Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,3,Mod(7,348)]
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, 0, 6]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("348.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 348.p (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: | \(9.48231319974\) |
Analytic rank: | \(0\) |
Dimension: | \(360\) |
Relative dimension: | \(60\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −1.99911 | + | 0.0596572i | −0.751509 | + | 1.56052i | 3.99288 | − | 0.238523i | −0.103650 | − | 0.454120i | 1.40925 | − | 3.16449i | 0.138351 | − | 0.287290i | −7.96798 | + | 0.715037i | −1.87047 | − | 2.34549i | 0.234299 | + | 0.901653i |
7.2 | −1.99866 | + | 0.0730893i | 0.751509 | − | 1.56052i | 3.98932 | − | 0.292162i | −0.987961 | − | 4.32854i | −1.38796 | + | 3.17389i | −2.47930 | + | 5.14832i | −7.95195 | + | 0.875510i | −1.87047 | − | 2.34549i | 2.29097 | + | 8.57909i |
7.3 | −1.98520 | + | 0.242860i | 0.751509 | − | 1.56052i | 3.88204 | − | 0.964252i | 0.211154 | + | 0.925126i | −1.11291 | + | 3.28046i | 3.86437 | − | 8.02445i | −7.47244 | + | 2.85703i | −1.87047 | − | 2.34549i | −0.643859 | − | 1.78528i |
7.4 | −1.90359 | − | 0.613475i | 0.751509 | − | 1.56052i | 3.24730 | + | 2.33561i | −0.704065 | − | 3.08471i | −2.38791 | + | 2.50956i | 4.78219 | − | 9.93031i | −4.74868 | − | 6.43817i | −1.87047 | − | 2.34549i | −0.552142 | + | 6.30394i |
7.5 | −1.89547 | + | 0.638129i | −0.751509 | + | 1.56052i | 3.18558 | − | 2.41911i | −1.33767 | − | 5.86070i | 0.428643 | − | 3.43748i | −0.670921 | + | 1.39318i | −4.49446 | + | 6.61814i | −1.87047 | − | 2.34549i | 6.27538 | + | 10.2551i |
7.6 | −1.88944 | + | 0.655768i | 0.751509 | − | 1.56052i | 3.13994 | − | 2.47806i | 0.548610 | + | 2.40362i | −0.396586 | + | 3.44133i | −5.75269 | + | 11.9456i | −4.30768 | + | 6.74121i | −1.87047 | − | 2.34549i | −2.61278 | − | 4.18172i |
7.7 | −1.87351 | + | 0.699984i | −0.751509 | + | 1.56052i | 3.02005 | − | 2.62285i | 1.93067 | + | 8.45881i | 0.315614 | − | 3.44969i | −4.21585 | + | 8.75430i | −3.82212 | + | 7.02790i | −1.87047 | − | 2.34549i | −9.53814 | − | 14.4962i |
7.8 | −1.87071 | − | 0.707423i | 0.751509 | − | 1.56052i | 2.99911 | + | 2.64677i | 0.521319 | + | 2.28405i | −2.50980 | + | 2.38765i | −2.29450 | + | 4.76459i | −3.73807 | − | 7.07296i | −1.87047 | − | 2.34549i | 0.640552 | − | 4.64158i |
7.9 | −1.81370 | − | 0.842913i | −0.751509 | + | 1.56052i | 2.57900 | + | 3.05758i | −0.188089 | − | 0.824073i | 2.67839 | − | 2.19686i | −4.83870 | + | 10.0477i | −2.10025 | − | 7.71939i | −1.87047 | − | 2.34549i | −0.353485 | + | 1.65316i |
7.10 | −1.80344 | − | 0.864632i | −0.751509 | + | 1.56052i | 2.50482 | + | 3.11863i | −1.76951 | − | 7.75272i | 2.70458 | − | 2.16454i | 3.35862 | − | 6.97425i | −1.82084 | − | 7.79003i | −1.87047 | − | 2.34549i | −3.51204 | + | 15.5116i |
