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Properties

Label 670.2.k.a
Level $670$
Weight $2$
Character orbit 670.k
Analytic conductor $5.350$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{2} + 6 q^{3} - 4 q^{4} - 4 q^{5} + 6 q^{6} + 13 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{2} + 6 q^{3} - 4 q^{4} - 4 q^{5} + 6 q^{6} + 13 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} - 14 q^{11} - 5 q^{12} - 6 q^{13} - 9 q^{14} + 6 q^{15} - 4 q^{16} - 24 q^{17} + 4 q^{18} + 24 q^{19} - 4 q^{20} - 10 q^{21} + 19 q^{22} - 15 q^{23} - 5 q^{24} - 4 q^{25} + 16 q^{26} + 54 q^{27} + 2 q^{28} - 40 q^{29} - 16 q^{30} + 15 q^{31} - 4 q^{32} - 40 q^{33} - 2 q^{34} - 9 q^{35} + 4 q^{36} - 6 q^{37} - 20 q^{38} + 21 q^{39} - 4 q^{40} + 8 q^{41} + 23 q^{42} - 54 q^{43} + 19 q^{44} - 7 q^{45} + 7 q^{46} - 23 q^{47} + 6 q^{48} - 7 q^{49} - 4 q^{50} + 61 q^{51} - 6 q^{52} + 17 q^{53} - 12 q^{54} - 3 q^{55} + 2 q^{56} - 18 q^{57} - 18 q^{58} - q^{59} + 6 q^{60} - 6 q^{61} - 7 q^{62} - 43 q^{63} - 4 q^{64} - 6 q^{65} - 18 q^{66} + 44 q^{67} - 2 q^{68} - 45 q^{69} - 9 q^{70} - 14 q^{71} - 7 q^{72} - 26 q^{73} + 38 q^{74} - 5 q^{75} + 24 q^{76} + 21 q^{77} - 45 q^{78} + 2 q^{79} - 4 q^{80} + 4 q^{81} + 8 q^{82} - 11 q^{83} - 10 q^{84} - 24 q^{85} + 45 q^{86} - 7 q^{87} - 14 q^{88} + 61 q^{89} - 7 q^{90} + 70 q^{91} + 18 q^{92} + 10 q^{93} + 43 q^{94} + 13 q^{95} - 16 q^{96} - 18 q^{97} + 26 q^{98} - 48 q^{99}+O(q^{100}) \) 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 −1.66896 − 1.07257i −0.654861 + 0.755750i −0.142315 − 0.989821i 0.282338 − 1.96370i 0.856255 + 1.87494i −0.959493 − 0.281733i 0.388762 + 0.851270i 0.841254 − 0.540641i
81.2 0.415415 + 0.909632i 0.122066 + 0.0784473i −0.654861 + 0.755750i −0.142315 − 0.989821i −0.0206500 + 0.143624i 0.187686 + 0.410976i −0.959493 − 0.281733i −1.23750 − 2.70974i 0.841254 − 0.540641i
81.3 0.415415 + 0.909632i 0.461640 + 0.296678i −0.654861 + 0.755750i −0.142315 − 0.989821i −0.0780956 + 0.543167i −1.43554 − 3.14338i −0.959493 − 0.281733i −1.12115 − 2.45498i 0.841254 − 0.540641i
81.4 0.415415 + 0.909632i 2.11891 + 1.36174i −0.654861 + 0.755750i −0.142315 − 0.989821i −0.358455 + 2.49311i 0.581873 + 1.27412i −0.959493 − 0.281733i 1.38919 + 3.04190i 0.841254 − 0.540641i
91.1 0.415415 − 0.909632i −1.66896 + 1.07257i −0.654861 − 0.755750i −0.142315 + 0.989821i 0.282338 + 1.96370i 0.856255 − 1.87494i −0.959493 + 0.281733i 0.388762 − 0.851270i 0.841254 + 0.540641i
91.2 0.415415 − 0.909632i 0.122066 − 0.0784473i −0.654861 − 0.755750i −0.142315 + 0.989821i −0.0206500 − 0.143624i 0.187686 − 0.410976i −0.959493 + 0.281733i −1.23750 + 2.70974i 0.841254 + 0.540641i
91.3 0.415415 − 0.909632i 0.461640 − 0.296678i −0.654861 − 0.755750i −0.142315 + 0.989821i −0.0780956 − 0.543167i −1.43554 + 3.14338i −0.959493 + 0.281733i −1.12115 + 2.45498i 0.841254 + 0.540641i
