# Properties

 Label 1288.1.bi.e Level $1288$ Weight $1$ Character orbit 1288.bi Analytic conductor $0.643$ Analytic rank $0$ Dimension $20$ Projective image $D_{22}$ CM discriminant -56 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1288 = 2^{3} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1288.bi (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.642795736271$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$11$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{44}^{18} q^{2} + ( \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{3} -\zeta_{44}^{14} q^{4} + ( -\zeta_{44}^{3} + \zeta_{44}^{15} ) q^{5} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{6} + \zeta_{44}^{6} q^{7} + \zeta_{44}^{10} q^{8} + ( -1 - \zeta_{44}^{2} - \zeta_{44}^{4} ) q^{9} +O(q^{10})$$ $$q + \zeta_{44}^{18} q^{2} + ( \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{3} -\zeta_{44}^{14} q^{4} + ( -\zeta_{44}^{3} + \zeta_{44}^{15} ) q^{5} + ( -\zeta_{44}^{7} - \zeta_{44}^{9} ) q^{6} + \zeta_{44}^{6} q^{7} + \zeta_{44}^{10} q^{8} + ( -1 - \zeta_{44}^{2} - \zeta_{44}^{4} ) q^{9} + ( -\zeta_{44}^{11} - \zeta_{44}^{21} ) q^{10} + ( \zeta_{44}^{3} + \zeta_{44}^{5} ) q^{12} + ( -\zeta_{44} + \zeta_{44}^{9} ) q^{13} -\zeta_{44}^{2} q^{14} + ( -\zeta_{44}^{4} - \zeta_{44}^{6} - \zeta_{44}^{14} - \zeta_{44}^{16} ) q^{15} -\zeta_{44}^{6} q^{16} + ( 1 - \zeta_{44}^{18} - \zeta_{44}^{20} ) q^{18} + ( \zeta_{44} - \zeta_{44}^{5} ) q^{19} + ( \zeta_{44}^{7} + \zeta_{44}^{17} ) q^{20} + ( \zeta_{44}^{17} + \zeta_{44}^{19} ) q^{21} + \zeta_{44}^{16} q^{23} + ( -\zeta_{44} + \zeta_{44}^{21} ) q^{24} + ( \zeta_{44}^{6} - \zeta_{44}^{8} - \zeta_{44}^{18} ) q^{25} + ( -\zeta_{44}^{5} - \zeta_{44}^{19} ) q^{26} + ( -\zeta_{44}^{11} - \zeta_{44}^{13} - \zeta_{44}^{15} - \zeta_{44}^{17} ) q^{27} -\zeta_{44}^{20} q^{28} + ( 1 + \zeta_{44}^{2} + \zeta_{44}^{10} + \zeta_{44}^{12} ) q^{30} + \zeta_{44}^{2} q^{32} + ( -\zeta_{44}^{9} + \zeta_{44}^{21} ) q^{35} + ( \zeta_{44}^{14} + \zeta_{44}^{16} + \zeta_{44}^{18} ) q^{36} + ( \zeta_{44} + \zeta_{44}^{19} ) q^{38} + ( -1 - \zeta_{44}^{12} - \zeta_{44}^{14} + \zeta_{44}^{20} ) q^{39} + ( -\zeta_{44}^{3} - \zeta_{44}^{13} ) q^{40} + ( -\zeta_{44}^{13} - \zeta_{44}^{15} ) q^{42} + ( \zeta_{44}^{3} + \zeta_{44}^{5} + \zeta_{44}^{7} - \zeta_{44}^{15} - \zeta_{44}^{17} - \zeta_{44}^{19} ) q^{45} -\zeta_{44}^{12} q^{46} + ( -\zeta_{44}^{17} - \zeta_{44}^{19} ) q^{48} + \zeta_{44}^{12} q^{49} + ( -\zeta_{44}^{2} + \zeta_{44}^{4} + \zeta_{44}^{14} ) q^{50} + ( \zeta_{44} + \zeta_{44}^{15} ) q^{52} + ( \zeta_{44}^{7} + \zeta_{44}^{9} + \zeta_{44}^{11} + \zeta_{44}^{13} ) q^{54} + \zeta_{44}^{16} q^{56} + ( \zeta_{44}^{12} + \zeta_{44}^{14} - \zeta_{44}^{