Newspace parameters
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.5272463770\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 3470x^{3} - 72162x^{2} - 343807x + 342288 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{14}\cdot 5\cdot 71 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 3470x^{3} - 72162x^{2} - 343807x + 342288 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 153\nu^{4} - 1509\nu^{3} - 509991\nu^{2} - 7510725\nu - 15417776 ) / 76960 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3501\nu^{4} + 155607\nu^{3} - 16205907\nu^{2} - 703403145\nu - 2865538992 ) / 76960 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25983\nu^{4} - 340467\nu^{3} - 84777729\nu^{2} - 914713875\nu - 1313859920 ) / 15392 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2294739\nu^{4} + 30957687\nu^{3} + 7648967853\nu^{2} + 77034917655\nu - 149155163152 ) / 76960 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 223\beta_{3} + 450\beta_{2} - 184652\beta _1 + 736465 ) / 1656288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 44\beta_{4} + 866\beta_{3} - 9\beta_{2} - 75205\beta _1 + 143796218 ) / 103518 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 19591\beta_{4} + 953977\beta_{3} + 1924974\beta_{2} - 560255060\beta _1 + 78836545783 ) / 1656288 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 323617\beta_{4} + 8317711\beta_{3} + 5074488\beta_{2} - 2220988010\beta _1 + 1081199708557 ) / 207036 \)
|
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 |
|
−342.118 | 0 | 84276.6 | 267526. | 0 | 2.26179e6 | −1.76220e7 | 0 | −9.15253e7 | |||||||||||||||||||||||||||||||||
| 1.2 | −225.004 | 0 | 17858.9 | −338163. | 0 | 626121. | 3.35462e6 | 0 | 7.60882e7 | ||||||||||||||||||||||||||||||||||
| 1.3 | −84.4187 | 0 | −25641.5 | 151347. | 0 | −2.00754e6 | 4.93085e6 | 0 | −1.27765e7 | ||||||||||||||||||||||||||||||||||
| 1.4 | 109.269 | 0 | −20828.4 | −93591.2 | 0 | 3.01703e6 | −5.85640e6 | 0 | −1.02266e7 | ||||||||||||||||||||||||||||||||||
| 1.5 | 269.272 | 0 | 39739.4 | 29607.0 | 0 | −1.74386e6 | 1.87720e6 | 0 | 7.97233e6 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
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 | 27.16.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 27.16.a.c | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.16.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 27.16.a.c | yes | 5 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 273T_{2}^{4} - 92358T_{2}^{3} - 21593520T_{2}^{2} + 1213667712T_{2} + 191201762304 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(27))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 191201762304 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 37\!\cdots\!25 \)
|
| $7$ |
\( T^{5} + \cdots - 14\!\cdots\!75 \)
|
| $11$ |
\( T^{5} + \cdots + 67\!\cdots\!25 \)
|
| $13$ |
\( T^{5} + \cdots - 75\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots - 47\!\cdots\!52 \)
|
| $19$ |
\( T^{5} + \cdots + 37\!\cdots\!24 \)
|
| $23$ |
\( T^{5} + \cdots + 24\!\cdots\!56 \)
|
| $29$ |
\( T^{5} + \cdots - 44\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 19\!\cdots\!21 \)
|
| $37$ |
\( T^{5} + \cdots - 38\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 32\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots - 72\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 76\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 19\!\cdots\!81 \)
|
| $59$ |
\( T^{5} + \cdots + 48\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 10\!\cdots\!08 \)
|
| $67$ |
\( T^{5} + \cdots - 44\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 60\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 25\!\cdots\!75 \)
|
| $79$ |
\( T^{5} + \cdots + 27\!\cdots\!28 \)
|
| $83$ |
\( T^{5} + \cdots + 19\!\cdots\!21 \)
|
| $89$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 34\!\cdots\!75 \)
|
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