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Properties

Label 786.2.k.a
Level $786$
Weight $2$
Character orbit 786.k
Analytic conductor $6.276$
Analytic rank $0$
Dimension $264$
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] = [786,2,Mod(47,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.k (of order \(26\), degree \(12\), 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: \(6.27624159887\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(22\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 22 q^{2} - q^{3} - 22 q^{4} - q^{6} + 2 q^{7} - 22 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 22 q^{2} - q^{3} - 22 q^{4} - q^{6} + 2 q^{7} - 22 q^{8} - q^{9} + 12 q^{12} + 4 q^{13} + 2 q^{14} + 35 q^{15} - 22 q^{16} - 29 q^{17} - q^{18} - 13 q^{20} - 2 q^{21} + 8 q^{23} - q^{24} + 16 q^{25} + 4 q^{26} - 28 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} - 22 q^{32} + 100 q^{33} + 10 q^{34} + 91 q^{35} - q^{36} - 27 q^{39} - 2 q^{42} - 25 q^{43} + 19 q^{45} + 8 q^{46} + 12 q^{47} - q^{48} - 8 q^{49} + 16 q^{50} + 52 q^{51} + 4 q^{52} + 11 q^{54} - 96 q^{55} + 2 q^{56} + 94 q^{57} - 22 q^{58} + 22 q^{60} - 12 q^{61} - 26 q^{62} + 74 q^{63} - 22 q^{64} - 56 q^{66} + 26 q^{67} + 10 q^{68} - 28 q^{69} - 36 q^{71} - 27 q^{72} + 20 q^{75} + 64 q^{78} - 91 q^{79} - 25 q^{81} + 24 q^{83} - 2 q^{84} - 104 q^{85} + 40 q^{86} + 144 q^{87} + 19 q^{90} - 28 q^{91} + 8 q^{92} + 100 q^{93} + 12 q^{94} + 64 q^{95} - q^{96} + 39 q^{97} - 8 q^{98} - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0.568065 + 0.822984i −1.72179 + 0.188243i −0.354605 + 0.935016i 0.0921752 − 0.175625i −1.13301 − 1.31007i −0.0166196 + 0.136875i −0.970942 + 0.239316i 2.92913 − 0.648231i 0.196898 − 0.0239078i
47.2 0.568065 + 0.822984i −1.70428 − 0.308937i −0.354605 + 0.935016i 0.344279 − 0.655968i −0.713890 − 1.57809i 0.379614 − 3.12640i −0.970942 + 0.239316i 2.80912 + 1.05303i 0.735424 − 0.0892966i
47.3 0.568065 + 0.822984i −1.68070 + 0.418622i −0.354605 + 0.935016i −1.59276 + 3.03475i −1.29927 − 1.14539i −0.240124 + 1.97760i −0.970942 + 0.239316i 2.64951 − 1.40716i −3.40234 + 0.413119i
47.4 0.568065 + 0.822984i −1.39333 − 1.02890i −0.354605 + 0.935016i −0.587519 + 1.11942i 0.0552731 − 1.73117i 0.201689 − 1.66105i −0.970942 + 0.239316i 0.882710 + 2.86720i −1.25502 + 0.152387i
47.5 0.568065 + 0.822984i −1.38572 + 1.03912i −0.354605 + 0.935016i 1.00301 − 1.91108i −1.64236 − 0.550142i 0.0228478 − 0.188169i −0.970942 + 0.239316i 0.840463 − 2.87986i 2.14256 − 0.260154i
47.6 0.568065 + 0.822984i −1.16418 − 1.28245i −0.354605 + 0.935016i −1.70551 + 3.24957i 0.394105 − 1.68662i −0.308572 + 2.54132i −0.970942 + 0.239316i −0.289357 + 2.98601i −3.64318 + 0.442362i
47.7 0.568065 + 0.822984i −1.05673 + 1.37234i −0.354605 + 0.935016i −0.933662 + 1.77894i −1.72971 − 0.0900955i 0.616394 − 5.07646i −0.970942 + 0.239316i −0.766634 − 2.90039i −1.99442 + 0.242167i
47.8 0.568065 + 0.822984i −0.967502 − 1.43664i −0.354605 + 0.935016i 1.25906 − 2.39895i 0.632729 − 1.61234i −0.372699 + 3.06945i −0.970942 + 0.239316i −1.12788 + 2.77991i 2.68952 − 0.326567i
