Newspace parameters
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 585.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.67124851824\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
| Defining polynomial: |
\( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 195) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} + 13\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 8\nu^{4} + 13\nu^{2} ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 4\nu^{3} + 65\nu^{2} + 20\nu ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 11\nu^{5} + 33\nu^{3} + 19\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{8} - 13\nu^{6} - 53\nu^{4} + 4\nu^{3} - 65\nu^{2} + 20\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 69\nu^{2} + 12 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} + 15\nu^{7} + 2\nu^{6} + 77\nu^{5} + 20\nu^{4} + 155\nu^{3} + 54\nu^{2} + 96\nu + 32 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} - 15\nu^{7} + 2\nu^{6} - 77\nu^{5} + 20\nu^{4} - 155\nu^{3} + 54\nu^{2} - 96\nu + 32 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{4} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} - 7\beta_{7} - 7\beta_{6} + 7\beta_{4} - 2\beta_{3} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{6} - 8\beta_{4} + 4\beta_{2} + 27\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{9} - 8\beta_{8} + 43\beta_{7} + 43\beta_{6} - 43\beta_{4} + 20\beta_{3} - 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55\beta_{6} + 4\beta_{5} + 55\beta_{4} - 44\beta_{2} - 151\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 51\beta_{9} + 51\beta_{8} - 253\beta_{7} - 257\beta_{6} + 257\beta_{4} - 154\beta_{3} + 351 \)
|
| \(\nu^{9}\) | \(=\) |
\( -4\beta_{9} + 4\beta_{8} - 364\beta_{6} - 60\beta_{5} - 364\beta_{4} + 352\beta_{2} + 865\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) | \(496\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 469.1 |
|
− | 2.47948i | 0 | −4.14785 | 1.72481 | − | 1.42303i | 0 | − | 0.949959i | 5.32555i | 0 | −3.52839 | − | 4.27665i | ||||||||||||||||||||||||||||||||||||||||||
| 469.2 | − | 2.26036i | 0 | −3.10922 | 2.23266 | − | 0.123438i | 0 | 4.96953i | 2.50723i | 0 | −0.279015 | − | 5.04661i | ||||||||||||||||||||||||||||||||||||||||||||
| 469.3 | − | 1.77159i | 0 | −1.13853 | −1.51036 | + | 1.64888i | 0 | − | 0.437634i | − | 1.52618i | 0 | 2.92114 | + | 2.67573i | ||||||||||||||||||||||||||||||||||||||||||
| 469.4 | − | 1.22308i | 0 | 0.504085 | 0.638796 | + | 2.14288i | 0 | 4.18388i | − | 3.06269i | 0 | 2.62091 | − | 0.781296i | |||||||||||||||||||||||||||||||||||||||||||
| 469.5 | − | 0.329386i | 0 | 1.89150 | −2.08591 | + | 0.805596i | 0 | − | 3.70203i | − | 1.28181i | 0 | 0.265352 | + | 0.687069i | ||||||||||||||||||||||||||||||||||||||||||
| 469.6 | 0.329386i | 0 | 1.89150 | −2.08591 | − | 0.805596i | 0 | 3.70203i | 1.28181i | 0 | 0.265352 | − | 0.687069i | |||||||||||||||||||||||||||||||||||||||||||||
| 469.7 | 1.22308i | 0 | 0.504085 | 0.638796 | − | 2.14288i | 0 | − | 4.18388i | 3.06269i | 0 | 2.62091 | + | 0.781296i | ||||||||||||||||||||||||||||||||||||||||||||
