Properties

Label 861.2.s.a
Level $861$
Weight $2$
Character orbit 861.s
Analytic conductor $6.875$
Analytic rank $0$
Dimension $106$
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] = [861,2,Mod(206,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.206");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.s (of order \(6\), degree \(2\), 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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(106\)
Relative dimension: \(53\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 106 q + 52 q^{4} - 5 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 106 q + 52 q^{4} - 5 q^{7} - 2 q^{9} - 6 q^{10} - 22 q^{12} - 20 q^{14} - 4 q^{15} - 42 q^{16} - 5 q^{18} + 3 q^{19} - 36 q^{20} + 8 q^{21} - 12 q^{22} + 18 q^{23} - 57 q^{25} + 42 q^{26} + 18 q^{27} - 14 q^{28} - 107 q^{30} - 21 q^{31} - 2 q^{33} - 48 q^{35} - 16 q^{36} - q^{37} + 4 q^{39} - 18 q^{40} + 106 q^{41} + 77 q^{42} - 6 q^{43} + 210 q^{44} + 24 q^{45} - 8 q^{46} + 16 q^{47} + 70 q^{48} - 3 q^{49} - 42 q^{51} + 6 q^{52} - 16 q^{54} - 60 q^{56} - 22 q^{57} + 10 q^{58} - 16 q^{59} - 4 q^{60} + 18 q^{61} - 104 q^{62} + 33 q^{63} - 84 q^{64} + 36 q^{65} + 8 q^{66} + 21 q^{67} + 36 q^{68} - 12 q^{69} + 38 q^{70} - 148 q^{72} + 21 q^{73} - 40 q^{75} - 100 q^{77} + 60 q^{78} - 11 q^{79} - 36 q^{80} - 2 q^{81} - 20 q^{83} + 118 q^{84} - 4 q^{85} + 90 q^{86} + 17 q^{87} - 14 q^{88} + 16 q^{89} + 44 q^{90} + 19 q^{91} - 87 q^{93} + 24 q^{94} - 156 q^{96} - 268 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1 −2.41357 + 1.39347i −1.72865 + 0.108517i 2.88354 4.99444i 0.0887849 + 0.153780i 4.02099 2.67074i 1.75882 + 1.97650i 10.4987i 2.97645 0.375176i −0.428577 0.247439i
206.2 −2.31389 + 1.33592i 1.10471 + 1.33402i 2.56938 4.45031i −1.13816 1.97134i −4.33832 1.61098i −2.63605 0.226330i 8.38632i −0.559244 + 2.94741i 5.26713 + 3.04098i
206.3 −2.27082 + 1.31106i −0.573901 1.63421i 2.43775 4.22230i 1.02980 + 1.78367i 3.44577 + 2.95857i −2.05759 + 1.66323i 7.53988i −2.34128 + 1.87575i −4.67700 2.70027i
206.4 −2.15448 + 1.24389i −1.08687 + 1.34860i 2.09453 3.62783i −1.86942 3.23792i 0.664120 4.25748i 1.06496 2.42195i 5.44590i −0.637447 2.93149i 8.05525 + 4.65070i
206.5 −2.13361 + 1.23184i 1.71477 0.244047i 2.03486 3.52448i 2.02087 + 3.50024i −3.35803 + 2.63302i −2.59807 0.500056i 5.09914i 2.88088 0.836968i −8.62348 4.97877i
206.6 −2.03071 + 1.17243i −1.00636 1.40969i 1.74919 3.02968i −0.645695 1.11838i 3.69640 + 1.68279i 2.64245 0.132132i 3.51348i −0.974466 + 2.83733i 2.62244 + 1.51407i
206.7 −1.94516 + 1.12304i 1.64499 0.542238i 1.52243 2.63692i −2.05023 3.55111i −2.59080 + 2.90212i 2.47054 + 0.946801i 2.34682i 2.41196 1.78395i 7.97605 + 4.60497i
206.8 −1.84438 + 1.06485i 0.697137 1.58556i 1.26783 2.19594i 1.12887 + 1.95526i 0.402604 + 3.66673i 2.05168 + 1.67051i 1.14079i −2.02800 2.21070i −4.16413 2.40416i
