Properties

Label 29.10.a.a
Level $29$
Weight $10$
Character orbit 29.a
Self dual yes
Analytic conductor $14.936$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9360392488\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 2901 x^{7} - 4592 x^{6} + 2830996 x^{5} + 7409504 x^{4} - 1038861888 x^{3} - 2974719488 x^{2} + 115773751296 x + 456378417152\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -27 - \beta_{4} ) q^{3} + ( 133 + 2 \beta_{1} + \beta_{2} ) q^{4} + ( -83 - 5 \beta_{1} - \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{5} + ( -261 + 51 \beta_{1} - 5 \beta_{2} + 11 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{6} + ( -796 + 81 \beta_{1} - 8 \beta_{2} + 16 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -1538 + 87 \beta_{1} - 5 \beta_{2} - \beta_{3} + 27 \beta_{4} + \beta_{5} + \beta_{6} - 5 \beta_{7} + \beta_{8} ) q^{8} + ( 3670 + 298 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} + 44 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + \beta_{7} - 7 \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -27 - \beta_{4} ) q^{3} + ( 133 + 2 \beta_{1} + \beta_{2} ) q^{4} + ( -83 - 5 \beta_{1} - \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{5} + ( -261 + 51 \beta_{1} - 5 \beta_{2} + 11 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{6} + ( -796 + 81 \beta_{1} - 8 \beta_{2} + 16 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -1538 + 87 \beta_{1} - 5 \beta_{2} - \beta_{3} + 27 \beta_{4} + \beta_{5} + \beta_{6} - 5 \beta_{7} + \beta_{8} ) q^{8} + ( 3670 + 298 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} + 44 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + \beta_{7} - 7 \beta_{8} ) q^{9} + ( 4206 + 459 \beta_{1} + 34 \beta_{2} - 4 \beta_{3} + 74 \beta_{4} - 6 \beta_{5} + 22 \beta_{6} + 20 \beta_{7} + 12 \beta_{8} ) q^{10} + ( -6595 + 886 \beta_{1} + 45 \beta_{2} - 30 \beta_{3} - 11 \beta_{4} - 8 \beta_{5} + \beta_{6} - 4 \beta_{7} + 13 \beta_{8} ) q^{11} + ( -14128 + 871 \beta_{1} + 7 \beta_{2} + 29 \beta_{3} - 247 \beta_{4} + 35 \beta_{5} + 3 \beta_{6} - 27 \beta_{7} - 45 \beta_{8} ) q^{12} + ( -18431 + 725 \beta_{1} + 72 \beta_{3} - 172 \beta_{4} + 24 \beta_{5} - 60 \beta_{6} - 50 \beta_{7} + 15 \beta_{8} ) q^{13} + ( -45072 + 1958 \beta_{1} - 60 \beta_{2} - 4 \beta_{3} - 392 \beta_{4} + 32 \beta_{5} - 8 \beta_{6} + 36 \beta_{7} - 28 \beta_{8} ) q^{14} + ( -76951 + 2871 \beta_{1} + 3 \beta_{2} - 194 \beta_{3} - 71 \beta_{4} - 56 \beta_{5} - 59 \beta_{6} + 79 \beta_{7} + 113 \beta_{8} ) q^{15} + ( -116079 + 1292 \beta_{1} - 183 \beta_{2} - 96 \beta_{3} - 378 \beta_{4} - 66 \beta_{5} + 54 \beta_{6} + 52 \beta_{7} - 20 \beta_{8} ) q^{16} + ( -43896 - 1311 \beta_{1} - 53 \beta_{2} + 298 \beta_{3} + 124 \beta_{4} - 90 \beta_{5} - 113 \beta_{6} + 7 \beta_{7} - 95 \beta_{8} ) q^{17} + ( -186596 - 7834 \beta_{1} - 284 \beta_{2} + 160 \beta_{3} + 264 \beta_{4} + 104 \beta_{5} + 240 \beta_{6} - 48 \beta_{7} + 192 \beta_{8} ) q^{18} + ( -250574 - 2924 \beta_{1} + 21 \beta_{2} - 242 \beta_{3} - 436 \beta_{4} - 160 \beta_{5} - 59 \beta_{6} - 280 \beta_{7} - 433 \beta_{8} ) q^{19} + ( -248897 - 10772 \beta_{1} - 207 \beta_{2} + 218 \beta_{3} + 618 \beta_{4} + 126 \beta_{5} + 222 \beta_{6} + 26 \beta_{7} + 118 \beta_{8} ) q^{20} + ( -194706 - 16971 \beta_{1} + 613 \beta_{2} + 174 \beta_{3} + 2292 \beta_{4} + 374 \beta_{5} + 177 \beta_{6} + 27 \beta_{7} + 255 \beta_{8} ) q^{21} + ( -590115 - 7503 \beta_{1} - 159 \beta_{2} - 748 \beta_{3} - 559 \beta_{4} - 123 \beta_{5} + 111 \beta_{6} - 340 \beta_{7} - 