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Properties

Label 670.2.k.b
Level $670$
Weight $2$
Character orbit 670.k
Analytic conductor $5.350$
Analytic rank $0$
Dimension $50$
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: \(50\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 5 q^{2} + 2 q^{3} - 5 q^{4} + 5 q^{5} + 2 q^{6} + q^{7} - 5 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 5 q^{2} + 2 q^{3} - 5 q^{4} + 5 q^{5} + 2 q^{6} + q^{7} - 5 q^{8} - q^{9} + 5 q^{10} + 30 q^{11} - 9 q^{12} + 20 q^{13} - 21 q^{14} - 2 q^{15} - 5 q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + 5 q^{20} + 23 q^{21} - 3 q^{22} - 5 q^{23} + 2 q^{24} - 5 q^{25} - 2 q^{26} + 2 q^{27} - 10 q^{28} + 26 q^{29} - 2 q^{30} + 3 q^{31} - 5 q^{32} + 38 q^{33} + 2 q^{34} - 12 q^{35} - q^{36} + 22 q^{37} - 40 q^{38} - 35 q^{39} + 5 q^{40} - 18 q^{41} - 10 q^{42} + 87 q^{43} - 3 q^{44} + 12 q^{45} + 17 q^{46} + 35 q^{47} + 2 q^{48} + 60 q^{49} - 5 q^{50} + 19 q^{51} - 24 q^{52} - 9 q^{53} + 2 q^{54} + 25 q^{55} + 12 q^{56} + 64 q^{57} + 4 q^{58} - 49 q^{59} - 2 q^{60} - 47 q^{61} - 19 q^{62} - 24 q^{63} - 5 q^{64} + 24 q^{65} - 6 q^{66} - 21 q^{67} + 2 q^{68} - 25 q^{69} - 12 q^{70} + 92 q^{71} - q^{72} + 18 q^{73} - 22 q^{74} - 9 q^{75} + 4 q^{76} - 65 q^{77} + 31 q^{78} - 58 q^{79} + 5 q^{80} - 77 q^{81} - 18 q^{82} - 49 q^{83} - 54 q^{84} - 2 q^{85} - q^{86} - 147 q^{87} + 30 q^{88} - 21 q^{89} + 12 q^{90} + 62 q^{91} + 6 q^{92} - 34 q^{93} - 9 q^{94} + 7 q^{95} + 2 q^{96} + 98 q^{97} - 39 q^{98} - 4 q^{99}+O(q^{100}) \) 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 −1.71676 − 1.10329i −0.654861 + 0.755750i 0.142315 + 0.989821i 0.290424 − 2.01994i −0.0216539 − 0.0474154i −0.959493 − 0.281733i 0.483759 + 1.05928i −0.841254 + 0.540641i
81.2 0.415415 + 0.909632i −0.351502 − 0.225896i −0.654861 + 0.755750i 0.142315 + 0.989821i 0.0594635 − 0.413578i −0.689278 − 1.50931i −0.959493 − 0.281733i −1.17372 − 2.57009i −0.841254 + 0.540641i
81.3 0.415415 + 0.909632i −0.326609 − 0.209899i −0.654861 + 0.755750i 0.142315 + 0.989821i 0.0552524 − 0.384289i 1.81684 + 3.97833i −0.959493 − 0.281733i −1.18363 − 2.59179i −0.841254 + 0.540641i
81.4 0.415415 + 0.909632i 1.59621 + 1.02582i −0.654861 + 0.755750i 0.142315 + 0.989821i −0.270030 + 1.87810i 0.123242 + 0.269862i −0.959493 − 0.281733i 0.249326 + 0.545947i −0.841254 + 0.540641i
81.5 0.415415 + 0.909632i 2.41302 + 1.55075i −0.654861 + 0.755750i 0.142315 + 0.989821i −0.408210 + 2.83916i 1.65229 + 3.61802i −0.959493 − 0.281733i 2.17157 + 4.75508i −0.841254 + 0.540641i
91.1 0.415415 − 0.909632i −1.71676 + 1.10329i −0.654861 − 0.755750i 0.142315 − 0.989821i 0.290424 + 2.01994i −0.0216539 + 0.0474154i −0.959493 + 0.281733i 0.483759 − 1.05928i −0.841254 − 0.540641i
91.2 0.415415 − 0.909632i −0.351502 + 0.225896i −0.654861 − 0.755750i 0.142315 − 0.989821i 0.0594635 + 0.413578i −0.689278 + 1.50931i −0.959493 + 0.281733i −1.17372 + 2.57009i −0.841254 − 0.540641i
