Properties

Label 324.6.a.e
Level 324
Weight 6
Character orbit 324.a
Self dual yes
Analytic conductor 51.964
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.9643576194\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( 4 + \beta_{2} ) q^{5} + ( -6 - \beta_{1} + \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( 4 + \beta_{2} ) q^{5} + ( -6 - \beta_{1} + \beta_{2} ) q^{7} + ( -36 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( 34 + \beta_{1} + 8 \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + ( 229 - 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} ) q^{17} + ( -86 + \beta_{1} + 17 \beta_{2} + 2 \beta_{3} + 7 \beta_{4} ) q^{19} + ( -78 - 6 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} + 15 \beta_{4} ) q^{23} + ( 955 + 12 \beta_{1} + 33 \beta_{2} - 22 \beta_{3} - 14 \beta_{4} ) q^{25} + ( 1198 + 27 \beta_{1} - 2 \beta_{2} + 15 \beta_{3} - 15 \beta_{4} ) q^{29} + ( -562 - 12 \beta_{1} - 6 \beta_{2} + 25 \beta_{3} - 7 \beta_{4} ) q^{31} + ( 3726 + 21 \beta_{1} - 7 \beta_{2} - 40 \beta_{3} - 30 \beta_{4} ) q^{35} + ( -1512 - 13 \beta_{1} - 23 \beta_{2} + 23 \beta_{3} + 49 \beta_{4} ) q^{37} + ( 3693 + 51 \beta_{1} - 23 \beta_{2} + 19 \beta_{3} + 45 \beta_{4} ) q^{41} + ( -256 - 54 \beta_{1} - 162 \beta_{2} - 24 \beta_{3} - 21 \beta_{4} ) q^{43} + ( 5026 - 87 \beta_{1} - 33 \beta_{2} + 38 \beta_{3} + 18 \beta_{4} ) q^{47} + ( 817 + 53 \beta_{1} - 62 \beta_{2} - \beta_{3} - 35 \beta_{4} ) q^{49} + ( 11704 - 27 \beta_{1} - 53 \beta_{2} - 15 \beta_{3} + 15 \beta_{4} ) q^{53} + ( 1542 + 66 \beta_{1} - 264 \beta_{2} - 49 \beta_{3} - 41 \beta_{4} ) q^{55} + ( 18076 - 138 \beta_{1} + 182 \beta_{2} - 50 \beta_{3} - 75 \beta_{4} ) q^{59} + ( -304 + 87 \beta_{1} + 120 \beta_{2} + 23 \beta_{3} + 49 \beta_{4} ) q^{61} + ( 29692 + 87 \beta_{1} + 140 \beta_{2} - 119 \beta_{3} - 45 \beta_{4} ) q^{65} + ( -2692 - 75 \beta_{1} - 195 \beta_{2} - 57 \beta_{3} + 138 \beta_{4} ) q^{67} + ( 22780 - 129 \beta_{1} + 347 \beta_{2} + 259 \beta_{3} + 81 \beta_{4} ) q^{71} + ( 1331 - 141 \beta_{1} + 690 \beta_{2} + 61 \beta_{3} - 133 \beta_{4} ) q^{73} + ( 42430 + 24 \beta_{1} - 403 \beta_{2} + 52 \beta_{3} - 132 \beta_{4} ) q^{77} + ( -6134 + 112 \beta_{1} + 122 \beta_{2} + 323 \beta_{3} + 91 \beta_{4} ) q^{79} + ( 45886 + 129 \beta_{1} + 21 \beta_{2} - 130 \beta_{3} + 240 \beta_{4} ) q^{83} + ( 9780 - 165 \beta_{1} + 1065 \beta_{2} - 197 \beta_{3} - 91 \beta_{4} ) q^{85} + ( 59962 + 375 \beta_{1} - 529 \beta_{2} + 55 \beta_{3} + 225 \beta_{4} ) q^{89} + ( 12562 - 24 \beta_{1} - 84 \beta_{2} - 309 \beta_{3} - 357 \beta_{4} ) q^{91} + ( 79024 + 438 \beta_{1} + 26 \beta_{2} - 386 \beta_{3} - 390 \beta_{4} ) q^{95} + ( -8219 - 605 \beta_{1} + 749 \beta_{2} + 43 \beta_{3} + 281 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 21q^{5} - 29q^{7} + O(q^{10}) \) \( 5q + 21q^{5} - 29q^{7} - 177q^{11} + 181q^{13} + 1140q^{17} - 416q^{19} - 399q^{23} + 4778q^{25} + 6033q^{29} - 2759q^{31} + 18573q^{35} - 7586q^{37} + 18435q^{41} - 1469q^{43} + 25155q^{47} + 4056q^{49} + 58422q^{53} + 7389q^{55} + 90537q^{59} - 1403q^{61} + 148407q^{65} - 13907q^{67} + 114684q^{71} + 7600q^{73} + 211983q^{77} - 29993q^{79} + 228951q^{83} + 49662q^{85} + 299166q^{89} + 62465q^{91} + 394764q^{95} - 40541q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 110 x^{3} + 39 x^{2} + 2214 x - 1944\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{4} + 20 \nu^{3} + 74 \nu^{2} - 1209 \nu - 1674 \)\()/27\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{4} - 46 \nu^{3} - 208 \nu^{2} + 1941 \nu - 3186 \)\()/54\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} + 92 \nu^{2} + 195 \nu - 1056 \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 92 \nu^{2} - 33 \nu + 993 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 2 \beta_{3} + 21\)\()/54\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + 7 \beta_{3} + 27 \beta_{2} + 27 \beta_{1} + 4830\)\()/108\)
