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: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{12} + 4x^{10} - 3x^{9} + 22x^{8} - 9x^{7} + 69x^{6} - 27x^{5} + 198x^{4} - 81x^{3} + 324x^{2} + 729 \) |
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 \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} + 4x^{10} - 3x^{9} + 22x^{8} - 9x^{7} + 69x^{6} - 27x^{5} + 198x^{4} - 81x^{3} + 324x^{2} + 729 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 10 \nu^{11} - 501 \nu^{10} + 572 \nu^{9} - 2514 \nu^{8} + 2678 \nu^{7} - 7881 \nu^{6} + 11532 \nu^{5} - 25200 \nu^{4} + 18216 \nu^{3} - 38205 \nu^{2} + 45765 \nu - 8262 ) / 78003 \) |
\(\beta_{3}\) | \(=\) | \( ( - 11 \nu^{11} + 444 \nu^{10} - 719 \nu^{9} + 1215 \nu^{8} - 3410 \nu^{7} + 7734 \nu^{6} - 11775 \nu^{5} + 17541 \nu^{4} - 21564 \nu^{3} + 43200 \nu^{2} - 49329 \nu + 22113 ) / 26001 \) |
\(\beta_{4}\) | \(=\) | \( ( 52 \nu^{11} - 669 \nu^{10} + 1648 \nu^{9} - 24 \nu^{8} + 7453 \nu^{7} - 8517 \nu^{6} + 22659 \nu^{5} - 22077 \nu^{4} + 66483 \nu^{3} - 18009 \nu^{2} + 126036 \nu - 19440 ) / 78003 \) |
\(\beta_{5}\) | \(=\) | \( ( 140 \nu^{11} - 690 \nu^{10} + 659 \nu^{9} - 4287 \nu^{8} + 2954 \nu^{7} - 15819 \nu^{6} + 3225 \nu^{5} - 31437 \nu^{4} + 13653 \nu^{3} - 106488 \nu^{2} - 68688 \nu - 118341 ) / 78003 \) |
\(\beta_{6}\) | \(=\) | \( ( - 2 \nu^{11} - 3 \nu^{10} + \nu^{9} - 33 \nu^{8} + \nu^{7} - 183 \nu^{6} + 168 \nu^{5} - 585 \nu^{4} + 306 \nu^{3} - 1323 \nu^{2} + 891 \nu - 2673 ) / 729 \) |
\(\beta_{7}\) | \(=\) | \( ( - 104 \nu^{11} + 54 \nu^{10} - 407 \nu^{9} + 690 \nu^{8} - 2387 \nu^{7} + 3231 \nu^{6} - 7761 \nu^{5} + 4671 \nu^{4} - 14517 \nu^{3} + 7128 \nu^{2} - 26730 \nu - 13122 ) / 26001 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{11} + 4\nu^{9} - 3\nu^{8} + 22\nu^{7} - 9\nu^{6} + 69\nu^{5} - 27\nu^{4} + 198\nu^{3} - 81\nu^{2} + 324\nu ) / 243 \) |
\(\beta_{9}\) | \(=\) | \( ( - 355 \nu^{11} + 30 \nu^{10} + 83 \nu^{9} - 651 \nu^{8} - 268 \nu^{7} - 4839 \nu^{6} - 852 \nu^{5} - 25011 \nu^{4} + 5310 \nu^{3} - 25893 \nu^{2} - 405 \nu - 137295 ) / 78003 \) |
\(\beta_{10}\) | \(=\) | \( ( - 167 \nu^{11} + 204 \nu^{10} - 848 \nu^{9} + 966 \nu^{8} - 2657 \nu^{7} + 4074 \nu^{6} - 8490 \nu^{5} + 6732 \nu^{4} - 13005 \nu^{3} + 16335 \nu^{2} - 28755 \nu + 2430 ) / 26001 \) |
\(\beta_{11}\) | \(=\) | \( ( - 877 \nu^{11} + 1227 \nu^{10} - 3571 \nu^{9} + 3516 \nu^{8} - 17638 \nu^{7} + 15582 \nu^{6} - 47751 \nu^{5} + 31824 \nu^{4} - 80388 \nu^{3} + 51219 \nu^{2} + \cdots - 116154 ) / 78003 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{11} - \beta_{9} + \beta_{8} + \beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 1 \) |
\(\nu^{4}\) | \(=\) | \( 3\beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{8} - 2\beta_{7} - 2\beta_{6} - \beta_{4} - 3\beta_{3} + \beta _1 - 5 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{7} + \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 8 \beta _1 - 6 \) |
\(\nu^{6}\) | \(=\) | \( - 3 \beta_{11} + \beta_{10} + 2 \beta_{9} + 9 \beta_{8} + 12 \beta_{7} - \beta_{6} - 3 \beta_{5} - 6 \beta_{4} - \beta_{3} + 8 \beta_{2} - 4 \beta _1 - 3 \) |
\(\nu^{7}\) | \(=\) | \( - 11 \beta_{11} + 16 \beta_{10} + 16 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} - 11 \beta_{4} - 7 \beta_{3} - 13 \beta_{2} + 8 \beta _1 - 10 \) |
\(\nu^{8}\) | \(=\) | \( - 15 \beta_{11} + 6 \beta_{10} - 16 \beta_{8} + 8 \beta_{7} + 15 \beta_{6} - 3 \beta_{5} + 13 \beta_{4} + \beta_{3} - 47 \beta_{2} - 12 \beta _1 + 29 \) |
\(\nu^{9}\) | \(=\) | \( 25 \beta_{11} - 75 \beta_{10} + 32 \beta_{9} + 13 \beta_{8} + 33 \beta_{7} - 15 \beta_{6} - 11 \beta_{5} - \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 3 \beta _1 + 49 \) |
\(\nu^{10}\) | \(=\) | \( 3 \beta_{11} - 19 \beta_{10} + 88 \beta_{9} + 28 \beta_{8} - 65 \beta_{7} + 7 \beta_{6} - 9 \beta_{5} - 55 \beta_{4} + 6 \beta_{3} - 90 \beta_{2} + 19 \beta _1 + 112 \) |
\(\nu^{11}\) | \(=\) | \( - 116 \beta_{11} + 17 \beta_{10} - 111 \beta_{9} - 72 \beta_{8} + 44 \beta_{7} + 127 \beta_{6} + 67 \beta_{5} + 96 \beta_{4} + 177 \beta_{3} - 87 \beta_{2} + 19 \beta _1 + 84 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).
