Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(98,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([10, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.98");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bw (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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
98.1 | −2.48860 | − | 1.26801i | −0.468497 | + | 0.152224i | 3.40974 | + | 4.69310i | 0.186560 | + | 0.256778i | 1.35893 | + | 0.215233i | 0.874200 | + | 1.71571i | −1.66075 | − | 10.4856i | −2.23073 | + | 1.62072i | −0.138678 | − | 0.875577i |
98.2 | −2.41178 | − | 1.22886i | 2.07708 | − | 0.674883i | 3.13099 | + | 4.30944i | −1.52572 | − | 2.09998i | −5.83878 | − | 0.924772i | 0.426803 | + | 0.837648i | −1.40867 | − | 8.89401i | 1.43173 | − | 1.04021i | 1.09912 | + | 6.93957i |
98.3 | −2.23979 | − | 1.14123i | −1.41006 | + | 0.458156i | 2.53867 | + | 3.49417i | −1.50993 | − | 2.07824i | 3.68109 | + | 0.583028i | −1.41798 | − | 2.78294i | −0.911939 | − | 5.75776i | −0.648687 | + | 0.471299i | 1.01017 | + | 6.37798i |
98.4 | −2.15546 | − | 1.09826i | −1.48696 | + | 0.483143i | 2.26425 | + | 3.11648i | 1.43058 | + | 1.96903i | 3.73570 | + | 0.591677i | 0.147708 | + | 0.289894i | −0.700928 | − | 4.42548i | −0.449421 | + | 0.326523i | −0.921056 | − | 5.81532i |
98.5 | −2.14272 | − | 1.09177i | 1.90221 | − | 0.618064i | 2.22372 | + | 3.06069i | 0.387835 | + | 0.533808i | −4.75068 | − | 0.752434i | −0.469341 | − | 0.921134i | −0.670845 | − | 4.23555i | 0.809335 | − | 0.588016i | −0.248225 | − | 1.56723i |
98.6 | −2.12020 | − | 1.08029i | 1.44369 | − | 0.469082i | 2.15263 | + | 2.96284i | 1.58283 | + | 2.17858i | −3.56765 | − | 0.565060i | 1.57322 | + | 3.08761i | −0.618773 | − | 3.90678i | −0.562857 | + | 0.408939i | −1.00241 | − | 6.32895i |
98.7 | −2.05137 | − | 1.04522i | −2.42667 | + | 0.788474i | 1.94005 | + | 2.67025i | −2.15303 | − | 2.96339i | 5.80214 | + | 0.918968i | −0.102778 | − | 0.201713i | −0.468428 | − | 2.95754i | 2.84000 | − | 2.06338i | 1.31925 | + | 8.32940i |
98.8 | −2.02197 | − | 1.03024i | −3.18205 | + | 1.03391i | 1.85138 | + | 2.54821i | 1.84897 | + | 2.54488i | 7.49918 | + | 1.18775i | −1.97207 | − | 3.87041i | −0.408164 | − | 2.57705i | 6.62941 | − | 4.81655i | −1.11670 | − | 7.05056i |
98.9 | −1.99089 | − | 1.01441i | 1.03710 | − | 0.336973i | 1.75905 | + | 2.42112i | −0.856452 | − | 1.17881i | −2.40658 | − | 0.381164i | −1.85571 | − | 3.64203i | −0.346980 | − | 2.19075i | −1.46503 | + | 1.06441i | 0.509312 | + | 3.21567i |
98.10 | −1.84687 | − | 0.941029i | −2.17884 | + | 0.707948i | 1.34984 | + | 1.85789i | −0.0779410 | − | 0.107277i | 4.69024 | + | 0.742862i | 1.68706 | + | 3.31104i | −0.0961350 | − | 0.606972i | 1.81911 | − | 1.32166i | 0.0429968 | + | 0.271471i |
98.11 | −1.74735 | − | 0.890321i | 2.46689 | − | 0.801542i | 1.08500 | + | 1.49337i | −2.53296 | − | 3.48632i | −5.02416 | − | 0.795749i | 0.332296 | + | 0.652168i | 0.0472716 | + | 0.298461i | 3.01604 | − | 2.19128i | 1.32203 | + | 8.34698i |
