Newspace parameters
| Level: | \( N \) | \(=\) | \( 34 = 2 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 34.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(17.5112184296\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3262740.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 421x + 3451 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 421x + 3451 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu^{2} + 48\nu - 1140 \)
|
| \(\beta_{2}\) | \(=\) |
\( 6\nu^{2} + 60\nu - 1706 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{2} + 3\beta _1 + 8 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} - 5\beta _1 + 1124 ) / 4 \)
|
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 |
|
−16.0000 | −137.240 | 256.000 | −1254.38 | 2195.83 | −6969.72 | −4096.00 | −848.306 | 20070.0 | |||||||||||||||||||||||||||
| 1.2 | −16.0000 | −36.8994 | 256.000 | 2047.04 | 590.391 | 1984.81 | −4096.00 | −18321.4 | −32752.7 | ||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 258.139 | 256.000 | 521.334 | −4130.22 | 11896.9 | −4096.00 | 46952.7 | −8341.35 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 34.10.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 306.10.a.g | 3 | ||
| 4.b | odd | 2 | 1 | 272.10.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 34.10.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 272.10.a.c | 3 | 4.b | odd | 2 | 1 | ||
| 306.10.a.g | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 84T_{3}^{2} - 39888T_{3} - 1307232 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(34))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{3} \)
|
| $3$ |
\( T^{3} - 84 T^{2} + \cdots - 1307232 \)
|
| $5$ |
\( T^{3} + \cdots + 1338661224 \)
|
| $7$ |
\( T^{3} + \cdots + 164576889568 \)
|
| $11$ |
\( T^{3} + \cdots - 1357228811808 \)
|
| $13$ |
\( T^{3} + \cdots - 804383490695096 \)
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| $17$ |
\( (T - 83521)^{3} \)
|
| $19$ |
\( T^{3} + \cdots + 10\!\cdots\!12 \)
|
| $23$ |
\( T^{3} + \cdots - 12\!\cdots\!28 \)
|
| $29$ |
\( T^{3} + \cdots + 17\!\cdots\!64 \)
|
| $31$ |
\( T^{3} + \cdots - 23\!\cdots\!92 \)
|
| $37$ |
\( T^{3} + \cdots - 29\!\cdots\!44 \)
|
| $41$ |
\( T^{3} + \cdots + 41\!\cdots\!68 \)
|
| $43$ |
\( T^{3} + \cdots - 11\!\cdots\!16 \)
|
| $47$ |
\( T^{3} + \cdots + 36\!\cdots\!04 \)
|
| $53$ |
\( T^{3} + \cdots + 22\!\cdots\!40 \)
|
| $59$ |
\( T^{3} + \cdots + 99\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 10\!\cdots\!20 \)
|
| $67$ |
\( T^{3} + \cdots - 90\!\cdots\!72 \)
|
| $71$ |
\( T^{3} + \cdots - 32\!\cdots\!44 \)
|
| $73$ |
\( T^{3} + \cdots + 37\!\cdots\!80 \)
|
| $79$ |
\( T^{3} + \cdots + 17\!\cdots\!60 \)
|
| $83$ |
\( T^{3} + \cdots + 17\!\cdots\!16 \)
|
| $89$ |
\( T^{3} + \cdots - 84\!\cdots\!36 \)
|
| $97$ |
\( T^{3} + \cdots + 26\!\cdots\!96 \)
|
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