188 \beta_{8} ) q^{22} + ( -188250 - 2689 \beta_{1} + 1306 \beta_{2} - 540 \beta_{3} + 3290 \beta_{4} + 104 \beta_{5} - 288 \beta_{6} + 895 \beta_{7} + 170 \beta_{8} ) q^{23} + ( -494179 - 9142 \beta_{1} + 371 \beta_{2} - 414 \beta_{3} + 2212 \beta_{4} + 96 \beta_{5} + 152 \beta_{6} + 518 \beta_{7} + 898 \beta_{8} ) q^{24} + ( -109638 - 25086 \beta_{1} + 781 \beta_{2} + 1754 \beta_{3} + 3552 \beta_{4} - 150 \beta_{5} - 567 \beta_{6} - 365 \beta_{7} - 544 \beta_{8} ) q^{25} + ( -473446 + 8841 \beta_{1} - 666 \beta_{2} + 1652 \beta_{3} - 4414 \beta_{4} - 398 \beta_{5} - 1506 \beta_{6} - 620 \beta_{7} - 1228 \beta_{8} ) q^{26} + ( -776183 - 12398 \beta_{1} + 1163 \beta_{2} - 1194 \beta_{3} + 1781 \beta_{4} + 452 \beta_{5} + 463 \beta_{6} - 26 \beta_{7} + 101 \beta_{8} ) q^{27} + ( -943690 + 23812 \beta_{1} - 1930 \beta_{2} - 136 \beta_{3} - 2896 \beta_{4} - 1824 \beta_{5} - 512 \beta_{6} - 72 \beta_{7} + 504 \beta_{8} ) q^{28} -707281 q^{29} + ( -1876815 + 60347 \beta_{1} - 2167 \beta_{2} - 2368 \beta_{3} - 20387 \beta_{4} + 809 \beta_{5} + 547 \beta_{6} - 2096 \beta_{7} - 2728 \beta_{8} ) q^{30} + ( -1329251 - 40435 \beta_{1} - 2858 \beta_{2} - 1400 \beta_{3} + 9179 \beta_{4} + 740 \beta_{5} + 2632 \beta_{6} + 617 \beta_{7} + 5046 \beta_{8} ) q^{31} + ( -146410 + 144767 \beta_{1} - 561 \beta_{2} - 1277 \beta_{3} - 12545 \beta_{4} - 675 \beta_{5} - 611 \beta_{6} + 3703 \beta_{7} + 541 \beta_{8} ) q^{32} + ( 193009 - 39514 \beta_{1} + 1073 \beta_{2} + 1822 \beta_{3} + 3232 \beta_{4} + 1258 \beta_{5} + 4869 \beta_{6} - 579 \beta_{7} - 1266 \beta_{8} ) q^{33} + ( 958818 + 46048 \beta_{1} - 4570 \beta_{2} + 9028 \beta_{3} - 12326 \beta_{4} + 2234 \beta_{5} - 434 \beta_{6} - 1396 \beta_{7} + 4 \beta_{8} ) q^{34} + ( -362900 + 60017 \beta_{1} - 2676 \beta_{2} - 908 \beta_{3} - 12180 \beta_{4} + 1996 \beta_{5} - 3410 \beta_{6} - 2519 \beta_{7} - 2836 \beta_{8} ) q^{35} + ( 3318546 + 151912 \beta_{1} + 6230 \beta_{2} + 1348 \beta_{3} + 5572 \beta_{4} + 1068 \beta_{5} + 1292 \beta_{6} + 4724 \beta_{7} + 2428 \beta_{8} ) q^{36} + ( 1610496 + 89172 \beta_{1} + 300 \beta_{2} - 7232 \beta_{3} + 9676 \beta_{4} - 9128 \beta_{5} + 1496 \beta_{6} + 1930 \beta_{7} - 6674 \beta_{8} ) q^{37} + ( 1641460 + 285320 \beta_{1} + 11824 \beta_{2} - 14420 \beta_{3} - 10160 \beta_{4} - 1128 \beta_{5} - 5176 \beta_{6} - 60 \beta_{7} + 7436 \beta_{8} ) q^{38} + ( 4568237 - 146713 \beta_{1} + 1622 \beta_{2} + 8160 \beta_{3} + 25979 \beta_{4} + 3456 \beta_{5} - 2380 \beta_{6} - 3595 \beta_{7} - 3200 \beta_{8} ) q^{39} + ( 5016804 + 104185 \beta_{1} - 5289 \beta_{2} + 8113 \beta_{3} - 6775 \beta_{4} + 1107 \beta_{5} - 7101 \beta_{6} - 5299 \beta_{7} - 5865 \beta_{8} ) q^{40} + ( 5825006 - 231453 \beta_{1} + 3331 \beta_{2} + 8622 \beta_{3} + 37736 \beta_{4} - 2870 \beta_{5} - 7457 \beta_{6} - 5601 \beta_{7} + 6847 \beta_{8} ) q^{41} + ( 11458538 - 6842 \beta_{1} + 25182 \beta_{2} + 3620 \beta_{3} + 38626 \beta_{4} - 414 \beta_{5} + 8310 \beta_{6} + 1308 \beta_{7} - 1932 \beta_{8} ) q^{42} + ( 2418533 - 225220 \beta_{1} - 13109 \beta_{2} - 18446 \beta_{3} + 63353 \beta_{4} - 4444 \beta_{5} + 3467 \beta_{6} + 16270 \beta_{7} + 16631 \beta_{8} ) q^{43} + ( 7880828 + 256841 \beta_{1} + 8677 \beta_{2} - 12637 \beta_{3} - 6945 \beta_{4} - 1755 \beta_{5} - 4059 \beta_{6} + 4331 \beta_{7} - 1667 \beta_{8} ) q^{44} + ( 6793548 - 676250 \beta_{1} - 6415 \beta_{2} + 14044 \beta_{3} + 36832 \beta_{4} + 696 \beta_{5} + 5915 \beta_{6} - 10012 \beta_{7} - 10115 \beta_{8} ) q^{45} + ( 2281574 - 335084 \beta_{1} - 18006 \beta_{2} - 1892 \beta_{3} - 42706 \beta_{4} + 17950 \beta_{5} + 13026 \beta_{6} - 13132 \beta_{7} - 3404 \beta_{8} ) q^{46} + ( 5001575 - 436393 \beta_{1} - 13375 \beta_{2} + 17254 \beta_{3} - 75281 \beta_{4} + 11792 \beta_{5} - 7341 \beta_{6} + 793 \beta_{7} - 12839 \beta_{8} ) q^{47} + ( 13659444 - 117033 \beta_{1} + 11589 \beta_{2} - 10821 \beta_{3} + 103083 \beta_{4} - 15447 \beta_{5} + 14457 \beta_{6} + 9879 \beta_{7} + 8973 \beta_{8} ) q^{48} + ( -2980267 - 574486 \beta_{1} - 34726 \beta_{2} - 20580 \beta_{3} - 21328 \beta_{4} - 2936 \beta_{5} + 18350 \beta_{6} + 24134 \beta_{7} + 5570 \beta_{8} ) q^{49} + ( 17328436 - 224580 \beta_{1} + 18932 \beta_{2} + 42632 \beta_{3} - 41924 \beta_{4} + 9644 \beta_{5} + 2700 \beta_{6} - 8800 \beta_{7} + 1656 \beta_{8} ) q^{50} + ( -3204794 - 517611 \beta_{1} - 16562 \beta_{2} - 1024 \beta_{3} - 72418 \beta_{4} + 1936 \beta_{5} - 21028 \beta_{6} - 11527 \beta_{7} - 6830 \beta_{8} ) q^{51} + ( 3281073 + 194084 \beta_{1} - 51273 \beta_{2} - 594 \beta_{3} - 40082 \beta_{4} - 2678 \beta_{5} - 6006 \beta_{6} + 11358 \beta_{7} - 6654 \beta_{8} ) q^{52} + ( -12351433 - 57519 \beta_{1} + 13908 \beta_{2} - 31500 \beta_{3} - 178340 \beta_{4} - 25760 \beta_{5} - 11320 \beta_{6} - 21780 \beta_{7} + 4313 \beta_{8} ) q^{53} + ( 7854421 + 483161 \beta_{1} + 39721 \beta_{2} - 41324 \beta_{3} + 68361 \beta_{4} - 4291 \beta_{5} + 7239 \beta_{6} + 2748 \beta_{7} + 6756 \beta_{8} ) q^{54} + ( -19278387 - 324309 \beta_{1} - 57868 \beta_{2} + 31832 \beta_{3} - 263569 \beta_{4} + 13508 \beta_{5} - 46438 \beta_{6} - 21259 \beta_{7} - 11542 \beta_{8} ) q^{55} + ( 7485780 + 515682 \beta_{1} + 19210 \beta_{2} + 24050 \beta_{3} - 42486 \beta_{4} - 17122 \beta_{5} + 4894 \beta_{6} - 31238 \beta_{7} - 6162 \beta_{8} ) q^{56} + ( 9535226 - 206534 \beta_{1} + 98519 \beta_{2} + 1928 \beta_{3} + 256620 \beta_{4} - 14312 \beta_{5} + 28209 \beta_{6} + 62340 \beta_{7} + 64851 \beta_{8} ) q^{57} + 707281 \beta_{1} q^{58} + ( -26187370 + 1163815 \beta_{1} + 24100 \beta_{2} + 8524 \beta_{3} - 332146 \beta_{4} + 72524 \beta_{5} - 20614 \beta_{6} + 22807 \beta_{7} - 3020 \beta_{8} ) q^{59} + ( -5366670 + 1620321 \beta_{1} - 63409 \beta_{2} - 39273 \beta_{3} + 328955 \beta_{4} - 15767 \beta_{5} - 64887 \beta_{6} - 3009 \beta_{7} + 633 \beta_{8} ) q^{60} + ( -26756236 - 88871 \beta_{1} + 68825 \beta_{2} + 29922 \beta_{3} - 191196 \beta_{4} + 10626 \beta_{5} + 47501 \beta_{6} - 16525 \beta_{7} - 52277 \beta_{8} ) q^{61} + ( 29115631 + 2207281 \beta_{1} + 175347 \beta_{2} + 11628 \beta_{3} + 29783 \beta_{4} - 41885 \beta_{5} + 80289 \beta_{6} + 36132 \beta_{7} - 66188 \beta_{8} ) q^{62} + ( -36493176 + 927178 \beta_{1} + 38048 \beta_{2} - 3320 \beta_{3} + 84484 \beta_{4} - 39948 \beta_{5} - 57352 \beta_{6} - 56998 \beta_{7} - 37172 \beta_{8} ) q^{63} + ( -37157023 - 481980 \beta_{1} - 257223 \beta_{2} + 80144 \beta_{3} + 188398 \beta_{4} + 74854 \beta_{5} - 42978 \beta_{6} - 48556 \beta_{7} - 1076 \beta_{8} ) q^{64} + ( -69493357 + 2137342 \beta_{1} + 53032 \beta_{2} - 138182 \beta_{3} + 44332 \beta_{4} + 402 \beta_{5} + 56966 \beta_{6} + 21595 \beta_{7} + 47501 \beta_{8} ) q^{65} + ( 24808720 + 324061 \beta_{1} + 53192 \beta_{2} - 7200 \beta_{3} + 879136 \beta_{4} - 40720 \beta_{5} + 45016 \beta_{6} + 84520 \beta_{7} + 87968 \beta_{8} ) q^{66} + ( -74697432 - 1452058 \beta_{1} + 112640 \beta_{2} - 37944 \beta_{3} + 440 \beta_{4} + 19172 \beta_{5} + 92104 \beta_{6} - 20150 \beta_{7} + 119928 \beta_{8} ) q^{67} + ( -7115786 + 977150 \beta_{1} - 161616 \beta_{2} + 76374 \beta_{3} + 