Properties

 Label 432.2.s.d Level 432 Weight 2 Character orbit 432.s Analytic conductor 3.450 Analytic rank 0 Dimension 2 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.s (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 4 - 2 \zeta_{6} ) q^{5} + ( 2 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 4 - 2 \zeta_{6} ) q^{5} + ( 2 + 2 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} + ( -1 + 2 \zeta_{6} ) q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} + ( 7 - 7 \zeta_{6} ) q^{25} + ( 2 + 2 \zeta_{6} ) q^{29} + 12 q^{35} + 2 q^{37} + ( 6 - 3 \zeta_{6} ) q^{41} + ( -3 - 3 \zeta_{6} ) q^{43} + ( -12 + 12 \zeta_{6} ) q^{47} + 5 \zeta_{6} q^{49} + ( -6 + 12 \zeta_{6} ) q^{55} -15 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + ( -8 - 8 \zeta_{6} ) q^{65} + ( 10 - 5 \zeta_{6} ) q^{67} -6 q^{71} -11 q^{73} + ( -12 + 6 \zeta_{6} ) q^{77} + ( -2 - 2 \zeta_{6} ) q^{79} + ( 12 - 12 \zeta_{6} ) q^{83} + 6 \zeta_{6} q^{85} + ( -8 + 16 \zeta_{6} ) q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{95} + ( -13 + 13 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{5} + 6q^{7} + O(q^{10})$$ $$2q + 6q^{5} + 6q^{7} - 3q^{11} - 4q^{13} + 7q^{25} + 6q^{29} + 24q^{35} + 4q^{37} + 9q^{41} - 9q^{43} - 12q^{47} + 5q^{49} - 15q^{59} - 8q^{61} - 24q^{65} + 15q^{67} - 12q^{71} - 22q^{73} - 18q^{77} - 6q^{79} + 12q^{83} + 6q^{85} - 6q^{95} - 13q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 3.00000 1.73205i 0 3.00000 + 1.73205i 0 0 0
287.1 0 0 0 3.00000 + 1.73205i 0 3.00000 1.73205i 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
36.h even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.s.d 2
3.b odd 2 1 144.2.s.d yes 2
4.b odd 2 1 432.2.s.c 2
8.b even 2 1 1728.2.s.b 2
8.d odd 2 1 1728.2.s.a 2
9.c even 3 1 144.2.s.a 2
9.c even 3 1 1296.2.c.d 2
9.d odd 6 1 432.2.s.c 2
9.d odd 6 1 1296.2.c.b 2
12.b even 2 1 144.2.s.a 2
24.f even 2 1 576.2.s.d 2
24.h odd 2 1 576.2.s.a 2
36.f odd 6 1 144.2.s.d yes 2
36.f odd 6 1 1296.2.c.b 2
36.h even 6 1 inner 432.2.s.d 2
36.h even 6 1 1296.2.c.d 2
72.j odd 6 1 1728.2.s.a 2
72.j odd 6 1 5184.2.c.c 2
72.l even 6 1 1728.2.s.b 2
72.l even 6 1 5184.2.c.a 2
72.n even 6 1 576.2.s.d 2
72.n even 6 1 5184.2.c.a 2
72.p odd 6 1 576.2.s.a 2
72.p odd 6 1 5184.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.a 2 9.c even 3 1
144.2.s.a 2 12.b even 2 1
144.2.s.d yes 2 3.b odd 2 1
144.2.s.d yes 2 36.f odd 6 1
432.2.s.c 2 4.b odd 2 1
432.2.s.c 2 9.d odd 6 1
432.2.s.d 2 1.a even 1 1 trivial
432.2.s.d 2 36.h even 6 1 inner
576.2.s.a 2 24.h odd 2 1
576.2.s.a 2 72.p odd 6 1
576.2.s.d 2 24.f even 2 1
576.2.s.d 2 72.n even 6 1
1296.2.c.b 2 9.d odd 6 1
1296.2.c.b 2 36.f odd 6 1
1296.2.c.d 2 9.c even 3 1
1296.2.c.d 2 36.h even 6 1
1728.2.s.a 2 8.d odd 2 1
1728.2.s.a 2 72.j odd 6 1
1728.2.s.b 2 8.b even 2 1
1728.2.s.b 2 72.l even 6 1
5184.2.c.a 2 72.l even 6 1
5184.2.c.a 2 72.n even 6 1
5184.2.c.c 2 72.j odd 6 1
5184.2.c.c 2 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 $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{2} - 6 T_{5} + 12$$ $$T_{7}^{2} - 6 T_{7} + 12$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4}$$
$7$ $$( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} )$$
$11$ $$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$1 - 31 T^{2} + 289 T^{4}$$
$19$ $$1 - 35 T^{2} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$1 - 6 T + 41 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$1 + 31 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4}$$
$43$ $$1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4}$$
$47$ $$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 11 T + 73 T^{2} )^{2}$$
$79$ $$1 + 6 T + 91 T^{2} + 474 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$1 + 14 T^{2} + 7921 T^{4}$$
$97$ $$1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4}$$