# Properties

 Label 384.10.a.e Level $384$ Weight $10$ Character orbit 384.a Self dual yes Analytic conductor $197.774$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$197.773761087$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 14124 x^{2} - 170336 x + 18391464$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{15}\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 81 q^{3} + ( -432 - \beta_{1} ) q^{5} + ( -1210 - \beta_{1} + \beta_{3} ) q^{7} + 6561 q^{9} +O(q^{10})$$ $$q + 81 q^{3} + ( -432 - \beta_{1} ) q^{5} + ( -1210 - \beta_{1} + \beta_{3} ) q^{7} + 6561 q^{9} + ( 3956 + 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -20610 - 38 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} ) q^{13} + ( -34992 - 81 \beta_{1} ) q^{15} + ( 41478 + 134 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 134976 + 262 \beta_{1} - 19 \beta_{2} - 20 \beta_{3} ) q^{19} + ( -98010 - 81 \beta_{1} + 81 \beta_{3} ) q^{21} + ( -182420 - 50 \beta_{1} - 56 \beta_{2} - 130 \beta_{3} ) q^{23} + ( 56203 + 644 \beta_{1} + 40 \beta_{2} - 50 \beta_{3} ) q^{25} + 531441 q^{27} + ( -462716 + 457 \beta_{1} - 94 \beta_{2} + 62 \beta_{3} ) q^{29} + ( -799490 + 55 \beta_{1} - 110 \beta_{2} - 471 \beta_{3} ) q^{31} + ( 320436 + 486 \beta_{1} + 243 \beta_{2} + 162 \beta_{3} ) q^{33} + ( 2143728 + 3018 \beta_{1} - 427 \beta_{2} - 862 \beta_{3} ) q^{35} + ( -1046998 + 5496 \beta_{1} - 134 \beta_{2} + 863 \beta_{3} ) q^{37} + ( -1669410 - 3078 \beta_{1} - 162 \beta_{2} - 567 \beta_{3} ) q^{39} + ( 56926 + 2954 \beta_{1} - 77 \beta_{2} - 2128 \beta_{3} ) q^{41} + ( 400088 - 2070 \beta_{1} + 835 \beta_{2} - 1992 \beta_{3} ) q^{43} + ( -2834352 - 6561 \beta_{1} ) q^{45} + ( -4513476 - 4542 \beta_{1} + 2268 \beta_{2} + 1886 \beta_{3} ) q^{47} + ( 14432205 + 13072 \beta_{1} - 1778 \beta_{2} - 4070 \beta_{3} ) q^{49} + ( 3359718 + 10854 \beta_{1} + 405 \beta_{2} + 324 \beta_{3} ) q^{51} + ( -7354572 + 12469 \beta_{1} + 2398 \beta_{2} + 3502 \beta_{3} ) q^{53} + ( -11476704 - 23972 \beta_{1} - 1990 \beta_{2} - 9940 \beta_{3} ) q^{55} + ( 10933056 + 21222 \beta_{1} - 1539 \beta_{2} - 1620 \beta_{3} ) q^{57} + ( 9575012 + 74736 \beta_{1} + 624 \beta_{2} - 4764 \beta_{3} ) q^{59} + ( 24941162 + 50092 \beta_{1} - 226 \beta_{2} + 7511 \beta_{3} ) q^{61} + ( -7938810 - 6561 \beta_{1} + 6561 \beta_{3} ) q^{63} + ( 78530208 + 32118 \beta_{1} + 5333 \beta_{2} + 9528 \beta_{3} ) q^{65} + ( -45929252 + 30672 \beta_{1} - 1208 \beta_{2} + 14196 \beta_{3} ) q^{67} + ( -14776020 - 4050 \beta_{1} - 4536 \beta_{2} - 10530 \beta_{3} ) q^{69} + ( -45467020 - 33330 \beta_{1} + 2444 \beta_{2} + 1526 \beta_{3} ) q^{71} + ( 63634790 + 84688 \beta_{1} + 258 \beta_{2} + 15230 \beta_{3} ) q^{73} + ( 4552443 + 52164 \beta_{1} + 3240 \beta_{2} - 4050 \beta_{3} ) q^{75} + ( 57641176 + 231688 \beta_{1} + 4018 \beta_{2} - 23116 \beta_{3} ) q^{77} + ( -207894546 - 183265 \beta_{1} - 7272 \beta_{2} + 19493 \beta_{3} ) q^{79} + 43046721 q^{81} + ( -171980988 - 176402 \beta_{1} - 12549 \beta_{2} + 35054 \beta_{3} ) q^{83} + ( -260347776 - 100062 \beta_{1} - 8588 \beta_{2} - 10908 \beta_{3} ) q^{85} + ( -37479996 + 37017 \beta_{1} - 7614 \beta_{2} + 5022 \beta_{3} ) q^{87} + ( 156906818 + 127676 \beta_{1} + 19190 \beta_{2} + 23480 \beta_{3} ) q^{89} + ( -254536908 - 101126 \beta_{1} - 9541 \beta_{2} - 3412 \beta_{3} ) q^{91} + ( -64758690 + 4455 \beta_{1} - 8910 \beta_{2} - 38151 \beta_{3} ) q^{93} + ( -541779552 - 83512 \beta_{1} + 4028 \beta_{2} + 83908 \beta_{3} ) q^{95} + ( -222346470 - 764780 \beta_{1} - 20230 \beta_{2} + 43420 \beta_{3} ) q^{97} + ( 25955316 + 39366 \beta_{1} + 19683 \beta_{2} + 13122 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 324q^{3} - 1728q^{5} - 4840q^{7} + 26244q^{9} + O(q^{10})$$ $$4q + 324q^{3} - 1728q^{5} - 4840q^{7} + 26244q^{9} + 15824q^{11} - 82440q^{13} - 139968q^{15} + 165912q^{17} + 539904q^{19} - 392040q^{21} - 729680q^{23} + 224812q^{25} + 2125764q^{27} - 1850864q^{29} - 3197960q^{31} + 1281744q^{33} + 8574912q^{35} - 4187992q^{37} - 6677640q^{39} + 227704q^{41} + 1600352q^{43} - 11337408q^{45} - 18053904q^{47} + 57728820q^{49} + 13438872q^{51} - 29418288q^{53} - 45906816q^{55} + 43732224q^{57} + 38300048q^{59} + 99764648q^{61} - 31755240q^{63} + 314120832q^{65} - 183717008q^{67} - 59104080q^{69} - 181868080q^{71} + 254539160q^{73} + 18209772q^{75} + 230564704q^{77} - 831578184q^{79} + 172186884q^{81} - 687923952q^{83} - 1041391104q^{85} - 149919984q^{87} + 627627272q^{89} - 1018147632q^{91} - 259034760q^{93} - 2167118208q^{95} - 889385880q^{97} + 103821264q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 14124 x^{2} - 170336 x + 18391464$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-8 \nu^{3} + 206 \nu^{2} + 126232 \nu - 432756$$$$)/2853$$ $$\beta_{2}$$ $$=$$ $$($$$$160 \nu^{3} - 4120 \nu^{2} - 1429088 \nu + 8655120$$$$)/2853$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{2} - 80 \nu - 28248$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 20 \beta_{1}$$$$)/384$$ $$\nu^{2}$$ $$=$$ $$($$$$72 \beta_{3} + 5 \beta_{2} + 100 \beta_{1} + 677952$$$$)/96$$ $$\nu^{3}$$ $$=$$ $$($$$$3708 \beta_{3} + 8147 \beta_{2} + 94468 \beta_{1} + 24528384$$$$)/192$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 119.418 31.4956 −103.810 −47.1038
0 81.0000 0 −1818.41 0 3817.40 0 6561.00 0
1.2 0 81.0000 0 −1657.87 0 −11369.1 0 6561.00 0
1.3 0 81.0000 0 397.737 0 7340.72 0 6561.00 0
1.4 0 81.0000 0 1350.54 0 −4629.00 0 6561.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.10.a.e yes 4
4.b odd 2 1 384.10.a.a 4
8.b even 2 1 384.10.a.d yes 4
8.d odd 2 1 384.10.a.h yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.a.a 4 4.b odd 2 1
