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Properties

Label 810.2.s.a
Level $810$
Weight $2$
Character orbit 810.s
Analytic conductor $6.468$
Analytic rank $0$
Dimension $216$
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(17,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([22, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.s (of order \(36\), degree \(12\), 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: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 24 q^{11} + 24 q^{20} - 24 q^{23} + 36 q^{25} + 108 q^{35} + 36 q^{38} + 72 q^{41} + 48 q^{47} + 48 q^{50} + 12 q^{56} + 36 q^{61} - 24 q^{65} - 72 q^{67} - 36 q^{68} + 240 q^{77} + 60 q^{83} - 72 q^{86} + 48 q^{92} + 60 q^{95} - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.906308 + 0.422618i 0 0.642788 − 0.766044i −2.23440 − 0.0864530i 0 −3.15874 − 0.276354i −0.258819 + 0.965926i 0 2.06159 − 0.865944i
17.2 −0.906308 + 0.422618i 0 0.642788 − 0.766044i −2.00343 + 0.993102i 0 0.000512095 0 4.48025e-5i −0.258819 + 0.965926i 0 1.39602 − 1.74674i
17.3 −0.906308 + 0.422618i 0 0.642788 − 0.766044i −0.993387 − 2.00329i 0 1.47317 + 0.128886i −0.258819 + 0.965926i 0 1.74694 + 1.39578i
17.4 −0.906308 + 0.422618i 0 0.642788 − 0.766044i −0.842674 + 2.07121i 0 1.73135 + 0.151473i −0.258819 + 0.965926i 0 −0.111608 − 2.23328i
17.5 −0.906308 + 0.422618i 0 0.642788 − 0.766044i 0.460985 − 2.18803i 0 −1.84790 − 0.161670i −0.258819 + 0.965926i 0 0.506909 + 2.17785i
17.6 −0.906308 + 0.422618i 0 0.642788 − 0.766044i 0.893056 + 2.04999i 0 −3.58615 − 0.313747i −0.258819 + 0.965926i 0 −1.67575 − 1.48050i
17.7 −0.906308 + 0.422618i 0 0.642788 − 0.766044i 0.975338 − 2.01214i 0 5.12168 + 0.448089i −0.258819 + 0.965926i 0 −0.0335884 + 2.23582i
17.8 −0.906308 + 0.422618i 0 0.642788 − 0.766044i 1.39590 + 1.74684i 0 1.62157 + 0.141869i −0.258819 + 0.965926i 0 −2.00336 − 0.993246i
17.9 −0.906308 + 0.422618i 0 0.642788 − 0.766044i 2.12538 + 0.694824i 0 −3.31761 − 0.290253i −0.258819 + 0.965926i 0 −2.21989 + 0.268498i
17.10 0.906308 − 0.422618i 0 0.642788 − 0.766044i −2.23426 − 0.0899855i 0 −1.89299 − 0.165615i 0.258819 − 0.965926i 0 −2.06295 + 0.862683i
17.11 0.906308 − 0.422618i 0 0.642788 − 0.766044i −2.23011 + 0.163105i 0 3.73281 + 0.326578i 0.258819 − 0.965926i 0 −1.95224 + 1.09031i
17.12 0.906308 − 0.422618i 0 0.642788 − 0.766044i −2.04220 + 0.910725i 0 −1.36563 − 0.119477i 0.258819 − 0.965926i 0 −1.46597 + 1.68847i
17.13 0.906308 − 0.422618i 0 0.642788 − 0.766044i −0.639439 − 2.14269i 0 3.64169 + 0.318607i 0.258819 − 0.965926i 0 −1.48507 − 1.67170i
17.14 0.906308 − 0.422618i 0 0.642788 − 0.766044i 0.0174118 + 2.23600i 0 −4.56637 − 0.399506i 0.258819 − 0.965926i 0 0.960755 + 2.01915i
17.15 0.906308 − 0.422618i 0 0.642788 − 0.766044i 1.28469 + 1.83018i 0 0.618317 + 0.0540957i 0.258819 − 0.965926i 0 1.93780 + 1.11577i
17.16 0.906308 − 0.422618i 0 0.642788 − 0.766044i 1.58821 − 1.57404i 0 1.23570 + 0.108110i 0.258819 − 0.965926i 0 0.774190 − 2.09777i
17.17 0.906308 − 0.422618i 0 0.642788 − 0.766044i 2.00011 − 0.999773i 0 −3.41173 − 0.298488i 0.258819 − 0.965926i 0 1.39020 − 1.75139i
17.18 0.906308 − 0.422618i 0 0.642788 − 0.766044i 2.03234 + 0.932518i 0 3.97033 + 0.347359i 0.258819 − 0.965926i 0 2.23603 − 0.0137560i
143.1 −0.906308 − 0.422618i 0 0.642788 + 0.766044i −2.23440 + 0.0864530i 0 −3.15874 + 0.276354i −0.258819 − 0.965926i 0 2.06159 + 0.865944i
143.2 −0.906308 − 0.422618i 0 0.642788 + 0.766044i −2.00343 − 0.993102i 0 0.000512095 0 4.48025e-5i −0.258819 − 0.965926i 0 1.39602 + 1.74674i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.s.a 216
3.b odd 2 1 270.2.r.a ✓ 216
5.c odd 4 1 inner 810.2.s.a 216
15.e even 4 1 270.2.r.a ✓ 216
27.e even 9 1 270.2.r.a ✓ 216
27.f odd 18 1 inner 810.2.s.a 216
135.q even 36 1 inner 810.2.s.a 216
135.r odd 36 1 270.2.r.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.r.a ✓ 216 3.b odd 2 1
270.2.r.a ✓ 216 15.e even 4 1
270.2.r.a ✓ 216 27.e even 9 1
270.2.r.a ✓ 216 135.r odd 36 1
810.2.s.a 216 1.a even 1 1 trivial
810.2.s.a 216 5.c odd 4 1 inner
810.2.s.a 216 27.f odd 18 1 inner
810.2.s.a 216 135.q even 36 1 inner

Hecke kernels

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