# Properties

 Label 384.8.a.n Level $384$ Weight $8$ Character orbit 384.a Self dual yes Analytic conductor $119.956$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 430 x^{2} - 2448 x + 12138$$ 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.

 $$f(q)$$ $$=$$ $$q -27 q^{3} + ( -48 - \beta_{2} ) q^{5} + ( 170 + \beta_{3} ) q^{7} + 729 q^{9} +O(q^{10})$$ $$q -27 q^{3} + ( -48 - \beta_{2} ) q^{5} + ( 170 + \beta_{3} ) q^{7} + 729 q^{9} + ( 1124 - \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{11} + ( 3210 - 6 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 1296 + 27 \beta_{2} ) q^{15} + ( -3738 + 13 \beta_{1} + 32 \beta_{2} - 30 \beta_{3} ) q^{17} + ( 5376 - 7 \beta_{1} - 56 \beta_{2} - 14 \beta_{3} ) q^{19} + ( -4590 - 27 \beta_{3} ) q^{21} + ( 13748 - 8 \beta_{1} - 180 \beta_{2} + 110 \beta_{3} ) q^{23} + ( 47683 - 56 \beta_{1} + 38 \beta_{2} + 98 \beta_{3} ) q^{25} -19683 q^{27} + ( -60596 + 70 \beta_{1} - 357 \beta_{2} + 90 \beta_{3} ) q^{29} + ( 37858 - 2 \beta_{1} - 276 \beta_{2} - 59 \beta_{3} ) q^{31} + ( -30348 + 27 \beta_{1} + 54 \beta_{2} + 216 \beta_{3} ) q^{33} + ( -68496 + 49 \beta_{1} - 370 \beta_{2} - 252 \beta_{3} ) q^{35} + ( -28322 - 98 \beta_{1} - 317 \beta_{2} + 287 \beta_{3} ) q^{37} + ( -86670 + 162 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{39} + ( -59794 + 99 \beta_{1} + 1372 \beta_{2} - 278 \beta_{3} ) q^{41} + ( -373832 + 191 \beta_{1} - 1148 \beta_{2} + 458 \beta_{3} ) q^{43} + ( -34992 - 729 \beta_{2} ) q^{45} + ( 193092 - 124 \beta_{1} - 1272 \beta_{2} - 1146 \beta_{3} ) q^{47} + ( -394275 - 34 \beta_{1} + 986 \beta_{2} - 102 \beta_{3} ) q^{49} + ( 100926 - 351 \beta_{1} - 864 \beta_{2} + 810 \beta_{3} ) q^{51} + ( -597444 + 138 \beta_{1} + 471 \beta_{2} - 206 \beta_{3} ) q^{53} + ( 647520 - 602 \beta_{1} - 636 \beta_{2} + 2016 \beta_{3} ) q^{55} + ( -145152 + 189 \beta_{1} + 1512 \beta_{2} + 378 \beta_{3} ) q^{57} + ( -35308 - 784 \beta_{1} + 1292 \beta_{2} - 1876 \beta_{3} ) q^{59} + ( -307826 - 454 \beta_{1} - 2337 \beta_{2} - 1833 \beta_{3} ) q^{61} + ( 123930 + 729 \beta_{3} ) q^{63} + ( -260256 - 595 \beta_{1} - 9972 \beta_{2} - 1330 \beta_{3} ) q^{65} + ( 110348 + 616 \beta_{1} + 1452 \beta_{2} + 1996 \beta_{3} ) q^{67} + ( -371196 + 216 \beta_{1} + 4860 \beta_{2} - 2970 \beta_{3} ) q^{69} + ( 376876 + 1268 \beta_{1} - 68 \beta_{2} - 3570 \beta_{3} ) q^{71} + ( -379210 + 2 \beta_{1} + 4382 \beta_{2} - 3554 \beta_{3} ) q^{73} + ( -1287441 + 1512 \beta_{1} - 1026 \beta_{2} - 2646 \beta_{3} ) q^{75} + ( -3085112 - 42 \beta_{1} - 7592 \beta_{2} + 2936 \beta_{3} ) q^{77} + ( 2385234 + 2088 \beta_{1} + 12964 \beta_{2} - 67 \beta_{3} ) q^{79} + 531441 q^{81} + ( 1146900 + 751 \beta_{1} + 11242 \beta_{2} + 2508 \beta_{3} ) q^{83} + ( -1595712 + 1596 \beta_{1} + 24254 \beta_{2} + 6972 \beta_{3} ) q^{85} + ( 1636092 - 1890 \beta_{1} + 9639 \beta_{2} - 2430 \beta_{3} ) q^{87} + ( 40594 - 330 \beta_{1} - 5160 \beta_{2} + 8860 \beta_{3} ) q^{89} + ( 1170276 - 2457 \beta_{1} + 6832 \beta_{2} + 2998 \beta_{3} ) q^{91} + ( -1022166 + 54 \beta_{1} + 7452 \beta_{2} + 1593 \beta_{3} ) q^{93} + ( 7305312 - 4508 \beta_{1} - 10780 \beta_{2} + 7644 \beta_{3} ) q^{95} + ( 681690 + 2730 \beta_{1} - 17900 \beta_{2} + 7600 \beta_{3} ) q^{97} + ( 819396 - 729 \beta_{1} - 1458 \beta_{2} - 5832 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 108q^{3} - 192q^{5} + 680q^{7} + 2916q^{9} + O(q^{10})$$ $$4q - 108q^{3} - 192q^{5} + 680q^{7} + 2916q^{9} + 4496q^{11} + 12840q^{13} + 5184q^{15} - 14952q^{17} + 21504q^{19} - 18360q^{21} + 54992q^{23} + 190732q^{25} - 78732q^{27} - 242384q^{29} + 151432q^{31} - 121392q^{33} - 273984q^{35} - 113288q^{37} - 346680q^{39} - 239176q^{41} - 1495328q^{43} - 139968q^{45} + 772368q^{47} - 1577100q^{49} + 403704q^{51} - 2389776q^{53} + 2590080q^{55} - 580608q^{57} - 141232q^{59} - 1231304q^{61} + 495720q^{63} - 1041024q^{65} + 441392q^{67} - 1484784q^{69} + 1507504q^{71} - 1516840q^{73} - 5149764q^{75} - 12340448q^{77} + 9540936q^{79} + 2125764q^{81} + 4587600q^{83} - 6382848q^{85} + 6544368q^{87} + 162376q^{89} + 4681104q^{91} - 4088664q^{93} + 29221248q^{95} + 2726760q^{97} + 3277584q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 430 x^{2} - 2448 x + 12138$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-96 \nu^{3} + 1360 \nu^{2} + 36928 \nu - 116144$$$$)/119$$ $$\beta_{2}$$ $$=$$ $$($$$$-200 \nu^{3} + 2516 \nu^{2} + 52816 \nu - 173740$$$$)/119$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{3} + 132 \nu^{2} + 1808 \nu - 13692$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{3} - 2 \beta_{2} + 7 \beta_{1}$$$$)/768$$ $$\nu^{2}$$ $$=$$ $$($$$$38 \beta_{3} - 34 \beta_{2} + 17 \beta_{1} + 41280$$$$)/192$$ $$\nu^{3}$$ $$=$$ $$($$$$173 \beta_{3} - 337 \beta_{2} + 338 \beta_{1} + 176256$$$$)/96$$

