Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(35,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([7, 7, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.35");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 348.t (of order \(14\), degree \(6\), 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: | \(2.77879399034\) |
Analytic rank: | \(0\) |
Dimension: | \(336\) |
Relative dimension: | \(56\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
35.1 | −1.41321 | − | 0.0532282i | −1.71646 | + | 0.231871i | 1.99433 | + | 0.150445i | 2.31270 | + | 1.84432i | 2.43806 | − | 0.236318i | 1.11294 | + | 0.254022i | −2.81041 | − | 0.318766i | 2.89247 | − | 0.795994i | −3.17016 | − | 2.72951i |
35.2 | −1.40439 | + | 0.166373i | 1.72952 | + | 0.0935630i | 1.94464 | − | 0.467305i | −1.09448 | − | 0.872821i | −2.44450 | + | 0.156346i | 0.256689 | + | 0.0585876i | −2.65329 | + | 0.979815i | 2.98249 | + | 0.323638i | 1.68230 | + | 1.04369i |
35.3 | −1.39037 | + | 0.258621i | −0.520219 | − | 1.65208i | 1.86623 | − | 0.719156i | 1.14035 | + | 0.909396i | 1.15056 | + | 2.16246i | −3.79504 | − | 0.866193i | −2.40875 | + | 1.48254i | −2.45874 | + | 1.71889i | −1.82069 | − | 0.969474i |
35.4 | −1.37991 | − | 0.309609i | 0.578538 | − | 1.63257i | 1.80828 | + | 0.854463i | −2.25006 | − | 1.79437i | −1.30379 | + | 2.07368i | 0.00145454 | 0.000331989i | −2.23071 | − | 1.73894i | −2.33059 | − | 1.88901i | 2.54933 | + | 3.17270i | |
35.5 | −1.32649 | + | 0.490331i | −1.67868 | + | 0.426655i | 1.51915 | − | 1.30084i | −2.71854 | − | 2.16797i | 2.01755 | − | 1.38906i | −4.49236 | − | 1.02535i | −1.37730 | + | 2.47043i | 2.63593 | − | 1.43244i | 4.66914 | + | 1.54280i |
35.6 | −1.31443 | − | 0.521790i | −0.468526 | + | 1.66748i | 1.45547 | + | 1.37172i | −0.227131 | − | 0.181131i | 1.48592 | − | 1.94732i | −2.72634 | − | 0.622270i | −1.19737 | − | 2.56248i | −2.56097 | − | 1.56251i | 0.204036 | + | 0.356598i |
35.7 | −1.29359 | − | 0.571517i | 1.49175 | + | 0.880152i | 1.34674 | + | 1.47861i | 1.76879 | + | 1.41056i | −1.42669 | − | 1.99112i | 4.04660 | + | 0.923610i | −0.897070 | − | 2.68240i | 1.45066 | + | 2.62594i | −1.48192 | − | 2.83558i |
35.8 | −1.28664 | + | 0.586989i | 0.441588 | + | 1.67481i | 1.31089 | − | 1.51049i | 2.87932 | + | 2.29618i | −1.55126 | − | 1.89568i | −0.962659 | − | 0.219721i | −0.800003 | + | 2.71293i | −2.61000 | + | 1.47916i | −5.05248 | − | 1.26423i |
35.9 | −1.23942 | + | 0.681054i | 1.07334 | − | 1.35939i | 1.07233 | − | 1.68823i | 1.51787 | + | 1.21046i | −0.404494 | + | 2.41586i | 4.22479 | + | 0.964281i | −0.179298 | + | 2.82274i | −0.695904 | − | 2.91817i | −2.70566 | − | 0.466519i |
35.10 | −1.21474 | − | 0.724167i | 1.36947 | − | 1.06044i | 0.951165 | + | 1.75934i | 3.09364 | + | 2.46710i | −2.43149 | + | 0.296432i | −2.60358 | − | 0.594250i | 0.118644 | − | 2.82594i | 0.750915 | − | 2.90450i | −1.97137 | − | 5.23718i |
35.11 | −1.19538 | − | 0.755688i | −1.14525 | − | 1.29939i | 0.857871 | + | 1.80667i | 0.278916 | + | 0.222428i | 0.387077 | + | 2.41871i | 0.870753 | + | 0.198744i | 0.339796 | − | 2.80794i | −0.376812 | + | 2.97624i | −0.165325 | − | 0.476660i |
