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Properties

Label 550.2.y.a
Level $550$
Weight $2$
Character orbit 550.y
Analytic conductor $4.392$
Analytic rank $0$
Dimension $120$
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] = [550,2,Mod(9,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.y (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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 10 q^{3} + 30 q^{4} + 8 q^{5} - 16 q^{6} + 10 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 10 q^{3} + 30 q^{4} + 8 q^{5} - 16 q^{6} + 10 q^{7} + 36 q^{9} - 4 q^{10} + 4 q^{11} - 10 q^{12} - 6 q^{15} - 30 q^{16} - 20 q^{19} + 2 q^{20} + 8 q^{21} - 20 q^{22} + 10 q^{23} - 4 q^{24} - 16 q^{25} - 10 q^{27} + 10 q^{28} - 12 q^{29} + 4 q^{30} - 20 q^{31} - 50 q^{33} + 38 q^{35} + 124 q^{36} - 30 q^{38} + 10 q^{39} + 4 q^{40} + 10 q^{41} - 10 q^{42} - 4 q^{44} + 34 q^{45} - 16 q^{46} - 20 q^{47} - 8 q^{49} - 24 q^{50} + 8 q^{51} - 10 q^{54} + 6 q^{55} - 60 q^{57} - 28 q^{59} - 14 q^{60} + 12 q^{61} - 50 q^{63} + 30 q^{64} + 54 q^{65} + 16 q^{66} - 50 q^{67} - 30 q^{69} - 16 q^{70} + 18 q^{71} - 20 q^{74} + 54 q^{75} - 40 q^{76} - 110 q^{77} + 30 q^{78} + 48 q^{79} - 2 q^{80} + 76 q^{81} + 20 q^{82} + 32 q^{84} - 68 q^{85} - 6 q^{86} - 20 q^{87} + 30 q^{89} - 30 q^{90} - 30 q^{91} + 10 q^{92} + 120 q^{93} + 64 q^{94} - 130 q^{95} + 4 q^{96} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.587785 − 0.809017i −1.73320 + 2.38555i −0.309017 + 0.951057i 0.787598 + 2.09277i 2.94870 2.68273 − 3.69246i 0.951057 − 0.309017i −1.75980 − 5.41610i 1.23015 − 1.86728i
9.2 −0.587785 − 0.809017i −1.33610 + 1.83898i −0.309017 + 0.951057i 0.428741 + 2.19458i 2.27310 −1.73726 + 2.39114i 0.951057 − 0.309017i −0.669638 − 2.06093i 1.52344 − 1.63680i
9.3 −0.587785 − 0.809017i −1.26540 + 1.74167i −0.309017 + 0.951057i 2.09268 − 0.787852i 2.15282 −0.656224 + 0.903215i 0.951057 − 0.309017i −0.505129 − 1.55463i −1.86743 − 1.22992i
9.4 −0.587785 − 0.809017i −1.21331 + 1.66998i −0.309017 + 0.951057i −0.682346 − 2.12941i 2.06421 1.98062 − 2.72610i 0.951057 − 0.309017i −0.389657 − 1.19924i −1.32166 + 1.80367i
9.5 −0.587785 − 0.809017i −1.08267 + 1.49017i −0.309017 + 0.951057i −1.16281 − 1.90994i 1.84195 −0.277911 + 0.382512i 0.951057 − 0.309017i −0.121371 − 0.373541i −0.861690 + 2.06337i
9.6 −0.587785 − 0.809017i −0.944276 + 1.29968i −0.309017 + 0.951057i −1.78549 + 1.34612i 1.60650 −1.73091 + 2.38239i 0.951057 − 0.309017i 0.129529 + 0.398648i 2.13852 + 0.653260i
9.7 −0.587785 − 0.809017i −0.0765005 + 0.105294i −0.309017 + 0.951057i 1.94826 + 1.09740i 0.130150 0.559340 − 0.769866i 0.951057 − 0.309017i 0.921816 + 2.83706i −0.257342 − 2.22121i