## 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
 −13.4480 3.20119 22.6791 −12.4323
0 −27.0000 0 −530.466 0 930.332 0 729.000 0
1.2 0 −27.0000 0 −170.323 0 −803.427 0 729.000 0
1.3 0 −27.0000 0 76.3562 0 439.494 0 729.000 0
1.4 0 −27.0000 0 432.433 0 113.601 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.n 4
4.b odd 2 1 384.8.a.r yes 4
8.b even 2 1 384.8.a.s yes 4
8.d odd 2 1 384.8.a.o yes 4

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

 $$T_{5}^{4} + 192 T_{5}^{3} - 233184 T_{5}^{2} - 22830080 T_{5} + 2983276800$$ $$T_{7}^{4} - 680 T_{7}^{3} - 627336 T_{7}^{2} + 407076832 T_{7} - 37318092656$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 27 + T )^{4}$$
$5$ $$2983276800 - 22830080 T - 233184 T^{2} + 192 T^{3} + T^{4}$$
$7$ $$-37318092656 + 407076832 T - 627336 T^{2} - 680 T^{3} + T^{4}$$
$11$ $$552050534693120 + 80873665280 T - 51757344 T^{2} - 4496 T^{3} + T^{4}$$
$13$ $$-2206787363930352 + 1704045801568 T - 118878120 T^{2} - 12840 T^{3} + T^{4}$$
$17$ $$501535254519216912 - 12612841818976 T - 1527439272 T^{2} + 14952 T^{3} + T^{4}$$
$19$ $$111267999147626496 + 4415480774656 T - 1128753024 T^{2} - 21504 T^{3} + T^{4}$$
$23$ $$-10957440785644181248 + 811363281794816 T - 12102667296 T^{2} - 54992 T^{3} + T^{4}$$
$29$ $$-3101636390659660800 - 8298669461030400 T - 34230604416 T^{2} + 242384 T^{3} + T^{4}$$
$31$ $$-76587946662315264624 + 2622223946440032 T - 16869354696 T^{2} - 151432 T^{3} + T^{4}$$
$37$ $$21\!\cdots\!80$$$$- 3518324458893280 T - 113704932456 T^{2} + 113288 T^{3} + T^{4}$$
$41$ $$51\!\cdots\!76$$$$+ 67627712660395296 T - 452081286120 T^{2} + 239176 T^{3} + T^{4}$$
$43$ $$-$$$$38\!\cdots\!68$$$$- 266866222586916864 T + 281905350144 T^{2} + 1495328 T^{3} + T^{4}$$
$47$ $$37\!\cdots\!92$$$$+ 1028659840610650880 T - 1611252798240 T^{2} - 772368 T^{3} + T^{4}$$
$53$ $$81\!\cdots\!76$$$$+ 678316436078621184 T + 1983300053376 T^{2} + 2389776 T^{3} + T^{4}$$
$59$ $$-$$$$12\!\cdots\!84$$$$+ 3421459186395628288 T - 5536960908960 T^{2} + 141232 T^{3} + T^{4}$$
$61$ $$70\!\cdots\!56$$$$- 3283458873252851936 T - 5254448671464 T^{2} + 1231304 T^{3} + T^{4}$$
$67$ $$86\!\cdots\!04$$$$+ 1184110925351609600 T - 5894428427424 T^{2} - 441392 T^{3} + T^{4}$$
$71$ $$52\!\cdots\!72$$$$+ 20540829503292883200 T - 18295169893920 T^{2} - 1507504 T^{3} + T^{4}$$
$73$ $$-$$$$15\!\cdots\!52$$$$- 25962509969892863072 T - 10236033852840 T^{2} + 1516840 T^{3} + T^{4}$$
$79$ $$-$$$$11\!\cdots\!48$$$$+$$$$47\!\cdots\!52$$$$T - 25872325295688 T^{2} - 9540936 T^{3} + T^{4}$$
$83$ $$-$$$$15\!\cdots\!92$$$$+$$$$10\!\cdots\!44$$$$T - 36667301759520 T^{2} - 4587600 T^{3} + T^{4}$$
$89$ $$-$$$$24\!\cdots\!04$$$$+$$$$23\!\cdots\!64$$$$T - 59295822522984 T^{2} - 162376 T^{3} + T^{4}$$
$97$ $$25\!\cdots\!00$$$$+ 26618308655581340000 T - 128502414095400 T^{2} - 2726760 T^{3} + T^{4}$$