# Properties

 Label 384.8.a.m Level $384$ Weight $8$ Character orbit 384.a Self dual yes Analytic conductor $119.956$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$119.955849786$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 620 x^{2} - 700 x + 83625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{15}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -27 q^{3} + ( -84 + \beta_{2} ) q^{5} + ( -170 - \beta_{3} ) q^{7} + 729 q^{9} +O(q^{10})$$ $$q -27 q^{3} + ( -84 + \beta_{2} ) q^{5} + ( -170 - \beta_{3} ) q^{7} + 729 q^{9} + ( -964 - \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{11} + ( -2670 + 2 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{13} + ( 2268 - 27 \beta_{2} ) q^{15} + ( 6558 + \beta_{1} - 20 \beta_{2} + 7 \beta_{3} ) q^{17} + ( 3864 - 3 \beta_{1} - 44 \beta_{2} + 19 \beta_{3} ) q^{19} + ( 4590 + 27 \beta_{3} ) q^{21} + ( 2828 - 76 \beta_{2} - 14 \beta_{3} ) q^{23} + ( 39763 - 4 \beta_{1} - 166 \beta_{2} + 58 \beta_{3} ) q^{25} -19683 q^{27} + ( 464 + 6 \beta_{1} + 61 \beta_{2} + 76 \beta_{3} ) q^{29} + ( -17938 + 22 \beta_{1} - 84 \beta_{2} - 111 \beta_{3} ) q^{31} + ( 26028 + 27 \beta_{1} - 162 \beta_{2} + 27 \beta_{3} ) q^{33} + ( 44760 + 57 \beta_{1} - 818 \beta_{2} + 221 \beta_{3} ) q^{35} + ( 45022 - 66 \beta_{1} + 885 \beta_{2} - 65 \beta_{3} ) q^{37} + ( 72090 - 54 \beta_{1} - 243 \beta_{2} - 27 \beta_{3} ) q^{39} + ( 2806 - 41 \beta_{1} - 8 \beta_{2} - 91 \beta_{3} ) q^{41} + ( -16672 - 61 \beta_{1} - 544 \beta_{2} - 111 \beta_{3} ) q^{43} + ( -61236 + 729 \beta_{2} ) q^{45} + ( 333612 - 4 \beta_{1} - 1432 \beta_{2} - 170 \beta_{3} ) q^{47} + ( 600285 - 78 \beta_{1} - 2226 \beta_{2} + 232 \beta_{3} ) q^{49} + ( -177066 - 27 \beta_{1} + 540 \beta_{2} - 189 \beta_{3} ) q^{51} + ( 216144 + 218 \beta_{1} + 1297 \beta_{2} + 8 \beta_{3} ) q^{53} + ( 826224 - 18 \beta_{1} - 1484 \beta_{2} + 1486 \beta_{3} ) q^{55} + ( -104328 + 81 \beta_{1} + 1188 \beta_{2} - 513 \beta_{3} ) q^{57} + ( -469612 + 56 \beta_{1} - 2156 \beta_{2} - 1428 \beta_{3} ) q^{59} + ( 475294 - 54 \beta_{1} - 2807 \beta_{2} - 1133 \beta_{3} ) q^{61} + ( -123930 - 729 \beta_{3} ) q^{63} + ( 1091736 + 9 \beta_{1} - 4000 \beta_{2} - 1533 \beta_{3} ) q^{65} + ( -1376372 - 336 \beta_{1} + 1876 \beta_{2} + 1476 \beta_{3} ) q^{67} + ( -76356 + 2052 \beta_{2} + 378 \beta_{3} ) q^{69} + ( 241924 + 516 \beta_{1} + 6740 \beta_{2} + 502 \beta_{3} ) q^{71} + ( 811190 + 446 \beta_{1} + 11002 \beta_{2} + 656 \beta_{3} ) q^{73} + ( -1073601 + 108 \beta_{1} + 4482 \beta_{2} - 1566 \beta_{3} ) q^{75} + ( 