# Properties

 Label 768.4.d.g Level $768$ Weight $4$ Character orbit 768.d Analytic conductor $45.313$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$45.3134668844$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 18 i q^{5} -8 q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + 18 i q^{5} -8 q^{7} -9 q^{9} -36 i q^{11} -10 i q^{13} -54 q^{15} + 18 q^{17} -100 i q^{19} -24 i q^{21} -72 q^{23} -199 q^{25} -27 i q^{27} -234 i q^{29} -16 q^{31} + 108 q^{33} -144 i q^{35} + 226 i q^{37} + 30 q^{39} -90 q^{41} -452 i q^{43} -162 i q^{45} + 432 q^{47} -279 q^{49} + 54 i q^{51} -414 i q^{53} + 648 q^{55} + 300 q^{57} + 684 i q^{59} + 422 i q^{61} + 72 q^{63} + 180 q^{65} + 332 i q^{67} -216 i q^{69} + 360 q^{71} -26 q^{73} -597 i q^{75} + 288 i q^{77} + 512 q^{79} + 81 q^{81} -1188 i q^{83} + 324 i q^{85} + 702 q^{87} + 630 q^{89} + 80 i q^{91} -48 i q^{93} + 1800 q^{95} -1054 q^{97} + 324 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{7} - 18q^{9} + O(q^{10})$$ $$2q - 16q^{7} - 18q^{9} - 108q^{15} + 36q^{17} - 144q^{23} - 398q^{25} - 32q^{31} + 216q^{33} + 60q^{39} - 180q^{41} + 864q^{47} - 558q^{49} + 1296q^{55} + 600q^{57} + 144q^{63} + 360q^{65} + 720q^{71} - 52q^{73} + 1024q^{79} + 162q^{81} + 1404q^{87} + 1260q^{89} + 3600q^{95} - 2108q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
385.1
 − 1.00000i 1.00000i
0 3.00000i 0 18.0000i 0 −8.00000 0 −9.00000 0
385.2 0 3.00000i 0 18.0000i 0 −8.00000 0 −9.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.d.g 2
4.b odd 2 1 768.4.d.j 2
8.b even 2 1 inner 768.4.d.g 2
8.d odd 2 1 768.4.d.j 2
16.e even 4 1 12.4.a.a 1
16.e even 4 1 192.4.a.f 1
16.f odd 4 1 48.4.a.a 1
16.f odd 4 1 192.4.a.l 1
48.i odd 4 1 36.4.a.a 1
48.i odd 4 1 576.4.a.b 1
48.k even 4 1 144.4.a.g 1
48.k even 4 1 576.4.a.a 1
80.i odd 4 1 300.4.d.e 2
80.j even 4 1 1200.4.f.d 2
80.k odd 4 1 1200.4.a.be 1
80.q even 4 1 300.4.a.b 1
80.s even 4 1 1200.4.f.d 2
80.t odd 4 1 300.4.d.e 2
112.j even 4 1 2352.4.a.bk 1
112.l odd 4 1 588.4.a.c 1
112.w even 12 2 588.4.i.d 2
112.x odd 12 2 588.4.i.e 2
144.w odd 12 2 324.4.e.a 2
144.x even 12 2 324.4.e.h 2
176.l odd 4 1 1452.4.a.d 1
208.m odd 4 1 2028.4.b.c 2
208.p even 4 1 2028.4.a.c 1
208.r odd 4 1 2028.4.b.c 2
240.bb even 4 1 900.4.d.c 2
240.bf even 4 1 900.4.d.c 2
240.bm odd 4 1 900.4.a.g 1
336.y even 4 1 1764.4.a.b 1
336.bo even 12 2 1764.4.k.o 2
336.bt odd 12 2 1764.4.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 16.e even 4 1
36.4.a.a 1 48.i odd 4 1
48.4.a.a 1 16.f odd 4 1
144.4.a.g 1 48.k even 4 1
192.4.a.f 1 16.e even 4 1
192.4.a.l 1 16.f odd 4 1
300.4.a.b 1 80.q even 4 1
300.4.d.e 2 80.i odd 4 1
300.4.d.e 2 80.t odd 4 1
324.4.e.a 2 144.w odd 12 2
324.4.e.h 2 144.x even 12 2
576.4.a.a 1 48.k even 4 1
576.4.a.b 1 48.i odd 4 1
588.4.a.c 1 112.l odd 4 1
588.4.i.d 2 112.w even 12 2
588.4.i.e 2 112.x odd 12 2
768.4.d.g 2 1.a even 1 1 trivial
768.4.d.g 2 8.b even 2 1 inner
768.4.d.j 2 4.b odd 2 1
768.4.d.j 2 8.d odd 2 1
900.4.a.g 1 240.bm odd 4 1
900.4.d.c 2 240.bb even 4 1
900.4.d.c 2 240.bf even 4 1
1200.4.a.be 1 80.k odd 4 1
1200.4.f.d 2 80.j even 4 1
1200.4.f.d 2 80.s even 4 1
1452.4.a.d 1 176.l odd 4 1
1764.4.a.b 1 336.y even 4 1
1764.4.k.b 2 336.bt odd 12 2
1764.4.k.o 2 336.bo even 12 2
2028.4.a.c 1 208.p even 4 1
2028.4.b.c 2 208.m odd 4 1
2028.4.b.c 2 208.r odd 4 1
2352.4.a.bk 1 112.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 324$$ $$T_{7} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$324 + T^{2}$$
$7$ $$( 8 + T )^{2}$$
$11$ $$1296 + T^{2}$$
$13$ $$100 + T^{2}$$
$17$ $$( -18 + T )^{2}$$
$19$ $$10000 + T^{2}$$
$23$ $$( 72 + T )^{2}$$
$29$ $$54756 + T^{2}$$
$31$ $$( 16 + T )^{2}$$
$37$ $$51076 + T^{2}$$
$41$ $$( 90 + T )^{2}$$
$43$ $$204304 + T^{2}$$
$47$ $$( -432 + T )^{2}$$
$53$ $$171396 + T^{2}$$
$59$ $$467856 + T^{2}$$
$61$ $$178084 + T^{2}$$
$67$ $$110224 + T^{2}$$
$71$ $$( -360 + T )^{2}$$
$73$ $$( 26 + T )^{2}$$
$79$ $$( -512 + T )^{2}$$
$83$ $$1411344 + T^{2}$$
$89$ $$( -630 + T )^{2}$$
$97$ $$( 1054 + T )^{2}$$