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Properties

Label 550.2.g.c
Level $550$
Weight $2$
Character orbit 550.g
Analytic conductor $4.392$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [550,2,Mod(291,550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 15 q^{2} - 4 q^{3} - 15 q^{4} + q^{6} - 3 q^{7} - 15 q^{8} - 17 q^{9} - 5 q^{10} + 7 q^{11} + q^{12} + 12 q^{13} - 3 q^{14} + 12 q^{15} - 15 q^{16} + q^{17} - 17 q^{18} + 3 q^{19} - 5 q^{20} - 4 q^{21}+ \cdots + 21 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −1.05249 + 3.23923i −0.809017 − 0.587785i 0.949639 − 2.02440i 2.75546 + 2.00196i −1.13526 + 0.824814i −0.809017 + 0.587785i −6.95785 − 5.05517i −1.63186 − 1.52873i
291.2 0.309017 − 0.951057i −0.807690 + 2.48582i −0.809017 − 0.587785i −2.21989 − 0.268513i 2.11456 + 1.53632i 3.23079 − 2.34731i −0.809017 + 0.587785i −3.09986 − 2.25218i −0.941354 + 2.02826i
291.3 0.309017 − 0.951057i −0.805027 + 2.47762i −0.809017 − 0.587785i 1.01112 + 1.99440i 2.10759 + 1.53125i 2.12811 − 1.54617i −0.809017 + 0.587785i −3.06348 − 2.22575i 2.20924 − 0.345330i
291.4 0.309017 − 0.951057i −0.391447 + 1.20475i −0.809017 − 0.587785i −1.60082 − 1.56121i 1.02482 + 0.744577i −1.25694 + 0.913224i −0.809017 + 0.587785i 1.12886 + 0.820164i −1.97948 + 1.04003i
291.5 0.309017 − 0.951057i −0.347673 + 1.07003i −0.809017 − 0.587785i 2.09898 + 0.770911i 0.910220 + 0.661314i 0.0285107 − 0.0207142i −0.809017 + 0.587785i 1.40297 + 1.01932i 1.38180 − 1.75802i
291.6 0.309017 − 0.951057i −0.337394 + 1.03839i −0.809017 − 0.587785i 1.58445 − 1.57782i 0.883308 + 0.641761i −3.75459 + 2.72787i −0.809017 + 0.587785i 1.46263 + 1.06266i −1.01097 − 1.99448i
291.7 0.309017 − 0.951057i −0.118638 + 0.365131i −0.809017 − 0.587785i −2.11479 − 0.726402i 0.310599 + 0.225664i −1.38138 + 1.00363i −0.809017 + 0.587785i 2.30781 + 1.67672i −1.34436 + 1.78682i
291.8 0.309017 − 0.951057i −0.0527535 + 0.162358i −0.809017 − 0.587785i 0.755692 − 2.10450i 0.138110 + 0.100343i 3.80372 − 2.76356i −0.809017 + 0.587785i 2.40347 + 1.74623i −1.76798 − 1.36903i
291.9 0.309017 − 0.951057i 0.133515 − 0.410916i −0.809017 − 0.587785i −1.52224 + 1.63792i −0.349546 − 0.253960i 1.47926 − 1.07475i −0.809017 + 0.587785i 2.27603 + 1.65363i 1.08735 + 1.95388i
291.10 0.309017 − 0.951057i 0.211038 − 0.649508i −0.809017 − 0.587785i 0.918435 + 2.03874i −0.552504 − 0.401418i −3.50766 + 2.54847i −0.809017 + 0.587785i 2.04973 + 1.48921i 2.22277 − 0.243478i
291.11 0.309017 − 0.951057i 0.485103 − 1.49299i −0.809017 − 0.587785i 1.93438 + 1.12168i −1.27001 − 0.922720i 0.831135 − 0.603855i −0.809017 + 0.587785i 0.433350 + 0.314847i 1.66454 − 1.49309i
291.12 0.309017 − 0.951057i 0.541268 − 1.66585i −0.809017 − 0.587785i 0.244213 − 2.22269i −1.41706 − 1.02955i −1.01444 + 0.737034i −0.809017 + 0.587785i −0.0550374 − 0.0399870i −2.03844 − 0.919110i
291.13 0.309017 − 0.951057i 0.808711 − 2.48896i −0.809017 − 0.587785i −1.57774 + 1.58453i −2.11723 − 1.53826i −3.91599 + 2.84513i −0.809017 + 0.587785i −3.11384 − 2.26234i 1.01943 + 1.99017i
291.14 0.309017 − 0.951057i 0.834075 − 2.56702i −0.809017 − 0.587785i −1.52907 − 1.63155i −2.18364 − 1.58651i 2.81132 − 2.04255i −0.809017 + 0.587785i −3.46686 − 2.51882i −2.02420 + 0.950058i
291.15 0.309017 − 0.951057i 1.01744 − 3.13136i −0.809017 − 0.587785i 2.18567 − 0.472054i −2.66369 − 1.93528i −0.773637 + 0.562080i −0.809017 + 0.587785i −6.34316 − 4.60857i 0.226460 − 2.22457i
361.1 0.309017 + 0.951057i −1.05249 − 3.23923i −0.809017 + 0.587785i 0.949639 + 2.02440i 2.75546 − 2.00196i −1.13526 − 0.824814i −0.809017 − 0.587785i −6.95785 + 5.05517i −1.63186 + 1.52873i
361.2 0.309017 + 0.951057i −0.807690 − 2.48582i −0.809017 + 0.587785i −2.21989 + 0.268513i 2.11456 − 1.53632i 3.23079 + 2.34731i −0.809017 − 0.587785i −3.09986 + 2.25218i −0.941354 − 2.02826i
361.3 0.309017 + 0.951057i −0.805027 − 2.47762i −0.809017 + 0.587785i 1.01112 − 1.99440i 2.10759 − 1.53125i 2.12811 + 1.54617i −0.809017 − 0.587785i −3.06348 + 2.22575i 2.20924 + 0.345330i
361.4 0.309017 + 0.951057i −0.391447 − 1.20475i −0.809017 + 0.587785i −1.60082 + 1.56121i 1.02482 − 0.744577i −1.25694 − 0.913224i −0.809017 − 0.587785i 1.12886 − 0.820164i −1.97948 − 1.04003i
361.5 0.309017 + 0.951057i −0.347673 − 1.07003i −0.809017 + 0.587785i 2.09898 − 0.770911i 0.910220 − 0.661314i 0.0285107 + 0.0207142i −0.809017 − 0.587785i 1.40297 − 1.01932i 1.38180 + 1.75802i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 291.15
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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.c ✓ 60
11.c even 5 1 550.2.j.c yes 60
25.d even 5 1 550.2.j.c yes 60
275.g even 5 1 inner 550.2.g.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.g.c ✓ 60 1.a even 1 1 trivial
550.2.g.c ✓ 60 275.g even 5 1 inner
550.2.j.c yes 60 11.c even 5 1
550.2.j.c yes 60 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 4 T_{3}^{59} + 39 T_{3}^{58} + 132 T_{3}^{57} + 817 T_{3}^{56} + 2444 T_{3}^{55} + \cdots + 2253001 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display