2244872 - 106 \beta_{1} - 18432 \beta_{2} + 766 \beta_{3} ) q^{77} + ( -1617954 - 168 \beta_{1} + 5756 \beta_{2} + 1051 \beta_{3} ) q^{79} + 531441 q^{81} + ( -4254900 + 439 \beta_{1} + 2906 \beta_{2} - 541 \beta_{3} ) q^{83} + ( -3030648 - 268 \beta_{1} + 12114 \beta_{2} - 3624 \beta_{3} ) q^{85} + ( -12528 - 162 \beta_{1} - 1647 \beta_{2} - 2052 \beta_{3} ) q^{87} + ( 3389954 - 610 \beta_{1} + 5680 \beta_{2} + 2450 \beta_{3} ) q^{89} + ( -1673076 + 1331 \beta_{1} + 17460 \beta_{2} + 6005 \beta_{3} ) q^{91} + ( 484326 - 594 \beta_{1} + 2268 \beta_{2} + 2997 \beta_{3} ) q^{93} + ( -5630976 - 1060 \beta_{1} + 21644 \beta_{2} - 4000 \beta_{3} ) q^{95} + ( 545130 - 550 \beta_{1} + 13700 \beta_{2} + 6970 \beta_{3} ) q^{97} + ( -702756 - 729 \beta_{1} + 4374 \beta_{2} - 729 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 108q^{3} - 336q^{5} - 680q^{7} + 2916q^{9} + O(q^{10})$$ $$4q - 108q^{3} - 336q^{5} - 680q^{7} + 2916q^{9} - 3856q^{11} - 10680q^{13} + 9072q^{15} + 26232q^{17} + 15456q^{19} + 18360q^{21} + 11312q^{23} + 159052q^{25} - 78732q^{27} + 1856q^{29} - 71752q^{31} + 104112q^{33} + 179040q^{35} + 180088q^{37} + 288360q^{39} + 11224q^{41} - 66688q^{43} - 244944q^{45} + 1334448q^{47} + 2401140q^{49} - 708264q^{51} + 864576q^{53} + 3304896q^{55} - 417312q^{57} - 1878448q^{59} + 1901176q^{61} - 495720q^{63} + 4366944q^{65} - 5505488q^{67} - 305424q^{69} + 967696q^{71} + 3244760q^{73} - 4294404q^{75} + 8979488q^{77} - 6471816q^{79} + 2125764q^{81} - 17019600q^{83} - 12122592q^{85} - 50112q^{87} + 13559816q^{89} - 6692304q^{91} + 1937304q^{93} - 22523904q^{95} + 2180520q^{97} - 2811024q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 620 x^{2} - 700 x + 83625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$16 \nu^{3} - 100 \nu^{2} + 680 \nu + 22600$$$$)/25$$ $$\beta_{2}$$ $$=$$ $$($$$$16 \nu^{3} - 220 \nu^{2} - 5320 \nu + 59800$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$48 \nu^{3} - 1020 \nu^{2} - 14760 \nu + 291000$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - 6 \beta_{2} + 3 \beta_{1}$$$$)/768$$ $$\nu^{2}$$ $$=$$ $$($$$$-25 \beta_{3} + 70 \beta_{2} + 5 \beta_{1} + 119040$$$$)/384$$ $$\nu^{3}$$ $$=$$ $$($$$$-355 \beta_{3} + 1130 \beta_{2} + 1135 \beta_{1} + 403200$$$$)/768$$

## 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
 12.7912 −17.7724 21.8313 −16.8500
0 −27.0000 0 −514.363 0 −1600.84 0 729.000 0
1.2 0 −27.0000 0 −282.264 0 1362.23 0 729.000 0
1.3 0 −27.0000 0 127.307 0 547.276 0 729.000 0
