Newspace parameters
Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 342.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.73088374913\) |
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
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Defining polynomial: | \( x^{12} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{3} \) |
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 \( x^{12} + 8x^{10} + 55x^{8} - 2x^{7} + 70x^{6} - 32x^{5} + 73x^{4} - 18x^{3} + 13x^{2} + 2x + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( - 411176 \nu^{11} + 3327904 \nu^{10} - 3541281 \nu^{9} + 25462287 \nu^{8} - 25192919 \nu^{7} + 174805454 \nu^{6} - 53789475 \nu^{5} + \cdots + 10915007 ) / 8057223 \) |
\(\beta_{2}\) | \(=\) | \( ( 485726 \nu^{11} - 2471284 \nu^{10} + 2275851 \nu^{9} - 19554387 \nu^{8} + 14692937 \nu^{7} - 134803199 \nu^{6} - 43318125 \nu^{5} - 171073498 \nu^{4} + \cdots - 3803162 ) / 8057223 \) |
\(\beta_{3}\) | \(=\) | \( ( 573380 \nu^{11} + 1123298 \nu^{10} + 5455884 \nu^{9} + 8537454 \nu^{8} + 37790027 \nu^{7} + 57094012 \nu^{6} + 80439177 \nu^{5} + 34029107 \nu^{4} + \cdots - 8741327 ) / 8057223 \) |
\(\beta_{4}\) | \(=\) | \( ( - 723523 \nu^{11} - 2374420 \nu^{10} - 5988987 \nu^{9} - 18519324 \nu^{8} - 41432173 \nu^{7} - 125252072 \nu^{6} - 57042816 \nu^{5} - 116563363 \nu^{4} + \cdots - 5341028 ) / 8057223 \) |
\(\beta_{5}\) | \(=\) | \( ( - 1042276 \nu^{11} + 503822 \nu^{10} - 5877465 \nu^{9} + 5217513 \nu^{8} - 37931593 \nu^{7} + 38234422 \nu^{6} + 58653528 \nu^{5} + 120977024 \nu^{4} + \cdots + 5590234 ) / 8057223 \) |
\(\beta_{6}\) | \(=\) | \( ( 475714 \nu^{11} - 219777 \nu^{10} + 3753182 \nu^{9} - 1727756 \nu^{8} + 25748182 \nu^{7} - 12842883 \nu^{6} + 30886544 \nu^{5} - 29151715 \nu^{4} + \cdots + 755148 ) / 2685741 \) |
\(\beta_{7}\) | \(=\) | \( ( 491336 \nu^{11} - 66931 \nu^{10} + 3789188 \nu^{9} - 896846 \nu^{8} + 25810123 \nu^{7} - 7407675 \nu^{6} + 26207109 \nu^{5} - 38561249 \nu^{4} + \cdots - 5226000 ) / 2685741 \) |
\(\beta_{8}\) | \(=\) | \( ( 496946 \nu^{11} - 348319 \nu^{10} + 4407278 \nu^{9} - 2829986 \nu^{8} + 30660580 \nu^{7} - 20565930 \nu^{6} + 58131969 \nu^{5} - 43917038 \nu^{4} + \cdots - 7544085 ) / 2685741 \) |
\(\beta_{9}\) | \(=\) | \( ( - 826298 \nu^{11} + 506575 \nu^{10} - 6411425 \nu^{9} + 3835658 \nu^{8} - 44063809 \nu^{7} + 27624861 \nu^{6} - 49514907 \nu^{5} + 48686057 \nu^{4} + \cdots - 4239471 ) / 2685741 \) |
\(\beta_{10}\) | \(=\) | \( ( - 5325568 \nu^{11} - 291295 \nu^{10} - 42073479 \nu^{9} - 1655457 \nu^{8} - 288320839 \nu^{7} - 261647 \nu^{6} - 340849530 \nu^{5} + 184170155 \nu^{4} + \cdots + 3040609 ) / 8057223 \) |