\(\nu^{3}\)\(=\)\((\)\(155 \beta_{4} + 283 \beta_{3} + 27 \beta_{2} + 189 \beta_{1} + 11814\)\()/108\)
\(\nu^{4}\)\(=\)\((\)\(304 \beta_{4} + 671 \beta_{3} + 1269 \beta_{2} + 1431 \beta_{1} + 181065\)\()/54\)

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
−6.24439
0.900358
4.43077
10.4099
−7.49663
0 0 0 −81.4540 0 −179.262 0 0 0
1.2 0 0 0 −26.3205 0 63.2575 0 0 0
1.3 0 0 0 −9.76844 0 136.668 0 0 0
1.4 0 0 0 28.1436 0 −151.408 0 0 0
1.5 0 0 0 110.399 0 101.745 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

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 324.6.a.e 5
3.b odd 2 1 324.6.a.d 5
9.c even 3 2 36.6.e.a 10
9.d odd 6 2 108.6.e.a 10
36.f odd 6 2 144.6.i.d 10
36.h even 6 2 432.6.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.e.a 10 9.c even 3 2
108.6.e.a 10 9.d odd 6 2
144.6.i.d 10 36.f odd 6 2
324.6.a.d 5 3.b odd 2 1
324.6.a.e 5 1.a even 1 1 trivial
432.6.i.d 10 36.h even 6 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 21 T_{5}^{4} - 9981 T_{5}^{3} - 56727 T_{5}^{2} + 7030800 T_{5} + 65069568 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(324))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 21 T + 5644 T^{2} - 319227 T^{3} + 11115175 T^{4} - 1519942932 T^{5} + 34734921875 T^{6} - 3117451171875 T^{7} + 172241210937500 T^{8} - 2002716064453125 T^{9} + 298023223876953125 T^{10} \)
$7$ \( 1 + 29 T + 40410 T^{2} + 2467167 T^{3} + 1121022921 T^{4} + 42673614444 T^{5} + 18841032233247 T^{6} + 696913612649583 T^{7} + 191848960616796630 T^{8} + 2313975722630748029 T^{9} + \)\(13\!\cdots\!07\)\( T^{10} \)
$11$ \( 1 + 177 T + 427561 T^{2} + 2122014 T^{3} + 77616186361 T^{4} - 7157256911361 T^{5} + 12500164429625411 T^{6} + 55039578127266414 T^{7} + \)\(17\!\cdots\!11\)\( T^{8} + \)\(11\!\cdots\!77\)\( T^{9} + \)\(10\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 - 181 T + 1045092 T^{2} - 87339735 T^{3} + 629954553255 T^{4} - 52722114809928 T^{5} + 233897715941708715 T^{6} - 12040524145591320015 T^{7} + \)\(53\!\cdots\!44\)\( T^{8} - \)\(34\!\cdots\!81\)\( T^{9} + \)\(70\!\cdots\!93\)\( T^{10} \)
$17$ \( 1 - 1140 T + 4980550 T^{2} - 3443850354 T^{3} + 10068870522169 T^{4} - 5069379208548852 T^{5} + 14296356292995309833 T^{6} - \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} - \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} \)
$19$ \( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + 399 T + 16236442 T^{2} - 15815242731 T^{3} + 114281668162057 T^{4} - 214964375471995932 T^{5} + \)\(73\!\cdots\!51\)\( T^{6} - \)\(65\!\cdots\!19\)\( T^{7} + \)\(43\!\cdots\!94\)\( T^{8} + \)\(68\!\cdots\!99\)\( T^{9} + \)\(11\!\cdots\!43\)\( T^{10} \)
$29$ \( 1 - 6033 T + 32744932 T^{2} - 82954489011 T^{3} + 288237740873407 T^{4} - 1430006647746298968 T^{5} + \)\(59\!\cdots\!43\)\( T^{6} - \)\(34\!\cdots\!11\)\( T^{7} + \)\(28\!\cdots\!68\)\( T^{8} - \)\(10\!\cdots\!33\)\( T^{9} + \)\(36\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 + 2759 T + 62514558 T^{2} - 8483492775 T^{3} + 1879317626697489 T^{4} - 2373615527060533968 T^{5} + \)\(53\!\cdots\!39\)\( T^{6} - \)\(69\!\cdots\!75\)\( T^{7} + \)\(14\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!59\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} \)
