Newspace parameters
Level: | \( N \) | \(=\) | \( 1134 = 2 \cdot 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1134.k (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.05503558921\) |
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
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Defining polynomial: | \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3^{4} \) |
Twist minimal: | no (minimal twist has level 126) |
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 \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561 \) :
\(\beta_{1}\) | \(=\) | \( ( 50 \nu^{15} + 1352 \nu^{14} - 6827 \nu^{13} + 7676 \nu^{12} + 27422 \nu^{11} - 107246 \nu^{10} + 107467 \nu^{9} + 206194 \nu^{8} - 757363 \nu^{7} + 724572 \nu^{6} + \cdots + 2825604 ) / 142155 \) |
\(\beta_{2}\) | \(=\) | \( ( 169 \nu^{15} - 866 \nu^{14} + 1319 \nu^{13} + 2308 \nu^{12} - 13199 \nu^{11} + 19055 \nu^{10} + 12131 \nu^{9} - 90010 \nu^{8} + 128722 \nu^{7} + 13734 \nu^{6} - 313347 \nu^{5} + \cdots - 244944 ) / 47385 \) |
\(\beta_{3}\) | \(=\) | \( ( 154 \nu^{15} - 1325 \nu^{14} + 3608 \nu^{13} - 224 \nu^{12} - 22478 \nu^{11} + 55022 \nu^{10} - 23518 \nu^{9} - 159688 \nu^{8} + 382978 \nu^{7} - 226785 \nu^{6} + \cdots - 1285227 ) / 47385 \) |
\(\beta_{4}\) | \(=\) | \( ( 1445 \nu^{15} - 9836 \nu^{14} + 21081 \nu^{13} + 15627 \nu^{12} - 172766 \nu^{11} + 334353 \nu^{10} + 13019 \nu^{9} - 1283242 \nu^{8} + 2419149 \nu^{7} - 693301 \nu^{6} + \cdots - 7117227 ) / 47385 \) |
\(\beta_{5}\) | \(=\) | \( ( 2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + 37829 \nu^{9} - 2521996 \nu^{8} + 4717386 \nu^{7} + \cdots - 14041269 ) / 47385 \) |
\(\beta_{6}\) | \(=\) | \( ( - 11192 \nu^{15} + 70123 \nu^{14} - 136087 \nu^{13} - 145859 \nu^{12} + 1215277 \nu^{11} - 2154880 \nu^{10} - 494788 \nu^{9} + 9027170 \nu^{8} - 15633236 \nu^{7} + \cdots + 42998607 ) / 142155 \) |
\(\beta_{7}\) | \(=\) | \( ( 16898 \nu^{15} - 108472 \nu^{14} + 217033 \nu^{13} + 208526 \nu^{12} - 1883968 \nu^{11} + 3436840 \nu^{10} + 565132 \nu^{9} - 13978340 \nu^{8} + 24892859 \nu^{7} + \cdots - 69445998 ) / 142155 \) |
\(\beta_{8}\) | \(=\) | \( ( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} - 771188 \nu^{10} - 200900 \nu^{9} + 3269857 \nu^{8} - 5585140 \nu^{7} + \cdots + 14963454 ) / 28431 \) |
\(\beta_{9}\) | \(=\) | \( ( 8586 \nu^{15} - 53254 \nu^{14} + 101261 \nu^{13} + 115567 \nu^{12} - 919066 \nu^{11} + 1601225 \nu^{10} + 428149 \nu^{9} - 6806990 \nu^{8} + 11591968 \nu^{7} + \cdots - 30865131 ) / 47385 \) |
\(\beta_{10}\) | \(=\) | \( ( - 26068 \nu^{15} + 165707 \nu^{14} - 325403 \nu^{13} - 334861 \nu^{12} + 2873363 \nu^{11} - 5145350 \nu^{10} - 1063457 \nu^{9} + 21324130 \nu^{8} + \cdots + 102032298 ) / 142155 \) |
\(\beta_{11}\) | \(=\) | \( ( - 31130 \nu^{15} + 194263 \nu^{14} - 374983 \nu^{13} - 405716 \nu^{12} + 3354958 \nu^{11} - 5936959 \nu^{10} - 1366837 \nu^{9} + 24841391 \nu^{8} + \cdots + 117273501 ) / 142155 \) |
\(\beta_{12}\) | \(=\) | \( ( 33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + 6268915 \nu^{10} + 1580329 \nu^{9} - 26474600 \nu^{8} + \cdots - 121811526 ) / 142155 \) |
