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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 4 x^{8} - 747 x^{7} + 3070 x^{6} + 180816 x^{5} - 678576 x^{4} - 15901072 x^{3} + \cdots + 220377344 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4 x^{8} - 747 x^{7} + 3070 x^{6} + 180816 x^{5} - 678576 x^{4} - 15901072 x^{3} + \cdots + 220377344 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 70 \nu^{7} - 887 \nu^{6} + 41172 \nu^{5} + 224824 \nu^{4} - 5776320 \nu^{3} + \cdots + 95597184 ) / 2096640 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + \nu - 168 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 70 \nu^{7} - 887 \nu^{6} + 41172 \nu^{5} + 224824 \nu^{4} - 6824640 \nu^{3} + \cdots + 481378944 ) / 2096640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13 \nu^{8} - 42 \nu^{7} + 7443 \nu^{6} + 17036 \nu^{5} - 1184248 \nu^{4} - 1820032 \nu^{3} + \cdots - 719363200 ) / 2096640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 9 \nu^{8} - 28 \nu^{7} + 6121 \nu^{6} + 14060 \nu^{5} - 1261200 \nu^{4} - 1978528 \nu^{3} + \cdots - 380460928 ) / 524160 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5 \nu^{8} - 14 \nu^{7} + 4799 \nu^{6} + 11084 \nu^{5} - 1600232 \nu^{4} - 2399104 \nu^{3} + \cdots - 4382651776 ) / 1048320 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{8} - 70 \nu^{7} - 887 \nu^{6} + 50532 \nu^{5} + 252904 \nu^{4} - 10943040 \nu^{3} + \cdots + 1037587584 ) / 299520 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta _1 + 168 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 261\beta _1 - 136 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{7} + 4\beta_{6} - 8\beta_{5} + 2\beta_{4} + 361\beta_{3} - 2\beta_{2} - 627\beta _1 + 43716 \)
|
| \(\nu^{5}\) | \(=\) |
\( 32 \beta_{8} + 12 \beta_{7} - 12 \beta_{6} + 24 \beta_{5} - 1110 \beta_{4} - 1471 \beta_{3} + \cdots - 93836 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 96 \beta_{8} - 1860 \beta_{7} + 2404 \beta_{6} - 5224 \beta_{5} + 2570 \beta_{4} + 124953 \beta_{3} + \cdots + 12996260 \)
|
| \(\nu^{7}\) | \(=\) |
\( 18976 \beta_{8} + 7644 \beta_{7} - 9980 \beta_{6} + 19544 \beta_{5} - 489734 \beta_{4} - 636287 \beta_{3} + \cdots - 43734524 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 74336 \beta_{8} - 709508 \beta_{7} + 1028516 \beta_{6} - 2455144 \beta_{5} + 1696842 \beta_{4} + \cdots + 4145413220 \)
|
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 |
|
−19.1706 | −27.0000 | 239.513 | −314.365 | 517.607 | −343.000 | −2137.77 | 729.000 | 6026.57 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −15.4520 | −27.0000 | 110.764 | 443.039 | 417.204 | −343.000 | 266.324 | 729.000 | −6845.85 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −10.5283 | −27.0000 | −17.1549 | −281.557 | 284.264 | −343.000 | 1528.23 | 729.000 | 2964.32 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.12110 | −27.0000 | −111.017 | 306.750 | 111.270 | −343.000 | 985.012 | 729.000 | −1264.15 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.603157 | −27.0000 | −127.636 | −106.086 | 16.2852 | −343.000 | 154.189 | 729.000 | 63.9867 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 8.66493 | −27.0000 | −52.9191 | 59.9733 | −233.953 | −343.000 | −1567.65 | 729.000 | 519.664 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 12.1110 | −27.0000 | 18.6762 | 97.4412 | −326.997 | −343.000 | −1324.02 | 729.000 | 1180.11 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 14.8187 | −27.0000 | 91.5926 | −406.952 | −400.104 | −343.000 | −539.509 | 729.000 | −6030.48 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 18.2806 | −27.0000 | 206.181 | 531.756 | −493.576 | −343.000 | 1429.19 | 729.000 | 9720.83 | ||||||||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.8.a.b | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 4 T_{2}^{8} - 747 T_{2}^{7} + 3070 T_{2}^{6} + 180816 T_{2}^{5} - 678576 T_{2}^{4} + \cdots + 220377344 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 4 T^{8} + \cdots + 220377344 \)
|
| $3$ |
\( (T + 27)^{9} \)
|
| $5$ |
\( T^{9} + \cdots - 16\!\cdots\!00 \)
|
| $7$ |
\( (T + 343)^{9} \)
|
| $11$ |
\( T^{9} + \cdots + 68\!\cdots\!08 \)
|
| $13$ |
\( (T + 2197)^{9} \)
|
| $17$ |
\( T^{9} + \cdots - 20\!\cdots\!76 \)
|
| $19$ |
\( T^{9} + \cdots - 51\!\cdots\!40 \)
|
| $23$ |
\( T^{9} + \cdots - 91\!\cdots\!64 \)
|
| $29$ |
\( T^{9} + \cdots + 91\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots + 17\!\cdots\!24 \)
|
| $37$ |
\( T^{9} + \cdots + 11\!\cdots\!20 \)
|
| $41$ |
\( T^{9} + \cdots - 93\!\cdots\!40 \)
|
| $43$ |
\( T^{9} + \cdots - 49\!\cdots\!52 \)
|
| $47$ |
\( T^{9} + \cdots - 14\!\cdots\!64 \)
|
| $53$ |
\( T^{9} + \cdots + 41\!\cdots\!20 \)
|
| $59$ |
\( T^{9} + \cdots + 65\!\cdots\!20 \)
|
| $61$ |
\( T^{9} + \cdots - 12\!\cdots\!80 \)
|
| $67$ |
\( T^{9} + \cdots - 70\!\cdots\!80 \)
|
| $71$ |
\( T^{9} + \cdots - 78\!\cdots\!52 \)
|
| $73$ |
\( T^{9} + \cdots - 31\!\cdots\!36 \)
|
| $79$ |
\( T^{9} + \cdots + 97\!\cdots\!60 \)
|
| $83$ |
\( T^{9} + \cdots - 20\!\cdots\!04 \)
|
| $89$ |
\( T^{9} + \cdots - 66\!\cdots\!00 \)
|
| $97$ |
\( T^{9} + \cdots + 27\!\cdots\!88 \)
|
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