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Properties

Label 51.3.j.a
Level $51$
Weight $3$
Character orbit 51.j
Analytic conductor $1.390$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [51,3,Mod(7,51)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(51, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 11])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("51.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 51.j (of order \(16\), degree \(8\), 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: \(1.38964934824\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{10} - 80 q^{11} - 48 q^{13} - 64 q^{14} + 16 q^{17} + 48 q^{19} + 224 q^{20} + 192 q^{22} + 112 q^{23} - 144 q^{24} + 80 q^{25} - 368 q^{26} - 240 q^{28} - 160 q^{29} - 192 q^{30} - 64 q^{31}+ \cdots + 240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.50885 + 3.64269i −0.962276 + 1.44015i −8.16414 − 8.16414i 0.501657 + 2.52200i −3.79408 − 5.67825i −0.668021 + 3.35837i 27.4872 − 11.3856i −1.14805 − 2.77164i −9.94380 − 1.97794i
7.2 −0.927907 + 2.24016i 0.962276 − 1.44015i −1.32890 − 1.32890i 1.42152 + 7.14649i 2.33327 + 3.49198i 0.118569 − 0.596086i −4.75061 + 1.96777i −1.14805 − 2.77164i −17.3284 − 3.44682i
7.3 −0.412445 + 0.995731i −0.962276 + 1.44015i 2.00706 + 2.00706i −0.354838 − 1.78389i −1.03711 − 1.55215i −2.27450 + 11.4347i −6.80922 + 2.82047i −1.14805 − 2.77164i 1.92263 + 0.382434i
7.4 0.217952 − 0.526183i 0.962276 − 1.44015i 2.59906 + 2.59906i −0.692512 − 3.48150i −0.548051 − 0.820216i −0.153043 + 0.769402i 4.03879 − 1.67292i −1.14805 − 2.77164i −1.98284 − 0.394411i
7.5 0.277861 − 0.670815i −0.962276 + 1.44015i 2.45564 + 2.45564i 1.19712 + 6.01834i 0.698694 + 1.04567i 2.29221 − 11.5237i 5.01286 − 2.07640i −1.14805 − 2.77164i 4.36982 + 0.869212i
7.6 1.27100 − 3.06846i 0.962276 − 1.44015i −4.97161 − 4.97161i 1.29074 + 6.48900i −3.19599 − 4.78314i −0.615844 + 3.09605i −9.30025 + 3.85229i −1.14805 − 2.77164i 21.5518 + 4.28692i
10.1 −1.20106 + 2.89963i −1.44015 − 0.962276i −4.13685 − 4.13685i −8.14921 + 1.62098i 4.51995 − 3.02014i 2.78848 + 0.554662i 5.36545 − 2.22244i 1.14805 + 2.77164i 5.08750 − 25.5766i
10.2 −0.699672 + 1.68916i 1.44015 + 0.962276i 0.464717 + 0.464717i −1.46623 + 0.291651i −2.63307 + 1.75936i 2.48667 + 0.494629i −7.86676 + 3.25852i 1.14805 + 2.77164i 0.533234 − 2.68075i
10.3 −0.129487 + 0.312609i −1.44015 − 0.962276i 2.74747 + 2.74747i 4.39054 − 0.873333i 0.487296 − 0.325601i 5.56021 + 1.10600i −2.46508 + 1.02107i 1.14805 + 2.77164i −0.295505 + 1.48561i
10.4 0.599063 − 1.44627i 1.44015 + 0.962276i 1.09562 + 1.09562i 1.92847 − 0.383596i 2.25445 − 1.50637i −10.3593 − 2.06060i 8.02596 − 3.32446i 1.14805 + 2.77164i 0.600492 − 3.01888i
10.5 1.13525 − 2.74074i −1.44015 − 0.962276i −3.39445 − 3.39445i 2.36138 − 0.469707i −4.27229 + 2.85465i −3.45573 − 0.687388i −2.19392 + 0.908751i 1.14805 + 2.77164i 1.39341 − 7.00517i
10.6 1.37830 − 3.32751i 1.44015 + 0.962276i −6.34417 − 6.34417i −5.25707 + 1.04570i 5.18693 − 3.46580i 12.7656 + 2.53924i −16.5444 + 6.85292i 1.14805 + 2.77164i −3.76625 + 18.9342i
22.1 −1.50885 − 3.64269i −0.962276 − 1.44015i −8.16414 + 8.16414i 0.501657 − 2.52200i −3.79408 + 5.67825i −0.668021 − 3.35837i 27.4872 + 11.3856i −1.14805 + 2.77164i −9.94380 + 1.97794i
22.2 −0.927907 − 2.24016i 0.962276 + 1.44015i −1.32890 + 1.32890i 1.42152 − 7.14649i 2.33327 − 3.49198i 0.118569 + 0.596086i −4.75061 − 1.96777i −1.14805 + 2.77164i −17.3284 + 3.44682i
22.3 −0.412445 − 0.995731i −0.962276 − 1.44015i 2.00706 − 2.00706i −0.354838 + 1.78389i −1.03711 + 1.55215i −2.27450 − 11.4347i −6.80922 − 2.82047i −1.14805 + 2.77164i 1.92263 − 0.382434i
22.4 0.217952 + 0.526183i 0.962276 + 1.44015i 2.59906 − 2.59906i −0.692512 + 3.48150i −0.548051 + 0.820216i −0.153043 − 0.769402i 4.03879 + 1.67292i −1.14805 + 2.77164i −1.98284 + 0.394411i
22.5 0.277861 + 0.670815i −0.962276 − 1.44015i 2.45564 − 2.45564i 1.19712 − 6.01834i 0.698694 − 1.04567i 2.29221 + 11.5237i 5.01286 + 2.07640i −1.14805 + 2.77164i 4.36982 − 0.869212i
22.6 1.27100 + 3.06846i 0.962276 + 1.44015i −4.97161 + 4.97161i 1.29074 − 6.48900i −3.19599 + 4.78314i −0.615844 − 3.09605i −9.30025 − 3.85229i −1.14805 + 2.77164i 21.5518 − 4.28692i
28.1 −2.49222 − 1.03231i 0.337906 + 1.69877i 2.31706 + 2.31706i −6.00641 − 4.01336i 0.911522 − 4.58253i −8.31178 + 5.55375i 0.746544 + 1.80232i −2.77164 + 1.14805i 10.8263 + 16.2026i
28.2 −1.58813 − 0.657825i 0.337906 + 1.69877i −0.739002 − 0.739002i 7.17579 + 4.79471i 0.580854 − 2.92015i 6.00636 − 4.01332i 3.31880 + 8.01229i −2.77164 + 1.14805i −8.24201 − 12.3350i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.3.j.a ✓ 48
3.b odd 2 1 153.3.p.c 48
17.e odd 16 1 inner 51.3.j.a ✓ 48
51.i even 16 1 153.3.p.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.3.j.a ✓ 48 1.a even 1 1 trivial
51.3.j.a ✓ 48 17.e odd 16 1 inner
153.3.p.c 48 3.b odd 2 1
153.3.p.c 48 51.i even 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(51, [\chi])\).