Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,2,Mod(5,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 168.ba (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: | \(1.34148675396\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.41416 | − | 0.0117470i | 1.70617 | − | 0.298296i | 1.99972 | + | 0.0332243i | 0.337879 | + | 0.195075i | −2.41631 | + | 0.401798i | 1.39526 | + | 2.24795i | −2.82755 | − | 0.0704754i | 2.82204 | − | 1.01789i | −0.475526 | − | 0.279837i |
5.2 | −1.37674 | − | 0.323405i | −0.609093 | + | 1.62142i | 1.79082 | + | 0.890488i | 2.24840 | + | 1.29811i | 1.36294 | − | 2.03529i | −2.53678 | + | 0.751482i | −2.17750 | − | 1.80513i | −2.25801 | − | 1.97519i | −2.67564 | − | 2.51430i |
5.3 | −1.33258 | − | 0.473539i | −1.52791 | − | 0.815778i | 1.55152 | + | 1.26205i | 0.461663 | + | 0.266541i | 1.64975 | + | 1.81061i | 0.489180 | − | 2.60014i | −1.46989 | − | 2.41649i | 1.66901 | + | 2.49287i | −0.488984 | − | 0.573802i |
5.4 | −1.30261 | + | 0.550645i | −1.09218 | − | 1.34430i | 1.39358 | − | 1.43455i | −2.46958 | − | 1.42581i | 2.16291 | + | 1.14970i | −1.02032 | + | 2.44110i | −1.02536 | + | 2.63603i | −0.614290 | + | 2.93643i | 4.00201 | + | 0.497414i |
5.5 | −1.26535 | + | 0.631567i | 0.528554 | − | 1.64943i | 1.20225 | − | 1.59831i | 2.66818 | + | 1.54047i | 0.372919 | + | 2.42094i | −1.46307 | − | 2.20441i | −0.511826 | + | 2.78173i | −2.44126 | − | 1.74363i | −4.34911 | − | 0.264112i |
5.6 | −1.07638 | − | 0.917276i | 1.52791 | + | 0.815778i | 0.317209 | + | 1.97468i | −0.461663 | − | 0.266541i | −0.896325 | − | 2.27961i | 0.489180 | − | 2.60014i | 1.46989 | − | 2.41649i | 1.66901 | + | 2.49287i | 0.252435 | + | 0.710374i |
5.7 | −0.998351 | + | 1.00165i | 0.390548 | + | 1.68745i | −0.00659077 | − | 1.99999i | 1.54900 | + | 0.894317i | −2.08013 | − | 1.29347i | 2.63573 | + | 0.230049i | 2.00986 | + | 1.99009i | −2.69494 | + | 1.31806i | −2.44224 | + | 0.658710i |
5.8 | −0.968446 | − | 1.03059i | 0.609093 | − | 1.62142i | −0.124224 | + | 1.99614i | −2.24840 | − | 1.29811i | −2.26089 | + | 0.942534i | −2.53678 | + | 0.751482i | 2.17750 | − | 1.80513i | −2.25801 | − | 1.97519i | 0.839632 | + | 3.57432i |
5.9 | −0.717256 | − | 1.21883i | −1.70617 | + | 0.298296i | −0.971089 | + | 1.74842i | −0.337879 | − | 0.195075i | 1.58733 | + | 1.86558i | 1.39526 | + | 2.24795i | 2.82755 | − | 0.0704754i | 2.82204 | − | 1.01789i | 0.00458308 | + | 0.551736i |
5.10 | −0.368276 | + | 1.36542i | −1.26610 | − | 1.18195i | −1.72875 | − | 1.00570i | 1.54900 | + | 0.894317i | 2.08013 | − | 1.29347i | 2.63573 | + | 0.230049i | 2.00986 | − | 1.99009i | 0.206000 | + | 2.99292i | −1.79158 | + | 1.78568i |
5.11 | −0.174432 | − | 1.40341i | 1.09218 | + | 1.34430i | −1.93915 | + | 0.489600i | 2.46958 | + | 1.42581i | 1.69610 | − | 1.76727i | −1.02032 | + | 2.44110i | 1.02536 | + | 2.63603i | −0.614290 | + | 2.93643i | 1.57023 | − | 3.71455i |
5.12 | −0.0857242 | − | 1.41161i | −0.528554 | + | 1.64943i | −1.98530 | + | 0.242019i | −2.66818 | − | 1.54047i | 2.37367 | + | 0.604718i | −1.46307 | − | 2.20441i | 0.511826 | + | 2.78173i | −2.44126 | − | 1.74363i | −1.94583 | + | 3.89849i |
5.13 | 0.0857242 | + | 1.41161i | 1.69273 | + | 0.366975i | −1.98530 | + | 0.242019i | 2.66818 | + | 1.54047i | −0.372919 | + | 2.42094i | −1.46307 | − | 2.20441i | −0.511826 | − | 2.78173i | 2.73066 | + | 1.24238i | −1.94583 | + | 3.89849i |
5.14 | 0.174432 | + | 1.40341i | 0.618109 | + | 1.61801i | −1.93915 | + | 0.489600i | −2.46958 | − | 1.42581i | −2.16291 | + | 1.14970i | −1.02032 | + | 2.44110i | −1.02536 | − | 2.63603i | −2.23588 | + | 2.00021i | 1.57023 | − | 3.71455i |
