# Properties

 Label 324.3.g.d Level $324$ Weight $3$ Character orbit 324.g 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.g (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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{10}$$ Twist minimal: yes 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{5} + ( -3 \zeta_{24}^{2} + \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{7} +O(q^{10})$$ $$q + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{5} + ( -3 \zeta_{24}^{2} + \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{7} + ( 9 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{11} + ( 1 - 6 \zeta_{24}^{2} - \zeta_{24}^{4} + 12 \zeta_{24}^{6} ) q^{13} + ( 21 \zeta_{24} - 21 \zeta_{24}^{3} - 18 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{17} + ( -7 + 30 \zeta_{24}^{2} - 15 \zeta_{24}^{6} ) q^{19} + ( 24 \zeta_{24} + 9 \zeta_{24}^{3} - 15 \zeta_{24}^{5} + 15 \zeta_{24}^{7} ) q^{23} + ( -9 \zeta_{24}^{2} - 7 \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{25} + ( 27 \zeta_{24} + 33 \zeta_{24}^{3} - 33 \zeta_{24}^{5} - 27 \zeta_{24}^{7} ) q^{29} + ( -8 - 18 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 36 \zeta_{24}^{6} ) q^{31} + ( 21 \zeta_{24} - 21 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{35} + ( 14 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{37} + ( 30 \zeta_{24} - 9 \zeta_{24}^{3} - 39 \zeta_{24}^{5} + 39 \zeta_{24}^{7} ) q^{41} + ( -27 \zeta_{24}^{2} + 13 \zeta_{24}^{4} - 27 \zeta_{24}^{6} ) q^{43} + ( 18 \zeta_{24} - 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{47} + ( 21 + 6 \zeta_{24}^{2} - 21 \zeta_{24}^{4} - 12 \zeta_{24}^{6} ) q^{49} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 27 \zeta_{24}^{5} - 30 \zeta_{24}^{7} ) q^{53} + ( 9 + 18 \zeta_{24}^{2} - 9 \zeta_{24}^{6} ) q^{55} + ( -30 \zeta_{24} + 18 \zeta_{24}^{3} + 48 \zeta_{24}^{5} - 48 \zeta_{24}^{7} ) q^{59} + ( 27 \zeta_{24}^{2} + 16 \zeta_{24}^{4} + 27 \zeta_{24}^{6} ) q^{61} + ( -18 \zeta_{24} + 21 \zeta_{24}^{3} - 21 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{65} + ( -5 - 9 \zeta_{24}^{2} + 5 \zeta_{24}^{4} + 18 \zeta_{24}^{6} ) q^{67} + ( -15 \zeta_{24} + 15 \zeta_{24}^{3} + 63 \zeta_{24}^{5} + 48 \zeta_{24}^{7} ) q^{71} + ( -64 - 54 \zeta_{24}^{2} + 27 \zeta_{24}^{6} ) q^{73} + ( -30 \zeta_{24} - 36 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{77} + ( 21 \zeta_{24}^{2} - 77 \zeta_{24}^{4} + 21 \zeta_{24}^{6} ) q^{79} + ( -90 \zeta_{24} - 72 \zeta_{24}^{3} + 72 \zeta_{24}^{5} + 90 \zeta_{24}^{7} ) q^{83} + ( -72 + 63 \zeta_{24}^{2} + 72 \zeta_{24}^{4} - 126 \zeta_{24}^{6} ) q^{85} + ( -60 \zeta_{24} + 60 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 51 \zeta_{24}^{7} ) q^{89} + ( 55 - 18 \zeta_{24}^{2} + 9 \zeta_{24}^{6} ) q^{91} + ( -66 \zeta_{24} + 45 \zeta_{24}^{3} + 111 \zeta_{24}^{5} - 111 \zeta_{24}^{7} ) q^{95} + ( 66 \zeta_{24}^{2} + 40 \zeta_{24}^{4} + 66 \zeta_{24}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{7} + O(q^{10})$$ $$8 q + 4 q^{7} + 4 q^{13} - 56 q^{19} - 28 q^{25} - 32 q^{31} + 112 q^{37} + 52 q^{43} + 84 q^{49} + 72 q^{55} + 64 q^{61} - 20 q^{67} - 512 q^{73} - 308 q^{79} - 288 q^{85} + 440 q^{91} + 160 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{24}$$

