Newspace parameters
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(31.4352286833\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{1177})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 591x^{2} + 592x + 85262 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 7^{2} \) |
| 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
\( x^{4} - 2x^{3} - 591x^{2} + 592x + 85262 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 599\nu - 300 ) / 167 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 3536\nu - 1769 ) / 167 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 82\nu^{2} - 383\nu - 24566 ) / 167 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 7 ) / 14 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 28\beta_{3} + \beta_{2} + 12\beta _1 + 4151 ) / 14 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 43\beta_{2} - 250\beta _1 + 889 ) / 2 \)
|
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 | −29.2088 | 0 | 98.5052 | 0 | 0 | 0 | 610.152 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −19.3093 | 0 | −49.9872 | 0 | 0 | 0 | 129.848 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 19.3093 | 0 | 49.9872 | 0 | 0 | 0 | 129.848 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 29.2088 | 0 | −98.5052 | 0 | 0 | 0 | 610.152 | 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 | 196.6.a.k | ✓ | 4 |
| 4.b | odd | 2 | 1 | 784.6.a.bi | 4 | ||
| 7.b | odd | 2 | 1 | inner | 196.6.a.k | ✓ | 4 |
| 7.c | even | 3 | 2 | 196.6.e.l | 8 | ||
| 7.d | odd | 6 | 2 | 196.6.e.l | 8 | ||
| 28.d | even | 2 | 1 | 784.6.a.bi | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 196.6.a.k | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 196.6.a.k | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 196.6.e.l | 8 | 7.c | even | 3 | 2 | ||
| 196.6.e.l | 8 | 7.d | odd | 6 | 2 | ||
| 784.6.a.bi | 4 | 4.b | odd | 2 | 1 | ||
| 784.6.a.bi | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 1226T_{3}^{2} + 318096 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(196))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 1226 T^{2} + 318096 \)
|
| $5$ |
\( T^{4} - 12202 T^{2} + 24245776 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 222 T - 45352)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 11601874944 \)
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| $17$ |
\( T^{4} + \cdots + 13393243063684 \)
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| $19$ |
\( T^{4} + \cdots + 79621037584 \)
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| $23$ |
\( (T^{2} - 1704 T - 2965168)^{2} \)
|
| $29$ |
\( (T^{2} - 10362 T + 24016784)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 14733805879296 \)
|
| $37$ |
\( (T^{2} - 6866 T - 30258128)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 20\!\cdots\!24 \)
|
| $43$ |
\( (T^{2} + 14498 T - 176356136)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $53$ |
\( (T^{2} - 264 T - 11286484)^{2} \)
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| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!44 \)
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| $61$ |
\( T^{4} + \cdots + 21\!\cdots\!56 \)
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| $67$ |
\( (T^{2} - 97692 T + 1918780416)^{2} \)
|
| $71$ |
\( (T^{2} - 10836 T - 321527808)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 95\!\cdots\!16 \)
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| $79$ |
\( (T^{2} + 58652 T - 1683365024)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 49\!\cdots\!24 \)
|
| $97$ |
\( T^{4} + \cdots + 79\!\cdots\!84 \)
|
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