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Properties

Label 420.2.bv.c
Level $420$
Weight $2$
Character orbit 420.bv
Analytic conductor $3.354$
Analytic rank $0$
Dimension $48$
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(53,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.53");
 
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.bv (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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75} + 20 q^{81} - 56 q^{85} - 24 q^{87} - 24 q^{91} + 6 q^{93} + 56 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 0 −1.72581 + 0.146940i 0 1.07475 + 1.96085i 0 −2.28090 + 1.34070i 0 2.95682 − 0.507180i 0
53.2 0 −1.66431 − 0.479661i 0 −2.05928 − 0.871404i 0 2.63382 − 0.250938i 0 2.53985 + 1.59661i 0
53.3 0 −1.56806 + 0.735650i 0 −1.07475 − 1.96085i 0 −2.28090 + 1.34070i 0 1.91764 − 2.30709i 0
53.4 0 −1.20150 + 1.24755i 0 2.05928 + 0.871404i 0 2.63382 − 0.250938i 0 −0.112777 − 2.99788i 0
53.5 0 −0.795708 − 1.53846i 0 0.746108 − 2.10792i 0 −2.40557 − 1.10146i 0 −1.73370 + 2.44833i 0
53.6 0 −0.302208 − 1.70548i 0 −1.92665 + 1.13491i 0 −0.225753 + 2.63610i 0 −2.81734 + 1.03082i 0
53.7 0 0.0801247 + 1.73020i 0 −0.746108 + 2.10792i 0 −2.40557 − 1.10146i 0 −2.98716 + 0.277263i 0
53.8 0 0.591022 + 1.62810i 0 1.92665 − 1.13491i 0 −0.225753 + 2.63610i 0 −2.30139 + 1.92448i 0
53.9 0 0.774760 − 1.54911i 0 2.17211 + 0.530994i 0 1.78762 − 1.95049i 0 −1.79949 − 2.40038i 0
53.10 0 1.30157 − 1.14276i 0 −2.15607 + 0.592743i 0 −2.24126 − 1.40596i 0 0.388181 − 2.97478i 0
53.11 0 1.44552 + 0.954190i 0 −2.17211 − 0.530994i 0 1.78762 − 1.95049i 0 1.17904 + 2.75860i 0
53.12 0 1.69858 + 0.338877i 0 2.15607 − 0.592743i 0 −2.24126 − 1.40596i 0 2.77033 + 1.15122i 0
137.1 0 −1.70548 + 0.302208i 0 −1.94619 + 1.10107i 0 −2.63610 − 0.225753i 0 2.81734 − 1.03082i 0
137.2 0 −1.54911 − 0.774760i 0 0.626199 − 2.14660i 0 1.95049 + 1.78762i 0 1.79949 + 2.40038i 0
137.3 0 −1.53846 + 0.795708i 0 2.19857 + 0.407811i 0 1.10146 − 2.40557i 0 1.73370 − 2.44833i 0
137.4 0 −1.14276 − 1.30157i 0 −1.59137 + 1.57084i 0 1.40596 − 2.24126i 0 −0.388181 + 2.97478i 0
137.5 0 −0.479661 + 1.66431i 0 −0.274984 + 2.21910i 0 0.250938 + 2.63382i 0 −2.53985 − 1.59661i 0
137.6 0 0.146940 + 1.72581i 0 −1.16077 − 1.91118i 0 −1.34070 − 2.28090i 0 −2.95682 + 0.507180i 0
137.7 0 0.338877 − 1.69858i 0 1.59137 − 1.57084i 0 1.40596 − 2.24126i 0 −2.77033 − 1.15122i 0
137.8 0 0.735650 + 1.56806i 0 1.16077 + 1.91118i 0 −1.34070 − 2.28090i 0 −1.91764 + 2.30709i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.12
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 420.2.bv.c ✓ 48
3.b odd 2 1 inner 420.2.bv.c ✓ 48
5.c odd 4 1 inner 420.2.bv.c ✓ 48
7.c even 3 1 inner 420.2.bv.c ✓ 48
15.e even 4 1 inner 420.2.bv.c ✓ 48
21.h odd 6 1 inner 420.2.bv.c ✓ 48
35.l odd 12 1 inner 420.2.bv.c ✓ 48
105.x even 12 1 inner 420.2.bv.c ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bv.c ✓ 48 1.a even 1 1 trivial
420.2.bv.c ✓ 48 3.b odd 2 1 inner
420.2.bv.c ✓ 48 5.c odd 4 1 inner
420.2.bv.c ✓ 48 7.c even 3 1 inner
420.2.bv.c ✓ 48 15.e even 4 1 inner
420.2.bv.c ✓ 48 21.h odd 6 1 inner
420.2.bv.c ✓ 48 35.l odd 12 1 inner
420.2.bv.c ✓ 48 105.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11}^{24} - 80 T_{11}^{22} + 4061 T_{11}^{20} - 123664 T_{11}^{18} + 2716141 T_{11}^{16} + \cdots + 888287400100 \) Copy content Toggle raw display
\( T_{17}^{48} - 1878 T_{17}^{44} + 2110815 T_{17}^{40} - 1556497350 T_{17}^{36} + 852322307473 T_{17}^{32} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display