# Properties

 Label 441.1.b.a Level $441$ Weight $1$ Character orbit 441.b Analytic conductor $0.220$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 441.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.220087670571$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.189.1 Artin image: $SD_{16}$ Artin field: Galois closure of 8.2.257298363.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{4} +O(q^{10})$$ $$q -\beta q^{2} - q^{4} -\beta q^{11} - q^{16} -2 q^{22} + \beta q^{23} + q^{25} + \beta q^{29} + \beta q^{32} + \beta q^{44} + 2 q^{46} -\beta q^{50} + \beta q^{53} + 2 q^{58} + q^{64} -2 q^{67} -\beta q^{71} + 2 q^{79} -\beta q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{16} - 4q^{22} + 2q^{25} + 4q^{46} + 4q^{58} + 2q^{64} - 4q^{67} + 4q^{79} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 1.41421i − 1.41421i
1.41421i 0 −1.00000 0 0 0 0 0 0
197.2 1.41421i 0 −1.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.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 441.1.b.a 2
3.b odd 2 1 inner 441.1.b.a 2
7.b odd 2 1 CM 441.1.b.a 2
7.c even 3 2 441.1.q.a 4
7.d odd 6 2 441.1.q.a 4
9.c even 3 2 3969.1.r.c 4
9.d odd 6 2 3969.1.r.c 4
21.c even 2 1 inner 441.1.b.a 2
21.g even 6 2 441.1.q.a 4
21.h odd 6 2 441.1.q.a 4
63.g even 3 2 3969.1.n.b 4
63.h even 3 2 3969.1.j.b 4
63.i even 6 2 3969.1.j.b 4
63.j odd 6 2 3969.1.j.b 4
63.k odd 6 2 3969.1.n.b 4
63.l odd 6 2 3969.1.r.c 4
63.n odd 6 2 3969.1.n.b 4
63.o even 6 2 3969.1.r.c 4
63.s even 6 2 3969.1.n.b 4
63.t odd 6 2 3969.1.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.b.a 2 1.a even 1 1 trivial
441.1.b.a 2 3.b odd 2 1 inner
441.1.b.a 2 7.b odd 2 1 CM
441.1.b.a 2 21.c even 2 1 inner
441.1.q.a 4 7.c even 3 2
441.1.q.a 4 7.d odd 6 2
441.1.q.a 4 21.g even 6 2
441.1.q.a 4 21.h odd 6 2
3969.1.j.b 4 63.h even 3 2
3969.1.j.b 4 63.i even 6 2
3969.1.j.b 4 63.j odd 6 2
3969.1.j.b 4 63.t odd 6 2
3969.1.n.b 4 63.g even 3 2
3969.1.n.b 4 63.k odd 6 2
3969.1.n.b 4 63.n odd 6 2
3969.1.n.b 4 63.s even 6 2
3969.1.r.c 4 9.c even 3 2
3969.1.r.c 4 9.d odd 6 2
3969.1.r.c 4 63.l odd 6 2
3969.1.r.c 4 63.o even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

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