# Properties

 Label 324.6.e.d Level $324$ Weight $6$ Character orbit 324.e Analytic conductor $51.964$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$51.9643576194$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4) 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.

 $$f(q)$$ $$=$$ $$q + 54 \zeta_{6} q^{5} + ( 88 - 88 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 54 \zeta_{6} q^{5} + ( 88 - 88 \zeta_{6} ) q^{7} + ( 540 - 540 \zeta_{6} ) q^{11} + 418 \zeta_{6} q^{13} -594 q^{17} + 836 q^{19} -4104 \zeta_{6} q^{23} + ( 209 - 209 \zeta_{6} ) q^{25} + ( -594 + 594 \zeta_{6} ) q^{29} -4256 \zeta_{6} q^{31} + 4752 q^{35} -298 q^{37} + 17226 \zeta_{6} q^{41} + ( 12100 - 12100 \zeta_{6} ) q^{43} + ( -1296 + 1296 \zeta_{6} ) q^{47} + 9063 \zeta_{6} q^{49} -19494 q^{53} + 29160 q^{55} -7668 \zeta_{6} q^{59} + ( 34738 - 34738 \zeta_{6} ) q^{61} + ( -22572 + 22572 \zeta_{6} ) q^{65} -21812 \zeta_{6} q^{67} + 46872 q^{71} + 67562 q^{73} -47520 \zeta_{6} q^{77} + ( 76912 - 76912 \zeta_{6} ) q^{79} + ( 67716 - 67716 \zeta_{6} ) q^{83} -32076 \zeta_{6} q^{85} -29754 q^{89} + 36784 q^{91} + 45144 \zeta_{6} q^{95} + ( 122398 - 122398 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 54 q^{5} + 88 q^{7} + O(q^{10})$$ $$2 q + 54 q^{5} + 88 q^{7} + 540 q^{11} + 418 q^{13} - 1188 q^{17} + 1672 q^{19} - 4104 q^{23} + 209 q^{25} - 594 q^{29} - 4256 q^{31} + 9504 q^{35} - 596 q^{37} + 17226 q^{41} + 12100 q^{43} - 1296 q^{47} + 9063 q^{49} - 38988 q^{53} + 58320 q^{55} - 7668 q^{59} + 34738 q^{61} - 22572 q^{65} - 21812 q^{67} + 93744 q^{71} + 135124 q^{73} - 47520 q^{77} + 76912 q^{79} + 67716 q^{83} - 32076 q^{85} - 59508 q^{89} + 73568 q^{91} + 45144 q^{95} + 122398 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 27.0000 46.7654i 0 44.0000 + 76.2102i 0 0 0
217.1 0 0 0 27.0000 + 46.7654i 0 44.0000 76.2102i 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 324.6.e.d 2
3.b odd 2 1 324.6.e.a 2
9.c even 3 1 36.6.a.a 1
9.c even 3 1 inner 324.6.e.d 2
9.d odd 6 1 4.6.a.a 1
9.d odd 6 1 324.6.e.a 2
36.f odd 6 1 144.6.a.c 1
36.h even 6 1 16.6.a.b 1
45.h odd 6 1 100.6.a.b 1
45.j even 6 1 900.6.a.h 1
45.k odd 12 2 900.6.d.a 2
45.l even 12 2 100.6.c.b 2
63.i even 6 1 196.6.e.d 2
63.j odd 6 1 196.6.e.g 2
63.n odd 6 1 196.6.e.g 2
63.o even 6 1 196.6.a.e 1
63.s even 6 1 196.6.e.d 2
72.j odd 6 1 64.6.a.f 1
72.l even 6 1 64.6.a.b 1
72.n even 6 1 576.6.a.bc 1
72.p odd 6 1 576.6.a.bd 1
99.g even 6 1 484.6.a.a 1
117.n odd 6 1 676.6.a.a 1
117.z even 12 2 676.6.d.a 2
144.u even 12 2 256.6.b.c 2
144.w odd 12 2 256.6.b.g 2
180.n even 6 1 400.6.a.d 1
180.v odd 12 2 400.6.c.f 2
252.s odd 6 1 784.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 9.d odd 6 1
16.6.a.b 1 36.h even 6 1
36.6.a.a 1 9.c even 3 1
64.6.a.b 1 72.l even 6 1
64.6.a.f 1 72.j odd 6 1
100.6.a.b 1 45.h odd 6 1
100.6.c.b 2 45.l even 12 2
144.6.a.c 1 36.f odd 6 1
196.6.a.e 1 63.o even 6 1
196.6.e.d 2 63.i even 6 1
196.6.e.d 2 63.s even 6 1
196.6.e.g 2 63.j odd 6 1
196.6.e.g 2 63.n odd 6 1
256.6.b.c 2 144.u even 12 2
256.6.b.g 2 144.w odd 12 2
324.6.e.a 2 3.b odd 2 1
324.6.e.a 2 9.d odd 6 1
324.6.e.d 2 1.a even 1 1 trivial
324.6.e.d 2 9.c even 3 1 inner
400.6.a.d 1 180.n even 6 1
400.6.c.f 2 180.v odd 12 2
484.6.a.a 1 99.g even 6 1
576.6.a.bc 1 72.n even 6 1
576.6.a.bd 1 72.p odd 6 1
676.6.a.a 1 117.n odd 6 1
676.6.d.a 2 117.z even 12 2
784.6.a.d 1 252.s odd 6 1
900.6.a.h 1 45.j even 6 1
900.6.d.a 2 45.k odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 54 T_{5} + 2916$$ $$T_{7}^{2} - 88 T_{7} + 7744$$ $$T_{11}^{2} - 540 T_{11} + 291600$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$2916 - 54 T + T^{2}$$
$7$ $$7744 - 88 T + T^{2}$$
$11$ $$291600 - 540 T + T^{2}$$
$13$ $$174724 - 418 T + T^{2}$$
$17$ $$( 594 + T )^{2}$$
$19$ $$( -836 + T )^{2}$$
$23$ $$16842816 + 4104 T + T^{2}$$
$29$ $$352836 + 594 T + T^{2}$$
$31$ $$18113536 + 4256 T + T^{2}$$
$37$ $$( 298 + T )^{2}$$
$41$ $$296735076 - 17226 T + T^{2}$$
$43$ $$146410000 - 12100 T + T^{2}$$
$47$ $$1679616 + 1296 T + T^{2}$$
$53$ $$( 19494 + T )^{2}$$
$59$ $$58798224 + 7668 T + T^{2}$$
$61$ $$1206728644 - 34738 T + T^{2}$$
$67$ $$475763344 + 21812 T + T^{2}$$
$71$ $$( -46872 + T )^{2}$$
$73$ $$( -67562 + T )^{2}$$
$79$ $$5915455744 - 76912 T + T^{2}$$
$83$ $$4585456656 - 67716 T + T^{2}$$
$89$ $$( 29754 + T )^{2}$$
$97$ $$14981270404 - 122398 T + T^{2}$$