# Properties

 Label 768.3.b.e Level $768$ Weight $3$ Character orbit 768.b Analytic conductor $20.926$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## 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: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 384) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{5} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{5} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} + 3 q^{9} + ( -4 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{11} + ( 4 - 8 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{13} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{15} + ( 2 - 4 \zeta_{24} + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{17} + ( -12 - 12 \zeta_{24} + 8 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{19} + ( 2 + 10 \zeta_{24} - 10 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{21} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{23} + ( 1 + 16 \zeta_{24}^{2} + 16 \zeta_{24}^{5} - 8 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{25} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{27} + ( 10 - 18 \zeta_{24} + 18 \zeta_{24}^{3} - 20 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 10 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{29} + ( -8 - 14 \zeta_{24} + 14 \zeta_{24}^{3} + 16 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 20 \zeta_{24}^{7} ) q^{31} + ( -12 \zeta_{24} + 8 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{33} + ( 12 + 4 \zeta_{24} - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 28 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{35} + ( 16 + 4 \zeta_{24} - 4 \zeta_{24}^{3} - 32 \zeta_{24}^{4} + 20 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{37} + ( 6 - 8 \zeta_{24} + 8 \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 16 \zeta_{24}^{5} + 12 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{39} + ( -10 + 28 \zeta_{24} + 40 \zeta_{24}^{2} + 28 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 20 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{41} + ( -28 + 20 \zeta_{24} + 24 \zeta_{24}^{2} + 20 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 24 \zeta_{24}^{7} ) q^{43} + ( 6 - 6 \zeta_{24} + 6 \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{45} + ( 8 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 16 \zeta_{24}^{4} - 20 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 24 \zeta_{24}^{7} ) q^{47} + ( -11 + 8 \zeta_{24} + 64 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 32 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{49} + ( -12 + 20 \zeta_{24} - 4 \zeta_{24}^{2} + 20 \zeta_{24}^{3} - 28 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{51} + ( 22 + 18 \zeta_{24} - 18 \zeta_{24}^{3} - 44 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 22 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{53} + ( -24 + 40 \zeta_{24} - 40 \zeta_{24}^{3} + 48 \zeta_{24}^{4} - 24 \zeta_{24}^{5} - 8 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{55} + ( -12 + 20 \zeta_{24} + 24 \zeta_{24}^{2} + 20 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{57} + ( 64 - 16 \zeta_{24} - 24 \zeta_{24}^{2} - 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{59} + ( 24 + 36 \zeta_{24} - 36 \zeta_{24}^{3} - 48 \zeta_{24}^{4} - 28 \zeta_{24}^{5} - 34 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{61} + ( 18 \zeta_{24} - 18 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{63} + ( -20 - 20 \zeta_{24} + 72 \zeta_{24}^{2} - 20 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 36 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{65} + ( -48 \zeta_{24} + 8 \zeta_{24}^{2} - 48 \zeta_{24}^{3} + 48 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{67} + ( 4 + 4 \zeta_{24} - 4 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 20 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{69} + ( 24 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 48 \zeta_{24}^{4} + 12 \zeta_{24}^{5} - 60 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{71} + ( 2 - 56 \zeta_{24} + 64 \zeta_{24}^{2} - 56 \zeta_{24}^{3} + 40 \zeta_{24}^{5} - 32 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{73} + ( -24 - 16 \zeta_{24} - 2 \zeta_{24}^{2} - 16 \zeta_{24}^{3} + 32 \zeta_{24}^{5} + \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{75} + ( 32 - 16 \zeta_{24} + 16 \zeta_{24}^{3} - 64 \zeta_{24}^{4} + 16 \zeta_{24}^{5} + 40 \zeta_{24}^{6} ) q^{77} + ( 8 - 14 \zeta_{24} + 14 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 58 \zeta_{24}^{5} - 10 \zeta_{24}^{6} + 44 \zeta_{24}^{7} ) q^{79} + 9 q^{81} + ( 68 + 12 \zeta_{24} + 12 \zeta_{24}^{3} + 44 \zeta_{24}^{5} - 56 \zeta_{24}^{7} ) q^{83} + ( 28 - 76 \zeta_{24} + 76 \zeta_{24}^{3} - 56 \zeta_{24}^{4} + 44 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 32 \zeta_{24}^{7} ) q^{85} + ( -10 - 34 \zeta_{24} + 34 \zeta_{24}^{3} + 20 \zeta_{24}^{4} + 14 \zeta_{24}^{5} + 30 \zeta_{24}^{6} - 20 \zeta_{24}^{7} ) q^{87} + ( 30 - 24 \zeta_{24} + 48 \zeta_{24}^{2} - 24 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 24 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{89} + ( 4 + 36 \zeta_{24} - 16 \zeta_{24}^{2} + 36 \zeta_{24}^{3} - 76 \zeta_{24}^{5} + 8 \zeta_{24}^{6} + 40 \zeta_{24}^{7} ) q^{91} + ( 6 - 34 \zeta_{24} + 34 \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 26 \zeta_{24}^{5} - 24 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{93} + ( -16 + 32 \zeta_{24} - 32 \zeta_{24}^{3} + 32 \zeta_{24}^{4} - 80 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{95} + ( 50 + 88 \zeta_{24} - 16 \zeta_{24}^{2} + 88 \zeta_{24}^{3} - 88 \zeta_{24}^{5} + 8 \zeta_{24}^{6} ) q^{97} + ( -12 + 12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{9} + O(q^{10})$$ $$8q + 24q^{9} - 32q^{11} + 16q^{17} - 96q^{19} + 8q^{25} + 96q^{35} - 80q^{41} - 224q^{43} - 88q^{49} - 96q^{51} - 96q^{57} + 512q^{59} - 160q^{65} + 16q^{73} - 192q^{75} + 72q^{81} + 544q^{83} + 240q^{89} + 32q^{91} + 400q^{97} - 96q^{99} + O(q^{100})$$

## 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
 −0.965926 − 0.258819i 0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i
0 −1.73205 0 4.29253i 0 2.75787i 0 3.00000 0
127.2 0 −1.73205 0 1.36433i 0 1.24213i 0 3.00000 0
127.3 0 −1.73205 0 1.36433i 0 1.24213i 0 3.00000 0
127.4 0 −1.73205 0 4.29253i 0 2.75787i 0 3.00000 0
127.5 0 1.73205 0 8.29253i 0 8.55583i 0 3.00000 0
127.6 0 1.73205 0 2.63567i 0 12.5558i 0 3.00000 0
127.7 0 1.73205 0 2.63567i 0 12.5558i 0 3.00000 0
127.8 0 1.73205 0 8.29253i 0 8.55583i 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 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.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.e 8
3.b odd 2 1 2304.3.b.t 8
4.b odd 2 1 768.3.b.f 8
8.b even 2 1 768.3.b.f 8
8.d odd 2 1 inner 768.3.b.e 8
12.b even 2 1 2304.3.b.q 8
16.e even 4 1 384.3.g.a 8
16.e even 4 1 384.3.g.b yes 8
16.f odd 4 1 384.3.g.a 8
16.f odd 4 1 384.3.g.b yes 8
24.f even 2 1 2304.3.b.t 8
24.h odd 2 1 2304.3.b.q 8
48.i odd 4 1 1152.3.g.c 8
48.i odd 4 1 1152.3.g.f 8
48.k even 4 1 1152.3.g.c 8
48.k even 4 1 1152.3.g.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.g.a 8 16.e even 4 1
384.3.g.a 8 16.f odd 4 1
384.3.g.b yes 8 16.e even 4 1
384.3.g.b yes 8 16.f odd 4 1
768.3.b.e 8 1.a even 1 1 trivial
768.3.b.e 8 8.d odd 2 1 inner
768.3.b.f 8 4.b odd 2 1
768.3.b.f 8 8.b even 2 1
1152.3.g.c 8 48.i odd 4 1
1152.3.g.c 8 48.k even 4 1
1152.3.g.f 8 48.i odd 4 1
1152.3.g.f 8 48.k even 4 1
2304.3.b.q 8 12.b even 2 1
2304.3.b.q 8 24.h odd 2 1
2304.3.b.t 8 3.b odd 2 1
2304.3.b.t 8 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{8} + 96 T_{5}^{6} + 2048 T_{5}^{4} + 12288 T_{5}^{2} + 16384$$ $$T_{11}^{2} + 8 T_{11} - 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -3 + T^{2} )^{4}$$
$5$ $$16384 + 12288 T^{2} + 2048 T^{4} + 96 T^{6} + T^{8}$$
$7$ $$135424 + 108288 T^{2} + 13664 T^{4} + 240 T^{6} + T^{8}$$
$11$ $$( -80 + 8 T + T^{2} )^{4}$$
$13$ $$1274204416 + 31661312 T^{2} + 255840 T^{4} + 848 T^{6} + T^{8}$$
$17$ $$( 70288 + 9824 T - 904 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$19$ $$( -149504 - 12288 T + 320 T^{2} + 48 T^{3} + T^{4} )^{2}$$
$23$ $$34668544 + 6930432 T^{2} + 218624 T^{4} + 960 T^{6} + T^{8}$$
$29$ $$604263694336 + 3050582016 T^{2} + 5390336 T^{4} + 3936 T^{6} + T^{8}$$
$31$ $$1115025664 + 1078935296 T^{2} + 3628896 T^{4} + 3440 T^{6} + T^{8}$$
$37$ $$306596978944 + 3833071872 T^{2} + 12519008 T^{4} + 7056 T^{6} + T^{8}$$
$41$ $$( -3231344 - 244960 T - 4168 T^{2} + 40 T^{3} + T^{4} )^{2}$$
$43$ $$( -3295232 - 126976 T + 1856 T^{2} + 112 T^{3} + T^{4} )^{2}$$
$47$ $$102236225536 + 1132249088 T^{2} + 4154880 T^{4} + 5312 T^{6} + T^{8}$$
$53$ $$9915747549184 + 41509859328 T^{2} + 33972224 T^{4} + 10080 T^{6} + T^{8}$$
$59$ $$( 9050368 - 806912 T + 22688 T^{2} - 256 T^{3} + T^{4} )^{2}$$
$61$ $$175366666641664 + 245715261696 T^{2} + 114745952 T^{4} + 20112 T^{6} + T^{8}$$
$67$ $$( 20793600 - 9312 T^{2} + T^{4} )^{2}$$
$71$ $$11090924732416 + 69089804288 T^{2} + 112264704 T^{4} + 22208 T^{6} + T^{8}$$
$73$ $$( -2276336 - 525344 T - 16104 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$79$ $$3915755776 + 5149011712 T^{2} + 109548384 T^{4} + 23152 T^{6} + T^{8}$$
$83$ $$( -9283328 + 161024 T + 17312 T^{2} - 272 T^{3} + T^{4} )^{2}$$
$89$ $$( -1654256 + 151584 T - 1384 T^{2} - 120 T^{3} + T^{4} )^{2}$$
$97$ $$( 161817616 + 2636000 T - 16360 T^{2} - 200 T^{3} + T^{4} )^{2}$$