Newspace parameters
Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 342.f (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.73088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(18\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{18} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} - 324 x^{6} - 1701 x^{5} + 243 x^{4} - 4374 x^{3} + 8748 x^{2} + 19683 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
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 \(1,\beta_1,\ldots,\beta_{17}\) 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 \( x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} - 324 x^{6} - 1701 x^{5} + 243 x^{4} - 4374 x^{3} + 8748 x^{2} + 19683 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 138 \nu^{17} - 3248 \nu^{16} + 2796 \nu^{15} - 5540 \nu^{14} + 15462 \nu^{13} + 11971 \nu^{12} + 18537 \nu^{11} + 24135 \nu^{10} - 73143 \nu^{9} - 39708 \nu^{8} + \cdots - 19081575 ) / 2574099 \) |
\(\beta_{3}\) | \(=\) | \( ( - 44 \nu^{17} - 351 \nu^{16} + 397 \nu^{15} - 774 \nu^{14} + 2455 \nu^{13} + 282 \nu^{12} + 2460 \nu^{11} + 2058 \nu^{10} - 11835 \nu^{9} - 3222 \nu^{8} - 34533 \nu^{7} + \cdots - 2755620 ) / 234009 \) |
\(\beta_{4}\) | \(=\) | \( ( 18 \nu^{17} - 85 \nu^{16} + 144 \nu^{15} - 267 \nu^{14} + 411 \nu^{13} - 57 \nu^{12} + 15 \nu^{11} + 802 \nu^{10} - 1272 \nu^{9} - 141 \nu^{8} - 5877 \nu^{7} + 1431 \nu^{6} + 3186 \nu^{5} + \cdots - 475308 ) / 78003 \) |
\(\beta_{5}\) | \(=\) | \( ( 1832 \nu^{17} - 15738 \nu^{16} + 19523 \nu^{15} - 39879 \nu^{14} + 80051 \nu^{13} - 12477 \nu^{12} + 47199 \nu^{11} + 83412 \nu^{10} - 298899 \nu^{9} + 74061 \nu^{8} + \cdots - 86769225 ) / 7722297 \) |
\(\beta_{6}\) | \(=\) | \( ( 1940 \nu^{17} - 13275 \nu^{16} + 17804 \nu^{15} - 34275 \nu^{14} + 69926 \nu^{13} - 6459 \nu^{12} + 37956 \nu^{11} + 93591 \nu^{10} - 230022 \nu^{9} + 103329 \nu^{8} + \cdots - 72689319 ) / 7722297 \) |
\(\beta_{7}\) | \(=\) | \( ( - 67 \nu^{17} + 515 \nu^{16} - 736 \nu^{15} + 1505 \nu^{14} - 2365 \nu^{13} - 142 \nu^{12} - 1749 \nu^{11} - 3888 \nu^{10} + 9603 \nu^{9} - 6606 \nu^{8} + 33966 \nu^{7} + \cdots + 2790612 ) / 234009 \) |
\(\beta_{8}\) | \(=\) | \( ( - 3248 \nu^{17} + 3348 \nu^{16} - 6368 \nu^{15} + 15600 \nu^{14} + 9073 \nu^{13} + 16881 \nu^{12} + 24135 \nu^{11} - 71901 \nu^{10} - 21078 \nu^{9} - 196371 \nu^{8} + \cdots - 5006043 ) / 7722297 \) |
\(\beta_{9}\) | \(=\) | \( ( - 372 \nu^{17} - 736 \nu^{16} + 432 \nu^{15} - 1369 \nu^{14} + 5703 \nu^{13} + 1649 \nu^{12} + 7989 \nu^{11} - 906 \nu^{10} - 26901 \nu^{9} - 32139 \nu^{8} - 45171 \nu^{7} + \cdots - 7103376 ) / 858033 \) |
\(\beta_{10}\) | \(=\) | \( ( 323 \nu^{17} - 699 \nu^{16} + 1193 \nu^{15} - 3204 \nu^{14} + 2150 \nu^{13} - 3252 \nu^{12} - 2067 \nu^{11} + 12834 \nu^{10} + 2070 \nu^{9} + 31050 \nu^{8} - 38745 \nu^{7} + \cdots - 2427570 ) / 702027 \) |
\(\beta_{11}\) | \(=\) | \( ( 4154 \nu^{17} + 3903 \nu^{16} - 6001 \nu^{15} - 3741 \nu^{14} - 47794 \nu^{13} + 3753 \nu^{12} - 47427 \nu^{11} + 47781 \nu^{10} + 213885 \nu^{9} + 240597 \nu^{8} + \cdots + 55296108 ) / 7722297 \) |
