# Properties

 Label 384.9.b.b Level $384$ Weight $9$ Character orbit 384.b Analytic conductor $156.433$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 384.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$156.433386263$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 4826 x^{6} + 8748877 x^{4} + 7060845096 x^{2} + 2140819627716$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{11}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{4} q^{7} + 2187 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{4} q^{7} + 2187 q^{9} + ( -44 \beta_{1} + \beta_{5} ) q^{11} + ( 22 \beta_{2} + \beta_{3} ) q^{13} + ( 2 \beta_{4} + \beta_{7} ) q^{15} + ( 34606 - \beta_{6} ) q^{17} + ( 2420 \beta_{1} - 5 \beta_{5} ) q^{19} + ( -27 \beta_{2} + 27 \beta_{3} ) q^{21} + ( 6 \beta_{4} + 4 \beta_{7} ) q^{23} + ( -355247 - 22 \beta_{6} ) q^{25} -2187 \beta_{1} q^{27} + ( -99 \beta_{2} + 110 \beta_{3} ) q^{29} + ( 77 \beta_{4} + 20 \beta_{7} ) q^{31} + ( 96876 + 27 \beta_{6} ) q^{33} + ( -6584 \beta_{1} + 57 \beta_{5} ) q^{35} + ( -1876 \beta_{2} + 477 \beta_{3} ) q^{37} + ( -35 \beta_{4} + 23 \beta_{7} ) q^{39} + ( 728354 + 23 \beta_{6} ) q^{41} + ( 49476 \beta_{1} - 145 \beta_{5} ) q^{43} + 2187 \beta_{2} q^{45} + ( 10 \beta_{4} + 128 \beta_{7} ) q^{47} + ( -2074991 + 204 \beta_{6} ) q^{49} + ( -34630 \beta_{1} - 81 \beta_{5} ) q^{51} + ( 1495 \beta_{2} + 842 \beta_{3} ) q^{53} + ( 2648 \beta_{4} + 404 \beta_{7} ) q^{55} + ( -5295780 - 135 \beta_{6} ) q^{57} + ( -98180 \beta_{1} + 848 \beta_{5} ) q^{59} + ( 400 \beta_{2} - 3413 \beta_{3} ) q^{61} -2187 \beta_{4} q^{63} + ( -16620384 - 449 \beta_{6} ) q^{65} + ( -150228 \beta_{1} - 580 \beta_{5} ) q^{67} + ( 8694 \beta_{2} + 54 \beta_{3} ) q^{69} + ( -3622 \beta_{4} + 636 \beta_{7} ) q^{71} + ( -7219970 - 340 \beta_{6} ) q^{73} + ( 354719 \beta_{1} - 1782 \beta_{5} ) q^{75} + ( -24748 \beta_{2} - 8516 \beta_{3} ) q^{77} + ( 9797 \beta_{4} - 1348 \beta_{7} ) q^{79} + 4782969 q^{81} + ( 714228 \beta_{1} + 1347 \beta_{5} ) q^{83} + ( 3630 \beta_{2} + 1840 \beta_{3} ) q^{85} + ( -8888 \beta_{4} + 11 \beta_{7} ) q^{87} + ( -32347346 + 1546 \beta_{6} ) q^{89} + ( 139112 \beta_{1} + 1923 \beta_{5} ) q^{91} + ( 44739 \beta_{2} - 999 \beta_{3} ) q^{93} + ( -17640 \beta_{4} - 4220 \beta_{7} ) q^{95} + ( -96436178 - 1190 \beta_{6} ) q^{97} + ( -96228 \beta_{1} + 2187 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 17496q^{9} + O(q^{10})$$ $$8q + 17496q^{9} + 276848q^{17} - 2841976q^{25} + 775008q^{33} + 5826832q^{41} - 16599928q^{49} - 42366240q^{57} - 132963072q^{65} - 57759760q^{73} + 38263752q^{81} - 258778768q^{89} - 771489424q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4826 x^{6} + 8748877 x^{4} + 7060845096 x^{2} + 2140819627716$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-22041 \nu^{7} - 80033094 \nu^{5} - 97666072335 \nu^{3} - 40034401078230 \nu$$$$)/ 84091362124$$ $$\beta_{2}$$ $$=$$ $$($$$$-135 \nu^{6} - 530927 \nu^{4} - 696017072 \nu^{2} - 304279869852$$$$)/1206926$$ $$\beta_{3}$$ $$=$$ $$($$$$402 \nu^{6} + 1366418 \nu^{4} + 1557414512 \nu^{2} + 595243216008$$$$)/603463$$ $$\beta_{4}$$ $$=$$ $$($$$$443391 \nu^{7} + 1598960541 \nu^{5} + 1920327097770 \nu^{3} + 770822874559080 \nu$$$$)/ 42045681062$$ $$\beta_{5}$$ $$=$$ $$($$$$461414 \nu^{7} + 2224833092 \nu^{5} + 3357073959114 \nu^{3} + 1626110629448292 \nu$$$$)/ 21022840531$$ $$\beta_{6}$$ $$=$$ $$($$$$-768 \nu^{6} - 2775168 \nu^{4} - 3337694208 \nu^{2} - 1335834521664$$$$)/86209$$ $$\beta_{7}$$ $$=$$ $$($$$$-1072761 \nu^{7} - 3866206107 \nu^{5} - 4636501305750 \nu^{3} - 1847566336831416 \nu$$$$)/ 3822334642$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 29 \beta_{4} + 96 \beta_{1}$$$$)/2592$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 8 \beta_{3} - 32 \beta_{2} - 463296$$$$)/384$$ $$\nu^{3}$$ $$=$$ $$($$$$-1384 \beta_{7} - 81 \beta_{5} - 48200 \beta_{4} - 464088 \beta_{1}$$$$)/3456$$ $$\nu^{4}$$ $$=$$ $$($$$$-2401 \beta_{6} - 20288 \beta_{3} + 70400 \beta_{2} + 556082112$$$$)/384$$ $$\nu^{5}$$ $$=$$ $$($$$$154880 \beta_{7} + 36195 \beta_{5} + 6632064 \beta_{4} + 104020488 \beta_{1}$$$$)/384$$ $$\nu^{6}$$ $$=$$ $$($$$$4286953 \beta_{6} + 38543040 \beta_{3} - 115319808 \beta_{2} - 663855941568$$$$)/384$$ $$\nu^{7}$$ $$=$$ $$($$$$-450207760 \beta_{7} - 274643145 \beta_{5} - 24462797648 \beta_{4} - 529545004824 \beta_{1}$$$$)/1152$$

## 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
 1.73205 + 33.5532i 1.73205 − 35.9608i 1.73205 + 35.9608i 1.73205 − 33.5532i −1.73205 − 33.5532i −1.73205 + 35.9608i −1.73205 − 35.9608i −1.73205 + 33.5532i
0 −46.7654 0 1205.32i 0 1133.44i 0 2187.00 0
319.2 0 −46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.3 0 −46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.4 0 −46.7654 0 1205.32i 0 1133.44i 0 2187.00 0
319.5 0 46.7654 0 1205.32i 0 1133.44i 0 2187.00 0
319.6 0 46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.7 0 46.7654 0 197.353i 0 3794.06i 0 2187.00 0
319.8 0 46.7654 0 1205.32i 0 1133.44i 0 2187.00 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 384.9.b.b 8
4.b odd 2 1 inner 384.9.b.b 8
8.b even 2 1 inner 384.9.b.b 8
8.d odd 2 1 inner 384.9.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.9.b.b 8 1.a even 1 1 trivial
384.9.b.b 8 4.b odd 2 1 inner
384.9.b.b 8 8.b even 2 1 inner
384.9.b.b 8 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 1491744 T_{5}^{2} + 56583532800$$ acting on $$S_{9}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -2187 + T^{2} )^{4}$$
$5$ $$( 56583532800 + 1491744 T^{2} + T^{4} )^{2}$$
$7$ $$( 18492829141248 + 15679584 T^{2} + T^{4} )^{2}$$
$11$ $$( 115520687528622336 - 696931680 T^{2} + T^{4} )^{2}$$
$13$ $$( 81910692156837888 + 786981504 T^{2} + T^{4} )^{2}$$
$17$ $$( 165051460 - 69212 T + T^{2} )^{4}$$
$19$ $$( 17802225831238560000 - 42855996000 T^{2} + T^{4} )^{2}$$
$23$ $$( 98677366453592690688 + 51799867776 T^{2} + T^{4} )^{2}$$
$29$ $$($$$$21\!\cdots\!00$$$$+ 566761587744 T^{2} + T^{4} )^{2}$$
$31$ $$($$$$26\!\cdots\!88$$$$+ 1369173987936 T^{2} + T^{4} )^{2}$$
$37$ $$($$$$56\!\cdots\!28$$$$+ 15049518744576 T^{2} + T^{4} )^{2}$$
$41$ $$( -15705528188 - 1456708 T + T^{2} )^{4}$$
$43$ $$($$$$35\!\cdots\!24$$$$- 25198148122464 T^{2} + T^{4} )^{2}$$
$47$ $$($$$$21\!\cdots\!12$$$$+ 52582489089408 T^{2} + T^{4} )^{2}$$
$53$ $$($$$$48\!\cdots\!00$$$$+ 37287632692512 T^{2} + T^{4} )^{2}$$
$59$ $$($$$$51\!\cdots\!00$$$$- 537373081395552 T^{2} + T^{4} )^{2}$$
$61$ $$($$$$49\!\cdots\!00$$$$+ 539483534602752 T^{2} + T^{4} )^{2}$$
$67$ $$($$$$44\!\cdots\!24$$$$- 330049356887136 T^{2} + T^{4} )^{2}$$
$71$ $$($$$$54\!\cdots\!92$$$$+ 1515456173420928 T^{2} + T^{4} )^{2}$$
$73$ $$( -67231781704700 + 14439940 T + T^{2} )^{4}$$
$79$ $$($$$$13\!\cdots\!00$$$$+ 7402940888084064 T^{2} + T^{4} )^{2}$$
$83$ $$($$$$24\!\cdots\!84$$$$- 3477726014338656 T^{2} + T^{4} )^{2}$$
$89$ $$( -1421500804154300 + 64694692 T + T^{2} )^{4}$$
$97$ $$( 7837779508054084 + 192872356 T + T^{2} )^{4}$$