Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(16,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([12, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bo (of order \(15\), 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 \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −0.291878 | − | 2.77703i | −0.717005 | + | 2.20672i | −5.67041 | + | 1.20528i | 0.453028 | + | 0.0962941i | 6.33739 | + | 1.34705i | 1.96421 | + | 0.417507i | 3.27642 | + | 10.0838i | −1.92844 | − | 1.40110i | 0.135183 | − | 1.28618i |
16.2 | −0.286555 | − | 2.72639i | 0.569725 | − | 1.75343i | −5.39478 | + | 1.14670i | 0.0387212 | + | 0.00823044i | −4.94379 | − | 1.05084i | −2.62129 | − | 0.557172i | 2.97796 | + | 9.16521i | −0.322888 | − | 0.234592i | 0.0113436 | − | 0.107927i |
16.3 | −0.264872 | − | 2.52009i | 0.00192934 | − | 0.00593790i | −4.32438 | + | 0.919176i | −4.03276 | − | 0.857189i | −0.0154750 | − | 0.00328932i | 4.85771 | + | 1.03254i | 1.89573 | + | 5.83447i | 2.42702 | + | 1.76333i | −1.09203 | + | 10.3899i |
16.4 | −0.257849 | − | 2.45327i | 0.296341 | − | 0.912045i | −3.99576 | + | 0.849325i | 2.49094 | + | 0.529465i | −2.31391 | − | 0.491836i | 1.88631 | + | 0.400947i | 1.58937 | + | 4.89158i | 1.68304 | + | 1.22280i | 0.656635 | − | 6.24747i |
16.5 | −0.256882 | − | 2.44407i | −0.696487 | + | 2.14357i | −3.95120 | + | 0.839854i | 3.14962 | + | 0.669472i | 5.41795 | + | 1.15162i | −4.25555 | − | 0.904545i | 1.54882 | + | 4.76677i | −1.68274 | − | 1.22258i | 0.827157 | − | 7.86987i |
16.6 | −0.252656 | − | 2.40386i | −0.0802338 | + | 0.246934i | −3.75840 | + | 0.798873i | 2.17761 | + | 0.462865i | 0.613866 | + | 0.130481i | −0.547109 | − | 0.116292i | 1.37611 | + | 4.23523i | 2.37251 | + | 1.72373i | 0.562476 | − | 5.35160i |
16.7 | −0.251744 | − | 2.39518i | −0.434628 | + | 1.33765i | −3.71722 | + | 0.790119i | −3.20140 | − | 0.680478i | 3.31332 | + | 0.704267i | −2.98490 | − | 0.634461i | 1.33981 | + | 4.12350i | 0.826655 | + | 0.600600i | −0.823936 | + | 7.83922i |
16.8 | −0.244617 | − | 2.32738i | 0.821511 | − | 2.52835i | −3.40056 | + | 0.722812i | −2.84180 | − | 0.604043i | −6.08538 | − | 1.29349i | −1.40696 | − | 0.299059i | 1.06777 | + | 3.28626i | −3.29062 | − | 2.39078i | −0.710683 | + | 6.76170i |
16.9 | −0.230388 | − | 2.19200i | 0.864314 | − | 2.66009i | −2.79548 | + | 0.594197i | 2.39029 | + | 0.508072i | −6.03003 | − | 1.28172i | 3.78799 | + | 0.805163i | 0.584332 | + | 1.79839i | −3.90197 | − | 2.83495i | 0.562998 | − | 5.35657i |
16.10 | −0.220377 | − | 2.09675i | 0.280558 | − | 0.863469i | −2.39150 | + | 0.508329i | −1.15357 | − | 0.245198i | −1.87231 | − | 0.397971i | −0.463869 | − | 0.0985984i | 0.289869 | + | 0.892125i | 1.76019 | + | 1.27885i | −0.259899 | + | 2.47278i |
16.11 | −0.215195 | − | 2.04744i | −0.621413 | + | 1.91251i | −2.18941 | + | 0.465374i | −1.64819 | − | 0.350333i | 4.04948 | + | 0.860744i | −0.0274548 | − | 0.00583569i | 0.151617 | + | 0.466630i | −0.844501 | − | 0.613566i | −0.362606 | + | 3.44996i |
