Newspace parameters
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.30580664368\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 10.0.1481180578947.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} - 2x^{9} + 6x^{8} - 11x^{7} + 22x^{6} - 45x^{5} + 66x^{4} - 99x^{3} + 162x^{2} - 162x + 243 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
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 \( x^{10} - 2x^{9} + 6x^{8} - 11x^{7} + 22x^{6} - 45x^{5} + 66x^{4} - 99x^{3} + 162x^{2} - 162x + 243 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{9} + 25\nu^{8} - 3\nu^{7} - 47\nu^{6} - 32\nu^{5} + 135\nu^{4} + 57\nu^{3} + 198\nu^{2} + 1188\nu + 648 ) / 1701 \) |
\(\beta_{3}\) | \(=\) | \( ( 4\nu^{9} - 26\nu^{8} + 51\nu^{7} + \nu^{6} + 124\nu^{5} - 153\nu^{4} + 606\nu^{3} - 720\nu^{2} - 351\nu - 810 ) / 1701 \) |
\(\beta_{4}\) | \(=\) | \( ( 2 \nu^{9} - 37 \nu^{8} + 42 \nu^{7} - 121 \nu^{6} + 218 \nu^{5} - 339 \nu^{4} + 528 \nu^{3} - 810 \nu^{2} + 891 \nu - 2997 ) / 1701 \) |
\(\beta_{5}\) | \(=\) | \( ( - 10 \nu^{9} + 23 \nu^{8} - 75 \nu^{7} + 92 \nu^{6} - 37 \nu^{5} + 372 \nu^{4} - 318 \nu^{3} + 729 \nu^{2} - 162 \nu - 810 ) / 1701 \) |
\(\beta_{6}\) | \(=\) | \( ( 11 \nu^{9} - 19 \nu^{8} + 51 \nu^{7} - 139 \nu^{6} + 425 \nu^{5} - 573 \nu^{4} + 1068 \nu^{3} - 1350 \nu^{2} + 3240 \nu - 2511 ) / 1701 \) |
\(\beta_{7}\) | \(=\) | \( ( - 14 \nu^{9} + 25 \nu^{8} - 15 \nu^{7} + 64 \nu^{6} - 5 \nu^{5} + 357 \nu^{4} - 321 \nu^{3} - 135 \nu^{2} - 162 \nu - 891 ) / 1701 \) |
\(\beta_{8}\) | \(=\) | \( ( 11\nu^{9} - 10\nu^{8} + 33\nu^{7} - 58\nu^{6} + 83\nu^{5} - 132\nu^{4} + 204\nu^{3} - 189\nu^{2} + 891\nu + 162 ) / 567 \) |
\(\beta_{9}\) | \(=\) | \( ( - 37 \nu^{9} + 68 \nu^{8} - 111 \nu^{7} + 281 \nu^{6} - 451 \nu^{5} + 1011 \nu^{4} - 1425 \nu^{3} + 2079 \nu^{2} - 3564 \nu + 3321 ) / 1701 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} - 2 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{8} - 2\beta_{7} + \beta_{5} + 3\beta_{3} + 3\beta_{2} \) |
\(\nu^{4}\) | \(=\) | \( -2\beta_{9} + \beta_{8} + 3\beta_{7} - 4\beta_{6} + 4\beta_{5} + 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + 5 \) |
\(\nu^{5}\) | \(=\) | \( 4\beta_{9} + 5\beta_{8} + 2\beta_{7} + 5\beta_{6} + 3\beta_{5} + 2\beta_{4} - 3\beta_{3} - 6\beta_{2} - 6\beta _1 + 5 \) |
\(\nu^{6}\) | \(=\) | \( 3\beta_{9} + \beta_{8} - \beta_{7} + 6\beta_{6} - \beta_{5} - 12\beta_{4} + 6\beta_{3} - 12\beta_{2} + 9\beta _1 - 12 \) |
\(\nu^{7}\) | \(=\) | \( 23\beta_{9} + 14\beta_{8} + 10\beta_{6} - 25\beta_{5} + 4\beta_{4} + 6\beta_{3} - 3\beta_{2} + 6\beta _1 - 35 \) |
\(\nu^{8}\) | \(=\) | \( 11 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 34 \beta_{6} - 27 \beta_{5} - 35 \beta_{4} - 15 \beta_{3} + 15 \beta_{2} - 36 \beta _1 - 56 \) |
\(\nu^{9}\) | \(=\) | \( - 63 \beta_{9} + 23 \beta_{8} + 22 \beta_{7} - 36 \beta_{6} + 52 \beta_{5} - 81 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 39 \beta _1 - 36 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).
