Properties

Label 768.2.c.h
Level 768
Weight 2
Character orbit 768.c
Analytic conductor 6.133
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 768.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{17} -2 \zeta_{8}^{2} q^{19} + 5 q^{25} + ( -\zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{41} -10 \zeta_{8}^{2} q^{43} + 7 q^{49} + ( -4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{51} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{57} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{59} -14 \zeta_{8}^{2} q^{67} -2 q^{73} + ( 5 \zeta_{8} + 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{75} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{89} -10 q^{97} + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} + 20q^{25} + 16q^{33} + 28q^{49} + 8q^{57} - 8q^{73} - 28q^{81} - 40q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −1.41421 1.00000i 0 0 0 0 0 1.00000 + 2.82843i 0
767.2 0 −1.41421 + 1.00000i 0 0 0 0 0 1.00000 2.82843i 0
767.3 0 1.41421 1.00000i 0 0 0 0 0 1.00000 2.82843i 0
767.4 0 1.41421 + 1.00000i 0 0 0 0 0 1.00000 + 2.82843i 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 768.2.c.h 4
3.b odd 2 1 inner 768.2.c.h 4
4.b odd 2 1 inner 768.2.c.h 4
8.b even 2 1 inner 768.2.c.h 4
8.d odd 2 1 CM 768.2.c.h 4
12.b even 2 1 inner 768.2.c.h 4
16.e even 4 1 24.2.f.a 2
16.e even 4 1 96.2.f.a 2
16.f odd 4 1 24.2.f.a 2
16.f odd 4 1 96.2.f.a 2
24.f even 2 1 inner 768.2.c.h 4
24.h odd 2 1 inner 768.2.c.h 4
48.i odd 4 1 24.2.f.a 2
48.i odd 4 1 96.2.f.a 2
48.k even 4 1 24.2.f.a 2
48.k even 4 1 96.2.f.a 2
80.i odd 4 1 600.2.m.a 4
80.i odd 4 1 2400.2.m.a 4
80.j even 4 1 600.2.m.a 4
80.j even 4 1 2400.2.m.a 4
80.k odd 4 1 600.2.b.a 2
80.k odd 4 1 2400.2.b.a 2
80.q even 4 1 600.2.b.a 2
80.q even 4 1 2400.2.b.a 2
80.s even 4 1 600.2.m.a 4
80.s even 4 1 2400.2.m.a 4
80.t odd 4 1 600.2.m.a 4
80.t odd 4 1 2400.2.m.a 4
144.u even 12 2 648.2.l.b 4
144.u even 12 2 2592.2.p.b 4
144.v odd 12 2 648.2.l.b 4
144.v odd 12 2 2592.2.p.b 4
144.w odd 12 2 648.2.l.b 4
144.w odd 12 2 2592.2.p.b 4
144.x even 12 2 648.2.l.b 4
144.x even 12 2 2592.2.p.b 4
240.t even 4 1 600.2.b.a 2
240.t even 4 1 2400.2.b.a 2
240.z odd 4 1 600.2.m.a 4
240.z odd 4 1 2400.2.m.a 4
240.bb even 4 1 600.2.m.a 4
240.bb even 4 1 2400.2.m.a 4
240.bd odd 4 1 600.2.m.a 4
240.bd odd 4 1 2400.2.m.a 4
240.bf even 4 1 600.2.m.a 4
240.bf even 4 1 2400.2.m.a 4
240.bm odd 4 1 600.2.b.a 2
240.bm odd 4 1 2400.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 16.e even 4 1
24.2.f.a 2 16.f odd 4 1
24.2.f.a 2 48.i odd 4 1
24.2.f.a 2 48.k even 4 1
96.2.f.a 2 16.e even 4 1
96.2.f.a 2 16.f odd 4 1
96.2.f.a 2 48.i odd 4 1
96.2.f.a 2 48.k even 4 1
600.2.b.a 2 80.k odd 4 1
600.2.b.a 2 80.q even 4 1
600.2.b.a 2 240.t even 4 1
600.2.b.a 2 240.bm odd 4 1
600.2.m.a 4 80.i odd 4 1
600.2.m.a 4 80.j even 4 1
600.2.m.a 4 80.s even 4 1
600.2.m.a 4 80.t odd 4 1
600.2.m.a 4 240.z odd 4 1
600.2.m.a 4 240.bb even 4 1
600.2.m.a 4 240.bd odd 4 1
600.2.m.a 4 240.bf even 4 1
648.2.l.b 4 144.u even 12 2
648.2.l.b 4 144.v odd 12 2
648.2.l.b 4 144.w odd 12 2
648.2.l.b 4 144.x even 12 2
768.2.c.h 4 1.a even 1 1 trivial
768.2.c.h 4 3.b odd 2 1 inner
768.2.c.h 4 4.b odd 2 1 inner
768.2.c.h 4 8.b even 2 1 inner
768.2.c.h 4 8.d odd 2 1 CM
768.2.c.h 4 12.b even 2 1 inner
768.2.c.h 4 24.f even 2 1 inner
768.2.c.h 4 24.h odd 2 1 inner
2400.2.b.a 2 80.k odd 4 1
2400.2.b.a 2 80.q even 4 1
2400.2.b.a 2 240.t even 4 1
2400.2.b.a 2 240.bm odd 4 1
2400.2.m.a 4 80.i odd 4 1
2400.2.m.a 4 80.j even 4 1
2400.2.m.a 4 80.s even 4 1
2400.2.m.a 4 80.t odd 4 1
2400.2.m.a 4 240.z odd 4 1
2400.2.m.a 4 240.bb even 4 1
2400.2.m.a 4 240.bd odd 4 1
2400.2.m.a 4 240.bf even 4 1
2592.2.p.b 4 144.u even 12 2
2592.2.p.b 4 144.v odd 12 2
2592.2.p.b 4 144.w odd 12 2
2592.2.p.b 4 144.x even 12 2

Hecke kernels

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

\( T_{5} \)
\( T_{7} \)
\( T_{11}^{2} - 8 \)
\( T_{13} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( ( 1 - 5 T^{2} )^{4} \)
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 + 14 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2}( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 34 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2}( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{4} \)
$59$ \( ( 1 - 82 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 62 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{4} \)
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