Newspace parameters
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(52.7930693068\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 449x^{4} + 37224x^{2} - 205776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 13) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 449x^{4} + 37224x^{2} - 205776 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 365\nu^{2} + 12324 ) / 240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 365\nu^{3} - 12084\nu ) / 240 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} - 150 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 445\nu^{3} + 34644\nu ) / 160 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 150 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} - 3\beta_{3} + 282\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 365\beta_{4} + 240\beta_{2} + 42426 \)
|
| \(\nu^{5}\) | \(=\) |
\( -730\beta_{5} - 1335\beta_{3} + 90846\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−18.4900 | −27.4160 | 213.880 | −70.5606 | 506.922 | −454.852 | −1587.93 | −1435.36 | 1304.67 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −10.0583 | 50.8656 | −26.8297 | −8.88672 | −511.623 | 510.452 | 1557.33 | 400.306 | 89.3858 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.43912 | −51.4496 | −122.051 | 488.312 | 125.492 | −616.243 | 609.904 | 460.056 | −1191.05 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.43912 | −51.4496 | −122.051 | −488.312 | −125.492 | 616.243 | −609.904 | 460.056 | −1191.05 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 10.0583 | 50.8656 | −26.8297 | 8.88672 | 511.623 | −510.452 | −1557.33 | 400.306 | 89.3858 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 18.4900 | −27.4160 | 213.880 | 70.5606 | −506.922 | 454.852 | 1587.93 | −1435.36 | 1304.67 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.8.a.d | 6 | |
| 13.b | even | 2 | 1 | inner | 169.8.a.d | 6 | |
| 13.d | odd | 4 | 2 | 13.8.b.a | ✓ | 6 | |
| 39.f | even | 4 | 2 | 117.8.b.b | 6 | ||
| 52.f | even | 4 | 2 | 208.8.f.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.8.b.a | ✓ | 6 | 13.d | odd | 4 | 2 | |
| 117.8.b.b | 6 | 39.f | even | 4 | 2 | ||
| 169.8.a.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 169.8.a.d | 6 | 13.b | even | 2 | 1 | inner | |
| 208.8.f.a | 6 | 52.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 449T_{2}^{4} + 37224T_{2}^{2} - 205776 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(169))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 449 T^{4} + \cdots - 205776 \)
|
| $3$ |
\( (T^{3} + 28 T^{2} + \cdots - 71748)^{2} \)
|
| $5$ |
\( T^{6} + \cdots - 93756690000 \)
|
| $7$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{3} + 6576 T^{2} + \cdots + 976631438250)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 62\!\cdots\!36 \)
|
| $23$ |
\( (T^{3} + \cdots + 38560806996192)^{2} \)
|
| $29$ |
\( (T^{3} + \cdots + 16\!\cdots\!80)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 33\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 30\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 11\!\cdots\!88)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 32\!\cdots\!96 \)
|
| $53$ |
\( (T^{3} + \cdots + 18\!\cdots\!92)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 36\!\cdots\!96 \)
|
| $61$ |
\( (T^{3} + \cdots + 10\!\cdots\!88)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 54\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots - 39\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 10\!\cdots\!04 \)
|
| $79$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 14\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots - 22\!\cdots\!96 \)
|
| $97$ |
\( T^{6} + \cdots - 17\!\cdots\!56 \)
|
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