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Properties

Label 840.2.cl.b
Level $840$
Weight $2$
Character orbit 840.cl
Analytic conductor $6.707$
Analytic rank $0$
Dimension $64$
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] = [840,2,Mod(451,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.cl (of order \(6\), 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{2} + 14 q^{4} - 32 q^{5} - 4 q^{8} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{2} + 14 q^{4} - 32 q^{5} - 4 q^{8} + 32 q^{9} + 2 q^{10} - 8 q^{11} + 6 q^{14} - 2 q^{16} - 2 q^{18} - 4 q^{20} + 20 q^{22} + 12 q^{24} - 32 q^{25} + 34 q^{26} - 10 q^{28} + 16 q^{31} - 18 q^{32} - 8 q^{34} + 4 q^{36} + 24 q^{37} - 2 q^{38} + 2 q^{40} - 4 q^{42} + 16 q^{43} - 26 q^{44} + 32 q^{45} + 38 q^{46} + 32 q^{48} - 8 q^{49} - 4 q^{50} + 2 q^{52} + 16 q^{55} - 36 q^{56} - 16 q^{57} + 28 q^{58} + 48 q^{59} - 12 q^{62} - 28 q^{64} + 4 q^{66} + 8 q^{67} - 60 q^{68} - 18 q^{70} - 2 q^{72} + 24 q^{73} - 42 q^{74} + 32 q^{76} - 8 q^{77} - 12 q^{78} - 2 q^{80} - 32 q^{81} + 46 q^{82} + 20 q^{84} - 84 q^{86} - 98 q^{88} + 4 q^{90} - 40 q^{91} + 92 q^{92} - 82 q^{94} - 74 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1 −1.40394 + 0.170191i −0.866025 − 0.500000i 1.94207 − 0.477874i −0.500000 − 0.866025i 1.30094 + 0.554578i 2.34568 + 1.22384i −2.64521 + 1.00143i 0.500000 + 0.866025i 0.849357 + 1.13075i
451.2 −1.37643 − 0.324719i 0.866025 + 0.500000i 1.78912 + 0.893905i −0.500000 − 0.866025i −1.02966 − 0.969429i 1.84817 − 1.89321i −2.17232 − 1.81136i 0.500000 + 0.866025i 0.407000 + 1.35438i
451.3 −1.36858 − 0.356342i −0.866025 − 0.500000i 1.74604 + 0.975368i −0.500000 − 0.866025i 1.00706 + 0.992893i 0.527768 − 2.59258i −2.04204 − 1.95706i 0.500000 + 0.866025i 0.375690 + 1.36340i
451.4 −1.36856 + 0.356440i 0.866025 + 0.500000i 1.74590 − 0.975618i −0.500000 − 0.866025i −1.36343 − 0.375592i 2.31357 + 1.28351i −2.04162 + 1.95750i 0.500000 + 0.866025i 0.992965 + 1.00699i
451.5 −1.27906 + 0.603333i −0.866025 − 0.500000i 1.27198 − 1.54340i −0.500000 − 0.866025i 1.40936 + 0.117027i −2.61080 + 0.428616i −0.695750 + 2.74152i 0.500000 + 0.866025i 1.16203 + 0.806030i
451.6 −1.27179 − 0.618501i 0.866025 + 0.500000i 1.23491 + 1.57321i −0.500000 − 0.866025i −0.792154 − 1.17153i −1.40315 + 2.24302i −0.597521 − 2.76459i 0.500000 + 0.866025i 0.100259 + 1.41066i
451.7 −1.25755 − 0.646975i −0.866025 − 0.500000i 1.16285 + 1.62720i −0.500000 − 0.866025i 0.765580 + 1.18907i −2.50998 + 0.836664i −0.409577 − 2.79862i 0.500000 + 0.866025i 0.0684768 + 1.41255i
451.8 −1.25023 + 0.660998i 0.866025 + 0.500000i 1.12616 − 1.65280i −0.500000 − 0.866025i −1.41323 − 0.0526748i −0.600185 − 2.57678i −0.315464 + 2.81078i 0.500000 + 0.866025i 1.19756 + 0.752234i
