Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(35.3862065541\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 37736x^{3} - 223354x^{2} + 285905207x + 4244770124 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{6} \) |
| 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
\( x^{5} - 2x^{4} - 37736x^{3} - 223354x^{2} + 285905207x + 4244770124 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{4} - 1021\nu^{3} + 143937\nu^{2} + 23519093\nu - 292333244 ) / 28584 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 11\nu - 15091 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -19\nu^{4} + 1837\nu^{3} + 564111\nu^{2} - 40072949\nu - 2342928196 ) / 57168 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 11\beta _1 + 15091 \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{4} - 3\beta_{3} - 19\beta_{2} + 22610\beta _1 + 170243 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2042\beta_{4} + 29400\beta_{3} - 1837\beta_{2} + 403518\beta _1 + 341200384 \)
|
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 |
|
−172.469 | −729.000 | 21553.5 | 32656.4 | 125730. | −130985. | −2.30444e6 | 531441. | −5.63221e6 | |||||||||||||||||||||||||||||||||
| 1.2 | −107.696 | −729.000 | 3406.54 | −59568.7 | 78510.7 | −147751. | 515378. | 531441. | 6.41534e6 | ||||||||||||||||||||||||||||||||||
| 1.3 | −26.1214 | −729.000 | −7509.67 | 30211.0 | 19042.5 | −259800. | 410149. | 531441. | −789152. | ||||||||||||||||||||||||||||||||||
| 1.4 | 95.5503 | −729.000 | 937.857 | 10418.4 | −69656.2 | 284641. | −693135. | 531441. | 995484. | ||||||||||||||||||||||||||||||||||
| 1.5 | 157.736 | −729.000 | 16688.8 | −3399.13 | −114990. | −413908. | 1.34025e6 | 531441. | −536166. | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(1\) |
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 | 33.14.a.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 99.14.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.14.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 99.14.a.d | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 53T_{2}^{4} - 36614T_{2}^{3} - 1456784T_{2}^{2} + 267355808T_{2} + 7312606720 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{5} + 53 T^{4} + \cdots + 7312606720 \)
$3$
\( (T + 729)^{5} \)
$5$
\( T^{5} - 10318 T^{4} + \cdots - 20\!\cdots\!00 \)
$7$
\( T^{5} + 667804 T^{4} + \cdots - 59\!\cdots\!44 \)
$11$
\( (T + 1771561)^{5} \)
$13$
\( T^{5} + 31377922 T^{4} + \cdots - 30\!\cdots\!56 \)
$17$
\( T^{5} - 360159766 T^{4} + \cdots + 17\!\cdots\!00 \)
$19$
\( T^{5} - 277365792 T^{4} + \cdots - 17\!\cdots\!88 \)
$23$
\( T^{5} + 999029152 T^{4} + \cdots + 14\!\cdots\!28 \)
$29$
\( T^{5} + 2221037718 T^{4} + \cdots - 12\!\cdots\!96 \)
$31$
\( T^{5} + 10330582000 T^{4} + \cdots + 21\!\cdots\!08 \)
$37$
\( T^{5} + 1590413370 T^{4} + \cdots + 17\!\cdots\!48 \)
$41$
\( T^{5} - 37939643902 T^{4} + \cdots - 15\!\cdots\!16 \)
$43$
\( T^{5} + 54578532624 T^{4} + \cdots + 42\!\cdots\!24 \)
$47$
\( T^{5} + 220524883048 T^{4} + \cdots + 71\!\cdots\!00 \)
$53$
\( T^{5} + 302789973354 T^{4} + \cdots - 93\!\cdots\!24 \)
$59$
\( T^{5} - 512143278932 T^{4} + \cdots + 69\!\cdots\!60 \)
$61$
\( T^{5} - 695442645846 T^{4} + \cdots - 16\!\cdots\!20 \)
$67$
\( T^{5} - 95503857604 T^{4} + \cdots - 19\!\cdots\!00 \)
$71$
\( T^{5} - 931692017328 T^{4} + \cdots - 36\!\cdots\!68 \)
$73$
\( T^{5} - 2888174540186 T^{4} + \cdots + 48\!\cdots\!08 \)
$79$
\( T^{5} - 177146521324 T^{4} + \cdots - 73\!\cdots\!64 \)
$83$
\( T^{5} + 1227400754244 T^{4} + \cdots + 10\!\cdots\!28 \)
$89$
\( T^{5} - 4944150911010 T^{4} + \cdots - 67\!\cdots\!36 \)
$97$
\( T^{5} - 3048798953578 T^{4} + \cdots - 71\!\cdots\!00 \)
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