Properties

 Label 384.10.d.a Level $384$ Weight $10$ Character orbit 384.d Analytic conductor $197.774$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$197.773761087$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 13062 x^{6} + 45211107 x^{4} + 45928424926 x^{2} + 852972309225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 81 \beta_{4} q^{3} + ( 140 \beta_{4} - \beta_{5} ) q^{5} + ( -1704 - 2 \beta_{1} + \beta_{3} ) q^{7} -6561 q^{9} +O(q^{10})$$ $$q + 81 \beta_{4} q^{3} + ( 140 \beta_{4} - \beta_{5} ) q^{5} + ( -1704 - 2 \beta_{1} + \beta_{3} ) q^{7} -6561 q^{9} + ( 9588 \beta_{4} + 5 \beta_{5} + 8 \beta_{6} + 5 \beta_{7} ) q^{11} + ( 35820 \beta_{4} - 5 \beta_{5} - 15 \beta_{6} + 4 \beta_{7} ) q^{13} + ( -11340 + 81 \beta_{1} ) q^{15} + ( 1038 + 99 \beta_{1} - 7 \beta_{2} + 60 \beta_{3} ) q^{17} + ( -58860 \beta_{4} - 33 \beta_{5} + 100 \beta_{6} - 61 \beta_{7} ) q^{19} + ( -138024 \beta_{4} - 162 \beta_{5} - 81 \beta_{6} ) q^{21} + ( -576576 - 24 \beta_{1} + 56 \beta_{2} - 110 \beta_{3} ) q^{23} + ( -594363 - 426 \beta_{1} + 158 \beta_{2} + 204 \beta_{3} ) q^{25} -531441 \beta_{4} q^{27} + ( -653500 \beta_{4} - 167 \beta_{5} + 366 \beta_{6} - 652 \beta_{7} ) q^{29} + ( -937416 + 1138 \beta_{1} - 172 \beta_{2} + 95 \beta_{3} ) q^{31} + ( -776628 - 405 \beta_{1} + 405 \beta_{2} + 648 \beta_{3} ) q^{33} + ( 4302960 \beta_{4} + 4177 \beta_{5} - 960 \beta_{6} + 665 \beta_{7} ) q^{35} + ( 3851156 \beta_{4} + 5173 \beta_{5} + 513 \beta_{6} + 828 \beta_{7} ) q^{37} + ( -2901420 + 405 \beta_{1} + 324 \beta_{2} - 1215 \beta_{3} ) q^{39} + ( -5439862 - 10817 \beta_{1} - 235 \beta_{2} + 1860 \beta_{3} ) q^{41} + ( -447468 \beta_{4} + 7247 \beta_{5} + 1564 \beta_{6} - 1509 \beta_{7} ) q^{43} + ( -918540 \beta_{4} + 6561 \beta_{5} ) q^{45} + ( 6172752 + 16820 \beta_{1} + 672 \beta_{2} + 3950 \beta_{3} ) q^{47} + ( -9263351 + 5288 \beta_{1} - 2492 \beta_{2} - 348 \beta_{3} ) q^{49} + ( 84078 \beta_{4} + 8019 \beta_{5} - 4860 \beta_{6} + 567 \beta_{7} ) q^{51} + ( -2656004 \beta_{4} - 897 \beta_{5} + 15570 \beta_{6} + 4020 \beta_{7} ) q^{53} + ( 2378832 + 44776 \beta_{1} - 5492 \beta_{2} + 3744 \beta_{3} ) q^{55} + ( 4767660 + 2673 \beta_{1} - 4941 \beta_{2} + 8100 \beta_{3} ) q^{57} + ( -19616076 \beta_{4} + 30648 \beta_{5} + 7944 \beta_{6} + 5328 \beta_{7} ) q^{59} + ( -5818932 \beta_{4} - 7253 \beta_{5} - 