Newspace parameters
| Level: | \( N \) | \(=\) | \( 2401 = 7^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2401.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.1720815253\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
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| Defining polynomial: |
\( x^{9} - 3x^{8} - 7x^{7} + 20x^{6} + 19x^{5} - 39x^{4} - 22x^{3} + 17x^{2} + 2x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 7x^{7} + 20x^{6} + 19x^{5} - 39x^{4} - 22x^{3} + 17x^{2} + 2x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 6\nu^{7} + 4\nu^{6} + 29\nu^{5} - 33\nu^{4} - 38\nu^{3} + 36\nu^{2} - 5 ) / 7 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{8} - 5\nu^{7} - 13\nu^{6} + 23\nu^{5} + 39\nu^{4} - 27\nu^{3} - 47\nu^{2} + 7\nu + 4 ) / 7 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} + 6\nu^{7} - 4\nu^{6} - 29\nu^{5} + 33\nu^{4} + 45\nu^{3} - 43\nu^{2} - 28\nu + 5 ) / 7 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{8} - 5\nu^{7} - 20\nu^{6} + 44\nu^{5} + 67\nu^{4} - 111\nu^{3} - 82\nu^{2} + 70\nu + 4 ) / 7 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{8} - 10\nu^{7} - 33\nu^{6} + 67\nu^{5} + 99\nu^{4} - 117\nu^{3} - 115\nu^{2} + 21\nu + 15 ) / 7 \)
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| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{8} + 10\nu^{7} + 33\nu^{6} - 60\nu^{5} - 120\nu^{4} + 96\nu^{3} + 178\nu^{2} - 22 ) / 7 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 5\nu^{8} - 16\nu^{7} - 29\nu^{6} + 96\nu^{5} + 66\nu^{4} - 162\nu^{3} - 65\nu^{2} + 42\nu - 11 ) / 7 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - \beta_{6} + \beta_{4} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{6} + 2\beta_{4} + \beta_{2} + 5\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 5\beta_{8} - 6\beta_{6} + \beta_{5} + 8\beta_{4} + \beta_{3} + 3\beta_{2} + 9\beta _1 + 16 \)
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| \(\nu^{5}\) | \(=\) |
\( 9\beta_{8} + \beta_{7} - 11\beta_{6} + 3\beta_{5} + 21\beta_{4} + 3\beta_{3} + 12\beta_{2} + 30\beta _1 + 31 \)
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| \(\nu^{6}\) | \(=\) |
\( 30\beta_{8} + 3\beta_{7} - 40\beta_{6} + 12\beta_{5} + 66\beta_{4} + 14\beta_{3} + 36\beta_{2} + 70\beta _1 + 106 \)
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| \(\nu^{7}\) | \(=\) |
\( 70\beta_{8} + 14\beta_{7} - 95\beta_{6} + 36\beta_{5} + 186\beta_{4} + 43\beta_{3} + 114\beta_{2} + 206\beta _1 + 261 \)
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| \(\nu^{8}\) | \(=\) |
\( 206 \beta_{8} + 43 \beta_{7} - 291 \beta_{6} + 114 \beta_{5} + 547 \beta_{4} + 148 \beta_{3} + \cdots + 782 \)
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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 |
|
−2.68304 | −1.53393 | 5.19873 | 3.13701 | 4.11559 | 0 | −8.58233 | −0.647074 | −8.41673 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.56249 | −2.72442 | 4.56636 | −0.891804 | 6.98131 | 0 | −6.57626 | 4.42249 | 2.28524 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.05695 | 1.13615 | 2.23103 | −0.313688 | −2.33701 | 0 | −0.475218 | −1.70915 | 0.645241 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.28070 | 1.47994 | −0.359817 | 0.730097 | −1.89535 | 0 | 3.02221 | −0.809792 | −0.935033 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.769458 | −2.86008 | −1.40793 | 3.92353 | 2.20071 | 0 | 2.62226 | 5.18004 | −3.01899 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.512316 | −2.00787 | −1.73753 | −2.80551 | 1.02866 | 0 | 1.91480 | 1.03153 | 1.43731 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.826405 | 2.52586 | −1.31706 | 0.884122 | 2.08738 | 0 | −2.74123 | 3.37997 | 0.730642 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.19536 | 1.29481 | −0.571112 | 0.778425 | 1.54777 | 0 | −3.07341 | −1.32346 | 0.930499 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.84319 | 0.689531 | 1.39734 | −1.44217 | 1.27093 | 0 | −1.11082 | −2.52455 | −2.65819 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
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 | 2401.2.a.e | ✓ | 9 |
| 7.b | odd | 2 | 1 | 2401.2.a.f | yes | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2401.2.a.e | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 2401.2.a.f | yes | 9 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2401))\):
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\( T_{2}^{9} + 6T_{2}^{8} + 5T_{2}^{7} - 29T_{2}^{6} - 50T_{2}^{5} + 27T_{2}^{4} + 83T_{2}^{3} + 12T_{2}^{2} - 35T_{2} - 13 \)
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\( T_{3}^{9} + 2T_{3}^{8} - 15T_{3}^{7} - 22T_{3}^{6} + 84T_{3}^{5} + 63T_{3}^{4} - 217T_{3}^{3} - 14T_{3}^{2} + 210T_{3} - 91 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 6 T^{8} + \cdots - 13 \)
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| $3$ |
\( T^{9} + 2 T^{8} + \cdots - 91 \)
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| $5$ |
\( T^{9} - 4 T^{8} + \cdots - 7 \)
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| $7$ |
\( T^{9} \)
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| $11$ |
\( T^{9} + 10 T^{8} + \cdots - 601 \)
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| $13$ |
\( T^{9} - 70 T^{7} + \cdots + 49 \)
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| $17$ |
\( T^{9} + 9 T^{8} + \cdots - 3227 \)
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| $19$ |
\( T^{9} + 10 T^{8} + \cdots + 1421 \)
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| $23$ |
\( T^{9} + 13 T^{8} + \cdots - 49937 \)
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| $29$ |
\( T^{9} + 31 T^{8} + \cdots + 41117 \)
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| $31$ |
\( T^{9} + 16 T^{8} + \cdots - 9562889 \)
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| $37$ |
\( T^{9} + 6 T^{8} + \cdots + 9511853 \)
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| $41$ |
\( T^{9} - 21 T^{8} + \cdots - 887047 \)
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| $43$ |
\( T^{9} - 4 T^{8} + \cdots - 1000819 \)
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| $47$ |
\( T^{9} - 11 T^{8} + \cdots - 1443841 \)
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| $53$ |
\( T^{9} + 3 T^{8} + \cdots - 371293 \)
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| $59$ |
\( T^{9} + 23 T^{8} + \cdots + 579971 \)
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| $61$ |
\( T^{9} - 39 T^{8} + \cdots - 3335927 \)
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| $67$ |
\( T^{9} + 3 T^{8} + \cdots - 221339 \)
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| $71$ |
\( T^{9} + 24 T^{8} + \cdots - 632227 \)
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| $73$ |
\( T^{9} - 26 T^{8} + \cdots + 10337537 \)
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| $79$ |
\( T^{9} + 34 T^{8} + \cdots + 6243049 \)
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| $83$ |
\( T^{9} - 35 T^{8} + \cdots - 5671211 \)
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| $89$ |
\( T^{9} + 31 T^{8} + \cdots + 65599037 \)
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| $97$ |
\( T^{9} + 49 T^{8} + \cdots + 356005531 \)
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