# Properties

 Label 1424.1.bt.a Level $1424$ Weight $1$ Character orbit 1424.bt Analytic conductor $0.711$ Analytic rank $0$ Dimension $20$ Projective image $D_{44}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1424 = 2^{4} \cdot 89$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1424.bt (of order $$44$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.710668577989$$ Analytic rank: $$0$$ Dimension: $$20$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{44}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{44} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{44}^{2} + \zeta_{44}^{8} ) q^{5} + \zeta_{44}^{19} q^{9} +O(q^{10})$$ $$q + ( \zeta_{44}^{2} + \zeta_{44}^{8} ) q^{5} + \zeta_{44}^{19} q^{9} + ( \zeta_{44}^{7} - \zeta_{44}^{12} ) q^{13} + ( -\zeta_{44} + \zeta_{44}^{3} ) q^{17} + ( \zeta_{44}^{4} + \zeta_{44}^{10} + \zeta_{44}^{16} ) q^{25} + ( -\zeta_{44}^{6} - \zeta_{44}^{15} ) q^{29} + ( \zeta_{44}^{13} + \zeta_{44}^{20} ) q^{37} + ( -\zeta_{44}^{11} + \zeta_{44}^{14} ) q^{41} + ( -\zeta_{44}^{5} + \zeta_{44}^{21} ) q^{45} -\zeta_{44}^{21} q^{49} + ( -\zeta_{44}^{13} - \zeta_{44}^{17} ) q^{53} + ( \zeta_{44}^{17} + \zeta_{44}^{18} ) q^{61} + ( \zeta_{44}^{9} - \zeta_{44}^{14} + \zeta_{44}^{15} - \zeta_{44}^{20} ) q^{65} + ( -\zeta_{44}^{10} - \zeta_{44}^{18} ) q^{73} -\zeta_{44}^{16} q^{81} + ( -\zeta_{44}^{3} + \zeta_{44}^{5} - \zeta_{44}^{9} + \zeta_{44}^{11} ) q^{85} + \zeta_{44}^{6} q^{89} + ( \zeta_{44} - \zeta_{44}^{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + O(q^{10})$$ $$20q + 2q^{13} - 2q^{25} - 2q^{29} - 2q^{37} + 2q^{41} + 2q^{61} - 4q^{73} + 2q^{81} + 2q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1424\mathbb{Z}\right)^\times$$.

 $$n$$ $$357$$ $$1247$$ $$1249$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{44}^{19}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.281733 + 0.959493i −0.540641 + 0.841254i 0.755750 + 0.654861i −0.281733 − 0.959493i 0.989821 + 0.142315i 0.909632 + 0.415415i −0.909632 + 0.415415i 0.540641 + 0.841254i −0.540641 − 0.841254i 0.909632 − 0.415415i −0.909632 − 0.415415i −0.989821 − 0.142315i 0.281733 + 0.959493i −0.755750 − 0.654861i 0.540641 − 0.841254i 0.281733 − 0.959493i −0.989821 + 0.142315i 0.755750 − 0.654861i −0.755750 + 0.654861i 0.989821 − 0.142315i
0 0 0 −1.49611 + 0.215109i 0 0 0 −0.755750 0.654861i 0
79.1 0 0 0 −0.557730 1.89945i 0 0 0 −0.989821 + 0.142315i 0
287.1 0 0 0 0.983568 + 0.449181i 0 0 0 0.540641 + 0.841254i 0
303.1 0 0 0 −1.49611 0.215109i 0 0 0 −0.755750 + 0.654861i 0
335.1 0 0 0 1.37491 + 1.19136i 0 0 0 −0.909632 + 0.415415i 0
351.1 0 0 0 −0.304632 + 0.474017i 0 0 0 −0.281733 + 0.959493i 0
463.1 0 0 0 −0.304632 0.474017i 0 0 0 0.281733 + 0.959493i 0
543.1 0 0 0 −0.557730 + 1.89945i 0 0 0 0.989821 + 0.142315i 0
703.1 0 0 0 −0.557730 + 1.89945i 0 0 0 −0.989821 0.142315i 0
783.1 0 0 0 −0.304632 0.474017i 0 0 0 −0.281733 0.959493i 0
895.1 0 0 0 −0.304632 + 0.474017i 0 0 0 0.281733 0.959493i 0
