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Properties

Label 840.2.cf.a
Level $840$
Weight $2$
Character orbit 840.cf
Analytic conductor $6.707$
Analytic rank $0$
Dimension $256$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(101,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.101"); 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.cf (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(128\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 30 q^{12} + 16 q^{16} - 10 q^{18} + 8 q^{22} + 128 q^{25} + 12 q^{28} - 40 q^{36} - 54 q^{42} + 16 q^{46} + 16 q^{49} + 36 q^{52} - 72 q^{54} - 16 q^{58} - 14 q^{60} + 80 q^{63} - 48 q^{64} - 60 q^{66}+ \cdots - 144 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.41345 − 0.0464010i −0.314700 + 1.70322i 1.99569 + 0.131171i 0.866025 − 0.500000i 0.523844 − 2.39282i 1.40407 − 2.24245i −2.81473 − 0.278006i −2.80193 − 1.07201i −1.24729 + 0.666542i
101.2 −1.40863 + 0.125583i 1.67472 + 0.441957i 1.96846 − 0.353800i −0.866025 + 0.500000i −2.41455 − 0.412236i −1.92276 + 1.81742i −2.72839 + 0.745579i 2.60935 + 1.48031i 1.15711 − 0.813072i
101.3 −1.40637 − 0.148734i 1.30218 − 1.14208i 1.95576 + 0.418349i 0.866025 − 0.500000i −2.00121 + 1.41250i −1.45671 − 2.20862i −2.68830 − 0.879241i 0.391324 − 2.97437i −1.29232 + 0.574378i
101.4 −1.40529 − 0.158613i 0.0400234 − 1.73159i 1.94968 + 0.445794i 0.866025 − 0.500000i −0.330897 + 2.42704i −0.593444 + 2.57834i −2.66916 − 0.935716i −2.99680 − 0.138608i −1.29632 + 0.565283i
101.5 −1.40319 + 0.176221i 0.803145 − 1.53459i 1.93789 − 0.494544i −0.866025 + 0.500000i −0.856539 + 2.29485i 1.62875 + 2.08499i −2.63208 + 1.03544i −1.70992 − 2.46499i 1.12709 − 0.854208i
101.6 −1.39955 + 0.203104i −1.71710 + 0.227111i 1.91750 − 0.568510i 0.866025 − 0.500000i 2.35704 − 0.666603i 2.64224 − 0.136357i −2.56817 + 1.18511i 2.89684 − 0.779944i −1.11050 + 0.875670i
101.7 −1.39458 − 0.234852i 0.707405 + 1.58101i 1.88969 + 0.655039i −0.866025 + 0.500000i −0.615228 − 2.37097i 2.21860 + 1.44146i −2.48148 − 1.35730i −1.99916 + 2.23682i 1.32517 − 0.493900i
101.8 −1.38593 + 0.281424i −1.72907 − 0.101633i 1.84160 − 0.780068i −0.866025 + 0.500000i 2.42497 − 0.345745i −0.542825 − 2.58947i −2.33280 + 1.59939i 2.97934 + 0.351460i 1.05954 − 0.936685i
101.9 −1.38253 + 0.297655i −0.830199 + 1.52012i 1.82280 − 0.823038i −0.866025 + 0.500000i 0.695306 − 2.34873i −1.41654 + 2.23460i −2.27511 + 1.68044i −1.62154 − 2.52401i 1.04848 − 0.949044i
101.10 −1.37981 − 0.310051i 1.73203 − 0.00870717i 1.80774 + 0.855622i 0.866025 − 0.500000i −2.39257 − 0.525004i 0.204495 + 2.63784i −2.22904 − 1.74109i 2.99985 − 0.0301621i −1.34997 + 0.421391i
