# Properties

 Label 384.10.a.d 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: $$x^{4} - 14124x^{2} - 170336x + 18391464$$ 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.

 $$f(q)$$ $$=$$ $$q - 81 q^{3} + (\beta_1 + 432) q^{5} + (\beta_{3} - \beta_1 - 1210) q^{7} + 6561 q^{9}+O(q^{10})$$ q - 81 * q^3 + (b1 + 432) * q^5 + (b3 - b1 - 1210) * q^7 + 6561 * q^9 $$q - 81 q^{3} + (\beta_1 + 432) q^{5} + (\beta_{3} - \beta_1 - 1210) q^{7} + 6561 q^{9} + ( - 2 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 3956) q^{11} + (7 \beta_{3} + 2 \beta_{2} + 38 \beta_1 + 20610) q^{13} + ( - 81 \beta_1 - 34992) q^{15} + (4 \beta_{3} + 5 \beta_{2} + 134 \beta_1 + 41478) q^{17} + (20 \beta_{3} + 19 \beta_{2} - 262 \beta_1 - 134976) q^{19} + ( - 81 \beta_{3} + 81 \beta_1 + 98010) q^{21} + ( - 130 \beta_{3} - 56 \beta_{2} - 50 \beta_1 - 182420) q^{23} + ( - 50 \beta_{3} + 40 \beta_{2} + 644 \beta_1 + 56203) q^{25} - 531441 q^{27} + ( - 62 \beta_{3} + 94 \beta_{2} - 457 \beta_1 + 462716) q^{29} + ( - 471 \beta_{3} - 110 \beta_{2} + 55 \beta_1 - 799490) q^{31} + (162 \beta_{3} + 243 \beta_{2} + 486 \beta_1 + 320436) q^{33} + (862 \beta_{3} + 427 \beta_{2} - 3018 \beta_1 - 2143728) q^{35} + ( - 863 \beta_{3} + 134 \beta_{2} - 5496 \beta_1 + 1046998) q^{37} + ( - 567 \beta_{3} - 162 \beta_{2} - 3078 \beta_1 - 1669410) q^{39} + ( - 2128 \beta_{3} - 77 \beta_{2} + 2954 \beta_1 + 56926) q^{41} + (1992 \beta_{3} - 835 \beta_{2} + 2070 \beta_1 - 400088) q^{43} + (6561 \beta_1 + 2834352) q^{45} + (1886 \beta_{3} + 2268 \beta_{2} - 4542 \beta_1 - 4513476) q^{47} + ( - 4070 \beta_{3} - 1778 \beta_{2} + 13072 \beta_1 + 14432205) q^{49} + ( - 324 \beta_{3} - 405 \beta_{2} - 10854 \beta_1 - 3359718) q^{51} + ( - 3502 \beta_{3} - 2398 \beta_{2} - 12469 \beta_1 + 7354572) q^{53} + ( - 9940 \beta_{3} - 1990 \beta_{2} - 23972 \beta_1 - 11476704) q^{55} + ( - 1620 \beta_{3} - 1539 \beta_{2} + 21222 \beta_1 + 10933056) q^{57} + (4764 \beta_{3} - 624 \beta_{2} - 74736 \beta_1 - 9575012) q^{59} + ( - 7511 \beta_{3} + 226 \beta_{2} - 50092 \beta_1 - 24941162) q^{61} + (6561 \beta_{3} - 6561 \beta_1 - 7938810) q^{63} + (9528 \beta_{3} + 5333 \beta_{2} + 32118 \beta_1 + 78530208) q^{65} + ( - 14196 \beta_{3} + 1208 \beta_{2} - 30672 \beta_1 + 45929252) q^{67} + (10530 \beta_{3} + 4536 \beta_{2} + 4050 \beta_1 + 14776020) q^{69} + (1526 \beta_{3} + 2444 \beta_{2} - 33330 \beta_1 - 45467020) q^{71} + (15230 \beta_{3} + 258 \beta_{2} + 84688 \beta_1 + 63634790) q^{73} + (4050 \beta_{3} - 3240 \beta_{2} - 52164 \beta_1 - 4552443) q^{75} + (23116 \beta_{3} - 4018 \beta_{2} - 231688 \beta_1 - 57641176) q^{77} + (19493 \beta_{3} - 7272 \beta_{2} - 183265 \beta_1 - 207894546) q^{79} + 43046721 q^{81} + ( - 35054 \beta_{3} + 12549 \beta_{2} + 176402 \beta_1 + 171980988) q^{83} + (10908 \beta_{3} + 8588 \beta_{2} + 100062 \beta_1 + 260347776) q^{85} + (5022 \beta_{3} - 7614 \beta_{2} + 37017 \beta_1 - 37479996) q^{87} + (23480 \beta_{3} + 19190 \beta_{2} + 127676 \beta_1 + 156906818) q^{89} + (3412 \beta_{3} + 9541 \beta_{2} + 101126 \beta_1 + 254536908) q^{91} + (38151 \beta_{3} + 8910 \beta_{2} - 4455 \beta_1 + 64758690) q^{93} + (83908 \beta_{3} + 4028 \beta_{2} - 83512 \beta_1 - 541779552) q^{95} + (43420 \beta_{3} - 20230 \beta_{2} - 764780 \beta_1 - 222346470) q^{97} + ( - 13122 \beta_{3} - 19683 \beta_{2} - 39366 \beta_1 - 25955316) q^{99}+O(q^{100})$$ q - 81 * q^3 + (b1 + 