Properties

Label 29.6.a.b
Level $29$
Weight $6$
Character orbit 29.a
Self dual yes
Analytic conductor $4.651$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65113077458\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 184 x^{5} + 584 x^{4} + 10145 x^{3} - 34491 x^{2} - 149754 x + 524902\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 4 + \beta_{5} ) q^{3} + ( 23 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( 3 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{6} + ( 23 + 6 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - 4 \beta_{6} ) q^{7} + ( 140 - 13 \beta_{1} - \beta_{2} + \beta_{3} + 8 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{8} + ( 139 + 15 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 7 \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 4 + \beta_{5} ) q^{3} + ( 23 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( 3 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{6} + ( 23 + 6 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - 4 \beta_{6} ) q^{7} + ( 140 - 13 \beta_{1} - \beta_{2} + \beta_{3} + 8 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{8} + ( 139 + 15 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 7 \beta_{6} ) q^{9} + ( 123 + 11 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{10} + ( 153 + 21 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} ) q^{11} + ( 24 + 15 \beta_{1} + 11 \beta_{2} - 15 \beta_{3} - 8 \beta_{4} + 22 \beta_{5} - 14 \beta_{6} ) q^{12} + ( 35 + 55 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} - 15 \beta_{4} + 20 \beta_{5} - 7 \beta_{6} ) q^{13} + ( -276 - 20 \beta_{1} - 26 \beta_{2} + 20 \beta_{3} - 28 \beta_{5} + 20 \beta_{6} ) q^{14} + ( -98 + 41 \beta_{1} - 25 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} + 39 \beta_{6} ) q^{15} + ( 79 - 101 \beta_{1} - \beta_{2} + 6 \beta_{3} + 32 \beta_{4} - 16 \beta_{5} - 8 \beta_{6} ) q^{16} + ( -106 - 45 \beta_{1} + 15 \beta_{2} - 17 \beta_{3} + 2 \beta_{4} + 21 \beta_{5} - 43 \beta_{6} ) q^{17} + ( -766 - 84 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 18 \beta_{4} + 10 \beta_{5} - 32 \beta_{6} ) q^{18} + ( 631 - 77 \beta_{1} + 18 \beta_{2} + 8 \beta_{3} + 36 \beta_{4} - 55 \beta_{5} + 35 \beta_{6} ) q^{19} + ( -843 - 165 \beta_{1} - 25 \beta_{2} + 52 \beta_{3} + 4 \beta_{4} - 76 \beta_{5} + 24 \beta_{6} ) q^{20} + ( -680 + 129 \beta_{1} + 65 \beta_{2} - 7 \beta_{3} - 114 \beta_{4} + 39 \beta_{5} - 33 \beta_{6} ) q^{21} + ( -851 - 74 \beta_{1} + 16 \beta_{2} - 73 \beta_{3} + 2 \beta_{4} + 19 \beta_{5} + 21 \beta_{6} ) q^{22} + ( -673 + 26 \beta_{1} + 13 \beta_{2} + 5 \beta_{3} - 52 \beta_{4} - 14 \beta_{5} - 24 \beta_{6} ) q^{23} + ( -581 + 31 \beta_{1} - 29 \beta_{2} - 34 \beta_{3} + 