Newspace parameters
Level: | \( N \) | \(=\) | \( 285 = 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 285.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.27573645761\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
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
\( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( -\nu^{8} + 3\nu^{7} - 9\nu^{6} + 15\nu^{5} - 45\nu^{4} + 135\nu^{3} - 86\nu^{2} + 24\nu - 72 ) / 234 \)
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\(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - 3\nu^{8} + 9\nu^{7} - 15\nu^{6} + 45\nu^{5} - 135\nu^{4} + 86\nu^{3} - 258\nu^{2} + 72\nu - 702 ) / 234 \)
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\(\beta_{4}\) | \(=\) |
\( ( 9 \nu^{9} - 9 \nu^{8} + 79 \nu^{7} - 12 \nu^{6} + 486 \nu^{5} - 132 \nu^{4} + 1035 \nu^{3} + 279 \nu^{2} + 1529 \nu - 420 ) / 468 \)
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\(\beta_{5}\) | \(=\) |
\( ( 28 \nu^{9} - 44 \nu^{8} + 275 \nu^{7} - 229 \nu^{6} + 1713 \nu^{5} - 1551 \nu^{4} + 4223 \nu^{3} - 2497 \nu^{2} + 6334 \nu - 3932 ) / 702 \)
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\(\beta_{6}\) | \(=\) |
\( ( 31 \nu^{9} - 5 \nu^{8} + 275 \nu^{7} + 158 \nu^{6} + 1830 \nu^{5} + 906 \nu^{4} + 4319 \nu^{3} + 2963 \nu^{2} + 6685 \nu + 2332 ) / 702 \)
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\(\beta_{7}\) | \(=\) |
\( ( - 34 \nu^{9} - \nu^{8} - 257 \nu^{7} - 248 \nu^{6} - 1740 \nu^{5} - 1176 \nu^{4} - 3254 \nu^{3} - 3479 \nu^{2} - 6541 \nu - 1306 ) / 702 \)
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\(\beta_{8}\) | \(=\) |
\( ( - 9 \nu^{9} + 9 \nu^{8} - 79 \nu^{7} + 12 \nu^{6} - 486 \nu^{5} + 132 \nu^{4} - 1035 \nu^{3} - 123 \nu^{2} - 1529 \nu + 420 ) / 156 \)
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\(\beta_{9}\) | \(=\) |
\( ( - 49 \nu^{9} + 56 \nu^{8} - 428 \nu^{7} + 163 \nu^{6} - 2595 \nu^{5} + 1389 \nu^{4} - 5228 \nu^{3} + 1423 \nu^{2} - 7909 \nu + 4160 ) / 702 \)
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\(\nu\) | \(=\) |
\( \beta_1 \)
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\(\nu^{2}\) | \(=\) |
\( \beta_{8} + 3\beta_{4} \)
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\(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{6} - \beta_{3} + 5\beta_{2} \)
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\(\nu^{4}\) | \(=\) |
\( \beta_{9} - 8\beta_{8} - 2\beta_{6} - 14\beta_{4} - 8\beta_{3} + \beta_{2} - 14 \)
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\(\nu^{5}\) | \(=\) |
\( 10\beta_{9} - 11\beta_{8} - \beta_{6} + 8\beta_{5} - 11\beta_{4} - 28\beta_{2} - 19\beta_1 \)
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\(\nu^{6}\) | \(=\) |
\( 9\beta_{9} + 3\beta_{7} + 12\beta_{6} + 57\beta_{3} - 24\beta_{2} + 76 \)
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\(\nu^{7}\) | \(=\) |
\( - 66 \beta_{9} + 96 \beta_{8} + 54 \beta_{7} + 78 \beta_{6} - 54 \beta_{5} + 96 \beta_{4} + 96 \beta_{3} - 12 \beta_{2} + 103 \beta _1 + 42 \)
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\(\nu^{8}\) | \(=\) |
\( -174\beta_{9} + 397\beta_{8} + 66\beta_{6} - 42\beta_{5} + 495\beta_{4} + 156\beta_{2} + 48\beta_1 \)
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\(\nu^{9}\) | \(=\) |
\( -108\beta_{9} - 355\beta_{7} - 463\beta_{6} - 769\beta_{3} + 1181\beta_{2} - 426 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).