101.11 −1.37748 − 0.320241i −1.53075 − 0.810433i 1.79489 + 0.882249i −0.866025 + 0.500000i 1.84904 + 1.60656i −2.04110 + 1.68343i −2.18989 − 1.79008i 1.68640 + 2.48114i 1.35305 − 0.411403i
101.12 −1.36499 − 0.369851i −1.18582 − 1.26247i 1.72642 + 1.00969i 0.866025 − 0.500000i 1.15172 + 2.16184i 2.63832 − 0.198131i −1.98312 − 2.01674i −0.187642 + 2.99413i −1.36705 + 0.362197i
101.13 −1.35626 − 0.400715i −1.62841 + 0.590144i 1.67886 + 1.08694i 0.866025 − 0.500000i 2.44502 − 0.147856i −2.64480 + 0.0708975i −1.84140 − 2.14691i 2.30346 − 1.92200i −1.37491 + 0.331098i
101.14 −1.35436 + 0.407060i 0.401248 + 1.68493i 1.66860 − 1.10262i 0.866025 − 0.500000i −1.22931 − 2.11868i −2.61061 + 0.429788i −1.81107 + 2.17257i −2.67800 + 1.35215i −0.969383 + 1.02971i
101.15 −1.32538 − 0.493337i 1.47720 − 0.904369i 1.51324 + 1.30771i −0.866025 + 0.500000i −2.40400 + 0.469871i 2.22730 − 1.42798i −1.36047 − 2.47974i 1.36423 − 2.67187i 1.39448 − 0.235446i
101.16 −1.32124 + 0.504319i 1.35380 − 1.08038i 1.49132 − 1.33265i −0.866025 + 0.500000i −1.24383 + 2.11019i −2.34427 − 1.22654i −1.29831 + 2.51285i 0.665543 − 2.92524i 0.892064 − 1.09737i
101.17 −1.30091 − 0.554645i −1.41157 + 1.00373i 1.38474 + 1.44309i −0.866025 + 0.500000i 2.39304 − 0.522843i 2.47291 + 0.940605i −1.00102 − 2.64537i 0.985054 − 2.83367i 1.40394 − 0.170119i
101.18 −1.29619 + 0.565579i 1.27809 + 1.16896i 1.36024 − 1.46620i −0.866025 + 0.500000i −2.31780 − 0.792343i −0.0356994 − 2.64551i −0.933887 + 2.66980i 0.267045 + 2.98809i 0.839748 − 1.13790i
101.19 −1.27612 + 0.609528i 1.29538 + 1.14978i 1.25695 − 1.55566i 0.866025 − 0.500000i −2.35388 − 0.677684i 2.04884 + 1.67400i −0.655798 + 2.75135i 0.356013 + 2.97880i −0.800386 + 1.16593i
101.20 −1.27220 − 0.617664i 0.00506025 − 1.73204i 1.23698 + 1.57158i −0.866025 + 0.500000i −1.07626 + 2.20038i −1.36209 − 2.26820i −0.602979 − 2.76341i −2.99995 − 0.0175291i 1.41059 − 0.101187i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.128
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba 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.cf.a ✓ 256
3.b odd 2 1 inner 840.2.cf.a ✓ 256
7.d odd 6 1 inner 840.2.cf.a ✓ 256
8.b even 2 1 inner 840.2.cf.a ✓ 256
21.g even 6 1 inner 840.2.cf.a ✓ 256
24.h odd 2 1 inner 840.2.cf.a ✓ 256
56.j odd 6 1 inner 840.2.cf.a ✓ 256
168.ba even 6 1 inner 840.2.cf.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.cf.a ✓ 256 1.a even 1 1 trivial
840.2.cf.a ✓ 256 3.b odd 2 1 inner
840.2.cf.a ✓ 256 7.d odd 6 1 inner
840.2.cf.a ✓ 256 8.b even 2 1 inner
840.2.cf.a ✓ 256 21.g even 6 1 inner
840.2.cf.a ✓ 256 24.h odd 2 1 inner
840.2.cf.a ✓ 256 56.j odd 6 1 inner
840.2.cf.a ✓ 256 168.ba even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(840, [\chi])\).