104 \beta_{4} + 108 \beta_{5} + 44 \beta_{6} ) q^{24} + ( 742 + 114 \beta_{1} + 21 \beta_{2} + 10 \beta_{3} + 31 \beta_{4} + 5 \beta_{5} - 30 \beta_{6} ) q^{25} + ( -2835 + 75 \beta_{1} - 96 \beta_{2} - 64 \beta_{3} - 22 \beta_{4} + 40 \beta_{5} + 42 \beta_{6} ) q^{26} + ( 873 + 101 \beta_{1} - 150 \beta_{2} + 100 \beta_{3} + 204 \beta_{4} - 66 \beta_{5} + 45 \beta_{6} ) q^{27} + ( -206 + 324 \beta_{1} + 32 \beta_{2} + 146 \beta_{3} - 64 \beta_{4} + 4 \beta_{5} - 12 \beta_{6} ) q^{28} + 841 q^{29} + ( -2617 + 550 \beta_{1} + 94 \beta_{2} - 27 \beta_{3} - 234 \beta_{4} + 257 \beta_{5} - 233 \beta_{6} ) q^{30} + ( 979 + 170 \beta_{1} - 23 \beta_{2} - 87 \beta_{3} - 76 \beta_{4} - 157 \beta_{5} - 32 \beta_{6} ) q^{31} + ( 1086 - 47 \beta_{1} + 169 \beta_{2} + \beta_{3} - 88 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{32} + ( 4991 - 42 \beta_{1} + 17 \beta_{2} + 76 \beta_{3} + 209 \beta_{4} + 75 \beta_{5} + 154 \beta_{6} ) q^{33} + ( 3016 - 148 \beta_{1} + 14 \beta_{2} - 128 \beta_{3} + 172 \beta_{4} - 94 \beta_{5} + 234 \beta_{6} ) q^{34} + ( 1319 - 844 \beta_{1} - 105 \beta_{2} - 105 \beta_{3} + 48 \beta_{4} - 270 \beta_{5} - 18 \beta_{6} ) q^{35} + ( -210 + 112 \beta_{1} + 112 \beta_{2} - 62 \beta_{3} - 140 \beta_{4} - 68 \beta_{5} - 72 \beta_{6} ) q^{36} + ( 676 + 26 \beta_{1} - 96 \beta_{2} - 110 \beta_{3} - 246 \beta_{4} - 56 \beta_{5} + 94 \beta_{6} ) q^{37} + ( 3976 - 1332 \beta_{1} + 80 \beta_{2} + 132 \beta_{3} + 328 \beta_{4} - 98 \beta_{5} - 142 \beta_{6} ) q^{38} + ( 4811 - 136 \beta_{1} + 175 \beta_{2} + 115 \beta_{3} - 124 \beta_{4} + 117 \beta_{5} - 150 \beta_{6} ) q^{39} + ( 3260 + 157 \beta_{1} - 167 \beta_{2} + 343 \beta_{3} + 240 \beta_{4} - 266 \beta_{5} - 74 \beta_{6} ) q^{40} + ( 3018 - 543 \beta_{1} + 169 \beta_{2} - 177 \beta_{3} - 256 \beta_{4} + 221 \beta_{5} + 139 \beta_{6} ) q^{41} + ( -7942 + 434 \beta_{1} - 622 \beta_{2} + 160 \beta_{3} + 52 \beta_{4} - 154 \beta_{5} + 206 \beta_{6} ) q^{42} + ( 2691 + 331 \beta_{1} - 58 \beta_{2} + 8 \beta_{3} - 184 \beta_{4} + 196 \beta_{5} - 405 \beta_{6} ) q^{43} + ( -958 + 191 \beta_{1} + 391 \beta_{2} - 449 \beta_{3} - 200 \beta_{4} + 686 \beta_{5} - 102 \beta_{6} ) q^{44} + ( 1479 - 119 \beta_{1} - 386 \beta_{2} + 218 \beta_{3} + 482 \beta_{4} + 335 \beta_{5} + 221 \beta_{6} ) q^{45} + ( -1986 + 648 \beta_{1} - 226 \beta_{2} + 218 \beta_{3} - 28 \beta_{4} - 98 \beta_{5} + 90 \beta_{6} ) q^{46} + ( 2512 - 1247 \beta_{1} + 105 \beta_{2} + 159 \beta_{3} - 32 \beta_{4} - 176 \beta_{5} + 11 \beta_{6} ) q^{47} + ( -2758 + 641 \beta_{1} + 121 \beta_{2} - 363 \beta_{3} + 136 \beta_{4} - 254 \beta_{5} + 314 \beta_{6} ) q^{48} + ( 5477 + 188 \beta_{1} + 280 \beta_{2} - 176 \beta_{3} + 80 \beta_{4} + 564 \beta_{5} - 340 \beta_{6} ) q^{49} + ( -5452 - 1044 \beta_{1} - 206 \beta_{2} + 44 \beta_{3} + 342 \beta_{4} - 514 \beta_{5} + 248 \beta_{6} ) q^{50} + ( 2997 - 240 \beta_{1} + 849 \beta_{2} - 415 \beta_{3} - 472 \beta_{4} - 524 \beta_{5} + 134 \beta_{6} ) q^{51} + ( -6233 + 3291 \beta_{1} + 307 \beta_{2} - 438 \beta_{3} - 540 \beta_{4} + 1068 \beta_{5} - 288 \beta_{6} ) q^{52} + ( -7861 - 801 \beta_{1} - 16 \beta_{2} + 447 \beta_{3} - 125 \beta_{4} + 430 \beta_{5} + 77 \beta_{6} ) q^{53} + ( -4995 + 326 \beta_{1} + 552 \beta_{2} + 295 \beta_{3} - 342 \beta_{4} - 821 \beta_{5} - 187 \beta_{6} ) q^{54} + ( -4245 + 1906 \beta_{1} - 871 \beta_{2} + 73 \beta_{3} + 452 \beta_{4} - 345 \beta_{5} + 248 \beta_{6} ) q^{55} + ( -11750 + 504 \beta_{1} - 344 \beta_{2} + 542 \beta_{3} + 16 \beta_{4} - 1212 \beta_{5} - 284 \beta_{6} ) q^{56} + ( -12325 + 47 \beta_{1} - 210 \beta_{2} - 266 \beta_{3} + 570 \beta_{4} + 841 \beta_{5} - 485 \beta_{6} ) q^{57} + ( 841 - 841 \beta_{1} ) q^{58} + ( 1513 + 976 \beta_{1} - 143 \beta_{2} + 545 \beta_{3} + 408 \beta_{4} - 304 \beta_{5} + 614 \beta_{6} ) q^{59} + ( -26912 + 2253 \beta_{1} - 719 \beta_{2} - 325 \beta_{3} - 488 \beta_{4} - 714 \beta_{5} + 74 \beta_{6} ) q^{60} + ( 15032 - 1235 \beta_{1} + 279 \beta_{2} + 135 \beta_{3} + 798 \beta_{4} - 861 \beta_{5} + 159 \beta_{6} ) q^{61} + ( -5731 + 122 \beta_{1} + 86 \beta_{2} + 23 \beta_{3} - 110 \beta_{4} + 1581 \beta_{5} - 293 \beta_{6} ) q^{62} + ( -12686 - 934 \beta_{1} + 892 \beta_{2} - 340 \beta_{3} - 1044 \beta_{4} + 98 \beta_{5} - 750 \beta_{6} ) q^{63} + ( -1141 - 315 \beta_{1} - 775 \beta_{2} + 176 \beta_{3} - 16 \beta_{4} - 236 \beta_{5} + 492 \beta_{6} ) q^{64} + ( -20920 - 529 \beta_{1} - 837 \beta_{2} - 494 \beta_{3} + 67 \beta_{4} - 34 \beta_{5} + 399 \beta_{6} ) q^{65} + ( 3439 - 6259 \beta_{1} + 242 \beta_{2} - 248 \beta_{3} + 514 \beta_{4} - 1098 \beta_{5} - 300 \beta_{6} ) q^{66} + ( 14418 + 974 \beta_{1} - 60 \beta_{2} + 364 \beta_{3} - 244 \beta_{4} + 306 \beta_{5} + 1190 \beta_{6} ) q^{67} + ( 13340 - 2098 \beta_{1} + 702 \beta_{2} - 634 \beta_{3} + 456 \beta_{4} + 1784 \beta_{5} + 56 \beta_{6} ) q^{68} + ( -15864 + 211 \beta_{1} + 515 \beta_{2} - 143 \beta_{3} - 1260 \beta_{4} - 369 \beta_{5} - 631 \beta_{6} ) q^{69} + ( 50672 - 1552 \beta_{1} + 1762 \beta_{2} - 108 \beta_{3} - 72 \beta_{4} + 2136 \beta_{5} - 552 \beta_{6} ) q^{70} + ( -6916 - 428 \beta_{1} - 1208 \beta_{2} - 164 \beta_{3} + 900 \beta_{4} - 626 \beta_{5} - 500 \beta_{6} ) q^{71} + ( 18990 + 1992 \beta_{1} - 856 \beta_{2} + 426 \beta_{3} + 1192 \beta_{4} + 4 \beta_{5} + 1276 \beta_{6} ) q^{72} + ( -1672 - 2024 \beta_{1} + 78 \beta_{2} + 656 \beta_{3} - 1170 \beta_{4} + 230 \beta_{5} + 128 \beta_{6} ) q^{73} + ( 1240 + 2428 \beta_{1} + 