Newspace parameters
| Level: | \( N \) | \(=\) | \( 6897 = 3 \cdot 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6897.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(55.0728222741\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{16} - 3 x^{15} - 18 x^{14} + 57 x^{13} + 121 x^{12} - 417 x^{11} - 374 x^{10} + 1494 x^{9} + 490 x^{8} + \cdots - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 627) |
| 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
\( x^{16} - 3 x^{15} - 18 x^{14} + 57 x^{13} + 121 x^{12} - 417 x^{11} - 374 x^{10} + 1494 x^{9} + 490 x^{8} + \cdots - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 49 \nu^{15} + 4513 \nu^{14} - 16093 \nu^{13} - 68058 \nu^{12} + 266742 \nu^{11} + 330931 \nu^{10} + \cdots + 10020 ) / 64790 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3387 \nu^{15} - 41222 \nu^{14} - 22990 \nu^{13} + 813789 \nu^{12} - 238320 \nu^{11} + \cdots + 1227455 ) / 323950 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 676 \nu^{15} - 1339 \nu^{14} - 11913 \nu^{13} + 16796 \nu^{12} + 85957 \nu^{11} - 38163 \nu^{10} + \cdots - 29690 ) / 32395 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1955 \nu^{15} + 10476 \nu^{14} + 15466 \nu^{13} - 166193 \nu^{12} + 92316 \nu^{11} + 899857 \nu^{10} + \cdots + 8135 ) / 64790 \)
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| \(\beta_{7}\) | \(=\) |
\( ( - 2004 \nu^{15} + 5963 \nu^{14} + 31559 \nu^{13} - 98135 \nu^{12} - 174426 \nu^{11} + 568926 \nu^{10} + \cdots + 62905 ) / 64790 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2017 \nu^{15} - 4369 \nu^{14} - 41382 \nu^{13} + 85500 \nu^{12} + 332198 \nu^{11} - 644293 \nu^{10} + \cdots + 164535 ) / 32395 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10271 \nu^{15} - 22961 \nu^{14} - 239305 \nu^{13} + 502892 \nu^{12} + 2202405 \nu^{11} + \cdots + 660390 ) / 161975 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 35281 \nu^{15} + 106141 \nu^{14} + 632225 \nu^{13} - 2041322 \nu^{12} - 4211260 \nu^{11} + \cdots - 862540 ) / 323950 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 47483 \nu^{15} + 148003 \nu^{14} + 759715 \nu^{13} - 2615616 \nu^{12} - 4190540 \nu^{11} + \cdots - 1476170 ) / 323950 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 30346 \nu^{15} - 93251 \nu^{14} - 533995 \nu^{13} + 1731812 \nu^{12} + 3521690 \nu^{11} + \cdots + 917790 ) / 161975 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 65381 \nu^{15} + 203141 \nu^{14} + 1123375 \nu^{13} - 3736122 \nu^{12} - 7063910 \nu^{11} + \cdots - 955190 ) / 323950 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 66171 \nu^{15} + 198476 \nu^{14} + 1239370 \nu^{13} - 3867887 \nu^{12} - 8846890 \nu^{11} + \cdots - 1781165 ) / 323950 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 93011 \nu^{15} + 279051 \nu^{14} + 1691855 \nu^{13} - 5239972 \nu^{12} - 11686980 \nu^{11} + \cdots - 1891440 ) / 323950 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{3} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + 13 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} - 9\beta_{7} + 8\beta_{6} - \beta_{5} - 8\beta_{3} + 27\beta _1 + 8 \)
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| \(\nu^{6}\) | \(=\) |
\( - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - 11 \beta_{9} + 9 \beta_{8} + 10 \beta_{7} + \cdots + 64 \)
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| \(\nu^{7}\) | \(=\) |
\( - 13 \beta_{15} + 11 \beta_{14} + 12 \beta_{13} - 2 \beta_{12} - \beta_{11} - 11 \beta_{10} - 2 \beta_{9} + \cdots + 50 \)
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| \(\nu^{8}\) | \(=\) |
\( \beta_{15} - 13 \beta_{14} + 13 \beta_{12} + 12 \beta_{11} + 14 \beta_{10} - 89 \beta_{9} + 66 \beta_{8} + \cdots + 341 \)