35.12 | −1.18856 | + | 0.766379i | 0.725197 | + | 1.57292i | 0.825327 | − | 1.82177i | −2.26138 | − | 1.80339i | −2.06739 | − | 1.31373i | −0.188704 | − | 0.0430704i | 0.415217 | + | 2.79778i | −1.94818 | + | 2.28136i | 4.06986 | + | 0.410356i |
35.13 | −1.03132 | + | 0.967665i | −1.50908 | − | 0.850111i | 0.127251 | − | 1.99595i | −0.0130888 | − | 0.0104380i | 2.37897 | − | 0.583541i | 1.19888 | + | 0.273637i | 1.80017 | + | 2.18160i | 1.55462 | + | 2.56576i | 0.0235993 | − | 0.00190066i |
35.14 | −1.00274 | − | 0.997254i | −1.59562 | + | 0.673802i | 0.0109695 | + | 1.99997i | −0.278916 | − | 0.222428i | 2.27194 | + | 0.915587i | −0.870753 | − | 0.198744i | 1.98348 | − | 2.01639i | 2.09198 | − | 2.15026i | 0.0578626 | + | 0.501188i |
35.15 | −0.976315 | − | 1.02314i | 0.773743 | + | 1.54962i | −0.0936199 | + | 1.99781i | −3.09364 | − | 2.46710i | 0.830057 | − | 2.30456i | 2.60358 | + | 0.594250i | 2.13543 | − | 1.85470i | −1.80264 | + | 2.39802i | 0.496188 | + | 5.57388i |
35.16 | −0.927728 | + | 1.06739i | −1.21862 | + | 1.23084i | −0.278641 | − | 1.98049i | −0.252365 | − | 0.201255i | −0.183246 | − | 2.44263i | 3.46126 | + | 0.790009i | 2.37246 | + | 1.53994i | −0.0299545 | − | 2.99985i | 0.448943 | − | 0.0826624i |
35.17 | −0.845038 | − | 1.13398i | 1.72591 | − | 0.145741i | −0.571822 | + | 1.91651i | −1.76879 | − | 1.41056i | −1.62373 | − | 1.83399i | −4.04660 | − | 0.923610i | 2.65650 | − | 0.971091i | 2.95752 | − | 0.503073i | −0.104856 | + | 3.19775i |
35.18 | −0.834189 | + | 1.14198i | 1.21862 | − | 1.23084i | −0.608258 | − | 1.90526i | −0.252365 | − | 0.201255i | 0.389050 | + | 2.41840i | −3.46126 | − | 0.790009i | 2.68318 | + | 0.894727i | −0.0299545 | − | 2.99985i | 0.440350 | − | 0.120313i |
35.19 | −0.801196 | − | 1.16537i | 0.301364 | − | 1.70563i | −0.716169 | + | 1.86738i | 0.227131 | + | 0.181131i | −2.22914 | + | 1.01535i | 2.72634 | + | 0.622270i | 2.74998 | − | 0.661535i | −2.81836 | − | 1.02803i | 0.0291078 | − | 0.409812i |
35.20 | −0.713912 | + | 1.22079i | 1.50908 | + | 0.850111i | −0.980658 | − | 1.74307i | −0.0130888 | − | 0.0104380i | −2.11516 | + | 1.23536i | −1.19888 | − | 0.273637i | 2.82803 | + | 0.0472247i | 1.55462 | + | 2.56576i | 0.0220869 | − | 0.00852690i |
See next 80 embeddings (of 336 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
29.e | even | 14 | 1 | inner |
87.h | odd | 14 | 1 | inner |
116.h | odd | 14 | 1 | inner |
348.t | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 348.2.t.a | ✓ | 336 |
3.b | odd | 2 | 1 | inner | 348.2.t.a | ✓ | 336 |
4.b | odd | 2 | 1 | inner | 348.2.t.a | ✓ | 336 |
12.b | even | 2 | 1 | inner | 348.2.t.a | ✓ | 336 |
29.e | even | 14 | 1 | inner | 348.2.t.a | ✓ | 336 |
87.h | odd | 14 | 1 | inner | 348.2.t.a | ✓ | 336 |
116.h | odd | 14 | 1 | inner | 348.2.t.a | ✓ | 336 |
348.t | even | 14 | 1 | inner | 348.2.t.a | ✓ | 336 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
348.2.t.a | ✓ | 336 | 1.a | even | 1 | 1 | trivial |
348.2.t.a | ✓ | 336 | 3.b | odd | 2 | 1 | inner |
348.2.t.a | ✓ | 336 | 4.b | odd | 2 | 1 | inner |
348.2.t.a | ✓ | 336 | 12.b | even | 2 | 1 | inner |
348.2.t.a | ✓ | 336 | 29.e | even | 14 | 1 | inner |
348.2.t.a | ✓ | 336 | 87.h | odd | 14 | 1 | inner |
348.2.t.a | ✓ | 336 | 116.h | odd | 14 | 1 | inner |
348.2.t.a | ✓ | 336 | 348.t | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(348, [\chi])\).