Properties

 Label 384.3.i.a Level $384$ Weight $3$ Character orbit 384.i Analytic conductor $10.463$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.i (of order $$4$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.629407744.1 Defining polynomial: $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{7} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{7} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} -2 \beta_{4} q^{11} + ( 11 + 2 \beta_{2} + 13 \beta_{3} + \beta_{6} + \beta_{7} ) q^{13} + ( 14 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{15} + ( \beta_{1} - \beta_{4} - 9 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{17} + ( 2 - 8 \beta_{2} - 6 \beta_{3} - 4 \beta_{6} - 4 \beta_{7} ) q^{19} + ( 5 - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{4} + 10 \beta_{5} ) q^{23} + ( 14 - 14 \beta_{2} + 17 \beta_{3} - 14 \beta_{6} ) q^{25} + ( 5 - 12 \beta_{3} - \beta_{4} - 2 \beta_{6} - 9 \beta_{7} ) q^{27} + ( -7 \beta_{1} - 7 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{29} + ( 9 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{31} + ( -8 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 6 \beta_{7} ) q^{33} + ( 4 \beta_{1} - 6 \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( -9 + 19 \beta_{3} - 5 \beta_{6} + 5 \beta_{7} ) q^{37} + ( 13 + 10 \beta_{1} - 13 \beta_{2} - 19 \beta_{3} + 10 \beta_{4} + 3 \beta_{5} - 13 \beta_{6} ) q^{39} + ( \beta_{1} + \beta_{4} + 7 \beta_{5} ) q^{41} + ( 34 - 26 \beta_{3} - 4 \beta_{6} + 4 \beta_{7} ) q^{43} + ( 24 + 17 \beta_{1} - 20 \beta_{2} + 4 \beta_{3} - 15 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} ) q^{45} + ( 14 \beta_{1} - 14 \beta_{4} - 24 \beta_{5} - 10 \beta_{6} - 10 \beta_{7} ) q^{47} + ( 41 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{49} + ( -6 + 10 \beta_{1} + 20 \beta_{2} + 14 \beta_{3} + 24 \beta_{5} + 27 \beta_{6} + 27 \beta_{7} ) q^{51} + ( 7 \beta_{4} - 15 \beta_{6} - 15 \beta_{7} ) q^{53} + ( -12 + 12 \beta_{2} - 28 \beta_{3} + 12 \beta_{6} ) q^{55} + ( -6 + 6 \beta_{1} + 6 \beta_{2} + 30 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} ) q^{57} + ( -10 \beta_{4} - 5 \beta_{6} - 5 \beta_{7} ) q^{59} + ( -31 + 10 \beta_{2} - 21 \beta_{3} + 5 \beta_{6} + 5 \beta_{7} ) q^{61} + ( -13 + 4 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 10 \beta_{6} + 9 \beta_{7} ) q^{63} + ( -29 \beta_{1} + 29 \beta_{4} + 5 \beta_{5} - 24 \beta_{6} - 24 \beta_{7} ) q^{65} + ( 44 - 30 \beta_{2} + 14 \beta_{3} - 15 \beta_{6} - 15 \beta_{7} ) q^{67} + ( 12 + 12 \beta_{3} - 12 \beta_{4} + 6 \beta_{6} + 30 \beta_{7} ) q^{69} + ( -16 \beta_{1} - 16 \beta_{4} - 26 \beta_{5} ) q^{71} + ( -14 + 14 \beta_{2} - 22 \beta_{3} + 14 \beta_{6} ) q^{73} + ( -39 + 42 \beta_{3} + 45 \beta_{4} - 3 \beta_{6} ) q^{75} + ( -4 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{77} + ( -17 - 13 \beta_{2} - 13 \beta_{3} - 13 \beta_{7} ) q^{79} + ( 23 + 22 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 22 \beta_{4} + 6 \beta_{5} + 28 \beta_{6} + 36 \beta_{7} ) q^{81} + ( -38 \beta_{1} + 2 \beta_{5} - 18 \beta_{6} - 18 \beta_{7} ) q^{83} + ( -8 - 36 \beta_{3} + 22 \beta_{6} - 22 \beta_{7} ) q^{85} + ( 42 \beta_{3} + 21 \beta_{5} ) q^{87} + ( -18 \beta_{1} - 18 \beta_{4} + 10 \beta_{5} ) q^{89} + ( -32 + 6 \beta_{3} + 13 \beta_{6} - 13 \beta_{7} ) q^{91} + ( -3 + 7 \beta_{1} - 10 \beta_{2} - 13 \beta_{3} + 3 \beta_{5} ) q^{93} + ( -22 \beta_{1} + 22 \beta_{4} + 26 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -60 - 8 \beta_{2} - 8 \beta_{3} - 8 \beta_{7} ) q^{97} + ( -24 - 14 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} + 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} + O(q^{10})$$ $$8q - 4q^{3} + 96q^{13} + 112q^{15} - 16q^{19} + 32q^{21} + 68q^{27} + 72q^{31} - 64q^{33} - 112q^{37} + 240q^{43} + 112q^{45} + 328q^{49} + 32q^{51} - 208q^{61} - 104q^{63} + 232q^{67} - 324q^{75} - 136q^{79} + 184q^{81} + 112q^{85} - 152q^{91} - 64q^{93} - 480q^{97} - 160q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 10 \nu^{3} + 24 \nu^{2} + 8 \nu$$$$)/24$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 4 \nu$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3} + 10 \nu$$$$)/6$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 8 \nu^{6} - 2 \nu^{5} - 8 \nu^{4} - 10 \nu^{3} + 8 \nu^{2} - 8 \nu - 32$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 8 \nu^{6} - 2 \nu^{5} + 8 \nu^{4} - 10 \nu^{3} - 8 \nu^{2} - 8 \nu + 32$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 2 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 4 \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} + \beta_{6} + 4 \beta_{3} + 4$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \beta_{4} + 3 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −1.38255 − 0.297594i −0.767178 + 1.18804i 1.38255 + 0.297594i 0.767178 − 1.18804i −1.38255 + 0.297594i −0.767178 − 1.18804i 1.38255 − 0.297594i 0.767178 + 1.18804i
0 −2.90783 0.737922i 0 −1.57472 1.57472i 0 3.64575i 0 7.91094 + 4.29150i 0
161.2 0 −1.13234 + 2.77809i 0 −6.28651 6.28651i 0 1.64575i 0 −6.43560 6.29150i 0
161.3 0 −0.737922 2.90783i 0 1.57472 + 1.57472i 0 3.64575i 0 −7.91094 + 4.29150i 0
161.4 0 2.77809 1.13234i 0 6.28651 + 6.28651i 0 1.64575i 0 6.43560 6.29150i 0
353.1 0 −2.90783 + 0.737922i 0 −1.57472 + 1.57472i 0 3.64575i 0 7.91094 4.29150i 0
353.2 0 −1.13234 2.77809i 0 −6.28651 + 6.28651i 0 1.64575i 0 −6.43560 + 6.29150i 0
353.3 0 −0.737922 + 2.90783i 0 1.57472 1.57472i 0 3.64575i 0 −7.91094 4.29150i 0
353.4 0 2.77809 + 1.13234i 0 6.28651 6.28651i 0 1.64575i 0 6.43560 + 6.29150i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.4 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
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.i.a 8
3.b odd 2 1 inner 384.3.i.a 8
4.b odd 2 1 384.3.i.b 8
8.b even 2 1 48.3.i.a 8
8.d odd 2 1 192.3.i.a 8
12.b even 2 1 384.3.i.b 8
16.e even 4 1 48.3.i.a 8
16.e even 4 1 inner 384.3.i.a 8
16.f odd 4 1 192.3.i.a 8
16.f odd 4 1 384.3.i.b 8
24.f even 2 1 192.3.i.a 8
24.h odd 2 1 48.3.i.a 8
48.i odd 4 1 48.3.i.a 8
48.i odd 4 1 inner 384.3.i.a 8
48.k even 4 1 192.3.i.a 8
48.k even 4 1 384.3.i.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.a 8 8.b even 2 1
48.3.i.a 8 16.e even 4 1
48.3.i.a 8 24.h odd 2 1
48.3.i.a 8 48.i odd 4 1
192.3.i.a 8 8.d odd 2 1
192.3.i.a 8 16.f odd 4 1
192.3.i.a 8 24.f even 2 1
192.3.i.a 8 48.k even 4 1
384.3.i.a 8 1.a even 1 1 trivial
384.3.i.a 8 3.b odd 2 1 inner
384.3.i.a 8 16.e even 4 1 inner
384.3.i.a 8 48.i odd 4 1 inner
384.3.i.b 8 4.b odd 2 1
384.3.i.b 8 12.b even 2 1
384.3.i.b 8 16.f odd 4 1
384.3.i.b 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{8} + 6272 T_{5}^{4} + 153664$$ $$T_{19}^{4} + 8 T_{19}^{3} + 32 T_{19}^{2} - 1728 T_{19} + 46656$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$6561 + 2916 T + 648 T^{2} - 108 T^{3} - 126 T^{4} - 12 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$153664 + 6272 T^{4} + T^{8}$$
$7$ $$( 36 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$16384 + 2048 T^{4} + T^{8}$$
$13$ $$( 75076 - 13152 T + 1152 T^{2} - 48 T^{3} + T^{4} )^{2}$$
$17$ $$( 103968 + 920 T^{2} + T^{4} )^{2}$$
$19$ $$( 46656 - 1728 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$23$ $$( 225792 - 1120 T^{2} + T^{4} )^{2}$$
$29$ $$29884728384 + 614656 T^{4} + T^{8}$$
$31$ $$( 74 - 18 T + T^{2} )^{4}$$
$37$ $$( 1764 + 2352 T + 1568 T^{2} + 56 T^{3} + T^{4} )^{2}$$
$41$ $$( 2592 - 664 T^{2} + T^{4} )^{2}$$
$43$ $$( 2483776 - 189120 T + 7200 T^{2} - 120 T^{3} + T^{4} )^{2}$$
$47$ $$( 14193792 + 8144 T^{2} + T^{4} )^{2}$$
$53$ $$18847994944 + 18284288 T^{4} + T^{8}$$
$59$ $$2624400000000 + 3520000 T^{4} + T^{8}$$
$61$ $$( 1004004 + 104208 T + 5408 T^{2} + 104 T^{3} + T^{4} )^{2}$$
$67$ $$( 2155024 + 170288 T + 6728 T^{2} - 116 T^{3} + T^{4} )^{2}$$
$71$ $$( 24668288 - 16560 T^{2} + T^{4} )^{2}$$
$73$ $$( 788544 + 3712 T^{2} + T^{4} )^{2}$$
$79$ $$( -894 + 34 T + T^{2} )^{4}$$
$83$ $$1048798070063104 + 135735296 T^{4} + T^{8}$$
$89$ $$( 184832 - 6240 T^{2} + T^{4} )^{2}$$
$97$ $$( 3152 + 120 T + T^{2} )^{4}$$