Properties

Label 768.4.a.d
Level $768$
Weight $4$
Character orbit 768.a
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{3} + 8 q^{5} + 12 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 8 q^{5} + 12 q^{7} + 9 q^{9} - 12 q^{11} - 20 q^{13} + 24 q^{15} + 62 q^{17} + 108 q^{19} + 36 q^{21} - 72 q^{23} - 61 q^{25} + 27 q^{27} + 128 q^{29} + 204 q^{31} - 36 q^{33} + 96 q^{35} + 228 q^{37} - 60 q^{39} + 22 q^{41} - 204 q^{43} + 72 q^{45} + 600 q^{47} - 199 q^{49} + 186 q^{51} - 256 q^{53} - 96 q^{55} + 324 q^{57} - 828 q^{59} + 84 q^{61} + 108 q^{63} - 160 q^{65} + 348 q^{67} - 216 q^{69} + 456 q^{71} - 822 q^{73} - 183 q^{75} - 144 q^{77} + 1356 q^{79} + 81 q^{81} + 108 q^{83} + 496 q^{85} + 384 q^{87} + 938 q^{89} - 240 q^{91} + 612 q^{93} + 864 q^{95} + 1278 q^{97} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 3.00000 0 8.00000 0 12.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.a.d 1
3.b odd 2 1 2304.4.a.f 1
4.b odd 2 1 768.4.a.b 1
8.b even 2 1 768.4.a.a 1
8.d odd 2 1 768.4.a.c 1
12.b even 2 1 2304.4.a.e 1
16.e even 4 2 384.4.d.a 2
16.f odd 4 2 384.4.d.b yes 2
24.f even 2 1 2304.4.a.k 1
24.h odd 2 1 2304.4.a.l 1
48.i odd 4 2 1152.4.d.b 2
48.k even 4 2 1152.4.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 16.e even 4 2
384.4.d.b yes 2 16.f odd 4 2
768.4.a.a 1 8.b even 2 1
768.4.a.b 1 4.b odd 2 1
768.4.a.c 1 8.d odd 2 1
768.4.a.d 1 1.a even 1 1 trivial
1152.4.d.b 2 48.i odd 4 2
1152.4.d.g 2 48.k even 4 2
2304.4.a.e 1 12.b even 2 1
2304.4.a.f 1 3.b odd 2 1
2304.4.a.k 1 24.f even 2 1
2304.4.a.l 1 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5} - 8 \) Copy content Toggle raw display
\( T_{7} - 12 \) Copy content Toggle raw display
\( T_{11} + 12 \) Copy content Toggle raw display
\( T_{19} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 8 \) Copy content Toggle raw display
$7$ \( T - 12 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T + 20 \) Copy content Toggle raw display
$17$ \( T - 62 \) Copy content Toggle raw display
$19$ \( T - 108 \) Copy content Toggle raw display
$23$ \( T + 72 \) Copy content Toggle raw display
$29$ \( T - 128 \) Copy content Toggle raw display
$31$ \( T - 204 \) Copy content Toggle raw display
$37$ \( T - 228 \) Copy content Toggle raw display
$41$ \( T - 22 \) Copy content Toggle raw display
$43$ \( T + 204 \) Copy content Toggle raw display
$47$ \( T - 600 \) Copy content Toggle raw display
$53$ \( T + 256 \) Copy content Toggle raw display
$59$ \( T + 828 \) Copy content Toggle raw display
$61$ \( T - 84 \) Copy content Toggle raw display
$67$ \( T - 348 \) Copy content Toggle raw display
$71$ \( T - 456 \) Copy content Toggle raw display
$73$ \( T + 822 \) Copy content Toggle raw display
$79$ \( T - 1356 \) Copy content Toggle raw display
$83$ \( T - 108 \) Copy content Toggle raw display
$89$ \( T - 938 \) Copy content Toggle raw display
$97$ \( T - 1278 \) Copy content Toggle raw display
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