# Properties

 Label 69.3.f.a Level $69$ Weight $3$ Character orbit 69.f Analytic conductor $1.880$ Analytic rank $0$ Dimension $80$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 69.f (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.88011382409$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$8$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 4q^{2} - 12q^{6} - 4q^{8} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 4q^{2} - 12q^{6} - 4q^{8} - 24q^{9} + 8q^{13} - 208q^{16} - 110q^{17} + 12q^{18} - 66q^{19} - 176q^{20} - 8q^{23} - 12q^{24} + 244q^{25} + 328q^{26} + 528q^{28} + 50q^{29} + 182q^{31} + 428q^{32} - 242q^{34} - 536q^{35} - 198q^{36} - 352q^{37} - 770q^{38} - 216q^{39} - 110q^{40} - 208q^{41} - 330q^{42} - 88q^{43} - 154q^{44} - 72q^{46} + 24q^{47} + 360q^{48} + 256q^{49} + 726q^{50} + 264q^{51} + 506q^{52} + 352q^{53} + 162q^{54} - 38q^{55} + 1210q^{56} + 528q^{57} - 306q^{58} + 776q^{59} + 330q^{60} - 308q^{61} + 392q^{62} - 288q^{64} - 22q^{67} - 108q^{69} + 344q^{70} - 80q^{71} - 12q^{72} + 46q^{73} - 374q^{74} + 72q^{75} - 946q^{76} - 728q^{77} - 144q^{78} - 572q^{79} - 2178q^{80} - 72q^{81} - 820q^{82} - 704q^{83} - 922q^{85} - 1100q^{86} + 192q^{87} - 528q^{88} - 264q^{89} + 330q^{92} + 24q^{93} + 874q^{94} + 622q^{95} - 468q^{96} + 792q^{97} - 724q^{98} - 330q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −2.41844 + 2.79102i 1.45709 0.936417i −1.37172 9.54055i 7.78774 3.55654i −0.910325 + 6.33145i −2.15122 7.32637i 17.5181 + 11.2582i 1.24625 2.72890i −8.90776 + 30.3371i
7.2 −2.14901 + 2.48009i −1.45709 + 0.936417i −0.963341 6.70018i −3.83816 + 1.75283i 0.808910 5.62609i −1.29720 4.41787i 7.64456 + 4.91286i 1.24625 2.72890i 3.90107 13.2858i
7.3 −1.71450 + 1.97864i 1.45709 0.936417i −0.406240 2.82546i −6.24106 + 2.85020i −0.645356 + 4.48855i 3.18154 + 10.8353i −2.52293 1.62139i 1.24625 2.72890i 5.06079 17.2355i
7.4 0.141760 0.163600i 1.45709 0.936417i 0.562590 + 3.91290i 2.81506 1.28559i 0.0533600 0.371127i 0.266428 + 0.907372i 1.44834 + 0.930792i 1.24625 2.72890i 0.188740 0.642789i
7.5 0.216767 0.250162i −1.45709 + 0.936417i 0.553666 + 3.85083i −5.14558 + 2.34991i −0.0815934 + 0.567494i 1.37186 + 4.67213i 2.19721 + 1.41206i 1.24625 2.72890i −0.527533 + 1.79661i
7.6 1.09815 1.26733i −1.45709 + 0.936417i 0.169064 + 1.17586i 7.84831 3.58420i −0.413354 + 2.87494i −1.72032 5.85888i 7.31870 + 4.70345i 1.24625 2.72890i 4.07623 13.8824i
7.7 1.89640 2.18856i 1.45709 0.936417i −0.624207 4.34145i −2.49041 + 1.13733i 0.713823 4.96475i 0.240886 + 0.820382i −0.940602 0.604488i 1.24625 2.72890i −2.23369 + 7.60725i
