# Properties

 Label 384.4.d.f Level $384$ Weight $4$ Character orbit 384.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1534132224.8 Defining polynomial: $$x^{8} + 18 x^{6} + 107 x^{4} + 210 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{22}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{6} q^{5} -\beta_{2} q^{7} -9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{6} q^{5} -\beta_{2} q^{7} -9 q^{9} + ( -4 \beta_{1} - \beta_{3} ) q^{11} + ( -\beta_{6} - \beta_{7} ) q^{13} + ( \beta_{2} + \beta_{4} ) q^{15} + ( 30 - \beta_{5} ) q^{17} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{19} + ( -2 \beta_{6} + \beta_{7} ) q^{21} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{23} + ( -43 - 2 \beta_{5} ) q^{25} -9 \beta_{1} q^{27} + ( 5 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 5 \beta_{2} + 4 \beta_{4} ) q^{31} + ( 36 - \beta_{5} ) q^{33} + ( 40 \beta_{1} + 3 \beta_{3} ) q^{35} + ( 17 \beta_{6} - \beta_{7} ) q^{37} + ( -6 \beta_{2} + 3 \beta_{4} ) q^{39} + ( -102 - 5 \beta_{5} ) q^{41} + ( -44 \beta_{1} - 3 \beta_{3} ) q^{43} + 9 \beta_{6} q^{45} + ( -14 \beta_{2} - 8 \beta_{4} ) q^{47} + ( 209 - 8 \beta_{5} ) q^{49} + ( 30 \beta_{1} + 9 \beta_{3} ) q^{51} + ( -13 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -4 \beta_{2} + 4 \beta_{4} ) q^{55} + ( -36 - 3 \beta_{5} ) q^{57} + ( -156 \beta_{1} + 16 \beta_{3} ) q^{59} + ( -49 \beta_{6} + 5 \beta_{7} ) q^{61} + 9 \beta_{2} q^{63} + ( -144 - 9 \beta_{5} ) q^{65} + ( 196 \beta_{1} - 12 \beta_{3} ) q^{67} + ( -24 \beta_{6} - 6 \beta_{7} ) q^{69} + ( 30 \beta_{2} + 12 \beta_{4} ) q^{71} + ( -242 - 8 \beta_{5} ) q^{73} + ( -43 \beta_{1} + 18 \beta_{3} ) q^{75} + ( 16 \beta_{6} - 12 \beta_{7} ) q^{77} + ( -11 \beta_{2} - 20 \beta_{4} ) q^{79} + 81 q^{81} + ( 284 \beta_{1} + 17 \beta_{3} ) q^{83} + ( 26 \beta_{6} + 8 \beta_{7} ) q^{85} + ( 9 \beta_{2} - 9 \beta_{4} ) q^{87} + ( 918 - 6 \beta_{5} ) q^{89} + ( -432 \beta_{1} - 63 \beta_{3} ) q^{91} + ( 38 \beta_{6} - \beta_{7} ) q^{93} + ( 4 \beta_{2} + 28 \beta_{4} ) q^{95} + ( -370 + 2 \beta_{5} ) q^{97} + ( 36 \beta_{1} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 72q^{9} + O(q^{10})$$ $$8q - 72q^{9} + 240q^{17} - 344q^{25} + 288q^{33} - 816q^{41} + 1672q^{49} - 288q^{57} - 1152q^{65} - 1936q^{73} + 648q^{81} + 7344q^{89} - 2960q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 107 x^{4} + 210 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} - 33 \nu^{3} - 87 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{6} + 6 \nu^{4} - 2 \nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{5} + 104 \nu^{3} + 328 \nu$$ $$\beta_{4}$$ $$=$$ $$8 \nu^{6} + 96 \nu^{4} + 272 \nu^{2} - 48$$ $$\beta_{5}$$ $$=$$ $$24 \nu^{6} + 288 \nu^{4} + 864 \nu^{2} + 72$$ $$\beta_{6}$$ $$=$$ $$2 \nu^{7} + 31 \nu^{5} + 155 \nu^{3} + 253 \nu$$ $$\beta_{7}$$ $$=$$ $$10 \nu^{7} + 131 \nu^{5} + 487 \nu^{3} + 377 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{7} + 10 \beta_{6} - 3 \beta_{3} + 16 \beta_{1}$$$$)/96$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 3 \beta_{4} - 216$$$$)/48$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} - 15 \beta_{6} + 6 \beta_{3} - 16 \beta_{1}$$$$)/24$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{5} + 