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Properties

Label 561.2.t.b
Level $561$
Weight $2$
Character orbit 561.t
Analytic conductor $4.480$
Analytic rank $0$
Dimension $128$
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] = [561,2,Mod(35,561)]
 
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([5, 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("561.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.t (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: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + q^{3} - 32 q^{4} + 5 q^{5} - 18 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + q^{3} - 32 q^{4} + 5 q^{5} - 18 q^{8} - q^{9} + 7 q^{11} - 16 q^{12} + 3 q^{15} - 28 q^{16} - 32 q^{17} + 18 q^{18} - 20 q^{20} + 4 q^{21} - 10 q^{22} + 17 q^{25} + 60 q^{26} - 32 q^{27} - 10 q^{28} - 28 q^{29} + 102 q^{30} + 4 q^{32} - 5 q^{33} - 18 q^{36} + 6 q^{37} + 210 q^{38} + 29 q^{39} + 30 q^{40} - 27 q^{41} + 17 q^{42} + 42 q^{44} - 42 q^{45} - 123 q^{48} + 20 q^{49} - 8 q^{50} + q^{51} - 180 q^{53} + 10 q^{54} + 23 q^{55} + 40 q^{57} - 46 q^{58} + 60 q^{59} - 122 q^{60} + 20 q^{61} - 52 q^{62} + 82 q^{63} + 6 q^{64} - 6 q^{65} - 82 q^{66} - 48 q^{67} - 32 q^{68} + 10 q^{69} - 4 q^{70} + 50 q^{71} - 20 q^{72} - 10 q^{73} - 162 q^{74} + 76 q^{75} + 42 q^{77} + 56 q^{78} - 80 q^{79} - 157 q^{81} - 26 q^{82} + 28 q^{83} + 110 q^{84} - 60 q^{86} - 12 q^{87} - 52 q^{88} + 24 q^{90} - 40 q^{91} + 100 q^{92} - 88 q^{93} + 80 q^{94} - 146 q^{95} + 145 q^{96} + 30 q^{97} - 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −0.829491 + 2.55291i −1.06964 + 1.36231i −4.21127 − 3.05966i −0.687075 + 0.223244i −2.59059 − 3.86071i −2.96112 + 4.07563i 6.96099 − 5.05745i −0.711755 − 2.91434i − 1.93922i
35.2 −0.774507 + 2.38369i 0.0106193 − 1.73202i −3.46407 − 2.51679i 2.00948 − 0.652918i 4.12036 + 1.36677i 2.45011 − 3.37228i 4.62682 − 3.36158i −2.99977 − 0.0367858i 5.29565i
35.3 −0.749419 + 2.30647i 1.01339 + 1.40465i −3.14016 − 2.28146i −3.89995 + 1.26717i −3.99924 + 1.28467i −0.405546 + 0.558187i 3.69141 − 2.68196i −0.946098 + 2.84691i − 9.94476i
35.4 −0.724626 + 2.23017i 1.53232 − 0.807471i −2.83054 − 2.05650i 4.00992 − 1.30290i 0.690443 + 4.00244i −1.71771 + 2.36422i 2.84324 − 2.06574i 1.69598 − 2.47460i 9.88692i
35.5 −0.712116 + 2.19167i −1.71028 + 0.273762i −2.67827 − 1.94588i 0.127735 − 0.0415037i 0.617921 − 3.94332i 0.965783 − 1.32929i 2.44327 − 1.77514i 2.85011 − 0.936420i 0.309509i
35.6 −0.632892 + 1.94784i 1.69516 + 0.355586i −1.77550 − 1.28998i −0.174261 + 0.0566207i −1.76548 + 3.07685i 2.86425 − 3.94230i 0.322507 − 0.234315i 2.74712 + 1.20555i − 0.375267i
35.7 −0.592794 + 1.82443i 1.43431 − 0.970955i −1.35912 − 0.987458i −1.50229 + 0.488124i 0.921191 + 3.19238i −1.53652 + 2.11484i −0.496683 + 0.360861i 1.11449 − 2.78530i − 3.03019i
