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Properties

Label 561.2.m.d
Level $561$
Weight $2$
Character orbit 561.m
Analytic conductor $4.480$
Analytic rank $0$
Dimension $24$
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(103,561)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.m (of order \(5\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} - 6 q^{3} - 7 q^{4} + 7 q^{5} + 3 q^{6} + 5 q^{7} - 10 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} - 6 q^{3} - 7 q^{4} + 7 q^{5} + 3 q^{6} + 5 q^{7} - 10 q^{8} - 6 q^{9} + 16 q^{10} - 7 q^{11} + 18 q^{12} + 6 q^{13} - 6 q^{14} + 7 q^{15} - 23 q^{16} - 6 q^{17} - 2 q^{18} + 6 q^{19} + 3 q^{20} - 10 q^{21} + 23 q^{22} - 78 q^{23} + 10 q^{24} + q^{25} + 10 q^{26} - 6 q^{27} - 13 q^{28} + 20 q^{29} + 11 q^{30} + 5 q^{31} - 22 q^{32} - 2 q^{33} - 2 q^{34} + 29 q^{35} - 7 q^{36} - 10 q^{37} + 2 q^{38} + 6 q^{39} + 44 q^{40} - 16 q^{41} + 19 q^{42} - 36 q^{43} + 3 q^{44} - 28 q^{45} + 23 q^{46} + 19 q^{47} + 12 q^{48} - 15 q^{49} + 34 q^{50} - 6 q^{51} - 6 q^{52} - 5 q^{53} - 2 q^{54} + 24 q^{55} - 50 q^{56} - 9 q^{57} - 79 q^{58} + 34 q^{59} + 3 q^{60} - 14 q^{61} + 36 q^{62} + 5 q^{63} - 20 q^{64} + 6 q^{65} + 3 q^{66} - 18 q^{67} - 2 q^{68} + 7 q^{69} + 46 q^{70} - 8 q^{71} + 10 q^{72} + 7 q^{73} + 47 q^{74} - 9 q^{75} + 58 q^{76} + 43 q^{77} - 50 q^{78} + 14 q^{79} - 51 q^{80} - 6 q^{81} - 88 q^{82} - 47 q^{83} + 22 q^{84} + 7 q^{85} - 13 q^{86} + 20 q^{87} + 115 q^{88} - 124 q^{89} - 19 q^{90} + 10 q^{91} - 19 q^{92} + 5 q^{93} + 28 q^{94} - 13 q^{95} - 27 q^{96} - 30 q^{97} - 14 q^{98} - 7 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −1.80508 − 1.31147i 0.309017 − 0.951057i 0.920329 + 2.83248i 0.625886 − 0.454733i −1.80508 + 1.31147i −0.926218 − 2.85061i 0.674481 − 2.07584i −0.809017 − 0.587785i −1.72614
103.2 −0.695370 − 0.505216i 0.309017 − 0.951057i −0.389738 − 1.19949i −1.95714 + 1.42194i −0.695370 + 0.505216i 0.322424 + 0.992318i −0.866204 + 2.66590i −0.809017 − 0.587785i 2.07932
103.3 0.0552438 + 0.0401370i 0.309017 − 0.951057i −0.616593 − 1.89768i 0.989888 − 0.719196i 0.0552438 − 0.0401370i −1.39118 − 4.28160i 0.0843066 − 0.259469i −0.809017 − 0.587785i 0.0835515
103.4 0.357186 + 0.259511i 0.309017 − 0.951057i −0.557798 − 1.71673i 3.46967 − 2.52087i 0.357186 − 0.259511i 0.982692 + 3.02442i 0.519138 − 1.59774i −0.809017 − 0.587785i 1.89351
103.5 1.23742 + 0.899036i 0.309017 − 0.951057i 0.104900 + 0.322850i −0.0100028 + 0.00726748i 1.23742 − 0.899036i −0.0399436 − 0.122934i 0.784854 − 2.41553i −0.809017 − 0.587785i −0.0189114
103.6 2.15962 + 1.56905i 0.309017 − 0.951057i 1.58398 + 4.87500i 2.54481 − 1.84891i 2.15962 − 1.56905i 0.625170 + 1.92407i −2.57854 + 7.93593i −0.809017 − 0.587785i 8.39687
256.1 −1.80508 + 1.31147i 0.309017 + 0.951057i 0.920329 − 2.83248i 0.625886 + 0.454733i −1.80508 − 1.31147i −0.926218 + 2.85061i 0.674481 + 2.07584i −0.809017 + 0.587785i −1.72614
