# Properties

 Label 768.3.h.d Level $768$ Weight $3$ Character orbit 768.h Analytic conductor $20.926$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + 6 q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + 6 q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + 10 \zeta_{8}^{2} q^{13} + ( 16 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{15} + ( 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{17} + 2 \zeta_{8}^{2} q^{19} + ( -12 \zeta_{8} + 6 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{21} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{23} + 7 q^{25} + ( -10 \zeta_{8} + 23 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{27} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{29} -22 q^{31} + ( 16 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{33} + ( -24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{35} + 6 \zeta_{8}^{2} q^{37} + ( -10 - 20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{39} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{41} -82 \zeta_{8}^{2} q^{43} + ( -28 \zeta_{8} + 32 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{45} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{47} -13 q^{49} + ( -16 \zeta_{8} - 64 \zeta_{8}^{2} + 16 \zeta_{8}^{3} ) q^{51} + ( 44 \zeta_{8} - 44 \zeta_{8}^{3} ) q^{53} + 32 q^{55} + ( -2 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{57} + ( -52 \zeta_{8} + 52 \zeta_{8}^{3} ) q^{59} -86 \zeta_{8}^{2} q^{61} + ( 42 - 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{63} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{65} + 2 \zeta_{8}^{2} q^{67} + ( 8 \zeta_{8} + 32 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{69} + ( -88 \zeta_{8} - 88 \zeta_{8}^{3} ) q^{71} -82 q^{73} + ( -14 \zeta_{8} + 7 \zeta_{8}^{2} + 14 \zeta_{8}^{3} ) q^{75} + ( -24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{77} + 10 q^{79} + ( 17 - 56 \zeta_{8} - 56 \zeta_{8}^{3} ) q^{81} + ( -52 \zeta_{8} + 52 \zeta_{8}^{3} ) q^{83} -128 \zeta_{8}^{2} q^{85} + ( -48 + 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{87} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{89} + 60 \zeta_{8}^{2} q^{91} + ( 44 \zeta_{8} - 22 \zeta_{8}^{2} - 44 \zeta_{8}^{3} ) q^{93} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} -94 q^{97} + ( -28 \zeta_{8} + 32 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 24q^{7} + 28q^{9} + O(q^{10})$$ $$4q + 24q^{7} + 28q^{9} + 64q^{15} + 28q^{25} - 88q^{31} + 64q^{33} - 40q^{39} - 52q^{49} + 128q^{55} - 8q^{57} + 168q^{63} - 328q^{73} + 40q^{79} + 68q^{81} - 192q^{87} - 376q^{97} + O(q^{100})$$