\(n\) | \(47\) | \(235\) |
\(\chi(n)\) | \(-1 + \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 |
|
0.500000 | + | 0.866025i | −1.44505 | − | 0.954904i | −0.500000 | + | 0.866025i | −0.286426 | + | 0.496105i | 0.104448 | − | 1.72890i | −1.38989 | − | 2.40736i | −1.00000 | 1.17632 | + | 2.75976i | −0.572852 | ||||||||||||||||||||||||||||||||||||||||
139.2 | 0.500000 | + | 0.866025i | −1.20109 | + | 1.24795i | −0.500000 | + | 0.866025i | 1.39132 | − | 2.40983i | −1.68130 | − | 0.416203i | −1.77657 | − | 3.07710i | −1.00000 | −0.114750 | − | 2.99780i | 2.78263 | |||||||||||||||||||||||||||||||||||||||||
139.3 | 0.500000 | + | 0.866025i | −0.416383 | − | 1.68126i | −0.500000 | + | 0.866025i | −0.229999 | + | 0.398369i | 1.24782 | − | 1.20123i | 2.38325 | + | 4.12791i | −1.00000 | −2.65325 | + | 1.40009i | −0.459997 | |||||||||||||||||||||||||||||||||||||||||
139.4 | 0.500000 | + | 0.866025i | 0.706634 | + | 1.58135i | −0.500000 | + | 0.866025i | 1.63705 | − | 2.83545i | −1.01617 | + | 1.40264i | −0.135712 | − | 0.235061i | −1.00000 | −2.00134 | + | 2.23487i | 3.27410 | |||||||||||||||||||||||||||||||||||||||||
139.5 | 0.500000 | + | 0.866025i | 1.07449 | − | 1.35848i | −0.500000 | + | 0.866025i | 1.11191 | − | 1.92588i | 1.71372 | + | 0.251297i | −0.920971 | − | 1.59517i | −1.00000 | −0.690936 | − | 2.91935i | 2.22381 | |||||||||||||||||||||||||||||||||||||||||
139.6 | 0.500000 | + | 0.866025i | 1.28140 | + | 1.16534i | −0.500000 | + | 0.866025i | −1.12385 | + | 1.94656i | −0.368518 | + | 1.69239i | 0.339893 | + | 0.588712i | −1.00000 | 0.283955 | + | 2.98653i | −2.24770 | |||||||||||||||||||||||||||||||||||||||||
277.1 | 0.500000 | − | 0.866025i | −1.44505 | + | 0.954904i | −0.500000 | − | 0.866025i | −0.286426 | − | 0.496105i | 0.104448 | + | 1.72890i | −1.38989 | + | 2.40736i | −1.00000 | 1.17632 | − | 2.75976i | −0.572852 | |||||||||||||||||||||||||||||||||||||||||
277.2 | 0.500000 | − | 0.866025i | −1.20109 | − | 1.24795i | −0.500000 | − | 0.866025i | 1.39132 | + | 2.40983i | −1.68130 | + | 0.416203i | −1.77657 | + | 3.07710i | −1.00000 | −0.114750 | + | 2.99780i | 2.78263 | |||||||||||||||||||||||||||||||||||||||||
277.3 | 0.500000 | − | 0.866025i | −0.416383 | + | 1.68126i | −0.500000 | − | 0.866025i | −0.229999 | − | 0.398369i | 1.24782 | + | 1.20123i | 2.38325 | − | 4.12791i | −1.00000 | −2.65325 | − | 1.40009i | −0.459997 | |||||||||||||||||||||||||||||||||||||||||
277.4 | 0.500000 | − | 0.866025i | 0.706634 | − | 1.58135i | −0.500000 | − | 0.866025i | 1.63705 | + | 2.83545i | −1.01617 | − | 1.40264i | −0.135712 | + | 0.235061i | −1.00000 | −2.00134 | − | 2.23487i | 3.27410 | |||||||||||||||||||||||||||||||||||||||||