9.8 −0.587785 − 0.809017i 0.123192 − 0.169560i −0.309017 + 0.951057i 2.12755 − 0.688128i −0.209587 2.33651 − 3.21594i 0.951057 − 0.309017i 0.913477 + 2.81139i −1.80725 − 1.31675i
9.9 −0.587785 − 0.809017i 0.497705 − 0.685032i −0.309017 + 0.951057i −2.20982 − 0.341604i −0.846746 −0.278395 + 0.383177i 0.951057 − 0.309017i 0.705493 + 2.17128i 1.02254 + 1.98857i
9.10 −0.587785 − 0.809017i 0.595858 − 0.820128i −0.309017 + 0.951057i 0.865795 − 2.06165i −1.01373 −1.56447 + 2.15330i 0.951057 − 0.309017i 0.609488 + 1.87581i −2.17681 + 0.511365i
9.11 −0.587785 − 0.809017i 1.01549 − 1.39770i −0.309017 + 0.951057i −0.237081 + 2.22346i −1.72766 −1.80512 + 2.48454i 0.951057 − 0.309017i 0.00469836 + 0.0144601i 1.93817 − 1.11512i
9.12 −0.587785 − 0.809017i 1.12564 − 1.54931i −0.309017 + 0.951057i −1.13118 − 1.92884i −1.91505 −1.07739 + 1.48290i 0.951057 − 0.309017i −0.206247 − 0.634762i −0.895571 + 2.04889i
9.13 −0.587785 − 0.809017i 1.42576 − 1.96239i −0.309017 + 0.951057i −1.70408 + 1.44779i −2.42565 2.55575 − 3.51769i 0.951057 − 0.309017i −0.891136 − 2.74263i 2.17292 + 0.527640i
9.14 −0.587785 − 0.809017i 1.58811 − 2.18585i −0.309017 + 0.951057i 1.62719 + 1.53370i −2.70186 0.548931 − 0.755538i 0.951057 − 0.309017i −1.32878 − 4.08957i 0.284351 − 2.21791i
9.15 −0.587785 − 0.809017i 1.76428 − 2.42832i −0.309017 + 0.951057i 1.57384 − 1.58840i −3.00157 0.272808 − 0.375488i 0.951057 − 0.309017i −1.85701 − 5.71530i −2.21013 − 0.339624i
9.16 0.587785 + 0.809017i −1.82098 + 2.50637i −0.309017 + 0.951057i 2.22096 + 0.259465i −3.09804 −0.707841 + 0.974260i −0.951057 + 0.309017i −2.03885 − 6.27493i 1.09554 + 1.94931i
9.17 0.587785 + 0.809017i −1.66132 + 2.28661i −0.309017 + 0.951057i −2.04526 − 0.903831i −2.82640 2.36170 − 3.25060i −0.951057 + 0.309017i −1.54155 − 4.74440i −0.470960 − 2.18591i
9.18 0.587785 + 0.809017i −1.51138 + 2.08024i −0.309017 + 0.951057i −1.63302 + 1.52750i −2.57132 −1.07757 + 1.48314i −0.951057 + 0.309017i −1.11607 − 3.43492i −2.19564 − 0.423299i
9.19 0.587785 + 0.809017i −1.41347 + 1.94547i −0.309017 + 0.951057i 0.229624 − 2.22425i −2.40474 −2.66913 + 3.67374i −0.951057 + 0.309017i −0.859918 − 2.64656i 1.93442 − 1.12161i
9.20 0.587785 + 0.809017i −0.899384 + 1.23790i −0.309017 + 0.951057i 0.556284 − 2.16577i −1.53012 0.898076 − 1.23610i −0.951057 + 0.309017i 0.203557 + 0.626484i 2.07912 − 0.822963i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.bb even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.y.a ✓ 120
11.c even 5 1 550.2.z.a yes 120
25.e even 10 1 550.2.z.a yes 120
275.bb even 10 1 inner 550.2.y.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.y.a ✓ 120 1.a even 1 1 trivial
550.2.y.a ✓ 120 275.bb even 10 1 inner
550.2.z.a yes 120 11.c even 5 1
550.2.z.a yes 120 25.e even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(550, [\chi])\).