Newspace parameters
| Level: | \( N \) | \(=\) | \( 1547 = 7 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1547.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(12.3528571927\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{14} - x^{13} - 21 x^{12} + 18 x^{11} + 170 x^{10} - 121 x^{9} - 672 x^{8} + 383 x^{7} + 1362 x^{6} + \cdots - 64 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - x^{13} - 21 x^{12} + 18 x^{11} + 170 x^{10} - 121 x^{9} - 672 x^{8} + 383 x^{7} + 1362 x^{6} + \cdots - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 459 \nu^{13} - 743 \nu^{12} - 9019 \nu^{11} + 13138 \nu^{10} + 66822 \nu^{9} - 83851 \nu^{8} + \cdots + 10848 ) / 1936 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 461 \nu^{13} + 277 \nu^{12} + 9401 \nu^{11} - 4130 \nu^{10} - 72434 \nu^{9} + 19605 \nu^{8} + \cdots - 18576 ) / 1936 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 137 \nu^{13} - 23 \nu^{12} - 2935 \nu^{11} + 52 \nu^{10} + 24084 \nu^{9} + 2985 \nu^{8} - 94890 \nu^{7} + \cdots + 13666 ) / 242 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1227 \nu^{13} - 395 \nu^{12} - 25511 \nu^{11} + 4502 \nu^{10} + 201614 \nu^{9} - 7363 \nu^{8} + \cdots + 86016 ) / 1936 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 549 \nu^{13} - 141 \nu^{12} + 11931 \nu^{11} + 4296 \nu^{10} - 99142 \nu^{9} - 44063 \nu^{8} + \cdots - 63368 ) / 968 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 839 \nu^{13} + 291 \nu^{12} + 17711 \nu^{11} - 3362 \nu^{10} - 142838 \nu^{9} + 5183 \nu^{8} + \cdots - 62984 ) / 968 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 839 \nu^{13} + 291 \nu^{12} + 17711 \nu^{11} - 3362 \nu^{10} - 142838 \nu^{9} + 5183 \nu^{8} + \cdots - 56208 ) / 968 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2261 \nu^{13} + 989 \nu^{12} + 47313 \nu^{11} - 12642 \nu^{10} - 378098 \nu^{9} + 36205 \nu^{8} + \cdots - 166800 ) / 1936 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1997 \nu^{13} + 549 \nu^{12} + 42385 \nu^{11} - 5250 \nu^{10} - 343954 \nu^{9} - 7531 \nu^{8} + \cdots - 173752 ) / 968 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2089 \nu^{13} + 409 \nu^{12} + 44469 \nu^{11} - 2282 \nu^{10} - 362042 \nu^{9} - 30775 \nu^{8} + \cdots - 188504 ) / 968 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} - \beta_{9} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + 8\beta_{3} + \beta_{2} + 30\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{13} + \beta_{12} + 11 \beta_{10} - 12 \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{5} + \cdots + 77 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 3 \beta_{13} + 3 \beta_{12} + \beta_{11} + 12 \beta_{10} - 12 \beta_{9} + 11 \beta_{8} - \beta_{7} + \cdots + 14 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 18 \beta_{13} + 15 \beta_{12} + 94 \beta_{10} - 102 \beta_{9} + 16 \beta_{8} - 17 \beta_{7} + \cdots + 463 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 55 \beta_{13} + 52 \beta_{12} + 14 \beta_{11} + 111 \beta_{10} - 108 \beta_{9} + 97 \beta_{8} + \cdots + 152 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 218 \beta_{13} + 163 \beta_{12} + 3 \beta_{11} + 740 \beta_{10} - 776 \beta_{9} + 180 \beta_{8} + \cdots + 2937 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 665 \beta_{13} + 604 \beta_{12} + 136 \beta_{11} + 946 \beta_{10} - 881 \beta_{9} + 804 \beta_{8} + \cdots + 1479 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 2224 \beta_{13} + 1564 \beta_{12} + 60 \beta_{11} + 5631 \beta_{10} - 5656 \beta_{9} + 1742 \beta_{8} + \cdots + 19273 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 6726 \beta_{13} + 5922 \beta_{12} + 1154 \beta_{11} + 7780 \beta_{10} - 6903 \beta_{9} + 6495 \beta_{8} + \cdots + 13409 \)
|
