Properties

Label 420.2.bk.a
Level $420$
Weight $2$
Character orbit 420.bk
Analytic conductor $3.354$
Analytic rank $0$
Dimension $96$
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] = [420,2,Mod(19,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bk (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: \(3.35371688489\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{9} + 18 q^{10} + 4 q^{14} - 8 q^{16} + 4 q^{25} - 8 q^{30} - 30 q^{40} + 16 q^{44} + 44 q^{46} - 8 q^{49} - 48 q^{50} - 52 q^{56} + 2 q^{60} - 144 q^{64} - 36 q^{66} - 42 q^{70} - 20 q^{74} - 36 q^{80} - 48 q^{81} - 20 q^{84} - 32 q^{85} - 32 q^{86} - 36 q^{94} - 60 q^{96}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.40539 0.157734i −0.866025 + 0.500000i 1.95024 + 0.443354i −2.23392 0.0978861i 1.29597 0.566094i −2.28831 1.32803i −2.67092 0.930704i 0.500000 0.866025i 3.12409 + 0.489933i
19.2 −1.40244 + 0.182077i 0.866025 0.500000i 1.93370 0.510706i −1.21850 1.87490i −1.12351 + 0.858905i −1.54335 + 2.14897i −2.61891 + 1.06832i 0.500000 0.866025i 2.05025 + 2.40758i
19.3 −1.39456 + 0.234979i −0.866025 + 0.500000i 1.88957 0.655384i 2.22645 0.207142i 1.09023 0.900776i 1.32770 + 2.28850i −2.48111 + 1.35798i 0.500000 0.866025i −3.05624 + 0.812041i
19.4 −1.34009 0.451826i 0.866025 0.500000i 1.59171 + 1.21098i 1.35968 1.77518i −1.38647 + 0.278754i 2.56678 0.641578i −1.58588 2.34200i 0.500000 0.866025i −2.62417 + 1.76458i
19.5 −1.33244 0.473912i 0.866025 0.500000i 1.55081 + 1.26292i −1.63361 + 1.52686i −1.39089 + 0.255802i −0.935228 2.47494i −1.46786 2.41772i 0.500000 0.866025i 2.90030 1.26027i
19.6 −1.27723 + 0.607202i −0.866025 + 0.500000i 1.26261 1.55107i −2.23451 + 0.0834732i 0.802509 1.16447i 2.60462 0.464731i −0.670831 + 2.74772i 0.500000 0.866025i 2.80329 1.46341i
19.7 −1.26422 + 0.633844i 0.866025 0.500000i 1.19648 1.60263i −0.558189 + 2.16528i −0.777921 + 1.18103i 0.242117 + 2.63465i −0.496796 + 2.78446i 0.500000 0.866025i −0.666777 3.09118i
19.8 −1.23847 0.682786i 0.866025 0.500000i 1.06761 + 1.69122i 1.44800 + 1.70391i −1.41394 + 0.0279240i −1.82043 + 1.91990i −0.167457 2.82347i 0.500000 0.866025i −0.629895 3.09891i
19.9 −1.21054 0.731152i −0.866025 + 0.500000i 0.930834 + 1.77018i 1.44800 + 1.70391i 1.41394 + 0.0279240i 1.82043 1.91990i 0.167457 2.82347i 0.500000 0.866025i −0.507051 3.12136i
19.10 −1.13701 + 0.840954i 0.866025 0.500000i 0.585592 1.91235i 2.13108 + 0.677136i −0.564204 + 1.29679i 2.06024 1.65994i 0.942373 + 2.66682i 0.500000 0.866025i −2.99250 + 1.02223i
19.11 −1.12697 + 0.854361i −0.866025 + 0.500000i 0.540133 1.92568i 0.789103 + 2.09220i 0.548806 1.30338i −1.62304 2.08943i 1.03651 + 2.63166i 0.500000 0.866025i −2.67680 1.68368i
19.12 −1.07664 0.916974i −0.866025 + 0.500000i 0.318317 + 1.97451i −1.63361 + 1.52686i 1.39089 + 0.255802i 0.935228 + 2.47494i 1.46786 2.41772i 0.500000 0.866025i 3.15891 0.145905i
19.13 −1.06134 0.934643i −0.866025 + 0.500000i 0.252886 + 1.98395i 1.35968 1.77518i 1.38647 + 0.278754i −2.56678 + 0.641578i 1.58588 2.34200i 0.500000 0.866025i −3.10224 + 0.613260i
19.14 −0.987366 + 1.01248i −0.866025 + 0.500000i −0.0502184 1.99937i −0.336963 2.21053i 0.348845 1.37051i −1.65799 + 2.06181i 2.07390 + 1.92326i 0.500000 0.866025i 2.57082 + 1.84144i
19.15 −0.839296 1.13824i 0.866025 0.500000i −0.591164 + 1.91063i −2.23392 0.0978861i −1.29597 0.566094i 2.28831 + 1.32803i 2.67092 0.930704i 0.500000 0.866025i 1.76351 + 2.62489i
19.16 −0.834650 + 1.14165i 0.866025 0.500000i −0.606717 1.90575i −1.74710 + 1.39558i −0.152005 + 1.40602i −2.20715 1.45893i 2.68209 + 0.897981i 0.500000 0.866025i −0.135049 3.15939i
19.17 −0.571370 + 1.29365i 0.866025 0.500000i −1.34707 1.47831i 0.335062 2.21082i 0.152005 + 1.40602i −2.20715 1.45893i 2.68209 0.897981i 0.500000 0.866025i 2.66859 + 1.69665i
19.18 −0.543538 1.30559i −0.866025 + 0.500000i −1.40913 + 1.41928i −1.21850 1.87490i 1.12351 + 0.858905i 1.54335 2.14897i 2.61891 + 1.06832i 0.500000 0.866025i −1.78555 + 2.60994i
19.19 −0.493780 1.32521i 0.866025 0.500000i −1.51236 + 1.30872i 2.22645 0.207142i −1.09023 0.900776i −1.32770 2.28850i 2.48111 + 1.35798i 0.500000 0.866025i −1.37388 2.84824i
19.20 −0.383148 + 1.36132i −0.866025 + 0.500000i −1.70640 1.04318i −2.08286 + 0.813448i −0.348845 1.37051i −1.65799 + 2.06181i 2.07390 1.92326i 0.500000 0.866025i −0.309322 3.14711i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.f even 6 1 inner
35.i odd 6 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bk.a 96
4.b odd 2 1 inner 420.2.bk.a 96
5.b even 2 1 inner 420.2.bk.a 96
7.d odd 6 1 inner 420.2.bk.a 96
20.d odd 2 1 inner 420.2.bk.a 96
28.f even 6 1 inner 420.2.bk.a 96
35.i odd 6 1 inner 420.2.bk.a 96
140.s even 6 1 inner 420.2.bk.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bk.a 96 1.a even 1 1 trivial
420.2.bk.a 96 4.b odd 2 1 inner
420.2.bk.a 96 5.b even 2 1 inner
420.2.bk.a 96 7.d odd 6 1 inner
420.2.bk.a 96 20.d odd 2 1 inner
420.2.bk.a 96 28.f even 6 1 inner
420.2.bk.a 96 35.i odd 6 1 inner
420.2.bk.a 96 140.s even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).