# Properties

 Label 441.2.c.b Level $441$ Weight $2$ Character orbit 441.c Analytic conductor $3.521$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 7^{2}$$ Twist minimal: yes 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 $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{16}^{2} + \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{2} + ( -1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{4} + ( 2 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{5} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{16}^{2} + \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{2} + ( -1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{4} + ( 2 \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{5} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{8} + ( -4 \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{10} + ( -2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{11} + ( -\zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{13} + 3 q^{16} + ( -3 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{17} + ( -2 \zeta_{16} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{19} + ( 7 \zeta_{16}^{3} - 7 \zeta_{16}^{5} ) q^{20} -2 q^{22} + 2 \zeta_{16}^{4} q^{23} + ( 5 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{25} + ( -\zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{26} + ( 2 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{29} + ( 2 \zeta_{16} - 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{31} + ( -\zeta_{16}^{2} - 3 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{32} + ( -\zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{34} + ( -4 - \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{37} + ( -2 \zeta_{16} + 8 \zeta_{16}^{3} - 8 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{38} + ( 6 \zeta_{16} - 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{40} + ( \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{41} + ( -4 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{43} + ( -2 \zeta_{16}^{2} - 6 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{44} + ( -2 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{46} + ( 4 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{47} + ( -6 \zeta_{16}^{2} + 7 \zeta_{16}^{4} - 6 \zeta_{16}^{6} ) q^{50} + ( 7 \zeta_{16} + 7 \zeta_{16}^{7} ) q^{52} + ( 5 \zeta_{16}^{2} + 5 \zeta_{16}^{6} ) q^{53} + ( -4 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{55} + ( 6 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{58} + ( 4 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{59} + ( 4 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{61} + ( 2 \zeta_{16} - 8 \zeta_{16}^{3} + 8 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{62} + ( 7 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{64} + ( -\zeta_{16}^{2} + 10 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{65} + ( -6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{67} + ( -7 \zeta_{16} + 7 \zeta_{16}^{7} ) q^{68} + ( -2 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{71} + ( 2 \zeta_{16} + 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{73} + ( 3 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 3 \zeta_{16}^{6} ) q^{74} + ( 14 \zeta_{16} + 14 \zeta_{16}^{7} ) q^{76} + ( -4 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{79} + ( 6 \zeta_{16} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{80} + ( -9 \zeta_{16} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} - 9 \zeta_{16}^{7} ) q^{82} + ( -8 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 8 \zeta_{16}^{7} ) q^{83} + ( -8 - 9 \zeta_{16}^{2} + 9 \zeta_{16}^{6} ) q^{85} + ( 10 \zeta_{16}^{2} - 16 \zeta_{16}^{4} + 10 \zeta_{16}^{6} ) q^{86} + ( -2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{88} + ( -4 \zeta_{16} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{89} + ( 4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{92} + ( -8 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{94} + ( -2 \zeta_{16}^{2} + 20 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{95} + ( -2 \zeta_{16} - 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + O(q^{10})$$ $$8q - 8q^{4} + 24q^{16} - 16q^{22} + 40q^{25} - 32q^{37} - 32q^{43} - 16q^{46} + 48q^{58} + 56q^{64} - 32q^{79} - 64q^{85} - 16q^{88} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 −0.382683 − 0.923880i 0.382683 + 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i
2.41421i 0 −3.82843 −3.37849 0 0 4.41421i 0 8.15640i
440.2 2.41421i 0 −3.82843 3.37849 0 0 4.41421i 0 8.15640i
440.3 0.414214i 0 1.82843 −2.93015 0 0 1.58579i 0 1.21371i
440.4 0.414214i 0 1.82843 2.93015 0 0 1.58579i 0 1.21371i
440.5 0.414214i 0 1.82843 −2.93015 0 0 1.58579i 0 1.21371i
440.6 0.414214i 0 1.82843 2.93015 0 0 1.58579i 0 1.21371i
440.7 2.41421i 0 −3.82843 −3.37849 0 0 4.41421i 0 8.15640i
440.8 2.41421i 0 −3.82843 3.37849 0 0 4.41421i 0 8.15640i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 440.8 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 inner
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 441.2.c.b 8
3.b odd 2 1 inner 441.2.c.b 8
4.b odd 2 1 7056.2.k.g 8
7.b odd 2 1 inner 441.2.c.b 8
7.c even 3 2 441.2.p.c 16
7.d odd 6 2 441.2.p.c 16
12.b even 2 1 7056.2.k.g 8
21.c even 2 1 inner 441.2.c.b 8
21.g even 6 2 441.2.p.c 16
21.h odd 6 2 441.2.p.c 16
28.d even 2 1 7056.2.k.g 8
84.h odd 2 1 7056.2.k.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.c.b 8 1.a even 1 1 trivial
441.2.c.b 8 3.b odd 2 1 inner
441.2.c.b 8 7.b odd 2 1 inner
441.2.c.b 8 21.c even 2 1 inner
441.2.p.c 16 7.c even 3 2
441.2.p.c 16 7.d odd 6 2
441.2.p.c 16 21.g even 6 2
441.2.p.c 16 21.h odd 6 2
7056.2.k.g 8 4.b odd 2 1
7056.2.k.g 8 12.b even 2 1
7056.2.k.g 8 28.d even 2 1
7056.2.k.g 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 6 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 98 - 20 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 16 + 24 T^{2} + T^{4} )^{2}$$
$13$ $$( 98 + 20 T^{2} + T^{4} )^{2}$$
$17$ $$( 98 - 52 T^{2} + T^{4} )^{2}$$
$19$ $$( 1568 + 80 T^{2} + T^{4} )^{2}$$
$23$ $$( 4 + T^{2} )^{4}$$
$29$ $$( 16 + 24 T^{2} + T^{4} )^{2}$$
$31$ $$( 1568 + 80 T^{2} + T^{4} )^{2}$$
$37$ $$( 14 + 8 T + T^{2} )^{4}$$
$41$ $$( 98 - 68 T^{2} + T^{4} )^{2}$$
$43$ $$( -56 + 8 T + T^{2} )^{4}$$
$47$ $$( 1568 - 80 T^{2} + T^{4} )^{2}$$
$53$ $$( 50 + T^{2} )^{4}$$
$59$ $$( 1568 - 80 T^{2} + T^{4} )^{2}$$
$61$ $$( 98 + 68 T^{2} + T^{4} )^{2}$$
$67$ $$( -72 + T^{2} )^{4}$$
$71$ $$( 16 + 24 T^{2} + T^{4} )^{2}$$
$73$ $$( 98 + 52 T^{2} + T^{4} )^{2}$$
$79$ $$( -16 + 8 T + T^{2} )^{4}$$
$83$ $$( 25088 - 320 T^{2} + T^{4} )^{2}$$
$89$ $$( 4802 - 164 T^{2} + T^{4} )^{2}$$
$97$ $$( 98 + 52 T^{2} + T^{4} )^{2}$$