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Properties

Label 670.2.k.c
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
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{2} - 2 q^{3} - 5 q^{4} - 5 q^{5} + 2 q^{6} - q^{7} + 5 q^{8} - q^{9} + 5 q^{10} - 12 q^{11} - 2 q^{12} + 24 q^{13} + 23 q^{14} - 2 q^{15} - 5 q^{16} + 31 q^{17} + q^{18} + 4 q^{19} - 5 q^{20} + 12 q^{21} + q^{22} + 27 q^{23} - 9 q^{24} - 5 q^{25} - 2 q^{26} - 14 q^{27} + 10 q^{28} - 36 q^{29} + 2 q^{30} - 13 q^{31} + 5 q^{32} - 42 q^{33} + 2 q^{34} + 21 q^{35} - q^{36} - 50 q^{37} - 26 q^{38} - 31 q^{39} + 5 q^{40} - 10 q^{41} + 21 q^{42} + 16 q^{43} - q^{44} - q^{45} + 17 q^{46} - 13 q^{47} - 2 q^{48} - 40 q^{49} + 5 q^{50} - 19 q^{51} - 20 q^{52} - 5 q^{53} + 47 q^{54} + 10 q^{55} + 12 q^{56} - 90 q^{57} - 8 q^{58} - 20 q^{59} - 2 q^{60} + 12 q^{61} + 13 q^{62} - 15 q^{63} - 5 q^{64} - 20 q^{65} + 20 q^{66} - 21 q^{67} - 24 q^{68} + 77 q^{69} - 21 q^{70} + 24 q^{71} + 12 q^{72} - 68 q^{73} - 16 q^{74} - 2 q^{75} + 4 q^{76} + 7 q^{77} + 53 q^{78} - 26 q^{79} - 5 q^{80} - 21 q^{81} + 21 q^{82} - 10 q^{83} + 12 q^{84} + 31 q^{85} - 27 q^{86} + 61 q^{87} + 12 q^{88} + 51 q^{89} + q^{90} - 10 q^{91} - 6 q^{92} - 38 q^{93} - 9 q^{94} + 15 q^{95} + 2 q^{96} + 72 q^{97} - 37 q^{98} - 32 q^{99}+O(q^{100}) \) 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 −2.42744 − 1.56002i −0.654861 + 0.755750i −0.142315 − 0.989821i −0.410650 + 2.85614i 1.43244 + 3.13661i 0.959493 + 0.281733i 2.21256 + 4.84483i −0.841254 + 0.540641i
81.2 −0.415415 − 0.909632i −0.902239 − 0.579833i −0.654861 + 0.755750i −0.142315 − 0.989821i −0.152632 + 1.06158i −1.45966 − 3.19620i 0.959493 + 0.281733i −0.768418 − 1.68260i −0.841254 + 0.540641i
81.3 −0.415415 − 0.909632i −0.628511 − 0.403919i −0.654861 + 0.755750i −0.142315 − 0.989821i −0.106325 + 0.739508i 1.26092 + 2.76102i 0.959493 + 0.281733i −1.01437 − 2.22116i −0.841254 + 0.540641i
81.4 −0.415415 − 0.909632i 0.490330 + 0.315116i −0.654861 + 0.755750i −0.142315 − 0.989821i 0.0829491 − 0.576924i −0.0825464 − 0.180751i 0.959493 + 0.281733i −1.10512 − 2.41987i −0.841254 + 0.540641i
81.5 −0.415415 − 0.909632i 2.36605 + 1.52057i −0.654861 + 0.755750i −0.142315 − 0.989821i 0.400265 − 2.78391i 1.07136 + 2.34595i 0.959493 + 0.281733i 2.03983 + 4.46661i −0.841254 + 0.540641i
91.1 −0.415415 + 0.909632i −2.42744 + 1.56002i −0.654861 − 0.755750i −0.142315 + 0.989821i −0.410650 − 2.85614i 1.43244 − 3.13661i 0.959493 − 0.281733i 2.21256 − 4.84483i −0.841254 − 0.540641i
91.2 −0.415415 + 0.909632i −0.902239 + 0.579833i −0.654861 − 0.755750i −0.142315 + 0.989821i −0.152632 − 1.06158i −1.45966 + 3.19620i 0.959493 − 0.281733i −0.768418 + 1.68260i −0.841254 − 0.540641i
91.3 −0.415415 + 0.909632i −0.628511 + 0.403919i −0.654861 − 0.755750i −0.142315 + 0.989821i −0.106325 − 0.739508i 1.26092 − 2.76102i 0.959493 − 0.281733i −1.01437 + 2.22116i −0.841254 − 0.540641i
