Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [454,2,Mod(1,454)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(454, 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("454.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 454 = 2 \cdot 227 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 454.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: | \(3.62520825177\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 14x^{5} + 11x^{4} + 44x^{3} - 19x^{2} - 27x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 14x^{5} + 11x^{4} + 44x^{3} - 19x^{2} - 27x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{6} - 16\nu^{5} + 87\nu^{4} + 191\nu^{3} - 441\nu^{2} - 430\nu + 353 ) / 73 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -16\nu^{6} + 18\nu^{5} + 258\nu^{4} - 5\nu^{3} - 900\nu^{2} - 155\nu + 634 ) / 73 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -30\nu^{6} + 52\nu^{5} + 429\nu^{4} - 201\nu^{3} - 1286\nu^{2} + 47\nu + 550 ) / 73 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 55\nu^{6} - 71\nu^{5} - 823\nu^{4} + 40\nu^{3} + 2455\nu^{2} + 583\nu - 1130 ) / 73 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 61\nu^{6} - 96\nu^{5} - 865\nu^{4} + 197\nu^{3} + 2537\nu^{2} + 486\nu - 1167 ) / 73 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 5\beta_{3} + 4\beta_{2} + 9\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{6} + 6\beta_{5} + 17\beta_{4} - 29\beta_{3} + 20\beta_{2} + 24\beta _1 + 56 \)
|
| \(\nu^{5}\) | \(=\) |
\( -33\beta_{6} + 39\beta_{5} + 54\beta_{4} - 103\beta_{3} + 80\beta_{2} + 126\beta _1 + 177 \)
|
| \(\nu^{6}\) | \(=\) |
\( -101\beta_{6} + 140\beta_{5} + 278\beta_{4} - 474\beta_{3} + 355\beta_{2} + 460\beta _1 + 858 \)
|
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 |
|
1.00000 | −3.13277 | 1.00000 | −2.56845 | −3.13277 | 0.452005 | 1.00000 | 6.81425 | −2.56845 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.02930 | 1.00000 | 3.57091 | −1.02930 | 0.302132 | 1.00000 | −1.94054 | 3.57091 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.841300 | 1.00000 | −0.689846 | −0.841300 | 1.97919 | 1.00000 | −2.29221 | −0.689846 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.51469 | 1.00000 | 1.76211 | 1.51469 | 3.39439 | 1.00000 | −0.705722 | 1.76211 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.56555 | 1.00000 | 1.10760 | 1.56555 | −0.825150 | 1.00000 | −0.549046 | 1.10760 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.70514 | 1.00000 | −4.10338 | 2.70514 | 2.89328 | 1.00000 | 4.31776 | −4.10338 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.21800 | 1.00000 | −0.0789406 | 3.21800 | −3.19584 | 1.00000 | 7.35551 | −0.0789406 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(227\) | \(1\) |
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 | 454.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 4086.2.a.p | 7 | ||
| 4.b | odd | 2 | 1 | 3632.2.a.o | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 454.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 3632.2.a.o | 7 | 4.b | odd | 2 | 1 | ||
| 4086.2.a.p | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 4T_{3}^{6} - 9T_{3}^{5} + 48T_{3}^{4} - 11T_{3}^{3} - 92T_{3}^{2} + 28T_{3} + 56 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(454))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} - 4 T^{6} + \cdots + 56 \)
|
| $5$ |
\( T^{7} + T^{6} - 20 T^{5} + \cdots - 4 \)
|
| $7$ |
\( T^{7} - 5 T^{6} + \cdots - 7 \)
|
| $11$ |
\( T^{7} - 10 T^{6} + \cdots - 3304 \)
|
| $13$ |
\( T^{7} + 5 T^{6} + \cdots - 4 \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots + 8608 \)
|
| $19$ |
\( T^{7} - 69 T^{5} + \cdots + 5056 \)
|
| $23$ |
\( T^{7} - 11 T^{6} + \cdots - 5747 \)
|
| $29$ |
\( T^{7} - 16 T^{6} + \cdots + 3416 \)
|
| $31$ |
\( T^{7} + 2 T^{6} + \cdots + 1568 \)
|
| $37$ |
\( T^{7} + 23 T^{6} + \cdots - 1216 \)
|
| $41$ |
\( T^{7} - 46 T^{5} + \cdots - 32 \)
|
| $43$ |
\( T^{7} - 73 T^{5} + \cdots - 4904 \)
|
| $47$ |
\( T^{7} + 3 T^{6} + \cdots + 77861 \)
|
| $53$ |
\( T^{7} + 16 T^{6} + \cdots + 8 \)
|
| $59$ |
\( T^{7} - 4 T^{6} + \cdots - 65408 \)
|
| $61$ |
\( T^{7} + 15 T^{6} + \cdots - 593812 \)
|
| $67$ |
\( T^{7} - T^{6} + \cdots - 14128 \)
|
| $71$ |
\( T^{7} - 33 T^{6} + \cdots - 543293 \)
|
| $73$ |
\( T^{7} + 11 T^{6} + \cdots - 13308673 \)
|
| $79$ |
\( T^{7} - 11 T^{6} + \cdots + 707 \)
|
| $83$ |
\( T^{7} - 17 T^{6} + \cdots - 370412 \)
|
| $89$ |
\( T^{7} + 9 T^{6} + \cdots + 283817 \)
|
| $97$ |
\( T^{7} + 17 T^{6} + \cdots - 11760263 \)
|
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