432) * q^5 + (b3 - b1 - 1210) * q^7 + 6561 * q^9 + (-2*b3 - 3*b2 - 6*b1 - 3956) * q^11 + (7*b3 + 2*b2 + 38*b1 + 20610) * q^13 + (-81*b1 - 34992) * q^15 + (4*b3 + 5*b2 + 134*b1 + 41478) * q^17 + (20*b3 + 19*b2 - 262*b1 - 134976) * q^19 + (-81*b3 + 81*b1 + 98010) * q^21 + (-130*b3 - 56*b2 - 50*b1 - 182420) * q^23 + (-50*b3 + 40*b2 + 644*b1 + 56203) * q^25 - 531441 * q^27 + (-62*b3 + 94*b2 - 457*b1 + 462716) * q^29 + (-471*b3 - 110*b2 + 55*b1 - 799490) * q^31 + (162*b3 + 243*b2 + 486*b1 + 320436) * q^33 + (862*b3 + 427*b2 - 3018*b1 - 2143728) * q^35 + (-863*b3 + 134*b2 - 5496*b1 + 1046998) * q^37 + (-567*b3 - 162*b2 - 3078*b1 - 1669410) * q^39 + (-2128*b3 - 77*b2 + 2954*b1 + 56926) * q^41 + (1992*b3 - 835*b2 + 2070*b1 - 400088) * q^43 + (6561*b1 + 2834352) * q^45 + (1886*b3 + 2268*b2 - 4542*b1 - 4513476) * q^47 + (-4070*b3 - 1778*b2 + 13072*b1 + 14432205) * q^49 + (-324*b3 - 405*b2 - 10854*b1 - 3359718) * q^51 + (-3502*b3 - 2398*b2 - 12469*b1 + 7354572) * q^53 + (-9940*b3 - 1990*b2 - 23972*b1 - 11476704) * q^55 + (-1620*b3 - 1539*b2 + 21222*b1 + 10933056) * q^57 + (4764*b3 - 624*b2 - 74736*b1 - 9575012) * q^59 + (-7511*b3 + 226*b2 - 50092*b1 - 24941162) * q^61 + (6561*b3 - 6561*b1 - 7938810) * q^63 + (9528*b3 + 5333*b2 + 32118*b1 + 78530208) * q^65 + (-14196*b3 + 1208*b2 - 30672*b1 + 45929252) * q^67 + (10530*b3 + 4536*b2 + 4050*b1 + 14776020) * q^69 + (1526*b3 + 2444*b2 - 33330*b1 - 45467020) * q^71 + (15230*b3 + 258*b2 + 84688*b1 + 63634790) * q^73 + (4050*b3 - 3240*b2 - 52164*b1 - 4552443) * q^75 + (23116*b3 - 4018*b2 - 231688*b1 - 57641176) * q^77 + (19493*b3 - 7272*b2 - 183265*b1 - 207894546) * q^79 + 43046721 * q^81 + (-35054*b3 + 12549*b2 + 176402*b1 + 171980988) * q^83 + (10908*b3 + 8588*b2 + 100062*b1 + 260347776) * q^85 + (5022*b3 - 7614*b2 + 37017*b1 - 37479996) * q^87 + (23480*b3 + 19190*b2 + 127676*b1 + 156906818) * q^89 + (3412*b3 + 9541*b2 + 101126*b1 + 254536908) * q^91 + (38151*b3 + 8910*b2 - 4455*b1 + 64758690) * q^93 + (83908*b3 + 4028*b2 - 83512*b1 - 541779552) * q^95 + (43420*b3 - 20230*b2 - 764780*b1 - 222346470) * q^97 + (-13122*b3 - 19683*b2 - 39366*b1 - 25955316) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10})$$ 4 * q - 324 * q^3 + 1728 * q^5 - 4840 * q^7 + 26244 * q^9 $$4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9} - 15824 q^{11} + 82440 q^{13} - 139968 q^{15} + 165912 q^{17} - 539904 q^{19} + 392040 q^{21} - 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} + 1850864 q^{29} - 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} + 4187992 q^{37} - 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} + 11337408 q^{45} - 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} + 29418288 q^{53} - 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} - 99764648 q^{61} - 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} + 59104080 q^{69} - 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} - 230564704 q^{77} - 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} + 1041391104 q^{85} - 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} + 259034760 q^{93} - 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100})$$ 4 * q - 324 * q^3 + 1728 * q^5 - 4840 * q^7 + 26244 * q^9 - 15824 * q^11 + 82440 * q^13 - 139968 * q^15 + 165912 * q^17 - 539904 * q^19 + 392040 * q^21 - 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 + 1850864 * q^29 - 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 + 4187992 * q^37 - 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 + 11337408 * q^45 - 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 + 29418288 * q^53 - 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 - 99764648 * q^61 - 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 + 59104080 * q^69 - 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 - 230564704 * q^77 - 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 + 1041391104 * q^85 - 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 + 259034760 * q^93 - 2167118208 * q^95 - 889385880 * q^97 - 103821264 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -8\nu^{3} + 206\nu^{2} + 126232\nu - 432756 ) / 2853$$ (-8*v^3 + 206*v^2 + 126232*v - 432756) / 2853 $$\beta_{2}$$ $$=$$ $$( 160\nu^{3} - 4120\nu^{2} - 1429088\nu + 8655120 ) / 2853$$ (160*v^3 - 4120*v^2 - 1429088*v + 8655120) / 2853 $$\beta_{3}$$ $$=$$ $$( 4\nu^{2} - 80\nu - 28248 ) / 3$$ (4*v^2 - 80*v - 28248) / 3
 $$\nu$$ $$=$$ $$( \beta_{2} + 20\beta_1 ) / 384$$ (b2 + 20*b1) / 384 $$\nu^{2}$$ $$=$$ $$( 72\beta_{3} + 5\beta_{2} + 100\beta _1 + 677952 ) / 96$$ (72*b3 + 5*b2 + 100*b1 + 677952) / 96 $$\nu^{3}$$ $$=$$ $$( 3708\beta_{3} + 8147\beta_{2} + 94468\beta _1 + 24528384 ) / 192$$ (3708*b3 + 8147*b2 + 94468*b1 + 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
 −47.1038 −103.810 31.4956 119.418
0 −81.0000 0 −1350.54 0 −4629.00 0 6561.00 0
1.2 0 −81.0000 0 −397.737 0 7340.72 0 6561.00 0
1.3 0 −81.0000 0 1657.87 0 −11369.1 0 6561.00 0
1.4 0 −81.0000 0 1818.41 0 3817.40 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.d yes 4
4.b odd 2 1 384.10.a.h yes 4
8.b even 2 1 384.10.a.e yes 4
8.d odd 2 1 384.10.a.a 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5}^{4} - 1728T_{5}^{3} - 2525664T_{5}^{2} + 3403197440T_{5} + 1619372294400$$ T5^4 - 1728*T5^3 - 2525664*T5^2 + 3403197440*T5 + 1619372294400 $$T_{7}^{4} + 4840T_{7}^{3} - 97858824T_{7}^{2} - 138919065824T_{7} + 1474756197184144$$ T7^4 + 4840*T7^3 - 97858824*T7^2 - 138919065824*T7 + 1474756197184144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T + 81)^{4}$$
$5$ $$T^{4} - 1728 T^{3} + \cdots + 1619372294400$$
$7$ $$T^{4} + 4840 T^{3} + \cdots + 14\!\cdots\!44$$
$11$ $$T^{4} + 15824 T^{3} + \cdots + 68\!\cdots\!60$$
$13$ $$T^{4} - 82440 T^{3} + \cdots - 51\!\cdots\!16$$
$17$ $$T^{4} - 165912 T^{3} + \cdots + 74\!\cdots\!48$$
$19$ $$T^{4} + 539904 T^{3} + \cdots - 25\!\cdots\!96$$
$23$ $$T^{4} + 729680 T^{3} + \cdots + 21\!\cdots\!44$$
$29$ $$T^{4} - 1850864 T^{3} + \cdots + 47\!\cdots\!88$$
$31$ $$T^{4} + 3197960 T^{3} + \cdots - 20\!\cdots\!92$$
$37$ $$T^{4} - 4187992 T^{3} + \cdots + 25\!\cdots\!40$$
$41$ $$T^{4} - 227704 T^{3} + \cdots + 19\!\cdots\!64$$
$43$ $$T^{4} + 1600352 T^{3} + \cdots + 54\!\cdots\!56$$
$47$ $$T^{4} + 18053904 T^{3} + \cdots + 30\!\cdots\!56$$
$53$ $$T^{4} - 29418288 T^{3} + \cdots - 10\!\cdots\!00$$
$59$ $$T^{4} + 38300048 T^{3} + \cdots + 11\!\cdots\!04$$
$61$ $$T^{4} + 99764648 T^{3} + \cdots - 15\!\cdots\!76$$
$67$ $$T^{4} - 183717008 T^{3} + \cdots + 10\!\cdots\!56$$
$71$ $$T^{4} + 181868080 T^{3} + \cdots - 15\!\cdots\!64$$
$73$ $$T^{4} - 254539160 T^{3} + \cdots - 13\!\cdots\!92$$
$79$ $$T^{4} + 831578184 T^{3} + \cdots - 70\!\cdots\!24$$
$83$ $$T^{4} - 687923952 T^{3} + \cdots - 31\!\cdots\!56$$
$89$ $$T^{4} - 627627272 T^{3} + \cdots + 34\!\cdots\!00$$
$97$ $$T^{4} + 889385880 T^{3} + \cdots + 12\!\cdots\!00$$