# Properties

 Label 768.6.d.c Level 768 Weight 6 Character orbit 768.d Analytic conductor 123.175 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 768.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$123.174773616$$ 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 6) 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.

 $$f(q)$$ $$=$$ $$q -9 i q^{3} + 66 i q^{5} -176 q^{7} -81 q^{9} +O(q^{10})$$ $$q -9 i q^{3} + 66 i q^{5} -176 q^{7} -81 q^{9} + 60 i q^{11} -658 i q^{13} + 594 q^{15} -414 q^{17} + 956 i q^{19} + 1584 i q^{21} -600 q^{23} -1231 q^{25} + 729 i q^{27} + 5574 i q^{29} -3592 q^{31} + 540 q^{33} -11616 i q^{35} + 8458 i q^{37} -5922 q^{39} -19194 q^{41} -13316 i q^{43} -5346 i q^{45} -19680 q^{47} + 14169 q^{49} + 3726 i q^{51} + 31266 i q^{53} -3960 q^{55} + 8604 q^{57} -26340 i q^{59} -31090 i q^{61} + 14256 q^{63} + 43428 q^{65} -16804 i q^{67} + 5400 i q^{69} -6120 q^{71} + 25558 q^{73} + 11079 i q^{75} -10560 i q^{77} + 74408 q^{79} + 6561 q^{81} -6468 i q^{83} -27324 i q^{85} + 50166 q^{87} + 32742 q^{89} + 115808 i q^{91} + 32328 i q^{93} -63096 q^{95} + 166082 q^{97} -4860 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 352q^{7} - 162q^{9} + O(q^{10})$$ $$2q - 352q^{7} - 162q^{9} + 1188q^{15} - 828q^{17} - 1200q^{23} - 2462q^{25} - 7184q^{31} + 1080q^{33} - 11844q^{39} - 38388q^{41} - 39360q^{47} + 28338q^{49} - 7920q^{55} + 17208q^{57} + 28512q^{63} + 86856q^{65} - 12240q^{71} + 51116q^{73} + 148816q^{79} + 13122q^{81} + 100332q^{87} + 65484q^{89} - 126192q^{95} + 332164q^{97} + O(q^{100})$$

## 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 9.00000i 0 66.0000i 0 −176.000 0 −81.0000 0
385.2 0 9.00000i 0 66.0000i 0 −176.000 0 −81.0000 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 768.6.d.c 2
4.b odd 2 1 768.6.d.p 2
8.b even 2 1 inner 768.6.d.c 2
8.d odd 2 1 768.6.d.p 2
16.e even 4 1 6.6.a.a 1
16.e even 4 1 192.6.a.o 1
16.f odd 4 1 48.6.a.c 1
16.f odd 4 1 192.6.a.g 1
48.i odd 4 1 18.6.a.b 1
48.i odd 4 1 576.6.a.j 1
48.k even 4 1 144.6.a.j 1
48.k even 4 1 576.6.a.i 1
80.i odd 4 1 150.6.c.b 2
80.q even 4 1 150.6.a.d 1
80.t odd 4 1 150.6.c.b 2
112.l odd 4 1 294.6.a.m 1
112.w even 12 2 294.6.e.g 2
112.x odd 12 2 294.6.e.a 2
144.w odd 12 2 162.6.c.h 2
144.x even 12 2 162.6.c.e 2
176.l odd 4 1 726.6.a.a 1
208.p even 4 1 1014.6.a.c 1
240.bb even 4 1 450.6.c.j 2
240.bf even 4 1 450.6.c.j 2
240.bm odd 4 1 450.6.a.m 1
336.y even 4 1 882.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 16.e even 4 1
18.6.a.b 1 48.i odd 4 1
48.6.a.c 1 16.f odd 4 1
144.6.a.j 1 48.k even 4 1
150.6.a.d 1 80.q even 4 1
150.6.c.b 2 80.i odd 4 1
150.6.c.b 2 80.t odd 4 1
162.6.c.e 2 144.x even 12 2
162.6.c.h 2 144.w odd 12 2
192.6.a.g 1 16.f odd 4 1
192.6.a.o 1 16.e even 4 1
294.6.a.m 1 112.l odd 4 1
294.6.e.a 2 112.x odd 12 2
294.6.e.g 2 112.w even 12 2
450.6.a.m 1 240.bm odd 4 1
450.6.c.j 2 240.bb even 4 1
450.6.c.j 2 240.bf even 4 1
576.6.a.i 1 48.k even 4 1
576.6.a.j 1 48.i odd 4 1
726.6.a.a 1 176.l odd 4 1
768.6.d.c 2 1.a even 1 1 trivial
768.6.d.c 2 8.b even 2 1 inner
768.6.d.p 2 4.b odd 2 1
768.6.d.p 2 8.d odd 2 1
882.6.a.a 1 336.y even 4 1
1014.6.a.c 1 208.p even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{2} + 4356$$ $$T_{7} + 176$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 81 T^{2}$$
$5$ $$1 - 1894 T^{2} + 9765625 T^{4}$$
$7$ $$( 1 + 176 T + 16807 T^{2} )^{2}$$
$11$ $$1 - 318502 T^{2} + 25937424601 T^{4}$$
$13$ $$1 - 309622 T^{2} + 137858491849 T^{4}$$
$17$ $$( 1 + 414 T + 1419857 T^{2} )^{2}$$
$19$ $$1 - 4038262 T^{2} + 6131066257801 T^{4}$$
$23$ $$( 1 + 600 T + 6436343 T^{2} )^{2}$$
$29$ $$1 - 9952822 T^{2} + 420707233300201 T^{4}$$
$31$ $$( 1 + 3592 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 67150150 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 19194 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 116701030 T^{2} + 21611482313284249 T^{4}$$
$47$ $$( 1 + 19680 T + 229345007 T^{2} )^{2}$$
$53$ $$1 + 141171770 T^{2} + 174887470365513049 T^{4}$$
$59$ $$1 - 736052998 T^{2} + 511116753300641401 T^{4}$$
$61$ $$1 - 722604502 T^{2} + 713342911662882601 T^{4}$$
$67$ $$1 - 2417875798 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 6120 T + 1804229351 T^{2} )^{2}$$
$73$ $$( 1 - 25558 T + 2073071593 T^{2} )^{2}$$
$79$ $$( 1 - 74408 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7836246262 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 32742 T + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 - 166082 T + 8587340257 T^{2} )^{2}$$