\(n\) | \(172\) | \(191\) | \(211\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
106.1 |
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−1.12375 | + | 1.94639i | 0.500000 | − | 0.866025i | −1.52562 | − | 2.64245i | −0.500000 | + | 0.866025i | 1.12375 | + | 1.94639i | 3.16638 | 2.36264 | −0.500000 | − | 0.866025i | −1.12375 | − | 1.94639i | ||||||||||||||||||||||||||||||||||
106.2 | −0.690702 | + | 1.19633i | 0.500000 | − | 0.866025i | 0.0458624 | + | 0.0794360i | −0.500000 | + | 0.866025i | 0.690702 | + | 1.19633i | −4.36264 | −2.88952 | −0.500000 | − | 0.866025i | −0.690702 | − | 1.19633i | |||||||||||||||||||||||||||||||||||
106.3 | 0.145349 | − | 0.251751i | 0.500000 | − | 0.866025i | 0.957748 | + | 1.65887i | −0.500000 | + | 0.866025i | −0.145349 | − | 0.251751i | −0.486575 | 1.13822 | −0.500000 | − | 0.866025i | 0.145349 | + | 0.251751i | |||||||||||||||||||||||||||||||||||
106.4 | 0.823305 | − | 1.42601i | 0.500000 | − | 0.866025i | −0.355663 | − | 0.616027i | −0.500000 | + | 0.866025i | −0.823305 | − | 1.42601i | 4.47988 | 2.12194 | −0.500000 | − | 0.866025i | 0.823305 | + | 1.42601i | |||||||||||||||||||||||||||||||||||
106.5 | 1.34580 | − | 2.33099i | 0.500000 | − | 0.866025i | −2.62233 | − | 4.54201i | −0.500000 | + | 0.866025i | −1.34580 | − | 2.33099i | −0.797044 | −8.73329 | −0.500000 | − | 0.866025i | 1.34580 | + | 2.33099i | |||||||||||||||||||||||||||||||||||
121.1 | −1.12375 | − | 1.94639i | 0.500000 | + | 0.866025i | −1.52562 | + | 2.64245i | −0.500000 | − | 0.866025i | 1.12375 | − | 1.94639i | 3.16638 | 2.36264 | −0.500000 | + | 0.866025i | −1.12375 | + | 1.94639i | |||||||||||||||||||||||||||||||||||
121.2 | −0.690702 | − | 1.19633i | 0.500000 | + | 0.866025i | 0.0458624 | − | 0.0794360i | −0.500000 | − | 0.866025i | 0.690702 | − | 1.19633i | −4.36264 | −2.88952 | −0.500000 | + | 0.866025i | −0.690702 | + | 1.19633i | |||||||||||||||||||||||||||||||||||
121.3 | 0.145349 | + | 0.251751i | 0.500000 | + | 0.866025i | 0.957748 | − | 1.65887i | −0.500000 | − | 0.866025i | −0.145349 | + | 0.251751i | −0.486575 | 1.13822 | −0.500000 | + | 0.866025i | 0.145349 | − | 0.251751i | |||||||||||||||||||||||||||||||||||
121.4 | 0.823305 | + | 1.42601i | 0.500000 | + | 0.866025i | −0.355663 | + | 0.616027i | −0.500000 | − | 0.866025i | −0.823305 | + | 1.42601i | 4.47988 | 2.12194 | −0.500000 | + | 0.866025i | 0.823305 | − | 1.42601i | |||||||||||||||||||||||||||||||||||