47.9 0.568065 + 0.822984i −0.847179 + 1.51072i −0.354605 + 0.935016i 1.58148 − 3.01326i −1.72455 + 0.160974i −0.0331220 + 0.272784i −0.970942 + 0.239316i −1.56457 − 2.55971i 3.37825 − 0.410194i
47.10 0.568065 + 0.822984i −0.714527 + 1.57780i −0.354605 + 0.935016i −0.664519 + 1.26614i −1.70440 + 0.308248i −0.600361 + 4.94442i −0.970942 + 0.239316i −1.97890 − 2.25476i −1.41950 + 0.172358i
47.11 0.568065 + 0.822984i −0.315085 − 1.70315i −0.354605 + 0.935016i −1.45975 + 2.78132i 1.22268 − 1.22681i 0.630777 − 5.19492i −0.970942 + 0.239316i −2.80144 + 1.07328i −3.11821 + 0.378619i
47.12 0.568065 + 0.822984i −0.147501 − 1.72576i −0.354605 + 0.935016i 0.681039 − 1.29761i 1.33648 − 1.10173i 0.155912 − 1.28405i −0.970942 + 0.239316i −2.95649 + 0.509104i 1.45479 − 0.176643i
47.13 0.568065 + 0.822984i 0.368280 + 1.69244i −0.354605 + 0.935016i −0.0973774 + 0.185537i −1.18365 + 1.26451i −0.196763 + 1.62049i −0.970942 + 0.239316i −2.72874 + 1.24659i −0.208011 + 0.0252571i
47.14 0.568065 + 0.822984i 0.848352 − 1.51007i −0.354605 + 0.935016i −1.07533 + 2.04887i 1.72468 − 0.159636i −0.231480 + 1.90641i −0.970942 + 0.239316i −1.56060 − 2.56213i −2.29705 + 0.278912i
47.15 0.568065 + 0.822984i 0.865326 + 1.50040i −0.354605 + 0.935016i −1.58383 + 3.01773i −0.743247 + 1.56448i 0.174605 − 1.43800i −0.970942 + 0.239316i −1.50242 + 2.59668i −3.38326 + 0.410802i
47.16 0.568065 + 0.822984i 1.24429 − 1.20489i −0.354605 + 0.935016i 1.70983 − 3.25781i 1.69844 + 0.339576i −0.521810 + 4.29749i −0.970942 + 0.239316i 0.0965036 − 2.99845i 3.65242 − 0.443484i
47.17 0.568065 + 0.822984i 1.38200 − 1.04407i −0.354605 + 0.935016i 1.22190 − 2.32814i 1.64432 + 0.544267i 0.356698 − 2.93767i −0.970942 + 0.239316i 0.819853 − 2.88580i 2.61015 − 0.316929i
47.18 0.568065 + 0.822984i 1.41464 + 0.999399i −0.354605 + 0.935016i 1.24441 − 2.37102i −0.0188834 + 1.73195i −0.350672 + 2.88804i −0.970942 + 0.239316i 1.00240 + 2.82758i 2.65821 − 0.322766i
47.19 0.568065 + 0.822984i 1.58755 − 0.692596i −0.354605 + 0.935016i −0.424926 + 0.809629i 1.47183 + 0.913087i 0.309294 − 2.54726i −0.970942 + 0.239316i 2.04062 − 2.19906i −0.907697 + 0.110214i
47.20 0.568065 + 0.822984i 1.59953 + 0.664461i −0.354605 + 0.935016i 1.51502 − 2.88664i 0.361795 + 1.69384i 0.495669 − 4.08220i −0.970942 + 0.239316i 2.11698 + 2.12565i 3.23629 − 0.392956i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
393.j even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.k.a ✓ 264
3.b odd 2 1 786.2.k.b yes 264
131.f odd 26 1 786.2.k.b yes 264
393.j even 26 1 inner 786.2.k.a ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.k.a ✓ 264 1.a even 1 1 trivial
786.2.k.a ✓ 264 393.j even 26 1 inner
786.2.k.b yes 264 3.b odd 2 1
786.2.k.b yes 264 131.f odd 26 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{264} - 63 T_{5}^{262} + 2333 T_{5}^{260} + 1014 T_{5}^{259} - 66098 T_{5}^{258} + \cdots + 72\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display