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Properties

Label 1011.2.d.b
Level $1011$
Weight $2$
Character orbit 1011.d
Analytic conductor $8.073$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1011,2,Mod(673,1011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1011.673");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1011 = 3 \cdot 337 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1011.d (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: \(8.07287564435\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{3} + 40 q^{4} + 2 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{3} + 40 q^{4} + 2 q^{7} + 30 q^{9} - 40 q^{12} - 10 q^{13} + 2 q^{14} + 44 q^{16} - 2 q^{21} - 10 q^{25} - 30 q^{27} + 8 q^{28} - 20 q^{32} + 40 q^{36} + 24 q^{37} + 10 q^{39} - 2 q^{42} - 26 q^{43} - 8 q^{47} - 44 q^{48} + 40 q^{49} - 4 q^{50} - 20 q^{52} + 38 q^{55} - 2 q^{56} + 2 q^{63} + 44 q^{64} - 10 q^{74} + 10 q^{75} - 8 q^{79} + 30 q^{81} - 2 q^{82} - 8 q^{84} - 66 q^{85} - 30 q^{86} - 56 q^{91} + 8 q^{94} - 52 q^{95} + 20 q^{96} + 76 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1 −2.78188 −1.00000 5.73885 2.79359i 2.78188 −0.692312 −10.4010 1.00000 − 7.77144i
673.2 −2.78188 −1.00000 5.73885 − 2.79359i 2.78188 −0.692312 −10.4010 1.00000 7.77144i
673.3 −2.33807 −1.00000 3.46656 1.23419i 2.33807 4.43718 −3.42891 1.00000 − 2.88561i
673.4 −2.33807 −1.00000 3.46656 − 1.23419i 2.33807 4.43718 −3.42891 1.00000 2.88561i
673.5 −2.25294 −1.00000 3.07574 0.773438i 2.25294 −3.18023 −2.42358 1.00000 − 1.74251i
673.6 −2.25294 −1.00000 3.07574 − 0.773438i 2.25294 −3.18023 −2.42358 1.00000 1.74251i
673.7 −1.70880 −1.00000 0.919991 4.07741i 1.70880 −0.230137 1.84552 1.00000 − 6.96747i
673.8 −1.70880 −1.00000 0.919991 − 4.07741i 1.70880 −0.230137 1.84552 1.00000 6.96747i
673.9 −1.63745 −1.00000 0.681245 1.66935i 1.63745 0.440556 2.15940 1.00000 − 2.73348i
673.10 −1.63745 −1.00000 0.681245 − 1.66935i 1.63745 0.440556 2.15940 1.00000 2.73348i
673.11 −0.883195 −1.00000 −1.21997 0.740962i 0.883195 −2.77542 2.84386 1.00000 − 0.654414i
673.12 −0.883195 −1.00000 −1.21997 − 0.740962i 0.883195 −2.77542 2.84386 1.00000 0.654414i
673.13 −0.417103 −1.00000 −1.82602 2.22540i 0.417103 4.05906 1.59585 1.00000 − 0.928221i
673.14 −0.417103 −1.00000 −1.82602 − 2.22540i 0.417103 4.05906 1.59585 1.00000 0.928221i
673.15 −0.221017 −1.00000 −1.95115 3.19466i 0.221017 −0.204417 0.873270 1.00000 − 0.706074i
673.16 −0.221017 −1.00000 −1.95115 − 3.19466i 0.221017 −0.204417 0.873270 1.00000 0.706074i
673.17 0.537581 −1.00000 −1.71101 − 1.10247i −0.537581 −3.97479 −1.99497 1.00000 − 0.592666i
673.18 0.537581 −1.00000 −1.71101 1.10247i −0.537581 −3.97479 −1.99497 1.00000 0.592666i
673.19 0.972859 −1.00000 −1.05354 1.98474i −0.972859 1.49347 −2.97067 1.00000 1.93087i
673.20 0.972859 −1.00000 −1.05354 − 1.98474i −0.972859 1.49347 −2.97067 1.00000 − 1.93087i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
337.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1011.2.d.b ✓ 30
337.b even 2 1 inner 1011.2.d.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1011.2.d.b ✓ 30 1.a even 1 1 trivial
1011.2.d.b ✓ 30 337.b even 2 1 inner