Properties

Label 806.2.w.a
Level $806$
Weight $2$
Character orbit 806.w
Analytic conductor $6.436$
Analytic rank $0$
Dimension $76$
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(439,806)]
 
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([5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.w (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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{3} + 38 q^{4} + 6 q^{7} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{3} + 38 q^{4} + 6 q^{7} + 80 q^{9} + 6 q^{11} - 2 q^{12} + 14 q^{13} + 2 q^{14} - 38 q^{16} - 6 q^{17} - 24 q^{18} - 30 q^{19} - 18 q^{21} - 12 q^{22} - 4 q^{23} + 50 q^{25} - 2 q^{26} - 28 q^{27} + 18 q^{29} + 10 q^{30} - 4 q^{31} + 6 q^{33} + 4 q^{35} + 40 q^{36} + 12 q^{38} + 18 q^{39} - 42 q^{41} + 18 q^{42} - 8 q^{43} + 6 q^{44} + 60 q^{45} + 2 q^{48} + 42 q^{49} + 8 q^{51} + 16 q^{52} + 10 q^{53} - 18 q^{54} - 32 q^{55} - 2 q^{56} + 12 q^{57} + 6 q^{58} + 12 q^{59} + 12 q^{61} + 8 q^{62} - 42 q^{63} - 76 q^{64} - 34 q^{65} + 16 q^{66} - 30 q^{67} - 12 q^{68} - 8 q^{69} + 24 q^{73} + 28 q^{74} - 26 q^{75} - 8 q^{77} - 10 q^{78} - 18 q^{79} + 76 q^{81} - 48 q^{83} + 24 q^{85} - 18 q^{86} + 22 q^{87} - 6 q^{88} - 12 q^{89} - 34 q^{90} + 54 q^{91} - 8 q^{92} - 32 q^{93} - 32 q^{95} + 108 q^{97} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1 −0.866025 + 0.500000i −3.09898 0.500000 0.866025i 3.82547 2.20864i 2.68380 1.54949i −1.26741 + 0.731739i 1.00000i 6.60368 −2.20864 + 3.82547i
439.2 −0.866025 + 0.500000i −3.07789 0.500000 0.866025i −0.798451 + 0.460986i 2.66553 1.53895i 4.50628 2.60170i 1.00000i 6.47342 0.460986 0.798451i
439.3 −0.866025 + 0.500000i −3.05151 0.500000 0.866025i −0.288325 + 0.166465i 2.64269 1.52576i −2.38358 + 1.37616i 1.00000i 6.31172 0.166465 0.288325i
439.4 −0.866025 + 0.500000i −2.43132 0.500000 0.866025i −3.48098 + 2.00974i 2.10559 1.21566i 0.459014 0.265012i 1.00000i 2.91134 2.00974 3.48098i
439.5 −0.866025 + 0.500000i −1.63443 0.500000 0.866025i 2.24632 1.29691i 1.41546 0.817215i 0.490644 0.283274i 1.00000i −0.328637 −1.29691 + 2.24632i
439.6 −0.866025 + 0.500000i −1.60467 0.500000 0.866025i 2.03633 1.17568i 1.38968 0.802334i 3.91920 2.26275i 1.00000i −0.425039 −1.17568 + 2.03633i
439.7 −0.866025 + 0.500000i −1.52216 0.500000 0.866025i −1.27474 + 0.735973i 1.31823 0.761079i −4.28830 + 2.47585i 1.00000i −0.683032 0.735973 1.27474i
439.8 −0.866025 + 0.500000i −1.20830 0.500000 0.866025i 0.897452 0.518144i 1.04642 0.604149i −0.236744 + 0.136684i 1.00000i −1.54002 −0.518144 + 0.897452i
439.9 −0.866025 + 0.500000i −0.339158 0.500000 0.866025i 1.66872 0.963438i 0.293719 0.169579i −0.304644 + 0.175886i 1.00000i −2.88497 −0.963438 + 1.66872i
439.10 −0.866025 + 0.500000i −0.0977685 0.500000 0.866025i −2.80209 + 1.61779i 0.0846700 0.0488843i 0.245298 0.141623i 1.00000i −2.99044 1.61779 2.80209i
439.11 −0.866025 + 0.500000i 0.597655 0.500000 0.866025i −2.25230 + 1.30037i −0.517585 + 0.298828i 4.23575 2.44551i 1.00000i −2.64281 1.30037 2.25230i
439.12 −0.866025 + 0.500000i 0.616372 0.500000 0.866025i −0.916907 + 0.529377i −0.533794 + 0.308186i 0.932513 0.538387i 1.00000i −2.62009 0.529377 0.916907i
439.13 −0.866025 + 0.500000i 1.00798 0.500000 0.866025i −0.178395 + 0.102997i −0.872939 + 0.503992i −1.66018 + 0.958507i 1.00000i −1.98397 0.102997 0.178395i
439.14 −0.866025 + 0.500000i 1.85261 0.500000 0.866025i 3.66076 2.11354i −1.60441 + 0.926305i 0.822470 0.474853i 1.00000i 0.432160 −2.11354 + 3.66076i
439.15 −0.866025 + 0.500000i 1.88976 0.500000 0.866025i −0.545912 + 0.315182i −1.63658 + 0.944882i −3.21000 + 1.85330i 1.00000i 0.571208 0.315182 0.545912i
439.16 −0.866025 + 0.500000i 2.16919 0.500000 0.866025i −3.78268 + 2.18393i −1.87857 + 1.08459i −2.57189 + 1.48488i 1.00000i 1.70537 2.18393 3.78268i
439.17 −0.866025 + 0.500000i 2.51146 0.500000 0.866025i 0.600246 0.346552i −2.17499 + 1.25573i 2.39883 1.38497i 1.00000i 3.30746 −0.346552 + 0.600246i
439.18 −0.866025 + 0.500000i 2.99236 0.500000 0.866025i −0.887634 + 0.512476i −2.59146 + 1.49618i 1.52289 0.879243i 1.00000i 5.95423 0.512476 0.887634i
439.19 −0.866025 + 0.500000i 3.42879 0.500000 0.866025i 2.27311 1.31238i −2.96942 + 1.71440i −2.97616 + 1.71829i 1.00000i 8.75662 −1.31238 + 2.27311i
439.20 0.866025 0.500000i −3.16068 0.500000 0.866025i −1.09564 + 0.632568i −2.73723 + 1.58034i 1.09699 0.633348i 1.00000i 6.98987 −0.632568 + 1.09564i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.v even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.w.a yes 76
13.e even 6 1 806.2.o.a 76
31.c even 3 1 806.2.o.a 76
403.v even 6 1 inner 806.2.w.a yes 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.o.a 76 13.e even 6 1
806.2.o.a 76 31.c even 3 1
806.2.w.a yes 76 1.a even 1 1 trivial
806.2.w.a yes 76 403.v even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).