Properties

Label 768.4.a.g
Level $768$
Weight $4$
Character orbit 768.a
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + 3 \beta q^{5} + \beta q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 3 \beta q^{5} + \beta q^{7} + 9 q^{9} - 16 \beta q^{13} - 9 \beta q^{15} - 90 q^{17} + 116 q^{19} - 3 \beta q^{21} + 30 \beta q^{23} - 17 q^{25} - 27 q^{27} - 75 \beta q^{29} - 87 \beta q^{31} + 36 q^{35} + 10 \beta q^{37} + 48 \beta q^{39} + 54 q^{41} + 20 q^{43} + 27 \beta q^{45} + 114 \beta q^{47} - 331 q^{49} + 270 q^{51} + 141 \beta q^{53} - 348 q^{57} + 324 q^{59} + 166 \beta q^{61} + 9 \beta q^{63} - 576 q^{65} - 116 q^{67} - 90 \beta q^{69} - 318 \beta q^{71} - 1106 q^{73} + 51 q^{75} - 43 \beta q^{79} + 81 q^{81} - 1152 q^{83} - 270 \beta q^{85} + 225 \beta q^{87} - 918 q^{89} - 192 q^{91} + 261 \beta q^{93} + 348 \beta q^{95} + 190 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 18 q^{9} - 180 q^{17} + 232 q^{19} - 34 q^{25} - 54 q^{27} + 72 q^{35} + 108 q^{41} + 40 q^{43} - 662 q^{49} + 540 q^{51} - 696 q^{57} + 648 q^{59} - 1152 q^{65} - 232 q^{67} - 2212 q^{73} + 102 q^{75} + 162 q^{81} - 2304 q^{83} - 1836 q^{89} - 384 q^{91} + 380 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 −3.00000 0 −10.3923 0 −3.46410 0 9.00000 0
1.2 0 −3.00000 0 10.3923 0 3.46410 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.a.g 2
3.b odd 2 1 2304.4.a.bg 2
4.b odd 2 1 768.4.a.n 2
8.b even 2 1 768.4.a.n 2
8.d odd 2 1 inner 768.4.a.g 2
12.b even 2 1 2304.4.a.be 2
16.e even 4 2 192.4.d.a 4
16.f odd 4 2 192.4.d.a 4
24.f even 2 1 2304.4.a.bg 2
24.h odd 2 1 2304.4.a.be 2
48.i odd 4 2 576.4.d.g 4
48.k even 4 2 576.4.d.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.a 4 16.e even 4 2
192.4.d.a 4 16.f odd 4 2
576.4.d.g 4 48.i odd 4 2
576.4.d.g 4 48.k even 4 2
768.4.a.g 2 1.a even 1 1 trivial
768.4.a.g 2 8.d odd 2 1 inner
768.4.a.n 2 4.b odd 2 1
768.4.a.n 2 8.b even 2 1
2304.4.a.be 2 12.b even 2 1
2304.4.a.be 2 24.h odd 2 1
2304.4.a.bg 2 3.b odd 2 1
2304.4.a.bg 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{2} - 108 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19} - 116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 108 \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3072 \) Copy content Toggle raw display
$17$ \( (T + 90)^{2} \) Copy content Toggle raw display
$19$ \( (T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10800 \) Copy content Toggle raw display
$29$ \( T^{2} - 67500 \) Copy content Toggle raw display
$31$ \( T^{2} - 90828 \) Copy content Toggle raw display
$37$ \( T^{2} - 1200 \) Copy content Toggle raw display
$41$ \( (T - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T - 20)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 155952 \) Copy content Toggle raw display
$53$ \( T^{2} - 238572 \) Copy content Toggle raw display
$59$ \( (T - 324)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 330672 \) Copy content Toggle raw display
$67$ \( (T + 116)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 1213488 \) Copy content Toggle raw display
$73$ \( (T + 1106)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 22188 \) Copy content Toggle raw display
$83$ \( (T + 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T + 918)^{2} \) Copy content Toggle raw display
$97$ \( (T - 190)^{2} \) Copy content Toggle raw display
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