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Properties

Label 840.2.db.f
Level $840$
Weight $2$
Character orbit 840.db
Analytic conductor $6.707$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(109,840)]
 
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([0, 3, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.109");
 
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.db (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: \(88\)
Relative dimension: \(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 44 q^{3} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 44 q^{3} - 44 q^{9} - 11 q^{10} + 16 q^{13} - 14 q^{14} - 20 q^{16} - 18 q^{20} - 4 q^{22} - 12 q^{25} + 4 q^{26} - 88 q^{27} - 36 q^{28} + 11 q^{30} + 24 q^{31} + 10 q^{32} - 24 q^{34} + 12 q^{35} - 40 q^{37} - 12 q^{38} + 8 q^{39} + 29 q^{40} - 40 q^{41} - 10 q^{42} - 32 q^{43} - 8 q^{44} + 10 q^{46} - 40 q^{48} + 8 q^{49} + 98 q^{50} + 6 q^{52} + 24 q^{53} + 16 q^{55} - 26 q^{56} - 10 q^{58} - 9 q^{60} - 64 q^{62} - 48 q^{64} + 4 q^{65} - 2 q^{66} + 70 q^{68} - 11 q^{70} - 48 q^{71} + 38 q^{74} + 12 q^{75} + 100 q^{76} - 16 q^{77} + 8 q^{78} - 8 q^{79} + 3 q^{80} - 44 q^{81} + 38 q^{82} + 64 q^{83} - 6 q^{84} + 32 q^{85} + 8 q^{88} - 8 q^{89} + 22 q^{90} + 92 q^{92} - 24 q^{93} + 26 q^{94} + 68 q^{95} - 10 q^{96} + 6 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41421 − 0.00425293i 0.500000 − 0.866025i 1.99996 + 0.0120291i 1.91427 + 1.15568i −0.710787 + 1.22261i 2.15205 + 1.53905i −2.82831 − 0.0255173i −0.500000 − 0.866025i −2.70225 − 1.64251i
109.2 −1.40886 + 0.122891i 0.500000 − 0.866025i 1.96980 − 0.346274i −1.82844 − 1.28717i −0.598005 + 1.28156i −2.38344 + 1.14858i −2.73262 + 0.729924i −0.500000 − 0.866025i 2.73420 + 1.58875i
109.3 −1.38819 + 0.270043i 0.500000 − 0.866025i 1.85415 − 0.749744i 0.905560 − 2.04450i −0.460231 + 1.33723i −0.273531 − 2.63157i −2.37146 + 1.54149i −0.500000 − 0.866025i −0.704989 + 3.08269i
109.4 −1.31535 − 0.519474i 0.500000 − 0.866025i 1.46029 + 1.36658i −1.48694 − 1.67003i −1.10755 + 0.879390i 2.61894 − 0.375689i −1.21089 − 2.55612i −0.500000 − 0.866025i 1.08831 + 2.96910i
109.5 −1.30694 + 0.540296i 0.500000 − 0.866025i 1.41616 − 1.41226i −2.00252 + 0.994937i −0.185558 + 1.40199i 2.64327 + 0.114618i −1.08779 + 2.61088i −0.500000 − 0.866025i 2.07961 − 2.38227i
109.6 −1.29016 + 0.579222i 0.500000 − 0.866025i 1.32900 − 1.49457i 1.96411 + 1.06877i −0.143457 + 1.40692i −2.54960 + 0.706767i −0.848931 + 2.69802i −0.500000 − 0.866025i −3.15306 − 0.241230i
109.7 −1.21971 − 0.715753i 0.500000 − 0.866025i 0.975395 + 1.74603i −0.806511 + 2.08556i −1.22972 + 0.698425i −0.391796 + 2.61658i 0.0600221 − 2.82779i −0.500000 − 0.866025i 2.47645 − 1.96651i
109.8 −1.18948 − 0.764942i 0.500000 − 0.866025i 0.829726 + 1.81977i 1.47935 − 1.67676i −1.25720 + 0.647649i −0.875614 + 2.49666i 0.405075 − 2.79927i −0.500000 − 0.866025i −3.04228 + 0.862863i
109.9 −1.15192 + 0.820411i 0.500000 − 0.866025i 0.653851 − 1.89010i −0.935996 + 2.03074i 0.134536 + 1.40780i −1.97421 − 1.76139i 0.797474 + 2.71368i −0.500000 − 0.866025i −0.587848 − 3.10716i
