Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [169,2,Mod(14,169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169, base_ring=CyclotomicField(26))
chi = DirichletCharacter(H, H._module([18]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("169.14");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 169.g (of order \(13\), degree \(12\), 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: | \(1.34947179416\) |
Analytic rank: | \(0\) |
Dimension: | \(156\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{13})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{13}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
14.1 | −1.55724 | + | 2.25606i | −1.88671 | + | 0.990222i | −1.95558 | − | 5.15643i | −0.621291 | − | 0.550416i | 0.704070 | − | 5.79854i | −0.145399 | − | 0.0358377i | 9.35521 | + | 2.30585i | 0.874940 | − | 1.26757i | 2.20927 | − | 0.544537i |
14.2 | −1.29190 | + | 1.87164i | 1.09718 | − | 0.575847i | −1.12482 | − | 2.96591i | 0.226698 | + | 0.200837i | −0.339674 | + | 2.79747i | 2.40457 | + | 0.592674i | 2.58801 | + | 0.637888i | −0.831980 | + | 1.20533i | −0.668766 | + | 0.164836i |
14.3 | −0.958835 | + | 1.38911i | 0.0179783 | − | 0.00943576i | −0.301058 | − | 0.793826i | −2.19225 | − | 1.94216i | −0.00413093 | + | 0.0340212i | −2.78022 | − | 0.685263i | −1.88632 | − | 0.464936i | −1.70396 | + | 2.46861i | 4.79989 | − | 1.18307i |
14.4 | −0.885893 | + | 1.28344i | −1.70070 | + | 0.892595i | −0.153196 | − | 0.403946i | 2.77189 | + | 2.45568i | 0.361046 | − | 2.97348i | 3.26613 | + | 0.805028i | −2.37420 | − | 0.585188i | 0.391453 | − | 0.567117i | −5.60732 | + | 1.38208i |
14.5 | −0.707334 | + | 1.02475i | 2.32028 | − | 1.21778i | 0.159418 | + | 0.420350i | 1.61135 | + | 1.42753i | −0.393297 | + | 3.23909i | −1.03213 | − | 0.254398i | −2.96148 | − | 0.729939i | 2.19654 | − | 3.18224i | −2.60263 | + | 0.641490i |
14.6 | −0.479816 | + | 0.695133i | −2.71090 | + | 1.42279i | 0.456223 | + | 1.20296i | −1.82534 | − | 1.61711i | 0.311704 | − | 2.56712i | −0.220084 | − | 0.0542458i | −2.69533 | − | 0.664340i | 3.62047 | − | 5.24515i | 1.99993 | − | 0.492940i |
14.7 | −0.0917982 | + | 0.132993i | −0.752776 | + | 0.395087i | 0.699950 | + | 1.84562i | 1.16867 | + | 1.03535i | 0.0165598 | − | 0.136382i | −2.31458 | − | 0.570492i | −0.623512 | − | 0.153682i | −1.29362 | + | 1.87413i | −0.244976 | + | 0.0603811i |
14.8 | 0.0291084 | − | 0.0421708i | 2.28283 | − | 1.19812i | 0.708279 | + | 1.86758i | −2.44658 | − | 2.16748i | 0.0159238 | − | 0.131144i | 1.95389 | + | 0.481590i | 0.198879 | + | 0.0490192i | 2.07162 | − | 3.00127i | −0.162620 | + | 0.0400823i |
14.9 | 0.481559 | − | 0.697658i | 1.04345 | − | 0.547645i | 0.454382 | + | 1.19811i | 0.674379 | + | 0.597448i | 0.120414 | − | 0.991695i | −0.515253 | − | 0.126998i | 2.70085 | + | 0.665700i | −0.915320 | + | 1.32607i | 0.741567 | − | 0.182780i |
14.10 | 0.711778 | − | 1.03119i | −1.78709 | + | 0.937940i | 0.152488 | + | 0.402078i | −0.453990 | − | 0.402200i | −0.304822 | + | 2.51044i | 4.78395 | + | 1.17914i | 2.95631 | + | 0.728665i | 0.609782 | − | 0.883421i | −0.737885 | + | 0.181872i |
