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Properties

Label 47.2.c.a
Level $47$
Weight $2$
Character orbit 47.c
Analytic conductor $0.375$
Analytic rank $0$
Dimension $66$
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] = [47,2,Mod(2,47)] 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("47.2"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(47, base_ring=CyclotomicField(46)) chi = DirichletCharacter(H, H._module([18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 47.c (of order \(23\), degree \(22\), 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: \(0.375296889500\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(3\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 21 q^{2} - 19 q^{3} - 19 q^{4} - 15 q^{5} - 3 q^{6} - 19 q^{7} - 5 q^{8} - 12 q^{9} + 3 q^{10} - 5 q^{11} + 16 q^{12} - 17 q^{13} + 6 q^{14} + 3 q^{15} + 3 q^{16} - 11 q^{17} - 3 q^{19} + 19 q^{20}+ \cdots - 101 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.59831 − 0.357129i −0.832772 + 0.506420i 4.69783 + 1.31627i 1.18390 + 3.33118i 2.34465 − 1.01843i 0.00809392 − 0.118329i −6.92511 − 3.00800i −0.943147 + 1.82019i −1.88648 − 9.07824i
2.2 −0.695664 − 0.0956168i 1.82754 − 1.11135i −1.45103 − 0.406559i 0.245360 + 0.690377i −1.37762 + 0.598383i −0.0288100 + 0.421187i 2.25869 + 0.981089i 0.724598 − 1.39841i −0.104677 − 0.503731i
2.3 1.33105 + 0.182949i −0.887464 + 0.539679i −0.187599 − 0.0525628i −0.103404 − 0.290951i −1.28000 + 0.555981i 0.104560 − 1.52862i −2.70476 − 1.17484i −0.883856 + 1.70576i −0.0844071 − 0.406189i
3.1 −2.08955 + 1.27068i 1.69211 − 2.39717i 1.83145 − 3.53455i 0.399150 + 1.92081i −0.489699 + 7.15914i 0.396775 − 0.111171i 0.330590 + 4.83305i −1.87856 − 5.28576i −3.27479 − 3.50645i
3.2 −0.966383 + 0.587670i −0.921199 + 1.30504i −0.331591 + 0.639941i 0.186227 + 0.896176i 0.123296 − 1.80253i 2.11146 − 0.591604i −0.210001 − 3.07010i 0.150113 + 0.422379i −0.706623 − 0.756608i
3.3 1.59587 − 0.970469i −1.51474 + 2.14590i 0.684855 − 1.32171i −0.704153 − 3.38857i −0.334798 + 4.89457i −2.45476 + 0.687792i 0.0651834 + 0.952947i −1.30580 − 3.67416i −4.41224 − 4.72435i
4.1 −1.46574 − 0.410682i 0.343609 − 0.663135i 0.270905 + 0.164741i 2.97152 − 2.41751i −0.775980 + 0.830872i −3.61730 − 0.497186i 1.74853 + 1.87222i 1.40836 + 1.99519i −5.34832 + 2.32310i
4.2 −1.38906 − 0.389195i −1.46968 + 2.83636i 0.0691651 + 0.0420602i −1.63627 + 1.33120i 3.14537 − 3.36787i 2.46168 + 0.338350i 1.88953 + 2.02319i −4.15493 − 5.88620i 2.79097 − 1.21229i
4.3 1.00038 + 0.280293i 0.0747175 − 0.144198i −0.786643 − 0.478368i −1.16821 + 0.950406i 0.115164 − 0.123310i −0.942512 − 0.129545i −2.07107 − 2.21758i 1.71483 + 2.42936i −1.43504 + 0.623327i
