Newspace parameters
Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 325.s (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.59513806569\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 65) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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 \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( - 20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409 ) / 5992 \) |
\(\beta_{2}\) | \(=\) | \( ( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409 ) / 5992 \) |
\(\beta_{3}\) | \(=\) | \( ( - 69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539 ) / 1712 \) |
\(\beta_{4}\) | \(=\) | \( ( 321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + \cdots - 3773 ) / 23968 \) |
\(\beta_{5}\) | \(=\) | \( ( 700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + \cdots + 26961 ) / 11984 \) |
\(\beta_{6}\) | \(=\) | \( ( - 409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} + \cdots - 2996 ) / 5992 \) |
\(\beta_{7}\) | \(=\) | \( ( 1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + \cdots - 30699 ) / 23968 \) |
\(\beta_{8}\) | \(=\) | \( ( 700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + \cdots - 26961 ) / 11984 \) |
\(\beta_{9}\) | \(=\) | \( ( - 419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + \cdots + 83703 ) / 23968 \) |
\(\beta_{10}\) | \(=\) | \( ( - 3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + \cdots + 81509 ) / 23968 \) |
\(\beta_{11}\) | \(=\) | \( ( 3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + \cdots + 39483 ) / 23968 \) |
\(\beta_{12}\) | \(=\) | \( ( 3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + \cdots - 39483 ) / 23968 \) |
\(\beta_{13}\) | \(=\) | \( ( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} + \cdots - 15221 ) / 11984 \) |
\(\beta_{14}\) | \(=\) | \( ( 2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + \cdots - 15221 ) / 11984 \) |
\(\beta_{15}\) | \(=\) | \( ( - 6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + \cdots + 1285 ) / 23968 \) |
\(\beta_{16}\) | \(=\) | \( ( 8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + \cdots + 23947 ) / 23968 \) |
\(\beta_{17}\) | \(=\) | \( ( - 4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} + \cdots + 11984 ) / 11984 \) |
\(\beta_{18}\) | \(=\) | \( ( 10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} + \cdots + 20195 ) / 23968 \) |
\(\beta_{19}\) | \(=\) | \( ( - 16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} + \cdots - 15757 ) / 23968 \) |
\(\nu\) | \(=\) | \( \beta_{2} + \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{12} + \beta_{10} - \beta_{3} + \beta _1 - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 7 \beta _1 + 8 \) |
\(\nu^{5}\) | \(=\) | \( - 9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + 17 \beta_{10} + 8 \beta_{9} + 10 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 18 \beta_{4} + 9 \beta_{3} + 37 \beta_{2} + 46 \beta _1 - 2 \) |