91.3 0.415415 − 0.909632i −0.326609 + 0.209899i −0.654861 − 0.755750i 0.142315 − 0.989821i 0.0552524 + 0.384289i 1.81684 − 3.97833i −0.959493 + 0.281733i −1.18363 + 2.59179i −0.841254 − 0.540641i
91.4 0.415415 − 0.909632i 1.59621 − 1.02582i −0.654861 − 0.755750i 0.142315 − 0.989821i −0.270030 − 1.87810i 0.123242 − 0.269862i −0.959493 + 0.281733i 0.249326 − 0.545947i −0.841254 − 0.540641i
91.5 0.415415 − 0.909632i 2.41302 − 1.55075i −0.654861 − 0.755750i 0.142315 − 0.989821i −0.408210 − 2.83916i 1.65229 − 3.61802i −0.959493 + 0.281733i 2.17157 − 4.75508i −0.841254 − 0.540641i
131.1 −0.959493 − 0.281733i −0.325974 + 2.26720i 0.841254 + 0.540641i −0.415415 − 0.909632i 0.951514 − 2.08352i 3.43296 + 1.00801i −0.654861 − 0.755750i −2.15545 − 0.632899i 0.142315 + 0.989821i
131.2 −0.959493 − 0.281733i −0.308976 + 2.14897i 0.841254 + 0.540641i −0.415415 − 0.909632i 0.901896 − 1.97488i −0.0356954 − 0.0104811i −0.654861 − 0.755750i −1.64415 − 0.482766i 0.142315 + 0.989821i
131.3 −0.959493 − 0.281733i −0.0437852 + 0.304533i 0.841254 + 0.540641i −0.415415 − 0.909632i 0.127808 − 0.279862i −4.20360 − 1.23429i −0.654861 − 0.755750i 2.78766 + 0.818530i 0.142315 + 0.989821i
131.4 −0.959493 − 0.281733i 0.0953649 − 0.663278i 0.841254 + 0.540641i −0.415415 − 0.909632i −0.278369 + 0.609543i 4.29211 + 1.26028i −0.654861 − 0.755750i 2.44764 + 0.718691i 0.142315 + 0.989821i
131.5 −0.959493 − 0.281733i 0.396978 − 2.76104i 0.841254 + 0.540641i −0.415415 − 0.909632i −1.15877 + 2.53736i −2.99731 − 0.880089i −0.654861 − 0.755750i −4.58727 − 1.34694i 0.142315 + 0.989821i
241.1 −0.654861 − 0.755750i −1.18032 + 2.58454i −0.142315 + 0.989821i 0.959493 + 0.281733i 2.72621 − 0.800487i −0.650120 − 0.750279i 0.841254 − 0.540641i −3.32211 − 3.83392i −0.415415 − 0.909632i
241.2 −0.654861 − 0.755750i −0.436391 + 0.955564i −0.142315 + 0.989821i 0.959493 + 0.281733i 1.00794 − 0.295959i 2.61745 + 3.02069i 0.841254 − 0.540641i 1.24192 + 1.43325i −0.415415 − 0.909632i
241.3 −0.654861 − 0.755750i −0.376919 + 0.825337i −0.142315 + 0.989821i 0.959493 + 0.281733i 0.870577 − 0.255624i −1.54779 − 1.78624i 0.841254 − 0.540641i 1.42547 + 1.64508i −0.415415 − 0.909632i
241.4 −0.654861 − 0.755750i 0.200632 − 0.439323i −0.142315 + 0.989821i 0.959493 + 0.281733i −0.463404 + 0.136068i −1.81243 − 2.09166i 0.841254 − 0.540641i 1.81183 + 2.09096i −0.415415 − 0.909632i
241.5 −0.654861 − 0.755750i 1.09406 − 2.39565i −0.142315 + 0.989821i 0.959493 + 0.281733i −2.52697 + 0.741986i 2.17965 + 2.51545i 0.841254 − 0.540641i −2.57762 − 2.97473i −0.415415 − 0.909632i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.5
Significant digits:
Format:

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.b ✓ 50
67.e even 11 1 inner 670.2.k.b ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.k.b ✓ 50 1.a even 1 1 trivial
670.2.k.b ✓ 50 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} - 2 T_{3}^{49} + 10 T_{3}^{48} - 22 T_{3}^{47} + 102 T_{3}^{46} - 140 T_{3}^{45} + \cdots + 64009 \) acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\). Copy content Toggle raw display