Properties

Label 550.2.j.c
Level $550$
Weight $2$
Character orbit 550.j
Analytic conductor $4.392$
Analytic rank $0$
Dimension $60$
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(81,550)]
 
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([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.j (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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 15 q^{2} + q^{3} - 15 q^{4} + 6 q^{6} - 3 q^{7} - 15 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 15 q^{2} + q^{3} - 15 q^{4} + 6 q^{6} - 3 q^{7} - 15 q^{8} - 12 q^{9} - 5 q^{10} - 3 q^{11} + q^{12} - 8 q^{13} + 12 q^{14} - 3 q^{15} - 15 q^{16} + 6 q^{17} - 17 q^{18} - 22 q^{19} + 10 q^{20} - 4 q^{21} + 7 q^{22} - 3 q^{23} - 4 q^{24} + 18 q^{25} + 12 q^{26} - 8 q^{27} - 3 q^{28} - 22 q^{29} - 8 q^{30} + 15 q^{31} + 60 q^{32} + 25 q^{33} + q^{34} + 11 q^{35} + 58 q^{36} + 94 q^{37} + 3 q^{38} - 12 q^{39} - 5 q^{40} - 11 q^{41} - 19 q^{42} + 40 q^{43} + 7 q^{44} - 20 q^{45} + 2 q^{46} - 24 q^{47} - 4 q^{48} - 30 q^{49} - 27 q^{50} - 48 q^{51} - 8 q^{52} - 14 q^{53} - 8 q^{54} - 7 q^{55} - 3 q^{56} + 30 q^{57} - 22 q^{58} - 42 q^{59} + 12 q^{60} + 19 q^{61} - 20 q^{62} - 24 q^{63} - 15 q^{64} - 7 q^{65} + 10 q^{66} - 40 q^{67} - 4 q^{68} - 2 q^{69} - 19 q^{70} + 19 q^{71} - 17 q^{72} - 16 q^{73} - 11 q^{74} + 25 q^{75} + 38 q^{76} - 23 q^{77} - 12 q^{78} + 70 q^{79} - 86 q^{81} - 11 q^{82} - 14 q^{83} + 46 q^{84} - 31 q^{85} + 68 q^{87} - 8 q^{88} + q^{89} + 45 q^{90} - 25 q^{91} - 3 q^{92} - 40 q^{93} + 66 q^{94} + 19 q^{95} + q^{96} + 66 q^{97} - 30 q^{98} + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0.309017 0.951057i −0.950235 2.92452i −0.809017 0.587785i 2.07322 0.837701i −3.07503 −0.250676 0.771502i −0.809017 + 0.587785i −5.22284 + 3.79461i −0.156039 2.23062i
81.2 0.309017 0.951057i −0.798378 2.45716i −0.809017 0.587785i −0.351676 + 2.20824i −2.58361 0.962145 + 2.96118i −0.809017 + 0.587785i −2.97315 + 2.16012i 1.99149 + 1.01685i
81.3 0.309017 0.951057i −0.632955 1.94804i −0.809017 0.587785i −0.818463 2.08089i −2.04829 −1.46197 4.49947i −0.809017 + 0.587785i −0.967161 + 0.702684i −2.23197 + 0.135373i
81.4 0.309017 0.951057i −0.600130 1.84701i −0.809017 0.587785i −2.13045 0.679120i −1.94206 0.275504 + 0.847915i −0.809017 + 0.587785i −0.624244 + 0.453540i −1.30423 + 1.81631i
81.5 0.309017 0.951057i −0.293907 0.904554i −0.809017 0.587785i −0.536306 + 2.17080i −0.951104 −0.974143 2.99810i −0.809017 + 0.587785i 1.69521 1.23165i 1.89883 + 1.18087i
81.6 0.309017 0.951057i −0.224929 0.692259i −0.809017 0.587785i 1.90463 1.17148i −0.727884 1.27708 + 3.93044i −0.809017 + 0.587785i 1.99842 1.45194i −0.525580 2.17342i
81.7 0.309017 0.951057i −0.0525279 0.161664i −0.809017 0.587785i −0.0858628 2.23442i −0.169984 0.376127 + 1.15760i −0.809017 + 0.587785i 2.40367 1.74637i −2.15159 0.608813i
