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Properties

Label 1050.3.c.c
Level $1050$
Weight $3$
Character orbit 1050.c
Analytic conductor $28.610$
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] = [1050,3,Mod(449,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.c (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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 210)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} + 32 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} + 32 q^{6} + 8 q^{9} + 128 q^{16} - 96 q^{19} + 56 q^{21} + 64 q^{24} - 320 q^{34} + 16 q^{36} - 312 q^{39} + 64 q^{46} - 224 q^{49} + 168 q^{51} + 64 q^{54} + 224 q^{61} + 256 q^{64} - 16 q^{69} - 192 q^{76} - 16 q^{79} - 248 q^{81} + 112 q^{84} - 112 q^{91} - 64 q^{94} + 128 q^{96} + 104 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1 −1.41421 −2.92855 − 0.650833i 2.00000 0 4.14160 + 0.920417i − 2.64575i −2.82843 8.15283 + 3.81200i 0
449.2 −1.41421 −2.92855 + 0.650833i 2.00000 0 4.14160 − 0.920417i 2.64575i −2.82843 8.15283 − 3.81200i 0
449.3 −1.41421 −2.88800 − 0.812085i 2.00000 0 4.08424 + 1.14846i 2.64575i −2.82843 7.68104 + 4.69059i 0
449.4 −1.41421 −2.88800 + 0.812085i 2.00000 0 4.08424 − 1.14846i − 2.64575i −2.82843 7.68104 − 4.69059i 0
449.5 −1.41421 −2.35293 − 1.86110i 2.00000 0 3.32755 + 2.63200i − 2.64575i −2.82843 2.07259 + 8.75810i 0
449.6 −1.41421 −2.35293 + 1.86110i 2.00000 0 3.32755 − 2.63200i 2.64575i −2.82843 2.07259 − 8.75810i 0
449.7 −1.41421 −1.60951 − 2.53169i 2.00000 0 2.27619 + 3.58035i − 2.64575i −2.82843 −3.81894 + 8.14958i 0
449.8 −1.41421 −1.60951 + 2.53169i 2.00000 0 2.27619 − 3.58035i 2.64575i −2.82843 −3.81894 − 8.14958i 0
449.9 −1.41421 −0.922735 − 2.85457i 2.00000 0 1.30494 + 4.03697i 2.64575i −2.82843 −7.29712 + 5.26802i 0
449.10 −1.41421 −0.922735 + 2.85457i 2.00000 0 1.30494 − 4.03697i − 2.64575i −2.82843 −7.29712 − 5.26802i 0
449.11 −1.41421 0.396304 − 2.97371i 2.00000 0 −0.560459 + 4.20546i 2.64575i −2.82843 −8.68589 − 2.35699i 0
449.12 −1.41421 0.396304 + 2.97371i 2.00000 0 −0.560459 − 4.20546i − 2.64575i −2.82843 −8.68589 + 2.35699i 0
449.13 −1.41421 2.05675 − 2.18398i 2.00000 0 −2.90869 + 3.08861i 2.64575i −2.82843 −0.539533 − 8.98381i 0
449.14 −1.41421 2.05675 + 2.18398i 2.00000 0 −2.90869 − 3.08861i − 2.64575i −2.82843 −0.539533 + 8.98381i 0
449.15 −1.41421 2.59182 − 1.51079i 2.00000 0 −3.66538 + 2.13658i 2.64575i −2.82843 4.43502 − 7.83139i 0
449.16 −1.41421 2.59182 + 1.51079i 2.00000 0 −3.66538 − 2.13658i − 2.64575i −2.82843 4.43502 + 7.83139i 0
449.17 1.41421 −2.59182 − 1.51079i 2.00000 0 −3.66538 − 2.13658i 2.64575i 2.82843 4.43502 + 7.83139i 0
449.18 1.41421 −2.59182 + 1.51079i 2.00000 0 −3.66538 + 2.13658i − 2.64575i 2.82843 4.43502 − 7.83139i 0
449.19 1.41421 −2.05675 − 2.18398i 2.00000 0 −2.90869 − 3.08861i 2.64575i 2.82843 −0.539533 + 8.98381i 0
449.20 1.41421 −2.05675 + 2.18398i 2.00000 0 −2.90869 + 3.08861i − 2.64575i 2.82843 −0.539533 − 8.98381i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.32
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.c.c 32
3.b odd 2 1 inner 1050.3.c.c 32
5.b even 2 1 inner 1050.3.c.c 32
5.c odd 4 1 210.3.e.a ✓ 16
5.c odd 4 1 1050.3.e.d 16
15.d odd 2 1 inner 1050.3.c.c 32
15.e even 4 1 210.3.e.a ✓ 16
15.e even 4 1 1050.3.e.d 16
20.e even 4 1 1680.3.l.c 16
60.l odd 4 1 1680.3.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.e.a ✓ 16 5.c odd 4 1
210.3.e.a ✓ 16 15.e even 4 1
1050.3.c.c 32 1.a even 1 1 trivial
1050.3.c.c 32 3.b odd 2 1 inner
1050.3.c.c 32 5.b even 2 1 inner
1050.3.c.c 32 15.d odd 2 1 inner
1050.3.e.d 16 5.c odd 4 1
1050.3.e.d 16 15.e even 4 1
1680.3.l.c 16 20.e even 4 1
1680.3.l.c 16 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 1348 T_{11}^{14} + 718702 T_{11}^{12} + 193917500 T_{11}^{10} + 28186215185 T_{11}^{8} + \cdots + 33\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display