7.11 | −1.73118 | + | 1.00150i | −0.751509 | + | 1.56052i | 1.99400 | − | 3.46756i | 1.06534 | + | 4.66756i | −0.261862 | − | 3.45419i | 4.00714 | − | 8.32092i | 0.0207712 | + | 7.99997i | −1.87047 | − | 2.34549i | −6.51886 | − | 7.01347i |
7.12 | −1.72275 | − | 1.01594i | 0.751509 | − | 1.56052i | 1.93573 | + | 3.50042i | 2.09993 | + | 9.20041i | −2.88006 | + | 1.92490i | −0.318726 | + | 0.661841i | 0.221455 | − | 7.99693i | −1.87047 | − | 2.34549i | 5.72942 | − | 17.9834i |
7.13 | −1.69008 | + | 1.06941i | 0.751509 | − | 1.56052i | 1.71274 | − | 3.61476i | −1.98094 | − | 8.67907i | 0.398723 | + | 3.44108i | 1.05784 | − | 2.19662i | 0.970976 | + | 7.94086i | −1.87047 | − | 2.34549i | 12.6294 | + | 12.5499i |
7.14 | −1.67048 | − | 1.09977i | −0.751509 | + | 1.56052i | 1.58101 | + | 3.67429i | 1.77906 | + | 7.79457i | 2.97160 | − | 1.78034i | 4.70222 | − | 9.76426i | 1.39982 | − | 7.87658i | −1.87047 | − | 2.34549i | 5.60034 | − | 14.9772i |
7.15 | −1.58178 | + | 1.22391i | 0.751509 | − | 1.56052i | 1.00407 | − | 3.87193i | 1.38979 | + | 6.08909i | 0.721225 | + | 3.38819i | 2.55132 | − | 5.29787i | 3.15070 | + | 7.35344i | −1.87047 | − | 2.34549i | −9.65087 | − | 7.93062i |
7.16 | −1.42923 | + | 1.39904i | −0.751509 | + | 1.56052i | 0.0853886 | − | 3.99909i | −1.39231 | − | 6.10010i | −1.10915 | − | 3.28173i | −3.13087 | + | 6.50132i | 5.47283 | + | 5.83507i | −1.87047 | − | 2.34549i | 10.5242 | + | 6.77055i |
7.17 | −1.37609 | − | 1.45134i | 0.751509 | − | 1.56052i | −0.212750 | + | 3.99434i | −1.52239 | − | 6.67004i | −3.29899 | + | 1.05673i | −3.08524 | + | 6.40657i | 6.08989 | − | 5.18780i | −1.87047 | − | 2.34549i | −7.58552 | + | 11.3881i |
7.18 | −1.34742 | − | 1.47799i | −0.751509 | + | 1.56052i | −0.368934 | + | 3.98295i | 1.46187 | + | 6.40485i | 3.31904 | − | 0.991951i | −3.03117 | + | 6.29428i | 6.38388 | − | 4.82141i | −1.87047 | − | 2.34549i | 7.49659 | − | 10.7906i |
7.19 | −1.10839 | − | 1.66477i | −0.751509 | + | 1.56052i | −1.54295 | + | 3.69044i | −0.213954 | − | 0.937392i | 3.43088 | − | 0.478576i | 2.52620 | − | 5.24571i | 7.85393 | − | 1.52179i | −1.87047 | − | 2.34549i | −1.32340 | + | 1.39518i |
7.20 | −1.09610 | + | 1.67289i | −0.751509 | + | 1.56052i | −1.59712 | − | 3.66732i | −0.224107 | − | 0.981875i | −1.78685 | − | 2.96769i | −2.00246 | + | 4.15814i | 7.88562 | + | 1.34796i | −1.87047 | − | 2.34549i | 1.88821 | + | 0.701331i |
See next 80 embeddings (of 360 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
29.d | even | 7 | 1 | inner |
116.j | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 348.3.p.a | ✓ | 360 |
4.b | odd | 2 | 1 | inner | 348.3.p.a | ✓ | 360 |
29.d | even | 7 | 1 | inner | 348.3.p.a | ✓ | 360 |
116.j | odd | 14 | 1 | inner | 348.3.p.a | ✓ | 360 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
348.3.p.a | ✓ | 360 | 1.a | even | 1 | 1 | trivial |
348.3.p.a | ✓ | 360 | 4.b | odd | 2 | 1 | inner |
348.3.p.a | ✓ | 360 | 29.d | even | 7 | 1 | inner |
348.3.p.a | ✓ | 360 | 116.j | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(348, [\chi])\).