Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(11,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([7, 40]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.bm (of order \(42\), 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: | \(12.0163796583\) |
Analytic rank: | \(0\) |
Dimension: | \(1320\) |
Relative dimension: | \(110\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −3.85226 | + | 0.879253i | 2.99955 | − | 0.0522460i | 10.4629 | − | 5.03868i | 0.270126 | + | 1.79217i | −11.5091 | + | 2.83862i | 2.04933 | − | 6.69330i | −23.5185 | + | 18.7554i | 8.99454 | − | 0.313429i | −2.61637 | − | 6.66639i |
11.2 | −3.76346 | + | 0.858985i | 0.379079 | + | 2.97595i | 9.82188 | − | 4.72997i | 1.26468 | + | 8.39059i | −3.98295 | − | 10.8742i | 2.79668 | + | 6.41705i | −20.8290 | + | 16.6106i | −8.71260 | + | 2.25625i | −11.9669 | − | 30.4913i |
11.3 | −3.69776 | + | 0.843989i | −2.64374 | − | 1.41796i | 9.35723 | − | 4.50620i | −0.222314 | − | 1.47496i | 10.9727 | + | 3.01199i | −6.33007 | + | 2.98835i | −18.9361 | + | 15.1010i | 4.97876 | + | 7.49746i | 2.06691 | + | 5.26641i |
11.4 | −3.69032 | + | 0.842292i | 1.91071 | − | 2.31283i | 9.30515 | − | 4.48112i | −0.842237 | − | 5.58787i | −5.10304 | + | 10.1445i | 1.93246 | + | 6.72797i | −18.7270 | + | 14.9343i | −1.69840 | − | 8.83829i | 7.81475 | + | 19.9117i |
11.5 | −3.67202 | + | 0.838116i | −2.44499 | − | 1.73840i | 9.17745 | − | 4.41963i | 0.632965 | + | 4.19945i | 10.4350 | + | 4.33425i | 6.89263 | + | 1.22136i | −18.2167 | + | 14.5273i | 2.95595 | + | 8.50073i | −5.84389 | − | 14.8900i |
11.6 | −3.64215 | + | 0.831296i | −2.05762 | + | 2.18316i | 8.97030 | − | 4.31987i | −1.36579 | − | 9.06141i | 5.67932 | − | 9.66188i | 6.99915 | − | 0.109390i | −17.3970 | + | 13.8736i | −0.532367 | − | 8.98424i | 12.5071 | + | 31.8676i |
11.7 | −3.49099 | + | 0.796796i | −2.43344 | + | 1.75453i | 7.94827 | − | 3.82768i | 0.762633 | + | 5.05974i | 7.09713 | − | 8.06399i | −2.91713 | − | 6.36320i | −13.4992 | + | 10.7653i | 2.84328 | − | 8.53907i | −6.69393 | − | 17.0558i |
11.8 | −3.48644 | + | 0.795757i | −0.837676 | + | 2.88068i | 7.91816 | − | 3.81319i | −0.550933 | − | 3.65520i | 0.628189 | − | 10.7099i | −6.83606 | + | 1.50610i | −13.3882 | + | 10.6767i | −7.59660 | − | 4.82615i | 4.82945 | + | 12.3052i |
11.9 | −3.37519 | + | 0.770366i | 1.97862 | − | 2.25501i | 7.19458 | − | 3.46473i | 0.651352 | + | 4.32144i | −4.94102 | + | 9.13536i | −6.47385 | + | 2.66257i | −10.7872 | + | 8.60252i | −1.17016 | − | 8.92360i | −5.52752 | − | 14.0839i |
11.10 | −3.37360 | + | 0.770003i | −0.382520 | − | 2.97551i | 7.18442 | − | 3.45983i | −0.509050 | − | 3.37733i | 3.58162 | + | 9.74366i | 4.00587 | − | 5.74047i | −10.7516 | + | 8.57412i | −8.70736 | + | 2.27638i | 4.31789 | + | 11.0018i |
11.11 | −3.33917 | + | 0.762143i | 1.42379 | + | 2.64061i | 6.96530 | − | 3.35431i | −0.320505 | − | 2.12641i | −6.76680 | − | 7.73230i | −1.05395 | − | 6.92020i | −9.99061 | + | 7.96725i | −4.94563 | + | 7.51936i | 2.69085 | + | 6.85617i |
