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Properties

Label 840.2.bz.a
Level $840$
Weight $2$
Character orbit 840.bz
Analytic conductor $6.707$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(19,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bz (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 48 q^{3} - 48 q^{9} + 13 q^{10} + 14 q^{14} + 4 q^{16} - 22 q^{20} + 96 q^{27} + 4 q^{28} - 5 q^{30} + 30 q^{32} - 8 q^{35} + 12 q^{38} - 23 q^{40} + 2 q^{42} - 16 q^{44} - 22 q^{46} - 8 q^{48} + 12 q^{50}+ \cdots + 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41393 − 0.0284169i −0.500000 − 0.866025i 1.99838 + 0.0803589i 2.20852 + 0.349926i 0.682354 + 1.23871i −2.63671 + 0.218493i −2.82329 − 0.170410i −0.500000 + 0.866025i −3.11274 − 0.557529i
19.2 −1.41004 − 0.108624i −0.500000 − 0.866025i 1.97640 + 0.306329i 0.386692 − 2.20238i 0.610946 + 1.27544i 0.514926 + 2.59516i −2.75352 − 0.646620i −0.500000 + 0.866025i −0.784481 + 3.06343i
19.3 −1.40279 − 0.179360i −0.500000 − 0.866025i 1.93566 + 0.503209i −2.21180 − 0.328536i 0.546067 + 1.30453i −0.383657 − 2.61779i −2.62508 − 1.05308i −0.500000 + 0.866025i 3.04377 + 0.857577i
19.4 −1.38434 + 0.289139i −0.500000 − 0.866025i 1.83280 − 0.800535i −2.21694 − 0.291884i 0.942572 + 1.05430i 2.63177 − 0.271606i −2.30575 + 1.63815i −0.500000 + 0.866025i 3.15339 − 0.236937i
19.5 −1.30868 − 0.536059i −0.500000 − 0.866025i 1.42528 + 1.40306i 0.561119 + 2.16452i 0.190099 + 1.40138i 2.02879 + 1.69824i −1.11311 − 2.60019i −0.500000 + 0.866025i 0.425985 − 3.13345i
19.6 −1.26622 + 0.629839i −0.500000 − 0.866025i 1.20660 − 1.59503i 0.606984 + 2.15211i 1.17856 + 0.781655i 0.404592 − 2.61463i −0.523212 + 2.77961i −0.500000 + 0.866025i −2.12406 − 2.34273i
19.7 −1.24612 + 0.668719i −0.500000 − 0.866025i 1.10563 − 1.66661i 0.412046 − 2.19778i 1.20219 + 0.744812i −1.77141 − 1.96523i −0.263253 + 2.81615i −0.500000 + 0.866025i 0.956236 + 3.01423i
19.8 −1.22956 − 0.698700i −0.500000 − 0.866025i 1.02364 + 1.71819i −1.94999 + 1.09432i 0.00968783 + 1.41418i −1.44154 + 2.21855i −0.0581233 − 2.82783i −0.500000 + 0.866025i 3.16223 + 0.0169322i
19.9 −1.21987 − 0.715480i −0.500000 − 0.866025i 0.976176 + 1.74559i 1.94999 − 1.09432i −0.00968783 + 1.41418i 1.44154 − 2.21855i 0.0581233 − 2.82783i −0.500000 + 0.866025i −3.16170 − 0.0602542i
19.10 −1.19515 + 0.756046i −0.500000 − 0.866025i 0.856789 − 1.80718i −1.90078 − 1.17773i 1.25233 + 0.657011i −2.10021 + 1.60907i 0.342319 + 2.80764i −0.500000 + 0.866025i 3.16214 − 0.0295053i
19.11 −1.11858 − 0.865320i −0.500000 − 0.866025i 0.502443 + 1.93586i −0.561119 − 2.16452i −0.190099 + 1.40138i −2.02879 − 1.69824i 1.11311 − 2.60019i −0.500000 + 0.866025i −1.24534 + 2.90674i
