Properties

Label 1476.2.x.b
Level $1476$
Weight $2$
Character orbit 1476.x
Analytic conductor $11.786$
Analytic rank $0$
Dimension $28$
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] = [1476,2,Mod(161,1476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1476, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 4, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1476.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1476.x (of order \(8\), degree \(4\), 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: \(11.7859193383\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{5} + 4 q^{11} + 12 q^{13} + 4 q^{17} - 16 q^{23} - 4 q^{29} + 20 q^{35} - 8 q^{37} + 12 q^{41} - 8 q^{43} - 16 q^{47} - 8 q^{53} - 40 q^{55} - 12 q^{61} - 8 q^{65} + 32 q^{67} + 40 q^{71} + 24 q^{73} + 24 q^{79} + 4 q^{85} + 36 q^{89} + 32 q^{91} - 20 q^{95} + 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 0 0 −2.81045 + 2.81045i 0 −1.84108 0.762601i 0 0 0
161.2 0 0 0 −1.96503 + 1.96503i 0 1.04671 + 0.433563i 0 0 0
161.3 0 0 0 −0.160559 + 0.160559i 0 −1.58587 0.656890i 0 0 0
161.4 0 0 0 0.659514 0.659514i 0 2.33734 + 0.968156i 0 0 0
161.5 0 0 0 1.08307 1.08307i 0 −3.46391 1.43480i 0 0 0
161.6 0 0 0 1.75916 1.75916i 0 4.68497 + 1.94058i 0 0 0
161.7 0 0 0 2.43430 2.43430i 0 −1.17816 0.488012i 0 0 0
413.1 0 0 0 −2.56900 + 2.56900i 0 0.926138 2.23590i 0 0 0
413.2 0 0 0 −0.991747 + 0.991747i 0 −1.53086 + 3.69582i 0 0 0
413.3 0 0 0 −0.790254 + 0.790254i 0 0.489069 1.18072i 0 0 0
413.4 0 0 0 −0.358870 + 0.358870i 0 0.130495 0.315043i 0 0 0
413.5 0 0 0 1.44287 1.44287i 0 −0.365476 + 0.882337i 0 0 0
413.6 0 0 0 1.95510 1.95510i 0 −1.33274 + 3.21753i 0 0 0
413.7 0 0 0 2.31190 2.31190i 0 1.68338 4.06403i 0 0 0
629.1 0 0 0 −2.56900 2.56900i 0 0.926138 + 2.23590i 0 0 0
629.2 0 0 0 −0.991747 0.991747i 0 −1.53086 3.69582i 0 0 0
629.3 0 0 0 −0.790254 0.790254i 0 0.489069 + 1.18072i 0 0 0
629.4 0 0 0 −0.358870 0.358870i 0 0.130495 + 0.315043i 0 0 0
629.5 0 0 0 1.44287 + 1.44287i 0 −0.365476 0.882337i 0 0 0
629.6 0 0 0 1.95510 + 1.95510i 0 −1.33274 3.21753i 0 0 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
123.i even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1476.2.x.b yes 28
3.b odd 2 1 1476.2.x.a 28
41.e odd 8 1 1476.2.x.a 28
123.i even 8 1 inner 1476.2.x.b yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1476.2.x.a 28 3.b odd 2 1
1476.2.x.a 28 41.e odd 8 1
1476.2.x.b yes 28 1.a even 1 1 trivial
1476.2.x.b yes 28 123.i even 8 1 inner