# Properties

 Label 441.3.d.d Level $441$ Weight $3$ Character orbit 441.d 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.d (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( 2 - 4 \zeta_{6} ) q^{5} -8 q^{8} +O(q^{10})$$ $$q + 2 q^{2} + ( 2 - 4 \zeta_{6} ) q^{5} -8 q^{8} + ( 4 - 8 \zeta_{6} ) q^{10} -10 q^{11} + ( 7 - 14 \zeta_{6} ) q^{13} -16 q^{16} + ( -4 + 8 \zeta_{6} ) q^{17} + ( 19 - 38 \zeta_{6} ) q^{19} -20 q^{22} -40 q^{23} + 13 q^{25} + ( 14 - 28 \zeta_{6} ) q^{26} -16 q^{29} + ( -3 + 6 \zeta_{6} ) q^{31} + ( -8 + 16 \zeta_{6} ) q^{34} + 5 q^{37} + ( 38 - 76 \zeta_{6} ) q^{38} + ( -16 + 32 \zeta_{6} ) q^{40} + ( -14 + 28 \zeta_{6} ) q^{41} -19 q^{43} -80 q^{46} + ( 30 - 60 \zeta_{6} ) q^{47} + 26 q^{50} + 32 q^{53} + ( -20 + 40 \zeta_{6} ) q^{55} -32 q^{58} + ( 24 - 48 \zeta_{6} ) q^{59} + ( 12 - 24 \zeta_{6} ) q^{61} + ( -6 + 12 \zeta_{6} ) q^{62} + 64 q^{64} -42 q^{65} + 59 q^{67} + 26 q^{71} + ( 11 - 22 \zeta_{6} ) q^{73} + 10 q^{74} + 47 q^{79} + ( -32 + 64 \zeta_{6} ) q^{80} + ( -28 + 56 \zeta_{6} ) q^{82} + ( 14 - 28 \zeta_{6} ) q^{83} + 24 q^{85} -38 q^{86} + 80 q^{88} + ( -68 + 136 \zeta_{6} ) q^{89} + ( 60 - 120 \zeta_{6} ) q^{94} -114 q^{95} + ( -28 + 56 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - 16q^{8} + O(q^{10})$$ $$2q + 4q^{2} - 16q^{8} - 20q^{11} - 32q^{16} - 40q^{22} - 80q^{23} + 26q^{25} - 32q^{29} + 10q^{37} - 38q^{43} - 160q^{46} + 52q^{50} + 64q^{53} - 64q^{58} + 128q^{64} - 84q^{65} + 118q^{67} + 52q^{71} + 20q^{74} + 94q^{79} + 48q^{85} - 76q^{86} + 160q^{88} - 228q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 0.5 + 0.866025i 0.5 − 0.866025i
2.00000 0 0 3.46410i 0 0 −8.00000 0 6.92820i
244.2 2.00000 0 0 3.46410i 0 0 −8.00000 0 6.92820i
 $$n$$: 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.b odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{3}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$12 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 10 + T )^{2}$$
$13$ $$147 + T^{2}$$
$17$ $$48 + T^{2}$$
$19$ $$1083 + T^{2}$$
$23$ $$( 40 + T )^{2}$$
$29$ $$( 16 + T )^{2}$$
$31$ $$27 + T^{2}$$
$37$ $$( -5 + T )^{2}$$
$41$ $$588 + T^{2}$$
$43$ $$( 19 + T )^{2}$$
$47$ $$2700 + T^{2}$$
$53$ $$( -32 + T )^{2}$$
$59$ $$1728 + T^{2}$$
$61$ $$432 + T^{2}$$
$67$ $$( -59 + T )^{2}$$
$71$ $$( -26 + T )^{2}$$
$73$ $$363 + T^{2}$$
$79$ $$( -47 + T )^{2}$$
$83$ $$588 + T^{2}$$
$89$ $$13872 + T^{2}$$
$97$ $$2352 + T^{2}$$