98.12 | −1.70563 | − | 0.869060i | 0.00252967 | 0.000821940i | 0.978327 | + | 1.34655i | −0.840861 | − | 1.15735i | −0.00502899 | 0.000796514i | 2.11792 | + | 4.15666i | 0.100489 | + | 0.634464i | −2.42705 | + | 1.76335i | 0.428392 | + | 2.70476i | ||
98.13 | −1.62952 | − | 0.830281i | 1.26524 | − | 0.411100i | 0.790393 | + | 1.08788i | 2.48239 | + | 3.41671i | −2.40305 | − | 0.380606i | 0.388582 | + | 0.762635i | 0.187479 | + | 1.18370i | −0.995232 | + | 0.723078i | −1.20826 | − | 7.62868i |
98.14 | −1.43787 | − | 0.732630i | 0.306682 | − | 0.0996469i | 0.355147 | + | 0.488818i | −1.47103 | − | 2.02469i | −0.513972 | − | 0.0814052i | 0.394499 | + | 0.774247i | 0.352363 | + | 2.22473i | −2.34293 | + | 1.70224i | 0.631789 | + | 3.98896i |
98.15 | −1.43559 | − | 0.731472i | 1.33143 | − | 0.432609i | 0.350309 | + | 0.482159i | 0.706009 | + | 0.971738i | −2.22784 | − | 0.352855i | −1.64721 | − | 3.23282i | 0.353880 | + | 2.23431i | −0.841484 | + | 0.611374i | −0.302743 | − | 1.91145i |
98.16 | −1.36503 | − | 0.695519i | −0.524500 | + | 0.170420i | 0.203996 | + | 0.280776i | 1.20659 | + | 1.66073i | 0.834490 | + | 0.132170i | −0.448015 | − | 0.879279i | 0.396143 | + | 2.50115i | −2.18099 | + | 1.58458i | −0.491967 | − | 3.10616i |
98.17 | −1.27921 | − | 0.651792i | −1.55213 | + | 0.504317i | 0.0359835 | + | 0.0495271i | −0.568553 | − | 0.782546i | 2.31421 | + | 0.366536i | −0.936934 | − | 1.83884i | 0.435435 | + | 2.74923i | −0.272281 | + | 0.197824i | 0.217243 | + | 1.37162i |
98.18 | −1.22315 | − | 0.623227i | 3.09508 | − | 1.00565i | −0.0678803 | − | 0.0934292i | 0.326859 | + | 0.449883i | −4.41250 | − | 0.698872i | −1.41286 | − | 2.77289i | 0.454299 | + | 2.86833i | 6.14113 | − | 4.46179i | −0.119419 | − | 0.753984i |
98.19 | −1.18411 | − | 0.603334i | 2.41859 | − | 0.785848i | −0.137466 | − | 0.189205i | 0.0804878 | + | 0.110782i | −3.33801 | − | 0.528688i | 1.85019 | + | 3.63121i | 0.464410 | + | 2.93217i | 2.80497 | − | 2.03793i | −0.0284679 | − | 0.179739i |
98.20 | −1.06364 | − | 0.541952i | −2.47514 | + | 0.804221i | −0.337952 | − | 0.465151i | 2.00512 | + | 2.75981i | 3.06851 | + | 0.486004i | 1.32683 | + | 2.60405i | 0.480858 | + | 3.03602i | 3.05249 | − | 2.21776i | −0.637042 | − | 4.02213i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
61.j | odd | 20 | 1 | inner |
671.bw | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.bw.a | ✓ | 480 |
11.b | odd | 2 | 1 | inner | 671.2.bw.a | ✓ | 480 |
61.j | odd | 20 | 1 | inner | 671.2.bw.a | ✓ | 480 |
671.bw | even | 20 | 1 | inner | 671.2.bw.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bw.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
671.2.bw.a | ✓ | 480 | 11.b | odd | 2 | 1 | inner |
671.2.bw.a | ✓ | 480 | 61.j | odd | 20 | 1 | inner |
671.2.bw.a | ✓ | 480 | 671.bw | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).