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: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 1077 x^{9} + 861 x^{8} + 405413 x^{7} - 153981 x^{6} - 63447399 x^{5} + \cdots + 38866516416 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 1077 x^{9} + 861 x^{8} + 405413 x^{7} - 153981 x^{6} - 63447399 x^{5} + \cdots + 38866516416 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 196 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 619674277691197 \nu^{10} + 422183527450322 \nu^{9} + \cdots - 14\!\cdots\!04 ) / 44\!\cdots\!96 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 653123426361581 \nu^{10} + \cdots - 19\!\cdots\!80 ) / 44\!\cdots\!96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 34855096667627 \nu^{10} - 245502335188618 \nu^{9} + \cdots + 14\!\cdots\!88 ) / 18\!\cdots\!04 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 619674277691197 \nu^{10} - 422183527450322 \nu^{9} + \cdots + 13\!\cdots\!04 ) / 22\!\cdots\!48 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 26\!\cdots\!17 \nu^{10} + \cdots + 79\!\cdots\!32 ) / 44\!\cdots\!96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 21\!\cdots\!03 \nu^{10} + \cdots + 48\!\cdots\!64 ) / 22\!\cdots\!48 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 23\!\cdots\!47 \nu^{10} + \cdots - 68\!\cdots\!92 ) / 22\!\cdots\!48 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 13\!\cdots\!13 \nu^{10} + 236943664719058 \nu^{9} + \cdots + 33\!\cdots\!60 ) / 55\!\cdots\!12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 196 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 2\beta_{3} - \beta_{2} + 324\beta _1 + 29 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{10} - 11 \beta_{9} - 5 \beta_{8} - 6 \beta_{7} + 7 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + \cdots + 63598 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{10} - 52 \beta_{9} - 10 \beta_{8} - 4 \beta_{7} + 567 \beta_{6} - 78 \beta_{5} + \cdots - 19717 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 905 \beta_{10} - 6965 \beta_{9} - 3065 \beta_{8} - 3414 \beta_{7} + 5165 \beta_{6} - 3709 \beta_{5} + \cdots + 23604838 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2008 \beta_{10} - 41112 \beta_{9} - 12744 \beta_{8} - 768 \beta_{7} + 268547 \beta_{6} + \cdots - 16494777 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 561195 \beta_{10} - 3529465 \beta_{9} - 1504103 \beta_{8} - 1538626 \beta_{7} + 2856389 \beta_{6} + \cdots + 9334779034 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1022174 \beta_{10} - 23607860 \beta_{9} - 9345862 \beta_{8} + 501828 \beta_{7} + 121055035 \beta_{6} + \cdots - 7574978985 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 301173001 \beta_{10} - 1668503161 \beta_{9} - 694034049 \beta_{8} - 651351894 \beta_{7} + \cdots + 3820071063302 \)
|
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 |
|
−21.8750 | 27.0000 | 350.517 | 374.828 | −590.626 | −343.000 | −4867.56 | 729.000 | −8199.38 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −19.8976 | 27.0000 | 267.914 | −73.1636 | −537.235 | −343.000 | −2783.95 | 729.000 | 1455.78 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −12.9710 | 27.0000 | 40.2456 | −477.091 | −350.216 | −343.000 | 1138.26 | 729.000 | 6188.32 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −10.6246 | 27.0000 | −15.1176 | 116.261 | −286.865 | −343.000 | 1520.57 | 729.000 | −1235.22 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −6.40236 | 27.0000 | −87.0098 | 397.948 | −172.864 | −343.000 | 1376.57 | 729.000 | −2547.80 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.502507 | 27.0000 | −127.747 | 49.2500 | 13.5677 | −343.000 | −128.515 | 729.000 | 24.7484 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.714411 | 27.0000 | −127.490 | −514.615 | 19.2891 | −343.000 | −182.524 | 729.000 | −367.646 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.4324 | 27.0000 | −19.1660 | 410.988 | 281.674 | −343.000 | −1535.29 | 729.000 | 4287.57 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 13.7480 | 27.0000 | 61.0063 | −70.9199 | 371.195 | −343.000 | −921.027 | 729.000 | −975.004 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 16.2711 | 27.0000 | 136.749 | −103.508 | 439.320 | −343.000 | 142.349 | 729.000 | −1684.19 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 20.1022 | 27.0000 | 276.099 | −279.978 | 542.760 | −343.000 | 2977.12 | 729.000 | −5628.18 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.d | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 10 T_{2}^{10} - 1032 T_{2}^{9} - 8712 T_{2}^{8} + 373739 T_{2}^{7} + 2617802 T_{2}^{6} + \cdots + 6467627264 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots + 6467627264 \)
|
| $3$ |
\( (T - 27)^{11} \)
|
| $5$ |
\( T^{11} + \cdots - 12\!\cdots\!00 \)
|
| $7$ |
\( (T + 343)^{11} \)
|
| $11$ |
\( T^{11} + \cdots + 17\!\cdots\!64 \)
|
| $13$ |
\( (T + 2197)^{11} \)
|
| $17$ |
\( T^{11} + \cdots + 20\!\cdots\!44 \)
|
| $19$ |
\( T^{11} + \cdots - 45\!\cdots\!88 \)
|
| $23$ |
\( T^{11} + \cdots + 10\!\cdots\!88 \)
|
| $29$ |
\( T^{11} + \cdots + 13\!\cdots\!88 \)
|
| $31$ |
\( T^{11} + \cdots + 45\!\cdots\!80 \)
|
| $37$ |
\( T^{11} + \cdots + 45\!\cdots\!28 \)
|
| $41$ |
\( T^{11} + \cdots - 43\!\cdots\!04 \)
|
| $43$ |
\( T^{11} + \cdots - 78\!\cdots\!64 \)
|
| $47$ |
\( T^{11} + \cdots - 70\!\cdots\!68 \)
|
| $53$ |
\( T^{11} + \cdots + 51\!\cdots\!28 \)
|
| $59$ |
\( T^{11} + \cdots + 29\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots - 21\!\cdots\!00 \)
|
| $67$ |
\( T^{11} + \cdots + 18\!\cdots\!36 \)
|
| $71$ |
\( T^{11} + \cdots - 34\!\cdots\!52 \)
|
| $73$ |
\( T^{11} + \cdots + 10\!\cdots\!20 \)
|
| $79$ |
\( T^{11} + \cdots + 34\!\cdots\!88 \)
|
| $83$ |
\( T^{11} + \cdots - 19\!\cdots\!84 \)
|
| $89$ |
\( T^{11} + \cdots - 41\!\cdots\!12 \)
|
| $97$ |
\( T^{11} + \cdots - 62\!\cdots\!52 \)
|
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