Properties

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