Newspace parameters
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 91.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.36917381052\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 31x^{3} - 15x^{2} + 151x + 165 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 31x^{3} - 15x^{2} + 151x + 165 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 22\nu^{2} + 31\nu + 53 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 28\nu^{2} + 19\nu + 104 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 26\nu^{2} + 35\nu + 105 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{4} + 2\beta_{2} + \beta _1 + 26 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{4} + 4\beta_{3} + 2\beta_{2} + 9\beta _1 + 27 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -80\beta_{4} + 24\beta_{3} + 64\beta_{2} + 45\beta _1 + 628 ) / 2 \)
|
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 |
|
−5.03322 | 1.22470 | 17.3333 | 9.50502 | −6.16419 | −7.00000 | −46.9764 | −25.5001 | −47.8408 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.63697 | −7.57514 | −5.32034 | −12.8053 | 12.4003 | −7.00000 | 21.8050 | 30.3828 | 20.9619 | ||||||||||||||||||||||||||||||||||
| 1.3 | 3.21660 | 5.91606 | 2.34651 | 13.3721 | 19.0296 | −7.00000 | −18.1850 | 7.99973 | 43.0126 | ||||||||||||||||||||||||||||||||||
| 1.4 | 5.19660 | 4.37659 | 19.0047 | −11.6183 | 22.7434 | −7.00000 | 57.1868 | −7.84546 | −60.3756 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.25698 | −8.94221 | 19.6359 | 17.5465 | −47.0090 | −7.00000 | 61.1697 | 52.9630 | 92.2419 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 91.4.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 819.4.a.i | 5 | ||
| 4.b | odd | 2 | 1 | 1456.4.a.x | 5 | ||
| 5.b | even | 2 | 1 | 2275.4.a.i | 5 | ||
| 7.b | odd | 2 | 1 | 637.4.a.e | 5 | ||
| 13.b | even | 2 | 1 | 1183.4.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.4.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 637.4.a.e | 5 | 7.b | odd | 2 | 1 | ||
| 819.4.a.i | 5 | 3.b | odd | 2 | 1 | ||
| 1183.4.a.f | 5 | 13.b | even | 2 | 1 | ||
| 1456.4.a.x | 5 | 4.b | odd | 2 | 1 | ||
| 2275.4.a.i | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 7T_{2}^{4} - 22T_{2}^{3} + 206T_{2}^{2} - 84T_{2} - 724 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 7 T^{4} + \cdots - 724 \)
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| $3$ |
\( T^{5} + 5 T^{4} + \cdots - 2148 \)
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| $5$ |
\( T^{5} - 16 T^{4} + \cdots - 331800 \)
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| $7$ |
\( (T + 7)^{5} \)
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| $11$ |
\( T^{5} - 83 T^{4} + \cdots + 918012 \)
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| $13$ |
\( (T + 13)^{5} \)
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| $17$ |
\( T^{5} + \cdots + 1344572016 \)
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| $19$ |
\( T^{5} - 170 T^{4} + \cdots + 424515950 \)
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| $23$ |
\( T^{5} + \cdots - 3281522805 \)
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| $29$ |
\( T^{5} - 376 T^{4} + \cdots + 517604540 \)
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| $31$ |
\( T^{5} + 7 T^{4} + \cdots + 482544063 \)
|
| $37$ |
\( T^{5} + \cdots + 55043010540 \)
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| $41$ |
\( T^{5} + \cdots + 2379566705040 \)
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| $43$ |
\( T^{5} + \cdots + 1735911107884 \)
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| $47$ |
\( T^{5} + \cdots + 938562407727 \)
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| $53$ |
\( T^{5} + \cdots - 6948120773700 \)
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| $59$ |
\( T^{5} + \cdots - 253582298400 \)
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| $61$ |
\( T^{5} + \cdots + 2593495819760 \)
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| $67$ |
\( T^{5} + \cdots - 2449811658480 \)
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| $71$ |
\( T^{5} + \cdots + 1516354514304 \)
|
| $73$ |
\( T^{5} + \cdots - 2743683978567 \)
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| $79$ |
\( T^{5} + \cdots + 145291873107345 \)
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| $83$ |
\( T^{5} + \cdots + 619165313806284 \)
|
| $89$ |
\( T^{5} + \cdots - 431632636884750 \)
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| $97$ |
\( T^{5} + \cdots + 10\!\cdots\!35 \)
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