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Properties

Label 861.2.i.f
Level $861$
Weight $2$
Character orbit 861.i
Analytic conductor $6.875$
Analytic rank $0$
Dimension $26$
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(247,861)]
 
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([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.247");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.i (of order \(3\), 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: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{2} - 13 q^{3} - 12 q^{4} + 8 q^{5} - 8 q^{6} + 5 q^{7} - 24 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{2} - 13 q^{3} - 12 q^{4} + 8 q^{5} - 8 q^{6} + 5 q^{7} - 24 q^{8} - 13 q^{9} + q^{10} + 10 q^{11} - 12 q^{12} - 32 q^{13} - 18 q^{14} - 16 q^{15} - 26 q^{16} + 12 q^{17} + 4 q^{18} + 11 q^{19} - 12 q^{20} - 4 q^{21} + 2 q^{22} + 15 q^{23} + 12 q^{24} - 15 q^{25} + 18 q^{26} + 26 q^{27} - 55 q^{28} - 16 q^{29} + q^{30} + 9 q^{31} + 23 q^{32} + 10 q^{33} + 14 q^{34} - 10 q^{35} + 24 q^{36} + 2 q^{37} + 20 q^{38} + 16 q^{39} + 49 q^{40} + 26 q^{41} + 12 q^{42} - 14 q^{43} + 22 q^{44} + 8 q^{45} + 4 q^{46} + 26 q^{47} + 52 q^{48} - 7 q^{49} - 30 q^{50} + 12 q^{51} + 24 q^{52} - 4 q^{53} + 4 q^{54} - 2 q^{55} + 9 q^{56} - 22 q^{57} - 39 q^{58} - 3 q^{59} + 6 q^{60} + 28 q^{61} - 14 q^{62} - q^{63} + 4 q^{64} + 20 q^{65} - q^{66} - 7 q^{67} + 55 q^{68} - 30 q^{69} + 8 q^{70} - 80 q^{71} + 12 q^{72} - 2 q^{73} - q^{74} - 15 q^{75} + 52 q^{76} + 28 q^{77} - 36 q^{78} - 13 q^{79} + 22 q^{80} - 13 q^{81} + 4 q^{82} - 28 q^{83} + 11 q^{84} + 96 q^{85} + 49 q^{86} + 8 q^{87} - 20 q^{88} + 35 q^{89} - 2 q^{90} + 4 q^{91} - 210 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} + 23 q^{96} - 128 q^{97} - 17 q^{98} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
247.1 −1.23629 + 2.14132i −0.500000 − 0.866025i −2.05684 − 3.56255i 2.12804 − 3.68587i 2.47258 2.44346 + 1.01464i 5.22624 −0.500000 + 0.866025i 5.26176 + 9.11363i
247.2 −1.07755 + 1.86637i −0.500000 − 0.866025i −1.32221 − 2.29014i −0.392348 + 0.679567i 2.15509 1.64896 − 2.06904i 1.38879 −0.500000 + 0.866025i −0.845546 − 1.46453i
247.3 −0.733243 + 1.27001i −0.500000 − 0.866025i −0.0752892 − 0.130405i −0.639727 + 1.10804i 1.46649 −0.417829 + 2.61255i −2.71215 −0.500000 + 0.866025i −0.938150 − 1.62492i
247.4 −0.323136 + 0.559688i −0.500000 − 0.866025i 0.791166 + 1.37034i −1.10366 + 1.91160i 0.646272 −2.63667 + 0.219072i −2.31516 −0.500000 + 0.866025i −0.713267 − 1.23541i
247.5 −0.279188 + 0.483567i −0.500000 − 0.866025i 0.844108 + 1.46204i 0.148598 − 0.257380i 0.558375 −0.852801 − 2.50454i −2.05941 −0.500000 + 0.866025i 0.0829736 + 0.143714i
247.6 −0.0982823 + 0.170230i −0.500000 − 0.866025i 0.980681 + 1.69859i 0.503330 − 0.871794i 0.196565 2.64572 − 0.0137631i −0.778664 −0.500000 + 0.866025i 0.0989369 + 0.171364i
