Newspace parameters
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.j (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
Defining polynomial: |
\( x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 355 \nu^{9} - 2199 \nu^{8} + 7398 \nu^{7} - 18528 \nu^{6} + 39184 \nu^{5} - 87134 \nu^{4} + 143476 \nu^{3} - 156238 \nu^{2} + 124656 \nu - 55404 ) / 40746 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 4282 \nu^{9} - 12311 \nu^{8} + 50191 \nu^{7} - 64307 \nu^{6} + 231068 \nu^{5} - 318192 \nu^{4} + 721480 \nu^{3} - 289782 \nu^{2} + 136368 \nu - 235800 ) / 448206 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 6550 \nu^{9} + 8818 \nu^{8} - 53189 \nu^{7} + 2209 \nu^{6} - 263193 \nu^{5} + 44032 \nu^{4} - 494008 \nu^{3} - 642880 \nu^{2} - 339018 \nu + 99432 ) / 448206 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 8029 \nu^{9} + 19682 \nu^{8} - 80242 \nu^{7} + 81275 \nu^{6} - 369416 \nu^{5} + 508704 \nu^{4} - 959878 \nu^{3} + 463284 \nu^{2} - 218016 \nu + 1426266 ) / 448206 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 9476 \nu^{9} + 17684 \nu^{8} - 89335 \nu^{7} + 36452 \nu^{6} - 354900 \nu^{5} + 123782 \nu^{4} - 621080 \nu^{3} - 732992 \nu^{2} + 907392 \nu - 408504 ) / 448206 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 10145 \nu^{9} - 7555 \nu^{8} - 72631 \nu^{7} - 139765 \nu^{6} - 519015 \nu^{5} - 700936 \nu^{4} - 1168190 \nu^{3} - 1844240 \nu^{2} - 2260776 \nu - 953850 ) / 448206 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 5301 \nu^{9} + 6361 \nu^{8} - 43172 \nu^{7} - 3447 \nu^{6} - 217077 \nu^{5} - 19472 \nu^{4} - 414542 \nu^{3} - 551312 \nu^{2} - 311802 \nu - 297390 ) / 149402 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 9385 \nu^{9} + 23127 \nu^{8} - 94287 \nu^{7} + 111339 \nu^{6} - 434076 \nu^{5} + 597744 \nu^{4} - 956678 \nu^{3} + 544374 \nu^{2} - 256176 \nu + 675982 ) / 149402 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{5} - 3\beta_{4} + \beta_{3} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 5\beta_{3} - \beta_{2} - 1 \)
|
\(\nu^{4}\) | \(=\) |
\( -7\beta_{8} + \beta_{7} + \beta_{6} + 14\beta_{4} - \beta _1 - 14 \)
|
\(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - 10\beta_{8} + 2\beta_{7} - 10\beta_{5} + 10\beta_{4} - 30\beta_{3} + 8\beta_{2} - 20\beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( 10\beta_{9} - 12\beta_{6} - 46\beta_{5} - 58\beta_{3} + 12\beta_{2} + 76 \)
|
\(\nu^{7}\) | \(=\) |
\( 82\beta_{8} - 22\beta_{7} - 56\beta_{6} - 84\beta_{4} + 110\beta _1 + 84 \)
|
\(\nu^{8}\) | \(=\) |
\( -78\beta_{9} + 304\beta_{8} - 78\beta_{7} + 304\beta_{5} - 446\beta_{4} + 414\beta_{3} - 104\beta_{2} + 110\beta_1 \)
|
\(\nu^{9}\) | \(=\) |
\( -182\beta_{9} + 382\beta_{6} + 622\beta_{5} + 1268\beta_{3} - 382\beta_{2} - 660 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) | \(496\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
406.1 |
|
−1.31604 | − | 2.27945i | 0 | −2.46394 | + | 4.26767i | 1.00000 | 0 | −0.544875 | + | 0.943751i | 7.70645 | 0 | −1.31604 | − | 2.27945i | ||||||||||||||||||||||||||||||||||||||||
406.2 | −0.905157 | − | 1.56778i | 0 | −0.638619 | + | 1.10612i | 1.00000 | 0 | 2.21251 | − | 3.83219i | −1.30843 | 0 | −0.905157 | − | 1.56778i | |||||||||||||||||||||||||||||||||||||||||
406.3 | −0.313396 | − | 0.542817i | 0 | 0.803566 | − | 1.39182i | 1.00000 | 0 | −2.21563 | + | 3.83759i | −2.26092 | 0 | −0.313396 | − | 0.542817i | |||||||||||||||||||||||||||||||||||||||||
406.4 | 0.473183 | + | 0.819577i | 0 | 0.552196 | − | 0.956432i | 1.00000 | 0 | 0.781529 | − | 1.35365i | 2.93789 | 0 | 0.473183 | + | 0.819577i | |||||||||||||||||||||||||||||||||||||||||