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| \(\nu^{9}\) | \(=\) |
\( - 117 \beta_{15} + 91 \beta_{14} + 102 \beta_{13} - 27 \beta_{12} - 13 \beta_{11} - 89 \beta_{10} + \cdots + 289 \)
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| \(\nu^{10}\) | \(=\) |
\( 16 \beta_{15} - 119 \beta_{14} - 2 \beta_{13} + 116 \beta_{12} + 105 \beta_{11} + 132 \beta_{10} + \cdots + 1915 \)
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| \(\nu^{11}\) | \(=\) |
\( - 908 \beta_{15} + 678 \beta_{14} + 762 \beta_{13} - 248 \beta_{12} - 116 \beta_{11} - 648 \beta_{10} + \cdots + 1617 \)
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| \(\nu^{12}\) | \(=\) |
\( 175 \beta_{15} - 949 \beta_{14} - 38 \beta_{13} + 896 \beta_{12} + 813 \beta_{11} + 1060 \beta_{10} + \cdots + 11143 \)
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| \(\nu^{13}\) | \(=\) |
\( - 6542 \beta_{15} + 4799 \beta_{14} + 5356 \beta_{13} - 1956 \beta_{12} - 897 \beta_{11} - 4509 \beta_{10} + \cdots + 8877 \)
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| \(\nu^{14}\) | \(=\) |
\( 1634 \beta_{15} - 7067 \beta_{14} - 465 \beta_{13} + 6465 \beta_{12} + 5910 \beta_{11} + 7854 \beta_{10} + \cdots + 66437 \)
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| \(\nu^{15}\) | \(=\) |
\( - 45227 \beta_{15} + 33007 \beta_{14} + 36463 \beta_{13} - 14319 \beta_{12} - 6511 \beta_{11} + \cdots + 47915 \)
|
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 |
|
−2.50967 | −1.00000 | 4.29846 | 4.26849 | 2.50967 | 2.85546 | −5.76838 | 1.00000 | −10.7125 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.41640 | −1.00000 | 3.83898 | −2.71140 | 2.41640 | −1.98097 | −4.44371 | 1.00000 | 6.55182 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.33517 | −1.00000 | 3.45302 | 2.73936 | 2.33517 | −3.55214 | −3.39305 | 1.00000 | −6.39688 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.94310 | −1.00000 | 1.77564 | −3.62893 | 1.94310 | −1.83871 | 0.435950 | 1.00000 | 7.05138 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.58029 | −1.00000 | 0.497326 | −0.196982 | 1.58029 | −1.13950 | 2.37467 | 1.00000 | 0.311290 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.24215 | −1.00000 | −0.457075 | 2.60359 | 1.24215 | 3.18969 | 3.05204 | 1.00000 | −3.23404 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.763195 | −1.00000 | −1.41753 | 1.56297 | 0.763195 | 2.33093 | 2.60825 | 1.00000 | −1.19285 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.308708 | −1.00000 | −1.90470 | 4.13307 | 0.308708 | −1.74454 | 1.20541 | 1.00000 | −1.27591 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.137096 | −1.00000 | −1.98120 | −1.00936 | 0.137096 | 3.80132 | 0.545809 | 1.00000 | 0.138380 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.210794 | −1.00000 | −1.95557 | −2.77906 | −0.210794 | 0.799054 | −0.833809 | 1.00000 | −0.585808 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.673455 | −1.00000 | −1.54646 | 1.68989 | −0.673455 | −1.64609 | −2.38838 | 1.00000 | 1.13806 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.41865 | −1.00000 | 0.0125585 | −1.57412 | −1.41865 | 2.36838 | −2.81948 | 1.00000 | −2.23312 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.46141 | −1.00000 | 0.135729 | 2.17265 | −1.46141 | 4.77062 | −2.72447 | 1.00000 | 3.17514 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.83257 | −1.00000 | 1.35830 | 0.910966 | −1.83257 | −5.13177 | −1.17596 | 1.00000 | 1.66941 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.06177 | −1.00000 | 2.25090 | −2.06163 | −2.06177 | −2.39666 | 0.517301 | 1.00000 | −4.25062 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.57713 | −1.00000 | 4.64162 | 1.88049 | −2.57713 | 1.31492 | 6.80781 | 1.00000 | 4.84626 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 6897.2.a.bl | 16 | |
