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Properties

Label 207.3.j.b
Level $207$
Weight $3$
Character orbit 207.j
Analytic conductor $5.640$
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] = [207,3,Mod(10,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.j (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: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 69)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} + 4 q^{8} + 8 q^{13} - 208 q^{16} + 110 q^{17} - 66 q^{19} + 176 q^{20} + 8 q^{23} + 244 q^{25} - 328 q^{26} + 528 q^{28} - 50 q^{29} + 182 q^{31} - 428 q^{32} - 242 q^{34} + 536 q^{35} - 352 q^{37} + 770 q^{38} - 110 q^{40} + 208 q^{41} - 88 q^{43} + 154 q^{44} - 72 q^{46} - 24 q^{47} + 256 q^{49} - 726 q^{50} + 506 q^{52} - 352 q^{53} - 38 q^{55} - 1210 q^{56} - 306 q^{58} - 776 q^{59} - 308 q^{61} - 392 q^{62} - 288 q^{64} - 22 q^{67} + 344 q^{70} + 80 q^{71} + 46 q^{73} + 374 q^{74} - 946 q^{76} + 728 q^{77} - 572 q^{79} + 2178 q^{80} - 820 q^{82} + 704 q^{83} - 922 q^{85} + 1100 q^{86} - 528 q^{88} + 264 q^{89} - 330 q^{92} + 874 q^{94} - 622 q^{95} + 792 q^{97} + 724 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.55609 − 2.94989i 0 −1.59897 + 11.1211i −1.23125 − 0.562293i 0 3.18330 − 10.8413i 23.7585 − 15.2686i 0 1.48849 + 5.06932i
10.2 −1.89640 − 2.18856i 0 −0.624207 + 4.34145i 2.49041 + 1.13733i 0 0.240886 − 0.820382i 0.940602 − 0.604488i 0 −2.23369 − 7.60725i
10.3 −1.09815 − 1.26733i 0 0.169064 − 1.17586i −7.84831 − 3.58420i 0 −1.72032 + 5.85888i −7.31870 + 4.70345i 0 4.07623 + 13.8824i
10.4 −0.216767 − 0.250162i 0 0.553666 − 3.85083i 5.14558 + 2.34991i 0 1.37186 − 4.67213i −2.19721 + 1.41206i 0 −0.527533 − 1.79661i
10.5 −0.141760 − 0.163600i 0 0.562590 − 3.91290i −2.81506 − 1.28559i 0 0.266428 − 0.907372i −1.44834 + 0.930792i 0 0.188740 + 0.642789i
10.6 1.71450 + 1.97864i 0 −0.406240 + 2.82546i 6.24106 + 2.85020i 0 3.18154 − 10.8353i 2.52293 − 1.62139i 0 5.06079 + 17.2355i
10.7 2.14901 + 2.48009i 0 −0.963341 + 6.70018i 3.83816 + 1.75283i 0 −1.29720 + 4.41787i −7.64456 + 4.91286i 0 3.90107 + 13.2858i
10.8 2.41844 + 2.79102i 0 −1.37172 + 9.54055i −7.78774 − 3.55654i 0 −2.15122 + 7.32637i −17.5181 + 11.2582i 0 −8.90776 − 30.3371i
19.1 −1.59491 − 3.49237i 0 −7.03348 + 8.11707i −2.60332 + 4.05084i 0 −1.41492 + 0.203435i 24.8304 + 7.29087i 0 18.2991 + 2.63102i
19.2 −1.26828 − 2.77715i 0 −3.48458 + 4.02142i 2.04339 − 3.17957i 0 −0.848344 + 0.121973i 3.87000 + 1.13633i 0 −11.4217 − 1.64220i
19.3 −0.823772 − 1.80381i 0 0.0443165 − 0.0511440i 3.67013 − 5.71084i 0 4.89199 − 0.703361i −7.73950 − 2.27252i 0 −13.3246 − 1.91579i
19.4 −0.325075 − 0.711815i 0 2.21844 − 2.56021i −2.02423 + 3.14977i 0 −11.0173 + 1.58404i −5.54689 − 1.62871i 0 2.90008 + 0.416968i
19.5 −0.0585879 − 0.128290i 0 2.60642 − 3.00797i −0.992973 + 1.54510i 0 8.01382 − 1.15221i −1.07988 − 0.317082i 0 0.256396 + 0.0368642i
19.6 0.818934 + 1.79322i 0 0.0744732 − 0.0859466i −0.591721 + 0.920735i 0 4.32938 − 0.622470i 7.78115 + 2.28475i 0 −2.13566 − 0.307061i
19.7 0.929697 + 2.03575i 0 −0.660506 + 0.762264i −3.71416 + 5.77935i 0 −11.7535 + 1.68990i 6.42351 + 1.88611i 0 −15.2184 − 2.18807i
19.8 1.23384 + 2.70174i 0 −3.15759 + 3.64405i 4.82215 − 7.50341i 0 1.37721 − 0.198013i −2.34192 − 0.687649i 0 26.2221 + 3.77016i
28.1 −3.43833 − 1.00958i 0 7.43783 + 4.78001i 7.37162 + 1.05988i 0 8.84686 − 4.04022i −11.3612 − 13.1115i 0 −24.2760 − 11.0865i
28.2 −2.88088 − 0.845903i 0 4.21891 + 2.71133i 2.07974 + 0.299021i 0 −5.63637 + 2.57404i −1.99576 − 2.30323i 0 −5.73853 − 2.62070i
28.3 −2.39170 − 0.702267i 0 1.86205 + 1.19667i −7.10421 − 1.02143i 0 −5.02746 + 2.29596i 2.91633 + 3.36562i 0 16.2738 + 7.43201i
28.4 −0.553915 − 0.162644i 0 −3.08465 − 1.98238i −5.41984 − 0.779256i 0 6.09328 − 2.78271i 2.89841 + 3.34495i 0 2.87539 + 1.31315i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.8
Significant digits:
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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 207.3.j.b 80
3.b odd 2 1 69.3.f.a ✓ 80
23.d odd 22 1 inner 207.3.j.b 80
69.g even 22 1 69.3.f.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.3.f.a ✓ 80 3.b odd 2 1
69.3.f.a ✓ 80 69.g even 22 1
207.3.j.b 80 1.a even 1 1 trivial
207.3.j.b 80 23.d odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 4 T_{2}^{79} + 24 T_{2}^{78} + 84 T_{2}^{77} + 618 T_{2}^{76} + 2660 T_{2}^{75} + \cdots + 16\!\cdots\!29 \) acting on \(S_{3}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display