\(n\) | \(47\) | \(235\) |
\(\chi(n)\) | \(\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 |
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−0.500000 | − | 0.866025i | −1.66620 | − | 0.473061i | −0.500000 | + | 0.866025i | 0.274896 | − | 0.476134i | 0.423416 | + | 1.67950i | 0.708031 | + | 1.22635i | 1.00000 | 2.55243 | + | 1.57642i | −0.549792 | ||||||||||||||||||||||||||||||||||
139.2 | −0.500000 | − | 0.866025i | −0.944171 | + | 1.45208i | −0.500000 | + | 0.866025i | 1.85973 | − | 3.22115i | 1.72963 | + | 0.0916356i | 2.00591 | + | 3.47433i | 1.00000 | −1.21708 | − | 2.74203i | −3.71947 | |||||||||||||||||||||||||||||||||||
139.3 | −0.500000 | − | 0.866025i | 0.365780 | + | 1.69299i | −0.500000 | + | 0.866025i | 0.231157 | − | 0.400376i | 1.28328 | − | 1.16327i | 0.165447 | + | 0.286563i | 1.00000 | −2.73241 | + | 1.23852i | −0.462315 | |||||||||||||||||||||||||||||||||||
139.4 | −0.500000 | − | 0.866025i | 1.07051 | − | 1.36162i | −0.500000 | + | 0.866025i | −1.08471 | + | 1.87878i | −1.71445 | − | 0.246277i | 1.41120 | + | 2.44426i | 1.00000 | −0.708023 | − | 2.91525i | 2.16943 | |||||||||||||||||||||||||||||||||||
139.5 | −0.500000 | − | 0.866025i | 1.67408 | − | 0.444362i | −0.500000 | + | 0.866025i | 1.21893 | − | 2.11124i | −1.22187 | − | 1.22761i | −1.79058 | − | 3.10138i | 1.00000 | 2.60509 | − | 1.48779i | −2.43785 | |||||||||||||||||||||||||||||||||||
277.1 | −0.500000 | + | 0.866025i | −1.66620 | + | 0.473061i | −0.500000 | − | 0.866025i | 0.274896 | + | 0.476134i | 0.423416 | − | 1.67950i | 0.708031 | − | 1.22635i | 1.00000 | 2.55243 | − | 1.57642i | −0.549792 | |||||||||||||||||||||||||||||||||||
277.2 | −0.500000 | + | 0.866025i | −0.944171 | − | 1.45208i | −0.500000 | − | 0.866025i | 1.85973 | + | 3.22115i | 1.72963 | − | 0.0916356i | 2.00591 | − | 3.47433i | 1.00000 | −1.21708 | + | 2.74203i | −3.71947 | |||||||||||||||||||||||||||||||||||
277.3 | −0.500000 | + | 0.866025i | 0.365780 | − | 1.69299i | −0.500000 | − | 0.866025i | 0.231157 | + | 0.400376i | 1.28328 | + | 1.16327i | 0.165447 | − | 0.286563i | 1.00000 | −2.73241 | − | 1.23852i | −0.462315 | |||||||||||||||||||||||||||||||||||
277.4 | −0.500000 | + | 0.866025i | 1.07051 | + | 1.36162i | −0.500000 | − | 0.866025i | −1.08471 | − | 1.87878i | −1.71445 | + | 0.246277i | 1.41120 | − | 2.44426i | 1.00000 | −0.708023 | + | 2.91525i | 2.16943 | |||||||||||||||||||||||||||||||||||
