# Properties

 Label 384.5.b.c Level $384$ Weight $5$ Character orbit 384.b Analytic conductor $39.694$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 384.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.6940658242$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{28}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} -\beta_{5} q^{7} + 27 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} -\beta_{5} q^{7} + 27 q^{9} + ( 7 \beta_{2} + \beta_{6} ) q^{11} + ( -2 \beta_{1} + 15 \beta_{3} ) q^{13} + ( 5 \beta_{4} + 4 \beta_{5} ) q^{15} + ( 174 + \beta_{7} ) q^{17} + ( 45 \beta_{2} + 3 \beta_{6} ) q^{19} + ( -3 \beta_{1} + 18 \beta_{3} ) q^{21} + ( -12 \beta_{4} - 18 \beta_{5} ) q^{23} + ( -447 - 2 \beta_{7} ) q^{25} -27 \beta_{2} q^{27} + ( -3 \beta_{1} + 89 \beta_{3} ) q^{29} + ( 36 \beta_{4} + 5 \beta_{5} ) q^{31} + ( -180 - 3 \beta_{7} ) q^{33} + ( -157 \beta_{2} - 7 \beta_{6} ) q^{35} + ( 20 \beta_{1} + 21 \beta_{3} ) q^{37} + ( 3 \beta_{4} - 21 \beta_{5} ) q^{39} + ( -126 + 9 \beta_{7} ) q^{41} + ( 153 \beta_{2} - 9 \beta_{6} ) q^{43} + ( 27 \beta_{1} - 27 \beta_{3} ) q^{45} + ( 96 \beta_{4} - 54 \beta_{5} ) q^{47} + ( 1297 - 4 \beta_{7} ) q^{49} + ( -177 \beta_{2} - 9 \beta_{6} ) q^{51} + ( -25 \beta_{1} - 5 \beta_{3} ) q^{53} + ( 36 \beta_{4} - 160 \beta_{5} ) q^{55} + ( -1188 - 9 \beta_{7} ) q^{57} + 252 \beta_{2} q^{59} + ( -88 \beta_{1} - 249 \beta_{3} ) q^{61} -27 \beta_{5} q^{63} + ( 2976 + 17 \beta_{7} ) q^{65} + ( -432 \beta_{2} + 12 \beta_{6} ) q^{67} + ( -90 \beta_{1} + 216 \beta_{3} ) q^{69} + ( 300 \beta_{4} + 18 \beta_{5} ) q^{71} + ( -1282 - 4 \beta_{7} ) q^{73} + ( 453 \beta_{2} + 18 \beta_{6} ) q^{75} + ( 148 \beta_{1} - 456 \beta_{3} ) q^{77} + ( 108 \beta_{4} - 243 \beta_{5} ) q^{79} + 729 q^{81} + ( -403 \beta_{2} + 35 \beta_{6} ) q^{83} + ( 238 \beta_{1} - 1182 \beta_{3} ) q^{85} + ( 71 \beta_{4} - 98 \beta_{5} ) q^{87} + ( -3954 - 10 \beta_{7} ) q^{89} + ( 873 \beta_{2} + 27 \beta_{6} ) q^{91} + ( 123 \beta_{1} + 234 \beta_{3} ) q^{93} + ( -12 \beta_{4} - 576 \beta_{5} ) q^{95} + ( -5650 - 10 \beta_{7} ) q^{97} + ( 189 \beta_{2} + 27 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 216q^{9} + O(q^{10})$$ $$8q + 216q^{9} + 1392q^{17} - 3576q^{25} - 1440q^{33} - 1008q^{41} + 10376q^{49} - 9504q^{57} + 23808q^{65} - 10256q^{73} + 5832q^{81} - 31632q^{89} - 45200q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$24 \nu^{6} + 108$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{7} - 3 \nu^{5} + 3 \nu^{3} + 24 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{7} - 24 \nu^{6} + 25 \nu^{5} - 40 \nu^{4} + 55 \nu^{3} - 120 \nu^{2} + 184 \nu - 208$$$$)/10$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{7} + 48 \nu^{6} + 25 \nu^{5} + 80 \nu^{4} + 55 \nu^{3} + 240 \nu^{2} + 184 \nu + 416$$$$)/10$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - 128 \nu^{6} + \nu^{5} - 384 \nu^{4} - \nu^{3} - 128 \nu^{2} - 8 \nu - 768$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$-36 \nu^{7} - 60 \nu^{5} - 132 \nu^{3} - 192 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 8 \beta_{5} + 16 \beta_{4} - 12 \beta_{3} + 32 \beta_{2}$$$$)/384$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{6} + 12 \beta_{5} - 12 \beta_{4} + \beta_{2} - 8 \beta_{1} - 