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Properties

Label 670.2.k.d
Level $670$
Weight $2$
Character orbit 670.k
Analytic conductor $5.350$
Analytic rank $0$
Dimension $60$
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] = [670,2,Mod(81,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(22))
 
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("670.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.k (of order \(11\), degree \(10\), 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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 2 q^{3} - 6 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 2 q^{3} - 6 q^{4} + 6 q^{5} - 2 q^{6} + 3 q^{7} + 6 q^{8} - 6 q^{9} - 6 q^{10} + 8 q^{11} + 2 q^{12} - 10 q^{13} + 19 q^{14} - 2 q^{15} - 6 q^{16} - 7 q^{17} + 6 q^{18} + 6 q^{20} - 31 q^{21} + 3 q^{22} + 5 q^{23} - 2 q^{24} - 6 q^{25} - 12 q^{26} - 10 q^{27} + 14 q^{28} + 30 q^{29} - 20 q^{30} - 9 q^{31} + 6 q^{32} + 52 q^{33} + 18 q^{34} + 8 q^{35} - 6 q^{36} + 26 q^{37} - 22 q^{38} - 31 q^{39} - 6 q^{40} + 44 q^{41} - 2 q^{42} + 15 q^{43} - 3 q^{44} + 6 q^{45} - 5 q^{46} + 53 q^{47} + 2 q^{48} + 59 q^{49} + 6 q^{50} + 11 q^{51} - 10 q^{52} + 29 q^{53} - 23 q^{54} - 8 q^{55} - 14 q^{56} + 8 q^{57} - 30 q^{58} - 28 q^{59} - 2 q^{60} + 31 q^{61} + 9 q^{62} - 114 q^{63} - 6 q^{64} + 10 q^{65} - 8 q^{66} - 24 q^{67} + 4 q^{68} - 31 q^{69} - 8 q^{70} - 42 q^{71} + 6 q^{72} - 56 q^{73} - 4 q^{74} + 2 q^{75} - 19 q^{77} - 79 q^{78} + 18 q^{79} + 6 q^{80} - 6 q^{81} - 33 q^{82} + 28 q^{83} + 24 q^{84} + 7 q^{85} + 7 q^{86} + 57 q^{87} - 8 q^{88} + 33 q^{89} - 6 q^{90} + 14 q^{91} - 6 q^{92} + 30 q^{93} - 9 q^{94} - 11 q^{95} + 20 q^{96} - 116 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −0.415415 − 0.909632i −2.76146 − 1.77468i −0.654861 + 0.755750i 0.142315 + 0.989821i −0.467156 + 3.24914i −0.988243 − 2.16395i 0.959493 + 0.281733i 3.22991 + 7.07253i 0.841254 − 0.540641i
81.2 −0.415415 − 0.909632i −1.38912 − 0.892736i −0.654861 + 0.755750i 0.142315 + 0.989821i −0.234998 + 1.63445i 0.644195 + 1.41059i 0.959493 + 0.281733i −0.113556 − 0.248652i 0.841254 − 0.540641i
81.3 −0.415415 − 0.909632i −1.36967 − 0.880231i −0.654861 + 0.755750i 0.142315 + 0.989821i −0.231706 + 1.61155i 0.357839 + 0.783557i 0.959493 + 0.281733i −0.145067 − 0.317653i 0.841254 − 0.540641i
81.4 −0.415415 − 0.909632i 0.643644 + 0.413645i −0.654861 + 0.755750i 0.142315 + 0.989821i 0.108885 − 0.757313i −0.632291 − 1.38452i 0.959493 + 0.281733i −1.00307 − 2.19642i 0.841254 − 0.540641i
81.5 −0.415415 − 0.909632i 1.45873 + 0.937467i −0.654861 + 0.755750i 0.142315 + 0.989821i 0.246773 − 1.71634i −1.51503 − 3.31745i 0.959493 + 0.281733i 0.00279444 + 0.00611896i 0.841254 − 0.540641i
81.6 −0.415415 − 0.909632i 1.73537 + 1.11526i −0.654861 + 0.755750i 0.142315 + 0.989821i 0.293573 − 2.04184i 1.19192 + 2.60993i 0.959493 + 0.281733i 0.521474 + 1.14187i 0.841254 − 0.540641i
91.1 −0.415415 + 0.909632i −2.76146 + 1.77468i −0.654861 − 0.755750i 0.142315 − 0.989821i −0.467156 − 3.24914i −0.988243 + 2.16395i 0.959493 − 0.281733i 3.22991 − 7.07253i 0.841254 + 0.540641i
