Properties

Label 768.4.a.h
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - 3 q^{3} + \beta q^{5} + 5 \beta q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + \beta q^{5} + 5 \beta q^{7} + 9 q^{9} + 20 q^{11} + 14 \beta q^{13} - 3 \beta q^{15} - 34 q^{17} + 52 q^{19} - 15 \beta q^{21} + 22 \beta q^{23} - 117 q^{25} - 27 q^{27} + 71 \beta q^{29} - 39 \beta q^{31} - 60 q^{33} + 40 q^{35} - 96 \beta q^{37} - 42 \beta q^{39} - 26 q^{41} + 252 q^{43} + 9 \beta q^{45} + 122 \beta q^{47} - 143 q^{49} + 102 q^{51} - 241 \beta q^{53} + 20 \beta q^{55} - 156 q^{57} + 364 q^{59} + 260 \beta q^{61} + 45 \beta q^{63} + 112 q^{65} + 628 q^{67} - 66 \beta q^{69} - 118 \beta q^{71} + 338 q^{73} + 351 q^{75} + 100 \beta q^{77} - 279 \beta q^{79} + 81 q^{81} + 1036 q^{83} - 34 \beta q^{85} - 213 \beta q^{87} + 234 q^{89} + 560 q^{91} + 117 \beta q^{93} + 52 \beta q^{95} - 178 q^{97} + 180 q^{99} +O(q^{100}) \) 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 \( 2 q - 6 q^{3} + 18 q^{9} + 40 q^{11} - 68 q^{17} + 104 q^{19} - 234 q^{25} - 54 q^{27} - 120 q^{33} + 80 q^{35} - 52 q^{41} + 504 q^{43} - 286 q^{49} + 204 q^{51} - 312 q^{57} + 728 q^{59} + 224 q^{65} + 1256 q^{67} + 676 q^{73} + 702 q^{75} + 162 q^{81} + 2072 q^{83} + 468 q^{89} + 1120 q^{91} - 356 q^{97} + 360 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −3.00000 0 −2.82843 0 −14.1421 0 9.00000 0
1.2 0 −3.00000 0 2.82843 0 14.1421 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.h 2
3.b odd 2 1 2304.4.a.bb 2
4.b odd 2 1 768.4.a.m 2
8.b even 2 1 768.4.a.m 2
8.d odd 2 1 inner 768.4.a.h 2
12.b even 2 1 2304.4.a.bh 2
16.e even 4 2 384.4.d.d 4
16.f odd 4 2 384.4.d.d 4
24.f even 2 1 2304.4.a.bb 2
24.h odd 2 1 2304.4.a.bh 2
48.i odd 4 2 1152.4.d.n 4
48.k even 4 2 1152.4.d.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.d 4 16.e even 4 2
384.4.d.d 4 16.f odd 4 2
768.4.a.h 2 1.a even 1 1 trivial
768.4.a.h 2 8.d odd 2 1 inner
768.4.a.m 2 4.b odd 2 1
768.4.a.m 2 8.b even 2 1
1152.4.d.n 4 48.i odd 4 2
1152.4.d.n 4 48.k even 4 2
2304.4.a.bb 2 3.b odd 2 1
2304.4.a.bb 2 24.f even 2 1
2304.4.a.bh 2 12.b even 2 1
2304.4.a.bh 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 200 \) Copy content Toggle raw display
\( T_{11} - 20 \) Copy content Toggle raw display
\( T_{19} - 52 \) 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
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} - 200 \) Copy content Toggle raw display
$11$ \( (T - 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 1568 \) Copy content Toggle raw display
$17$ \( (T + 34)^{2} \) Copy content Toggle raw display
$19$ \( (T - 52)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3872 \) Copy content Toggle raw display
$29$ \( T^{2} - 40328 \) Copy content Toggle raw display
$31$ \( T^{2} - 12168 \) Copy content Toggle raw display
$37$ \( T^{2} - 73728 \) Copy content Toggle raw display
$41$ \( (T + 26)^{2} \) Copy content Toggle raw display
$43$ \( (T - 252)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 119072 \) Copy content Toggle raw display
$53$ \( T^{2} - 464648 \) Copy content Toggle raw display
$59$ \( (T - 364)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 540800 \) Copy content Toggle raw display
$67$ \( (T - 628)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 111392 \) Copy content Toggle raw display
$73$ \( (T - 338)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 622728 \) Copy content Toggle raw display
$83$ \( (T - 1036)^{2} \) Copy content Toggle raw display
$89$ \( (T - 234)^{2} \) Copy content Toggle raw display
$97$ \( (T + 178)^{2} \) Copy content Toggle raw display
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