Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [770,2,Mod(573,770)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("770.573");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 770.l (of order \(4\), 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: | \(6.14848095564\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
573.1 | −0.707107 | + | 0.707107i | −2.03841 | + | 2.03841i | − | 1.00000i | −2.07139 | + | 0.842227i | − | 2.88275i | −2.55056 | − | 0.703324i | 0.707107 | + | 0.707107i | − | 5.31025i | 0.869149 | − | 2.06024i | |||
573.2 | −0.707107 | + | 0.707107i | −2.02780 | + | 2.02780i | − | 1.00000i | −0.515374 | − | 2.17587i | − | 2.86774i | 1.43371 | − | 2.22362i | 0.707107 | + | 0.707107i | − | 5.22391i | 1.90299 | + | 1.17414i | |||
573.3 | −0.707107 | + | 0.707107i | −0.903547 | + | 0.903547i | − | 1.00000i | −1.41673 | − | 1.72999i | − | 1.27781i | 1.36406 | + | 2.26701i | 0.707107 | + | 0.707107i | 1.36720i | 2.22507 | + | 0.221508i | ||||
573.4 | −0.707107 | + | 0.707107i | −0.839228 | + | 0.839228i | − | 1.00000i | 0.690407 | − | 2.12681i | − | 1.18685i | −2.60331 | + | 0.472008i | 0.707107 | + | 0.707107i | 1.59139i | 1.01569 | + | 1.99208i | ||||
573.5 | −0.707107 | + | 0.707107i | −0.0715208 | + | 0.0715208i | − | 1.00000i | 1.95282 | − | 1.08927i | − | 0.101146i | 2.63480 | − | 0.240478i | 0.707107 | + | 0.707107i | 2.98977i | −0.610623 | + | 2.15108i | ||||
573.6 | −0.707107 | + | 0.707107i | 0.0715208 | − | 0.0715208i | − | 1.00000i | −1.95282 | + | 1.08927i | 0.101146i | 0.240478 | − | 2.63480i | 0.707107 | + | 0.707107i | 2.98977i | 0.610623 | − | 2.15108i | |||||
573.7 | −0.707107 | + | 0.707107i | 0.839228 | − | 0.839228i | − | 1.00000i | −0.690407 | + | 2.12681i | 1.18685i | −0.472008 | + | 2.60331i | 0.707107 | + | 0.707107i | 1.59139i | −1.01569 | − | 1.99208i | |||||
573.8 | −0.707107 | + | 0.707107i | 0.903547 | − | 0.903547i | − | 1.00000i | 1.41673 | + | 1.72999i | 1.27781i | −2.26701 | − | 1.36406i | 0.707107 | + | 0.707107i | 1.36720i | −2.22507 | − | 0.221508i | |||||
573.9 | −0.707107 | + | 0.707107i | 2.02780 | − | 2.02780i | − | 1.00000i | 0.515374 | + | 2.17587i | 2.86774i | 2.22362 | − | 1.43371i | 0.707107 | + | 0.707107i | − | 5.22391i | −1.90299 | − | 1.17414i | ||||
573.10 | −0.707107 | + | 0.707107i | 2.03841 | − | 2.03841i | − | 1.00000i | 2.07139 | − | 0.842227i | 2.88275i | 0.703324 | + | 2.55056i | 0.707107 | + | 0.707107i | − | 5.31025i | −0.869149 | + | 2.06024i | ||||
573.11 | 0.707107 | − | 0.707107i | −2.23110 | + | 2.23110i | − | 1.00000i | −0.288161 | − | 2.21742i | 3.15525i | 2.06987 | + | 1.64792i | −0.707107 | − | 0.707107i | − | 6.95559i | −1.77171 | − | 1.36419i | ||||
573.12 | 0.707107 | − | 0.707107i | −1.65110 | + | 1.65110i | − | 1.00000i | 2.15340 | + | 0.602381i | 2.33501i | 0.00754262 | + | 2.64574i | −0.707107 | − | 0.707107i | − | 2.45228i | 1.94863 | − | 1.09674i | ||||
573.13 | 0.707107 | − | 0.707107i | −1.49597 | + | 1.49597i | − | 1.00000i | −0.446287 | + | 2.19108i | 2.11563i | −1.17733 | − | 2.36936i | −0.707107 | − | 0.707107i | − | 1.47587i | 1.23375 | + | 1.86490i | ||||
573.14 | 0.707107 | − | 0.707107i | −0.949748 | + | 0.949748i | − | 1.00000i | −2.23301 | + | 0.116862i | 1.34315i | 2.47882 | − | 0.924898i | −0.707107 | − | 0.707107i | 1.19596i | −1.49634 | + | 1.66161i | |||||
573.15 | 0.707107 | − | 0.707107i | −0.602672 | + | 0.602672i | − | 1.00000i | −0.490670 | − | 2.18157i | 0.852308i | −2.60076 | + | 0.485856i | −0.707107 | − | 0.707107i | 2.27357i | −1.88956 | − | 1.19565i | |||||
573.16 | 0.707107 | − | 0.707107i | 0.602672 | − | 0.602672i | − | 1.00000i | 0.490670 | + | 2.18157i | − | 0.852308i | −0.485856 | + | 2.60076i | −0.707107 | − | 0.707107i | 2.27357i | 1.88956 | + | 1.19565i | ||||
573.17 | 0.707107 | − | 0.707107i | 0.949748 | − | 0.949748i | − | 1.00000i | 2.23301 | − | 0.116862i | − | 1.34315i | 0.924898 | − | 2.47882i | −0.707107 | − | 0.707107i | 1.19596i | 1.49634 | − | 1.66161i | ||||
573.18 | 0.707107 | − | 0.707107i | 1.49597 | − | 1.49597i | − | 1.00000i | 0.446287 | − | 2.19108i | − | 2.11563i | 2.36936 | + | 1.17733i | −0.707107 | − | 0.707107i | − | 1.47587i | −1.23375 | − | 1.86490i | |||
573.19 | 0.707107 | − | 0.707107i | 1.65110 | − | 1.65110i | − | 1.00000i | −2.15340 | − | 0.602381i | − | 2.33501i | −2.64574 | − | 0.00754262i | −0.707107 | − | 0.707107i | − | 2.45228i | −1.94863 | + | 1.09674i | |||
573.20 | 0.707107 | − | 0.707107i | 2.23110 | − | 2.23110i | − | 1.00000i | 0.288161 | + | 2.21742i | − | 3.15525i | −1.64792 | − | 2.06987i | −0.707107 | − | 0.707107i | − | 6.95559i | 1.77171 | + | 1.36419i | |||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 770.2.l.c | ✓ | 40 |
5.c | odd | 4 | 1 | inner | 770.2.l.c | ✓ | 40 |
7.b | odd | 2 | 1 | inner | 770.2.l.c | ✓ | 40 |
35.f | even | 4 | 1 | inner | 770.2.l.c | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
770.2.l.c | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
770.2.l.c | ✓ | 40 | 5.c | odd | 4 | 1 | inner |
770.2.l.c | ✓ | 40 | 7.b | odd | 2 | 1 | inner |
770.2.l.c | ✓ | 40 | 35.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{40} + 294 T_{3}^{36} + 32981 T_{3}^{32} + 1774604 T_{3}^{28} + 47378156 T_{3}^{24} + 599438976 T_{3}^{20} + 3200847280 T_{3}^{16} + 7743540160 T_{3}^{12} + 8026477632 T_{3}^{8} + \cdots + 262144 \)
acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).