$41$ \( 1 - 18435 T + 457528267 T^{2} - 6389512835910 T^{3} + 92046695080587061 T^{4} - \)\(99\!\cdots\!97\)\( T^{5} + \)\(10\!\cdots\!61\)\( T^{6} - \)\(85\!\cdots\!10\)\( T^{7} + \)\(71\!\cdots\!67\)\( T^{8} - \)\(33\!\cdots\!35\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 + 1469 T + 274021497 T^{2} + 2209484086710 T^{3} + 59712901022608185 T^{4} + \)\(32\!\cdots\!87\)\( T^{5} + \)\(87\!\cdots\!55\)\( T^{6} + \)\(47\!\cdots\!90\)\( T^{7} + \)\(87\!\cdots\!79\)\( T^{8} + \)\(68\!\cdots\!69\)\( T^{9} + \)\(68\!\cdots\!43\)\( T^{10} \)
$47$ \( 1 - 25155 T + 1034020258 T^{2} - 20179979725617 T^{3} + 464012894081647969 T^{4} - \)\(66\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!83\)\( T^{6} - \)\(10\!\cdots\!33\)\( T^{7} + \)\(12\!\cdots\!94\)\( T^{8} - \)\(69\!\cdots\!55\)\( T^{9} + \)\(63\!\cdots\!07\)\( T^{10} \)
$53$ \( 1 - 58422 T + 3354568213 T^{2} - 110313236959296 T^{3} + 3390725554692289246 T^{4} - \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} - \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} - \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} \)
$59$ \( 1 - 90537 T + 5365830529 T^{2} - 236000523803862 T^{3} + 8392018437003638425 T^{4} - \)\(24\!\cdots\!43\)\( T^{5} + \)\(59\!\cdots\!75\)\( T^{6} - \)\(12\!\cdots\!62\)\( T^{7} + \)\(19\!\cdots\!71\)\( T^{8} - \)\(23\!\cdots\!37\)\( T^{9} + \)\(18\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 + 1403 T + 3538874292 T^{2} + 2256100311765 T^{3} + 5454602684103950271 T^{4} + \)\(18\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!71\)\( T^{6} + \)\(16\!\cdots\!65\)\( T^{7} + \)\(21\!\cdots\!92\)\( T^{8} + \)\(71\!\cdots\!03\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 + 13907 T + 4070090193 T^{2} - 10411373671926 T^{3} + 6695736735425021001 T^{4} - \)\(80\!\cdots\!07\)\( T^{5} + \)\(90\!\cdots\!07\)\( T^{6} - \)\(18\!\cdots\!74\)\( T^{7} + \)\(10\!\cdots\!99\)\( T^{8} + \)\(46\!\cdots\!07\)\( T^{9} + \)\(44\!\cdots\!07\)\( T^{10} \)
$71$ \( 1 - 114684 T + 7758380659 T^{2} - 426246123888336 T^{3} + 19260501229393543450 T^{4} - \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} - \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \)
$79$ \( 1 + 29993 T + 6251931678 T^{2} - 85699438105257 T^{3} + 15422207141619743649 T^{4} - \)\(73\!\cdots\!92\)\( T^{5} + \)\(47\!\cdots\!51\)\( T^{6} - \)\(81\!\cdots\!57\)\( T^{7} + \)\(18\!\cdots\!22\)\( T^{8} + \)\(26\!\cdots\!93\)\( T^{9} + \)\(27\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 - 228951 T + 31015128418 T^{2} - 2949331181889681 T^{3} + \)\(22\!\cdots\!25\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(88\!\cdots\!75\)\( T^{6} - \)\(45\!\cdots\!69\)\( T^{7} + \)\(18\!\cdots\!26\)\( T^{8} - \)\(55\!\cdots\!51\)\( T^{9} + \)\(94\!\cdots\!43\)\( T^{10} \)
$89$ \( 1 - 299166 T + 52616244181 T^{2} - 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} - \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} - \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} - \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + 40541 T + 19537068819 T^{2} + 1527692999826186 T^{3} + \)\(22\!\cdots\!25\)\( T^{4} + \)\(18\!\cdots\!11\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} + \)\(11\!\cdots\!14\)\( T^{7} + \)\(12\!\cdots\!67\)\( T^{8} + \)\(22\!\cdots\!41\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} \)
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