Properties

Label 768.2.j.d
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.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{3} + ( 2 - 2 \zeta_{8}^{2} ) q^{5} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8} q^{3} + ( 2 - 2 \zeta_{8}^{2} ) q^{5} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} + 4 \zeta_{8}^{3} q^{11} + ( 3 + 3 \zeta_{8}^{2} ) q^{13} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{15} -6 q^{17} + 2 \zeta_{8} q^{19} + ( -3 + 3 \zeta_{8}^{2} ) q^{21} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} -3 \zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + ( 4 + 4 \zeta_{8}^{2} ) q^{29} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{31} -4 q^{33} + 12 \zeta_{8} q^{35} + ( 3 - 3 \zeta_{8}^{2} ) q^{37} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{39} -10 \zeta_{8}^{2} q^{41} -6 \zeta_{8}^{3} q^{43} + ( 2 + 2 \zeta_{8}^{2} ) q^{45} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} -11 q^{49} -6 \zeta_{8} q^{51} + ( 4 - 4 \zeta_{8}^{2} ) q^{53} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{55} + 2 \zeta_{8}^{2} q^{57} + ( 3 + 3 \zeta_{8}^{2} ) q^{61} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{63} + 12 q^{65} -4 \zeta_{8} q^{67} + ( 2 - 2 \zeta_{8}^{2} ) q^{69} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} -3 \zeta_{8}^{3} q^{75} + ( -12 - 12 \zeta_{8}^{2} ) q^{77} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} - q^{81} + 16 \zeta_{8} q^{83} + ( -12 + 12 \zeta_{8}^{2} ) q^{85} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{87} -14 \zeta_{8}^{2} q^{89} + 18 \zeta_{8}^{3} q^{91} + ( 3 + 3 \zeta_{8}^{2} ) q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} -4 q^{97} -4 \zeta_{8} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{5} + O(q^{10}) \) \( 4q + 8q^{5} + 12q^{13} - 24q^{17} - 12q^{21} + 16q^{29} - 16q^{33} + 12q^{37} + 8q^{45} - 44q^{49} + 16q^{53} + 12q^{61} + 48q^{65} + 8q^{69} - 48q^{77} - 4q^{81} - 48q^{85} + 12q^{93} - 16q^{97} + O(q^{100}) \)

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 2.00000 + 2.00000i 0 4.24264i 0 1.00000i 0
193.2 0 0.707107 0.707107i 0 2.00000 + 2.00000i 0 4.24264i 0 1.00000i 0
577.1 0 −0.707107 0.707107i 0 2.00000 2.00000i 0 4.24264i 0 1.00000i 0
577.2 0 0.707107 + 0.707107i 0 2.00000 2.00000i 0 4.24264i 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.d yes 4
3.b odd 2 1 2304.2.k.a 4
4.b odd 2 1 inner 768.2.j.d yes 4
8.b even 2 1 768.2.j.a 4
8.d odd 2 1 768.2.j.a 4
12.b even 2 1 2304.2.k.a 4
16.e even 4 1 768.2.j.a 4
16.e even 4 1 inner 768.2.j.d yes 4
16.f odd 4 1 768.2.j.a 4
16.f odd 4 1 inner 768.2.j.d yes 4
24.f even 2 1 2304.2.k.d 4
24.h odd 2 1 2304.2.k.d 4
32.g even 8 1 3072.2.a.d 2
32.g even 8 1 3072.2.a.f 2
32.g even 8 2 3072.2.d.d 4
32.h odd 8 1 3072.2.a.d 2
32.h odd 8 1 3072.2.a.f 2
32.h odd 8 2 3072.2.d.d 4
48.i odd 4 1 2304.2.k.a 4
48.i odd 4 1 2304.2.k.d 4
48.k even 4 1 2304.2.k.a 4
48.k even 4 1 2304.2.k.d 4
96.o even 8 1 9216.2.a.e 2
96.o even 8 1 9216.2.a.q 2
96.p odd 8 1 9216.2.a.e 2
96.p odd 8 1 9216.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.a 4 8.b even 2 1
768.2.j.a 4 8.d odd 2 1
768.2.j.a 4 16.e even 4 1
768.2.j.a 4 16.f odd 4 1
768.2.j.d yes 4 1.a even 1 1 trivial
768.2.j.d yes 4 4.b odd 2 1 inner
768.2.j.d yes 4 16.e even 4 1 inner
768.2.j.d yes 4 16.f odd 4 1 inner
2304.2.k.a 4 3.b odd 2 1
2304.2.k.a 4 12.b even 2 1
2304.2.k.a 4 48.i odd 4 1
2304.2.k.a 4 48.k even 4 1
2304.2.k.d 4 24.f even 2 1
2304.2.k.d 4 24.h odd 2 1
2304.2.k.d 4 48.i odd 4 1
2304.2.k.d 4 48.k even 4 1
3072.2.a.d 2 32.g even 8 1
3072.2.a.d 2 32.h odd 8 1
3072.2.a.f 2 32.g even 8 1
3072.2.a.f 2 32.h odd 8 1
3072.2.d.d 4 32.g even 8 2
3072.2.d.d 4 32.h odd 8 2
9216.2.a.e 2 96.o even 8 1
9216.2.a.e 2 96.p odd 8 1
9216.2.a.q 2 96.o even 8 1
9216.2.a.q 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 \(S_{2}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} - 4 T_{5} + 8 \)
\( T_{13}^{2} - 6 T_{13} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + T^{4} \)
$5$ \( ( 8 - 4 T + T^{2} )^{2} \)
$7$ \( ( 18 + T^{2} )^{2} \)
$11$ \( 256 + T^{4} \)
$13$ \( ( 18 - 6 T + T^{2} )^{2} \)
$17$ \( ( 6 + T )^{4} \)
$19$ \( 16 + T^{4} \)
$23$ \( ( 8 + T^{2} )^{2} \)
$29$ \( ( 32 - 8 T + T^{2} )^{2} \)
$31$ \( ( -18 + T^{2} )^{2} \)
$37$ \( ( 18 - 6 T + T^{2} )^{2} \)
$41$ \( ( 100 + T^{2} )^{2} \)
$43$ \( 1296 + T^{4} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( 32 - 8 T + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 18 - 6 T + T^{2} )^{2} \)
$67$ \( 256 + T^{4} \)
$71$ \( ( 8 + T^{2} )^{2} \)
$73$ \( ( 256 + T^{2} )^{2} \)
$79$ \( ( -18 + T^{2} )^{2} \)
$83$ \( 65536 + T^{4} \)
$89$ \( ( 196 + T^{2} )^{2} \)
$97$ \( ( 4 + T )^{4} \)
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