Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(137,671)]
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, 14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.137");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.bp (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
137.1 | −2.57449 | + | 1.14624i | 0.529240 | 3.97588 | − | 4.41566i | 1.19623 | − | 0.532596i | −1.36252 | + | 0.606635i | −1.38263 | + | 2.39478i | −3.43277 | + | 10.5650i | −2.71990 | −2.46920 | + | 2.74232i | ||||
137.2 | −2.43783 | + | 1.08539i | −2.39080 | 3.42669 | − | 3.80572i | −2.00788 | + | 0.893966i | 5.82838 | − | 2.59496i | −0.136549 | + | 0.236510i | −2.57374 | + | 7.92116i | 2.71594 | 3.92457 | − | 4.35868i | ||||
137.3 | −2.35494 | + | 1.04849i | 3.34787 | 3.10816 | − | 3.45196i | 0.0183215 | − | 0.00815726i | −7.88404 | + | 3.51020i | −1.21247 | + | 2.10006i | −2.10702 | + | 6.48474i | 8.20825 | −0.0345933 | + | 0.0384197i | ||||
137.4 | −2.21929 | + | 0.988093i | −3.12919 | 2.61068 | − | 2.89945i | 1.80666 | − | 0.804378i | 6.94460 | − | 3.09194i | −0.128366 | + | 0.222336i | −1.42753 | + | 4.39348i | 6.79186 | −3.21472 | + | 3.57030i | ||||
137.5 | −2.17011 | + | 0.966197i | 0.867144 | 2.43760 | − | 2.70723i | 1.49944 | − | 0.667593i | −1.88180 | + | 0.837832i | 1.81831 | − | 3.14940i | −1.20602 | + | 3.71175i | −2.24806 | −2.60893 | + | 2.89751i | ||||
137.6 | −2.14577 | + | 0.955358i | 0.623440 | 2.35336 | − | 2.61367i | −3.77163 | + | 1.67924i | −1.33776 | + | 0.595608i | −1.91296 | + | 3.31334i | −1.10111 | + | 3.38887i | −2.61132 | 6.48878 | − | 7.20652i | ||||
137.7 | −2.09682 | + | 0.933564i | −1.39103 | 2.18685 | − | 2.42874i | 2.66607 | − | 1.18701i | 2.91674 | − | 1.29862i | 0.00498853 | − | 0.00864039i | −0.899498 | + | 2.76837i | −1.06503 | −4.48212 | + | 4.97790i | ||||
137.8 | −2.08703 | + | 0.929204i | 2.16609 | 2.15400 | − | 2.39226i | −1.33270 | + | 0.593358i | −4.52069 | + | 2.01274i | 0.762723 | − | 1.32108i | −0.860639 | + | 2.64877i | 1.69194 | 2.23004 | − | 2.47671i | ||||
137.9 | −2.06451 | + | 0.919179i | 1.96864 | 2.07905 | − | 2.30902i | 3.94607 | − | 1.75690i | −4.06427 | + | 1.80953i | 0.857271 | − | 1.48484i | −0.773131 | + | 2.37945i | 0.875529 | −6.53179 | + | 7.25429i | ||||
137.10 | −2.00024 | + | 0.890563i | −0.229366 | 1.86958 | − | 2.07638i | −0.920880 | + | 0.410002i | 0.458785 | − | 0.204264i | −0.400791 | + | 0.694190i | −0.537254 | + | 1.65350i | −2.94739 | 1.47685 | − | 1.64020i | ||||
137.11 | −1.99641 | + | 0.888861i | −1.51549 | 1.85734 | − | 2.06278i | −2.27536 | + | 1.01305i | 3.02555 | − | 1.34706i | 1.72939 | − | 2.99539i | −0.523869 | + | 1.61230i | −0.703282 | 3.64209 | − | 4.04495i | ||||
137.12 | −1.49033 | + | 0.663538i | 1.79780 | 0.442541 | − | 0.491491i | −1.22283 | + | 0.544439i | −2.67931 | + | 1.19291i | −1.13085 | + | 1.95869i | 0.674833 | − | 2.07692i | 0.232067 | 1.46116 | − | 1.62279i | ||||
137.13 | −1.45198 | + | 0.646464i | −2.30627 | 0.352077 | − | 0.391021i | 2.10166 | − | 0.935721i | 3.34866 | − | 1.49092i | 2.53048 | − | 4.38292i | 0.723870 | − | 2.22784i | 2.31886 | −2.44667 | + | 2.71730i | ||||
137.14 | −1.41158 | + | 0.628476i | −2.57569 | 0.259313 | − | 0.287996i | −0.484850 | + | 0.215869i | 3.63579 | − | 1.61876i | −1.23772 | + | 2.14379i | 0.769923 | − | 2.36958i | 3.63417 | 0.548735 | − | 0.609432i | ||||
137.15 | −1.38776 | + | 0.617870i | −2.32214 | 0.205851 | − | 0.228621i | −0.499363 | + | 0.222331i | 3.22257 | − | 1.43478i | −2.30818 | + | 3.99788i | 0.794437 | − | 2.44502i | 2.39234 | 0.555624 | − | 0.617083i | ||||
137.16 | −1.34365 | + | 0.598232i | −0.290905 | 0.109257 | − | 0.121342i | 0.756690 | − | 0.336900i | 0.390875 | − | 0.174029i | 1.27131 | − | 2.20198i | 0.834798 | − | 2.56924i | −2.91537 | −0.815183 | + | 0.905353i | ||||
137.17 | −1.33715 | + | 0.595335i | 3.06873 | 0.0952715 | − | 0.105810i | 0.742728 | − | 0.330684i | −4.10334 | + | 1.82692i | 1.44889 | − | 2.50954i | 0.840209 | − | 2.58590i | 6.41711 | −0.796268 | + | 0.884345i | ||||
137.18 | −1.26633 | + | 0.563806i | 1.92528 | −0.0525500 | + | 0.0583627i | 1.36638 | − | 0.608352i | −2.43804 | + | 1.08548i | −1.74446 | + | 3.02150i | 0.890340 | − | 2.74018i | 0.706700 | −1.38729 | + | 1.54075i | ||||
137.19 | −1.22445 | + | 0.545161i | −2.78243 | −0.136178 | + | 0.151241i | −3.78811 | + | 1.68658i | 3.40696 | − | 1.51687i | 0.359396 | − | 0.622493i | 0.912662 | − | 2.80889i | 4.74193 | 3.71891 | − | 4.13027i | ||||
137.20 | −1.08503 | + | 0.483086i | 2.37673 | −0.394344 | + | 0.437964i | −3.17227 | + | 1.41238i | −2.57882 | + | 1.14816i | 1.55889 | − | 2.70008i | 0.950348 | − | 2.92487i | 2.64883 | 2.75970 | − | 3.06496i | ||||
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
671.bp | 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.bp.a | yes | 480 |
11.c | even | 5 | 1 | 671.2.bl.a | ✓ | 480 | |
61.i | even | 15 | 1 | 671.2.bl.a | ✓ | 480 | |
671.bp | even | 15 | 1 | inner | 671.2.bp.a | yes | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bl.a | ✓ | 480 | 11.c | even | 5 | 1 | |
671.2.bl.a | ✓ | 480 | 61.i | even | 15 | 1 | |
671.2.bp.a | yes | 480 | 1.a | even | 1 | 1 | trivial |
671.2.bp.a | yes | 480 | 671.bp | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).