\(\beta_{11}\) | \(=\) | \( ( - 5636270 \nu^{11} - 3730295 \nu^{10} - 45709146 \nu^{9} - 29404869 \nu^{8} - 313840082 \nu^{7} - 190033528 \nu^{6} - 412733472 \nu^{5} + \cdots - 28561402 ) / 8057223 \) |
\(\nu\) | \(=\) | \( ( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} - \beta_{2} ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{11} - 2\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 8\beta_{6} + \beta_{5} + 2\beta_{4} - 5\beta_{3} + 1 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( -8\beta_{11} + 11\beta_{10} + 8\beta_{7} - 5\beta_{6} - 3\beta_{5} - 2\beta_{4} + 2\beta_{2} - 8\beta _1 - 3 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( 21 \beta_{10} - 16 \beta_{9} - 6 \beta_{8} + 17 \beta_{7} + 46 \beta_{6} + \beta_{5} + \beta_{3} - 12 \beta_{2} + 8 \beta _1 - 46 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( 56 \beta_{11} - 77 \beta_{10} + 2 \beta_{9} - 56 \beta_{8} - \beta_{7} + 36 \beta_{6} + 54 \beta_{5} + 8 \beta_{4} + 33 \beta_{3} + 31 \beta _1 + 23 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( 41 \beta_{11} - 41 \beta_{10} + 31 \beta_{9} - 151 \beta_{7} + 10 \beta_{6} - 55 \beta_{5} - 76 \beta_{4} + 227 \beta_{3} + 76 \beta_{2} - 45 \beta _1 + 244 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( - 6 \beta_{10} - 4 \beta_{9} + 379 \beta_{8} - 365 \beta_{7} - 56 \beta_{6} - 204 \beta_{5} - 204 \beta_{3} - 43 \beta_{2} + 155 \beta _1 + 56 ) / 3 \) |
\(\nu^{8}\) | \(=\) | \( - 97 \beta_{11} - 215 \beta_{10} + 178 \beta_{9} + 97 \beta_{8} + 68 \beta_{7} - 689 \beta_{6} + 94 \beta_{5} + 166 \beta_{4} - 532 \beta_{3} - 27 \beta _1 + 121 \) |
\(\nu^{9}\) | \(=\) | \( ( - 2548 \beta_{11} + 3490 \beta_{10} - 81 \beta_{9} + 2557 \beta_{7} - 1360 \beta_{6} - 1020 \beta_{5} - 250 \beta_{4} - 258 \beta_{3} + 250 \beta_{2} - 2380 \beta _1 - 1515 ) / 3 \) |
\(\nu^{10}\) | \(=\) | \( ( 6288 \beta_{10} - 4928 \beta_{9} - 2077 \beta_{8} + 5645 \beta_{7} + 13253 \beta_{6} + 621 \beta_{5} + 621 \beta_{3} - 3296 \beta_{2} + 2371 \beta _1 - 13253 ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( 17111 \beta_{11} - 22451 \beta_{10} + 294 \beta_{9} - 17111 \beta_{8} - 621 \beta_{7} + 13154 \beta_{6} + 15773 \beta_{5} + 1469 \beta_{4} + 11381 \beta_{3} + 9095 \beta _1 + 6678 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(325\) |
\(\chi(n)\) | \(-\beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
115.1 |
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−0.500000 | + | 0.866025i | −1.70951 | + | 0.278542i | −0.500000 | − | 0.866025i | −1.71495 | − | 2.97037i | 0.613529 | − | 1.61975i | −0.0414111 | + | 0.0717262i | 1.00000 | 2.84483 | − | 0.952340i | 3.42989 | ||||||||||||||||||||||||||||||||||||||||
