Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,4,Mod(649,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.649");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 675.b (of order \(2\), degree \(1\), 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: | \(39.8262892539\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.12559936.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{3} + 49x^{2} - 42x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 135) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 6x^{3} + 49x^{2} - 42x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} + 49\nu^{4} + 21\nu^{3} - 9\nu^{2} + 1544 ) / 334 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -49\nu^{5} - 21\nu^{4} - 9\nu^{3} + 147\nu^{2} - 2338\nu + 1056 ) / 1002 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\nu^{5} + 107\nu^{4} + 189\nu^{3} - 81\nu^{2} + 1872 ) / 334 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -245\nu^{5} - 105\nu^{4} - 45\nu^{3} + 1737\nu^{2} - 11690\nu + 5280 ) / 1002 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 63\nu^{5} + 27\nu^{4} - 60\nu^{3} - 523\nu^{2} + 2505\nu - 1143 ) / 167 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 - 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{5} - 14\beta_{4} + 7\beta_{3} + 25\beta_{2} - 14\beta _1 + 25 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{3} + 9\beta _1 - 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 49\beta_{5} + 116\beta_{4} + 49\beta_{3} - 265\beta_{2} - 116\beta _1 + 265 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 5.45876i | 0 | −21.7980 | 0 | 0 | 11.8065i | 75.3201i | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 649.2 | − | 2.58488i | 0 | 1.31841 | 0 | 0 | − | 22.8935i | − | 24.0869i | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 649.3 | − | 2.12612i | 0 | 3.47962 | 0 | 0 | − | 30.7000i | − | 24.4070i | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 649.4 | 2.12612i | 0 | 3.47962 | 0 | 0 | 30.7000i | 24.4070i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 649.5 | 2.58488i | 0 | 1.31841 | 0 | 0 | 22.8935i | 24.0869i | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 649.6 | 5.45876i | 0 | −21.7980 | 0 | 0 | − | 11.8065i | − | 75.3201i | 0 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 675.4.b.n | 6 | |
| 3.b | odd | 2 | 1 | 675.4.b.m | 6 | ||
| 5.b | even | 2 | 1 | inner | 675.4.b.n | 6 | |
| 5.c | odd | 4 | 1 | 135.4.a.h | yes | 3 | |
| 5.c | odd | 4 | 1 | 675.4.a.p | 3 | ||
| 15.d | odd | 2 | 1 | 675.4.b.m | 6 | ||
| 15.e | even | 4 | 1 | 135.4.a.e | ✓ | 3 | |
| 15.e | even | 4 | 1 | 675.4.a.s | 3 | ||
| 20.e | even | 4 | 1 | 2160.4.a.bq | 3 | ||
| 45.k | odd | 12 | 2 | 405.4.e.q | 6 | ||
| 45.l | even | 12 | 2 | 405.4.e.v | 6 | ||
| 60.l | odd | 4 | 1 | 2160.4.a.bi | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.e | ✓ | 3 | 15.e | even | 4 | 1 | |
| 135.4.a.h | yes | 3 | 5.c | odd | 4 | 1 | |
| 405.4.e.q | 6 | 45.k | odd | 12 | 2 | ||
| 405.4.e.v | 6 | 45.l | even | 12 | 2 | ||
| 675.4.a.p | 3 | 5.c | odd | 4 | 1 | ||
| 675.4.a.s | 3 | 15.e | even | 4 | 1 | ||
| 675.4.b.m | 6 | 3.b | odd | 2 | 1 | ||
| 675.4.b.m | 6 | 15.d | odd | 2 | 1 | ||
| 675.4.b.n | 6 | 1.a | even | 1 | 1 | trivial | |
| 675.4.b.n | 6 | 5.b | even | 2 | 1 | inner | |
| 2160.4.a.bi | 3 | 60.l | odd | 4 | 1 | ||
| 2160.4.a.bq | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(675, [\chi])\):
|
\( T_{2}^{6} + 41T_{2}^{4} + 364T_{2}^{2} + 900 \)
|
|
\( T_{7}^{6} + 1606T_{7}^{4} + 698409T_{7}^{2} + 68856804 \)
|
|
\( T_{11}^{3} - 5T_{11}^{2} - 2888T_{11} + 31260 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 41 T^{4} + \cdots + 900 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 1606 T^{4} + \cdots + 68856804 \)
|
| $11$ |
\( (T^{3} - 5 T^{2} + \cdots + 31260)^{2} \)
|
| $13$ |
\( T^{6} + 1587 T^{4} + \cdots + 41280625 \)
|
| $17$ |
\( T^{6} + \cdots + 1743897600 \)
|
| $19$ |
\( (T^{3} - 50 T^{2} + \cdots + 368012)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 306362250000 \)
|
| $29$ |
\( (T^{3} + 115 T^{2} + \cdots - 6440340)^{2} \)
|
| $31$ |
\( (T^{3} + 115 T^{2} + \cdots - 938304)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 513801865171204 \)
|
| $41$ |
\( (T^{3} - 580 T^{2} + \cdots - 3917280)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 28707478180096 \)
|
| $47$ |
\( T^{6} + \cdots + 207021450297600 \)
|
| $53$ |
\( T^{6} + \cdots + 160233065222400 \)
|
| $59$ |
\( (T^{3} + 380 T^{2} + \cdots - 5205120)^{2} \)
|
| $61$ |
\( (T^{3} + 152 T^{2} + \cdots - 5069066)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 431363822490000 \)
|
| $71$ |
\( (T^{3} - 40 T^{2} + \cdots + 216071280)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 270535190786116 \)
|
| $79$ |
\( (T^{3} + 1013 T^{2} + \cdots - 90596925)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + 1020 T^{2} + \cdots - 125064000)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 752435512191364 \)
|
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