451.9 −1.02347 − 0.975962i −0.866025 − 0.500000i 0.0949970 + 1.99774i −0.500000 − 0.866025i 0.398373 + 1.35694i 2.26482 + 1.36769i 1.85249 − 2.13735i 0.500000 + 0.866025i −0.333471 + 1.37434i
451.10 −0.870485 + 1.11457i 0.866025 + 0.500000i −0.484512 − 1.94042i −0.500000 − 0.866025i −1.31114 + 0.530000i −2.64173 + 0.145753i 2.58449 + 1.14909i 0.500000 + 0.866025i 1.40048 + 0.196579i
451.11 −0.826059 + 1.14788i −0.866025 − 0.500000i −0.635253 − 1.89643i −0.500000 − 0.866025i 1.28933 − 0.581063i 2.28120 − 1.34019i 2.70163 + 0.837371i 0.500000 + 0.866025i 1.40712 + 0.141448i
451.12 −0.531729 − 1.31044i 0.866025 + 0.500000i −1.43453 + 1.39360i −0.500000 − 0.866025i 0.194731 − 1.40074i −2.63542 + 0.233548i 2.58902 + 1.13885i 0.500000 + 0.866025i −0.869013 + 1.11571i
451.13 −0.406054 + 1.35467i 0.866025 + 0.500000i −1.67024 − 1.10014i −0.500000 − 0.866025i −1.02899 + 0.970148i 0.685304 + 2.55546i 2.16852 − 1.81590i 0.500000 + 0.866025i 1.37620 − 0.325680i
451.14 −0.394283 − 1.35814i 0.866025 + 0.500000i −1.68908 + 1.07098i −0.500000 − 0.866025i 0.337610 − 1.37332i 1.68178 + 2.04245i 2.12052 + 1.87174i 0.500000 + 0.866025i −0.979041 + 1.02053i
451.15 −0.320728 + 1.37736i −0.866025 − 0.500000i −1.79427 − 0.883518i −0.500000 − 0.866025i 0.966441 − 1.03247i −2.01115 − 1.71910i 1.79240 − 2.18799i 0.500000 + 0.866025i 1.35320 − 0.410924i
451.16 −0.0878302 − 1.41148i −0.866025 − 0.500000i −1.98457 + 0.247942i −0.500000 − 0.866025i −0.629679 + 1.26630i 2.57235 + 0.618893i 0.524271 + 2.77941i 0.500000 + 0.866025i −1.17847 + 0.781805i
451.17 0.296636 − 1.38275i 0.866025 + 0.500000i −1.82401 − 0.820348i −0.500000 − 0.866025i 0.948271 − 1.04918i −1.36890 − 2.26409i −1.67541 + 2.27882i 0.500000 + 0.866025i −1.34582 + 0.434483i
451.18 0.314849 + 1.37872i 0.866025 + 0.500000i −1.80174 + 0.868177i −0.500000 − 0.866025i −0.416693 + 1.35143i −0.345366 − 2.62311i −1.76425 − 2.21075i 0.500000 + 0.866025i 1.03658 − 0.962027i
451.19 0.474798 + 1.33213i −0.866025 − 0.500000i −1.54913 + 1.26498i −0.500000 − 0.866025i 0.254878 − 1.39106i −1.87381 + 1.86785i −2.42065 − 1.46304i 0.500000 + 0.866025i 0.916259 − 1.07725i
451.20 0.750940 − 1.19837i −0.866025 − 0.500000i −0.872179 − 1.79981i −0.500000 − 0.866025i −1.24952 + 0.662349i −0.662766 + 2.56139i −2.81179 − 0.306353i 0.500000 + 0.866025i −1.41329 − 0.0511481i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.cl.b yes 64
7.d odd 6 1 840.2.cl.a ✓ 64
8.d odd 2 1 840.2.cl.a ✓ 64
56.m even 6 1 inner 840.2.cl.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.cl.a ✓ 64 7.d odd 6 1
840.2.cl.a ✓ 64 8.d odd 2 1
840.2.cl.b yes 64 1.a even 1 1 trivial
840.2.cl.b yes 64 56.m even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{32} - 232 T_{13}^{30} - 16 T_{13}^{29} + 23656 T_{13}^{28} + 2560 T_{13}^{27} + \cdots - 13496322239 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display