400 \beta_{2} - 300 \beta_{3} - 1828 \beta_{4} + 3380 \beta_{5} - 1184 \beta_{6} ) q^{74} + ( 4632 + 2538 \beta_{1} + 606 \beta_{2} + 86 \beta_{3} + 308 \beta_{4} + 750 \beta_{5} - 18 \beta_{6} ) q^{75} + ( 54382 - 7766 \beta_{1} + 422 \beta_{2} + 528 \beta_{3} + 1280 \beta_{4} - 2164 \beta_{5} + 244 \beta_{6} ) q^{76} + ( -16494 + 6147 \beta_{1} + 453 \beta_{2} - 43 \beta_{3} - 1406 \beta_{4} - 279 \beta_{5} - 1607 \beta_{6} ) q^{77} + ( 9793 - 7878 \beta_{1} - 1422 \beta_{2} + 1027 \beta_{3} + 970 \beta_{4} - 3155 \beta_{5} + 1323 \beta_{6} ) q^{78} + ( 29930 - 375 \beta_{1} + 707 \beta_{2} - 807 \beta_{3} + 368 \beta_{4} - 910 \beta_{5} - 545 \beta_{6} ) q^{79} + ( 19057 + 1725 \beta_{1} + 345 \beta_{2} + 962 \beta_{3} + 176 \beta_{4} - 2192 \beta_{5} - 72 \beta_{6} ) q^{80} + ( -18650 + 3353 \beta_{1} - 1998 \beta_{2} + 910 \beta_{3} + 2610 \beta_{4} + 139 \beta_{5} + 1893 \beta_{6} ) q^{81} + ( 31162 - 4318 \beta_{1} + 202 \beta_{2} - 1360 \beta_{3} - 392 \beta_{4} + 2706 \beta_{5} - 746 \beta_{6} ) q^{82} + ( -18805 + 1222 \beta_{1} - 7 \beta_{2} - 1663 \beta_{3} + 228 \beta_{4} - 1810 \beta_{5} + 620 \beta_{6} ) q^{83} + ( -7142 + 13524 \beta_{1} - 356 \beta_{2} + 874 \beta_{3} - 1224 \beta_{4} + 1272 \beta_{5} - 1000 \beta_{6} ) q^{84} + ( -18370 + 4087 \beta_{1} - 567 \beta_{2} - 401 \beta_{3} - 908 \beta_{4} - 3409 \beta_{5} - 255 \beta_{6} ) q^{85} + ( -9099 - 418 \beta_{1} - 904 \beta_{2} + 183 \beta_{3} - 782 \beta_{4} - 1433 \beta_{5} + 1937 \beta_{6} ) q^{86} + ( 3364 + 841 \beta_{5} ) q^{87} + ( 19701 + 1767 \beta_{1} - 973 \beta_{2} - 1972 \beta_{3} + 1096 \beta_{4} + 2736 \beta_{5} + 208 \beta_{6} ) q^{88} + ( -13052 + 9087 \beta_{1} + 169 \beta_{2} + 687 \beta_{3} + 1892 \beta_{4} + 83 \beta_{5} + 697 \beta_{6} ) q^{89} + ( 5570 + 2332 \beta_{1} + 1976 \beta_{2} - 1050 \beta_{3} - 2272 \beta_{4} - 1412 \beta_{5} - 624 \beta_{6} ) q^{90} + ( 24833 + 7078 \beta_{1} + 1867 \beta_{2} - 701 \beta_{3} - 1852 \beta_{4} + 3492 \beta_{5} - 1576 \beta_{6} ) q^{91} + ( -17590 + 4982 \beta_{1} - 998 \beta_{2} + 1400 \beta_{3} - 240 \beta_{4} - 616 \beta_{5} + 176 \beta_{6} ) q^{92} + ( -63250 - 10706 \beta_{1} + 1220 \beta_{2} + 87 \beta_{3} - 1759 \beta_{4} + 1759 \beta_{5} - 1500 \beta_{6} ) q^{93} + ( 65645 - 7694 \beta_{1} + 138 \beta_{2} + 1799 \beta_{3} + 1042 \beta_{4} - 2529 \beta_{5} + 265 \beta_{6} ) q^{94} + ( -16309 - 4987 \beta_{1} + 140 \beta_{2} + 2342 \beta_{3} + 1148 \beta_{4} - 329 \beta_{5} - 279 \beta_{6} ) q^{95} + ( -17961 + 2189 \beta_{1} + 1841 \beta_{2} - 1156 \beta_{3} - 1936 \beta_{4} + 2316 \beta_{5} - 3580 \beta_{6} ) q^{96} + ( -13744 - 7193 \beta_{1} - 43 \beta_{2} + 535 \beta_{3} - 1176 \beta_{4} + 475 \beta_{5} + 865 \beta_{6} ) q^{97} + ( -951 - 8233 \beta_{1} - 784 \beta_{2} - 2464 \beta_{3} + 2112 \beta_{4} - 1720 \beta_{5} + 2552 \beta_{6} ) q^{98} + ( 43573 + 3445 \beta_{1} - 2318 \beta_{2} + 1740 \beta_{3} + 3564 \beta_{4} + 2369 \beta_{5} + 2061 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9} + O(q^{10}) \) \( 7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9} + 922 q^{10} + 1106 q^{11} + 214 q^{12} + 408 q^{13} - 2008 q^{14} - 614 q^{15} + 242 q^{16} - 874 q^{17} - 5598 q^{18} + 4288 q^{19} - 6350 q^{20} - 4200 q^{21} - 6114 q^{22} - 4532 q^{23} - 4318 q^{24} + 5527 q^{25} - 19806 q^{26} + 5942 q^{27} - 496 q^{28} + 5887 q^{29} - 16734 q^{30} + 7794 q^{31} + 7898 q^{32} + 34410 q^{33} + 20840 q^{34} + 7088 q^{35} - 572 q^{36} + 5086 q^{37} + 23732 q^{38} + 33394 q^{39} + 22906 q^{40} + 19826 q^{41} - 55440 q^{42} + 19498 q^{43} - 6074 q^{44} + 7854 q^{45} - 12404 q^{46} + 14278 q^{47} - 16406 q^{48} + 38431 q^{49} - 41066 q^{50} + 23892 q^{51} - 34302 q^{52} - 58644 q^{53} - 31194 q^{54} - 25574 q^{55} - 79560 q^{56} - 88540 q^{57} + 3364 q^{58} + 12888 q^{59} - 180822 q^{60} + 102866 q^{61} - 42654 q^{62} - 88632 q^{63} - 10170 q^{64} - 149206 q^{65} + 7710 q^{66} + 102996 q^{67} + 85100 q^{68} - 107244 q^{69} + 349480 q^{70} - 51596 q^{71} + 135568 q^{72} - 17566 q^{73} + 12132 q^{74} + 39356 q^{75} + 360740 q^{76} - 94104 q^{77} + 46386 q^{78} + 212058 q^{79} + 142510 q^{80} - 128285 q^{81} + 201924 q^{82} - 122928 q^{83} - 12328 q^{84} - 109336 q^{85} - 63290 q^{86} + 21866 q^{87} + 136666 q^{88} - 66510 q^{89} + 56084 q^{90} + 194368 q^{91} - 110108 q^{92} - 474274 q^{93} + 438926 q^{94} - 131676 q^{95} - 117018 q^{96} - 118182 q^{97} - 29132 q^{98} + 300668 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 184 x^{5} + 584 x^{4} + 10145 x^{3} - 34491 x^{2} - 149754 x + 524902\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 54 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 6 \nu^{5} - 114 \nu^{4} + 618 \nu^{3} + 2727 \nu^{2} - 10724 \nu - 11462 \)\()/240\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 5 \nu^{4} - 119 \nu^{3} + 547 \nu^{2} + 2890 \nu - 11242 \)\()/96\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 10 \nu^{5} - 218 \nu^{4} - 1094 \nu^{3} + 13231 \nu^{2} + 21356 \nu - 185766 \)\()/960\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{5} - 226 \nu^{4} - 238 \nu^{3} + 14279 \nu^{2} + 5916 \nu - 208054 \)\()/960\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 54\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} + 6 \beta_{5} - 8 \beta_{4} - \beta_{3} + 4 \beta_{2} + 71 \beta_{1} - 41\)
\(\nu^{4}\)\(=\)\(-16 \beta_{6} + 8 \beta_{5} + 2 \beta_{3} + 105 \beta_{2} - 95 \beta_{1} + 3846\)
