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Properties

Label 810.2.p.a
Level $810$
Weight $2$
Character orbit 810.p
Analytic conductor $6.468$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,2,Mod(19,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.p (of order \(18\), 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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{5} + 12 q^{11} - 6 q^{14} - 12 q^{20} - 18 q^{25} + 72 q^{26} - 6 q^{29} + 36 q^{31} - 18 q^{35} + 12 q^{41} + 12 q^{44} + 18 q^{49} + 12 q^{50} + 6 q^{56} + 84 q^{59} - 18 q^{61} + 54 q^{64} + 6 q^{65} + 48 q^{74} - 72 q^{79} + 36 q^{86} + 132 q^{89} - 36 q^{94} + 18 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i −2.12627 − 0.692069i 0 −1.94767 + 2.32114i 0.866025 + 0.500000i 0 0.0768969 + 2.23475i
19.2 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i −2.00523 + 0.989469i 0 0.650202 − 0.774881i 0.866025 + 0.500000i 0 1.61563 + 1.54588i
19.3 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i −1.27128 − 1.83952i 0 2.50497 − 2.98531i 0.866025 + 0.500000i 0 −1.29378 + 1.82377i
19.4 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i −0.326956 + 2.21204i 0 2.21166 − 2.63576i 0.866025 + 0.500000i 0 2.19046 − 0.449323i
19.5 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i −0.298468 + 2.21606i 0 −2.57922 + 3.07380i 0.866025 + 0.500000i 0 2.18450 − 0.477468i
19.6 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i 1.09596 + 1.94907i 0 0.582960 − 0.694744i 0.866025 + 0.500000i 0 1.45668 − 1.69649i
19.7 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i 1.17383 − 1.90318i 0 −0.974601 + 1.16148i 0.866025 + 0.500000i 0 −2.18988 − 0.452116i
19.8 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i 1.78992 − 1.34021i 0 −1.09389 + 1.30365i 0.866025 + 0.500000i 0 −1.87158 − 1.22360i
19.9 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i 2.23453 − 0.0828267i 0 −1.70233 + 2.02876i 0.866025 + 0.500000i 0 −0.842087 − 2.07145i
19.10 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −2.14036 − 0.647200i 0 1.09389 − 1.30365i −0.866025 − 0.500000i 0 −0.123877 − 2.23263i
19.11 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −2.12810 + 0.686424i 0 1.70233 − 2.02876i −0.866025 − 0.500000i 0 −1.37288 − 1.76499i
19.12 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −1.75397 − 1.38693i 0 0.974601 − 1.16148i −0.866025 − 0.500000i 0 0.703398 − 2.12255i
19.13 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −0.363249 + 2.20637i 0 −0.582960 + 0.694744i −0.866025 − 0.500000i 0 −2.19754 + 0.413280i
19.14 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 0.565460 − 2.16339i 0 −2.50497 + 2.98531i −0.866025 − 0.500000i 0 2.22632 − 0.208565i
19.15 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 1.03841 + 1.98033i 0 2.57922 − 3.07380i −0.866025 − 0.500000i 0 −1.50575 + 1.65310i
19.16 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 1.06380 + 1.96681i 0 −2.21166 + 2.63576i −0.866025 − 0.500000i 0 −1.48435 + 1.67233i
19.17 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 1.76134 − 1.37756i 0 1.94767 − 2.32114i −0.866025 − 0.500000i 0 1.89690 + 1.18397i
19.18 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 2.22272 + 0.243967i 0 −0.650202 + 0.774881i −0.866025 − 0.500000i 0 0.530960 + 2.17211i
199.1 −0.642788 − 0.766044i 0 −0.173648 + 0.984808i −2.21219 + 0.325886i 0 −0.531122 + 0.0936511i 0.866025 − 0.500000i 0 1.67161 + 1.48516i
199.2 −0.642788 − 0.766044i 0 −0.173648 + 0.984808i −1.96492 − 1.06728i 0 4.83804 − 0.853077i 0.866025 − 0.500000i 0 0.445446 + 2.19125i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.e even 9 1 inner
135.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.p.a 108
3.b odd 2 1 270.2.p.a ✓ 108
5.b even 2 1 inner 810.2.p.a 108
15.d odd 2 1 270.2.p.a ✓ 108
27.e even 9 1 inner 810.2.p.a 108
27.f odd 18 1 270.2.p.a ✓ 108
135.n odd 18 1 270.2.p.a ✓ 108
135.p even 18 1 inner 810.2.p.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.p.a ✓ 108 3.b odd 2 1
270.2.p.a ✓ 108 15.d odd 2 1
270.2.p.a ✓ 108 27.f odd 18 1
270.2.p.a ✓ 108 135.n odd 18 1
810.2.p.a 108 1.a even 1 1 trivial
810.2.p.a 108 5.b even 2 1 inner
810.2.p.a 108 27.e even 9 1 inner
810.2.p.a 108 135.p even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(810, [\chi])\).