Newspace parameters
Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 570.q (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.55147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 12.0.89539436150784.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) :
\(\beta_{1}\) | \(=\) | \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + 144 \nu^{3} - 48 \nu^{2} + 48 \nu + 748 ) / 460 \) |
\(\beta_{2}\) | \(=\) | \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612 ) / 460 \) |
\(\beta_{3}\) | \(=\) | \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1370 \nu^{2} + 12 \nu - 12 ) / 460 \) |
\(\beta_{4}\) | \(=\) | \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142 ) / 230 \) |
\(\beta_{5}\) | \(=\) | \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} - 110 \nu^{4} + 50 \nu^{3} + 1140 \nu^{2} - 12 \nu + 12 ) / 230 \) |
\(\beta_{6}\) | \(=\) | \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + 278 \nu^{4} - 118 \nu^{3} - 1982 \nu^{2} + 32 \nu - 32 ) / 460 \) |
\(\beta_{7}\) | \(=\) | \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + 1614 \nu^{4} + 3424 \nu^{3} + 484 \nu^{2} - 484 \nu - 420 ) / 460 \) |
\(\beta_{8}\) | \(=\) | \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12 ) / 20 \) |
\(\beta_{9}\) | \(=\) | \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476 ) / 460 \) |
\(\beta_{10}\) | \(=\) | \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + 2510 \nu^{4} + 2790 \nu^{3} - 1294 \nu^{2} + 1060 \nu - 1060 ) / 460 \) |
\(\beta_{11}\) | \(=\) | \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612 ) / 230 \) |
\(\nu\) | \(=\) | \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{5} - 2\beta_{3} \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) |
\(\nu^{4}\) | \(=\) | \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) |
\(\nu^{6}\) | \(=\) | \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) |
\(\nu^{7}\) | \(=\) | \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 33 \beta_{5} + 39 \beta_{4} + 39 \beta_{3} + 11 \beta_{2} - 33 \beta_1 \) |
\(\nu^{8}\) | \(=\) | \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) |
\(\nu^{9}\) | \(=\) | \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) |
\(\nu^{10}\) | \(=\) | \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) |
\(\nu^{11}\) | \(=\) | \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(211\) | \(457\) |
\(\chi(n)\) | \(1\) | \(-1 + \beta_{8}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
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−0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −0.837733 | + | 2.07321i | 0.500000 | + | 0.866025i | − | 0.785680i | − | 1.00000i | 0.500000 | + | 0.866025i | 1.76210 | − | 1.37659i | ||||||||||||||||||||||||||||||||||||||
49.2 | −0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −0.445186 | − | 2.19130i | 0.500000 | + | 0.866025i | − | 4.67513i | − | 1.00000i | 0.500000 | + | 0.866025i | −0.710109 | + | 2.12032i | |||||||||||||||||||||||||||||||||||||||
49.3 | −0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 2.14894 | + | 0.618092i | 0.500000 | + | 0.866025i | − | 3.53919i | − | 1.00000i | 0.500000 | + | 0.866025i | −1.55199 | − | 1.60976i | |||||||||||||||||||||||||||||||||||||||
49.4 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −1.60976 | − | 1.55199i | 0.500000 | + | 0.866025i | 3.53919i | 1.00000i | 0.500000 | + | 0.866025i | −0.618092 | − | 2.14894i | |||||||||||||||||||||||||||||||||||||||||
49.5 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −1.37659 | + | 1.76210i | 0.500000 | + | 0.866025i | 0.785680i | 1.00000i | 0.500000 | + | 0.866025i | −2.07321 | + | 0.837733i | |||||||||||||||||||||||||||||||||||||||||
49.6 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 2.12032 | − | 0.710109i | 0.500000 | + | 0.866025i | 4.67513i | 1.00000i | 0.500000 | + | 0.866025i | 2.19130 | + | 0.445186i | |||||||||||||||||||||||||||||||||||||||||
