# Properties

 Label 441.3.d.c Level $441$ Weight $3$ Character orbit 441.d Analytic conductor $12.016$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$2\cdot 7$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -4 q^{4} +O(q^{10})$$ $$q -4 q^{4} + ( 7 - 14 \zeta_{6} ) q^{13} + 16 q^{16} + ( -21 + 42 \zeta_{6} ) q^{19} + 25 q^{25} + ( -35 + 70 \zeta_{6} ) q^{31} + 73 q^{37} + 61 q^{43} + ( -28 + 56 \zeta_{6} ) q^{52} + ( -56 + 112 \zeta_{6} ) q^{61} -64 q^{64} -13 q^{67} + ( 63 - 126 \zeta_{6} ) q^{73} + ( 84 - 168 \zeta_{6} ) q^{76} + 11 q^{79} + ( -112 + 224 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} + O(q^{10})$$ $$2q - 8q^{4} + 32q^{16} + 50q^{25} + 146q^{37} + 122q^{43} - 128q^{64} - 26q^{67} + 22q^{79} + O(q^{100})$$

## 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
0 0 −4.00000 0 0 0 0 0 0
244.2 0 0 −4.00000 0 0 0 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 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.c 2
3.b odd 2 1 CM 441.3.d.c 2
7.b odd 2 1 inner 441.3.d.c 2
7.c even 3 1 63.3.m.b 2
7.c even 3 1 441.3.m.c 2
7.d odd 6 1 63.3.m.b 2
7.d odd 6 1 441.3.m.c 2
21.c even 2 1 inner 441.3.d.c 2
21.g even 6 1 63.3.m.b 2
21.g even 6 1 441.3.m.c 2
21.h odd 6 1 63.3.m.b 2
21.h odd 6 1 441.3.m.c 2
28.f even 6 1 1008.3.cg.d 2
28.g odd 6 1 1008.3.cg.d 2
84.j odd 6 1 1008.3.cg.d 2
84.n even 6 1 1008.3.cg.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.b 2 7.c even 3 1
63.3.m.b 2 7.d odd 6 1
63.3.m.b 2 21.g even 6 1
63.3.m.b 2 21.h odd 6 1
441.3.d.c 2 1.a even 1 1 trivial
441.3.d.c 2 3.b odd 2 1 CM
441.3.d.c 2 7.b odd 2 1 inner
441.3.d.c 2 21.c even 2 1 inner
441.3.m.c 2 7.c even 3 1
441.3.m.c 2 7.d odd 6 1
441.3.m.c 2 21.g even 6 1
441.3.m.c 2 21.h odd 6 1
1008.3.cg.d 2 28.f even 6 1
1008.3.cg.d 2 28.g odd 6 1
1008.3.cg.d 2 84.j odd 6 1
1008.3.cg.d 2 84.n even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$147 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$1323 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$3675 + T^{2}$$
$37$ $$( -73 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( -61 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$9408 + T^{2}$$
$67$ $$( 13 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$11907 + T^{2}$$
$79$ $$( -11 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$37632 + T^{2}$$