256.2 −0.695370 + 0.505216i 0.309017 + 0.951057i −0.389738 + 1.19949i −1.95714 − 1.42194i −0.695370 − 0.505216i 0.322424 − 0.992318i −0.866204 − 2.66590i −0.809017 + 0.587785i 2.07932
256.3 0.0552438 − 0.0401370i 0.309017 + 0.951057i −0.616593 + 1.89768i 0.989888 + 0.719196i 0.0552438 + 0.0401370i −1.39118 + 4.28160i 0.0843066 + 0.259469i −0.809017 + 0.587785i 0.0835515
256.4 0.357186 − 0.259511i 0.309017 + 0.951057i −0.557798 + 1.71673i 3.46967 + 2.52087i 0.357186 + 0.259511i 0.982692 − 3.02442i 0.519138 + 1.59774i −0.809017 + 0.587785i 1.89351
256.5 1.23742 − 0.899036i 0.309017 + 0.951057i 0.104900 − 0.322850i −0.0100028 − 0.00726748i 1.23742 + 0.899036i −0.0399436 + 0.122934i 0.784854 + 2.41553i −0.809017 + 0.587785i −0.0189114
256.6 2.15962 − 1.56905i 0.309017 + 0.951057i 1.58398 − 4.87500i 2.54481 + 1.84891i 2.15962 + 1.56905i 0.625170 − 1.92407i −2.57854 − 7.93593i −0.809017 + 0.587785i 8.39687
460.1 −0.658651 + 2.02712i −0.809017 + 0.587785i −2.05736 − 1.49476i −0.186477 − 0.573916i −0.658651 − 2.02712i −0.165183 − 0.120012i 0.936396 − 0.680332i 0.309017 − 0.951057i 1.28622
460.2 −0.532612 + 1.63921i −0.809017 + 0.587785i −0.785303 − 0.570556i −1.21172 − 3.72929i −0.532612 − 1.63921i 2.86209 + 2.07943i −1.43527 + 1.04278i 0.309017 − 0.951057i 6.75848
460.3 −0.134444 + 0.413775i −0.809017 + 0.587785i 1.46490 + 1.06431i 0.472638 + 1.45463i −0.134444 − 0.413775i −2.28531 − 1.66038i −1.34129 + 0.974502i 0.309017 − 0.951057i −0.665433
460.4 0.178680 − 0.549921i −0.809017 + 0.587785i 1.34755 + 0.979051i 0.0913190 + 0.281051i 0.178680 + 0.549921i 0.546666 + 0.397176i 1.71476 − 1.24585i 0.309017 − 0.951057i 0.170873
460.5 0.569659 − 1.75323i −0.809017 + 0.587785i −1.13127 − 0.821915i −0.208666 − 0.642208i 0.569659 + 1.75323i 1.38205 + 1.00412i 0.897329 − 0.651948i 0.309017 − 0.951057i −1.24481
460.6 0.768351 − 2.36474i −0.809017 + 0.587785i −3.38360 − 2.45833i −1.12021 − 3.44766i 0.768351 + 2.36474i 0.586739 + 0.426291i −4.38997 + 3.18950i 0.309017 − 0.951057i −9.01354
511.1 −0.658651 − 2.02712i −0.809017 − 0.587785i −2.05736 + 1.49476i −0.186477 + 0.573916i −0.658651 + 2.02712i −0.165183 + 0.120012i 0.936396 + 0.680332i 0.309017 + 0.951057i 1.28622
511.2 −0.532612 − 1.63921i −0.809017 − 0.587785i −0.785303 + 0.570556i −1.21172 + 3.72929i −0.532612 + 1.63921i 2.86209 − 2.07943i −1.43527 − 1.04278i 0.309017 + 0.951057i 6.75848
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.m.d ✓ 24
11.c even 5 1 inner 561.2.m.d ✓ 24
11.c even 5 1 6171.2.a.bk 12
11.d odd 10 1 6171.2.a.bl 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.m.d ✓ 24 1.a even 1 1 trivial
561.2.m.d ✓ 24 11.c even 5 1 inner
6171.2.a.bk 12 11.c even 5 1
6171.2.a.bl 12 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 3 T_{2}^{23} + 14 T_{2}^{22} - 27 T_{2}^{21} + 94 T_{2}^{20} - 146 T_{2}^{19} + 546 T_{2}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display