Properties

Label 384.10.a.p
Level $384$
Weight $10$
Character orbit 384.a
Self dual yes
Analytic conductor $197.774$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 3854x^{3} + 12258x^{2} + 2877633x + 16772643 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{29}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 81 q^{3} + ( - \beta_1 + 154) q^{5} + (\beta_{2} - 8) q^{7} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 q^{3} + ( - \beta_1 + 154) q^{5} + (\beta_{2} - 8) q^{7} + 6561 q^{9} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 8219) q^{11} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 15 \beta_1 + 4504) q^{13} + ( - 81 \beta_1 + 12474) q^{15} + ( - 8 \beta_{4} - 7 \beta_{3} + 22 \beta_{2} - 75 \beta_1 - 263) q^{17} + (17 \beta_{3} + 30 \beta_{2} - 115 \beta_1 - 132997) q^{19} + (81 \beta_{2} - 648) q^{21} + (14 \beta_{4} + 4 \beta_{3} + 74 \beta_{2} - 210 \beta_1 + 73876) q^{23} + ( - 35 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} - 159 \beta_1 + 164637) q^{25} + 531441 q^{27} + (32 \beta_{4} + 90 \beta_{3} - 494 \beta_{2} + 325 \beta_1 + 1478480) q^{29} + (106 \beta_{4} - 34 \beta_{3} + 401 \beta_{2} + 2060 \beta_1 + 1830602) q^{31} + (81 \beta_{4} + 81 \beta_{3} + 162 \beta_{2} + 665739) q^{33} + (13 \beta_{4} - 13 \beta_{3} + 846 \beta_{2} + 38 \beta_1 - 633891) q^{35} + (56 \beta_{4} - 126 \beta_{3} - 685 \beta_{2} - 3835 \beta_1 + 2383104) q^{37} + ( - 162 \beta_{4} + 162 \beta_{3} + 81 \beta_{2} + 1215 \beta_1 + 364824) q^{39} + ( - 226 \beta_{4} - 149 \beta_{3} + 1946 \beta_{2} + \cdots - 1696401) q^{41}+ \cdots + (6561 \beta_{4} + 6561 \beta_{3} + 13122 \beta_{2} + \cdots + 53924859) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 405 q^{3} + 772 q^{5} - 38 q^{7} + 32805 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 405 q^{3} + 772 q^{5} - 38 q^{7} + 32805 q^{9} + 41100 q^{11} + 22486 q^{13} + 62532 q^{15} - 1130 q^{17} - 664712 q^{19} - 3078 q^{21} + 369972 q^{23} + 823383 q^{25} + 2657205 q^{27} + 7390736 q^{29} + 9149938 q^{31} + 3329100 q^{33} - 3167800 q^{35} + 11922058 q^{37} + 1821366 q^{39} - 8471746 q^{41} - 8948896 q^{43} + 5065092 q^{45} - 5051660 q^{47} + 39616113 q^{49} - 91530 q^{51} - 31431984 q^{53} - 67216 q^{55} - 53841672 q^{57} + 204260948 q^{59} + 190850874 q^{61} - 249318 q^{63} - 165466760 q^{65} + 274483500 q^{67} + 29967732 q^{69} + 162722908 q^{71} - 508927538 q^{73} + 66694023 q^{75} + 428895960 q^{77} + 491411266 q^{79} + 215233605 q^{81} + 766279260 q^{83} + 713985400 q^{85} + 598649616 q^{87} - 954097990 q^{89} + 503505932 q^{91} + 741144978 q^{93} + 968680288 q^{95} - 677085326 q^{97} + 269657100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 3854x^{3} + 12258x^{2} + 2877633x + 16772643 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 76\nu^{4} + 3226\nu^{3} - 148062\nu^{2} - 5556366\nu - 30411504 ) / 2835 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{4} + 466\nu^{3} + 90\nu^{2} - 979830\nu - 12122892 ) / 567 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 116\nu^{4} + 5198\nu^{3} - 214362\nu^{2} - 8540586\nu - 62948961 ) / 567 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 316\nu^{4} + 13546\nu^{3} - 638094\nu^{2} - 23384574\nu - 91076940 ) / 567 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 6\beta_{3} - 6\beta_{2} - 65\beta _1 + 1206 ) / 6144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -133\beta_{4} + 6\beta_{3} + 54\beta_{2} + 2705\beta _1 + 9473934 ) / 6144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1465\beta_{4} + 3441\beta_{3} - 900\beta_{2} - 56480\beta _1 - 7766949 ) / 1536 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -108685\beta_{4} - 33474\beta_{3} - 45162\beta_{2} + 2584145\beta _1 + 5580603894 ) / 1536 \) 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
−56.6510
44.2229
−6.33623
−24.0924
43.8567
0 81.0000 0 −1765.98 0 255.174 0 6561.00 0
1.2 0 81.0000 0 −1251.33 0 561.535 0 6561.00 0
1.3 0 81.0000 0 805.708 0 −10630.5 0 6561.00 0
1.4 0 81.0000 0 857.587 0 11220.9 0 6561.00 0
1.5 0 81.0000 0 2126.01 0 −1445.12 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.p yes 5
4.b odd 2 1 384.10.a.l yes 5
8.b even 2 1 384.10.a.i 5
8.d odd 2 1 384.10.a.m yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.a.i 5 8.b even 2 1
384.10.a.l yes 5 4.b odd 2 1
384.10.a.m yes 5 8.d odd 2 1
384.10.a.p yes 5 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5}^{5} - 772T_{5}^{4} - 4996512T_{5}^{3} + 2911922816T_{5}^{2} + 4908777191680T_{5} - 3246201269068800 \) Copy content Toggle raw display
\( T_{7}^{5} + 38T_{7}^{4} - 120691352T_{7}^{3} - 74139909200T_{7}^{2} + 123569166920528T_{7} - 24699995411682080 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T - 81)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 772 T^{4} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{5} + 38 T^{4} + \cdots - 24\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{5} - 41100 T^{4} + \cdots - 16\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{5} - 22486 T^{4} + \cdots - 19\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{5} + 1130 T^{4} + \cdots - 65\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{5} + 664712 T^{4} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{5} - 369972 T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{5} - 7390736 T^{4} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} - 9149938 T^{4} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{5} - 11922058 T^{4} + \cdots - 67\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{5} + 8471746 T^{4} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{5} + 8948896 T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{5} + 5051660 T^{4} + \cdots - 16\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{5} + 31431984 T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{5} - 204260948 T^{4} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{5} - 190850874 T^{4} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{5} - 274483500 T^{4} + \cdots + 50\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{5} - 162722908 T^{4} + \cdots - 44\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{5} + 508927538 T^{4} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{5} - 491411266 T^{4} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} - 766279260 T^{4} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{5} + 954097990 T^{4} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{5} + 677085326 T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
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