Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,8,Mod(1,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.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: | \(85.2811119572\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 2x^{6} - 521x^{5} + 332x^{4} + 75999x^{3} + 5166x^{2} - 2726135x + 1976424 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 521x^{5} + 332x^{4} + 75999x^{3} + 5166x^{2} - 2726135x + 1976424 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 227\nu + 56 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 55\nu^{5} + 166\nu^{4} - 20010\nu^{3} + 9811\nu^{2} + 1439011\nu - 646088 ) / 25600 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} - 2\nu - 149 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 5\nu^{5} + 1778\nu^{4} - 2110\nu^{3} - 267527\nu^{2} + 137673\nu + 5847976 ) / 25600 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\nu^{6} - 45\nu^{5} - 4706\nu^{4} + 10990\nu^{3} + 472239\nu^{2} - 423761\nu - 5732072 ) / 25600 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta _1 + 149 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 8\beta_{2} + 231\beta _1 + 242 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{6} + 56\beta_{5} + 295\beta_{4} + 8\beta_{3} + 32\beta_{2} + 1012\beta _1 + 34723 \)
|
| \(\nu^{5}\) | \(=\) |
\( 128\beta_{6} + 288\beta_{5} + 1228\beta_{4} + 544\beta_{3} + 3152\beta_{2} + 61885\beta _1 + 137508 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9696\beta_{6} + 25136\beta_{5} + 86301\beta_{4} + 5648\beta_{3} + 18592\beta_{2} + 407990\beta _1 + 9300289 \)
|
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 |
|
−17.1648 | 27.0000 | 166.631 | −70.4970 | −463.450 | 343.000 | −663.086 | 729.000 | 1210.07 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −13.4293 | 27.0000 | 52.3457 | 290.415 | −362.591 | 343.000 | 1015.98 | 729.000 | −3900.07 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −9.39348 | 27.0000 | −39.7625 | −275.883 | −253.624 | 343.000 | 1575.87 | 729.000 | 2591.51 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.262816 | 27.0000 | −127.931 | 176.284 | −7.09604 | 343.000 | 67.2628 | 729.000 | −46.3302 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.42366 | 27.0000 | −98.5839 | −320.812 | 146.439 | 343.000 | −1228.91 | 729.000 | −1739.97 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 12.5171 | 27.0000 | 28.6776 | 212.515 | 337.961 | 343.000 | −1243.23 | 729.000 | 2660.07 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 17.3096 | 27.0000 | 171.624 | −342.022 | 467.360 | 343.000 | 755.107 | 729.000 | −5920.27 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 273.8.a.a | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 5T_{2}^{6} - 512T_{2}^{5} - 2268T_{2}^{4} + 72112T_{2}^{3} + 229936T_{2}^{2} - 2489088T_{2} - 668736 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 5 T^{6} + \cdots - 668736 \)
|
| $3$ |
\( (T - 27)^{7} \)
|
| $5$ |
\( T^{7} + \cdots - 23\!\cdots\!00 \)
|
| $7$ |
\( (T - 343)^{7} \)
|
| $11$ |
\( T^{7} + \cdots + 29\!\cdots\!68 \)
|
| $13$ |
\( (T - 2197)^{7} \)
|
| $17$ |
\( T^{7} + \cdots - 25\!\cdots\!56 \)
|
| $19$ |
\( T^{7} + \cdots - 93\!\cdots\!32 \)
|
| $23$ |
\( T^{7} + \cdots + 29\!\cdots\!76 \)
|
| $29$ |
\( T^{7} + \cdots - 34\!\cdots\!44 \)
|
| $31$ |
\( T^{7} + \cdots - 18\!\cdots\!92 \)
|
| $37$ |
\( T^{7} + \cdots + 16\!\cdots\!40 \)
|
| $41$ |
\( T^{7} + \cdots + 29\!\cdots\!60 \)
|
| $43$ |
\( T^{7} + \cdots - 54\!\cdots\!08 \)
|
| $47$ |
\( T^{7} + \cdots - 48\!\cdots\!36 \)
|
| $53$ |
\( T^{7} + \cdots - 41\!\cdots\!60 \)
|
| $59$ |
\( T^{7} + \cdots - 10\!\cdots\!56 \)
|
| $61$ |
\( T^{7} + \cdots + 37\!\cdots\!80 \)
|
| $67$ |
\( T^{7} + \cdots - 39\!\cdots\!40 \)
|
| $71$ |
\( T^{7} + \cdots + 37\!\cdots\!04 \)
|
| $73$ |
\( T^{7} + \cdots - 38\!\cdots\!88 \)
|
| $79$ |
\( T^{7} + \cdots + 61\!\cdots\!28 \)
|
| $83$ |
\( T^{7} + \cdots - 48\!\cdots\!52 \)
|
| $89$ |
\( T^{7} + \cdots + 15\!\cdots\!60 \)
|
| $97$ |
\( T^{7} + \cdots - 80\!\cdots\!92 \)
|
| show more | |
| show less |