Newspace parameters
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(61.2274649949\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 1162x^{2} + 318096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 1162x^{2} + 318096 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 1478\nu ) / 141 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} - 3938\nu ) / 141 \)
|
| \(\beta_{3}\) | \(=\) |
\( 28\nu^{2} - 16268 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta_1 ) / 14 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 16268 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 739\beta_{2} + 1969\beta_1 ) / 14 \)
|
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 |
|
0 | −88.6706 | 0 | 60.5215 | 0 | 0 | 0 | 5675.48 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.03728 | 0 | −347.645 | 0 | 0 | 0 | −2137.48 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 7.03728 | 0 | 347.645 | 0 | 0 | 0 | −2137.48 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 88.6706 | 0 | −60.5215 | 0 | 0 | 0 | 5675.48 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 196.8.a.c | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 196.8.a.c | ✓ | 4 |
| 7.c | even | 3 | 2 | 196.8.e.e | 8 | ||
| 7.d | odd | 6 | 2 | 196.8.e.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 196.8.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 196.8.a.c | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 196.8.e.e | 8 | 7.c | even | 3 | 2 | ||
| 196.8.e.e | 8 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 7912T_{3}^{2} + 389376 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(196))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 7912 T^{2} + 389376 \)
|
| $5$ |
\( T^{4} - 124520 T^{2} + 442681600 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 3648 T - 11933584)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 899204335933696 \)
|
| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $23$ |
\( (T^{2} + 58920 T - 2565734400)^{2} \)
|
| $29$ |
\( (T^{2} - 100116 T - 1400899996)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} - 215220 T - 112030623900)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} - 90320 T - 166208248400)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 26\!\cdots\!96 \)
|
| $53$ |
\( (T^{2} - 1045260 T - 534141507100)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} + \cdots + 9306143719600)^{2} \)
|
| $71$ |
\( (T^{2} + \cdots + 5168396140544)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots - 14786225395136)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 36\!\cdots\!76 \)
|
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