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Properties

Label 495.2.ba.c
Level $495$
Weight $2$
Character orbit 495.ba
Analytic conductor $3.953$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(64,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.ba (of order \(10\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} + 4 q^{5} - 12 q^{10} + 4 q^{14} - 44 q^{16} - 16 q^{19} - 46 q^{20} + 14 q^{25} + 76 q^{26} - 20 q^{31} - 24 q^{34} + 40 q^{35} - 72 q^{40} - 60 q^{41} + 48 q^{44} + 108 q^{46} - 28 q^{49} + 38 q^{50} - 20 q^{55} - 24 q^{56} - 60 q^{59} + 40 q^{61} + 64 q^{64} - 20 q^{65} + 86 q^{70} + 32 q^{71} + 32 q^{74} - 136 q^{76} - 52 q^{79} - 42 q^{80} - 70 q^{85} + 104 q^{86} - 40 q^{89} - 40 q^{91} + 72 q^{94} + 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.46515 − 0.800975i 0 3.81735 + 2.77347i 2.15417 − 0.599639i 0 2.00926 − 2.76550i −4.14176 − 5.70065i 0 −5.79063 − 0.247234i
64.2 −2.10559 − 0.684148i 0 2.34742 + 1.70550i 1.27465 + 1.83719i 0 −0.936570 + 1.28908i −1.17323 − 1.61482i 0 −1.42699 − 4.74042i
64.3 −2.00341 − 0.650947i 0 1.97188 + 1.43266i −1.72286 + 1.42539i 0 −1.94929 + 2.68296i −0.541555 − 0.745386i 0 4.37946 − 1.73415i
64.4 −1.20873 − 0.392740i 0 −0.311255 − 0.226140i −2.20885 − 0.347855i 0 0.284071 − 0.390991i 1.78148 + 2.45199i 0 2.53328 + 1.28796i
64.5 −0.464400 − 0.150893i 0 −1.42514 − 1.03542i −1.35773 − 1.77667i 0 3.00296 − 4.13322i 1.07962 + 1.48598i 0 0.362444 + 1.02996i
64.6 −0.283476 − 0.0921069i 0 −1.54616 − 1.12335i 1.92187 + 1.14299i 0 −1.36429 + 1.87778i 0.685226 + 0.943133i 0 −0.439527 − 0.501026i
64.7 0.283476 + 0.0921069i 0 −1.54616 − 1.12335i −0.882995 + 2.05434i 0 1.36429 − 1.87778i −0.685226 − 0.943133i 0 −0.439527 + 0.501026i
64.8 0.464400 + 0.150893i 0 −1.42514 − 1.03542i 0.0541280 − 2.23541i 0 −3.00296 + 4.13322i −1.07962 − 1.48598i 0 0.362444 − 1.02996i
64.9 1.20873 + 0.392740i 0 −0.311255 − 0.226140i 1.58253 − 1.57975i 0 −0.284071 + 0.390991i −1.78148 − 2.45199i 0 2.53328 − 1.28796i
64.10 2.00341 + 0.650947i 0 1.97188 + 1.43266i 2.23165 + 0.140493i 0 1.94929 − 2.68296i 0.541555 + 0.745386i 0 4.37946 + 1.73415i
64.11 2.10559 + 0.684148i 0 2.34742 + 1.70550i 0.0486578 + 2.23554i 0 0.936570 − 1.28908i 1.17323 + 1.61482i 0 −1.42699 + 4.74042i
64.12 2.46515 + 0.800975i 0 3.81735 + 2.77347i −2.09522 + 0.781069i 0 −2.00926 + 2.76550i 4.14176 + 5.70065i 0 −5.79063 + 0.247234i
289.1 −1.60225 − 2.20531i 0 −1.67816 + 5.16484i 0.762627 + 2.10200i 0 −0.462529 − 0.150285i 8.89393 − 2.88981i 0 3.41365 − 5.04977i
