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Properties

Label 806.2.f.a
Level $806$
Weight $2$
Character orbit 806.f
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(191,806)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.f (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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 19 q^{2} - q^{3} - 19 q^{4} + 2 q^{6} - 8 q^{7} + 38 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 19 q^{2} - q^{3} - 19 q^{4} + 2 q^{6} - 8 q^{7} + 38 q^{8} - 16 q^{9} + 4 q^{11} - q^{12} - 2 q^{13} + 4 q^{14} - 13 q^{15} - 19 q^{16} + 6 q^{17} - 16 q^{18} - 14 q^{19} - 2 q^{22} + 2 q^{23} - q^{24} - 25 q^{25} - 2 q^{26} + 32 q^{27} + 4 q^{28} - 12 q^{29} - 13 q^{30} - 19 q^{31} - 19 q^{32} - q^{33} - 3 q^{34} - 16 q^{35} + 32 q^{36} + q^{37} + 7 q^{38} - 19 q^{39} - 6 q^{41} - 6 q^{43} - 2 q^{44} - 50 q^{45} + 2 q^{46} + 16 q^{47} + 2 q^{48} + 58 q^{49} - 25 q^{50} + 5 q^{51} + 4 q^{52} + 3 q^{53} - 16 q^{54} + 2 q^{55} - 8 q^{56} - 14 q^{57} - 12 q^{58} + 16 q^{59} + 26 q^{60} - 2 q^{61} + 23 q^{62} + 6 q^{63} + 38 q^{64} - q^{65} + 2 q^{66} - 3 q^{68} + 34 q^{69} + 32 q^{70} - 13 q^{71} - 16 q^{72} - 16 q^{73} - 2 q^{74} + 30 q^{75} + 7 q^{76} + 8 q^{77} + 2 q^{78} + 15 q^{79} + 5 q^{81} + 3 q^{82} - 47 q^{83} - 6 q^{85} + 3 q^{86} - 52 q^{87} + 4 q^{88} - 42 q^{89} + 25 q^{90} - 4 q^{91} - 4 q^{92} - 13 q^{93} - 8 q^{94} - q^{95} - q^{96} - 2 q^{97} - 29 q^{98} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 −0.500000 + 0.866025i −1.69109 − 2.92905i −0.500000 − 0.866025i −1.21966 − 2.11251i 3.38217 −1.27179 1.00000 −4.21955 + 7.30847i 2.43932
191.2 −0.500000 + 0.866025i −1.41228 − 2.44614i −0.500000 − 0.866025i 1.35138 + 2.34065i 2.82456 2.29126 1.00000 −2.48906 + 4.31117i −2.70275
191.3 −0.500000 + 0.866025i −1.26924 − 2.19839i −0.500000 − 0.866025i 1.64266 + 2.84517i 2.53848 −2.87996 1.00000 −1.72194 + 2.98248i −3.28532
191.4 −0.500000 + 0.866025i −1.12900 − 1.95548i −0.500000 − 0.866025i −1.32531 − 2.29550i 2.25799 3.94877 1.00000 −1.04927 + 1.81738i 2.65062
191.5 −0.500000 + 0.866025i −0.878327 − 1.52131i −0.500000 − 0.866025i 0.798076 + 1.38231i 1.75665 −2.77611 1.00000 −0.0429178 + 0.0743358i −1.59615
191.6 −0.500000 + 0.866025i −0.807532 − 1.39869i −0.500000 − 0.866025i −0.846852 − 1.46679i 1.61506 0.162444 1.00000 0.195785 − 0.339109i 1.69370
191.7 −0.500000 + 0.866025i −0.697231 − 1.20764i −0.500000 − 0.866025i −0.896886 − 1.55345i 1.39446 −1.79694 1.00000 0.527739 − 0.914071i 1.79377
191.8 −0.500000 + 0.866025i −0.336731 − 0.583236i −0.500000 − 0.866025i 0.845726 + 1.46484i 0.673463 2.89574 1.00000 1.27322 − 2.20529i −1.69145
191.9 −0.500000 + 0.866025i −0.173119 − 0.299850i −0.500000 − 0.866025i −2.00131 − 3.46637i 0.346237 −4.72203 1.00000 1.44006 − 2.49426i 4.00261
191.10 −0.500000 + 0.866025i 0.0271667 + 0.0470541i −0.500000 − 0.866025i −2.00925 − 3.48012i −0.0543334 3.59501 1.00000 1.49852 − 2.59552i 4.01850
191.11 −0.500000 + 0.866025i 0.148372 + 0.256988i −0.500000 − 0.866025i −0.177420 − 0.307301i −0.296744 −0.979552 1.00000 1.45597 − 2.52182i 0.354840
191.12 −0.500000 + 0.866025i 0.372534 + 0.645248i −0.500000 − 0.866025i 0.536147 + 0.928634i −0.745069 −2.77032 1.00000 1.22244 − 2.11732i −1.07229
191.13 −0.500000 + 0.866025i 0.570203 + 0.987621i −0.500000 − 0.866025i 1.89822 + 3.28781i −1.14041 −2.59626 1.00000 0.849737 − 1.47179i −3.79644
191.14 −0.500000 + 0.866025i 0.820502 + 1.42115i −0.500000 − 0.866025i 1.56144 + 2.70449i −1.64100 3.49508 1.00000 0.153551 − 0.265959i −3.12287
191.15 −0.500000 + 0.866025i 0.883630 + 1.53049i −0.500000 − 0.866025i −0.882039 − 1.52774i −1.76726 −2.14258 1.00000 −0.0616023 + 0.106698i 1.76408
191.16 −0.500000 + 0.866025i 1.15557 + 2.00151i −0.500000 − 0.866025i −0.706502 − 1.22370i −2.31115 3.65708 1.00000 −1.17070 + 2.02771i 1.41300
191.17 −0.500000 + 0.866025i 1.17053 + 2.02742i −0.500000 − 0.866025i 0.854897 + 1.48073i −2.34107 2.62400 1.00000 −1.24030 + 2.14826i −1.70979
191.18 −0.500000 + 0.866025i 1.23067 + 2.13158i −0.500000 − 0.866025i −0.890819 − 1.54294i −2.46134 0.185663 1.00000 −1.52909 + 2.64847i 1.78164
191.19 −0.500000 + 0.866025i 1.51536 + 2.62467i −0.500000 − 0.866025i 1.46750 + 2.54179i −3.03071 −4.91951 1.00000 −3.09261 + 5.35655i −2.93501
211.1 −0.500000 − 0.866025i −1.69109 + 2.92905i −0.500000 + 0.866025i −1.21966 + 2.11251i 3.38217 −1.27179 1.00000 −4.21955 − 7.30847i 2.43932
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.19
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{38} + T_{3}^{37} + 37 T_{3}^{36} + 22 T_{3}^{35} + 790 T_{3}^{34} + 311 T_{3}^{33} + 11244 T_{3}^{32} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display