Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2835,2,Mod(1,2835)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2835, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2835.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2835 = 3^{4} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2835.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: | \(22.6375889730\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.458011800.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 7x^{4} + 11x^{3} + 19x^{2} + 2x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 315) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 7x^{4} + 11x^{3} + 19x^{2} + 2x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 4\nu^{4} + 3\nu^{3} - 13\nu^{2} - 8\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} - 3\nu^{3} + 14\nu^{2} + 6\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 7\nu^{3} + 12\nu^{2} + 16\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 3\beta_{3} + \beta_{2} + 9\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 13\beta_{4} + 14\beta_{3} + 4\beta_{2} + 30\beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 48\beta_{4} + 51\beta_{3} + 19\beta_{2} + 113\beta _1 + 88 \)
|
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 |
|
−2.70717 | 0 | 5.32877 | 1.00000 | 0 | 1.00000 | −9.01155 | 0 | −2.70717 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.26284 | 0 | −0.405224 | 1.00000 | 0 | 1.00000 | 3.03742 | 0 | −1.26284 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.737247 | 0 | −1.45647 | 1.00000 | 0 | 1.00000 | −2.54827 | 0 | 0.737247 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.49267 | 0 | 0.228076 | 1.00000 | 0 | 1.00000 | −2.64491 | 0 | 1.49267 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.18216 | 0 | 2.76183 | 1.00000 | 0 | 1.00000 | 1.66244 | 0 | 2.18216 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.55793 | 0 | 4.54302 | 1.00000 | 0 | 1.00000 | 6.50486 | 0 | 2.55793 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 2835.2.a.v | 6 | |
| 3.b | odd | 2 | 1 | 2835.2.a.u | 6 | ||
| 9.c | even | 3 | 2 | 945.2.i.e | 12 | ||
| 9.d | odd | 6 | 2 | 315.2.i.e | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.2.i.e | ✓ | 12 | 9.d | odd | 6 | 2 | |
| 945.2.i.e | 12 | 9.c | even | 3 | 2 | ||
| 2835.2.a.u | 6 | 3.b | odd | 2 | 1 | ||
| 2835.2.a.v | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2835))\):
|
\( T_{2}^{6} - 3T_{2}^{5} - 7T_{2}^{4} + 27T_{2}^{3} - 5T_{2}^{2} - 36T_{2} + 21 \)
|
|
\( T_{11}^{6} - 7T_{11}^{5} - 19T_{11}^{4} + 271T_{11}^{3} - 785T_{11}^{2} + 870T_{11} - 315 \)
|
|
\( T_{13}^{6} - 10T_{13}^{5} + 9T_{13}^{4} + 145T_{13}^{3} - 291T_{13}^{2} - 349T_{13} + 823 \)
|
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\( T_{17}^{6} + 7T_{17}^{5} - 51T_{17}^{4} - 401T_{17}^{3} + 133T_{17}^{2} + 2964T_{17} - 501 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 21 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 7 T^{5} + \cdots - 315 \)
|
| $13$ |
\( T^{6} - 10 T^{5} + \cdots + 823 \)
|
| $17$ |
\( T^{6} + 7 T^{5} + \cdots - 501 \)
|
| $19$ |
\( T^{6} - 15 T^{5} + \cdots + 400 \)
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| $23$ |
\( T^{6} - 14 T^{5} + \cdots - 216 \)
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| $29$ |
\( T^{6} + 13 T^{5} + \cdots - 60 \)
|
| $31$ |
\( T^{6} - 10 T^{5} + \cdots - 4400 \)
|
| $37$ |
\( T^{6} - 21 T^{5} + \cdots + 42472 \)
|
| $41$ |
\( T^{6} + 4 T^{5} + \cdots + 2100 \)
|
| $43$ |
\( T^{6} - 13 T^{5} + \cdots - 6476 \)
|
| $47$ |
\( T^{6} - 8 T^{5} + \cdots - 9303 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots - 21480 \)
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| $59$ |
\( T^{6} - 21 T^{5} + \cdots + 75000 \)
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| $61$ |
\( T^{6} - 2 T^{5} + \cdots - 4592 \)
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| $67$ |
\( T^{6} - 6 T^{5} + \cdots - 267848 \)
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| $71$ |
\( T^{6} - 29 T^{5} + \cdots + 6219 \)
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| $73$ |
\( T^{6} - 8 T^{5} + \cdots + 86137 \)
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| $79$ |
\( T^{6} + 22 T^{5} + \cdots + 134569 \)
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| $83$ |
\( T^{6} - 5 T^{5} + \cdots - 19365 \)
|
| $89$ |
\( T^{6} + T^{5} + \cdots - 36492 \)
|
| $97$ |
\( T^{6} - 32 T^{5} + \cdots - 9761 \)
|
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