16.12 | −0.198811 | − | 1.89156i | −0.586899 | + | 1.80629i | −1.58217 | + | 0.336301i | 2.82418 | + | 0.600299i | 3.53339 | + | 0.751044i | 3.66803 | + | 0.779664i | −0.224800 | − | 0.691864i | −0.491182 | − | 0.356865i | 0.574022 | − | 5.46145i |
16.13 | −0.191354 | − | 1.82061i | 1.04429 | − | 3.21399i | −1.32172 | + | 0.280941i | 2.18932 | + | 0.465354i | −6.05127 | − | 1.28624i | −2.96533 | − | 0.630300i | −0.366996 | − | 1.12950i | −6.81216 | − | 4.94933i | 0.428295 | − | 4.07495i |
16.14 | −0.184476 | − | 1.75517i | −0.941366 | + | 2.89723i | −1.09029 | + | 0.231748i | −1.83535 | − | 0.390115i | 5.25878 | + | 1.11779i | 2.80280 | + | 0.595753i | −0.482840 | − | 1.48603i | −5.08070 | − | 3.69135i | −0.346141 | + | 3.29331i |
16.15 | −0.165121 | − | 1.57102i | −0.227252 | + | 0.699410i | −0.484557 | + | 0.102996i | −1.08369 | − | 0.230346i | 1.13631 | + | 0.241531i | −2.76074 | − | 0.586814i | −0.734475 | − | 2.26048i | 1.98952 | + | 1.44547i | −0.182938 | + | 1.74054i |
16.16 | −0.153909 | − | 1.46434i | −1.03247 | + | 3.17760i | −0.164315 | + | 0.0349261i | 1.62503 | + | 0.345410i | 4.81201 | + | 1.02282i | −2.52407 | − | 0.536509i | −0.833565 | − | 2.56545i | −6.60413 | − | 4.79818i | 0.255693 | − | 2.43275i |
16.17 | −0.152164 | − | 1.44775i | 0.233842 | − | 0.719692i | −0.116521 | + | 0.0247673i | −0.866362 | − | 0.184151i | −1.07751 | − | 0.229033i | 4.25982 | + | 0.905453i | −0.846098 | − | 2.60402i | 1.96378 | + | 1.42677i | −0.134774 | + | 1.28229i |
16.18 | −0.151974 | − | 1.44594i | 0.480498 | − | 1.47882i | −0.111352 | + | 0.0236685i | 1.55918 | + | 0.331414i | −2.21131 | − | 0.470028i | −4.52491 | − | 0.961798i | −0.847417 | − | 2.60808i | 0.471021 | + | 0.342217i | 0.242249 | − | 2.30485i |
16.19 | −0.151234 | − | 1.43889i | 0.0840938 | − | 0.258814i | −0.0912417 | + | 0.0193940i | −0.0300022 | − | 0.00637716i | −0.385123 | − | 0.0818605i | 0.0224112 | + | 0.00476365i | −0.852478 | − | 2.62366i | 2.36714 | + | 1.71983i | −0.00463871 | + | 0.0441343i |
16.20 | −0.139041 | − | 1.32289i | −0.409404 | + | 1.26002i | 0.225600 | − | 0.0479528i | 3.78591 | + | 0.804720i | 1.72378 | + | 0.366401i | 2.34108 | + | 0.497611i | −0.916896 | − | 2.82192i | 1.00702 | + | 0.731643i | 0.538156 | − | 5.12021i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
671.bo | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.bo.a | yes | 480 |
11.c | even | 5 | 1 | 671.2.bk.a | ✓ | 480 | |
61.i | even | 15 | 1 | 671.2.bk.a | ✓ | 480 | |
671.bo | even | 15 | 1 | inner | 671.2.bo.a | yes | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bk.a | ✓ | 480 | 11.c | even | 5 | 1 | |
671.2.bk.a | ✓ | 480 | 61.i | even | 15 | 1 | |
671.2.bo.a | yes | 480 | 1.a | even | 1 | 1 | trivial |
671.2.bo.a | yes | 480 | 671.bo | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).