81.8 0.309017 0.951057i 0.0548441 + 0.168793i −0.809017 0.587785i −2.22390 + 0.232910i 0.177479 0.806547 + 2.48230i −0.809017 + 0.587785i 2.40157 1.74484i −0.465714 + 2.18703i
81.9 0.309017 0.951057i 0.166113 + 0.511243i −0.809017 0.587785i −1.05185 + 1.97322i 0.537553 −0.668877 2.05859i −0.809017 + 0.587785i 2.19328 1.59351i 1.55160 + 1.61013i
81.10 0.309017 0.951057i 0.423436 + 1.30320i −0.809017 0.587785i 2.09937 + 0.769843i 1.37027 0.201752 + 0.620927i −0.809017 + 0.587785i 0.908011 0.659709i 1.38090 1.75872i
81.11 0.309017 0.951057i 0.487764 + 1.50118i −0.809017 0.587785i 1.30223 + 1.81775i 1.57844 0.881273 + 2.71228i −0.809017 + 0.587785i 0.411412 0.298908i 2.13119 0.676779i
81.12 0.309017 0.951057i 0.615443 + 1.89414i −0.809017 0.587785i 2.20137 0.392380i 1.99161 −1.33176 4.09873i −0.809017 + 0.587785i −0.781936 + 0.568109i 0.307086 2.21488i
81.13 0.309017 0.951057i 0.753141 + 2.31793i −0.809017 0.587785i −2.23552 + 0.0494187i 2.43722 −1.11298 3.42540i −0.809017 + 0.587785i −2.37853 + 1.72810i −0.643814 + 2.14138i
81.14 0.309017 0.951057i 0.867391 + 2.66956i −0.809017 0.587785i −1.93587 + 1.11912i 2.80694 1.24292 + 3.82531i −0.809017 + 0.587785i −3.94711 + 2.86774i 0.466129 + 2.18694i
81.15 0.309017 0.951057i 0.993948 + 3.05906i −0.809017 0.587785i 0.671040 2.13300i 3.21648 0.704109 + 2.16703i −0.809017 + 0.587785i −5.94284 + 4.31773i −1.82124 1.29733i
141.1 −0.809017 + 0.587785i −2.66369 1.93528i 0.309017 0.951057i −1.49078 + 1.66661i 3.29250 −0.773637 0.562080i 0.309017 + 0.951057i 2.42287 + 7.45683i 0.226460 2.22457i
141.2 −0.809017 + 0.587785i −2.18364 1.58651i 0.309017 0.951057i 2.19604 + 0.421183i 2.69912 2.81132 + 2.04255i 0.309017 + 0.951057i 1.32422 + 4.07554i −2.02420 + 0.950058i
141.3 −0.809017 + 0.587785i −2.11723 1.53826i 0.309017 0.951057i 0.345051 2.20928i 2.61704 −3.91599 2.84513i 0.309017 + 0.951057i 1.18938 + 3.66054i 1.01943 + 1.99017i
141.4 −0.809017 + 0.587785i −1.41706 1.02955i 0.309017 0.951057i 1.10889 + 1.94174i 1.75158 −1.01444 0.737034i 0.309017 + 0.951057i 0.0210224 + 0.0647003i −2.03844 0.919110i
141.5 −0.809017 + 0.587785i −1.27001 0.922720i 0.309017 0.951057i −2.22425 + 0.229544i 1.56982 0.831135 + 0.603855i 0.309017 + 0.951057i −0.165525 0.509434i 1.66454 1.49309i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.j even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.2.j.c yes 60
11.c even 5 1 550.2.g.c 60
25.d even 5 1 550.2.g.c 60
275.j even 5 1 inner 550.2.j.c yes 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.g.c 60 11.c even 5 1
550.2.g.c 60 25.d even 5 1
550.2.j.c yes 60 1.a even 1 1 trivial
550.2.j.c yes 60 275.j even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - T_{3}^{59} + 29 T_{3}^{58} - 18 T_{3}^{57} + 537 T_{3}^{56} - 296 T_{3}^{55} + 8385 T_{3}^{54} - 4022 T_{3}^{53} + 113560 T_{3}^{52} - 48860 T_{3}^{51} + 1254655 T_{3}^{50} - 472271 T_{3}^{49} + 11873047 T_{3}^{48} + \cdots + 2253001 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display