Properties

Label 513.2.cd.a
Level $513$
Weight $2$
Character orbit 513.cd
Analytic conductor $4.096$
Analytic rank $0$
Dimension $108$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(116,513)] 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("513.116"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.cd (of order \(18\), degree \(6\), 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: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 9 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} - 12 q^{10} + 9 q^{11} - 6 q^{13} + 9 q^{14} + 9 q^{16} + 27 q^{17} - 15 q^{19} + 18 q^{20} - 6 q^{22} + 9 q^{23} - 3 q^{25} - 72 q^{26} - 36 q^{28} - 18 q^{29}+ \cdots - 27 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −1.90556 − 1.59895i 0 0.727205 + 4.12418i 2.43276 − 2.89925i 0 1.63251 2.72112 − 4.71311i 0 −9.27152 + 1.63482i
116.2 −1.78700 − 1.49947i 0 0.597656 + 3.38948i −0.797436 + 0.950347i 0 −0.314886 1.68164 − 2.91269i 0 2.85003 − 0.502538i
116.3 −1.46425 − 1.22865i 0 0.287148 + 1.62850i 1.02278 − 1.21890i 0 0.784390 −0.331042 + 0.573381i 0 −2.99522 + 0.528138i
116.4 −1.37859 − 1.15678i 0 0.215090 + 1.21983i −0.265173 + 0.316021i 0 −4.97178 −0.685071 + 1.18658i 0 0.731132 − 0.128918i
116.5 −0.742125 − 0.622716i 0 −0.184323 − 1.04535i −0.194712 + 0.232049i 0 −1.08691 −1.48294 + 2.56853i 0 0.289001 − 0.0509587i
116.6 −0.704973 − 0.591543i 0 −0.200232 − 1.13557i −0.899185 + 1.07161i 0 1.63939 −1.45086 + 2.51296i 0 1.26780 − 0.223548i
116.7 −0.400522 − 0.336078i 0 −0.299827 − 1.70040i −2.05581 + 2.45002i 0 2.01117 −0.974224 + 1.68741i 0 1.64680 − 0.290374i
116.8 −0.388217 − 0.325752i 0 −0.302699 − 1.71669i 2.48025 − 2.95585i 0 −2.21821 −0.948484 + 1.64282i 0 −1.92575 + 0.339561i
116.9 −0.0307038 − 0.0257635i 0 −0.347017 − 1.96803i 1.41680 − 1.68848i 0 2.99977 −0.0801297 + 0.138789i 0 −0.0870023 + 0.0153409i
116.10 0.373949 + 0.313780i 0 −0.305917 − 1.73494i −1.23004 + 1.46591i 0 −3.29328 0.918148 − 1.59028i 0 −0.919944 + 0.162211i
116.11 0.586300 + 0.491964i 0 −0.245577 − 1.39274i −1.29387 + 1.54197i 0 −1.87387 1.30656 − 2.26302i 0 −1.51719 + 0.267522i
116.12 0.965067 + 0.809788i 0 −0.0716975 − 0.406617i 0.180362 − 0.214947i 0 3.15185 1.51989 − 2.63252i 0 0.348122 − 0.0613834i
116.13 1.09154 + 0.915910i 0 0.00526991 + 0.0298872i 2.61333 − 3.11445i 0 −3.25194 1.40328 − 2.43055i 0 5.70511 − 1.00597i
116.14 1.20013 + 1.00703i 0 0.0789109 + 0.447526i −1.56758 + 1.86817i 0 4.63240 1.21069 − 2.09698i 0 −3.76261 + 0.663450i
116.15 1.49018 + 1.25041i 0 0.309818 + 1.75707i 1.38825 − 1.65445i 0 −0.126291 0.209926 − 0.363602i 0 4.13748 − 0.729550i
116.16 1.73314 + 1.45428i 0 0.541559 + 3.07134i −1.93467 + 2.30565i 0 −4.02280 −1.26552 + 2.19195i 0 −6.70611 + 1.18247i
116.17 1.91073 + 1.60329i 0 0.733045 + 4.15731i 1.20081 − 1.43107i 0 2.44537 −2.77045 + 4.79855i 0 4.58886 − 0.809141i
116.18 2.06424 + 1.73210i 0 0.913607 + 5.18132i −0.404473 + 0.482032i 0 −1.01627 −4.39400 + 7.61064i 0 −1.66986 + 0.294441i
260.1 −2.32498 − 0.846225i 0 3.15737 + 2.64934i 0.319347 − 0.877398i 0 −4.79570 −2.62469 − 4.54609i 0 −1.48495 + 1.76970i
260.2 −2.25663 − 0.821348i 0 2.88570 + 2.42139i 0.520210 − 1.42926i 0 3.51071 −2.12170 − 3.67490i 0 −2.34785 + 2.79805i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 116.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.bd even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.cd.a 108
3.b odd 2 1 171.2.bd.a yes 108
9.c even 3 1 171.2.x.a ✓ 108
9.d odd 6 1 513.2.bo.a 108
19.f odd 18 1 513.2.bo.a 108
57.j even 18 1 171.2.x.a ✓ 108
171.bd even 18 1 inner 513.2.cd.a 108
171.be odd 18 1 171.2.bd.a yes 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.x.a ✓ 108 9.c even 3 1
171.2.x.a ✓ 108 57.j even 18 1
171.2.bd.a yes 108 3.b odd 2 1
171.2.bd.a yes 108 171.be odd 18 1
513.2.bo.a 108 9.d odd 6 1
513.2.bo.a 108 19.f odd 18 1
513.2.cd.a 108 1.a even 1 1 trivial
513.2.cd.a 108 171.bd even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(513, [\chi])\).