5.15 | 0.368276 | − | 1.36542i | −0.390548 | − | 1.68745i | −1.72875 | − | 1.00570i | −1.54900 | − | 0.894317i | −2.44790 | − | 0.0881824i | 2.63573 | + | 0.230049i | −2.00986 | + | 1.99009i | −2.69494 | + | 1.31806i | −1.79158 | + | 1.78568i |
5.16 | 0.717256 | + | 1.21883i | 1.11142 | − | 1.32844i | −0.971089 | + | 1.74842i | 0.337879 | + | 0.195075i | 2.41631 | + | 0.401798i | 1.39526 | + | 2.24795i | −2.82755 | + | 0.0704754i | −0.529502 | − | 2.95290i | 0.00458308 | + | 0.551736i |
5.17 | 0.968446 | + | 1.03059i | −1.70874 | − | 0.283220i | −0.124224 | + | 1.99614i | 2.24840 | + | 1.29811i | −1.36294 | − | 2.03529i | −2.53678 | + | 0.751482i | −2.17750 | + | 1.80513i | 2.83957 | + | 0.967897i | 0.839632 | + | 3.57432i |
5.18 | 0.998351 | − | 1.00165i | 1.26610 | + | 1.18195i | −0.00659077 | − | 1.99999i | −1.54900 | − | 0.894317i | 2.44790 | − | 0.0881824i | 2.63573 | + | 0.230049i | −2.00986 | − | 1.99009i | 0.206000 | + | 2.99292i | −2.44224 | + | 0.658710i |
5.19 | 1.07638 | + | 0.917276i | −0.0574700 | + | 1.73110i | 0.317209 | + | 1.97468i | 0.461663 | + | 0.266541i | −1.64975 | + | 1.81061i | 0.489180 | − | 2.60014i | −1.46989 | + | 2.41649i | −2.99339 | − | 0.198972i | 0.252435 | + | 0.710374i |
5.20 | 1.26535 | − | 0.631567i | −1.69273 | − | 0.366975i | 1.20225 | − | 1.59831i | −2.66818 | − | 1.54047i | −2.37367 | + | 0.604718i | −1.46307 | − | 2.20441i | 0.511826 | − | 2.78173i | 2.73066 | + | 1.24238i | −4.34911 | − | 0.264112i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
8.b | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
24.h | odd | 2 | 1 | inner |
56.j | odd | 6 | 1 | inner |
168.ba | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 168.2.ba.c | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 168.2.ba.c | ✓ | 48 |
4.b | odd | 2 | 1 | 672.2.bi.c | 48 | ||
7.d | odd | 6 | 1 | inner | 168.2.ba.c | ✓ | 48 |
8.b | even | 2 | 1 | inner | 168.2.ba.c | ✓ | 48 |
8.d | odd | 2 | 1 | 672.2.bi.c | 48 | ||
12.b | even | 2 | 1 | 672.2.bi.c | 48 | ||
21.g | even | 6 | 1 | inner | 168.2.ba.c | ✓ | 48 |
24.f | even | 2 | 1 | 672.2.bi.c | 48 | ||
24.h | odd | 2 | 1 | inner | 168.2.ba.c | ✓ | 48 |
28.f | even | 6 | 1 | 672.2.bi.c | 48 | ||
56.j | odd | 6 | 1 | inner | 168.2.ba.c | ✓ | 48 |
56.m | even | 6 | 1 | 672.2.bi.c | 48 | ||
84.j | odd | 6 | 1 | 672.2.bi.c | 48 | ||
168.ba | even | 6 | 1 | inner | 168.2.ba.c | ✓ | 48 |
168.be | odd | 6 | 1 | 672.2.bi.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.2.ba.c | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
168.2.ba.c | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
168.2.ba.c | ✓ | 48 | 7.d | odd | 6 | 1 | inner |
168.2.ba.c | ✓ | 48 | 8.b | even | 2 | 1 | inner |
168.2.ba.c | ✓ | 48 | 21.g | even | 6 | 1 | inner |
168.2.ba.c | ✓ | 48 | 24.h | odd | 2 | 1 | inner |
168.2.ba.c | ✓ | 48 | 56.j | odd | 6 | 1 | inner |
168.2.ba.c | ✓ | 48 | 168.ba | even | 6 | 1 | inner |
672.2.bi.c | 48 | 4.b | odd | 2 | 1 | ||
672.2.bi.c | 48 | 8.d | odd | 2 | 1 | ||
672.2.bi.c | 48 | 12.b | even | 2 | 1 | ||
672.2.bi.c | 48 | 24.f | even | 2 | 1 | ||
672.2.bi.c | 48 | 28.f | even | 6 | 1 | ||
672.2.bi.c | 48 | 56.m | even | 6 | 1 | ||
672.2.bi.c | 48 | 84.j | odd | 6 | 1 | ||
672.2.bi.c | 48 | 168.be | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 28 T_{5}^{22} + 498 T_{5}^{20} - 5472 T_{5}^{18} + 44115 T_{5}^{16} - 241512 T_{5}^{14} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).