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Properties

Label 561.2.m.e
Level $561$
Weight $2$
Character orbit 561.m
Analytic conductor $4.480$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + q^{2} + 10 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 5 q^{7} - 16 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + q^{2} + 10 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 5 q^{7} - 16 q^{8} - 10 q^{9} + 8 q^{10} + 16 q^{11} - 50 q^{12} - 4 q^{13} - 4 q^{14} + q^{15} - 61 q^{16} - 10 q^{17} - 4 q^{18} - 8 q^{19} - 33 q^{20} + 10 q^{21} + 19 q^{22} - 32 q^{23} - 4 q^{24} - 8 q^{25} - 20 q^{26} + 10 q^{27} + 5 q^{28} + 18 q^{29} - 23 q^{30} - 31 q^{31} + 94 q^{32} + 9 q^{33} + 6 q^{34} - 9 q^{35} - 5 q^{36} - 38 q^{37} + 4 q^{39} + 78 q^{40} - 33 q^{41} - q^{42} + 92 q^{43} - 75 q^{44} + 14 q^{45} - 45 q^{46} + 17 q^{47} - 14 q^{48} - 59 q^{49} + 30 q^{50} + 10 q^{51} - 114 q^{52} - 3 q^{53} - 6 q^{54} + 41 q^{55} + 78 q^{56} - 2 q^{57} + 95 q^{58} - 62 q^{59} + 33 q^{60} + 34 q^{61} + 12 q^{62} + 5 q^{63} - 110 q^{64} + 52 q^{65} - 29 q^{66} + 42 q^{67} - 20 q^{68} + 2 q^{69} - 162 q^{70} + 42 q^{71} + 4 q^{72} + 15 q^{73} - 3 q^{74} + 23 q^{75} + 162 q^{76} - 21 q^{77} - 30 q^{78} + 6 q^{79} + 139 q^{80} - 10 q^{81} - 40 q^{82} - 23 q^{83} - 10 q^{84} - q^{85} + 13 q^{86} - 8 q^{87} + 11 q^{88} + 88 q^{89} - 27 q^{90} + 44 q^{91} - 71 q^{92} + 31 q^{93} - 12 q^{94} - 9 q^{95} + 31 q^{96} - 42 q^{97} + 162 q^{98} - 9 q^{99}+O(q^{100}) \) 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 −2.19595 − 1.59545i −0.309017 + 0.951057i 1.65870 + 5.10494i −3.19694 + 2.32271i 2.19595 − 1.59545i −1.27011 − 3.90899i 2.82471 − 8.69357i −0.809017 − 0.587785i 10.7261
103.2 −2.17490 − 1.58016i −0.309017 + 0.951057i 1.61527 + 4.97129i 0.471761 − 0.342754i 2.17490 − 1.58016i 1.52142 + 4.68244i 2.68090 − 8.25095i −0.809017 − 0.587785i −1.56764
103.3 −1.16729 − 0.848089i −0.309017 + 0.951057i 0.0252874 + 0.0778266i −1.49693 + 1.08758i 1.16729 − 0.848089i 0.351568 + 1.08201i −0.855248 + 2.63218i −0.809017 − 0.587785i 2.66973
103.4 −1.02375 − 0.743800i −0.309017 + 0.951057i −0.123202 − 0.379177i −0.752868 + 0.546991i 1.02375 − 0.743800i −1.51124 − 4.65113i −0.937981 + 2.88681i −0.809017 − 0.587785i 1.17760
103.5 −0.218887 − 0.159030i −0.309017 + 0.951057i −0.595413 − 1.83249i −0.315583 + 0.229284i 0.218887 − 0.159030i 0.650970 + 2.00348i −0.328309 + 1.01043i −0.809017 − 0.587785i 0.105540
103.6 0.0801710 + 0.0582476i −0.309017 + 0.951057i −0.614999 − 1.89277i 2.72119 − 1.97706i −0.0801710 + 0.0582476i −0.507758 − 1.56272i 0.122190 − 0.376061i −0.809017 − 0.587785i 0.333320
103.7 0.755357 + 0.548799i −0.309017 + 0.951057i −0.348650 − 1.07304i −3.32292 + 2.41424i −0.755357 + 0.548799i 1.26971 + 3.90775i 0.902566 − 2.77781i −0.809017 − 0.587785i −3.83492