\(\beta_{13}\) | \(=\) | \( ( 5368 \nu^{15} - 32959 \nu^{14} + 62025 \nu^{13} + 72930 \nu^{12} - 567820 \nu^{11} + 981351 \nu^{10} + 280045 \nu^{9} - 4204439 \nu^{8} + 7108452 \nu^{7} + \cdots - 18826182 ) / 15795 \) |
\(\beta_{14}\) | \(=\) | \( ( - 62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} - 12322064 \nu^{10} - 2527469 \nu^{9} + 51016561 \nu^{8} + \cdots + 244346949 ) / 142155 \) |
\(\beta_{15}\) | \(=\) | \( ( 5105 \nu^{15} - 31804 \nu^{14} + 61039 \nu^{13} + 67583 \nu^{12} - 549514 \nu^{11} + 965497 \nu^{10} + 239686 \nu^{9} - 4072523 \nu^{8} + 6992981 \nu^{7} + \cdots - 18816948 ) / 10935 \) |
\(\nu\) | \(=\) | \( ( -\beta_{15} + 2\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 + 2 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta _1 + 3 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 8 \beta_{2} + \beta _1 - 4 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( 5 \beta_{15} - 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 11 \beta _1 + 8 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( - 10 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 30 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} + \beta_{3} + 16 \beta_{2} + 21 \beta _1 - 1 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( - 7 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + 15 \beta_{8} + 30 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 12 \beta_{4} + 32 \beta_{3} - 21 \beta_{2} + 51 \beta _1 + 20 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( - 36 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} + 37 \beta_{12} + 24 \beta_{11} - 21 \beta_{10} + 33 \beta_{9} + 49 \beta_{8} + 93 \beta_{7} - \beta_{5} - 39 \beta_{4} - 17 \beta_{3} + 46 \beta_{2} + 90 \beta _1 - 3 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( ( - 42 \beta_{15} + 68 \beta_{14} + 10 \beta_{13} + 66 \beta_{12} + 50 \beta_{11} - 25 \beta_{10} + 42 \beta_{9} - 71 \beta_{8} + 179 \beta_{7} + 61 \beta_{6} + 18 \beta_{5} + 50 \beta_{4} + 31 \beta_{3} - 16 \beta_{2} + 163 \beta _1 + 94 ) / 3 \) |
\(\nu^{9}\) | \(=\) | \( ( - 44 \beta_{15} + 32 \beta_{14} + 216 \beta_{12} + 72 \beta_{11} + 14 \beta_{10} + 40 \beta_{9} + 50 \beta_{8} - 19 \beta_{7} + 66 \beta_{6} + 126 \beta_{5} + 66 \beta_{4} - 64 \beta_{3} + 110 \beta_{2} + 67 \beta _1 + 168 ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( - 58 \beta_{15} - 18 \beta_{14} + 90 \beta_{13} + 172 \beta_{12} + 126 \beta_{11} + 162 \beta_{10} + 62 \beta_{9} + 14 \beta_{8} - 161 \beta_{7} - 8 \beta_{6} + 167 \beta_{5} + 158 \beta_{4} + 153 \beta_{3} - 144 \beta_{2} - 61 \beta _1 + 188 ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( - 126 \beta_{15} - 351 \beta_{14} - 100 \beta_{13} + 231 \beta_{12} + 13 \beta_{11} + 333 \beta_{10} + 48 \beta_{9} + 80 \beta_{8} - 482 \beta_{7} - 81 \beta_{6} + 141 \beta_{5} - 219 \beta_{4} - 138 \beta_{3} + 86 \beta_{2} + \cdots + 217 ) / 3 \) |
