Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(146,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.146");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.bn (of order \(6\), degree \(2\), 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: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
146.1 | −2.30434 | + | 1.33041i | −0.643938 | + | 1.60790i | 2.54000 | − | 4.39941i | −3.38484 | −0.655319 | − | 4.56186i | −1.66594 | + | 2.05540i | 8.19533i | −2.17069 | − | 2.07078i | 7.79984 | − | 4.50324i | ||||
146.2 | −2.30434 | + | 1.33041i | 0.643938 | − | 1.60790i | 2.54000 | − | 4.39941i | 3.38484 | 0.655319 | + | 4.56186i | −0.947057 | + | 2.47044i | 8.19533i | −2.17069 | − | 2.07078i | −7.79984 | + | 4.50324i | ||||
146.3 | −2.01386 | + | 1.16270i | −1.06546 | − | 1.36557i | 1.70376 | − | 2.95099i | −0.993521 | 3.73345 | + | 1.51126i | −0.593544 | − | 2.57831i | 3.27304i | −0.729580 | + | 2.90993i | 2.00081 | − | 1.15517i | ||||
146.4 | −2.01386 | + | 1.16270i | 1.06546 | + | 1.36557i | 1.70376 | − | 2.95099i | 0.993521 | −3.73345 | − | 1.51126i | 2.52966 | − | 0.775133i | 3.27304i | −0.729580 | + | 2.90993i | −2.00081 | + | 1.15517i | ||||
146.5 | −1.88922 | + | 1.09074i | −1.60830 | − | 0.642938i | 1.37944 | − | 2.38926i | −0.155426 | 3.73972 | − | 0.539589i | 0.959965 | + | 2.46545i | 1.65550i | 2.17326 | + | 2.06808i | 0.293635 | − | 0.169530i | ||||
146.6 | −1.88922 | + | 1.09074i | 1.60830 | + | 0.642938i | 1.37944 | − | 2.38926i | 0.155426 | −3.73972 | + | 0.539589i | −2.61513 | + | 0.401373i | 1.65550i | 2.17326 | + | 2.06808i | −0.293635 | + | 0.169530i | ||||
146.7 | −1.43804 | + | 0.830256i | −1.23680 | + | 1.21257i | 0.378649 | − | 0.655839i | −1.88889 | 0.771823 | − | 2.77059i | 0.896028 | − | 2.48940i | − | 2.06352i | 0.0593312 | − | 2.99941i | 2.71631 | − | 1.56826i | |||
146.8 | −1.43804 | + | 0.830256i | 1.23680 | − | 1.21257i | 0.378649 | − | 0.655839i | 1.88889 | −0.771823 | + | 2.77059i | 1.70787 | − | 2.02068i | − | 2.06352i | 0.0593312 | − | 2.99941i | −2.71631 | + | 1.56826i | |||
146.9 | −1.29226 | + | 0.746087i | −1.56373 | + | 0.744804i | 0.113292 | − | 0.196227i | 2.73189 | 1.46506 | − | 2.12916i | 1.84288 | + | 1.89837i | − | 2.64625i | 1.89053 | − | 2.32935i | −3.53032 | + | 2.03823i | |||
146.10 | −1.29226 | + | 0.746087i | 1.56373 | − | 0.744804i | 0.113292 | − | 0.196227i | −2.73189 | −1.46506 | + | 2.12916i | −2.56547 | − | 0.646794i | − | 2.64625i | 1.89053 | − | 2.32935i | 3.53032 | − | 2.03823i | |||
146.11 | −0.828363 | + | 0.478255i | −1.28843 | − | 1.15756i | −0.542544 | + | 0.939713i | 3.79208 | 1.62090 | + | 0.342680i | −2.21378 | − | 1.44885i | − | 2.95092i | 0.320110 | + | 2.98287i | −3.14122 | + | 1.81358i | |||
146.12 | −0.828363 | + | 0.478255i | 1.28843 | + | 1.15756i | −0.542544 | + | 0.939713i | −3.79208 | −1.62090 | − | 0.342680i | 2.36163 | + | 1.19277i | − | 2.95092i | 0.320110 | + | 2.98287i | 3.14122 | − | 1.81358i | |||