| 469.8 | 1.77159i | 0 | −1.13853 | −1.51036 | − | 1.64888i | 0 | 0.437634i | 1.52618i | 0 | 2.92114 | − | 2.67573i | |||||||||||||||||||||||||||||||||||||||||||||
| 469.9 | 2.26036i | 0 | −3.10922 | 2.23266 | + | 0.123438i | 0 | − | 4.96953i | − | 2.50723i | 0 | −0.279015 | + | 5.04661i | |||||||||||||||||||||||||||||||||||||||||||
| 469.10 | 2.47948i | 0 | −4.14785 | 1.72481 | + | 1.42303i | 0 | 0.949959i | − | 5.32555i | 0 | −3.52839 | + | 4.27665i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 585.2.c.c | 10 | |
| 3.b | odd | 2 | 1 | 195.2.c.b | ✓ | 10 | |
| 5.b | even | 2 | 1 | inner | 585.2.c.c | 10 | |
| 5.c | odd | 4 | 1 | 2925.2.a.bl | 5 | ||
| 5.c | odd | 4 | 1 | 2925.2.a.bm | 5 | ||
| 12.b | even | 2 | 1 | 3120.2.l.p | 10 | ||
| 15.d | odd | 2 | 1 | 195.2.c.b | ✓ | 10 | |
| 15.e | even | 4 | 1 | 975.2.a.r | 5 | ||
| 15.e | even | 4 | 1 | 975.2.a.s | 5 | ||
| 60.h | even | 2 | 1 | 3120.2.l.p | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.2.c.b | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 195.2.c.b | ✓ | 10 | 15.d | odd | 2 | 1 | |
| 585.2.c.c | 10 | 1.a | even | 1 | 1 | trivial | |
| 585.2.c.c | 10 | 5.b | even | 2 | 1 | inner | |
| 975.2.a.r | 5 | 15.e | even | 4 | 1 | ||
| 975.2.a.s | 5 | 15.e | even | 4 | 1 | ||
| 2925.2.a.bl | 5 | 5.c | odd | 4 | 1 | ||
| 2925.2.a.bm | 5 | 5.c | odd | 4 | 1 | ||
| 3120.2.l.p | 10 | 12.b | even | 2 | 1 | ||
| 3120.2.l.p | 10 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 16T_{2}^{8} + 90T_{2}^{6} + 208T_{2}^{4} + 169T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} + 16 T^{8} + 90 T^{6} + 208 T^{4} + \cdots + 16 \)
$3$
\( T^{10} \)
$5$
\( T^{10} - 2 T^{9} - 3 T^{8} + 8 T^{7} + \cdots + 3125 \)
$7$
\( T^{10} + 57 T^{8} + 1072 T^{6} + \cdots + 1024 \)
$11$
\( (T^{5} + 5 T^{4} - 24 T^{3} - 100 T^{2} + \cdots + 452)^{2} \)
$13$
\( (T^{2} + 1)^{5} \)
$17$
\( T^{10} + 73 T^{8} + 1744 T^{6} + \cdots + 6400 \)
$19$
\( (T^{5} + 8 T^{4} - 40 T^{3} - 448 T^{2} + \cdots + 1280)^{2} \)
$23$
\( T^{10} + 161 T^{8} + 8096 T^{6} + \cdots + 102400 \)
$29$
\( (T^{5} - 8 T^{4} - 24 T^{3} + 192 T^{2} + \cdots - 640)^{2} \)
$31$
\( (T^{5} - 12 T^{4} - 40 T^{3} + 736 T^{2} + \cdots + 128)^{2} \)
$37$
\( T^{10} + 209 T^{8} + 11312 T^{6} + \cdots + 350464 \)
$41$
\( (T^{5} + 5 T^{4} - 48 T^{3} - 204 T^{2} + \cdots + 196)^{2} \)
$43$
\( T^{10} + 128 T^{8} + 4960 T^{6} + \cdots + 200704 \)
$47$
\( T^{10} + 204 T^{8} + 10360 T^{6} + \cdots + 64 \)
$53$
\( T^{10} + 217 T^{8} + 2608 T^{6} + \cdots + 256 \)
$59$
\( (T^{5} - 8 T^{4} - 108 T^{3} + 472 T^{2} + \cdots - 400)^{2} \)
$61$
\( (T^{5} - 13 T^{4} + 128 T^{2} - 256)^{2} \)
$67$
\( T^{10} + 320 T^{8} + \cdots + 15745024 \)
$71$
\( (T^{5} + 5 T^{4} - 208 T^{3} - 1220 T^{2} + \cdots + 28868)^{2} \)
$73$
\( T^{10} + 436 T^{8} + \cdots + 223323136 \)
$79$
\( (T^{5} + T^{4} - 248 T^{3} - 728 T^{2} + \cdots + 400)^{2} \)
$83$
\( T^{10} + 588 T^{8} + \cdots + 3685946944 \)
$89$
\( (T^{5} - 19 T^{4} - 56 T^{3} + 1108 T^{2} + \cdots + 4900)^{2} \)
$97$
\( T^{10} + 649 T^{8} + \cdots + 1260534016 \)
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