16} - \zeta_{44}^{18} ) q^{57} + ( \zeta_{44}^{3} - \zeta_{44}^{7} ) q^{59} + ( -\zeta_{44}^{6} - \zeta_{44}^{8} + \zeta_{44}^{18} + \zeta_{44}^{20} ) q^{60} + ( -\zeta_{44}^{6} - \zeta_{44}^{8} - \zeta_{44}^{10} ) q^{63} + \zeta_{44}^{20} q^{64} + ( -\zeta_{44}^{2} + \zeta_{44}^{4} - \zeta_{44}^{12} - \zeta_{44}^{16} ) q^{65} + ( -\zeta_{44}^{5} - \zeta_{44}^{7} ) q^{69} + ( \zeta_{44}^{5} - \zeta_{44}^{17} ) q^{70} + ( -1 + \zeta_{44}^{14} ) q^{71} + ( -\zeta_{44}^{10} - \zeta_{44}^{12} - \zeta_{44}^{14} ) q^{72} + ( \zeta_{44}^{7} + \zeta_{44}^{9} + \zeta_{44}^{17} - \zeta_{44}^{21} ) q^{75} + ( -\zeta_{44}^{15} + \zeta_{44}^{19} ) q^{76} + ( \zeta_{44}^{8} + \zeta_{44}^{10} - \zeta_{44}^{16} - \zeta_{44}^{18} ) q^{78} + ( \zeta_{44}^{2} - \zeta_{44}^{8} ) q^{79} + ( \zeta_{44}^{9} - \zeta_{44}^{21} ) q^{80} + ( 1 + \zeta_{44}^{2} + \zeta_{44}^{4} + \zeta_{44}^{6} + \zeta_{44}^{8} ) q^{81} + ( \zeta_{44}^{5} - \zeta_{44}^{21} ) q^{83} + ( \zeta_{44}^{9} + \zeta_{44}^{11} ) q^{84} + ( -\zeta_{44} - \zeta_{44}^{3} + \zeta_{44}^{11} + \zeta_{44}^{13} + \zeta_{44}^{15} + \zeta_{44}^{21} ) q^{90} + ( -\zeta_{44}^{7} + \zeta_{44}^{15} ) q^{91} + \zeta_{44}^{8} q^{92} + ( -\zeta_{44}^{4} + \zeta_{44}^{8} + \zeta_{44}^{16} - \zeta_{44}^{20} ) q^{95} + ( \zeta_{44}^{13} + \zeta_{44}^{15} ) q^{96} -\zeta_{44}^{8} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{2} - 2q^{4} + 2q^{7} + 2q^{8} - 20q^{9} + O(q^{10})$$ $$20q + 2q^{2} - 2q^{4} + 2q^{7} + 2q^{8} - 20q^{9} - 2q^{14} - 2q^{16} + 20q^{18} - 2q^{23} + 2q^{25} + 2q^{28} + 22q^{30} + 2q^{32} + 2q^{36} - 22q^{39} + 2q^{46} - 2q^{49} - 2q^{50} - 2q^{56} - 2q^{63} - 2q^{64} - 18q^{71} - 2q^{72} + 4q^{79} + 20q^{81} - 2q^{92} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1288\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$645$$ $$967$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{44}^{18}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −0.540641 + 0.841254i 0.540641 − 0.841254i 0.281733 + 0.959493i −0.281733 − 0.959493i −0.909632 + 0.415415i 0.909632 − 0.415415i 0.755750 + 0.654861i −0.755750 − 0.654861i 0.989821 − 0.142315i −0.989821 + 0.142315i 0.989821 + 0.142315i −0.989821 − 0.142315i 0.755750 − 0.654861i −0.755750 + 0.654861i 0.281733 − 0.959493i −0.281733 + 0.959493i −0.540641 − 0.841254i 0.540641 + 0.841254i −0.909632 − 0.415415i 0.909632 + 0.415415i
0.654861 + 0.755750i −0.909632 0.584585i −0.142315 + 0.989821i −0.234072 + 0.512546i −0.153882 1.07028i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.0702757 + 0.153882i −0.540641 + 0.158746i
13.2 0.654861 + 0.755750i 0.909632 + 0.584585i −0.142315 + 0.989821i 0.234072 0.512546i 0.153882 + 1.07028i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.0702757 + 0.153882i 0.540641 0.158746i
