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Properties

Label 420.2.bo.a
Level $420$
Weight $2$
Character orbit 420.bo
Analytic conductor $3.354$
Analytic rank $0$
Dimension $32$
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] = [420,2,Mod(73,420)]
 
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, 0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.73");
 
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.bo (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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{5} + 8 q^{11} + 8 q^{15} - 8 q^{21} + 16 q^{23} - 4 q^{25} + 24 q^{31} + 12 q^{33} + 20 q^{35} + 20 q^{37} - 24 q^{43} + 12 q^{47} - 8 q^{51} + 40 q^{53} - 16 q^{57} - 24 q^{61} - 12 q^{63} - 52 q^{65} - 16 q^{71} - 60 q^{73} - 48 q^{75} - 84 q^{77} + 16 q^{81} - 8 q^{85} - 48 q^{87} + 40 q^{91} + 8 q^{93} - 36 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0 −0.258819 − 0.965926i 0 −2.13305 + 0.670893i 0 1.80016 + 1.93892i 0 −0.866025 + 0.500000i 0
73.2 0 −0.258819 − 0.965926i 0 −1.57194 + 1.59028i 0 −1.62725 − 2.08616i 0 −0.866025 + 0.500000i 0
73.3 0 −0.258819 − 0.965926i 0 0.541965 − 2.16939i 0 2.42432 − 1.05955i 0 −0.866025 + 0.500000i 0
73.4 0 −0.258819 − 0.965926i 0 1.40421 + 1.74017i 0 1.06666 + 2.42120i 0 −0.866025 + 0.500000i 0
73.5 0 0.258819 + 0.965926i 0 −2.12960 − 0.681768i 0 1.94431 + 1.79435i 0 −0.866025 + 0.500000i 0
73.6 0 0.258819 + 0.965926i 0 −1.25559 − 1.85027i 0 −2.64359 − 0.106918i 0 −0.866025 + 0.500000i 0
73.7 0 0.258819 + 0.965926i 0 −0.0833091 + 2.23452i 0 −1.70817 + 2.02043i 0 −0.866025 + 0.500000i 0
73.8 0 0.258819 + 0.965926i 0 2.22732 + 0.197620i 0 2.20765 − 1.45817i 0 −0.866025 + 0.500000i 0
157.1 0 −0.965926 + 0.258819i 0 −2.16320 + 0.566202i 0 2.08616 − 1.62725i 0 0.866025 − 0.500000i 0
157.2 0 −0.965926 + 0.258819i 0 −1.64754 + 1.51183i 0 −1.93892 + 1.80016i 0 0.866025 − 0.500000i 0
157.3 0 −0.965926 + 0.258819i 0 −0.804928 − 2.08617i 0 −2.42120 + 1.06666i 0 0.866025 − 0.500000i 0
157.4 0 −0.965926 + 0.258819i 0 2.14973 + 0.615342i 0 1.05955 + 2.42432i 0 0.866025 − 0.500000i 0
157.5 0 0.965926 − 0.258819i 0 −1.97680 − 1.04511i 0 −2.02043 − 1.70817i 0 0.866025 − 0.500000i 0
157.6 0 0.965926 − 0.258819i 0 −0.474372 + 2.18517i 0 −1.79435 + 1.94431i 0 0.866025 − 0.500000i 0
157.7 0 0.965926 − 0.258819i 0 0.942515 − 2.02772i 0 1.45817 + 2.20765i 0 0.866025 − 0.500000i 0
157.8 0 0.965926 − 0.258819i 0 0.974584 + 2.01251i 0 0.106918 − 2.64359i 0 0.866025 − 0.500000i 0
313.1 0 −0.965926 − 0.258819i 0 −2.16320 − 0.566202i 0 2.08616 + 1.62725i 0 0.866025 + 0.500000i 0
313.2 0 −0.965926 − 0.258819i 0 −1.64754 − 1.51183i 0 −1.93892 − 1.80016i 0 0.866025 + 0.500000i 0
313.3 0 −0.965926 − 0.258819i 0 −0.804928 + 2.08617i 0 −2.42120 − 1.06666i 0 0.866025 + 0.500000i 0
313.4 0 −0.965926 − 0.258819i 0 2.14973 − 0.615342i 0 1.05955 − 2.42432i 0 0.866025 + 0.500000i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
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
35.k 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.bo.a ✓ 32
3.b odd 2 1 1260.2.dq.c 32
5.b even 2 1 2100.2.ce.e 32
5.c odd 4 1 inner 420.2.bo.a ✓ 32
5.c odd 4 1 2100.2.ce.e 32
7.c even 3 1 2940.2.x.c 32
7.d odd 6 1 inner 420.2.bo.a ✓ 32
7.d odd 6 1 2940.2.x.c 32
15.e even 4 1 1260.2.dq.c 32
21.g even 6 1 1260.2.dq.c 32
35.i odd 6 1 2100.2.ce.e 32
35.k even 12 1 inner 420.2.bo.a ✓ 32
35.k even 12 1 2100.2.ce.e 32
35.k even 12 1 2940.2.x.c 32
35.l odd 12 1 2940.2.x.c 32
105.w odd 12 1 1260.2.dq.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bo.a ✓ 32 1.a even 1 1 trivial
420.2.bo.a ✓ 32 5.c odd 4 1 inner
420.2.bo.a ✓ 32 7.d odd 6 1 inner
420.2.bo.a ✓ 32 35.k even 12 1 inner
1260.2.dq.c 32 3.b odd 2 1
1260.2.dq.c 32 15.e even 4 1
1260.2.dq.c 32 21.g even 6 1
1260.2.dq.c 32 105.w odd 12 1
2100.2.ce.e 32 5.b even 2 1
2100.2.ce.e 32 5.c odd 4 1
2100.2.ce.e 32 35.i odd 6 1
2100.2.ce.e 32 35.k even 12 1
2940.2.x.c 32 7.c even 3 1
2940.2.x.c 32 7.d odd 6 1
2940.2.x.c 32 35.k even 12 1
2940.2.x.c 32 35.l odd 12 1

Hecke kernels

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