# Properties

 Label 441.3.m.e 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -6 + 3 \zeta_{6} ) q^{5} + 7 q^{8} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -6 + 3 \zeta_{6} ) q^{5} + 7 q^{8} + ( -3 - 3 \zeta_{6} ) q^{10} + ( -11 + 11 \zeta_{6} ) q^{11} + ( -4 + 8 \zeta_{6} ) q^{13} -5 \zeta_{6} q^{16} + ( 14 + 14 \zeta_{6} ) q^{17} + ( 4 - 2 \zeta_{6} ) q^{19} + ( -9 + 18 \zeta_{6} ) q^{20} -11 q^{22} + 28 \zeta_{6} q^{23} + ( 2 - 2 \zeta_{6} ) q^{25} + ( -8 + 4 \zeta_{6} ) q^{26} -25 q^{29} + ( 19 + 19 \zeta_{6} ) q^{31} + ( 33 - 33 \zeta_{6} ) q^{32} + ( -14 + 28 \zeta_{6} ) q^{34} + 58 \zeta_{6} q^{37} + ( 2 + 2 \zeta_{6} ) q^{38} + ( -42 + 21 \zeta_{6} ) q^{40} + ( 2 - 4 \zeta_{6} ) q^{41} + 26 q^{43} + 33 \zeta_{6} q^{44} + ( -28 + 28 \zeta_{6} ) q^{46} + ( -88 + 44 \zeta_{6} ) q^{47} + 2 q^{50} + ( 12 + 12 \zeta_{6} ) q^{52} + ( 31 - 31 \zeta_{6} ) q^{53} + ( 33 - 66 \zeta_{6} ) q^{55} -25 \zeta_{6} q^{58} + ( -5 - 5 \zeta_{6} ) q^{59} + ( -16 + 8 \zeta_{6} ) q^{61} + ( -19 + 38 \zeta_{6} ) q^{62} + 13 q^{64} -36 \zeta_{6} q^{65} + ( 52 - 52 \zeta_{6} ) q^{67} + ( 84 - 42 \zeta_{6} ) q^{68} -64 q^{71} + ( -4 - 4 \zeta_{6} ) q^{73} + ( -58 + 58 \zeta_{6} ) q^{74} + ( 6 - 12 \zeta_{6} ) q^{76} -17 \zeta_{6} q^{79} + ( 15 + 15 \zeta_{6} ) q^{80} + ( 4 - 2 \zeta_{6} ) q^{82} + ( 31 - 62 \zeta_{6} ) q^{83} -126 q^{85} + 26 \zeta_{6} q^{86} + ( -77 + 77 \zeta_{6} ) q^{88} + ( -92 + 46 \zeta_{6} ) q^{89} + 84 q^{92} + ( -44 - 44 \zeta_{6} ) q^{94} + ( -18 + 18 \zeta_{6} ) q^{95} + ( -53 + 106 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{4} - 9q^{5} + 14q^{8} + O(q^{10})$$ $$2q + q^{2} + 3q^{4} - 9q^{5} + 14q^{8} - 9q^{10} - 11q^{11} - 5q^{16} + 42q^{17} + 6q^{19} - 22q^{22} + 28q^{23} + 2q^{25} - 12q^{26} - 50q^{29} + 57q^{31} + 33q^{32} + 58q^{37} + 6q^{38} - 63q^{40} + 52q^{43} + 33q^{44} - 28q^{46} - 132q^{47} + 4q^{50} + 36q^{52} + 31q^{53} - 25q^{58} - 15q^{59} - 24q^{61} + 26q^{64} - 36q^{65} + 52q^{67} + 126q^{68} - 128q^{71} - 12q^{73} - 58q^{74} - 17q^{79} + 45q^{80} + 6q^{82} - 252q^{85} + 26q^{86} - 77q^{88} - 138q^{89} + 168q^{92} - 132q^{94} - 18q^{95} + O(q^{100})$$

## 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
0.500000 0.866025i 0 1.50000 + 2.59808i −4.50000 2.59808i 0 0 7.00000 0 −4.50000 + 2.59808i
325.1 0.500000 + 0.866025i 0 1.50000 2.59808i −4.50000 + 2.59808i 0 0 7.00000 0 −4.50000 2.59808i
 $$n$$: 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.e 2
3.b odd 2 1 147.3.f.c 2
7.b odd 2 1 63.3.m.c 2
7.c even 3 1 63.3.m.c 2
7.c even 3 1 441.3.d.b 2
7.d odd 6 1 441.3.d.b 2
7.d odd 6 1 inner 441.3.m.e 2
21.c even 2 1 21.3.f.b 2
21.g even 6 1 147.3.d.b 2
21.g even 6 1 147.3.f.c 2
21.h odd 6 1 21.3.f.b 2
21.h odd 6 1 147.3.d.b 2
28.d even 2 1 1008.3.cg.g 2
28.g odd 6 1 1008.3.cg.g 2
84.h odd 2 1 336.3.bh.a 2
84.j odd 6 1 2352.3.f.d 2
84.n even 6 1 336.3.bh.a 2
84.n even 6 1 2352.3.f.d 2
105.g even 2 1 525.3.o.g 2
105.k odd 4 2 525.3.s.c 4
105.o odd 6 1 525.3.o.g 2
105.x even 12 2 525.3.s.c 4

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

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{5}^{2} + 9 T_{5} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$27 + 9 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$121 + 11 T + T^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$588 - 42 T + T^{2}$$
$19$ $$12 - 6 T + T^{2}$$
$23$ $$784 - 28 T + T^{2}$$
$29$ $$( 25 + T )^{2}$$
$31$ $$1083 - 57 T + T^{2}$$
$37$ $$3364 - 58 T + T^{2}$$
$41$ $$12 + T^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$5808 + 132 T + T^{2}$$
$53$ $$961 - 31 T + T^{2}$$
$59$ $$75 + 15 T + T^{2}$$
$61$ $$192 + 24 T + T^{2}$$
$67$ $$2704 - 52 T + T^{2}$$
$71$ $$( 64 + T )^{2}$$
$73$ $$48 + 12 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$2883 + T^{2}$$
$89$ $$6348 + 138 T + T^{2}$$
$97$ $$8427 + T^{2}$$