# Properties

 Label 441.3.m.g Level $441$ Weight $3$ Character orbit 441.m Analytic conductor $12.016$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 441.m (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0163796583$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{4} + ( 6 - 3 \zeta_{6} ) q^{5} -3 q^{8} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{4} + ( 6 - 3 \zeta_{6} ) q^{5} -3 q^{8} + ( 9 + 9 \zeta_{6} ) q^{10} + ( 15 - 15 \zeta_{6} ) q^{11} + ( -8 + 16 \zeta_{6} ) q^{13} + 11 \zeta_{6} q^{16} + ( 6 + 6 \zeta_{6} ) q^{17} + ( 12 - 6 \zeta_{6} ) q^{19} + ( -15 + 30 \zeta_{6} ) q^{20} + 45 q^{22} + ( 2 - 2 \zeta_{6} ) q^{25} + ( -48 + 24 \zeta_{6} ) q^{26} + 9 q^{29} + ( 7 + 7 \zeta_{6} ) q^{31} + ( -45 + 45 \zeta_{6} ) q^{32} + ( -18 + 36 \zeta_{6} ) q^{34} -10 \zeta_{6} q^{37} + ( 18 + 18 \zeta_{6} ) q^{38} + ( -18 + 9 \zeta_{6} ) q^{40} + ( 6 - 12 \zeta_{6} ) q^{41} -74 q^{43} + 75 \zeta_{6} q^{44} + 6 q^{50} + ( -40 - 40 \zeta_{6} ) q^{52} + ( 33 - 33 \zeta_{6} ) q^{53} + ( 45 - 90 \zeta_{6} ) q^{55} + 27 \zeta_{6} q^{58} + ( 9 + 9 \zeta_{6} ) q^{59} + ( -104 + 52 \zeta_{6} ) q^{61} + ( -21 + 42 \zeta_{6} ) q^{62} -91 q^{64} + 72 \zeta_{6} q^{65} + ( 76 - 76 \zeta_{6} ) q^{67} + ( -60 + 30 \zeta_{6} ) q^{68} -84 q^{71} + ( 36 + 36 \zeta_{6} ) q^{73} + ( 30 - 30 \zeta_{6} ) q^{74} + ( -30 + 60 \zeta_{6} ) q^{76} + 43 \zeta_{6} q^{79} + ( 33 + 33 \zeta_{6} ) q^{80} + ( 36 - 18 \zeta_{6} ) q^{82} + ( 69 - 138 \zeta_{6} ) q^{83} + 54 q^{85} -222 \zeta_{6} q^{86} + ( -45 + 45 \zeta_{6} ) q^{88} + ( -84 + 42 \zeta_{6} ) q^{89} + ( 54 - 54 \zeta_{6} ) q^{95} + ( 107 - 214 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} - 5q^{4} + 9q^{5} - 6q^{8} + O(q^{10})$$ $$2q + 3q^{2} - 5q^{4} + 9q^{5} - 6q^{8} + 27q^{10} + 15q^{11} + 11q^{16} + 18q^{17} + 18q^{19} + 90q^{22} + 2q^{25} - 72q^{26} + 18q^{29} + 21q^{31} - 45q^{32} - 10q^{37} + 54q^{38} - 27q^{40} - 148q^{43} + 75q^{44} + 12q^{50} - 120q^{52} + 33q^{53} + 27q^{58} + 27q^{59} - 156q^{61} - 182q^{64} + 72q^{65} + 76q^{67} - 90q^{68} - 168q^{71} + 108q^{73} + 30q^{74} + 43q^{79} + 99q^{80} + 54q^{82} + 108q^{85} - 222q^{86} - 45q^{88} - 126q^{89} + 54q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 2.59808i 0 −2.50000 4.33013i 4.50000 + 2.59808i 0 0 −3.00000 0 13.5000 7.79423i
325.1 1.50000 + 2.59808i 0 −2.50000 + 4.33013i 4.50000 2.59808i 0 0 −3.00000 0 13.5000 + 7.79423i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.m.g 2
3.b odd 2 1 147.3.f.a 2
7.b odd 2 1 63.3.m.d 2
7.c even 3 1 63.3.m.d 2
7.c even 3 1 441.3.d.a 2
7.d odd 6 1 441.3.d.a 2
7.d odd 6 1 inner 441.3.m.g 2
21.c even 2 1 21.3.f.a 2
21.g even 6 1 147.3.d.c 2
21.g even 6 1 147.3.f.a 2
21.h odd 6 1 21.3.f.a 2
21.h odd 6 1 147.3.d.c 2
28.d even 2 1 1008.3.cg.a 2
28.g odd 6 1 1008.3.cg.a 2
84.h odd 2 1 336.3.bh.d 2
84.j odd 6 1 2352.3.f.a 2
84.n even 6 1 336.3.bh.d 2
84.n even 6 1 2352.3.f.a 2
105.g even 2 1 525.3.o.h 2
105.k odd 4 2 525.3.s.e 4
105.o odd 6 1 525.3.o.h 2
105.x even 12 2 525.3.s.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 21.c even 2 1
21.3.f.a 2 21.h odd 6 1
63.3.m.d 2 7.b odd 2 1
63.3.m.d 2 7.c even 3 1
147.3.d.c 2 21.g even 6 1
147.3.d.c 2 21.h odd 6 1
147.3.f.a 2 3.b odd 2 1
147.3.f.a 2 21.g even 6 1
336.3.bh.d 2 84.h odd 2 1
336.3.bh.d 2 84.n even 6 1
441.3.d.a 2 7.c even 3 1
441.3.d.a 2 7.d odd 6 1
441.3.m.g 2 1.a even 1 1 trivial
441.3.m.g 2 7.d odd 6 1 inner
525.3.o.h 2 105.g even 2 1
525.3.o.h 2 105.o odd 6 1
525.3.s.e 4 105.k odd 4 2
525.3.s.e 4 105.x even 12 2
1008.3.cg.a 2 28.d even 2 1
1008.3.cg.a 2 28.g odd 6 1
2352.3.f.a 2 84.j odd 6 1
2352.3.f.a 2 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 $$S_{3}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} - 3 T_{2} + 9$$ $$T_{5}^{2} - 9 T_{5} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$27 - 9 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$225 - 15 T + T^{2}$$
$13$ $$192 + T^{2}$$
$17$ $$108 - 18 T + T^{2}$$
$19$ $$108 - 18 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$147 - 21 T + T^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$108 + T^{2}$$
$43$ $$( 74 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$1089 - 33 T + T^{2}$$
$59$ $$243 - 27 T + T^{2}$$
$61$ $$8112 + 156 T + T^{2}$$
$67$ $$5776 - 76 T + T^{2}$$
$71$ $$( 84 + T )^{2}$$
$73$ $$3888 - 108 T + T^{2}$$
$79$ $$1849 - 43 T + T^{2}$$
$83$ $$14283 + T^{2}$$
$89$ $$5292 + 126 T + T^{2}$$
$97$ $$34347 + T^{2}$$