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Properties

Label 671.2.e.a
Level $671$
Weight $2$
Character orbit 671.e
Analytic conductor $5.358$
Analytic rank $0$
Dimension $52$
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(474,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.474");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.e (of order \(3\), degree \(2\), 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: \(52\)
Relative dimension: \(26\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 2 q^{2} + 2 q^{3} - 30 q^{4} - 6 q^{6} - 7 q^{7} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 2 q^{2} + 2 q^{3} - 30 q^{4} - 6 q^{6} - 7 q^{7} + 50 q^{9} - 2 q^{10} + 52 q^{11} + 11 q^{12} - 3 q^{13} - 3 q^{14} - 12 q^{15} - 34 q^{16} - 8 q^{17} - 12 q^{18} + 5 q^{19} - 22 q^{20} - 2 q^{21} - 2 q^{22} - 2 q^{24} - 16 q^{25} - 37 q^{26} + 8 q^{27} + 104 q^{28} + 10 q^{29} - 20 q^{31} + 8 q^{32} + 2 q^{33} - 12 q^{34} + 18 q^{35} - 60 q^{36} + 4 q^{37} - 20 q^{38} + 23 q^{39} - 38 q^{40} - 6 q^{41} - 56 q^{42} - 22 q^{43} - 30 q^{44} + 25 q^{46} - 26 q^{48} - 45 q^{49} - 16 q^{50} - 38 q^{51} - 18 q^{53} - 46 q^{54} + 16 q^{56} + 36 q^{57} - 56 q^{58} - 13 q^{59} + 40 q^{60} + 14 q^{61} + 26 q^{62} - 39 q^{63} + 164 q^{64} + 15 q^{65} - 6 q^{66} + 7 q^{67} - 39 q^{68} + 20 q^{69} + 162 q^{70} - 8 q^{71} + 6 q^{72} + 27 q^{73} - 13 q^{74} + 16 q^{75} + 20 q^{76} - 7 q^{77} + 20 q^{78} + 5 q^{79} - 8 q^{80} - 68 q^{81} + 6 q^{82} - 22 q^{83} - 28 q^{84} - 36 q^{85} + 13 q^{86} - 25 q^{87} + 36 q^{89} + 41 q^{90} - 42 q^{91} - 17 q^{92} - 26 q^{93} - 14 q^{94} - 40 q^{95} - 3 q^{96} - 47 q^{97} - 2 q^{98} + 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1 −1.39981 − 2.42454i 2.03714 −2.91894 + 5.05576i −0.880908 − 1.52578i −2.85161 − 4.93914i −2.33090 − 4.03724i 10.7446 1.14995 −2.46621 + 4.27160i
474.2 −1.28918 − 2.23293i −3.07366 −2.32398 + 4.02525i 0.692759 + 1.19989i 3.96250 + 6.86325i −0.857264 − 1.48482i 6.82738 6.44736 1.78618 − 3.09376i
474.3 −1.21768 − 2.10909i −0.119390 −1.96550 + 3.40434i 0.0110850 + 0.0191998i 0.145379 + 0.251804i 0.177168 + 0.306863i 4.70267 −2.98575 0.0269960 − 0.0467585i
474.4 −1.10662 − 1.91672i 2.91652 −1.44921 + 2.51010i 1.12816 + 1.95402i −3.22747 − 5.59015i 0.991463 + 1.71726i 1.98840 5.50610 2.49688 − 4.32472i
474.5 −1.07290 − 1.85831i −1.75555 −1.30222 + 2.25550i −0.834005 − 1.44454i 1.88352 + 3.26235i 2.55493 + 4.42527i 1.29698 0.0819409 −1.78960 + 3.09968i
474.6 −0.968326 − 1.67719i −1.30850 −0.875312 + 1.51608i −1.56023 − 2.70239i 1.26706 + 2.19461i −1.74364 − 3.02008i −0.482955 −1.28782 −3.02161 + 5.23359i
