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: | \(1\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{20} - 3 x^{19} - 26 x^{18} + 81 x^{17} + 273 x^{16} - 901 x^{15} - 1480 x^{14} + 5366 x^{13} + \cdots - 304 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 3 x^{19} - 26 x^{18} + 81 x^{17} + 273 x^{16} - 901 x^{15} - 1480 x^{14} + 5366 x^{13} + \cdots - 304 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 423497524 \nu^{19} + 3266642890 \nu^{18} + 5790771544 \nu^{17} - 83894721727 \nu^{16} + \cdots + 492160582228 ) / 130282473450 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1005767612 \nu^{19} - 1170163925 \nu^{18} - 36224821187 \nu^{17} + 53323527236 \nu^{16} + \cdots + 3063752066416 ) / 260564946900 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 911433107 \nu^{19} + 4807581805 \nu^{18} - 33440887092 \nu^{17} - 143947264289 \nu^{16} + \cdots - 4641936363704 ) / 173709964600 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2530894069 \nu^{19} - 7204561705 \nu^{18} - 67086759384 \nu^{17} + 192304334717 \nu^{16} + \cdots + 4418760453472 ) / 173709964600 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9506645341 \nu^{19} - 8211818425 \nu^{18} - 303282472216 \nu^{17} + 250581619573 \nu^{16} + \cdots + 1781009620088 ) / 521129893800 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 11184631 \nu^{19} + 21220669 \nu^{18} + 320406838 \nu^{17} - 581582761 \nu^{16} + \cdots - 5543772272 ) / 478100820 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 6603161 \nu^{19} + 11866643 \nu^{18} + 194576795 \nu^{17} - 327260909 \nu^{16} + \cdots - 5344660096 ) / 239050410 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 18236093 \nu^{19} - 43523648 \nu^{18} - 495359087 \nu^{17} + 1156716695 \nu^{16} + \cdots + 6046399132 ) / 478100820 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 20209164563 \nu^{19} - 70443993875 \nu^{18} - 512966072588 \nu^{17} + 1888325975639 \nu^{16} + \cdots + 14962867353184 ) / 521129893800 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 4952028385 \nu^{19} + 13632097957 \nu^{18} + 134472896896 \nu^{17} - 376062079969 \nu^{16} + \cdots - 4920395997632 ) / 104225978760 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 11774531 \nu^{19} + 26220511 \nu^{18} + 328917554 \nu^{17} - 707384315 \nu^{16} + \cdots - 4426747024 ) / 159366940 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 42375055 \nu^{19} - 100964512 \nu^{18} - 1161704911 \nu^{17} + 2697286879 \nu^{16} + \cdots + 13304767112 ) / 478100820 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 53413231663 \nu^{19} - 114087222685 \nu^{18} - 1516653583918 \nu^{17} + 3136945784959 \nu^{16} + \cdots + 31009596069344 ) / 521129893800 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 3853515217 \nu^{19} - 10158822691 \nu^{18} - 105285358650 \nu^{17} + 277052639393 \nu^{16} + \cdots + 3103862166792 ) / 34741992920 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 36297628586 \nu^{19} + 100723321805 \nu^{18} + 972310057151 \nu^{17} + \cdots - 22866359732728 ) / 260564946900 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 7273316599 \nu^{19} + 17997162178 \nu^{18} + 197447498503 \nu^{17} - 482782498093 \nu^{16} + \cdots - 4207274875640 ) / 52112989380 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 44826264101 \nu^{19} + 112601169725 \nu^{18} + 1214186790476 \nu^{17} + \cdots - 28781422579468 ) / 260564946900 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{14} + \beta_{13} - \beta_{10} - \beta_{8} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{19} - \beta_{18} - \beta_{17} - \beta_{15} - \beta_{14} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{17} - \beta_{15} + 8 \beta_{14} + 8 \beta_{13} + 2 \beta_{12} - \beta_{11} - 11 \beta_{10} + \cdots + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{19} - 10 \beta_{18} - 10 \beta_{17} - \beta_{16} - 11 \beta_{15} - 10 \beta_{14} + \beta_{13} + \cdots + 64 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{19} + \beta_{18} - 25 \beta_{17} + 2 \beta_{16} - 14 \beta_{15} + 57 \beta_{14} + 56 \beta_{13} + \cdots + 92 \)
