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Properties

Label 840.2.dr.a
Level $840$
Weight $2$
Character orbit 840.dr
Analytic conductor $6.707$
Analytic rank $0$
Dimension $384$
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(157,840)]
 
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, 6, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.157");
 
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.dr (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: \(384\)
Relative dimension: \(96\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 24 q^{8} + 16 q^{16} + 44 q^{28} + 36 q^{38} - 60 q^{40} + 20 q^{42} + 48 q^{46} + 144 q^{47} - 72 q^{50} - 108 q^{52} - 32 q^{56} - 8 q^{58} - 20 q^{60} + 72 q^{66} - 132 q^{68} + 80 q^{70} - 128 q^{71} - 12 q^{72} + 48 q^{78} - 108 q^{80} + 192 q^{81} + 60 q^{82} + 64 q^{86} - 68 q^{88} - 72 q^{92} - 48 q^{95} + 168 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 −1.41408 − 0.0193689i 0.258819 + 0.965926i 1.99925 + 0.0547783i −2.22880 − 0.180086i −0.347282 − 1.37091i −2.02901 + 1.69797i −2.82604 − 0.116184i −0.866025 + 0.500000i 3.14822 + 0.297826i
157.2 −1.41382 − 0.0333408i 0.258819 + 0.965926i 1.99778 + 0.0942759i −1.14546 + 1.92039i −0.333719 − 1.37427i 0.906917 − 2.48546i −2.82135 − 0.199897i −0.866025 + 0.500000i 1.68351 − 2.67690i
157.3 −1.41061 + 0.100867i −0.258819 − 0.965926i 1.97965 − 0.284570i 0.867435 + 2.06096i 0.462524 + 1.33644i 2.48525 + 0.907478i −2.76382 + 0.601100i −0.866025 + 0.500000i −1.43150 − 2.81972i
157.4 −1.40978 − 0.111933i 0.258819 + 0.965926i 1.97494 + 0.315600i 1.06054 − 1.96857i −0.256758 − 1.39071i 2.60104 + 0.484368i −2.74890 − 0.665987i −0.866025 + 0.500000i −1.71547 + 2.65653i
157.5 −1.40478 + 0.163043i 0.258819 + 0.965926i 1.94683 − 0.458080i 1.24511 + 1.85734i −0.521072 − 1.31472i −1.62706 + 2.08631i −2.66019 + 0.960920i −0.866025 + 0.500000i −2.05194 − 2.40615i
157.6 −1.40382 + 0.171145i −0.258819 − 0.965926i 1.94142 − 0.480513i −1.89572 − 1.18585i 0.528649 + 1.31169i −2.64322 + 0.115610i −2.64316 + 1.00682i −0.866025 + 0.500000i 2.86420 + 1.34027i
157.7 −1.39382 + 0.239275i −0.258819 − 0.965926i 1.88550 − 0.667014i −2.22472 + 0.224956i 0.591870 + 1.28440i 0.337612 − 2.62412i −2.46845 + 1.38085i −0.866025 + 0.500000i 3.04705 − 0.845870i
157.8 −1.37384 − 0.335510i −0.258819 − 0.965926i 1.77487 + 0.921873i 1.24923 − 1.85457i 0.0314978 + 1.41386i −2.32694 + 1.25911i −2.12908 − 1.86199i −0.866025 + 0.500000i −2.33846 + 2.12876i
157.9 −1.35484 + 0.405458i −0.258819 − 0.965926i 1.67121 − 1.09867i 0.855872 − 2.06579i 0.742302 + 1.20374i 1.44346 + 2.21730i −1.81876 + 2.16613i −0.866025 + 0.500000i −0.321983 + 3.14584i
157.10 −1.33351 + 0.470892i −0.258819 − 0.965926i 1.55652 − 1.25588i 2.22150 + 0.254825i 0.799986 + 1.16620i −1.20758 − 2.35409i −1.48426 + 2.40769i −0.866025 + 0.500000i −3.08240 + 0.706274i
