# Properties

 Label 384.9.b.a 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: $$x^{8} - 4x^{7} + 200x^{6} - 586x^{5} + 10949x^{4} - 20926x^{3} + 78946x^{2} - 68580x + 222900$$ x^8 - 4*x^7 + 200*x^6 - 586*x^5 + 10949*x^4 - 20926*x^3 + 78946*x^2 - 68580*x + 222900 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{16}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{5} q^{5} - 7 \beta_{4} q^{7} + 2187 q^{9}+O(q^{10})$$ q - b3 * q^3 - b5 * q^5 - 7*b4 * q^7 + 2187 * q^9 $$q - \beta_{3} q^{3} - \beta_{5} q^{5} - 7 \beta_{4} q^{7} + 2187 q^{9} + ( - 61 \beta_{3} - \beta_{2}) q^{11} + ( - 19 \beta_{6} + 22 \beta_{5}) q^{13} + (27 \beta_{4} - \beta_1) q^{15} + ( - \beta_{7} + 11630) q^{17} + ( - 667 \beta_{3} + \beta_{2}) q^{19} + (63 \beta_{6} + 63 \beta_{5}) q^{21} + ( - 1186 \beta_{4} - 4 \beta_1) q^{23} + ( - 2 \beta_{7} + 49489) q^{25} - 2187 \beta_{3} q^{27} + ( - 490 \beta_{6} - 1077 \beta_{5}) q^{29} + (1183 \beta_{4} + 44 \beta_1) q^{31} + ( - 9 \beta_{7} + 133164) q^{33} + ( - 6601 \beta_{3} + 63 \beta_{2}) q^{35} + ( - 1383 \beta_{6} + 80 \beta_{5}) q^{37} + (3510 \beta_{4} + 41 \beta_1) q^{39} + (47 \beta_{7} + 372322) q^{41} + (11049 \beta_{3} - 139 \beta_{2}) q^{43} - 2187 \beta_{5} q^{45} + ( - 2874 \beta_{4} + 256 \beta_1) q^{47} + 2018065 q^{49} + ( - 11657 \beta_{3} - 243 \beta_{2}) q^{51} + (4802 \beta_{6} + 1889 \beta_{5}) q^{53} + ( - 33428 \beta_{4} + 44 \beta_1) q^{55} + (9 \beta_{7} + 1458972) q^{57} + (101684 \beta_{3} + 760 \beta_{2}) q^{59} + ( - 8497 \beta_{6} + 13780 \beta_{5}) q^{61} - 15309 \beta_{4} q^{63} + ( - 89 \beta_{7} + 9628128) q^{65} + ( - 79896 \beta_{3} - 196 \beta_{2}) q^{67} + (11646 \beta_{6} + 2898 \beta_{5}) q^{69} + (73282 \beta_{4} - 252 \beta_1) q^{71} + ( - 176 \beta_{7} + 23923582) q^{73} + ( - 49543 \beta_{3} - 486 \beta_{2}) q^{75} + (10444 \beta_{6} - 49028 \beta_{5}) q^{77} + ( - 76433 \beta_{4} + 836 \beta_1) q^{79} + 4782969 q^{81} + ( - 227655 \beta_{3} + 1461 \beta_{2}) q^{83} + (32240 \beta_{6} + 46098 \beta_{5}) q^{85} + (134919 \beta_{4} - 587 \beta_1) q^{87} + ( - 278 \beta_{7} + 52859054) q^{89} + ( - 859327 \beta_{3} - 2583 \beta_{2}) q^{91} + ( - 21339 \beta_{6} + 74889 \beta_{5}) q^{93} + (53084 \beta_{4} - 772 \beta_1) q^{95} + (1154 \beta_{7} + 3921070) q^{97} + ( - 133407 \beta_{3} - 2187 \beta_{2}) q^{99}+O(q^{100})$$ q - b3 * q^3 - b5 * q^5 - 7*b4 * q^7 + 2187 * q^9 + (-61*b3 - b2) * q^11 + (-19*b6 + 22*b5) * q^13 + (27*b4 - b1) * q^15 + (-b7 + 11630) * q^17 + (-667*b3 + b2) * q^19 + (63*b6 + 