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.
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 |
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 \(S_{2}^{\mathrm{new}}(495, [\chi])\).