# Properties

 Label 324.3.f.h Level $324$ Weight $3$ Character orbit 324.f Analytic conductor $8.828$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + ( - 5 \zeta_{6} + 10) q^{7} - 8 q^{8} +O(q^{10})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 + (-7*z + 7) * q^5 + (-5*z + 10) * q^7 - 8 * q^8 $$q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + ( - 5 \zeta_{6} + 10) q^{7} - 8 q^{8} - 14 \zeta_{6} q^{10} + ( - 5 \zeta_{6} + 10) q^{11} + (20 \zeta_{6} - 20) q^{13} + ( - 20 \zeta_{6} + 10) q^{14} + (16 \zeta_{6} - 16) q^{16} + 8 q^{17} + ( - 12 \zeta_{6} + 6) q^{19} - 28 q^{20} + ( - 20 \zeta_{6} + 10) q^{22} + (2 \zeta_{6} + 2) q^{23} - 24 \zeta_{6} q^{25} + 40 \zeta_{6} q^{26} + ( - 20 \zeta_{6} - 20) q^{28} + 10 \zeta_{6} q^{29} + ( - 31 \zeta_{6} - 31) q^{31} + 32 \zeta_{6} q^{32} + ( - 16 \zeta_{6} + 16) q^{34} + ( - 70 \zeta_{6} + 35) q^{35} - 10 q^{37} + ( - 12 \zeta_{6} - 12) q^{38} + (56 \zeta_{6} - 56) q^{40} + (50 \zeta_{6} - 50) q^{41} + (10 \zeta_{6} - 20) q^{43} + ( - 20 \zeta_{6} - 20) q^{44} + ( - 4 \zeta_{6} + 8) q^{46} + ( - 50 \zeta_{6} + 100) q^{47} + ( - 26 \zeta_{6} + 26) q^{49} - 48 q^{50} + 80 q^{52} + 47 q^{53} + ( - 70 \zeta_{6} + 35) q^{55} + (40 \zeta_{6} - 80) q^{56} + 20 q^{58} + (20 \zeta_{6} + 20) q^{59} + 64 \zeta_{6} q^{61} + (62 \zeta_{6} - 124) q^{62} + 64 q^{64} + 140 \zeta_{6} q^{65} + (50 \zeta_{6} + 50) q^{67} - 32 \zeta_{6} q^{68} + ( - 70 \zeta_{6} - 70) q^{70} - 55 q^{73} + (20 \zeta_{6} - 20) q^{74} + (24 \zeta_{6} - 48) q^{76} + ( - 75 \zeta_{6} + 75) q^{77} + (4 \zeta_{6} - 8) q^{79} + 112 \zeta_{6} q^{80} + 100 \zeta_{6} q^{82} + ( - 17 \zeta_{6} + 34) q^{83} + ( - 56 \zeta_{6} + 56) q^{85} + (40 \zeta_{6} - 20) q^{86} + (40 \zeta_{6} - 80) q^{88} - 10 q^{89} + (200 \zeta_{6} - 100) q^{91} + ( - 16 \zeta_{6} + 8) q^{92} + ( - 200 \zeta_{6} + 100) q^{94} + ( - 42 \zeta_{6} - 42) q^{95} + 25 \zeta_{6} q^{97} - 52 \zeta_{6} q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 - 4*z * q^4 + (-7*z + 7) * q^5 + (-5*z + 10) * q^7 - 8 * q^8 - 14*z * q^10 + (-5*z + 10) * q^11 + (20*z - 20) * q^13 + (-20*z + 10) * q^14 + (16*z - 16) * q^16 + 8 * q^17 + (-12*z + 6) * q^19 - 28 * q^20 + (-20*z + 10) * q^22 + (2*z + 2) * q^23 - 24*z * q^25 + 40*z * q^26 + (-20*z - 20) * q^28 + 10*z * q^29 + (-31*z - 31) * q^31 + 32*z * q^32 + (-16*z + 16) * q^34 + (-70*z + 35) * q^35 - 10 * q^37 + (-12*z - 12) * q^38 + (56*z - 56) * q^40 + (50*z - 50) * q^41 + (10*z - 20) * q^43 + (-20*z - 20) * q^44 + (-4*z + 8) * q^46 + (-50*z + 100) * q^47 + (-26*z + 26) * q^49 - 48 * q^50 + 80 * q^52 + 47 * q^53 + (-70*z + 35) * q^55 + (40*z - 80) * q^56 + 20 * q^58 + (20*z + 20) * q^59 + 64*z * q^61 + (62*z - 124) * q^62 + 64 * q^64 + 140*z * q^65 + (50*z + 50) * q^67 - 32*z * q^68 + (-70*z - 70) * q^70 - 55 * q^73 + (20*z - 20) * q^74 + (24*z - 48) * q^76 + (-75*z + 75) * q^77 + (4*z - 8) * q^79 + 112*z * q^80 + 100*z * q^82 + (-17*z + 34) * q^83 + (-56*z + 56) * q^85 + (40*z - 20) * q^86 + (40*z - 80) * q^88 - 10 * q^89 + (200*z - 100) * q^91 + (-16*z + 8) * q^92 + (-200*z + 100) * q^94 + (-42*z - 42) * q^95 + 25*z * q^97 - 52*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} + 7 q^{5} + 15 q^{7} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 + 7 * q^5 + 15 * q^7 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} + 7 q^{5} + 15 q^{7} - 16 q^{8} - 14 q^{10} + 15 q^{11} - 20 q^{13} - 