19.12 −1.08973 + 0.901383i −0.500000 − 0.866025i 0.375017 − 1.96453i 1.56223 − 1.59982i 1.32548 + 0.493041i 2.63556 + 0.232039i 1.36212 + 2.47883i −0.500000 + 0.866025i −0.260352 + 3.15154i
19.13 −1.06845 + 0.926509i −0.500000 − 0.866025i 0.283163 − 1.97985i 1.15715 + 1.91338i 1.33660 + 0.462049i −1.30898 + 2.29925i 1.53181 + 2.37772i −0.500000 + 0.866025i −3.00911 − 0.972235i
19.14 −0.856727 − 1.12518i −0.500000 − 0.866025i −0.532038 + 1.92794i 2.21180 + 0.328536i −0.546067 + 1.30453i 0.383657 + 2.61779i 2.62508 − 1.05308i −0.500000 + 0.866025i −1.52525 − 2.77013i
19.15 −0.799089 − 1.16681i −0.500000 − 0.866025i −0.722912 + 1.86478i −0.386692 + 2.20238i −0.610946 + 1.27544i −0.514926 − 2.59516i 2.75352 − 0.646620i −0.500000 + 0.866025i 2.87877 − 1.30870i
19.16 −0.731574 − 1.21029i −0.500000 − 0.866025i −0.929600 + 1.77083i −2.20852 − 0.349926i −0.682354 + 1.23871i 2.63671 − 0.218493i 2.82329 − 0.170410i −0.500000 + 0.866025i 1.19218 + 2.92894i
19.17 −0.726830 + 1.21314i −0.500000 − 0.866025i −0.943436 − 1.76350i −1.44110 + 1.70975i 1.41403 + 0.0228815i 2.38144 − 1.15271i 2.82510 + 0.137241i −0.500000 + 0.866025i −1.02673 − 2.99096i
19.18 −0.601011 + 1.28015i −0.500000 − 0.866025i −1.27757 − 1.53877i −2.19934 + 0.403590i 1.40915 − 0.119585i −2.62822 − 0.304041i 2.73769 − 0.710669i −0.500000 + 0.866025i 0.805173 − 3.05805i
19.19 −0.466600 + 1.33502i −0.500000 − 0.866025i −1.56457 − 1.24584i −0.363037 − 2.20640i 1.38946 − 0.263423i 0.635767 − 2.56823i 2.39326 − 1.50742i −0.500000 + 0.866025i 3.11499 + 0.544845i
19.20 −0.441768 − 1.34344i −0.500000 − 0.866025i −1.60968 + 1.18698i 2.21694 + 0.291884i −0.942572 + 1.05430i −2.63177 + 0.271606i 2.30575 + 1.63815i −0.500000 + 0.866025i −0.587242 − 3.10727i
See all 96 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
35.i odd 6 1 inner
280.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.bz.a ✓ 96
5.b even 2 1 840.2.bz.b yes 96
7.d odd 6 1 840.2.bz.b yes 96
8.d odd 2 1 inner 840.2.bz.a ✓ 96
35.i odd 6 1 inner 840.2.bz.a ✓ 96
40.e odd 2 1 840.2.bz.b yes 96
56.m even 6 1 840.2.bz.b yes 96
280.ba even 6 1 inner 840.2.bz.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bz.a ✓ 96 1.a even 1 1 trivial
840.2.bz.a ✓ 96 8.d odd 2 1 inner
840.2.bz.a ✓ 96 35.i odd 6 1 inner
840.2.bz.a ✓ 96 280.ba even 6 1 inner
840.2.bz.b yes 96 5.b even 2 1
840.2.bz.b yes 96 7.d odd 6 1
840.2.bz.b yes 96 40.e odd 2 1
840.2.bz.b yes 96 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{48} + 216 T_{17}^{46} + 176 T_{17}^{45} + 26976 T_{17}^{44} + 34492 T_{17}^{43} + \cdots + 10\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display