Properties

Label 768.1.i.a
Level $768$
Weight $1$
Character orbit 768.i
Analytic conductor $0.383$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 768.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.18432.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} + ( -1 - \zeta_{8}^{2} ) q^{13} + ( -1 + \zeta_{8}^{2} ) q^{21} -\zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{31} + ( 1 - \zeta_{8}^{2} ) q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{39} - q^{49} + ( 1 + \zeta_{8}^{2} ) q^{61} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{63} -2 \zeta_{8} q^{67} -\zeta_{8}^{3} q^{75} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{79} - q^{81} -2 \zeta_{8}^{3} q^{91} + ( 1 + \zeta_{8}^{2} ) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{13} - 4q^{21} + 4q^{37} - 4q^{49} + 4q^{61} - 4q^{81} + 4q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 −0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
65.2 0 0.707107 + 0.707107i 0 0 0 1.41421i 0 1.00000i 0
449.1 0 −0.707107 + 0.707107i 0 0 0 1.41421i 0 1.00000i 0
449.2 0 0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
48.i odd 4 1 inner
48.k even 4 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + 2 T_{13} + 2 \) acting on \(S_{1}^{\mathrm{new}}(768, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 2 + 2 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -2 + T^{2} )^{2} \)
$37$ \( ( 2 - 2 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 2 - 2 T + T^{2} )^{2} \)
$67$ \( 16 + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( -2 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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