\(\nu^{6}\) | \(=\) | \( - 13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 38 \beta_{12} + 25 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 37 \beta_{8} + 10 \beta_{7} - 13 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} + \cdots - 39 \) |
\(\nu^{7}\) | \(=\) | \( 66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} - 121 \beta_{11} - 120 \beta_{10} - 52 \beta_{9} - 76 \beta_{8} - 52 \beta_{7} - 24 \beta_{6} + 28 \beta_{5} + 138 \beta_{4} - 69 \beta_{3} - 236 \beta_{2} + \cdots + 21 \) |
\(\nu^{8}\) | \(=\) | \( 120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} - 56 \beta_{13} - 252 \beta_{12} - 229 \beta_{11} - 397 \beta_{10} + 76 \beta_{9} - 332 \beta_{8} - 84 \beta_{7} + 112 \beta_{6} + \cdots + 213 \) |
\(\nu^{9}\) | \(=\) | \( - 462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + 818 \beta_{11} + 813 \beta_{10} + 325 \beta_{9} + 537 \beta_{8} + 325 \beta_{7} + 122 \beta_{6} - 113 \beta_{5} + \cdots - 170 \) |
\(\nu^{10}\) | \(=\) | \( - 974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + 546 \beta_{13} + 1707 \beta_{12} + 1862 \beta_{11} + 2910 \beta_{10} - 539 \beta_{9} + 2648 \beta_{8} + \cdots - 1255 \) |
\(\nu^{11}\) | \(=\) | \( 3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} - 5456 \beta_{11} - 5476 \beta_{10} - 2042 \beta_{9} - 3716 \beta_{8} - 2042 \beta_{7} - 676 \beta_{6} + \cdots + 1267 \) |
\(\nu^{12}\) | \(=\) | \( 7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} - 4610 \beta_{13} - 11688 \beta_{12} - 14278 \beta_{11} - 20988 \beta_{10} + 3766 \beta_{9} + \cdots + 7797 \) |
\(\nu^{13}\) | \(=\) | \( - 22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + 60422 \beta_{12} + 36438 \beta_{11} + 37058 \beta_{10} + 13074 \beta_{9} + 25620 \beta_{8} + 13074 \beta_{7} + \cdots - 9130 \) |
\(\nu^{14}\) | \(=\) | \( - 54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} - 100084 \beta_{14} + 36202 \beta_{13} + 80555 \beta_{12} + 105990 \beta_{11} + 149815 \beta_{10} + \cdots - 50290 \) |
\(\nu^{15}\) | \(=\) | \( 156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} - 412141 \beta_{12} - 245006 \beta_{11} - 252616 \beta_{10} - 85481 \beta_{9} - 176993 \beta_{8} + \cdots + 64782 \) |
\(\nu^{16}\) | \(=\) | \( 398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + 709463 \beta_{14} - 272912 \beta_{13} - 557750 \beta_{12} - 771768 \beta_{11} - 1062637 \beta_{10} + \cdots + 333034 \) |
\(\nu^{17}\) | \(=\) | \( - 1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + 2825568 \beta_{12} + 1660831 \beta_{11} + 1734115 \beta_{10} + 569378 \beta_{9} + \cdots - 456258 \) |
\(\nu^{18}\) | \(=\) | \( - 2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} - 5013336 \beta_{14} + 2007386 \beta_{13} + 3875098 \beta_{12} + \cdots - 2246355 \) |
\(\nu^{19}\) | \(=\) | \( 7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} - 19469181 \beta_{12} - 11344809 \beta_{11} - 11973534 \beta_{10} + \cdots + 3202109 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
\(n\) | \(27\) | \(301\) |
\(\chi(n)\) | \(-\beta_{11} - \beta_{12}\) | \(-\beta_{11}\) |
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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32.1 |