146.13 | −0.755524 | + | 0.436202i | −0.0385029 | + | 1.73162i | −0.619456 | + | 1.07293i | 0.294723 | −0.726247 | − | 1.32508i | −0.863899 | + | 2.50074i | − | 2.82564i | −2.99704 | − | 0.133345i | −0.222671 | + | 0.128559i | |||
146.14 | −0.755524 | + | 0.436202i | 0.0385029 | − | 1.73162i | −0.619456 | + | 1.07293i | −0.294723 | 0.726247 | + | 1.32508i | −1.73375 | + | 1.99853i | − | 2.82564i | −2.99704 | − | 0.133345i | 0.222671 | − | 0.128559i | |||
146.15 | −0.265128 | + | 0.153072i | −1.51585 | − | 0.837970i | −0.953138 | + | 1.65088i | −2.83438 | 0.530164 | − | 0.00986484i | 2.54431 | − | 0.725603i | − | 1.19588i | 1.59561 | + | 2.54048i | 0.751474 | − | 0.433863i | |||
146.16 | −0.265128 | + | 0.153072i | 1.51585 | + | 0.837970i | −0.953138 | + | 1.65088i | 2.83438 | −0.530164 | + | 0.00986484i | −0.643764 | − | 2.56624i | − | 1.19588i | 1.59561 | + | 2.54048i | −0.751474 | + | 0.433863i | |||
146.17 | 0.265128 | − | 0.153072i | −0.0322229 | + | 1.73175i | −0.953138 | + | 1.65088i | 2.83438 | 0.256539 | + | 0.464068i | 2.54431 | − | 0.725603i | 1.19588i | −2.99792 | − | 0.111604i | 0.751474 | − | 0.433863i | ||||
146.18 | 0.265128 | − | 0.153072i | 0.0322229 | − | 1.73175i | −0.953138 | + | 1.65088i | −2.83438 | −0.256539 | − | 0.464068i | −0.643764 | − | 2.56624i | 1.19588i | −2.99792 | − | 0.111604i | −0.751474 | + | 0.433863i | ||||
146.19 | 0.755524 | − | 0.436202i | −1.51888 | − | 0.832467i | −0.619456 | + | 1.07293i | −0.294723 | −1.51067 | + | 0.0335901i | −0.863899 | + | 2.50074i | 2.82564i | 1.61400 | + | 2.52884i | −0.222671 | + | 0.128559i | ||||
146.20 | 0.755524 | − | 0.436202i | 1.51888 | + | 0.832467i | −0.619456 | + | 1.07293i | 0.294723 | 1.51067 | − | 0.0335901i | −1.73375 | + | 1.99853i | 2.82564i | 1.61400 | + | 2.52884i | 0.222671 | − | 0.128559i | ||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
21.c | even | 2 | 1 | inner |
39.i | odd | 6 | 1 | inner |
91.n | odd | 6 | 1 | inner |
273.bn | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.bn.c | ✓ | 64 |
3.b | odd | 2 | 1 | inner | 273.2.bn.c | ✓ | 64 |
7.b | odd | 2 | 1 | inner | 273.2.bn.c | ✓ | 64 |
13.c | even | 3 | 1 | inner | 273.2.bn.c | ✓ | 64 |
21.c | even | 2 | 1 | inner | 273.2.bn.c | ✓ | 64 |
39.i | odd | 6 | 1 | inner | 273.2.bn.c | ✓ | 64 |
91.n | odd | 6 | 1 | inner | 273.2.bn.c | ✓ | 64 |
273.bn | even | 6 | 1 | inner | 273.2.bn.c | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bn.c | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
273.2.bn.c | ✓ | 64 | 3.b | odd | 2 | 1 | inner |
273.2.bn.c | ✓ | 64 | 7.b | odd | 2 | 1 | inner |
273.2.bn.c | ✓ | 64 | 13.c | even | 3 | 1 | inner |
273.2.bn.c | ✓ | 64 | 21.c | even | 2 | 1 | inner |
273.2.bn.c | ✓ | 64 | 39.i | odd | 6 | 1 | inner |
273.2.bn.c | ✓ | 64 | 91.n | odd | 6 | 1 | inner |
273.2.bn.c | ✓ | 64 | 273.bn | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):
\( T_{2}^{32} - 24 T_{2}^{30} + 346 T_{2}^{28} - 3260 T_{2}^{26} + 22737 T_{2}^{24} - 118297 T_{2}^{22} + \cdots + 5329 \) |
\( T_{19}^{32} - 172 T_{19}^{30} + 17833 T_{19}^{28} - 1204554 T_{19}^{26} + 60174841 T_{19}^{24} + \cdots + 98\!\cdots\!56 \) |