Properties

Label 384.10.a.h
Level $384$
Weight $10$
Character orbit 384.a
Self dual yes
Analytic conductor $197.774$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 14124 x^{2} - 170336 x + 18391464\)
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.

\(f(q)\) \(=\) \( q + 81 q^{3} + ( 432 + \beta_{1} ) q^{5} + ( 1210 + \beta_{1} - \beta_{3} ) q^{7} + 6561 q^{9} +O(q^{10})\) \( q + 81 q^{3} + ( 432 + \beta_{1} ) q^{5} + ( 1210 + \beta_{1} - \beta_{3} ) q^{7} + 6561 q^{9} + ( 3956 + 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 20610 + 38 \beta_{1} + 2 \beta_{2} + 7 \beta_{3} ) q^{13} + ( 34992 + 81 \beta_{1} ) q^{15} + ( 41478 + 134 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 134976 + 262 \beta_{1} - 19 \beta_{2} - 20 \beta_{3} ) q^{19} + ( 98010 + 81 \beta_{1} - 81 \beta_{3} ) q^{21} + ( 182420 + 50 \beta_{1} + 56 \beta_{2} + 130 \beta_{3} ) q^{23} + ( 56203 + 644 \beta_{1} + 40 \beta_{2} - 50 \beta_{3} ) q^{25} + 531441 q^{27} + ( 462716 - 457 \beta_{1} + 94 \beta_{2} - 62 \beta_{3} ) q^{29} + ( 799490 - 55 \beta_{1} + 110 \beta_{2} + 471 \beta_{3} ) q^{31} + ( 320436 + 486 \beta_{1} + 243 \beta_{2} + 162 \beta_{3} ) q^{33} + ( 2143728 + 3018 \beta_{1} - 427 \beta_{2} - 862 \beta_{3} ) q^{35} + ( 1046998 - 5496 \beta_{1} + 134 \beta_{2} - 863 \beta_{3} ) q^{37} + ( 1669410 + 3078 \beta_{1} + 162 \beta_{2} + 567 \beta_{3} ) q^{39} + ( 56926 + 2954 \beta_{1} - 77 \beta_{2} - 2128 \beta_{3} ) q^{41} + ( 400088 - 2070 \beta_{1} + 835 \beta_{2} - 1992 \beta_{3} ) q^{43} + ( 2834352 + 6561 \beta_{1} ) q^{45} + ( 4513476 + 4542 \beta_{1} - 2268 \beta_{2} - 1886 \beta_{3} ) q^{47} + ( 14432205 + 13072 \beta_{1} - 1778 \beta_{2} - 4070 \beta_{3} ) q^{49} + ( 3359718 + 10854 \beta_{1} + 405 \beta_{2} + 324 \beta_{3} ) q^{51} + ( 7354572 - 12469 \beta_{1} - 2398 \beta_{2} - 3502 \beta_{3} ) q^{53} + ( 11476704 + 23972 \beta_{1} + 1990 \beta_{2} + 9940 \beta_{3} ) q^{55} + ( 10933056 + 21222 \beta_{1} - 1539 \beta_{2} - 1620 \beta_{3} ) q^{57} + ( 9575012 + 74736 \beta_{1} + 624 \beta_{2} - 4764 \beta_{3} ) q^{59} + ( -24941162 - 50092 \beta_{1} + 226 \beta_{2} - 7511 \beta_{3} ) q^{61} + ( 7938810 + 6561 \beta_{1} - 6561 \beta_{3} ) q^{63} + ( 78530208 + 32118 \beta_{1} + 5333 \beta_{2} + 9528 \beta_{3} ) q^{65} + ( -45929252 + 30672 \beta_{1} - 1208 \beta_{2} + 14196 \beta_{3} ) q^{67} + ( 14776020 + 4050 \beta_{1} + 4536 \beta_{2} + 10530 \beta_{3} ) q^{69} + ( 45467020 + 33330 \beta_{1} - 2444 \beta_{2} - 1526 \beta_{3} ) q^{71} + ( 63634790 + 84688 \beta_{1} + 258 \beta_{2} + 15230 \beta_{3} ) q^{73} + ( 4552443 + 52164 \beta_{1} + 3240 \beta_{2} - 4050 \beta_{3} ) q^{75} + ( -57641176 - 231688 \beta_{1} - 4018 \beta_{2} + 23116 \beta_{3} ) q^{77} + ( 207894546 + 183265 \beta_{1} + 7272 \beta_{2} - 19493 \beta_{3} ) q^{79} + 