1.4 0 −27.0000 0 333.320 0 −988.659 0 729.000 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 384.8.a.m 4
4.b odd 2 1 384.8.a.q yes 4
8.b even 2 1 384.8.a.t yes 4
8.d odd 2 1 384.8.a.p yes 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5}^{4} + 336 T_{5}^{3} - 179328 T_{5}^{2} - 33072640 T_{5} + 6160819200$$ $$T_{7}^{4} + 680 T_{7}^{3} - 2616456 T_{7}^{2} - 1091635168 T_{7} +$$$$11\!\cdots\!44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 27 + T )^{4}$$
$5$ $$6160819200 - 33072640 T - 179328 T^{2} + 336 T^{3} + T^{4}$$
$7$ $$1179914752144 - 1091635168 T - 2616456 T^{2} + 680 T^{3} + T^{4}$$
$11$ $$616855133477120 - 92451285760 T - 47418144 T^{2} + 3856 T^{3} + T^{4}$$
$13$ $$-1135121051964912 - 1092466557728 T - 130456680 T^{2} + 10680 T^{3} + T^{4}$$
$17$ $$-85261472264688 + 456617575456 T - 40749672 T^{2} - 26232 T^{3} + T^{4}$$
$19$ $$66184194227060736 + 15442631512064 T - 1652181504 T^{2} - 15456 T^{3} + T^{4}$$
$23$ $$78506732047722752 + 19501261404416 T - 1649434656 T^{2} - 11312 T^{3} + T^{4}$$
$29$ $$3510543429849657600 + 892236262302720 T - 18595712736 T^{2} - 1856 T^{3} + T^{4}$$
$31$ $$28\!\cdots\!16$$$$- 1414228573341792 T - 46853382216 T^{2} + 71752 T^{3} + T^{4}$$
$37$ $$11\!\cdots\!80$$$$+ 45208903974156320 T - 367286474856 T^{2} - 180088 T^{3} + T^{4}$$
$41$ $$99\!\cdots\!16$$$$- 5527704361317984 T - 94782752040 T^{2} - 11224 T^{3} + T^{4}$$
$43$ $$-$$$$14\!\cdots\!28$$$$- 38694299846713344 T - 240423117696 T^{2} + 66688 T^{3} + T^{4}$$
$47$ $$-$$$$31\!\cdots\!48$$$$+ 242095834182836480 T + 161444890080 T^{2} - 1334448 T^{3} + T^{4}$$
$53$ $$93\!\cdots\!16$$$$+ 75129531842036736 T - 1858173310944 T^{2} - 864576 T^{3} + T^{4}$$
$59$ $$34\!\cdots\!96$$$$- 5922536324035566848 T - 5003969676960 T^{2} + 1878448 T^{3} + T^{4}$$
$61$ $$64\!\cdots\!76$$$$+ 1369768940648072224 T - 3789284339304 T^{2} - 1901176 T^{3} + T^{4}$$
$67$ $$-$$$$14\!\cdots\!56$$$$- 21045834946226912000 T + 1396391412576 T^{2} + 5505488 T^{3} + T^{4}$$
$71$ $$27\!\cdots\!92$$$$+ 11621312350171994880 T - 20031637873440 T^{2} - 967696 T^{3} + T^{4}$$
$73$ $$21\!\cdots\!28$$$$+ 60876432342660625312 T - 30431014778280 T^{2} - 3244760 T^{3} + T^{4}$$
$79$ $$-$$$$53\!\cdots\!88$$$$- 33551667235350446432 T + 5100752087352 T^{2} + 6471816 T^{3} + T^{4}$$
$83$ $$84\!\cdots\!48$$$$+$$$$21\!\cdots\!64$$$$T + 99098563131360 T^{2} + 17019600 T^{3} + T^{4}$$
$89$ $$-$$$$39\!\cdots\!44$$$$+ 8127212659654219744 T + 34822069903896 T^{2} - 13559816 T^{3} + T^{4}$$
$97$ $$25\!\cdots\!00$$$$-$$$$14\!\cdots\!00$$$$T - 169078534713000 T^{2} - 2180520 T^{3} + T^{4}$$