# Properties

 Label 324.5.c.a Level 324 Weight 5 Character orbit 324.c Analytic conductor 33.492 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 324.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.4918680392$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{20}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -3 + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -3 + \beta_{3} ) q^{7} -\beta_{2} q^{11} + ( 2 + \beta_{3} + \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{17} + ( 70 + \beta_{6} ) q^{19} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + \beta_{7} ) q^{23} + ( -88 + 4 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{25} + ( -8 \beta_{1} - 5 \beta_{2} + \beta_{4} + \beta_{7} ) q^{29} + ( -46 - 4 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{31} + ( -15 \beta_{1} + 6 \beta_{2} - \beta_{4} + 2 \beta_{7} ) q^{35} + ( -5 - 19 \beta_{3} - 5 \beta_{5} - \beta_{6} ) q^{37} + ( 26 \beta_{1} - \beta_{2} - 3 \beta_{4} + 2 \beta_{7} ) q^{41} + ( 15 - 3 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{43} + ( 11 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} + 4 \beta_{7} ) q^{47} + ( 71 - 19 \beta_{3} - 11 \beta_{5} + 2 \beta_{6} ) q^{49} + ( 61 \beta_{1} - 25 \beta_{2} + 5 \beta_{4} + 5 \beta_{7} ) q^{53} + ( -212 + 34 \beta_{3} + 13 \beta_{5} - \beta_{6} ) q^{55} + ( -63 \beta_{1} - 20 \beta_{2} - \beta_{4} + 7 \beta_{7} ) q^{59} + ( 480 - 53 \beta_{3} + 19 \beta_{5} + 2 \beta_{6} ) q^{61} + ( 106 \beta_{1} + 41 \beta_{2} + 3 \beta_{4} + 9 \beta_{7} ) q^{65} + ( -16 + 96 \beta_{3} - 3 \beta_{5} ) q^{67} + ( 35 \beta_{1} + 17 \beta_{2} + 9 \beta_{4} + 13 \beta_{7} ) q^{71} + ( -953 + 83 \beta_{3} - \beta_{5} - 8 \beta_{6} ) q^{73} + ( -105 \beta_{1} + 18 \beta_{2} - 14 \beta_{4} + 6 \beta_{7} ) q^{77} + ( 568 + 90 \beta_{3} - 3 \beta_{5} + 7 \beta_{6} ) q^{79} + ( -229 \beta_{1} - 12 \beta_{2} - 7 \beta_{4} + 12 \beta_{7} ) q^{83} + ( 731 - 97 \beta_{3} - 7 \beta_{5} + 7 \beta_{6} ) q^{85} + ( -241 \beta_{1} - 19 \beta_{2} + 3 \beta_{4} + 7 \beta_{7} ) q^{89} + ( 1940 - 162 \beta_{3} + 7 \beta_{5} + 7 \beta_{6} ) q^{91} + ( 384 \beta_{1} - 14 \beta_{2} + 20 \beta_{4} + 16 \beta_{7} ) q^{95} + ( -1822 - 17 \beta_{3} + \beta_{5} - 9 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 26q^{7} + O(q^{10})$$ $$8q - 26q^{7} + 10q^{13} + 562q^{19} - 706q^{25} - 374q^{31} + 16q^{37} + 136q^{43} + 654q^{49} - 1818q^{55} + 3874q^{61} - 308q^{67} - 7802q^{73} + 4390q^{79} + 6084q^{85} + 15830q^{91} - 14564q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 6 x^{6} + 121 x^{5} + 1104 x^{4} - 1647 x^{3} + 6529 x^{2} + 85254 x + 440076$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$17 \nu^{7} - 142 \nu^{6} + 