# Properties

 Label 384.8.d.c Level $384$ Weight $8$ Character orbit 384.d Analytic conductor $119.956$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$119.955849786$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225$$ x^8 + 449*x^6 + 50632*x^4 + 69129*x^2 + 18225 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - 27 \beta_1 q^{3} + (\beta_{5} + 28 \beta_1) q^{5} + ( - \beta_{3} - 360) q^{7} - 729 q^{9}+O(q^{10})$$ q - 27*b1 * q^3 + (b5 + 28*b1) * q^5 + (-b3 - 360) * q^7 - 729 * q^9 $$q - 27 \beta_1 q^{3} + (\beta_{5} + 28 \beta_1) q^{5} + ( - \beta_{3} - 360) q^{7} - 729 q^{9} + ( - \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 36 \beta_1) q^{11} + ( - \beta_{7} - 4 \beta_{6} - 17 \beta_{5} + 540 \beta_1) q^{13} + ( - 27 \beta_{2} + 756) q^{15} + ( - 11 \beta_{4} - 13 \beta_{3} - 33 \beta_{2} + 2862) q^{17} + (5 \beta_{7} - 3 \beta_{6} - 39 \beta_{5} + 4068 \beta_1) q^{19} + ( - 27 \beta_{7} + 9720 \beta_1) q^{21} + (24 \beta_{4} + 22 \beta_{3} + 156 \beta_{2} - 25920) q^{23} + ( - 10 \beta_{4} - 74 \beta_{3} + 198 \beta_{2} - 25587) q^{25} + 19683 \beta_1 q^{27} + ( - 58 \beta_{7} - 52 \beta_{6} + 59 \beta_{5} + 61972 \beta_1) q^{29} + ( - 60 \beta_{4} - 59 \beta_{3} + 528 \beta_{2} + 2232) q^{31} + (81 \beta_{4} + 27 \beta_{3} - 81 \beta_{2} + 972) q^{33} + ( - 21 \beta_{7} - 105 \beta_{6} - 1329 \beta_{5} - 35280 \beta_1) q^{35} + ( - 177 \beta_{7} + 228 \beta_{6} + 397 \beta_{5} + \cdots + 115844 \beta_1) q^{37}+ \cdots + (729 \beta_{7} - 2187 \beta_{6} - 2187 \beta_{5} - 26244 \beta_1) q^{99}+O(q^{100})$$ q - 27*b1 * q^3 + (b5 + 28*b1) * q^5 + (-b3 - 360) * q^7 - 729 * q^9 + (-b7 + 3*b6 + 3*b5 + 36*b1) * q^11 + (-b7 - 4*b6 - 17*b5 + 540*b1) * q^13 + (-27*b2 + 756) * q^15 + (-11*b4 - 13*b3 - 33*b2 + 2862) * q^17 + (5*b7 - 3*b6 - 39*b5 + 4068*b1) * q^19 + (-27*b7 + 9720*b1) * q^21 + (24*b4 + 22*b3 + 156*b2 - 25920) * q^23 + (-10*b4 - 74*b3 + 198*b2 - 25587) * q^25 + 19683*b1 * q^27 + (-58*b7 - 52*b6 + 59*b5 + 61972*b1) * q^29 + (-60*b4 - 59*b3 + 528*b2 + 2232) * q^31 + (81*b4 + 27*b3 - 81*b2 + 972) * q^33 + (-21*b7 - 105*b6 - 1329*b5 - 35280*b1) * q^35 + (-177*b7 + 228*b6 + 397*b5 + 115844*b1) * q^37 + (-108*b4 + 27*b3 + 459*b2 + 14580) * q^39 + (73*b4 - 313*b3 - 941*b2 + 85882) * q^41 + (77*b7 + 189*b6 - 495*b5 - 205884*b1) * q^43 + (-729*b5 - 20412*b1) * q^45 + (288*b4 + 194*b3 + 720*b2 + 248400) * q^47 + (-140*b4 + 728*b3 + 784*b2 + 601945) * q^49 + (-351*b7 + 297*b6 - 891*b5 - 77274*b1) * q^51 + (642*b7 - 276*b6 - 1899*b5 + 443276*b1) * q^53 + (60*b4 - 684*b3 + 1128*b2 - 259632) * q^55 + (-81*b4 - 135*b3 + 1053*b2 + 109836) * q^57 + (72*b7 - 720*b6 + 3672*b5 + 131364*b1) * q^59 + (1213*b7 - 332*b6 + 3571*b5 + 422460*b1) * q^61 + (729*b3 + 262440) * q^63 + (-195*b4 + 1923*b3 - 865*b2 + 1608976) * q^65 + (2220*b7 + 84*b6 - 3444*b5 - 902268*b1) * q^67 + (594*b7 - 648*b6 + 4212*b5 + 699840*b1) * q^69 + (312*b4 + 1322*b3 - 4668*b2 + 792000) * q^71 + (-700*b4 - 1112*b3 + 3040*b2 + 1115094) * q^73 + (-1998*b7 + 270*b6 + 5346*b5 + 690849*b1) * q^75 + (-1408*b7 + 1676*b6 - 9860*b5 - 301664*b1) * q^77 + (-1680*b4 - 3815*b3 - 7716*b2 - 156456) * q^79 + 531441 * q^81 + (-4207*b7 + 357*b6 - 4323*b5 - 434844*b1) * q^83 + (-944*b7 - 320*b6 - 2706*b5 + 3425480*b1) * q^85 + (-1404*b4 + 1566*b3 - 1593*b2 + 1673244) * q^87 + (2566*b4 - 1078*b3 - 17742*b2 + 555926) * q^89 + (3961*b7 - 543*b6 + 32253*b5 - 2505600*b1) * q^91 + (-1593*b7 + 1620*b6 + 14256*b5 - 60264*b1) * q^93 + (720*b4 + 3264*b3 - 7020*b2 + 3950928) * q^95 + (3982*b4 - 4870*b3 + 4306*b2 - 1769698) * q^97 + (729*b7 - 2187*b6 - 2187*b5 - 26244*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2880 q^{7} - 5832 q^{9}+O(q^{10})$$ 8 * q - 2880 * q^7 - 5832 * q^9 $$8 q - 2880 q^{7} - 5832 q^{9} + 6048 q^{15} + 22896 q^{17} - 207360 q^{23} - 204696 q^{25} + 17856 q^{31} + 7776 q^{33} + 116640 q^{39} + 687056 q^{41} + 1987200 q^{47} + 4815560 q^{49} - 2077056 q^{55} + 878688 q^{57} + 2099520 q^{63} + 12871808 q^{65} + 6336000 q^{71} + 8920752 q^{73} - 1251648 q^{79} + 4251528 q^{81} + 13385952 q^{87} + 4447408 q^{89} + 31607424 q^{95} - 14157584 q^{97}+O(q^{100})$$ 8 * q - 2880 * q^7 - 5832 * q^9 + 6048 * q^15 + 22896 * q^17 - 207360 * q^23 - 204696 * q^25 + 17856 * q^31 + 7776 * q^33 + 116640 * q^39 + 687056 * q^41 + 1987200 * q^47 + 4815560 * q^49 - 2077056 * q^55 + 878688 * q^57 + 2099520 * q^63 + 12871808 * q^65 + 6336000 * q^71 + 8920752 * q^73 - 1251648 * q^79 + 4251528 * q^81 + 13385952 * q^87 + 4447408 * q^89 + 31607424 * q^95 - 14157584 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 449x^{6} + 50632x^{4} + 69129x^{2} + 18225$$ :