474.7 −0.894433 − 1.54920i −0.575368 −0.600022 + 1.03927i 2.08743 + 3.61553i 0.514629 + 0.891363i 0.371978 + 0.644284i −1.43101 −2.66895 3.73413 − 6.46770i
474.8 −0.738467 − 1.27906i 2.69547 −0.0906660 + 0.157038i −1.97155 − 3.41482i −1.99052 − 3.44767i −0.812537 − 1.40736i −2.68605 4.26557 −2.91185 + 5.04347i
474.9 −0.602444 − 1.04346i 2.88926 0.274123 − 0.474794i 1.22195 + 2.11647i −1.74062 − 3.01484i −1.97926 − 3.42819i −3.07035 5.34781 1.47231 − 2.55011i
474.10 −0.536790 − 0.929747i −2.63709 0.423714 − 0.733894i 1.18192 + 2.04715i 1.41556 + 2.45183i −0.752816 − 1.30392i −3.05694 3.95425 1.26889 − 2.19778i
474.11 −0.372490 − 0.645171i −1.26258 0.722503 − 1.25141i −0.0314823 − 0.0545290i 0.470297 + 0.814579i 1.61404 + 2.79560i −2.56646 −1.40590 −0.0234537 + 0.0406230i
474.12 −0.320705 − 0.555478i 0.649053 0.794296 − 1.37576i 0.442759 + 0.766881i −0.208155 − 0.360534i −1.50175 − 2.60110i −2.30176 −2.57873 0.283990 − 0.491885i
474.13 −0.310878 − 0.538456i 0.974785 0.806710 − 1.39726i −0.223659 − 0.387389i −0.303039 − 0.524879i 1.06871 + 1.85105i −2.24666 −2.04979 −0.139061 + 0.240861i
474.14 0.131418 + 0.227623i 2.69076 0.965459 − 1.67222i −1.45219 − 2.51526i 0.353614 + 0.612478i 1.92643 + 3.33667i 1.03319 4.24017 0.381687 − 0.661101i
474.15 0.172443 + 0.298681i 1.48381 0.940527 − 1.62904i 1.21544 + 2.10521i 0.255873 + 0.443185i 0.147655 + 0.255746i 1.33852 −0.798307 −0.419190 + 0.726058i
474.16 0.196695 + 0.340685i −1.57016 0.922622 − 1.59803i −1.67457 − 2.90045i −0.308843 − 0.534931i −1.04200 − 1.80479i 1.51268 −0.534592 0.658760 − 1.14101i
474.17 0.408544 + 0.707618i −2.11340 0.666184 − 1.15387i −0.472681 − 0.818707i −0.863417 − 1.49548i 1.09515 + 1.89685i 2.72284 1.46647 0.386222 − 0.668955i
474.18 0.566722 + 0.981591i 2.34130 0.357653 − 0.619474i −0.833325 − 1.44336i 1.32687 + 2.29820i −0.538269 − 0.932309i 3.07765 2.48171 0.944526 − 1.63597i
474.19 0.716721 + 1.24140i −0.188236 −0.0273779 + 0.0474198i 1.37339 + 2.37878i −0.134913 − 0.233676i 1.70991 + 2.96164i 2.78839 −2.96457 −1.96867 + 3.40984i
474.20 0.772502 + 1.33801i −0.117047 −0.193519 + 0.335184i −0.0594005 − 0.102885i −0.0904189 − 0.156610i −2.59858 − 4.50087i 2.49203 −2.98630 0.0917739 − 0.158957i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.e.a ✓ 52
61.c even 3 1 inner 671.2.e.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.e.a ✓ 52 1.a even 1 1 trivial
671.2.e.a ✓ 52 61.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{52} + 2 T_{2}^{51} + 43 T_{2}^{50} + 78 T_{2}^{49} + 1018 T_{2}^{48} + 1724 T_{2}^{47} + \cdots + 2531281 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display