911.1 0 0 0 1.37491 + 1.19136i 0 0 0 0.909632 0.415415i 0
943.1 0 0 0 −1.49611 0.215109i 0 0 0 0.755750 0.654861i 0
959.1 0 0 0 0.983568 + 0.449181i 0 0 0 −0.540641 0.841254i 0
1167.1 0 0 0 −0.557730 1.89945i 0 0 0 0.989821 0.142315i 0
1199.1 0 0 0 −1.49611 + 0.215109i 0 0 0 0.755750 + 0.654861i 0
1263.1 0 0 0 1.37491 1.19136i 0 0 0 0.909632 + 0.415415i 0
1295.1 0 0 0 0.983568 0.449181i 0 0 0 0.540641 0.841254i 0
1375.1 0 0 0 0.983568 0.449181i 0 0 0 −0.540641 + 0.841254i 0
1407.1 0 0 0 1.37491 1.19136i 0 0 0 −0.909632 0.415415i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1407.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
89.g even 44 1 inner
356.n odd 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1424.1.bt.a 20
4.b odd 2 1 CM 1424.1.bt.a 20
89.g even 44 1 inner 1424.1.bt.a 20
356.n odd 44 1 inner 1424.1.bt.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1424.1.bt.a 20 1.a even 1 1 trivial
1424.1.bt.a 20 4.b odd 2 1 CM
1424.1.bt.a 20 89.g even 44 1 inner
1424.1.bt.a 20 356.n odd 44 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1424, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20}$$
$5$ $$( 11 + 11 T + 11 T^{2} - 33 T^{3} + 22 T^{5} + T^{10} )^{2}$$
$7$ $$T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 - 12 T + 105 T^{2} - 484 T^{3} + 1218 T^{4} - 1702 T^{5} + 1324 T^{6} - 484 T^{7} - 178 T^{8} + 420 T^{9} - 331 T^{10} + 122 T^{11} + 93 T^{12} - 8 T^{14} + 8 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$17$ $$1 + 19 T^{2} + 119 T^{4} - 203 T^{6} + 444 T^{8} - 474 T^{10} + 234 T^{12} - 64 T^{14} + 16 T^{16} - 4 T^{18} + T^{20}$$
$19$ $$T^{20}$$
$23$ $$T^{20}$$
$29$ $$1 - 10 T + 17 T^{2} + 44 T^{3} + 338 T^{4} + 316 T^{5} + 400 T^{6} + 110 T^{7} - 90 T^{8} + 460 T^{9} + 505 T^{10} + 274 T^{11} + 93 T^{12} - 44 T^{13} - 52 T^{14} - 30 T^{15} - 4 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$31$ $$T^{20}$$
$37$ $$1 - 10 T + 50 T^{2} - 25 T^{4} + 52 T^{5} + 730 T^{6} + 748 T^{7} + 383 T^{8} - 2 T^{9} + 472 T^{10} + 472 T^{11} + 236 T^{12} + 58 T^{14} + 58 T^{15} + 29 T^{16} + 2 T^{18} + 2 T^{19} + T^{20}$$
$41$ $$1 - 12 T + 17 T^{2} + 154 T^{3} - 135 T^{4} - 404 T^{5} + 554 T^{6} - 22 T^{7} - 90 T^{8} + 46 T^{9} + 230 T^{10} - 252 T^{11} + 368 T^{12} - 242 T^{13} + 223 T^{14} - 102 T^{15} + 73 T^{16} - 22 T^{17} + 13 T^{18} - 2 T^{19} + T^{20}$$
$43$ $$T^{20}$$
$47$ $$T^{20}$$
$53$ $$1 + 19 T^{2} + 119 T^{4} - 203 T^{6} + 444 T^{8} - 474 T^{10} + 234 T^{12} - 64 T^{14} + 16 T^{16} - 4 T^{18} + T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 + 10 T + 61 T^{2} + 66 T^{3} - 69 T^{4} + 36 T^{5} + 37 T^{6} - 660 T^{7} + 944 T^{8} - 614 T^{9} + 142 T^{10} + 166 T^{11} + 16 T^{12} - 88 T^{13} + 80 T^{14} - 36 T^{15} - 4 T^{16} + 2 T^{18} - 2 T^{19} + T^{20}$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$( 1 - 5 T + 3 T^{2} + 7 T^{3} + 20 T^{4} + 10 T^{5} + 16 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$97$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$