Newspace parameters
Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1110.x (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.86339462436\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{16} + 60x^{14} + 1362x^{12} + 15028x^{10} + 86441x^{8} + 260376x^{6} + 382684x^{4} + 224224x^{2} + 38416 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
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 \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} + 60x^{14} + 1362x^{12} + 15028x^{10} + 86441x^{8} + 260376x^{6} + 382684x^{4} + 224224x^{2} + 38416 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 11 \nu^{14} + 613 \nu^{12} + 12389 \nu^{10} + 113739 \nu^{8} + 490228 \nu^{6} + 982004 \nu^{4} + 849840 \nu^{2} - 34944 \nu + 206192 ) / 69888 \) |
\(\beta_{3}\) | \(=\) | \( ( - 16872 \nu^{15} - 85673 \nu^{14} - 845916 \nu^{13} - 4862599 \nu^{12} - 13686912 \nu^{11} - 100872023 \nu^{10} - 65555292 \nu^{9} + \cdots - 1892498384 ) / 1061598720 \) |
\(\beta_{4}\) | \(=\) | \( ( - 1843 \nu^{15} + 87690 \nu^{14} - 55469 \nu^{13} + 5110410 \nu^{12} + 441587 \nu^{11} + 110660190 \nu^{10} + 28000317 \nu^{9} + 1128794190 \nu^{8} + \cdots + 3013123680 ) / 303313920 \) |
\(\beta_{5}\) | \(=\) | \( ( 1406 \nu^{15} + 70493 \nu^{13} + 1140576 \nu^{11} + 5462941 \nu^{9} - 21516416 \nu^{7} - 230818172 \nu^{5} - 501688272 \nu^{3} - 208036752 \nu ) / 44233280 \) |
\(\beta_{6}\) | \(=\) | \( ( - 1843 \nu^{15} - 87690 \nu^{14} - 55469 \nu^{13} - 5110410 \nu^{12} + 441587 \nu^{11} - 110660190 \nu^{10} + 28000317 \nu^{9} - 1128794190 \nu^{8} + \cdots - 3013123680 ) / 303313920 \) |
\(\beta_{7}\) | \(=\) | \( ( - 52443 \nu^{15} - 10192 \nu^{14} - 3509229 \nu^{13} - 343196 \nu^{12} - 91328373 \nu^{11} + 666008 \nu^{10} - 1178556723 \nu^{9} + 128790228 \nu^{8} + \cdots + 9551468864 ) / 1061598720 \) |
\(\beta_{8}\) | \(=\) | \( ( 12239 \nu^{15} - 23772 \nu^{14} + 694657 \nu^{13} - 1327536 \nu^{12} + 14410289 \nu^{11} - 26856732 \nu^{10} + 136836879 \nu^{9} - 245232792 \nu^{8} + \cdots - 92593536 ) / 151656960 \) |
\(\beta_{9}\) | \(=\) | \( ( 263 \nu^{15} + 15241 \nu^{13} + 328169 \nu^{11} + 3345303 \nu^{9} + 17160772 \nu^{7} + 44457716 \nu^{5} + 52527696 \nu^{3} + 17328752 \nu + 1712256 ) / 3424512 \) |
\(\beta_{10}\) | \(=\) | \( ( 12239 \nu^{15} + 23772 \nu^{14} + 694657 \nu^{13} + 1327536 \nu^{12} + 14410289 \nu^{11} + 26856732 \nu^{10} + 136836879 \nu^{9} + 245232792 \nu^{8} + \cdots + 92593536 ) / 151656960 \) |
\(\beta_{11}\) | \(=\) | \( ( - 17481 \nu^{15} - 1169743 \nu^{13} - 30442791 \nu^{11} - 392852241 \nu^{9} - 2634019244 \nu^{7} - 8742547388 \nu^{5} - 12512903888 \nu^{3} + \cdots - 5058208848 \nu ) / 176933120 \) |
\(\beta_{12}\) | \(=\) | \( ( - 407712 \nu^{15} + 191303 \nu^{14} - 23678916 \nu^{13} + 10175809 \nu^{12} - 509843112 \nu^{11} + 188699273 \nu^{10} - 5151114132 \nu^{9} + \cdots - 2614850896 ) / 2123197440 \) |
\(\beta_{13}\) | \(=\) | \( ( 407712 \nu^{15} + 191303 \nu^{14} + 23678916 \nu^{13} + 10175809 \nu^{12} + 509843112 \nu^{11} + 188699273 \nu^{10} + 5151114132 \nu^{9} + \cdots - 2614850896 ) / 2123197440 \) |
\(\beta_{14}\) | \(=\) | \( ( - 152793 \nu^{15} + 28329 \nu^{14} - 8812379 \nu^{13} + 1614627 \nu^{12} - 188046063 \nu^{11} + 33560079 \nu^{10} - 1883030733 \nu^{9} + \cdots - 1258733168 ) / 353866240 \) |
