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Properties

Label 2300.3.d.b
Level $2300$
Weight $3$
Character orbit 2300.d
Analytic conductor $62.670$
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] = [2300,3,Mod(1149,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2300.d (of order \(2\), degree \(1\), not 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: \(62.6704608029\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 460)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 128 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 128 q^{9} - 180 q^{29} + 20 q^{31} - 40 q^{39} + 372 q^{41} - 4 q^{49} + 180 q^{59} + 464 q^{69} - 476 q^{71} + 1408 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1149.1 0 − 3.41677i 0 0 0 11.7428 0 −2.67432 0
1149.2 0 3.41677i 0 0 0 11.7428 0 −2.67432 0
1149.3 0 − 5.14043i 0 0 0 7.88014 0 −17.4240 0
1149.4 0 5.14043i 0 0 0 7.88014 0 −17.4240 0
1149.5 0 − 2.47022i 0 0 0 −8.25674 0 2.89803 0
1149.6 0 2.47022i 0 0 0 −8.25674 0 2.89803 0
1149.7 0 − 0.886481i 0 0 0 9.10808 0 8.21415 0
1149.8 0 0.886481i 0 0 0 9.10808 0 8.21415 0
1149.9 0 − 4.53300i 0 0 0 −5.73038 0 −11.5481 0
1149.10 0 4.53300i 0 0 0 −5.73038 0 −11.5481 0
1149.11 0 − 5.89296i 0 0 0 −1.21480 0 −25.7269 0
1149.12 0 5.89296i 0 0 0 −1.21480 0 −25.7269 0
1149.13 0 − 0.613622i 0 0 0 2.35348 0 8.62347 0
1149.14 0 0.613622i 0 0 0 2.35348 0 8.62347 0
1149.15 0 − 1.83366i 0 0 0 −0.167846 0 5.63770 0
1149.16 0 1.83366i 0 0 0 −0.167846 0 5.63770 0
1149.17 0 − 1.83366i 0 0 0 0.167846 0 5.63770 0
1149.18 0 1.83366i 0 0 0 0.167846 0 5.63770 0
1149.19 0 − 0.613622i 0 0 0 −2.35348 0 8.62347 0
1149.20 0 0.613622i 0 0 0 −2.35348 0 8.62347 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1149.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.3.d.b 32
5.b even 2 1 inner 2300.3.d.b 32
5.c odd 4 1 460.3.f.a ✓ 16
5.c odd 4 1 2300.3.f.e 16
15.e even 4 1 4140.3.d.a 16
20.e even 4 1 1840.3.k.c 16
23.b odd 2 1 inner 2300.3.d.b 32
115.c odd 2 1 inner 2300.3.d.b 32
115.e even 4 1 460.3.f.a ✓ 16
115.e even 4 1 2300.3.f.e 16
345.l odd 4 1 4140.3.d.a 16
460.k odd 4 1 1840.3.k.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.3.f.a ✓ 16 5.c odd 4 1
460.3.f.a ✓ 16 115.e even 4 1
1840.3.k.c 16 20.e even 4 1
1840.3.k.c 16 460.k odd 4 1
2300.3.d.b 32 1.a even 1 1 trivial
2300.3.d.b 32 5.b even 2 1 inner
2300.3.d.b 32 23.b odd 2 1 inner
2300.3.d.b 32 115.c odd 2 1 inner
2300.3.f.e 16 5.c odd 4 1
2300.3.f.e 16 115.e even 4 1
4140.3.d.a 16 15.e even 4 1
4140.3.d.a 16 345.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 104 T_{3}^{14} + 4152 T_{3}^{12} + 80474 T_{3}^{10} + 792224 T_{3}^{8} + 3830456 T_{3}^{6} + \cdots + 1336336 \) acting on \(S_{3}^{\mathrm{new}}(2300, [\chi])\). Copy content Toggle raw display