Properties

Label 384.10.a.f
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
−34.7879
45.8607
−25.0977
14.0249
0 81.0000 0 −2550.24 0 11484.8 0 6561.00 0
1.2 0 81.0000 0 −250.262 0 −1655.62 0 6561.00 0
1.3 0 81.0000 0 −126.806 0 −5939.66 0 6561.00 0
1.4 0 81.0000 0 2687.31 0 950.516 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.f yes 4
4.b odd 2 1 384.10.a.b 4
8.b even 2 1 384.10.a.c yes 4
8.d odd 2 1 384.10.a.g yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.a.b 4 4.b odd 2 1
384.10.a.c yes 4 8.b even 2 1
384.10.a.f yes 4 1.a even 1 1 trivial
384.10.a.g yes 4 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(384))\):

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

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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