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: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 3 x^{10} - 997 x^{9} + 3111 x^{8} + 336848 x^{7} - 938632 x^{6} - 44941024 x^{5} + \cdots + 35492366336 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{4} \) |
| 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
\( x^{11} - 3 x^{10} - 997 x^{9} + 3111 x^{8} + 336848 x^{7} - 938632 x^{6} - 44941024 x^{5} + \cdots + 35492366336 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 182 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 168230457307 \nu^{10} - 2730448142683 \nu^{9} + 116344624277403 \nu^{8} + \cdots + 24\!\cdots\!16 ) / 39\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 168230457307 \nu^{10} - 2730448142683 \nu^{9} + 116344624277403 \nu^{8} + \cdots - 80\!\cdots\!84 ) / 13\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23140441307 \nu^{10} + 209797547963 \nu^{9} - 24192506582523 \nu^{8} + \cdots - 36\!\cdots\!56 ) / 20\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 926984520857 \nu^{10} - 5286454208483 \nu^{9} + 974898294422103 \nu^{8} + \cdots + 40\!\cdots\!16 ) / 49\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 31841965571 \nu^{10} + 258910657597 \nu^{9} + 28369848691107 \nu^{8} + \cdots - 30\!\cdots\!72 ) / 12\!\cdots\!04 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 9126184624867 \nu^{10} - 14765991748243 \nu^{9} + \cdots + 19\!\cdots\!76 ) / 79\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 11455537770919 \nu^{10} + 6938602759289 \nu^{9} + \cdots + 18\!\cdots\!92 ) / 79\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 14327943224581 \nu^{10} - 51762714138911 \nu^{9} + \cdots - 17\!\cdots\!28 ) / 98\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 182 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{3} - 2\beta_{2} + 320\beta _1 - 118 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 8 \beta_{3} + \cdots + 58182 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12 \beta_{10} - 12 \beta_{9} + 16 \beta_{8} + 52 \beta_{7} + 12 \beta_{6} - 16 \beta_{5} + 469 \beta_{4} + \cdots - 119670 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1322 \beta_{10} - 2612 \beta_{9} + 1406 \beta_{8} - 148 \beta_{7} + 1864 \beta_{6} + 962 \beta_{5} + \cdots + 21268930 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9332 \beta_{10} - 5412 \beta_{9} + 6948 \beta_{8} + 38136 \beta_{7} + 12808 \beta_{6} - 18560 \beta_{5} + \cdots - 72674194 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 702650 \beta_{10} - 1341692 \beta_{9} + 750754 \beta_{8} - 190760 \beta_{7} + 705292 \beta_{6} + \cdots + 8171903342 \)
|
| \(\nu^{9}\) | \(=\) |
\( 5441772 \beta_{10} - 1275100 \beta_{9} + 1679160 \beta_{8} + 20941740 \beta_{7} + 8490548 \beta_{6} + \cdots - 37424525934 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 346141322 \beta_{10} - 635221540 \beta_{9} + 362218486 \beta_{8} - 148216748 \beta_{7} + \cdots + 3215834894122 \)
|
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 |
|
−18.8734 | −27.0000 | 228.205 | 504.493 | 509.581 | 343.000 | −1891.20 | 729.000 | −9521.49 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.9189 | −27.0000 | 158.250 | −101.209 | 456.811 | 343.000 | −511.789 | 729.000 | 1712.35 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.86080 | −27.0000 | −66.2078 | −365.241 | 212.242 | 343.000 | 1526.63 | 729.000 | 2871.09 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −6.68992 | −27.0000 | −83.2450 | −113.852 | 180.628 | 343.000 | 1413.21 | 729.000 | 761.657 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.59612 | −27.0000 | −121.260 | 321.571 | 70.0954 | 343.000 | 647.110 | 729.000 | −834.839 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.85917 | −27.0000 | −113.107 | −18.0803 | −104.198 | 343.000 | −930.473 | 729.000 | −69.7751 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.48708 | −27.0000 | −107.866 | −25.4017 | −121.151 | 343.000 | −1058.35 | 729.000 | −113.979 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 12.2566 | −27.0000 | 22.2232 | 381.734 | −330.927 | 343.000 | −1296.46 | 729.000 | 4678.74 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 15.4290 | −27.0000 | 110.055 | −200.286 | −416.584 | 343.000 | −276.867 | 729.000 | −3090.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 19.9937 | −27.0000 | 271.749 | 268.242 | −539.830 | 343.000 | 2874.07 | 729.000 | 5363.15 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 21.9136 | −27.0000 | 352.204 | −481.971 | −591.666 | 343.000 | 4913.11 | 729.000 | −10561.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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.e | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.e | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 25 T_{2}^{10} - 717 T_{2}^{9} + 19197 T_{2}^{8} + 151664 T_{2}^{7} - 4660952 T_{2}^{6} + \cdots + 62547158016 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots + 62547158016 \)
|
| $3$ |
\( (T + 27)^{11} \)
|
| $5$ |
\( T^{11} + \cdots + 30\!\cdots\!00 \)
|
| $7$ |
\( (T - 343)^{11} \)
|
| $11$ |
\( T^{11} + \cdots - 94\!\cdots\!36 \)
|
| $13$ |
\( (T - 2197)^{11} \)
|
| $17$ |
\( T^{11} + \cdots + 72\!\cdots\!80 \)
|
| $19$ |
\( T^{11} + \cdots + 26\!\cdots\!84 \)
|
| $23$ |
\( T^{11} + \cdots + 88\!\cdots\!04 \)
|
| $29$ |
\( T^{11} + \cdots - 23\!\cdots\!80 \)
|
| $31$ |
\( T^{11} + \cdots - 21\!\cdots\!00 \)
|
| $37$ |
\( T^{11} + \cdots + 70\!\cdots\!80 \)
|
| $41$ |
\( T^{11} + \cdots - 25\!\cdots\!28 \)
|
| $43$ |
\( T^{11} + \cdots - 15\!\cdots\!16 \)
|
| $47$ |
\( T^{11} + \cdots + 17\!\cdots\!48 \)
|
| $53$ |
\( T^{11} + \cdots + 11\!\cdots\!96 \)
|
| $59$ |
\( T^{11} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots - 27\!\cdots\!60 \)
|
| $67$ |
\( T^{11} + \cdots - 17\!\cdots\!60 \)
|
| $71$ |
\( T^{11} + \cdots + 49\!\cdots\!32 \)
|
| $73$ |
\( T^{11} + \cdots + 40\!\cdots\!00 \)
|
| $79$ |
\( T^{11} + \cdots + 80\!\cdots\!44 \)
|
| $83$ |
\( T^{11} + \cdots - 14\!\cdots\!28 \)
|
| $89$ |
\( T^{11} + \cdots - 19\!\cdots\!36 \)
|
| $97$ |
\( T^{11} + \cdots + 34\!\cdots\!24 \)
|
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