# Properties

 Label 324.3.f.o Level 324 Weight 3 Character orbit 324.f Analytic conductor 8.828 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.207360000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{1} + \beta_{7} ) q^{4} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{1} + \beta_{7} ) q^{4} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{8} + ( -2 + 8 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{10} + ( 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{11} + ( \beta_{1} + 2 \beta_{5} + 4 \beta_{7} ) q^{13} + ( 3 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{14} + ( 14 \beta_{1} + 2 \beta_{5} ) q^{16} + ( 10 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{6} ) q^{17} + ( 3 - 6 \beta_{1} ) q^{19} + ( 8 \beta_{3} + 4 \beta_{4} ) q^{20} + ( 20 - 10 \beta_{1} + 6 \beta_{5} + 4 \beta_{7} ) q^{22} + ( -7 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} ) q^{23} + ( -3 + 3 \beta_{1} + 8 \beta_{5} + 4 \beta_{7} ) q^{25} + ( -4 \beta_{2} + 7 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} ) q^{26} + ( 2 - 31 \beta_{1} - 3 \beta_{5} - 2 \beta_{7} ) q^{28} + ( -4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{29} + ( 16 - 8 \beta_{1} + 8 \beta_{5} ) q^{31} + ( 12 \beta_{3} + 4 \beta_{6} ) q^{32} + ( 6 - 38 \beta_{1} + 8 \beta_{5} + 2 \beta_{7} ) q^{34} + ( 4 \beta_{2} - 24 \beta_{3} - 10 \beta_{4} + 3 \beta_{6} ) q^{35} + ( -7 - 2 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{38} + ( -36 + 28 \beta_{1} + 8 \beta_{5} + 4 \beta_{7} ) q^{40} + ( 16 \beta_{2} - 8 \beta_{6} ) q^{41} + ( 20 + 20 \beta_{1} + 4 \beta_{7} ) q^{43} + ( -24 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} + 4 \beta_{6} ) q^{44} + ( 22 + 28 \beta_{1} - 10 \beta_{5} - 6 \beta_{7} ) q^{46} + ( 3 \beta_{2} - 21 \beta_{3} - \beta_{4} + 10 \beta_{6} ) q^{47} + ( 14 \beta_{1} + 4 \beta_{5} + 8 \beta_{7} ) q^{49} + ( -\beta_{2} + 8 \beta_{4} + 8 \beta_{6} ) q^{50} + ( -31 + 60 \beta_{1} + 3 \beta_{5} - \beta_{7} ) q^{52} + ( -12 \beta_{2} + 28 \beta_{3} - 8 \beta_{4} - 10 \beta_{6} ) q^{53} + ( -20 + 40 \beta_{1} + 4 \beta_{5} + 4 \beta_{7} ) q^{55} + ( -30 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} ) q^{56} + ( 32 - 24 \beta_{1} - 8 \beta_{5} ) q^{58} + ( -19 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} - 5 \beta_{6} ) q^{59} + ( -5 + 5 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} ) q^{61} + ( -16 \beta_{2} + 16 \beta_{6} ) q^{62} + ( -44 + 12 \beta_{5} + 12 \beta_{7} ) q^{64} + ( 11 \beta_{2} + 19 \beta_{3} + 11 \beta_{4} + 4 \beta_{6} ) q^{65} + ( -38 + 19 \beta_{1} - 16 \beta_{5} ) q^{67} + ( -8 \beta_{2} - 36 \beta_{3} + 4 \beta_{4} + 12 \beta_{6} ) q^{68} + ( 70 - 74 \beta_{1} - 20 \beta_{5} - 14 \beta_{7} ) q^{70} + ( 24 \beta_{2} - 48 \beta_{3} - 12 \beta_{4} - 6 \beta_{6} ) q^{71} + ( 29 + 4 \beta_{5} - 4 \beta_{7} ) q^{73} + ( 5 \beta_{2} - \beta_{3} + 4 \beta_{4} - 8 \beta_{6} ) q^{74} + ( 3 + 3 \beta_{1} - 6 \beta_{5} - 3 \beta_{7} ) q^{76} + ( 43 \beta_{2} - 33 \beta_{3} - 11 \beta_{4} - 5 \beta_{6} ) q^{77} + ( -55 - 55 \beta_{1} - 14 \beta_{7} ) q^{79} + ( 32 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{6} ) q^{80} + ( 64 - 16 \beta_{1} + 16 \beta_{5} ) q^{82} + ( -6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} ) q^{83} + ( -52 \beta_{1} - 12 \beta_{5} - 24 \beta_{7} ) q^{85} + ( -24 \beta_{2} + 48 \beta_{3} + 8 \beta_{4} - 8 \beta_{6} ) q^{86} + ( -100 + 80 \beta_{1} - 12 \beta_{5} + 4 \beta_{7} ) q^{88} + ( 10 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{6} ) q^{89} + ( 61 - 122 \beta_{1} - 8 \beta_{5} - 8 \beta_{7} ) q^{91} + ( -16 \beta_{2} + 48 \beta_{3} - 12 \beta_{4} - 8 \beta_{6} ) q^{92} + ( -52 - 10 \beta_{1} - 18 \beta_{5} - 20 \beta_{7} ) q^{94} + ( -9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} ) q^{95} + ( 31 - 31 \beta_{1} - 32 \beta_{5} - 16 \beta_{7} ) q^{97} + ( -8 \beta_{2} + 26 \beta_{3} + 16 \beta_{4} - 8 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} - 12q^{7} + O(q^{10})$$ $$8q - 4q^{4} - 12q^{7} + 16q^{10} + 4q^{13} + 56q^{16} + 120q^{22} - 12q^{25} - 108q^{28} + 96q^{31} - 104q^{34} - 56q^{37} - 176q^{40} + 240q^{43} + 288q^{46} + 56q^{49} - 8q^{52} + 160q^{58} - 20q^{61} - 352q^{64} - 228q^{67} + 264q^{70} + 232q^{73} + 36q^{76} - 660q^{79} + 448q^{82} - 208q^{85} - 480q^{88} - 456q^{94} + 124q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{6} - 16 \nu^{4} - 96 \nu^{2} - 8$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} - 24 \nu^{3} + 24 \nu$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} + 32 \nu^{5} + 160 \nu^{3} + 184 \nu$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} - 16 \nu^{5} - 64 \nu^{3} + 264 \nu$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{6} - 48 \nu^{4} - 224 \nu^{2} - 312$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} - 16 \nu^{5} - 80 \nu^{3} + 24 \nu$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-11 \nu^{6} - 48 \nu^{4} - 224 \nu^{2} + 120$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{6} - \beta_{4} + 3 \beta_{3} - \beta_{2}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{5} - 9 \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{6} + 5 \beta_{4} - 3 \beta_{3} - 7 \beta_{2}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{7} - 4 \beta_{5} + 14 \beta_{1} - 14$$ $$\nu^{5}$$ $$=$$ $$($$$$-22 \beta_{6} - 4 \beta_{4} - 12 \beta_{3} + 56 \beta_{2}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-32 \beta_{7} + 32 \beta_{5} + 216$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$136 \beta_{6} - 116 \beta_{4} + 156 \beta_{3} - 116 \beta_{2}$$$$)/3$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$-1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.437016 − 0.756934i −1.14412 + 1.98168i 1.14412 − 1.98168i −0.437016 + 0.756934i 0.437016 + 0.756934i −1.14412 − 1.98168i 1.14412 + 1.98168i −0.437016 − 0.756934i
−1.85123 0.756934i 0 2.85410 + 2.80252i −3.70246 6.41285i 0 −8.20820 4.73901i −3.16228 7.34847i 0 2.00000 + 14.6742i
55.2 −0.270091 + 1.98168i 0 −3.85410 1.07047i −0.540182 0.935622i 0 5.20820 + 3.00696i 3.16228 7.34847i 0 2.00000 0.817763i
55.3 0.270091 1.98168i 0 −3.85410 1.07047i 0.540182 + 0.935622i 0 5.20820 + 3.00696i −3.16228 + 7.34847i 0 2.00000 0.817763i
55.4 1.85123 + 0.756934i 0 2.85410 + 2.80252i 3.70246 + 6.41285i 0 −8.20820 4.73901i 3.16228 + 7.34847i 0 2.00000 + 14.6742i
271.1 −1.85123 + 0.756934i 0 2.85410 2.80252i −3.70246 + 6.41285i 0 −8.20820 + 4.73901i −3.16228 + 7.34847i 0 2.00000 14.6742i
271.2 −0.270091 1.98168i 0 −3.85410 + 1.07047i −0.540182 + 0.935622i 0 5.20820 3.00696i 3.16228 + 7.34847i 0 2.00000 + 0.817763i
271.3 0.270091 + 1.98168i 0 −3.85410 + 1.07047i 0.540182 0.935622i 0 5.20820 3.00696i −3.16228 7.34847i 0 2.00000 + 0.817763i
271.4 1.85123 0.756934i 0 2.85410 2.80252i 3.70246 6.41285i 0 −8.20820 + 4.73901i 3.16228 7.34847i 0 2.00000 14.6742i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.4 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
36.f odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.o 8
3.b odd 2 1 inner 324.3.f.o 8
4.b odd 2 1 324.3.f.p 8
9.c even 3 1 108.3.d.d 8
9.c even 3 1 324.3.f.p 8
9.d odd 6 1 108.3.d.d 8
9.d odd 6 1 324.3.f.p 8
12.b even 2 1 324.3.f.p 8
36.f odd 6 1 108.3.d.d 8
36.f odd 6 1 inner 324.3.f.o 8
36.h even 6 1 108.3.d.d 8
36.h even 6 1 inner 324.3.f.o 8
72.j odd 6 1 1728.3.g.l 8
72.l even 6 1 1728.3.g.l 8