247.7 0.220740 − 0.382333i −0.500000 − 0.866025i 0.902547 + 1.56326i 1.91947 − 3.32462i −0.441481 −2.64147 + 0.150533i 1.67988 −0.500000 + 0.866025i −0.847410 − 1.46776i
247.8 0.233526 − 0.404478i −0.500000 − 0.866025i 0.890932 + 1.54314i −0.188849 + 0.327096i −0.467051 2.44636 + 1.00764i 1.76632 −0.500000 + 0.866025i 0.0882022 + 0.152771i
247.9 0.608512 − 1.05397i −0.500000 − 0.866025i 0.259425 + 0.449338i 1.72572 − 2.98904i −1.21702 −0.228004 + 2.63591i 3.06550 −0.500000 + 0.866025i −2.10025 − 3.63774i
247.10 0.922933 − 1.59857i −0.500000 − 0.866025i −0.703612 − 1.21869i 0.706554 − 1.22379i −1.84587 −1.88875 − 1.85273i 1.09419 −0.500000 + 0.866025i −1.30420 − 2.25895i
247.11 1.10049 − 1.90611i −0.500000 − 0.866025i −1.42216 − 2.46325i −1.38811 + 2.40427i −2.20098 0.691542 + 2.55378i −1.85834 −0.500000 + 0.866025i 3.05519 + 5.29175i
247.12 1.29452 − 2.24217i −0.500000 − 0.866025i −2.35155 − 4.07301i 1.72607 − 2.98964i −2.58904 −0.431309 − 2.61036i −6.99843 −0.500000 + 0.866025i −4.46885 − 7.74027i
247.13 1.36697 − 2.36766i −0.500000 − 0.866025i −2.73720 − 4.74097i −1.14509 + 1.98336i −2.73393 1.72079 − 2.00970i −9.49877 −0.500000 + 0.866025i 3.13061 + 5.42237i
739.1 −1.23629 − 2.14132i −0.500000 + 0.866025i −2.05684 + 3.56255i 2.12804 + 3.68587i 2.47258 2.44346 − 1.01464i 5.22624 −0.500000 − 0.866025i 5.26176 − 9.11363i
739.2 −1.07755 − 1.86637i −0.500000 + 0.866025i −1.32221 + 2.29014i −0.392348 − 0.679567i 2.15509 1.64896 + 2.06904i 1.38879 −0.500000 − 0.866025i −0.845546 + 1.46453i
739.3 −0.733243 − 1.27001i −0.500000 + 0.866025i −0.0752892 + 0.130405i −0.639727 − 1.10804i 1.46649 −0.417829 − 2.61255i −2.71215 −0.500000 − 0.866025i −0.938150 + 1.62492i
739.4 −0.323136 − 0.559688i −0.500000 + 0.866025i 0.791166 − 1.37034i −1.10366 − 1.91160i 0.646272 −2.63667 − 0.219072i −2.31516 −0.500000 − 0.866025i −0.713267 + 1.23541i
739.5 −0.279188 − 0.483567i −0.500000 + 0.866025i 0.844108 − 1.46204i 0.148598 + 0.257380i 0.558375 −0.852801 + 2.50454i −2.05941 −0.500000 − 0.866025i 0.0829736 − 0.143714i
739.6 −0.0982823 − 0.170230i −0.500000 + 0.866025i 0.980681 − 1.69859i 0.503330 + 0.871794i 0.196565 2.64572 + 0.0137631i −0.778664 −0.500000 − 0.866025i 0.0989369 − 0.171364i
739.7 0.220740 + 0.382333i −0.500000 + 0.866025i 0.902547 − 1.56326i 1.91947 + 3.32462i −0.441481 −2.64147 − 0.150533i 1.67988 −0.500000 − 0.866025i −0.847410 + 1.46776i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 247.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.i.f ✓ 26
7.c even 3 1 inner 861.2.i.f ✓ 26
7.c even 3 1 6027.2.a.bi 13
7.d odd 6 1 6027.2.a.bh 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.i.f ✓ 26 1.a even 1 1 trivial
861.2.i.f ✓ 26 7.c even 3 1 inner
6027.2.a.bh 13 7.d odd 6 1
6027.2.a.bi 13 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} - 4 T_{2}^{25} + 27 T_{2}^{24} - 68 T_{2}^{23} + 319 T_{2}^{22} - 671 T_{2}^{21} + 2470 T_{2}^{20} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\). Copy content Toggle raw display