43046721 q^{81} + ( -171980988 - 176402 \beta_{1} - 12549 \beta_{2} + 35054 \beta_{3} ) q^{83} + ( 260347776 + 100062 \beta_{1} + 8588 \beta_{2} + 10908 \beta_{3} ) q^{85} + ( 37479996 - 37017 \beta_{1} + 7614 \beta_{2} - 5022 \beta_{3} ) q^{87} + ( 156906818 + 127676 \beta_{1} + 19190 \beta_{2} + 23480 \beta_{3} ) q^{89} + ( -254536908 - 101126 \beta_{1} - 9541 \beta_{2} - 3412 \beta_{3} ) q^{91} + ( 64758690 - 4455 \beta_{1} + 8910 \beta_{2} + 38151 \beta_{3} ) q^{93} + ( 541779552 + 83512 \beta_{1} - 4028 \beta_{2} - 83908 \beta_{3} ) q^{95} + ( -222346470 - 764780 \beta_{1} - 20230 \beta_{2} + 43420 \beta_{3} ) q^{97} + ( 25955316 + 39366 \beta_{1} + 19683 \beta_{2} + 13122 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 324q^{3} + 1728q^{5} + 4840q^{7} + 26244q^{9} + O(q^{10}) \) \( 4q + 324q^{3} + 1728q^{5} + 4840q^{7} + 26244q^{9} + 15824q^{11} + 82440q^{13} + 139968q^{15} + 165912q^{17} + 539904q^{19} + 392040q^{21} + 729680q^{23} + 224812q^{25} + 2125764q^{27} + 1850864q^{29} + 3197960q^{31} + 1281744q^{33} + 8574912q^{35} + 4187992q^{37} + 6677640q^{39} + 227704q^{41} + 1600352q^{43} + 11337408q^{45} + 18053904q^{47} + 57728820q^{49} + 13438872q^{51} + 29418288q^{53} + 45906816q^{55} + 43732224q^{57} + 38300048q^{59} - 99764648q^{61} + 31755240q^{63} + 314120832q^{65} - 183717008q^{67} + 59104080q^{69} + 181868080q^{71} + 254539160q^{73} + 18209772q^{75} - 230564704q^{77} + 831578184q^{79} + 172186884q^{81} - 687923952q^{83} + 1041391104q^{85} + 149919984q^{87} + 627627272q^{89} - 1018147632q^{91} + 259034760q^{93} + 2167118208q^{95} - 889385880q^{97} + 103821264q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 14124 x^{2} - 170336 x + 18391464\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -8 \nu^{3} + 206 \nu^{2} + 126232 \nu - 432756 \)\()/2853\)
\(\beta_{2}\)\(=\)\((\)\( 160 \nu^{3} - 4120 \nu^{2} - 1429088 \nu + 8655120 \)\()/2853\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{2} - 80 \nu - 28248 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 20 \beta_{1}\)\()/384\)
\(\nu^{2}\)\(=\)\((\)\(72 \beta_{3} + 5 \beta_{2} + 100 \beta_{1} + 677952\)\()/96\)
\(\nu^{3}\)\(=\)\((\)\(3708 \beta_{3} + 8147 \beta_{2} + 94468 \beta_{1} + 24528384\)\()/192\)

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
−47.1038
−103.810
31.4956
119.418
0 81.0000 0 −1350.54 0 4629.00 0 6561.00 0
1.2 0 81.0000 0 −397.737 0 −7340.72 0 6561.00 0
1.3 0 81.0000 0 1657.87 0 11369.1 0 6561.00 0
1.4 0 81.0000 0 1818.41 0 −3817.40 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.h yes 4
4.b odd 2 1 384.10.a.d yes 4
8.b even 2 1 384.10.a.a 4
8.d odd 2 1 384.10.a.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.a.a 4 8.b even 2 1
384.10.a.d yes 4 4.b odd 2 1
384.10.a.e yes 4 8.d odd 2 1
