Properties

Label 768.2.c.e
Level $768$
Weight $2$
Character orbit 768.c
Analytic conductor $6.133$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{3} + 2 \beta q^{5} + 2 \beta q^{7} + ( -1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{3} + 2 \beta q^{5} + 2 \beta q^{7} + ( -1 - 2 \beta ) q^{9} + 2 q^{11} -4 q^{13} + ( 4 + 2 \beta ) q^{15} + 4 \beta q^{17} + 2 \beta q^{19} + ( 4 + 2 \beta ) q^{21} + 8 q^{23} -3 q^{25} + ( -5 - \beta ) q^{27} -2 \beta q^{29} + 6 \beta q^{31} + ( 2 - 2 \beta ) q^{33} -8 q^{35} -4 q^{37} + ( -4 + 4 \beta ) q^{39} + 2 \beta q^{43} + ( 8 - 2 \beta ) q^{45} - q^{49} + ( 8 + 4 \beta ) q^{51} -6 \beta q^{53} + 4 \beta q^{55} + ( 4 + 2 \beta ) q^{57} + 6 q^{59} -4 q^{61} + ( 8 - 2 \beta ) q^{63} -8 \beta q^{65} -10 \beta q^{67} + ( 8 - 8 \beta ) q^{69} + 8 q^{71} + 10 q^{73} + ( -3 + 3 \beta ) q^{75} + 4 \beta q^{77} -2 \beta q^{79} + ( -7 + 4 \beta ) q^{81} + 6 q^{83} -16 q^{85} + ( -4 - 2 \beta ) q^{87} + 4 \beta q^{89} -8 \beta q^{91} + ( 12 + 6 \beta ) q^{93} -8 q^{95} -6 q^{97} + ( -2 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{9} + 4q^{11} - 8q^{13} + 8q^{15} + 8q^{21} + 16q^{23} - 6q^{25} - 10q^{27} + 4q^{33} - 16q^{35} - 8q^{37} - 8q^{39} + 16q^{45} - 2q^{49} + 16q^{51} + 8q^{57} + 12q^{59} - 8q^{61} + 16q^{63} + 16q^{69} + 16q^{71} + 20q^{73} - 6q^{75} - 14q^{81} + 12q^{83} - 32q^{85} - 8q^{87} + 24q^{93} - 16q^{95} - 12q^{97} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
1.41421i
1.41421i
0 1.00000 1.41421i 0 2.82843i 0 2.82843i 0 −1.00000 2.82843i 0
767.2 0 1.00000 + 1.41421i 0 2.82843i 0 2.82843i 0 −1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.c.e 2
3.b odd 2 1 768.2.c.a 2
4.b odd 2 1 768.2.c.a 2
8.b even 2 1 768.2.c.b 2
8.d odd 2 1 768.2.c.f 2
12.b even 2 1 inner 768.2.c.e 2
16.e even 4 2 384.2.f.d yes 4
16.f odd 4 2 384.2.f.b 4
24.f even 2 1 768.2.c.b 2
24.h odd 2 1 768.2.c.f 2
48.i odd 4 2 384.2.f.b 4
48.k even 4 2 384.2.f.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.f.b 4 16.f odd 4 2
384.2.f.b 4 48.i odd 4 2
384.2.f.d yes 4 16.e even 4 2
384.2.f.d yes 4 48.k even 4 2
768.2.c.a 2 3.b odd 2 1
768.2.c.a 2 4.b odd 2 1
768.2.c.b 2 8.b even 2 1
768.2.c.b 2 24.f even 2 1
768.2.c.e 2 1.a even 1 1 trivial
768.2.c.e 2 12.b even 2 1 inner
768.2.c.f 2 8.d odd 2 1
768.2.c.f 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 \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 8 \)
\( T_{7}^{2} + 8 \)
\( T_{11} - 2 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 2 T + T^{2} \)
$5$ \( 8 + T^{2} \)
$7$ \( 8 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 32 + T^{2} \)
$19$ \( 8 + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( 8 + T^{2} \)
$31$ \( 72 + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 8 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 72 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( 200 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( 8 + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 32 + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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