## 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
 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i
0 0 0 −5.01910 2.89778i 0 3.09808 + 5.36603i 0 0 0
53.2 0 0 0 −1.34486 0.776457i 0 −2.09808 3.63397i 0 0 0
53.3 0 0 0 1.34486 + 0.776457i 0 −2.09808 3.63397i 0 0 0
53.4 0 0 0 5.01910 + 2.89778i 0 3.09808 + 5.36603i 0 0 0
269.1 0 0 0 −5.01910 + 2.89778i 0 3.09808 5.36603i 0 0 0
269.2 0 0 0 −1.34486 + 0.776457i 0 −2.09808 + 3.63397i 0 0 0
269.3 0 0 0 1.34486 0.776457i 0 −2.09808 + 3.63397i 0 0 0
269.4 0 0 0 5.01910 2.89778i 0 3.09808 5.36603i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.4 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
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 324.3.g.d 8
3.b odd 2 1 inner 324.3.g.d 8
4.b odd 2 1 1296.3.q.n 8
9.c even 3 1 324.3.c.a 4
9.c even 3 1 inner 324.3.g.d 8
9.d odd 6 1 324.3.c.a 4
9.d odd 6 1 inner 324.3.g.d 8
12.b even 2 1 1296.3.q.n 8
36.f odd 6 1 1296.3.e.f 4
36.f odd 6 1 1296.3.q.n 8
36.h even 6 1 1296.3.e.f 4
36.h even 6 1 1296.3.q.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.c.a 4 9.c even 3 1
324.3.c.a 4 9.d odd 6 1
324.3.g.d 8 1.a even 1 1 trivial
324.3.g.d 8 3.b odd 2 1 inner
324.3.g.d 8 9.c even 3 1 inner
324.3.g.d 8 9.d odd 6 1 inner
1296.3.e.f 4 36.f odd 6 1
1296.3.e.f 4 36.h even 6 1
1296.3.q.n 8 4.b odd 2 1
1296.3.q.n 8 12.b even 2 1
1296.3.q.n 8 36.f odd 6 1
1296.3.q.n 8 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 $$S_{3}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{8} - 36 T_{5}^{6} + 1215 T_{5}^{4} - 2916 T_{5}^{2} + 6561$$ $$T_{7}^{4} - 2 T_{7}^{3} + 30 T_{7}^{2} + 52 T_{7} + 676$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$6561 - 2916 T^{2} + 1215 T^{4} - 36 T^{6} + T^{8}$$
$7$ $$( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$104976 - 81648 T^{2} + 63180 T^{4} - 252 T^{6} + T^{8}$$
$13$ $$( 11449 + 214 T + 111 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$17$ $$( 558009 + 1548 T^{2} + T^{4} )^{2}$$
$19$ $$( -626 + 14 T + T^{2} )^{4}$$
$23$ $$512249392656 - 1262523024 T^{2} + 2395980 T^{4} - 1764 T^{6} + T^{8}$$
$29$ $$3986287771761 - 7403277852 T^{2} + 11752695 T^{4} - 3708 T^{6} + T^{8}$$
$31$ $$( 824464 - 14528 T + 1164 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$37$ $$( 169 - 28 T + T^{2} )^{4}$$
$41$ $$1536953616 - 196176816 T^{2} + 25000812 T^{4} - 5004 T^{6} + T^{8}$$
$43$ $$( 4072324 + 52468 T + 2694 T^{2} - 26 T^{3} + T^{4} )^{2}$$
$47$ $$892616806656 - 3673320192 T^{2} + 14171760 T^{4} - 3888 T^{6} + T^{8}$$
$53$ $$( 1127844 + 3276 T^{2} + T^{4} )^{2}$$
$59$ $$470025421056 - 4837480704 T^{2} + 49101552 T^{4} - 7056 T^{6} + T^{8}$$
$61$ $$( 3728761 + 61792 T + 2955 T^{2} - 32 T^{3} + T^{4} )^{2}$$
$67$ $$( 47524 - 2180 T + 318 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$71$ $$( 171396 + 12996 T^{2} + T^{4} )^{2}$$
$73$ $$( 1909 + 128 T + T^{2} )^{4}$$
$79$ $$( 21215236 + 709324 T + 19110 T^{2} + 154 T^{3} + T^{4} )^{2}$$
$83$ $$14281868906496 - 102852965376 T^{2} + 736931520 T^{4} - 27216 T^{6} + T^{8}$$
$89$ $$( 2313441 + 12564 T^{2} + T^{4} )^{2}$$
$97$ $$( 131515024 + 917440 T + 17868 T^{2} - 80 T^{3} + T^{4} )^{2}$$