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Properties

Label 840.2.db.e
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
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 44 q^{3} - 44 q^{9} - 3 q^{10} - 16 q^{13} - 14 q^{14} - 20 q^{16} - 22 q^{20} + 4 q^{22} - 12 q^{25} + 4 q^{26} + 88 q^{27} + 36 q^{28} - 3 q^{30} + 24 q^{31} - 10 q^{32} - 24 q^{34} - 12 q^{35} + 40 q^{37} + 12 q^{38} + 8 q^{39} + 13 q^{40} - 40 q^{41} + 10 q^{42} + 32 q^{43} - 8 q^{44} + 10 q^{46} + 40 q^{48} + 8 q^{49} + 14 q^{50} - 6 q^{52} - 24 q^{53} + 16 q^{55} - 26 q^{56} + 10 q^{58} + 11 q^{60} + 64 q^{62} - 48 q^{64} + 4 q^{65} - 2 q^{66} - 70 q^{68} - 39 q^{70} - 48 q^{71} + 38 q^{74} - 12 q^{75} + 100 q^{76} + 16 q^{77} - 8 q^{78} - 8 q^{79} + 17 q^{80} - 44 q^{81} - 38 q^{82} - 64 q^{83} - 6 q^{84} - 32 q^{85} - 8 q^{88} - 8 q^{89} + 6 q^{90} - 92 q^{92} + 24 q^{93} + 26 q^{94} + 68 q^{95} - 10 q^{96} - 6 q^{98}+O(q^{100}) \) 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 −1.41349 + 0.0453635i −0.500000 + 0.866025i 1.99588 − 0.128241i −0.729565 − 2.11370i 0.667457 − 1.24680i 0.634691 + 2.56850i −2.81534 + 0.271807i −0.500000 − 0.866025i 1.12711 + 2.95459i
109.2 −1.41067 + 0.100092i −0.500000 + 0.866025i 1.97996 − 0.282394i −2.11085 + 0.737787i 0.618651 − 1.27172i 1.72467 − 2.00637i −2.76480 + 0.596544i −0.500000 − 0.866025i 2.90385 − 1.25205i
109.3 −1.39623 + 0.224790i −0.500000 + 0.866025i 1.89894 − 0.627719i 0.919633 + 2.03820i 0.503443 − 1.32157i 2.58511 + 0.563191i −2.51026 + 1.30331i −0.500000 − 0.866025i −1.74219 − 2.63909i
109.4 −1.33151 − 0.476542i −0.500000 + 0.866025i 1.54582 + 1.26904i 0.542606 − 2.16923i 1.07845 − 0.914847i 2.20647 − 1.45996i −1.45351 − 2.42638i −0.500000 − 0.866025i −1.75621 + 2.62977i
109.5 −1.31176 + 0.528472i −0.500000 + 0.866025i 1.44144 − 1.38646i −0.619196 + 2.14863i 0.198211 − 1.40025i −0.143755 + 2.64184i −1.15812 + 2.58046i −0.500000 − 0.866025i −0.323251 − 3.14571i
109.6 −1.28646 + 0.587389i −0.500000 + 0.866025i 1.30995 − 1.51130i 0.935996 − 2.03074i 0.134536 − 1.40780i −1.97421 − 1.76139i −0.797474 + 2.71368i −0.500000 − 0.866025i −0.0112858 + 3.16226i
109.7 −1.28309 − 0.594707i −0.500000 + 0.866025i 1.29265 + 1.52613i −0.575628 + 2.16071i 1.15658 − 0.813837i −2.58650 + 0.556790i −0.750988 − 2.72691i −0.500000 − 0.866025i 2.02357 − 2.43005i
109.8 −1.19618 − 0.754422i −0.500000 + 0.866025i 0.861695 + 1.80485i −1.15281 − 1.91599i 1.25144 − 0.658712i −2.12917 − 1.57055i 0.330875 − 2.80901i −0.500000 − 0.866025i −0.0664982 + 3.16158i
109.9 −1.14670 + 0.827696i −0.500000 + 0.866025i 0.629837 − 1.89824i −1.96411 − 1.06877i −0.143457 − 1.40692i −2.54960 + 0.706767i 0.848931 + 2.69802i −0.500000 − 0.866025i 3.13686 − 0.400123i