292278 \beta_{4} + 18082 \beta_{5} - 5662 \beta_{6} + 40198 \beta_{7} + 28762 \beta_{8} ) q^{68} + ( -78545868 - 1577081 \beta_{1} - 149423 \beta_{2} + 93130 \beta_{3} + 214988 \beta_{4} + 106642 \beta_{5} + 25541 \beta_{6} - 112849 \beta_{7} - 124475 \beta_{8} ) q^{69} + ( -40655628 + 769118 \beta_{1} - 54944 \beta_{2} - 93868 \beta_{3} - 262716 \beta_{4} - 7316 \beta_{5} - 125732 \beta_{6} - 8324 \beta_{7} + 7580 \beta_{8} ) q^{70} + ( -52791986 + 2165872 \beta_{1} + 54480 \beta_{2} + 96 \beta_{3} - 148578 \beta_{4} - 130176 \beta_{5} + 114576 \beta_{6} + 91792 \beta_{7} - 44688 \beta_{8} ) q^{71} + ( -2157496 - 1569666 \beta_{1} - 203166 \beta_{2} + 30142 \beta_{3} + 266270 \beta_{4} - 2294 \beta_{5} - 50006 \beta_{6} + 2470 \beta_{7} - 114670 \beta_{8} ) q^{72} + ( -47182012 - 558930 \beta_{1} - 25102 \beta_{2} + 142332 \beta_{3} + 33816 \beta_{4} - 164828 \beta_{5} - 95962 \beta_{6} - 684 \beta_{7} - 71828 \beta_{8} ) q^{73} + ( -59205544 - 997104 \beta_{1} - 53056 \beta_{2} - 188424 \beta_{3} - 437960 \beta_{4} + 74600 \beta_{5} + 83112 \beta_{6} - 27352 \beta_{7} + 155192 \beta_{8} ) q^{74} + ( -101515956 - 372964 \beta_{1} - 139952 \beta_{2} + 9956 \beta_{3} - 325476 \beta_{4} + 70584 \beta_{5} - 204224 \beta_{6} - 63086 \beta_{7} - 31222 \beta_{8} ) q^{75} + ( -61303180 - 4922548 \beta_{1} - 61552 \beta_{2} - 181676 \beta_{3} - 903652 \beta_{4} + 72548 \beta_{5} - 6524 \beta_{6} - 45340 \beta_{7} + 30684 \beta_{8} ) q^{76} + ( -20308958 - 95257 \beta_{1} + 158087 \beta_{2} + 161758 \beta_{3} + 296520 \beta_{4} - 86570 \beta_{5} + 29979 \beta_{6} - 63819 \beta_{7} + 60469 \beta_{8} ) q^{77} + ( 103173079 - 5846971 \beta_{1} + 227131 \beta_{2} + 127084 \beta_{3} - 42217 \beta_{4} + 19139 \beta_{5} - 6879 \beta_{6} + 8532 \beta_{7} + 22788 \beta_{8} ) q^{78} + ( -18670643 + 749607 \beta_{1} + 325605 \beta_{2} - 116134 \beta_{3} - 294151 \beta_{4} - 114796 \beta_{5} + 52343 \beta_{6} + 331809 \beta_{7} + 25073 \beta_{8} ) q^{79} + ( 62678939 + 954082 \beta_{1} - 34187 \beta_{2} + 9926 \beta_{3} - 960072 \beta_{4} - 45764 \beta_{5} - 285116 \beta_{6} - 43462 \beta_{7} - 58322 \beta_{8} ) q^{80} + ( -101782313 - 2234720 \beta_{1} - 244153 \beta_{2} - 100116 \beta_{3} + 408500 \beta_{4} + 234680 \beta_{5} - 54815 \beta_{6} + 21430 \beta_{7} + 158303 \beta_{8} ) q^{81} + ( 163453062 - 8638778 \beta_{1} + 610618 \beta_{2} + 313572 \beta_{3} - 1695306 \beta_{4} + 3046 \beta_{5} + 73570 \beta_{6} - 156164 \beta_{7} - 253244 \beta_{8} ) q^{82} + ( -52076434 + 4049549 \beta_{1} - 15816 \beta_{2} + 53164 \beta_{3} - 585694 \beta_{4} + 159008 \beta_{5} + 902 \beta_{6} - 20567 \beta_{7} - 7184 \beta_{8} ) q^{83} + ( 105654192 - 9430070 \beta_{1} - 285462 \beta_{2} - 51986 \beta_{3} + 467502 \beta_{4} - 168118 \beta_{5} + 145034 \beta_{6} - 9250 \beta_{7} + 39874 \beta_{8} ) q^{84} + ( -30113032 - 1963013 \beta_{1} + 448581 \beta_{2} - 325826 \beta_{3} + 859792 \beta_{4} - 57658 \beta_{5} + 388489 \beta_{6} + 75503 \beta_{7} + 61537 \beta_{8} ) q^{85} + ( 162783349 + 303321 \beta_{1} + 408689 \beta_{2} - 71812 \beta_{3} - 1761247 \beta_{4} + 137477 \beta_{5} + 429599 \beta_{6} - 31068 \beta_{7} - 254116 \beta_{8} ) q^{86} + ( 19096587 + 707281 \beta_{4} ) q^{87} + ( 130682251 - 7384598 \beta_{1} - 276383 \beta_{2} + 84546 \beta_{3} - 361456 \beta_{4} + 162748 \beta_{5} - 96252 \beta_{6} + 42894 \beta_{7} + 105050 \beta_{8} ) q^{88} + ( -75866946 + 9870751 \beta_{1} - 1381031 \beta_{2} + 188462 \beta_{3} + 2293404 \beta_{4} + 233770 \beta_{5} - 251495 \beta_{6} - 55833 \beta_{7} - 539 \beta_{8} ) q^{89} + ( 446637088 - 2995754 \beta_{1} + 1005104 \beta_{2} + 116304 \beta_{3} + 966164 \beta_{4} - 89644 \beta_{5} + 56860 \beta_{6} + 200200 \beta_{7} + 239408 \beta_{8} ) q^{90} + ( 1173028 - 952661 \beta_{1} + 666284 \beta_{2} - 9572 \beta_{3} - 404136 \beta_{4} - 352704 \beta_{5} + 215806 \beta_{6} - 247865 \beta_{7} + 186132 \beta_{8} ) q^{91} + ( 302734526 + 7576482 \beta_{1} + 73772 \beta_{2} - 211758 \beta_{3} + 1242818 \beta_{4} - 432410 \beta_{5} - 100090 \beta_{6} - 9182 \beta_{7} + 125950 \beta_{8} ) q^{92} + ( -96091269 + 7093327 \beta_{1} - 1300727 \beta_{2} - 360632 \beta_{3} + 3188764 \beta_{4} - 210284 \beta_{5} - 255541 \beta_{6} - 166204 \beta_{7} - 82310 \beta_{8} ) q^{93} + ( 269621683 - 17119 \beta_{1} - 520605 \beta_{2} + 301352 \beta_{3} + 1241407 \beta_{4} + 59219 \beta_{5} - 469327 \beta_{6} + 106328 \beta_{7} + 144096 \beta_{8} ) q^{94} + ( 7656826 + 24201092 \beta_{1} - 841035 \beta_{2} + 199502 \beta_{3} - 179160 \beta_{4} - 144576 \beta_{5} - 775175 \beta_{6} - 174826 \beta_{7} - 442343 \beta_{8} ) q^{95} + ( 345890645 - 11611162 \beta_{1} + 430571 \beta_{2} + 199290 \beta_{3} - 1406288 \beta_{4} + 39948 \beta_{5} + 627908 \beta_{6} - 280906 \beta_{7} - 424574 \beta_{8} ) q^{96} + ( 19287570 + 10830253 \beta_{1} + 637349 \beta_{2} - 344738 \beta_{3} + 576056 \beta_{4} + 464778 \beta_{5} + 3937 \beta_{6} + 218181 \beta_{7} - 151279 \beta_{8} ) q^{97} + ( 364199072 + 17538803 \beta_{1} - 21728 \beta_{2} - 222704 \beta_{3} + 1342880 \beta_{4} + 68544 \beta_{5} + 284272 \beta_{6} + 337200 \beta_{7} + 158336 \beta_{8} ) q^{98} + ( 34295796 + 13221890 \beta_{1} - 234173 \beta_{2} - 95698 \beta_{3} - 1132594 \beta_{4} + 3852 \beta_{5} - 456137 \beta_{6} + 388654 \beta_{7} - 27859 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 244q^{3} + 1194q^{4} - 738q^{5} - 2330q^{6} - 7128q^{7} - 13776q^{8} + 33017q^{9} + O(q^{10}) \) \( 9q - 244q^{3} + 1194q^{4} - 738q^{5} - 2330q^{6} - 7128q^{7} - 13776q^{8} + 33017q^{9} + 37812q^{10} - 59512q^{11} - 127348q^{12} - 165758q^{13} - 406080q^{14} - 693178q^{15} - 1044958q^{16} - 394814q^{17} - 1676576q^{18} - 2256606q^{19} - 2237578q^{20} - 1750168q^{21} - 5311718q^{22} - 1699500q^{23} - 4446318q^{24} - 983481q^{25} - 4264740q^{26} - 6987958q^{27} - 8491636q^{28} - 6365529q^{29} - 16907854q^{30} - 11929632q^{31} - 1346192q^{32} + 1750252q^{33} + 8655764q^{34} - 3275324q^{35} + 29848532q^{36} + 14454898q^{37} + 14709736q^{38} + 41155042q^{39} + 45167060q^{40} + 52495202q^{41} + 103102340q^{42} + 21819888q^{43} + 70837004q^{44} + 61248326q^{45} + 20628012q^{46} + 44968948q^{47} + 122982540q^{48} - 26826775q^{49} + 155997680q^{50} - 28882428q^{51} + 29562122q^{52} - 111394302q^{53} + 70575802q^{54} - 173560742q^{55} + 67419136q^{56} + 85769252q^{57} - 236142720q^{59} - 47991000q^{60} - 241129054q^{61} + 261343278q^{62} - 328513060q^{63} - 333112958q^{64} - 625660884q^{65} + 223958776q^{66} - 672046492q^{67} - 63179948q^{68} - 705827600q^{69} - 366389016q^{70} - 475841956q^{71} - 18937608q^{72} - 424813822q^{73} - 532689728q^{74} - 913708498q^{75} - 552478056q^{76} - 182224776q^{77} + 928127886q^{78} - 170801148q^{79} + 562655678q^{80} - 914585851q^{81} + 1468192652q^{82} - 468898296q^{83} + 952386216q^{84} - 271552972q^{85} + 1462277802q^{86} + 172576564q^{87} + 1176890862q^{88} - 676036598q^{89} + 4017858752q^{90} + 9763884q^{91} + 2724990708q^{92} - 858755220q^{93} + 2429128614q^{94} + 69331732q^{95} + 3111862050q^{96} + 170708754q^{97} + 3278517600q^{98} + 305494078q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 2901 x^{7} - 4592 x^{6} + 2830996 x^{5} + 7409504 x^{4} - 1038861888 x^{3} - 2974719488 x^{2} + 115773751296 x + 456378417152\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 645 \)