72.n even 6 1 1728.3.g.l 8
72.p odd 6 1 1728.3.g.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.d 8 9.c even 3 1
108.3.d.d 8 9.d odd 6 1
108.3.d.d 8 36.f odd 6 1
108.3.d.d 8 36.h even 6 1
324.3.f.o 8 1.a even 1 1 trivial
324.3.f.o 8 3.b odd 2 1 inner
324.3.f.o 8 36.f odd 6 1 inner
324.3.f.o 8 36.h even 6 1 inner
324.3.f.p 8 4.b odd 2 1
324.3.f.p 8 9.c even 3 1
324.3.f.p 8 9.d odd 6 1
324.3.f.p 8 12.b even 2 1
1728.3.g.l 8 72.j odd 6 1
1728.3.g.l 8 72.l even 6 1
1728.3.g.l 8 72.n even 6 1
1728.3.g.l 8 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{8} + 56 T_{5}^{6} + 3072 T_{5}^{4} + 3584 T_{5}^{2} + 4096$$ $$T_{7}^{4} + 6 T_{7}^{3} - 45 T_{7}^{2} - 342 T_{7} + 3249$$ $$T_{11}^{8} - 360 T_{11}^{6} + 115200 T_{11}^{4} - 5184000 T_{11}^{2} + 207360000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} - 12 T^{4} + 32 T^{6} + 256 T^{8}$$
$3$ 1
$5$ $$1 - 44 T^{2} + 922 T^{4} + 10384 T^{6} - 572429 T^{8} + 6490000 T^{10} + 360156250 T^{12} - 10742187500 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 + 6 T + 53 T^{2} + 246 T^{3} - 132 T^{4} + 12054 T^{5} + 127253 T^{6} + 705894 T^{7} + 5764801 T^{8} )^{2}$$
$11$ $$1 + 124 T^{2} + 250 T^{4} - 1755344 T^{6} - 210683021 T^{8} - 25699991504 T^{10} + 53589720250 T^{12} + 389165118713404 T^{14} + 45949729863572161 T^{16}$$
$13$ $$( 1 - 2 T - 155 T^{2} + 358 T^{3} - 3956 T^{4} + 60502 T^{5} - 4426955 T^{6} - 9653618 T^{7} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 695 T^{2} + 130321 T^{4} )^{4}$$
$23$ $$1 + 604 T^{2} - 268070 T^{4} + 44215216 T^{6} + 220143229459 T^{8} + 12373230260656 T^{10} - 20992825824277670 T^{12} + 13236433156940273884 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 - 2468 T^{2} + 3338026 T^{4} - 3303260048 T^{6} + 2770686029299 T^{8} - 2336333070009488 T^{10} + 1669835532870554986 T^{12} -$$$$87\!\cdots\!88$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$( 1 - 48 T + 1922 T^{2} - 55392 T^{3} + 1146243 T^{4} - 53231712 T^{5} + 1775007362 T^{6} - 42600176688 T^{7} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$1 - 3140 T^{2} + 4692298 T^{4} + 1520450800 T^{6} - 9144788425517 T^{8} + 4296430573058800 T^{10} + 37467648682754010058 T^{12} -$$$$70\!\cdots\!40$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 - 120 T + 9458 T^{2} - 558960 T^{3} + 27153363 T^{4} - 1033517040 T^{5} + 32335019858 T^{6} - 758563565880 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 + 4444 T^{2} + 5312410 T^{4} + 20786205616 T^{6} + 88238287231699 T^{8} + 101430052606488496 T^{10} +$$$$12\!\cdots\!10$$$$T^{12} +$$$$51\!\cdots\!04$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 + 6076 T^{2} + 12975610 T^{4} - 1777570256 T^{6} - 20949143941901 T^{8} - 21539460494814416 T^{10} +$$$$19\!\cdots\!10$$$$T^{12} +$$$$10\!\cdots\!56$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$( 1 + 10 T - 7187 T^{2} - 1550 T^{3} + 38882428 T^{4} - 5767550 T^{5} - 99510059267 T^{6} + 515203743610 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 + 114 T + 10553 T^{2} + 709194 T^{3} + 37996068 T^{4} + 3183571866 T^{5} + 212654779913 T^{6} + 10312255567266 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4}$$
$79$ $$( 1 + 330 T + 54917 T^{2} + 6143610 T^{3} + 534190908 T^{4} + 38342270010 T^{5} + 2139021598277 T^{6} + 80218860321930 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 + 26116 T^{2} + 416905450 T^{4} + 4445553374224 T^{6} + 35678198362101619 T^{8} +$$$$21\!\cdots\!04$$$$T^{10} +$$$$93\!\cdots\!50$$$$T^{12} +$$$$27\!\cdots\!76$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 62 T - 4415 T^{2} + 654658 T^{3} - 56486396 T^{4} + 6159677122 T^{5} - 390856775615 T^{6} - 51644264305598 T^{7} + 7837433594376961 T^{8} )^{2}$$