121.5 | 1.34580 | + | 2.33099i | 0.500000 | + | 0.866025i | −2.62233 | + | 4.54201i | −0.500000 | − | 0.866025i | −1.34580 | + | 2.33099i | −0.797044 | −8.73329 | −0.500000 | + | 0.866025i | 1.34580 | − | 2.33099i | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 285.2.i.f | ✓ | 10 |
3.b | odd | 2 | 1 | 855.2.k.i | 10 | ||
19.c | even | 3 | 1 | inner | 285.2.i.f | ✓ | 10 |
19.c | even | 3 | 1 | 5415.2.a.y | 5 | ||
19.d | odd | 6 | 1 | 5415.2.a.z | 5 | ||
57.h | odd | 6 | 1 | 855.2.k.i | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
285.2.i.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
285.2.i.f | ✓ | 10 | 19.c | even | 3 | 1 | inner |
855.2.k.i | 10 | 3.b | odd | 2 | 1 | ||
855.2.k.i | 10 | 57.h | odd | 6 | 1 | ||
5415.2.a.y | 5 | 19.c | even | 3 | 1 | ||
5415.2.a.z | 5 | 19.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - T_{2}^{9} + 9T_{2}^{8} - 2T_{2}^{7} + 56T_{2}^{6} - 18T_{2}^{5} + 125T_{2}^{4} + T_{2}^{3} + 189T_{2}^{2} - 52T_{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} - T^{9} + 9 T^{8} - 2 T^{7} + \cdots + 16 \)
$3$
\( (T^{2} - T + 1)^{5} \)
$5$
\( (T^{2} + T + 1)^{5} \)
$7$
\( (T^{5} - 2 T^{4} - 23 T^{3} + 36 T^{2} + \cdots + 24)^{2} \)
$11$
\( (T^{5} - 5 T^{4} - 32 T^{3} + 148 T^{2} + \cdots - 992)^{2} \)
$13$
\( T^{10} - 8 T^{9} + 63 T^{8} + \cdots + 16384 \)
$17$
\( T^{10} + 10 T^{9} + 135 T^{8} + \cdots + 25240576 \)
$19$
\( T^{10} - 5 T^{9} + 18 T^{8} + \cdots + 2476099 \)
$23$
\( T^{10} - 2 T^{9} + 72 T^{8} + \cdots + 147456 \)
$29$
\( T^{10} - 7 T^{9} + 111 T^{8} + \cdots + 9048064 \)
$31$
\( (T^{5} + 9 T^{4} - 30 T^{3} - 222 T^{2} + \cdots + 117)^{2} \)
$37$
\( (T^{5} - 6 T^{4} - 87 T^{3} + 472 T^{2} + \cdots - 9024)^{2} \)
$41$
\( T^{10} + 12 T^{9} + 120 T^{8} + \cdots + 36864 \)
$43$
\( T^{10} - 8 T^{9} + 63 T^{8} + \cdots + 16384 \)
$47$
\( T^{10} + 6 T^{9} + 189 T^{8} + \cdots + 4562496 \)
$53$
\( T^{10} + 8 T^{9} + 159 T^{8} + \cdots + 18731584 \)
$59$
\( T^{10} - T^{9} + 177 T^{8} + \cdots + 20647936 \)
$61$
\( T^{10} + 7 T^{9} + 222 T^{8} + \cdots + 591267856 \)
$67$
\( T^{10} - 14 T^{9} + 363 T^{8} + \cdots + 70157376 \)
$71$
\( T^{10} - 27 T^{9} + 507 T^{8} + \cdots + 37161216 \)
$73$
\( T^{10} + 26 T^{9} + \cdots + 2135179264 \)
$79$
\( T^{10} + 23 T^{9} + \cdots + 11935125504 \)
$83$
\( (T^{5} + 12 T^{4} - 21 T^{3} - 660 T^{2} + \cdots - 2304)^{2} \)
$89$
\( T^{10} - 9 T^{9} + 213 T^{8} + \cdots + 126157824 \)
$97$
\( (T^{2} + 2 T + 4)^{5} \)
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