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Properties

Label 495.2.bj.a
Level $495$
Weight $2$
Character orbit 495.bj
Analytic conductor $3.953$
Analytic rank $0$
Dimension $32$
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(28,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 15, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.bj (of order \(20\), degree \(8\), 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: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{2} + 2 q^{5} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{2} + 2 q^{5} + 10 q^{8} + 24 q^{11} - 10 q^{13} - 8 q^{16} - 16 q^{20} + 10 q^{22} + 24 q^{23} + 16 q^{25} - 20 q^{26} + 50 q^{28} - 28 q^{31} + 10 q^{35} - 8 q^{37} - 10 q^{38} - 50 q^{40} - 40 q^{41} + 60 q^{46} + 28 q^{47} + 50 q^{50} - 50 q^{52} + 24 q^{53} - 64 q^{55} + 80 q^{56} - 50 q^{58} - 60 q^{61} - 100 q^{62} - 8 q^{67} + 30 q^{68} + 30 q^{70} - 24 q^{71} + 50 q^{73} - 70 q^{77} - 98 q^{80} - 10 q^{82} - 90 q^{83} + 30 q^{85} - 100 q^{86} + 170 q^{88} + 20 q^{91} + 68 q^{92} + 40 q^{95} - 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −0.665529 + 1.30617i 0 −0.0875924 − 0.120561i 1.11862 − 1.93615i 0 −4.16343 − 0.659422i −2.68004 + 0.424477i 0 1.78448 + 2.74968i
28.2 0.261423 − 0.513072i 0 0.980670 + 1.34978i 1.76986 + 1.36660i 0 1.17850 + 0.186656i 2.08639 − 0.330452i 0 1.16385 − 0.550801i
28.3 0.474334 − 0.930933i 0 0.533928 + 0.734888i −2.23541 + 0.0540419i 0 −0.543058 − 0.0860119i 3.00128 − 0.475357i 0 −1.01002 + 2.10665i
28.4 1.15100 − 2.25897i 0 −2.60258 − 3.58214i −1.91314 + 1.15755i 0 −1.78576 − 0.282837i −6.07934 + 0.962874i 0 0.412838 + 5.65406i
73.1 −0.380541 − 2.40264i 0 −3.72576 + 1.21057i 0.622076 + 2.14779i 0 1.17208 − 2.30033i 2.11764 + 4.15610i 0 4.92366 − 2.31195i
73.2 −0.193716 − 1.22307i 0 0.443732 − 0.144177i 0.536423 − 2.17077i 0 1.57455 − 3.09022i −1.38667 − 2.72149i 0 −2.75893 − 0.235572i
73.3 0.0763931 + 0.482327i 0 1.67531 − 0.544341i −1.47656 + 1.67922i 0 −1.32402 + 2.59854i 0.833936 + 1.63669i 0 −0.922733 − 0.583903i
73.4 0.237790 + 1.50135i 0 −0.295389 + 0.0959778i 1.71486 − 1.43501i 0 0.0869260 − 0.170602i 1.16585 + 2.28811i 0 2.56223 + 2.23337i
118.1 −1.50135 − 0.237790i 0 0.295389 + 0.0959778i 1.89470 + 1.18749i 0 0.170602 − 0.0869260i 2.28811 + 1.16585i 0 −2.56223 − 2.23337i
118.2 −0.482327 − 0.0763931i 0 −1.67531 − 0.544341i −2.05331 − 0.885381i 0 −2.59854 + 1.32402i 1.63669 + 0.833936i 0 0.922733 + 0.583903i
118.3 1.22307 + 0.193716i 0 −0.443732 − 0.144177i 2.23029 − 0.160637i 0 3.09022 − 1.57455i −2.72149 − 1.38667i 0 2.75893 + 0.235572i
118.4 2.40264 + 0.380541i 0 3.72576 + 1.21057i −1.85044 + 1.25533i 0 2.30033 − 1.17208i 4.15610 + 2.11764i 0 −4.92366 + 2.31195i
127.1 −1.30617 − 0.665529i 0 0.0875924 + 0.120561i 0.233059 − 2.22389i 0 0.659422 − 4.16343i 0.424477 + 2.68004i 0 −1.78448 + 2.74968i
127.2 0.513072 + 0.261423i 0 −0.980670 − 1.34978i −2.23511 + 0.0653109i 0 −0.186656 + 1.17850i −0.330452 − 2.08639i 0 −1.16385 − 0.550801i
