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Properties

Label 819.2.gh.c
Level $819$
Weight $2$
Character orbit 819.gh
Analytic conductor $6.540$
Analytic rank $0$
Dimension $36$
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] = [819,2,Mod(19,819)] 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("819.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(819, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.gh (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} - 4 q^{11} - 42 q^{14} + 12 q^{16} + 4 q^{17} - 24 q^{19} + 14 q^{20} + 4 q^{22} + 12 q^{23} - 24 q^{25} + 28 q^{26} - 12 q^{28} - 8 q^{29} - 6 q^{31} - 46 q^{32} + 24 q^{34} + 10 q^{35}+ \cdots - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.45222 − 0.657070i 0 3.84958 + 2.22256i 2.20549 − 0.590961i 0 2.58986 + 0.540936i −4.38935 − 4.38935i 0 −5.79666
19.2 −2.02536 − 0.542694i 0 2.07553 + 1.19831i −1.89270 + 0.507149i 0 −1.04247 + 2.43172i −0.588043 − 0.588043i 0 4.10864
19.3 −1.64664 − 0.441217i 0 0.784712 + 0.453054i −1.35835 + 0.363968i 0 0.501823 − 2.59772i 1.31861 + 1.31861i 0 2.39731
19.4 −0.562494 − 0.150720i 0 −1.43837 − 0.830442i 0.672922 − 0.180309i 0 −2.49458 − 0.881516i 1.50746 + 1.50746i 0 −0.405691
19.5 0.511829 + 0.137144i 0 −1.48889 − 0.859611i 2.03763 − 0.545981i 0 0.917679 + 2.48150i −1.39354 − 1.39354i 0 1.11780
19.6 0.759625 + 0.203541i 0 −1.19645 − 0.690770i −2.02645 + 0.542987i 0 1.65876 − 2.06120i −1.88042 − 1.88042i 0 −1.64986
19.7 1.50246 + 0.402582i 0 0.363252 + 0.209723i −2.78312 + 0.745735i 0 0.599883 + 2.57685i −1.73841 − 1.73841i 0 −4.48173
19.8 1.53002 + 0.409967i 0 0.440828 + 0.254512i 3.63307 − 0.973479i 0 −1.31124 − 2.29797i −1.66997 − 1.66997i 0 5.95776
19.9 2.38279 + 0.638467i 0 3.53801 + 2.04267i −0.488495 + 0.130892i 0 −2.15177 + 1.53945i 3.63751 + 3.63751i 0 −1.24755
262.1 −0.705851 − 2.63427i 0 −4.70911 + 2.71881i 0.914933 − 3.41458i 0 2.62078 − 0.362619i 6.62917 + 6.62917i 0 −9.64073
262.2 −0.569735 − 2.12628i 0 −2.46442 + 1.42283i −0.837395 + 3.12520i 0 0.780325 − 2.52806i 1.31631 + 1.31631i 0 7.12215
262.3 −0.255728 − 0.954388i 0 0.886590 − 0.511873i 0.244406 − 0.912136i 0 2.46785 − 0.953787i −2.11257 − 2.11257i 0 −0.933034
262.4 −0.157691 − 0.588511i 0 1.41057 − 0.814394i −0.529856 + 1.97745i 0 −2.23905 − 1.40948i −1.56335 − 1.56335i 0 1.24730
262.5 −0.0407083 − 0.151925i 0 1.71063 − 0.987631i 0.570893 − 2.13060i 0 0.961303 + 2.46493i −0.442117 − 0.442117i 0 −0.346933
262.6 0.273028 + 1.01896i 0 0.768324 − 0.443592i −1.01943 + 3.80456i 0 1.44977 + 2.21318i 2.15363 + 2.15363i 0 −4.15502
262.7 0.339011 + 1.26521i 0 0.246231 − 0.142161i −0.109857 + 0.409991i 0 −2.64485 − 0.0688957i 2.11573 + 2.11573i 0 −0.555967
262.8 0.478662 + 1.78639i 0 −1.23003 + 0.710156i 0.0199621 − 0.0744995i 0 1.85948 − 1.88211i 0.758075 + 0.758075i 0 0.142640
262.9 0.639011 + 2.38482i 0 −3.54699 + 2.04786i 0.746344 − 2.78539i 0 −2.52355 + 0.794791i −3.65872 − 3.65872i 0 7.11959
388.1 −2.45222 + 0.657070i 0 3.84958 − 2.22256i 2.20549 + 0.590961i 0 2.58986 − 0.540936i −4.38935 + 4.38935i 0 −5.79666
388.2 −2.02536 + 0.542694i 0 2.07553 − 1.19831i −1.89270 − 0.507149i 0 −1.04247 − 2.43172i −0.588043 + 0.588043i 0 4.10864
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.gh.c 36
3.b odd 2 1 273.2.cg.a yes 36
7.d odd 6 1 819.2.et.c 36
13.f odd 12 1 819.2.et.c 36
21.g even 6 1 273.2.bt.a ✓ 36
39.k even 12 1 273.2.bt.a ✓ 36
91.w even 12 1 inner 819.2.gh.c 36
273.ch odd 12 1 273.2.cg.a yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.a ✓ 36 21.g even 6 1
273.2.bt.a ✓ 36 39.k even 12 1
273.2.cg.a yes 36 3.b odd 2 1
273.2.cg.a yes 36 273.ch odd 12 1
819.2.et.c 36 7.d odd 6 1
819.2.et.c 36 13.f odd 12 1
819.2.gh.c 36 1.a even 1 1 trivial
819.2.gh.c 36 91.w even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 63 T_{2}^{32} + 38 T_{2}^{31} - 224 T_{2}^{29} + 3079 T_{2}^{28} - 1750 T_{2}^{27} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display