Properties

Label 768.2.j.b
Level $768$
Weight $2$
Character orbit 768.j
Analytic conductor $6.133$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
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
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{2} q^{9} + ( -1 + \zeta_{8}^{2} ) q^{13} + 6 q^{17} + 6 \zeta_{8}^{3} q^{19} + ( -1 - \zeta_{8}^{2} ) q^{21} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{23} -5 \zeta_{8}^{2} q^{25} + \zeta_{8} q^{27} + ( -6 + 6 \zeta_{8}^{2} ) q^{29} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( -5 - 5 \zeta_{8}^{2} ) q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{39} + 6 \zeta_{8}^{2} q^{41} + 6 \zeta_{8} q^{43} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{47} + 5 q^{49} + 6 \zeta_{8}^{3} q^{51} + ( 6 + 6 \zeta_{8}^{2} ) q^{53} -6 \zeta_{8}^{2} q^{57} + ( -5 + 5 \zeta_{8}^{2} ) q^{61} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{63} + 4 \zeta_{8}^{3} q^{67} + ( -6 - 6 \zeta_{8}^{2} ) q^{69} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + 5 \zeta_{8} q^{75} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{79} - q^{81} -12 \zeta_{8}^{3} q^{83} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{87} + 6 \zeta_{8}^{2} q^{89} -2 \zeta_{8} q^{91} + ( 1 - \zeta_{8}^{2} ) q^{93} -12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 4 q^{13} + 24 q^{17} - 4 q^{21} - 24 q^{29} - 20 q^{37} + 20 q^{49} + 24 q^{53} - 20 q^{61} - 24 q^{69} - 4 q^{81} + 4 q^{93} - 48 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
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
193.2 0 0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
577.1 0 −0.707107 0.707107i 0 0 0 1.41421i 0 1.00000i 0
577.2 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
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.j.b 4
3.b odd 2 1 2304.2.k.b 4
4.b odd 2 1 inner 768.2.j.b 4
8.b even 2 1 768.2.j.c yes 4
8.d odd 2 1 768.2.j.c yes 4
12.b even 2 1 2304.2.k.b 4
16.e even 4 1 inner 768.2.j.b 4
16.e even 4 1 768.2.j.c yes 4
16.f odd 4 1 inner 768.2.j.b 4
16.f odd 4 1 768.2.j.c yes 4
24.f even 2 1 2304.2.k.c 4
24.h odd 2 1 2304.2.k.c 4
32.g even 8 1 3072.2.a.a 2
32.g even 8 1 3072.2.a.g 2
32.g even 8 2 3072.2.d.a 4
32.h odd 8 1 3072.2.a.a 2
32.h odd 8 1 3072.2.a.g 2
32.h odd 8 2 3072.2.d.a 4
48.i odd 4 1 2304.2.k.b 4
48.i odd 4 1 2304.2.k.c 4
48.k even 4 1 2304.2.k.b 4
48.k even 4 1 2304.2.k.c 4
96.o even 8 1 9216.2.a.n 2
96.o even 8 1 9216.2.a.o 2
96.p odd 8 1 9216.2.a.n 2
96.p odd 8 1 9216.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.b 4 1.a even 1 1 trivial
768.2.j.b 4 4.b odd 2 1 inner
768.2.j.b 4 16.e even 4 1 inner
768.2.j.b 4 16.f odd 4 1 inner
768.2.j.c yes 4 8.b even 2 1
768.2.j.c yes 4 8.d odd 2 1
768.2.j.c yes 4 16.e even 4 1
768.2.j.c yes 4 16.f odd 4 1
2304.2.k.b 4 3.b odd 2 1
2304.2.k.b 4 12.b even 2 1
2304.2.k.b 4 48.i odd 4 1
2304.2.k.b 4 48.k even 4 1
2304.2.k.c 4 24.f even 2 1
2304.2.k.c 4 24.h odd 2 1
2304.2.k.c 4 48.i odd 4 1
2304.2.k.c 4 48.k even 4 1
3072.2.a.a 2 32.g even 8 1
3072.2.a.a 2 32.h odd 8 1
3072.2.a.g 2 32.g even 8 1
3072.2.a.g 2 32.h odd 8 1
3072.2.d.a 4 32.g even 8 2
3072.2.d.a 4 32.h odd 8 2
9216.2.a.n 2 96.o even 8 1
9216.2.a.n 2 96.p odd 8 1
9216.2.a.o 2 96.o even 8 1
9216.2.a.o 2 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5} \)
\( T_{13}^{2} + 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 2 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 2 + 2 T + T^{2} )^{2} \)
$17$ \( ( -6 + T )^{4} \)
$19$ \( 1296 + T^{4} \)
$23$ \( ( 72 + T^{2} )^{2} \)
$29$ \( ( 72 + 12 T + T^{2} )^{2} \)
$31$ \( ( -2 + T^{2} )^{2} \)
$37$ \( ( 50 + 10 T + T^{2} )^{2} \)
$41$ \( ( 36 + T^{2} )^{2} \)
$43$ \( 1296 + T^{4} \)
$47$ \( ( -72 + T^{2} )^{2} \)
$53$ \( ( 72 - 12 T + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 50 + 10 T + T^{2} )^{2} \)
$67$ \( 256 + T^{4} \)
$71$ \( ( 72 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( -2 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( 36 + T^{2} )^{2} \)
$97$ \( ( 12 + T )^{4} \)
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