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Properties

Label 550.2.g.b
Level $550$
Weight $2$
Character orbit 550.g
Analytic conductor $4.392$
Analytic rank $0$
Dimension $48$
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(291,550)]
 
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([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.291");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.g (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: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{2} + 2 q^{3} - 12 q^{4} - q^{5} - 7 q^{6} + 4 q^{7} + 12 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{2} + 2 q^{3} - 12 q^{4} - q^{5} - 7 q^{6} + 4 q^{7} + 12 q^{8} - 12 q^{9} + 6 q^{10} - 3 q^{11} + 7 q^{12} + 12 q^{13} + q^{14} - 3 q^{15} - 12 q^{16} + 6 q^{17} + 12 q^{18} + 10 q^{19} - q^{20} + 3 q^{21} + 3 q^{22} + 11 q^{23} - 2 q^{24} - 23 q^{25} - 12 q^{26} + 35 q^{27} - 6 q^{28} - 40 q^{29} - 2 q^{30} + 56 q^{31} - 48 q^{32} + 43 q^{33} - 6 q^{34} + 34 q^{35} - 17 q^{36} - 20 q^{37} + 25 q^{38} + 15 q^{39} - 4 q^{40} + 24 q^{41} + 32 q^{42} - 70 q^{43} + 7 q^{44} - 23 q^{45} - 11 q^{46} + 20 q^{47} - 18 q^{48} - 40 q^{49} - 12 q^{50} + 21 q^{51} + 2 q^{52} + 52 q^{53} + 5 q^{54} - 37 q^{55} + q^{56} + 9 q^{57} + 40 q^{58} - 35 q^{59} + 12 q^{60} - 39 q^{61} + 9 q^{62} + 48 q^{63} - 12 q^{64} + 33 q^{65} - 13 q^{66} + q^{67} - 4 q^{68} - 45 q^{69} - 9 q^{70} - 30 q^{71} - 58 q^{72} + 8 q^{73} - 45 q^{74} + 35 q^{75} + 30 q^{76} + 9 q^{77} - 20 q^{78} + 2 q^{79} + 4 q^{80} - 51 q^{81} + 11 q^{82} + 6 q^{83} + 3 q^{84} - 19 q^{85} - 40 q^{86} + 34 q^{87} - 2 q^{88} - 45 q^{89} - 77 q^{90} - 8 q^{91} + 16 q^{92} + 77 q^{93} - 5 q^{94} - 59 q^{95} - 2 q^{96} + 44 q^{97} + 5 q^{98} - 13 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
291.1 −0.309017 + 0.951057i −0.995851 + 3.06491i −0.809017 − 0.587785i −1.21089 + 1.87983i −2.60717 − 1.89422i 1.00522 − 0.730337i 0.809017 − 0.587785i −5.97492 − 4.34104i −1.41364 − 1.73252i
291.2 −0.309017 + 0.951057i −0.923670 + 2.84277i −0.809017 − 0.587785i −2.01563 − 0.968116i −2.41820 − 1.75693i −3.32144 + 2.41317i 0.809017 − 0.587785i −4.80110 − 3.48820i 1.54360 − 1.61781i
291.3 −0.309017 + 0.951057i −0.798301 + 2.45692i −0.809017 − 0.587785i 2.07109 − 0.842959i −2.08998 − 1.51846i 1.82811 − 1.32820i 0.809017 − 0.587785i −2.97210 − 2.15936i 0.161699 + 2.23021i
291.4 −0.309017 + 0.951057i −0.641254 + 1.97358i −0.809017 − 0.587785i 0.685191 − 2.12850i −1.67882 − 1.21974i 0.505636 − 0.367366i 0.809017 − 0.587785i −1.05675 − 0.767770i 1.81259 + 1.30940i
291.5 −0.309017 + 0.951057i −0.409325 + 1.25977i −0.809017 − 0.587785i −0.579880 − 2.15957i −1.07163 − 0.778582i 0.443375 − 0.322131i 0.809017 − 0.587785i 1.00757 + 0.732043i 2.23307 + 0.115845i
291.6 −0.309017 + 0.951057i −0.267643 + 0.823719i −0.809017 − 0.587785i 0.0515787 + 2.23547i −0.700698 − 0.509087i 3.31149 − 2.40594i 0.809017 − 0.587785i 1.82017 + 1.32243i −2.14200 − 0.641745i
291.7 −0.309017 + 0.951057i −0.258812 + 0.796543i −0.809017 − 0.587785i 2.21616 + 0.297713i −0.677580 − 0.492291i −3.97188 + 2.88574i 0.809017 − 0.587785i 1.85955 + 1.35105i −0.967973 + 2.01570i
