# Properties

 Label 324.4.e.e Level 324 Weight 4 Character orbit 324.e Analytic conductor 19.117 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 37 - 37 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 37 - 37 \zeta_{6} ) q^{7} + 19 \zeta_{6} q^{13} -163 q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} -308 \zeta_{6} q^{31} + 323 q^{37} + ( 520 - 520 \zeta_{6} ) q^{43} -1026 \zeta_{6} q^{49} + ( -719 + 719 \zeta_{6} ) q^{61} + 127 \zeta_{6} q^{67} -919 q^{73} + ( 1387 - 1387 \zeta_{6} ) q^{79} + 703 q^{91} + ( 523 - 523 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 37q^{7} + O(q^{10})$$ $$2q + 37q^{7} + 19q^{13} - 326q^{19} + 125q^{25} - 308q^{31} + 646q^{37} + 520q^{43} - 1026q^{49} - 719q^{61} + 127q^{67} - 1838q^{73} + 1387q^{79} + 1406q^{91} + 523q^{97} + O(q^{100})$$

## 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 18.5000 + 32.0429i 0 0 0
217.1 0 0 0 0 0 18.5000 32.0429i 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 324.4.e.e 2
3.b odd 2 1 CM 324.4.e.e 2
9.c even 3 1 108.4.a.b 1
9.c even 3 1 inner 324.4.e.e 2
9.d odd 6 1 108.4.a.b 1
9.d odd 6 1 inner 324.4.e.e 2
36.f odd 6 1 432.4.a.h 1
36.h even 6 1 432.4.a.h 1
72.j odd 6 1 1728.4.a.o 1
72.l even 6 1 1728.4.a.r 1
72.n even 6 1 1728.4.a.o 1
72.p odd 6 1 1728.4.a.r 1

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

 $$T_{5}$$ $$T_{7}^{2} - 37 T_{7} + 1369$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 125 T^{2} + 15625 T^{4}$$
$7$ $$( 1 - 20 T + 343 T^{2} )( 1 - 17 T + 343 T^{2} )$$
$11$ $$1 - 1331 T^{2} + 1771561 T^{4}$$
$13$ $$( 1 - 89 T + 2197 T^{2} )( 1 + 70 T + 2197 T^{2} )$$
$17$ $$( 1 + 4913 T^{2} )^{2}$$
$19$ $$( 1 + 163 T + 6859 T^{2} )^{2}$$
$23$ $$1 - 12167 T^{2} + 148035889 T^{4}$$
$29$ $$1 - 24389 T^{2} + 594823321 T^{4}$$
$31$ $$( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} )$$
$37$ $$( 1 - 323 T + 50653 T^{2} )^{2}$$
$41$ $$1 - 68921 T^{2} + 4750104241 T^{4}$$
$43$ $$( 1 - 449 T + 79507 T^{2} )( 1 - 71 T + 79507 T^{2} )$$
$47$ $$1 - 103823 T^{2} + 10779215329 T^{4}$$
$53$ $$( 1 + 148877 T^{2} )^{2}$$
$59$ $$1 - 205379 T^{2} + 42180533641 T^{4}$$
$61$ $$( 1 - 182 T + 226981 T^{2} )( 1 + 901 T + 226981 T^{2} )$$
$67$ $$( 1 - 1007 T + 300763 T^{2} )( 1 + 880 T + 300763 T^{2} )$$
$71$ $$( 1 + 357911 T^{2} )^{2}$$
$73$ $$( 1 + 919 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 - 884 T + 493039 T^{2} )( 1 - 503 T + 493039 T^{2} )$$
$83$ $$1 - 571787 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 704969 T^{2} )^{2}$$
$97$ $$( 1 - 1853 T + 912673 T^{2} )( 1 + 1330 T + 912673 T^{2} )$$