 $$\beta_{1}$$ $$=$$ $$( 131\nu^{7} + 58684\nu^{5} + 6584732\nu^{3} + 5063679\nu ) / 1626480$$ (131*v^7 + 58684*v^5 + 6584732*v^3 + 5063679*v) / 1626480 $$\beta_{2}$$ $$=$$ $$( 65\nu^{6} + 27156\nu^{4} + 2829796\nu^{2} + 2486853 ) / 9036$$ (65*v^6 + 27156*v^4 + 2829796*v^2 + 2486853) / 9036 $$\beta_{3}$$ $$=$$ $$( 57\nu^{6} + 24308\nu^{4} + 2496836\nu^{2} - 8497827 ) / 9036$$ (57*v^6 + 24308*v^4 + 2496836*v^2 - 8497827) / 9036 $$\beta_{4}$$ $$=$$ $$( 553\nu^{6} + 249076\nu^{4} + 28104132\nu^{2} + 20876013 ) / 4518$$ (553*v^6 + 249076*v^4 + 28104132*v^2 + 20876013) / 4518 $$\beta_{5}$$ $$=$$ $$( -713\nu^{7} - 320092\nu^{5} - 36039416\nu^{3} - 40729437\nu ) / 101655$$ (-713*v^7 - 320092*v^5 - 36039416*v^3 - 40729437*v) / 101655 $$\beta_{6}$$ $$=$$ $$( 7231\nu^{7} + 3248924\nu^{5} + 366814612\nu^{3} + 540861579\nu ) / 101655$$ (7231*v^7 + 3248924*v^5 + 366814612*v^3 + 540861579*v) / 101655 $$\beta_{7}$$ $$=$$ $$( 9022\nu^{7} + 4049168\nu^{5} + 455154004\nu^{3} + 338539158\nu ) / 101655$$ (9022*v^7 + 4049168*v^5 + 455154004*v^3 + 338539158*v) / 101655
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{6} - 25\beta_{5} - 192\beta_1 ) / 768$$ (-b7 - b6 - 25*b5 - 192*b1) / 768 $$\nu^{2}$$ $$=$$ $$( \beta_{4} - 73\beta_{3} + 47\beta_{2} - 86208 ) / 768$$ (b4 - 73*b3 + 47*b2 - 86208) / 768 $$\nu^{3}$$ $$=$$ $$( 71\beta_{7} + 143\beta_{6} + 2783\beta_{5} + 37824\beta_1 ) / 384$$ (71*b7 + 143*b6 + 2783*b5 + 37824*b1) / 384 $$\nu^{4}$$ $$=$$ $$( -31\beta_{4} + 16303\beta_{3} - 13769\beta_{2} + 19264704 ) / 768$$ (-31*b4 + 16303*b3 - 13769*b2 + 19264704) / 768 $$\nu^{5}$$ $$=$$ $$( -22159\beta_{7} - 63991\beta_{6} - 1252807\beta_{5} - 28166592\beta_1 ) / 768$$ (-22159*b7 - 63991*b6 - 1252807*b5 - 28166592*b1) / 768 $$\nu^{6}$$ $$=$$ $$( -3823\beta_{4} - 454133\beta_{3} + 476635\beta_{2} - 540598368 ) / 96$$ (-3823*b4 - 454133*b3 + 476635*b2 - 540598368) / 96 $$\nu^{7}$$ $$=$$ $$( 2827561\beta_{7} + 14328841\beta_{6} + 282409921\beta_{5} + 8832273600\beta_1 ) / 768$$ (2827561*b7 + 14328841*b6 + 282409921*b5 + 8832273600*b1) / 768

