# Properties

 Label 1564.1.w.c Level $1564$ Weight $1$ Character orbit 1564.w Analytic conductor $0.781$ Analytic rank $0$ Dimension $20$ Projective image $D_{22}$ CM discriminant -68 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1564 = 2^{2} \cdot 17 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1564.w (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.780537679758$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{44}^{10} q^{2} + ( -\zeta_{44}^{7} - \zeta_{44}^{21} ) q^{3} + \zeta_{44}^{20} q^{4} + ( \zeta_{44}^{9} - \zeta_{44}^{17} ) q^{6} + ( -\zeta_{44}^{5} + \zeta_{44}^{9} ) q^{7} -\zeta_{44}^{8} q^{8} + ( -\zeta_{44}^{6} + \zeta_{44}^{14} - \zeta_{44}^{20} ) q^{9} +O(q^{10})$$ $$q + \zeta_{44}^{10} q^{2} + ( -\zeta_{44}^{7} - \zeta_{44}^{21} ) q^{3} + \zeta_{44}^{20} q^{4} + ( \zeta_{44}^{9} - \zeta_{44}^{17} ) q^{6} + ( -\zeta_{44}^{5} + \zeta_{44}^{9} ) q^{7} -\zeta_{44}^{8} q^{8} + ( -\zeta_{44}^{6} + \zeta_{44}^{14} - \zeta_{44}^{20} ) q^{9} + ( -\zeta_{44}^{9} - \zeta_{44}^{15} ) q^{11} + ( \zeta_{44}^{5} + \zeta_{44}^{19} ) q^{12} + ( \zeta_{44}^{2} + \zeta_{44}^{6} ) q^{13} + ( -\zeta_{44}^{15} + \zeta_{44}^{19} ) q^{14} -\zeta_{44}^{18} q^{16} -\zeta_{44}^{2} q^{17} + ( -\zeta_{44}^{2} + \zeta_{44}^{8} - \zeta_{44}^{16} ) q^{18} + ( -\zeta_{44}^{4} + \zeta_{44}^{8} + \zeta_{44}^{12} - \zeta_{44}^{16} ) q^{21} + ( \zeta_{44}^{3} - \zeta_{44}^{19} ) q^{22} -\zeta_{44}^{13} q^{23} + ( -\zeta_{44}^{7} + \zeta_{44}^{15} ) q^{24} -\zeta_{44}^{10} q^{25} + ( \zeta_{44}^{12} + \zeta_{44}^{16} ) q^{26} + ( -\zeta_{44}^{5} + \zeta_{44}^{13} - \zeta_{44}^{19} - \zeta_{44}^{21} ) q^{27} + ( \zeta_{44}^{3} - \zeta_{44}^{7} ) q^{28} + ( \zeta_{44}^{5} + \zeta_{44}^{11} ) q^{31} + \zeta_{44}^{6} q^{32} + ( -1 - \zeta_{44}^{8} - \zeta_{44}^{14} + \zeta_{44}^{16} ) q^{33} -\zeta_{44}^{12} q^{34} + ( \zeta_{44}^{4} - \zeta_{44}^{12} + \zeta_{44}^{18} ) q^{36} + ( \zeta_{44} + \zeta_{44}^{5} - \zeta_{44}^{9} - \zeta_{44}^{13} ) q^{39} + ( -1 + \zeta_{44}^{4} - \zeta_{44}^{14} + \zeta_{44}^{18} ) q^{42} + ( \zeta_{44}^{7} + \zeta_{44}^{13} ) q^{44} + \zeta_{44} q^{46} + ( -\zeta_{44}^{3} - \zeta_{44}^{17} ) q^{48} + ( \zeta_{44}^{10} - \zeta_{44}^{14} + \zeta_{44}^{18} ) q^{49} -\zeta_{44}^{20} q^{50} + ( -\zeta_{44} + \zeta_{44}^{9} ) q^{51} + ( -1 - \zeta_{44}^{4} ) q^{52} + ( \zeta_{44}^{4} - \zeta_{44}^{10} ) q^{53} + ( -\zeta_{44} + \zeta_{44}^{7} + \zeta_{44}^{9} - \zeta_{44}^{15} ) q^{54} + ( \zeta_{44}^{13} - \zeta_{44}^{17} ) q^{56} + ( \zeta_{44}^{15} + \zeta_{44}^{21} ) q^{62} + ( -\zeta_{44} - \zeta_{44}^{3} + \zeta_{44}^{7} + \zeta_{44}^{11} - \zeta_{44}^{15} - \zeta_{44}^{19} ) q^{63} + \zeta_{44}^{16} q^{64} + ( \zeta_{44}^{2} - \zeta_{44}^{4} - \zeta_{44}^{10} - \zeta_{44}^{18} ) q^{66} + q^{68} + ( -\zeta_{44}^{12} + \zeta_{44}^{20} ) q^{69} + ( 1 - \zeta_{44}^{6} + \zeta_{44}^{14} ) q^{72} + ( -\zeta_{44}^{9} + \zeta_{44}^{17} ) q^{75} + ( \zeta_{44}^{2} + \zeta_{44}^{14} - \zeta_{44}^{18} + \zeta_{44}^{20} ) q^{77} + ( \zeta_{44} + \zeta_{44}^{11} + \zeta_{44}^{15} - \zeta_{44}^{19} ) q^{78} + ( -\zeta_{44}^{3} - \zeta_{44}^{5} ) q^{79} + ( -\zeta_{44}^{4} - \zeta_{44}^{6} + \zeta_{44}^{12} - \zeta_{44}^{18} - \zeta_{44}^{20} ) q^{81} + ( \zeta_{44}^{2} - \zeta_{44}^{6} - \zeta_{44}^{10} + \zeta_{44}^{14} ) q^{84} + ( -\zeta_{44} + \zeta_{44}^{17} ) q^{88} + ( -1 + \zeta_{44}^{6} ) q^{89} + ( -\zeta_{44}^{7} + \zeta_{44}^{15} ) q^{91} + \zeta_{44}^{11} q^{92} + ( \zeta_{44}^{4} + \zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{18} ) q^{93} + ( \zeta_{44}^{5} - \zeta_{44}^{13} ) q^{96} + ( \zeta_{44}^{2} - \zeta_{44}^{6} + \zeta_{44}^{20} ) q^{98} + ( \zeta_{44} - \zeta_{44}^{13} + \zeta_{44}^{15} + \zeta_{44}^{21} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{2} - 2q^{4} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$20q + 2q^{2} - 2q^{4} + 2q^{8} + 2q^{9} + 4q^{13} - 2q^{16} - 2q^{17} - 2q^{18} - 2q^{25} - 4q^{26} + 2q^{32} - 22q^{33} + 2q^{34} + 2q^{36} - 22q^{42} + 2q^{49} + 2q^{50} - 18q^{52} - 4q^{53} - 2q^{64} + 20q^{68} + 20q^{72} - 2q^{81} - 18q^{89} - 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1564\mathbb{Z}\right)^\times$$.

 $$n$$ $$783$$ $$921$$ $$1293$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{44}^{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 −0.281733 − 0.959493i 0.281733 + 0.959493i −0.755750 + 0.654861i 0.755750 − 0.654861i −0.755750 − 0.654861i 0.755750 + 0.654861i 0.540641 − 0.841254i −0.540641 + 0.841254i −0.281733 + 0.959493i 0.281733 − 0.959493i −0.989821 + 0.142315i 0.989821 − 0.142315i −0.909632 + 0.415415i 0.909632 − 0.415415i −0.909632 − 0.415415i 0.909632 + 0.415415i −0.989821 − 0.142315i 0.989821 + 0.142315i 0.540641 + 0.841254i −0.540641 − 0.841254i
0.959493 + 0.281733i −1.19136 + 1.37491i 0.841254 + 0.540641i 0 −1.53046 + 0.983568i 0.449181 + 0.983568i 0.654861 + 0.755750i −0.328708 2.28621i 0
271.2 0.959493 + 0.281733i 1.19136 1.37491i 0.841254 + 0.540641i 0 1.53046 0.983568i −0.449181 0.983568i 0.654861 + 0.755750i −0.328708 2.28621i 0
407.1 0.654861 0.755750i −0.474017 + 0.304632i −0.142315 0.989821i 0 −0.0801894 + 0.557730i −1.89945 + 0.557730i −0.841254 0.540641i −0.283524 + 0.620830i 0
407.2 0.654861 0.755750i 0.474017 0.304632i −0.142315 0.989821i 0 0.0801894 0.557730i 1.89945 0.557730i −0.841254 0.540641i −0.283524 + 0.620830i 0
611.1 0.654861 + 0.755750i −0.474017 0.304632i −0.142315 + 0.989821i 0 −0.0801894 0.557730i −1.89945 0.557730i −0.841254 + 0.540641i −0.283524 0.620830i 0
611.2 0.654861 + 0.755750i 0.474017 + 0.304632i −0.142315 + 0.989821i 0 0.0801894 + 0.557730i 1.89945 + 0.557730i −0.841254 + 0.540641i −0.283524 0.620830i 0
679.1 −0.841254 + 0.540641i −0.215109 + 1.49611i 0.415415 0.909632i 0 −0.627899 1.37491i −1.19136 1.37491i 0.142315 + 0.989821i −1.23259 0.361922i 0
679.2 −0.841254 + 0.540641i 0.215109 1.49611i 0.415415 0.909632i 0 0.627899 + 1.37491i 1.19136 + 1.37491i 0.142315 + 0.989821i −1.23259 0.361922i 0
883.1 0.959493 0.281733i −1.19136 1.37491i 0.841254 0.540641i 0 −1.53046 0.983568i 0.449181 0.983568i 0.654861 0.755750i −0.328708 + 2.28621i 0
883.2 0.959493 0.281733i 1.19136 + 1.37491i 0.841254 0.540641i 0 1.53046 + 0.983568i −0.449181 + 0.983568i 0.654861 0.755750i −0.328708 + 2.28621i 0
951.1 0.142315 0.989821i −0.449181 0.983568i −0.959493 0.281733i 0 −1.03748 + 0.304632i 0.474017 + 0.304632i −0.415415 + 0.909632i −0.110783 + 0.127850i 0
951.2 0.142315 0.989821i 0.449181 + 0.983568i −0.959493 0.281733i 0 1.03748 0.304632i −0.474017 0.304632i −0.415415 + 0.909632i −0.110783 + 0.127850i 0
