Properties

Label 768.3.h.b
Level $768$
Weight $3$
Character orbit 768.h
Analytic conductor $20.926$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 i q^{3} + 2 q^{7} -9 q^{9} +O(q^{10})\) \( q -3 i q^{3} + 2 q^{7} -9 q^{9} + 22 i q^{13} + 26 i q^{19} -6 i q^{21} -25 q^{25} + 27 i q^{27} + 46 q^{31} + 26 i q^{37} + 66 q^{39} + 22 i q^{43} -45 q^{49} + 78 q^{57} -74 i q^{61} -18 q^{63} + 122 i q^{67} + 46 q^{73} + 75 i q^{75} + 142 q^{79} + 81 q^{81} + 44 i q^{91} -138 i q^{93} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} - 18q^{9} + O(q^{10}) \) \( 2q + 4q^{7} - 18q^{9} - 50q^{25} + 92q^{31} + 132q^{39} - 90q^{49} + 156q^{57} - 36q^{63} + 92q^{73} + 284q^{79} + 162q^{81} + 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.00000i
1.00000i
0 3.00000i 0 0 0 2.00000 0 −9.00000 0
641.2 0 3.00000i 0 0 0 2.00000 0 −9.00000 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 \(\Q(\sqrt{-3}) \)
8.b 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.h.b 2
3.b odd 2 1 CM 768.3.h.b 2
4.b odd 2 1 768.3.h.a 2
8.b even 2 1 inner 768.3.h.b 2
8.d odd 2 1 768.3.h.a 2
12.b even 2 1 768.3.h.a 2
16.e even 4 1 48.3.e.a 1
16.e even 4 1 192.3.e.a 1
16.f odd 4 1 12.3.c.a 1
16.f odd 4 1 192.3.e.b 1
24.f even 2 1 768.3.h.a 2
24.h odd 2 1 inner 768.3.h.b 2
48.i odd 4 1 48.3.e.a 1
48.i odd 4 1 192.3.e.a 1
48.k even 4 1 12.3.c.a 1
48.k even 4 1 192.3.e.b 1
80.i odd 4 1 1200.3.c.c 2
80.j even 4 1 300.3.b.a 2
80.k odd 4 1 300.3.g.b 1
80.q even 4 1 1200.3.l.b 1
80.s even 4 1 300.3.b.a 2
80.t odd 4 1 1200.3.c.c 2
112.j even 4 1 588.3.c.c 1
112.u odd 12 2 588.3.p.c 2
112.v even 12 2 588.3.p.b 2
144.u even 12 2 324.3.g.b 2
144.v odd 12 2 324.3.g.b 2
144.w odd 12 2 1296.3.q.b 2
144.x even 12 2 1296.3.q.b 2
176.i even 4 1 1452.3.e.b 1
240.t even 4 1 300.3.g.b 1
240.z odd 4 1 300.3.b.a 2
240.bb even 4 1 1200.3.c.c 2
240.bd odd 4 1 300.3.b.a 2
240.bf even 4 1 1200.3.c.c 2
240.bm odd 4 1 1200.3.l.b 1
336.v odd 4 1 588.3.c.c 1
336.br odd 12 2 588.3.p.b 2
336.bu even 12 2 588.3.p.c 2
528.s odd 4 1 1452.3.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 16.f odd 4 1
12.3.c.a 1 48.k even 4 1
48.3.e.a 1 16.e even 4 1
48.3.e.a 1 48.i odd 4 1
192.3.e.a 1 16.e even 4 1
192.3.e.a 1 48.i odd 4 1
192.3.e.b 1 16.f odd 4 1
192.3.e.b 1 48.k even 4 1
300.3.b.a 2 80.j even 4 1
300.3.b.a 2 80.s even 4 1
300.3.b.a 2 240.z odd 4 1
300.3.b.a 2 240.bd odd 4 1
300.3.g.b 1 80.k odd 4 1
300.3.g.b 1 240.t even 4 1
324.3.g.b 2 144.u even 12 2
324.3.g.b 2 144.v odd 12 2
588.3.c.c 1 112.j even 4 1
588.3.c.c 1 336.v odd 4 1
588.3.p.b 2 112.v even 12 2
588.3.p.b 2 336.br odd 12 2
588.3.p.c 2 112.u odd 12 2
588.3.p.c 2 336.bu even 12 2
768.3.h.a 2 4.b odd 2 1
768.3.h.a 2 8.d odd 2 1
768.3.h.a 2 12.b even 2 1
768.3.h.a 2 24.f even 2 1
768.3.h.b 2 1.a even 1 1 trivial
768.3.h.b 2 3.b odd 2 1 CM
768.3.h.b 2 8.b even 2 1 inner
768.3.h.b 2 24.h odd 2 1 inner
1200.3.c.c 2 80.i odd 4 1
1200.3.c.c 2 80.t odd 4 1
1200.3.c.c 2 240.bb even 4 1
1200.3.c.c 2 240.bf even 4 1
1200.3.l.b 1 80.q even 4 1
1200.3.l.b 1 240.bm odd 4 1
1296.3.q.b 2 144.w odd 12 2
1296.3.q.b 2 144.x even 12 2
1452.3.e.b 1 176.i even 4 1
1452.3.e.b 1 528.s odd 4 1

Hecke kernels

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

\( T_{5} \)
\( T_{7} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 484 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 676 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -46 + T )^{2} \)
$37$ \( 676 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 484 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 5476 + T^{2} \)
$67$ \( 14884 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -46 + T )^{2} \)
$79$ \( ( -142 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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