109.10 −1.12138 + 0.861691i −0.500000 + 0.866025i 0.514977 − 1.93256i 2.00252 − 0.994937i −0.185558 − 1.40199i 2.64327 + 0.114618i 1.08779 + 2.61088i −0.500000 − 0.866025i −1.38826 + 2.84126i
109.11 −1.02283 − 0.976638i −0.500000 + 0.866025i 0.0923560 + 1.99787i −2.22688 + 0.202473i 1.35721 − 0.397476i 2.64103 − 0.158007i 1.85673 − 2.13367i −0.500000 − 0.866025i 2.47546 + 1.96776i
109.12 −0.927960 + 1.06719i −0.500000 + 0.866025i −0.277779 − 1.98062i −0.905560 + 2.04450i −0.460231 − 1.33723i −0.273531 − 2.63157i 2.37146 + 1.54149i −0.500000 − 0.866025i −1.34154 − 2.86361i
109.13 −0.893688 − 1.09605i −0.500000 + 0.866025i −0.402642 + 1.95905i 2.13859 − 0.653026i 1.39605 − 0.225933i 0.335123 − 2.62444i 2.50705 − 1.30947i −0.500000 − 0.866025i −2.62698 − 1.76039i
109.14 −0.810859 + 1.15867i −0.500000 + 0.866025i −0.685016 − 1.87903i 1.82844 + 1.28717i −0.598005 − 1.28156i −2.38344 + 1.14858i 2.73262 + 0.729924i −0.500000 − 0.866025i −2.97401 + 1.07483i
109.15 −0.774118 − 1.18353i −0.500000 + 0.866025i −0.801484 + 1.83238i 2.17778 + 0.507213i 1.41203 − 0.0786408i 1.19331 + 2.36136i 2.78912 − 0.469899i −0.500000 − 0.866025i −1.08556 − 2.97011i
109.16 −0.703420 + 1.22687i −0.500000 + 0.866025i −1.01040 − 1.72600i −1.91427 − 1.15568i −0.710787 − 1.22261i 2.15205 + 1.53905i 2.82831 − 0.0255173i −0.500000 − 0.866025i 2.76439 − 1.53562i
109.17 −0.637908 − 1.26217i −0.500000 + 0.866025i −1.18615 + 1.61030i −0.649632 + 2.13962i 1.41203 + 0.0786408i −1.19331 − 2.36136i 2.78912 + 0.469899i −0.500000 − 0.866025i 3.11497 − 0.544935i
109.18 −0.502361 − 1.32198i −0.500000 + 0.866025i −1.49527 + 1.32822i −1.63483 + 1.52556i 1.39605 + 0.225933i −0.335123 + 2.62444i 2.50705 + 1.30947i −0.500000 − 0.866025i 2.83803 + 1.39483i
109.19 −0.334379 − 1.37411i −0.500000 + 0.866025i −1.77638 + 0.918951i 1.28879 − 1.82730i 1.35721 + 0.397476i −2.64103 + 0.158007i 1.85673 + 2.13367i −0.500000 − 0.866025i −2.94186 − 1.15993i
109.20 −0.207798 + 1.39886i −0.500000 + 0.866025i −1.91364 − 0.581361i 1.48694 + 1.67003i −1.10755 − 0.879390i 2.61894 − 0.375689i 1.21089 − 2.55612i −0.500000 − 0.866025i −2.64513 + 1.73300i
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.e ✓ 88
5.b even 2 1 840.2.db.f yes 88
7.c even 3 1 inner 840.2.db.e ✓ 88
8.b even 2 1 840.2.db.f yes 88
35.j even 6 1 840.2.db.f yes 88
40.f even 2 1 inner 840.2.db.e ✓ 88
56.p even 6 1 840.2.db.f yes 88
280.bf even 6 1 inner 840.2.db.e ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.db.e ✓ 88 1.a even 1 1 trivial
840.2.db.e ✓ 88 7.c even 3 1 inner
840.2.db.e ✓ 88 40.f even 2 1 inner
840.2.db.e ✓ 88 280.bf even 6 1 inner
840.2.db.f yes 88 5.b even 2 1
840.2.db.f yes 88 8.b even 2 1
840.2.db.f yes 88 35.j even 6 1
840.2.db.f yes 88 56.p even 6 1

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