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Properties

Label 1620.2.u.a
Level $1620$
Weight $2$
Character orbit 1620.u
Analytic conductor $12.936$
Analytic rank $0$
Dimension $30$
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] = [1620,2,Mod(181,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.u (of order \(9\), degree \(6\), 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: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{13} - 12 q^{17} - 18 q^{23} - 30 q^{29} - 12 q^{31} - 6 q^{35} + 24 q^{37} + 27 q^{41} - 6 q^{43} + 45 q^{47} + 48 q^{49} + 36 q^{53} + 21 q^{59} + 36 q^{61} - 6 q^{65} - 42 q^{67} - 12 q^{71} + 21 q^{73} + 6 q^{77} + 18 q^{79} + 21 q^{83} + 21 q^{85} + 3 q^{89} - 9 q^{91} - 21 q^{95} - 39 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 0 0 0 −0.173648 − 0.984808i 0 −3.99038 − 3.34833i 0 0 0
181.2 0 0 0 −0.173648 − 0.984808i 0 −1.29898 − 1.08998i 0 0 0
181.3 0 0 0 −0.173648 − 0.984808i 0 −0.224025 − 0.187979i 0 0 0
181.4 0 0 0 −0.173648 − 0.984808i 0 1.97942 + 1.66093i 0 0 0
181.5 0 0 0 −0.173648 − 0.984808i 0 2.00188 + 1.67977i 0 0 0
361.1 0 0 0 0.939693 − 0.342020i 0 −0.505301 − 2.86570i 0 0 0
361.2 0 0 0 0.939693 − 0.342020i 0 −0.327622 − 1.85804i 0 0 0
361.3 0 0 0 0.939693 − 0.342020i 0 0.0249994 + 0.141778i 0 0 0
361.4 0 0 0 0.939693 − 0.342020i 0 0.105751 + 0.599742i 0 0 0
361.5 0 0 0 0.939693 − 0.342020i 0 0.354876 + 2.01260i 0 0 0
721.1 0 0 0 −0.766044 + 0.642788i 0 −3.23226 + 1.17645i 0 0 0
721.2 0 0 0 −0.766044 + 0.642788i 0 −1.86438 + 0.678579i 0 0 0
721.3 0 0 0 −0.766044 + 0.642788i 0 0.948207 − 0.345119i 0 0 0
721.4 0 0 0 −0.766044 + 0.642788i 0 1.61329 − 0.587189i 0 0 0
721.5 0 0 0 −0.766044 + 0.642788i 0 4.41453 − 1.60676i 0 0 0
901.1 0 0 0 −0.766044 − 0.642788i 0 −3.23226 − 1.17645i 0 0 0
901.2 0 0 0 −0.766044 − 0.642788i 0 −1.86438 − 0.678579i 0 0 0
901.3 0 0 0 −0.766044 − 0.642788i 0 0.948207 + 0.345119i 0 0 0
901.4 0 0 0 −0.766044 − 0.642788i 0 1.61329 + 0.587189i 0 0 0
901.5 0 0 0 −0.766044 − 0.642788i 0 4.41453 + 1.60676i 0 0 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.u.a 30
3.b odd 2 1 540.2.u.a ✓ 30
27.e even 9 1 inner 1620.2.u.a 30
27.f odd 18 1 540.2.u.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.u.a ✓ 30 3.b odd 2 1
540.2.u.a ✓ 30 27.f odd 18 1
1620.2.u.a 30 1.a even 1 1 trivial
1620.2.u.a 30 27.e even 9 1 inner