↑

Properties

Label 561.2.s.a
Level $561$
Weight $2$
Character orbit 561.s
Analytic conductor $4.480$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [561,2,Mod(16,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.s (of order \(10\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 36 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 36 q^{4} + 36 q^{9} + 4 q^{15} - 20 q^{16} + 10 q^{17} + 20 q^{19} + 32 q^{21} - 2 q^{25} - 16 q^{26} + 8 q^{30} - 40 q^{32} + 2 q^{33} + 72 q^{34} - 56 q^{35} + 36 q^{36} + 132 q^{38} - 32 q^{42} - 80 q^{43} - 44 q^{47} + 28 q^{49} + 84 q^{50} - 16 q^{51} - 84 q^{52} + 44 q^{53} - 14 q^{55} + 4 q^{59} - 24 q^{60} + 56 q^{64} + 60 q^{66} - 40 q^{67} + 12 q^{68} + 30 q^{69} - 140 q^{70} + 8 q^{76} + 4 q^{77} - 36 q^{81} - 48 q^{83} - 120 q^{84} + 106 q^{85} + 8 q^{86} - 72 q^{87} + 24 q^{89} + 72 q^{93} + 244 q^{94} - 344 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.828842 − 2.55091i −0.587785 + 0.809017i −4.20215 + 3.05304i 1.62502 + 0.528000i 2.55091 + 0.828842i −2.42873 − 3.34286i 6.93108 + 5.03572i −0.309017 − 0.951057i − 4.58291i
16.2 −0.828842 − 2.55091i 0.587785 − 0.809017i −4.20215 + 3.05304i −1.62502 − 0.528000i −2.55091 − 0.828842i 2.42873 + 3.34286i 6.93108 + 5.03572i −0.309017 − 0.951057i 4.58291i
16.3 −0.749654 − 2.30720i −0.587785 + 0.809017i −3.14315 + 2.28363i −0.242616 − 0.0788306i 2.30720 + 0.749654i 0.558808 + 0.769133i 3.69982 + 2.68808i −0.309017 − 0.951057i 0.618858i
16.4 −0.749654 − 2.30720i 0.587785 − 0.809017i −3.14315 + 2.28363i 0.242616 + 0.0788306i −2.30720 − 0.749654i −0.558808 − 0.769133i 3.69982 + 2.68808i −0.309017 − 0.951057i − 0.618858i
16.5 −0.664392 − 2.04479i −0.587785 + 0.809017i −2.12171 + 1.54152i −3.41949 − 1.11106i 2.04479 + 0.664392i −2.09344 − 2.88138i 1.08292 + 0.786788i −0.309017 − 0.951057i 7.73032i
16.6 −0.664392 − 2.04479i 0.587785 − 0.809017i −2.12171 + 1.54152i 3.41949 + 1.11106i −2.04479 − 0.664392i 2.09344 + 2.88138i 1.08292 + 0.786788i −0.309017 − 0.951057i − 7.73032i
16.7 −0.557601 − 1.71612i −0.587785 + 0.809017i −1.01611 + 0.738248i −2.28408 − 0.742141i 1.71612 + 0.557601i 2.37775 + 3.27269i −1.08613 − 0.789120i −0.309017 − 0.951057i 4.33356i
16.8 −0.557601 − 1.71612i 0.587785 − 0.809017i −1.01611 + 0.738248i 2.28408 + 0.742141i −1.71612 − 0.557601i −2.37775 − 3.27269i −1.08613 − 0.789120i −0.309017 − 0.951057i − 4.33356i
16.9 −0.473603 − 1.45760i −0.587785 + 0.809017i −0.282265 + 0.205078i 0.665302 + 0.216170i 1.45760 + 0.473603i −1.73774 − 2.39180i −2.04721 − 1.48739i −0.309017 − 0.951057i − 1.07212i
16.10 −0.473603 − 1.45760i 0.587785 − 0.809017i −0.282265 + 0.205078i −0.665302 − 0.216170i −1.45760 − 0.473603i 1.73774 + 2.39180i −2.04721 − 1.48739i −0.309017 − 0.951057i 1.07212i
16.11 −0.429555 − 1.32203i −0.587785 + 0.809017i 0.0547763 − 0.0397973i 1.66294 + 0.540323i 1.32203 + 0.429555i 1.54006 + 2.11971i −2.32532 − 1.68945i −0.309017 − 0.951057i − 2.43057i
16.12 −0.429555 − 1.32203i 0.587785 − 0.809017i 0.0547763 − 0.0397973i −1.66294 − 0.540323i −1.32203 − 0.429555i −1.54006 − 2.11971i −2.32532 − 1.68945i −0.309017 − 0.951057i 2.43057i
16.13 −0.389122 − 1.19760i −0.587785 + 0.809017i 0.335216 − 0.243549i 4.04992 + 1.31590i 1.19760 + 0.389122i −0.309195 − 0.425571i −2.45958 − 1.78699i −0.309017 − 0.951057i − 5.36221i
16.14 −0.389122 − 1.19760i 0.587785 − 0.809017i 0.335216 − 0.243549i −4.04992 − 1.31590i −1.19760 − 0.389122i 0.309195 + 0.425571i −2.45958 − 1.78699i −0.309017 − 0.951057i 5.36221i
16.15 −0.130881 − 0.402810i −0.587785 + 0.809017i 1.47291 − 1.07013i −3.08420 − 1.00212i 0.402810 + 0.130881i 1.34024 + 1.84468i −1.30914 − 0.951143i −0.309017 − 0.951057i 1.37351i
16.16 −0.130881 − 0.402810i 0.587785 − 0.809017i 1.47291 − 1.07013i 3.08420 + 1.00212i −0.402810 − 0.130881i −1.34024 − 1.84468i −1.30914 − 0.951143i −0.309017 − 0.951057i − 1.37351i
16.17 −0.108654 − 0.334404i −0.587785 + 0.809017i 1.51801 − 1.10290i −1.31299 − 0.426616i 0.334404 + 0.108654i −0.627587 − 0.863800i −1.10268 − 0.801140i −0.309017 − 0.951057i 0.485422i
16.18 −0.108654 − 0.334404i 0.587785 − 0.809017i 1.51801 − 1.10290i 1.31299 + 0.426616i −0.334404 − 0.108654i 0.627587 + 0.863800i −1.10268 − 0.801140i −0.309017 − 0.951057i − 0.485422i
16.19 0.203584 + 0.626567i −0.587785 + 0.809017i 1.26689 − 0.920453i 0.710023 + 0.230700i −0.626567 − 0.203584i −2.79700 − 3.84974i 1.90062 + 1.38088i −0.309017 − 0.951057i 0.491843i
16.20 0.203584 + 0.626567i 0.587785 − 0.809017i 1.26689 − 0.920453i −0.710023 − 0.230700i 0.626567 + 0.203584i 2.79700 + 3.84974i 1.90062 + 1.38088i −0.309017 − 0.951057i − 0.491843i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.b even 2 1 inner
187.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.s.a ✓ 144
11.c even 5 1 inner 561.2.s.a ✓ 144
17.b even 2 1 inner 561.2.s.a ✓ 144
187.j even 10 1 inner 561.2.s.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.s.a ✓ 144 1.a even 1 1 trivial
561.2.s.a ✓ 144 11.c even 5 1 inner
561.2.s.a ✓ 144 17.b even 2 1 inner
561.2.s.a ✓ 144 187.j even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(561, [\chi])\).