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−1.12540 | + | 1.94926i | −0.514229 | − | 1.91913i | −1.53307 | − | 2.65535i | 0 | 4.31958 | + | 1.15743i | 1.10607 | − | 0.638592i | 2.39966 | −0.820542 | + | 0.473740i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
32.2 | −0.511309 | + | 0.885613i | 0.721300 | + | 2.69193i | 0.477126 | + | 0.826407i | 0 | −2.75281 | − | 0.737614i | −0.834479 | + | 0.481787i | −3.02107 | −4.12812 | + | 2.38337i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
32.3 | −0.0656513 | + | 0.113711i | 0.0890070 | + | 0.332179i | 0.991380 | + | 1.71712i | 0 | −0.0436159 | − | 0.0116869i | 2.40874 | − | 1.39069i | −0.522947 | 2.49566 | − | 1.44087i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
32.4 | 0.792369 | − | 1.37242i | 0.0510678 | + | 0.190588i | −0.255697 | − | 0.442881i | 0 | 0.302032 | + | 0.0809291i | 0.474866 | − | 0.274164i | 2.35905 | 2.56436 | − | 1.48053i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
32.5 | 1.04397 | − | 1.80821i | −0.713171 | − | 2.66159i | −1.17974 | − | 2.04338i | 0 | −5.55724 | − | 1.48906i | −2.52122 | + | 1.45563i | −0.750585 | −3.97738 | + | 2.29634i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.1 | −0.759023 | − | 1.31467i | 0.653367 | + | 0.175069i | −0.152233 | + | 0.263675i | 0 | −0.265763 | − | 0.991842i | 2.24723 | + | 1.29744i | −2.57390 | −2.20184 | − | 1.27123i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.2 | 0.137404 | + | 0.237991i | 2.28256 | + | 0.611610i | 0.962240 | − | 1.66665i | 0 | 0.168076 | + | 0.627267i | 0.334376 | + | 0.193052i | 1.07848 | 2.23793 | + | 1.29207i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.3 | 0.246951 | + | 0.427732i | −0.908353 | − | 0.243392i | 0.878030 | − | 1.52079i | 0 | −0.120212 | − | 0.448637i | −3.18307 | − | 1.83775i | 1.85513 | −1.83221 | − | 1.05783i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.4 | 0.915816 | + | 1.58624i | −1.91432 | − | 0.512942i | −0.677439 | + | 1.17336i | 0 | −0.939520 | − | 3.50634i | 3.06478 | + | 1.76945i | 1.18163 | 0.803451 | + | 0.463873i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
132.5 | 1.32488 | + | 2.29475i | 1.25278 | + | 0.335680i | −2.51060 | + | 4.34849i | 0 | 0.889471 | + | 3.31955i | −0.0972962 | − | 0.0561740i | −8.00544 | −1.14131 | − | 0.658935i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
193.1 | −1.12540 | − | 1.94926i | −0.514229 | + | 1.91913i | −1.53307 | + | 2.65535i | 0 | 4.31958 | − | 1.15743i | 1.10607 | + | 0.638592i | 2.39966 | −0.820542 | − | 0.473740i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
193.2 | −0.511309 | − | 0.885613i | 0.721300 | − | 2.69193i | 0.477126 | − | 0.826407i | 0 | −2.75281 | + | 0.737614i | −0.834479 | − | 0.481787i | −3.02107 | −4.12812 | − | 2.38337i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
193.3 | −0.0656513 | − | 0.113711i | 0.0890070 | − | 0.332179i | 0.991380 | − | 1.71712i | 0 | −0.0436159 | + | 0.0116869i | 2.40874 | + | 1.39069i | −0.522947 | 2.49566 | + | 1.44087i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
