Newspace parameters
Level: | \( N \) | \(=\) | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1638.p (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.0794958511\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 10.0.23207289578928.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 182) |
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 \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) :
\(\beta_{1}\) | \(=\) | \( ( 2\nu^{9} - \nu^{8} - 21\nu^{7} + 35\nu^{6} + 68\nu^{5} - 111\nu^{4} - 201\nu^{3} + 351\nu^{2} + 405\nu - 891 ) / 81 \) |
\(\beta_{2}\) | \(=\) | \( ( -7\nu^{9} + 2\nu^{8} + 45\nu^{7} - 28\nu^{6} - 136\nu^{5} + 99\nu^{4} + 393\nu^{3} - 342\nu^{2} - 675\nu + 567 ) / 81 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{8} + 2\nu^{7} - 8\nu^{6} - 5\nu^{5} + 26\nu^{4} + 13\nu^{3} - 69\nu^{2} - 24\nu + 162 ) / 9 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{8} - 2\nu^{7} + 8\nu^{6} + 5\nu^{5} - 26\nu^{4} - 13\nu^{3} + 78\nu^{2} + 33\nu - 171 ) / 9 \) |
\(\beta_{5}\) | \(=\) | \( ( -4\nu^{9} + 20\nu^{8} + 15\nu^{7} - 88\nu^{6} - 37\nu^{5} + 276\nu^{4} + 87\nu^{3} - 783\nu^{2} + 108\nu + 972 ) / 81 \) |
\(\beta_{6}\) | \(=\) | \( ( 8\nu^{9} + 5\nu^{8} - 30\nu^{7} - 31\nu^{6} + 92\nu^{5} + 96\nu^{4} - 273\nu^{3} - 243\nu^{2} + 459\nu + 729 ) / 81 \) |
\(\beta_{7}\) | \(=\) | \( ( 8\nu^{9} - 4\nu^{8} - 57\nu^{7} + 59\nu^{6} + 164\nu^{5} - 174\nu^{4} - 480\nu^{3} + 567\nu^{2} + 891\nu - 1215 ) / 81 \) |
\(\beta_{8}\) | \(=\) | \( ( -6\nu^{9} + 2\nu^{8} + 35\nu^{7} - 24\nu^{6} - 109\nu^{5} + 77\nu^{4} + 306\nu^{3} - 273\nu^{2} - 531\nu + 432 ) / 27 \) |
\(\beta_{9}\) | \(=\) | \( ( - 23 \nu^{9} + 37 \nu^{8} + 141 \nu^{7} - 245 \nu^{6} - 410 \nu^{5} + 780 \nu^{4} + 1119 \nu^{3} - 2421 \nu^{2} - 1620 \nu + 3969 ) / 81 \) |
\(\nu\) | \(=\) | \( ( \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{2} + 1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} + 2 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( - 2 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 10 \beta_{2} - 6 \beta _1 + 4 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( -\beta_{9} - 2\beta_{8} + 9\beta_{7} - 10\beta_{6} + \beta_{5} - 2\beta_{4} + 9\beta_{3} + 5\beta_{2} - 6\beta _1 - 7 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( - 2 \beta_{9} - 19 \beta_{8} - 12 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 7 \beta_{4} - 9 \beta_{3} + 22 \beta_{2} - 9 \beta _1 + 1 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( -\beta_{9} - 8\beta_{8} + 24\beta_{7} - 13\beta_{6} + \beta_{5} - 17\beta_{4} + 35\beta_{2} - 3\beta _1 + 41 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( 19 \beta_{9} - 28 \beta_{8} - 21 \beta_{7} + \beta_{6} - 31 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 19 \beta_{2} + 51 \beta _1 - 23 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( ( - 49 \beta_{9} + 46 \beta_{8} - 21 \beta_{7} + 41 \beta_{6} + 64 \beta_{5} - 32 \beta_{4} - 12 \beta_{3} + 47 \beta_{2} + 63 \beta _1 + 185 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( - 23 \beta_{9} + 29 \beta_{8} - 45 \beta_{7} + 151 \beta_{6} + 2 \beta_{5} + 50 \beta_{4} + 3 \beta_{3} + 118 \beta_{2} + 111 \beta _1 - 104 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).