\(\nu^{5}\)\(=\)\(-318 \beta_{6} + 754 \beta_{5} - 856 \beta_{4} - 109 \beta_{3} + 454 \beta_{2} + 5631 \beta_{1} - 3945\)
\(\nu^{6}\)\(=\)\(-2496 \beta_{6} + 1728 \beta_{5} - 192 \beta_{4} + 432 \beta_{3} + 9495 \beta_{2} - 7471 \beta_{1} + 304316\)

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
9.56883
7.83842
4.90786
3.60554
−4.83960
−8.92709
−9.15396
−8.56883 18.3274 41.4249 −84.3249 −157.045 216.816 −80.7606 92.8952 722.566
1.2 −6.83842 −24.5656 14.7640 −54.3066 167.990 −37.6697 117.867 360.469 371.372
1.3 −3.90786 29.3989 −16.7287 64.0801 −114.887 −138.793 190.425 621.298 −250.416
1.4 −2.60554 −13.2844 −25.2112 69.0035 34.6131 156.573 149.066 −66.5241 −179.792
1.5 5.83960 15.9679 2.10095 31.5616 93.2461 106.304 −174.599 11.9736 184.307
1.6 9.92709 15.4219 66.5471 −58.0818 153.094 −210.388 342.952 −5.16616 −576.583
1.7 10.1540 −15.2661 71.1029 64.0682 −155.012 91.1564 397.049 −9.94539 650.545
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 29.6.a.b 7
3.b odd 2 1 261.6.a.e 7
4.b odd 2 1 464.6.a.k 7
5.b even 2 1 725.6.a.b 7
29.b even 2 1 841.6.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.b 7 1.a even 1 1 trivial
261.6.a.e 7 3.b odd 2 1
464.6.a.k 7 4.b odd 2 1
725.6.a.b 7 5.b even 2 1
841.6.a.b 7 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4 T_{2}^{6} - 181 T_{2}^{5} + 346 T_{2}^{4} + 10616 T_{2}^{3} + 2416 T_{2}^{2} - 186896 T_{2} - 351200 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -351200 - 186896 T + 2416 T^{2} + 10616 T^{3} + 346 T^{4} - 181 T^{5} - 4 T^{6} + T^{7} \)
$3$ \( 661023756 - 22844001 T - 7496290 T^{2} + 280279 T^{3} + 26056 T^{4} - 1015 T^{5} - 26 T^{6} + T^{7} \)
$5$ \( 2378174390186 - 71107482707 T - 1882536724 T^{2} + 54141495 T^{3} + 456170 T^{4} - 13189 T^{5} - 32 T^{6} + T^{7} \)
$7$ \( 361848785235968 + 2579467705856 T - 180798693888 T^{2} + 679110080 T^{3} + 12004416 T^{4} - 61112 T^{5} - 184 T^{6} + T^{7} \)
$11$ \( 515865611322413044 + 3479126648563423 T - 17338617711626 T^{2} - 73256939689 T^{3} + 323786496 T^{4} + 15177 T^{5} - 1106 T^{6} + T^{7} \)
$13$ \( 10011422243374034 + 35719316041077 T - 6217250606588 T^{2} - 8368783321 T^{3} + 449335138 T^{4} - 1099773 T^{5} - 408 T^{6} + T^{7} \)
$17$ \( -\)\(17\!\cdots\!28\)\( - 4762080661877200832 T + 4400022913305568 T^{2} + 11750005682096 T^{3} - 3880384296 T^{4} - 6477972 T^{5} + 874 T^{6} + T^{7} \)
$19$ \( \)\(15\!\cdots\!92\)\( - 384177380729022144 T - 22726079248952448 T^{2} - 2802099264912 T^{3} + 19511173648 T^{4} - 1439924 T^{5} - 4288 T^{6} + T^{7} \)
$23$ \( -1541996530463904256 - 37102640165875136 T + 139547314402368 T^{2} + 1885764458736 T^{3} - 7019218112 T^{4} + 2107804 T^{5} + 4532 T^{6} + T^{7} \)
$29$ \( ( -841 + T )^{7} \)
$31$ \( \)\(64\!\cdots\!48\)\( - \)\(87\!\cdots\!01\)\( T - 2325896554397613794 T^{2} + 2013745795872983 T^{3} + 398022767396 T^{4} - 75300839 T^{5} - 7794 T^{6} + T^{7} \)