\(\beta_{15}\) | \(=\) | \( ( - 539909 \nu^{15} - 84987 \nu^{14} - 31161847 \nu^{13} - 4843881 \nu^{12} - 665870579 \nu^{11} - 100680237 \nu^{10} - 6686136129 \nu^{9} + \cdots + 4306998864 ) / 1061598720 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{15} - \beta_{14} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - 2\beta_{3} - 2\beta_{2} - \beta _1 - 7 \) |
\(\nu^{3}\) | \(=\) | \( - \beta_{15} - \beta_{14} - 2 \beta_{11} - 2 \beta_{10} - 13 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 12 \beta _1 + 7 \) |
\(\nu^{4}\) | \(=\) | \( - 18 \beta_{15} + 18 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 28 \beta_{10} - 18 \beta_{9} + 28 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 48 \beta_{2} + 24 \beta _1 + 95 \) |
\(\nu^{5}\) | \(=\) | \( 27 \beta_{15} + 27 \beta_{14} + 44 \beta_{11} + 70 \beta_{10} + 303 \beta_{9} + 70 \beta_{8} + 34 \beta_{6} - 139 \beta_{5} + 34 \beta_{4} + 216 \beta _1 - 165 \) |
\(\nu^{6}\) | \(=\) | \( 328 \beta_{15} - 328 \beta_{14} + 54 \beta_{13} + 54 \beta_{12} - 63 \beta_{11} + 690 \beta_{10} + 328 \beta_{9} - 690 \beta_{8} + 126 \beta_{7} - 54 \beta_{6} + 243 \beta_{5} + 54 \beta_{4} + 486 \beta_{3} - 1044 \beta_{2} + \cdots - 1631 \) |
\(\nu^{7}\) | \(=\) | \( - 639 \beta_{15} - 639 \beta_{14} - 72 \beta_{13} + 72 \beta_{12} - 964 \beta_{11} - 1846 \beta_{10} - 6395 \beta_{9} - 1846 \beta_{8} - 458 \beta_{6} + 3619 \beta_{5} - 458 \beta_{4} - 4404 \beta _1 + 3517 \) |
\(\nu^{8}\) | \(=\) | \( - 6356 \beta_{15} + 6356 \beta_{14} - 1086 \beta_{13} - 1086 \beta_{12} + 2027 \beta_{11} - 16442 \beta_{10} - 6356 \beta_{9} + 16442 \beta_{8} - 4054 \beta_{7} + 1206 \beta_{6} - 8747 \beta_{5} + \cdots + 31135 \) |
\(\nu^{9}\) | \(=\) | \( 14483 \beta_{15} + 14483 \beta_{14} + 2848 \beta_{13} - 2848 \beta_{12} + 21832 \beta_{11} + 44942 \beta_{10} + 135095 \beta_{9} + 44942 \beta_{8} + 5690 \beta_{6} - 90523 \beta_{5} + 5690 \beta_{4} + \cdots - 74789 \) |
\(\nu^{10}\) | \(=\) | \( 130184 \beta_{15} - 130184 \beta_{14} + 19958 \beta_{13} + 19958 \beta_{12} - 53735 \beta_{11} + 384754 \beta_{10} + 130184 \beta_{9} - 384754 \beta_{8} + 107470 \beta_{7} - 26118 \beta_{6} + \cdots - 632007 \) |
\(\nu^{11}\) | \(=\) | \( - 323863 \beta_{15} - 323863 \beta_{14} - 81352 \beta_{13} + 81352 \beta_{12} - 501140 \beta_{11} - 1058558 \beta_{10} - 2896595 \beta_{9} - 1058558 \beta_{8} - 65386 \beta_{6} + \cdots + 1610229 \) |
\(\nu^{12}\) | \(=\) | \( - 2771924 \beta_{15} + 2771924 \beta_{14} - 358558 \beta_{13} - 358558 \beta_{12} + 1317035 \beta_{11} - 8899826 \beta_{10} - 2771924 \beta_{9} + 8899826 \beta_{8} - 2634070 \beta_{7} + \cdots + 13344047 \) |
\(\nu^{13}\) | \(=\) | \( 7228043 \beta_{15} + 7228043 \beta_{14} + 2067696 \beta_{13} - 2067696 \beta_{12} + 11521008 \beta_{11} + 24521326 \beta_{10} + 63035871 \beta_{9} + 24521326 \beta_{8} + \cdots - 35131957 \) |
\(\nu^{14}\) | \(=\) | \( 60481488 \beta_{15} - 60481488 \beta_{14} + 6504422 \beta_{13} + 6504422 \beta_{12} - 31115103 \beta_{11} + 204305602 \beta_{10} + 60481488 \beta_{9} - 204305602 \beta_{8} + \cdots - 289019607 \) |
\(\nu^{15}\) | \(=\) | \( - 161646111 \beta_{15} - 161646111 \beta_{14} - 49841816 \beta_{13} + 49841816 \beta_{12} - 264166636 \beta_{11} - 562800062 \beta_{10} - 1388334603 \beta_{9} + \cdots + 774990357 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).
