Newspace parameters
| Level: | \( N \) | \(=\) | \( 113 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 113.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.66721583065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} - 96 x^{14} + 367 x^{13} + 3634 x^{12} - 12955 x^{11} - 69186 x^{10} + 219721 x^{9} + \cdots + 49152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 4 x^{15} - 96 x^{14} + 367 x^{13} + 3634 x^{12} - 12955 x^{11} - 69186 x^{10} + 219721 x^{9} + \cdots + 49152 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 51953074573 \nu^{15} + 119310095463 \nu^{14} + 4771062410967 \nu^{13} + \cdots + 22\!\cdots\!12 ) / 18\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 642151953 \nu^{15} - 1981579333 \nu^{14} - 63320578707 \nu^{13} + 183232059550 \nu^{12} + \cdots + 132744653948928 ) / 20277691402240 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 804251442367 \nu^{15} + 3443740561917 \nu^{14} + 76589250485613 \nu^{13} + \cdots - 28\!\cdots\!32 ) / 18\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 74669730607 \nu^{15} + 296133978765 \nu^{14} + 7174472982077 \nu^{13} + \cdots - 46\!\cdots\!52 ) / 12\!\cdots\!92 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 209001310021 \nu^{15} - 887253323551 \nu^{14} - 19883894579599 \nu^{13} + \cdots + 39\!\cdots\!36 ) / 30\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1272129693859 \nu^{15} + 5352611729049 \nu^{14} + 119390423746761 \nu^{13} + \cdots - 44\!\cdots\!84 ) / 18\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 322343958533 \nu^{15} - 1308842877843 \nu^{14} - 31019078927667 \nu^{13} + \cdots + 17\!\cdots\!88 ) / 46\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 436280959281 \nu^{15} - 1720437534291 \nu^{14} - 42219486181699 \nu^{13} + \cdots + 23\!\cdots\!96 ) / 61\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 443864466531 \nu^{15} - 1595003201561 \nu^{14} - 43179542036649 \nu^{13} + \cdots + 85\!\cdots\!36 ) / 61\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 89421927247 \nu^{15} - 356978773789 \nu^{14} - 8589825208109 \nu^{13} + \cdots + 35\!\cdots\!20 ) / 12\!\cdots\!92 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 1343421813149 \nu^{15} + 4994302631559 \nu^{14} + 130082588059671 \nu^{13} + \cdots + 27\!\cdots\!76 ) / 18\!\cdots\!80 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 1566194323603 \nu^{15} + 7149220576953 \nu^{14} + 147992037499017 \nu^{13} + \cdots - 11\!\cdots\!88 ) / 18\!\cdots\!80 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 924930548063 \nu^{15} - 3759734212013 \nu^{14} - 88714162021277 \nu^{13} + \cdots + 27\!\cdots\!08 ) / 61\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{12} + \beta_{9} - \beta_{3} + 21\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{14} + \beta_{13} - 2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots + 277 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7 \beta_{15} - \beta_{13} - 35 \beta_{12} - 6 \beta_{11} - 7 \beta_{10} + 37 \beta_{9} - 6 \beta_{7} + \cdots + 41 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{15} + 48 \beta_{14} + 22 \beta_{13} - 27 \beta_{12} - 88 \beta_{11} - 48 \beta_{10} + \cdots + 6513 \)
|
| \(\nu^{7}\) | \(=\) |
\( 452 \beta_{15} + 55 \beta_{14} - 17 \beta_{13} - 1053 \beta_{12} - 362 \beta_{11} - 395 \beta_{10} + \cdots + 730 \)
|
| \(\nu^{8}\) | \(=\) |
\( 333 \beta_{15} + 1885 \beta_{14} + 330 \beta_{13} - 1482 \beta_{12} - 3220 \beta_{11} - 1690 \beta_{10} + \cdots + 160314 \)
|
| \(\nu^{9}\) | \(=\) |
\( 19441 \beta_{15} + 3873 \beta_{14} + 214 \beta_{13} - 30452 \beta_{12} - 15192 \beta_{11} - 15500 \beta_{10} + \cdots + 18073 \)
|
| \(\nu^{10}\) | \(=\) |
\( 21773 \beta_{15} + 67745 \beta_{14} + 2966 \beta_{13} - 58333 \beta_{12} - 109908 \beta_{11} + \cdots + 4050264 \)
|
| \(\nu^{11}\) | \(=\) |
\( 710161 \beta_{15} + 181718 \beta_{14} + 26127 \beta_{13} - 870422 \beta_{12} - 553382 \beta_{11} + \cdots + 649585 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1037876 \beta_{15} + 2295699 \beta_{14} - 31721 \beta_{13} - 2031531 \beta_{12} - 3600078 \beta_{11} + \cdots + 104164647 \)
|
| \(\nu^{13}\) | \(=\) |
\( 23836841 \beta_{15} + 7154497 \beta_{14} + 1217430 \beta_{13} - 24823693 \beta_{12} - 18733696 \beta_{11} + \cdots + 26051299 \)
|
| \(\nu^{14}\) | \(=\) |
\( 42225721 \beta_{15} + 74614632 \beta_{14} - 2769707 \beta_{13} - 66721371 \beta_{12} - 114606470 \beta_{11} + \cdots + 2715847332 \)
|
| \(\nu^{15}\) | \(=\) |
\( 761192422 \beta_{15} + 256020120 \beta_{14} + 44867310 \beta_{13} - 708676436 \beta_{12} + \cdots + 1017460510 \)
|
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 |
|