## 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
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −2.82843 1.00000i 0 −5.65685 0 6.00000 0 7.00000 + 5.65685i 0
641.2 0 −2.82843 + 1.00000i 0 −5.65685 0 6.00000 0 7.00000 5.65685i 0
641.3 0 2.82843 1.00000i 0 5.65685 0 6.00000 0 7.00000 5.65685i 0
641.4 0 2.82843 + 1.00000i 0 5.65685 0 6.00000 0 7.00000 + 5.65685i 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 inner
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.d 4
3.b odd 2 1 inner 768.3.h.d 4
4.b odd 2 1 768.3.h.c 4
8.b even 2 1 inner 768.3.h.d 4
8.d odd 2 1 768.3.h.c 4
12.b even 2 1 768.3.h.c 4
16.e even 4 1 24.3.e.a 2
16.e even 4 1 192.3.e.c 2
16.f odd 4 1 48.3.e.b 2
16.f odd 4 1 192.3.e.d 2
24.f even 2 1 768.3.h.c 4
24.h odd 2 1 inner 768.3.h.d 4
48.i odd 4 1 24.3.e.a 2
48.i odd 4 1 192.3.e.c 2
48.k even 4 1 48.3.e.b 2
48.k even 4 1 192.3.e.d 2
80.i odd 4 1 600.3.c.a 4
80.j even 4 1 1200.3.c.i 4
80.k odd 4 1 1200.3.l.n 2
80.q even 4 1 600.3.l.b 2
80.s even 4 1 1200.3.c.i 4
80.t odd 4 1 600.3.c.a 4
112.l odd 4 1 1176.3.d.a 2
144.u even 12 2 1296.3.q.e 4
144.v odd 12 2 1296.3.q.e 4
144.w odd 12 2 648.3.m.d 4
144.x even 12 2 648.3.m.d 4
240.t even 4 1 1200.3.l.n 2
240.z odd 4 1 1200.3.c.i 4
240.bb even 4 1 600.3.c.a 4
240.bd odd 4 1 1200.3.c.i 4
240.bf even 4 1 600.3.c.a 4
240.bm odd 4 1 600.3.l.b 2
336.y even 4 1 1176.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 16.e even 4 1
24.3.e.a 2 48.i odd 4 1
48.3.e.b 2 16.f odd 4 1
48.3.e.b 2 48.k even 4 1
192.3.e.c 2 16.e even 4 1
192.3.e.c 2 48.i odd 4 1
192.3.e.d 2 16.f odd 4 1
192.3.e.d 2 48.k even 4 1
600.3.c.a 4 80.i odd 4 1
600.3.c.a 4 80.t odd 4 1
600.3.c.a 4 240.bb even 4 1
600.3.c.a 4 240.bf even 4 1
600.3.l.b 2 80.q even 4 1
600.3.l.b 2 240.bm odd 4 1
648.3.m.d 4 144.w odd 12 2
648.3.m.d 4 144.x even 12 2
768.3.h.c 4 4.b odd 2 1
768.3.h.c 4 8.d odd 2 1
768.3.h.c 4 12.b even 2 1
768.3.h.c 4 24.f even 2 1
768.3.h.d 4 1.a even 1 1 trivial
768.3.h.d 4 3.b odd 2 1 inner
768.3.h.d 4 8.b even 2 1 inner
768.3.h.d 4 24.h odd 2 1 inner
1176.3.d.a 2 112.l odd 4 1
1176.3.d.a 2 336.y even 4 1
1200.3.c.i 4 80.j even 4 1
1200.3.c.i 4 80.s even 4 1
1200.3.c.i 4 240.z odd 4 1
1200.3.c.i 4 240.bd odd 4 1
1200.3.l.n 2 80.k odd 4 1
1200.3.l.n 2 240.t even 4 1
1296.3.q.e 4 144.u even 12 2
1296.3.q.e 4 144.v odd 12 2

## Hecke kernels

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

 $$T_{5}^{2} - 32$$ $$T_{7} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 14 T^{2} + T^{4}$$
$5$ $$( -32 + T^{2} )^{2}$$
$7$ $$( -6 + T )^{4}$$
$11$ $$( -32 + T^{2} )^{2}$$
$13$ $$( 100 + T^{2} )^{2}$$
$17$ $$( 512 + T^{2} )^{2}$$
$19$ $$( 4 + T^{2} )^{2}$$
$23$ $$( 128 + T^{2} )^{2}$$
$29$ $$( -288 + T^{2} )^{2}$$
$31$ $$( 22 + T )^{4}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( 1152 + T^{2} )^{2}$$
$43$ $$( 6724 + T^{2} )^{2}$$
$47$ $$( 4608 + T^{2} )^{2}$$
$53$ $$( -3872 + T^{2} )^{2}$$
$59$ $$( -5408 + T^{2} )^{2}$$
$61$ $$( 7396 + T^{2} )^{2}$$
$67$ $$( 4 + T^{2} )^{2}$$
$71$ $$( 15488 + T^{2} )^{2}$$
$73$ $$( 82 + T )^{4}$$
$79$ $$( -10 + T )^{4}$$
$83$ $$( -5408 + T^{2} )^{2}$$
$89$ $$( 1152 + T^{2} )^{2}$$
$97$ $$( 94 + T )^{4}$$