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Properties

Label 620.2.k.d
Level $620$
Weight $2$
Character orbit 620.k
Analytic conductor $4.951$
Analytic rank $0$
Dimension $88$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [620,2,Mod(63,620)] 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("620.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(620, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 620 = 2^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 620.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [88,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.95072492532\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 2 q^{2} + 2 q^{3} - 2 q^{5} + 4 q^{7} - 4 q^{8} - 10 q^{10} - 12 q^{12} - 6 q^{13} + 22 q^{14} - 2 q^{15} + 8 q^{16} - 2 q^{17} - 24 q^{18} + 8 q^{19} + 8 q^{20} - 16 q^{21} - 16 q^{22} + 10 q^{23}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 −1.41272 + 0.0649557i 1.92422 − 1.92422i 1.99156 − 0.183528i 0.843724 + 2.07078i −2.59339 + 2.84337i 0.570845 + 0.570845i −2.80160 + 0.388638i − 4.40523i −1.32646 − 2.87063i
63.2 −1.38578 − 0.282181i −0.166563 + 0.166563i 1.84075 + 0.782079i 2.17151 − 0.533433i 0.277819 − 0.183818i −1.23462 − 1.23462i −2.33018 − 1.60321i 2.94451i −3.15975 + 0.126459i
63.3 −1.34347 − 0.441684i 0.641688 − 0.641688i 1.60983 + 1.18678i −1.08300 − 1.95630i −1.14551 + 0.578666i −2.33699 − 2.33699i −1.63858 − 2.30544i 2.17647i 0.590919 + 3.10658i
63.4 −1.33983 + 0.452602i 1.88048 − 1.88048i 1.59030 − 1.21282i −0.710626 − 2.12014i −1.66842 + 3.37064i −3.61243 − 3.61243i −1.58181 + 2.34475i − 4.07241i 1.91170 + 2.51901i
63.5 −1.31190 + 0.528131i −0.327159 + 0.327159i 1.44215 − 1.38571i 2.20286 + 0.383933i 0.256416 − 0.601982i 3.52415 + 3.52415i −1.16012 + 2.57956i 2.78593i −3.09270 + 0.659719i
63.6 −1.25756 − 0.646956i −1.39122 + 1.39122i 1.16290 + 1.62717i −2.16579 − 0.556189i 2.64960 − 0.849482i 2.15391 + 2.15391i −0.409703 − 2.79860i − 0.871005i 2.36378 + 2.10061i
63.7 −1.24956 − 0.662261i 2.43487 − 2.43487i 1.12282 + 1.65508i −2.08204 + 0.815549i −4.65505 + 1.43001i 0.144148 + 0.144148i −0.306944 − 2.81172i − 8.85723i 3.14175 + 0.359772i
63.8 −1.20437 − 0.741280i −0.980406 + 0.980406i 0.901009 + 1.78555i 1.98090 + 1.03732i 1.90753 − 0.454015i −0.362721 − 0.362721i 0.238444 − 2.81836i 1.07761i −1.61679 − 2.71772i
63.9 −1.19714 + 0.752899i −2.32623 + 2.32623i 0.866287 − 1.80265i 2.15511 + 0.596241i 1.03341 − 4.53624i −0.483723 − 0.483723i 0.320145 + 2.81025i − 7.82269i −3.02888 + 0.908795i
63.10 −1.19230 + 0.760533i −1.88745 + 1.88745i 0.843178 − 1.81357i −1.84022 + 1.27027i 0.814944 − 3.68588i 2.29449 + 2.29449i 0.373959 + 2.80360i − 4.12491i 1.22801 − 2.91410i
63.11 −1.18032 + 0.779004i 1.18900 − 1.18900i 0.786304 − 1.83895i −1.22109 + 1.87322i −0.477163 + 2.32964i −0.636756 − 0.636756i 0.504458 + 2.78308i 0.172553i −0.0179735 − 3.16223i
63.12 −1.14410 − 0.831286i −1.99964 + 1.99964i 0.617928 + 1.90215i 0.571972 + 2.16168i 3.95005 − 0.625514i −2.91050 − 2.91050i 0.874256 − 2.68992i − 4.99708i 1.14258 − 2.94865i
63.13 −0.875177 − 1.11089i 1.34701 − 1.34701i −0.468132 + 1.94444i 2.18673 + 0.467153i −2.67524 − 0.317502i 0.689709 + 0.689709i 2.56975 − 1.18169i − 0.628865i −1.39482 − 2.83804i
63.14 −0.581290 − 1.28923i −1.01606 + 1.01606i −1.32420 + 1.49883i −1.31596 + 1.80783i 1.90056 + 0.719306i 0.314666 + 0.314666i 2.70207 + 0.835942i 0.935236i 3.09565 + 0.645699i
63.15 −0.548768 + 1.30340i 0.497186 − 0.497186i −1.39771 − 1.43053i 0.723535 + 2.11577i 0.375193 + 0.920873i 2.15915 + 2.15915i 2.63157 − 1.03674i 2.50561i −3.15475 − 0.218014i
63.16 −0.507403 − 1.32005i −2.16272 + 2.16272i −1.48508 + 1.33960i −1.19840 − 1.88781i 3.95227 + 1.75753i −0.840075 − 0.840075i 2.52188 + 1.28068i − 6.35469i −1.88394 + 2.53984i
63.17 −0.365915 + 1.36606i −1.52745 + 1.52745i −1.73221 − 0.999720i 2.21529 − 0.304123i −1.52766 − 2.64549i 0.438981 + 0.438981i 1.99951 − 2.00049i − 1.66619i −0.395159 + 3.13749i
63.18 −0.349716 + 1.37029i 1.22822 − 1.22822i −1.75540 − 0.958426i 0.150350 − 2.23101i 1.25349 + 2.11255i 0.510097 + 0.510097i 1.92721 − 2.07023i − 0.0170517i 3.00455 + 0.986242i
63.19 −0.290983 − 1.38395i 1.83695 − 1.83695i −1.83066 + 0.805414i 0.457111 + 2.18885i −3.07678 − 2.00773i −2.15608 − 2.15608i 1.64735 + 2.29919i − 3.74877i 2.89625 − 1.26954i
63.20 −0.227844 + 1.39574i −1.05245 + 1.05245i −1.89617 − 0.636022i −2.21931 − 0.273263i −1.22915 − 1.70874i −1.90067 − 1.90067i 1.31975 − 2.50165i 0.784701i 0.887061 − 3.03531i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.44
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 620.2.k.d yes 88
4.b odd 2 1 620.2.k.c ✓ 88
5.c odd 4 1 620.2.k.c ✓ 88
20.e even 4 1 inner 620.2.k.d yes 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
620.2.k.c ✓ 88 4.b odd 2 1
620.2.k.c ✓ 88 5.c odd 4 1
620.2.k.d yes 88 1.a even 1 1 trivial
620.2.k.d yes 88 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} - 2 T_{3}^{87} + 2 T_{3}^{86} + 616 T_{3}^{84} - 1232 T_{3}^{83} + 1232 T_{3}^{82} + \cdots + 4294967296 \) acting on \(S_{2}^{\mathrm{new}}(620, [\chi])\). Copy content Toggle raw display