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Properties

Label 806.2.bk.a
Level $806$
Weight $2$
Character orbit 806.bk
Analytic conductor $6.436$
Analytic rank $0$
Dimension $48$
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(131,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bk (of order \(15\), degree \(8\), 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: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} - 3 q^{3} - 12 q^{4} + 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} - 3 q^{3} - 12 q^{4} + 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + q^{9} + 4 q^{10} + 9 q^{11} - 3 q^{12} + 6 q^{13} - 7 q^{14} - 17 q^{15} - 12 q^{16} + q^{17} - 14 q^{18} + 9 q^{19} - q^{20} + 35 q^{21} - 6 q^{22} + 13 q^{23} + 2 q^{24} + q^{25} - 24 q^{26} + 39 q^{27} - 2 q^{28} - 6 q^{29} - 2 q^{30} - 18 q^{31} + 48 q^{32} + 35 q^{33} - 4 q^{34} - 24 q^{35} - 4 q^{36} - 17 q^{37} - 46 q^{38} - 4 q^{39} - q^{40} + 24 q^{41} + 5 q^{43} - 26 q^{44} - 11 q^{45} - 7 q^{46} + 60 q^{47} + 2 q^{48} - 5 q^{49} + 21 q^{50} - 78 q^{51} + 6 q^{52} - 43 q^{53} + 39 q^{54} + 10 q^{55} + 18 q^{56} + 8 q^{57} + 19 q^{58} - 17 q^{59} - 17 q^{60} - 6 q^{61} + 7 q^{62} - 80 q^{63} - 12 q^{64} + 4 q^{65} - 15 q^{66} + 38 q^{67} + 6 q^{68} - 102 q^{69} - 24 q^{70} + 16 q^{72} - 12 q^{73} + 3 q^{74} - 48 q^{75} + 14 q^{76} + 6 q^{78} + 34 q^{79} - 11 q^{80} - 55 q^{81} - 46 q^{82} + 45 q^{83} + 15 q^{84} - 3 q^{85} - 16 q^{87} + 4 q^{88} + 62 q^{89} - 11 q^{90} + 4 q^{91} - 12 q^{92} - 28 q^{93} - 70 q^{94} - 13 q^{95} - 3 q^{96} - 31 q^{97} + 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 0.309017 + 0.951057i −1.96859 + 0.418437i −0.809017 + 0.587785i 0.221640 − 0.383892i −1.00628 − 1.74294i −1.97085 − 0.877479i −0.809017 − 0.587785i 0.959618 − 0.427249i 0.433593 + 0.0921631i
131.2 0.309017 + 0.951057i −1.38485 + 0.294359i −0.809017 + 0.587785i 0.435123 − 0.753656i −0.707894 − 1.22611i 2.16099 + 0.962137i −0.809017 − 0.587785i −0.909474 + 0.404924i 0.851230 + 0.180934i
131.3 0.309017 + 0.951057i −1.21192 + 0.257602i −0.809017 + 0.587785i −1.62102 + 2.80768i −0.619499 − 1.07300i −0.623786 − 0.277727i −0.809017 − 0.587785i −1.33824 + 0.595822i −3.17119 − 0.674057i
131.4 0.309017 + 0.951057i 0.0885670 − 0.0188255i −0.809017 + 0.587785i 1.81159 − 3.13776i 0.0452728 + 0.0784148i −4.60614 − 2.05078i −0.809017 − 0.587785i −2.73315 + 1.21688i 3.54400 + 0.753301i
131.5 0.309017 + 0.951057i 1.57588 − 0.334964i −0.809017 + 0.587785i −0.548331 + 0.949737i 0.805544 + 1.39524i −1.65921 − 0.738726i −0.809017 − 0.587785i −0.369434 + 0.164483i −1.07270 − 0.228009i
131.6 0.309017 + 0.951057i 2.29638 − 0.488112i −0.809017 + 0.587785i 1.37013 − 2.37313i 1.17384 + 2.03316i 1.56665 + 0.697519i −0.809017 − 0.587785i 2.29449 − 1.02157i 2.68037 + 0.569731i
183.1 −0.809017 + 0.587785i −0.293618 + 2.79359i 0.309017 − 0.951057i 0.761658 − 1.31923i −1.40449 − 2.43265i 2.76197 − 0.587076i 0.309017 + 0.951057i −4.78350 − 1.01676i 0.159230 + 1.51497i
