Properties

Label 867.2.k.b
Level $867$
Weight $2$
Character orbit 867.k
Analytic conductor $6.923$
Analytic rank $0$
Dimension $416$
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] = [867,2,Mod(52,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.52");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.k (of order \(17\), degree \(16\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(416\)
Relative dimension: \(26\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416 q + q^{2} + 26 q^{3} - 29 q^{4} + 30 q^{5} - q^{6} + 9 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 416 q + q^{2} + 26 q^{3} - 29 q^{4} + 30 q^{5} - q^{6} + 9 q^{8} - 26 q^{9} + 27 q^{10} + 29 q^{12} - 4 q^{13} - 34 q^{14} - 30 q^{15} + 7 q^{16} - 2 q^{17} + q^{18} - 2 q^{19} - 20 q^{20} + 2 q^{22} - 26 q^{23} - 9 q^{24} + 42 q^{25} - 6 q^{26} + 26 q^{27} - 6 q^{28} - 4 q^{29} - 10 q^{30} + 2 q^{31} + 9 q^{32} - 169 q^{34} - 30 q^{35} - 29 q^{36} + 48 q^{38} + 4 q^{39} - 86 q^{40} + 2 q^{41} - 26 q^{43} - 95 q^{44} - 4 q^{45} + 36 q^{46} + 16 q^{47} + 27 q^{48} - 8 q^{49} - 156 q^{50} + 2 q^{51} + 85 q^{52} - 23 q^{53} + 16 q^{54} - 2 q^{55} - 8 q^{56} + 2 q^{57} + 79 q^{58} - 102 q^{59} + 20 q^{60} - 12 q^{61} + 8 q^{62} + 17 q^{63} - 25 q^{64} - 140 q^{65} - 36 q^{66} - 104 q^{67} - 5 q^{68} - 8 q^{69} - 126 q^{70} + 105 q^{71} - 25 q^{72} + 2 q^{73} + 59 q^{74} - 416 q^{75} + 45 q^{76} + 172 q^{77} + 6 q^{78} + 39 q^{79} + 130 q^{80} - 26 q^{81} - 52 q^{82} + 14 q^{83} + 6 q^{84} + 82 q^{85} - 6 q^{86} + 4 q^{87} + 26 q^{88} - 34 q^{89} + 10 q^{90} - 142 q^{91} - 100 q^{92} - 2 q^{93} + 139 q^{94} + 10 q^{95} - 60 q^{96} + 2 q^{97} + 11 q^{98} - 17 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −1.24250 2.49528i −0.932472 0.361242i −3.47735 + 4.60476i −2.30932 + 0.431686i 0.257200 + 2.77563i −3.05225 2.78249i 10.3307 + 1.93114i 0.739009 + 0.673696i 3.94651 + 5.22603i
52.2 −1.08825 2.18549i −0.932472 0.361242i −2.38683 + 3.16067i 1.24788 0.233270i 0.225268 + 2.43103i 0.673724 + 0.614181i 4.70534 + 0.879580i 0.739009 + 0.673696i −1.86781 2.47338i
52.3 −0.997929 2.00411i −0.932472 0.361242i −1.81533 + 2.40389i 3.37711 0.631290i 0.206573 + 2.22927i −1.88697 1.72020i 2.22784 + 0.416455i 0.739009 + 0.673696i −4.63529 6.13812i
52.4 −0.961507 1.93097i −0.932472 0.361242i −1.59886 + 2.11724i −3.25210 + 0.607924i 0.199033 + 2.14791i 1.43576 + 1.30887i 1.38487 + 0.258878i 0.739009 + 0.673696i 4.30080 + 5.69518i
52.5 −0.935806 1.87935i −0.932472 0.361242i −1.45096 + 1.92139i −1.18576 + 0.221657i 0.193713 + 2.09050i −2.04458 1.86388i 0.841374 + 0.157280i 0.739009 + 0.673696i 1.52622 + 2.02104i
52.6 −0.726082 1.45817i −0.932472 0.361242i −0.393792 + 0.521465i −2.58661 + 0.483521i 0.150300 + 1.62199i 1.36635 + 1.24560i −2.15610 0.403045i 0.739009 + 0.673696i 2.58315 + 3.42064i
52.7 −0.659046 1.32354i −0.932472 0.361242i −0.112156 + 0.148519i 0.975426 0.182339i 0.136423 + 1.47224i −2.91012 2.65293i −2.63626 0.492803i 0.739009 + 0.673696i −0.884185 1.17085i
