Newspace parameters
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(27.1048655484\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 13) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 168.1 |
|
− | 4.56155i | −1.63068 | 11.1922 | − | 103.462i | 7.43845i | − | 126.309i | − | 197.024i | −240.341 | −471.948 | ||||||||||||||||||||||||||
| 168.2 | − | 0.438447i | −26.3693 | 31.8078 | 61.4621i | 11.5616i | 162.309i | − | 27.9763i | 452.341 | 26.9479 | |||||||||||||||||||||||||||||
| 168.3 | 0.438447i | −26.3693 | 31.8078 | − | 61.4621i | − | 11.5616i | − | 162.309i | 27.9763i | 452.341 | 26.9479 | ||||||||||||||||||||||||||||
| 168.4 | 4.56155i | −1.63068 | 11.1922 | 103.462i | − | 7.43845i | 126.309i | 197.024i | −240.341 | −471.948 | ||||||||||||||||||||||||||||||
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 | 169.6.b.a | 4 | |
| 13.b | even | 2 | 1 | inner | 169.6.b.a | 4 | |
| 13.d | odd | 4 | 1 | 13.6.a.a | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 169.6.a.a | 2 | ||
| 39.f | even | 4 | 1 | 117.6.a.c | 2 | ||
| 52.f | even | 4 | 1 | 208.6.a.h | 2 | ||
| 65.f | even | 4 | 1 | 325.6.b.b | 4 | ||
| 65.g | odd | 4 | 1 | 325.6.a.b | 2 | ||
| 65.k | even | 4 | 1 | 325.6.b.b | 4 | ||
| 91.i | even | 4 | 1 | 637.6.a.a | 2 | ||
| 104.j | odd | 4 | 1 | 832.6.a.p | 2 | ||
| 104.m | even | 4 | 1 | 832.6.a.i | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.6.a.a | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 117.6.a.c | 2 | 39.f | even | 4 | 1 | ||
| 169.6.a.a | 2 | 13.d | odd | 4 | 1 | ||
| 169.6.b.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 169.6.b.a | 4 | 13.b | even | 2 | 1 | inner | |
| 208.6.a.h | 2 | 52.f | even | 4 | 1 | ||
| 325.6.a.b | 2 | 65.g | odd | 4 | 1 | ||
| 325.6.b.b | 4 | 65.f | even | 4 | 1 | ||
| 325.6.b.b | 4 | 65.k | even | 4 | 1 | ||
| 637.6.a.a | 2 | 91.i | even | 4 | 1 | ||
| 832.6.a.i | 2 | 104.m | even | 4 | 1 | ||
| 832.6.a.p | 2 | 104.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 21T_{2}^{2} + 4 \)
acting on \(S_{6}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 21T^{2} + 4 \)
|
| $3$ |
\( (T^{2} + 28 T + 43)^{2} \)
|
| $5$ |
\( T^{4} + 14482 T^{2} + 40436881 \)
|
| $7$ |
\( T^{4} + 42298 T^{2} + 420291001 \)
|
| $11$ |
\( T^{4} + 130664 T^{2} + 28686736 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} - 2630 T + 1659593)^{2} \)
|
| $19$ |
\( T^{4} + 55336 T^{2} + 441168016 \)
|
| $23$ |
\( (T^{2} - 2624 T - 6800144)^{2} \)
|
| $29$ |
\( (T^{2} + 812 T + 155044)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 197307196305664 \)
|
| $37$ |
\( T^{4} + \cdots + 32\!\cdots\!81 \)
|
| $41$ |
\( T^{4} + \cdots + 684577521664 \)
|
| $43$ |
\( (T^{2} + 2420 T - 93138877)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!41 \)
|
| $53$ |
\( (T^{2} + 43720 T + 280413712)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 35\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} - 1984 T - 366282304)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 14\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!81 \)
|
| $73$ |
\( T^{4} + \cdots + 63\!\cdots\!96 \)
|
| $79$ |
\( (T^{2} + 55296 T + 361555696)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 46\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!56 \)
|
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