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Properties

Label 1848.2.f.b
Level $1848$
Weight $2$
Character orbit 1848.f
Analytic conductor $14.756$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1848,2,Mod(1121,1848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1848, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("1848.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1848.f (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: \(14.7563542935\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} - 6 q^{9} + 4 q^{11} + 6 q^{15} + 4 q^{17} - 40 q^{25} + 4 q^{27} + 8 q^{29} - 8 q^{31} + 6 q^{33} - 4 q^{35} + 4 q^{37} + 4 q^{39} - 4 q^{41} + 18 q^{45} - 36 q^{49} + 16 q^{55} + 4 q^{57} + 4 q^{63} - 16 q^{65} + 36 q^{67} + 10 q^{69} + 40 q^{75} + 4 q^{77} - 26 q^{81} - 28 q^{83} + 48 q^{87} - 22 q^{93} + 48 q^{95} + 36 q^{97} - 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1 0 −1.69679 − 0.347701i 0 2.12201i 0 1.00000i 0 2.75821 + 1.17995i 0
1121.2 0 −1.69679 + 0.347701i 0 − 2.12201i 0 − 1.00000i 0 2.75821 − 1.17995i 0
1121.3 0 −1.68367 − 0.406499i 0 − 2.55990i 0 1.00000i 0 2.66952 + 1.36882i 0
1121.4 0 −1.68367 + 0.406499i 0 2.55990i 0 − 1.00000i 0 2.66952 − 1.36882i 0
1121.5 0 −1.48566 − 0.890394i 0 − 3.79118i 0 1.00000i 0 1.41440 + 2.64565i 0
1121.6 0 −1.48566 + 0.890394i 0 3.79118i 0 − 1.00000i 0 1.41440 − 2.64565i 0
1121.7 0 −1.42140 − 0.989759i 0 1.87784i 0 − 1.00000i 0 1.04075 + 2.81369i 0
1121.8 0 −1.42140 + 0.989759i 0 − 1.87784i 0 1.00000i 0 1.04075 − 2.81369i 0
1121.9 0 −1.14696 − 1.29788i 0 − 1.59454i 0 − 1.00000i 0 −0.368968 + 2.97722i 0
1121.10 0 −1.14696 + 1.29788i 0 1.59454i 0 1.00000i 0 −0.368968 − 2.97722i 0
1121.11 0 −0.961861 − 1.44043i 0 4.24689i 0 1.00000i 0 −1.14965 + 2.77098i 0
1121.12 0 −0.961861 + 1.44043i 0 − 4.24689i 0 − 1.00000i 0 −1.14965 − 2.77098i 0
1121.13 0 −0.856137 − 1.50567i 0 − 0.624815i 0 1.00000i 0 −1.53406 + 2.57811i 0
1121.14 0 −0.856137 + 1.50567i 0 0.624815i 0 − 1.00000i 0 −1.53406 − 2.57811i 0
1121.15 0 −0.606925 − 1.62223i 0 0.476296i 0 − 1.00000i 0 −2.26328 + 1.96915i 0
1121.16 0 −0.606925 + 1.62223i 0 − 0.476296i 0 1.00000i 0 −2.26328 − 1.96915i 0
1121.17 0 −0.374165 − 1.69115i 0 − 3.82600i 0 − 1.00000i 0 −2.72000 + 1.26554i 0
1121.18 0 −0.374165 + 1.69115i 0 3.82600i 0 1.00000i 0 −2.72000 − 1.26554i 0
1121.19 0 0.202579 − 1.72016i 0 − 0.521499i 0 1.00000i 0 −2.91792 − 0.696938i 0
1121.20 0 0.202579 + 1.72016i 0 0.521499i 0 − 1.00000i 0 −2.91792 + 0.696938i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1848.2.f.b yes 36
3.b odd 2 1 1848.2.f.a ✓ 36
11.b odd 2 1 1848.2.f.a ✓ 36
33.d even 2 1 inner 1848.2.f.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.2.f.a ✓ 36 3.b odd 2 1
1848.2.f.a ✓ 36 11.b odd 2 1
1848.2.f.b yes 36 1.a even 1 1 trivial
1848.2.f.b yes 36 33.d even 2 1 inner