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Properties

Label 1610.2.e.d
Level $1610$
Weight $2$
Character orbit 1610.e
Analytic conductor $12.856$
Analytic rank $0$
Dimension $22$
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] = [1610,2,Mod(1289,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1610.1289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.e (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: \(12.8559147254\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{4} + 8 q^{6} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{4} + 8 q^{6} - 38 q^{9} - 2 q^{10} + 20 q^{11} - 22 q^{14} - 8 q^{15} + 22 q^{16} + 8 q^{21} - 8 q^{24} - 2 q^{25} + 20 q^{26} - 24 q^{29} - 24 q^{30} - 32 q^{34} - 2 q^{35} + 38 q^{36} - 24 q^{39} + 2 q^{40} + 16 q^{41} - 20 q^{44} - 40 q^{45} - 22 q^{46} - 22 q^{49} + 4 q^{50} + 8 q^{51} - 20 q^{54} - 8 q^{55} + 22 q^{56} + 12 q^{59} + 8 q^{60} - 20 q^{61} - 22 q^{64} - 16 q^{65} + 56 q^{66} + 8 q^{69} + 28 q^{71} - 28 q^{74} - 28 q^{75} + 24 q^{79} + 94 q^{81} - 8 q^{84} + 28 q^{85} + 60 q^{86} + 12 q^{89} + 30 q^{90} + 20 q^{91} - 52 q^{94} - 52 q^{95} + 8 q^{96} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1289.1 − 1.00000i − 3.37679i −1.00000 2.17944 + 0.500052i −3.37679 − 1.00000i 1.00000i −8.40272 0.500052 − 2.17944i
1289.2 − 1.00000i − 2.22125i −1.00000 1.30169 − 1.81813i −2.22125 − 1.00000i 1.00000i −1.93397 −1.81813 − 1.30169i
1289.3 − 1.00000i − 1.39348i −1.00000 −2.17605 − 0.514590i −1.39348 − 1.00000i 1.00000i 1.05822 −0.514590 + 2.17605i
1289.4 − 1.00000i − 1.31996i −1.00000 0.795975 − 2.08960i −1.31996 − 1.00000i 1.00000i 1.25771 −2.08960 − 0.795975i
1289.5 − 1.00000i − 0.0748459i −1.00000 −1.17523 + 1.90233i −0.0748459 − 1.00000i 1.00000i 2.99440 1.90233 + 1.17523i
1289.6 − 1.00000i 0.233058i −1.00000 1.41505 + 1.73137i 0.233058 − 1.00000i 1.00000i 2.94568 1.73137 − 1.41505i
1289.7 − 1.00000i 1.26489i −1.00000 −1.46795 − 1.68675i 1.26489 − 1.00000i 1.00000i 1.40006 −1.68675 + 1.46795i
1289.8 − 1.00000i 2.26372i −1.00000 −0.617916 + 2.14900i 2.26372 − 1.00000i 1.00000i −2.12441 2.14900 + 0.617916i
1289.9 − 1.00000i 2.58400i −1.00000 2.07538 + 0.832333i 2.58400 − 1.00000i 1.00000i −3.67703 0.832333 − 2.07538i
1289.10 − 1.00000i 2.65080i −1.00000 −2.22446 + 0.227556i 2.65080 − 1.00000i 1.00000i −4.02674 0.227556 + 2.22446i
1289.11 − 1.00000i 3.38987i −1.00000 −0.105933 − 2.23356i 3.38987 − 1.00000i 1.00000i −8.49120 −2.23356 + 0.105933i
1289.12 1.00000i − 3.38987i −1.00000 −0.105933 + 2.23356i 3.38987 1.00000i − 1.00000i −8.49120 −2.23356 − 0.105933i
1289.13 1.00000i − 2.65080i −1.00000 −2.22446 − 0.227556i 2.65080 1.00000i − 1.00000i −4.02674 0.227556 − 2.22446i
1289.14 1.00000i − 2.58400i −1.00000 2.07538 − 0.832333i 2.58400 1.00000i − 1.00000i −3.67703 0.832333 + 2.07538i
1289.15 1.00000i − 2.26372i −1.00000 −0.617916 − 2.14900i 2.26372 1.00000i − 1.00000i −2.12441 2.14900 − 0.617916i
1289.16 1.00000i − 1.26489i −1.00000 −1.46795 + 1.68675i 1.26489 1.00000i − 1.00000i 1.40006 −1.68675 − 1.46795i
1289.17 1.00000i − 0.233058i −1.00000 1.41505 − 1.73137i 0.233058 1.00000i − 1.00000i 2.94568 1.73137 + 1.41505i
1289.18 1.00000i 0.0748459i −1.00000 −1.17523 − 1.90233i −0.0748459 1.00000i − 1.00000i 2.99440 1.90233 − 1.17523i
1289.19 1.00000i 1.31996i −1.00000 0.795975 + 2.08960i −1.31996 1.00000i − 1.00000i 1.25771 −2.08960 + 0.795975i
1289.20 1.00000i 1.39348i −1.00000 −2.17605 + 0.514590i −1.39348 1.00000i − 1.00000i 1.05822 −0.514590 − 2.17605i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1289.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1610.2.e.d ✓ 22
5.b even 2 1 inner 1610.2.e.d ✓ 22
5.c odd 4 1 8050.2.a.ci 11
5.c odd 4 1 8050.2.a.cj 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1610.2.e.d ✓ 22 1.a even 1 1 trivial
1610.2.e.d ✓ 22 5.b even 2 1 inner
8050.2.a.ci 11 5.c odd 4 1
8050.2.a.cj 11 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 52 T_{3}^{20} + 1144 T_{3}^{18} + 13924 T_{3}^{16} + 102924 T_{3}^{14} + 477092 T_{3}^{12} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(1610, [\chi])\). Copy content Toggle raw display