Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2401,2,Mod(1,2401)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2401, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2401.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2401 = 7^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2401.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: | \(19.1720815253\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{21})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 6x^{3} + 8x^{2} - 8x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 49) |
| 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 \(\nu = \zeta_{21} + \zeta_{21}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + \nu^{2} + 5\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} - \beta_{2} + 10\beta_1 \)
|
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 |
|
−1.91115 | −1.73068 | 1.65248 | 2.12712 | 3.30759 | 0 | 0.664166 | −0.00473965 | −4.06524 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.65248 | 0.466104 | 0.730682 | −3.88881 | −0.770226 | 0 | 2.09752 | −2.78275 | 6.42617 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.730682 | −1.14946 | −1.46610 | 0.258668 | 0.839890 | 0 | 2.53262 | −1.67874 | −0.189004 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.149460 | −2.91115 | −1.97766 | 2.19679 | 0.435100 | 0 | 0.594502 | 5.47477 | −0.328332 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.46610 | 0.977662 | 0.149460 | 0.921795 | 1.43335 | 0 | −2.71308 | −2.04418 | 1.35145 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.97766 | −2.65248 | 1.91115 | −1.61556 | −5.24570 | 0 | −0.175724 | 4.03564 | −3.19504 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
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 | 2401.2.a.c | 6 | |
| 7.b | odd | 2 | 1 | 2401.2.a.d | 6 | ||
| 49.e | even | 7 | 2 | 49.2.e.b | ✓ | 12 | |
| 49.f | odd | 14 | 2 | 343.2.e.b | 12 | ||
| 49.g | even | 21 | 2 | 343.2.g.b | 12 | ||
| 49.g | even | 21 | 2 | 343.2.g.d | 12 | ||
| 49.h | odd | 42 | 2 | 343.2.g.a | 12 | ||
| 49.h | odd | 42 | 2 | 343.2.g.c | 12 | ||
| 147.l | odd | 14 | 2 | 441.2.u.b | 12 | ||
| 196.k | odd | 14 | 2 | 784.2.u.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.2.e.b | ✓ | 12 | 49.e | even | 7 | 2 | |
| 343.2.e.b | 12 | 49.f | odd | 14 | 2 | ||
| 343.2.g.a | 12 | 49.h | odd | 42 | 2 | ||
| 343.2.g.b | 12 | 49.g | even | 21 | 2 | ||
| 343.2.g.c | 12 | 49.h | odd | 42 | 2 | ||
| 343.2.g.d | 12 | 49.g | even | 21 | 2 | ||
| 441.2.u.b | 12 | 147.l | odd | 14 | 2 | ||
| 784.2.u.b | 12 | 196.k | odd | 14 | 2 | ||
| 2401.2.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 2401.2.a.d | 6 | 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2401))\):
|
\( T_{2}^{6} + T_{2}^{5} - 6T_{2}^{4} - 6T_{2}^{3} + 8T_{2}^{2} + 8T_{2} + 1 \)
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\( T_{3}^{6} + 7T_{3}^{5} + 14T_{3}^{4} - 21T_{3}^{2} - 7T_{3} + 7 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 6 T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{6} + 7 T^{5} + \cdots + 7 \)
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| $5$ |
\( T^{6} - 14 T^{4} + \cdots + 7 \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} - 10 T^{5} + \cdots + 43 \)
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| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 7 \)
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| $17$ |
\( T^{6} - 56 T^{4} + \cdots - 1757 \)
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| $19$ |
\( T^{6} + 7 T^{5} + \cdots + 2107 \)
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| $23$ |
\( T^{6} + T^{5} - 13 T^{4} + \cdots + 1 \)
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| $29$ |
\( T^{6} - 12 T^{5} + \cdots - 41 \)
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| $31$ |
\( T^{6} + 7 T^{5} + \cdots - 8183 \)
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| $37$ |
\( T^{6} - 13 T^{5} + \cdots + 17179 \)
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| $41$ |
\( T^{6} + 14 T^{5} + \cdots - 3227 \)
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| $43$ |
\( T^{6} - 19 T^{5} + \cdots - 14153 \)
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| $47$ |
\( T^{6} + 21 T^{5} + \cdots - 54971 \)
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| $53$ |
\( T^{6} + 18 T^{5} + \cdots - 1728 \)
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| $59$ |
\( T^{6} - 7 T^{5} + \cdots + 7357 \)
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| $61$ |
\( T^{6} + 28 T^{5} + \cdots - 16163 \)
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| $67$ |
\( T^{6} - 24 T^{5} + \cdots - 293 \)
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| $71$ |
\( T^{6} - 33 T^{5} + \cdots - 559397 \)
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| $73$ |
\( T^{6} - 7 T^{5} + \cdots + 21469 \)
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| $79$ |
\( T^{6} + 8 T^{5} + \cdots - 21629 \)
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| $83$ |
\( T^{6} + 21 T^{5} + \cdots - 10871 \)
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| $89$ |
\( T^{6} + 21 T^{5} + \cdots + 299593 \)
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| $97$ |
\( T^{6} + 14 T^{5} + \cdots + 18571 \)
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