349.1 −0.415415 0.909632i −0.540641 + 0.158746i −0.654861 + 0.755750i 1.66538 + 1.07028i 0.368991 + 0.425839i 0.142315 + 0.989821i 0.959493 + 0.281733i −0.574161 + 0.368991i 0.281733 1.95949i
349.2 −0.415415 0.909632i 0.540641 0.158746i −0.654861 + 0.755750i −1.66538 1.07028i −0.368991 0.425839i 0.142315 + 0.989821i 0.959493 + 0.281733i −0.574161 + 0.368991i −0.281733 + 1.95949i
629.1 0.142315 0.989821i −0.755750 1.65486i −0.959493 0.281733i −0.708089 0.817178i −1.74557 + 0.512546i −0.841254 0.540641i −0.415415 + 0.909632i −1.51255 + 1.74557i −0.909632 + 0.584585i
629.2 0.142315 0.989821i 0.755750 + 1.65486i −0.959493 0.281733i 0.708089 + 0.817178i 1.74557 0.512546i −0.841254 0.540641i −0.415415 + 0.909632i −1.51255 + 1.74557i 0.909632 0.584585i
685.1 0.959493 + 0.281733i −0.989821 + 1.14231i 0.841254 + 0.540641i 0.258908 1.80075i −1.27155 + 0.817178i −0.415415 0.909632i 0.654861 + 0.755750i −0.182822 1.27155i 0.755750 1.65486i
685.2 0.959493 + 0.281733i 0.989821 1.14231i 0.841254 + 0.540641i −0.258908 + 1.80075i 1.27155 0.817178i −0.415415 0.909632i 0.654861 + 0.755750i −0.182822 1.27155i −0.755750 + 1.65486i
853.1 −0.841254 0.540641i −0.281733 1.95949i 0.415415 + 0.909632i −1.45027 0.425839i −0.822373 + 1.80075i 0.654861 0.755750i 0.142315 0.989821i −2.80075 + 0.822373i 0.989821 + 1.14231i
853.2 −0.841254 0.540641i 0.281733 + 1.95949i 0.415415 + 0.909632i 1.45027 + 0.425839i 0.822373 1.80075i 0.654861 0.755750i 0.142315 0.989821i −2.80075 + 0.822373i −0.989821 1.14231i
909.1 −0.841254 + 0.540641i −0.281733 + 1.95949i 0.415415 0.909632i −1.45027 + 0.425839i −0.822373 1.80075i 0.654861 + 0.755750i 0.142315 + 0.989821i −2.80075 0.822373i 0.989821 1.14231i
909.2 −0.841254 + 0.540641i 0.281733 1.95949i 0.415415 0.909632i 1.45027 0.425839i 0.822373 + 1.80075i 0.654861 + 0.755750i 0.142315 + 0.989821i −2.80075 0.822373i −0.989821 + 1.14231i
1021.1 0.959493 0.281733i −0.989821 1.14231i 0.841254 0.540641i 0.258908 + 1.80075i −1.27155 0.817178i −0.415415 + 0.909632i 0.654861 0.755750i −0.182822 + 1.27155i 0.755750 + 1.65486i
1021.2 0.959493 0.281733i 0.989821 + 1.14231i 0.841254 0.540641i −0.258908 1.80075i 1.27155 + 0.817178i −0.415415 + 0.909632i 0.654861 0.755750i −0.182822 + 1.27155i −0.755750 1.65486i
1133.1 −0.415415 + 0.909632i −0.540641 0.158746i −0.654861 0.755750i 1.66538 1.07028i 0.368991 0.425839i 0.142315 0.989821i 0.959493 0.281733i −0.574161 0.368991i 0.281733 + 1.95949i
1133.2 −0.415415 + 0.909632i 0.540641 + 0.158746i −0.654861 0.755750i −1.66538 + 1.07028i −0.368991 + 0.425839i 0.142315 0.989821i 0.959493 0.281733i −0.574161 0.368991i −0.281733 1.95949i
