# Properties

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

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## 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: $$x^{8} + 449 x^{6} + 50632 x^{4} + 69129 x^{2} + 18225$$ 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.

 $$f(q)$$ $$=$$ $$q + 27 \beta_{1} q^{3} + ( 28 \beta_{1} + \beta_{5} ) q^{5} + ( 360 + \beta_{3} ) q^{7} -729 q^{9} +O(q^{10})$$ $$q + 27 \beta_{1} q^{3} + ( 28 \beta_{1} + \beta_{5} ) q^{5} + ( 360 + \beta_{3} ) q^{7} -729 q^{9} + ( -36 \beta_{1} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{11} + ( 540 \beta_{1} - 17 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{13} + ( -756 + 27 \beta_{2} ) q^{15} + ( 2862 - 33 \beta_{2} - 13 \beta_{3} - 11 \beta_{4} ) q^{17} + ( -4068 \beta_{1} + 39 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} ) q^{19} + ( 9720 \beta_{1} - 27 \beta_{7} ) q^{21} + ( 25920 - 156 \beta_{2} - 22 \beta_{3} - 24 \beta_{4} ) q^{23} + ( -25587 + 198 \beta_{2} - 74 \beta_{3} - 10 \beta_{4} ) q^{25} -19683 \beta_{1} q^{27} + ( 61972 \beta_{1} + 59 \beta_{5} - 52 \beta_{6} - 58 \beta_{7} ) q^{29} + ( -2232 - 528 \beta_{2} + 59 \beta_{3} + 60 \beta_{4} ) q^{31} + ( 972 - 81 \beta_{2} + 27 \beta_{3} + 81 \beta_{4} ) q^{33} + ( 35280 \beta_{1} + 1329 \beta_{5} + 105 \beta_{6} + 21 \beta_{7} ) q^{35} + ( 115844 \beta_{1} + 397 \beta_{5} + 228 \beta_{6} - 177 \beta_{7} ) q^{37} + ( -14580 - 459 \beta_{2} - 27 \beta_{3} + 108 \beta_{4} ) q^{39} + ( 85882 - 941 \beta_{2} - 313 \beta_{3} + 73 \beta_{4} ) q^{41} + ( 205884 \beta_{1} + 495 \beta_{5} - 189 \beta_{6} - 77 \beta_{7} ) q^{43} + ( -20412 \beta_{1} - 729 \beta_{5} ) q^{45} + ( -248400 - 720 \beta_{2} - 194 \beta_{3} - 288 \beta_{4} ) q^{47} + ( 601945 + 784 \beta_{2} + 728 \beta_{3} - 140 \beta_{4} ) q^{49} + ( 77274 \beta_{1} + 891 \beta_{5} - 297 \beta_{6} + 351 \beta_{7} ) q^{51} + ( 443276 \beta_{1} - 1899 \beta_{5} - 276 \beta_{6} + 642 \beta_{7} ) q^{53} + ( 259632 - 1128 \beta_{2} + 684 \beta_{3} - 60 \beta_{4} ) q^{55} + ( 109836 + 1053 \beta_{2} - 135 \beta_{3} - 81 \beta_{4} ) q^{57} + ( -131364 \beta_{1} - 3672 \beta_{5} + 720 \beta_{6} - 72 \beta_{7} ) q^{59} + ( 422460 \beta_{1} + 3571 \beta_{5} - 332 \beta_{6} + 1213 \beta_{7} ) q^{61} + ( -262440 - 729 \beta_{3} ) q^{63} + ( 1608976 - 865 \beta_{2} + 1923 \beta_{3} - 195 \beta_{4} ) q^{65} + ( 902268 \beta_{1} + 3444 \beta_{5} - 84 \beta_{6} - 2220 \beta_{7} ) q^{67} + ( 699840 \beta_{1} + 4212 \beta_{5} - 648 \beta_{6} + 594 \beta_{7} ) q^{69} + ( -792000 + 4668 \beta_{2} - 1322 \beta_{3} - 