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 $2$

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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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 13632q^{7} - 52488q^{9} + O(q^{10}) \) \( 8q - 13632q^{7} - 52488q^{9} - 90720q^{15} + 8304q^{17} - 4612608q^{23} - 4754904q^{25} - 7499328q^{31} - 6213024q^{33} - 23211360q^{39} - 43518896q^{41} + 49382016q^{47} - 74106808q^{49} + 19030656q^{55} + 38141280q^{57} + 89439552q^{63} - 110270336q^{65} - 741751296q^{71} - 1507903440q^{73} + 1008373440q^{79} + 344373768q^{81} + 423468000q^{87} - 1337034448q^{89} - 543950208q^{95} - 904817936q^{97} + O(q^{100}) \)

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:

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

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