349.1 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −0.837733 | − | 2.07321i | 0.500000 | − | 0.866025i | 0.785680i | 1.00000i | 0.500000 | − | 0.866025i | 1.76210 | + | 1.37659i | |||||||||||||||||||||||||||||||||||||||||
349.2 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −0.445186 | + | 2.19130i | 0.500000 | − | 0.866025i | 4.67513i | 1.00000i | 0.500000 | − | 0.866025i | −0.710109 | − | 2.12032i | |||||||||||||||||||||||||||||||||||||||||
349.3 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 2.14894 | − | 0.618092i | 0.500000 | − | 0.866025i | 3.53919i | 1.00000i | 0.500000 | − | 0.866025i | −1.55199 | + | 1.60976i | |||||||||||||||||||||||||||||||||||||||||
349.4 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −1.60976 | + | 1.55199i | 0.500000 | − | 0.866025i | − | 3.53919i | − | 1.00000i | 0.500000 | − | 0.866025i | −0.618092 | + | 2.14894i | |||||||||||||||||||||||||||||||||||||||
349.5 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −1.37659 | − | 1.76210i | 0.500000 | − | 0.866025i | − | 0.785680i | − | 1.00000i | 0.500000 | − | 0.866025i | −2.07321 | − | 0.837733i | |||||||||||||||||||||||||||||||||||||||
349.6 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 2.12032 | + | 0.710109i | 0.500000 | − | 0.866025i | − | 4.67513i | − | 1.00000i | 0.500000 | − | 0.866025i | 2.19130 | − | 0.445186i | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
95.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 570.2.q.b | ✓ | 12 |
3.b | odd | 2 | 1 | 1710.2.t.b | 12 | ||
5.b | even | 2 | 1 | inner | 570.2.q.b | ✓ | 12 |
15.d | odd | 2 | 1 | 1710.2.t.b | 12 | ||
19.c | even | 3 | 1 | inner | 570.2.q.b | ✓ | 12 |
57.h | odd | 6 | 1 | 1710.2.t.b | 12 | ||
95.i | even | 6 | 1 | inner | 570.2.q.b | ✓ | 12 |
285.n | odd | 6 | 1 | 1710.2.t.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
570.2.q.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
570.2.q.b | ✓ | 12 | 5.b | even | 2 | 1 | inner |
570.2.q.b | ✓ | 12 | 19.c | even | 3 | 1 | inner |
570.2.q.b | ✓ | 12 | 95.i | even | 6 | 1 | inner |
1710.2.t.b | 12 | 3.b | odd | 2 | 1 | ||
1710.2.t.b | 12 | 15.d | odd | 2 | 1 | ||
1710.2.t.b | 12 | 57.h | odd | 6 | 1 | ||
1710.2.t.b | 12 | 285.n | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 35T_{7}^{4} + 295T_{7}^{2} + 169 \)
acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{3} \)
$3$
\( (T^{4} - T^{2} + 1)^{3} \)
$5$
\( T^{12} + T^{10} - 32 T^{9} + 6 T^{8} + \cdots + 15625 \)
$7$
\( (T^{6} + 35 T^{4} + 295 T^{2} + 169)^{2} \)
$11$
\( (T^{3} + T^{2} - 13 T - 5)^{4} \)
$13$
\( T^{12} - 12 T^{10} + 112 T^{8} + \cdots + 256 \)
$17$
\( T^{12} - 44 T^{10} + 1336 T^{8} + \cdots + 6250000 \)
$19$
\( (T^{6} - 3 T^{5} + 24 T^{4} - 25 T^{3} + \cdots + 6859)^{2} \)
$23$
\( T^{12} - 195 T^{10} + \cdots + 58120048561 \)
$29$
\( (T^{6} + 8 T^{5} + 80 T^{4} + 132 T^{3} + \cdots + 16900)^{2} \)
$31$
\( (T^{3} - 16 T + 16)^{4} \)
$37$
\( (T^{6} + 27 T^{4} + 107 T^{2} + 1)^{2} \)
$41$
\( (T^{6} - 5 T^{5} + 96 T^{4} + 633 T^{3} + \cdots + 19321)^{2} \)
$43$
\( T^{12} - 80 T^{10} + 4992 T^{8} + \cdots + 65536 \)
$47$
\( T^{12} - 264 T^{10} + \cdots + 147763360000 \)
$53$
\( T^{12} - 207 T^{10} + \cdots + 332150625 \)
$59$
\( (T^{6} - 4 T^{5} + 104 T^{4} + \cdots + 204304)^{2} \)
$61$
\( (T^{6} + 14 T^{5} + 214 T^{4} + \cdots + 77284)^{2} \)
$67$
\( T^{12} - 208 T^{10} + \cdots + 102627966736 \)
$71$
\( (T^{6} - 22 T^{5} + 346 T^{4} + \cdots + 42436)^{2} \)
$73$
\( T^{12} - 180 T^{10} + \cdots + 623201296 \)
$79$
\( (T^{6} + 18 T^{5} + 232 T^{4} + \cdots + 10816)^{2} \)
$83$
\( (T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 16384)^{2} \)
$89$
\( (T^{6} - 3 T^{5} + 286 T^{4} + \cdots + 4133089)^{2} \)
$97$
\( T^{12} - 200 T^{10} + \cdots + 3906250000 \)
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