52.8 −0.658174 1.32179i −0.932472 0.361242i −0.108673 + 0.143906i 0.955429 0.178601i 0.136243 + 1.47029i 3.01818 + 2.75144i −2.64116 0.493719i 0.739009 + 0.673696i −0.864912 1.14533i
52.9 −0.539836 1.08414i −0.932472 0.361242i 0.321338 0.425520i 3.76892 0.704533i 0.111747 + 1.20594i 1.49402 + 1.36198i −3.01576 0.563744i 0.739009 + 0.673696i −2.79841 3.70569i
52.10 −0.337548 0.677887i −0.932472 0.361242i 0.859676 1.13840i 0.183475 0.0342974i 0.0698728 + 0.754048i −1.37226 1.25098i −2.55065 0.476800i 0.739009 + 0.673696i −0.0851814 0.112798i
52.11 −0.145936 0.293078i −0.932472 0.361242i 1.14067 1.51049i −0.751566 + 0.140492i 0.0302088 + 0.326005i 1.21855 + 1.11086i −1.25281 0.234191i 0.739009 + 0.673696i 0.150855 + 0.199765i
52.12 −0.132827 0.266752i −0.932472 0.361242i 1.15176 1.52517i −3.36291 + 0.628637i 0.0274953 + 0.296721i 3.04508 + 2.77596i −1.14566 0.214161i 0.739009 + 0.673696i 0.614374 + 0.813562i
52.13 0.0157622 + 0.0316547i −0.932472 0.361242i 1.20452 1.59504i 2.54516 0.475773i −0.00326279 0.0352111i −3.39509 3.09503i 0.138996 + 0.0259828i 0.739009 + 0.673696i 0.0551778 + 0.0730672i
52.14 0.143755 + 0.288699i −0.932472 0.361242i 1.14259 1.51303i 2.53423 0.473730i −0.0297574 0.321134i 1.31400 + 1.19787i 1.23510 + 0.230880i 0.739009 + 0.673696i 0.501073 + 0.663528i
52.15 0.322135 + 0.646934i −0.932472 0.361242i 0.890516 1.17923i −3.94269 + 0.737017i −0.0666823 0.719617i −1.05780 0.964314i 2.47054 + 0.461824i 0.739009 + 0.673696i −1.74688 2.31324i
52.16 0.327859 + 0.658430i −0.932472 0.361242i 0.879231 1.16429i −1.66435 + 0.311121i −0.0678672 0.732404i 0.510905 + 0.465751i 2.50090 + 0.467500i 0.739009 + 0.673696i −0.750524 0.993855i
52.17 0.469706 + 0.943297i −0.932472 0.361242i 0.536083 0.709889i −0.721450 + 0.134862i −0.0972297 1.04928i −2.59800 2.36839i 2.99310 + 0.559507i 0.739009 + 0.673696i −0.466085 0.617196i
52.18 0.535145 + 1.07472i −0.932472 0.361242i 0.336633 0.445775i 1.14944 0.214867i −0.110776 1.19546i 1.79749 + 1.63863i 3.01951 + 0.564444i 0.739009 + 0.673696i 0.846039 + 1.12034i
52.19 0.550627 + 1.10581i −0.932472 0.361242i 0.285648 0.378259i 4.03495 0.754264i −0.113980 1.23004i 1.35721 + 1.23726i 3.00413 + 0.561569i 0.739009 + 0.673696i 3.05583 + 4.04657i
52.20 0.916970 + 1.84152i −0.932472 0.361242i −1.34511 + 1.78121i −0.902792 + 0.168761i −0.189814 2.04842i 2.83356 + 2.58313i −0.469224 0.0877133i 0.739009 + 0.673696i −1.13861 1.50776i
See next 80 embeddings (of 416 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 52.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
289.f even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.k.b 416
289.f even 17 1 inner 867.2.k.b 416
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.k.b 416 1.a even 1 1 trivial
867.2.k.b 416 289.f even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{416} - T_{2}^{415} + 41 T_{2}^{414} - 45 T_{2}^{413} + 913 T_{2}^{412} - 1093 T_{2}^{411} + 14884 T_{2}^{410} - 19054 T_{2}^{409} + 200281 T_{2}^{408} - 269259 T_{2}^{407} + 2367488 T_{2}^{406} - 3306896 T_{2}^{405} + \cdots + 40\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\). Copy content Toggle raw display