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: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3 x^{9} - 894 x^{8} + 1570 x^{7} + 272284 x^{6} - 175620 x^{5} - 32565946 x^{4} + \cdots - 2214651064 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3 x^{9} - 894 x^{8} + 1570 x^{7} + 272284 x^{6} - 175620 x^{5} - 32565946 x^{4} + \cdots - 2214651064 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 179 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 342389 \nu^{9} + 5358920 \nu^{8} + 237195886 \nu^{7} - 3668757552 \nu^{6} + \cdots + 20\!\cdots\!88 ) / 6511476674560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 342389 \nu^{9} - 5358920 \nu^{8} - 237195886 \nu^{7} + 3668757552 \nu^{6} + \cdots - 13\!\cdots\!48 ) / 6511476674560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2013943 \nu^{9} + 31442680 \nu^{8} + 1385885882 \nu^{7} - 22772371184 \nu^{6} + \cdots - 21\!\cdots\!44 ) / 1627869168640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6364957 \nu^{9} - 100014760 \nu^{8} - 4455973118 \nu^{7} + 62238506576 \nu^{6} + \cdots + 59\!\cdots\!76 ) / 3255738337280 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4837 \nu^{9} + 75655 \nu^{8} + 3344813 \nu^{7} - 52611171 \nu^{6} - 646766071 \nu^{5} + \cdots + 1396916297504 ) / 1422962560 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 890275 \nu^{9} + 14339932 \nu^{8} + 664766470 \nu^{7} - 10170295204 \nu^{6} + \cdots + 609463130658152 ) / 162786916864 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2227697 \nu^{9} - 18706510 \nu^{8} - 1750581608 \nu^{7} + 11715378606 \nu^{6} + \cdots + 139005983230456 ) / 406967292160 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 179 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta_{2} + 291\beta _1 + 249 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} + 2\beta_{6} - 8\beta_{5} - \beta_{4} + 3\beta_{3} + 399\beta_{2} + 1092\beta _1 + 51912 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16\beta_{7} + 4\beta_{6} - 22\beta_{5} + 534\beta_{4} + 166\beta_{3} + 832\beta_{2} + 96803\beta _1 + 152356 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 24 \beta_{8} + 2184 \beta_{7} + 748 \beta_{6} - 4746 \beta_{5} - 398 \beta_{4} + 386 \beta_{3} + \cdots + 17262263 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 192 \beta_{9} + 2912 \beta_{8} + 6864 \beta_{7} + 2212 \beta_{6} - 18254 \beta_{5} + 234655 \beta_{4} + \cdots + 66077949 \)
|
| \(\nu^{8}\) | \(=\) |
\( 22720 \beta_{9} + 19304 \beta_{8} + 927468 \beta_{7} + 195694 \beta_{6} - 2213162 \beta_{5} + \cdots + 6146597348 \)
|
| \(\nu^{9}\) | \(=\) |
\( 222592 \beta_{9} + 2576640 \beta_{8} + 2074880 \beta_{7} + 424984 \beta_{6} - 10696540 \beta_{5} + \cdots + 26303688784 \)
|
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 |
|
−20.5296 | −27.0000 | 293.463 | −7.55834 | 554.298 | 343.000 | −3396.88 | 729.000 | 155.169 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −15.1851 | −27.0000 | 102.588 | −429.294 | 409.998 | 343.000 | 385.889 | 729.000 | 6518.88 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −12.9926 | −27.0000 | 40.8071 | 98.9946 | 350.800 | 343.000 | 1132.86 | 729.000 | −1286.20 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −11.7793 | −27.0000 | 10.7531 | 266.841 | 318.042 | 343.000 | 1381.09 | 729.000 | −3143.21 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.80411 | −27.0000 | −124.745 | 426.633 | 48.7108 | 343.000 | 455.979 | 729.000 | −769.691 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.54633 | −27.0000 | −125.609 | −65.9083 | −41.7509 | 343.000 | −392.163 | 729.000 | −101.916 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.23135 | −27.0000 | −100.633 | −461.300 | −141.246 | 343.000 | −1196.06 | 729.000 | −2413.22 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 12.3276 | −27.0000 | 23.9705 | 55.2827 | −332.846 | 343.000 | −1282.44 | 729.000 | 681.504 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 17.1669 | −27.0000 | 166.703 | 407.021 | −463.507 | 343.000 | 664.409 | 729.000 | 6987.29 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 19.0185 | −27.0000 | 233.703 | −167.711 | −513.499 | 343.000 | 2010.31 | 729.000 | −3189.61 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.c | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 7 T_{2}^{9} - 876 T_{2}^{8} - 5570 T_{2}^{7} + 258200 T_{2}^{6} + 1440864 T_{2}^{5} + \cdots - 2802460672 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 2802460672 \)
|
| $3$ |
\( (T + 27)^{10} \)
|
| $5$ |
\( T^{10} + \cdots - 41\!\cdots\!00 \)
|
| $7$ |
\( (T - 343)^{10} \)
|
| $11$ |
\( T^{10} + \cdots - 49\!\cdots\!00 \)
|
| $13$ |
\( (T + 2197)^{10} \)
|
| $17$ |
\( T^{10} + \cdots - 63\!\cdots\!20 \)
|
| $19$ |
\( T^{10} + \cdots + 44\!\cdots\!24 \)
|
| $23$ |
\( T^{10} + \cdots + 15\!\cdots\!40 \)
|
| $29$ |
\( T^{10} + \cdots - 74\!\cdots\!96 \)
|
| $31$ |
\( T^{10} + \cdots + 66\!\cdots\!80 \)
|
| $37$ |
\( T^{10} + \cdots - 11\!\cdots\!84 \)
|
| $41$ |
\( T^{10} + \cdots - 39\!\cdots\!28 \)
|
| $43$ |
\( T^{10} + \cdots + 64\!\cdots\!92 \)
|
| $47$ |
\( T^{10} + \cdots - 62\!\cdots\!64 \)
|
| $53$ |
\( T^{10} + \cdots - 36\!\cdots\!32 \)
|
| $59$ |
\( T^{10} + \cdots + 96\!\cdots\!80 \)
|
| $61$ |
\( T^{10} + \cdots + 91\!\cdots\!96 \)
|
| $67$ |
\( T^{10} + \cdots - 20\!\cdots\!52 \)
|
| $71$ |
\( T^{10} + \cdots + 35\!\cdots\!04 \)
|
| $73$ |
\( T^{10} + \cdots - 16\!\cdots\!80 \)
|
| $79$ |
\( T^{10} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots - 46\!\cdots\!68 \)
|
| $89$ |
\( T^{10} + \cdots + 76\!\cdots\!20 \)
|
| $97$ |
\( T^{10} + \cdots + 58\!\cdots\!12 \)
|
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