312 \beta_{4} ) q^{71} + ( 1115094 + 3040 \beta_{2} - 1112 \beta_{3} - 700 \beta_{4} ) q^{73} + ( -690849 \beta_{1} - 5346 \beta_{5} - 270 \beta_{6} + 1998 \beta_{7} ) q^{75} + ( -301664 \beta_{1} - 9860 \beta_{5} + 1676 \beta_{6} - 1408 \beta_{7} ) q^{77} + ( 156456 + 7716 \beta_{2} + 3815 \beta_{3} + 1680 \beta_{4} ) q^{79} + 531441 q^{81} + ( 434844 \beta_{1} + 4323 \beta_{5} - 357 \beta_{6} + 4207 \beta_{7} ) q^{83} + ( 3425480 \beta_{1} - 2706 \beta_{5} - 320 \beta_{6} - 944 \beta_{7} ) q^{85} + ( -1673244 + 1593 \beta_{2} - 1566 \beta_{3} + 1404 \beta_{4} ) q^{87} + ( 555926 - 17742 \beta_{2} - 1078 \beta_{3} + 2566 \beta_{4} ) q^{89} + ( 2505600 \beta_{1} - 32253 \beta_{5} + 543 \beta_{6} - 3961 \beta_{7} ) q^{91} + ( -60264 \beta_{1} + 14256 \beta_{5} + 1620 \beta_{6} - 1593 \beta_{7} ) q^{93} + ( -3950928 + 7020 \beta_{2} - 3264 \beta_{3} - 720 \beta_{4} ) q^{95} + ( -1769698 + 4306 \beta_{2} - 4870 \beta_{3} + 3982 \beta_{4} ) q^{97} + ( 26244 \beta_{1} + 2187 \beta_{5} + 2187 \beta_{6} - 729 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2880 q^{7} - 5832 q^{9} + O(q^{10})$$ $$8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$131 \nu^{7} + 58684 \nu^{5} + 6584732 \nu^{3} + 5063679 \nu$$$$)/1626480$$ $$\beta_{2}$$ $$=$$ $$($$$$65 \nu^{6} + 27156 \nu^{4} + 2829796 \nu^{2} + 2486853$$$$)/9036$$ $$\beta_{3}$$ $$=$$ $$($$$$57 \nu^{6} + 24308 \nu^{4} + 2496836 \nu^{2} - 8497827$$$$)/9036$$ $$\beta_{4}$$ $$=$$ $$($$$$553 \nu^{6} + 249076 \nu^{4} + 28104132 \nu^{2} + 20876013$$$$)/4518$$ $$\beta_{5}$$ $$=$$ $$($$$$-713 \nu^{7} - 320092 \nu^{5} - 36039416 \nu^{3} - 40729437 \nu$$$$)/101655$$ $$\beta_{6}$$ $$=$$ $$($$$$7231 \nu^{7} + 3248924 \nu^{5} + 366814612 \nu^{3} + 540861579 \nu$$$$)/101655$$ $$\beta_{7}$$ $$=$$ $$($$$$9022 \nu^{7} + 4049168 \nu^{5} + 455154004 \nu^{3} + 338539158 \nu$$$$)/101655$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - 25 \beta_{5} - 192 \beta_{1}$$$$)/768$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - 73 \beta_{3} + 47 \beta_{2} - 86208$$$$)/768$$ $$\nu^{3}$$ $$=$$ $$($$$$71 \beta_{7} + 143 \beta_{6} + 2783 \beta_{5} + 37824 \beta_{1}$$$$)/384$$ $$\nu^{4}$$ $$=$$ $$($$$$-31 \beta_{4} + 16303 \beta_{3} - 13769 \beta_{2} + 19264704$$$$)/768$$ $$\nu^{5}$$ $$=$$ $$($$$$-22159 \beta_{7} - 63991 \beta_{6} - 1252807 \beta_{5} - 28166592 \beta_{1}$$$$)/768$$ $$\nu^{6}$$ $$=$$ $$($$$$-3823 \beta_{4} - 454133 \beta_{3} + 476635 \beta_{2} - 540598368$$$$)/96$$ $$\nu^{7}$$ $$=$$ $$($$$$2827561 \beta_{7} + 14328841 \beta_{6} + 282409921 \beta_{5} + 8832273600 \beta_{1}$$$$)/768$$

