# Properties

 Label 1449.1.bq.b Level $1449$ Weight $1$ Character orbit 1449.bq Analytic conductor $0.723$ Analytic rank $0$ Dimension $20$ Projective image $D_{22}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1449 = 3^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1449.bq (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.723145203305$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{22})$$ Coefficient field: $$\Q(\zeta_{44})$$ Defining polynomial: $$x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{22}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{22} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{44}^{5} - \zeta_{44}^{7} ) q^{2} + ( \zeta_{44}^{10} + \zeta_{44}^{12} + \zeta_{44}^{14} ) q^{4} + \zeta_{44}^{2} q^{7} + ( -\zeta_{44}^{15} - \zeta_{44}^{17} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{44}^{5} - \zeta_{44}^{7} ) q^{2} + ( \zeta_{44}^{10} + \zeta_{44}^{12} + \zeta_{44}^{14} ) q^{4} + \zeta_{44}^{2} q^{7} + ( -\zeta_{44}^{15} - \zeta_{44}^{17} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{8} + ( -\zeta_{44} + \zeta_{44}^{9} ) q^{11} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{14} + ( -1 - \zeta_{44}^{2} - \zeta_{44}^{4} - \zeta_{44}^{6} + \zeta_{44}^{20} ) q^{16} + ( \zeta_{44}^{6} + \zeta_{44}^{8} - \zeta_{44}^{14} - \zeta_{44}^{16} ) q^{22} + \zeta_{44}^{3} q^{23} -\zeta_{44}^{6} q^{25} + ( \zeta_{44}^{12} + \zeta_{44}^{14} + \zeta_{44}^{16} ) q^{28} + ( \zeta_{44}^{5} + \zeta_{44}^{15} ) q^{29} + ( \zeta_{44}^{3} + \zeta_{44}^{5} + \zeta_{44}^{7} + \zeta_{44}^{9} + \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{32} + ( \zeta_{44}^{6} + \zeta_{44}^{10} ) q^{37} + ( -\zeta_{44}^{12} + \zeta_{44}^{18} ) q^{43} + ( -\zeta_{44} - \zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{15} + \zeta_{44}^{19} + \zeta_{44}^{21} ) q^{44} + ( -\zeta_{44}^{8} - \zeta_{44}^{10} ) q^{46} + \zeta_{44}^{4} q^{49} + ( \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{50} + ( \zeta_{44}^{7} - \zeta_{44}^{19} ) q^{53} + ( \zeta_{44} - \zeta_{44}^{17} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{56} + ( 1 - \zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{20} ) q^{58} + ( -\zeta_{44}^{8} - \zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{14} - \zeta_{44}^{16} - \zeta_{44}^{18} - \zeta_{44}^{20} ) q^{64} + ( \zeta_{44}^{16} - \zeta_{44}^{18} ) q^{67} + ( -\zeta_{44}^{13} + \zeta_{44}^{21} ) q^{71} + ( -\zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{15} - \zeta_{44}^{17} ) q^{74} + ( -\zeta_{44}^{3} + \zeta_{44}^{11} ) q^{77} + ( \zeta_{44}^{4} - \zeta_{44}^{14} ) q^{79} + ( \zeta_{44} + \zeta_{44}^{3} + \zeta_{44}^{17} + \zeta_{44}^{19} ) q^{86} + ( -1 + \zeta_{44}^{2} + \zeta_{44}^{4} + \zeta_{44}^{6} + \zeta_{44}^{8} + \zeta_{44}^{16} + \zeta_{44}^{18} + \zeta_{44}^{20} ) q^{88} + ( \zeta_{44}^{13} + \zeta_{44}^{15} + \zeta_{44}^{17} ) q^{92} + ( -\zeta_{44}^{9} - \zeta_{44}^{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{4} + 2q^{7} + O(q^{10})$$ $$20q + 2q^{4} + 2q^{7} - 24q^{16} - 2q^{25} - 2q^{28} + 4q^{37} + 4q^{43} - 2q^{49} + 22q^{58} + 2q^{64} - 4q^{67} - 4q^{79} - 22q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times$$.

