Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,4,Mod(173,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1, 5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.173");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.bm (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: | \(14.8684813214\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
173.1 | 0 | −5.14829 | − | 0.703660i | 0 | 5.50907 | 0 | −15.5186 | + | 10.1081i | 0 | 26.0097 | + | 7.24529i | 0 | ||||||||||||
173.2 | 0 | −5.09076 | + | 1.04121i | 0 | −1.90049 | 0 | 3.04981 | − | 18.2674i | 0 | 24.8318 | − | 10.6011i | 0 | ||||||||||||
173.3 | 0 | −4.84961 | − | 1.86582i | 0 | −21.0452 | 0 | 9.48976 | + | 15.9042i | 0 | 20.0374 | + | 18.0970i | 0 | ||||||||||||
173.4 | 0 | −4.40362 | + | 2.75828i | 0 | 3.45780 | 0 | 13.4196 | + | 12.7638i | 0 | 11.7838 | − | 24.2928i | 0 | ||||||||||||
173.5 | 0 | −4.18792 | − | 3.07593i | 0 | 16.5659 | 0 | 17.5228 | − | 5.99607i | 0 | 8.07729 | + | 25.7635i | 0 | ||||||||||||
173.6 | 0 | −3.93325 | − | 3.39552i | 0 | −12.5831 | 0 | −6.65587 | − | 17.2829i | 0 | 3.94093 | + | 26.7108i | 0 | ||||||||||||
173.7 | 0 | −3.15513 | + | 4.12858i | 0 | 18.3795 | 0 | −11.4907 | − | 14.5246i | 0 | −7.09031 | − | 26.0524i | 0 | ||||||||||||
173.8 | 0 | −2.94236 | + | 4.28282i | 0 | −17.3719 | 0 | −18.1816 | + | 3.52543i | 0 | −9.68509 | − | 25.2032i | 0 | ||||||||||||
173.9 | 0 | −1.46664 | − | 4.98488i | 0 | −2.27383 | 0 | 17.2174 | + | 6.82357i | 0 | −22.6980 | + | 14.6220i | 0 | ||||||||||||
173.10 | 0 | −1.17608 | − | 5.06131i | 0 | 10.3118 | 0 | −12.2405 | + | 13.8985i | 0 | −24.2336 | + | 11.9051i | 0 | ||||||||||||
173.11 | 0 | −1.17430 | + | 5.06172i | 0 | −6.49181 | 0 | 18.4099 | − | 2.01918i | 0 | −24.2420 | − | 11.8880i | 0 | ||||||||||||
173.12 | 0 | −1.15557 | − | 5.06603i | 0 | −7.54500 | 0 | −17.0316 | − | 7.27489i | 0 | −24.3293 | + | 11.7083i | 0 | ||||||||||||
173.13 | 0 | −0.596161 | + | 5.16184i | 0 | 18.4760 | 0 | −4.17729 | + | 18.0430i | 0 | −26.2892 | − | 6.15457i | 0 | ||||||||||||
173.14 | 0 | 0.222952 | + | 5.19137i | 0 | −3.95888 | 0 | 0.649207 | − | 18.5089i | 0 | −26.9006 | + | 2.31485i | 0 | ||||||||||||
173.15 | 0 | 2.56423 | − | 4.51937i | 0 | 17.7911 | 0 | −5.94198 | − | 17.5412i | 0 | −13.8494 | − | 23.1774i | 0 | ||||||||||||
173.16 | 0 | 2.64863 | + | 4.47043i | 0 | −7.63023 | 0 | −0.685930 | + | 18.5076i | 0 | −12.9695 | + | 23.6810i | 0 | ||||||||||||
173.17 | 0 | 2.92280 | − | 4.29619i | 0 | −6.03810 | 0 | 10.8562 | − | 15.0047i | 0 | −9.91446 | − | 25.1138i | 0 | ||||||||||||
173.18 | 0 | 2.97726 | − | 4.25863i | 0 | −12.7833 | 0 | 2.83671 | + | 18.3017i | 0 | −9.27184 | − | 25.3581i | 0 | ||||||||||||
173.19 | 0 | 3.66543 | + | 3.68302i | 0 | 6.67787 | 0 | −16.9163 | − | 7.53913i | 0 | −0.129304 | + | 26.9997i | 0 | ||||||||||||
173.20 | 0 | 4.25643 | + | 2.98040i | 0 | 13.0880 | 0 | 15.3663 | − | 10.3381i | 0 | 9.23438 | + | 25.3718i | 0 | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.4.bm.a | yes | 48 |
3.b | odd | 2 | 1 | 756.4.bm.a | 48 | ||
7.d | odd | 6 | 1 | 252.4.w.a | ✓ | 48 | |
9.c | even | 3 | 1 | 756.4.w.a | 48 | ||
9.d | odd | 6 | 1 | 252.4.w.a | ✓ | 48 | |
21.g | even | 6 | 1 | 756.4.w.a | 48 | ||
63.k | odd | 6 | 1 | 756.4.bm.a | 48 | ||
63.s | even | 6 | 1 | inner | 252.4.bm.a | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.4.w.a | ✓ | 48 | 7.d | odd | 6 | 1 | |
252.4.w.a | ✓ | 48 | 9.d | odd | 6 | 1 | |
252.4.bm.a | yes | 48 | 1.a | even | 1 | 1 | trivial |
252.4.bm.a | yes | 48 | 63.s | even | 6 | 1 | inner |
756.4.w.a | 48 | 9.c | even | 3 | 1 | ||
756.4.w.a | 48 | 21.g | even | 6 | 1 | ||
756.4.bm.a | 48 | 3.b | odd | 2 | 1 | ||
756.4.bm.a | 48 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(252, [\chi])\).