193.4 | 0.792369 | + | 1.37242i | 0.0510678 | − | 0.190588i | −0.255697 | + | 0.442881i | 0 | 0.302032 | − | 0.0809291i | 0.474866 | + | 0.274164i | 2.35905 | 2.56436 | + | 1.48053i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
193.5 | 1.04397 | + | 1.80821i | −0.713171 | + | 2.66159i | −1.17974 | + | 2.04338i | 0 | −5.55724 | + | 1.48906i | −2.52122 | − | 1.45563i | −0.750585 | −3.97738 | − | 2.29634i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.1 | −0.759023 | + | 1.31467i | 0.653367 | − | 0.175069i | −0.152233 | − | 0.263675i | 0 | −0.265763 | + | 0.991842i | 2.24723 | − | 1.29744i | −2.57390 | −2.20184 | + | 1.27123i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.2 | 0.137404 | − | 0.237991i | 2.28256 | − | 0.611610i | 0.962240 | + | 1.66665i | 0 | 0.168076 | − | 0.627267i | 0.334376 | − | 0.193052i | 1.07848 | 2.23793 | − | 1.29207i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.3 | 0.246951 | − | 0.427732i | −0.908353 | + | 0.243392i | 0.878030 | + | 1.52079i | 0 | −0.120212 | + | 0.448637i | −3.18307 | + | 1.83775i | 1.85513 | −1.83221 | + | 1.05783i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.4 | 0.915816 | − | 1.58624i | −1.91432 | + | 0.512942i | −0.677439 | − | 1.17336i | 0 | −0.939520 | + | 3.50634i | 3.06478 | − | 1.76945i | 1.18163 | 0.803451 | − | 0.463873i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.5 | 1.32488 | − | 2.29475i | 1.25278 | − | 0.335680i | −2.51060 | − | 4.34849i | 0 | 0.889471 | − | 3.31955i | −0.0972962 | + | 0.0561740i | −8.00544 | −1.14131 | + | 0.658935i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.o | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 325.2.s.b | 20 | |
5.b | even | 2 | 1 | 65.2.o.a | ✓ | 20 | |
5.c | odd | 4 | 1 | 65.2.t.a | yes | 20 | |
5.c | odd | 4 | 1 | 325.2.x.b | 20 | ||
13.f | odd | 12 | 1 | 325.2.x.b | 20 | ||
15.d | odd | 2 | 1 | 585.2.cf.a | 20 | ||
15.e | even | 4 | 1 | 585.2.dp.a | 20 | ||
65.d | even | 2 | 1 | 845.2.o.g | 20 | ||
65.f | even | 4 | 1 | 845.2.o.e | 20 | ||
65.g | odd | 4 | 1 | 845.2.t.e | 20 | ||
65.g | odd | 4 | 1 | 845.2.t.f | 20 | ||
65.h | odd | 4 | 1 | 845.2.t.g | 20 | ||
65.k | even | 4 | 1 | 845.2.o.f | 20 | ||
65.l | even | 6 | 1 | 845.2.k.d | 20 | ||
65.l | even | 6 | 1 | 845.2.o.f | 20 | ||
65.n | even | 6 | 1 | 845.2.k.e | 20 | ||
65.n | even | 6 | 1 | 845.2.o.e | 20 | ||
65.o | even | 12 | 1 | inner | 325.2.s.b | 20 | |
65.o | even | 12 | 1 | 845.2.k.d | 20 | ||
65.o | even | 12 | 1 | 845.2.o.g | 20 | ||
65.q | odd | 12 | 1 | 845.2.f.e | 20 | ||
65.q | odd | 12 | 1 | 845.2.t.f | 20 | ||
65.r | odd | 12 | 1 | 845.2.f.d | 20 | ||
65.r | odd | 12 | 1 | 845.2.t.e | 20 | ||
65.s | odd | 12 | 1 | 65.2.t.a | yes | 20 | |
65.s | odd | 12 | 1 | 845.2.f.d | 20 | ||
65.s | odd | 12 | 1 | 845.2.f.e | 20 | ||
65.s | odd | 12 | 1 | 845.2.t.g | 20 | ||
65.t | even | 12 | 1 | 65.2.o.a | ✓ | 20 | |
65.t | even | 12 | 1 | 845.2.k.e | 20 | ||
195.bc | odd | 12 | 1 | 585.2.cf.a | 20 | ||
195.bh | even | 12 | 1 | 585.2.dp.a | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.o.a | ✓ | 20 | 5.b | even | 2 | 1 | |