63*b5) * q^21 + (-1186*b4 - 4*b1) * q^23 + (-2*b7 + 49489) * q^25 - 2187*b3 * q^27 + (-490*b6 - 1077*b5) * q^29 + (1183*b4 + 44*b1) * q^31 + (-9*b7 + 133164) * q^33 + (-6601*b3 + 63*b2) * q^35 + (-1383*b6 + 80*b5) * q^37 + (3510*b4 + 41*b1) * q^39 + (47*b7 + 372322) * q^41 + (11049*b3 - 139*b2) * q^43 - 2187*b5 * q^45 + (-2874*b4 + 256*b1) * q^47 + 2018065 * q^49 + (-11657*b3 - 243*b2) * q^51 + (4802*b6 + 1889*b5) * q^53 + (-33428*b4 + 44*b1) * q^55 + (9*b7 + 1458972) * q^57 + (101684*b3 + 760*b2) * q^59 + (-8497*b6 + 13780*b5) * q^61 - 15309*b4 * q^63 + (-89*b7 + 9628128) * q^65 + (-79896*b3 - 196*b2) * q^67 + (11646*b6 + 2898*b5) * q^69 + (73282*b4 - 252*b1) * q^71 + (-176*b7 + 23923582) * q^73 + (-49543*b3 - 486*b2) * q^75 + (10444*b6 - 49028*b5) * q^77 + (-76433*b4 + 836*b1) * q^79 + 4782969 * q^81 + (-227655*b3 + 1461*b2) * q^83 + (32240*b6 + 46098*b5) * q^85 + (134919*b4 - 587*b1) * q^87 + (-278*b7 + 52859054) * q^89 + (-859327*b3 - 2583*b2) * q^91 + (-21339*b6 + 74889*b5) * q^93 + (53084*b4 - 772*b1) * q^95 + (1154*b7 + 3921070) * q^97 + (-133407*b3 - 2187*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 17496 q^{9}+O(q^{10})$$ 8 * q + 17496 * q^9 $$8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97}+O(q^{100})$$ 8 * q + 17496 * q^9 + 93040 * q^17 + 395912 * q^25 + 1065312 * q^33 + 2978576 * q^41 + 16144520 * q^49 + 11671776 * q^57 + 77025024 * q^65 + 191388656 * q^73 + 38263752 * q^81 + 422872432 * q^89 + 31368560 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 200x^{6} - 586x^{5} + 10949x^{4} - 20926x^{3} + 78946x^{2} - 68580x + 222900$$ :

 $$\beta_{1}$$ $$=$$ $$( -432\nu^{6} + 1296\nu^{5} - 76896\nu^{4} + 151632\nu^{3} - 3639600\nu^{2} + 3564000\nu - 12572496 ) / 403$$ (-432*v^6 + 1296*v^5 - 76896*v^4 + 151632*v^3 - 3639600*v^2 + 3564000*v - 12572496) / 403 $$\beta_{2}$$ $$=$$ $$( 422 \nu^{7} - 466693 \nu^{6} + 1475675 \nu^{5} - 68583127 \nu^{4} + 138435443 \nu^{3} - 536132632 \nu^{2} + 473860092 \nu + 89182264770 ) / 5257910$$ (422*v^7 - 466693*v^6 + 1475675*v^5 - 68583127*v^4 + 138435443*v^3 - 536132632*v^2 + 473860092*v + 89182264770) / 5257910 $$\beta_{3}$$ $$=$$ $$( - 3798 \nu^{7} + 13293 \nu^{6} - 720243 \nu^{5} + 1767375 \nu^{4} - 35892171 \nu^{3} + 52077528 \nu^{2} - 94544604 \nu + 38651310 ) / 5257910$$ (-3798*v^7 + 13293*v^6 - 720243*v^5 + 1767375*v^4 - 35892171*v^3 + 52077528*v^2 - 94544604*v + 38651310) / 5257910 $$\beta_{4}$$ $$=$$ $$( - 1288 \nu^{7} + 4508 \nu^{6} - 269884 \nu^{5} + 663440 \nu^{4} - 16358356 \nu^{3} + 23876348 \nu^{2} - 