\(\nu^{12}\) | \(=\) | \( ( - 209 \beta_{15} - 418 \beta_{14} - 251 \beta_{13} - 170 \beta_{12} + 68 \beta_{11} + 416 \beta_{10} + 301 \beta_{9} - 320 \beta_{8} + 325 \beta_{7} - 36 \beta_{6} - 298 \beta_{5} + 153 \beta_{4} - 357 \beta_{3} + 376 \beta_{2} + \cdots - 97 ) / 3 \) |
\(\nu^{13}\) | \(=\) | \( ( 385 \beta_{15} - 609 \beta_{14} - 1440 \beta_{13} + 402 \beta_{12} - 186 \beta_{11} + 414 \beta_{10} - 155 \beta_{9} - 478 \beta_{8} - 220 \beta_{7} + 98 \beta_{6} - 597 \beta_{5} + 277 \beta_{4} - 1199 \beta_{3} + 1540 \beta_{2} + \cdots + 1243 ) / 3 \) |
\(\nu^{14}\) | \(=\) | \( ( 1281 \beta_{15} - 511 \beta_{14} - 835 \beta_{13} + 960 \beta_{12} + 1042 \beta_{11} + 812 \beta_{10} - 27 \beta_{9} + 3809 \beta_{8} + 541 \beta_{7} - 1481 \beta_{6} - 882 \beta_{5} + 1178 \beta_{4} - 1454 \beta_{3} + \cdots - 1641 ) / 3 \) |
\(\nu^{15}\) | \(=\) | \( ( 2826 \beta_{15} - 1179 \beta_{14} - 1012 \beta_{13} + 2828 \beta_{12} + 1522 \beta_{11} + 2838 \beta_{10} - 2730 \beta_{9} + 6361 \beta_{8} - 11 \beta_{7} - 2559 \beta_{6} - 431 \beta_{5} - 2124 \beta_{4} + 809 \beta_{3} + \cdots + 208 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(407\) |
\(\chi(n)\) | \(1 - \beta_{8}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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647.1 |
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−0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.80966 | − | 3.13442i | 0 | 2.14611 | + | 1.54733i | 1.00000i | 0 | 3.13442 | + | 1.80966i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | −0.450129 | − | 0.779646i | 0 | −2.62906 | − | 0.296732i | 1.00000i | 0 | 0.779646 | + | 0.450129i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.3 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.483662 | + | 0.837727i | 0 | −0.238876 | + | 2.63495i | 1.00000i | 0 | −0.837727 | − | 0.483662i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.4 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.77612 | + | 3.07634i | 0 | −1.14420 | − | 2.38554i | 1.00000i | 0 | −3.07634 | − | 1.77612i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.5 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −1.82207 | − | 3.15592i | 0 | −1.04503 | + | 2.43062i | − | 1.00000i | 0 | −3.15592 | − | 1.82207i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.6 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | −0.0338034 | − | 0.0585493i | 0 | 1.44425 | − | 2.21679i | − | 1.00000i | 0 | −0.0585493 | − | 0.0338034i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.7 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.714925 | + | 1.23829i | 0 | 2.10995 | − | 1.59628i | − | 1.00000i | 0 | 1.23829 | + | 0.714925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
