# Properties

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

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$20.9264843029$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 12) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + 2 \zeta_{12}^{3} q^{13} + ( 2 - 4 \zeta_{12}^{2} ) q^{15} + 10 q^{17} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{19} -12 \zeta_{12}^{3} q^{21} + ( -16 + 32 \zeta_{12}^{2} ) q^{23} + 21 q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -26 \zeta_{12}^{3} q^{29} + ( -4 + 8 \zeta_{12}^{2} ) q^{31} -12 q^{33} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{35} -26 \zeta_{12}^{3} q^{37} + ( 2 - 4 \zeta_{12}^{2} ) q^{39} -58 q^{41} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{43} + 6 \zeta_{12}^{3} q^{45} + ( -40 + 80 \zeta_{12}^{2} ) q^{47} + q^{49} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{51} + 74 \zeta_{12}^{3} q^{53} + ( -8 + 16 \zeta_{12}^{2} ) q^{55} + 36 q^{57} + ( -104 \zeta_{12} + 52 \zeta_{12}^{3} ) q^{59} + 26 \zeta_{12}^{3} q^{61} + ( -12 + 24 \zeta_{12}^{2} ) q^{63} -4 q^{65} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{67} -48 \zeta_{12}^{3} q^{69} + 46 q^{73} + ( -42 \zeta_{12} + 21 \zeta_{12}^{3} ) q^{75} + 48 \zeta_{12}^{3} q^{77} + ( -68 + 136 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{83} + 20 \zeta_{12}^{3} q^{85} + ( -26 + 52 \zeta_{12}^{2} ) q^{87} -82 q^{89} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{91} -12 \zeta_{12}^{3} q^{93} + ( 24 - 48 \zeta_{12}^{2} ) q^{95} + 2 q^{97} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 40q^{17} + 84q^{25} - 48q^{33} - 232q^{41} + 4q^{49} + 144q^{57} - 16q^{65} + 184q^{73} + 36q^{81} - 328q^{89} + 8q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
0 −1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.2 0 −1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.3 0 1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.4 0 1.73205 0 2.00000i 0 6.92820i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.b.c 4
3.b odd 2 1 2304.3.b.l 4
4.b odd 2 1 inner 768.3.b.c 4
8.b even 2 1 inner 768.3.b.c 4
8.d odd 2 1 inner 768.3.b.c 4
12.b even 2 1 2304.3.b.l 4
16.e even 4 1 12.3.d.a 2
16.e even 4 1 192.3.g.b 2
16.f odd 4 1 12.3.d.a 2
16.f odd 4 1 192.3.g.b 2
24.f even 2 1 2304.3.b.l 4
24.h odd 2 1 2304.3.b.l 4
48.i odd 4 1 36.3.d.c 2
48.i odd 4 1 576.3.g.e 2
48.k even 4 1 36.3.d.c 2
48.k even 4 1 576.3.g.e 2
80.i odd 4 1 300.3.f.a 4
80.j even 4 1 300.3.f.a 4
80.k odd 4 1 300.3.c.b 2
80.q even 4 1 300.3.c.b 2
80.s even 4 1 300.3.f.a 4
80.t odd 4 1 300.3.f.a 4
112.j even 4 1 588.3.g.b 2
112.l odd 4 1 588.3.g.b 2
144.u even 12 1 324.3.f.a 2
144.u even 12 1 324.3.f.g 2
144.v odd 12 1 324.3.f.d 2
144.v odd 12 1 324.3.f.j 2
144.w odd 12 1 324.3.f.a 2
144.w odd 12 1 324.3.f.g 2
144.x even 12 1 324.3.f.d 2
144.x even 12 1 324.3.f.j 2
240.t even 4 1 900.3.c.e 2
240.z odd 4 1 900.3.f.c 4
240.bb even 4 1 900.3.f.c 4
240.bd odd 4 1 900.3.f.c 4
240.bf even 4 1 900.3.f.c 4
240.bm odd 4 1 900.3.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 16.e even 4 1
12.3.d.a 2 16.f odd 4 1
36.3.d.c 2 48.i odd 4 1
36.3.d.c 2 48.k even 4 1
192.3.g.b 2 16.e even 4 1
192.3.g.b 2 16.f odd 4 1
300.3.c.b 2 80.k odd 4 1
300.3.c.b 2 80.q even 4 1
300.3.f.a 4 80.i odd 4 1
300.3.f.a 4 80.j even 4 1
300.3.f.a 4 80.s even 4 1
300.3.f.a 4 80.t odd 4 1
324.3.f.a 2 144.u even 12 1
324.3.f.a 2 144.w odd 12 1
324.3.f.d 2 144.v odd 12 1
324.3.f.d 2 144.x even 12 1
324.3.f.g 2 144.u even 12 1
324.3.f.g 2 144.w odd 12 1
324.3.f.j 2 144.v odd 12 1
324.3.f.j 2 144.x even 12 1
576.3.g.e 2 48.i odd 4 1
576.3.g.e 2 48.k even 4 1
588.3.g.b 2 112.j even 4 1
588.3.g.b 2 112.l odd 4 1
768.3.b.c 4 1.a even 1 1 trivial
768.3.b.c 4 4.b odd 2 1 inner
768.3.b.c 4 8.b even 2 1 inner
768.3.b.c 4 8.d odd 2 1 inner
900.3.c.e 2 240.t even 4 1
900.3.c.e 2 240.bm odd 4 1
900.3.f.c 4 240.z odd 4 1
900.3.f.c 4 240.bb even 4 1
900.3.f.c 4 240.bd odd 4 1
900.3.f.c 4 240.bf even 4 1
2304.3.b.l 4 3.b odd 2 1
2304.3.b.l 4 12.b even 2 1
2304.3.b.l 4 24.f even 2 1
2304.3.b.l 4 24.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 4$$ $$T_{11}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 4 + T^{2} )^{2}$$
$7$ $$( 48 + T^{2} )^{2}$$
$11$ $$( -48 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -10 + T )^{4}$$
$19$ $$( -432 + T^{2} )^{2}$$
$23$ $$( 768 + T^{2} )^{2}$$
$29$ $$( 676 + T^{2} )^{2}$$
$31$ $$( 48 + T^{2} )^{2}$$
$37$ $$( 676 + T^{2} )^{2}$$
$41$ $$( 58 + T )^{4}$$
$43$ $$( -2352 + T^{2} )^{2}$$
$47$ $$( 4800 + T^{2} )^{2}$$
$53$ $$( 5476 + T^{2} )^{2}$$
$59$ $$( -8112 + T^{2} )^{2}$$
$61$ $$( 676 + T^{2} )^{2}$$
$67$ $$( -48 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( -46 + T )^{4}$$
$79$ $$( 13872 + T^{2} )^{2}$$
$83$ $$( -2352 + T^{2} )^{2}$$
$89$ $$( 82 + T )^{4}$$
$97$ $$( -2 + T )^{4}$$
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