157.11 −1.32940 + 0.482377i 0.258819 + 0.965926i 1.53462 − 1.28255i 0.414466 − 2.19732i −0.810015 − 1.15926i −2.29207 − 1.32151i −1.42146 + 2.44529i −0.866025 + 0.500000i 0.508945 + 3.12105i
157.12 −1.30720 − 0.539659i −0.258819 − 0.965926i 1.41754 + 1.41088i −1.06642 − 1.96539i −0.182943 + 1.40233i 2.29821 − 1.31081i −1.09160 − 2.60929i −0.866025 + 0.500000i 0.333389 + 3.14465i
157.13 −1.30124 − 0.553872i −0.258819 − 0.965926i 1.38645 + 1.44144i −1.84602 + 1.26183i −0.198214 + 1.40025i 0.392327 + 2.61650i −1.00573 − 2.64358i −0.866025 + 0.500000i 3.10101 − 0.619482i
157.14 −1.27555 − 0.610717i 0.258819 + 0.965926i 1.25405 + 1.55800i 2.21536 + 0.303647i 0.259771 − 1.39015i −1.57334 − 2.12711i −0.648108 − 2.75317i −0.866025 + 0.500000i −2.64035 − 1.74027i
157.15 −1.26553 − 0.631217i 0.258819 + 0.965926i 1.20313 + 1.59765i −1.04145 − 1.97873i 0.282165 − 1.38578i −1.29588 − 2.30666i −0.514136 − 2.78131i −0.866025 + 0.500000i 0.0689703 + 3.16153i
157.16 −1.26302 + 0.636214i 0.258819 + 0.965926i 1.19046 − 1.60711i 2.21262 − 0.322994i −0.941431 − 1.05532i 1.84092 − 1.90026i −0.481119 + 2.78721i −0.866025 + 0.500000i −2.58910 + 1.81565i
157.17 −1.26124 − 0.639745i 0.258819 + 0.965926i 1.18145 + 1.61374i 2.23137 − 0.144861i 0.291513 − 1.38384i −0.328562 + 2.62527i −0.457713 − 2.79115i −0.866025 + 0.500000i −2.90697 − 1.24480i
157.18 −1.24110 − 0.677992i −0.258819 − 0.965926i 1.08065 + 1.68291i 1.72589 − 1.42172i −0.333670 + 1.37429i −0.551080 − 2.58772i −0.200197 − 2.82133i −0.866025 + 0.500000i −3.10592 + 0.594357i
157.19 −1.16764 + 0.797882i 0.258819 + 0.965926i 0.726767 − 1.86328i −2.02375 + 0.951014i −1.07290 − 0.921347i 1.71268 + 2.01661i 0.638075 + 2.75551i −0.866025 + 0.500000i 1.60422 − 2.72516i
157.20 −1.13237 − 0.847192i −0.258819 − 0.965926i 0.564532 + 1.91867i 0.310203 + 2.21445i −0.525245 + 1.31306i 0.853984 − 2.50414i 0.986224 − 2.65092i −0.866025 + 0.500000i 1.52480 − 2.77038i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
35.k even 12 1 inner
40.i odd 4 1 inner
56.j odd 6 1 inner
280.bv 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.dr.a ✓ 384
5.c odd 4 1 inner 840.2.dr.a ✓ 384
7.d odd 6 1 inner 840.2.dr.a ✓ 384
8.b even 2 1 inner 840.2.dr.a ✓ 384
35.k even 12 1 inner 840.2.dr.a ✓ 384
40.i odd 4 1 inner 840.2.dr.a ✓ 384
56.j odd 6 1 inner 840.2.dr.a ✓ 384
280.bv even 12 1 inner 840.2.dr.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.dr.a ✓ 384 1.a even 1 1 trivial
840.2.dr.a ✓ 384 5.c odd 4 1 inner
840.2.dr.a ✓ 384 7.d odd 6 1 inner
840.2.dr.a ✓ 384 8.b even 2 1 inner
840.2.dr.a ✓ 384 35.k even 12 1 inner
840.2.dr.a ✓ 384 40.i odd 4 1 inner
840.2.dr.a ✓ 384 56.j odd 6 1 inner
840.2.dr.a ✓ 384 280.bv even 12 1 inner

Hecke kernels

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