384.10.a.h 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5}^{4} - 1728 T_{5}^{3} - 2525664 T_{5}^{2} + 3403197440 T_{5} + \)\(16\!\cdots\!00\)\( \)
\( T_{7}^{4} - 4840 T_{7}^{3} - 97858824 T_{7}^{2} + 138919065824 T_{7} + \)\(14\!\cdots\!44\)\( \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -81 + T )^{4} \)
$5$ \( 1619372294400 + 3403197440 T - 2525664 T^{2} - 1728 T^{3} + T^{4} \)
$7$ \( 1474756197184144 + 138919065824 T - 97858824 T^{2} - 4840 T^{3} + T^{4} \)
$11$ \( 6858181745009404160 + 1103792080640 T - 6159760416 T^{2} - 15824 T^{3} + T^{4} \)
$13$ \( -514638491031365616 - 183723133968416 T - 9910153704 T^{2} - 82440 T^{3} + T^{4} \)
$17$ \( \)\(74\!\cdots\!48\)\( + 1451814354814624 T - 71608865448 T^{2} - 165912 T^{3} + T^{4} \)
$19$ \( -\)\(25\!\cdots\!96\)\( + 111265942104473600 T - 412602051456 T^{2} - 539904 T^{3} + T^{4} \)
$23$ \( \)\(21\!\cdots\!44\)\( + 851951504698601216 T - 3334463529504 T^{2} - 729680 T^{3} + T^{4} \)
$29$ \( \)\(47\!\cdots\!88\)\( + 3667397931699343872 T - 6132162551424 T^{2} - 1850864 T^{3} + T^{4} \)
$31$ \( -\)\(20\!\cdots\!92\)\( + 63299272042313217888 T - 24992419229640 T^{2} - 3197960 T^{3} + T^{4} \)
$37$ \( \)\(25\!\cdots\!40\)\( - \)\(64\!\cdots\!20\)\( T - 203243447922216 T^{2} - 4187992 T^{3} + T^{4} \)
$41$ \( \)\(19\!\cdots\!64\)\( - \)\(22\!\cdots\!48\)\( T - 494045022312552 T^{2} - 227704 T^{3} + T^{4} \)
$43$ \( \)\(54\!\cdots\!56\)\( + \)\(49\!\cdots\!48\)\( T - 970945230014976 T^{2} - 1600352 T^{3} + T^{4} \)
$47$ \( \)\(30\!\cdots\!56\)\( + \)\(29\!\cdots\!24\)\( T - 3577671621321504 T^{2} - 18053904 T^{3} + T^{4} \)
$53$ \( -\)\(10\!\cdots\!00\)\( + \)\(21\!\cdots\!60\)\( T - 4952579056155264 T^{2} - 29418288 T^{3} + T^{4} \)
$59$ \( \)\(11\!\cdots\!04\)\( + \)\(35\!\cdots\!56\)\( T - 22171735492406688 T^{2} - 38300048 T^{3} + T^{4} \)
$61$ \( -\)\(15\!\cdots\!76\)\( - \)\(98\!\cdots\!96\)\( T - 11706470425376808 T^{2} + 99764648 T^{3} + T^{4} \)
$67$ \( \)\(10\!\cdots\!56\)\( - \)\(17\!\cdots\!92\)\( T - 13843775367055776 T^{2} + 183717008 T^{3} + T^{4} \)
$71$ \( -\)\(15\!\cdots\!64\)\( + \)\(44\!\cdots\!60\)\( T + 4165003675633632 T^{2} - 181868080 T^{3} + T^{4} \)
$73$ \( -\)\(13\!\cdots\!92\)\( + \)\(57\!\cdots\!76\)\( T - 26793790810227240 T^{2} - 254539160 T^{3} + T^{4} \)
$79$ \( -\)\(70\!\cdots\!24\)\( + \)\(65\!\cdots\!24\)\( T + 61673089702752696 T^{2} - 831578184 T^{3} + T^{4} \)
$83$ \( -\)\(31\!\cdots\!56\)\( - \)\(19\!\cdots\!64\)\( T - 178092834287762208 T^{2} + 687923952 T^{3} + T^{4} \)
$89$ \( \)\(34\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T - 191915279205961704 T^{2} - 627627272 T^{3} + T^{4} \)
$97$ \( \)\(12\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T - 2283188040832386600 T^{2} + 889385880 T^{3} + T^{4} \)
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