\(\beta_{3}\)\(=\)\((\)\(-24206118505 \nu^{8} + 3628782688552 \nu^{7} - 51380960778211 \nu^{6} - 5585746755804376 \nu^{5} + 114728804711338636 \nu^{4} + 2036284603062787776 \nu^{3} - 37031257559876553664 \nu^{2} - 119824273783332350976 \nu - 318309112134922022912\)\()/ 6207629273111834624 \)
\(\beta_{4}\)\(=\)\((\)\(-30645811893 \nu^{8} + 46114646956 \nu^{7} + 72238283274393 \nu^{6} + 39514496384020 \nu^{5} - 52730446038313316 \nu^{4} - 138810329723380464 \nu^{3} + 12232140633130223040 \nu^{2} + 51171846561640967424 \nu - 566945984913808912384\)\()/ 3103814636555917312 \)
\(\beta_{5}\)\(=\)\((\)\(-145943679107 \nu^{8} + 3948353615584 \nu^{7} + 511126104077119 \nu^{6} - 9649471699062160 \nu^{5} - 583997383883102588 \nu^{4} + 7225812409972194912 \nu^{3} + 237917267859066078656 \nu^{2} - 1672141488408091220480 \nu - 23835185471020802486272\)\()/ 6207629273111834624 \)
\(\beta_{6}\)\(=\)\((\)\(-206831158817 \nu^{8} + 10132200890616 \nu^{7} + 426846090538933 \nu^{6} - 19178886358511208 \nu^{5} - 250864100373963860 \nu^{4} + 9643203691587552256 \nu^{3} + 26711124769445420608 \nu^{2} - 984362576950249740288 \nu + 596537573429976887296\)\()/ 6207629273111834624 \)
\(\beta_{7}\)\(=\)\((\)\(-40920390157 \nu^{8} + 422294149298 \nu^{7} + 97500534227889 \nu^{6} - 819303572581354 \nu^{5} - 73700257565341764 \nu^{4} + 583798768217020680 \nu^{3} + 18409968314351361408 \nu^{2} - 182922462425474090112 \nu - 1072681155315824624640\)\()/ 775953659138979328 \)
\(\beta_{8}\)\(=\)\((\)\(346626955361 \nu^{8} + 3949803218648 \nu^{7} - 990199083095925 \nu^{6} - 11663314406222248 \nu^{5} + 849024072423653588 \nu^{4} + 9807347762134578240 \nu^{3} - 194758365437806467648 \nu^{2} - 1789026083597896539136 \nu + 155905174428934848512\)\()/ 6207629273111834624 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 645\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + 5 \beta_{7} - \beta_{6} - \beta_{5} - 27 \beta_{4} + \beta_{3} + 5 \beta_{2} + 937 \beta_{1} + 1538\)
\(\nu^{4}\)\(=\)\(-20 \beta_{8} + 52 \beta_{7} + 54 \beta_{6} - 66 \beta_{5} - 378 \beta_{4} - 96 \beta_{3} + 1353 \beta_{2} + 4364 \beta_{1} + 612497\)
\(\nu^{5}\)\(=\)\(-2589 \beta_{8} + 6537 \beta_{7} - 1437 \beta_{6} - 1373 \beta_{5} - 42751 \beta_{4} + 3325 \beta_{3} + 10801 \beta_{2} + 987777 \beta_{1} + 3296234\)
\(\nu^{6}\)\(=\)\(-52276 \beta_{8} + 84564 \beta_{7} + 95262 \beta_{6} - 94106 \beta_{5} - 779282 \beta_{4} - 165616 \beta_{3} + 1633593 \beta_{2} + 7544132 \beta_{1} + 650555745\)
\(\nu^{7}\)\(=\)\(-4158157 \beta_{8} + 7523417 \beta_{7} - 1124589 \beta_{6} - 1517101 \beta_{5} - 58467439 \beta_{4} + 5509229 \beta_{3} + 17313905 \beta_{2} + 1093110241 \beta_{1} + 5557992858\)
\(\nu^{8}\)\(=\)\(-93877892 \beta_{8} + 106963044 \beta_{7} + 132621294 \beta_{6} - 107787978 \beta_{5} - 1308606562 \beta_{4} - 217160544 \beta_{3} + 1939154697 \beta_{2} + 11416575668 \beta_{1} + 724198092865\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
35.6552
33.5115
19.7716
15.4003
−4.12402
−12.4485
−24.3559
−30.2923
−33.1179
−35.6552 190.523 759.292 −886.258 −6793.13 −3878.27 −8817.24 16616.0 31599.7
1.2 −33.5115 −215.845 611.021 −1510.04 7233.29 2235.57 −3318.35 26906.1 50603.9
1.3 −19.7716 15.1371 −121.083 339.780 −299.284 3392.81 12517.1 −19453.9 −6718.00
1.4 −15.4003 −207.293 −274.831 1546.00 3192.38 3958.22 12117.4 23287.4 −23808.9
