Properties

Label 768.1.e.c
Level $768$
Weight $1$
Character orbit 768.e
Analytic conductor $0.383$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -8, -24, 12
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.150994944.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{3} - q^{9} +O(q^{10})\) \( q -i q^{3} - q^{9} -2 i q^{11} + q^{25} + i q^{27} -2 q^{33} - q^{49} + 2 i q^{59} + 2 q^{73} -i q^{75} + q^{81} + 2 i q^{83} + 2 q^{97} + 2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{25} - 4q^{33} - 2q^{49} + 4q^{73} + 2q^{81} + 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.00000i
1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
257.2 0 1.00000i 0 0 0 0 0 −1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.1.e.c 2
3.b odd 2 1 inner 768.1.e.c 2
4.b odd 2 1 inner 768.1.e.c 2
8.b even 2 1 inner 768.1.e.c 2
8.d odd 2 1 CM 768.1.e.c 2
12.b even 2 1 RM 768.1.e.c 2
16.e even 4 1 384.1.h.a 1
16.e even 4 1 384.1.h.b yes 1
16.f odd 4 1 384.1.h.a 1
16.f odd 4 1 384.1.h.b yes 1
24.f even 2 1 inner 768.1.e.c 2
24.h odd 2 1 CM 768.1.e.c 2
32.g even 8 4 3072.1.i.f 4
32.h odd 8 4 3072.1.i.f 4
48.i odd 4 1 384.1.h.a 1
48.i odd 4 1 384.1.h.b yes 1
48.k even 4 1 384.1.h.a 1
48.k even 4 1 384.1.h.b yes 1
96.o even 8 4 3072.1.i.f 4
96.p odd 8 4 3072.1.i.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.1.h.a 1 16.e even 4 1
384.1.h.a 1 16.f odd 4 1
384.1.h.a 1 48.i odd 4 1
384.1.h.a 1 48.k even 4 1
384.1.h.b yes 1 16.e even 4 1
384.1.h.b yes 1 16.f odd 4 1
384.1.h.b yes 1 48.i odd 4 1
384.1.h.b yes 1 48.k even 4 1
768.1.e.c 2 1.a even 1 1 trivial
768.1.e.c 2 3.b odd 2 1 inner
768.1.e.c 2 4.b odd 2 1 inner
768.1.e.c 2 8.b even 2 1 inner
768.1.e.c 2 8.d odd 2 1 CM
768.1.e.c 2 12.b even 2 1 RM
768.1.e.c 2 24.f even 2 1 inner
768.1.e.c 2 24.h odd 2 1 CM
3072.1.i.f 4 32.g even 8 4
3072.1.i.f 4 32.h odd 8 4
3072.1.i.f 4 96.o even 8 4
3072.1.i.f 4 96.p odd 8 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(768, [\chi])\):

\( T_{11}^{2} + 4 \)
\( T_{19} \)

Hecke characteristic polynomials

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