91.4 0.415415 − 0.909632i 2.11891 − 1.36174i −0.654861 − 0.755750i −0.142315 + 0.989821i −0.358455 − 2.49311i 0.581873 − 1.27412i −0.959493 + 0.281733i 1.38919 − 3.04190i 0.841254 + 0.540641i
131.1 −0.959493 − 0.281733i −0.239559 + 1.66617i 0.841254 + 0.540641i 0.415415 + 0.909632i 0.699269 − 1.53119i 1.11485 + 0.327349i −0.654861 − 0.755750i 0.159748 + 0.0469063i −0.142315 − 0.989821i
131.2 −0.959493 − 0.281733i −0.00917542 + 0.0638164i 0.841254 + 0.540641i 0.415415 + 0.909632i 0.0267829 − 0.0586464i 3.50046 + 1.02783i −0.654861 − 0.755750i 2.87449 + 0.844027i −0.142315 − 0.989821i
131.3 −0.959493 − 0.281733i 0.257399 − 1.79025i 0.841254 + 0.540641i 0.415415 + 0.909632i −0.751345 + 1.64522i −2.18307 − 0.641008i −0.654861 − 0.755750i −0.260268 − 0.0764216i −0.142315 − 0.989821i
131.4 −0.959493 − 0.281733i 0.329018 − 2.28837i 0.841254 + 0.540641i 0.415415 + 0.909632i −0.960398 + 2.10298i 0.750268 + 0.220299i −0.654861 − 0.755750i −2.24990 − 0.660629i −0.142315 − 0.989821i
241.1 −0.654861 − 0.755750i −1.25726 + 2.75302i −0.142315 + 0.989821i −0.959493 − 0.281733i 2.90393 − 0.852670i −0.0691570 − 0.0798114i 0.841254 − 0.540641i −4.03383 − 4.65529i 0.415415 + 0.909632i
241.2 −0.654861 − 0.755750i −0.711574 + 1.55813i −0.142315 + 0.989821i −0.959493 − 0.281733i 1.64354 − 0.482586i 2.53447 + 2.92494i 0.841254 − 0.540641i 0.0431516 + 0.0497996i 0.415415 + 0.909632i
241.3 −0.654861 − 0.755750i 0.188442 − 0.412631i −0.142315 + 0.989821i −0.959493 − 0.281733i −0.435249 + 0.127801i −1.95329 − 2.25422i 0.841254 − 0.540641i 1.82983 + 2.11173i 0.415415 + 0.909632i
241.4 −0.654861 − 0.755750i 0.368865 − 0.807701i −0.142315 + 0.989821i −0.959493 − 0.281733i −0.851975 + 0.250162i 0.703347 + 0.811706i 0.841254 − 0.540641i 1.44826 + 1.67138i 0.415415 + 0.909632i
411.1 0.841254 + 0.540641i −1.57680 − 0.462990i 0.415415 + 0.909632i −0.654861 + 0.755750i −1.07618 − 1.24197i −0.776855 − 0.499255i −0.142315 + 0.989821i −0.251829 − 0.161841i −0.959493 + 0.281733i
411.2 0.841254 + 0.540641i −0.498110 − 0.146258i 0.415415 + 0.909632i −0.654861 + 0.755750i −0.339963 − 0.392339i 0.533833 + 0.343074i −0.142315 + 0.989821i −2.29704 − 1.47622i −0.959493 + 0.281733i
411.3 0.841254 + 0.540641i 1.52430 + 0.447574i 0.415415 + 0.909632i −0.654861 + 0.755750i 1.04034 + 1.20062i 3.31920 + 2.13312i −0.142315 + 0.989821i −0.400602 − 0.257451i −0.959493 + 0.281733i
411.4 0.841254 + 0.540641i 2.99367 + 0.879021i 0.415415 + 0.909632i −0.654861 + 0.755750i 2.04320 + 2.35798i −0.745344 − 0.479004i −0.142315 + 0.989821i 5.66563 + 3.64108i −0.959493 + 0.281733i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
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.a ✓ 40
67.e even 11 1 inner 670.2.k.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.k.a ✓ 40 1.a even 1 1 trivial
670.2.k.a ✓ 40 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 6 T_{3}^{39} + 22 T_{3}^{38} - 90 T_{3}^{37} + 322 T_{3}^{36} - 888 T_{3}^{35} + 2334 T_{3}^{34} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\). Copy content Toggle raw display