1.5 4.12402 158.305 −494.992 522.723 652.853 −9369.41 −4152.85 5377.54 2155.72
1.6 12.4485 −4.54310 −357.034 208.913 −56.5549 10778.0 −10818.2 −19662.4 2600.65
1.7 24.3559 100.461 81.2084 −2213.18 2446.80 −1575.25 −10492.3 −9590.68 −53903.9
1.8 30.2923 −209.271 405.624 2211.65 −6339.31 −7440.75 −3222.36 24111.4 66995.9
1.9 33.1179 −71.4733 584.794 −957.580 −2367.05 −5228.90 2410.80 −14574.6 −31713.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.10.a.a 9
3.b odd 2 1 261.10.a.b 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.a.a 9 1.a even 1 1 trivial
261.10.a.b 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{9} - \cdots\) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -456378417152 + 115773751296 T + 2974719488 T^{2} - 1038861888 T^{3} - 7409504 T^{4} + 2830996 T^{5} + 4592 T^{6} - 2901 T^{7} + T^{9} \)
$3$ \( 139448153485716570 + 22417373996705511 T - 1910524100708538 T^{2} - 17850271400190 T^{3} + 373609654362 T^{4} + 1888155720 T^{5} - 17687790 T^{6} - 75314 T^{7} + 244 T^{8} + T^{9} \)
$5$ \( -\)\(35\!\cdots\!50\)\( + \)\(26\!\cdots\!25\)\( T - \)\(30\!\cdots\!00\)\( T^{2} - 10536908148328369600 T^{3} + 10726588387887300 T^{4} + 17365035539354 T^{5} - 5671880168 T^{6} - 8025000 T^{7} + 738 T^{8} + T^{9} \)
$7$ \( \)\(72\!\cdots\!48\)\( + \)\(17\!\cdots\!00\)\( T - \)\(30\!\cdots\!68\)\( T^{2} - \)\(38\!\cdots\!44\)\( T^{3} + 32088973731469462784 T^{4} + 3743981545421104 T^{5} - 1113278556304 T^{6} - 142773652 T^{7} + 7128 T^{8} + T^{9} \)
$11$ \( \)\(40\!\cdots\!78\)\( + \)\(47\!\cdots\!51\)\( T - \)\(27\!\cdots\!90\)\( T^{2} - \)\(15\!\cdots\!18\)\( T^{3} + \)\(53\!\cdots\!38\)\( T^{4} + 14519616472836973928 T^{5} - 319537372361426 T^{6} - 6201052530 T^{7} + 59512 T^{8} + T^{9} \)
$13$ \( -\)\(14\!\cdots\!94\)\( + \)\(26\!\cdots\!41\)\( T + \)\(10\!\cdots\!08\)\( T^{2} + \)\(42\!\cdots\!32\)\( T^{3} + \)\(55\!\cdots\!68\)\( T^{4} - 95991206323824331574 T^{5} - 6922586494877528 T^{6} - 30198684088 T^{7} + 165758 T^{8} + T^{9} \)
$17$ \( -\)\(19\!\cdots\!24\)\( + \)\(10\!\cdots\!40\)\( T - \)\(12\!\cdots\!76\)\( T^{2} - \)\(32\!\cdots\!28\)\( T^{3} + \)\(92\!\cdots\!24\)\( T^{4} + \)\(24\!\cdots\!44\)\( T^{5} - 135065461973618136 T^{6} - 349913413980 T^{7} + 394814 T^{8} + T^{9} \)
$19$ \( \)\(11\!\cdots\!60\)\( + \)\(28\!\cdots\!28\)\( T + \)\(22\!\cdots\!16\)\( T^{2} + \)\(64\!\cdots\!84\)\( T^{3} + \)\(28\!\cdots\!24\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{5} - 2443896578326262208 T^{6} + 385040818832 T^{7} + 2256606 T^{8} + T^{9} \)
$23$ \( \)\(27\!\cdots\!32\)\( - \)\(19\!\cdots\!44\)\( T - \)\(12\!\cdots\!84\)\( T^{2} - \)\(32\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!16\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} - 15301772097554110336 T^{6} - 8422474262444 T^{7} + 1699500 T^{8} + T^{9} \)
$29$ \( ( 707281 + T )^{9} \)
$31$ \( -\)\(54\!\cdots\!58\)\( - \)\(60\!\cdots\!09\)\( T - \)\(65\!\cdots\!30\)\( T^{2} + \)\(12\!\cdots\!98\)\( T^{3} + \)\(39\!\cdots\!06\)\( T^{4} - \)\(11\!\cdots\!52\)\( T^{5} - \)\(14\!\cdots\!54\)\( T^{6} - 86690058353006 T^{7} + 11929632 T^{8} + T^{9} \)
$37$ \( \)\(15\!\cdots\!20\)\( - \)\(21\!\cdots\!12\)\( T + \)\(65\!\cdots\!72\)\( T^{2} + \)\(53\!\cdots\!08\)\( T^{3} - \)\(16\!\cdots\!56\)\( T^{4} + \)\(84\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!24\)\( T^{6} - 628702071725920 T^{7} - 14454898 T^{8} + T^{9} \)