635 \nu^{5} - 1115 \nu^{4} + 17923 \nu^{3} - 159143 \nu^{2} + 464580 \nu - 386178$$$$)/104247$$ $$\beta_{2}$$ $$=$$ $$($$$$53 \nu^{7} - 367 \nu^{6} + 5765 \nu^{5} - 70703 \nu^{4} + 113944 \nu^{3} + 106240 \nu^{2} - 1050579 \nu - 42441945$$$$)/104247$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{6} + 34 \nu^{5} + 116 \nu^{4} + 197 \nu^{3} - 223 \nu^{2} + 25974 \nu + 60657$$$$)/729$$ $$\beta_{4}$$ $$=$$ $$($$$$118 \nu^{7} + 49 \nu^{6} - 1775 \nu^{5} + 11717 \nu^{4} + 118754 \nu^{3} + 166664 \nu^{2} + 172422 \nu + 2470650$$$$)/34749$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{7} - 38 \nu^{6} + 73 \nu^{5} + 410 \nu^{4} + 428 \nu^{3} - 18433 \nu^{2} + 18954 \nu + 178665$$$$)/729$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} + 2 \nu^{6} + 143 \nu^{5} + 661 \nu^{4} + 4441 \nu^{3} + 15841 \nu^{2} + 96174 \nu + 445521$$$$)/729$$ $$\beta_{7}$$ $$=$$ $$($$$$-1052 \nu^{7} - 1433 \nu^{6} + 40423 \nu^{5} - 237610 \nu^{4} - 608323 \nu^{3} + 281948 \nu^{2} + 16720785 \nu - 71046885$$$$)/104247$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 18 \beta_{3} - 3 \beta_{2} + 22 \beta_{1} + 186$$$$)/486$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{7} + 3 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} - \beta_{2} - 116 \beta_{1} - 60$$$$)/162$$ $$\nu^{3}$$ $$=$$ $$($$$$19 \beta_{7} + 24 \beta_{6} + 27 \beta_{5} + 56 \beta_{4} - 228 \beta_{3} - 37 \beta_{2} + 125 \beta_{1} - 7947$$$$)/162$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{7} + 111 \beta_{6} + 96 \beta_{5} - 64 \beta_{4} - 570 \beta_{3} - 220 \beta_{2} - 219 \beta_{1} - 120225$$$$)/162$$ $$\nu^{5}$$ $$=$$ $$($$$$-217 \beta_{7} + 930 \beta_{6} + 285 \beta_{5} - 2155 \beta_{4} - 2505 \beta_{3} + 515 \beta_{2} + 2208 \beta_{1} - 206598$$$$)/162$$ $$\nu^{6}$$ $$=$$ $$($$$$-4159 \beta_{7} + 651 \beta_{6} - 6591 \beta_{5} - 13543 \beta_{4} + 15891 \beta_{3} + 5291 \beta_{2} + 68865 \beta_{1} + 432933$$$$)/162$$ $$\nu^{7}$$ $$=$$ $$($$$$-26773 \beta_{7} - 19995 \beta_{6} - 32457 \beta_{5} - 47548 \beta_{4} + 237246 \beta_{3} + 67505 \beta_{2} + 53688 \beta_{1} + 13250454$$$$)/162$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −3.41053 − 2.74723i −3.05006 − 3.25531i 4.23522 + 4.06612i 3.72537 + 4.42407i 3.72537 − 4.42407i 4.23522 − 4.06612i −3.05006 + 3.25531i −3.41053 + 2.74723i
0 0 0 40.2664i 0 14.7738 0 0 0
161.2 0 0 0 31.6564i 0 −75.3660 0 0 0
161.3 0 0 0 12.2819i 0 −14.2840 0 0 0
161.4 0 0 0 8.86801i 0 61.8763 0 0 0
161.5 0 0 0 8.86801i 0 61.8763 0 0 0
161.6 0 0 0 12.2819i 0 −14.2840 0 0 0
161.7 0 0 0 31.6564i 0 −75.3660 0 0 0
161.8 0 0 0 40.2664i 0 14.7738 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.5.c.a 8
3.b odd 2 1 inner 324.5.c.a 8
4.b odd 2 1 1296.5.e.g 8
