Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,2,Mod(41,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.l (of order \(12\), degree \(4\), 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: | \(2.08409549276\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | −2.58894 | − | 0.693705i | 1.11469 | − | 1.32570i | 4.48934 | + | 2.59192i | 1.87884 | − | 3.25425i | −3.80551 | + | 2.65888i | −0.557243 | − | 0.965174i | −6.03415 | − | 6.03415i | −0.514937 | − | 2.95548i | −7.12170 | + | 7.12170i |
41.2 | −2.53829 | − | 0.680131i | 1.14019 | + | 1.30383i | 4.24826 | + | 2.45274i | −0.621203 | + | 1.07596i | −2.00736 | − | 4.08496i | 2.07180 | + | 3.58847i | −5.39881 | − | 5.39881i | −0.399921 | + | 2.97322i | 2.30858 | − | 2.30858i |
41.3 | −2.50426 | − | 0.671015i | −1.40115 | − | 1.01822i | 4.08901 | + | 2.36079i | −1.37767 | + | 2.38620i | 2.82561 | + | 3.49008i | −1.08986 | − | 1.88769i | −4.98934 | − | 4.98934i | 0.926458 | + | 2.85336i | 5.05123 | − | 5.05123i |
41.4 | −2.28686 | − | 0.612761i | −1.11376 | + | 1.32648i | 3.12219 | + | 1.80260i | 0.548233 | − | 0.949568i | 3.35982 | − | 2.35100i | 0.0705617 | + | 0.122216i | −2.68724 | − | 2.68724i | −0.519088 | − | 2.95475i | −1.83559 | + | 1.83559i |
41.5 | −2.04933 | − | 0.549115i | 0.850029 | − | 1.50912i | 2.16616 | + | 1.25063i | −1.19443 | + | 2.06881i | −2.57067 | + | 2.62592i | 0.190553 | + | 0.330047i | −0.752008 | − | 0.752008i | −1.55490 | − | 2.56559i | 3.58379 | − | 3.58379i |
41.6 | −1.80586 | − | 0.483880i | −1.12121 | − | 1.32018i | 1.29495 | + | 0.747640i | 0.957523 | − | 1.65848i | 1.38595 | + | 2.92660i | 2.42523 | + | 4.20063i | 0.667230 | + | 0.667230i | −0.485769 | + | 2.96041i | −2.53166 | + | 2.53166i |
41.7 | −1.78585 | − | 0.478518i | 1.70849 | + | 0.284740i | 1.22825 | + | 0.709129i | −1.14932 | + | 1.99067i | −2.91485 | − | 1.32605i | −1.59562 | − | 2.76369i | 0.760534 | + | 0.760534i | 2.83785 | + | 0.972949i | 3.00508 | − | 3.00508i |
41.8 | −1.45552 | − | 0.390005i | −1.72711 | − | 0.130672i | 0.234384 | + | 0.135322i | 1.15386 | − | 1.99855i | 2.46289 | + | 0.863780i | −1.71329 | − | 2.96750i | 1.84266 | + | 1.84266i | 2.96585 | + | 0.451370i | −2.45892 | + | 2.45892i |
41.9 | −1.27699 | − | 0.342169i | −0.592542 | + | 1.62754i | −0.218425 | − | 0.126108i | −1.41847 | + | 2.45685i | 1.31356 | − | 1.87561i | −0.670736 | − | 1.16175i | 2.10542 | + | 2.10542i | −2.29779 | − | 1.92877i | 2.65203 | − | 2.65203i |
41.10 | −1.04682 | − | 0.280495i | 1.72928 | − | 0.0978759i | −0.714895 | − | 0.412745i | 1.56226 | − | 2.70591i | −1.83770 | − | 0.382596i | 0.431040 | + | 0.746582i | 2.16525 | + | 2.16525i | 2.98084 | − | 0.338510i | −2.39439 | + | 2.39439i |
41.11 | −0.983600 | − | 0.263555i | 0.777862 | + | 1.54756i | −0.834044 | − | 0.481535i | 0.359924 | − | 0.623407i | −0.357240 | − | 1.72718i | 0.867014 | + | 1.50171i | 2.13354 | + | 2.13354i | −1.78986 | + | 2.40757i | −0.518323 | + | 0.518323i |
41.12 | −0.726652 | − | 0.194706i | 0.184317 | − | 1.72222i | −1.24194 | − | 0.717034i | 0.993651 | − | 1.72105i | −0.469260 | + | 1.21556i | −0.796821 | − | 1.38013i | 1.82674 | + | 1.82674i | −2.93205 | − | 0.634868i | −1.05714 | + | 1.05714i |
