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: \( x^{4} - 2070x^{2} - 13768x + 561570 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + 81 q^{3} + ( - \beta_{2} - 60) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1210) q^{7} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 q^{3} + ( - \beta_{2} - 60) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1210) q^{7} + 6561 q^{9} + ( - 4 \beta_{3} + 13 \beta_{2} - 5 \beta_1 - 24916) q^{11} + ( - 19 \beta_{3} + 23 \beta_{2} - 5 \beta_1 - 15210) q^{13} + ( - 81 \beta_{2} - 4860) q^{15} + (10 \beta_{3} - 83 \beta_{2} - 41 \beta_1 - 108738) q^{17} + ( - 30 \beta_{3} + 29 \beta_{2} + 55 \beta_1 + 157944) q^{19} + (81 \beta_{3} + 162 \beta_{2} + 81 \beta_1 + 98010) q^{21} + (174 \beta_{3} + 216 \beta_{2} - 146 \beta_1 + 187348) q^{23} + (494 \beta_{3} - 150 \beta_{2} + 698 \beta_1 + 1497883) q^{25} + 531441 q^{27} + (202 \beta_{3} + 41 \beta_{2} - 556 \beta_1 - 1977136) q^{29} + ( - 171 \beta_{3} - 1072 \beta_{2} + 663 \beta_1 + 2837810) q^{31} + ( - 324 \beta_{3} + 1053 \beta_{2} - 405 \beta_1 - 2018196) q^{33} + ( - 1096 \beta_{3} - 6049 \beta_{2} - 1607 \beta_1 - 6391752) q^{35} + ( - 2949 \beta_{3} + 1123 \beta_{2} - 1571 \beta_1 - 3398230) q^{37} + ( - 1539 \beta_{3} + 1863 \beta_{2} - 405 \beta_1 - 1232010) q^{39} + (1018 \beta_{3} + 9263 \beta_{2} - 1171 \beta_1 - 4709722) q^{41} + ( - 30 \beta_{3} - 8681 \beta_{2} + 1509 \beta_1 + 3544480) q^{43} + ( - 6561 \beta_{2} - 393660) q^{45} + ( - 3450 \beta_{3} - 2440 \beta_{2} - 6662 \beta_1 - 9444780) q^{47} + (1774 \beta_{3} + 24828 \beta_{2} + 6296 \beta_1 + 2352333) q^{49} + (810 \beta_{3} - 6723 \beta_{2} - 3321 \beta_1 - 8807778) q^{51} + ( - 2334 \beta_{3} + 31753 \beta_{2} + 1480 \beta_1 + 28834128) q^{53} + ( - 6008 \beta_{3} + 52550 \beta_{2} - 7966 \beta_1 - 46145136) q^{55} + ( - 2430 \beta_{3} + 2349 \beta_{2} + 4455 \beta_1 + 12793464) q^{57} + (1924 \beta_{3} - 19436 \beta_{2} - 4356 \beta_1 - 28757020) q^{59} + (4043 \beta_{3} + 38587 \beta_{2} + 5041 \beta_1 + 43307162) q^{61} + (6561 \beta_{3} + 13122 \beta_{2} + 6561 \beta_1 + 7938810) q^{63} + ( - 9058 \beta_{3} + 65809 \beta_{2} - 15741 \beta_1 - 82019496) q^{65} + ( - 42716 \beta_{3} - 34324 \beta_{2} - 17388 \beta_1 - 57946276) q^{67} + (14094 \beta_{3} + 17496 \beta_{2} - 11826 \beta_1 + 15175188) q^{69} + (39230 \beta_{3} + 20804 \beta_{2} + 51562 \beta_1 + 49369052) q^{71} + (33290 \beta_{3} - 38004 \beta_{2} - 8672 \beta_1 + 11157350) q^{73} + (40014 \beta_{3} - 12150 \beta_{2} + 56538 \beta_1 + 121328523) q^{75} + ( - 8096 \beta_{3} - 22982 \beta_{2} - 14662 \beta_1 - 77029480) q^{77} + ( - 25099 \beta_{3} + 98534 \beta_{2} - 7939 \beta_1 - 88943646) q^{79} + 43046721 q^{81} + (60312 \beta_{3} - 79319 \beta_{2} + 20983 \beta_1 - 151903332) q^{83} + (39004 \beta_{3} + 235986 \beta_{2} + 69288 \beta_1 + 271837848) q^{85} + (16362 \beta_{3} + 3321 \beta_{2} - 45036 \beta_1 - 160148016) q^{87} + ( - 114996 \beta_{3} - 68962 \beta_{2} + \cdots - 289286606) q^{89}+ \cdots + ( - 26244 \beta_{3} + 85293 \beta_{2} - 32805 \beta_1 - 163473876) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9} - 99664 q^{11} - 60840 q^{13} - 19440 q^{15} - 434952 q^{17} + 631776 q^{19} + 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} + 2125764 q^{27} - 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} - 25567008 q^{35} - 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} + 14177920 q^{43} - 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} - 35231112 q^{51} + 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} - 115028080 q^{59} + 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} - 231785104 q^{67} + 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} + 485314092 q^{75} - 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} - 607613328 q^{83} + 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} - 847629840 q^{91} + 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} - 653895504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2070x^{2} - 13768x + 561570 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -88\nu^{3} - 100\nu^{2} + 76928\nu + 1012188 ) / 483 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{3} + 1396\nu^{2} - 15488\nu - 1362252 ) / 483 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -160\nu^{3} + 4736\nu^{2} + 246656\nu - 3249600 ) / 483 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{3} - 7\beta_{2} - 3\beta_1 ) / 768 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + 19\beta_{2} - 9\beta _1 + 99360 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 428\beta_{3} - 1573\beta_{2} - 1689\beta _1 + 1982592 ) / 192 \) Copy content Toggle raw display

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 \(S_{10}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5}^{4} + 240T_{5}^{3} - 6873216T_{5}^{2} - 2588500480T_{5} - 217486924800 \) Copy content Toggle raw display
\( T_{7}^{4} - 4840T_{7}^{3} - 73699080T_{7}^{2} - 39372619552T_{7} + 107350205672080 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 240 T^{3} + \cdots - 217486924800 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 107350205672080 \) Copy content Toggle raw display
$11$ \( T^{4} + 99664 T^{3} + \cdots - 54\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{4} + 60840 T^{3} + \cdots - 34\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{4} + 434952 T^{3} + \cdots - 64\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{4} - 631776 T^{3} + \cdots - 21\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{4} - 749392 T^{3} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + 7908544 T^{3} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} - 11351240 T^{3} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + 13592920 T^{3} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{4} + 18838888 T^{3} + \cdots + 83\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{4} - 14177920 T^{3} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + 37779120 T^{3} + \cdots + 56\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} - 115336512 T^{3} + \cdots - 30\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{4} + 115028080 T^{3} + \cdots + 76\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{4} - 173228648 T^{3} + \cdots - 31\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{4} + 231785104 T^{3} + \cdots - 96\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{4} - 197476208 T^{3} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{4} - 44629400 T^{3} + \cdots + 67\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{4} + 355774584 T^{3} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{4} + 607613328 T^{3} + \cdots - 66\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + 1157146424 T^{3} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{4} + 1599536472 T^{3} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
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