288$$$$)/384$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 60 \beta_{3}$$$$)/192$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 3 \beta_{2} - 24 \beta_{1} - 96$$$$)/384$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} + 8 \beta_{5} + 16 \beta_{4} - 132 \beta_{3} - 352 \beta_{2}$$$$)/384$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{1} - 108$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} - 56 \beta_{5} - 112 \beta_{4} - 156 \beta_{3} + 416 \beta_{2}$$$$)/384$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 1.09445 + 0.895644i −0.228425 + 1.39564i −0.228425 − 1.39564i 1.09445 − 0.895644i 0.228425 − 1.39564i −1.09445 − 0.895644i −1.09445 + 0.895644i 0.228425 + 1.39564i
0 −5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
319.2 0 −5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.3 0 −5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.4 0 −5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
319.5 0 5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
319.6 0 5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.7 0 5.19615 0 23.7490i 0 9.38251i 0 27.0000 0
319.8 0 5.19615 0 39.7490i 0 46.0431i 0 27.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 319.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
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 384.5.b.c 8
3.b odd 2 1 1152.5.b.k 8
4.b odd 2 1 inner 384.5.b.c 8
8.b even 2 1 inner 384.5.b.c 8
8.d odd 2 1 inner 384.5.b.c 8
12.b even 2 1 1152.5.b.k 8
16.e even 4 1 768.5.g.c 4
16.e even 4 1 768.5.g.g 4
16.f odd 4 1 768.5.g.c 4
16.f odd 4 1 768.5.g.g 4
24.f even 2 1 1152.5.b.k 8
24.h odd 2 1 1152.5.b.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.c 8 1.a even 1 1 trivial
384.5.b.c 8 4.b odd 2 1 inner
384.5.b.c 8 8.b even 2 1 inner
384.5.b.c 8 8.d odd 2 1 inner
768.5.g.c 4 16.e even 4 1
768.5.g.c 4 16.f odd 4 1
768.5.g.g 4 16.e even 4 1
768.5.g.g 4 16.f odd 4 1
1152.5.b.k 8 3.b odd 2 1
1152.5.b.k 8 12.b even 2 1
1152.5.b.k 8 24.f even 2 1
1152.5.b.k 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 2144 T_{5}^{2} + 891136$$ acting on $$S_{5}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -27 + T^{2} )^{4}$$
$5$ $$( 891136 + 2144 T^{2} + T^{4} )^{2}$$
$7$ $$( 186624 + 2208 T^{2} + T^{4} )^{2}$$
$11$ $$( 412252416 - 45408 T^{2} + T^{4} )^{2}$$
$13$ $$( 107495424 + 36864 T^{2} + T^{4} )^{2}$$
$17$ $$( -34236 - 348 T + T^{2} )^{4}$$
$19$ $$( 19955517696 - 491616 T^{2} + T^{4} )^{2}$$
$23$ $$( 36790308864 + 825984 T^{2} + T^{4} )^{2}$$
$29$ $$( 247876528384 + 1032032 T^{2} + T^{4} )^{2}$$
$31$ $$( 189245880576 + 1389216 T^{2} + T^{4} )^{2}$$
$37$ $$( 140607000576 + 862848 T^{2} + T^{4} )^{2}$$
$41$ $$( -5209596 + 252 T + T^{2} )^{4}$$
$43$ $$( 1176686901504 - 4797792 T^{2} + T^{4} )^{2}$$
$47$ $$( 54724012314624 + 17165952 T^{2} + T^{4} )^{2}$$
$53$ $$( 394886560000 + 1263200 T^{2} + T^{4} )^{2}$$
$59$ $$( -1714608 + T^{2} )^{4}$$
$61$ $$( 14729384300544 + 23548032 T^{2} + T^{4} )^{2}$$
$67$ $$( 4145361152256 - 16458336 T^{2} + T^{4} )^{2}$$
$71$ $$( 424196534145024 + 94718592 T^{2} + T^{4} )^{2}$$
$73$ $$( 611332 + 2564 T + T^{2} )^{4}$$
$79$ $$( 3797136795711744 + 147736224 T^{2} + T^{4} )^{2}$$
$83$ $$( 470880972843264 - 61970016 T^{2} + T^{4} )^{2}$$
$89$ $$( 9182916 + 7908 T + T^{2} )^{4}$$
$97$ $$( 25471300 + 11300 T + T^{2} )^{4}$$