# Properties

 Label 324.4.e.h Level $324$ Weight $4$ Character orbit 324.e Analytic conductor $19.117$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$19.1166188419$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 18 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7}+O(q^{10})$$ q + 18*z * q^5 + (8*z - 8) * q^7 $$q + 18 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + (36 \zeta_{6} - 36) q^{11} + 10 \zeta_{6} q^{13} + 18 q^{17} - 100 q^{19} - 72 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} + ( - 234 \zeta_{6} + 234) q^{29} + 16 \zeta_{6} q^{31} - 144 q^{35} - 226 q^{37} - 90 \zeta_{6} q^{41} + (452 \zeta_{6} - 452) q^{43} + (432 \zeta_{6} - 432) q^{47} + 279 \zeta_{6} q^{49} + 414 q^{53} - 648 q^{55} + 684 \zeta_{6} q^{59} + (422 \zeta_{6} - 422) q^{61} + (180 \zeta_{6} - 180) q^{65} - 332 \zeta_{6} q^{67} - 360 q^{71} + 26 q^{73} - 288 \zeta_{6} q^{77} + (512 \zeta_{6} - 512) q^{79} + ( - 1188 \zeta_{6} + 1188) q^{83} + 324 \zeta_{6} q^{85} - 630 q^{89} - 80 q^{91} - 1800 \zeta_{6} q^{95} + ( - 1054 \zeta_{6} + 1054) q^{97} +O(q^{100})$$ q + 18*z * q^5 + (8*z - 8) * q^7 + (36*z - 36) * q^11 + 10*z * q^13 + 18 * q^17 - 100 * q^19 - 72*z * q^23 + (199*z - 199) * q^25 + (-234*z + 234) * q^29 + 16*z * q^31 - 144 * q^35 - 226 * q^37 - 90*z * q^41 + (452*z - 452) * q^43 + (432*z - 432) * q^47 + 279*z * q^49 + 414 * q^53 - 648 * q^55 + 684*z * q^59 + (422*z - 422) * q^61 + (180*z - 180) * q^65 - 332*z * q^67 - 360 * q^71 + 26 * q^73 - 288*z * q^77 + (512*z - 512) * q^79 + (-1188*z + 1188) * q^83 + 324*z * q^85 - 630 * q^89 - 80 * q^91 - 1800*z * q^95 + (-1054*z + 1054) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 18 q^{5} - 8 q^{7}+O(q^{10})$$ 2 * q + 18 * q^5 - 8 * q^7 $$2 q + 18 q^{5} - 8 q^{7} - 36 q^{11} + 10 q^{13} + 36 q^{17} - 200 q^{19} - 72 q^{23} - 199 q^{25} + 234 q^{29} + 16 q^{31} - 288 q^{35} - 452 q^{37} - 90 q^{41} - 452 q^{43} - 432 q^{47} + 279 q^{49} + 828 q^{53} - 1296 q^{55} + 684 q^{59} - 422 q^{61} - 180 q^{65} - 332 q^{67} - 720 q^{71} + 52 q^{73} - 288 q^{77} - 512 q^{79} + 1188 q^{83} + 324 q^{85} - 1260 q^{89} - 160 q^{91} - 1800 q^{95} + 1054 q^{97}+O(q^{100})$$ 2 * q + 18 * q^5 - 8 * q^7 - 36 * q^11 + 10 * q^13 + 36 * q^17 - 200 * q^19 - 72 * q^23 - 199 * q^25 + 234 * q^29 + 16 * q^31 - 288 * q^35 - 452 * q^37 - 90 * q^41 - 452 * q^43 - 432 * q^47 + 279 * q^49 + 828 * q^53 - 1296 * q^55 + 684 * q^59 - 422 * q^61 - 180 * q^65 - 332 * q^67 - 720 * q^71 + 52 * q^73 - 288 * q^77 - 512 * q^79 + 1188 * q^83 + 324 * q^85 - 1260 * q^89 - 160 * q^91 - 1800 * q^95 + 1054 * q^97

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 9.00000 15.5885i 0 −4.00000 6.92820i 0 0 0
