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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [69,3,Mod(7,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 19]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 69.f (of order \(22\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 7.8
Significant digits:
Format:

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 \(S_{3}^{\mathrm{new}}(69, [\chi])\).