9.c even 3 1 36.5.g.a 8
9.c even 3 1 108.5.g.a 8
9.d odd 6 1 36.5.g.a 8
9.d odd 6 1 108.5.g.a 8
12.b even 2 1 1296.5.e.g 8
36.f odd 6 1 144.5.q.c 8
36.f odd 6 1 432.5.q.c 8
36.h even 6 1 144.5.q.c 8
36.h even 6 1 432.5.q.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a 8 9.c even 3 1
36.5.g.a 8 9.d odd 6 1
108.5.g.a 8 9.c even 3 1
108.5.g.a 8 9.d odd 6 1
144.5.q.c 8 36.f odd 6 1
144.5.q.c 8 36.h even 6 1
324.5.c.a 8 1.a even 1 1 trivial
324.5.c.a 8 3.b odd 2 1 inner
432.5.q.c 8 36.f odd 6 1
432.5.q.c 8 36.h even 6 1
1296.5.e.g 8 4.b odd 2 1
1296.5.e.g 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 2853 T_{5}^{6} + 2238759 T_{5}^{4} + 403999407 T_{5}^{2} + 19274879556$$ acting on $$S_{5}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2147 T^{2} + 2477509 T^{4} - 2147976218 T^{6} + 1511855308306 T^{8} - 839053210156250 T^{10} + 378037872314453125 T^{12} -$$$$12\!\cdots\!75$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16}$$
$7$ $$( 1 + 13 T + 4723 T^{2} + 93076 T^{3} + 12134350 T^{4} + 223475476 T^{5} + 27227155123 T^{6} + 179936733613 T^{7} + 33232930569601 T^{8} )^{2}$$
$11$ $$1 - 67376 T^{2} + 2511150250 T^{4} - 60712318203824 T^{6} + 1049763807787064539 T^{8} -$$$$13\!\cdots\!44$$$$T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$66\!\cdots\!16$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16}$$
$13$ $$( 1 - 5 T + 42079 T^{2} - 3738530 T^{3} + 1399773016 T^{4} - 106776155330 T^{5} + 34325133008959 T^{6} - 116490425612405 T^{7} + 665416609183179841 T^{8} )^{2}$$
$17$ $$1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} +$$$$63\!\cdots\!86$$$$T^{8} -$$$$47\!\cdots\!55$$$$T^{10} +$$$$25\!\cdots\!74$$$$T^{12} -$$$$97\!\cdots\!25$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16}$$
$19$ $$( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$
$23$ $$1 - 103955 T^{2} + 204060092029 T^{4} - 14985633296690750 T^{6} +$$$$21\!\cdots\!46$$$$T^{8} -$$$$11\!\cdots\!50$$$$T^{10} +$$$$12\!\cdots\!69$$$$T^{12} -$$$$49\!\cdots\!55$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16}$$
$29$ $$1 - 3708803 T^{2} + 6181658247973 T^{4} - 6420936064399980986 T^{6} +$$$$50\!\cdots\!30$$$$T^{8} -$$$$32\!\cdots\!46$$$$T^{10} +$$$$15\!\cdots\!33$$$$T^{12} -$$$$46\!\cdots\!43$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16}$$
$31$ $$( 1 + 187 T + 2550973 T^{2} + 331939744 T^{3} + 3082958795560 T^{4} + 306553324318624 T^{5} + 2175702008453980093 T^{6} +$$$$14\!\cdots\!07$$$$T^{7} +$$$$72\!\cdots\!81$$$$T^{8} )^{2}$$
$37$ $$( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$
$41$ $$1 - 12649388 T^{2} + 74446575673498 T^{4} -$$$$29\!\cdots\!76$$$$T^{6} +$$$$89\!\cdots\!55$$$$T^{8} -$$$$23\!\cdots\!96$$$$T^{10} +$$$$47\!\cdots\!18$$$$T^{12} -$$$$64\!\cdots\!68$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16}$$