647.8 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 1.14095 | + | 1.97618i | 0 | −2.64314 | − | 0.117551i | − | 1.00000i | 0 | 1.97618 | + | 1.14095i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.1 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.80966 | + | 3.13442i | 0 | 2.14611 | − | 1.54733i | − | 1.00000i | 0 | 3.13442 | − | 1.80966i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.450129 | + | 0.779646i | 0 | −2.62906 | + | 0.296732i | − | 1.00000i | 0 | 0.779646 | − | 0.450129i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.3 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.483662 | − | 0.837727i | 0 | −0.238876 | − | 2.63495i | − | 1.00000i | 0 | −0.837727 | + | 0.483662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.4 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.77612 | − | 3.07634i | 0 | −1.14420 | + | 2.38554i | − | 1.00000i | 0 | −3.07634 | + | 1.77612i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.5 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.82207 | + | 3.15592i | 0 | −1.04503 | − | 2.43062i | 1.00000i | 0 | −3.15592 | + | 1.82207i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.6 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.0338034 | + | 0.0585493i | 0 | 1.44425 | + | 2.21679i | 1.00000i | 0 | −0.0585493 | + | 0.0338034i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.7 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.714925 | − | 1.23829i | 0 | 2.10995 | + | 1.59628i | 1.00000i | 0 | 1.23829 | − | 0.714925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
971.8 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.14095 | − | 1.97618i | 0 | −2.64314 | + | 0.117551i | 1.00000i | 0 | 1.97618 | − | 1.14095i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
21.g | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} + 24 T_{5}^{14} - 24 T_{5}^{13} + 423 T_{5}^{12} - 450 T_{5}^{11} + 3582 T_{5}^{10} - 5814 T_{5}^{9} + 22536 T_{5}^{8} - 25002 T_{5}^{7} + 42201 T_{5}^{6} - 19494 T_{5}^{5} + 32724 T_{5}^{4} - 11826 T_{5}^{3} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{4} \)
$3$
\( T^{16} \)
$5$
\( T^{16} + 24 T^{14} - 24 T^{13} + 423 T^{12} + \cdots + 81 \)
$7$
\( T^{16} + 4 T^{15} + 9 T^{14} + \cdots + 5764801 \)
$11$
\( T^{16} + 12 T^{15} + 18 T^{14} + \cdots + 61732449 \)
$13$
\( T^{16} + 150 T^{14} + \cdots + 390971529 \)
$17$
\( T^{16} + 18 T^{15} + 231 T^{14} + \cdots + 56070144 \)
$19$
\( T^{16} - 72 T^{14} + 4167 T^{12} + \cdots + 9199089 \)
$23$
\( T^{16} - 6 T^{15} - 54 T^{14} + \cdots + 187388721 \)
$29$
\( T^{16} + 108 T^{14} + 4518 T^{12} + \cdots + 1108809 \)
$31$
\( T^{16} + 6 T^{15} - 84 T^{14} + \cdots + 65610000 \)
$37$
\( T^{16} + 2 T^{15} + \cdots + 32746159681 \)
$41$
\( (T^{8} - 6 T^{7} - 69 T^{6} + 102 T^{5} + \cdots + 9)^{2} \)
$43$
\( (T^{8} - 2 T^{7} - 131 T^{6} + 88 T^{5} + \cdots + 54769)^{2} \)
$47$
\( T^{16} - 18 T^{15} + \cdots + 588203099136 \)
$53$
\( T^{16} + 36 T^{15} + \cdots + 36759242529 \)
$59$
\( T^{16} + 30 T^{15} + \cdots + 216504090000 \)
$61$
\( T^{16} - 60 T^{15} + \cdots + 547560000 \)
$67$
\( T^{16} - 14 T^{15} + \cdots + 2603856784 \)
$71$
\( T^{16} + 486 T^{14} + \cdots + 65610000 \)
$73$
\( T^{16} - 150 T^{14} + \cdots + 71115489 \)
$79$
\( T^{16} + 16 T^{15} + \cdots + 970422010000 \)
$83$
\( (T^{8} - 177 T^{6} + 156 T^{5} + \cdots - 30879)^{2} \)
$89$
\( T^{16} + \cdots + 131145120363321 \)
$97$
\( T^{16} + 798 T^{14} + \cdots + 9120206721024 \)
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