Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,3,Mod(10,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([0, 23]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.10");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.s (of order \(28\), 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: | \(7.11173489980\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{28})\) |
Twist minimal: | no (minimal twist has level 87) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −2.89410 | + | 1.81848i | 0 | 3.33338 | − | 6.92184i | 0.541618 | + | 0.123621i | 0 | 3.21833 | − | 1.54986i | 1.40933 | + | 12.5082i | 0 | −1.79230 | + | 0.627151i | ||||||
10.2 | −2.72695 | + | 1.71346i | 0 | 2.76480 | − | 5.74117i | 2.24899 | + | 0.513318i | 0 | −4.13335 | + | 1.99052i | 0.855404 | + | 7.59192i | 0 | −7.01245 | + | 2.45376i | ||||||
10.3 | −1.12909 | + | 0.709452i | 0 | −0.964020 | + | 2.00181i | −1.86346 | − | 0.425323i | 0 | 5.50650 | − | 2.65179i | −0.928933 | − | 8.24451i | 0 | 2.40576 | − | 0.841811i | ||||||
10.4 | −0.689063 | + | 0.432967i | 0 | −1.44819 | + | 3.00719i | −5.51213 | − | 1.25811i | 0 | 10.2092 | − | 4.91649i | −0.668589 | − | 5.93389i | 0 | 4.34292 | − | 1.51965i | ||||||
10.5 | −0.293906 | + | 0.184673i | 0 | −1.68326 | + | 3.49532i | 4.09256 | + | 0.934101i | 0 | −7.74224 | + | 3.72847i | −0.306229 | − | 2.71785i | 0 | −1.37533 | + | 0.481249i | ||||||
10.6 | 0.153173 | − | 0.0962451i | 0 | −1.72134 | + | 3.57439i | −4.03972 | − | 0.922040i | 0 | −4.21732 | + | 2.03096i | 0.161373 | + | 1.43223i | 0 | −0.707519 | + | 0.247572i | ||||||
10.7 | 1.34844 | − | 0.847278i | 0 | −0.635136 | + | 1.31887i | 8.55782 | + | 1.95327i | 0 | −3.33059 | + | 1.60393i | 0.974241 | + | 8.64663i | 0 | 13.1946 | − | 4.61700i | ||||||
10.8 | 1.96972 | − | 1.23766i | 0 | 0.612471 | − | 1.27181i | 3.10034 | + | 0.707633i | 0 | 5.42647 | − | 2.61325i | 0.674177 | + | 5.98349i | 0 | 6.98262 | − | 2.44332i | ||||||
10.9 | 2.30831 | − | 1.45041i | 0 | 1.48908 | − | 3.09211i | −9.07338 | − | 2.07094i | 0 | −7.59823 | + | 3.65912i | 0.173376 | + | 1.53876i | 0 | −23.9479 | + | 8.37974i | ||||||
10.10 | 2.75540 | − | 1.73133i | 0 | 2.85918 | − | 5.93716i | −2.33652 | − | 0.533296i | 0 | 9.68828 | − | 4.66563i | −0.943583 | − | 8.37453i | 0 | −7.36138 | + | 2.57586i | ||||||
19.1 | −2.00834 | − | 3.19625i | 0 | −4.44708 | + | 9.23445i | −2.63960 | − | 0.602470i | 0 | 3.87010 | − | 1.86374i | 23.4424 | − | 2.64133i | 0 | 3.37555 | + | 9.64678i | ||||||
19.2 | −1.78356 | − | 2.83852i | 0 | −3.14057 | + | 6.52145i | 2.74140 | + | 0.625707i | 0 | 2.88031 | − | 1.38708i | 10.7875 | − | 1.21546i | 0 | −3.11337 | − | 8.89750i | ||||||
19.3 | −1.21026 | − | 1.92611i | 0 | −0.509654 | + | 1.05831i | −6.81765 | − | 1.55608i | 0 | −7.31790 | + | 3.52411i | −6.38668 | + | 0.719606i | 0 | 5.25391 | + | 15.0148i | ||||||
19.4 | −0.895405 | − | 1.42503i | 0 | 0.506578 | − | 1.05192i | −2.22038 | − | 0.506787i | 0 | −0.486644 | + | 0.234355i | −8.64224 | + | 0.973747i | 0 | 1.26595 | + | 3.61788i | ||||||
19.5 | 0.285100 | + | 0.453733i | 0 | 1.61094 | − | 3.34516i | −3.27824 | − | 0.748238i | 0 | 1.91324 | − | 0.921368i | 4.10709 | − | 0.462758i | 0 | −0.595126 | − | 1.70077i | ||||||
19.6 | 0.307001 | + | 0.488590i | 0 | 1.59106 | − | 3.30388i | 6.66842 | + | 1.52202i | 0 | −0.0882299 | + | 0.0424893i | 4.39633 | − | 0.495347i | 0 | 1.30357 | + | 3.72538i | ||||||
19.7 | 1.05332 | + | 1.67635i | 0 | 0.0348658 | − | 0.0723996i | −7.39573 | − | 1.68803i | 0 | 5.02392 | − | 2.41939i | 8.02753 | − | 0.904486i | 0 | −4.96036 | − | 14.1759i | ||||||
19.8 | 1.19609 | + | 1.90356i | 0 | −0.457390 | + | 0.949779i | 4.88061 | + | 1.11397i | 0 | −3.36949 | + | 1.62266i | 6.58101 | − | 0.741502i | 0 | 3.71713 | + | 10.6229i | ||||||
19.9 | 1.91360 | + | 3.04547i | 0 | −3.87752 | + | 8.05174i | −7.06239 | − | 1.61194i | 0 | 2.38032 | − | 1.14630i | −17.6447 | + | 1.98809i | 0 | −8.60544 | − | 24.5929i | ||||||
19.10 | 1.94439 | + | 3.09448i | 0 | −4.05961 | + | 8.42988i | 1.11394 | + | 0.254250i | 0 | −11.8327 | + | 5.69831i | −19.4529 | + | 2.19181i | 0 | 1.37917 | + | 3.94144i | ||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.f | odd | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 261.3.s.b | 120 | |
3.b | odd | 2 | 1 | 87.3.l.a | ✓ | 120 | |
29.f | odd | 28 | 1 | inner | 261.3.s.b | 120 | |
87.k | even | 28 | 1 | 87.3.l.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
87.3.l.a | ✓ | 120 | 3.b | odd | 2 | 1 | |
87.3.l.a | ✓ | 120 | 87.k | even | 28 | 1 | |
261.3.s.b | 120 | 1.a | even | 1 | 1 | trivial | |
261.3.s.b | 120 | 29.f | odd | 28 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 8 T_{2}^{119} + 32 T_{2}^{118} + 44 T_{2}^{117} - 458 T_{2}^{116} - 2880 T_{2}^{115} + \cdots + 18\!\cdots\!81 \) acting on \(S_{3}^{\mathrm{new}}(261, [\chi])\).