Properties

 Label 441.2.e.c Level 441 Weight 2 Character orbit 441.e Analytic conductor 3.521 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{4} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{4} + 7 q^{13} -4 \zeta_{6} q^{16} -7 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -7 + 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{37} + 5 q^{43} + ( 14 - 14 \zeta_{6} ) q^{52} + 14 \zeta_{6} q^{61} -8 q^{64} + ( -11 + 11 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{73} -14 q^{76} + 13 \zeta_{6} q^{79} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} + 14q^{13} - 4q^{16} - 7q^{19} + 5q^{25} - 7q^{31} + q^{37} + 10q^{43} + 14q^{52} + 14q^{61} - 16q^{64} - 11q^{67} - 7q^{73} - 28q^{76} + 13q^{79} - 28q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 1.00000 + 1.73205i 0 0 0 0 0 0
361.1 0 0 1.00000 1.73205i 0 0 0 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 $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.c 2
3.b odd 2 1 CM 441.2.e.c 2
7.b odd 2 1 63.2.e.a 2
7.c even 3 1 441.2.a.e 1
7.c even 3 1 inner 441.2.e.c 2
7.d odd 6 1 63.2.e.a 2
7.d odd 6 1 441.2.a.d 1
21.c even 2 1 63.2.e.a 2
21.g even 6 1 63.2.e.a 2
21.g even 6 1 441.2.a.d 1
21.h odd 6 1 441.2.a.e 1
21.h odd 6 1 inner 441.2.e.c 2
28.d even 2 1 1008.2.s.j 2
28.f even 6 1 1008.2.s.j 2
28.f even 6 1 7056.2.a.y 1
28.g odd 6 1 7056.2.a.bf 1
63.i even 6 1 567.2.g.d 2
63.k odd 6 1 567.2.h.c 2
63.l odd 6 1 567.2.g.d 2
63.l odd 6 1 567.2.h.c 2
63.o even 6 1 567.2.g.d 2
63.o even 6 1 567.2.h.c 2
63.s even 6 1 567.2.h.c 2
63.t odd 6 1 567.2.g.d 2
84.h odd 2 1 1008.2.s.j 2
84.j odd 6 1 1008.2.s.j 2
84.j odd 6 1 7056.2.a.y 1
84.n even 6 1 7056.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 7.b odd 2 1
63.2.e.a 2 7.d odd 6 1
63.2.e.a 2 21.c even 2 1
63.2.e.a 2 21.g even 6 1
441.2.a.d 1 7.d odd 6 1
441.2.a.d 1 21.g even 6 1
441.2.a.e 1 7.c even 3 1
441.2.a.e 1 21.h odd 6 1
441.2.e.c 2 1.a even 1 1 trivial
441.2.e.c 2 3.b odd 2 1 CM
441.2.e.c 2 7.c even 3 1 inner
441.2.e.c 2 21.h odd 6 1 inner
567.2.g.d 2 63.i even 6 1
567.2.g.d 2 63.l odd 6 1
567.2.g.d 2 63.o even 6 1
567.2.g.d 2 63.t odd 6 1
567.2.h.c 2 63.k odd 6 1
567.2.h.c 2 63.l odd 6 1
567.2.h.c 2 63.o even 6 1
567.2.h.c 2 63.s even 6 1
1008.2.s.j 2 28.d even 2 1
1008.2.s.j 2 28.f even 6 1
1008.2.s.j 2 84.h odd 2 1
1008.2.s.j 2 84.j odd 6 1
7056.2.a.y 1 28.f even 6 1
7056.2.a.y 1 84.j odd 6 1
7056.2.a.bf 1 28.g odd 6 1
7056.2.a.bf 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}$$ $$T_{5}$$ $$T_{13} - 7$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ 1
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$