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: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} - 1257 x^{10} + 5348 x^{9} + 596302 x^{8} - 1496552 x^{7} - 132333694 x^{6} + \cdots + 2527639352128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} - 1257 x^{10} + 5348 x^{9} + 596302 x^{8} - 1496552 x^{7} - 132333694 x^{6} + \cdots + 2527639352128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 212 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4869069651297 \nu^{11} + \cdots + 20\!\cdots\!64 ) / 10\!\cdots\!44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 64356085636259 \nu^{11} + 539194033346557 \nu^{10} + \cdots + 84\!\cdots\!04 ) / 52\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10446563891478 \nu^{11} + \cdots + 20\!\cdots\!68 ) / 65\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 64356085636259 \nu^{11} + 539194033346557 \nu^{10} + \cdots + 89\!\cdots\!44 ) / 26\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 215194752131829 \nu^{11} + \cdots + 87\!\cdots\!16 ) / 52\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 255205489511603 \nu^{11} + \cdots + 28\!\cdots\!52 ) / 52\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 11\!\cdots\!53 \nu^{11} + \cdots - 84\!\cdots\!12 ) / 52\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 26\!\cdots\!63 \nu^{11} + \cdots - 44\!\cdots\!72 ) / 52\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 919550424313709 \nu^{11} + \cdots + 24\!\cdots\!56 ) / 13\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 212 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 2\beta_{4} + 2\beta_{2} + 324\beta _1 + 405 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{8} + 2\beta_{7} + 5\beta_{6} - 12\beta_{4} - 2\beta_{3} + 461\beta_{2} + 1249\beta _1 + 68673 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 10 \beta_{11} - 2 \beta_{10} - 10 \beta_{9} + 4 \beta_{8} + 16 \beta_{7} + 554 \beta_{6} + \cdots + 255494 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 118 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} - 1172 \beta_{8} + 1384 \beta_{7} + 3942 \beta_{6} + \cdots + 25638062 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 7634 \beta_{11} - 650 \beta_{10} - 6226 \beta_{9} + 2196 \beta_{8} + 15984 \beta_{7} + \cdots + 141218395 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 103758 \beta_{11} + 17226 \beta_{10} - 21958 \beta_{9} - 546278 \beta_{8} + 768298 \beta_{7} + \cdots + 10290514203 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4398356 \beta_{11} + 138332 \beta_{10} - 3052500 \beta_{9} + 529048 \beta_{8} + 11015504 \beta_{7} + \cdots + 75923423196 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 66728876 \beta_{11} + 17258948 \beta_{10} - 18213596 \beta_{9} - 239812552 \beta_{8} + \cdots + 4322296825160 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 2314967844 \beta_{11} + 283565804 \beta_{10} - 1415485860 \beta_{9} - 120172216 \beta_{8} + \cdots + 40329782511969 \)
|
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 |
|
−21.2897 | 27.0000 | 325.250 | −428.332 | −574.821 | −343.000 | −4199.38 | 729.000 | 9119.05 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −17.2684 | 27.0000 | 170.199 | 377.434 | −466.248 | −343.000 | −728.717 | 729.000 | −6517.69 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −17.0448 | 27.0000 | 162.524 | −12.3548 | −460.209 | −343.000 | −588.457 | 729.000 | 210.585 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −11.4113 | 27.0000 | 2.21847 | −235.708 | −308.106 | −343.000 | 1435.33 | 729.000 | 2689.74 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −8.46580 | 27.0000 | −56.3302 | −42.3509 | −228.577 | −343.000 | 1560.50 | 729.000 | 358.535 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.563077 | 27.0000 | −127.683 | 73.7060 | −15.2031 | −343.000 | 143.969 | 729.000 | −41.5022 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.63440 | 27.0000 | −106.522 | 258.476 | 125.129 | −343.000 | −1086.87 | 729.000 | 1197.88 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.1990 | 27.0000 | −23.9812 | −420.340 | 275.372 | −343.000 | −1550.05 | 729.000 | −4287.03 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 12.3574 | 27.0000 | 24.7044 | −353.308 | 333.649 | −343.000 | −1276.46 | 729.000 | −4365.95 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 14.2296 | 27.0000 | 74.4801 | 344.792 | 384.198 | −343.000 | −761.564 | 729.000 | 4906.24 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 20.1816 | 27.0000 | 279.297 | −163.479 | 544.903 | −343.000 | 3053.43 | 729.000 | −3299.28 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 20.4412 | 27.0000 | 289.843 | 478.466 | 551.913 | −343.000 | 3308.27 | 729.000 | 9780.42 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.g | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.g | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 6 T_{2}^{11} - 1257 T_{2}^{10} + 7332 T_{2}^{9} + 587374 T_{2}^{8} - 3314364 T_{2}^{7} + \cdots + 1168730841088 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 1168730841088 \)
|
| $3$ |
\( (T - 27)^{12} \)
|
| $5$ |
\( T^{12} + \cdots - 15\!\cdots\!00 \)
|
| $7$ |
\( (T + 343)^{12} \)
|
| $11$ |
\( T^{12} + \cdots - 44\!\cdots\!00 \)
|
| $13$ |
\( (T - 2197)^{12} \)
|
| $17$ |
\( T^{12} + \cdots - 41\!\cdots\!60 \)
|
| $19$ |
\( T^{12} + \cdots + 15\!\cdots\!32 \)
|
| $23$ |
\( T^{12} + \cdots + 46\!\cdots\!60 \)
|
| $29$ |
\( T^{12} + \cdots + 10\!\cdots\!08 \)
|
| $31$ |
\( T^{12} + \cdots - 15\!\cdots\!40 \)
|
| $37$ |
\( T^{12} + \cdots + 94\!\cdots\!68 \)
|
| $41$ |
\( T^{12} + \cdots - 19\!\cdots\!44 \)
|
| $43$ |
\( T^{12} + \cdots - 61\!\cdots\!68 \)
|
| $47$ |
\( T^{12} + \cdots + 27\!\cdots\!52 \)
|
| $53$ |
\( T^{12} + \cdots - 45\!\cdots\!36 \)
|
| $59$ |
\( T^{12} + \cdots + 62\!\cdots\!20 \)
|
| $61$ |
\( T^{12} + \cdots - 20\!\cdots\!16 \)
|
| $67$ |
\( T^{12} + \cdots - 35\!\cdots\!28 \)
|
| $71$ |
\( T^{12} + \cdots + 59\!\cdots\!64 \)
|
| $73$ |
\( T^{12} + \cdots - 11\!\cdots\!60 \)
|
| $79$ |
\( T^{12} + \cdots + 11\!\cdots\!80 \)
|
| $83$ |
\( T^{12} + \cdots - 61\!\cdots\!08 \)
|
| $89$ |
\( T^{12} + \cdots + 30\!\cdots\!20 \)
|
| $97$ |
\( T^{12} + \cdots - 93\!\cdots\!48 \)
|
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