206.9 −1.70805 + 0.986142i −1.18189 + 1.26615i 0.944954 1.63671i 1.73937 + 3.01268i 0.770120 3.32816i 2.64574 0.00746536i 0.217133i −0.206274 2.99290i −5.94186 3.43054i
206.10 −1.59373 + 0.920138i 0.561910 + 1.63837i 0.693309 1.20085i 0.958800 + 1.66069i −2.40306 2.09408i −1.44254 2.21790i 1.12879i −2.36851 + 1.84123i −3.05613 1.76446i
206.11 −1.51321 + 0.873651i −1.66589 0.474148i 0.526532 0.911981i −1.16247 2.01346i 2.93508 0.737920i −2.59569 0.512229i 1.65458i 2.55037 + 1.57976i 3.51813 + 2.03119i
206.12 −1.49330 + 0.862157i 1.14383 + 1.30063i 0.486628 0.842865i −0.428510 0.742200i −2.82944 0.956067i −0.337814 + 2.62410i 1.77043i −0.383285 + 2.97541i 1.27979 + 0.738885i
206.13 −1.44529 + 0.834437i 0.474014 1.66593i 0.392570 0.679952i 1.92742 + 3.33839i 0.705023 + 2.80328i −1.41003 2.23871i 2.02745i −2.55062 1.57935i −5.57136 3.21663i
206.14 −1.44370 + 0.833520i 1.51624 + 0.837276i 0.389510 0.674651i 0.984574 + 1.70533i −2.88687 + 0.0550382i 0.232807 + 2.63549i 2.03542i 1.59794 + 2.53901i −2.84285 1.64132i
206.15 −1.28760 + 0.743398i −0.267036 1.71134i 0.105281 0.182352i −1.62942 2.82224i 1.61604 + 2.00502i −0.433422 2.61001i 2.66053i −2.85738 + 0.913979i 4.19609 + 2.42262i
206.16 −1.27500 + 0.736123i 1.57218 0.726806i 0.0837529 0.145064i −0.207914 0.360118i −1.46951 + 2.08400i 2.35740 1.20110i 2.69788i 1.94350 2.28534i 0.530182 + 0.306101i
206.17 −1.21231 + 0.699930i −1.63502 + 0.571591i −0.0201967 + 0.0349817i −1.06109 1.83786i 1.58208 1.83735i 1.41763 2.23390i 2.85626i 2.34657 1.86912i 2.57275 + 1.48538i
206.18 −1.09044 + 0.629568i −0.568895 + 1.63596i −0.207288 + 0.359033i −0.914757 1.58441i −0.409598 2.14208i −2.62661 + 0.317703i 3.04028i −2.35272 1.86138i 1.99498 + 1.15180i
206.19 −0.977958 + 0.564624i 0.835884 1.51700i −0.362399 + 0.627693i −0.0432421 0.0748975i 0.0390773 + 1.95553i −2.31142 + 1.28737i 3.07697i −1.60260 2.53608i 0.0845779 + 0.0488311i
206.20 −0.651390 + 0.376080i −1.69203 0.370192i −0.717127 + 1.24210i 0.355703 + 0.616096i 1.24139 0.395199i 1.69688 + 2.02992i 2.58311i 2.72592 + 1.25275i −0.463404 0.267546i
See next 80 embeddings (of 106 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 206.53
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.s.a 106
3.b odd 2 1 861.2.s.b yes 106
7.d odd 6 1 861.2.s.b yes 106
21.g even 6 1 inner 861.2.s.a 106
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.s.a 106 1.a even 1 1 trivial
861.2.s.a 106 21.g even 6 1 inner
861.2.s.b yes 106 3.b odd 2 1
861.2.s.b yes 106 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{106} - 79 T_{2}^{104} + 3315 T_{2}^{102} - 96054 T_{2}^{100} - 30 T_{2}^{99} + 2134915 T_{2}^{98} + 2370 T_{2}^{97} - 38486201 T_{2}^{96} - 95778 T_{2}^{95} + 582396323 T_{2}^{94} + 2621442 T_{2}^{93} + \cdots + 407027712 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display