# Properties

 Label 384.4.f.g Level $384$ Weight $4$ Character orbit 384.f Analytic conductor $22.657$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{4} ) q^{3} + ( -2 \beta_{3} + \beta_{6} ) q^{5} + ( -3 \beta_{2} + 2 \beta_{5} ) q^{7} + ( -15 - 3 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{4} ) q^{3} + ( -2 \beta_{3} + \beta_{6} ) q^{5} + ( -3 \beta_{2} + 2 \beta_{5} ) q^{7} + ( -15 - 3 \beta_{5} ) q^{9} + ( -8 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{11} + ( -7 \beta_{1} - 4 \beta_{3} - 8 \beta_{4} ) q^{13} + ( -24 + 7 \beta_{2} + 6 \beta_{5} + \beta_{7} ) q^{15} + ( -16 \beta_{2} + 2 \beta_{5} ) q^{17} + ( 7 \beta_{3} - 16 \beta_{6} ) q^{19} + ( 9 \beta_{1} + 10 \beta_{3} - 8 \beta_{4} + 9 \beta_{6} ) q^{21} + ( -80 - 2 \beta_{7} ) q^{23} + ( 27 - 8 \beta_{7} ) q^{25} + ( -30 \beta_{3} - 3 \beta_{4} ) q^{27} + ( 14 \beta_{3} - 23 \beta_{6} ) q^{29} + ( -7 \beta_{2} - 6 \beta_{5} ) q^{31} + ( 42 + 16 \beta_{2} + 3 \beta_{5} - 8 \beta_{7} ) q^{33} + ( -8 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} ) q^{35} + ( 31 \beta_{1} - 28 \beta_{3} - 56 \beta_{4} ) q^{37} + ( 168 + 14 \beta_{2} + 12 \beta_{5} - 7 \beta_{7} ) q^{39} + ( -16 \beta_{2} + 24 \beta_{5} ) q^{41} + ( 49 \beta_{3} + 16 \beta_{6} ) q^{43} + ( -42 \beta_{1} + 6 \beta_{3} - 48 \beta_{4} - 15 \beta_{6} ) q^{45} + 16 \beta_{7} q^{47} + ( -97 - 24 \beta_{7} ) q^{49} + ( 48 \beta_{1} + 10 \beta_{3} - 8 \beta_{4} + 48 \beta_{6} ) q^{51} + ( 70 \beta_{3} + 45 \beta_{6} ) q^{53} + ( -78 \beta_{2} - 44 \beta_{5} ) q^{55} + ( 84 - 112 \beta_{2} - 21 \beta_{5} - 16 \beta_{7} ) q^{57} + ( 5 \beta_{3} + 10 \beta_{4} ) q^{59} + ( -7 \beta_{1} - 68 \beta_{3} - 136 \beta_{4} ) q^{61} + ( 336 + 45 \beta_{2} - 30 \beta_{5} + 18 \beta_{7} ) q^{63} + ( -112 \beta_{2} - 76 \beta_{5} ) q^{65} + ( -91 \beta_{3} + 32 \beta_{6} ) q^{67} + ( 42 \beta_{1} - 80 \beta_{3} - 80 \beta_{4} - 12 \beta_{6} ) q^{69} + ( -80 - 50 \beta_{7} ) q^{71} + ( -154 - 24 \beta_{7} ) q^{73} + ( 168 \beta_{1} + 27 \beta_{3} + 27 \beta_{4} - 48 \beta_{6} ) q^{75} + ( 68 \beta_{3} + 46 \beta_{6} ) q^{77} + ( 189 \beta_{2} + 2 \beta_{5} ) q^{79} + ( -279 + 90 \beta_{5} ) q^{81} + ( -168 \beta_{1} - 37 \beta_{3} - 74 \beta_{4} ) q^{83} + ( -164 \beta_{1} - 48 \beta_{3} - 96 \beta_{4} ) q^{85} + ( 168 - 161 \beta_{2} - 42 \beta_{5} - 23 \beta_{7} ) q^{87} + ( 128 \beta_{2} - 26 \beta_{5} ) q^{89} + ( -28 \beta_{3} - 16 \beta_{6} ) q^{91} + ( 21 \beta_{1} - 30 \beta_{3} + 24 \beta_{4} + 21 \beta_{6} ) q^{93} + ( -1232 + 78 \beta_{7} ) q^{95} + ( 714 - 16 \beta_{7} ) q^{97} + ( 120 \beta_{1} + 57 \beta_{3} + 30 \beta_{4} - 96 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 120 q^{9} + O(q^{10})$$ $$8 q - 120 q^{9} - 192 q^{15} - 640 q^{23} + 216 q^{25} + 336 q^{33} + 1344 q^{39} - 776 q^{49} + 672 q^{57} + 2688 q^{63} - 640 q^{71} - 1232 q^{73} - 2232 q^{81} + 1344 q^{87} - 9856 q^{95} + 5712 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{6} + 96 \nu^{2}$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-8 \nu^{7} - 2 \nu^{5} - 182 \nu^{3} - 38 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{7} + 2 \nu^{5} - 182 \nu^{3} + 38 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{7} - 5 \nu^{6} - \nu^{5} + 91 \nu^{3} - 110 \nu^{2} - 19 \nu$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$12 \nu^{7} + 2 \nu^{5} + 278 \nu^{3} + 58 \nu$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$-12 \nu^{7} + 2 \nu^{5} - 278 \nu^{3} + 58 \nu$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{4} + 92$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{4} + 2 \beta_{3} + 5 \beta_{1}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{6} + 2 \beta_{5} + 3 \beta_{3} + 3 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{7} - 92$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{6} - 19 \beta_{5} + 29 \beta_{3} - 29 \beta_{2}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-48 \beta_{4} - 24 \beta_{3} - 55 \beta_{1}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$91 \beta_{6} - 91 \beta_{5} - 139 \beta_{3} - 139 \beta_{2}$$$$)/8$$