115.2 | −0.500000 | + | 0.866025i | −1.05505 | + | 1.37363i | −0.500000 | − | 0.866025i | 1.72549 | + | 2.98863i | −0.662079 | − | 1.60052i | −1.90190 | + | 3.29418i | 1.00000 | −0.773746 | − | 2.89850i | −3.45098 | |||||||||||||||||||||||||||||||||||||||||
115.3 | −0.500000 | + | 0.866025i | 0.147638 | − | 1.72575i | −0.500000 | − | 0.866025i | −0.573032 | − | 0.992520i | 1.42072 | + | 0.990732i | 0.817447 | − | 1.41586i | 1.00000 | −2.95641 | − | 0.509572i | 1.14606 | |||||||||||||||||||||||||||||||||||||||||
115.4 | −0.500000 | + | 0.866025i | 0.178697 | + | 1.72281i | −0.500000 | − | 0.866025i | −1.95772 | − | 3.39088i | −1.58134 | − | 0.706648i | −1.65131 | + | 2.86015i | 1.00000 | −2.93613 | + | 0.615721i | 3.91545 | |||||||||||||||||||||||||||||||||||||||||
115.5 | −0.500000 | + | 0.866025i | 1.71412 | + | 0.248580i | −0.500000 | − | 0.866025i | 2.19415 | + | 3.80037i | −1.07234 | + | 1.36018i | 1.88373 | − | 3.26272i | 1.00000 | 2.87642 | + | 0.852192i | −4.38829 | |||||||||||||||||||||||||||||||||||||||||
115.6 | −0.500000 | + | 0.866025i | 1.72410 | − | 0.165767i | −0.500000 | − | 0.866025i | 0.326066 | + | 0.564763i | −0.718491 | + | 1.57600i | −2.10656 | + | 3.64868i | 1.00000 | 2.94504 | − | 0.571599i | −0.652132 | |||||||||||||||||||||||||||||||||||||||||
229.1 | −0.500000 | − | 0.866025i | −1.70951 | − | 0.278542i | −0.500000 | + | 0.866025i | −1.71495 | + | 2.97037i | 0.613529 | + | 1.61975i | −0.0414111 | − | 0.0717262i | 1.00000 | 2.84483 | + | 0.952340i | 3.42989 | |||||||||||||||||||||||||||||||||||||||||
229.2 | −0.500000 | − | 0.866025i | −1.05505 | − | 1.37363i | −0.500000 | + | 0.866025i | 1.72549 | − | 2.98863i | −0.662079 | + | 1.60052i | −1.90190 | − | 3.29418i | 1.00000 | −0.773746 | + | 2.89850i | −3.45098 | |||||||||||||||||||||||||||||||||||||||||
229.3 | −0.500000 | − | 0.866025i | 0.147638 | + | 1.72575i | −0.500000 | + | 0.866025i | −0.573032 | + | 0.992520i | 1.42072 | − | 0.990732i | 0.817447 | + | 1.41586i | 1.00000 | −2.95641 | + | 0.509572i | 1.14606 | |||||||||||||||||||||||||||||||||||||||||
229.4 | −0.500000 | − | 0.866025i | 0.178697 | − | 1.72281i | −0.500000 | + | 0.866025i | −1.95772 | + | 3.39088i | −1.58134 | + | 0.706648i | −1.65131 | − | 2.86015i | 1.00000 | −2.93613 | − | 0.615721i | 3.91545 | |||||||||||||||||||||||||||||||||||||||||
229.5 | −0.500000 | − | 0.866025i | 1.71412 | − | 0.248580i | −0.500000 | + | 0.866025i | 2.19415 | − | 3.80037i | −1.07234 | − | 1.36018i | 1.88373 | + | 3.26272i | 1.00000 | 2.87642 | − | 0.852192i | −4.38829 | |||||||||||||||||||||||||||||||||||||||||