## 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
 14.6646i 0.596953i − 1.01122i − 15.2503i 15.2503i 1.01122i − 0.596953i − 14.6646i
0 27.0000i 0 344.306i 0 −1669.75 0 −729.000 0
193.2 0 27.0000i 0 135.999i 0 678.568 0 −729.000 0
193.3 0 27.0000i 0 69.8856i 0 860.192 0 −729.000 0
193.4 0 27.0000i 0 522.419i 0 −1309.01 0 −729.000 0
193.5 0 27.0000i 0 522.419i 0 −1309.01 0 −729.000 0
193.6 0 27.0000i 0 69.8856i 0 860.192 0 −729.000 0
193.7 0 27.0000i 0 135.999i 0 678.568 0 −729.000 0
193.8 0 27.0000i 0 344.306i 0 −1669.75 0 −729.000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.d.c 8
4.b odd 2 1 384.8.d.d yes 8
8.b even 2 1 inner 384.8.d.c 8
8.d odd 2 1 384.8.d.d yes 8

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{8} + 414848T_{5}^{6} + 41596678144T_{5}^{4} + 791786461593600T_{5}^{2} + 2922622746624000000$$ T5^8 + 414848*T5^6 + 41596678144*T5^4 + 791786461593600*T5^2 + 2922622746624000000 $$T_{7}^{4} + 1440T_{7}^{3} - 1814176T_{7}^{2} - 1624601088T_{7} + 1275801663744$$ T7^4 + 1440*T7^3 - 1814176*T7^2 - 1624601088*T7 + 1275801663744

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 729)^{4}$$
$5$ $$T^{8} + 414848 T^{6} + \cdots + 29\!\cdots\!00$$
$7$ $$(T^{4} + 1440 T^{3} + \cdots + 1275801663744)^{2}$$
$11$ $$T^{8} + 83236928 T^{6} + \cdots + 49\!\cdots\!96$$
$13$ $$T^{8} + 269496640 T^{6} + \cdots + 11\!\cdots\!00$$
$17$ $$(T^{4} - 11448 T^{3} + \cdots + 58\!\cdots\!48)^{2}$$
$19$ $$T^{8} + 878694464 T^{6} + \cdots + 11\!\cdots\!04$$
$23$ $$(T^{4} + 103680 T^{3} + \cdots + 11\!\cdots\!96)^{2}$$
$29$ $$T^{8} + 69547739776 T^{6} + \cdots + 58\!\cdots\!64$$
$31$ $$(T^{4} - 8928 T^{3} + \cdots - 45\!\cdots\!92)^{2}$$
$37$ $$T^{8} + 646806181952 T^{6} + \cdots + 78\!\cdots\!16$$
$41$ $$(T^{4} - 343528 T^{3} + \cdots - 32\!\cdots\!92)^{2}$$
$43$ $$T^{8} + 703896484928 T^{6} + \cdots + 68\!\cdots\!96$$
$47$ $$(T^{4} - 993600 T^{3} + \cdots - 24\!\cdots\!40)^{2}$$
$53$ $$T^{8} + 4748734157440 T^{6} + \cdots + 55\!\cdots\!16$$
$59$ $$T^{8} + 10826005492800 T^{6} + \cdots + 19\!\cdots\!96$$
$61$ $$T^{8} + 12824477047872 T^{6} + \cdots + 29\!\cdots\!44$$
$67$ $$T^{8} + 35890312491072 T^{6} + \cdots + 16\!\cdots\!56$$
$71$ $$(T^{4} - 3168000 T^{3} + \cdots + 15\!\cdots\!48)^{2}$$
$73$ $$(T^{4} - 4460376 T^{3} + \cdots - 79\!\cdots\!08)^{2}$$
$79$ $$(T^{4} + 625824 T^{3} + \cdots - 11\!\cdots\!64)^{2}$$
$83$ $$T^{8} + 93693006217280 T^{6} + \cdots + 36\!\cdots\!84$$
$89$ $$(T^{4} - 2223704 T^{3} + \cdots + 69\!\cdots\!52)^{2}$$
$97$ $$(T^{4} + 7078792 T^{3} + \cdots - 29\!\cdots\!64)^{2}$$