# Properties

 Label 441.3.m.d 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -8 + 4 \zeta_{6} ) q^{5} + 7 q^{8} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -8 + 4 \zeta_{6} ) q^{5} + 7 q^{8} + ( -4 - 4 \zeta_{6} ) q^{10} + ( 10 - 10 \zeta_{6} ) q^{11} + ( 4 - 8 \zeta_{6} ) q^{13} -5 \zeta_{6} q^{16} + ( 24 - 12 \zeta_{6} ) q^{19} + ( -12 + 24 \zeta_{6} ) q^{20} + 10 q^{22} -14 \zeta_{6} q^{23} + ( 23 - 23 \zeta_{6} ) q^{25} + ( 8 - 4 \zeta_{6} ) q^{26} + 38 q^{29} + ( 16 + 16 \zeta_{6} ) q^{31} + ( 33 - 33 \zeta_{6} ) q^{32} -26 \zeta_{6} q^{37} + ( 12 + 12 \zeta_{6} ) q^{38} + ( -56 + 28 \zeta_{6} ) q^{40} + ( 40 - 80 \zeta_{6} ) q^{41} + 26 q^{43} -30 \zeta_{6} q^{44} + ( 14 - 14 \zeta_{6} ) q^{46} + ( 32 - 16 \zeta_{6} ) q^{47} + 23 q^{50} + ( -12 - 12 \zeta_{6} ) q^{52} + ( 10 - 10 \zeta_{6} ) q^{53} + ( -40 + 80 \zeta_{6} ) q^{55} + 38 \zeta_{6} q^{58} + ( -44 - 44 \zeta_{6} ) q^{59} + ( -40 + 20 \zeta_{6} ) q^{61} + ( -16 + 32 \zeta_{6} ) q^{62} + 13 q^{64} + 48 \zeta_{6} q^{65} + ( -74 + 74 \zeta_{6} ) q^{67} + 62 q^{71} + ( -24 - 24 \zeta_{6} ) q^{73} + ( 26 - 26 \zeta_{6} ) q^{74} + ( 36 - 72 \zeta_{6} ) q^{76} + 46 \zeta_{6} q^{79} + ( 20 + 20 \zeta_{6} ) q^{80} + ( 80 - 40 \zeta_{6} ) q^{82} + ( -52 + 104 \zeta_{6} ) q^{83} + 26 \zeta_{6} q^{86} + ( 70 - 70 \zeta_{6} ) q^{88} + ( -48 + 24 \zeta_{6} ) q^{89} -42 q^{92} + ( 16 + 16 \zeta_{6} ) q^{94} + ( -144 + 144 \zeta_{6} ) q^{95} + ( 32 - 64 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{4} - 12q^{5} + 14q^{8} + O(q^{10})$$ $$2q + q^{2} + 3q^{4} - 12q^{5} + 14q^{8} - 12q^{10} + 10q^{11} - 5q^{16} + 36q^{19} + 20q^{22} - 14q^{23} + 23q^{25} + 12q^{26} + 76q^{29} + 48q^{31} + 33q^{32} - 26q^{37} + 36q^{38} - 84q^{40} + 52q^{43} - 30q^{44} + 14q^{46} + 48q^{47} + 46q^{50} - 36q^{52} + 10q^{53} + 38q^{58} - 132q^{59} - 60q^{61} + 26q^{64} + 48q^{65} - 74q^{67} + 124q^{71} - 72q^{73} + 26q^{74} + 46q^{79} + 60q^{80} + 120q^{82} + 26q^{86} + 70q^{88} - 72q^{89} - 84q^{92} + 48q^{94} - 144q^{95} + O(q^{100})$$

## 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 −6.00000 3.46410i 0 0 7.00000 0 −6.00000 + 3.46410i
325.1 0.500000 + 0.866025i 0 1.50000 2.59808i −6.00000 + 3.46410i 0 0 7.00000 0 −6.00000 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.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.d 2
3.b odd 2 1 147.3.f.d 2
7.b odd 2 1 441.3.m.f 2
7.c even 3 1 63.3.d.b 2
7.c even 3 1 441.3.m.f 2
7.d odd 6 1 63.3.d.b 2
7.d odd 6 1 inner 441.3.m.d 2
21.c even 2 1 147.3.f.b 2
21.g even 6 1 21.3.d.a 2
21.g even 6 1 147.3.f.d 2
21.h odd 6 1 21.3.d.a 2
21.h odd 6 1 147.3.f.b 2
28.f even 6 1 1008.3.f.d 2
28.g odd 6 1 1008.3.f.d 2
84.j odd 6 1 336.3.f.a 2
84.n even 6 1 336.3.f.a 2
105.o odd 6 1 525.3.h.a 2
105.p even 6 1 525.3.h.a 2
105.w odd 12 2 525.3.e.a 4
105.x even 12 2 525.3.e.a 4
168.s odd 6 1 1344.3.f.c 2
168.v even 6 1 1344.3.f.b 2
168.ba even 6 1 1344.3.f.c 2
168.be odd 6 1 1344.3.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 21.g even 6 1
21.3.d.a 2 21.h odd 6 1
63.3.d.b 2 7.c even 3 1
63.3.d.b 2 7.d odd 6 1
147.3.f.b 2 21.c even 2 1
147.3.f.b 2 21.h odd 6 1
147.3.f.d 2 3.b odd 2 1
147.3.f.d 2 21.g even 6 1
336.3.f.a 2 84.j odd 6 1
336.3.f.a 2 84.n even 6 1
441.3.m.d 2 1.a even 1 1 trivial
441.3.m.d 2 7.d odd 6 1 inner
441.3.m.f 2 7.b odd 2 1
441.3.m.f 2 7.c even 3 1
525.3.e.a 4 105.w odd 12 2
525.3.e.a 4 105.x even 12 2
525.3.h.a 2 105.o odd 6 1
525.3.h.a 2 105.p even 6 1
1008.3.f.d 2 28.f even 6 1
1008.3.f.d 2 28.g odd 6 1
1344.3.f.b 2 168.v even 6 1
1344.3.f.b 2 168.be odd 6 1
1344.3.f.c 2 168.s odd 6 1
1344.3.f.c 2 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{5}^{2} + 12 T_{5} + 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$48 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$100 - 10 T + T^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$432 - 36 T + T^{2}$$
$23$ $$196 + 14 T + T^{2}$$
$29$ $$( -38 + T )^{2}$$
$31$ $$768 - 48 T + T^{2}$$
$37$ $$676 + 26 T + T^{2}$$
$41$ $$4800 + T^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$768 - 48 T + T^{2}$$
$53$ $$100 - 10 T + T^{2}$$
$59$ $$5808 + 132 T + T^{2}$$
$61$ $$1200 + 60 T + T^{2}$$
$67$ $$5476 + 74 T + T^{2}$$
$71$ $$( -62 + T )^{2}$$
$73$ $$1728 + 72 T + T^{2}$$
$79$ $$2116 - 46 T + T^{2}$$
$83$ $$8112 + T^{2}$$
$89$ $$1728 + 72 T + T^{2}$$
$97$ $$3072 + T^{2}$$