109.10 −1.14150 − 0.834859i 0.500000 − 0.866025i 0.606022 + 1.90597i −2.03889 + 0.918118i −1.29376 + 0.571134i −2.25627 − 1.38175i 0.899447 − 2.68160i −0.500000 − 0.866025i 3.09388 + 0.654155i
109.11 −1.11355 + 0.871783i 0.500000 − 0.866025i 0.479989 − 1.94155i 0.619196 − 2.14863i 0.198211 + 1.40025i −0.143755 + 2.64184i 1.15812 + 2.58046i −0.500000 − 0.866025i 1.18363 + 2.93241i
109.12 −0.967550 − 1.03143i 0.500000 − 0.866025i −0.127696 + 1.99592i −0.671266 − 2.13293i −1.37702 + 0.322207i −0.635552 − 2.56828i 2.18220 − 1.79944i −0.500000 − 0.866025i −1.55049 + 2.75608i
109.13 −0.892791 + 1.09678i 0.500000 − 0.866025i −0.405849 − 1.95839i −0.919633 − 2.03820i 0.503443 + 1.32157i 2.58511 + 0.563191i 2.51026 + 1.30331i −0.500000 − 0.866025i 3.05650 + 0.811055i
109.14 −0.792016 + 1.17163i 0.500000 − 0.866025i −0.745421 − 1.85590i 2.11085 − 0.737787i 0.618651 + 1.27172i 1.72467 − 2.00637i 2.76480 + 0.596544i −0.500000 − 0.866025i −0.807412 + 3.05746i
109.15 −0.790870 − 1.17240i 0.500000 − 0.866025i −0.749050 + 1.85443i 1.06003 + 1.96884i −1.41076 + 0.0987126i 1.50681 − 2.17475i 2.76654 − 0.588428i −0.500000 − 0.866025i 1.46992 − 2.79988i
109.16 −0.746029 + 1.20143i 0.500000 − 0.866025i −0.886882 − 1.79261i 0.729565 + 2.11370i 0.667457 + 1.24680i 0.634691 + 2.56850i 2.81534 + 0.271807i −0.500000 − 0.866025i −3.08375 − 0.700359i
109.17 −0.619895 − 1.27111i 0.500000 − 0.866025i −1.23146 + 1.57591i 1.17505 + 1.90243i −1.41076 − 0.0987126i −1.50681 + 2.17475i 2.76654 + 0.588428i −0.500000 − 0.866025i 1.68980 − 2.67293i
109.18 −0.409470 − 1.35364i 0.500000 − 0.866025i −1.66467 + 1.10855i −1.51154 − 1.64780i −1.37702 − 0.322207i 0.635552 + 2.56828i 2.18220 + 1.79944i −0.500000 − 0.866025i −1.61159 + 2.72080i
109.19 −0.253055 + 1.39139i 0.500000 − 0.866025i −1.87193 − 0.704196i −0.542606 + 2.16923i 1.07845 + 0.914847i 2.20647 − 1.45996i 1.45351 − 2.42638i −0.500000 − 0.866025i −2.88094 − 1.30391i
109.20 −0.152261 − 1.40599i 0.500000 − 0.866025i −1.95363 + 0.428156i 1.81456 − 1.30667i −1.29376 − 0.571134i 2.25627 + 1.38175i 0.899447 + 2.68160i −0.500000 − 0.866025i −2.11345 − 2.35230i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
40.f even 2 1 inner
280.bf 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.db.f yes 88
5.b even 2 1 840.2.db.e ✓ 88
7.c even 3 1 inner 840.2.db.f yes 88
8.b even 2 1 840.2.db.e ✓ 88
35.j even 6 1 840.2.db.e ✓ 88
40.f even 2 1 inner 840.2.db.f yes 88
56.p even 6 1 840.2.db.e ✓ 88
280.bf even 6 1 inner 840.2.db.f yes 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.db.e ✓ 88 5.b even 2 1
840.2.db.e ✓ 88 8.b even 2 1
840.2.db.e ✓ 88 35.j even 6 1
840.2.db.e ✓ 88 56.p even 6 1
840.2.db.f yes 88 1.a even 1 1 trivial
840.2.db.f yes 88 7.c even 3 1 inner
840.2.db.f yes 88 40.f even 2 1 inner
840.2.db.f yes 88 280.bf even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{88} - 282 T_{11}^{86} + 43117 T_{11}^{84} - 4543838 T_{11}^{82} + 364970322 T_{11}^{80} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
\( T_{13}^{22} - 4 T_{13}^{21} - 153 T_{13}^{20} + 644 T_{13}^{19} + 9364 T_{13}^{18} - 41720 T_{13}^{17} + \cdots + 2800753424 \) Copy content Toggle raw display