# Properties

 Label 768.2.j.d Level $768$ Weight $2$ Character orbit 768.j Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{3} + ( 2 - 2 \zeta_{8}^{2} ) q^{5} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{3} + ( 2 - 2 \zeta_{8}^{2} ) q^{5} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} + 4 \zeta_{8}^{3} q^{11} + ( 3 + 3 \zeta_{8}^{2} ) q^{13} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{15} -6 q^{17} + 2 \zeta_{8} q^{19} + ( -3 + 3 \zeta_{8}^{2} ) q^{21} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} -3 \zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + ( 4 + 4 \zeta_{8}^{2} ) q^{29} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{31} -4 q^{33} + 12 \zeta_{8} q^{35} + ( 3 - 3 \zeta_{8}^{2} ) q^{37} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{39} -10 \zeta_{8}^{2} q^{41} -6 \zeta_{8}^{3} q^{43} + ( 2 + 2 \zeta_{8}^{2} ) q^{45} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} -11 q^{49} -6 \zeta_{8} q^{51} + ( 4 - 4 \zeta_{8}^{2} ) q^{53} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{55} + 2 \zeta_{8}^{2} q^{57} + ( 3 + 3 \zeta_{8}^{2} ) q^{61} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{63} + 12 q^{65} -4 \zeta_{8} q^{67} + ( 2 - 2 \zeta_{8}^{2} ) q^{69} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} -3 \zeta_{8}^{3} q^{75} + ( -12 - 12 \zeta_{8}^{2} ) q^{77} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} - q^{81} + 16 \zeta_{8} q^{83} + ( -12 + 12 \zeta_{8}^{2} ) q^{85} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{87} -14 \zeta_{8}^{2} q^{89} + 18 \zeta_{8}^{3} q^{91} + ( 3 + 3 \zeta_{8}^{2} ) q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} -4 q^{97} -4 \zeta_{8} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{5} + O(q^{10})$$ $$4q + 8q^{5} + 12q^{13} - 24q^{17} - 12q^{21} + 16q^{29} - 16q^{33} + 12q^{37} + 8q^{45} - 44q^{49} + 16q^{53} + 12q^{61} + 48q^{65} + 8q^{69} - 48q^{77} - 4q^{81} - 48q^{85} + 12q^{93} - 16q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 −0.707107 + 0.707107i 0 2.00000 + 2.00000i 0 4.24264i 0 1.00000i 0
193.2 0 0.707107 0.707107i 0 2.00000 + 2.00000i 0 4.24264i 0 1.00000i 0
577.1 0 −0.707107 0.707107i 0 2.00000 2.00000i 0 4.24264i 0 1.00000i 0
577.2 0 0.707107 + 0.707107i 0 2.00000 2.00000i 0 4.24264i 0 1.00000i 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
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.j.d yes 4
3.b odd 2 1 2304.2.k.a 4
4.b odd 2 1 inner 768.2.j.d yes 4
8.b even 2 1 768.2.j.a 4
8.d odd 2 1 768.2.j.a 4
12.b even 2 1 2304.2.k.a 4
16.e even 4 1 768.2.j.a 4
16.e even 4 1 inner 768.2.j.d yes 4
16.f odd 4 1 768.2.j.a 4
16.f odd 4 1 inner 768.2.j.d yes 4
24.f even 2 1 2304.2.k.d 4
24.h odd 2 1 2304.2.k.d 4
32.g even 8 1 3072.2.a.d 2
32.g even 8 1 3072.2.a.f 2
32.g even 8 2 3072.2.d.d 4
32.h odd 8 1 3072.2.a.d 2
32.h odd 8 1 3072.2.a.f 2
32.h odd 8 2 3072.2.d.d 4
48.i odd 4 1 2304.2.k.a 4
48.i odd 4 1 2304.2.k.d 4
48.k even 4 1 2304.2.k.a 4
48.k even 4 1 2304.2.k.d 4
96.o even 8 1 9216.2.a.e 2
96.o even 8 1 9216.2.a.q 2
96.p odd 8 1 9216.2.a.e 2
96.p odd 8 1 9216.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.a 4 8.b even 2 1
768.2.j.a 4 8.d odd 2 1
768.2.j.a 4 16.e even 4 1
768.2.j.a 4 16.f odd 4 1
768.2.j.d yes 4 1.a even 1 1 trivial
768.2.j.d yes 4 4.b odd 2 1 inner
768.2.j.d yes 4 16.e even 4 1 inner
768.2.j.d yes 4 16.f odd 4 1 inner
2304.2.k.a 4 3.b odd 2 1
2304.2.k.a 4 12.b even 2 1
2304.2.k.a 4 48.i odd 4 1
2304.2.k.a 4 48.k even 4 1
2304.2.k.d 4 24.f even 2 1
2304.2.k.d 4 24.h odd 2 1
2304.2.k.d 4 48.i odd 4 1
2304.2.k.d 4 48.k even 4 1
3072.2.a.d 2 32.g even 8 1
3072.2.a.d 2 32.h odd 8 1
3072.2.a.f 2 32.g even 8 1
3072.2.a.f 2 32.h odd 8 1
3072.2.d.d 4 32.g even 8 2
3072.2.d.d 4 32.h odd 8 2
9216.2.a.e 2 96.o even 8 1
9216.2.a.e 2 96.p odd 8 1
9216.2.a.q 2 96.o even 8 1
9216.2.a.q 2 96.p odd 8 1

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 8$$ $$T_{13}^{2} - 6 T_{13} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$( 8 - 4 T + T^{2} )^{2}$$
$7$ $$( 18 + T^{2} )^{2}$$
$11$ $$256 + T^{4}$$
$13$ $$( 18 - 6 T + T^{2} )^{2}$$
$17$ $$( 6 + T )^{4}$$
$19$ $$16 + T^{4}$$
$23$ $$( 8 + T^{2} )^{2}$$
$29$ $$( 32 - 8 T + T^{2} )^{2}$$
$31$ $$( -18 + T^{2} )^{2}$$
$37$ $$( 18 - 6 T + T^{2} )^{2}$$
$41$ $$( 100 + T^{2} )^{2}$$
$43$ $$1296 + T^{4}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$( 32 - 8 T + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 18 - 6 T + T^{2} )^{2}$$
$67$ $$256 + T^{4}$$
$71$ $$( 8 + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( -18 + T^{2} )^{2}$$
$83$ $$65536 + T^{4}$$
$89$ $$( 196 + T^{2} )^{2}$$
$97$ $$( 4 + T )^{4}$$