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Properties

Label 2205.2.g.b
Level $2205$
Weight $2$
Character orbit 2205.g
Analytic conductor $17.607$
Analytic rank $0$
Dimension $24$
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] = [2205,2,Mod(2204,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.2204");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.g (of order \(2\), degree \(1\), 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: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} - 24 q^{25} + 48 q^{46} + 48 q^{64} + 120 q^{79} - 72 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2204.1 −2.56937 0 4.60168 −0.884697 − 2.05361i 0 0 −6.68468 0 2.27312 + 5.27649i
2204.2 −2.56937 0 4.60168 0.884697 + 2.05361i 0 0 −6.68468 0 −2.27312 − 5.27649i
2204.3 −2.56937 0 4.60168 0.884697 − 2.05361i 0 0 −6.68468 0 −2.27312 + 5.27649i
2204.4 −2.56937 0 4.60168 −0.884697 + 2.05361i 0 0 −6.68468 0 2.27312 − 5.27649i
2204.5 −1.91314 0 1.66012 −0.630583 − 2.14531i 0 0 0.650234 0 1.20640 + 4.10429i
2204.6 −1.91314 0 1.66012 0.630583 + 2.14531i 0 0 0.650234 0 −1.20640 − 4.10429i
2204.7 −1.91314 0 1.66012 0.630583 − 2.14531i 0 0 0.650234 0 −1.20640 + 4.10429i
2204.8 −1.91314 0 1.66012 −0.630583 + 2.14531i 0 0 0.650234 0 1.20640 − 4.10429i
2204.9 −1.31841 0 −0.261802 2.19538 − 0.424645i 0 0 2.98198 0 −2.89440 + 0.559855i
2204.10 −1.31841 0 −0.261802 −2.19538 + 0.424645i 0 0 2.98198 0 2.89440 − 0.559855i
2204.11 −1.31841 0 −0.261802 −2.19538 − 0.424645i 0 0 2.98198 0 2.89440 + 0.559855i
2204.12 −1.31841 0 −0.261802 2.19538 + 0.424645i 0 0 2.98198 0 −2.89440 − 0.559855i
2204.13 1.31841 0 −0.261802 2.19538 + 0.424645i 0 0 −2.98198 0 2.89440 + 0.559855i
2204.14 1.31841 0 −0.261802 −2.19538 − 0.424645i 0 0 −2.98198 0 −2.89440 − 0.559855i
2204.15 1.31841 0 −0.261802 −2.19538 + 0.424645i 0 0 −2.98198 0 −2.89440 + 0.559855i
2204.16 1.31841 0 −0.261802 2.19538 − 0.424645i 0 0 −2.98198 0 2.89440 − 0.559855i
2204.17 1.91314 0 1.66012 −0.630583 + 2.14531i 0 0 −0.650234 0 −1.20640 + 4.10429i
2204.18 1.91314 0 1.66012 0.630583 − 2.14531i 0 0 −0.650234 0 1.20640 − 4.10429i
2204.19 1.91314 0 1.66012 0.630583 + 2.14531i 0 0 −0.650234 0 1.20640 + 4.10429i
2204.20 1.91314 0 1.66012 −0.630583 − 2.14531i 0 0 −0.650234 0 −1.20640 − 4.10429i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2204.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.2.g.b 24
3.b odd 2 1 inner 2205.2.g.b 24
5.b even 2 1 inner 2205.2.g.b 24
7.b odd 2 1 inner 2205.2.g.b 24
7.c even 3 1 315.2.bb.b ✓ 24
7.d odd 6 1 315.2.bb.b ✓ 24
15.d odd 2 1 inner 2205.2.g.b 24
21.c even 2 1 inner 2205.2.g.b 24
21.g even 6 1 315.2.bb.b ✓ 24
21.h odd 6 1 315.2.bb.b ✓ 24
35.c odd 2 1 inner 2205.2.g.b 24
35.i odd 6 1 315.2.bb.b ✓ 24
35.j even 6 1 315.2.bb.b ✓ 24
35.k even 12 2 1575.2.bk.i 24
35.l odd 12 2 1575.2.bk.i 24
105.g even 2 1 inner 2205.2.g.b 24
105.o odd 6 1 315.2.bb.b ✓ 24
105.p even 6 1 315.2.bb.b ✓ 24
105.w odd 12 2 1575.2.bk.i 24
105.x even 12 2 1575.2.bk.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.b ✓ 24 7.c even 3 1
315.2.bb.b ✓ 24 7.d odd 6 1
315.2.bb.b ✓ 24 21.g even 6 1
315.2.bb.b ✓ 24 21.h odd 6 1
315.2.bb.b ✓ 24 35.i odd 6 1
315.2.bb.b ✓ 24 35.j even 6 1
315.2.bb.b ✓ 24 105.o odd 6 1
315.2.bb.b ✓ 24 105.p even 6 1
1575.2.bk.i 24 35.k even 12 2
1575.2.bk.i 24 35.l odd 12 2
1575.2.bk.i 24 105.w odd 12 2
1575.2.bk.i 24 105.x even 12 2
2205.2.g.b 24 1.a even 1 1 trivial
2205.2.g.b 24 3.b odd 2 1 inner
2205.2.g.b 24 5.b even 2 1 inner
2205.2.g.b 24 7.b odd 2 1 inner
2205.2.g.b 24 15.d odd 2 1 inner
2205.2.g.b 24 21.c even 2 1 inner
2205.2.g.b 24 35.c odd 2 1 inner
2205.2.g.b 24 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 12T_{2}^{4} + 42T_{2}^{2} - 42 \) acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\). Copy content Toggle raw display