Properties

Label 671.2.bd.a
Level $671$
Weight $2$
Character orbit 671.bd
Analytic conductor $5.358$
Analytic rank $0$
Dimension $240$
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] = [671,2,Mod(102,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.102");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bd (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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - q^{3} + 57 q^{4} - 5 q^{7} + 5 q^{8} - 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - q^{3} + 57 q^{4} - 5 q^{7} + 5 q^{8} - 57 q^{9} - 20 q^{10} - 5 q^{11} - 3 q^{12} - 10 q^{13} - 2 q^{14} - 4 q^{15} - 37 q^{16} + 10 q^{18} + 7 q^{19} - 7 q^{20} + q^{22} - 15 q^{23} - 55 q^{24} + 196 q^{25} + 11 q^{27} - 5 q^{28} + 10 q^{29} + 5 q^{31} - 10 q^{33} + q^{34} - 10 q^{35} - 134 q^{36} - 15 q^{37} + 44 q^{39} - 25 q^{40} - 52 q^{41} - 2 q^{42} - 5 q^{43} + 50 q^{44} - 56 q^{45} - 62 q^{46} + 3 q^{47} + 32 q^{48} + 71 q^{49} - 55 q^{50} + 30 q^{51} + 62 q^{52} - 115 q^{54} - 30 q^{55} + 33 q^{56} + 9 q^{57} + 43 q^{58} - 11 q^{60} - 29 q^{61} - 30 q^{62} + 25 q^{63} - 13 q^{64} - 11 q^{65} - 55 q^{66} + 10 q^{67} - 25 q^{68} - 68 q^{70} - 75 q^{71} + 140 q^{72} - 14 q^{73} - 42 q^{74} - 26 q^{75} - 12 q^{76} - 18 q^{77} + 30 q^{78} - q^{80} - 89 q^{81} - 50 q^{82} - 61 q^{83} - 15 q^{84} - 8 q^{86} - 90 q^{87} + 10 q^{88} + 60 q^{89} + 120 q^{90} - 5 q^{92} + 130 q^{93} + 65 q^{94} - q^{95} - 25 q^{97} + 105 q^{98} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
102.1 −1.62186 + 2.23229i 0.327661 0.238060i −1.73468 5.33881i −2.62339 1.11754i 2.05763 2.83209i 9.48276 + 3.08114i −0.876362 + 2.69716i 4.25477 5.85618i
102.2 −1.53609 + 2.11425i 2.00887 1.45953i −1.49243 4.59324i −0.0405784 6.48921i −1.49840 + 2.06237i 7.03287 + 2.28512i 0.978276 3.01082i 0.0623321 0.0857928i
102.3 −1.52483 + 2.09875i −1.26756 + 0.920933i −1.46160 4.49834i 2.66995 4.06454i 0.972481 1.33851i 6.73513 + 2.18838i −0.168472 + 0.518503i −4.07122 + 5.60356i
102.4 −1.49902 + 2.06323i 2.00706 1.45822i −1.39181 4.28355i 3.60749 6.32693i 1.54400 2.12513i 6.07335 + 1.97335i 0.974855 3.00029i −5.40771 + 7.44307i
102.5 −1.45402 + 2.00129i 0.314204 0.228283i −1.27295 3.91773i 2.04936 0.960741i −0.786859 + 1.08302i 4.98608 + 1.62008i −0.880440 + 2.70971i −2.97982 + 4.10137i
102.6 −1.45197 + 1.99847i −2.09315 + 1.52076i −1.26762 3.90133i −1.20752 6.39120i 1.89653 2.61035i 4.93856 + 1.60464i 1.14150 3.51319i 1.75329 2.41320i
102.7 −1.42256 + 1.95798i −2.47668 + 1.79941i −1.19200 3.66859i −1.88066 7.40907i −2.72530 + 3.75106i 4.27522 + 1.38910i 1.96900 6.05996i 2.67535 3.68230i
102.8 −1.40937 + 1.93983i 0.438999 0.318952i −1.15858 3.56575i −4.27798 1.30110i −1.80714 + 2.48731i 3.98901 + 1.29611i −0.836061 + 2.57313i 6.02924 8.29854i
102.9 −1.25583 + 1.72851i 1.68142 1.22162i −0.792581 2.43931i −1.35471 4.44049i −0.563963 + 0.776229i 1.14775 + 0.372928i 0.407758 1.25495i 1.70129 2.34162i
102.10 −1.22626 + 1.68780i −1.09421 + 0.794989i −0.726927 2.23725i 0.967708 2.82167i −0.537161 + 0.739339i 0.699180 + 0.227177i −0.361767 + 1.11340i −1.18666 + 1.63330i
102.11 −1.18673 + 1.63340i 0.570475 0.414474i −0.641616 1.97469i −0.927575 1.42368i 1.04506 1.43840i 0.146532 + 0.0476111i −0.773398 + 2.38028i 1.10078 1.51510i
102.12 −1.18452 + 1.63036i −2.40841 + 1.74981i −0.636935 1.96029i 4.34190 5.99927i −0.657519 + 0.904997i 0.117230 + 0.0380904i 1.81155 5.57538i −5.14309 + 7.07886i
102.13 −1.10903 + 1.52644i 2.26374 1.64471i −0.482058 1.48362i −0.790541 5.27950i 2.48480 3.42003i −0.789602 0.256557i 1.49243 4.59322i 0.876731 1.20672i
102.14 −1.01923 + 1.40285i 0.332528 0.241596i −0.311121 0.957531i 2.08625 0.712728i −2.32616 + 3.20168i −1.63792 0.532192i −0.874845 + 2.69249i −2.12637 + 2.92669i
102.15 −0.937766 + 1.29072i −1.39323 + 1.01224i −0.168530 0.518682i −1.42360 2.74753i 0.165676 0.228034i −2.20716 0.717149i −0.0105864 + 0.0325816i 1.33501 1.83748i
102.16 −0.921245 + 1.26799i −0.946289 + 0.687519i −0.141060 0.434137i −3.39867 1.83325i 1.02358 1.40883i −2.30078 0.747569i −0.504270 + 1.55198i 3.13101 4.30946i
102.17 −0.821787 + 1.13109i 2.50821 1.82232i 0.0139969 + 0.0430781i 1.93065 4.33457i −2.03656 + 2.80309i −2.71959 0.883647i 2.04321 6.28834i −1.58658 + 2.18374i
102.18 −0.811356 + 1.11674i −0.683410 + 0.496527i 0.0292341 + 0.0899733i 2.32240 1.16605i 2.50690 3.45045i −2.74980 0.893464i −0.706540 + 2.17451i −1.88429 + 2.59350i
102.19 −0.680409 + 0.936503i −1.18938 + 0.864135i 0.203953 + 0.627703i −1.32564 1.70182i −2.96018 + 4.07434i −2.92847 0.951517i −0.259156 + 0.797600i 0.901977 1.24146i
102.20 −0.646017 + 0.889167i 0.881531 0.640469i 0.244755 + 0.753279i 3.06178 1.19758i 0.560983 0.772127i −2.91846 0.948266i −0.560156 + 1.72398i −1.97796 + 2.72243i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 102.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.bd even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bd.a 240
11.c even 5 1 671.2.bf.a yes 240
61.g even 10 1 671.2.bf.a yes 240
671.bd even 10 1 inner 671.2.bd.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bd.a 240 1.a even 1 1 trivial
671.2.bd.a 240 671.bd even 10 1 inner
671.2.bf.a yes 240 11.c even 5 1
671.2.bf.a yes 240 61.g even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).