Newspace parameters
| Level: | \( N \) | \(=\) | \( 1040 = 2^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1040.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(61.3619864060\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 97x^{4} + 192x^{3} + 2324x^{2} - 3158x - 13948 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 520) |
| 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
\( x^{6} - 2x^{5} - 97x^{4} + 192x^{3} + 2324x^{2} - 3158x - 13948 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 40\nu^{5} + 14\nu^{4} - 2831\nu^{3} - 497\nu^{2} + 30318\nu + 33328 ) / 10161 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -12\nu^{5} - 230\nu^{4} + 1188\nu^{3} + 20584\nu^{2} - 40030\nu - 315280 ) / 3387 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 76\nu^{5} + 704\nu^{4} - 6395\nu^{3} - 41927\nu^{2} + 170730\nu + 298381 ) / 10161 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -14\nu^{5} + 108\nu^{4} + 1386\nu^{3} - 8350\nu^{2} - 29014\nu + 94687 ) / 1129 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 179\nu^{5} + 232\nu^{4} - 15463\nu^{3} - 15010\nu^{2} + 240162\nu + 276494 ) / 10161 \)
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| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3\beta _1 + 6 ) / 16 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} + 7\beta_{3} + 9\beta_{2} - 11\beta _1 + 530 ) / 16 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -126\beta_{5} - 31\beta_{4} + 39\beta_{3} - 39\beta_{2} + 357\beta _1 + 82 ) / 16 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 266\beta_{5} + 181\beta_{4} + 555\beta_{3} + 653\beta_{2} - 1087\beta _1 + 25634 ) / 16 \)
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| \(\nu^{5}\) | \(=\) |
\( ( -7470\beta_{5} - 1487\beta_{4} + 1895\beta_{3} - 2119\beta_{2} + 27301\beta _1 - 14462 ) / 16 \)
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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 |
|
0 | −8.95492 | 0 | 5.00000 | 0 | −19.2741 | 0 | 53.1905 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.85320 | 0 | 5.00000 | 0 | 12.3750 | 0 | 19.9663 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.15206 | 0 | 5.00000 | 0 | −31.6241 | 0 | −25.6728 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.733633 | 0 | 5.00000 | 0 | 9.91107 | 0 | −26.4618 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 6.82343 | 0 | 5.00000 | 0 | 22.9556 | 0 | 19.5592 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 9.40311 | 0 | 5.00000 | 0 | −32.3435 | 0 | 61.4186 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 1040.4.a.ba | 6 | |
| 4.b | odd | 2 | 1 | 520.4.a.k | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.4.a.k | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 1040.4.a.ba | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1040))\):
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\( T_{3}^{6} - 132T_{3}^{4} - 36T_{3}^{3} + 4056T_{3}^{2} + 1632T_{3} - 3328 \)
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\( T_{7}^{6} + 38T_{7}^{5} - 876T_{7}^{4} - 32360T_{7}^{3} + 304416T_{7}^{2} + 6152448T_{7} - 55504896 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 132 T^{4} + \cdots - 3328 \)
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| $5$ |
\( (T - 5)^{6} \)
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| $7$ |
\( T^{6} + 38 T^{5} + \cdots - 55504896 \)
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| $11$ |
\( T^{6} + \cdots - 2433014016 \)
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| $13$ |
\( (T - 13)^{6} \)
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| $17$ |
\( T^{6} + \cdots + 13443260352 \)
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| $19$ |
\( T^{6} + \cdots - 180467368448 \)
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| $23$ |
\( T^{6} + \cdots + 5449660416 \)
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| $29$ |
\( T^{6} + \cdots + 1474831403072 \)
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| $31$ |
\( T^{6} + \cdots + 20293855694592 \)
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| $37$ |
\( T^{6} + \cdots + 29769067393600 \)
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| $41$ |
\( T^{6} + \cdots + 3484024714816 \)
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| $43$ |
\( T^{6} + \cdots - 145712050154752 \)
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| $47$ |
\( T^{6} + \cdots - 56068283359232 \)
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| $53$ |
\( T^{6} + \cdots - 129657317252416 \)
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| $59$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
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| $61$ |
\( T^{6} + \cdots + 60\!\cdots\!48 \)
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| $67$ |
\( T^{6} + \cdots + 15159561301504 \)
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| $71$ |
\( T^{6} + \cdots + 17\!\cdots\!68 \)
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| $73$ |
\( T^{6} + \cdots + 14\!\cdots\!16 \)
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| $79$ |
\( T^{6} + \cdots - 37\!\cdots\!16 \)
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| $83$ |
\( T^{6} + \cdots - 98\!\cdots\!76 \)
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| $89$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 10\!\cdots\!72 \)
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