277.5 | −0.500000 | + | 0.866025i | 1.67408 | + | 0.444362i | −0.500000 | − | 0.866025i | 1.21893 | + | 2.11124i | −1.22187 | + | 1.22761i | −1.79058 | + | 3.10138i | 1.00000 | 2.60509 | + | 1.48779i | −2.43785 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 414.2.e.c | ✓ | 10 |
3.b | odd | 2 | 1 | 1242.2.e.c | 10 | ||
9.c | even | 3 | 1 | inner | 414.2.e.c | ✓ | 10 |
9.c | even | 3 | 1 | 3726.2.a.t | 5 | ||
9.d | odd | 6 | 1 | 1242.2.e.c | 10 | ||
9.d | odd | 6 | 1 | 3726.2.a.s | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.2.e.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
414.2.e.c | ✓ | 10 | 9.c | even | 3 | 1 | inner |
1242.2.e.c | 10 | 3.b | odd | 2 | 1 | ||
1242.2.e.c | 10 | 9.d | odd | 6 | 1 | ||
3726.2.a.s | 5 | 9.d | odd | 6 | 1 | ||
3726.2.a.t | 5 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 5 T_{5}^{9} + 25 T_{5}^{8} - 46 T_{5}^{7} + 136 T_{5}^{6} - 205 T_{5}^{5} + 554 T_{5}^{4} - 483 T_{5}^{3} + 326 T_{5}^{2} - 105 T_{5} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + T + 1)^{5} \)
$3$
\( T^{10} - T^{9} + 2 T^{7} - 2 T^{6} + \cdots + 243 \)
$5$
\( T^{10} - 5 T^{9} + 25 T^{8} - 46 T^{7} + \cdots + 25 \)
$7$
\( T^{10} - 5 T^{9} + 32 T^{8} - 89 T^{7} + \cdots + 361 \)
$11$
\( T^{10} - 3 T^{9} + 46 T^{8} + \cdots + 710649 \)
$13$
\( T^{10} - 8 T^{9} + 68 T^{8} - 122 T^{7} + \cdots + 361 \)
$17$
\( (T^{5} + T^{4} - 45 T^{3} - 14 T^{2} + \cdots - 501)^{2} \)
$19$
\( (T^{5} + T^{4} - 93 T^{3} - 113 T^{2} + \cdots + 2043)^{2} \)
$23$
\( (T^{2} + T + 1)^{5} \)
$29$
\( T^{10} - 18 T^{9} + 241 T^{8} + \cdots + 463761 \)
$31$
\( T^{10} - 8 T^{9} + 190 T^{8} + \cdots + 112550881 \)
$37$
\( (T^{5} + 6 T^{4} - 55 T^{3} - 255 T^{2} + \cdots + 2043)^{2} \)
$41$
\( T^{10} - 24 T^{9} + \cdots + 157527601 \)
$43$
\( T^{10} + 11 T^{9} + 179 T^{8} + \cdots + 1058841 \)
$47$
\( T^{10} - 9 T^{9} + 91 T^{8} + \cdots + 305809 \)
$53$
\( (T^{5} + 29 T^{4} + 323 T^{3} + 1717 T^{2} + \cdots + 3947)^{2} \)
$59$
\( T^{10} - 21 T^{9} + \cdots + 8393674689 \)
$61$
\( T^{10} - 17 T^{9} + \cdots + 151560721 \)
$67$
\( T^{10} - 3 T^{9} + 204 T^{8} + \cdots + 393824025 \)
$71$
\( (T^{5} + 9 T^{4} - 202 T^{3} - 1875 T^{2} + \cdots + 17079)^{2} \)
$73$
\( (T^{5} + 7 T^{4} - 97 T^{3} - 769 T^{2} + \cdots - 89)^{2} \)
$79$
\( T^{10} - 15 T^{9} + 145 T^{8} + \cdots + 1089 \)
$83$
\( T^{10} - 21 T^{9} + 323 T^{8} + \cdots + 2253001 \)
$89$
\( (T^{5} + 9 T^{4} - 115 T^{3} - 273 T^{2} + \cdots - 849)^{2} \)
$97$
\( T^{10} + 32 T^{9} + \cdots + 1093955625 \)
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