$41$ \( \)\(15\!\cdots\!00\)\( - \)\(54\!\cdots\!00\)\( T - \)\(85\!\cdots\!40\)\( T^{2} + \)\(27\!\cdots\!28\)\( T^{3} - \)\(14\!\cdots\!52\)\( T^{4} - \)\(58\!\cdots\!48\)\( T^{5} + \)\(57\!\cdots\!36\)\( T^{6} - 365081076405100 T^{7} - 52495202 T^{8} + T^{9} \)
$43$ \( \)\(15\!\cdots\!26\)\( - \)\(95\!\cdots\!65\)\( T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(17\!\cdots\!50\)\( T^{3} - \)\(55\!\cdots\!50\)\( T^{4} + \)\(18\!\cdots\!60\)\( T^{5} + \)\(66\!\cdots\!98\)\( T^{6} - 2650033496940858 T^{7} - 21819888 T^{8} + T^{9} \)
$47$ \( -\)\(26\!\cdots\!66\)\( + \)\(14\!\cdots\!91\)\( T - \)\(29\!\cdots\!26\)\( T^{2} - \)\(48\!\cdots\!74\)\( T^{3} - \)\(21\!\cdots\!98\)\( T^{4} + \)\(19\!\cdots\!80\)\( T^{5} + \)\(80\!\cdots\!62\)\( T^{6} - 2476928632257790 T^{7} - 44968948 T^{8} + T^{9} \)
$53$ \( -\)\(22\!\cdots\!22\)\( - \)\(57\!\cdots\!51\)\( T - \)\(25\!\cdots\!16\)\( T^{2} + \)\(47\!\cdots\!44\)\( T^{3} + \)\(24\!\cdots\!48\)\( T^{4} + \)\(13\!\cdots\!46\)\( T^{5} - \)\(10\!\cdots\!68\)\( T^{6} - 9262930538115296 T^{7} + 111394302 T^{8} + T^{9} \)
$59$ \( \)\(74\!\cdots\!00\)\( - \)\(57\!\cdots\!00\)\( T - \)\(11\!\cdots\!00\)\( T^{2} + \)\(51\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!16\)\( T^{4} - \)\(79\!\cdots\!56\)\( T^{5} - \)\(98\!\cdots\!12\)\( T^{6} - 29196037479600076 T^{7} + 236142720 T^{8} + T^{9} \)
$61$ \( -\)\(35\!\cdots\!36\)\( - \)\(15\!\cdots\!08\)\( T + \)\(15\!\cdots\!24\)\( T^{2} + \)\(15\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!40\)\( T^{4} - \)\(33\!\cdots\!12\)\( T^{5} - \)\(58\!\cdots\!12\)\( T^{6} - 10323430343939724 T^{7} + 241129054 T^{8} + T^{9} \)
$67$ \( -\)\(13\!\cdots\!92\)\( - \)\(21\!\cdots\!80\)\( T + \)\(88\!\cdots\!84\)\( T^{2} + \)\(15\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} - \)\(83\!\cdots\!16\)\( T^{5} - \)\(34\!\cdots\!08\)\( T^{6} + 78996421322342032 T^{7} + 672046492 T^{8} + T^{9} \)
$71$ \( \)\(12\!\cdots\!84\)\( + \)\(98\!\cdots\!92\)\( T - \)\(18\!\cdots\!12\)\( T^{2} - \)\(77\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!80\)\( T^{4} + \)\(35\!\cdots\!24\)\( T^{5} - \)\(59\!\cdots\!68\)\( T^{6} - 104058715533546120 T^{7} + 475841956 T^{8} + T^{9} \)
$73$ \( -\)\(71\!\cdots\!00\)\( + \)\(51\!\cdots\!40\)\( T - \)\(68\!\cdots\!52\)\( T^{2} - \)\(36\!\cdots\!72\)\( T^{3} + \)\(41\!\cdots\!76\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{5} - \)\(75\!\cdots\!60\)\( T^{6} - 177137511254621536 T^{7} + 424813822 T^{8} + T^{9} \)
$79$ \( -\)\(17\!\cdots\!50\)\( + \)\(16\!\cdots\!15\)\( T + \)\(96\!\cdots\!66\)\( T^{2} - \)\(97\!\cdots\!22\)\( T^{3} - \)\(12\!\cdots\!30\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} - \)\(22\!\cdots\!98\)\( T^{6} - 721169615347863814 T^{7} + 170801148 T^{8} + T^{9} \)
$83$ \( -\)\(96\!\cdots\!16\)\( - \)\(38\!\cdots\!64\)\( T - \)\(20\!\cdots\!28\)\( T^{2} + \)\(38\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!16\)\( T^{4} - \)\(34\!\cdots\!36\)\( T^{5} - \)\(82\!\cdots\!28\)\( T^{6} - 114693159038698124 T^{7} + 468898296 T^{8} + T^{9} \)
$89$ \( -\)\(53\!\cdots\!80\)\( + \)\(22\!\cdots\!28\)\( T - \)\(44\!\cdots\!12\)\( T^{2} - \)\(31\!\cdots\!52\)\( T^{3} + \)\(41\!\cdots\!00\)\( T^{4} + \)\(12\!\cdots\!76\)\( T^{5} - \)\(10\!\cdots\!36\)\( T^{6} - 1911346892599658156 T^{7} + 676036598 T^{8} + T^{9} \)
$97$ \( -\)\(49\!\cdots\!32\)\( + \)\(76\!\cdots\!20\)\( T + \)\(20\!\cdots\!80\)\( T^{2} - \)\(10\!\cdots\!36\)\( T^{3} - \)\(81\!\cdots\!52\)\( T^{4} + \)\(34\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!92\)\( T^{6} - 3616874167824063660 T^{7} - 170708754 T^{8} + T^{9} \)
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