Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [435,4,Mod(1,435)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, 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("435.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 435 = 3 \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 435.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: | \(25.6658308525\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 23\nu^{4} + 67\nu^{3} + 128\nu^{2} - 131\nu - 94 ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} - 2\nu^{5} - 99\nu^{4} + \nu^{3} + 814\nu^{2} + 157\nu - 1262 ) / 20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - \nu^{5} + 38\nu^{4} + 43\nu^{3} - 363\nu^{2} - 304\nu + 744 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 17\nu^{6} + 2\nu^{5} - 581\nu^{4} - 381\nu^{3} + 4946\nu^{2} + 3183\nu - 8898 ) / 40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 18\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 6\beta_{5} - \beta_{4} - 2\beta_{3} + 24\beta_{2} + 42\beta _1 + 166 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{6} + 38\beta_{5} + 19\beta_{4} - 32\beta_{3} + 69\beta_{2} + 388\beta _1 + 246 \)
|
| \(\nu^{6}\) | \(=\) |
\( 70\beta_{6} + 223\beta_{5} - 14\beta_{4} - 87\beta_{3} + 566\beta_{2} + 1315\beta _1 + 3348 \)
|
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 |
|
−5.12367 | 3.00000 | 18.2520 | −5.00000 | −15.3710 | −21.7657 | −52.5281 | 9.00000 | 25.6184 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.07921 | 3.00000 | 1.48151 | −5.00000 | −9.23762 | 23.1532 | 20.0718 | 9.00000 | 15.3960 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.26184 | 3.00000 | −6.40776 | −5.00000 | −3.78552 | −13.4311 | 18.1803 | 9.00000 | 6.30920 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.10304 | 3.00000 | −6.78329 | −5.00000 | −3.30913 | 1.72550 | 16.3066 | 9.00000 | 5.51522 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.08033 | 3.00000 | −3.67221 | −5.00000 | 6.24100 | 15.0555 | −24.2821 | 9.00000 | −10.4017 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.52131 | 3.00000 | 4.39964 | −5.00000 | 10.5639 | −8.86571 | −12.6780 | 9.00000 | −17.6066 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.96612 | 3.00000 | 7.73008 | −5.00000 | 11.8983 | −32.8717 | −1.07055 | 9.00000 | −19.8306 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 435.4.a.j | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1305.4.a.m | 7 | ||
| 5.b | even | 2 | 1 | 2175.4.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.j | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1305.4.a.m | 7 | 3.b | odd | 2 | 1 | ||
| 2175.4.a.m | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + T_{2}^{6} - 35T_{2}^{5} - 18T_{2}^{4} + 329T_{2}^{3} + 167T_{2}^{2} - 767T_{2} - 638 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(435))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots - 638 \)
|
| $3$ |
\( (T - 3)^{7} \)
|
| $5$ |
\( (T + 5)^{7} \)
|
| $7$ |
\( T^{7} + 37 T^{6} + \cdots - 51243488 \)
|
| $11$ |
\( T^{7} + 11 T^{6} + \cdots - 344198304 \)
|
| $13$ |
\( T^{7} + \cdots + 724528410064 \)
|
| $17$ |
\( T^{7} + \cdots - 1088280942608 \)
|
| $19$ |
\( T^{7} + \cdots + 206799749120 \)
|
| $23$ |
\( T^{7} + \cdots - 199055640559616 \)
|
| $29$ |
\( (T + 29)^{7} \)
|
| $31$ |
\( T^{7} + \cdots - 2046710061056 \)
|
| $37$ |
\( T^{7} + \cdots + 48\!\cdots\!52 \)
|
| $41$ |
\( T^{7} + \cdots + 24\!\cdots\!48 \)
|
| $43$ |
\( T^{7} + \cdots - 24\!\cdots\!48 \)
|
| $47$ |
\( T^{7} + \cdots - 41\!\cdots\!36 \)
|
| $53$ |
\( T^{7} + \cdots - 40\!\cdots\!68 \)
|
| $59$ |
\( T^{7} + \cdots - 19\!\cdots\!80 \)
|
| $61$ |
\( T^{7} + \cdots + 36\!\cdots\!16 \)
|
| $67$ |
\( T^{7} + \cdots + 43\!\cdots\!52 \)
|
| $71$ |
\( T^{7} + \cdots + 57\!\cdots\!52 \)
|
| $73$ |
\( T^{7} + \cdots - 24\!\cdots\!56 \)
|
| $79$ |
\( T^{7} + \cdots + 16\!\cdots\!60 \)
|
| $83$ |
\( T^{7} + \cdots + 20\!\cdots\!24 \)
|
| $89$ |
\( T^{7} + \cdots - 53\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 27\!\cdots\!76 \)
|
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