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Properties

Label 840.2.dq.a
Level $840$
Weight $2$
Character orbit 840.dq
Analytic conductor $6.707$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(137,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 6, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.137");
 
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.dq (of order \(12\), degree \(4\), 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: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 4 q^{7} + 4 q^{21} + 8 q^{25} + 24 q^{27} + 8 q^{37} + 16 q^{45} - 24 q^{51} + 16 q^{55} + 16 q^{57} + 24 q^{61} - 36 q^{63} + 16 q^{67} - 8 q^{73} + 24 q^{75} - 12 q^{81} + 16 q^{85} - 64 q^{91} - 24 q^{93} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 0 −1.73201 + 0.0116350i 0 −2.05448 + 0.882675i 0 2.59078 + 0.536528i 0 2.99973 − 0.0403038i 0
137.2 0 −1.72189 − 0.187379i 0 1.63008 − 1.53064i 0 −2.64389 + 0.0990965i 0 2.92978 + 0.645289i 0
137.3 0 −1.69133 − 0.373370i 0 2.16538 − 0.557787i 0 −1.05167 + 2.42775i 0 2.72119 + 1.26298i 0
137.4 0 −1.67775 + 0.430299i 0 −1.24772 − 1.85559i 0 −0.357979 − 2.62142i 0 2.62969 − 1.44387i 0
137.5 0 −1.67651 − 0.435119i 0 1.69922 + 1.45350i 0 2.56127 + 0.663230i 0 2.62134 + 1.45896i 0
137.6 0 −1.60590 − 0.648916i 0 −2.17918 − 0.501171i 0 −2.43518 − 1.03437i 0 2.15782 + 2.08419i 0
137.7 0 −1.58057 + 0.708369i 0 1.01338 + 1.99325i 0 −1.70094 + 2.02652i 0 1.99643 − 2.23926i 0
137.8 0 −1.52860 + 0.814487i 0 −0.474297 + 2.18519i 0 0.0232428 − 2.64565i 0 1.67322 − 2.49005i 0
137.9 0 −1.51563 + 0.838379i 0 0.711435 − 2.11987i 0 2.04365 − 1.68032i 0 1.59424 − 2.54134i 0
137.10 0 −1.42415 − 0.985801i 0 0.536640 + 2.17072i 0 −0.891494 − 2.49103i 0 1.05639 + 2.80785i 0
137.11 0 −1.39150 − 1.03137i 0 1.04643 − 1.97610i 0 2.54345 − 0.728611i 0 0.872532 + 2.87031i 0
137.12 0 −1.26053 + 1.18789i 0 −1.83884 + 1.27227i 0 −0.371330 + 2.61956i 0 0.177853 − 2.99472i 0
137.13 0 −1.22807 + 1.22141i 0 −1.76661 − 1.37080i 0 −2.21781 + 1.44268i 0 0.0163257 − 2.99996i 0
137.14 0 −0.937692 − 1.45627i 0 −2.17292 + 0.527641i 0 2.05378 − 1.66793i 0 −1.24147 + 2.73107i 0
137.15 0 −0.900844 + 1.47935i 0 2.10229 + 0.761811i 0 −2.53120 − 0.770075i 0 −1.37696 − 2.66533i 0
137.16 0 −0.817689 − 1.52689i 0 −0.322646 − 2.21267i 0 −2.62555 − 0.326304i 0 −1.66277 + 2.49704i 0
137.17 0 −0.809013 − 1.53150i 0 −0.763856 − 2.10155i 0 1.49496 + 2.18291i 0 −1.69100 + 2.47801i 0
137.18 0 −0.788932 + 1.54194i 0 1.10559 − 1.94362i 0 2.00560 + 1.72557i 0 −1.75517 − 2.43298i 0
137.19 0 −0.727817 − 1.57171i 0 −2.13557 + 0.662834i 0 −0.440810 + 2.60877i 0 −1.94057 + 2.28784i 0
137.20 0 −0.563748 + 1.63774i 0 2.23252 − 0.125945i 0 −1.81854 − 1.92170i 0 −2.36438 − 1.84654i 0
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.dq.a ✓ 192
3.b odd 2 1 inner 840.2.dq.a ✓ 192
5.c odd 4 1 inner 840.2.dq.a ✓ 192
7.c even 3 1 inner 840.2.dq.a ✓ 192
15.e even 4 1 inner 840.2.dq.a ✓ 192
21.h odd 6 1 inner 840.2.dq.a ✓ 192
35.l odd 12 1 inner 840.2.dq.a ✓ 192
105.x even 12 1 inner 840.2.dq.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.dq.a ✓ 192 1.a even 1 1 trivial
840.2.dq.a ✓ 192 3.b odd 2 1 inner
840.2.dq.a ✓ 192 5.c odd 4 1 inner
840.2.dq.a ✓ 192 7.c even 3 1 inner
840.2.dq.a ✓ 192 15.e even 4 1 inner
840.2.dq.a ✓ 192 21.h odd 6 1 inner
840.2.dq.a ✓ 192 35.l odd 12 1 inner
840.2.dq.a ✓ 192 105.x even 12 1 inner

Hecke kernels

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