Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9075,2,Mod(1,9075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, 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("9075.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9075.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: | \(72.4642398343\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.725.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.35567 | −1.00000 | 3.54920 | 0 | 2.35567 | 0.193527 | −3.64941 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.47726 | −1.00000 | 0.182297 | 0 | 1.47726 | −2.29496 | 2.68522 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −0.262360 | −1.00000 | −1.93117 | 0 | 0.262360 | −3.19353 | 1.03138 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.09529 | −1.00000 | −0.800331 | 0 | −1.09529 | −0.705037 | −3.06719 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 9075.2.a.cm | 4 | |
| 5.b | even | 2 | 1 | 1815.2.a.w | 4 | ||
| 11.b | odd | 2 | 1 | 9075.2.a.di | 4 | ||
| 11.d | odd | 10 | 2 | 825.2.n.g | 8 | ||
| 15.d | odd | 2 | 1 | 5445.2.a.bf | 4 | ||
| 55.d | odd | 2 | 1 | 1815.2.a.p | 4 | ||
| 55.h | odd | 10 | 2 | 165.2.m.d | ✓ | 8 | |
| 55.l | even | 20 | 4 | 825.2.bx.f | 16 | ||
| 165.d | even | 2 | 1 | 5445.2.a.bt | 4 | ||
| 165.r | even | 10 | 2 | 495.2.n.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.m.d | ✓ | 8 | 55.h | odd | 10 | 2 | |
| 495.2.n.a | 8 | 165.r | even | 10 | 2 | ||
| 825.2.n.g | 8 | 11.d | odd | 10 | 2 | ||
| 825.2.bx.f | 16 | 55.l | even | 20 | 4 | ||
| 1815.2.a.p | 4 | 55.d | odd | 2 | 1 | ||
| 1815.2.a.w | 4 | 5.b | even | 2 | 1 | ||
| 5445.2.a.bf | 4 | 15.d | odd | 2 | 1 | ||
| 5445.2.a.bt | 4 | 165.d | even | 2 | 1 | ||
| 9075.2.a.cm | 4 | 1.a | even | 1 | 1 | trivial | |
| 9075.2.a.di | 4 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):
|
\( T_{2}^{4} + 3T_{2}^{3} - 4T_{2} - 1 \)
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\( T_{7}^{4} + 6T_{7}^{3} + 10T_{7}^{2} + 3T_{7} - 1 \)
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\( T_{13}^{4} + 7T_{13}^{3} + 10T_{13}^{2} - 6T_{13} - 11 \)
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\( T_{17}^{4} + 10T_{17}^{3} + 16T_{17}^{2} - 10T_{17} - 1 \)
|
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\( T_{19}^{4} - 9T_{19}^{3} + 22T_{19}^{2} - 16T_{19} + 1 \)
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\( T_{23}^{4} - 3T_{23}^{3} - 55T_{23}^{2} + 129T_{23} + 449 \)
|
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\( T_{37}^{4} - 3T_{37}^{3} - 121T_{37}^{2} + 213T_{37} + 3541 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3 T^{3} + \cdots - 1 \)
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| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 6 T^{3} + \cdots - 1 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} + 7 T^{3} + \cdots - 11 \)
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| $17$ |
\( T^{4} + 10 T^{3} + \cdots - 1 \)
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| $19$ |
\( T^{4} - 9 T^{3} + \cdots + 1 \)
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| $23$ |
\( T^{4} - 3 T^{3} + \cdots + 449 \)
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| $29$ |
\( T^{4} - 15 T^{3} + \cdots + 499 \)
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| $31$ |
\( T^{4} + 13 T^{3} + \cdots - 1249 \)
|
| $37$ |
\( T^{4} - 3 T^{3} + \cdots + 3541 \)
|
| $41$ |
\( T^{4} - 22 T^{3} + \cdots + 41 \)
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| $43$ |
\( T^{4} - 110 T^{2} + \cdots + 275 \)
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| $47$ |
\( T^{4} - 2 T^{3} + \cdots + 31 \)
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| $53$ |
\( T^{4} + 10 T^{3} + \cdots - 641 \)
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| $59$ |
\( T^{4} + 21 T^{3} + \cdots - 589 \)
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| $61$ |
\( T^{4} - 11 T^{3} + \cdots + 1151 \)
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| $67$ |
\( T^{4} + T^{3} + \cdots + 619 \)
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| $71$ |
\( T^{4} + 13 T^{3} + \cdots + 281 \)
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| $73$ |
\( T^{4} + T^{3} + \cdots + 931 \)
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| $79$ |
\( T^{4} + 4 T^{3} + \cdots - 169 \)
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| $83$ |
\( T^{4} + 3 T^{3} + \cdots + 41 \)
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| $89$ |
\( T^{4} + 10 T^{3} + \cdots + 209 \)
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| $97$ |
\( T^{4} - 22 T^{3} + \cdots - 1159 \)
|
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