$43$ $$( 1 - 68 T + 12955228 T^{2} - 545371076 T^{3} + 65226299853025 T^{4} - 1864515179999876 T^{5} +$$$$15\!\cdots\!28$$$$T^{6} -$$$$27\!\cdots\!68$$$$T^{7} +$$$$13\!\cdots\!01$$$$T^{8} )^{2}$$
$47$ $$1 - 19981955 T^{2} + 227810016714829 T^{4} -$$$$17\!\cdots\!90$$$$T^{6} +$$$$10\!\cdots\!46$$$$T^{8} -$$$$42\!\cdots\!90$$$$T^{10} +$$$$12\!\cdots\!09$$$$T^{12} -$$$$26\!\cdots\!55$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16}$$
$53$ $$1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} +$$$$67\!\cdots\!26$$$$T^{8} -$$$$52\!\cdots\!60$$$$T^{10} +$$$$44\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16}$$
$59$ $$1 - 31340696 T^{2} + 646808461063090 T^{4} -$$$$10\!\cdots\!44$$$$T^{6} +$$$$14\!\cdots\!19$$$$T^{8} -$$$$15\!\cdots\!24$$$$T^{10} +$$$$13\!\cdots\!90$$$$T^{12} -$$$$99\!\cdots\!56$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16}$$
$61$ $$( 1 - 1937 T + 14281603 T^{2} + 60991858006 T^{3} - 94182960283700 T^{4} + 844483568245653046 T^{5} +$$$$27\!\cdots\!43$$$$T^{6} -$$$$51\!\cdots\!77$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$
$67$ $$( 1 + 154 T + 33859570 T^{2} - 10195927724 T^{3} + 608674748042419 T^{4} - 205459373273578604 T^{5} +$$$$13\!\cdots\!70$$$$T^{6} +$$$$12\!\cdots\!94$$$$T^{7} +$$$$16\!\cdots\!81$$$$T^{8} )^{2}$$
$71$ $$1 - 68871716 T^{2} + 3244147638477940 T^{4} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$30\!\cdots\!74$$$$T^{8} -$$$$73\!\cdots\!64$$$$T^{10} +$$$$13\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$
$73$ $$( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} +$$$$47\!\cdots\!70$$$$T^{6} +$$$$89\!\cdots\!21$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$
$79$ $$( 1 - 2195 T + 92542939 T^{2} - 195388180760 T^{3} + 5006999286743146 T^{4} - 7610385467044641560 T^{5} +$$$$14\!\cdots\!79$$$$T^{6} -$$$$12\!\cdots\!95$$$$T^{7} +$$$$23\!\cdots\!21$$$$T^{8} )^{2}$$
$83$ $$1 - 81188291 T^{2} + 8014262915096365 T^{4} -$$$$34\!\cdots\!54$$$$T^{6} +$$$$22\!\cdots\!34$$$$T^{8} -$$$$78\!\cdots\!14$$$$T^{10} +$$$$40\!\cdots\!65$$$$T^{12} -$$$$92\!\cdots\!11$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16}$$
$89$ $$1 - 294759296 T^{2} + 46567064448316540 T^{4} -$$$$48\!\cdots\!04$$$$T^{6} +$$$$35\!\cdots\!14$$$$T^{8} -$$$$19\!\cdots\!24$$$$T^{10} +$$$$72\!\cdots\!40$$$$T^{12} -$$$$17\!\cdots\!36$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16}$$
$97$ $$( 1 + 7282 T + 336254488 T^{2} + 1720884578884 T^{3} + 43500636050571385 T^{4} +$$$$15\!\cdots\!04$$$$T^{5} +$$$$26\!\cdots\!68$$$$T^{6} +$$$$50\!\cdots\!62$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{2}$$