\(n\) | \(379\) | \(703\) | \(911\) |
\(\chi(n)\) | \(\beta_{2}\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
919.1 |
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0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.69274 | + | 2.93192i | 0 | −2.60522 | + | 0.461340i | −1.00000 | 0 | −3.38549 | ||||||||||||||||||||||||||||||||||||||||
919.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.451759 | + | 0.782469i | 0 | −0.686411 | − | 2.55516i | −1.00000 | 0 | −0.903518 | |||||||||||||||||||||||||||||||||||||||||
919.3 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.561520 | − | 0.972581i | 0 | 2.64005 | + | 0.173642i | −1.00000 | 0 | 1.12304 | |||||||||||||||||||||||||||||||||||||||||
919.4 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 0.917780 | − | 1.58964i | 0 | 1.05365 | − | 2.42690i | −1.00000 | 0 | 1.83556 | |||||||||||||||||||||||||||||||||||||||||
919.5 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.66520 | − | 2.88422i | 0 | −0.402064 | + | 2.61502i | −1.00000 | 0 | 3.33041 | |||||||||||||||||||||||||||||||||||||||||
991.1 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.69274 | − | 2.93192i | 0 | −2.60522 | − | 0.461340i | −1.00000 | 0 | −3.38549 | |||||||||||||||||||||||||||||||||||||||||
991.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.451759 | − | 0.782469i | 0 | −0.686411 | + | 2.55516i | −1.00000 | 0 | −0.903518 | |||||||||||||||||||||||||||||||||||||||||
991.3 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0.561520 | + | 0.972581i | 0 | 2.64005 | − | 0.173642i | −1.00000 | 0 | 1.12304 | |||||||||||||||||||||||||||||||||||||||||
991.4 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 0.917780 | + | 1.58964i | 0 | 1.05365 | + | 2.42690i | −1.00000 | 0 | 1.83556 | |||||||||||||||||||||||||||||||||||||||||
991.5 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.66520 | + | 2.88422i | 0 | −0.402064 | − | 2.61502i | −1.00000 | 0 | 3.33041 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1638.2.p.k | 10 | |
3.b | odd | 2 | 1 | 182.2.h.d | yes | 10 | |
7.c | even | 3 | 1 | 1638.2.m.j | 10 | ||
13.c | even | 3 | 1 | 1638.2.m.j | 10 | ||
21.c | even | 2 | 1 | 1274.2.h.s | 10 | ||
21.g | even | 6 | 1 | 1274.2.e.s | 10 | ||
21.g | even | 6 | 1 | 1274.2.g.q | 10 | ||
21.h | odd | 6 | 1 | 182.2.e.d | ✓ | 10 | |
21.h | odd | 6 | 1 | 1274.2.g.p | 10 | ||
39.i | odd | 6 | 1 | 182.2.e.d | ✓ | 10 | |
91.g | even | 3 | 1 | inner | 1638.2.p.k | 10 | |
273.r | even | 6 | 1 | 1274.2.g.q | 10 | ||
273.s | odd | 6 | 1 | 1274.2.g.p | 10 | ||
273.bf | even | 6 | 1 | 1274.2.h.s | 10 | ||
