Newspace parameters
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.65113077458\) |
Analytic rank: | \(0\) |
Dimension: | \(7\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
Defining polynomial: |
\( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{5} \) |
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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 54 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 6\nu^{5} - 114\nu^{4} + 618\nu^{3} + 2727\nu^{2} - 10724\nu - 11462 ) / 240 \)
|
\(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} - 119\nu^{3} + 547\nu^{2} + 2890\nu - 11242 ) / 96 \)
|
\(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 10\nu^{5} - 218\nu^{4} - 1094\nu^{3} + 13231\nu^{2} + 21356\nu - 185766 ) / 960 \)
|
\(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 226\nu^{4} - 238\nu^{3} + 14279\nu^{2} + 5916\nu - 208054 ) / 960 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 54 \)
|
\(\nu^{3}\) | \(=\) |
\( -2\beta_{6} + 6\beta_{5} - 8\beta_{4} - \beta_{3} + 4\beta_{2} + 71\beta _1 - 41 \)
|
\(\nu^{4}\) | \(=\) |
\( -16\beta_{6} + 8\beta_{5} + 2\beta_{3} + 105\beta_{2} - 95\beta _1 + 3846 \)
|
\(\nu^{5}\) | \(=\) |
\( -318\beta_{6} + 754\beta_{5} - 856\beta_{4} - 109\beta_{3} + 454\beta_{2} + 5631\beta _1 - 3945 \)
|
\(\nu^{6}\) | \(=\) |
\( -2496\beta_{6} + 1728\beta_{5} - 192\beta_{4} + 432\beta_{3} + 9495\beta_{2} - 7471\beta _1 + 304316 \)
|
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 |
|
−8.56883 | 18.3274 | 41.4249 | −84.3249 | −157.045 | 216.816 | −80.7606 | 92.8952 | 722.566 | |||||||||||||||||||||||||||||||||||||||
1.2 | −6.83842 | −24.5656 | 14.7640 | −54.3066 | 167.990 | −37.6697 | 117.867 | 360.469 | 371.372 | ||||||||||||||||||||||||||||||||||||||||
1.3 | −3.90786 | 29.3989 | −16.7287 | 64.0801 | −114.887 | −138.793 | 190.425 | 621.298 | −250.416 | ||||||||||||||||||||||||||||||||||||||||
1.4 | −2.60554 | −13.2844 | −25.2112 | 69.0035 | 34.6131 | 156.573 | 149.066 | −66.5241 | −179.792 | ||||||||||||||||||||||||||||||||||||||||
1.5 | 5.83960 | 15.9679 | 2.10095 | 31.5616 | 93.2461 | 106.304 | −174.599 | 11.9736 | 184.307 | ||||||||||||||||||||||||||||||||||||||||
1.6 | 9.92709 | 15.4219 | 66.5471 | −58.0818 | 153.094 | −210.388 | 342.952 | −5.16616 | −576.583 | ||||||||||||||||||||||||||||||||||||||||
1.7 | 10.1540 | −15.2661 | 71.1029 | 64.0682 | −155.012 | 91.1564 | 397.049 | −9.94539 | 650.545 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(29\) | \(-1\) |
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 | 29.6.a.b | ✓ | 7 |
3.b | odd | 2 | 1 | 261.6.a.e | 7 | ||
4.b | odd | 2 | 1 | 464.6.a.k | 7 | ||
5.b | even | 2 | 1 | 725.6.a.b | 7 | ||
29.b | even | 2 | 1 | 841.6.a.b | 7 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.6.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
261.6.a.e | 7 | 3.b | odd | 2 | 1 | ||
464.6.a.k | 7 | 4.b | odd | 2 | 1 | ||
725.6.a.b | 7 | 5.b | even | 2 | 1 | ||
841.6.a.b | 7 | 29.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 4T_{2}^{6} - 181T_{2}^{5} + 346T_{2}^{4} + 10616T_{2}^{3} + 2416T_{2}^{2} - 186896T_{2} - 351200 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{7} - 4 T^{6} - 181 T^{5} + \cdots - 351200 \)
$3$
\( T^{7} - 26 T^{6} + \cdots + 661023756 \)
$5$
\( T^{7} - 32 T^{6} + \cdots + 2378174390186 \)
$7$
\( T^{7} + \cdots + 361848785235968 \)
$11$
\( T^{7} - 1106 T^{6} + \cdots + 51\!\cdots\!44 \)
$13$
\( T^{7} - 408 T^{6} + \cdots + 10\!\cdots\!34 \)
$17$
\( T^{7} + 874 T^{6} + \cdots - 17\!\cdots\!28 \)
$19$
\( T^{7} - 4288 T^{6} + \cdots + 15\!\cdots\!92 \)
$23$
\( T^{7} + 4532 T^{6} + \cdots - 15\!\cdots\!56 \)
$29$
\( (T - 841)^{7} \)
$31$
\( T^{7} - 7794 T^{6} + \cdots + 64\!\cdots\!48 \)
$37$
\( T^{7} - 5086 T^{6} + \cdots - 16\!\cdots\!56 \)
$41$
\( T^{7} - 19826 T^{6} + \cdots - 56\!\cdots\!72 \)
$43$
\( T^{7} - 19498 T^{6} + \cdots - 58\!\cdots\!60 \)
$47$
\( T^{7} - 14278 T^{6} + \cdots - 14\!\cdots\!36 \)
$53$
\( T^{7} + 58644 T^{6} + \cdots + 84\!\cdots\!94 \)
$59$
\( T^{7} - 12888 T^{6} + \cdots + 15\!\cdots\!00 \)
$61$
\( T^{7} - 102866 T^{6} + \cdots + 45\!\cdots\!80 \)
$67$
\( T^{7} - 102996 T^{6} + \cdots - 11\!\cdots\!60 \)
$71$
\( T^{7} + 51596 T^{6} + \cdots - 20\!\cdots\!52 \)
$73$
\( T^{7} + 17566 T^{6} + \cdots + 43\!\cdots\!96 \)
$79$
\( T^{7} - 212058 T^{6} + \cdots - 28\!\cdots\!00 \)
$83$
\( T^{7} + 122928 T^{6} + \cdots - 97\!\cdots\!52 \)
$89$
\( T^{7} + 66510 T^{6} + \cdots - 79\!\cdots\!20 \)
$97$
\( T^{7} + 118182 T^{6} + \cdots + 62\!\cdots\!52 \)
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