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Properties

Label 806.2.x.a
Level $806$
Weight $2$
Character orbit 806.x
Analytic conductor $6.436$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [806,2,Mod(233,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.233");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.x (of order \(10\), degree \(4\), 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: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 4 q^{3} + 40 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 4 q^{3} + 40 q^{4} - 48 q^{9} - 4 q^{12} + 14 q^{13} + 6 q^{14} - 40 q^{16} - 22 q^{17} - 12 q^{23} - 216 q^{25} + 24 q^{26} + 34 q^{27} + 68 q^{30} + 4 q^{35} - 192 q^{36} + 28 q^{38} + 30 q^{39} + 12 q^{42} + 68 q^{43} + 4 q^{48} + 18 q^{49} - 10 q^{51} - 14 q^{52} + 12 q^{53} - 48 q^{55} + 4 q^{56} + 4 q^{61} - 52 q^{62} + 40 q^{64} + 4 q^{65} + 48 q^{66} - 28 q^{68} - 12 q^{69} + 4 q^{74} + 30 q^{75} - 46 q^{77} - 18 q^{78} - 64 q^{79} - 48 q^{81} + 96 q^{87} + 20 q^{88} - 72 q^{90} + 94 q^{91} - 8 q^{92} + 12 q^{94} + 44 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1 −0.587785 − 0.809017i −2.60953 − 1.89593i −0.309017 + 0.951057i − 2.84178i 3.22555i −3.14900 − 1.02317i 0.951057 − 0.309017i 2.28802 + 7.04181i −2.29904 + 1.67035i
233.2 −0.587785 − 0.809017i −2.26862 − 1.64825i −0.309017 + 0.951057i − 4.12891i 2.80417i 2.73976 + 0.890201i 0.951057 − 0.309017i 1.50287 + 4.62536i −3.34036 + 2.42691i
233.3 −0.587785 − 0.809017i −2.07917 − 1.51061i −0.309017 + 0.951057i 1.81907i 2.57000i −1.84528 − 0.599567i 0.951057 − 0.309017i 1.11398 + 3.42847i 1.47166 − 1.06922i
233.4 −0.587785 − 0.809017i −1.95332 − 1.41917i −0.309017 + 0.951057i 0.284568i 2.41444i 0.0320518 + 0.0104142i 0.951057 − 0.309017i 0.874367 + 2.69103i 0.230220 − 0.167265i
233.5 −0.587785 − 0.809017i −1.37673 − 1.00025i −0.309017 + 0.951057i − 0.387790i 1.70173i 3.83297 + 1.24541i 0.951057 − 0.309017i −0.0321770 − 0.0990306i −0.313729 + 0.227937i
233.6 −0.587785 − 0.809017i −1.33656 − 0.971071i −0.309017 + 0.951057i 3.86052i 1.65208i −0.401844 − 0.130567i 0.951057 − 0.309017i −0.0836250 − 0.257371i 3.12323 − 2.26916i
233.7 −0.587785 − 0.809017i −1.25142 − 0.909207i −0.309017 + 0.951057i − 1.24516i 1.54683i 0.832333 + 0.270441i 0.951057 − 0.309017i −0.187667 − 0.577578i −1.00736 + 0.731886i
233.8 −0.587785 − 0.809017i −0.503518 − 0.365828i −0.309017 + 0.951057i 2.20987i 0.622383i −4.22532 − 1.37289i 0.951057 − 0.309017i −0.807350 − 2.48477i 1.78782 − 1.29893i
233.9 −0.587785 − 0.809017i −0.371624 − 0.270001i −0.309017 + 0.951057i − 2.90619i 0.459353i −4.41370 − 1.43410i 0.951057 − 0.309017i −0.861847 − 2.65249i −2.35116 + 1.70822i
233.10 −0.587785 − 0.809017i 0.193642 + 0.140689i −0.309017 + 0.951057i − 0.119048i − 0.239354i 1.73854 + 0.564886i 0.951057 − 0.309017i −0.909347 − 2.79868i −0.0963117 + 0.0699745i
233.11 −0.587785 − 0.809017i 0.320261 + 0.232683i −0.309017 + 0.951057i 2.25971i − 0.395864i 2.89626 + 0.941053i 0.951057 − 0.309017i −0.878625 − 2.70413i 1.82815 − 1.32823i
233.12 −0.587785 − 0.809017i 0.569861 + 0.414028i −0.309017 + 0.951057i − 4.13784i − 0.704387i 0.911375 + 0.296124i 0.951057 − 0.309017i −0.773729 − 2.38129i −3.34758 + 2.43216i
233.13 −0.587785 − 0.809017i 0.708307 + 0.514615i −0.309017 + 0.951057i 0.866942i − 0.875516i −0.321887 − 0.104587i 0.951057 − 0.309017i −0.690181 − 2.12416i 0.701371 − 0.509576i
233.14 −0.587785 − 0.809017i 1.16488 + 0.846337i −0.309017 + 0.951057i 2.70080i − 1.43988i −2.14836 − 0.698044i 0.951057 − 0.309017i −0.286384 − 0.881400i 2.18499 − 1.58749i
233.15 −0.587785 − 0.809017i 1.22688 + 0.891380i −0.309017 + 0.951057i 3.34201i − 1.51651i 3.96009 + 1.28671i 0.951057 − 0.309017i −0.216377 − 0.665939i 2.70374 − 1.96438i
233.16 −0.587785 − 0.809017i 1.80623 + 1.31230i −0.309017 + 0.951057i − 3.17543i − 2.23262i −1.91532 − 0.622326i 0.951057 − 0.309017i 0.613272 + 1.88746i −2.56897 + 1.86647i
233.17 −0.587785 − 0.809017i 2.10239 + 1.52748i −0.309017 + 0.951057i − 0.319086i − 2.59870i −3.05362 − 0.992180i 0.951057 − 0.309017i 1.15981 + 3.56953i −0.258146 + 0.187554i
233.18 −0.587785 − 0.809017i 2.28155 + 1.65764i −0.309017 + 0.951057i − 0.936333i − 2.82015i 0.329788 + 0.107155i 0.951057 − 0.309017i 1.53064 + 4.71082i −0.757509 + 0.550363i
233.19 −0.587785 − 0.809017i 2.33113 + 1.69367i −0.309017 + 0.951057i − 1.23723i − 2.88144i 4.97215 + 1.61555i 0.951057 − 0.309017i 1.63862 + 5.04315i −1.00094 + 0.727228i
233.20 −0.587785 − 0.809017i 2.66340 + 1.93507i −0.309017 + 0.951057i 4.09131i − 3.29214i −1.35877 − 0.441493i 0.951057 − 0.309017i 2.42214 + 7.45458i 3.30994 − 2.40481i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
31.d even 5 1 inner
403.y even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.x.a ✓ 160
13.b even 2 1 inner 806.2.x.a ✓ 160
31.d even 5 1 inner 806.2.x.a ✓ 160
403.y even 10 1 inner 806.2.x.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.x.a ✓ 160 1.a even 1 1 trivial
806.2.x.a ✓ 160 13.b even 2 1 inner
806.2.x.a ✓ 160 31.d even 5 1 inner
806.2.x.a ✓ 160 403.y even 10 1 inner

Hecke kernels

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