289.2 −1.38939 − 1.91233i 0 −1.10857 + 3.41183i 1.97379 − 1.05079i 0 1.63033 + 0.529727i 3.56862 − 1.15951i 0 −4.75182 − 2.31457i
289.3 −1.12116 − 1.54314i 0 −0.506258 + 1.55810i −1.25493 − 1.85071i 0 −4.23918 − 1.37739i −0.656175 + 0.213204i 0 −1.44894 + 4.01149i
289.4 −0.472206 − 0.649936i 0 0.418596 − 1.28830i 1.33989 + 1.79016i 0 0.483617 + 0.157137i −2.56307 + 0.832793i 0 0.530788 − 1.71617i
289.5 −0.460901 − 0.634375i 0 0.428031 − 1.31734i −0.741158 − 2.10966i 0 1.44371 + 0.469090i −2.52448 + 0.820252i 0 −0.996719 + 1.44252i
289.6 −0.169760 − 0.233654i 0 0.592258 − 1.82278i −1.62175 + 1.53945i 0 −3.48791 − 1.13329i −1.07580 + 0.349548i 0 0.635009 + 0.117592i
289.7 0.169760 + 0.233654i 0 0.592258 − 1.82278i 0.962959 − 2.01810i 0 3.48791 + 1.13329i 1.07580 − 0.349548i 0 0.635009 − 0.117592i
289.8 0.460901 + 0.634375i 0 0.428031 − 1.31734i −2.23544 − 0.0529614i 0 −1.44371 − 0.469090i 2.52448 − 0.820252i 0 −0.996719 − 1.44252i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.ba.c 48
3.b odd 2 1 165.2.s.a ✓ 48
5.b even 2 1 inner 495.2.ba.c 48
11.c even 5 1 inner 495.2.ba.c 48
15.d odd 2 1 165.2.s.a ✓ 48
15.e even 4 1 825.2.n.o 24
15.e even 4 1 825.2.n.p 24
33.f even 10 1 1815.2.c.k 24
33.h odd 10 1 165.2.s.a ✓ 48
33.h odd 10 1 1815.2.c.j 24
55.j even 10 1 inner 495.2.ba.c 48
165.o odd 10 1 165.2.s.a ✓ 48
165.o odd 10 1 1815.2.c.j 24
165.r even 10 1 1815.2.c.k 24
165.u odd 20 1 9075.2.a.dx 12
165.u odd 20 1 9075.2.a.ea 12
165.v even 20 1 825.2.n.o 24
165.v even 20 1 825.2.n.p 24
165.v even 20 1 9075.2.a.dy 12
165.v even 20 1 9075.2.a.dz 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.s.a ✓ 48 3.b odd 2 1
165.2.s.a ✓ 48 15.d odd 2 1
165.2.s.a ✓ 48 33.h odd 10 1
165.2.s.a ✓ 48 165.o odd 10 1
495.2.ba.c 48 1.a even 1 1 trivial
495.2.ba.c 48 5.b even 2 1 inner
495.2.ba.c 48 11.c even 5 1 inner
495.2.ba.c 48 55.j even 10 1 inner
825.2.n.o 24 15.e even 4 1
825.2.n.o 24 165.v even 20 1
825.2.n.p 24 15.e even 4 1
825.2.n.p 24 165.v even 20 1
1815.2.c.j 24 33.h odd 10 1
1815.2.c.j 24 165.o odd 10 1
1815.2.c.k 24 33.f even 10 1
1815.2.c.k 24 165.r even 10 1
9075.2.a.dx 12 165.u odd 20 1
9075.2.a.dy 12 165.v even 20 1
9075.2.a.dz 12 165.v even 20 1
9075.2.a.ea 12 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 18 T_{2}^{46} + 215 T_{2}^{44} - 2162 T_{2}^{42} + 19181 T_{2}^{40} - 134722 T_{2}^{38} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display