Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(136,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.136");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), degree \(2\), not 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: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.84779568.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 135) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -11\nu^{5} + 143\nu^{4} + 21\nu^{3} + 1672\nu^{2} - 1056\nu + 13728 ) / 3760 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -17\nu^{5} + 221\nu^{4} - 993\nu^{3} + 2584\nu^{2} - 1632\nu + 17456 ) / 3760 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 39\nu^{5} - 37\nu^{4} + 481\nu^{3} + 182\nu^{2} + 5624\nu + 208 ) / 3760 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -61\nu^{5} + 88\nu^{4} - 1144\nu^{3} + 1047\nu^{2} - 13376\nu + 8448 ) / 1880 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -31\nu^{5} + 27\nu^{4} - 351\nu^{3} + 200\nu^{2} - 4104\nu + 2592 ) / 752 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} + \beta_{4} + 23\beta_{3} - 23 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{2} + 17\beta _1 - 11 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -23\beta_{5} - 3\beta_{4} - 93\beta_{3} + 3\beta_{2} + 23\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -233\beta_{5} + 131\beta_{4} - 227\beta_{3} + 227 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
−1.29244 | + | 2.23857i | 0 | 0.659207 | + | 1.14178i | 2.50000 | + | 4.33013i | 0 | 11.4468 | − | 19.8264i | −24.0869 | 0 | −12.9244 | ||||||||||||||||||||||||||||
| 136.2 | 1.06306 | − | 1.84127i | 0 | 1.73981 | + | 3.01344i | 2.50000 | + | 4.33013i | 0 | −15.3500 | + | 26.5870i | 24.4070 | 0 | 10.6306 | |||||||||||||||||||||||||||||
| 136.3 | 2.72938 | − | 4.72742i | 0 | −10.8990 | − | 18.8776i | 2.50000 | + | 4.33013i | 0 | 5.90326 | − | 10.2247i | −75.3201 | 0 | 27.2938 | |||||||||||||||||||||||||||||
| 271.1 | −1.29244 | − | 2.23857i | 0 | 0.659207 | − | 1.14178i | 2.50000 | − | 4.33013i | 0 | 11.4468 | + | 19.8264i | −24.0869 | 0 | −12.9244 | |||||||||||||||||||||||||||||
| 271.2 | 1.06306 | + | 1.84127i | 0 | 1.73981 | − | 3.01344i | 2.50000 | − | 4.33013i | 0 | −15.3500 | − | 26.5870i | 24.4070 | 0 | 10.6306 | |||||||||||||||||||||||||||||
| 271.3 | 2.72938 | + | 4.72742i | 0 | −10.8990 | + | 18.8776i | 2.50000 | − | 4.33013i | 0 | 5.90326 | + | 10.2247i | −75.3201 | 0 | 27.2938 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.e.v | 6 | |
| 3.b | odd | 2 | 1 | 405.4.e.q | 6 | ||
| 9.c | even | 3 | 1 | 135.4.a.e | ✓ | 3 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.v | 6 | |
| 9.d | odd | 6 | 1 | 135.4.a.h | yes | 3 | |
| 9.d | odd | 6 | 1 | 405.4.e.q | 6 | ||
| 36.f | odd | 6 | 1 | 2160.4.a.bi | 3 | ||
| 36.h | even | 6 | 1 | 2160.4.a.bq | 3 | ||
| 45.h | odd | 6 | 1 | 675.4.a.p | 3 | ||
| 45.j | even | 6 | 1 | 675.4.a.s | 3 | ||
| 45.k | odd | 12 | 2 | 675.4.b.m | 6 | ||
| 45.l | even | 12 | 2 | 675.4.b.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.e | ✓ | 3 | 9.c | even | 3 | 1 | |
| 135.4.a.h | yes | 3 | 9.d | odd | 6 | 1 | |
| 405.4.e.q | 6 | 3.b | odd | 2 | 1 | ||
| 405.4.e.q | 6 | 9.d | odd | 6 | 1 | ||
| 405.4.e.v | 6 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.v | 6 | 9.c | even | 3 | 1 | inner | |
| 675.4.a.p | 3 | 45.h | odd | 6 | 1 | ||
| 675.4.a.s | 3 | 45.j | even | 6 | 1 | ||
| 675.4.b.m | 6 | 45.k | odd | 12 | 2 | ||
| 675.4.b.n | 6 | 45.l | even | 12 | 2 | ||
| 2160.4.a.bi | 3 | 36.f | odd | 6 | 1 | ||
| 2160.4.a.bq | 3 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{6} - 5T_{2}^{5} + 33T_{2}^{4} - 20T_{2}^{3} + 214T_{2}^{2} - 240T_{2} + 900 \)
|
|
\( T_{7}^{6} - 4T_{7}^{5} + 811T_{7}^{4} - 13416T_{7}^{3} + 665217T_{7}^{2} - 6596910T_{7} + 68856804 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 5 T^{5} + \cdots + 900 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots + 68856804 \)
|
| $11$ |
\( T^{6} - 5 T^{5} + \cdots + 977187600 \)
|
| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 41280625 \)
|
| $17$ |
\( (T^{3} + 155 T^{2} + \cdots + 41760)^{2} \)
|
| $19$ |
\( (T^{3} + 50 T^{2} + \cdots - 368012)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 306362250000 \)
|
| $29$ |
\( T^{6} + \cdots + 41477979315600 \)
|
| $31$ |
\( T^{6} + \cdots + 880414396416 \)
|
| $37$ |
\( (T^{3} + 384 T^{2} + \cdots - 22667198)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 15345082598400 \)
|
| $43$ |
\( T^{6} + \cdots + 28707478180096 \)
|
| $47$ |
\( T^{6} + \cdots + 207021450297600 \)
|
| $53$ |
\( (T^{3} - 400 T^{2} + \cdots + 12658320)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 27093274214400 \)
|
| $61$ |
\( T^{6} + \cdots + 25695430112356 \)
|
| $67$ |
\( T^{6} + \cdots + 431363822490000 \)
|
| $71$ |
\( (T^{3} + 40 T^{2} + \cdots - 216071280)^{2} \)
|
| $73$ |
\( (T^{3} + 980 T^{2} + \cdots + 16447954)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 82\!\cdots\!25 \)
|
| $83$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + 1020 T^{2} + \cdots - 125064000)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 752435512191364 \)
|
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