9309 \beta_{6} - 628 \beta_{7} ) q^{61} + ( 11179944 + 13122 \beta_{1} - 6561 \beta_{3} ) q^{63} + ( -13783792 + 30179 \beta_{1} + 4497 \beta_{2} - 24444 \beta_{3} ) q^{65} + ( -53533068 \beta_{4} + 117668 \beta_{5} + 5112 \beta_{6} - 7700 \beta_{7} ) q^{67} + ( -46702656 \beta_{4} - 1944 \beta_{5} + 8910 \beta_{6} - 4536 \beta_{7} ) q^{69} + ( -92718912 + 38296 \beta_{1} + 24920 \beta_{2} + 1406 \beta_{3} ) q^{71} + ( -188487930 - 33352 \beta_{1} + 21044 \beta_{2} - 8796 \beta_{3} ) q^{73} + ( -48143403 \beta_{4} - 34506 \beta_{5} - 16524 \beta_{6} - 12798 \beta_{7} ) q^{75} + ( -180436192 \beta_{4} - 70564 \beta_{5} - 30900 \beta_{6} - 14476 \beta_{7} ) q^{77} + ( 126046680 + 120814 \beta_{1} - 2512 \beta_{2} - 76033 \beta_{3} ) q^{79} + 43046721 q^{81} + ( -31850604 \beta_{4} + 195603 \beta_{5} + 37712 \beta_{6} + 20443 \beta_{7} ) q^{83} + ( -274245016 \beta_{4} - 6894 \beta_{5} - 110928 \beta_{6} + 2624 \beta_{7} ) q^{85} + ( 52933500 + 13527 \beta_{1} - 52812 \beta_{2} + 29646 \beta_{3} ) q^{87} + ( -167129306 + 98298 \beta_{1} - 31186 \beta_{2} + 106728 \beta_{3} ) q^{89} + ( 257374848 \beta_{4} - 223165 \beta_{5} - 54172 \beta_{6} + 19271 \beta_{7} ) q^{91} + ( -75930696 \beta_{4} + 92178 \beta_{5} - 7695 \beta_{6} + 13932 \beta_{7} ) q^{93} + ( -67993776 - 207324 \beta_{1} - 6384 \beta_{2} + 218928 \beta_{3} ) q^{95} + ( -113102242 + 91610 \beta_{1} - 2234 \beta_{2} - 135600 \beta_{3} ) q^{97} + ( -62906868 \beta_{4} - 32805 \beta_{5} - 52488 \beta_{6} - 32805 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 13632 q^{7} - 52488 q^{9} + O(q^{10})$$ $$8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 13062 x^{6} + 45211107 x^{4} + 45928424926 x^{2} + 852972309225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4766 \nu^{6} + 51851374 \nu^{4} + 88099997114 \nu^{2} - 43229725983072$$$$)/ 34840779813$$ $$\beta_{2}$$ $$=$$ $$($$$$169786 \nu^{6} + 1579828010 \nu^{4} + 1426285970446 \nu^{2} - 1771419698208576$$$$)/ 243885458691$$ $$\beta_{3}$$ $$=$$ $$($$$$199366 \nu^{6} + 2241395438 \nu^{4} + 5124570485866 \nu^{2} + 1439961094183104$$$$)/ 243885458691$$ $$\beta_{4}$$ $$=$$ $$($$$$28496 \nu^{7} + 372522607 \nu^{5} + 1264029013547 \nu^{3} + 1137344409469551 \nu$$$$)/ 4843958573246310$$ $$\beta_{5}$$ $$=$$ $$($$$$-797877302 \nu^{7} - 9758638510954 \nu^{5} - 29157793972297394 \nu^{3} - 22780987632578534232 \nu$$$$)/ 75081357885317805$$ $$\beta_{6}$$ $$=$$ $$($$$$-1012635742 \nu^{7} - 15512580206174 \nu^{5} - 70850434762666054 \nu^{3} - 96437550563928427332 \nu$$$$)/ 75081357885317805$$ $$\beta_{7}$$ $$=$$ $$($$$$575969794 \nu^{7} + 7049643741998 \nu^{5} + 21095692147663558 \nu^{3} + 18573275336880625944 \nu$$$$)/ 10725908269331115$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 5 \beta_{5} - 96 \beta_{4}$$$$)/192$$ $$\nu^{2}$$ $$=$$ $$($$$$48 \beta_{3} + 13 \beta_{2} - 353 \beta_{1} - 626976$$$$)/192$$ $$\nu^{3}$$ $$=$$ $$($$$$-5251 \beta_{7} - 2988 \beta_{6} - 36371 \beta_{5} - 24620160 \beta_{4}$$$$)/192$$ $$\nu^{4}$$ $$=$$ $$($$$$-307416 \beta_{3} - 258409 \beta_{2} + 3152165 \beta_{1} + 3849294240$$$$)/192$$ $$\nu^{5}$$ $$=$$ $$($$$$17617919 \beta_{7} + 13863726 \beta_{6} + 145754053 \beta_{5} + 134259999792 \beta_{4}$$$$)/96$$ $$\nu^{6}$$ $$=$$ $$($$$$1228613316 \beta_{3} + 1285518437 \beta_{2} - 13182475621 \beta_{1} - 14273447764896$$$$)/96$$ $$\nu^{7}$$ $$=$$ $$($$$$-267619114645 \beta_{7} - 229933464768 \beta_{6} - 2397041072585 \beta_{5} - 2381735984318112 \beta_{4}$$$$)/192$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 43.3581i − 4.34998i 53.8781i − 90.8862i 90.8862i − 53.8781i 4.34998i − 43.3581i
0 81.0000i 0 1783.68i 0 −4965.03 0 −6561.00 0
193.2 0 81.0000i 0 1428.10i 0 6382.13 0 −6561.00 0
193.3 0 81.0000i 0 473.533i 0 −574.939 0 −6561.00 0
193.4 0 81.0000i 0 2178.24i 0 −7658.16 0 −6561.00 0
193.5 0 81.0000i 0 2178.24i 0 −7658.16 0 −6561.00 0
193.6 0 81.0000i 0 473.533i 0 −574.939 0 −6561.00 0
193.7 0 81.0000i 0 1428.10i 0 6382.13 0 −6561.00 0
193.8 0 81.0000i 0 1783.68i 0 −4965.03 0 −6561.00 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

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

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

 $$T_{5}^{8} + 10189952 T_{5}^{6} +$$$$33\!\cdots\!76$$$$T_{5}^{4} +$$$$37\!\cdots\!00$$$$T_{5}^{2} +$$$$69\!\cdots\!00$$ $$T_{7}^{4} + 6816 T_{7}^{3} - 38951584 T_{7}^{2} - 267125408256 T_{7} -$$$$13\!\cdots\!64$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 6561 + T^{2} )^{4}$$
$5$ $$69\!\cdots\!00$$$$+ 37796298115527475200 T^{2} + 33495402213376 T^{4} + 10189952 T^{6} + T^{8}$$
$7$ $$( -139519152670464 - 267125408256 T - 38951584 T^{2} + 6816 T^{3} + T^{4} )^{2}$$
$11$ $$95\!\cdots\!76$$$$+$$$$58\!\cdots\!32$$$$T^{2} + 46409298987583088128 T^{4} + 11978643008 T^{6} + T^{8}$$
