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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.860280160.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 9x^{3} + 23x^{2} - 22x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1815) |
| 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
\( x^{6} - x^{5} - 10x^{4} + 9x^{3} + 23x^{2} - 22x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 9\nu^{3} - \nu^{2} + 16\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 10\nu^{3} - 8\nu^{2} + 20\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 22 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{4} + 9\beta_{3} + 10\beta_{2} + 29\beta _1 + 6 \)
|
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 |
|
−2.73519 | −1.00000 | 5.48125 | 0 | 2.73519 | 4.20410 | −9.52186 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.53672 | −1.00000 | 0.361523 | 0 | 1.53672 | −2.86503 | 2.51789 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.19557 | −1.00000 | −0.570614 | 0 | 1.19557 | 3.76208 | 3.07335 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0838261 | −1.00000 | −1.99297 | 0 | −0.0838261 | −2.34295 | −0.334715 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.95455 | −1.00000 | 1.82026 | 0 | −1.95455 | 2.58348 | −0.351308 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.42911 | −1.00000 | 3.90056 | 0 | −2.42911 | −1.34168 | 4.61665 | 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.do | 6 | |
| 5.b | even | 2 | 1 | 9075.2.a.ds | 6 | ||
| 5.c | odd | 4 | 2 | 1815.2.c.i | yes | 12 | |
| 11.b | odd | 2 | 1 | 9075.2.a.dr | 6 | ||
| 55.d | odd | 2 | 1 | 9075.2.a.dp | 6 | ||
| 55.e | even | 4 | 2 | 1815.2.c.h | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.2.c.h | ✓ | 12 | 55.e | even | 4 | 2 | |
| 1815.2.c.i | yes | 12 | 5.c | odd | 4 | 2 | |
| 9075.2.a.do | 6 | 1.a | even | 1 | 1 | trivial | |
| 9075.2.a.dp | 6 | 55.d | odd | 2 | 1 | ||
| 9075.2.a.dr | 6 | 11.b | odd | 2 | 1 | ||
| 9075.2.a.ds | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):
|
\( T_{2}^{6} + T_{2}^{5} - 10T_{2}^{4} - 9T_{2}^{3} + 23T_{2}^{2} + 22T_{2} - 2 \)
|
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\( T_{7}^{6} - 4T_{7}^{5} - 19T_{7}^{4} + 62T_{7}^{3} + 136T_{7}^{2} - 232T_{7} - 368 \)
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\( T_{13}^{6} + 2T_{13}^{5} - 47T_{13}^{4} - 110T_{13}^{3} + 300T_{13}^{2} + 432T_{13} + 32 \)
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\( T_{17}^{6} + 2T_{17}^{5} - 68T_{17}^{4} - 106T_{17}^{3} + 1025T_{17}^{2} + 2036T_{17} + 46 \)
|
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\( T_{19}^{6} + 2T_{19}^{5} - 47T_{19}^{4} - 64T_{19}^{3} + 368T_{19}^{2} + 128T_{19} - 512 \)
|
|
\( T_{23}^{6} + 12T_{23}^{5} - 21T_{23}^{4} - 564T_{23}^{3} - 528T_{23}^{2} + 4656T_{23} - 1072 \)
|
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\( T_{37}^{6} + 24T_{37}^{5} + 126T_{37}^{4} - 858T_{37}^{3} - 7687T_{37}^{2} + 4710T_{37} + 98540 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 10 T^{4} + \cdots - 2 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots - 368 \)
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| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} + 2 T^{5} + \cdots + 32 \)
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| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 46 \)
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| $19$ |
\( T^{6} + 2 T^{5} + \cdots - 512 \)
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| $23$ |
\( T^{6} + 12 T^{5} + \cdots - 1072 \)
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| $29$ |
\( T^{6} - 12 T^{5} + \cdots - 7688 \)
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| $31$ |
\( T^{6} - 18 T^{5} + \cdots - 2656 \)
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| $37$ |
\( T^{6} + 24 T^{5} + \cdots + 98540 \)
|
| $41$ |
\( T^{6} + 6 T^{5} + \cdots - 52544 \)
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| $43$ |
\( T^{6} + 10 T^{5} + \cdots + 16960 \)
|
| $47$ |
\( T^{6} + 8 T^{5} + \cdots - 160 \)
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| $53$ |
\( T^{6} + 18 T^{5} + \cdots - 9188 \)
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| $59$ |
\( T^{6} + 18 T^{5} + \cdots - 8224 \)
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| $61$ |
\( T^{6} - 4 T^{5} + \cdots - 104284 \)
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| $67$ |
\( T^{6} + 12 T^{5} + \cdots - 464 \)
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| $71$ |
\( T^{6} - 130 T^{4} + \cdots + 11744 \)
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| $73$ |
\( T^{6} + 14 T^{5} + \cdots + 44288 \)
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| $79$ |
\( T^{6} - 2 T^{5} + \cdots - 196624 \)
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| $83$ |
\( T^{6} - 20 T^{5} + \cdots + 266240 \)
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| $89$ |
\( T^{6} + 16 T^{5} + \cdots - 80776 \)
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| $97$ |
\( T^{6} + 12 T^{5} + \cdots + 23572 \)
|
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