| 11.b | odd | 2 | 1 | 6897.2.a.bm | 16 | ||
| 11.d | odd | 10 | 2 | 627.2.j.b | ✓ | 32 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 627.2.j.b | ✓ | 32 | 11.d | odd | 10 | 2 | |
| 6897.2.a.bl | 16 | 1.a | even | 1 | 1 | trivial | |
| 6897.2.a.bm | 16 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\):
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\( T_{2}^{16} + 3 T_{2}^{15} - 18 T_{2}^{14} - 57 T_{2}^{13} + 121 T_{2}^{12} + 417 T_{2}^{11} - 374 T_{2}^{10} + \cdots - 5 \)
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\( T_{5}^{16} - 8 T_{5}^{15} - 18 T_{5}^{14} + 278 T_{5}^{13} - 158 T_{5}^{12} - 3571 T_{5}^{11} + \cdots - 21824 \)
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\( T_{7}^{16} - 2 T_{7}^{15} - 62 T_{7}^{14} + 117 T_{7}^{13} + 1467 T_{7}^{12} - 2395 T_{7}^{11} + \cdots + 498880 \)
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\( T_{13}^{16} + 5 T_{13}^{15} - 90 T_{13}^{14} - 446 T_{13}^{13} + 3157 T_{13}^{12} + 15821 T_{13}^{11} + \cdots - 13718000 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 3 T^{15} + \cdots - 5 \)
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| $3$ |
\( (T + 1)^{16} \)
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| $5$ |
\( T^{16} - 8 T^{15} + \cdots - 21824 \)
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| $7$ |
\( T^{16} - 2 T^{15} + \cdots + 498880 \)
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| $11$ |
\( T^{16} \)
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| $13$ |
\( T^{16} + 5 T^{15} + \cdots - 13718000 \)
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| $17$ |
\( T^{16} + 4 T^{15} + \cdots - 136720 \)
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| $19$ |
\( (T + 1)^{16} \)
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| $23$ |
\( T^{16} + \cdots + 104808521 \)
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| $29$ |
\( T^{16} + \cdots - 482548291 \)
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| $31$ |
\( T^{16} - 19 T^{15} + \cdots - 92549 \)
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| $37$ |
\( T^{16} + \cdots + 3997251920 \)
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| $41$ |
\( T^{16} + \cdots - 5575549905 \)
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| $43$ |
\( T^{16} + \cdots + 20863912000 \)
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| $47$ |
\( T^{16} + \cdots + 22306902145 \)
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| $53$ |
\( T^{16} + \cdots + 229797265351 \)
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| $59$ |
\( T^{16} + \cdots - 18177777920 \)
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| $61$ |
\( T^{16} + \cdots - 63134655305 \)
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| $67$ |
\( T^{16} + \cdots + 4161239681555 \)
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| $71$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
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| $73$ |
\( T^{16} + \cdots + 4053802819 \)
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| $79$ |
\( T^{16} + \cdots + 5142126873025 \)
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| $83$ |
\( T^{16} + \cdots + 364089172775 \)
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| $89$ |
\( T^{16} + \cdots - 21342579920 \)
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| $97$ |
\( T^{16} + \cdots - 1227850230896 \)
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