14.11 | 1.00858 | − | 1.46118i | −2.49701 | + | 1.31053i | −0.408612 | − | 1.07742i | 2.96132 | + | 2.62350i | −0.603512 | + | 4.97037i | −1.91869 | − | 0.472914i | 1.46133 | + | 0.360186i | 2.81336 | − | 4.07585i | 6.82015 | − | 1.68102i |
14.12 | 1.13519 | − | 1.64460i | 0.811870 | − | 0.426102i | −0.706856 | − | 1.86382i | −2.38631 | − | 2.11409i | 0.220855 | − | 1.81891i | −0.234049 | − | 0.0576880i | 0.0128813 | + | 0.00317496i | −1.22662 | + | 1.77707i | −6.18573 | + | 1.52465i |
14.13 | 1.39315 | − | 2.01832i | 0.422612 | − | 0.221804i | −1.42355 | − | 3.75360i | 1.09172 | + | 0.967184i | 0.141089 | − | 1.16197i | 0.0247635 | + | 0.00610365i | −4.79682 | − | 1.18231i | −1.57479 | + | 2.28148i | 3.47302 | − | 0.856022i |
27.1 | −0.952850 | − | 2.51246i | −0.297229 | + | 0.430610i | −3.90751 | + | 3.46175i | 0.365684 | + | 3.01168i | 1.36511 | + | 0.336468i | −2.17334 | − | 1.14066i | 7.66220 | + | 4.02143i | 0.966734 | + | 2.54907i | 7.21828 | − | 3.78845i |
27.2 | −0.814971 | − | 2.14890i | 0.722310 | − | 1.04645i | −2.45658 | + | 2.17634i | −0.370802 | − | 3.05383i | −2.83737 | − | 0.699350i | 0.780099 | + | 0.409428i | 2.60879 | + | 1.36920i | 0.490496 | + | 1.29333i | −6.26019 | + | 3.28560i |
27.3 | −0.568511 | − | 1.49904i | −1.77394 | + | 2.57000i | −0.426895 | + | 0.378196i | 0.00472042 | + | 0.0388761i | 4.86104 | + | 1.19814i | −3.29128 | − | 1.72740i | −2.02954 | − | 1.06519i | −2.39421 | − | 6.31302i | 0.0555933 | − | 0.0291776i |
27.4 | −0.553905 | − | 1.46053i | −0.577862 | + | 0.837177i | −0.329307 | + | 0.291741i | 0.268400 | + | 2.21047i | 1.54280 | + | 0.380266i | 3.47659 | + | 1.82465i | −2.15772 | − | 1.13246i | 0.696873 | + | 1.83750i | 3.07979 | − | 1.61640i |
27.5 | −0.235915 | − | 0.622057i | 1.17724 | − | 1.70553i | 1.16572 | − | 1.03274i | 0.100994 | + | 0.831764i | −1.33867 | − | 0.329951i | 1.56980 | + | 0.823895i | −2.09560 | − | 1.09986i | −0.459118 | − | 1.21059i | 0.493578 | − | 0.259050i |
27.6 | −0.168406 | − | 0.444051i | 0.0744765 | − | 0.107898i | 1.32820 | − | 1.17668i | −0.265542 | − | 2.18694i | −0.0604545 | − | 0.0149007i | −3.17143 | − | 1.66450i | −1.58721 | − | 0.833034i | 1.05772 | + | 2.78898i | −0.926393 | + | 0.486209i |
27.7 | −0.000492135 | − | 0.00129765i | −1.38011 | + | 1.99943i | 1.49702 | − | 1.32624i | −0.444348 | − | 3.65953i | 0.00327377 | 0.000806912i | 3.94802 | + | 2.07208i | −0.00491548 | − | 0.00257984i | −1.02922 | − | 2.71382i | −0.00453012 | + | 0.00237759i | |
See next 80 embeddings (of 156 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
169.g | even | 13 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 169.2.g.a | ✓ | 156 |
169.g | even | 13 | 1 | inner | 169.2.g.a | ✓ | 156 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
169.2.g.a | ✓ | 156 | 1.a | even | 1 | 1 | trivial |
169.2.g.a | ✓ | 156 | 169.g | even | 13 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(169, [\chi])\).