19 \beta_{4} - 8 \beta_{2} + 1320$$$$)/48$$ $$\nu^{5}$$ $$=$$ $$($$$$-74 \beta_{7} + 370 \beta_{6} - 177 \beta_{3} + 176 \beta_{1}$$$$)/96$$ $$\nu^{6}$$ $$=$$ $$($$$$19 \beta_{5} - 60 \beta_{4} + 48 \beta_{2} - 4104$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$470 \beta_{7} - 2302 \beta_{6} + 1263 \beta_{3} + 208 \beta_{1}$$$$)/96$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.0690906i 2.63019i − 2.21597i − 2.48330i 2.48330i 2.21597i − 2.63019i − 0.0690906i
0 3.00000i 0 17.4288i 0 2.99032 0 −9.00000 0
193.2 0 3.00000i 0 5.67763i 0 33.0917 0 −9.00000 0
193.3 0 3.00000i 0 5.67763i 0 −33.0917 0 −9.00000 0
193.4 0 3.00000i 0 17.4288i 0 −2.99032 0 −9.00000 0
193.5 0 3.00000i 0 17.4288i 0 −2.99032 0 −9.00000 0
193.6 0 3.00000i 0 5.67763i 0 −33.0917 0 −9.00000 0
193.7 0 3.00000i 0 5.67763i 0 33.0917 0 −9.00000 0
193.8 0 3.00000i 0 17.4288i 0 2.99032 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.8 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 384.4.d.f 8
3.b odd 2 1 1152.4.d.p 8
4.b odd 2 1 inner 384.4.d.f 8
8.b even 2 1 inner 384.4.d.f 8
8.d odd 2 1 inner 384.4.d.f 8
12.b even 2 1 1152.4.d.p 8
16.e even 4 1 768.4.a.u 4
16.e even 4 1 768.4.a.v 4
16.f odd 4 1 768.4.a.u 4
16.f odd 4 1 768.4.a.v 4
24.f even 2 1 1152.4.d.p 8
24.h odd 2 1 1152.4.d.p 8
48.i odd 4 1 2304.4.a.by 4
48.i odd 4 1 2304.4.a.cb 4
48.k even 4 1 2304.4.a.by 4
48.k even 4 1 2304.4.a.cb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.f 8 1.a even 1 1 trivial
384.4.d.f 8 4.b odd 2 1 inner
384.4.d.f 8 8.b even 2 1 inner
384.4.d.f 8 8.d odd 2 1 inner
768.4.a.u 4 16.e even 4 1
768.4.a.u 4 16.f odd 4 1
768.4.a.v 4 16.e even 4 1
768.4.a.v 4 16.f odd 4 1
1152.4.d.p 8 3.b odd 2 1
1152.4.d.p 8 12.b even 2 1
1152.4.d.p 8 24.f even 2 1
1152.4.d.p 8 24.h odd 2 1
2304.4.a.by 4 48.i odd 4 1
2304.4.a.by 4 48.k even 4 1
2304.4.a.cb 4 48.i odd 4 1
2304.4.a.cb 4 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 $$S_{4}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{4} + 336 T_{5}^{2} + 9792$$ $$T_{7}^{4} - 1104 T_{7}^{2} + 9792$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 + T^{2} )^{4}$$
$5$ $$( 9792 + 336 T^{2} + T^{4} )^{2}$$
$7$ $$( 9792 - 1104 T^{2} + T^{4} )^{2}$$
$11$ $$( 135424 + 1312 T^{2} + T^{4} )^{2}$$
$13$ $$( 12690432 + 8640 T^{2} + T^{4} )^{2}$$
$17$ $$( -3708 - 60 T + T^{2} )^{4}$$
$19$ $$( 19927296 + 9504 T^{2} + T^{4} )^{2}$$
$23$ $$( 621831168 - 53568 T^{2} + T^{4} )^{2}$$
$29$ $$( 419577408 + 41040 T^{2} + T^{4} )^{2}$$
$31$ $$( 461095488 - 55248 T^{2} + T^{4} )^{2}$$
$37$ $$( 2487324672 + 107136 T^{2} + T^{4} )^{2}$$
$41$ $$( -104796 + 204 T + T^{2} )^{4}$$
$43$ $$( 164249856 + 44064 T^{2} + T^{4} )^{2}$$
$47$ $$( 21332616192 - 302400 T^{2} + T^{4} )^{2}$$
$53$ $$( 806557248 + 87888 T^{2} + T^{4} )^{2}$$
$59$ $$( 7735554304 + 700192 T^{2} + T^{4} )^{2}$$
$61$ $$( 270509248512 + 1040256 T^{2} + T^{4} )^{2}$$
$67$ $$( 73992704256 + 838944 T^{2} + T^{4} )^{2}$$
$71$ $$( 297070322688 - 1104192 T^{2} + T^{4} )^{2}$$
$73$ $$( -236348 + 484 T + T^{2} )^{4}$$
$79$ $$( 1904357952 - 1039824 T^{2} + T^{4} )^{2}$$
$83$ $$( 334010020096 + 1747744 T^{2} + T^{4} )^{2}$$
$89$ $$( 676836 - 1836 T + T^{2} )^{4}$$
$97$ $$( 118468 + 740 T + T^{2} )^{4}$$