↑

Properties

Label 561.2.f.a
Level $561$
Weight $2$
Character orbit 561.f
Analytic conductor $4.480$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [561,2,Mod(494,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.494");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.f (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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{3} + 32 q^{4} - 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{3} + 32 q^{4} - 12 q^{8} + q^{9} - 7 q^{11} - 4 q^{12} - 3 q^{15} + 8 q^{16} + 32 q^{17} - 18 q^{18} - 4 q^{21} + 10 q^{22} - 12 q^{25} + 2 q^{27} + 28 q^{29} + 48 q^{30} - 10 q^{31} - 4 q^{32} + 30 q^{33} - 22 q^{36} - 6 q^{37} - 14 q^{39} - 18 q^{41} + 8 q^{42} - 42 q^{44} - 3 q^{45} - 22 q^{48} - 20 q^{49} + 8 q^{50} - q^{51} - 10 q^{54} - 23 q^{55} - 24 q^{58} + 22 q^{60} + 52 q^{62} + 28 q^{63} - 16 q^{64} + 6 q^{65} + 42 q^{66} + 18 q^{67} + 32 q^{68} + 5 q^{69} + 24 q^{70} - 40 q^{72} - 108 q^{74} - 6 q^{75} - 42 q^{77} - 6 q^{78} - 23 q^{81} - 4 q^{82} + 32 q^{83} - 70 q^{84} + 12 q^{87} - 8 q^{88} - 14 q^{90} + 40 q^{91} + 13 q^{93} + 6 q^{95} + 90 q^{96} - 30 q^{97} + 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
494.1 −2.64151 −1.38140 − 1.04486i 4.97758 2.07119i 3.64898 + 2.76002i − 1.29197i −7.86530 0.816523 + 2.88674i − 5.47107i
494.2 −2.64151 −1.38140 + 1.04486i 4.97758 − 2.07119i 3.64898 − 2.76002i 1.29197i −7.86530 0.816523 − 2.88674i 5.47107i
494.3 −2.30738 0.783608 − 1.54465i 3.32403 − 2.50446i −1.80809 + 3.56411i − 4.65961i −3.05504 −1.77192 − 2.42081i 5.77874i
494.4 −2.30738 0.783608 + 1.54465i 3.32403 2.50446i −1.80809 − 3.56411i 4.65961i −3.05504 −1.77192 + 2.42081i − 5.77874i
494.5 −2.24036 1.63481 − 0.572172i 3.01922 2.28561i −3.66258 + 1.28187i 2.48940i −2.28343 2.34524 − 1.87079i − 5.12059i
494.6 −2.24036 1.63481 + 0.572172i 3.01922 − 2.28561i −3.66258 − 1.28187i − 2.48940i −2.28343 2.34524 + 1.87079i 5.12059i
494.7 −1.97679 −1.10632 − 1.33269i 1.90770 − 3.63272i 2.18696 + 2.63444i 1.32210i 0.182458 −0.552110 + 2.94876i 7.18113i
494.8 −1.97679 −1.10632 + 1.33269i 1.90770 3.63272i 2.18696 − 2.63444i − 1.32210i 0.182458 −0.552110 − 2.94876i − 7.18113i
494.9 −1.48594 0.858679 − 1.50422i 0.208021 − 0.363104i −1.27595 + 2.23518i 2.96071i 2.66278 −1.52534 − 2.58328i 0.539552i
494.10 −1.48594 0.858679 + 1.50422i 0.208021 0.363104i −1.27595 − 2.23518i − 2.96071i 2.66278 −1.52534 + 2.58328i − 0.539552i
494.11 −0.939626 −1.61783 − 0.618559i −1.11710 − 2.64495i 1.52016 + 0.581214i − 1.45500i 2.92891 2.23477 + 2.00145i 2.48526i
494.12 −0.939626 −1.61783 + 0.618559i −1.11710 2.64495i 1.52016 − 0.581214i 1.45500i 2.92891 2.23477 − 2.00145i − 2.48526i
494.13 −0.434349 1.59295 − 0.680081i −1.81134 2.04726i −0.691896 + 0.295393i − 3.34808i 1.65545 2.07498 − 2.16667i − 0.889226i
494.14 −0.434349 1.59295 + 0.680081i −1.81134 − 2.04726i −0.691896 − 0.295393i 3.34808i 1.65545 2.07498 + 2.16667i 0.889226i
494.15 −0.428952 −1.41888 − 0.993373i −1.81600 0.857302i 0.608629 + 0.426109i − 1.07087i 1.63688 1.02642 + 2.81895i − 0.367741i
494.16 −0.428952 −1.41888 + 0.993373i −1.81600 − 0.857302i 0.608629 − 0.426109i 1.07087i 1.63688 1.02642 − 2.81895i 0.367741i
494.17 0.322911 −0.225945 − 1.71725i −1.89573 − 1.08454i −0.0729601 − 0.554520i 4.56232i −1.25797 −2.89790 + 0.776008i − 0.350210i
494.18 0.322911 −0.225945 + 1.71725i −1.89573 1.08454i −0.0729601 + 0.554520i − 4.56232i −1.25797 −2.89790 − 0.776008i 0.350210i
494.19 0.880971 1.21289 − 1.23649i −1.22389 − 3.32075i 1.06852 − 1.08931i 0.460813i −2.84015 −0.0577963 − 2.99944i − 2.92549i
494.20 0.880971 1.21289 + 1.23649i −1.22389 3.32075i 1.06852 + 1.08931i − 0.460813i −2.84015 −0.0577963 + 2.99944i 2.92549i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 494.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.f.a ✓ 32
3.b odd 2 1 561.2.f.b yes 32
11.b odd 2 1 561.2.f.b yes 32
33.d even 2 1 inner 561.2.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.f.a ✓ 32 1.a even 1 1 trivial
561.2.f.a ✓ 32 33.d even 2 1 inner
561.2.f.b yes 32 3.b odd 2 1
561.2.f.b yes 32 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 24 T_{2}^{14} + 2 T_{2}^{13} + 231 T_{2}^{12} - 38 T_{2}^{11} - 1142 T_{2}^{10} + 268 T_{2}^{9} + \cdots + 72 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display