# Properties

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

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## 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: $$x^{4} - 2070 x^{2} - 13768 x + 561570$$ 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.

 $$f(q)$$ $$=$$ $$q + 81 q^{3} + ( 60 + \beta_{2} ) q^{5} + ( -1210 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{7} + 6561 q^{9} +O(q^{10})$$ $$q + 81 q^{3} + ( 60 + \beta_{2} ) q^{5} + ( -1210 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{7} + 6561 q^{9} + ( -24916 - 5 \beta_{1} + 13 \beta_{2} - 4 \beta_{3} ) q^{11} + ( 15210 + 5 \beta_{1} - 23 \beta_{2} + 19 \beta_{3} ) q^{13} + ( 4860 + 81 \beta_{2} ) q^{15} + ( -108738 - 41 \beta_{1} - 83 \beta_{2} + 10 \beta_{3} ) q^{17} + ( 157944 + 55 \beta_{1} + 29 \beta_{2} - 30 \beta_{3} ) q^{19} + ( -98010 - 81 \beta_{1} - 162 \beta_{2} - 81 \beta_{3} ) q^{21} + ( -187348 + 146 \beta_{1} - 216 \beta_{2} - 174 \beta_{3} ) q^{23} + ( 1497883 + 698 \beta_{1} - 150 \beta_{2} + 494 \beta_{3} ) q^{25} + 531441 q^{27} + ( 1977136 + 556 \beta_{1} - 41 \beta_{2} - 202 \beta_{3} ) q^{29} + ( -2837810 - 663 \beta_{1} + 1072 \beta_{2} + 171 \beta_{3} ) q^{31} + ( -2018196 - 405 \beta_{1} + 1053 \beta_{2} - 324 \beta_{3} ) q^{33} + ( -6391752 - 1607 \beta_{1} - 6049 \beta_{2} - 1096 \beta_{3} ) q^{35} + ( 3398230 + 1571 \beta_{1} - 1123 \beta_{2} + 2949 \beta_{3} ) q^{37} + ( 1232010 + 405 \beta_{1} - 1863 \beta_{2} + 1539 \beta_{3} ) q^{39} + ( -4709722 - 1171 \beta_{1} + 9263 \beta_{2} + 1018 \beta_{3} ) q^{41} + ( 3544480 + 1509 \beta_{1} - 8681 \beta_{2} - 30 \beta_{3} ) q^{43} + ( 393660 + 6561 \beta_{2} ) q^{45} + ( 9444780 + 6662 \beta_{1} + 2440 \beta_{2} + 3450 \beta_{3} ) q^{47} + ( 2352333 + 6296 \beta_{1} + 24828 \beta_{2} + 1774 \beta_{3} ) q^{49} + ( -8807778 - 3321 \beta_{1} - 6723 \beta_{2} + 810 \beta_{3} ) q^{51} + ( -28834128 - 1480 \beta_{1} - 31753 \beta_{2} + 2334 \beta_{3} ) q^{53} + ( 46145136 + 7966 \beta_{1} - 52550 \beta_{2} + 6008 \beta_{3} ) q^{55} + ( 12793464 + 4455 \beta_{1} + 2349 \beta_{2} - 2430 \beta_{3} ) q^{57} + ( -28757020 - 4356 \beta_{1} - 19436 \beta_{2} + 1924 \beta_{3} ) q^{59} + ( -43307162 - 5041 \beta_{1} - 38587 \beta_{2} - 4043 \beta_{3} ) q^{61} + ( -7938810 - 6561 \beta_{1} - 13122 \beta_{2} - 6561 \beta_{3} ) q^{63} + ( -82019496 - 15741 \beta_{1} + 65809 \beta_{2} - 9058 \beta_{3} ) q^{65} + ( -57946276 - 17388 \beta_{1} - 34324 \beta_{2} - 42716 \beta_{3} ) q^{67} + ( -15175188 + 11826 \beta_{1} - 17496 \beta_{2} - 14094 \beta_{3} ) q^{69} + ( -49369052 - 51562 \beta_{1} - 20804 \beta_{2} - 39230 \beta_{3} ) q^{71} + ( 11157350 - 8672 \beta_{1} - 38004 \beta_{2} + 33290 \beta_{3} ) q^{73} + ( 121328523 + 56538 \beta_{1} - 12150 \beta_{2} + 40014 \beta_{3} ) q^{75} + ( 77029480 + 14662 \beta_{1} + 22982 \beta_{2} + 8096 \beta_{3} ) q^{77} + ( 88943646 + 7939 \beta_{1} - 98534 \beta_{2} + 25099 \beta_{3} ) q^{79} + 43046721 q^{81} + ( -151903332 + 20983 \beta_{1} - 79319 \beta_{2} + 60312 \beta_{3} ) q^{83} + ( -271837848 - 69288 \beta_{1} - 235986 \beta_{2} - 39004 \beta_{3} ) q^{85} + ( 160148016 + 45036 \beta_{1} - 3321 \beta_{2} - 16362 \beta_{3} ) q^{87} + ( -289286606 - 6142 \beta_{1} - 68962 \beta_{2} - 114996 \beta_{3} ) q^{89} + ( -211907460 + 8969 \beta_{1} + 17875 \beta_{2} + 74790 \beta_{3} ) q^{91} + ( -229862610 - 53703 \beta_{1} + 86832 \beta_{2} + 13851 \beta_{3} ) q^{93} + ( 82424832 + 36352 \beta_{1} + 322864 \beta_{2} + 9556 \beta_{3} ) q^{95} + ( -399884118 + 49562 \beta_{1} + 41750 \beta_{2} + 36136 \beta_{3} ) q^{97} + ( -163473876 - 32805 \beta_{1} + 85293 \beta_{2} - 26244 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 324q^{3} + 240q^{5} - 4840q^{7} + 26244q^{9} + O(q^{10})$$ $$4q + 324q^{3} + 240q^{5} - 4840q^{7} + 26244q^{9} - 99664q^{11} + 60840q^{13} + 19440q^{15} - 434952q^{17} + 631776q^{19} - 392040q^{21} - 749392q^{23} + 5991532q^{25} + 2125764q^{27} + 7908544q^{29} - 11351240q^{31} - 8072784q^{33} - 25567008q^{35} + 13592920q^{37} + 4928040q^{39} - 18838888q^{41} + 14177920q^{43} + 1574640q^{45} + 37779120q^{47} + 9409332q^{49} - 35231112q^{51} - 115336512q^{53} + 184580544q^{55} + 51173856q^{57} - 115028080q^{59} - 173228648q^{61} - 31755240q^{63} - 328077984q^{65} - 231785104q^{67} - 60700752q^{69} - 197476208q^{71} + 44629400q^{73} + 485314092q^{75} + 308117920q^{77} + 355774584q^{79} + 172186884q^{81} - 607613328q^{83} - 1087351392q^{85} + 640592064q^{87} - 1157146424q^{89} - 847629840q^{91} - 919450440q^{93} + 329699328q^{95} - 1599536472q^{97} - 653895504q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2070 x^{2} - 13768 x + 561570$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-88 \nu^{3} - 100 \nu^{2} + 76928 \nu + 1012188$$$$)/483$$ $$\beta_{2}$$ $$=$$ $$($$$$-8 \nu^{3} + 1396 \nu^{2} - 15488 \nu - 1362252$$$$)/483$$ $$\beta_{3}$$ $$=$$ $$($$$$-160 \nu^{3} + 4736 \nu^{2} + 246656 \nu - 3249600$$$$)/483$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} - 7 \beta_{2} - 3 \beta_{1}$$$$)/768$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + 19 \beta_{2} - 9 \beta_{1} + 99360$$$$)/96$$ $$\nu^{3}$$ $$=$$ $$($$$$428 \beta_{3} - 1573 \beta_{2} - 1689 \beta_{1} + 1982592$$$$)/192$$

## 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
 14.0249 −25.0977 45.8607 −34.7879
0 81.0000 0 −2687.31 0 −950.516 0 6561.00 0
1.2 0 81.0000 0 126.806 0 5939.66 0 6561.00 0
1.3 0 81.0000 0 250.262 0 1655.62 0 6561.00 0
1.4 0 81.0000 0 2550.24 0 −11484.8 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.g yes 4
