Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(95,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 20, 21]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.95");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 462.bc (of order \(30\), 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: | \(3.68908857338\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
95.1 | 0.104528 | − | 0.994522i | −1.64414 | + | 0.544800i | −0.978148 | − | 0.207912i | 1.25353 | − | 2.81547i | 0.369957 | + | 1.69208i | 1.28350 | + | 2.31358i | −0.309017 | + | 0.951057i | 2.40638 | − | 1.79146i | −2.66902 | − | 1.54096i |
95.2 | 0.104528 | − | 0.994522i | −1.63004 | + | 0.585637i | −0.978148 | − | 0.207912i | 0.112013 | − | 0.251585i | 0.412043 | + | 1.68233i | −2.57741 | − | 0.597448i | −0.309017 | + | 0.951057i | 2.31406 | − | 1.90922i | −0.238498 | − | 0.137697i |
95.3 | 0.104528 | − | 0.994522i | −1.45000 | − | 0.947367i | −0.978148 | − | 0.207912i | 0.625478 | − | 1.40485i | −1.09374 | + | 1.34303i | −1.00131 | − | 2.44895i | −0.309017 | + | 0.951057i | 1.20499 | + | 2.74736i | −1.33177 | − | 0.768898i |
95.4 | 0.104528 | − | 0.994522i | −1.41798 | − | 0.994648i | −0.978148 | − | 0.207912i | −1.12093 | + | 2.51765i | −1.13742 | + | 1.30625i | 2.40939 | − | 1.09308i | −0.309017 | + | 0.951057i | 1.02135 | + | 2.82079i | 2.38669 | + | 1.37795i |
95.5 | 0.104528 | − | 0.994522i | −1.18732 | − | 1.26106i | −0.978148 | − | 0.207912i | −0.157991 | + | 0.354854i | −1.37826 | + | 1.04900i | 0.106147 | + | 2.64362i | −0.309017 | + | 0.951057i | −0.180521 | + | 2.99456i | 0.336396 | + | 0.194218i |
95.6 | 0.104528 | − | 0.994522i | −0.612442 | + | 1.62016i | −0.978148 | − | 0.207912i | −1.31655 | + | 2.95701i | 1.54727 | + | 0.778440i | −1.86309 | − | 1.87854i | −0.309017 | + | 0.951057i | −2.24983 | − | 1.98451i | 2.80319 | + | 1.61843i |
95.7 | 0.104528 | − | 0.994522i | −0.611589 | + | 1.62048i | −0.978148 | − | 0.207912i | 0.684905 | − | 1.53832i | 1.54768 | + | 0.777625i | 1.54934 | − | 2.14465i | −0.309017 | + | 0.951057i | −2.25192 | − | 1.98214i | −1.45830 | − | 0.841952i |
95.8 | 0.104528 | − | 0.994522i | 0.0618530 | − | 1.73095i | −0.978148 | − | 0.207912i | 1.30191 | − | 2.92414i | −1.71500 | − | 0.242447i | −2.04556 | + | 1.67800i | −0.309017 | + | 0.951057i | −2.99235 | − | 0.214128i | −2.77204 | − | 1.60044i |
95.9 | 0.104528 | − | 0.994522i | 0.333518 | + | 1.69964i | −0.978148 | − | 0.207912i | 0.0756668 | − | 0.169950i | 1.72519 | − | 0.154030i | 1.63703 | + | 2.07849i | −0.309017 | + | 0.951057i | −2.77753 | + | 1.13372i | −0.161110 | − | 0.0930169i |
95.10 | 0.104528 | − | 0.994522i | 0.398305 | − | 1.68563i | −0.978148 | − | 0.207912i | −1.53261 | + | 3.44230i | −1.63476 | − | 0.572320i | −2.01818 | + | 1.71083i | −0.309017 | + | 0.951057i | −2.68271 | − | 1.34279i | 3.26324 | + | 1.88403i |
95.11 | 0.104528 | − | 0.994522i | 0.812842 | − | 1.52947i | −0.978148 | − | 0.207912i | 0.565716 | − | 1.27062i | −1.43613 | − | 0.968262i | 0.339893 | − | 2.62383i | −0.309017 | + | 0.951057i | −1.67858 | − | 2.48644i | −1.20453 | − | 0.695433i |
95.12 | 0.104528 | − | 0.994522i | 0.891795 | + | 1.48482i | −0.978148 | − | 0.207912i | −0.333330 | + | 0.748672i | 1.56991 | − | 0.731703i | −2.01015 | + | 1.72026i | −0.309017 | + | 0.951057i | −1.40940 | + | 2.64832i | 0.709728 | + | 0.409762i |
