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Properties

Label 1050.3.c.b
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: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 16 q^{6} - 16 q^{9} + 128 q^{16} + 48 q^{19} + 56 q^{21} - 32 q^{24} + 48 q^{31} + 256 q^{34} - 32 q^{36} + 192 q^{39} + 160 q^{46} - 224 q^{49} + 288 q^{51} - 80 q^{54} - 112 q^{61} + 256 q^{64} - 192 q^{66} + 344 q^{69} + 96 q^{76} - 256 q^{79} + 160 q^{81} + 112 q^{84} - 448 q^{91} + 416 q^{94} - 64 q^{96} - 304 q^{99}+O(q^{100}) \) 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 −2.79495 − 1.09006i 2.00000 0 3.95266 + 1.54158i 2.64575i −2.82843 6.62354 + 6.09333i 0
449.2 −1.41421 −2.79495 + 1.09006i 2.00000 0 3.95266 − 1.54158i − 2.64575i −2.82843 6.62354 − 6.09333i 0
449.3 −1.41421 −2.16802 − 2.07357i 2.00000 0 3.06604 + 2.93247i 2.64575i −2.82843 0.400623 + 8.99108i 0
449.4 −1.41421 −2.16802 + 2.07357i 2.00000 0 3.06604 − 2.93247i − 2.64575i −2.82843 0.400623 − 8.99108i 0
449.5 −1.41421 −0.577861 − 2.94382i 2.00000 0 0.817218 + 4.16319i 2.64575i −2.82843 −8.33215 + 3.40224i 0
449.6 −1.41421 −0.577861 + 2.94382i 2.00000 0 0.817218 − 4.16319i − 2.64575i −2.82843 −8.33215 − 3.40224i 0
449.7 −1.41421 −0.236961 − 2.99063i 2.00000 0 0.335113 + 4.22939i − 2.64575i −2.82843 −8.88770 + 1.41732i 0
449.8 −1.41421 −0.236961 + 2.99063i 2.00000 0 0.335113 − 4.22939i 2.64575i −2.82843 −8.88770 − 1.41732i 0
449.9 −1.41421 0.836130 − 2.88113i 2.00000 0 −1.18247 + 4.07453i 2.64575i −2.82843 −7.60177 − 4.81799i 0
449.10 −1.41421 0.836130 + 2.88113i 2.00000 0 −1.18247 − 4.07453i − 2.64575i −2.82843 −7.60177 + 4.81799i 0
449.11 −1.41421 2.24820 − 1.98636i 2.00000 0 −3.17943 + 2.80913i − 2.64575i −2.82843 1.10878 − 8.93144i 0
449.12 −1.41421 2.24820 + 1.98636i 2.00000 0 −3.17943 − 2.80913i 2.64575i −2.82843 1.10878 + 8.93144i 0
449.13 −1.41421 2.53883 − 1.59823i 2.00000 0 −3.59045 + 2.26024i 2.64575i −2.82843 3.89133 − 8.11526i 0
449.14 −1.41421 2.53883 + 1.59823i 2.00000 0 −3.59045 − 2.26024i − 2.64575i −2.82843 3.89133 + 8.11526i 0
449.15 −1.41421 2.98306 − 0.318317i 2.00000 0 −4.21869 + 0.450168i − 2.64575i −2.82843 8.79735 − 1.89912i 0
449.16 −1.41421 2.98306 + 0.318317i 2.00000 0 −4.21869 − 0.450168i 2.64575i −2.82843 8.79735 + 1.89912i 0
449.17 1.41421 −2.98306 − 0.318317i 2.00000 0 −4.21869 − 0.450168i − 2.64575i 2.82843 8.79735 + 1.89912i 0
449.18 1.41421 −2.98306 + 0.318317i 2.00000 0 −4.21869 + 0.450168i 2.64575i 2.82843 8.79735 − 1.89912i 0
449.19 1.41421 −2.53883 − 1.59823i 2.00000 0 −3.59045 − 2.26024i 2.64575i 2.82843 3.89133 + 8.11526i 0
449.20 1.41421 −2.53883 + 1.59823i 2.00000 0 −3.59045 + 2.26024i − 2.64575i 2.82843 3.89133 − 8.11526i 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.b 32
3.b odd 2 1 inner 1050.3.c.b 32
5.b even 2 1 inner 1050.3.c.b 32
5.c odd 4 1 1050.3.e.b ✓ 16
5.c odd 4 1 1050.3.e.c yes 16
15.d odd 2 1 inner 1050.3.c.b 32
15.e even 4 1 1050.3.e.b ✓ 16
15.e even 4 1 1050.3.e.c yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.c.b 32 1.a even 1 1 trivial
1050.3.c.b 32 3.b odd 2 1 inner
1050.3.c.b 32 5.b even 2 1 inner
1050.3.c.b 32 15.d odd 2 1 inner
1050.3.e.b ✓ 16 5.c odd 4 1
1050.3.e.b ✓ 16 15.e even 4 1
1050.3.e.c yes 16 5.c odd 4 1
1050.3.e.c yes 16 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 1168 T_{11}^{14} + 519460 T_{11}^{12} + 115450208 T_{11}^{10} + 13781338958 T_{11}^{8} + \cdots + 917436908786841 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display