Newspace parameters
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(16.1075214316\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.432216027.2 |
|
|
|
| Defining polynomial: |
\( x^{6} + 11x^{4} - 14x^{3} + 121x^{2} - 77x + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 11x^{4} - 14x^{3} + 121x^{2} - 77x + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 7 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 11\nu^{3} - 7\nu^{2} + 121\nu ) / 77 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} + 11\nu^{2} - 7\nu + 70 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 10\nu^{3} + 18\nu^{2} - 110\nu + 70 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 7\beta_{3} - \beta_{2} - \beta _1 - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 11\beta_{2} + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{5} + 11\beta_{4} - 77\beta_{3} + 18\beta_1 \)
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| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} + 126\beta_{3} - 128\beta_{2} - 128\beta _1 - 126 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
| \(n\) | \(92\) | \(106\) | \(157\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{3}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.79899 | + | 3.11595i | −1.50000 | − | 2.59808i | −2.47277 | − | 4.28296i | −0.896985 | + | 1.55362i | 10.7940 | 18.4959 | − | 0.950110i | −10.9899 | −4.50000 | + | 7.79423i | −3.22734 | − | 5.58992i | ||||||||||||||||||||||
| 79.2 | 0.331419 | − | 0.574035i | −1.50000 | − | 2.59808i | 3.78032 | + | 6.54771i | 5.49426 | − | 9.51633i | −1.98851 | 15.8128 | + | 9.64127i | 10.3142 | −4.50000 | + | 7.79423i | −3.64180 | − | 6.30779i | |||||||||||||||||||||||
| 79.3 | 1.46758 | − | 2.54192i | −1.50000 | − | 2.59808i | −0.307557 | − | 0.532705i | 8.90273 | − | 15.4200i | −8.80545 | −15.3087 | − | 10.4232i | 21.6758 | −4.50000 | + | 7.79423i | −26.1309 | − | 45.2600i | |||||||||||||||||||||||
| 235.1 | −1.79899 | − | 3.11595i | −1.50000 | + | 2.59808i | −2.47277 | + | 4.28296i | −0.896985 | − | 1.55362i | 10.7940 | 18.4959 | + | 0.950110i | −10.9899 | −4.50000 | − | 7.79423i | −3.22734 | + | 5.58992i | |||||||||||||||||||||||
| 235.2 | 0.331419 | + | 0.574035i | −1.50000 | + | 2.59808i | 3.78032 | − | 6.54771i | 5.49426 | + | 9.51633i | −1.98851 | 15.8128 | − | 9.64127i | 10.3142 | −4.50000 | − | 7.79423i | −3.64180 | + | 6.30779i | |||||||||||||||||||||||
| 235.3 | 1.46758 | + | 2.54192i | −1.50000 | + | 2.59808i | −0.307557 | + | 0.532705i | 8.90273 | + | 15.4200i | −8.80545 | −15.3087 | + | 10.4232i | 21.6758 | −4.50000 | − | 7.79423i | −26.1309 | + | 45.2600i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 273.4.i.b | ✓ | 6 |
| 7.c | even | 3 | 1 | inner | 273.4.i.b | ✓ | 6 |
| 7.c | even | 3 | 1 | 1911.4.a.j | 3 | ||
| 7.d | odd | 6 | 1 | 1911.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.i.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 273.4.i.b | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 1911.4.a.i | 3 | 7.d | odd | 6 | 1 | ||
| 1911.4.a.j | 3 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 11T_{2}^{4} - 14T_{2}^{3} + 121T_{2}^{2} - 77T_{2} + 49 \)
acting on \(S_{4}^{\mathrm{new}}(273, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 11 T^{4} + \cdots + 49 \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} - 27 T^{5} + \cdots + 123201 \)
|
| $7$ |
\( T^{6} - 38 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} - 19 T^{5} + \cdots + 4272489 \)
|
| $13$ |
\( (T + 13)^{6} \)
|
| $17$ |
\( T^{6} + 25 T^{5} + \cdots + 90801841 \)
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| $19$ |
\( T^{6} + \cdots + 3204578881 \)
|
| $23$ |
\( T^{6} + \cdots + 11277802809 \)
|
| $29$ |
\( (T^{3} - 274 T^{2} + \cdots + 1429589)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 2582799337881 \)
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| $37$ |
\( T^{6} + \cdots + 35972633202961 \)
|
| $41$ |
\( (T^{3} + 66 T^{2} + \cdots - 5892039)^{2} \)
|
| $43$ |
\( (T^{3} - 598 T^{2} + \cdots - 349639)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 113153597967424 \)
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| $53$ |
\( T^{6} + \cdots + 24055993805481 \)
|
| $59$ |
\( T^{6} + \cdots + 226911713560921 \)
|
| $61$ |
\( T^{6} + \cdots + 929348211655369 \)
|
| $67$ |
\( T^{6} + \cdots + 29921545463481 \)
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| $71$ |
\( (T^{3} - 1039 T^{2} + \cdots + 183869749)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 50777153904481 \)
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| $79$ |
\( T^{6} + \cdots + 31\!\cdots\!29 \)
|
| $83$ |
\( (T^{3} - 836 T^{2} + \cdots - 41702703)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 13\!\cdots\!01 \)
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| $97$ |
\( (T^{3} - 267 T^{2} + \cdots + 710857157)^{2} \)
|
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