\(\beta_{12}\) | \(=\) | \( ( 4646 \nu^{17} + 1410 \nu^{16} + 611 \nu^{15} - 8502 \nu^{14} - 29536 \nu^{13} - 3258 \nu^{12} - 36294 \nu^{11} + 64287 \nu^{10} + 123705 \nu^{9} + 261036 \nu^{8} + \cdots + 32896854 ) / 7722297 \) |
\(\beta_{13}\) | \(=\) | \( ( 538 \nu^{17} - 985 \nu^{16} + 2017 \nu^{15} - 3138 \nu^{14} + 1642 \nu^{13} - 2898 \nu^{12} + 741 \nu^{11} + 14731 \nu^{10} - 4089 \nu^{9} + 36678 \nu^{8} - 39735 \nu^{7} + \cdots - 2093688 ) / 858033 \) |
\(\beta_{14}\) | \(=\) | \( ( 6235 \nu^{17} - 14928 \nu^{16} + 21007 \nu^{15} - 52014 \nu^{14} + 30850 \nu^{13} - 21834 \nu^{12} - 10695 \nu^{11} + 257778 \nu^{10} - 162882 \nu^{9} + 469125 \nu^{8} + \cdots - 31893021 ) / 7722297 \) |
\(\beta_{15}\) | \(=\) | \( ( - 8599 \nu^{17} + 9192 \nu^{16} - 9736 \nu^{15} + 48816 \nu^{14} + 45905 \nu^{13} + 27537 \nu^{12} + 87042 \nu^{11} - 246780 \nu^{10} - 256617 \nu^{9} - 571374 \nu^{8} + \cdots - 53741151 ) / 7722297 \) |
\(\beta_{16}\) | \(=\) | \( ( - 3391 \nu^{17} + 477 \nu^{16} - 5620 \nu^{15} + 8307 \nu^{14} + 17675 \nu^{13} + 14505 \nu^{12} + 43011 \nu^{11} - 45996 \nu^{10} - 62559 \nu^{9} - 253791 \nu^{8} + \cdots - 22447368 ) / 2574099 \) |
\(\beta_{17}\) | \(=\) | \( ( - 3391 \nu^{17} + 477 \nu^{16} - 5620 \nu^{15} + 8307 \nu^{14} + 17675 \nu^{13} + 14505 \nu^{12} + 43011 \nu^{11} - 45996 \nu^{10} - 62559 \nu^{9} - 253791 \nu^{8} + \cdots - 22447368 ) / 2574099 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{17} - \beta_{16} \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{14} + \beta_{10} - \beta_{9} - \beta_{8} - 2\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2\beta _1 + 2 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{17} - \beta_{16} + 2 \beta_{15} + \beta_{14} - 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} - 2 \beta_{3} + \beta_{2} + 3 \beta _1 + 3 \) |
\(\nu^{5}\) | \(=\) | \( - \beta_{17} - 2 \beta_{16} - \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 7 \beta _1 - 4 \) |
\(\nu^{6}\) | \(=\) | \( - 2 \beta_{17} + 2 \beta_{16} - 3 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} - 6 \beta_{11} + 3 \beta_{10} + \beta_{9} - 9 \beta_{8} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{2} + 5 \beta _1 + 9 \) |
\(\nu^{7}\) | \(=\) | \( - 2 \beta_{17} - \beta_{16} + 3 \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 4 \beta_{10} + 8 \beta_{9} + 14 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} + 10 \beta_{5} - \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + 7 \beta _1 - 1 \) |
\(\nu^{8}\) | \(=\) | \( 4 \beta_{17} - 4 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} - 9 \beta_{13} - 9 \beta_{11} + 5 \beta_{10} - 23 \beta_{9} + 12 \beta_{8} - 21 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 27 \beta_{4} + 2 \beta_{3} - \beta_{2} - 14 \beta _1 - 21 \) |
\(\nu^{9}\) | \(=\) | \( 18 \beta_{17} - 3 \beta_{16} - 24 \beta_{15} - 28 \beta_{14} + 27 \beta_{13} - 3 \beta_{11} + 7 \beta_{10} - 31 \beta_{9} + 56 \beta_{8} - 6 \beta_{7} + 16 \beta_{6} - 8 \beta_{5} + 15 \beta_{4} + \beta_{3} - 2 \beta_{2} - 29 \beta _1 - 40 \) |