406.5 | 1.06141 | + | 1.83842i | 0 | −1.25320 | + | 2.17061i | 1.00000 | 0 | −0.733534 | + | 1.27052i | −1.07500 | 0 | 1.06141 | + | 1.83842i | |||||||||||||||||||||||||||||||||||||||||
451.1 | −1.31604 | + | 2.27945i | 0 | −2.46394 | − | 4.26767i | 1.00000 | 0 | −0.544875 | − | 0.943751i | 7.70645 | 0 | −1.31604 | + | 2.27945i | |||||||||||||||||||||||||||||||||||||||||
451.2 | −0.905157 | + | 1.56778i | 0 | −0.638619 | − | 1.10612i | 1.00000 | 0 | 2.21251 | + | 3.83219i | −1.30843 | 0 | −0.905157 | + | 1.56778i | |||||||||||||||||||||||||||||||||||||||||
451.3 | −0.313396 | + | 0.542817i | 0 | 0.803566 | + | 1.39182i | 1.00000 | 0 | −2.21563 | − | 3.83759i | −2.26092 | 0 | −0.313396 | + | 0.542817i | |||||||||||||||||||||||||||||||||||||||||
451.4 | 0.473183 | − | 0.819577i | 0 | 0.552196 | + | 0.956432i | 1.00000 | 0 | 0.781529 | + | 1.35365i | 2.93789 | 0 | 0.473183 | − | 0.819577i | |||||||||||||||||||||||||||||||||||||||||
451.5 | 1.06141 | − | 1.83842i | 0 | −1.25320 | − | 2.17061i | 1.00000 | 0 | −0.733534 | − | 1.27052i | −1.07500 | 0 | 1.06141 | − | 1.83842i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.j.h | ✓ | 10 |
3.b | odd | 2 | 1 | 585.2.j.i | yes | 10 | |
13.c | even | 3 | 1 | inner | 585.2.j.h | ✓ | 10 |
13.c | even | 3 | 1 | 7605.2.a.co | 5 | ||
13.e | even | 6 | 1 | 7605.2.a.cl | 5 | ||
39.h | odd | 6 | 1 | 7605.2.a.cn | 5 | ||
39.i | odd | 6 | 1 | 585.2.j.i | yes | 10 | |
39.i | odd | 6 | 1 | 7605.2.a.cm | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.j.h | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
585.2.j.h | ✓ | 10 | 13.c | even | 3 | 1 | inner |
585.2.j.i | yes | 10 | 3.b | odd | 2 | 1 | |
585.2.j.i | yes | 10 | 39.i | odd | 6 | 1 | |
7605.2.a.cl | 5 | 13.e | even | 6 | 1 | ||
7605.2.a.cm | 5 | 39.i | odd | 6 | 1 | ||
7605.2.a.cn | 5 | 39.h | odd | 6 | 1 | ||
7605.2.a.co | 5 | 13.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 2T_{2}^{9} + 10T_{2}^{8} + 8T_{2}^{7} + 50T_{2}^{6} + 42T_{2}^{5} + 124T_{2}^{4} + 12T_{2}^{3} + 96T_{2}^{2} + 36T_{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} + 2 T^{9} + 10 T^{8} + 8 T^{7} + \cdots + 36 \)
$3$
\( T^{10} \)
$5$
\( (T - 1)^{10} \)
$7$
\( T^{10} + T^{9} + 23 T^{8} + 22 T^{7} + \cdots + 2401 \)
$11$
\( T^{10} + 8 T^{9} + 64 T^{8} + 128 T^{7} + \cdots + 576 \)
$13$
\( T^{10} - T^{9} - 21 T^{8} + \cdots + 371293 \)
$17$
\( T^{10} + 50 T^{8} - 28 T^{7} + \cdots + 412164 \)
$19$
\( T^{10} - 4 T^{9} + 24 T^{8} - 40 T^{7} + \cdots + 144 \)
$23$
\( T^{10} - 6 T^{9} + 56 T^{8} + \cdots + 5184 \)
$29$
\( T^{10} + 16 T^{9} + 198 T^{8} + \cdots + 66564 \)
$31$
\( (T^{5} - 9 T^{4} - 30 T^{3} + 178 T^{2} + \cdots + 603)^{2} \)
$37$
\( T^{10} - 4 T^{9} + 80 T^{8} + \cdots + 20736 \)
$41$
\( T^{10} + 6 T^{9} + 134 T^{8} + \cdots + 141562404 \)
$43$
\( T^{10} + 15 T^{9} + 195 T^{8} + \cdots + 4239481 \)
$47$
\( (T^{5} - 10 T^{4} - 90 T^{3} + 752 T^{2} + \cdots - 7434)^{2} \)
$53$
\( (T^{5} + 20 T^{4} - 1872 T^{2} + \cdots - 15552)^{2} \)
$59$
\( T^{10} + 12 T^{9} + 186 T^{8} + \cdots + 7584516 \)
$61$
\( T^{10} + 11 T^{9} + 135 T^{8} + \cdots + 9138529 \)
$67$
\( T^{10} + 5 T^{9} + 69 T^{8} - 212 T^{7} + \cdots + 289 \)
$71$
\( T^{10} + 10 T^{9} + 122 T^{8} + \cdots + 26244 \)
$73$
\( (T^{5} - T^{4} - 236 T^{3} + 144 T^{2} + \cdots + 9693)^{2} \)
$79$
\( (T^{5} + 17 T^{4} - 26 T^{3} - 1086 T^{2} + \cdots + 2349)^{2} \)
$83$
\( (T^{5} - 16 T^{4} - 116 T^{3} + 2060 T^{2} + \cdots - 6264)^{2} \)
$89$
\( T^{10} + 4 T^{9} + 118 T^{8} + \cdots + 2916 \)
$97$
\( T^{10} - 11 T^{9} + 251 T^{8} + \cdots + 58967041 \)
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