Properties

Label 1860.2.i.b
Level $1860$
Weight $2$
Character orbit 1860.i
Analytic conductor $14.852$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1860,2,Mod(929,1860)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1860, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1860.929");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.i (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: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9} - 8 q^{19} + 64 q^{25} + 24 q^{31} + 8 q^{39} - 16 q^{45} - 160 q^{49} + 68 q^{51} - 24 q^{69} + 52 q^{81}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1 0 −1.72938 0.0961633i 0 −1.38738 + 1.75362i 0 3.39946i 0 2.98151 + 0.332606i 0
929.2 0 −1.72938 0.0961633i 0 1.38738 + 1.75362i 0 3.39946i 0 2.98151 + 0.332606i 0
929.3 0 −1.72938 + 0.0961633i 0 −1.38738 1.75362i 0 3.39946i 0 2.98151 0.332606i 0
929.4 0 −1.72938 + 0.0961633i 0 1.38738 1.75362i 0 3.39946i 0 2.98151 0.332606i 0
929.5 0 −1.62488 0.599792i 0 −2.11867 + 0.714996i 0 1.63784i 0 2.28050 + 1.94918i 0
929.6 0 −1.62488 0.599792i 0 2.11867 + 0.714996i 0 1.63784i 0 2.28050 + 1.94918i 0
929.7 0 −1.62488 + 0.599792i 0 −2.11867 0.714996i 0 1.63784i 0 2.28050 1.94918i 0
929.8 0 −1.62488 + 0.599792i 0 2.11867 0.714996i 0 1.63784i 0 2.28050 1.94918i 0
929.9 0 −1.37690 1.05079i 0 −2.02481 0.948759i 0 3.69636i 0 0.791696 + 2.89365i 0
929.10 0 −1.37690 1.05079i 0 2.02481 0.948759i 0 3.69636i 0 0.791696 + 2.89365i 0
929.11 0 −1.37690 + 1.05079i 0 −2.02481 + 0.948759i 0 3.69636i 0 0.791696 2.89365i 0
929.12 0 −1.37690 + 1.05079i 0 2.02481 + 0.948759i 0 3.69636i 0 0.791696 2.89365i 0
929.13 0 −0.910753 1.47327i 0 −1.00177 + 1.99911i 0 4.87923i 0 −1.34106 + 2.68357i 0
929.14 0 −0.910753 1.47327i 0 1.00177 + 1.99911i 0 4.87923i 0 −1.34106 + 2.68357i 0
929.15 0 −0.910753 + 1.47327i 0 −1.00177 1.99911i 0 4.87923i 0 −1.34106 2.68357i 0
929.16 0 −0.910753 + 1.47327i 0 1.00177 1.99911i 0 4.87923i 0 −1.34106 2.68357i 0
929.17 0 −0.577293 1.63301i 0 −2.10556 0.752745i 0 0.552709i 0 −2.33347 + 1.88545i 0
929.18 0 −0.577293 1.63301i 0 2.10556 0.752745i 0 0.552709i 0 −2.33347 + 1.88545i 0
929.19 0 −0.577293 + 1.63301i 0 −2.10556 + 0.752745i 0 0.552709i 0 −2.33347 1.88545i 0
929.20 0 −0.577293 + 1.63301i 0 2.10556 + 0.752745i 0 0.552709i 0 −2.33347 1.88545i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.48
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
31.b odd 2 1 inner
93.c even 2 1 inner
155.c odd 2 1 inner
465.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.i.b 48
3.b odd 2 1 inner 1860.2.i.b 48
5.b even 2 1 inner 1860.2.i.b 48
15.d odd 2 1 inner 1860.2.i.b 48
31.b odd 2 1 inner 1860.2.i.b 48
93.c even 2 1 inner 1860.2.i.b 48
155.c odd 2 1 inner 1860.2.i.b 48
465.g even 2 1 inner 1860.2.i.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.i.b 48 1.a even 1 1 trivial
1860.2.i.b 48 3.b odd 2 1 inner
1860.2.i.b 48 5.b even 2 1 inner
1860.2.i.b 48 15.d odd 2 1 inner
1860.2.i.b 48 31.b odd 2 1 inner
1860.2.i.b 48 93.c even 2 1 inner
1860.2.i.b 48 155.c odd 2 1 inner
1860.2.i.b 48 465.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 62T_{7}^{10} + 1425T_{7}^{8} + 15108T_{7}^{6} + 72416T_{7}^{4} + 121444T_{7}^{2} + 30760 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\). Copy content Toggle raw display