## Character values

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

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −0.323042 − 0.323042i 1.54779 − 1.54779i −0.323042 + 0.323042i 1.54779 + 1.54779i −1.54779 + 1.54779i 0.323042 + 0.323042i −1.54779 − 1.54779i 0.323042 − 0.323042i
0 −2.44949 4.58258i 0 2.31464 0 29.6636i 0 −15.0000 + 22.4499i 0
191.2 0 −2.44949 4.58258i 0 17.2813 0 0.269691i 0 −15.0000 + 22.4499i 0
191.3 0 −2.44949 + 4.58258i 0 2.31464 0 29.6636i 0 −15.0000 22.4499i 0
191.4 0 −2.44949 + 4.58258i 0 17.2813 0 0.269691i 0 −15.0000 22.4499i 0
191.5 0 2.44949 4.58258i 0 −17.2813 0 0.269691i 0 −15.0000 22.4499i 0
191.6 0 2.44949 4.58258i 0 −2.31464 0 29.6636i 0 −15.0000 22.4499i 0
191.7 0 2.44949 + 4.58258i 0 −17.2813 0 0.269691i 0 −15.0000 + 22.4499i 0
191.8 0 2.44949 + 4.58258i 0 −2.31464 0 29.6636i 0 −15.0000 + 22.4499i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.8 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.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.4.f.g 8
3.b odd 2 1 384.4.f.h yes 8
4.b odd 2 1 384.4.f.h yes 8
8.b even 2 1 inner 384.4.f.g 8
8.d odd 2 1 384.4.f.h yes 8
12.b even 2 1 inner 384.4.f.g 8
16.e even 4 1 768.4.c.n 4
16.e even 4 1 768.4.c.o 4
16.f odd 4 1 768.4.c.m 4
16.f odd 4 1 768.4.c.p 4
24.f even 2 1 inner 384.4.f.g 8
24.h odd 2 1 384.4.f.h yes 8
48.i odd 4 1 768.4.c.m 4
48.i odd 4 1 768.4.c.p 4
48.k even 4 1 768.4.c.n 4
48.k even 4 1 768.4.c.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.g 8 1.a even 1 1 trivial
384.4.f.g 8 8.b even 2 1 inner
384.4.f.g 8 12.b even 2 1 inner
384.4.f.g 8 24.f even 2 1 inner
384.4.f.h yes 8 3.b odd 2 1
384.4.f.h yes 8 4.b odd 2 1
384.4.f.h yes 8 8.d odd 2 1
384.4.f.h yes 8 24.h odd 2 1
768.4.c.m 4 16.f odd 4 1
768.4.c.m 4 48.i odd 4 1
768.4.c.n 4 16.e even 4 1
768.4.c.n 4 48.k even 4 1
768.4.c.o 4 16.e even 4 1
768.4.c.o 4 48.k even 4 1
768.4.c.p 4 16.f odd 4 1
768.4.c.p 4 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{4} - 304 T_{5}^{2} + 1600$$ $$T_{23}^{2} + 160 T_{23} + 5056$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 729 + 30 T^{2} + T^{4} )^{2}$$
$5$ $$( 1600 - 304 T^{2} + T^{4} )^{2}$$
$7$ $$( 64 + 880 T^{2} + T^{4} )^{2}$$
$11$ $$( 883600 + 2216 T^{2} + T^{4} )^{2}$$
$13$ $$( 313600 + 4256 T^{2} + T^{4} )^{2}$$
$17$ $$( 35046400 + 12736 T^{2} + T^{4} )^{2}$$
$19$ $$( 173185600 - 31024 T^{2} + T^{4} )^{2}$$
$23$ $$( 5056 + 160 T + T^{2} )^{4}$$
$29$ $$( 621006400 - 68656 T^{2} + T^{4} )^{2}$$
$31$ $$( 705600 + 6384 T^{2} + T^{4} )^{2}$$
$37$ $$( 2548230400 + 162464 T^{2} + T^{4} )^{2}$$
$41$ $$( 681836544 + 76800 T^{2} + T^{4} )^{2}$$
$43$ $$( 1873850944 - 143920 T^{2} + T^{4} )^{2}$$
$47$ $$( -86016 + T^{2} )^{4}$$
$53$ $$( 17640000 - 462000 T^{2} + T^{4} )^{2}$$
$59$ $$( 2100 + T^{2} )^{4}$$
$61$ $$( 150258567424 + 778400 T^{2} + T^{4} )^{2}$$
$67$ $$( 19993960000 - 512176 T^{2} + T^{4} )^{2}$$
$71$ $$( -833600 + 160 T + T^{2} )^{4}$$
$73$ $$( -169820 + 308 T + T^{2} )^{4}$$
$79$ $$( 734586126400 + 1715056 T^{2} + T^{4} )^{2}$$
$83$ $$( 113291481744 + 1133160 T^{2} + T^{4} )^{2}$$
$89$ $$( 126280729600 + 862144 T^{2} + T^{4} )^{2}$$
$97$ $$( 423780 - 1428 T + T^{2} )^{4}$$