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Properties

Label 513.2.t.a
Level $513$
Weight $2$
Character orbit 513.t
Analytic conductor $4.096$
Analytic rank $0$
Dimension $36$
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(179,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.179"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.t (of order \(6\), degree \(2\), 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: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{2} + 30 q^{4} + 3 q^{5} - q^{7} + 12 q^{8} - 6 q^{10} + 9 q^{11} + 3 q^{14} + 18 q^{16} - 27 q^{17} + q^{19} - 9 q^{20} - 6 q^{22} + 11 q^{25} + 2 q^{28} + 12 q^{29} - 12 q^{31} + 30 q^{32}+ \cdots - 84 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −2.58252 0 4.66943 1.09060 − 0.629658i 0 0.0839377 + 0.145384i −6.89386 0 −2.81650 + 1.62611i
179.2 −2.29322 0 3.25888 −0.0873194 + 0.0504139i 0 −0.977625 − 1.69330i −2.88689 0 0.200243 − 0.115610i
179.3 −1.83651 0 1.37279 1.22285 − 0.706010i 0 0.660133 + 1.14338i 1.15189 0 −2.24577 + 1.29660i
179.4 −1.75508 0 1.08031 −3.41857 + 1.97371i 0 0.109801 + 0.190182i 1.61413 0 5.99987 − 3.46402i
179.5 −1.24245 0 −0.456316 −1.59664 + 0.921821i 0 −0.955231 − 1.65451i 3.05185 0 1.98375 − 1.14532i
179.6 −1.24204 0 −0.457348 3.19785 − 1.84628i 0 1.96257 + 3.39927i 3.05211 0 −3.97185 + 2.29315i
179.7 −0.675805 0 −1.54329 2.09151 − 1.20754i 0 −2.30620 − 3.99445i 2.39457 0 −1.41346 + 0.816059i
179.8 −0.280205 0 −1.92149 −2.62675 + 1.51656i 0 0.223737 + 0.387523i 1.09882 0 0.736029 − 0.424947i
179.9 0.173194 0 −1.97000 2.80886 − 1.62170i 0 −0.506772 − 0.877755i −0.687581 0 0.486478 − 0.280868i
179.10 0.676405 0 −1.54248 0.120039 − 0.0693047i 0 0.877021 + 1.51904i −2.39615 0 0.0811952 − 0.0468781i
179.11 0.691402 0 −1.52196 0.0926192 − 0.0534737i 0 1.77987 + 3.08283i −2.43509 0 0.0640371 − 0.0369718i
179.12 0.808184 0 −1.34684 −1.00246 + 0.578771i 0 −0.183020 − 0.317000i −2.70486 0 −0.810173 + 0.467754i
179.13 1.55091 0 0.405316 1.63223 − 0.942368i 0 −2.41078 − 4.17560i −2.47321 0 2.53144 − 1.46153i
179.14 1.59456 0 0.542617 −3.08644 + 1.78196i 0 −1.39322 − 2.41313i −2.32388 0 −4.92151 + 2.84144i
179.15 2.03532 0 2.14253 0.682489 − 0.394035i 0 1.40930 + 2.44098i 0.290094 0 1.38908 − 0.801988i
179.16 2.15080 0 2.62595 2.61989 − 1.51259i 0 1.86959 + 3.23822i 1.34630 0 5.63486 − 3.25329i
179.17 2.59353 0 4.72638 0.664721 − 0.383777i 0 −1.86632 − 3.23257i 7.07094 0 1.72397 − 0.995336i
179.18 2.63354 0 4.93552 −2.90548 + 1.67748i 0 1.12321 + 1.94546i 7.73082 0 −7.65169 + 4.41771i
278.1 −2.58252 0 4.66943 1.09060 + 0.629658i 0 0.0839377 − 0.145384i −6.89386 0 −2.81650 − 1.62611i
278.2 −2.29322 0 3.25888 −0.0873194 − 0.0504139i 0 −0.977625 + 1.69330i −2.88689 0 0.200243 + 0.115610i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.t.a 36
3.b odd 2 1 171.2.t.a yes 36
9.c even 3 1 171.2.k.a ✓ 36
9.d odd 6 1 513.2.k.a 36
19.d odd 6 1 513.2.k.a 36
57.f even 6 1 171.2.k.a ✓ 36
171.i odd 6 1 171.2.t.a yes 36
171.t even 6 1 inner 513.2.t.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.k.a ✓ 36 9.c even 3 1
171.2.k.a ✓ 36 57.f even 6 1
171.2.t.a yes 36 3.b odd 2 1
171.2.t.a yes 36 171.i odd 6 1
513.2.k.a 36 9.d odd 6 1
513.2.k.a 36 19.d odd 6 1
513.2.t.a 36 1.a even 1 1 trivial
513.2.t.a 36 171.t even 6 1 inner

Hecke kernels

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