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Properties

Label 621.2.s.a
Level $621$
Weight $2$
Character orbit 621.s
Analytic conductor $4.959$
Analytic rank $0$
Dimension $440$
Inner twists $4$

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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] = [621,2,Mod(17,621)] 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("621.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(621, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([55, 21])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 621 = 3^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 621.s (of order \(66\), degree \(20\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.95870996552\)
Analytic rank: \(0\)
Dimension: \(440\)
Relative dimension: \(22\) over \(\Q(\zeta_{66})\)
Twist minimal: no (minimal twist has level 207)
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q + 27 q^{2} - 29 q^{4} + 33 q^{5} - 11 q^{7} - 44 q^{10} + 33 q^{11} - 9 q^{13} + 33 q^{14} + 3 q^{16} - 44 q^{19} + 33 q^{20} + 27 q^{23} + 11 q^{25} - 44 q^{28} - 27 q^{29} - 3 q^{31} + 33 q^{32}+ \cdots + 22 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.254294 − 2.66308i 0 −5.06350 + 0.975909i 0.0783683 + 1.64515i 0 0.195151 + 0.487463i 2.37916 + 8.10268i 0 4.36126 − 0.627054i
17.2 −0.239165 − 2.50465i 0 −4.25222 + 0.819549i −0.0687260 − 1.44274i 0 0.409323 + 1.02244i 1.65196 + 5.62607i 0 −3.59711 + 0.517187i
17.3 −0.212136 − 2.22159i 0 −2.92660 + 0.564056i 0.162247 + 3.40597i 0 −1.13587 − 2.83725i 0.616455 + 2.09945i 0 7.53225 − 1.08297i
17.4 −0.208137 − 2.17971i 0 −2.74397 + 0.528857i −0.0111710 − 0.234508i 0 1.82068 + 4.54785i 0.490101 + 1.66913i 0 −0.508836 + 0.0731595i
17.5 −0.167850 − 1.75780i 0 −1.09783 + 0.211590i −0.135764 − 2.85004i 0 −1.56436 − 3.90758i −0.438761 − 1.49428i 0 −4.98701 + 0.717024i
17.6 −0.148626 − 1.55648i 0 −0.436691 + 0.0841653i 0.112059 + 2.35241i 0 −0.570741 − 1.42564i −0.685107 − 2.33326i 0 3.64483 − 0.524048i
17.7 −0.143627 − 1.50413i 0 −0.277929 + 0.0535665i −0.0496215 − 1.04168i 0 0.658540 + 1.64495i −0.730892 − 2.48919i 0 −1.55970 + 0.224251i
17.8 −0.109078 − 1.14232i 0 0.670864 − 0.129299i −0.119182 − 2.50194i 0 −0.568097 − 1.41904i −0.867461 − 2.95430i 0 −2.84501 + 0.409051i
17.9 −0.100945 − 1.05715i 0 0.856485 − 0.165074i 0.172794 + 3.62740i 0 1.20269 + 3.00418i −0.859342 − 2.92665i 0 3.81726 − 0.548839i
17.10 −0.0557976 − 0.584339i 0 1.62552 − 0.313293i −0.176147 − 3.69777i 0 0.153622 + 0.383728i −0.604522 − 2.05881i 0 −2.15093 + 0.309256i
17.11 −0.00358223 − 0.0375148i 0 1.96246 − 0.378234i 0.124574 + 2.61513i 0 −1.01237 − 2.52879i −0.0424538 − 0.144584i 0 0.0976597 − 0.0140413i
17.12 0.00723894 + 0.0758096i 0 1.95816 − 0.377405i −0.00947493 − 0.198903i 0 −1.26788 − 3.16702i 0.0856963 + 0.291855i 0 0.0150102 − 0.00215814i
17.13 0.0253850 + 0.265844i 0 1.89383 − 0.365006i 0.158686 + 3.33122i 0 0.652180 + 1.62907i 0.295585 + 1.00667i 0 −0.881557 + 0.126749i
17.14 0.0488316 + 0.511387i 0 1.70472 − 0.328559i −0.0300003 − 0.629785i 0 1.52357 + 3.80569i 0.540725 + 1.84154i 0 0.320599 − 0.0460952i
17.15 0.0769467 + 0.805822i 0 1.32043 − 0.254492i −0.0837687 − 1.75852i 0 0.477405 + 1.19250i 0.762796 + 2.59784i 0 1.41061 − 0.202815i
17.16 0.122757 + 1.28557i 0 0.326236 − 0.0628768i −0.0711660 − 1.49396i 0 −1.58054 − 3.94799i 0.848549 + 2.88989i 0 1.91185 − 0.274883i
17.17 0.133932 + 1.40260i 0 0.0145101 − 0.00279659i −0.0155613 − 0.326673i 0 0.756236 + 1.88899i 0.799777 + 2.72379i 0 0.456107 − 0.0655783i
17.18 0.181306 + 1.89872i 0 −1.60840 + 0.309993i 0.131387 + 2.75816i 0 0.491986 + 1.22892i 0.194526 + 0.662495i 0 −5.21315 + 0.749537i
17.19 0.200433 + 2.09903i 0 −2.40188 + 0.462925i 0.0430034 + 0.902752i 0 −0.981910 − 2.45269i −0.265001 − 0.902511i 0 −1.88628 + 0.271206i
17.20 0.216863 + 2.27109i 0 −3.14695 + 0.606526i −0.137206 − 2.88031i 0 0.443169 + 1.10698i −0.774430 − 2.63747i 0 6.51168 − 0.936238i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.d odd 22 1 inner
207.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 621.2.s.a 440
3.b odd 2 1 207.2.o.a ✓ 440
9.c even 3 1 207.2.o.a ✓ 440
9.d odd 6 1 inner 621.2.s.a 440
23.d odd 22 1 inner 621.2.s.a 440
69.g even 22 1 207.2.o.a ✓ 440
207.o even 66 1 inner 621.2.s.a 440
207.p odd 66 1 207.2.o.a ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.o.a ✓ 440 3.b odd 2 1
207.2.o.a ✓ 440 9.c even 3 1
207.2.o.a ✓ 440 69.g even 22 1
207.2.o.a ✓ 440 207.p odd 66 1
621.2.s.a 440 1.a even 1 1 trivial
621.2.s.a 440 9.d odd 6 1 inner
621.2.s.a 440 23.d odd 22 1 inner
621.2.s.a 440 207.o even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(621, [\chi])\).