$13$ $$12\!\cdots\!64$$$$+$$$$76\!\cdots\!76$$$$T^{2} + 97589328147224286720 T^{4} + 27798823744 T^{6} + T^{8}$$
$17$ $$( -$$$$30\!\cdots\!80$$$$+ 49338488226303264 T - 211211009384 T^{2} - 4152 T^{3} + T^{4} )^{2}$$
$19$ $$14\!\cdots\!00$$$$+$$$$21\!\cdots\!96$$$$T^{2} +$$$$10\!\cdots\!40$$$$T^{4} + 1840089688640 T^{6} + T^{8}$$
$23$ $$( -$$$$24\!\cdots\!20$$$$- 147383466145431552 T + 1076395085696 T^{2} + 2306304 T^{3} + T^{4} )^{2}$$
$29$ $$21\!\cdots\!36$$$$+$$$$70\!\cdots\!12$$$$T^{2} +$$$$51\!\cdots\!08$$$$T^{4} + 126446858508928 T^{6} + T^{8}$$
$31$ $$( -$$$$12\!\cdots\!76$$$$- 24857816523885746688 T - 5083681907104 T^{2} + 3749664 T^{3} + T^{4} )^{2}$$
$37$ $$45\!\cdots\!00$$$$+$$$$24\!\cdots\!44$$$$T^{2} +$$$$13\!\cdots\!80$$$$T^{4} + 489154292072000 T^{6} + T^{8}$$
$41$ $$($$$$11\!\cdots\!00$$$$-$$$$69\!\cdots\!04$$$$T - 529708328936360 T^{2} + 21759448 T^{3} + T^{4} )^{2}$$
$43$ $$14\!\cdots\!56$$$$+$$$$34\!\cdots\!28$$$$T^{2} +$$$$48\!\cdots\!48$$$$T^{4} + 1381436801509952 T^{6} + T^{8}$$
$47$ $$( -$$$$77\!\cdots\!80$$$$+$$$$61\!\cdots\!76$$$$T - 2066485216488064 T^{2} - 24691008 T^{3} + T^{4} )^{2}$$
$53$ $$46\!\cdots\!64$$$$+$$$$53\!\cdots\!88$$$$T^{2} +$$$$18\!\cdots\!60$$$$T^{4} + 23777254891395712 T^{6} + T^{8}$$
$59$ $$94\!\cdots\!00$$$$+$$$$33\!\cdots\!16$$$$T^{2} +$$$$14\!\cdots\!08$$$$T^{4} + 21321177912687168 T^{6} + T^{8}$$
$61$ $$45\!\cdots\!96$$$$+$$$$17\!\cdots\!60$$$$T^{2} +$$$$18\!\cdots\!28$$$$T^{4} + 7421211199909440 T^{6} + T^{8}$$
$67$ $$14\!\cdots\!16$$$$+$$$$22\!\cdots\!12$$$$T^{2} +$$$$99\!\cdots\!28$$$$T^{4} + 174012682174452288 T^{6} + T^{8}$$
$71$ $$( -$$$$21\!\cdots\!16$$$$-$$$$25\!\cdots\!32$$$$T - 42584318695350400 T^{2} + 370875648 T^{3} + T^{4} )^{2}$$
$73$ $$( -$$$$50\!\cdots\!00$$$$+$$$$55\!\cdots\!40$$$$T + 147101942187473112 T^{2} + 753951720 T^{3} + T^{4} )^{2}$$
$79$ $$($$$$11\!\cdots\!08$$$$+$$$$93\!\cdots\!52$$$$T - 191737101483224992 T^{2} - 504186720 T^{3} + T^{4} )^{2}$$
$83$ $$34\!\cdots\!96$$$$+$$$$39\!\cdots\!12$$$$T^{2} +$$$$81\!\cdots\!68$$$$T^{4} + 553752814633245248 T^{6} + T^{8}$$
$89$ $$($$$$15\!\cdots\!60$$$$-$$$$10\!\cdots\!76$$$$T - 484471894678056488 T^{2} + 668517224 T^{3} + T^{4} )^{2}$$
$97$ $$($$$$71\!\cdots\!20$$$$-$$$$10\!\cdots\!64$$$$T - 679913722622028008 T^{2} + 452408968 T^{3} + T^{4} )^{2}$$