41.13 | −0.466927 | − | 0.125113i | −0.517906 | − | 1.65281i | −1.52968 | − | 0.883163i | −1.57483 | + | 2.72769i | 0.0350373 | + | 0.836538i | 0.454558 | + | 0.787318i | 1.28738 | + | 1.28738i | −2.46355 | + | 1.71200i | 1.07660 | − | 1.07660i |
41.14 | −0.0776028 | − | 0.0207936i | −1.64649 | + | 0.537659i | −1.72646 | − | 0.996773i | 0.0760639 | − | 0.131746i | 0.138952 | − | 0.00748738i | 0.766676 | + | 1.32792i | 0.226870 | + | 0.226870i | 2.42185 | − | 1.77050i | −0.00864225 | + | 0.00864225i |
41.15 | 0.188778 | + | 0.0505829i | 1.73139 | + | 0.0476698i | −1.69897 | − | 0.980902i | −1.89314 | + | 3.27902i | 0.324438 | + | 0.0965779i | 2.26829 | + | 3.92879i | −0.547502 | − | 0.547502i | 2.99546 | + | 0.165070i | −0.523246 | + | 0.523246i |
41.16 | 0.220291 | + | 0.0590268i | 1.37820 | − | 1.04907i | −1.68701 | − | 0.973994i | −0.486226 | + | 0.842168i | 0.365530 | − | 0.149750i | −2.17228 | − | 3.76249i | −0.636670 | − | 0.636670i | 0.798892 | − | 2.89167i | −0.156822 | + | 0.156822i |
41.17 | 0.469013 | + | 0.125672i | −0.397710 | + | 1.68577i | −1.52787 | − | 0.882117i | 2.09952 | − | 3.63648i | −0.398385 | + | 0.740668i | −1.45885 | − | 2.52681i | −1.29242 | − | 1.29242i | −2.68365 | − | 1.34090i | 1.44170 | − | 1.44170i |
41.18 | 0.703401 | + | 0.188476i | −1.41578 | − | 0.997788i | −1.27280 | − | 0.734852i | −0.0672512 | + | 0.116482i | −0.807798 | − | 0.968684i | 0.205252 | + | 0.355506i | −1.78664 | − | 1.78664i | 1.00884 | + | 2.82529i | −0.0692586 | + | 0.0692586i |
41.19 | 0.984157 | + | 0.263704i | 1.58668 | + | 0.694591i | −0.833025 | − | 0.480947i | 0.783948 | − | 1.35784i | 1.37837 | + | 1.10200i | 0.259733 | + | 0.449871i | −2.13391 | − | 2.13391i | 2.03509 | + | 2.20418i | 1.12960 | − | 1.12960i |
41.20 | 1.05023 | + | 0.281409i | −1.42135 | + | 0.989831i | −0.708253 | − | 0.408910i | −1.55901 | + | 2.70029i | −1.77130 | + | 0.639572i | −1.67740 | − | 2.90535i | −2.16641 | − | 2.16641i | 1.04047 | − | 2.81379i | −2.39721 | + | 2.39721i |
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
29.c | odd | 4 | 1 | inner |
261.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 261.2.l.a | ✓ | 112 |
3.b | odd | 2 | 1 | 783.2.m.a | 112 | ||
9.c | even | 3 | 1 | 783.2.m.a | 112 | ||
9.d | odd | 6 | 1 | inner | 261.2.l.a | ✓ | 112 |
29.c | odd | 4 | 1 | inner | 261.2.l.a | ✓ | 112 |
87.f | even | 4 | 1 | 783.2.m.a | 112 | ||
261.l | even | 12 | 1 | inner | 261.2.l.a | ✓ | 112 |
261.m | odd | 12 | 1 | 783.2.m.a | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
261.2.l.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
261.2.l.a | ✓ | 112 | 9.d | odd | 6 | 1 | inner |
261.2.l.a | ✓ | 112 | 29.c | odd | 4 | 1 | inner |
261.2.l.a | ✓ | 112 | 261.l | even | 12 | 1 | inner |
783.2.m.a | 112 | 3.b | odd | 2 | 1 | ||
783.2.m.a | 112 | 9.c | even | 3 | 1 | ||
783.2.m.a | 112 | 87.f | even | 4 | 1 | ||
783.2.m.a | 112 | 261.m | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(261, [\chi])\).