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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 33742x^{4} + 162324x^{3} + 293102344x^{2} - 383417792x - 721869968512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 33742x^{4} + 162324x^{3} + 293102344x^{2} - 383417792x - 721869968512 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 6\nu - 11249 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -107\nu^{5} - 3003\nu^{4} + 2807060\nu^{3} + 52913996\nu^{2} - 11923560576\nu - 238899045056 ) / 5623296 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -407\nu^{5} + 27993\nu^{4} + 11557604\nu^{3} - 843453316\nu^{2} - 53394471552\nu + 3646483043392 ) / 5623296 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -253\nu^{5} + 20115\nu^{4} + 7111180\nu^{3} - 601732268\nu^{2} - 31584491136\nu + 2633153552576 ) / 2409984 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 6\beta _1 + 11249 \)
|
| \(\nu^{3}\) | \(=\) |
\( -18\beta_{5} + 29\beta_{4} - 11\beta_{3} - 41\beta_{2} + 16381\beta _1 - 67036 \)
|
| \(\nu^{4}\) | \(=\) |
\( 402\beta_{5} - 505\beta_{4} - 297\beta_{3} + 27421\beta_{2} - 320885\beta _1 + 184087444 \)
|
| \(\nu^{5}\) | \(=\) |
\( -483498\beta_{5} + 774965\beta_{4} - 332795\beta_{3} - 1350661\beta_{2} + 324346009\beta _1 - 3594936192 \)
|
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 |
|
−139.593 | −729.000 | 11294.2 | −15834.3 | 101763. | 276059. | −433047. | 531441. | 2.21035e6 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −93.5104 | −729.000 | 552.193 | 14973.8 | 68169.1 | −516871. | 714401. | 531441. | −1.40020e6 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −62.1750 | −729.000 | −4326.27 | 60284.7 | 45325.6 | 589111. | 778323. | 531441. | −3.74820e6 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 74.8479 | −729.000 | −2589.80 | 32694.1 | −54564.1 | −118538. | −806994. | 531441. | 2.44709e6 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 76.8139 | −729.000 | −2291.63 | −53039.8 | −55997.3 | −524972. | −805288. | 531441. | −4.07420e6 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 154.617 | −729.000 | 15714.3 | −60010.5 | −112716. | 579363. | 1.16307e6 | 531441. | −9.27862e6 | |||||||||||||||||||||||||||||||||||||
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.e | ✓ | 6 |
| 3.b | odd | 2 | 1 | 99.14.a.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.14.a.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 99.14.a.f | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 11T_{2}^{5} - 33692T_{2}^{4} + 107492T_{2}^{3} + 293266640T_{2}^{2} - 789859840T_{2} - 721463635968 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - 11 T^{5} + \cdots - 721463635968 \)
$3$
\( (T + 729)^{6} \)
$5$
\( T^{6} + 20932 T^{5} + \cdots - 14\!\cdots\!00 \)
$7$
\( T^{6} - 284152 T^{5} + \cdots - 30\!\cdots\!92 \)
$11$
\( (T - 1771561)^{6} \)
$13$
\( T^{6} + 10016732 T^{5} + \cdots - 60\!\cdots\!56 \)
$17$
\( T^{6} - 67290212 T^{5} + \cdots - 31\!\cdots\!52 \)
$19$
\( T^{6} + 86262088 T^{5} + \cdots - 19\!\cdots\!00 \)
$23$
\( T^{6} - 1208367520 T^{5} + \cdots + 89\!\cdots\!16 \)
$29$
\( T^{6} - 2525702652 T^{5} + \cdots + 12\!\cdots\!20 \)
$31$
\( T^{6} - 9935999560 T^{5} + \cdots + 98\!\cdots\!32 \)
$37$
\( T^{6} + 5010959596 T^{5} + \cdots + 63\!\cdots\!08 \)
$41$
\( T^{6} + 44262563620 T^{5} + \cdots + 26\!\cdots\!16 \)
$43$
\( T^{6} - 41074648472 T^{5} + \cdots - 14\!\cdots\!00 \)
$47$
\( T^{6} - 149258240752 T^{5} + \cdots - 17\!\cdots\!76 \)
$53$
\( T^{6} - 357253697580 T^{5} + \cdots + 54\!\cdots\!12 \)
$59$
\( T^{6} - 14313635176 T^{5} + \cdots + 75\!\cdots\!20 \)
$61$
\( T^{6} + 496086757804 T^{5} + \cdots + 71\!\cdots\!88 \)
$67$
\( T^{6} + 771785141856 T^{5} + \cdots + 78\!\cdots\!12 \)
$71$
\( T^{6} + 775009869936 T^{5} + \cdots - 11\!\cdots\!60 \)
$73$
\( T^{6} + 482995319444 T^{5} + \cdots + 36\!\cdots\!52 \)
$79$
\( T^{6} - 3012132622488 T^{5} + \cdots - 11\!\cdots\!00 \)
$83$
\( T^{6} + 6855454677456 T^{5} + \cdots - 58\!\cdots\!08 \)
$89$
\( T^{6} + 10551236825748 T^{5} + \cdots - 30\!\cdots\!40 \)
$97$
\( T^{6} - 20589100190868 T^{5} + \cdots + 29\!\cdots\!24 \)
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