6.1 −1.94127 + 0.843214i −0.0788221 + 1.15234i 1.69243 − 1.81215i −3.12146 + 1.89820i −0.818653 − 2.30347i 0.453878 + 2.18418i −0.339896 + 0.956375i 1.65039 + 0.226840i 4.45902 − 6.31699i
6.2 −0.210933 + 0.0916210i −0.129273 + 1.88991i −1.32901 + 1.42302i 2.07900 − 1.26427i −0.145887 − 0.410487i −0.778015 − 3.74401i 0.303978 − 0.855313i −0.582977 − 0.0801283i −0.322695 + 0.457155i
6.3 0.469654 − 0.203999i 0.153075 − 2.23788i −1.18615 + 1.27005i −0.316902 + 0.192713i −0.384634 − 1.08226i 0.556995 + 2.68041i −0.640935 + 1.80342i −2.01262 − 0.276628i −0.109521 + 0.155156i
7.1 −0.166930 + 2.44043i 0.317531 − 0.0889680i −3.94645 − 0.542427i 1.04338 − 1.47813i 0.164115 + 0.789763i −0.892148 − 0.955258i 0.987175 − 4.75055i −2.47035 + 1.50225i 3.43311 + 2.79304i
7.2 −4.24620e−5 0 0.000620773i −0.680374 + 0.190632i 1.98137 + 0.272333i −0.130589 + 0.185003i −8.94490e−5 0 0.000430452i −1.88596 − 2.01937i −0.000506380 0.00243684i −2.13669 + 1.29935i −0.000109300 0 8.89220e-5i
7.3 0.157658 − 2.30488i −1.88340 + 0.527704i −3.30625 − 0.454434i 1.67149 − 2.36797i 0.919361 + 4.42421i 3.09844 + 3.31762i −0.628599 + 3.02498i 0.705464 − 0.429002i −5.19436 − 4.22592i
8.1 −1.94127 − 0.843214i −0.0788221 − 1.15234i 1.69243 + 1.81215i −3.12146 − 1.89820i −0.818653 + 2.30347i 0.453878 − 2.18418i −0.339896 − 0.956375i 1.65039 − 0.226840i 4.45902 + 6.31699i
8.2 −0.210933 − 0.0916210i −0.129273 − 1.88991i −1.32901 − 1.42302i 2.07900 + 1.26427i −0.145887 + 0.410487i −0.778015 + 3.74401i 0.303978 + 0.855313i −0.582977 + 0.0801283i −0.322695 − 0.457155i
8.3 0.469654 + 0.203999i 0.153075 + 2.23788i −1.18615 − 1.27005i −0.316902 − 0.192713i −0.384634 + 1.08226i 0.556995 − 2.68041i −0.640935 − 1.80342i −2.01262 + 0.276628i −0.109521 − 0.155156i
9.1 −0.910119 + 1.75645i 0.226030 + 0.635987i −1.10344 − 1.56321i −0.813179 + 0.353213i −1.32279 − 0.181814i 0.694533 − 0.422355i −0.169663 + 0.0233196i 1.97374 − 1.60576i 0.119688 − 1.74977i
9.2 −0.210869 + 0.406959i −1.04014 − 2.92667i 1.03221 + 1.46231i 2.03255 − 0.882862i 1.41037 + 0.193851i −2.61080 + 1.58766i −1.72092 + 0.236535i −5.15637 + 4.19502i −0.0693139 + 1.01333i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.c even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.2.c.a ✓ 66
3.b odd 2 1 423.2.i.a 66
4.b odd 2 1 752.2.m.c 66
47.c even 23 1 inner 47.2.c.a ✓ 66
47.c even 23 1 2209.2.a.l 33
47.d odd 46 1 2209.2.a.m 33
141.h odd 46 1 423.2.i.a 66
188.g odd 46 1 752.2.m.c 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.2.c.a ✓ 66 1.a even 1 1 trivial
47.2.c.a ✓ 66 47.c even 23 1 inner
423.2.i.a 66 3.b odd 2 1
423.2.i.a 66 141.h odd 46 1
752.2.m.c 66 4.b odd 2 1
752.2.m.c 66 188.g odd 46 1
2209.2.a.l 33 47.c even 23 1
2209.2.a.m 33 47.d odd 46 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(47, [\chi])\).