65.2.o.a | ✓ | 20 | 65.t | even | 12 | 1 | |
65.2.t.a | yes | 20 | 5.c | odd | 4 | 1 | |
65.2.t.a | yes | 20 | 65.s | odd | 12 | 1 | |
325.2.s.b | 20 | 1.a | even | 1 | 1 | trivial | |
325.2.s.b | 20 | 65.o | even | 12 | 1 | inner | |
325.2.x.b | 20 | 5.c | odd | 4 | 1 | ||
325.2.x.b | 20 | 13.f | odd | 12 | 1 | ||
585.2.cf.a | 20 | 15.d | odd | 2 | 1 | ||
585.2.cf.a | 20 | 195.bc | odd | 12 | 1 | ||
585.2.dp.a | 20 | 15.e | even | 4 | 1 | ||
585.2.dp.a | 20 | 195.bh | even | 12 | 1 | ||
845.2.f.d | 20 | 65.r | odd | 12 | 1 | ||
845.2.f.d | 20 | 65.s | odd | 12 | 1 | ||
845.2.f.e | 20 | 65.q | odd | 12 | 1 | ||
845.2.f.e | 20 | 65.s | odd | 12 | 1 | ||
845.2.k.d | 20 | 65.l | even | 6 | 1 | ||
845.2.k.d | 20 | 65.o | even | 12 | 1 | ||
845.2.k.e | 20 | 65.n | even | 6 | 1 | ||
845.2.k.e | 20 | 65.t | even | 12 | 1 | ||
845.2.o.e | 20 | 65.f | even | 4 | 1 | ||
845.2.o.e | 20 | 65.n | even | 6 | 1 | ||
845.2.o.f | 20 | 65.k | even | 4 | 1 | ||
845.2.o.f | 20 | 65.l | even | 6 | 1 | ||
845.2.o.g | 20 | 65.d | even | 2 | 1 | ||
845.2.o.g | 20 | 65.o | even | 12 | 1 | ||
845.2.t.e | 20 | 65.g | odd | 4 | 1 | ||
845.2.t.e | 20 | 65.r | odd | 12 | 1 | ||
845.2.t.f | 20 | 65.g | odd | 4 | 1 | ||
845.2.t.f | 20 | 65.q | odd | 12 | 1 | ||
845.2.t.g | 20 | 65.h | odd | 4 | 1 | ||
845.2.t.g | 20 | 65.s | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 4 T_{2}^{19} + 21 T_{2}^{18} - 48 T_{2}^{17} + 170 T_{2}^{16} - 322 T_{2}^{15} + 893 T_{2}^{14} - 1258 T_{2}^{13} + 2677 T_{2}^{12} - 2864 T_{2}^{11} + 5468 T_{2}^{10} - 4404 T_{2}^{9} + 6569 T_{2}^{8} - 3252 T_{2}^{7} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{20} - 4 T^{19} + 21 T^{18} - 48 T^{17} + \cdots + 1 \)
$3$
\( T^{20} - 2 T^{19} + 8 T^{18} - 20 T^{17} + \cdots + 16 \)
$5$
\( T^{20} \)
$7$
\( T^{20} - 6 T^{19} - 8 T^{18} + 120 T^{17} + \cdots + 64 \)
$11$
\( T^{20} + 16 T^{19} + 140 T^{18} + \cdots + 256 \)
$13$
\( T^{20} + 2 T^{19} + \cdots + 137858491849 \)
$17$
\( T^{20} - 10 T^{19} - 7 T^{18} + \cdots + 1168561 \)
$19$
\( T^{20} - 20 T^{19} + \cdots + 1583721616 \)
$23$
\( T^{20} - 2 T^{19} - 52 T^{18} + 344 T^{17} + \cdots + 144 \)
$29$
\( T^{20} - 173 T^{18} + \cdots + 206213167449 \)
$31$
\( T^{20} - 104 T^{17} + 6072 T^{16} + \cdots + 2166784 \)
$37$
\( T^{20} + 42 T^{19} + \cdots + 4508182449 \)
$41$
\( T^{20} - 10 T^{19} + \cdots + 3748255729 \)
$43$
\( T^{20} - 22 T^{19} + \cdots + 1370772640000 \)
$47$
\( T^{20} + 368 T^{18} + \cdots + 807469056 \)
$53$
\( T^{20} - 10 T^{19} + \cdots + 2978634160384 \)
$59$
\( T^{20} - 16 T^{19} + 320 T^{18} + \cdots + 33856 \)
$61$
\( T^{20} + 16 T^{19} + \cdots + 826457355409 \)
$67$
\( T^{20} - 58 T^{19} + \cdots + 15478905336976 \)
$71$
\( T^{20} + 16 T^{19} + \cdots + 11\!\cdots\!44 \)
$73$
\( (T^{10} + 36 T^{9} + 249 T^{8} + \cdots + 253113232)^{2} \)
$79$
\( T^{20} + 808 T^{18} + \cdots + 75\!\cdots\!36 \)
$83$
\( T^{20} + 448 T^{18} + \cdots + 11512611864576 \)
$89$
\( T^{20} - 6 T^{19} + \cdots + 329648222500 \)
$97$
\( T^{20} - 22 T^{19} + \cdots + 28\!\cdots\!36 \)
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