Properties

Label 768.2.a.i
Level $768$
Weight $2$
Character orbit 768.a
Self dual yes
Analytic conductor $6.133$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(6.13251087523\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} -\beta q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} -\beta q^{7} + q^{9} -4 q^{11} -2 \beta q^{13} -\beta q^{15} -2 q^{17} -4 q^{19} + \beta q^{21} + 2 \beta q^{23} + 3 q^{25} - q^{27} -\beta q^{29} + 3 \beta q^{31} + 4 q^{33} -8 q^{35} + 2 \beta q^{39} -10 q^{41} -12 q^{43} + \beta q^{45} -2 \beta q^{47} + q^{49} + 2 q^{51} -\beta q^{53} -4 \beta q^{55} + 4 q^{57} + 4 q^{59} + 4 \beta q^{61} -\beta q^{63} -16 q^{65} -4 q^{67} -2 \beta q^{69} -2 \beta q^{71} + 2 q^{73} -3 q^{75} + 4 \beta q^{77} + 3 \beta q^{79} + q^{81} + 4 q^{83} -2 \beta q^{85} + \beta q^{87} -6 q^{89} + 16 q^{91} -3 \beta q^{93} -4 \beta q^{95} + 14 q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} - 8q^{11} - 4q^{17} - 8q^{19} + 6q^{25} - 2q^{27} + 8q^{33} - 16q^{35} - 20q^{41} - 24q^{43} + 2q^{49} + 4q^{51} + 8q^{57} + 8q^{59} - 32q^{65} - 8q^{67} + 4q^{73} - 6q^{75} + 2q^{81} + 8q^{83} - 12q^{89} + 32q^{91} + 28q^{97} - 8q^{99} + O(q^{100}) \)

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 −1.00000 0 −2.82843 0 2.82843 0 1.00000 0
1.2 0 −1.00000 0 2.82843 0 −2.82843 0 1.00000 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 768.2.a.i 2
3.b odd 2 1 2304.2.a.x 2
4.b odd 2 1 768.2.a.l 2
8.b even 2 1 768.2.a.l 2
8.d odd 2 1 inner 768.2.a.i 2
12.b even 2 1 2304.2.a.r 2
16.e even 4 2 384.2.d.c 4
16.f odd 4 2 384.2.d.c 4
24.f even 2 1 2304.2.a.x 2
24.h odd 2 1 2304.2.a.r 2
48.i odd 4 2 1152.2.d.h 4
48.k even 4 2 1152.2.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.c 4 16.e even 4 2
384.2.d.c 4 16.f odd 4 2
768.2.a.i 2 1.a even 1 1 trivial
768.2.a.i 2 8.d odd 2 1 inner
768.2.a.l 2 4.b odd 2 1
768.2.a.l 2 8.b even 2 1
1152.2.d.h 4 48.i odd 4 2
1152.2.d.h 4 48.k even 4 2
2304.2.a.r 2 12.b even 2 1
2304.2.a.r 2 24.h odd 2 1
2304.2.a.x 2 3.b odd 2 1
2304.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{2} - 8 \)
\( T_{7}^{2} - 8 \)
\( T_{11} + 4 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( -32 + T^{2} \)
$29$ \( -8 + T^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( ( 12 + T )^{2} \)
$47$ \( -32 + T^{2} \)
$53$ \( -8 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -128 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( -72 + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( ( -14 + T )^{2} \)
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