\(\nu^{10}\) | \(=\) | \( - 17 \beta_{17} + 17 \beta_{16} + 50 \beta_{15} + 70 \beta_{14} + 18 \beta_{13} - 33 \beta_{12} + 33 \beta_{11} + 61 \beta_{10} + 40 \beta_{9} - 3 \beta_{8} + 57 \beta_{7} - 45 \beta_{5} + 9 \beta_{4} - 32 \beta_{3} + 58 \beta_{2} + 114 \beta _1 + 48 \) |
\(\nu^{11}\) | \(=\) | \( 62 \beta_{17} + 7 \beta_{16} - \beta_{14} + 136 \beta_{13} + 87 \beta_{12} + 3 \beta_{11} - 44 \beta_{10} + 41 \beta_{9} + 29 \beta_{8} - 42 \beta_{7} + 4 \beta_{6} - 29 \beta_{5} - 38 \beta_{4} - 8 \beta_{3} - 71 \beta_{2} + 43 \beta _1 + 104 \) |
\(\nu^{12}\) | \(=\) | \( - 11 \beta_{17} - 43 \beta_{16} - 81 \beta_{15} - 66 \beta_{14} - 57 \beta_{13} + 132 \beta_{12} - 78 \beta_{11} - 42 \beta_{10} - 44 \beta_{9} + 279 \beta_{8} - 99 \beta_{7} + 12 \beta_{6} - 75 \beta_{5} + 24 \beta_{4} + 36 \beta_{3} + \cdots - 135 \) |
\(\nu^{13}\) | \(=\) | \( 133 \beta_{17} - 109 \beta_{16} + 48 \beta_{15} + 50 \beta_{14} + 11 \beta_{13} + 3 \beta_{12} + 135 \beta_{11} + 112 \beta_{10} - 190 \beta_{9} + 482 \beta_{8} + 84 \beta_{7} - 47 \beta_{6} + 190 \beta_{5} + 116 \beta_{4} + 19 \beta_{3} + \cdots + 575 \) |
\(\nu^{14}\) | \(=\) | \( 94 \beta_{17} + 41 \beta_{16} + 178 \beta_{15} + 2 \beta_{14} + 162 \beta_{13} - 432 \beta_{12} + 99 \beta_{11} + 104 \beta_{10} - 770 \beta_{9} - 456 \beta_{8} + 195 \beta_{7} - 291 \beta_{6} + 336 \beta_{5} - 45 \beta_{4} + 47 \beta_{3} + \cdots + 159 \) |
\(\nu^{15}\) | \(=\) | \( 1215 \beta_{17} - 1416 \beta_{16} + 57 \beta_{15} - 127 \beta_{14} + 540 \beta_{13} - 270 \beta_{12} - 111 \beta_{11} - 164 \beta_{10} + 257 \beta_{9} + 308 \beta_{8} - 357 \beta_{7} + 115 \beta_{6} - 800 \beta_{5} + 555 \beta_{4} + \cdots + 509 \) |
\(\nu^{16}\) | \(=\) | \( - 287 \beta_{17} - 388 \beta_{16} + 518 \beta_{15} - 812 \beta_{14} + 252 \beta_{13} + 777 \beta_{12} + 708 \beta_{11} + 1465 \beta_{10} + 1048 \beta_{9} + 582 \beta_{8} + 138 \beta_{7} - 1017 \beta_{6} - 873 \beta_{5} + \cdots + 1416 \) |
\(\nu^{17}\) | \(=\) | \( 1187 \beta_{17} - 929 \beta_{16} + 2898 \beta_{15} + 539 \beta_{14} + 478 \beta_{13} - 1668 \beta_{12} + 4431 \beta_{11} + 694 \beta_{10} + 842 \beta_{9} - 4957 \beta_{8} + 1524 \beta_{7} + 481 \beta_{6} + 1609 \beta_{5} + \cdots + 3299 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(325\) |
\(\chi(n)\) | \(-\beta_{8}\) | \(-1 + \beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
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−1.00000 | −1.71861 | + | 0.215338i | 1.00000 | 2.13008 | − | 3.68941i | 1.71861 | − | 0.215338i | −0.603898 | + | 1.04598i | −1.00000 | 2.90726 | − | 0.740167i | −2.13008 | + | 3.68941i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.2 | −1.00000 | −1.60512 | + | 0.650828i | 1.00000 | −0.706161 | + | 1.22311i | 1.60512 | − | 0.650828i | 1.53389 | − | 2.65678i | −1.00000 | 2.15285 | − | 2.08932i | 0.706161 | − | 1.22311i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.3 | −1.00000 | −1.17768 | − | 1.27006i | 1.00000 | −0.0748074 | + | 0.129570i | 1.17768 | + | 1.27006i | −0.733568 | + | 1.27058i | −1.00000 | −0.226122 | + | 2.99147i | 0.0748074 | − | 0.129570i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.4 | −1.00000 | −0.894515 | + | 1.48319i | 1.00000 | −0.365468 | + | 0.633010i | 0.894515 | − | 1.48319i | −1.79034 | + | 3.10095i | −1.00000 | −1.39969 | − | 2.65347i | 0.365468 | − | 0.633010i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.5 | −1.00000 | 0.423085 | − | 1.67958i | 1.00000 | −1.97181 | + | 3.41528i | −0.423085 | + | 1.67958i | 1.02646 | − | 1.77787i | −1.00000 | −2.64200 | − | 1.42121i | 1.97181 | − | 3.41528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.6 | −1.00000 | 0.794053 | + | 1.53931i | 1.00000 | −1.46844 | + | 2.54341i | −0.794053 | − | 1.53931i | 0.360554 | − | 0.624499i | −1.00000 | −1.73896 | + | 2.44459i | 1.46844 | − | 2.54341i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.7 | −1.00000 | 1.09515 | − | 1.34188i | 1.00000 | 0.789777 | − | 1.36793i | −1.09515 | + | 1.34188i | 2.31561 | − | 4.01075i | −1.00000 | −0.601280 | − | 2.93913i | −0.789777 | + | 1.36793i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.8 | −1.00000 | 1.44298 | + | 0.958029i | 1.00000 | 1.19924 | − | 2.07714i | −1.44298 | − | 0.958029i | 0.959469 | − | 1.66185i | −1.00000 | 1.16436 | + | 2.76483i | −1.19924 | + | 2.07714i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
7.9 | −1.00000 | 1.64067 | − | 0.555168i | 1.00000 | 0.467593 | − | 0.809895i | −1.64067 | + | 0.555168i | −0.568176 | + | 0.984110i | −1.00000 | 2.38358 | − | 1.82169i | −0.467593 | + | 0.809895i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.1 | −1.00000 | −1.71861 | − | 0.215338i | 1.00000 | 2.13008 | + | 3.68941i | 1.71861 | + | 0.215338i | −0.603898 | − | 1.04598i | −1.00000 | 2.90726 | + | 0.740167i | −2.13008 | − | 3.68941i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.2 | −1.00000 | −1.60512 | − | 0.650828i | 1.00000 | −0.706161 | − | 1.22311i | 1.60512 | + | 0.650828i | 1.53389 | + | 2.65678i | −1.00000 | 2.15285 | + | 2.08932i | 0.706161 | + | 1.22311i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.3 | −1.00000 | −1.17768 | + | 1.27006i | 1.00000 | −0.0748074 | − | 0.129570i | 1.17768 | − | 1.27006i | −0.733568 | − | 1.27058i | −1.00000 | −0.226122 | − | 2.99147i | 0.0748074 | + | 0.129570i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.4 | −1.00000 | −0.894515 | − | 1.48319i | 1.00000 | −0.365468 | − | 0.633010i | 0.894515 | + | 1.48319i | −1.79034 | − | 3.10095i | −1.00000 | −1.39969 | + | 2.65347i | 0.365468 | + | 0.633010i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.5 | −1.00000 | 0.423085 | + | 1.67958i | 1.00000 | −1.97181 | − | 3.41528i | −0.423085 | − | 1.67958i | 1.02646 | + | 1.77787i | −1.00000 | −2.64200 | + | 1.42121i | 1.97181 | + | 3.41528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.6 | −1.00000 | 0.794053 | − | 1.53931i | 1.00000 | −1.46844 | − | 2.54341i | −0.794053 | + | 1.53931i | 0.360554 | + | 0.624499i | −1.00000 | −1.73896 | − | 2.44459i | 1.46844 | + | 2.54341i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.7 | −1.00000 | 1.09515 | + | 1.34188i | 1.00000 | 0.789777 | + | 1.36793i | −1.09515 | − | 1.34188i | 2.31561 | + | 4.01075i | −1.00000 | −0.601280 | + | 2.93913i | −0.789777 | − | 1.36793i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.8 | −1.00000 | 1.44298 | − | 0.958029i | 1.00000 | 1.19924 | + | 2.07714i | −1.44298 | + | 0.958029i | 0.959469 | + | 1.66185i | −1.00000 | 1.16436 | − | 2.76483i | −1.19924 | − | 2.07714i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.9 | −1.00000 | 1.64067 | + | 0.555168i | 1.00000 | 0.467593 | + | 0.809895i | −1.64067 | − | 0.555168i | −0.568176 | − | 0.984110i | −1.00000 | 2.38358 | + | 1.82169i | −0.467593 | − | 0.809895i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.h | 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.f.g | ✓ | 18 |
3.b | odd | 2 | 1 | 1026.2.f.g | 18 | ||
9.c | even | 3 | 1 | 342.2.h.g | yes | 18 | |
9.d | odd | 6 | 1 | 1026.2.h.g | 18 | ||
19.c | even | 3 | 1 | 342.2.h.g | yes | 18 | |
57.h | odd | 6 | 1 | 1026.2.h.g | 18 | ||
171.h | even | 3 | 1 | inner | 342.2.f.g | ✓ | 18 |
171.j | odd | 6 | 1 | 1026.2.f.g | 18 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
342.2.f.g | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
342.2.f.g | ✓ | 18 | 171.h | even | 3 | 1 | inner |
342.2.h.g | yes | 18 | 9.c | even | 3 | 1 | |
342.2.h.g | yes | 18 | 19.c | even | 3 | 1 | |
1026.2.f.g | 18 | 3.b | odd | 2 | 1 | ||
1026.2.f.g | 18 | 171.j | odd | 6 | 1 | ||
1026.2.h.g | 18 | 9.d | odd | 6 | 1 | ||
1026.2.h.g | 18 | 57.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{18} + 27 T_{5}^{16} + 4 T_{5}^{15} + 537 T_{5}^{14} + 45 T_{5}^{13} + 4414 T_{5}^{12} - 870 T_{5}^{11} + 26208 T_{5}^{10} - 3330 T_{5}^{9} + 64179 T_{5}^{8} + 2376 T_{5}^{7} + 113373 T_{5}^{6} + 8586 T_{5}^{5} + 73386 T_{5}^{4} + \cdots + 729 \)
acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 1)^{18} \)
$3$
\( T^{18} - 2 T^{16} + 6 T^{15} + \cdots + 19683 \)
$5$
\( T^{18} + 27 T^{16} + 4 T^{15} + 537 T^{14} + \cdots + 729 \)
$7$
\( T^{18} - 5 T^{17} + 41 T^{16} + \cdots + 84681 \)
$11$
\( T^{18} - T^{17} + 45 T^{16} + 114 T^{15} + \cdots + 81 \)
$13$
\( (T^{9} + T^{8} - 66 T^{7} - 82 T^{6} + \cdots + 1776)^{2} \)
$17$
\( T^{18} + 5 T^{17} + 84 T^{16} + \cdots + 130439241 \)
$19$
\( T^{18} - 9 T^{17} + \cdots + 322687697779 \)
$23$
\( (T^{9} - 2 T^{8} - 89 T^{7} + 146 T^{6} + \cdots - 1728)^{2} \)
$29$
\( T^{18} + 9 T^{17} + \cdots + 640923532929 \)
$31$
\( T^{18} - 4 T^{17} + \cdots + 2690808129 \)
$37$
\( (T^{9} - 10 T^{8} - 61 T^{7} + 1060 T^{6} + \cdots - 2416)^{2} \)
$41$
\( T^{18} - T^{17} + 87 T^{16} + \cdots + 1679616 \)
$43$
\( (T^{9} + 7 T^{8} - 195 T^{7} + \cdots - 299856)^{2} \)
$47$
\( T^{18} - 19 T^{17} + \cdots + 1579423076001 \)
$53$
\( T^{18} + 10 T^{17} + \cdots + 20372138361 \)
$59$
\( T^{18} + 5 T^{17} + \cdots + 13854720729249 \)
$61$
\( T^{18} - 18 T^{17} + \cdots + 50781270409 \)
$67$
\( (T^{9} + 22 T^{8} - 228 T^{7} + \cdots - 18567936)^{2} \)
$71$
\( T^{18} - 11 T^{17} + \cdots + 120758121 \)
$73$
\( T^{18} + \cdots + 490005727861329 \)
$79$
\( (T^{9} + 2 T^{8} - 370 T^{7} + \cdots + 53092528)^{2} \)
$83$
\( T^{18} + 7 T^{17} + \cdots + 25101031868649 \)
$89$
\( T^{18} - T^{17} + \cdots + 28991407990689 \)
$97$
\( (T^{9} - 504 T^{7} - 785 T^{6} + \cdots - 21665872)^{2} \)
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