# Properties

 Label 324.4.e.f 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7}+O(q^{10})$$ q + 3*z * q^5 + (-4*z + 4) * q^7 $$q + 3 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + (24 \zeta_{6} - 24) q^{11} + 25 \zeta_{6} q^{13} + 21 q^{17} - 52 q^{19} + 168 \zeta_{6} q^{23} + ( - 116 \zeta_{6} + 116) q^{25} + (177 \zeta_{6} - 177) q^{29} + 124 \zeta_{6} q^{31} + 12 q^{35} - 265 q^{37} + 426 \zeta_{6} q^{41} + ( - 160 \zeta_{6} + 160) q^{43} + (540 \zeta_{6} - 540) q^{47} + 327 \zeta_{6} q^{49} + 258 q^{53} - 72 q^{55} + 528 \zeta_{6} q^{59} + ( - 505 \zeta_{6} + 505) q^{61} + (75 \zeta_{6} - 75) q^{65} + 244 \zeta_{6} q^{67} - 204 q^{71} - 397 q^{73} + 96 \zeta_{6} q^{77} + (200 \zeta_{6} - 200) q^{79} + (540 \zeta_{6} - 540) q^{83} + 63 \zeta_{6} q^{85} + 453 q^{89} + 100 q^{91} - 156 \zeta_{6} q^{95} + (290 \zeta_{6} - 290) q^{97} +O(q^{100})$$ q + 3*z * q^5 + (-4*z + 4) * q^7 + (24*z - 24) * q^11 + 25*z * q^13 + 21 * q^17 - 52 * q^19 + 168*z * q^23 + (-116*z + 116) * q^25 + (177*z - 177) * q^29 + 124*z * q^31 + 12 * q^35 - 265 * q^37 + 426*z * q^41 + (-160*z + 160) * q^43 + (540*z - 540) * q^47 + 327*z * q^49 + 258 * q^53 - 72 * q^55 + 528*z * q^59 + (-505*z + 505) * q^61 + (75*z - 75) * q^65 + 244*z * q^67 - 204 * q^71 - 397 * q^73 + 96*z * q^77 + (200*z - 200) * q^79 + (540*z - 540) * q^83 + 63*z * q^85 + 453 * q^89 + 100 * q^91 - 156*z * q^95 + (290*z - 290) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + 3 * q^5 + 4 * q^7 $$2 q + 3 q^{5} + 4 q^{7} - 24 q^{11} + 25 q^{13} + 42 q^{17} - 104 q^{19} + 168 q^{23} + 116 q^{25} - 177 q^{29} + 124 q^{31} + 24 q^{35} - 530 q^{37} + 426 q^{41} + 160 q^{43} - 540 q^{47} + 327 q^{49} + 516 q^{53} - 144 q^{55} + 528 q^{59} + 505 q^{61} - 75 q^{65} + 244 q^{67} - 408 q^{71} - 794 q^{73} + 96 q^{77} - 200 q^{79} - 540 q^{83} + 63 q^{85} + 906 q^{89} + 200 q^{91} - 156 q^{95} - 290 q^{97}+O(q^{100})$$ 2 * q + 3 * q^5 + 4 * q^7 - 24 * q^11 + 25 * q^13 + 42 * q^17 - 104 * q^19 + 168 * q^23 + 116 * q^25 - 177 * q^29 + 124 * q^31 + 24 * q^35 - 530 * q^37 + 426 * q^41 + 160 * q^43 - 540 * q^47 + 327 * q^49 + 516 * q^53 - 144 * q^55 + 528 * q^59 + 505 * q^61 - 75 * q^65 + 244 * q^67 - 408 * q^71 - 794 * q^73 + 96 * q^77 - 200 * q^79 - 540 * q^83 + 63 * q^85 + 906 * q^89 + 200 * q^91 - 156 * q^95 - 290 * 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 1.50000 2.59808i 0 2.00000 + 3.46410i 0 0 0
217.1 0 0 0 1.50000 + 2.59808i 0 2.00000 3.46410i 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.f 2
3.b odd 2 1 324.4.e.c 2
9.c even 3 1 324.4.a.a 1
9.c even 3 1 inner 324.4.e.f 2
9.d odd 6 1 324.4.a.b yes 1
9.d odd 6 1 324.4.e.c 2
36.f odd 6 1 1296.4.a.d 1
36.h even 6 1 1296.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.4.a.a 1 9.c even 3 1
324.4.a.b yes 1 9.d odd 6 1
324.4.e.c 2 3.b odd 2 1
324.4.e.c 2 9.d odd 6 1
324.4.e.f 2 1.a even 1 1 trivial
324.4.e.f 2 9.c even 3 1 inner
1296.4.a.d 1 36.f odd 6 1
1296.4.a.e 1 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 $$S_{4}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2} + 24T + 576$$
$13$ $$T^{2} - 25T + 625$$
$17$ $$(T - 21)^{2}$$
$19$ $$(T + 52)^{2}$$
$23$ $$T^{2} - 168T + 28224$$
$29$ $$T^{2} + 177T + 31329$$
$31$ $$T^{2} - 124T + 15376$$
$37$ $$(T + 265)^{2}$$
$41$ $$T^{2} - 426T + 181476$$
$43$ $$T^{2} - 160T + 25600$$
$47$ $$T^{2} + 540T + 291600$$
$53$ $$(T - 258)^{2}$$
$59$ $$T^{2} - 528T + 278784$$
$61$ $$T^{2} - 505T + 255025$$
$67$ $$T^{2} - 244T + 59536$$
$71$ $$(T + 204)^{2}$$
$73$ $$(T + 397)^{2}$$
$79$ $$T^{2} + 200T + 40000$$
$83$ $$T^{2} + 540T + 291600$$
$89$ $$(T - 453)^{2}$$
$97$ $$T^{2} + 290T + 84100$$