183.2 −0.809017 + 0.587785i −0.0436310 + 0.415121i 0.309017 − 0.951057i 0.140134 − 0.242718i −0.208704 − 0.361485i −3.04555 + 0.647351i 0.309017 + 0.951057i 2.76402 + 0.587511i 0.0292959 + 0.278732i
183.3 −0.809017 + 0.587785i 0.0111189 − 0.105790i 0.309017 − 0.951057i 1.73191 − 2.99976i 0.0531862 + 0.0921212i 1.80328 − 0.383300i 0.309017 + 0.951057i 2.92337 + 0.621383i 0.362068 + 3.44485i
183.4 −0.809017 + 0.587785i 0.0697599 − 0.663721i 0.309017 − 0.951057i −1.63333 + 2.82901i 0.333688 + 0.577965i 2.91544 − 0.619696i 0.309017 + 0.951057i 2.49878 + 0.531133i −0.341459 − 3.24876i
183.5 −0.809017 + 0.587785i 0.183429 − 1.74521i 0.309017 − 0.951057i 1.05902 − 1.83428i 0.877412 + 1.51972i −1.11849 + 0.237742i 0.309017 + 0.951057i −0.0776733 − 0.0165100i 0.221396 + 2.10644i
183.6 −0.809017 + 0.587785i 0.242072 − 2.30316i 0.309017 − 0.951057i −0.145851 + 0.252622i 1.15792 + 2.00558i −0.00860407 + 0.00182885i 0.309017 + 0.951057i −2.31151 − 0.491326i −0.0304913 − 0.290105i
235.1 −0.809017 + 0.587785i −2.77678 + 1.23630i 0.309017 − 0.951057i 0.889497 + 1.54065i 1.51978 − 2.63234i −3.07828 − 3.41877i 0.309017 + 0.951057i 4.17468 − 4.63645i −1.62519 − 0.723582i
235.2 −0.809017 + 0.587785i −1.52551 + 0.679200i 0.309017 − 0.951057i −0.714838 − 1.23814i 0.834939 − 1.44616i 0.0603139 + 0.0669853i 0.309017 + 0.951057i −0.141527 + 0.157182i 1.30607 + 0.581502i
235.3 −0.809017 + 0.587785i −0.738232 + 0.328682i 0.309017 − 0.951057i 2.07909 + 3.60109i 0.404048 − 0.699831i 1.02682 + 1.14040i 0.309017 + 0.951057i −1.57044 + 1.74415i −3.79869 − 1.69128i
235.4 −0.809017 + 0.587785i 0.285958 − 0.127317i 0.309017 − 0.951057i 0.615324 + 1.06577i −0.156510 + 0.271083i 3.16837 + 3.51883i 0.309017 + 0.951057i −1.94183 + 2.15662i −1.12425 − 0.500550i
235.5 −0.809017 + 0.587785i 0.765246 − 0.340710i 0.309017 − 0.951057i −1.66197 − 2.87862i −0.418833 + 0.725440i −2.47157 − 2.74496i 0.309017 + 0.951057i −1.53787 + 1.70798i 3.03657 + 1.35197i
235.6 −0.809017 + 0.587785i 2.51117 − 1.11805i 0.309017 − 0.951057i −0.311632 − 0.539762i −1.37441 + 2.38055i −0.968626 − 1.07577i 0.309017 + 0.951057i 3.04857 − 3.38577i 0.569380 + 0.253504i
391.1 −0.809017 − 0.587785i −2.77678 − 1.23630i 0.309017 + 0.951057i 0.889497 − 1.54065i 1.51978 + 2.63234i −3.07828 + 3.41877i 0.309017 − 0.951057i 4.17468 + 4.63645i −1.62519 + 0.723582i
391.2 −0.809017 − 0.587785i −1.52551 − 0.679200i 0.309017 + 0.951057i −0.714838 + 1.23814i 0.834939 + 1.44616i 0.0603139 − 0.0669853i 0.309017 − 0.951057i −0.141527 − 0.157182i 1.30607 − 0.581502i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bk.a ✓ 48
31.g even 15 1 inner 806.2.bk.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bk.a ✓ 48 1.a even 1 1 trivial
806.2.bk.a ✓ 48 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 3 T_{3}^{47} - 5 T_{3}^{46} - 29 T_{3}^{45} + 9 T_{3}^{44} + 138 T_{3}^{43} + 318 T_{3}^{42} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display