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Properties

Label 627.2.j.b
Level $627$
Weight $2$
Character orbit 627.j
Analytic conductor $5.007$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [627,2,Mod(58,627)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("627.58"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(627, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 8, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] 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: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} + 8 q^{3} + q^{4} - 4 q^{5} - q^{6} + q^{7} - 13 q^{8} - 8 q^{9} + 10 q^{10} + 12 q^{11} - 26 q^{12} - 10 q^{13} + 18 q^{14} + 4 q^{15} - 19 q^{16} - 12 q^{17} - 4 q^{18} - 8 q^{19} + 15 q^{20}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 −1.95491 + 1.42032i −0.309017 − 0.951057i 1.18631 − 3.65109i 2.19357 + 1.59372i 1.95491 + 1.42032i 0.612153 − 1.88401i 1.37318 + 4.22622i −0.809017 + 0.587785i −6.55182
58.2 −1.88919 + 1.37258i −0.309017 − 0.951057i 1.06704 − 3.28402i −2.21619 − 1.61016i 1.88919 + 1.37258i 1.09767 − 3.37829i 1.04851 + 3.22699i −0.809017 + 0.587785i 6.39688
58.3 −1.00492 + 0.730115i −0.309017 − 0.951057i −0.141244 + 0.434704i −2.10635 − 1.53035i 1.00492 + 0.730115i −0.985669 + 3.03358i −0.943134 − 2.90267i −0.809017 + 0.587785i 3.23404
58.4 −0.249750 + 0.181454i −0.309017 − 0.951057i −0.588585 + 1.81148i −3.34373 − 2.42936i 0.249750 + 0.181454i 0.539094 − 1.65916i −0.372492 − 1.14641i −0.809017 + 0.587785i 1.27591
58.5 −0.110913 + 0.0805833i −0.309017 − 0.951057i −0.612226 + 1.88424i 0.816590 + 0.593287i 0.110913 + 0.0805833i −1.17467 + 3.61527i −0.168664 − 0.519095i −0.809017 + 0.587785i −0.138380
58.6 1.14771 − 0.833860i −0.309017 − 0.951057i 0.00388078 − 0.0119438i 1.27349 + 0.925244i −1.14771 − 0.833860i −0.731871 + 2.25247i 0.871266 + 2.68148i −0.809017 + 0.587785i 2.23312
58.7 1.66801 − 1.21188i −0.309017 − 0.951057i 0.695567 − 2.14073i 1.66790 + 1.21180i −1.66801 − 1.21188i 0.740608 − 2.27936i −0.159855 − 0.491983i −0.809017 + 0.587785i 4.25062
58.8 2.08495 − 1.51480i −0.309017 − 0.951057i 1.43434 − 4.41444i −1.52134 − 1.10532i −2.08495 − 1.51480i −0.406332 + 1.25056i −2.10373 − 6.47461i −0.809017 + 0.587785i −4.84626
115.1 −0.566294 − 1.74287i 0.809017 + 0.587785i −1.09889 + 0.798389i 0.281504 − 0.866380i 0.566294 − 1.74287i −4.15169 + 3.01638i −0.951370 − 0.691211i 0.309017 + 0.951057i −1.66941
115.2 −0.451602 − 1.38989i 0.809017 + 0.587785i −0.109807 + 0.0797795i 0.671386 − 2.06631i 0.451602 − 1.38989i 3.85952 − 2.80410i −2.20414 − 1.60140i 0.309017 + 0.951057i −3.17514
115.3 −0.208109 − 0.640494i 0.809017 + 0.587785i 1.25111 − 0.908986i 0.522205 − 1.60718i 0.208109 − 0.640494i −1.33171 + 0.967547i −1.93224 − 1.40385i 0.309017 + 0.951057i −1.13806
115.4 −0.0651389 − 0.200477i 0.809017 + 0.587785i 1.58209 − 1.14945i −0.858776 + 2.64304i 0.0651389 − 0.200477i 0.646449 − 0.469672i −0.674565 − 0.490100i 0.309017 + 0.951057i 0.585808
115.5 0.235840 + 0.725842i 0.809017 + 0.587785i 1.14681 − 0.833205i 0.482985 − 1.48647i −0.235840 + 0.725842i 1.88576 − 1.37009i 2.11011 + 1.53309i 0.309017 + 0.951057i 1.19285
115.6 0.488337 + 1.50295i 0.809017 + 0.587785i −0.402345 + 0.292321i −0.0608709 + 0.187341i −0.488337 + 1.50295i −0.921878 + 0.669784i 1.92114 + 1.39579i 0.309017 + 0.951057i −0.311290
115.7 0.600451 + 1.84800i 0.809017 + 0.587785i −1.43652 + 1.04370i −1.12140 + 3.45132i −0.600451 + 1.84800i −1.48755 + 1.08077i 0.352691 + 0.256245i 0.309017 + 0.951057i −7.05138
115.8 0.775532 + 2.38684i 0.809017 + 0.587785i −3.47753 + 2.52657i 1.31904 − 4.05958i −0.775532 + 2.38684i 2.31012 − 1.67840i −4.66672 − 3.39057i 0.309017 + 0.951057i 10.7125
229.1 −0.566294 + 1.74287i 0.809017 − 0.587785i −1.09889 − 0.798389i 0.281504 + 0.866380i 0.566294 + 1.74287i −4.15169 − 3.01638i −0.951370 + 0.691211i 0.309017 − 0.951057i −1.66941
229.2 −0.451602 + 1.38989i 0.809017 − 0.587785i −0.109807 − 0.0797795i 0.671386 + 2.06631i 0.451602 + 1.38989i 3.85952 + 2.80410i −2.20414 + 1.60140i 0.309017 − 0.951057i −3.17514
229.3 −0.208109 + 0.640494i 0.809017 − 0.587785i 1.25111 + 0.908986i 0.522205 + 1.60718i 0.208109 + 0.640494i −1.33171 − 0.967547i −1.93224 + 1.40385i 0.309017 − 0.951057i −1.13806
229.4 −0.0651389 + 0.200477i 0.809017 − 0.587785i 1.58209 + 1.14945i −0.858776 − 2.64304i 0.0651389 + 0.200477i 0.646449 + 0.469672i −0.674565 + 0.490100i 0.309017 − 0.951057i 0.585808
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.8
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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 627.2.j.b ✓ 32
11.c even 5 1 inner 627.2.j.b ✓ 32
11.c even 5 1 6897.2.a.bm 16
11.d odd 10 1 6897.2.a.bl 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.j.b ✓ 32 1.a even 1 1 trivial
627.2.j.b ✓ 32 11.c even 5 1 inner
6897.2.a.bl 16 11.d odd 10 1
6897.2.a.bm 16 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - T_{2}^{31} + 8 T_{2}^{30} + 2 T_{2}^{29} + 64 T_{2}^{28} - 37 T_{2}^{27} + 597 T_{2}^{26} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(627, [\chi])\). Copy content Toggle raw display