# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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