384.10.a.d yes 4 8.b even 2 1
384.10.a.e yes 4 1.a even 1 1 trivial
384.10.a.h yes 4 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 $$S_{10}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5}^{4} + 1728 T_{5}^{3} - 2525664 T_{5}^{2} - 3403197440 T_{5} +$$$$16\!\cdots\!00$$ $$T_{7}^{4} + 4840 T_{7}^{3} - 97858824 T_{7}^{2} - 138919065824 T_{7} +$$$$14\!\cdots\!44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -81 + T )^{4}$$
$5$ $$1619372294400 - 3403197440 T - 2525664 T^{2} + 1728 T^{3} + T^{4}$$
$7$ $$1474756197184144 - 138919065824 T - 97858824 T^{2} + 4840 T^{3} + T^{4}$$
$11$ $$6858181745009404160 + 1103792080640 T - 6159760416 T^{2} - 15824 T^{3} + T^{4}$$
$13$ $$-514638491031365616 + 183723133968416 T - 9910153704 T^{2} + 82440 T^{3} + T^{4}$$
$17$ $$74\!\cdots\!48$$$$+ 1451814354814624 T - 71608865448 T^{2} - 165912 T^{3} + T^{4}$$
$19$ $$-$$$$25\!\cdots\!96$$$$+ 111265942104473600 T - 412602051456 T^{2} - 539904 T^{3} + T^{4}$$
$23$ $$21\!\cdots\!44$$$$- 851951504698601216 T - 3334463529504 T^{2} + 729680 T^{3} + T^{4}$$
$29$ $$47\!\cdots\!88$$$$- 3667397931699343872 T - 6132162551424 T^{2} + 1850864 T^{3} + T^{4}$$
$31$ $$-$$$$20\!\cdots\!92$$$$- 63299272042313217888 T - 24992419229640 T^{2} + 3197960 T^{3} + T^{4}$$
$37$ $$25\!\cdots\!40$$$$+$$$$64\!\cdots\!20$$$$T - 203243447922216 T^{2} + 4187992 T^{3} + T^{4}$$
$41$ $$19\!\cdots\!64$$$$-$$$$22\!\cdots\!48$$$$T - 494045022312552 T^{2} - 227704 T^{3} + T^{4}$$
$43$ $$54\!\cdots\!56$$$$+$$$$49\!\cdots\!48$$$$T - 970945230014976 T^{2} - 1600352 T^{3} + T^{4}$$
$47$ $$30\!\cdots\!56$$$$-$$$$29\!\cdots\!24$$$$T - 3577671621321504 T^{2} + 18053904 T^{3} + T^{4}$$
$53$ $$-$$$$10\!\cdots\!00$$$$-$$$$21\!\cdots\!60$$$$T - 4952579056155264 T^{2} + 29418288 T^{3} + T^{4}$$
$59$ $$11\!\cdots\!04$$$$+$$$$35\!\cdots\!56$$$$T - 22171735492406688 T^{2} - 38300048 T^{3} + T^{4}$$
$61$ $$-$$$$15\!\cdots\!76$$$$+$$$$98\!\cdots\!96$$$$T - 11706470425376808 T^{2} - 99764648 T^{3} + T^{4}$$
$67$ $$10\!\cdots\!56$$$$-$$$$17\!\cdots\!92$$$$T - 13843775367055776 T^{2} + 183717008 T^{3} + T^{4}$$
$71$ $$-$$$$15\!\cdots\!64$$$$-$$$$44\!\cdots\!60$$$$T + 4165003675633632 T^{2} + 181868080 T^{3} + T^{4}$$
$73$ $$-$$$$13\!\cdots\!92$$$$+$$$$57\!\cdots\!76$$$$T - 26793790810227240 T^{2} - 254539160 T^{3} + T^{4}$$
$79$ $$-$$$$70\!\cdots\!24$$$$-$$$$65\!\cdots\!24$$$$T + 61673089702752696 T^{2} + 831578184 T^{3} + T^{4}$$
$83$ $$-$$$$31\!\cdots\!56$$$$-$$$$19\!\cdots\!64$$$$T - 178092834287762208 T^{2} + 687923952 T^{3} + T^{4}$$
$89$ $$34\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T - 191915279205961704 T^{2} - 627627272 T^{3} + T^{4}$$
$97$ $$12\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T - 2283188040832386600 T^{2} + 889385880 T^{3} + T^{4}$$