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Properties

Label 806.2.h.a
Level $806$
Weight $2$
Character orbit 806.h
Analytic conductor $6.436$
Analytic rank $0$
Dimension $38$
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] = [806,2,Mod(87,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.87");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.h (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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 19 q^{2} + 2 q^{3} - 19 q^{4} - q^{6} + 4 q^{7} + 38 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 19 q^{2} + 2 q^{3} - 19 q^{4} - q^{6} + 4 q^{7} + 38 q^{8} + 32 q^{9} - 2 q^{11} - q^{12} + 4 q^{13} + 4 q^{14} - 13 q^{15} - 19 q^{16} - 3 q^{17} - 16 q^{18} + 7 q^{19} + 4 q^{22} + 2 q^{23} + 2 q^{24} - 25 q^{25} - 2 q^{26} + 32 q^{27} - 8 q^{28} - 12 q^{29} - 13 q^{30} - 19 q^{31} - 19 q^{32} - q^{33} - 3 q^{34} - 16 q^{35} - 16 q^{36} - 2 q^{37} + 7 q^{38} - 19 q^{39} + 3 q^{41} + 3 q^{43} - 2 q^{44} + 25 q^{45} + 2 q^{46} + 16 q^{47} - q^{48} - 29 q^{49} + 50 q^{50} + 5 q^{51} - 2 q^{52} + 3 q^{53} - 16 q^{54} - 4 q^{55} + 4 q^{56} - 14 q^{57} - 12 q^{58} - 8 q^{59} + 26 q^{60} - 2 q^{61} - 4 q^{62} + 6 q^{63} + 38 q^{64} - 31 q^{65} + 2 q^{66} + 6 q^{68} - 17 q^{69} + 32 q^{70} + 26 q^{71} + 32 q^{72} - 16 q^{73} + q^{74} - 15 q^{75} - 14 q^{76} + 8 q^{77} + 2 q^{78} + 15 q^{79} - 10 q^{81} - 6 q^{82} - 47 q^{83} - 6 q^{85} + 3 q^{86} + 26 q^{87} - 2 q^{88} - 42 q^{89} + 25 q^{90} - 4 q^{91} - 4 q^{92} + 38 q^{93} - 8 q^{94} - q^{95} - q^{96} - 2 q^{97} + 58 q^{98} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1 −0.500000 + 0.866025i −3.03071 −0.500000 − 0.866025i 1.46750 − 2.54179i 1.51536 − 2.62467i 2.45976 − 4.26042i 1.00000 6.18522 1.46750 + 2.54179i
87.2 −0.500000 + 0.866025i −2.46134 −0.500000 − 0.866025i −0.890819 + 1.54294i 1.23067 − 2.13158i −0.0928315 + 0.160789i 1.00000 3.05819 −0.890819 − 1.54294i
87.3 −0.500000 + 0.866025i −2.34107 −0.500000 − 0.866025i 0.854897 − 1.48073i 1.17053 − 2.02742i −1.31200 + 2.27245i 1.00000 2.48059 0.854897 + 1.48073i
87.4 −0.500000 + 0.866025i −2.31115 −0.500000 − 0.866025i −0.706502 + 1.22370i 1.15557 − 2.00151i −1.82854 + 3.16713i 1.00000 2.34140 −0.706502 − 1.22370i
87.5 −0.500000 + 0.866025i −1.76726 −0.500000 − 0.866025i −0.882039 + 1.52774i 0.883630 − 1.53049i 1.07129 − 1.85553i 1.00000 0.123205 −0.882039 − 1.52774i
87.6 −0.500000 + 0.866025i −1.64100 −0.500000 − 0.866025i 1.56144 − 2.70449i 0.820502 − 1.42115i −1.74754 + 3.02683i 1.00000 −0.307103 1.56144 + 2.70449i
87.7 −0.500000 + 0.866025i −1.14041 −0.500000 − 0.866025i 1.89822 − 3.28781i 0.570203 − 0.987621i 1.29813 − 2.24843i 1.00000 −1.69947 1.89822 + 3.28781i