$37$ \( -\)\(16\!\cdots\!56\)\( - \)\(12\!\cdots\!48\)\( T + 30949163252985802240 T^{2} + 20164120745134336 T^{3} + 445110854496 T^{4} - 273984608 T^{5} - 5086 T^{6} + T^{7} \)
$41$ \( -\)\(56\!\cdots\!72\)\( - \)\(80\!\cdots\!60\)\( T - \)\(18\!\cdots\!92\)\( T^{2} + 15992490422438592 T^{3} + 4322757774032 T^{4} - 217946616 T^{5} - 19826 T^{6} + T^{7} \)
$43$ \( -\)\(58\!\cdots\!60\)\( - \)\(11\!\cdots\!41\)\( T - \)\(26\!\cdots\!82\)\( T^{2} + 25211595416791175 T^{3} + 5437038516432 T^{4} - 316513743 T^{5} - 19498 T^{6} + T^{7} \)
$47$ \( -\)\(14\!\cdots\!36\)\( - \)\(31\!\cdots\!69\)\( T + 37913247078045875370 T^{2} + 9699630620043943 T^{3} - 1210354130884 T^{4} - 403592367 T^{5} - 14278 T^{6} + T^{7} \)
$53$ \( \)\(84\!\cdots\!94\)\( - \)\(61\!\cdots\!75\)\( T + \)\(86\!\cdots\!32\)\( T^{2} - 4330722850397705 T^{3} - 39591011680062 T^{4} - 55959141 T^{5} + 58644 T^{6} + T^{7} \)
$59$ \( \)\(15\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( T - \)\(13\!\cdots\!80\)\( T^{2} + 1325580183274664304 T^{3} + 24300942873424 T^{4} - 2216676980 T^{5} - 12888 T^{6} + T^{7} \)
$61$ \( \)\(45\!\cdots\!80\)\( - \)\(25\!\cdots\!08\)\( T + \)\(46\!\cdots\!72\)\( T^{2} - 3470412083411730640 T^{3} + 73658693252072 T^{4} + 2180773148 T^{5} - 102866 T^{6} + T^{7} \)
$67$ \( -\)\(11\!\cdots\!60\)\( + \)\(23\!\cdots\!68\)\( T - \)\(67\!\cdots\!24\)\( T^{2} - 7624284713722041344 T^{3} + 328821833213824 T^{4} - 647318752 T^{5} - 102996 T^{6} + T^{7} \)
$71$ \( -\)\(20\!\cdots\!52\)\( + \)\(47\!\cdots\!44\)\( T + \)\(29\!\cdots\!88\)\( T^{2} - 5025269352663123344 T^{3} - 372698633777536 T^{4} - 5405760892 T^{5} + 51596 T^{6} + T^{7} \)
$73$ \( \)\(43\!\cdots\!96\)\( + \)\(24\!\cdots\!32\)\( T + \)\(27\!\cdots\!20\)\( T^{2} + 8121415099482870016 T^{3} - 149044648658016 T^{4} - 6051106208 T^{5} + 17566 T^{6} + T^{7} \)
$79$ \( -\)\(28\!\cdots\!00\)\( + \)\(15\!\cdots\!99\)\( T + \)\(37\!\cdots\!62\)\( T^{2} - 29939303716339944633 T^{3} + 170160611431108 T^{4} + 12042119713 T^{5} - 212058 T^{6} + T^{7} \)
$83$ \( -\)\(97\!\cdots\!52\)\( + \)\(31\!\cdots\!56\)\( T + \)\(95\!\cdots\!52\)\( T^{2} + 42724404510698882672 T^{3} - 2173461254983760 T^{4} - 15465968916 T^{5} + 122928 T^{6} + T^{7} \)
$89$ \( -\)\(79\!\cdots\!20\)\( - \)\(17\!\cdots\!88\)\( T + \)\(75\!\cdots\!92\)\( T^{2} + \)\(15\!\cdots\!96\)\( T^{3} - 1270895508816816 T^{4} - 23989383192 T^{5} + 66510 T^{6} + T^{7} \)
$97$ \( \)\(62\!\cdots\!52\)\( + \)\(96\!\cdots\!96\)\( T + \)\(35\!\cdots\!72\)\( T^{2} + 3500459353391389888 T^{3} - 1404596434864624 T^{4} - 10035608344 T^{5} + 118182 T^{6} + T^{7} \)
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