7.8 2.55609 2.94989i −1.45709 + 0.936417i −1.59897 11.1211i 1.23125 0.562293i −0.962140 + 6.69183i 3.18330 + 10.8413i −23.7585 15.2686i 1.24625 2.72890i 1.48849 5.06932i
10.1 −2.41844 2.79102i 1.45709 + 0.936417i −1.37172 + 9.54055i 7.78774 + 3.55654i −0.910325 6.33145i −2.15122 + 7.32637i 17.5181 11.2582i 1.24625 + 2.72890i −8.90776 30.3371i
10.2 −2.14901 2.48009i −1.45709 0.936417i −0.963341 + 6.70018i −3.83816 1.75283i 0.808910 + 5.62609i −1.29720 + 4.41787i 7.64456 4.91286i 1.24625 + 2.72890i 3.90107 + 13.2858i
10.3 −1.71450 1.97864i 1.45709 + 0.936417i −0.406240 + 2.82546i −6.24106 2.85020i −0.645356 4.48855i 3.18154 10.8353i −2.52293 + 1.62139i 1.24625 + 2.72890i 5.06079 + 17.2355i
10.4 0.141760 + 0.163600i 1.45709 + 0.936417i 0.562590 3.91290i 2.81506 + 1.28559i 0.0533600 + 0.371127i 0.266428 0.907372i 1.44834 0.930792i 1.24625 + 2.72890i 0.188740 + 0.642789i
10.5 0.216767 + 0.250162i −1.45709 0.936417i 0.553666 3.85083i −5.14558 2.34991i −0.0815934 0.567494i 1.37186 4.67213i 2.19721 1.41206i 1.24625 + 2.72890i −0.527533 1.79661i
10.6 1.09815 + 1.26733i −1.45709 0.936417i 0.169064 1.17586i 7.84831 + 3.58420i −0.413354 2.87494i −1.72032 + 5.85888i 7.31870 4.70345i 1.24625 + 2.72890i 4.07623 + 13.8824i
10.7 1.89640 + 2.18856i 1.45709 + 0.936417i −0.624207 + 4.34145i −2.49041 1.13733i 0.713823 + 4.96475i 0.240886 0.820382i −0.940602 + 0.604488i 1.24625 + 2.72890i −2.23369 7.60725i
10.8 2.55609 + 2.94989i −1.45709 0.936417i −1.59897 + 11.1211i 1.23125 + 0.562293i −0.962140 6.69183i 3.18330 10.8413i −23.7585 + 15.2686i 1.24625 + 2.72890i 1.48849 + 5.06932i
19.1 −1.23384 2.70174i −1.66189 + 0.487975i −3.15759 + 3.64405i −4.82215 + 7.50341i 3.36890 + 3.88791i 1.37721 0.198013i 2.34192 + 0.687649i 2.52376 1.62192i 26.2221 + 3.77016i
19.2 −0.929697 2.03575i −1.66189 + 0.487975i −0.660506 + 0.762264i 3.71416 5.77935i 2.53845 + 2.92953i −11.7535 + 1.68990i −6.42351 1.88611i 2.52376 1.62192i −15.2184 2.18807i
19.3 −0.818934 1.79322i 1.66189 0.487975i 0.0744732 0.0859466i 0.591721 0.920735i −2.23602 2.58051i 4.32938 0.622470i −7.78115 2.28475i 2.52376 1.62192i −2.13566 0.307061i
19.4 0.0585879 + 0.128290i −1.66189 + 0.487975i 2.60642 3.00797i 0.992973 1.54510i −0.159969 0.184614i 8.01382 1.15221i 1.07988 + 0.317082i 2.52376 1.62192i 0.256396 + 0.0368642i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.3.f.a 80
3.b odd 2 1 207.3.j.b 80
23.d odd 22 1 inner 69.3.f.a 80
69.g even 22 1 207.3.j.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.3.f.a 80 1.a even 1 1 trivial
69.3.f.a 80 23.d odd 22 1 inner
207.3.j.b 80 3.b odd 2 1
207.3.j.b 80 69.g even 22 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(69, [\chi])$$.