217.1 0 0 0 9.00000 + 15.5885i 0 −4.00000 + 6.92820i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.e.h 2
3.b odd 2 1 324.4.e.a 2
9.c even 3 1 12.4.a.a 1
9.c even 3 1 inner 324.4.e.h 2
9.d odd 6 1 36.4.a.a 1
9.d odd 6 1 324.4.e.a 2
36.f odd 6 1 48.4.a.a 1
36.h even 6 1 144.4.a.g 1
45.h odd 6 1 900.4.a.g 1
45.j even 6 1 300.4.a.b 1
45.k odd 12 2 300.4.d.e 2
45.l even 12 2 900.4.d.c 2
63.g even 3 1 588.4.i.d 2
63.h even 3 1 588.4.i.d 2
63.i even 6 1 1764.4.k.o 2
63.j odd 6 1 1764.4.k.b 2
63.k odd 6 1 588.4.i.e 2
63.l odd 6 1 588.4.a.c 1
63.n odd 6 1 1764.4.k.b 2
63.o even 6 1 1764.4.a.b 1
63.s even 6 1 1764.4.k.o 2
63.t odd 6 1 588.4.i.e 2
72.j odd 6 1 576.4.a.b 1
72.l even 6 1 576.4.a.a 1
72.n even 6 1 192.4.a.f 1
72.p odd 6 1 192.4.a.l 1
99.h odd 6 1 1452.4.a.d 1
117.t even 6 1 2028.4.a.c 1
117.y odd 12 2 2028.4.b.c 2
144.v odd 12 2 768.4.d.j 2
144.x even 12 2 768.4.d.g 2
180.p odd 6 1 1200.4.a.be 1
180.x even 12 2 1200.4.f.d 2
252.bi even 6 1 2352.4.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 9.c even 3 1
36.4.a.a 1 9.d odd 6 1
48.4.a.a 1 36.f odd 6 1
144.4.a.g 1 36.h even 6 1
192.4.a.f 1 72.n even 6 1
192.4.a.l 1 72.p odd 6 1
300.4.a.b 1 45.j even 6 1
300.4.d.e 2 45.k odd 12 2
324.4.e.a 2 3.b odd 2 1
324.4.e.a 2 9.d odd 6 1
324.4.e.h 2 1.a even 1 1 trivial
324.4.e.h 2 9.c even 3 1 inner
576.4.a.a 1 72.l even 6 1
576.4.a.b 1 72.j odd 6 1
588.4.a.c 1 63.l odd 6 1
588.4.i.d 2 63.g even 3 1
588.4.i.d 2 63.h even 3 1
588.4.i.e 2 63.k odd 6 1
588.4.i.e 2 63.t odd 6 1
768.4.d.g 2 144.x even 12 2
768.4.d.j 2 144.v odd 12 2
900.4.a.g 1 45.h odd 6 1
900.4.d.c 2 45.l even 12 2
1200.4.a.be 1 180.p odd 6 1
1200.4.f.d 2 180.x even 12 2
1452.4.a.d 1 99.h odd 6 1
1764.4.a.b 1 63.o even 6 1
1764.4.k.b 2 63.j odd 6 1
1764.4.k.b 2 63.n odd 6 1
1764.4.k.o 2 63.i even 6 1
1764.4.k.o 2 63.s even 6 1
2028.4.a.c 1 117.t even 6 1
2028.4.b.c 2 117.y odd 12 2
2352.4.a.bk 1 252.bi even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 18T_{5} + 324$$ T5^2 - 18*T5 + 324 $$T_{7}^{2} + 8T_{7} + 64$$ T7^2 + 8*T7 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 18T + 324$$
$7$ $$T^{2} + 8T + 64$$
$11$ $$T^{2} + 36T + 1296$$
$13$ $$T^{2} - 10T + 100$$
$17$ $$(T - 18)^{2}$$
$19$ $$(T + 100)^{2}$$
$23$ $$T^{2} + 72T + 5184$$
$29$ $$T^{2} - 234T + 54756$$
$31$ $$T^{2} - 16T + 256$$
$37$ $$(T + 226)^{2}$$
$41$ $$T^{2} + 90T + 8100$$
$43$ $$T^{2} + 452T + 204304$$
$47$ $$T^{2} + 432T + 186624$$
$53$ $$(T - 414)^{2}$$
$59$ $$T^{2} - 684T + 467856$$
$61$ $$T^{2} + 422T + 178084$$
$67$ $$T^{2} + 332T + 110224$$
$71$ $$(T + 360)^{2}$$
$73$ $$(T - 26)^{2}$$
$79$ $$T^{2} + 512T + 262144$$
$83$ $$T^{2} - 1188 T + 1411344$$
$89$ $$(T + 630)^{2}$$
$97$ $$T^{2} - 1054 T + 1110916$$