103.8 1.47113 + 1.06884i −0.309017 + 0.951057i 0.403780 + 1.24271i 2.74551 − 1.99473i −1.47113 + 1.06884i −1.10835 − 3.41116i 0.389604 − 1.19908i −0.809017 − 0.587785i 6.17107
103.9 1.94015 + 1.40960i −0.309017 + 0.951057i 1.15917 + 3.56756i 1.37228 − 0.997022i −1.94015 + 1.40960i 0.881334 + 2.71247i −1.29773 + 3.99400i −0.809017 − 0.587785i 4.06783
103.10 2.22496 + 1.61653i −0.309017 + 0.951057i 1.71925 + 5.29131i −1.96158 + 1.42517i −2.22496 + 1.61653i −0.704583 − 2.16848i −3.02857 + 9.32097i −0.809017 − 0.587785i −6.66826
256.1 −2.19595 + 1.59545i −0.309017 − 0.951057i 1.65870 − 5.10494i −3.19694 − 2.32271i 2.19595 + 1.59545i −1.27011 + 3.90899i 2.82471 + 8.69357i −0.809017 + 0.587785i 10.7261
256.2 −2.17490 + 1.58016i −0.309017 − 0.951057i 1.61527 − 4.97129i 0.471761 + 0.342754i 2.17490 + 1.58016i 1.52142 − 4.68244i 2.68090 + 8.25095i −0.809017 + 0.587785i −1.56764
256.3 −1.16729 + 0.848089i −0.309017 − 0.951057i 0.0252874 − 0.0778266i −1.49693 − 1.08758i 1.16729 + 0.848089i 0.351568 − 1.08201i −0.855248 − 2.63218i −0.809017 + 0.587785i 2.66973
256.4 −1.02375 + 0.743800i −0.309017 − 0.951057i −0.123202 + 0.379177i −0.752868 − 0.546991i 1.02375 + 0.743800i −1.51124 + 4.65113i −0.937981 − 2.88681i −0.809017 + 0.587785i 1.17760
256.5 −0.218887 + 0.159030i −0.309017 − 0.951057i −0.595413 + 1.83249i −0.315583 − 0.229284i 0.218887 + 0.159030i 0.650970 − 2.00348i −0.328309 − 1.01043i −0.809017 + 0.587785i 0.105540
256.6 0.0801710 − 0.0582476i −0.309017 − 0.951057i −0.614999 + 1.89277i 2.72119 + 1.97706i −0.0801710 − 0.0582476i −0.507758 + 1.56272i 0.122190 + 0.376061i −0.809017 + 0.587785i 0.333320
256.7 0.755357 − 0.548799i −0.309017 − 0.951057i −0.348650 + 1.07304i −3.32292 − 2.41424i −0.755357 − 0.548799i 1.26971 − 3.90775i 0.902566 + 2.77781i −0.809017 + 0.587785i −3.83492
256.8 1.47113 − 1.06884i −0.309017 − 0.951057i 0.403780 − 1.24271i 2.74551 + 1.99473i −1.47113 − 1.06884i −1.10835 + 3.41116i 0.389604 + 1.19908i −0.809017 + 0.587785i 6.17107
256.9 1.94015 − 1.40960i −0.309017 − 0.951057i 1.15917 − 3.56756i 1.37228 + 0.997022i −1.94015 − 1.40960i 0.881334 − 2.71247i −1.29773 − 3.99400i −0.809017 + 0.587785i 4.06783
256.10 2.22496 − 1.61653i −0.309017 − 0.951057i 1.71925 − 5.29131i −1.96158 − 1.42517i −2.22496 − 1.61653i −0.704583 + 2.16848i −3.02857 − 9.32097i −0.809017 + 0.587785i −6.66826
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.10
Significant digits:
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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.e ✓ 40
11.c even 5 1 inner 561.2.m.e ✓ 40
11.c even 5 1 6171.2.a.bs 20
11.d odd 10 1 6171.2.a.bp 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.m.e ✓ 40 1.a even 1 1 trivial
561.2.m.e ✓ 40 11.c even 5 1 inner
6171.2.a.bp 20 11.d odd 10 1
6171.2.a.bs 20 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - T_{2}^{39} + 13 T_{2}^{38} - 2 T_{2}^{37} + 139 T_{2}^{36} - 88 T_{2}^{35} + 1523 T_{2}^{34} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display