Newspace parameters
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(107.127453922\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 12623x^{2} - 303924x + 1814436 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 52) |
| 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
\( x^{4} - 2x^{3} - 12623x^{2} - 303924x + 1814436 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 55\nu - 6294 ) / 36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 233\nu - 6438 ) / 36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 19\nu^{2} - 11370\nu - 121284 ) / 108 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 4 ) / 8 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 55\beta_{2} + 233\beta _1 + 50572 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 864\beta_{3} + 12415\beta_{2} - 6943\beta _1 + 1976620 ) / 8 \)
|
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 | −144.718 | 0 | −2497.02 | 0 | −9767.71 | 0 | 1260.40 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −62.9670 | 0 | 1026.55 | 0 | 4497.03 | 0 | −15718.2 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 96.3430 | 0 | 223.782 | 0 | −1984.14 | 0 | −10401.0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 258.342 | 0 | −700.318 | 0 | −2996.18 | 0 | 47057.8 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 208.10.a.f | 4 | |
| 4.b | odd | 2 | 1 | 52.10.a.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.10.a.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 208.10.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 147T_{3}^{3} - 39661T_{3}^{2} + 1937115T_{3} + 226804788 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 147 T^{3} + \cdots + 226804788 \)
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| $5$ |
\( T^{4} + \cdots + 401718852990 \)
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| $7$ |
\( T^{4} + \cdots - 261130689389948 \)
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| $11$ |
\( T^{4} + \cdots + 37\!\cdots\!72 \)
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| $13$ |
\( (T - 28561)^{4} \)
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| $17$ |
\( T^{4} + \cdots - 27\!\cdots\!38 \)
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| $19$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 23\!\cdots\!84 \)
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| $29$ |
\( T^{4} + \cdots + 22\!\cdots\!28 \)
|
| $31$ |
\( T^{4} + \cdots + 94\!\cdots\!04 \)
|
| $37$ |
\( T^{4} + \cdots - 12\!\cdots\!58 \)
|
| $41$ |
\( T^{4} + \cdots - 10\!\cdots\!68 \)
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| $43$ |
\( T^{4} + \cdots - 38\!\cdots\!68 \)
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| $47$ |
\( T^{4} + \cdots - 65\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots - 44\!\cdots\!32 \)
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| $59$ |
\( T^{4} + \cdots + 81\!\cdots\!16 \)
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| $61$ |
\( T^{4} + \cdots - 10\!\cdots\!72 \)
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| $67$ |
\( T^{4} + \cdots - 46\!\cdots\!28 \)
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| $71$ |
\( T^{4} + \cdots + 51\!\cdots\!68 \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!24 \)
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| $79$ |
\( T^{4} + \cdots + 33\!\cdots\!40 \)
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| $83$ |
\( T^{4} + \cdots - 76\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 22\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 31\!\cdots\!28 \)
|
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