1087.1 −0.415415 + 0.909632i −1.89945 0.557730i −0.654861 0.755750i 0 1.29639 1.49611i 0.215109 1.49611i 0.959493 0.281733i 2.45561 + 1.57812i 0
1087.2 −0.415415 + 0.909632i 1.89945 + 0.557730i −0.654861 0.755750i 0 −1.29639 + 1.49611i −0.215109 + 1.49611i 0.959493 0.281733i 2.45561 + 1.57812i 0
1223.1 −0.415415 0.909632i −1.89945 + 0.557730i −0.654861 + 0.755750i 0 1.29639 + 1.49611i 0.215109 + 1.49611i 0.959493 + 0.281733i 2.45561 1.57812i 0
1223.2 −0.415415 0.909632i 1.89945 0.557730i −0.654861 + 0.755750i 0 −1.29639 1.49611i −0.215109 1.49611i 0.959493 + 0.281733i 2.45561 1.57812i 0
1291.1 0.142315 + 0.989821i −0.449181 + 0.983568i −0.959493 + 0.281733i 0 −1.03748 0.304632i 0.474017 0.304632i −0.415415 0.909632i −0.110783 0.127850i 0
1291.2 0.142315 + 0.989821i 0.449181 0.983568i −0.959493 + 0.281733i 0 1.03748 + 0.304632i −0.474017 + 0.304632i −0.415415 0.909632i −0.110783 0.127850i 0
1359.1 −0.841254 0.540641i −0.215109 1.49611i 0.415415 + 0.909632i 0 −0.627899 + 1.37491i −1.19136 + 1.37491i 0.142315 0.989821i −1.23259 + 0.361922i 0
1359.2 −0.841254 0.540641i 0.215109 + 1.49611i 0.415415 + 0.909632i 0 0.627899 1.37491i 1.19136 1.37491i 0.142315 0.989821i −1.23259 + 0.361922i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1359.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
4.b odd 2 1 inner
17.b even 2 1 inner
23.c even 11 1 inner
92.g odd 22 1 inner
391.n even 22 1 inner
1564.w odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1564.1.w.c 20
4.b odd 2 1 inner 1564.1.w.c 20
17.b even 2 1 inner 1564.1.w.c 20
23.c even 11 1 inner 1564.1.w.c 20
68.d odd 2 1 CM 1564.1.w.c 20
92.g odd 22 1 inner 1564.1.w.c 20
391.n even 22 1 inner 1564.1.w.c 20
1564.w odd 22 1 inner 1564.1.w.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1564.1.w.c 20 1.a even 1 1 trivial
1564.1.w.c 20 4.b odd 2 1 inner
1564.1.w.c 20 17.b even 2 1 inner
1564.1.w.c 20 23.c even 11 1 inner
1564.1.w.c 20 68.d odd 2 1 CM
1564.1.w.c 20 92.g odd 22 1 inner
1564.1.w.c 20 391.n even 22 1 inner
1564.1.w.c 20 1564.w odd 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} + 22 T_{3}^{12} + 462 T_{3}^{10} + 1452 T_{3}^{8} + 1573 T_{3}^{6} + 847 T_{3}^{4} - 121 T_{3}^{2} + 121$$ acting on $$S_{1}^{\mathrm{new}}(1564, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$3$ $$121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$121 - 121 T^{2} + 847 T^{4} + 1573 T^{6} + 1452 T^{8} + 462 T^{10} + 22 T^{12} + T^{20}$$
$11$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$13$ $$( 1 + 5 T + 3 T^{2} - 7 T^{3} + 20 T^{4} - 10 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$17$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$19$ $$T^{20}$$
$23$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20}$$
$29$ $$T^{20}$$
$31$ $$121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20}$$
$37$ $$T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$T^{20}$$
$53$ $$( 1 - 5 T + 14 T^{2} - 4 T^{3} - 2 T^{4} - T^{5} + 5 T^{6} + 8 T^{7} + 4 T^{8} + 2 T^{9} + T^{10} )^{2}$$
$59$ $$T^{20}$$
$61$ $$T^{20}$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$T^{20}$$
$79$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$( 1 + 5 T + 25 T^{2} + 70 T^{3} + 130 T^{4} + 166 T^{5} + 148 T^{6} + 91 T^{7} + 37 T^{8} + 9 T^{9} + T^{10} )^{2}$$
$97$ $$T^{20}$$