|
| \(\nu^{8}\) | \(=\) |
\( 79 \beta_{19} - 80 \beta_{18} - 78 \beta_{17} - 15 \beta_{16} - 96 \beta_{15} - 77 \beta_{14} + \cdots + 344 \)
|
| \(\nu^{9}\) | \(=\) |
\( 13 \beta_{19} + 18 \beta_{18} - 231 \beta_{17} + 32 \beta_{16} - 137 \beta_{15} + 400 \beta_{14} + \cdots + 711 \)
|
| \(\nu^{10}\) | \(=\) |
\( 587 \beta_{19} - 604 \beta_{18} - 567 \beta_{17} - 157 \beta_{16} - 774 \beta_{15} - 551 \beta_{14} + \cdots + 1978 \)
|
| \(\nu^{11}\) | \(=\) |
\( 118 \beta_{19} + 211 \beta_{18} - 1915 \beta_{17} + 343 \beta_{16} - 1168 \beta_{15} + 2818 \beta_{14} + \cdots + 5344 \)
|
| \(\nu^{12}\) | \(=\) |
\( 4287 \beta_{19} - 4482 \beta_{18} - 4028 \beta_{17} - 1415 \beta_{16} - 6008 \beta_{15} - 3864 \beta_{14} + \cdots + 11982 \)
|
| \(\nu^{13}\) | \(=\) |
\( 931 \beta_{19} + 2062 \beta_{18} - 15089 \beta_{17} + 3119 \beta_{16} - 9310 \beta_{15} + 19998 \beta_{14} + \cdots + 39716 \)
|
| \(\nu^{14}\) | \(=\) |
\( 31190 \beta_{19} - 33084 \beta_{18} - 28429 \beta_{17} - 11791 \beta_{16} - 45647 \beta_{15} + \cdots + 75557 \)
|
| \(\nu^{15}\) | \(=\) |
\( 6859 \beta_{19} + 18294 \beta_{18} - 115663 \beta_{17} + 26079 \beta_{16} - 71516 \beta_{15} + \cdots + 293526 \)
|
| \(\nu^{16}\) | \(=\) |
\( 226950 \beta_{19} - 243763 \beta_{18} - 200584 \beta_{17} - 93877 \beta_{16} - 342209 \beta_{15} + \cdots + 491360 \)
|
| \(\nu^{17}\) | \(=\) |
\( 48686 \beta_{19} + 153211 \beta_{18} - 872118 \beta_{17} + 207935 \beta_{16} - 537506 \beta_{15} + \cdots + 2161810 \)
|
| \(\nu^{18}\) | \(=\) |
\( 1652875 \beta_{19} - 1794142 \beta_{18} - 1418036 \beta_{17} - 726781 \beta_{16} - 2542822 \beta_{15} + \cdots + 3271412 \)
|
| \(\nu^{19}\) | \(=\) |
\( 338154 \beta_{19} + 1236598 \beta_{18} - 6506717 \beta_{17} + 1611049 \beta_{16} - 3985742 \beta_{15} + \cdots + 15880366 \)
|
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 |
|
−2.69319 | −1.00000 | 5.25328 | −0.486629 | 2.69319 | 2.49782 | −8.76171 | 1.00000 | 1.31059 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.68761 | −1.00000 | 5.22325 | −3.78906 | 2.68761 | 3.38908 | −8.66284 | 1.00000 | 10.1835 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.24972 | −1.00000 | 3.06124 | 2.43671 | 2.24972 | 4.40799 | −2.38750 | 1.00000 | −5.48193 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.16908 | −1.00000 | 2.70492 | 0.270542 | 2.16908 | −3.04129 | −1.52903 | 1.00000 | −0.586828 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.89014 | −1.00000 | 1.57264 | 0.892902 | 1.89014 | 2.76785 | 0.807768 | 1.00000 | −1.68771 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.64839 | −1.00000 | 0.717181 | 0.941331 | 1.64839 | −2.54422 | 2.11458 | 1.00000 | −1.55168 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.29472 | −1.00000 | −0.323702 | −2.21563 | 1.29472 | −0.848670 | 3.00854 | 1.00000 | 2.86861 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.898242 | −1.00000 | −1.19316 | 2.38103 | 0.898242 | −1.17240 | 2.86823 | 1.00000 | −2.13874 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.717677 | −1.00000 | −1.48494 | −2.61938 | 0.717677 | 1.81905 | 2.50106 | 1.00000 | 1.87987 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.495394 | −1.00000 | −1.75458 | −2.74216 | 0.495394 | 4.80343 | 