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Properties

Label 495.3.j.c
Level $495$
Weight $3$
Character orbit 495.j
Analytic conductor $13.488$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,3,Mod(298,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) 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("495.298"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 495.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4877730858\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{10} - 88 q^{13} - 296 q^{16} + 168 q^{25} + 248 q^{28} - 32 q^{31} - 24 q^{37} + 296 q^{40} - 48 q^{43} + 48 q^{46} + 64 q^{52} + 104 q^{58} + 576 q^{61} - 544 q^{67} - 1048 q^{70} - 408 q^{73}+ \cdots + 712 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1 −2.79494 + 2.79494i 0 − 11.6234i 3.96554 − 3.04540i 0 −8.48340 + 8.48340i 21.3068 + 21.3068i 0 −2.57174 + 19.5952i
298.2 −2.44305 + 2.44305i 0 − 7.93698i −4.99234 − 0.276657i 0 3.71159 − 3.71159i 9.61822 + 9.61822i 0 12.8724 − 11.5206i
298.3 −2.40357 + 2.40357i 0 − 7.55431i 2.32654 + 4.42574i 0 3.28861 − 3.28861i 8.54304 + 8.54304i 0 −16.2296 − 5.04559i
298.4 −2.02576 + 2.02576i 0 − 4.20738i 2.30178 − 4.43867i 0 5.06405 − 5.06405i 0.420106 + 0.420106i 0 4.32882 + 13.6545i
298.5 −1.89153 + 1.89153i 0 − 3.15579i −3.98330 − 3.02214i 0 −9.02252 + 9.02252i −1.59685 − 1.59685i 0 13.2510 − 1.81805i
298.6 −1.40545 + 1.40545i 0 0.0494143i 4.99874 − 0.112178i 0 1.33355 − 1.33355i −5.69125 − 5.69125i 0 −6.86783 + 7.18315i
298.7 −0.768581 + 0.768581i 0 2.81857i −2.92182 + 4.05746i 0 8.07326 − 8.07326i −5.24062 − 5.24062i 0 −0.872829 − 5.36414i
298.8 −0.762894 + 0.762894i 0 2.83599i 4.94835 + 0.716812i 0 −5.64620 + 5.64620i −5.21513 − 5.21513i 0 −4.32192 + 3.22822i
298.9 −0.692901 + 0.692901i 0 3.03978i −4.55660 − 2.05850i 0 3.68149 − 3.68149i −4.87787 − 4.87787i 0 4.58361 − 1.73093i
298.10 −0.364637 + 0.364637i 0 3.73408i −0.0133418 − 4.99998i 0 −2.00042 + 2.00042i −2.82013 − 2.82013i 0 1.82804 + 1.81831i
298.11 0.364637 − 0.364637i 0 3.73408i 0.0133418 + 4.99998i 0 −2.00042 + 2.00042i 2.82013 + 2.82013i 0 1.82804 + 1.81831i
298.12 0.692901 − 0.692901i 0 3.03978i 4.55660 + 2.05850i 0 3.68149 − 3.68149i 4.87787 + 4.87787i 0 4.58361 − 1.73093i
298.13 0.762894 − 0.762894i 0 2.83599i −4.94835 − 0.716812i 0 −5.64620 + 5.64620i 5.21513 + 5.21513i 0 −4.32192 + 3.22822i
298.14 0.768581 − 0.768581i 0 2.81857i 2.92182 − 4.05746i 0 8.07326 − 8.07326i 5.24062 + 5.24062i 0 −0.872829 − 5.36414i
298.15 1.40545 − 1.40545i 0 0.0494143i −4.99874 + 0.112178i 0 1.33355 − 1.33355i 5.69125 + 5.69125i 0 −6.86783 + 7.18315i
298.16 1.89153 − 1.89153i 0 − 3.15579i 3.98330 + 3.02214i 0 −9.02252 + 9.02252i 1.59685 + 1.59685i 0 13.2510 − 1.81805i
298.17 2.02576 − 2.02576i 0 − 4.20738i −2.30178 + 4.43867i 0 5.06405 − 5.06405i −0.420106 − 0.420106i 0 4.32882 + 13.6545i
298.18 2.40357 − 2.40357i 0 − 7.55431i −2.32654 − 4.42574i 0 3.28861 − 3.28861i −8.54304 − 8.54304i 0 −16.2296 − 5.04559i
298.19 2.44305 − 2.44305i 0 − 7.93698i 4.99234 + 0.276657i 0 3.71159 − 3.71159i −9.61822 − 9.61822i 0 12.8724 − 11.5206i
298.20 2.79494 − 2.79494i 0 − 11.6234i −3.96554 + 3.04540i 0 −8.48340 + 8.48340i −21.3068 − 21.3068i 0 −2.57174 + 19.5952i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.3.j.c ✓ 40
3.b odd 2 1 inner 495.3.j.c ✓ 40
5.c odd 4 1 inner 495.3.j.c ✓ 40
15.e even 4 1 inner 495.3.j.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.3.j.c ✓ 40 1.a even 1 1 trivial
495.3.j.c ✓ 40 3.b odd 2 1 inner
495.3.j.c ✓ 40 5.c odd 4 1 inner
495.3.j.c ✓ 40 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 658 T_{2}^{36} + 163925 T_{2}^{32} + 19652208 T_{2}^{28} + 1180899290 T_{2}^{24} + \cdots + 30821664721 \) acting on \(S_{3}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display