Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1183,2,Mod(1,1183)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1183.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1183 = 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1183.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: | \(9.44630255912\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1279733.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 3\nu^{2} + 5\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 7\beta_{2} + 18\beta _1 + 8 \)
|
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 |
|
−2.63777 | −2.71083 | 4.95781 | 2.08281 | 7.15053 | 1.00000 | −7.80201 | 4.34860 | −5.49396 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.93488 | −2.54570 | 1.74376 | −0.312100 | 4.92562 | 1.00000 | 0.495793 | 3.48058 | 0.603875 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.10591 | 1.33192 | −0.776957 | 1.90785 | −1.47298 | 1.00000 | 3.07107 | −1.22600 | −2.10992 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.312100 | 1.10066 | −1.90259 | −1.93488 | −0.343514 | 1.00000 | 1.21800 | −1.78856 | 0.603875 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.90785 | −1.08494 | 1.63989 | −1.10591 | −2.06989 | 1.00000 | −0.687029 | −1.82292 | −2.10992 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.08281 | −0.0911085 | 2.33809 | −2.63777 | −0.189762 | 1.00000 | 0.704173 | −2.99170 | −5.49396 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1183.2.a.n | ✓ | 6 |
| 7.b | odd | 2 | 1 | 8281.2.a.cb | 6 | ||
| 13.b | even | 2 | 1 | 1183.2.a.o | yes | 6 | |
| 13.d | odd | 4 | 2 | 1183.2.c.h | 12 | ||
| 91.b | odd | 2 | 1 | 8281.2.a.cg | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1183.2.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1183.2.a.o | yes | 6 | 13.b | even | 2 | 1 | |
| 1183.2.c.h | 12 | 13.d | odd | 4 | 2 | ||
| 8281.2.a.cb | 6 | 7.b | odd | 2 | 1 | ||
| 8281.2.a.cg | 6 | 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):
|
\( T_{2}^{6} + 2T_{2}^{5} - 8T_{2}^{4} - 15T_{2}^{3} + 14T_{2}^{2} + 28T_{2} + 7 \)
|
|
\( T_{11}^{6} + 8T_{11}^{5} - 16T_{11}^{4} - 197T_{11}^{3} - 28T_{11}^{2} + 1204T_{11} + 889 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 7 \)
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| $3$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 7 \)
|
| $7$ |
\( (T - 1)^{6} \)
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| $11$ |
\( T^{6} + 8 T^{5} + \cdots + 889 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + 23 T^{5} + \cdots - 7351 \)
|
| $19$ |
\( T^{6} - 13 T^{5} + \cdots + 581 \)
|
| $23$ |
\( T^{6} + 18 T^{5} + \cdots - 587 \)
|
| $29$ |
\( T^{6} + 15 T^{5} + \cdots + 3569 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 21463 \)
|
| $37$ |
\( T^{6} - 13 T^{5} + \cdots - 7 \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots + 503 \)
|
| $43$ |
\( T^{6} + 18 T^{5} + \cdots + 181 \)
|
| $47$ |
\( T^{6} - 16 T^{5} + \cdots + 30233 \)
|
| $53$ |
\( T^{6} + 25 T^{5} + \cdots + 24193 \)
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| $59$ |
\( T^{6} + 18 T^{5} + \cdots + 92911 \)
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| $61$ |
\( T^{6} - 16 T^{5} + \cdots - 12979 \)
|
| $67$ |
\( T^{6} + 16 T^{5} + \cdots + 26747 \)
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| $71$ |
\( T^{6} + 25 T^{5} + \cdots + 563899 \)
|
| $73$ |
\( T^{6} - 5 T^{5} + \cdots - 45367 \)
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| $79$ |
\( T^{6} - 2 T^{5} + \cdots - 10277 \)
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| $83$ |
\( T^{6} - 7 T^{5} + \cdots - 41203 \)
|
| $89$ |
\( T^{6} - 10 T^{5} + \cdots - 222257 \)
|
| $97$ |
\( T^{6} - 5 T^{5} + \cdots - 43931 \)
|
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