## 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
 15.2503i 1.01122i − 0.596953i − 14.6646i 14.6646i 0.596953i − 1.01122i − 15.2503i
0 27.0000i 0 522.419i 0 1309.01 0 −729.000 0
193.2 0 27.0000i 0 69.8856i 0 −860.192 0 −729.000 0
193.3 0 27.0000i 0 135.999i 0 −678.568 0 −729.000 0
193.4 0 27.0000i 0 344.306i 0 1669.75 0 −729.000 0
193.5 0 27.0000i 0 344.306i 0 1669.75 0 −729.000 0
193.6 0 27.0000i 0 135.999i 0 −678.568 0 −729.000 0
193.7 0 27.0000i 0 69.8856i 0 −860.192 0 −729.000 0
193.8 0 27.0000i 0 522.419i 0 1309.01 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.d yes 8
4.b odd 2 1 384.8.d.c 8
8.b even 2 1 inner 384.8.d.d yes 8
8.d odd 2 1 384.8.d.c 8

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

## Hecke kernels

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

 $$T_{5}^{8} + 414848 T_{5}^{6} + 41596678144 T_{5}^{4} +$$$$79\!\cdots\!00$$$$T_{5}^{2} +$$$$29\!\cdots\!00$$ $$T_{7}^{4} - 1440 T_{7}^{3} - 1814176 T_{7}^{2} + 1624601088 T_{7} +$$$$12\!\cdots\!44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 729 + T^{2} )^{4}$$
$5$ $$2922622746624000000 + 791786461593600 T^{2} + 41596678144 T^{4} + 414848 T^{6} + T^{8}$$
$7$ $$( 1275801663744 + 1624601088 T - 1814176 T^{2} - 1440 T^{3} + T^{4} )^{2}$$
$11$ $$49\!\cdots\!96$$$$+$$$$18\!\cdots\!12$$$$T^{2} + 2182692914595328 T^{4} + 83236928 T^{6} + T^{8}$$
$13$ $$11\!\cdots\!00$$$$+$$$$88\!\cdots\!64$$$$T^{2} + 24158081822553600 T^{4} + 269496640 T^{6} + T^{8}$$
$17$ $$( 58515537111378448 - 4730747963616 T - 965892200 T^{2} - 11448 T^{3} + T^{4} )^{2}$$
$19$ $$11\!\cdots\!04$$$$+$$$$30\!\cdots\!16$$$$T^{2} + 259431105120601600 T^{4} + 878694464 T^{6} + T^{8}$$
$23$ $$( 11244826934750416896 + 345149893656576 T - 4195709056 T^{2} - 103680 T^{3} + T^{4} )^{2}$$
$29$ $$58\!\cdots\!64$$$$+$$$$17\!\cdots\!40$$$$T^{2} +$$$$17\!\cdots\!44$$$$T^{4} + 69547739776 T^{6} + T^{8}$$
$31$ $$( -45978504782700033792 - 8873154899928576 T - 78069394336 T^{2} + 8928 T^{3} + T^{4} )^{2}$$
$37$ $$78\!\cdots\!16$$$$+$$$$96\!\cdots\!12$$$$T^{2} +$$$$13\!\cdots\!44$$$$T^{4} + 646806181952 T^{6} + T^{8}$$
$41$ $$( -$$$$32\!\cdots\!92$$$$+ 236193874814141280 T - 413267660072 T^{2} - 343528 T^{3} + T^{4} )^{2}$$
$43$ $$68\!\cdots\!96$$$$+$$$$18\!\cdots\!52$$$$T^{2} +$$$$17\!\cdots\!28$$$$T^{4} + 703896484928 T^{6} + T^{8}$$
$47$ $$( -$$$$24\!\cdots\!40$$$$- 255204023770681344 T - 148171776640 T^{2} + 993600 T^{3} + T^{4} )^{2}$$
$53$ $$55\!\cdots\!16$$$$+$$$$35\!\cdots\!60$$$$T^{2} +$$$$68\!\cdots\!08$$$$T^{4} + 4748734157440 T^{6} + T^{8}$$
$59$ $$19\!\cdots\!96$$$$+$$$$55\!\cdots\!00$$$$T^{2} +$$$$39\!\cdots\!28$$$$T^{4} + 10826005492800 T^{6} + T^{8}$$
$61$ $$29\!\cdots\!44$$$$+$$$$37\!\cdots\!08$$$$T^{2} +$$$$46\!\cdots\!60$$$$T^{4} + 12824477047872 T^{6} + T^{8}$$
$67$ $$16\!\cdots\!56$$$$+$$$$13\!\cdots\!08$$$$T^{2} +$$$$36\!\cdots\!28$$$$T^{4} + 35890312491072 T^{6} + T^{8}$$
$71$ $$($$$$15\!\cdots\!48$$$$- 656207839784779776 T - 5587298078848 T^{2} + 3168000 T^{3} + T^{4} )^{2}$$
$73$ $$( -$$$$79\!\cdots\!08$$$$+ 10171226233350440352 T + 1057825098072 T^{2} - 4460376 T^{3} + T^{4} )^{2}$$
$79$ $$( -$$$$11\!\cdots\!64$$$$+ 84013681756559970816 T - 52116620924320 T^{2} - 625824 T^{3} + T^{4} )^{2}$$
$83$ $$36\!\cdots\!84$$$$+$$$$22\!\cdots\!84$$$$T^{2} +$$$$26\!\cdots\!76$$$$T^{4} + 93693006217280 T^{6} + T^{8}$$
$89$ $$($$$$69\!\cdots\!52$$$$+$$$$25\!\cdots\!72$$$$T - 94251186593960 T^{2} - 2223704 T^{3} + T^{4} )^{2}$$
$97$ $$( -$$$$29\!\cdots\!64$$$$-$$$$15\!\cdots\!64$$$$T - 152508641417960 T^{2} + 7078792 T^{3} + T^{4} )^{2}$$
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