Properties

Label 768.4.a.l
Level $768$
Weight $4$
Character orbit 768.a
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\beta = 6\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9} - 48 q^{11} + 4 \beta q^{13} + 3 \beta q^{15} - 42 q^{17} + 92 q^{19} + 3 \beta q^{21} - 2 \beta q^{23} + 271 q^{25} + 27 q^{27} - \beta q^{29} - 7 \beta q^{31} - 144 q^{33} + 396 q^{35} + 10 \beta q^{37} + 12 \beta q^{39} + 6 q^{41} + 92 q^{43} + 9 \beta q^{45} + 2 \beta q^{47} + 53 q^{49} - 126 q^{51} - 25 \beta q^{53} - 48 \beta q^{55} + 276 q^{57} - 516 q^{59} - 18 \beta q^{61} + 9 \beta q^{63} + 1584 q^{65} - 524 q^{67} - 6 \beta q^{69} + 50 \beta q^{71} + 430 q^{73} + 813 q^{75} - 48 \beta q^{77} - 59 \beta q^{79} + 81 q^{81} - 432 q^{83} - 42 \beta q^{85} - 3 \beta q^{87} - 630 q^{89} + 1584 q^{91} - 21 \beta q^{93} + 92 \beta q^{95} + 862 q^{97} - 432 q^{99} +O(q^{100}) \) 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 \( 2 q + 6 q^{3} + 18 q^{9} - 96 q^{11} - 84 q^{17} + 184 q^{19} + 542 q^{25} + 54 q^{27} - 288 q^{33} + 792 q^{35} + 12 q^{41} + 184 q^{43} + 106 q^{49} - 252 q^{51} + 552 q^{57} - 1032 q^{59} + 3168 q^{65} - 1048 q^{67} + 860 q^{73} + 1626 q^{75} + 162 q^{81} - 864 q^{83} - 1260 q^{89} + 3168 q^{91} + 1724 q^{97} - 864 q^{99}+O(q^{100}) \) 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
−3.31662
3.31662
0 3.00000 0 −19.8997 0 −19.8997 0 9.00000 0
1.2 0 3.00000 0 19.8997 0 19.8997 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.l 2
3.b odd 2 1 2304.4.a.bl 2
4.b odd 2 1 768.4.a.i 2
8.b even 2 1 768.4.a.i 2
8.d odd 2 1 inner 768.4.a.l 2
12.b even 2 1 2304.4.a.y 2
16.e even 4 2 192.4.d.b 4
16.f odd 4 2 192.4.d.b 4
24.f even 2 1 2304.4.a.bl 2
24.h odd 2 1 2304.4.a.y 2
48.i odd 4 2 576.4.d.f 4
48.k even 4 2 576.4.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.b 4 16.e even 4 2
192.4.d.b 4 16.f odd 4 2
576.4.d.f 4 48.i odd 4 2
576.4.d.f 4 48.k even 4 2
768.4.a.i 2 4.b odd 2 1
768.4.a.i 2 8.b even 2 1
768.4.a.l 2 1.a even 1 1 trivial
768.4.a.l 2 8.d odd 2 1 inner
2304.4.a.y 2 12.b even 2 1
2304.4.a.y 2 24.h odd 2 1
2304.4.a.bl 2 3.b odd 2 1
2304.4.a.bl 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{2} - 396 \) Copy content Toggle raw display
\( T_{7}^{2} - 396 \) Copy content Toggle raw display
\( T_{11} + 48 \) Copy content Toggle raw display
\( T_{19} - 92 \) 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
$5$ \( T^{2} - 396 \) Copy content Toggle raw display
$7$ \( T^{2} - 396 \) Copy content Toggle raw display
$11$ \( (T + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6336 \) Copy content Toggle raw display
$17$ \( (T + 42)^{2} \) Copy content Toggle raw display
$19$ \( (T - 92)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 1584 \) Copy content Toggle raw display
$29$ \( T^{2} - 396 \) Copy content Toggle raw display
$31$ \( T^{2} - 19404 \) Copy content Toggle raw display
$37$ \( T^{2} - 39600 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 92)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 1584 \) Copy content Toggle raw display
$53$ \( T^{2} - 247500 \) Copy content Toggle raw display
$59$ \( (T + 516)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 128304 \) Copy content Toggle raw display
$67$ \( (T + 524)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 990000 \) Copy content Toggle raw display
$73$ \( (T - 430)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1378476 \) Copy content Toggle raw display
$83$ \( (T + 432)^{2} \) Copy content Toggle raw display
$89$ \( (T + 630)^{2} \) Copy content Toggle raw display
$97$ \( (T - 862)^{2} \) Copy content Toggle raw display
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