35.8 −0.470364 + 1.44763i −0.826421 − 1.52218i −0.256358 − 0.186255i −1.48075 + 0.481124i 2.59227 − 0.480375i 1.37219 − 1.88866i −2.07265 + 1.50587i −1.63406 + 2.51592i − 2.36988i
35.9 −0.434373 + 1.33686i −0.731390 + 1.57005i 0.0195094 + 0.0141744i 0.960724 − 0.312158i −1.78125 − 1.65976i −1.04774 + 1.44209i −2.30183 + 1.67238i −1.93014 − 2.29664i 1.41995i
35.10 −0.366928 + 1.12929i −1.45003 − 0.947325i 0.477377 + 0.346835i −1.62123 + 0.526770i 1.60186 − 1.28990i −1.71765 + 2.36414i −2.48810 + 1.80771i 1.20515 + 2.74729i − 2.02412i
35.11 −0.302978 + 0.932471i 0.708024 − 1.58073i 0.840327 + 0.610533i 2.29872 − 0.746899i 1.25947 + 1.13914i 0.597382 − 0.822225i −2.41032 + 1.75120i −1.99740 − 2.23839i 2.36978i
35.12 −0.262327 + 0.807361i 0.552888 + 1.64144i 1.03502 + 0.751985i 3.72554 − 1.21050i −1.47027 + 0.0157861i 1.14092 − 1.57035i −2.25220 + 1.63632i −2.38863 + 1.81506i 3.32541i
35.13 −0.256090 + 0.788164i −1.38592 + 1.03885i 1.06241 + 0.771889i −3.00156 + 0.975267i −0.463864 − 1.35838i 2.49004 − 3.42725i −2.22135 + 1.61391i 0.841573 − 2.87954i − 2.61548i
35.14 −0.221264 + 0.680981i 1.50287 + 0.861029i 1.20326 + 0.874217i −2.64427 + 0.859177i −0.918877 + 0.832914i −0.370024 + 0.509294i −2.02012 + 1.46770i 1.51726 + 2.58804i − 1.99081i
35.15 −0.129491 + 0.398533i −1.68094 − 0.417667i 1.47597 + 1.07236i 3.38767 − 1.10072i 0.384121 − 0.615826i −1.86099 + 2.56144i −1.29652 + 0.941978i 2.65111 + 1.40415i 1.49263i
35.16 −0.0208478 + 0.0641630i 0.921915 − 1.46631i 1.61435 + 1.17290i −2.36387 + 0.768067i 0.0748631 + 0.0897223i −2.85115 + 3.92427i −0.218073 + 0.158439i −1.30014 − 2.70363i − 0.167685i
35.17 −0.00310989 + 0.00957126i −0.268445 + 1.71112i 1.61795 + 1.17551i −0.108062 + 0.0351115i −0.0155428 − 0.00789076i −0.710527 + 0.977956i −0.0325664 + 0.0236609i −2.85587 − 0.918684i − 0.00114348i
35.18 0.0937814 − 0.288629i −1.18602 − 1.26228i 1.54352 + 1.12143i 1.65658 − 0.538256i −0.475558 + 0.223942i 2.36826 − 3.25963i 0.959478 − 0.697102i −0.186713 + 2.99418i − 0.528616i
35.19 0.123694 − 0.380690i 1.46979 + 0.916356i 1.48841 + 1.08139i 1.51020 − 0.490694i 0.530652 − 0.446188i −0.448902 + 0.617860i 1.24345 − 0.903420i 1.32058 + 2.69371i − 0.635615i
35.20 0.178100 − 0.548135i 1.55679 − 0.759219i 1.34930 + 0.980325i −1.17019 + 0.380218i −0.138891 − 0.988546i 1.58812 − 2.18586i 1.71020 − 1.24254i 1.84717 − 2.36388i 0.709140i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f 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.t.b yes 128
3.b odd 2 1 561.2.t.a ✓ 128
11.d odd 10 1 561.2.t.a ✓ 128
33.f even 10 1 inner 561.2.t.b yes 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.t.a ✓ 128 3.b odd 2 1
561.2.t.a ✓ 128 11.d odd 10 1
561.2.t.b yes 128 1.a even 1 1 trivial
561.2.t.b yes 128 33.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} + 48 T_{2}^{126} + 6 T_{2}^{125} + 1271 T_{2}^{124} + 268 T_{2}^{123} + 24613 T_{2}^{122} + \cdots + 3041536 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display