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Properties

Label 806.2.k.d
Level $806$
Weight $2$
Character orbit 806.k
Analytic conductor $6.436$
Analytic rank $0$
Dimension $36$
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(157,806)]
 
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([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.k (of order \(5\), 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: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 9 q^{2} + q^{3} - 9 q^{4} - 2 q^{5} - 6 q^{6} - 2 q^{7} + 9 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 9 q^{2} + q^{3} - 9 q^{4} - 2 q^{5} - 6 q^{6} - 2 q^{7} + 9 q^{8} - 12 q^{9} + 7 q^{10} - 5 q^{11} + q^{12} - 9 q^{13} + 2 q^{14} + 17 q^{15} - 9 q^{16} - 5 q^{17} + 7 q^{18} - q^{19} + 8 q^{20} - 9 q^{21} + 5 q^{22} + 6 q^{23} + 4 q^{24} + 22 q^{25} - 36 q^{26} - 23 q^{27} - 2 q^{28} - 19 q^{29} - 12 q^{30} + 12 q^{31} - 36 q^{32} - 21 q^{33} - 18 q^{35} + 38 q^{36} - 14 q^{37} + 31 q^{38} - 4 q^{39} - 8 q^{40} + 22 q^{41} - 21 q^{42} + 17 q^{43} + 5 q^{44} + 13 q^{45} + 9 q^{46} - 33 q^{47} - 4 q^{48} - 27 q^{49} - 7 q^{50} + 57 q^{51} - 9 q^{52} - 49 q^{53} + 23 q^{54} + 28 q^{55} - 8 q^{56} + 64 q^{57} - 26 q^{58} + 8 q^{59} + 17 q^{60} - 20 q^{61} + 13 q^{62} + 50 q^{63} - 9 q^{64} - 7 q^{65} - 9 q^{66} + 38 q^{67} + 10 q^{68} + 58 q^{69} + 18 q^{70} - 26 q^{71} + 7 q^{72} - 64 q^{73} - 11 q^{74} + 7 q^{75} - 31 q^{76} + 73 q^{77} - q^{78} + 4 q^{79} - 7 q^{80} - 14 q^{81} + 38 q^{82} + 10 q^{83} + 21 q^{84} - 9 q^{85} + 28 q^{86} - 86 q^{87} + 51 q^{89} + 27 q^{90} - 2 q^{91} + 6 q^{92} + 31 q^{93} - 2 q^{94} - 83 q^{95} - q^{96} - 23 q^{97} - 128 q^{98} + 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 0.809017 + 0.587785i −2.78492 + 2.02336i 0.309017 + 0.951057i 0.872775 −3.44235 −0.605327 − 1.86300i −0.309017 + 0.951057i 2.73473 − 8.41663i 0.706090 + 0.513004i
157.2 0.809017 + 0.587785i −1.60842 + 1.16859i 0.309017 + 0.951057i −2.00721 −1.98812 1.42217 + 4.37699i −0.309017 + 0.951057i 0.294377 − 0.905998i −1.62387 − 1.17981i
157.3 0.809017 + 0.587785i −0.967150 + 0.702676i 0.309017 + 0.951057i −0.301016 −1.19546 −1.08034 − 3.32496i −0.309017 + 0.951057i −0.485425 + 1.49398i −0.243527 − 0.176933i
157.4 0.809017 + 0.587785i −0.560668 + 0.407349i 0.309017 + 0.951057i 3.83429 −0.693024 1.24542 + 3.83301i −0.309017 + 0.951057i −0.778636 + 2.39639i 3.10200 + 2.25374i
157.5 0.809017 + 0.587785i −0.409950 + 0.297846i 0.309017 + 0.951057i −1.87580 −0.506726 −0.644104 − 1.98235i −0.309017 + 0.951057i −0.847704 + 2.60897i −1.51756 − 1.10257i
157.6 0.809017 + 0.587785i 0.316382 − 0.229865i 0.309017 + 0.951057i 1.40957 0.391069 −0.0557577 − 0.171605i −0.309017 + 0.951057i −0.879791 + 2.70772i 1.14037 + 0.828525i
