# Properties

 Label 441.2.e.e Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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 21) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{10} + ( -2 + 2 \zeta_{6} ) q^{11} - q^{13} + 4 \zeta_{6} q^{16} + \zeta_{6} q^{19} -4 q^{20} -4 q^{22} + ( 1 - \zeta_{6} ) q^{25} -2 \zeta_{6} q^{26} -4 q^{29} + ( 9 - 9 \zeta_{6} ) q^{31} + ( -8 + 8 \zeta_{6} ) q^{32} -3 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} -10 q^{41} + 5 q^{43} -4 \zeta_{6} q^{44} + 6 \zeta_{6} q^{47} + 2 q^{50} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 12 - 12 \zeta_{6} ) q^{53} -4 q^{55} -8 \zeta_{6} q^{58} + ( 12 - 12 \zeta_{6} ) q^{59} + 10 \zeta_{6} q^{61} + 18 q^{62} -8 q^{64} -2 \zeta_{6} q^{65} + ( 5 - 5 \zeta_{6} ) q^{67} + 6 q^{71} + ( -3 + 3 \zeta_{6} ) q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} -2 q^{76} + \zeta_{6} q^{79} + ( -8 + 8 \zeta_{6} ) q^{80} -20 \zeta_{6} q^{82} + 6 q^{83} + 10 \zeta_{6} q^{86} -16 \zeta_{6} q^{89} + ( -12 + 12 \zeta_{6} ) q^{94} + ( -2 + 2 \zeta_{6} ) q^{95} + 6 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{4} + 2q^{5} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{4} + 2q^{5} - 4q^{10} - 2q^{11} - 2q^{13} + 4q^{16} + q^{19} - 8q^{20} - 8q^{22} + q^{25} - 2q^{26} - 8q^{29} + 9q^{31} - 8q^{32} - 3q^{37} - 2q^{38} - 20q^{41} + 10q^{43} - 4q^{44} + 6q^{47} + 4q^{50} + 2q^{52} + 12q^{53} - 8q^{55} - 8q^{58} + 12q^{59} + 10q^{61} + 36q^{62} - 16q^{64} - 2q^{65} + 5q^{67} + 12q^{71} - 3q^{73} + 6q^{74} - 4q^{76} + q^{79} - 8q^{80} - 20q^{82} + 12q^{83} + 10q^{86} - 16q^{89} - 12q^{94} - 2q^{95} + 12q^{97} + O(q^{100})$$

## 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
1.00000 1.73205i 0 −1.00000 1.73205i 1.00000 1.73205i 0 0 0 0 −2.00000 3.46410i
361.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i 1.00000 + 1.73205i 0 0 0 0 −2.00000 + 3.46410i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.e.e 2
3.b odd 2 1 147.2.e.a 2
7.b odd 2 1 63.2.e.b 2
7.c even 3 1 441.2.a.a 1
7.c even 3 1 inner 441.2.e.e 2
7.d odd 6 1 63.2.e.b 2
7.d odd 6 1 441.2.a.b 1
12.b even 2 1 2352.2.q.c 2
21.c even 2 1 21.2.e.a 2
21.g even 6 1 21.2.e.a 2
21.g even 6 1 147.2.a.c 1
21.h odd 6 1 147.2.a.b 1
21.h odd 6 1 147.2.e.a 2
28.d even 2 1 1008.2.s.d 2
28.f even 6 1 1008.2.s.d 2
28.f even 6 1 7056.2.a.bp 1
28.g odd 6 1 7056.2.a.m 1
63.i even 6 1 567.2.g.a 2
63.k odd 6 1 567.2.h.a 2
63.l odd 6 1 567.2.g.f 2
63.l odd 6 1 567.2.h.a 2
63.o even 6 1 567.2.g.a 2
63.o even 6 1 567.2.h.f 2
63.s even 6 1 567.2.h.f 2
63.t odd 6 1 567.2.g.f 2
84.h odd 2 1 336.2.q.f 2
84.j odd 6 1 336.2.q.f 2
84.j odd 6 1 2352.2.a.d 1
84.n even 6 1 2352.2.a.w 1
84.n even 6 1 2352.2.q.c 2
105.g even 2 1 525.2.i.e 2
105.k odd 4 2 525.2.r.e 4
105.o odd 6 1 3675.2.a.c 1
105.p even 6 1 525.2.i.e 2
105.p even 6 1 3675.2.a.a 1
105.w odd 12 2 525.2.r.e 4
168.e odd 2 1 1344.2.q.c 2
168.i even 2 1 1344.2.q.m 2
168.s odd 6 1 9408.2.a.bz 1
168.v even 6 1 9408.2.a.k 1
168.ba even 6 1 1344.2.q.m 2
168.ba even 6 1 9408.2.a.bg 1
168.be odd 6 1 1344.2.q.c 2
168.be odd 6 1 9408.2.a.cv 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 21.c even 2 1
21.2.e.a 2 21.g even 6 1
63.2.e.b 2 7.b odd 2 1
63.2.e.b 2 7.d odd 6 1
147.2.a.b 1 21.h odd 6 1
147.2.a.c 1 21.g even 6 1
147.2.e.a 2 3.b odd 2 1
147.2.e.a 2 21.h odd 6 1
336.2.q.f 2 84.h odd 2 1
336.2.q.f 2 84.j odd 6 1
441.2.a.a 1 7.c even 3 1
441.2.a.b 1 7.d odd 6 1
441.2.e.e 2 1.a even 1 1 trivial
441.2.e.e 2 7.c even 3 1 inner
525.2.i.e 2 105.g even 2 1
525.2.i.e 2 105.p even 6 1
525.2.r.e 4 105.k odd 4 2
525.2.r.e 4 105.w odd 12 2
567.2.g.a 2 63.i even 6 1
567.2.g.a 2 63.o even 6 1
567.2.g.f 2 63.l odd 6 1
567.2.g.f 2 63.t odd 6 1
567.2.h.a 2 63.k odd 6 1
567.2.h.a 2 63.l odd 6 1
567.2.h.f 2 63.o even 6 1
567.2.h.f 2 63.s even 6 1
1008.2.s.d 2 28.d even 2 1
1008.2.s.d 2 28.f even 6 1
1344.2.q.c 2 168.e odd 2 1
1344.2.q.c 2 168.be odd 6 1
1344.2.q.m 2 168.i even 2 1
1344.2.q.m 2 168.ba even 6 1
2352.2.a.d 1 84.j odd 6 1
2352.2.a.w 1 84.n even 6 1
2352.2.q.c 2 12.b even 2 1
2352.2.q.c 2 84.n even 6 1
3675.2.a.a 1 105.p even 6 1
3675.2.a.c 1 105.o odd 6 1
7056.2.a.m 1 28.g odd 6 1
7056.2.a.bp 1 28.f even 6 1
9408.2.a.k 1 168.v even 6 1
9408.2.a.bg 1 168.ba even 6 1
9408.2.a.bz 1 168.s odd 6 1
9408.2.a.cv 1 168.be odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2 T_{2} + 4$$ $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{13} + 1$$