Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,2,Mod(139,252)]
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([3, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.139");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.bi (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: | \(2.01223013094\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(40\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 | −1.41265 | − | 0.0664123i | −0.758625 | − | 1.55708i | 1.99118 | + | 0.187635i | 2.66647 | + | 1.53948i | 0.968265 | + | 2.24999i | 1.15806 | + | 2.37885i | −2.80038 | − | 0.397302i | −1.84898 | + | 2.36247i | −3.66455 | − | 2.35184i |
139.2 | −1.41265 | − | 0.0664123i | 0.758625 | + | 1.55708i | 1.99118 | + | 0.187635i | −2.66647 | − | 1.53948i | −0.968265 | − | 2.24999i | −1.48111 | − | 2.19233i | −2.80038 | − | 0.397302i | −1.84898 | + | 2.36247i | 3.66455 | + | 2.35184i |
139.3 | −1.38071 | + | 0.306010i | −1.38170 | + | 1.04446i | 1.81272 | − | 0.845022i | 1.82425 | + | 1.05323i | 1.58811 | − | 1.86491i | 0.655553 | − | 2.56325i | −2.24425 | + | 1.72144i | 0.818200 | − | 2.88627i | −2.84106 | − | 0.895969i |
139.4 | −1.38071 | + | 0.306010i | 1.38170 | − | 1.04446i | 1.81272 | − | 0.845022i | −1.82425 | − | 1.05323i | −1.58811 | + | 1.86491i | 2.54762 | + | 0.713899i | −2.24425 | + | 1.72144i | 0.818200 | − | 2.88627i | 2.84106 | + | 0.895969i |
139.5 | −1.33347 | + | 0.471009i | −1.61200 | − | 0.633613i | 1.55630 | − | 1.25616i | −3.16954 | − | 1.82994i | 2.44799 | + | 0.0856414i | −2.26193 | + | 1.37246i | −1.48363 | + | 2.40808i | 2.19707 | + | 2.04277i | 5.08842 | + | 0.947288i |
139.6 | −1.33347 | + | 0.471009i | 1.61200 | + | 0.633613i | 1.55630 | − | 1.25616i | 3.16954 | + | 1.82994i | −2.44799 | − | 0.0856414i | −2.31955 | + | 1.27266i | −1.48363 | + | 2.40808i | 2.19707 | + | 2.04277i | −5.08842 | − | 0.947288i |
139.7 | −1.28849 | − | 0.582915i | −1.60547 | + | 0.649977i | 1.32042 | + | 1.50216i | −1.35495 | − | 0.782280i | 2.44751 | + | 0.0983632i | 2.11799 | + | 1.58559i | −0.825716 | − | 2.70522i | 2.15506 | − | 2.08703i | 1.28984 | + | 1.79778i |
139.8 | −1.28849 | − | 0.582915i | 1.60547 | − | 0.649977i | 1.32042 | + | 1.50216i | 1.35495 | + | 0.782280i | −2.44751 | − | 0.0983632i | −0.314165 | − | 2.62703i | −0.825716 | − | 2.70522i | 2.15506 | − | 2.08703i | −1.28984 | − | 1.79778i |
139.9 | −1.13617 | − | 0.842090i | −1.59413 | − | 0.677299i | 0.581768 | + | 1.91352i | 1.43245 | + | 0.827025i | 1.24086 | + | 2.11193i | −2.30627 | − | 1.29658i | 0.950366 | − | 2.66398i | 2.08253 | + | 2.15941i | −0.931077 | − | 2.14589i |
139.10 | −1.13617 | − | 0.842090i | 1.59413 | + | 0.677299i | 0.581768 | + | 1.91352i | −1.43245 | − | 0.827025i | −1.24086 | − | 2.11193i | −0.0302588 | + | 2.64558i | 0.950366 | − | 2.66398i | 2.08253 | + | 2.15941i | 0.931077 | + | 2.14589i |
139.11 | −1.07529 | + | 0.918559i | −0.476707 | − | 1.66516i | 0.312499 | − | 1.97544i | 0.0753642 | + | 0.0435115i | 2.04214 | + | 1.35264i | 0.0425387 | − | 2.64541i | 1.47853 | + | 2.41122i | −2.54550 | + | 1.58759i | −0.121006 | + | 0.0224389i |
139.12 | −1.07529 | + | 0.918559i | 0.476707 | + | 1.66516i | 0.312499 | − | 1.97544i | −0.0753642 | − | 0.0435115i | −2.04214 | − | 1.35264i | 2.31226 | + | 1.28587i | 1.47853 | + | 2.41122i | −2.54550 | + | 1.58759i | 0.121006 | − | 0.0224389i |
139.13 | −0.801205 | − | 1.16536i | −0.420223 | + | 1.68030i | −0.716141 | + | 1.86739i | 2.25662 | + | 1.30286i | 2.29485 | − | 0.856554i | −2.51837 | + | 0.811054i | 2.74996 | − | 0.661599i | −2.64683 | − | 1.41220i | −0.289711 | − | 3.67364i |