1.86000 | 1.00000 | 1.35845 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −0.414444 | −1.00000 | −1.82824 | −1.59002 | 0.414444 | −4.26239 | 1.58659 | 1.00000 | 0.658974 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.430104 | −1.00000 | −1.81501 | 3.94826 | −0.430104 | −0.533954 | −1.64085 | 1.00000 | 1.69816 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.647095 | −1.00000 | −1.58127 | 1.50920 | −0.647095 | −1.48159 | −2.31742 | 1.00000 | 0.976598 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 0.983433 | −1.00000 | −1.03286 | −3.73944 | −0.983433 | −3.08350 | −2.98261 | 1.00000 | −3.67749 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.50892 | −1.00000 | 0.276838 | 3.14626 | −1.50892 | 2.92143 | −2.60011 | 1.00000 | 4.74745 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 1.60139 | −1.00000 | 0.564462 | −4.14025 | −1.60139 | −1.59113 | −2.29886 | 1.00000 | −6.63017 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 1.80169 | −1.00000 | 1.24610 | 0.711705 | −1.80169 | 3.18787 | −1.35829 | 1.00000 | 1.28227 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.10369 | −1.00000 | 2.42552 | −1.56098 | −2.10369 | −0.929014 | 0.895153 | 1.00000 | −3.28382 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.37753 | −1.00000 | 3.65266 | 1.70548 | −2.37753 | −4.50539 | 3.92925 | 1.00000 | 4.05484 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 2.70475 | −1.00000 | 5.31567 | −4.05988 | −2.70475 | 1.19905 | 8.96805 | 1.00000 | −10.9810 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.bn | 20 | |
| 11.b | odd | 2 | 1 | 6897.2.a.bo | 20 | ||
| 11.d | odd | 10 | 2 | 627.2.j.c | ✓ | 40 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 627.2.j.c | ✓ | 40 | 11.d | odd | 10 | 2 | |
| 6897.2.a.bn | 20 | 1.a | even | 1 | 1 | trivial | |
| 6897.2.a.bo | 20 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\):
|
\( T_{2}^{20} + 3 T_{2}^{19} - 26 T_{2}^{18} - 81 T_{2}^{17} + 273 T_{2}^{16} + 901 T_{2}^{15} - 1480 T_{2}^{14} + \cdots - 304 \)
|
|
\( T_{5}^{20} + 9 T_{5}^{19} - 25 T_{5}^{18} - 411 T_{5}^{17} - 104 T_{5}^{16} + 7516 T_{5}^{15} + \cdots + 137439 \)
|
|
\( T_{7}^{20} - 3 T_{7}^{19} - 77 T_{7}^{18} + 212 T_{7}^{17} + 2466 T_{7}^{16} - 6028 T_{7}^{15} + \cdots - 5372525 \)
|
|
\( T_{13}^{20} + 9 T_{13}^{19} - 118 T_{13}^{18} - 1286 T_{13}^{17} + 4505 T_{13}^{16} + 72607 T_{13}^{15} + \cdots - 1321663744 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 3 T^{19} + \cdots - 304 \)
|
| $3$ |
\( (T + 1)^{20} \)
|
| $5$ |
\( T^{20} + 9 T^{19} + \cdots + 137439 \)
|
| $7$ |
\( T^{20} - 3 T^{19} + \cdots - 5372525 \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + \cdots - 1321663744 \)
|
| $17$ |
\( T^{20} - 7 T^{19} + \cdots - 76576509 \)
|
| $19$ |
\( (T - 1)^{20} \)
|
| $23$ |
\( T^{20} + \cdots - 40609248811 \)
|
| $29$ |
\( T^{20} + \cdots + 91183890242304 \)
|
| $31$ |
\( T^{20} + \cdots - 78190222336 \)
|
| $37$ |
\( T^{20} + \cdots + 178501747297280 \)
|
| $41$ |
\( T^{20} + \cdots - 35\!\cdots\!80 \)
|
| $43$ |
\( T^{20} + \cdots - 17\!\cdots\!95 \)
|
| $47$ |
\( T^{20} + \cdots - 2663592921 \)
|
| $53$ |
\( T^{20} + \cdots + 911322907335680 \)
|
| $59$ |
\( T^{20} + \cdots + 19\!\cdots\!76 \)
|
| $61$ |
\( T^{20} + \cdots - 30393031612581 \)
|
| $67$ |
\( T^{20} + \cdots - 616349384672256 \)
|
| $71$ |
\( T^{20} + \cdots + 10\!\cdots\!80 \)
|
| $73$ |
\( T^{20} + \cdots - 9979063682256 \)
|
| $79$ |
\( T^{20} + \cdots - 503868586436864 \)
|
| $83$ |
\( T^{20} + \cdots - 24\!\cdots\!69 \)
|
| $89$ |
\( T^{20} + \cdots + 25\!\cdots\!96 \)
|
| $97$ |
\( T^{20} + \cdots + 127136531162880 \)
|
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