229.6 | −0.500000 | − | 0.866025i | 1.72410 | + | 0.165767i | −0.500000 | + | 0.866025i | 0.326066 | − | 0.564763i | −0.718491 | − | 1.57600i | −2.10656 | − | 3.64868i | 1.00000 | 2.94504 | + | 0.571599i | −0.652132 | |||||||||||||||||||||||||||||||||||||||||
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 | 342.2.e.c | ✓ | 12 |
3.b | odd | 2 | 1 | 1026.2.e.d | 12 | ||
9.c | even | 3 | 1 | inner | 342.2.e.c | ✓ | 12 |
9.c | even | 3 | 1 | 3078.2.a.x | 6 | ||
9.d | odd | 6 | 1 | 1026.2.e.d | 12 | ||
9.d | odd | 6 | 1 | 3078.2.a.v | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
342.2.e.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
342.2.e.c | ✓ | 12 | 9.c | even | 3 | 1 | inner |
1026.2.e.d | 12 | 3.b | odd | 2 | 1 | ||
1026.2.e.d | 12 | 9.d | odd | 6 | 1 | ||
3078.2.a.v | 6 | 9.d | odd | 6 | 1 | ||
3078.2.a.x | 6 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 30 T_{5}^{10} + 16 T_{5}^{9} + 672 T_{5}^{8} + 336 T_{5}^{7} + 6600 T_{5}^{6} + 3936 T_{5}^{5} + 48192 T_{5}^{4} + 19456 T_{5}^{3} + 43872 T_{5}^{2} - 14592 T_{5} + 23104 \)
acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + T + 1)^{6} \)
$3$
\( T^{12} - 2 T^{11} - 2 T^{9} - 8 T^{8} + \cdots + 729 \)
$5$
\( T^{12} + 30 T^{10} + 16 T^{9} + \cdots + 23104 \)
$7$
\( T^{12} + 6 T^{11} + 48 T^{10} + 144 T^{9} + \cdots + 729 \)
$11$
\( T^{12} + 6 T^{11} + 60 T^{10} + \cdots + 760384 \)
$13$
\( T^{12} + 6 T^{11} + 84 T^{10} + \cdots + 19035769 \)
$17$
\( (T^{6} - 66 T^{4} - 18 T^{3} + 549 T^{2} + \cdots + 405)^{2} \)
$19$
\( (T + 1)^{12} \)
$23$
\( T^{12} + 12 T^{11} + 138 T^{10} + \cdots + 66049 \)
$29$
\( T^{12} - 6 T^{11} + 66 T^{10} + \cdots + 72361 \)
$31$
\( T^{12} + 6 T^{11} + 138 T^{10} + \cdots + 202094656 \)
$37$
\( (T^{6} - 6 T^{5} - 135 T^{4} + 628 T^{3} + \cdots - 90329)^{2} \)
$41$
\( T^{12} - 6 T^{11} + 66 T^{10} + \cdots + 1600 \)
$43$
\( T^{12} + 12 T^{11} + 210 T^{10} + \cdots + 72863296 \)
$47$
\( T^{12} + 6 T^{11} + \cdots + 2953161649 \)
$53$
\( (T^{6} + 18 T^{5} + 36 T^{4} - 664 T^{3} + \cdots + 1927)^{2} \)
$59$
\( T^{12} - 12 T^{11} + 234 T^{10} + \cdots + 20331081 \)
$61$
\( T^{12} + 12 T^{11} + 174 T^{10} + \cdots + 4494400 \)
$67$
\( T^{12} + 18 T^{11} + \cdots + 275017385241 \)
$71$
\( (T^{6} - 18 T^{5} - 30 T^{4} + 1232 T^{3} + \cdots - 50840)^{2} \)
$73$
\( (T^{6} - 24 T^{5} + 54 T^{4} + 2622 T^{3} + \cdots - 59967)^{2} \)
$79$
\( T^{12} + 18 T^{11} + \cdots + 1126273600 \)
$83$
\( T^{12} - 12 T^{11} + 138 T^{10} + \cdots + 10497600 \)
$89$
\( (T^{6} - 6 T^{5} - 102 T^{4} + 252 T^{3} + \cdots - 5400)^{2} \)
$97$
\( T^{12} - 18 T^{11} + \cdots + 5833925283904 \)
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