1189.1 0.654861 0.755750i −0.909632 + 0.584585i −0.142315 0.989821i −0.234072 0.512546i −0.153882 + 1.07028i 0.959493 0.281733i −0.841254 0.540641i 0.0702757 0.153882i −0.540641 0.158746i
1189.2 0.654861 0.755750i 0.909632 0.584585i −0.142315 0.989821i 0.234072 + 0.512546i 0.153882 1.07028i 0.959493 0.281733i −0.841254 0.540641i 0.0702757 0.153882i 0.540641 + 0.158746i
1245.1 0.142315 + 0.989821i −0.755750 + 1.65486i −0.959493 + 0.281733i −0.708089 + 0.817178i −1.74557 0.512546i −0.841254 + 0.540641i −0.415415 0.909632i −1.51255 1.74557i −0.909632 0.584585i
1245.2 0.142315 + 0.989821i 0.755750 1.65486i −0.959493 + 0.281733i 0.708089 0.817178i 1.74557 + 0.512546i −0.841254 + 0.540641i −0.415415 0.909632i −1.51255 1.74557i 0.909632 + 0.584585i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1245.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
8.b even 2 1 inner
23.c even 11 1 inner
161.l odd 22 1 inner
184.p even 22 1 inner
1288.bi odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.1.bi.e 20
7.b odd 2 1 inner 1288.1.bi.e 20
8.b even 2 1 inner 1288.1.bi.e 20
23.c even 11 1 inner 1288.1.bi.e 20
56.h odd 2 1 CM 1288.1.bi.e 20
161.l odd 22 1 inner 1288.1.bi.e 20
184.p even 22 1 inner 1288.1.bi.e 20
1288.bi odd 22 1 inner 1288.1.bi.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.1.bi.e 20 1.a even 1 1 trivial
1288.1.bi.e 20 7.b odd 2 1 inner
1288.1.bi.e 20 8.b even 2 1 inner
1288.1.bi.e 20 23.c even 11 1 inner
1288.1.bi.e 20 56.h odd 2 1 CM
1288.1.bi.e 20 161.l odd 22 1 inner
1288.1.bi.e 20 184.p even 22 1 inner
1288.1.bi.e 20 1288.bi odd 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1288, [\chi])$$:

 $$T_{3}^{20} + \cdots$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$3$ $$121 - 605 T^{2} + 1089 T^{4} + 484 T^{8} + 462 T^{10} + 330 T^{12} + 165 T^{14} + 55 T^{16} + 11 T^{18} + T^{20}$$
$5$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$11$ $$T^{20}$$
$13$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$121 + 484 T^{2} + 1210 T^{4} + 121 T^{6} + 605 T^{8} - 264 T^{10} + 55 T^{14} + T^{20}$$
$23$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$29$ $$T^{20}$$
$31$ $$T^{20}$$
$37$ $$T^{20}$$
$41$ $$T^{20}$$
$43$ $$T^{20}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$121 + 242 T^{2} + 1331 T^{4} + 1331 T^{6} + 121 T^{8} - 22 T^{10} + 154 T^{12} - 22 T^{14} + T^{20}$$
$61$ $$T^{20}$$
$67$ $$T^{20}$$
$71$ $$( 1 + 5 T + 25 T^{2} + 70 T^{3} + 130 T^{4} + 166 T^{5} + 148 T^{6} + 91 T^{7} + 37 T^{8} + 9 T^{9} + T^{10} )^{2}$$
$73$ $$T^{20}$$
$79$ $$( 1 + 5 T + 3 T^{2} - 7 T^{3} + 20 T^{4} - 10 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$83$ $$121 + 242 T^{2} + 1331 T^{4} + 1331 T^{6} + 121 T^{8} - 22 T^{10} + 154 T^{12} - 22 T^{14} + T^{20}$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$