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Properties

Label 495.2.bj.c
Level $495$
Weight $2$
Character orbit 495.bj
Analytic conductor $3.953$
Analytic rank $0$
Dimension $96$
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: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 165)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{5} - 20 q^{7} - 8 q^{11} + 8 q^{16} + 20 q^{17} + 60 q^{20} - 32 q^{22} - 32 q^{23} - 32 q^{25} - 60 q^{28} + 16 q^{31} + 8 q^{37} - 56 q^{38} + 120 q^{41} - 200 q^{46} - 60 q^{47} - 80 q^{50} + 40 q^{52} - 36 q^{53} + 80 q^{55} + 80 q^{56} + 44 q^{58} + 40 q^{61} - 80 q^{62} - 48 q^{67} - 80 q^{68} - 92 q^{70} - 32 q^{71} - 60 q^{73} + 24 q^{77} + 80 q^{80} + 32 q^{82} + 200 q^{83} - 80 q^{85} + 80 q^{86} - 144 q^{88} + 56 q^{91} - 20 q^{92} - 60 q^{95} + 32 q^{97}+O(q^{100}) \) 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 −1.18575 + 2.32716i 0 −2.83412 − 3.90083i −1.30674 − 1.81450i 0 −1.46263 − 0.231658i 7.27904 − 1.15289i 0 5.77211 − 0.889461i
28.2 −0.991111 + 1.94517i 0 −1.62580 − 2.23772i −2.23403 + 0.0954087i 0 0.0532647 + 0.00843630i 1.65161 − 0.261589i 0 2.02859 − 4.44012i
28.3 −0.895950 + 1.75840i 0 −1.11368 − 1.53285i 1.84639 + 1.26129i 0 1.95292 + 0.309312i −0.205250 + 0.0325083i 0 −3.87213 + 2.11664i
28.4 −0.590928 + 1.15976i 0 0.179719 + 0.247362i 1.66621 − 1.49122i 0 2.49573 + 0.395285i −2.96429 + 0.469498i 0 0.744848 + 2.81361i
28.5 −0.331042 + 0.649706i 0 0.863041 + 1.18787i −1.12140 + 1.93454i 0 −1.77896 − 0.281760i −2.49788 + 0.395626i 0 −0.885655 − 1.36900i
28.6 −0.170021 + 0.333685i 0 1.09313 + 1.50457i −0.0891995 + 2.23429i 0 0.999786 + 0.158351i −1.42769 + 0.226124i 0 −0.730383 − 0.409641i
28.7 0.0863333 − 0.169439i 0 1.15431 + 1.58878i −1.73786 − 1.40707i 0 −4.37063 − 0.692240i 0.744505 − 0.117918i 0 −0.388447 + 0.172984i
28.8 0.298067 − 0.584989i 0 0.922202 + 1.26930i 1.28833 − 1.82762i 0 2.25936 + 0.357848i 2.31434 − 0.366555i 0 −0.685133 − 1.29841i
28.9 0.757027 − 1.48575i 0 −0.458789 − 0.631469i 2.23602 + 0.0151940i 0 −2.35072 − 0.372318i 2.00841 − 0.318101i 0 1.71530 − 3.31066i
28.10 0.908700 − 1.78343i 0 −1.17930 − 1.62316i −0.626380 − 2.14654i 0 4.18741 + 0.663220i −0.0125370 + 0.00198567i 0 −4.39739 − 0.833463i
28.11 0.986271 − 1.93567i 0 −1.59850 − 2.20015i 0.401727 + 2.19969i 0 3.95619 + 0.626599i −1.54390 + 0.244529i 0 4.65407 + 1.39188i
28.12 1.12840 − 2.21461i 0 −2.45566 − 3.37992i 0.676946 − 2.13114i 0 −3.91204 − 0.619606i −5.34635 + 0.846779i 0 −3.95578 − 3.90395i
73.1 −0.408966 − 2.58211i 0 −4.59791 + 1.49395i −2.00042 − 0.999158i 0 −0.975676 + 1.91487i 3.36420 + 6.60261i 0 −1.76183 + 5.57392i
73.2 −0.374072 − 2.36180i 0 −3.53603 + 1.14893i 1.34914 − 1.78321i 0 −0.373384 + 0.732807i 1.86506 + 3.66039i 0 −4.71624 − 2.51934i
73.3 −0.229620 − 1.44976i 0 −0.146976 + 0.0477555i −0.283867 + 2.21798i 0 −1.53330 + 3.00927i −1.22978 − 2.41359i 0 3.28072 − 0.0977516i
73.4 −0.201671 − 1.27330i 0 0.321488 − 0.104458i 2.23487 − 0.0732929i 0 −2.05499 + 4.03315i −1.36839 − 2.68561i 0 −0.544032 − 2.83088i
73.5 −0.178502 − 1.12702i 0 0.663811 − 0.215685i −1.98355 + 1.03225i 0 0.694338 − 1.36271i −1.39764 − 2.74302i 0 1.51742 + 2.05123i
73.6 −0.0522232 − 0.329725i 0 1.79612 − 0.583595i 2.09365 + 0.785250i 0 1.33085 − 2.61195i −0.589341 − 1.15665i 0 0.149579 − 0.731337i
73.7 0.0220781 + 0.139396i 0 1.88317 − 0.611879i −0.421082 − 2.19606i 0 −0.796462 + 1.56314i 0.255016 + 0.500497i 0 0.296825 − 0.107182i
73.8 0.123435 + 0.779339i 0 1.30998 − 0.425638i 1.47874 + 1.67730i 0 0.409798 − 0.804274i 1.20986 + 2.37448i 0 −1.12466 + 1.35948i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.12
Significant digits:
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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.c 96
3.b odd 2 1 165.2.w.a ✓ 96
5.c odd 4 1 inner 495.2.bj.c 96
11.d odd 10 1 inner 495.2.bj.c 96
15.d odd 2 1 825.2.cw.b 96
15.e even 4 1 165.2.w.a ✓ 96
15.e even 4 1 825.2.cw.b 96
33.f even 10 1 165.2.w.a ✓ 96
55.l even 20 1 inner 495.2.bj.c 96
165.r even 10 1 825.2.cw.b 96
165.u odd 20 1 165.2.w.a ✓ 96
165.u odd 20 1 825.2.cw.b 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.w.a ✓ 96 3.b odd 2 1
165.2.w.a ✓ 96 15.e even 4 1
165.2.w.a ✓ 96 33.f even 10 1
165.2.w.a ✓ 96 165.u odd 20 1
495.2.bj.c 96 1.a even 1 1 trivial
495.2.bj.c 96 5.c odd 4 1 inner
495.2.bj.c 96 11.d odd 10 1 inner
495.2.bj.c 96 55.l even 20 1 inner
825.2.cw.b 96 15.d odd 2 1
825.2.cw.b 96 15.e even 4 1
825.2.cw.b 96 165.r even 10 1
825.2.cw.b 96 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 86 T_{2}^{92} - 180 T_{2}^{89} + 4943 T_{2}^{88} - 20 T_{2}^{87} + 15800 T_{2}^{85} + \cdots + 43046721 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display