273.bm | odd | 6 | 1 | 182.2.h.d | yes | 10 | |
273.bn | even | 6 | 1 | 1274.2.e.s | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
182.2.e.d | ✓ | 10 | 21.h | odd | 6 | 1 | |
182.2.e.d | ✓ | 10 | 39.i | odd | 6 | 1 | |
182.2.h.d | yes | 10 | 3.b | odd | 2 | 1 | |
182.2.h.d | yes | 10 | 273.bm | odd | 6 | 1 | |
1274.2.e.s | 10 | 21.g | even | 6 | 1 | ||
1274.2.e.s | 10 | 273.bn | even | 6 | 1 | ||
1274.2.g.p | 10 | 21.h | odd | 6 | 1 | ||
1274.2.g.p | 10 | 273.s | odd | 6 | 1 | ||
1274.2.g.q | 10 | 21.g | even | 6 | 1 | ||
1274.2.g.q | 10 | 273.r | even | 6 | 1 | ||
1274.2.h.s | 10 | 21.c | even | 2 | 1 | ||
1274.2.h.s | 10 | 273.bf | even | 6 | 1 | ||
1638.2.m.j | 10 | 7.c | even | 3 | 1 | ||
1638.2.m.j | 10 | 13.c | even | 3 | 1 | ||
1638.2.p.k | 10 | 1.a | even | 1 | 1 | trivial | |
1638.2.p.k | 10 | 91.g | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 2 T_{5}^{9} + 16 T_{5}^{8} - 26 T_{5}^{7} + 187 T_{5}^{6} - 293 T_{5}^{5} + 667 T_{5}^{4} - 329 T_{5}^{3} + 574 T_{5}^{2} - 147 T_{5} + 441 \)
acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T + 1)^{5} \)
$3$
\( T^{10} \)
$5$
\( T^{10} - 2 T^{9} + 16 T^{8} - 26 T^{7} + \cdots + 441 \)
$7$
\( T^{10} + 4 T^{8} - 4 T^{7} + \cdots + 16807 \)
$11$
\( (T^{5} + 4 T^{4} - 24 T^{3} - 65 T^{2} + \cdots - 9)^{2} \)
$13$
\( T^{10} + 6 T^{9} + 10 T^{8} + \cdots + 371293 \)
$17$
\( T^{10} - 3 T^{9} + 75 T^{8} + \cdots + 6718464 \)
$19$
\( (T^{5} - 5 T^{4} - 78 T^{3} + 390 T^{2} + \cdots - 7089)^{2} \)
$23$
\( T^{10} + 4 T^{9} + 19 T^{8} + 22 T^{7} + \cdots + 144 \)
$29$
\( T^{10} + T^{9} + 49 T^{8} + \cdots + 103041 \)
$31$
\( T^{10} + 21 T^{9} + 316 T^{8} + \cdots + 32228329 \)
$37$
\( T^{10} - 10 T^{9} + 157 T^{8} + \cdots + 144 \)
$41$
\( T^{10} - 9 T^{9} + 132 T^{8} + \cdots + 7733961 \)
$43$
\( T^{10} - 9 T^{9} + 130 T^{8} + \cdots + 6538249 \)
$47$
\( T^{10} - 13 T^{9} + 166 T^{8} + \cdots + 9529569 \)
$53$
\( T^{10} - 15 T^{9} + 294 T^{8} + \cdots + 5948721 \)
$59$
\( T^{10} + T^{9} + 130 T^{8} + \cdots + 52012944 \)
$61$
\( (T^{5} - 15 T^{4} - 79 T^{3} + 2378 T^{2} + \cdots + 22213)^{2} \)
$67$
\( (T^{5} - 220 T^{3} - 139 T^{2} + \cdots - 4703)^{2} \)
$71$
\( T^{10} - T^{9} + 214 T^{8} + \cdots + 1069878681 \)
$73$
\( T^{10} + 4 T^{9} + 132 T^{8} + \cdots + 41744521 \)
$79$
\( T^{10} + T^{9} + 186 T^{8} + \cdots + 66049 \)
$83$
\( (T^{5} + 48 T^{4} + 837 T^{3} + \cdots + 20412)^{2} \)
$89$
\( T^{10} - 9 T^{9} + 171 T^{8} + \cdots + 44836416 \)
$97$
\( T^{10} + 19 T^{9} + 318 T^{8} + \cdots + 42523441 \)
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