4.b odd 2 1 384.10.a.c yes 4
8.b even 2 1 384.10.a.b 4
8.d odd 2 1 384.10.a.f yes 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 240 T_{5}^{3} - 6873216 T_{5}^{2} + 2588500480 T_{5} - 217486924800$$ $$T_{7}^{4} + 4840 T_{7}^{3} - 73699080 T_{7}^{2} + 39372619552 T_{7} +$$$$10\!\cdots\!80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -81 + T )^{4}$$
$5$ $$-217486924800 + 2588500480 T - 6873216 T^{2} - 240 T^{3} + T^{4}$$
$7$ $$107350205672080 + 39372619552 T - 73699080 T^{2} + 4840 T^{3} + T^{4}$$
$11$ $$-5433574901626624 - 5488526384896 T + 1157609952 T^{2} + 99664 T^{3} + T^{4}$$
$13$ $$-3420928154350378224 + 1239819999141472 T - 17703868584 T^{2} - 60840 T^{3} + T^{4}$$
$17$ $$-$$$$64\!\cdots\!80$$$$- 32138794839108064 T - 80161247592 T^{2} + 434952 T^{3} + T^{4}$$
$19$ $$-$$$$21\!\cdots\!60$$$$+ 140394999467118592 T - 126027840000 T^{2} - 631776 T^{3} + T^{4}$$
$23$ $$85\!\cdots\!00$$$$+ 787832151529799936 T - 3857314584096 T^{2} + 749392 T^{3} + T^{4}$$
$29$ $$-$$$$23\!\cdots\!00$$$$+$$$$12\!\cdots\!00$$$$T + 403507342368 T^{2} - 7908544 T^{3} + T^{4}$$
$31$ $$-$$$$39\!\cdots\!00$$$$-$$$$18\!\cdots\!96$$$$T + 9640601683128 T^{2} + 11351240 T^{3} + T^{4}$$
$37$ $$-$$$$18\!\cdots\!52$$$$+$$$$53\!\cdots\!08$$$$T - 306478230344232 T^{2} - 13592920 T^{3} + T^{4}$$
$41$ $$83\!\cdots\!80$$$$-$$$$34\!\cdots\!16$$$$T - 652651509193128 T^{2} + 18838888 T^{3} + T^{4}$$
$43$ $$20\!\cdots\!36$$$$-$$$$16\!\cdots\!16$$$$T - 592933010898816 T^{2} - 14177920 T^{3} + T^{4}$$
$47$ $$56\!\cdots\!60$$$$+$$$$34\!\cdots\!04$$$$T - 1413954708273696 T^{2} - 37779120 T^{3} + T^{4}$$
$53$ $$-$$$$30\!\cdots\!88$$$$-$$$$35\!\cdots\!24$$$$T - 2411576760925920 T^{2} + 115336512 T^{3} + T^{4}$$
$59$ $$76\!\cdots\!28$$$$-$$$$95\!\cdots\!04$$$$T + 992149114231392 T^{2} + 115028080 T^{3} + T^{4}$$
$61$ $$-$$$$31\!\cdots\!40$$$$-$$$$22\!\cdots\!84$$$$T + 148756829450328 T^{2} + 173228648 T^{3} + T^{4}$$
$67$ $$-$$$$96\!\cdots\!32$$$$-$$$$17\!\cdots\!08$$$$T - 60903047243628960 T^{2} + 231785104 T^{3} + T^{4}$$
$71$ $$35\!\cdots\!52$$$$-$$$$11\!\cdots\!64$$$$T - 115813756628210976 T^{2} + 197476208 T^{3} + T^{4}$$
$73$ $$67\!\cdots\!08$$$$+$$$$38\!\cdots\!44$$$$T - 79671951811596840 T^{2} - 44629400 T^{3} + T^{4}$$
$79$ $$61\!\cdots\!44$$$$+$$$$60\!\cdots\!68$$$$T - 47613229363470024 T^{2} - 355774584 T^{3} + T^{4}$$
$83$ $$-$$$$66\!\cdots\!96$$$$-$$$$29\!\cdots\!84$$$$T - 58527612471023904 T^{2} + 607613328 T^{3} + T^{4}$$
$89$ $$-$$$$10\!\cdots\!80$$$$-$$$$46\!\cdots\!24$$$$T - 152887737068929896 T^{2} + 1157146424 T^{3} + T^{4}$$
$97$ $$57\!\cdots\!00$$$$+$$$$15\!\cdots\!40$$$$T + 834512009695893336 T^{2} + 1599536472 T^{3} + T^{4}$$
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