16 q^{16} + 16 q^{17} - 56 q^{20} + 6 q^{23} - 24 q^{25} + 40 q^{26} - 60 q^{28} + 10 q^{29} - 93 q^{31} + 32 q^{32} + 16 q^{34} - 20 q^{37} - 36 q^{38} - 56 q^{40} - 50 q^{41} - 30 q^{43} - 60 q^{44} + 12 q^{46} + 150 q^{47} + 26 q^{49} - 96 q^{50} + 160 q^{52} + 94 q^{53} - 120 q^{56} + 40 q^{58} + 60 q^{59} + 64 q^{61} - 186 q^{62} + 128 q^{64} + 140 q^{65} + 150 q^{67} - 32 q^{68} - 210 q^{70} - 110 q^{73} - 20 q^{74} - 72 q^{76} + 75 q^{77} - 12 q^{79} + 112 q^{80} + 100 q^{82} + 51 q^{83} + 56 q^{85} - 120 q^{88} - 20 q^{89} - 126 q^{95} + 25 q^{97} - 52 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 + 7 * q^5 + 15 * q^7 - 16 * q^8 - 14 * q^10 + 15 * q^11 - 20 * q^13 - 16 * q^16 + 16 * q^17 - 56 * q^20 + 6 * q^23 - 24 * q^25 + 40 * q^26 - 60 * q^28 + 10 * q^29 - 93 * q^31 + 32 * q^32 + 16 * q^34 - 20 * q^37 - 36 * q^38 - 56 * q^40 - 50 * q^41 - 30 * q^43 - 60 * q^44 + 12 * q^46 + 150 * q^47 + 26 * q^49 - 96 * q^50 + 160 * q^52 + 94 * q^53 - 120 * q^56 + 40 * q^58 + 60 * q^59 + 64 * q^61 - 186 * q^62 + 128 * q^64 + 140 * q^65 + 150 * q^67 - 32 * q^68 - 210 * q^70 - 110 * q^73 - 20 * q^74 - 72 * q^76 + 75 * q^77 - 12 * q^79 + 112 * q^80 + 100 * q^82 + 51 * q^83 + 56 * q^85 - 120 * q^88 - 20 * q^89 - 126 * q^95 + 25 * q^97 - 52 * q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 3.50000 + 6.06218i 0 7.50000 + 4.33013i −8.00000 0 −7.00000 + 12.1244i
271.1 1.00000 1.73205i 0 −2.00000 3.46410i 3.50000 6.06218i 0 7.50000 4.33013i −8.00000 0 −7.00000 12.1244i
 $$n$$: 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
36.f odd 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.h 2
3.b odd 2 1 324.3.f.c 2
4.b odd 2 1 324.3.f.b 2
9.c even 3 1 108.3.d.b yes 2
9.c even 3 1 324.3.f.b 2
9.d odd 6 1 108.3.d.a 2
9.d odd 6 1 324.3.f.i 2
12.b even 2 1 324.3.f.i 2
36.f odd 6 1 108.3.d.b yes 2
36.f odd 6 1 inner 324.3.f.h 2
36.h even 6 1 108.3.d.a 2
36.h even 6 1 324.3.f.c 2
72.j odd 6 1 1728.3.g.a 2
72.l even 6 1 1728.3.g.a 2
72.n even 6 1 1728.3.g.f 2
72.p odd 6 1 1728.3.g.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 9.d odd 6 1
108.3.d.a 2 36.h even 6 1
108.3.d.b yes 2 9.c even 3 1
108.3.d.b yes 2 36.f odd 6 1
324.3.f.b 2 4.b odd 2 1
324.3.f.b 2 9.c even 3 1
324.3.f.c 2 3.b odd 2 1
324.3.f.c 2 36.h even 6 1
324.3.f.h 2 1.a even 1 1 trivial
324.3.f.h 2 36.f odd 6 1 inner
324.3.f.i 2 9.d odd 6 1
324.3.f.i 2 12.b even 2 1
1728.3.g.a 2 72.j odd 6 1
1728.3.g.a 2 72.l even 6 1
1728.3.g.f 2 72.n even 6 1
1728.3.g.f 2 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 $$S_{3}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 7T_{5} + 49$$ T5^2 - 7*T5 + 49 $$T_{7}^{2} - 15T_{7} + 75$$ T7^2 - 15*T7 + 75 $$T_{11}^{2} - 15T_{11} + 75$$ T11^2 - 15*T11 + 75

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 7T + 49$$
$7$ $$T^{2} - 15T + 75$$
$11$ $$T^{2} - 15T + 75$$
$13$ $$T^{2} + 20T + 400$$
$17$ $$(T - 8)^{2}$$
$19$ $$T^{2} + 108$$
$23$ $$T^{2} - 6T + 12$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$T^{2} + 93T + 2883$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 50T + 2500$$
$43$ $$T^{2} + 30T + 300$$
$47$ $$T^{2} - 150T + 7500$$
$53$ $$(T - 47)^{2}$$
$59$ $$T^{2} - 60T + 1200$$
$61$ $$T^{2} - 64T + 4096$$
$67$ $$T^{2} - 150T + 7500$$
$71$ $$T^{2}$$
$73$ $$(T + 55)^{2}$$
$79$ $$T^{2} + 12T + 48$$
$83$ $$T^{2} - 51T + 867$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 25T + 625$$