87.8 −0.500000 + 0.866025i −0.745069 −0.500000 − 0.866025i 0.536147 − 0.928634i 0.372534 − 0.645248i 1.38516 − 2.39916i 1.00000 −2.44487 0.536147 + 0.928634i
87.9 −0.500000 + 0.866025i −0.296744 −0.500000 − 0.866025i −0.177420 + 0.307301i 0.148372 − 0.256988i 0.489776 − 0.848317i 1.00000 −2.91194 −0.177420 − 0.307301i
87.10 −0.500000 + 0.866025i −0.0543334 −0.500000 − 0.866025i −2.00925 + 3.48012i 0.0271667 − 0.0470541i −1.79750 + 3.11337i 1.00000 −2.99705 −2.00925 − 3.48012i
87.11 −0.500000 + 0.866025i 0.346237 −0.500000 − 0.866025i −2.00131 + 3.46637i −0.173119 + 0.299850i 2.36101 − 4.08939i 1.00000 −2.88012 −2.00131 − 3.46637i
87.12 −0.500000 + 0.866025i 0.673463 −0.500000 − 0.866025i 0.845726 − 1.46484i −0.336731 + 0.583236i −1.44787 + 2.50779i 1.00000 −2.54645 0.845726 + 1.46484i
87.13 −0.500000 + 0.866025i 1.39446 −0.500000 − 0.866025i −0.896886 + 1.55345i −0.697231 + 1.20764i 0.898469 − 1.55619i 1.00000 −1.05548 −0.896886 − 1.55345i
87.14 −0.500000 + 0.866025i 1.61506 −0.500000 − 0.866025i −0.846852 + 1.46679i −0.807532 + 1.39869i −0.0812222 + 0.140681i 1.00000 −0.391569 −0.846852 − 1.46679i
87.15 −0.500000 + 0.866025i 1.75665 −0.500000 − 0.866025i 0.798076 − 1.38231i −0.878327 + 1.52131i 1.38805 − 2.40418i 1.00000 0.0858356 0.798076 + 1.38231i
87.16 −0.500000 + 0.866025i 2.25799 −0.500000 − 0.866025i −1.32531 + 2.29550i −1.12900 + 1.95548i −1.97438 + 3.41973i 1.00000 2.09853 −1.32531 − 2.29550i
87.17 −0.500000 + 0.866025i 2.53848 −0.500000 − 0.866025i 1.64266 − 2.84517i −1.26924 + 2.19839i 1.43998 − 2.49412i 1.00000 3.44387 1.64266 + 2.84517i
87.18 −0.500000 + 0.866025i 2.82456 −0.500000 − 0.866025i 1.35138 − 2.34065i −1.41228 + 2.44614i −1.14563 + 1.98429i 1.00000 4.97811 1.35138 + 2.34065i
87.19 −0.500000 + 0.866025i 3.38217 −0.500000 − 0.866025i −1.21966 + 2.11251i −1.69109 + 2.92905i 0.635895 − 1.10140i 1.00000 8.43910 −1.21966 − 2.11251i
315.1 −0.500000 − 0.866025i −3.03071 −0.500000 + 0.866025i 1.46750 + 2.54179i 1.51536 + 2.62467i 2.45976 + 4.26042i 1.00000 6.18522 1.46750 − 2.54179i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.h.a yes 38
13.c even 3 1 806.2.f.a ✓ 38
31.c even 3 1 806.2.f.a ✓ 38
403.g even 3 1 inner 806.2.h.a yes 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.f.a ✓ 38 13.c even 3 1
806.2.f.a ✓ 38 31.c even 3 1
806.2.h.a yes 38 1.a even 1 1 trivial
806.2.h.a yes 38 403.g even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{19} - T_{3}^{18} - 36 T_{3}^{17} + 29 T_{3}^{16} + 535 T_{3}^{15} - 337 T_{3}^{14} - 4260 T_{3}^{13} + \cdots - 81 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display