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 |
|
−2.73449 | 2.01064 | 5.47744 | −2.91387 | −5.49809 | −1.00000 | −9.50904 | 1.04269 | 7.96796 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.49389 | −1.95087 | 4.21946 | −0.130680 | 4.86524 | −1.00000 | −5.53509 | 0.805889 | 0.325900 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.94832 | 2.63220 | 1.79596 | 4.26177 | −5.12838 | −1.00000 | 0.397533 | 3.92850 | −8.30329 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.84525 | 0.190787 | 1.40495 | 0.404876 | −0.352050 | −1.00000 | 1.09801 | −2.96360 | −0.747099 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.03606 | 3.24035 | −0.926589 | −2.65021 | −3.35718 | −1.00000 | 3.03211 | 7.49986 | 2.74577 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.763968 | −0.659044 | −1.41635 | 0.942629 | 0.503488 | −1.00000 | 2.60998 | −2.56566 | −0.720138 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.741427 | 0.0297094 | −1.45029 | −3.52109 | −0.0220274 | −1.00000 | 2.55814 | −2.99912 | 2.61063 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.251178 | −0.591218 | −1.93691 | 4.10786 | −0.148501 | −1.00000 | −0.988866 | −2.65046 | 1.03181 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.928177 | −0.220764 | −1.13849 | −2.93688 | −0.204908 | −1.00000 | −2.91307 | −2.95126 | −2.72594 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.21849 | −2.84736 | −0.515287 | 0.858057 | −3.46948 | −1.00000 | −3.06485 | 5.10748 | 1.04553 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.45104 | 2.30173 | 0.105519 | 1.10695 | 3.33990 | −1.00000 | −2.74897 | 2.29795 | 1.60623 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.75738 | 3.17512 | 1.08839 | 3.71259 | 5.57990 | −1.00000 | −1.60204 | 7.08138 | 6.52444 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.42571 | −1.40624 | 3.88409 | 3.84382 | −3.41113 | −1.00000 | 4.57025 | −1.02249 | 9.32400 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.53142 | 2.09496 | 4.40809 | 0.914190 | 5.30322 | −1.00000 | 6.09589 | 1.38885 | 2.31420 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
| \(13\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 1547.2.a.h | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1547.2.a.h | ✓ | 14 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1547))\):
|
\( T_{2}^{14} + T_{2}^{13} - 21 T_{2}^{12} - 18 T_{2}^{11} + 170 T_{2}^{10} + 121 T_{2}^{9} - 672 T_{2}^{8} + \cdots - 64 \)
|
|
\( T_{3}^{14} - 8 T_{3}^{13} + 4 T_{3}^{12} + 112 T_{3}^{11} - 222 T_{3}^{10} - 455 T_{3}^{9} + 1309 T_{3}^{8} + \cdots + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + T^{13} + \cdots - 64 \)
|
| $3$ |
\( T^{14} - 8 T^{13} + \cdots + 1 \)
|
| $5$ |
\( T^{14} - 8 T^{13} + \cdots - 864 \)
|
| $7$ |
\( (T + 1)^{14} \)
|
| $11$ |
\( T^{14} + T^{13} + \cdots - 2036624 \)
|
| $13$ |
\( (T - 1)^{14} \)
|
| $17$ |
\( (T + 1)^{14} \)
|
| $19$ |
\( T^{14} - 131 T^{12} + \cdots - 366992 \)
|
| $23$ |
\( T^{14} - 3 T^{13} + \cdots - 32722944 \)
|
| $29$ |
\( T^{14} - 4 T^{13} + \cdots + 2048 \)
|
| $31$ |
\( T^{14} - 2 T^{13} + \cdots + 1949216 \)
|
| $37$ |
\( T^{14} - 33 T^{13} + \cdots + 4885248 \)
|
| $41$ |
\( T^{14} + \cdots + 13399947744 \)
|
| $43$ |
\( T^{14} + \cdots - 105573537 \)
|
| $47$ |
\( T^{14} + \cdots - 1058174992 \)
|
| $53$ |
\( T^{14} + \cdots - 23532088987 \)
|
| $59$ |
\( T^{14} + \cdots - 107821312 \)
|
| $61$ |
\( T^{14} + \cdots - 5483333508 \)
|
| $67$ |
\( T^{14} + \cdots + 165549109952 \)
|
| $71$ |
\( T^{14} + \cdots + 19446007808 \)
|
| $73$ |
\( T^{14} + \cdots + 14959110784 \)
|
| $79$ |
\( T^{14} + \cdots - 1492381152256 \)
|
| $83$ |
\( T^{14} + \cdots + 3360225630096 \)
|
| $89$ |
\( T^{14} + \cdots + 134640710704 \)
|
| $97$ |
\( T^{14} + \cdots - 98640492192 \)
|
| show more | |
| show less |