Properties

Label 495.2.u.b
Level $495$
Weight $2$
Character orbit 495.u
Analytic conductor $3.953$
Analytic rank $0$
Dimension $68$
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(34,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.u (of order \(6\), degree \(2\), 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: \(68\)
Relative dimension: \(34\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 40 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 40 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{9} + 34 q^{11} - 10 q^{14} - 28 q^{15} - 52 q^{16} - 40 q^{19} + q^{20} - 14 q^{21} + 20 q^{24} + 2 q^{25} - 24 q^{26} - 6 q^{29} + 22 q^{30} - 36 q^{31} + 34 q^{34} - 16 q^{36} - 12 q^{39} + 15 q^{40} + 6 q^{41} + 80 q^{44} + 50 q^{45} + 84 q^{46} + 60 q^{49} - 19 q^{50} + 48 q^{51} - 50 q^{54} - 4 q^{55} + 30 q^{56} - 103 q^{60} - 62 q^{61} - 164 q^{64} - 17 q^{65} + 2 q^{66} - 20 q^{69} + 28 q^{70} - 60 q^{71} + 14 q^{74} + 11 q^{75} - 86 q^{76} + 32 q^{79} + 50 q^{80} - 112 q^{81} - 36 q^{84} + q^{85} + 14 q^{86} + 20 q^{89} + 101 q^{90} + 104 q^{91} + 38 q^{94} + 6 q^{95} + 40 q^{96} + 4 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.40159 1.38656i 1.71658 0.230960i 2.84508 + 4.92782i −2.23202 0.134408i −4.44276 1.82547i −1.56749 0.904992i 10.2332i 2.89331 0.792926i 5.17404 + 3.41762i
34.2 −2.30901 1.33311i −1.40726 1.00976i 2.55435 + 4.42426i 0.456072 2.18906i 1.90327 + 4.20757i 1.67661 + 0.967991i 8.28845i 0.960776 + 2.84199i −3.97133 + 4.44658i
34.3 −2.24330 1.29517i −1.29168 + 1.15393i 2.35494 + 4.07887i 0.619124 + 2.14865i 4.39217 0.915665i 3.40015 + 1.96308i 7.01949i 0.336889 2.98102i 1.39398 5.62194i
34.4 −2.19044 1.26465i −0.0214382 + 1.73192i 2.19869 + 3.80824i 2.17302 0.527243i 2.23723 3.76655i −2.80859 1.62154i 6.06370i −2.99908 0.0742583i −5.42665 1.59322i
34.5 −1.85631 1.07174i 0.637067 1.61064i 1.29725 + 2.24690i 0.372737 + 2.20478i −2.90877 + 2.30706i −2.53525 1.46372i 1.27429i −2.18829 2.05216i 1.67104 4.49223i
34.6 −1.82274 1.05236i 0.984983 + 1.42471i 1.21492 + 2.10430i −2.10780 + 0.746458i −0.296057 3.63344i 4.35713 + 2.51559i 0.904689i −1.05962 + 2.80664i 4.62750 + 0.857560i
34.7 −1.65897 0.957807i −0.608637 1.62159i 0.834788 + 1.44590i 2.19568 + 0.423084i −0.543461 + 3.27313i 0.188368 + 0.108754i 0.632963i −2.25912 + 1.97392i −3.23733 2.80492i
34.8 −1.60656 0.927550i 1.08333 1.35144i 0.720699 + 1.24829i −0.438671 2.19262i −2.99397 + 1.16633i 3.43672 + 1.98419i 1.03626i −0.652777 2.92812i −1.32901 + 3.92947i
34.9 −1.52649 0.881322i 0.294428 + 1.70684i 0.553458 + 0.958617i −1.99480 1.01034i 1.05484 2.86497i −2.73887 1.58129i 1.57419i −2.82662 + 1.00508i 2.15461 + 3.30034i
34.10 −1.24119 0.716601i −1.16940 1.27769i 0.0270352 + 0.0468263i −1.13845 1.92456i 0.535852 + 2.42386i −3.82108 2.20610i 2.78891i −0.265002 + 2.98827i 0.0338859 + 3.20456i
34.11 −1.20576 0.696146i −1.69928 0.335329i −0.0307612 0.0532800i −1.62199 + 1.53920i 1.81549 + 1.58727i 0.764219 + 0.441222i 2.87024i 2.77511 + 1.13964i 3.02724 0.726768i
34.12 −0.988594 0.570765i 1.72906 0.101785i −0.348454 0.603540i 1.84721 + 1.26008i −1.76743 0.886262i 2.48032 + 1.43202i 3.07860i 2.97928 0.351984i −1.10693 2.30003i
34.13 −0.832096 0.480411i −1.52769 + 0.816195i −0.538410 0.932554i 1.99967 + 1.00066i 1.66329 + 0.0547647i 0.0448556 + 0.0258974i 2.95628i 1.66765 2.49378i −1.18319 1.79331i
34.14 −0.457873 0.264353i −1.43793 + 0.965588i −0.860235 1.48997i 0.369572 2.20532i 0.913645 0.0619954i −0.575675 0.332366i 1.96704i 1.13528 2.77689i −0.752199 + 0.912057i
34.15 −0.454347 0.262317i 1.11658 + 1.32411i −0.862379 1.49369i 1.60826 1.55355i −0.159977 0.894500i −0.133177 0.0768899i 1.95414i −0.506517 + 2.95693i −1.13823 + 0.283976i
34.16 −0.184653 0.106609i 1.37287 + 1.05604i −0.977269 1.69268i −1.25926 + 1.84777i −0.140920 0.341362i −4.07041 2.35005i 0.843180i 0.769543 + 2.89962i 0.429516 0.206947i
34.17 −0.156402 0.0902987i 0.787143 1.54286i −0.983692 1.70380i 1.29279 1.82447i −0.262429 + 0.170228i −2.00416 1.15710i 0.716500i −1.76081 2.42890i −0.366942 + 0.168615i
34.18 0.156402 + 0.0902987i −0.787143 + 1.54286i −0.983692 1.70380i −2.22643 + 0.207348i −0.262429 + 0.170228i 2.00416 + 1.15710i 0.716500i −1.76081 2.42890i −0.366942 0.168615i
34.19 0.184653 + 0.106609i −1.37287 1.05604i −0.977269 1.69268i 2.22985 0.166669i −0.140920 0.341362i 4.07041 + 2.35005i 0.843180i 0.769543 + 2.89962i 0.429516 + 0.206947i
34.20 0.454347 + 0.262317i −1.11658 1.32411i −0.862379 1.49369i −2.14954 + 0.616016i −0.159977 0.894500i 0.133177 + 0.0768899i 1.95414i −0.506517 + 2.95693i −1.13823 0.283976i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.u.b 68
5.b even 2 1 inner 495.2.u.b 68
9.c even 3 1 inner 495.2.u.b 68
45.j even 6 1 inner 495.2.u.b 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.u.b 68 1.a even 1 1 trivial
495.2.u.b 68 5.b even 2 1 inner
495.2.u.b 68 9.c even 3 1 inner
495.2.u.b 68 45.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{68} - 54 T_{2}^{66} + 1599 T_{2}^{64} - 32724 T_{2}^{62} + 511335 T_{2}^{60} - 6414184 T_{2}^{58} + 66645475 T_{2}^{56} - 585175242 T_{2}^{54} + 4402996496 T_{2}^{52} - 28663491964 T_{2}^{50} + \cdots + 1048576 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display