# Properties

 Label 324.6.e.c Level 324 Weight 6 Character orbit 324.e Analytic conductor 51.964 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 25 - 25 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 25 - 25 \zeta_{6} ) q^{7} + 427 \zeta_{6} q^{13} -1711 q^{19} + ( 3125 - 3125 \zeta_{6} ) q^{25} + 10324 \zeta_{6} q^{31} -6661 q^{37} + ( 3352 - 3352 \zeta_{6} ) q^{43} + 16182 \zeta_{6} q^{49} + ( -56927 + 56927 \zeta_{6} ) q^{61} + 37939 \zeta_{6} q^{67} + 79577 q^{73} + ( -90857 + 90857 \zeta_{6} ) q^{79} + 10675 q^{91} + ( -177725 + 177725 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 25q^{7} + O(q^{10})$$ $$2q + 25q^{7} + 427q^{13} - 3422q^{19} + 3125q^{25} + 10324q^{31} - 13322q^{37} + 3352q^{43} + 16182q^{49} - 56927q^{61} + 37939q^{67} + 159154q^{73} - 90857q^{79} + 21350q^{91} - 177725q^{97} + O(q^{100})$$

## 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 0 0 12.5000 + 21.6506i 0 0 0
217.1 0 0 0 0 0 12.5000 21.6506i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
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 324.6.e.c 2
3.b odd 2 1 CM 324.6.e.c 2
9.c even 3 1 108.6.a.a 1
9.c even 3 1 inner 324.6.e.c 2
9.d odd 6 1 108.6.a.a 1
9.d odd 6 1 inner 324.6.e.c 2
36.f odd 6 1 432.6.a.e 1
36.h even 6 1 432.6.a.e 1

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

 $$T_{5}$$ $$T_{7}^{2} - 25 T_{7} + 625$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 3125 T^{2} + 9765625 T^{4}$$
$7$ $$( 1 - 236 T + 16807 T^{2} )( 1 + 211 T + 16807 T^{2} )$$
$11$ $$1 - 161051 T^{2} + 25937424601 T^{4}$$
$13$ $$( 1 - 1202 T + 371293 T^{2} )( 1 + 775 T + 371293 T^{2} )$$
$17$ $$( 1 + 1419857 T^{2} )^{2}$$
$19$ $$( 1 + 1711 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 6436343 T^{2} + 41426511213649 T^{4}$$
$29$ $$1 - 20511149 T^{2} + 420707233300201 T^{4}$$
$31$ $$( 1 - 7601 T + 28629151 T^{2} )( 1 - 2723 T + 28629151 T^{2} )$$
$37$ $$( 1 + 6661 T + 69343957 T^{2} )^{2}$$
$41$ $$1 - 115856201 T^{2} + 13422659310152401 T^{4}$$
$43$ $$( 1 - 22475 T + 147008443 T^{2} )( 1 + 19123 T + 147008443 T^{2} )$$
$47$ $$1 - 229345007 T^{2} + 52599132235830049 T^{4}$$
$53$ $$( 1 + 418195493 T^{2} )^{2}$$
$59$ $$1 - 714924299 T^{2} + 511116753300641401 T^{4}$$
$61$ $$( 1 + 18301 T + 844596301 T^{2} )( 1 + 38626 T + 844596301 T^{2} )$$
$67$ $$( 1 - 73475 T + 1350125107 T^{2} )( 1 + 35536 T + 1350125107 T^{2} )$$
$71$ $$( 1 + 1804229351 T^{2} )^{2}$$
$73$ $$( 1 - 79577 T + 2073071593 T^{2} )^{2}$$
$79$ $$( 1 - 9707 T + 3077056399 T^{2} )( 1 + 100564 T + 3077056399 T^{2} )$$
$83$ $$1 - 3939040643 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 + 43339 T + 8587340257 T^{2} )( 1 + 134386 T + 8587340257 T^{2} )$$