277.5 | 0.500000 | − | 0.866025i | 1.07449 | + | 1.35848i | −0.500000 | − | 0.866025i | 1.11191 | + | 1.92588i | 1.71372 | − | 0.251297i | −0.920971 | + | 1.59517i | −1.00000 | −0.690936 | + | 2.91935i | 2.22381 | |||||||||||||||||||||||||||||||||||||||||
277.6 | 0.500000 | − | 0.866025i | 1.28140 | − | 1.16534i | −0.500000 | − | 0.866025i | −1.12385 | − | 1.94656i | −0.368518 | − | 1.69239i | 0.339893 | − | 0.588712i | −1.00000 | 0.283955 | − | 2.98653i | −2.24770 | |||||||||||||||||||||||||||||||||||||||||
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.e | ✓ | 12 |
3.b | odd | 2 | 1 | 1242.2.e.e | 12 | ||
9.c | even | 3 | 1 | inner | 414.2.e.e | ✓ | 12 |
9.c | even | 3 | 1 | 3726.2.a.w | 6 | ||
9.d | odd | 6 | 1 | 1242.2.e.e | 12 | ||
9.d | odd | 6 | 1 | 3726.2.a.x | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.2.e.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
414.2.e.e | ✓ | 12 | 9.c | even | 3 | 1 | inner |
1242.2.e.e | 12 | 3.b | odd | 2 | 1 | ||
1242.2.e.e | 12 | 9.d | odd | 6 | 1 | ||
3726.2.a.w | 6 | 9.c | even | 3 | 1 | ||
3726.2.a.x | 6 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 5 T_{5}^{11} + 27 T_{5}^{10} - 56 T_{5}^{9} + 182 T_{5}^{8} - 235 T_{5}^{7} + 844 T_{5}^{6} - 525 T_{5}^{5} + 1432 T_{5}^{4} + 1299 T_{5}^{3} + 1365 T_{5}^{2} + 468 T_{5} + 144 \)
acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T + 1)^{6} \)
$3$
\( T^{12} + 4 T^{10} + 3 T^{9} + 22 T^{8} + \cdots + 729 \)
$5$
\( T^{12} - 5 T^{11} + 27 T^{10} - 56 T^{9} + \cdots + 144 \)
$7$
\( T^{12} + 3 T^{11} + 28 T^{10} + 99 T^{9} + \cdots + 256 \)
$11$
\( T^{12} - 6 T^{11} + 50 T^{10} - 126 T^{9} + \cdots + 729 \)
$13$
\( T^{12} + 6 T^{11} + 58 T^{10} + \cdots + 35344 \)
$17$
\( (T^{6} + 4 T^{5} - 14 T^{4} - 57 T^{3} + \cdots + 81)^{2} \)
$19$
\( (T^{6} - 2 T^{5} - 42 T^{4} + 130 T^{3} + \cdots + 765)^{2} \)
$23$
\( (T^{2} - T + 1)^{6} \)
$29$
\( T^{12} - 12 T^{11} + 113 T^{10} + \cdots + 104976 \)
$31$
\( T^{12} + 6 T^{11} + 104 T^{10} + \cdots + 1144900 \)
$37$
\( (T^{6} - 4 T^{5} - 91 T^{4} + 65 T^{3} + \cdots + 4008)^{2} \)
$41$
\( T^{12} - 15 T^{11} + 227 T^{10} + \cdots + 1846881 \)
$43$
\( T^{12} + 14 T^{11} + \cdots + 7951110561 \)
$47$
\( T^{12} - 9 T^{11} + \cdots + 28242146916 \)
$53$
\( (T^{6} + 5 T^{5} - 97 T^{4} - 763 T^{3} + \cdots - 870)^{2} \)
$59$
\( T^{12} - 18 T^{11} + \cdots + 1322704161 \)
$61$
\( T^{12} + 3 T^{11} + 208 T^{10} + \cdots + 18164644 \)
$67$
\( T^{12} - 8 T^{11} + 252 T^{10} + \cdots + 140493609 \)
$71$
\( (T^{6} + 9 T^{5} - 326 T^{4} + \cdots - 271728)^{2} \)
$73$
\( (T^{6} + 16 T^{5} - 98 T^{4} - 1702 T^{3} + \cdots - 32237)^{2} \)
$79$
\( T^{12} + 7 T^{11} + 179 T^{10} + \cdots + 324 \)
$83$
\( T^{12} - 3 T^{11} + 277 T^{10} + \cdots + 150111504 \)
$89$
\( (T^{6} + 21 T^{5} - 113 T^{4} + \cdots + 644958)^{2} \)
$97$
\( T^{12} - 13 T^{11} + \cdots + 1043321287761 \)
show more
show less