 $$n$$ $$442$$ $$829$$ $$1289$$ $$\chi(n)$$ $$-\zeta_{44}^{6}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.540641 + 0.841254i −0.540641 − 0.841254i 0.281733 + 0.959493i −0.281733 − 0.959493i 0.281733 − 0.959493i −0.281733 + 0.959493i −0.909632 + 0.415415i 0.909632 − 0.415415i 0.989821 + 0.142315i −0.989821 − 0.142315i 0.540641 − 0.841254i −0.540641 + 0.841254i −0.909632 − 0.415415i 0.909632 + 0.415415i −0.755750 + 0.654861i 0.755750 − 0.654861i 0.989821 − 0.142315i −0.989821 + 0.142315i −0.755750 − 0.654861i 0.755750 + 0.654861i
−1.03748 + 0.304632i 0 0.142315 0.0914602i 0 0 −0.415415 + 0.909632i 0.588302 0.678936i 0 0
55.2 1.03748 0.304632i 0 0.142315 0.0914602i 0 0 −0.415415 + 0.909632i −0.588302 + 0.678936i 0 0
118.1 −0.0801894 0.557730i 0 0.654861 0.192284i 0 0 −0.841254 + 0.540641i −0.393828 0.862362i 0 0
118.2 0.0801894 + 0.557730i 0 0.654861 0.192284i 0 0 −0.841254 + 0.540641i 0.393828 + 0.862362i 0 0
307.1 −0.0801894 + 0.557730i 0 0.654861 + 0.192284i 0 0 −0.841254 0.540641i −0.393828 + 0.862362i 0 0
307.2 0.0801894 0.557730i 0 0.654861 + 0.192284i 0 0 −0.841254 0.540641i 0.393828 0.862362i 0 0
370.1 −1.53046 0.983568i 0 0.959493 + 2.10100i 0 0 0.654861 0.755750i 0.339098 2.35848i 0 0
370.2 1.53046 + 0.983568i 0 0.959493 + 2.10100i 0 0 0.654861 0.755750i −0.339098 + 2.35848i 0 0
496.1 −1.29639 1.49611i 0 −0.415415 + 2.88927i 0 0 0.959493 + 0.281733i 3.19584 2.05384i 0 0
496.2 1.29639 + 1.49611i 0 −0.415415 + 2.88927i 0 0 0.959493 + 0.281733i −3.19584 + 2.05384i 0 0
685.1 −1.03748 0.304632i 0 0.142315 + 0.0914602i 0 0 −0.415415 0.909632i 0.588302 + 0.678936i 0 0
685.2 1.03748 + 0.304632i 0 0.142315 + 0.0914602i 0 0 −0.415415 0.909632i −0.588302 0.678936i 0 0
748.1 −1.53046 + 0.983568i 0 0.959493 2.10100i 0 0 0.654861 + 0.755750i 0.339098 + 2.35848i 0 0
748.2 1.53046 0.983568i 0 0.959493 2.10100i 0 0 0.654861 + 0.755750i −0.339098 2.35848i 0 0
811.1 −0.627899 + 1.37491i 0 −0.841254 0.970858i 0 0 0.142315 0.989821i 0.412791 0.121206i 0 0
811.2 0.627899 1.37491i 0 −0.841254 0.970858i 0 0 0.142315 0.989821i −0.412791 + 0.121206i 0 0
1189.1 −1.29639 + 1.49611i 0 −0.415415 2.88927i 0 0 0.959493 0.281733i 3.19584 + 2.05384i 0 0
1189.2 1.29639 1.49611i 0 −0.415415 2.88927i 0 0 0.959493 0.281733i −3.19584 2.05384i 0 0
1315.1 −0.627899 1.37491i 0 −0.841254 + 0.970858i 0 0 0.142315 + 0.989821i 0.412791 + 0.121206i 0 0
1315.2 0.627899 + 1.37491i 0 −0.841254 + 0.970858i 0 0 0.142315 + 0.989821i −0.412791 0.121206i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1315.2 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
23.c even 11 1 inner
69.h odd 22 1 inner
161.l odd 22 1 inner
483.v even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1449.1.bq.b 20
3.b odd 2 1 inner 1449.1.bq.b 20
7.b odd 2 1 CM 1449.1.bq.b 20
21.c even 2 1 inner 1449.1.bq.b 20
23.c even 11 1 inner 1449.1.bq.b 20
69.h odd 22 1 inner 1449.1.bq.b 20
161.l odd 22 1 inner 1449.1.bq.b 20
483.v even 22 1 inner 1449.1.bq.b 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1449.1.bq.b 20 1.a even 1 1 trivial
1449.1.bq.b 20 3.b odd 2 1 inner
1449.1.bq.b 20 7.b odd 2 1 CM
1449.1.bq.b 20 21.c even 2 1 inner
1449.1.bq.b 20 23.c even 11 1 inner
1449.1.bq.b 20 69.h odd 22 1 inner
1449.1.bq.b 20 161.l odd 22 1 inner
1449.1.bq.b 20 483.v even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1449, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$121 + 605 T^{2} + 484 T^{4} - 968 T^{6} + 484 T^{8} + 99 T^{10} + 165 T^{12} + 22 T^{16} + T^{20}$$
$3$ $$T^{20}$$
$5$ $$T^{20}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$11$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$29$ $$121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20}$$
$31$ $$T^{20}$$
$37$ $$( 1 - 6 T + 14 T^{2} - 7 T^{3} + 9 T^{4} + 12 T^{5} - 6 T^{6} + 3 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$41$ $$T^{20}$$
$43$ $$( 1 + 5 T + 14 T^{2} + 4 T^{3} - 2 T^{4} + T^{5} + 5 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$47$ $$T^{20}$$
$53$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$59$ $$T^{20}$$
$61$ $$T^{20}$$
$67$ $$( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$71$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$73$ $$T^{20}$$
$79$ $$( 1 + 6 T + 14 T^{2} + 7 T^{3} + 9 T^{4} - 12 T^{5} - 6 T^{6} - 3 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$