157.7 0.809017 + 0.587785i 1.36007 − 0.988151i 0.309017 + 0.951057i −3.29215 1.68114 1.21160 + 3.72892i −0.309017 + 0.951057i −0.0536938 + 0.165253i −2.66341 − 1.93508i
157.8 0.809017 + 0.587785i 2.12267 − 1.54221i 0.309017 + 0.951057i 2.33665 2.62377 −1.29852 − 3.99645i −0.309017 + 0.951057i 1.20027 − 3.69404i 1.89039 + 1.37345i
157.9 0.809017 + 0.587785i 2.22297 − 1.61508i 0.309017 + 0.951057i 1.87700 2.74774 0.422900 + 1.30155i −0.309017 + 0.951057i 1.40605 − 4.32737i 1.51853 + 1.10327i
287.1 −0.309017 + 0.951057i −0.958886 − 2.95115i −0.809017 − 0.587785i −3.48695 3.10302 −3.97741 − 2.88976i 0.809017 − 0.587785i −5.36276 + 3.89628i 1.07753 − 3.31628i
287.2 −0.309017 + 0.951057i −0.517111 − 1.59150i −0.809017 − 0.587785i −1.50951 1.67341 2.14003 + 1.55482i 0.809017 − 0.587785i 0.161573 − 0.117390i 0.466464 − 1.43563i
287.3 −0.309017 + 0.951057i −0.434808 − 1.33820i −0.809017 − 0.587785i 3.57466 1.40707 −1.10184 − 0.800536i 0.809017 − 0.587785i 0.825324 − 0.599633i −1.10463 + 3.39970i
287.4 −0.309017 + 0.951057i −0.0769082 − 0.236699i −0.809017 − 0.587785i −0.0897343 0.248880 −1.70424 − 1.23820i 0.809017 − 0.587785i 2.37694 − 1.72695i 0.0277294 − 0.0853424i
287.5 −0.309017 + 0.951057i 0.195831 + 0.602707i −0.809017 − 0.587785i −4.23670 −0.633723 3.91262 + 2.84268i 0.809017 − 0.587785i 2.10215 − 1.52730i 1.30921 − 4.02934i
287.6 −0.309017 + 0.951057i 0.258034 + 0.794148i −0.809017 − 0.587785i −2.05709 −0.835016 −1.39796 − 1.01568i 0.809017 − 0.587785i 1.86296 − 1.35352i 0.635675 − 1.95641i
287.7 −0.309017 + 0.951057i 0.352503 + 1.08489i −0.809017 − 0.587785i 2.25533 −1.14072 1.77240 + 1.28772i 0.809017 − 0.587785i 1.37432 − 0.998502i −0.696937 + 2.14495i
287.8 −0.309017 + 0.951057i 0.980999 + 3.01920i −0.809017 − 0.587785i 1.13148 −3.17458 −3.62444 − 2.63331i 0.809017 − 0.587785i −5.72619 + 4.16032i −0.349645 + 1.07610i
287.9 −0.309017 + 0.951057i 1.00936 + 3.10650i −0.809017 − 0.587785i 0.564407 −3.26637 2.36282 + 1.71669i 0.809017 − 0.587785i −6.20448 + 4.50782i −0.174411 + 0.536783i
469.1 −0.309017 − 0.951057i −0.958886 + 2.95115i −0.809017 + 0.587785i −3.48695 3.10302 −3.97741 + 2.88976i 0.809017 + 0.587785i −5.36276 − 3.89628i 1.07753 + 3.31628i
469.2 −0.309017 − 0.951057i −0.517111 + 1.59150i −0.809017 + 0.587785i −1.50951 1.67341 2.14003 − 1.55482i 0.809017 + 0.587785i 0.161573 + 0.117390i 0.466464 + 1.43563i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.k.d ✓ 36
31.d even 5 1 inner 806.2.k.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.k.d ✓ 36 1.a even 1 1 trivial
806.2.k.d ✓ 36 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - T_{3}^{35} + 20 T_{3}^{34} - 4 T_{3}^{33} + 244 T_{3}^{32} - 185 T_{3}^{31} + 2937 T_{3}^{30} + \cdots + 24025 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display