177074040 \nu + 84579636 ) / 976469$$ (-1288*v^7 + 4508*v^6 - 269884*v^5 + 663440*v^4 - 16358356*v^3 + 23876348*v^2 - 177074040*v + 84579636) / 976469 $$\beta_{5}$$ $$=$$ $$( 10064 \nu^{7} - 49762 \nu^{6} + 2067878 \nu^{5} - 8127346 \nu^{4} + 120250730 \nu^{3} - 342411844 \nu^{2} + 1396161240 \nu - 1195854900 ) / 3307395$$ (10064*v^7 - 49762*v^6 + 2067878*v^5 - 8127346*v^4 + 120250730*v^3 - 342411844*v^2 + 1396161240*v - 1195854900) / 3307395 $$\beta_{6}$$ $$=$$ $$( - 10064 \nu^{7} - 81080 \nu^{6} - 1675352 \nu^{5} - 20265368 \nu^{4} - 64119512 \nu^{3} - 1244701616 \nu^{2} + 162952032 \nu - 4311284880 ) / 3307395$$ (-10064*v^7 - 81080*v^6 - 1675352*v^5 - 20265368*v^4 - 64119512*v^3 - 1244701616*v^2 + 162952032*v - 4311284880) / 3307395 $$\beta_{7}$$ $$=$$ $$( 3072 \nu^{7} - 10752 \nu^{6} + 609024 \nu^{5} - 1495680 \nu^{4} + 31491840 \nu^{3} - 45747456 \nu^{2} + 82874880 \nu - 33862464 ) / 2423$$ (3072*v^7 - 10752*v^6 + 609024*v^5 - 1495680*v^4 + 31491840*v^3 - 45747456*v^2 + 82874880*v - 33862464) / 2423
 $$\nu$$ $$=$$ $$( -\beta_{6} + 8\beta_{5} + 12\beta_{4} + 16\beta_{3} + 432 ) / 864$$ (-b6 + 8*b5 + 12*b4 + 16*b3 + 432) / 864 $$\nu^{2}$$ $$=$$ $$( 36\beta_{6} + 144\beta_{5} + 144\beta_{4} + 195\beta_{3} + 27\beta_{2} - 4\beta _1 - 497664 ) / 10368$$ (36*b6 + 144*b5 + 144*b4 + 195*b3 + 27*b2 - 4*b1 - 497664) / 10368 $$\nu^{3}$$ $$=$$ $$( - 9 \beta_{7} + 652 \beta_{6} - 4784 \beta_{5} - 10992 \beta_{4} - 18621 \beta_{3} + 27 \beta_{2} - 4 \beta _1 - 499392 ) / 6912$$ (-9*b7 + 652*b6 - 4784*b5 - 10992*b4 - 18621*b3 + 27*b2 - 4*b1 - 499392) / 6912 $$\nu^{4}$$ $$=$$ $$( - 27 \beta_{7} - 9312 \beta_{6} - 25728 \beta_{5} - 33120 \beta_{4} - 56340 \beta_{3} - 2484 \beta_{2} + 616 \beta _1 + 42814656 ) / 10368$$ (-27*b7 - 9312*b6 - 25728*b5 - 33120*b4 - 56340*b3 - 2484*b2 + 616*b1 + 42814656) / 10368 $$\nu^{5}$$ $$=$$ $$( 4275 \beta_{7} - 212796 \beta_{6} + 1199088 \beta_{5} + 2870928 \beta_{4} + 8207181 \beta_{3} - 12555 \beta_{2} + 3100 \beta _1 + 216571968 ) / 20736$$ (4275*b7 - 212796*b6 + 1199088*b5 + 2870928*b4 + 8207181*b3 - 12555*b2 + 3100*b1 + 216571968) / 20736 $$\nu^{6}$$ $$=$$ $$( 540 \beta_{7} + 106610 \beta_{6} + 286520 \beta_{5} + 365772 \beta_{4} + 1039705 \beta_{3} + 17505 \beta_{2} - 6923 \beta _1 - 302102784 ) / 864$$ (540*b7 + 106610*b6 + 286520*b5 + 365772*b4 + 1039705*b3 + 17505*b2 - 6923*b1 - 302102784) / 864 $$\nu^{7}$$ $$=$$ $$( - 535311 \beta_{7} + 23742708 \beta_{6} - 92469072 \beta_{5} - 236122704 \beta_{4} - 1026485487 \beta_{3} + 1514457 \beta_{2} - 592396 \beta _1 - 26136385344 ) / 20736$$ (-535311*b7 + 23742708*b6 - 92469072*b5 - 236122704*b4 - 1026485487*b3 + 1514457*b2 - 592396*b1 - 26136385344) / 20736

