Properties

 Label 324.5.g.d Level 324 Weight 5 Character orbit 324.g Analytic conductor 33.492 Analytic rank 0 Dimension 4 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 324.g (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$33.4918680392$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 9 \zeta_{12} q^{5} -5 \zeta_{12}^{2} q^{7} +O(q^{10})$$ $$q + 9 \zeta_{12} q^{5} -5 \zeta_{12}^{2} q^{7} + ( -117 \zeta_{12} + 117 \zeta_{12}^{3} ) q^{11} + ( 34 - 34 \zeta_{12}^{2} ) q^{13} -450 \zeta_{12}^{3} q^{17} -64 q^{19} + 612 \zeta_{12} q^{23} -544 \zeta_{12}^{2} q^{25} + ( -1062 \zeta_{12} + 1062 \zeta_{12}^{3} ) q^{29} + ( 697 - 697 \zeta_{12}^{2} ) q^{31} -45 \zeta_{12}^{3} q^{35} -748 q^{37} + 684 \zeta_{12} q^{41} -2618 \zeta_{12}^{2} q^{43} + ( 2646 \zeta_{12} - 2646 \zeta_{12}^{3} ) q^{47} + ( 2376 - 2376 \zeta_{12}^{2} ) q^{49} + 1071 \zeta_{12}^{3} q^{53} -1053 q^{55} -5814 \zeta_{12} q^{59} -6404 \zeta_{12}^{2} q^{61} + ( 306 \zeta_{12} - 306 \zeta_{12}^{3} ) q^{65} + ( 5218 - 5218 \zeta_{12}^{2} ) q^{67} + 6570 \zeta_{12}^{3} q^{71} -4519 q^{73} + 585 \zeta_{12} q^{77} -7502 \zeta_{12}^{2} q^{79} + ( 5481 \zeta_{12} - 5481 \zeta_{12}^{3} ) q^{83} + ( 4050 - 4050 \zeta_{12}^{2} ) q^{85} -8874 \zeta_{12}^{3} q^{89} -170 q^{91} -576 \zeta_{12} q^{95} -10571 \zeta_{12}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{7} + O(q^{10})$$ $$4q - 10q^{7} + 68q^{13} - 256q^{19} - 1088q^{25} + 1394q^{31} - 2992q^{37} - 5236q^{43} + 4752q^{49} - 4212q^{55} - 12808q^{61} + 10436q^{67} - 18076q^{73} - 15004q^{79} + 8100q^{85} - 680q^{91} - 21142q^{97} + O(q^{100})$$

Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{12}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 −7.79423 4.50000i 0 −2.50000 4.33013i 0 0 0
53.2 0 0 0 7.79423 + 4.50000i 0 −2.50000 4.33013i 0 0 0
269.1 0 0 0 −7.79423 + 4.50000i 0 −2.50000 + 4.33013i 0 0 0
269.2 0 0 0 7.79423 4.50000i 0 −2.50000 + 4.33013i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

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

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{4} - 81 T_{5}^{2} + 6561$$ $$T_{7}^{2} + 5 T_{7} + 25$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 1169 T^{2} + 975936 T^{4} + 456640625 T^{6} + 152587890625 T^{8}$$
$7$ $$( 1 + 5 T - 2376 T^{2} + 12005 T^{3} + 5764801 T^{4} )^{2}$$
$11$ $$1 + 15593 T^{2} + 28782768 T^{4} + 3342498031433 T^{6} + 45949729863572161 T^{8}$$
$13$ $$( 1 - 34 T - 27405 T^{2} - 971074 T^{3} + 815730721 T^{4} )^{2}$$
$17$ $$( 1 + 35458 T^{2} + 6975757441 T^{4} )^{2}$$
$19$ $$( 1 + 64 T + 130321 T^{2} )^{4}$$
$23$ $$1 + 185138 T^{2} - 44034906237 T^{4} + 14498339192953778 T^{6} +$$$$61\!\cdots\!61$$$$T^{8}$$
$29$ $$1 + 286718 T^{2} - 418039201437 T^{4} + 143429651031351998 T^{6} +$$$$25\!\cdots\!21$$$$T^{8}$$
$31$ $$( 1 - 697 T - 437712 T^{2} - 643694137 T^{3} + 852891037441 T^{4} )^{2}$$
$37$ $$( 1 + 748 T + 1874161 T^{2} )^{4}$$
$41$ $$1 + 5183666 T^{2} + 18885467970435 T^{4} + 41391185422736737586 T^{6} +$$$$63\!\cdots\!41$$$$T^{8}$$
$43$ $$( 1 + 2618 T + 3435123 T^{2} + 8950421018 T^{3} + 11688200277601 T^{4} )^{2}$$
$47$ $$1 + 2758046 T^{2} - 16204468923645 T^{4} + 65672623932323279006 T^{6} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$( 1 - 14633921 T^{2} + 62259690411361 T^{4} )^{2}$$
$59$ $$1 - 9567874 T^{2} - 55286224724445 T^{4} -$$$$14\!\cdots\!54$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8}$$
$61$ $$( 1 + 6404 T + 27165375 T^{2} + 88668765764 T^{3} + 191707312997281 T^{4} )^{2}$$
$67$ $$( 1 - 5218 T + 7076403 T^{2} - 105148549378 T^{3} + 406067677556641 T^{4} )^{2}$$
$71$ $$( 1 - 7658462 T^{2} + 645753531245761 T^{4} )^{2}$$
$73$ $$( 1 + 4519 T + 28398241 T^{2} )^{4}$$
$79$ $$( 1 + 7502 T + 17329923 T^{2} + 292203507662 T^{3} + 1517108809906561 T^{4} )^{2}$$
$83$ $$1 + 64875281 T^{2} + 1956509852689920 T^{4} +$$$$14\!\cdots\!21$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 - 46736606 T^{2} + 3936588805702081 T^{4} )^{2}$$
$97$ $$( 1 + 10571 T + 23216760 T^{2} + 935843029451 T^{3} + 7837433594376961 T^{4} )^{2}$$