# Properties

 Label 324.4.e.g 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) 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.

 $$f(q)$$ $$=$$ $$q + 9 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 9 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + ( 63 - 63 \zeta_{6} ) q^{11} + 28 \zeta_{6} q^{13} -72 q^{17} + 98 q^{19} + 126 \zeta_{6} q^{23} + ( 44 - 44 \zeta_{6} ) q^{25} + ( -126 + 126 \zeta_{6} ) q^{29} + 259 \zeta_{6} q^{31} + 9 q^{35} + 386 q^{37} -450 \zeta_{6} q^{41} + ( 34 - 34 \zeta_{6} ) q^{43} + ( -54 + 54 \zeta_{6} ) q^{47} + 342 \zeta_{6} q^{49} + 693 q^{53} + 567 q^{55} + 180 \zeta_{6} q^{59} + ( 280 - 280 \zeta_{6} ) q^{61} + ( -252 + 252 \zeta_{6} ) q^{65} + 586 \zeta_{6} q^{67} -504 q^{71} + 161 q^{73} -63 \zeta_{6} q^{77} + ( -440 + 440 \zeta_{6} ) q^{79} + ( 999 - 999 \zeta_{6} ) q^{83} -648 \zeta_{6} q^{85} -882 q^{89} + 28 q^{91} + 882 \zeta_{6} q^{95} + ( 721 - 721 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 9q^{5} + q^{7} + O(q^{10})$$ $$2q + 9q^{5} + q^{7} + 63q^{11} + 28q^{13} - 144q^{17} + 196q^{19} + 126q^{23} + 44q^{25} - 126q^{29} + 259q^{31} + 18q^{35} + 772q^{37} - 450q^{41} + 34q^{43} - 54q^{47} + 342q^{49} + 1386q^{53} + 1134q^{55} + 180q^{59} + 280q^{61} - 252q^{65} + 586q^{67} - 1008q^{71} + 322q^{73} - 63q^{77} - 440q^{79} + 999q^{83} - 648q^{85} - 1764q^{89} + 56q^{91} + 882q^{95} + 721q^{97} + O(q^{100})$$

## 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 4.50000 7.79423i 0 0.500000 + 0.866025i 0 0 0
217.1 0 0 0 4.50000 + 7.79423i 0 0.500000 0.866025i 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.g 2
3.b odd 2 1 324.4.e.b 2
9.c even 3 1 108.4.a.a 1
9.c even 3 1 inner 324.4.e.g 2
9.d odd 6 1 108.4.a.d yes 1
9.d odd 6 1 324.4.e.b 2
36.f odd 6 1 432.4.a.c 1
36.h even 6 1 432.4.a.l 1
72.j odd 6 1 1728.4.a.g 1
72.l even 6 1 1728.4.a.h 1
72.n even 6 1 1728.4.a.y 1
72.p odd 6 1 1728.4.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.a 1 9.c even 3 1
108.4.a.d yes 1 9.d odd 6 1
324.4.e.b 2 3.b odd 2 1
324.4.e.b 2 9.d odd 6 1
324.4.e.g 2 1.a even 1 1 trivial
324.4.e.g 2 9.c even 3 1 inner
432.4.a.c 1 36.f odd 6 1
432.4.a.l 1 36.h even 6 1
1728.4.a.g 1 72.j odd 6 1
1728.4.a.h 1 72.l even 6 1
1728.4.a.y 1 72.n even 6 1
1728.4.a.z 1 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 $$S_{4}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{2} - 9 T_{5} + 81$$ $$T_{7}^{2} - T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 9 T - 44 T^{2} - 1125 T^{3} + 15625 T^{4}$$
$7$ $$1 - T - 342 T^{2} - 343 T^{3} + 117649 T^{4}$$
$11$ $$1 - 63 T + 2638 T^{2} - 83853 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 28 T - 1413 T^{2} - 61516 T^{3} + 4826809 T^{4}$$
$17$ $$( 1 + 72 T + 4913 T^{2} )^{2}$$
$19$ $$( 1 - 98 T + 6859 T^{2} )^{2}$$
$23$ $$1 - 126 T + 3709 T^{2} - 1533042 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 126 T - 8513 T^{2} + 3073014 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 259 T + 37290 T^{2} - 7715869 T^{3} + 887503681 T^{4}$$
$37$ $$( 1 - 386 T + 50653 T^{2} )^{2}$$
$41$ $$1 + 450 T + 133579 T^{2} + 31014450 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 34 T - 78351 T^{2} - 2703238 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 54 T - 100907 T^{2} + 5606442 T^{3} + 10779215329 T^{4}$$
$53$ $$( 1 - 693 T + 148877 T^{2} )^{2}$$
$59$ $$1 - 180 T - 172979 T^{2} - 36968220 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 280 T - 148581 T^{2} - 63554680 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 586 T + 42633 T^{2} - 176247118 T^{3} + 90458382169 T^{4}$$
$71$ $$( 1 + 504 T + 357911 T^{2} )^{2}$$
$73$ $$( 1 - 161 T + 389017 T^{2} )^{2}$$
$79$ $$1 + 440 T - 299439 T^{2} + 216937160 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 999 T + 426214 T^{2} - 571215213 T^{3} + 326940373369 T^{4}$$
$89$ $$( 1 + 882 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 721 T - 392832 T^{2} - 658037233 T^{3} + 832972004929 T^{4}$$