta_{3} ) q^{69} + ( -34 \beta_{1} + 16 \beta_{2} ) q^{71} + 10 q^{73} + ( 24 \beta_{1} + 15 \beta_{2} ) q^{75} + ( 32 + 4 \beta_{3} ) q^{77} + ( -17 \beta_{1} + 48 \beta_{2} ) q^{79} + 9 q^{81} + ( -12 \beta_{1} + 44 \beta_{2} ) q^{83} + ( -104 - 18 \beta_{3} ) q^{85} + ( -15 \beta_{1} + 20 \beta_{2} ) q^{87} + ( 34 - 8 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 16 \beta_{2} ) q^{91} + ( -48 + 9 \beta_{3} ) q^{93} + ( 28 \beta_{1} + 48 \beta_{2} ) q^{95} + ( -66 + 8 \beta_{3} ) q^{97} + ( -12 \beta_{1} - 12 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{5} - 12 q^{9} + O(q^{10}) \) \( 4 q + 16 q^{5} - 12 q^{9} + 16 q^{13} - 8 q^{17} + 60 q^{25} + 80 q^{29} - 48 q^{33} + 16 q^{37} - 72 q^{41} - 48 q^{45} + 164 q^{49} + 144 q^{53} - 48 q^{57} - 176 q^{61} - 128 q^{65} - 192 q^{69} + 40 q^{73} + 128 q^{77} + 36 q^{81} - 416 q^{85} + 136 q^{89} - 192 q^{93} - 264 q^{97} + O(q^{100}) \)

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

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

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
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 1.73205i 0 −0.898979 0 2.82843i 0 −3.00000 0
511.2 0 1.73205i 0 8.89898 0 2.82843i 0 −3.00000 0
511.3 0 1.73205i 0 −0.898979 0 2.82843i 0 −3.00000 0
511.4 0 1.73205i 0 8.89898 0 2.82843i 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
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 768.3.g.g 4
3.b odd 2 1 2304.3.g.o 4
4.b odd 2 1 inner 768.3.g.g 4
8.b even 2 1 768.3.g.c 4
8.d odd 2 1 768.3.g.c 4
12.b even 2 1 2304.3.g.o 4
16.e even 4 2 384.3.b.c 8
16.f odd 4 2 384.3.b.c 8
24.f even 2 1 2304.3.g.x 4
24.h odd 2 1 2304.3.g.x 4
48.i odd 4 2 1152.3.b.j 8
48.k even 4 2 1152.3.b.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.c 8 16.e even 4 2
384.3.b.c 8 16.f odd 4 2
768.3.g.c 4 8.b even 2 1
768.3.g.c 4 8.d odd 2 1
768.3.g.g 4 1.a even 1 1 trivial
768.3.g.g 4 4.b odd 2 1 inner
1152.3.b.j 8 48.i odd 4 2
1152.3.b.j 8 48.k even 4 2
2304.3.g.o 4 3.b odd 2 1
2304.3.g.o 4 12.b even 2 1
2304.3.g.x 4 24.f even 2 1
2304.3.g.x 4 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} - 8 T_{5} - 8 \)
\( T_{7}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -8 - 8 T + T^{2} )^{2} \)
$7$ \( ( 8 + T^{2} )^{2} \)
$11$ \( 6400 + 352 T^{2} + T^{4} \)
$13$ \( ( -80 - 8 T + T^{2} )^{2} \)
$17$ \( ( -380 + 4 T + T^{2} )^{2} \)
$19$ \( 6400 + 352 T^{2} + T^{4} \)
$23$ \( 541696 + 1600 T^{2} + T^{4} \)
$29$ \( ( -200 - 40 T + T^{2} )^{2} \)
$31$ \( 14400 + 2832 T^{2} + T^{4} \)
$37$ \( ( -1520 - 8 T + T^{2} )^{2} \)
$41$ \( ( -60 + 36 T + T^{2} )^{2} \)
$43$ \( 9935104 + 6496 T^{2} + T^{4} \)
$47$ \( 7750656 + 6720 T^{2} + T^{4} \)
$53$ \( ( 120 - 72 T + T^{2} )^{2} \)
$59$ \( ( 1200 + T^{2} )^{2} \)
$61$ \( ( 1552 + 88 T + T^{2} )^{2} \)
$67$ \( 92416 + 8800 T^{2} + T^{4} \)
$71$ \( 71910400 + 20032 T^{2} + T^{4} \)
$73$ \( ( -10 + T )^{4} \)
$79$ \( 21160000 + 18448 T^{2} + T^{4} \)
$83$ \( 21678336 + 13920 T^{2} + T^{4} \)
$89$ \( ( -380 - 68 T + T^{2} )^{2} \)
$97$ \( ( 2820 + 132 T + T^{2} )^{2} \)
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