# Properties

 Label 441.1.v.a Level $441$ Weight $1$ Character orbit 441.v Analytic conductor $0.220$ Analytic rank $0$ Dimension $6$ Projective image $D_{14}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 441.v (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.220087670571$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{14}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{14} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{14}^{4} q^{4} -\zeta_{14}^{2} q^{7} +O(q^{10})$$ $$q -\zeta_{14}^{4} q^{4} -\zeta_{14}^{2} q^{7} + ( \zeta_{14} - \zeta_{14}^{3} ) q^{13} -\zeta_{14} q^{16} + ( \zeta_{14}^{2} + \zeta_{14}^{5} ) q^{19} -\zeta_{14}^{5} q^{25} + \zeta_{14}^{6} q^{28} + ( \zeta_{14}^{3} + \zeta_{14}^{4} ) q^{31} + ( -1 - \zeta_{14}^{6} ) q^{37} + ( \zeta_{14}^{3} - \zeta_{14}^{6} ) q^{43} + \zeta_{14}^{4} q^{49} + ( -1 - \zeta_{14}^{5} ) q^{52} + ( -1 + \zeta_{14}^{6} ) q^{61} + \zeta_{14}^{5} q^{64} + ( -\zeta_{14} + \zeta_{14}^{6} ) q^{67} + ( \zeta_{14} + \zeta_{14}^{2} ) q^{73} + ( \zeta_{14}^{2} - \zeta_{14}^{6} ) q^{76} + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{79} + ( -\zeta_{14}^{3} + \zeta_{14}^{5} ) q^{91} + ( -\zeta_{14}^{2} - \zeta_{14}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{4} + q^{7} + O(q^{10})$$ $$6 q + q^{4} + q^{7} - q^{16} - q^{25} - q^{28} - 5 q^{37} + 2 q^{43} - q^{49} - 7 q^{52} - 7 q^{61} + q^{64} - 2 q^{67} - 2 q^{79} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\zeta_{14}^{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.900969 − 0.433884i 0.222521 − 0.974928i −0.623490 − 0.781831i −0.623490 + 0.781831i 0.222521 + 0.974928i 0.900969 + 0.433884i
0 0 0.222521 + 0.974928i 0 0 −0.623490 + 0.781831i 0 0 0
118.1 0 0 −0.623490 0.781831i 0 0 0.900969 + 0.433884i 0 0 0
181.1 0 0 0.900969 + 0.433884i 0 0 0.222521 0.974928i 0 0 0
307.1 0 0 0.900969 0.433884i 0 0 0.222521 + 0.974928i 0 0 0
370.1 0 0 −0.623490 + 0.781831i 0 0 0.900969 0.433884i 0 0 0
433.1 0 0 0.222521 0.974928i 0 0 −0.623490 0.781831i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 433.1 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 $$\Q(\sqrt{-3})$$
49.f odd 14 1 inner
147.k even 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.1.v.a 6
3.b odd 2 1 CM 441.1.v.a 6
7.b odd 2 1 3087.1.v.a 6
7.c even 3 2 3087.1.bj.b 12
7.d odd 6 2 3087.1.bj.a 12
9.c even 3 2 3969.1.bz.a 12
9.d odd 6 2 3969.1.bz.a 12
21.c even 2 1 3087.1.v.a 6
21.g even 6 2 3087.1.bj.a 12
21.h odd 6 2 3087.1.bj.b 12
49.e even 7 1 3087.1.v.a 6
49.f odd 14 1 inner 441.1.v.a 6
49.g even 21 2 3087.1.bj.a 12
49.h odd 42 2 3087.1.bj.b 12
147.k even 14 1 inner 441.1.v.a 6
147.l odd 14 1 3087.1.v.a 6
147.n odd 42 2 3087.1.bj.a 12
147.o even 42 2 3087.1.bj.b 12
441.bh even 42 2 3969.1.bz.a 12
441.bk odd 42 2 3969.1.bz.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.v.a 6 1.a even 1 1 trivial
441.1.v.a 6 3.b odd 2 1 CM
441.1.v.a 6 49.f odd 14 1 inner
441.1.v.a 6 147.k even 14 1 inner
3087.1.v.a 6 7.b odd 2 1
3087.1.v.a 6 21.c even 2 1
3087.1.v.a 6 49.e even 7 1
3087.1.v.a 6 147.l odd 14 1
3087.1.bj.a 12 7.d odd 6 2
3087.1.bj.a 12 21.g even 6 2
3087.1.bj.a 12 49.g even 21 2
3087.1.bj.a 12 147.n odd 42 2
3087.1.bj.b 12 7.c even 3 2
3087.1.bj.b 12 21.h odd 6 2
3087.1.bj.b 12 49.h odd 42 2
3087.1.bj.b 12 147.o even 42 2
3969.1.bz.a 12 9.c even 3 2
3969.1.bz.a 12 9.d odd 6 2
3969.1.bz.a 12 441.bh even 42 2
3969.1.bz.a 12 441.bk odd 42 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$7 - 7 T + 7 T^{3} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$7 + 14 T^{2} + 7 T^{4} + T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$7 + 14 T^{2} + 7 T^{4} + T^{6}$$
$37$ $$1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6}$$
$41$ $$T^{6}$$
$43$ $$1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$7 + 21 T + 35 T^{2} + 35 T^{3} + 21 T^{4} + 7 T^{5} + T^{6}$$
$67$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$71$ $$T^{6}$$
$73$ $$7 + 14 T + 7 T^{2} + T^{6}$$
$79$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$7 + 14 T^{2} + 7 T^{4} + T^{6}$$