## 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
 1.36603 + 9.69292i 1.36603 + 2.01178i 1.36603 − 2.01178i 1.36603 − 9.69292i −0.366025 + 2.01178i −0.366025 + 9.69292i −0.366025 − 9.69292i −0.366025 − 2.01178i
0 −46.7654 0 721.475i 0 1935.65i 0 2187.00 0
319.2 0 −46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.3 0 −46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.4 0 −46.7654 0 721.475i 0 1935.65i 0 2187.00 0
319.5 0 46.7654 0 721.475i 0 1935.65i 0 2187.00 0
319.6 0 46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.7 0 46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.8 0 46.7654 0 721.475i 0 1935.65i 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.a 8
4.b odd 2 1 inner 384.9.b.a 8
8.b even 2 1 inner 384.9.b.a 8
8.d odd 2 1 inner 384.9.b.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 682272T_{5}^{2} + 84192825600$$ acting on $$S_{9}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} - 2187)^{4}$$
$5$ $$(T^{4} + 682272 T^{2} + \cdots + 84192825600)^{2}$$
$7$ $$(T^{2} + 3746736)^{4}$$
$11$ $$(T^{4} - 612159840 T^{2} + \cdots + 84\!\cdots\!16)^{2}$$
$13$ $$(T^{4} + 1922702976 T^{2} + \cdots + 99\!\cdots\!44)^{2}$$
$17$ $$(T^{2} - 23260 T - 7909979324)^{4}$$
$19$ $$(T^{4} - 2542536288 T^{2} + \cdots + 45\!\cdots\!00)^{2}$$
$23$ $$(T^{4} + 237198222720 T^{2} + \cdots + 93\!\cdots\!84)^{2}$$
$29$ $$(T^{4} + 1490396210208 T^{2} + \cdots + 27\!\cdots\!00)^{2}$$
$31$ $$(T^{4} + 2886948519264 T^{2} + \cdots + 15\!\cdots\!00)^{2}$$
$37$ $$(T^{4} + 7501371886080 T^{2} + \cdots + 57\!\cdots\!56)^{2}$$
$41$ $$(T^{2} - 744644 T - 17633303147132)^{4}$$
$43$ $$(T^{4} - 12049696420704 T^{2} + \cdots + 30\!\cdots\!04)^{2}$$
$47$ $$(T^{4} + 91745074569600 T^{2} + \cdots + 19\!\cdots\!84)^{2}$$
$53$ $$(T^{4} + 88167062209824 T^{2} + \cdots + 14\!\cdots\!44)^{2}$$
$59$ $$(T^{4} - 389367404122464 T^{2} + \cdots + 22\!\cdots\!24)^{2}$$
$61$ $$(T^{4} + 463016972620800 T^{2} + \cdots + 66\!\cdots\!56)^{2}$$
$67$ $$(T^{4} - 50799407453280 T^{2} + \cdots + 62\!\cdots\!36)^{2}$$
$71$ $$(T^{4} + 908938263991680 T^{2} + \cdots + 13\!\cdots\!24)^{2}$$
$73$ $$(T^{2} - 47847164 T + 323128538436100)^{4}$$
$79$ $$(T^{4} + \cdots + 12\!\cdots\!00)^{2}$$
$83$ $$(T^{4} + \cdots + 27\!\cdots\!00)^{2}$$
$89$ $$(T^{2} - 105718108 T + 21\!\cdots\!00)^{4}$$
$97$ $$(T^{2} - 7842140 T - 10\!\cdots\!84)^{4}$$