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: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 1134 x^{10} - 344 x^{9} + 459124 x^{8} + 472004 x^{7} - 80671090 x^{6} - 190466980 x^{5} + \cdots + 21853094368 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{12} - 1134 x^{10} - 344 x^{9} + 459124 x^{8} + 472004 x^{7} - 80671090 x^{6} - 190466980 x^{5} + \cdots + 21853094368 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu^{2} - 326\nu - 653 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12\!\cdots\!51 \nu^{11} + \cdots + 26\!\cdots\!08 ) / 68\!\cdots\!04 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 56\!\cdots\!93 \nu^{11} + \cdots - 32\!\cdots\!40 ) / 85\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 88\!\cdots\!69 \nu^{11} + \cdots - 79\!\cdots\!84 ) / 68\!\cdots\!04 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 92\!\cdots\!83 \nu^{11} + \cdots - 12\!\cdots\!36 ) / 68\!\cdots\!04 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 11\!\cdots\!67 \nu^{11} + \cdots - 22\!\cdots\!88 ) / 68\!\cdots\!04 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!77 \nu^{11} + \cdots - 18\!\cdots\!04 ) / 68\!\cdots\!04 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 38\!\cdots\!63 \nu^{11} + \cdots + 15\!\cdots\!88 ) / 17\!\cdots\!76 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 27\!\cdots\!31 \nu^{11} + \cdots - 45\!\cdots\!76 ) / 68\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 189 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_{2} + 326\beta _1 + 86 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{11} + \beta_{10} - \beta_{9} + 3 \beta_{8} - 2 \beta_{5} + 24 \beta_{4} + 5 \beta_{3} + \cdots + 61277 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 25 \beta_{11} - 2 \beta_{10} + 19 \beta_{9} + 39 \beta_{8} - 41 \beta_{7} - 21 \beta_{6} + \cdots - 34084 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1479 \beta_{11} + 317 \beta_{10} - 742 \beta_{9} + 1780 \beta_{8} + 161 \beta_{7} - 205 \beta_{6} + \cdots + 23082141 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 19184 \beta_{11} - 240 \beta_{10} + 16964 \beta_{9} + 33564 \beta_{8} - 31836 \beta_{7} + \cdots - 35169850 \)
|
| \(\nu^{8}\) | \(=\) |
\( 820026 \beta_{11} + 41217 \beta_{10} - 448145 \beta_{9} + 785599 \beta_{8} + 129672 \beta_{7} + \cdots + 9325732189 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 11256009 \beta_{11} + 475914 \beta_{10} + 10989079 \beta_{9} + 20786183 \beta_{8} + \cdots - 19434194908 \)
|
| \(\nu^{10}\) | \(=\) |
\( 417090451 \beta_{11} - 23663483 \beta_{10} - 248436498 \beta_{9} + 314599716 \beta_{8} + \cdots + 3920879025021 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 5991821948 \beta_{11} + 521535952 \beta_{10} + 6212656104 \beta_{9} + 11320546900 \beta_{8} + \cdots - 9051641516402 \)
|
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 |
|
−22.2408 | −27.0000 | 366.655 | −166.673 | 600.502 | −343.000 | −5307.88 | 729.000 | 3706.94 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.9813 | −27.0000 | 160.364 | 425.977 | 458.495 | −343.000 | −549.584 | 729.000 | −7233.64 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −16.7551 | −27.0000 | 152.733 | −241.001 | 452.388 | −343.000 | −414.409 | 729.000 | 4038.00 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −14.8461 | −27.0000 | 92.4055 | 140.379 | 400.844 | −343.000 | 528.438 | 729.000 | −2084.07 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −6.06892 | −27.0000 | −91.1682 | 206.192 | 163.861 | −343.000 | 1330.11 | 729.000 | −1251.37 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.04596 | −27.0000 | −126.906 | −495.718 | 28.2408 | −343.000 | 266.621 | 729.000 | 518.500 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.28141 | −27.0000 | −109.670 | 504.993 | −115.598 | −343.000 | −1017.56 | 729.000 | 2162.08 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.83770 | −27.0000 | −104.597 | −215.029 | −130.618 | −343.000 | −1125.23 | 729.000 | −1040.25 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 7.44398 | −27.0000 | −72.5872 | −304.122 | −200.987 | −343.000 | −1493.17 | 729.000 | −2263.88 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 10.9448 | −27.0000 | −8.21135 | 267.840 | −295.510 | −343.000 | −1490.81 | 729.000 | 2931.45 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 18.0885 | −27.0000 | 199.195 | 173.985 | −488.390 | −343.000 | 1287.81 | 729.000 | 3147.14 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 20.3417 | −27.0000 | 285.787 | −419.822 | −549.227 | −343.000 | 3209.66 | 729.000 | −8539.91 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.f | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.f | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 12 T_{2}^{11} - 1068 T_{2}^{10} - 10776 T_{2}^{9} + 411685 T_{2}^{8} + 3078084 T_{2}^{7} + \cdots + 370283931136 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 370283931136 \)
|
| $3$ |
\( (T + 27)^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 15\!\cdots\!00 \)
|
| $7$ |
\( (T + 343)^{12} \)
|
| $11$ |
\( T^{12} + \cdots + 66\!\cdots\!40 \)
|
| $13$ |
\( (T - 2197)^{12} \)
|
| $17$ |
\( T^{12} + \cdots + 22\!\cdots\!96 \)
|
| $19$ |
\( T^{12} + \cdots + 11\!\cdots\!20 \)
|
| $23$ |
\( T^{12} + \cdots + 13\!\cdots\!44 \)
|
| $29$ |
\( T^{12} + \cdots - 34\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 55\!\cdots\!52 \)
|
| $37$ |
\( T^{12} + \cdots - 30\!\cdots\!20 \)
|
| $41$ |
\( T^{12} + \cdots + 51\!\cdots\!52 \)
|
| $43$ |
\( T^{12} + \cdots + 61\!\cdots\!36 \)
|
| $47$ |
\( T^{12} + \cdots + 53\!\cdots\!44 \)
|
| $53$ |
\( T^{12} + \cdots - 12\!\cdots\!68 \)
|
| $59$ |
\( T^{12} + \cdots + 14\!\cdots\!92 \)
|
| $61$ |
\( T^{12} + \cdots + 11\!\cdots\!88 \)
|
| $67$ |
\( T^{12} + \cdots + 17\!\cdots\!20 \)
|
| $71$ |
\( T^{12} + \cdots + 56\!\cdots\!40 \)
|
| $73$ |
\( T^{12} + \cdots + 16\!\cdots\!28 \)
|
| $79$ |
\( T^{12} + \cdots + 38\!\cdots\!60 \)
|
| $83$ |
\( T^{12} + \cdots - 13\!\cdots\!64 \)
|
| $89$ |
\( T^{12} + \cdots + 34\!\cdots\!12 \)
|
| $97$ |
\( T^{12} + \cdots - 91\!\cdots\!04 \)
|
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