91.4 −0.415415 + 0.909632i 0.490330 − 0.315116i −0.654861 − 0.755750i −0.142315 + 0.989821i 0.0829491 + 0.576924i −0.0825464 + 0.180751i 0.959493 − 0.281733i −1.10512 + 2.41987i −0.841254 − 0.540641i
91.5 −0.415415 + 0.909632i 2.36605 − 1.52057i −0.654861 − 0.755750i −0.142315 + 0.989821i 0.400265 + 2.78391i 1.07136 − 2.34595i 0.959493 − 0.281733i 2.03983 − 4.46661i −0.841254 − 0.540641i
131.1 0.959493 + 0.281733i −0.442767 + 3.07951i 0.841254 + 0.540641i 0.415415 + 0.909632i −1.29243 + 2.83003i 0.952113 + 0.279566i 0.654861 + 0.755750i −6.40889 − 1.88182i 0.142315 + 0.989821i
131.2 0.959493 + 0.281733i −0.174494 + 1.21364i 0.841254 + 0.540641i 0.415415 + 0.909632i −0.509347 + 1.11531i 3.07626 + 0.903273i 0.654861 + 0.755750i 1.43602 + 0.421653i 0.142315 + 0.989821i
131.3 0.959493 + 0.281733i −0.0950182 + 0.660866i 0.841254 + 0.540641i 0.415415 + 0.909632i −0.277357 + 0.607326i −2.14257 − 0.629115i 0.654861 + 0.755750i 2.45076 + 0.719609i 0.142315 + 0.989821i
131.4 0.959493 + 0.281733i 0.221751 − 1.54231i 0.841254 + 0.540641i 0.415415 + 0.909632i 0.647288 − 1.41736i 3.39232 + 0.996075i 0.654861 + 0.755750i 0.548925 + 0.161179i 0.142315 + 0.989821i
131.5 0.959493 + 0.281733i 0.251083 − 1.74632i 0.841254 + 0.540641i 0.415415 + 0.909632i 0.732908 − 1.60485i −0.408871 − 0.120055i 0.654861 + 0.755750i −0.108119 − 0.0317466i 0.142315 + 0.989821i
241.1 0.654861 + 0.755750i −1.07719 + 2.35871i −0.142315 + 0.989821i −0.959493 − 0.281733i −2.48800 + 0.730544i −3.32684 − 3.83938i −0.841254 + 0.540641i −2.43861 − 2.81430i −0.415415 − 0.909632i
241.2 0.654861 + 0.755750i −0.821254 + 1.79830i −0.142315 + 0.989821i −0.959493 − 0.281733i −1.89687 + 0.556971i 0.137050 + 0.158164i −0.841254 + 0.540641i −0.594827 − 0.686467i −0.415415 − 0.909632i
241.3 0.654861 + 0.755750i −0.00970382 + 0.0212484i −0.142315 + 0.989821i −0.959493 − 0.281733i −0.0224131 + 0.00658109i −1.24834 − 1.44066i −0.841254 + 0.540641i 1.96422 + 2.26684i −0.415415 − 0.909632i
241.4 0.654861 + 0.755750i 0.876999 − 1.92036i −0.142315 + 0.989821i −0.959493 − 0.281733i 2.02562 − 0.594777i −2.33007 − 2.68905i −0.841254 + 0.540641i −0.954073 − 1.10106i −0.415415 − 0.909632i
241.5 0.654861 + 0.755750i 0.912907 − 1.99899i −0.142315 + 0.989821i −0.959493 − 0.281733i 2.10856 − 0.619129i 2.29392 + 2.64733i −0.841254 + 0.540641i −1.19797 − 1.38253i −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.c ✓ 50
67.e even 11 1 inner 670.2.k.c ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.k.c ✓ 50 1.a even 1 1 trivial
670.2.k.c ✓ 50 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{50} + 2 T_{3}^{49} + 10 T_{3}^{48} + 26 T_{3}^{47} + 96 T_{3}^{46} + 346 T_{3}^{45} + \cdots + 64009 \) acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\). Copy content Toggle raw display