127.3 0.930933 + 0.474334i 0 −0.533928 − 0.734888i 1.77672 + 1.35766i 0 0.0860119 − 0.543058i −0.475357 − 3.00128i 0 1.01002 + 2.10665i
127.4 2.25897 + 1.15100i 0 2.60258 + 3.58214i 0.867371 + 2.06099i 0 0.282837 − 1.78576i 0.962874 + 6.07934i 0 −0.412838 + 5.65406i
172.1 −1.50135 + 0.237790i 0 0.295389 − 0.0959778i 1.89470 − 1.18749i 0 0.170602 + 0.0869260i 2.28811 − 1.16585i 0 −2.56223 + 2.23337i
172.2 −0.482327 + 0.0763931i 0 −1.67531 + 0.544341i −2.05331 + 0.885381i 0 −2.59854 − 1.32402i 1.63669 − 0.833936i 0 0.922733 − 0.583903i
172.3 1.22307 − 0.193716i 0 −0.443732 + 0.144177i 2.23029 + 0.160637i 0 3.09022 + 1.57455i −2.72149 + 1.38667i 0 2.75893 − 0.235572i
172.4 2.40264 − 0.380541i 0 3.72576 − 1.21057i −1.85044 − 1.25533i 0 2.30033 + 1.17208i 4.15610 − 2.11764i 0 −4.92366 − 2.31195i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.bj.a 32
3.b odd 2 1 55.2.l.a ✓ 32
5.c odd 4 1 inner 495.2.bj.a 32
11.d odd 10 1 inner 495.2.bj.a 32
12.b even 2 1 880.2.cm.a 32
15.d odd 2 1 275.2.bm.b 32
15.e even 4 1 55.2.l.a ✓ 32
15.e even 4 1 275.2.bm.b 32
33.d even 2 1 605.2.m.e 32
33.f even 10 1 55.2.l.a ✓ 32
33.f even 10 1 605.2.e.b 32
33.f even 10 1 605.2.m.c 32
33.f even 10 1 605.2.m.d 32
33.h odd 10 1 605.2.e.b 32
33.h odd 10 1 605.2.m.c 32
33.h odd 10 1 605.2.m.d 32
33.h odd 10 1 605.2.m.e 32
55.l even 20 1 inner 495.2.bj.a 32
60.l odd 4 1 880.2.cm.a 32
132.n odd 10 1 880.2.cm.a 32
165.l odd 4 1 605.2.m.e 32
165.r even 10 1 275.2.bm.b 32
165.u odd 20 1 55.2.l.a ✓ 32
165.u odd 20 1 275.2.bm.b 32
165.u odd 20 1 605.2.e.b 32
165.u odd 20 1 605.2.m.c 32
165.u odd 20 1 605.2.m.d 32
165.v even 20 1 605.2.e.b 32
165.v even 20 1 605.2.m.c 32
165.v even 20 1 605.2.m.d 32
165.v even 20 1 605.2.m.e 32
660.bv even 20 1 880.2.cm.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.l.a ✓ 32 3.b odd 2 1
55.2.l.a ✓ 32 15.e even 4 1
55.2.l.a ✓ 32 33.f even 10 1
55.2.l.a ✓ 32 165.u odd 20 1
275.2.bm.b 32 15.d odd 2 1
275.2.bm.b 32 15.e even 4 1
275.2.bm.b 32 165.r even 10 1
275.2.bm.b 32 165.u odd 20 1
495.2.bj.a 32 1.a even 1 1 trivial
495.2.bj.a 32 5.c odd 4 1 inner
495.2.bj.a 32 11.d odd 10 1 inner
495.2.bj.a 32 55.l even 20 1 inner
605.2.e.b 32 33.f even 10 1
605.2.e.b 32 33.h odd 10 1
605.2.e.b 32 165.u odd 20 1
605.2.e.b 32 165.v even 20 1
605.2.m.c 32 33.f even 10 1
605.2.m.c 32 33.h odd 10 1
605.2.m.c 32 165.u odd 20 1
605.2.m.c 32 165.v even 20 1
605.2.m.d 32 33.f even 10 1
605.2.m.d 32 33.h odd 10 1
605.2.m.d 32 165.u odd 20 1
605.2.m.d 32 165.v even 20 1
605.2.m.e 32 33.d even 2 1
605.2.m.e 32 33.h odd 10 1
605.2.m.e 32 165.l odd 4 1
605.2.m.e 32 165.v even 20 1
880.2.cm.a 32 12.b even 2 1
880.2.cm.a 32 60.l odd 4 1
880.2.cm.a 32 132.n odd 10 1
880.2.cm.a 32 660.bv even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 10 T_{2}^{31} + 50 T_{2}^{30} - 170 T_{2}^{29} + 430 T_{2}^{28} - 800 T_{2}^{27} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display