291.8 −0.309017 + 0.951057i 0.149643 − 0.460553i −0.809017 − 0.587785i −1.86556 − 1.23275i 0.391770 + 0.284637i −2.43894 + 1.77199i 0.809017 − 0.587785i 2.23734 + 1.62552i 1.74891 − 1.39331i
291.9 −0.309017 + 0.951057i 0.443623 − 1.36533i −0.809017 − 0.587785i 1.09508 + 1.94956i 1.16142 + 0.843820i 1.24150 − 0.902004i 0.809017 − 0.587785i 0.759725 + 0.551973i −2.19254 + 0.439039i
291.10 −0.309017 + 0.951057i 0.580062 − 1.78525i −0.809017 − 0.587785i 1.77295 − 1.36258i 1.51862 + 1.10334i 2.04516 − 1.48590i 0.809017 − 0.587785i −0.423587 − 0.307754i 0.748021 + 2.10724i
291.11 −0.309017 + 0.951057i 0.675366 − 2.07856i −0.809017 − 0.587785i −1.42897 − 1.71989i 1.76813 + 1.28462i 2.88568 − 2.09657i 0.809017 − 0.587785i −1.43725 − 1.04423i 2.07729 − 0.827559i
291.12 −0.309017 + 0.951057i 0.710094 − 2.18544i −0.809017 − 0.587785i −1.60015 + 1.56190i 1.85905 + 1.35068i −1.41589 + 1.02870i 0.809017 − 0.587785i −1.84488 − 1.34039i −0.990981 − 2.00448i
361.1 −0.309017 − 0.951057i −0.995851 − 3.06491i −0.809017 + 0.587785i −1.21089 − 1.87983i −2.60717 + 1.89422i 1.00522 + 0.730337i 0.809017 + 0.587785i −5.97492 + 4.34104i −1.41364 + 1.73252i
361.2 −0.309017 − 0.951057i −0.923670 − 2.84277i −0.809017 + 0.587785i −2.01563 + 0.968116i −2.41820 + 1.75693i −3.32144 − 2.41317i 0.809017 + 0.587785i −4.80110 + 3.48820i 1.54360 + 1.61781i
361.3 −0.309017 − 0.951057i −0.798301 − 2.45692i −0.809017 + 0.587785i 2.07109 + 0.842959i −2.08998 + 1.51846i 1.82811 + 1.32820i 0.809017 + 0.587785i −2.97210 + 2.15936i 0.161699 − 2.23021i
361.4 −0.309017 − 0.951057i −0.641254 − 1.97358i −0.809017 + 0.587785i 0.685191 + 2.12850i −1.67882 + 1.21974i 0.505636 + 0.367366i 0.809017 + 0.587785i −1.05675 + 0.767770i 1.81259 − 1.30940i
361.5 −0.309017 − 0.951057i −0.409325 − 1.25977i −0.809017 + 0.587785i −0.579880 + 2.15957i −1.07163 + 0.778582i 0.443375 + 0.322131i 0.809017 + 0.587785i 1.00757 − 0.732043i 2.23307 − 0.115845i
361.6 −0.309017 − 0.951057i −0.267643 − 0.823719i −0.809017 + 0.587785i 0.0515787 − 2.23547i −0.700698 + 0.509087i 3.31149 + 2.40594i 0.809017 + 0.587785i 1.82017 − 1.32243i −2.14200 + 0.641745i
361.7 −0.309017 − 0.951057i −0.258812 − 0.796543i −0.809017 + 0.587785i 2.21616 − 0.297713i −0.677580 + 0.492291i −3.97188 − 2.88574i 0.809017 + 0.587785i 1.85955 − 1.35105i −0.967973 − 2.01570i
361.8 −0.309017 − 0.951057i 0.149643 + 0.460553i −0.809017 + 0.587785i −1.86556 + 1.23275i 0.391770 − 0.284637i −2.43894 − 1.77199i 0.809017 + 0.587785i 2.23734 − 1.62552i 1.74891 + 1.39331i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 291.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.g 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.g.b ✓ 48
11.c even 5 1 550.2.j.b yes 48
25.d even 5 1 550.2.j.b yes 48
275.g even 5 1 inner 550.2.g.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.g.b ✓ 48 1.a even 1 1 trivial
550.2.g.b ✓ 48 275.g even 5 1 inner
550.2.j.b yes 48 11.c even 5 1
550.2.j.b yes 48 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 2 T_{3}^{47} + 26 T_{3}^{46} - 75 T_{3}^{45} + 493 T_{3}^{44} - 1170 T_{3}^{43} + \cdots + 5033760601 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display