11.12 | −3.09773 | + | 0.707036i | 1.26663 | − | 2.71949i | 5.49213 | − | 2.64487i | 1.21443 | + | 8.05721i | −2.00090 | + | 9.31980i | 6.59924 | − | 2.33453i | −5.20637 | + | 4.15194i | −5.79129 | − | 6.88919i | −9.45870 | − | 24.1004i |
11.13 | −3.07710 | + | 0.702327i | −0.451137 | − | 2.96589i | 5.37138 | − | 2.58672i | −0.770464 | − | 5.11169i | 3.47121 | + | 8.80947i | −6.20114 | − | 3.24744i | −4.84097 | + | 3.86055i | −8.59295 | + | 2.67604i | 5.96087 | + | 15.1881i |
11.14 | −3.06589 | + | 0.699769i | 1.56870 | + | 2.55718i | 5.30611 | − | 2.55529i | −0.653283 | − | 4.33425i | −6.59889 | − | 6.74230i | 6.93916 | + | 0.920911i | −4.64521 | + | 3.70443i | −4.07837 | + | 8.02290i | 5.03586 | + | 12.8312i |
11.15 | −3.04906 | + | 0.695927i | 2.99824 | + | 0.102770i | 5.20855 | − | 2.50831i | −1.48219 | − | 9.83366i | −9.21332 | + | 1.77320i | −6.90823 | − | 1.12979i | −4.35496 | + | 3.47296i | 8.97888 | + | 0.616258i | 11.3628 | + | 28.9519i |
11.16 | −3.03462 | + | 0.692632i | 2.93053 | + | 0.641862i | 5.12530 | − | 2.46821i | 0.0346843 | + | 0.230115i | −9.33762 | + | 0.0819742i | 3.59973 | + | 6.00350i | −4.10947 | + | 3.27719i | 8.17603 | + | 3.76199i | −0.264639 | − | 0.674289i |
11.17 | −2.98855 | + | 0.682116i | −2.92784 | + | 0.654027i | 4.86226 | − | 2.34154i | −0.365541 | − | 2.42521i | 8.30387 | − | 3.95172i | −1.60483 | + | 6.81355i | −3.34737 | + | 2.66944i | 8.14450 | − | 3.82977i | 2.74671 | + | 6.99851i |
11.18 | −2.98443 | + | 0.681177i | 2.79051 | + | 1.10138i | 4.83894 | − | 2.33031i | 1.02417 | + | 6.79491i | −9.07833 | − | 1.38614i | −6.89710 | + | 1.19581i | −3.28082 | + | 2.61637i | 6.57395 | + | 6.14681i | −7.68509 | − | 19.5813i |
11.19 | −2.97265 | + | 0.678489i | −2.86246 | + | 0.897940i | 4.77245 | − | 2.29829i | 0.403378 | + | 2.67624i | 7.89987 | − | 4.61141i | 6.95934 | − | 0.753351i | −3.09193 | + | 2.46573i | 7.38741 | − | 5.14064i | −3.01490 | − | 7.68183i |
11.20 | −2.91113 | + | 0.664446i | −0.844404 | − | 2.87871i | 4.42931 | − | 2.13304i | 0.930053 | + | 6.17050i | 4.37092 | + | 7.81924i | −0.314731 | + | 6.99292i | −2.13882 | + | 1.70566i | −7.57396 | + | 4.86159i | −6.80747 | − | 17.3451i |
See next 80 embeddings (of 1320 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
441.bm | odd | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.3.bm.a | yes | 1320 |
9.d | odd | 6 | 1 | 441.3.bi.a | ✓ | 1320 | |
49.g | even | 21 | 1 | 441.3.bi.a | ✓ | 1320 | |
441.bm | odd | 42 | 1 | inner | 441.3.bm.a | yes | 1320 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.3.bi.a | ✓ | 1320 | 9.d | odd | 6 | 1 | |
441.3.bi.a | ✓ | 1320 | 49.g | even | 21 | 1 | |
441.3.bm.a | yes | 1320 | 1.a | even | 1 | 1 | trivial |
441.3.bm.a | yes | 1320 | 441.bm | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(441, [\chi])\).