Newspace parameters
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 261.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.8601769712\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 172x^{5} - 59x^{4} + 8297x^{3} + 9726x^{2} - 69644x + 47968 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 87) |
| 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
\( x^{7} - x^{6} - 172x^{5} - 59x^{4} + 8297x^{3} + 9726x^{2} - 69644x + 47968 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 49 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -37\nu^{6} - 215\nu^{5} + 5940\nu^{4} + 29115\nu^{3} - 213765\nu^{2} - 850354\nu + 609460 ) / 38492 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} + 1838\nu^{5} - 14526\nu^{4} - 215107\nu^{3} + 1177554\nu^{2} + 6549648\nu - 11598056 ) / 38492 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 115\nu^{6} - 112\nu^{5} - 17942\nu^{4} - 21311\nu^{3} + 777540\nu^{2} + 2103584\nu - 3961392 ) / 38492 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -271\nu^{6} + 766\nu^{5} + 41946\nu^{4} - 32789\nu^{3} - 1751122\nu^{2} - 1915604\nu + 6585104 ) / 76984 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} - 3\beta_{5} - 2\beta_{3} + 4\beta_{2} + 78\beta _1 + 90 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{6} + 5\beta_{5} - 3\beta_{4} - 14\beta_{3} + 95\beta_{2} + 317\beta _1 + 3803 \)
|
| \(\nu^{5}\) | \(=\) |
\( -172\beta_{6} - 312\beta_{5} - 2\beta_{4} - 340\beta_{3} + 563\beta_{2} + 6726\beta _1 + 14971 \)
|
| \(\nu^{6}\) | \(=\) |
\( 710\beta_{6} + 255\beta_{5} - 470\beta_{4} - 2886\beta_{3} + 9350\beta_{2} + 38648\beta _1 + 327740 \)
|
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 |
|
−10.3875 | 0 | 75.8996 | 35.5449 | 0 | 145.852 | −456.006 | 0 | −369.222 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.51585 | 0 | 58.5514 | −75.4366 | 0 | −140.849 | −252.660 | 0 | 717.844 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.12395 | 0 | 18.7506 | −15.0033 | 0 | 149.361 | 94.3878 | 0 | 106.883 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.12758 | 0 | −30.7286 | −102.387 | 0 | 73.6139 | 70.7315 | 0 | 115.450 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.164259 | 0 | −31.9730 | 46.4998 | 0 | 0.553329 | 10.5082 | 0 | −7.63802 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.89724 | 0 | 15.5719 | 103.837 | 0 | −200.475 | −113.309 | 0 | 716.189 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.42188 | 0 | 38.9280 | −57.0545 | 0 | 139.944 | 58.3467 | 0 | −480.506 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 261.6.a.d | 7 | |
| 3.b | odd | 2 | 1 | 87.6.a.c | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.6.a.c | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 261.6.a.d | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 13T_{2}^{6} - 100T_{2}^{5} - 1559T_{2}^{4} + 1345T_{2}^{3} + 44764T_{2}^{2} + 53432T_{2} + 7576 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 13 T^{6} + \cdots + 7576 \)
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| $3$ |
\( T^{7} \)
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| $5$ |
\( T^{7} + \cdots - 1134714673184 \)
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| $7$ |
\( T^{7} + \cdots - 3506414197856 \)
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| $11$ |
\( T^{7} + \cdots + 36\!\cdots\!28 \)
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| $13$ |
\( T^{7} + \cdots + 17\!\cdots\!12 \)
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| $17$ |
\( T^{7} + \cdots + 25\!\cdots\!58 \)
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| $19$ |
\( T^{7} + \cdots - 16\!\cdots\!52 \)
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| $23$ |
\( T^{7} + \cdots - 28\!\cdots\!12 \)
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| $29$ |
\( (T - 841)^{7} \)
|
| $31$ |
\( T^{7} + \cdots + 16\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots - 10\!\cdots\!84 \)
|
| $41$ |
\( T^{7} + \cdots - 41\!\cdots\!88 \)
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| $43$ |
\( T^{7} + \cdots - 12\!\cdots\!68 \)
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| $47$ |
\( T^{7} + \cdots - 83\!\cdots\!96 \)
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| $53$ |
\( T^{7} + \cdots - 25\!\cdots\!00 \)
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| $59$ |
\( T^{7} + \cdots - 12\!\cdots\!84 \)
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| $61$ |
\( T^{7} + \cdots - 17\!\cdots\!28 \)
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| $67$ |
\( T^{7} + \cdots + 20\!\cdots\!04 \)
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| $71$ |
\( T^{7} + \cdots - 38\!\cdots\!92 \)
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| $73$ |
\( T^{7} + \cdots + 71\!\cdots\!72 \)
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| $79$ |
\( T^{7} + \cdots - 45\!\cdots\!04 \)
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| $83$ |
\( T^{7} + \cdots + 19\!\cdots\!84 \)
|
| $89$ |
\( T^{7} + \cdots - 43\!\cdots\!92 \)
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| $97$ |
\( T^{7} + \cdots + 16\!\cdots\!24 \)
|
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