95.13 | 0.104528 | − | 0.994522i | 1.28109 | − | 1.16568i | −0.978148 | − | 0.207912i | −0.572012 | + | 1.28476i | −1.02538 | − | 1.39592i | 2.51605 | + | 0.818231i | −0.309017 | + | 0.951057i | 0.282395 | − | 2.98668i | 1.21793 | + | 0.703172i |
95.14 | 0.104528 | − | 0.994522i | 1.39020 | + | 1.03313i | −0.978148 | − | 0.207912i | −1.35789 | + | 3.04988i | 1.17278 | − | 1.27459i | 1.07443 | − | 2.41777i | −0.309017 | + | 0.951057i | 0.865302 | + | 2.87250i | 2.89123 | + | 1.66925i |
95.15 | 0.104528 | − | 0.994522i | 1.67644 | + | 0.435353i | −0.978148 | − | 0.207912i | 0.789465 | − | 1.77317i | 0.608205 | − | 1.62175i | −2.28945 | + | 1.32606i | −0.309017 | + | 0.951057i | 2.62093 | + | 1.45969i | −1.68093 | − | 0.970486i |
95.16 | 0.104528 | − | 0.994522i | 1.70747 | + | 0.290781i | −0.978148 | − | 0.207912i | 1.75629 | − | 3.94468i | 0.467667 | − | 1.66772i | 2.64495 | + | 0.0650804i | −0.309017 | + | 0.951057i | 2.83089 | + | 0.992999i | −3.73949 | − | 2.15900i |
107.1 | 0.104528 | + | 0.994522i | −1.64414 | − | 0.544800i | −0.978148 | + | 0.207912i | 1.25353 | + | 2.81547i | 0.369957 | − | 1.69208i | 1.28350 | − | 2.31358i | −0.309017 | − | 0.951057i | 2.40638 | + | 1.79146i | −2.66902 | + | 1.54096i |
107.2 | 0.104528 | + | 0.994522i | −1.63004 | − | 0.585637i | −0.978148 | + | 0.207912i | 0.112013 | + | 0.251585i | 0.412043 | − | 1.68233i | −2.57741 | + | 0.597448i | −0.309017 | − | 0.951057i | 2.31406 | + | 1.90922i | −0.238498 | + | 0.137697i |
107.3 | 0.104528 | + | 0.994522i | −1.45000 | + | 0.947367i | −0.978148 | + | 0.207912i | 0.625478 | + | 1.40485i | −1.09374 | − | 1.34303i | −1.00131 | + | 2.44895i | −0.309017 | − | 0.951057i | 1.20499 | − | 2.74736i | −1.33177 | + | 0.768898i |
107.4 | 0.104528 | + | 0.994522i | −1.41798 | + | 0.994648i | −0.978148 | + | 0.207912i | −1.12093 | − | 2.51765i | −1.13742 | − | 1.30625i | 2.40939 | + | 1.09308i | −0.309017 | − | 0.951057i | 1.02135 | − | 2.82079i | 2.38669 | − | 1.37795i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
33.f | even | 10 | 1 | inner |
231.be | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 462.2.bc.a | ✓ | 128 |
3.b | odd | 2 | 1 | 462.2.bc.b | yes | 128 | |
7.c | even | 3 | 1 | inner | 462.2.bc.a | ✓ | 128 |
11.d | odd | 10 | 1 | 462.2.bc.b | yes | 128 | |
21.h | odd | 6 | 1 | 462.2.bc.b | yes | 128 | |
33.f | even | 10 | 1 | inner | 462.2.bc.a | ✓ | 128 |
77.o | odd | 30 | 1 | 462.2.bc.b | yes | 128 | |
231.be | even | 30 | 1 | inner | 462.2.bc.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
462.2.bc.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
462.2.bc.a | ✓ | 128 | 7.c | even | 3 | 1 | inner |
462.2.bc.a | ✓ | 128 | 33.f | even | 10 | 1 | inner |
462.2.bc.a | ✓ | 128 | 231.be | even | 30 | 1 | inner |
462.2.bc.b | yes | 128 | 3.b | odd | 2 | 1 | |
462.2.bc.b | yes | 128 | 11.d | odd | 10 | 1 | |
462.2.bc.b | yes | 128 | 21.h | odd | 6 | 1 | |
462.2.bc.b | yes | 128 | 77.o | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} + 46 T_{5}^{126} + 50 T_{5}^{125} + 762 T_{5}^{124} + 2300 T_{5}^{123} + \cdots + 21\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).