Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [572,4,Mod(1,572)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("572.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 572 = 2^{2} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 572.a (trivial) |
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: | yes |
| Analytic conductor: | \(33.7490925233\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 122x^{5} - 46x^{4} + 3838x^{3} + 2390x^{2} - 9135x + 3938 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 122x^{5} - 46x^{4} + 3838x^{3} + 2390x^{2} - 9135x + 3938 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} + 3\nu^{5} - 847\nu^{4} - 745\nu^{3} + 26061\nu^{2} + 31699\nu - 39654 ) / 180 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} + 2\nu^{5} - 363\nu^{4} - 395\nu^{3} + 11149\nu^{2} + 15456\nu - 16501 ) / 45 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\nu^{6} + 17\nu^{5} - 1633\nu^{4} - 2295\nu^{3} + 51859\nu^{2} + 73201\nu - 79186 ) / 180 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -19\nu^{6} - 11\nu^{5} + 2299\nu^{4} + 2325\nu^{3} - 70477\nu^{2} - 93523\nu + 99358 ) / 180 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 41\nu^{6} + 29\nu^{5} - 4961\nu^{4} - 5395\nu^{3} + 152143\nu^{2} + 205157\nu - 221302 ) / 180 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta_{2} + 35 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 3\beta_{3} + 2\beta_{2} + 58\beta _1 + 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{6} + 68\beta_{5} - 3\beta_{4} + 60\beta_{3} + 58\beta_{2} + 30\beta _1 + 2071 \)
|
| \(\nu^{5}\) | \(=\) |
\( 106\beta_{6} + 134\beta_{5} - 227\beta_{3} + 132\beta_{2} + 3529\beta _1 + 2197 \)
|
| \(\nu^{6}\) | \(=\) |
\( 666\beta_{6} + 4554\beta_{5} - 363\beta_{4} + 3315\beta_{3} + 3477\beta_{2} + 3762\beta _1 + 126925 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −8.45681 | 0 | −2.36903 | 0 | −4.66811 | 0 | 44.5176 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.32514 | 0 | 4.58098 | 0 | 20.7177 | 0 | 26.6577 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.736718 | 0 | 14.9534 | 0 | −10.7463 | 0 | −26.4572 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.698822 | 0 | −10.5788 | 0 | −24.7543 | 0 | −26.5116 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.18753 | 0 | −20.2032 | 0 | 33.4822 | 0 | −22.2147 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.34396 | 0 | −5.09273 | 0 | 4.09723 | 0 | 26.9337 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 7.68600 | 0 | 15.7093 | 0 | 19.8717 | 0 | 32.0746 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 572.4.a.d | ✓ | 7 |
| 4.b | odd | 2 | 1 | 2288.4.a.p | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 572.4.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2288.4.a.p | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 122T_{3}^{5} + 46T_{3}^{4} + 3838T_{3}^{3} - 2390T_{3}^{2} - 9135T_{3} - 3938 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(572))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 122 T^{5} + \cdots - 3938 \)
|
| $5$ |
\( T^{7} + 3 T^{6} + \cdots - 2774800 \)
|
| $7$ |
\( T^{7} - 38 T^{6} + \cdots + 70134206 \)
|
| $11$ |
\( (T - 11)^{7} \)
|
| $13$ |
\( (T + 13)^{7} \)
|
| $17$ |
\( T^{7} + \cdots + 5108851072 \)
|
| $19$ |
\( T^{7} + \cdots + 73640990304 \)
|
| $23$ |
\( T^{7} + \cdots - 1243577671784 \)
|
| $29$ |
\( T^{7} + \cdots + 118984424858368 \)
|
| $31$ |
\( T^{7} + \cdots - 23\!\cdots\!60 \)
|
| $37$ |
\( T^{7} + \cdots + 621014376102400 \)
|
| $41$ |
\( T^{7} + \cdots + 174267301464400 \)
|
| $43$ |
\( T^{7} + \cdots + 11584742737984 \)
|
| $47$ |
\( T^{7} + \cdots + 16\!\cdots\!84 \)
|
| $53$ |
\( T^{7} + \cdots - 44\!\cdots\!28 \)
|
| $59$ |
\( T^{7} + \cdots - 90\!\cdots\!40 \)
|
| $61$ |
\( T^{7} + \cdots - 14\!\cdots\!36 \)
|
| $67$ |
\( T^{7} + \cdots - 42\!\cdots\!00 \)
|
| $71$ |
\( T^{7} + \cdots - 17\!\cdots\!36 \)
|
| $73$ |
\( T^{7} + \cdots + 12\!\cdots\!52 \)
|
| $79$ |
\( T^{7} + \cdots - 40\!\cdots\!04 \)
|
| $83$ |
\( T^{7} + \cdots - 24\!\cdots\!68 \)
|
| $89$ |
\( T^{7} + \cdots + 81\!\cdots\!40 \)
|
| $97$ |
\( T^{7} + \cdots - 15\!\cdots\!88 \)
|
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