91.2 −0.415415 + 0.909632i −1.38912 + 0.892736i −0.654861 − 0.755750i 0.142315 − 0.989821i −0.234998 − 1.63445i 0.644195 − 1.41059i 0.959493 − 0.281733i −0.113556 + 0.248652i 0.841254 + 0.540641i
91.3 −0.415415 + 0.909632i −1.36967 + 0.880231i −0.654861 − 0.755750i 0.142315 − 0.989821i −0.231706 − 1.61155i 0.357839 − 0.783557i 0.959493 − 0.281733i −0.145067 + 0.317653i 0.841254 + 0.540641i
91.4 −0.415415 + 0.909632i 0.643644 − 0.413645i −0.654861 − 0.755750i 0.142315 − 0.989821i 0.108885 + 0.757313i −0.632291 + 1.38452i 0.959493 − 0.281733i −1.00307 + 2.19642i 0.841254 + 0.540641i
91.5 −0.415415 + 0.909632i 1.45873 − 0.937467i −0.654861 − 0.755750i 0.142315 − 0.989821i 0.246773 + 1.71634i −1.51503 + 3.31745i 0.959493 − 0.281733i 0.00279444 − 0.00611896i 0.841254 + 0.540641i
91.6 −0.415415 + 0.909632i 1.73537 − 1.11526i −0.654861 − 0.755750i 0.142315 − 0.989821i 0.293573 + 2.04184i 1.19192 − 2.60993i 0.959493 − 0.281733i 0.521474 − 1.14187i 0.841254 + 0.540641i
131.1 0.959493 + 0.281733i −0.310022 + 2.15625i 0.841254 + 0.540641i −0.415415 − 0.909632i −0.904951 + 1.98157i −4.27707 − 1.25586i 0.654861 + 0.755750i −1.67483 − 0.491775i −0.142315 − 0.989821i
131.2 0.959493 + 0.281733i −0.226460 + 1.57507i 0.841254 + 0.540641i −0.415415 − 0.909632i −0.661034 + 1.44746i 4.68496 + 1.37563i 0.654861 + 0.755750i 0.448931 + 0.131818i −0.142315 − 0.989821i
131.3 0.959493 + 0.281733i −0.0953368 + 0.663082i 0.841254 + 0.540641i −0.415415 − 0.909632i −0.278287 + 0.609363i 0.588113 + 0.172686i 0.654861 + 0.755750i 2.44789 + 0.718765i −0.142315 − 0.989821i
131.4 0.959493 + 0.281733i 0.127868 − 0.889342i 0.841254 + 0.540641i −0.415415 − 0.909632i 0.373245 − 0.817293i 0.528655 + 0.155227i 0.654861 + 0.755750i 2.10390 + 0.617761i −0.142315 − 0.989821i
131.5 0.959493 + 0.281733i 0.360255 − 2.50563i 0.841254 + 0.540641i −0.415415 − 0.909632i 1.05158 − 2.30263i −2.03462 − 0.597417i 0.654861 + 0.755750i −3.26990 − 0.960129i −0.142315 − 0.989821i
131.6 0.959493 + 0.281733i 0.428326 − 2.97908i 0.841254 + 0.540641i −0.415415 − 0.909632i 1.25028 − 2.73773i 4.88455 + 1.43423i 0.654861 + 0.755750i −5.81295 − 1.70684i −0.142315 − 0.989821i
241.1 0.654861 + 0.755750i −1.32956 + 2.91133i −0.142315 + 0.989821i 0.959493 + 0.281733i −3.07092 + 0.901702i 0.260261 + 0.300358i −0.841254 + 0.540641i −4.74354 − 5.47434i 0.415415 + 0.909632i
241.2 0.654861 + 0.755750i −0.930346 + 2.03717i −0.142315 + 0.989821i 0.959493 + 0.281733i −2.14884 + 0.630956i 1.98390 + 2.28954i −0.841254 + 0.540641i −1.31995 − 1.52330i 0.415415 + 0.909632i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.k.d ✓ 60
67.e even 11 1 inner 670.2.k.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.k.d ✓ 60 1.a even 1 1 trivial
670.2.k.d ✓ 60 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - 2 T_{3}^{59} + 14 T_{3}^{58} - 26 T_{3}^{57} + 140 T_{3}^{56} - 318 T_{3}^{55} + \cdots + 19562929 \) acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\). Copy content Toggle raw display