Properties

Label 768.3.e.k
Level $768$
Weight $3$
Character orbit 768.e
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{3} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{3} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{9} -14 \zeta_{8}^{2} q^{11} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{17} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{19} + 25 q^{25} + ( 10 \zeta_{8} + 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{27} + ( 14 - 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{33} + ( 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{41} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{43} -49 q^{49} + ( 24 \zeta_{8} - 96 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{51} + ( 48 + 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{57} -82 \zeta_{8}^{2} q^{59} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{67} -142 q^{73} + ( 50 \zeta_{8} + 25 \zeta_{8}^{2} - 50 \zeta_{8}^{3} ) q^{75} + ( 17 + 56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{81} + 158 \zeta_{8}^{2} q^{83} + ( -72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{89} -94 q^{97} + ( 56 \zeta_{8} - 98 \zeta_{8}^{2} - 56 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 28q^{9} + O(q^{10}) \) \( 4q + 28q^{9} + 100q^{25} + 56q^{33} - 196q^{49} + 192q^{57} - 568q^{73} + 68q^{81} - 376q^{97} + O(q^{100}) \)

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
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −2.82843 1.00000i 0 0 0 0 0 7.00000 + 5.65685i 0
257.2 0 −2.82843 + 1.00000i 0 0 0 0 0 7.00000 5.65685i 0
257.3 0 2.82843 1.00000i 0 0 0 0 0 7.00000 5.65685i 0
257.4 0 2.82843 + 1.00000i 0 0 0 0 0 7.00000 + 5.65685i 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 \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.e.k 4
3.b odd 2 1 inner 768.3.e.k 4
4.b odd 2 1 inner 768.3.e.k 4
8.b even 2 1 inner 768.3.e.k 4
8.d odd 2 1 CM 768.3.e.k 4
12.b even 2 1 inner 768.3.e.k 4
16.e even 4 1 384.3.h.b 2
16.e even 4 1 384.3.h.c yes 2
16.f odd 4 1 384.3.h.b 2
16.f odd 4 1 384.3.h.c yes 2
24.f even 2 1 inner 768.3.e.k 4
24.h odd 2 1 inner 768.3.e.k 4
48.i odd 4 1 384.3.h.b 2
48.i odd 4 1 384.3.h.c yes 2
48.k even 4 1 384.3.h.b 2
48.k even 4 1 384.3.h.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.b 2 16.e even 4 1
384.3.h.b 2 16.f odd 4 1
384.3.h.b 2 48.i odd 4 1
384.3.h.b 2 48.k even 4 1
384.3.h.c yes 2 16.e even 4 1
384.3.h.c yes 2 16.f odd 4 1
384.3.h.c yes 2 48.i odd 4 1
384.3.h.c yes 2 48.k even 4 1
768.3.e.k 4 1.a even 1 1 trivial
768.3.e.k 4 3.b odd 2 1 inner
768.3.e.k 4 4.b odd 2 1 inner
768.3.e.k 4 8.b even 2 1 inner
768.3.e.k 4 8.d odd 2 1 CM
768.3.e.k 4 12.b even 2 1 inner
768.3.e.k 4 24.f even 2 1 inner
768.3.e.k 4 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5} \)
\( T_{7} \)
\( T_{11}^{2} + 196 \)
\( T_{19}^{2} - 288 \)
\( T_{37} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 14 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 196 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 1152 + T^{2} )^{2} \)
$19$ \( ( -288 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 4608 + T^{2} )^{2} \)
$43$ \( ( -7200 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 6724 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -14112 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 142 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 24964 + T^{2} )^{2} \)
$89$ \( ( 10368 + T^{2} )^{2} \)
$97$ \( ( 94 + T )^{4} \)
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