−5.11584 | −1.77429 | 18.1718 | 12.8336 | 9.07699 | 2.62499 | −52.0375 | −23.8519 | −65.6545 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.10236 | −2.78568 | 18.0341 | −21.6644 | 14.2135 | 5.41338 | −51.1975 | −19.2400 | 110.540 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.58271 | 9.22947 | 13.0012 | 8.21844 | −42.2960 | 35.4416 | −22.9192 | 58.1831 | −37.6627 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.83279 | 6.02006 | 0.0246961 | −19.4184 | −17.0536 | 17.0039 | 22.5924 | 9.24110 | 55.0081 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.53150 | 4.14870 | −1.59149 | 19.0369 | −10.5025 | −7.79017 | 24.2809 | −9.78827 | −48.1920 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.24067 | −4.67519 | −2.97940 | −6.97810 | 10.4755 | −24.8897 | 24.6012 | −5.14263 | 15.6356 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.217566 | 10.1097 | −7.95267 | 4.73202 | −2.19952 | −1.33603 | 3.47075 | 75.2058 | −1.02953 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.103636 | −8.04380 | −7.98926 | −17.8495 | 0.833628 | 15.4098 | 1.65706 | 37.7027 | 1.84985 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.289902 | 0.0487587 | −7.91596 | 4.90092 | 0.0141353 | 29.5520 | −4.61407 | −26.9976 | 1.42079 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.724432 | −7.56189 | −7.47520 | 6.50611 | −5.47808 | −0.691224 | −11.2107 | 30.1822 | 4.71323 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.78221 | 5.83924 | −0.259288 | 15.7899 | 16.2460 | 9.09828 | −22.9791 | 7.09672 | 43.9309 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 3.61878 | 6.75045 | 5.09559 | −5.11403 | 24.4284 | 17.6836 | −10.5104 | 18.5686 | −18.5066 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 4.30089 | −8.18200 | 10.4977 | 10.0046 | −35.1899 | 22.9737 | 10.7423 | 39.9451 | 43.0288 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 4.74179 | 8.22348 | 14.4846 | −6.43602 | 38.9940 | −24.8490 | 30.7486 | 40.6257 | −30.5183 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 4.88274 | 0.158994 | 15.8412 | 12.5273 | 0.776328 | −11.3028 | 38.2865 | −26.9747 | 61.1674 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 5.38631 | 0.493997 | 21.0124 | −11.0893 | 2.66082 | 33.6576 | 70.0888 | −26.7560 | −59.7307 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(113\) | \( +1 \) |
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 | 113.4.a.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | 1017.4.a.e | 16 | ||
| 4.b | odd | 2 | 1 | 1808.4.a.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 113.4.a.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 1017.4.a.e | 16 | 3.b | odd | 2 | 1 | ||
| 1808.4.a.g | 16 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 4 T_{2}^{15} - 96 T_{2}^{14} + 367 T_{2}^{13} + 3634 T_{2}^{12} - 12955 T_{2}^{11} + \cdots + 49152 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(113))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 4 T^{15} + \cdots + 49152 \)
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| $3$ |
\( T^{16} - 18 T^{15} + \cdots + 33268784 \)
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| $5$ |
\( T^{16} + \cdots - 11\!\cdots\!52 \)
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| $7$ |
\( T^{16} + \cdots + 24\!\cdots\!80 \)
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| $11$ |
\( T^{16} + \cdots + 44\!\cdots\!80 \)
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| $13$ |
\( T^{16} + \cdots + 66\!\cdots\!72 \)
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| $17$ |
\( T^{16} + \cdots - 29\!\cdots\!04 \)
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| $19$ |
\( T^{16} + \cdots - 98\!\cdots\!84 \)
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| $23$ |
\( T^{16} + \cdots + 72\!\cdots\!64 \)
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| $29$ |
\( T^{16} + \cdots + 15\!\cdots\!16 \)
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| $31$ |
\( T^{16} + \cdots + 34\!\cdots\!88 \)
|
| $37$ |
\( T^{16} + \cdots + 85\!\cdots\!04 \)
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| $41$ |
\( T^{16} + \cdots - 16\!\cdots\!48 \)
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| $43$ |
\( T^{16} + \cdots + 21\!\cdots\!24 \)
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| $47$ |
\( T^{16} + \cdots - 68\!\cdots\!96 \)
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| $53$ |
\( T^{16} + \cdots + 68\!\cdots\!00 \)
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| $59$ |
\( T^{16} + \cdots + 25\!\cdots\!52 \)
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| $61$ |
\( T^{16} + \cdots - 11\!\cdots\!84 \)
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| $67$ |
\( T^{16} + \cdots + 16\!\cdots\!72 \)
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| $71$ |
\( T^{16} + \cdots + 40\!\cdots\!08 \)
|
| $73$ |
\( T^{16} + \cdots - 16\!\cdots\!96 \)
|
| $79$ |
\( T^{16} + \cdots + 53\!\cdots\!00 \)
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| $83$ |
\( T^{16} + \cdots - 24\!\cdots\!68 \)
|
| $89$ |
\( T^{16} + \cdots + 39\!\cdots\!16 \)
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| $97$ |
\( T^{16} + \cdots - 36\!\cdots\!16 \)
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