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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 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.

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

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 $$S_{4}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} - 37T_{7} + 1369$$ T7^2 - 37*T7 + 1369

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 37T + 1369$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 19T + 361$$
$17$ $$T^{2}$$
$19$ $$(T + 163)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 308T + 94864$$
$37$ $$(T - 323)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 520T + 270400$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 719T + 516961$$
$67$ $$T^{2} - 127T + 16129$$
$71$ $$T^{2}$$
$73$ $$(T + 919)^{2}$$
$79$ $$T^{2} - 1387 T + 1923769$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 523T + 273529$$