# 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 $8$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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