139.14 | −0.801205 | − | 1.16536i | 0.420223 | − | 1.68030i | −0.716141 | + | 1.86739i | −2.25662 | − | 1.30286i | −2.29485 | + | 0.856554i | −1.96158 | + | 1.77545i | 2.74996 | − | 0.661599i | −2.64683 | − | 1.41220i | 0.289711 | + | 3.67364i |
139.15 | −0.608631 | − | 1.27655i | −0.420223 | + | 1.68030i | −1.25914 | + | 1.55389i | −2.25662 | − | 1.30286i | 2.40074 | − | 0.486250i | 1.96158 | − | 1.77545i | 2.74996 | + | 0.661599i | −2.64683 | − | 1.41220i | −0.289711 | + | 3.67364i |
139.16 | −0.608631 | − | 1.27655i | 0.420223 | − | 1.68030i | −1.25914 | + | 1.55389i | 2.25662 | + | 1.30286i | −2.40074 | + | 0.486250i | 2.51837 | − | 0.811054i | 2.74996 | + | 0.661599i | −2.64683 | − | 1.41220i | 0.289711 | − | 3.67364i |
139.17 | −0.557437 | + | 1.29972i | −0.677739 | + | 1.59395i | −1.37853 | − | 1.44902i | −1.39056 | − | 0.802839i | −1.69389 | − | 1.76939i | −2.55603 | − | 0.683151i | 2.65176 | − | 0.983961i | −2.08134 | − | 2.16056i | 1.81861 | − | 1.35980i |
139.18 | −0.557437 | + | 1.29972i | 0.677739 | − | 1.59395i | −1.37853 | − | 1.44902i | 1.39056 | + | 0.802839i | 1.69389 | + | 1.76939i | −0.686390 | + | 2.55517i | 2.65176 | − | 0.983961i | −2.08134 | − | 2.16056i | −1.81861 | + | 1.35980i |
139.19 | −0.161186 | − | 1.40500i | −1.59413 | − | 0.677299i | −1.94804 | + | 0.452933i | −1.43245 | − | 0.827025i | −0.694651 | + | 2.34893i | 0.0302588 | − | 2.64558i | 0.950366 | + | 2.66398i | 2.08253 | + | 2.15941i | −0.931077 | + | 2.14589i |
139.20 | −0.161186 | − | 1.40500i | 1.59413 | + | 0.677299i | −1.94804 | + | 0.452933i | 1.43245 | + | 0.827025i | 0.694651 | − | 2.34893i | 2.30627 | + | 1.29658i | 0.950366 | + | 2.66398i | 2.08253 | + | 2.15941i | 0.931077 | − | 2.14589i |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
28.d | even | 2 | 1 | inner |
36.f | odd | 6 | 1 | inner |
63.l | odd | 6 | 1 | inner |
252.bi | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.2.bi.c | ✓ | 80 |
3.b | odd | 2 | 1 | 756.2.bi.c | 80 | ||
4.b | odd | 2 | 1 | inner | 252.2.bi.c | ✓ | 80 |
7.b | odd | 2 | 1 | inner | 252.2.bi.c | ✓ | 80 |
9.c | even | 3 | 1 | inner | 252.2.bi.c | ✓ | 80 |
9.d | odd | 6 | 1 | 756.2.bi.c | 80 | ||
12.b | even | 2 | 1 | 756.2.bi.c | 80 | ||
21.c | even | 2 | 1 | 756.2.bi.c | 80 | ||
28.d | even | 2 | 1 | inner | 252.2.bi.c | ✓ | 80 |
36.f | odd | 6 | 1 | inner | 252.2.bi.c | ✓ | 80 |
36.h | even | 6 | 1 | 756.2.bi.c | 80 | ||
63.l | odd | 6 | 1 | inner | 252.2.bi.c | ✓ | 80 |
63.o | even | 6 | 1 | 756.2.bi.c | 80 | ||
84.h | odd | 2 | 1 | 756.2.bi.c | 80 | ||
252.s | odd | 6 | 1 | 756.2.bi.c | 80 | ||
252.bi | even | 6 | 1 | inner | 252.2.bi.c | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.bi.c | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
252.2.bi.c | ✓ | 80 | 4.b | odd | 2 | 1 | inner |
252.2.bi.c | ✓ | 80 | 7.b | odd | 2 | 1 | inner |
252.2.bi.c | ✓ | 80 | 9.c | even | 3 | 1 | inner |
252.2.bi.c | ✓ | 80 | 28.d | even | 2 | 1 | inner |
252.2.bi.c | ✓ | 80 | 36.f | odd | 6 | 1 | inner |
252.2.bi.c | ✓ | 80 | 63.l | odd | 6 | 1 | inner |
252.2.bi.c | ✓ | 80 | 252.bi | even | 6 | 1 | inner |
756.2.bi.c | 80 | 3.b | odd | 2 | 1 | ||
756.2.bi.c | 80 | 9.d | odd | 6 | 1 | ||
756.2.bi.c | 80 | 12.b | even | 2 | 1 | ||
756.2.bi.c | 80 | 21.c | even | 2 | 1 | ||
756.2.bi.c | 80 | 36.h | even | 6 | 1 | ||
756.2.bi.c | 80 | 63.o | even | 6 | 1 | ||
756.2.bi.c | 80 | 84.h | odd | 2 | 1 | ||
756.2.bi.c | 80 | 252.s | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{40} - 58 T_{5}^{38} + 1986 T_{5}^{36} - 45000 T_{5}^{34} + 757017 T_{5}^{32} - 9662658 T_{5}^{30} + 96876664 T_{5}^{28} - 766064542 T_{5}^{26} + 4848544452 T_{5}^{24} - 24482028372 T_{5}^{22} + \cdots + 3370896 \)
acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).