\(n\) | \(371\) | \(631\) | \(667\) |
\(\chi(n)\) | \(1\) | \(\beta_{9}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
751.1 |
|
−0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | − | 1.00000i | −2.37879 | + | 4.12019i | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
751.2 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | − | 1.00000i | −0.267768 | + | 0.463788i | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
751.3 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | − | 1.00000i | 0.435667 | − | 0.754597i | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
751.4 | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | − | 1.00000i | 0.844871 | − | 1.46336i | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
751.5 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | 1.00000i | −1.62842 | + | 2.82051i | − | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
751.6 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | 1.00000i | −0.979923 | + | 1.69728i | − | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
751.7 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | 1.00000i | 1.08063 | − | 1.87170i | − | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
751.8 | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | 1.00000i | 1.89374 | − | 3.28006i | − | 1.00000i | −0.500000 | − | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.1 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | 1.00000i | −2.37879 | − | 4.12019i | − | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.2 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | 1.00000i | −0.267768 | − | 0.463788i | − | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.3 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | 1.00000i | 0.435667 | + | 0.754597i | − | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.4 | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | 1.00000i | 0.844871 | + | 1.46336i | − | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.5 | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | − | 1.00000i | −1.62842 | − | 2.82051i | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.6 | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | − | 1.00000i | −0.979923 | − | 1.69728i | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.7 | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | − | 1.00000i | 1.08063 | + | 1.87170i | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
841.8 | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | − | 1.00000i | 1.89374 | + | 3.28006i | 1.00000i | −0.500000 | + | 0.866025i | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1110.2.x.d | ✓ | 16 |
37.e | even | 6 | 1 | inner | 1110.2.x.d | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1110.2.x.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
1110.2.x.d | ✓ | 16 | 37.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 2 T_{7}^{15} + 32 T_{7}^{14} + 8 T_{7}^{13} + 623 T_{7}^{12} + 70 T_{7}^{11} + 6068 T_{7}^{10} - 4080 T_{7}^{9} + 37621 T_{7}^{8} - 20766 T_{7}^{7} + 121838 T_{7}^{6} - 85068 T_{7}^{5} + 256600 T_{7}^{4} + \cdots + 38416 \)
acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{4} \)
$3$
\( (T^{2} + T + 1)^{8} \)
$5$
\( (T^{4} - T^{2} + 1)^{4} \)
$7$
\( T^{16} + 2 T^{15} + 32 T^{14} + \cdots + 38416 \)
$11$
\( (T^{8} + 4 T^{7} - 62 T^{6} - 232 T^{5} + \cdots + 24336)^{2} \)
$13$
\( T^{16} + 6 T^{15} - 56 T^{14} + \cdots + 37417689 \)
$17$
\( T^{16} - 6 T^{15} - 14 T^{14} + \cdots + 82944 \)
$19$
\( T^{16} + 12 T^{15} - 21 T^{14} + \cdots + 26873856 \)
$23$
\( T^{16} + 160 T^{14} + 9172 T^{12} + \cdots + 5308416 \)
$29$
\( T^{16} + 222 T^{14} + \cdots + 10585940544 \)
$31$
\( T^{16} + 198 T^{14} + \cdots + 201867264 \)
$37$
\( T^{16} - 12 T^{15} + \cdots + 3512479453921 \)
$41$
\( T^{16} + \cdots + 215838879399936 \)
$43$
\( T^{16} + 142 T^{14} + 5869 T^{12} + \cdots + 9216 \)
$47$
\( (T^{8} - 34 T^{7} + 344 T^{6} + \cdots - 178848)^{2} \)
$53$
\( T^{16} + 12 T^{15} + \cdots + 20517855952896 \)
$59$
\( T^{16} - 6 T^{15} + \cdots + 14648745024 \)
$61$
\( T^{16} - 12 T^{15} + \cdots + 940868960256 \)
$67$
\( T^{16} + 36 T^{15} + \cdots + 1874890000 \)
$71$
\( T^{16} - 6 T^{15} + \cdots + 2190240000 \)
$73$
\( (T^{8} + 8 T^{7} - 253 T^{6} + \cdots - 914048)^{2} \)
$79$
\( T^{16} + 24 T^{15} + \cdots + 239802172416 \)
$83$
\( T^{16} + \cdots + 180864626782464 \)
$89$
\( T^{16} - 24 T^{15} + \cdots + 7020380160000 \)
$97$
\( T^{16} + 270 T^{14} + 20893 T^{12} + \cdots + 5760000 \)
show more
show less