Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6272,2,Mod(1,6272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6272, 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("6272.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6272 = 2^{7} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6272.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: | \(50.0821721477\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.26849792.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 13\nu^{4} + 54\nu^{3} - 209\nu^{2} - 669\nu + 288 ) / 101 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{5} - 16\nu^{4} + 120\nu^{3} + 265\nu^{2} - 140\nu - 370 ) / 101 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 24\nu^{5} + 9\nu^{4} - 320\nu^{3} - 168\nu^{2} + 407\nu + 44 ) / 101 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -36\nu^{5} - 64\nu^{4} + 480\nu^{3} + 858\nu^{2} - 560\nu - 571 ) / 101 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 48\nu^{5} + 18\nu^{4} - 640\nu^{3} - 336\nu^{2} + 1016\nu + 88 ) / 101 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + 4\beta_{2} + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{5} - 19\beta_{3} + 3\beta_{2} + 3\beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} - 16\beta_{4} - 8\beta_{3} + 48\beta_{2} + 88 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 116\beta_{5} - \beta_{4} - 208\beta_{3} + 50\beta_{2} + 40\beta _1 + 53 ) / 2 \)
|
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 |
|
0 | −2.99093 | 0 | 4.40514 | 0 | 0 | 0 | 5.94567 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.63158 | 0 | 1.21736 | 0 | 0 | 0 | 3.92520 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.359354 | 0 | −1.05486 | 0 | 0 | 0 | −2.87086 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.359354 | 0 | 1.05486 | 0 | 0 | 0 | −2.87086 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.63158 | 0 | −1.21736 | 0 | 0 | 0 | 3.92520 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.99093 | 0 | −4.40514 | 0 | 0 | 0 | 5.94567 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6272.2.a.bw | ✓ | 6 |
| 4.b | odd | 2 | 1 | 6272.2.a.bz | yes | 6 | |
| 7.b | odd | 2 | 1 | inner | 6272.2.a.bw | ✓ | 6 |
| 8.b | even | 2 | 1 | 6272.2.a.by | yes | 6 | |
| 8.d | odd | 2 | 1 | 6272.2.a.bx | yes | 6 | |
| 28.d | even | 2 | 1 | 6272.2.a.bz | yes | 6 | |
| 56.e | even | 2 | 1 | 6272.2.a.bx | yes | 6 | |
| 56.h | odd | 2 | 1 | 6272.2.a.by | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6272.2.a.bw | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6272.2.a.bw | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 6272.2.a.bx | yes | 6 | 8.d | odd | 2 | 1 | |
| 6272.2.a.bx | yes | 6 | 56.e | even | 2 | 1 | |
| 6272.2.a.by | yes | 6 | 8.b | even | 2 | 1 | |
| 6272.2.a.by | yes | 6 | 56.h | odd | 2 | 1 | |
| 6272.2.a.bz | yes | 6 | 4.b | odd | 2 | 1 | |
| 6272.2.a.bz | yes | 6 | 28.d | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6272))\):
|
\( T_{3}^{6} - 16T_{3}^{4} + 64T_{3}^{2} - 8 \)
|
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\( T_{5}^{6} - 22T_{5}^{4} + 52T_{5}^{2} - 32 \)
|
|
\( T_{11}^{3} + 2T_{11}^{2} - 20T_{11} + 16 \)
|
|
\( T_{13}^{6} - 78T_{13}^{4} + 1700T_{13}^{2} - 8192 \)
|
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\( T_{23}^{3} + 8T_{23}^{2} - 4T_{23} - 64 \)
|
|
\( T_{29}^{3} - 16T_{29} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 16 T^{4} + \cdots - 8 \)
|
| $5$ |
\( T^{6} - 22 T^{4} + \cdots - 32 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} + 2 T^{2} - 20 T + 16)^{2} \)
|
| $13$ |
\( T^{6} - 78 T^{4} + \cdots - 8192 \)
|
| $17$ |
\( (T^{2} - 18)^{3} \)
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| $19$ |
\( T^{6} - 88 T^{4} + \cdots - 19208 \)
|
| $23$ |
\( (T^{3} + 8 T^{2} - 4 T - 64)^{2} \)
|
| $29$ |
\( (T^{3} - 16 T + 8)^{2} \)
|
| $31$ |
\( T^{6} - 96 T^{4} + \cdots - 25088 \)
|
| $37$ |
\( (T^{3} - 12 T^{2} + 32 T - 8)^{2} \)
|
| $41$ |
\( T^{6} - 118 T^{4} + \cdots - 22472 \)
|
| $43$ |
\( (T^{3} - 22 T^{2} + \cdots - 208)^{2} \)
|
| $47$ |
\( T^{6} - 160 T^{4} + \cdots - 512 \)
|
| $53$ |
\( (T^{3} - 18 T^{2} + \cdots + 232)^{2} \)
|
| $59$ |
\( T^{6} - 168 T^{4} + \cdots - 66248 \)
|
| $61$ |
\( T^{6} - 294 T^{4} + \cdots - 53792 \)
|
| $67$ |
\( (T^{3} - 14 T^{2} + \cdots + 928)^{2} \)
|
| $71$ |
\( (T^{3} + 4 T^{2} + \cdots - 256)^{2} \)
|
| $73$ |
\( T^{6} - 70 T^{4} + \cdots - 392 \)
|
| $79$ |
\( (T^{3} - 16 T^{2} + \cdots + 512)^{2} \)
|
| $83$ |
\( T^{6} - 208 T^{4} + \cdots - 10952 \)
|
| $89$ |
\( T^{6} - 182 T^{4} + \cdots - 27848 \)
|
| $97$ |
\( T^{6} - 278 T^{4} + \cdots - 22472 \)
|
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