Properties

Label 768.2.f.d
Level $768$
Weight $2$
Character orbit 768.f
Analytic conductor $6.133$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 48)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 + 4 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 + 4 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + 2 \zeta_{12}^{3} q^{13} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{19} + 6 \zeta_{12}^{3} q^{21} -5 q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -6 + 12 \zeta_{12}^{2} ) q^{31} -10 \zeta_{12}^{3} q^{37} + ( -2 + 4 \zeta_{12}^{2} ) q^{39} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{43} -5 q^{49} + 6 q^{57} -14 \zeta_{12}^{3} q^{61} + ( -6 + 12 \zeta_{12}^{2} ) q^{63} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} -10 q^{73} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{75} + ( 10 - 20 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{91} + 18 \zeta_{12}^{3} q^{93} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + O(q^{10}) \) \( 4 q + 12 q^{9} - 20 q^{25} - 20 q^{49} + 24 q^{57} - 40 q^{73} + 36 q^{81} - 56 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −1.73205 0 0 0 3.46410i 0 3.00000 0
383.2 0 −1.73205 0 0 0 3.46410i 0 3.00000 0
383.3 0 1.73205 0 0 0 3.46410i 0 3.00000 0
383.4 0 1.73205 0 0 0 3.46410i 0 3.00000 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.f.d 4
3.b odd 2 1 CM 768.2.f.d 4
4.b odd 2 1 inner 768.2.f.d 4
8.b even 2 1 inner 768.2.f.d 4
8.d odd 2 1 inner 768.2.f.d 4
12.b even 2 1 inner 768.2.f.d 4
16.e even 4 1 48.2.c.a 2
16.e even 4 1 192.2.c.a 2
16.f odd 4 1 48.2.c.a 2
16.f odd 4 1 192.2.c.a 2
24.f even 2 1 inner 768.2.f.d 4
24.h odd 2 1 inner 768.2.f.d 4
48.i odd 4 1 48.2.c.a 2
48.i odd 4 1 192.2.c.a 2
48.k even 4 1 48.2.c.a 2
48.k even 4 1 192.2.c.a 2
80.i odd 4 1 1200.2.o.i 4
80.j even 4 1 1200.2.o.i 4
80.k odd 4 1 1200.2.h.e 2
80.q even 4 1 1200.2.h.e 2
80.s even 4 1 1200.2.o.i 4
80.t odd 4 1 1200.2.o.i 4
112.j even 4 1 2352.2.h.c 2
112.l odd 4 1 2352.2.h.c 2
144.u even 12 1 1296.2.s.b 2
144.u even 12 1 1296.2.s.e 2
144.v odd 12 1 1296.2.s.b 2
144.v odd 12 1 1296.2.s.e 2
144.w odd 12 1 1296.2.s.b 2
144.w odd 12 1 1296.2.s.e 2
144.x even 12 1 1296.2.s.b 2
144.x even 12 1 1296.2.s.e 2
240.t even 4 1 1200.2.h.e 2
240.z odd 4 1 1200.2.o.i 4
240.bb even 4 1 1200.2.o.i 4
240.bd odd 4 1 1200.2.o.i 4
240.bf even 4 1 1200.2.o.i 4
240.bm odd 4 1 1200.2.h.e 2
336.v odd 4 1 2352.2.h.c 2
336.y even 4 1 2352.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 16.e even 4 1
48.2.c.a 2 16.f odd 4 1
48.2.c.a 2 48.i odd 4 1
48.2.c.a 2 48.k even 4 1
192.2.c.a 2 16.e even 4 1
192.2.c.a 2 16.f odd 4 1
192.2.c.a 2 48.i odd 4 1
192.2.c.a 2 48.k even 4 1
768.2.f.d 4 1.a even 1 1 trivial
768.2.f.d 4 3.b odd 2 1 CM
768.2.f.d 4 4.b odd 2 1 inner
768.2.f.d 4 8.b even 2 1 inner
768.2.f.d 4 8.d odd 2 1 inner
768.2.f.d 4 12.b even 2 1 inner
768.2.f.d 4 24.f even 2 1 inner
768.2.f.d 4 24.h odd 2 1 inner
1200.2.h.e 2 80.k odd 4 1
1200.2.h.e 2 80.q even 4 1
1200.2.h.e 2 240.t even 4 1
1200.2.h.e 2 240.bm odd 4 1
1200.2.o.i 4 80.i odd 4 1
1200.2.o.i 4 80.j even 4 1
1200.2.o.i 4 80.s even 4 1
1200.2.o.i 4 80.t odd 4 1
1200.2.o.i 4 240.z odd 4 1
1200.2.o.i 4 240.bb even 4 1
1200.2.o.i 4 240.bd odd 4 1
1200.2.o.i 4 240.bf even 4 1
1296.2.s.b 2 144.u even 12 1
1296.2.s.b 2 144.v odd 12 1
1296.2.s.b 2 144.w odd 12 1
1296.2.s.b 2 144.x even 12 1
1296.2.s.e 2 144.u even 12 1
1296.2.s.e 2 144.v odd 12 1
1296.2.s.e 2 144.w odd 12 1
1296.2.s.e 2 144.x even 12 1
2352.2.h.c 2 112.j even 4 1
2352.2.h.c 2 112.l odd 4 1
2352.2.h.c 2 336.v odd 4 1
2352.2.h.c 2 336.y even 4 1

Hecke kernels

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

\( T_{5} \)
\( T_{19}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 12 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -12 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 108 + T^{2} )^{2} \)
$37$ \( ( 100 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -108 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 196 + T^{2} )^{2} \)
$67$ \( ( -12 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 10 + T )^{4} \)
$79$ \( ( 300 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 14 + T )^{4} \)
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