Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [805,2,Mod(206,805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("805.206");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 805 = 5 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 805.p (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: | \(6.42795736271\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
206.1 | −1.37415 | − | 2.38010i | −0.708535 | − | 0.409073i | −2.77657 | + | 4.80917i | −0.500000 | − | 0.866025i | 2.24851i | 1.75590 | − | 1.97910i | 9.76512 | −1.16532 | − | 2.01839i | −1.37415 | + | 2.38010i | ||||
206.2 | −1.32261 | − | 2.29083i | 2.39292 | + | 1.38155i | −2.49861 | + | 4.32773i | −0.500000 | − | 0.866025i | − | 7.30903i | −0.971243 | − | 2.46103i | 7.92835 | 2.31736 | + | 4.01379i | −1.32261 | + | 2.29083i | |||
206.3 | −1.30476 | − | 2.25991i | −1.67396 | − | 0.966460i | −2.40479 | + | 4.16521i | −0.500000 | − | 0.866025i | 5.04398i | 0.0791150 | + | 2.64457i | 7.33162 | 0.368089 | + | 0.637550i | −1.30476 | + | 2.25991i | ||||
206.4 | −1.06469 | − | 1.84410i | −0.246709 | − | 0.142438i | −1.26714 | + | 2.19475i | −0.500000 | − | 0.866025i | 0.606609i | −2.37090 | − | 1.17423i | 1.13767 | −1.45942 | − | 2.52779i | −1.06469 | + | 1.84410i | ||||
206.5 | −1.00378 | − | 1.73861i | 2.46971 | + | 1.42589i | −1.01517 | + | 1.75832i | −0.500000 | − | 0.866025i | − | 5.72513i | 2.64463 | + | 0.0770837i | 0.0608924 | 2.56631 | + | 4.44498i | −1.00378 | + | 1.73861i | |||
206.6 | −0.992940 | − | 1.71982i | −1.81894 | − | 1.05016i | −0.971861 | + | 1.68331i | −0.500000 | − | 0.866025i | 4.17100i | 0.654076 | + | 2.56363i | −0.111761 | 0.705687 | + | 1.22229i | −0.992940 | + | 1.71982i | ||||
206.7 | −0.958524 | − | 1.66021i | 1.30460 | + | 0.753212i | −0.837536 | + | 1.45066i | −0.500000 | − | 0.866025i | − | 2.88789i | −0.697059 | + | 2.55228i | −0.622902 | −0.365342 | − | 0.632791i | −0.958524 | + | 1.66021i | |||
206.8 | −0.827267 | − | 1.43287i | 1.10377 | + | 0.637262i | −0.368741 | + | 0.638678i | −0.500000 | − | 0.866025i | − | 2.10874i | 2.51706 | − | 0.815116i | −2.08888 | −0.687794 | − | 1.19129i | −0.827267 | + | 1.43287i | |||
206.9 | −0.789539 | − | 1.36752i | −0.576533 | − | 0.332861i | −0.246743 | + | 0.427371i | −0.500000 | − | 0.866025i | 1.05123i | −0.114652 | − | 2.64327i | −2.37890 | −1.27841 | − | 2.21427i | −0.789539 | + | 1.36752i | ||||
206.10 | −0.495730 | − | 0.858629i | −2.78618 | − | 1.60860i | 0.508504 | − | 0.880756i | −0.500000 | − | 0.866025i | 3.18973i | −1.47466 | − | 2.19667i | −2.99124 | 3.67520 | + | 6.36563i | −0.495730 | + | 0.858629i | ||||
206.11 | −0.477169 | − | 0.826482i | −0.426860 | − | 0.246448i | 0.544619 | − | 0.943307i | −0.500000 | − | 0.866025i | 0.470389i | 2.20137 | + | 1.46764i | −2.94818 | −1.37853 | − | 2.38768i | −0.477169 | + | 0.826482i | ||||
206.12 | −0.471261 | − | 0.816248i | −2.12288 | − | 1.22564i | 0.555826 | − | 0.962719i | −0.500000 | − | 0.866025i | 2.31039i | −2.30347 | + | 1.30154i | −2.93280 | 1.50441 | + | 2.60571i | −0.471261 | + | 0.816248i | ||||
206.13 | −0.322496 | − | 0.558580i | 0.702468 | + | 0.405570i | 0.791992 | − | 1.37177i | −0.500000 | − | 0.866025i | − | 0.523179i | −2.11060 | + | 1.59542i | −2.31164 | −1.17103 | − | 2.02828i | −0.322496 | + | 0.558580i | |||
206.14 | −0.321106 | − | 0.556172i | 2.14301 | + | 1.23727i | 0.793782 | − | 1.37487i | −0.500000 | − | 0.866025i | − | 1.58918i | −1.70587 | − | 2.02238i | −2.30398 | 1.56167 | + | 2.70489i | −0.321106 | + | 0.556172i | |||
206.15 | −0.0204929 | − | 0.0354947i | 1.19870 | + | 0.692069i | 0.999160 | − | 1.73060i | −0.500000 | − | 0.866025i | − | 0.0567299i | 1.05882 | + | 2.42465i | −0.163874 | −0.542081 | − | 0.938912i | −0.0204929 | + | 0.0354947i | |||
206.16 | 0.0296987 | + | 0.0514396i | −1.50278 | − | 0.867630i | 0.998236 | − | 1.72900i | −0.500000 | − | 0.866025i | − | 0.103070i | 0.0361863 | − | 2.64550i | 0.237380 | 0.00556458 | + | 0.00963813i | 0.0296987 | − | 0.0514396i | |||
206.17 | 0.0605633 | + | 0.104899i | 2.74857 | + | 1.58689i | 0.992664 | − | 1.71934i | −0.500000 | − | 0.866025i | 0.384428i | 2.51580 | − | 0.818987i | 0.482729 | 3.53641 | + | 6.12525i | 0.0605633 | − | 0.104899i | ||||
206.18 | 0.157097 | + | 0.272100i | −0.498094 | − | 0.287574i | 0.950641 | − | 1.64656i | −0.500000 | − | 0.866025i | − | 0.180708i | 0.939845 | − | 2.47319i | 1.22576 | −1.33460 | − | 2.31160i | 0.157097 | − | 0.272100i | |||
206.19 | 0.353959 | + | 0.613075i | −1.84959 | − | 1.06786i | 0.749426 | − | 1.29804i | −0.500000 | − | 0.866025i | − | 1.51192i | −1.78038 | + | 1.95710i | 2.47690 | 0.780653 | + | 1.35213i | 0.353959 | − | 0.613075i | |||
206.20 | 0.361068 | + | 0.625389i | 2.25889 | + | 1.30417i | 0.739259 | − | 1.28043i | −0.500000 | − | 0.866025i | 1.88358i | −1.90529 | + | 1.83572i | 2.51197 | 1.90173 | + | 3.29389i | 0.361068 | − | 0.625389i | ||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
161.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 805.2.p.a | ✓ | 64 |
7.d | odd | 6 | 1 | 805.2.p.b | yes | 64 | |
23.b | odd | 2 | 1 | 805.2.p.b | yes | 64 | |
161.g | even | 6 | 1 | inner | 805.2.p.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
805.2.p.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
805.2.p.a | ✓ | 64 | 161.g | even | 6 | 1 | inner |
805.2.p.b | yes | 64 | 7.d | odd | 6 | 1 | |
805.2.p.b | yes | 64 | 23.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{64} - 12 T_{11}^{63} - 115 T_{11}^{62} + 1956 T_{11}^{61} + 7038 T_{11}^{60} + \cdots + 87\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\).