Properties

Label 9.6.c.a
Level 9
Weight 6
Character orbit 9.c
Analytic conductor 1.443
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 9.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.44345437832\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{2} \) \( + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{3} \) \( + ( -13 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{4} \) \( + ( 20 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{5} \) \( + ( 24 - 3 \beta_{1} + 2 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{6} \) \( + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 11 \beta_{4} - 10 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{7} \) \( + ( -95 + 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{8} \) \( + ( -72 + 27 \beta_{1} - 3 \beta_{2} + 12 \beta_{4} - 24 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} ) q^{2} \) \( + ( -3 + 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{3} \) \( + ( -13 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{4} \) \( + ( 20 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{5} \) \( + ( 24 - 3 \beta_{1} + 2 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{6} \) \( + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 11 \beta_{4} - 10 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{7} \) \( + ( -95 + 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{8} \) \( + ( -72 + 27 \beta_{1} - 3 \beta_{2} + 12 \beta_{4} - 24 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{9} \) \( + ( 18 - 6 \beta_{2} - 8 \beta_{3} + \beta_{4} + 62 \beta_{5} + 13 \beta_{6} - 7 \beta_{7} ) q^{10} \) \( + ( 109 - 109 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 12 \beta_{6} + 2 \beta_{7} ) q^{11} \) \( + ( 327 + 54 \beta_{1} + 10 \beta_{3} + 45 \beta_{4} + 63 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} ) q^{12} \) \( + ( -62 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 69 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{13} \) \( + ( 344 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} - 2 \beta_{5} - 9 \beta_{6} - 7 \beta_{7} ) q^{14} \) \( + ( 12 - 546 \beta_{1} - 3 \beta_{2} - 20 \beta_{3} - 93 \beta_{4} - 41 \beta_{5} + 13 \beta_{6} - 18 \beta_{7} ) q^{15} \) \( + ( -97 + 97 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 108 \beta_{4} - 117 \beta_{5} - 6 \beta_{6} + 12 \beta_{7} ) q^{16} \) \( + ( -531 - 19 \beta_{2} + 51 \beta_{3} - 35 \beta_{4} + 41 \beta_{5} + 3 \beta_{6} + 16 \beta_{7} ) q^{17} \) \( + ( -1521 + 990 \beta_{1} + 27 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} - 39 \beta_{5} + 33 \beta_{6} - 9 \beta_{7} ) q^{18} \) \( + ( 119 + 39 \beta_{2} + 45 \beta_{3} - 3 \beta_{4} - 123 \beta_{5} - 81 \beta_{6} + 42 \beta_{7} ) q^{19} \) \( + ( 1676 - 1676 \beta_{1} - 38 \beta_{2} + 54 \beta_{3} - 2 \beta_{4} - 24 \beta_{5} - 60 \beta_{6} - 16 \beta_{7} ) q^{20} \) \( + ( 864 + 426 \beta_{1} - 9 \beta_{2} - 36 \beta_{3} + 34 \beta_{5} + 25 \beta_{6} - 63 \beta_{7} ) q^{21} \) \( + ( 261 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} - 57 \beta_{4} + 24 \beta_{5} + 56 \beta_{6} + 32 \beta_{7} ) q^{22} \) \( + ( 2216 \beta_{1} - 35 \beta_{2} - 84 \beta_{3} - 69 \beta_{4} + 49 \beta_{5} + 63 \beta_{6} + 14 \beta_{7} ) q^{23} \) \( + ( 1551 - 2913 \beta_{1} + 21 \beta_{2} + 67 \beta_{3} + 306 \beta_{4} + 163 \beta_{5} - 68 \beta_{6} + 30 \beta_{7} ) q^{24} \) \( + ( -331 + 331 \beta_{1} + 55 \beta_{2} - 70 \beta_{3} + 370 \beta_{4} + 410 \beta_{5} + 95 \beta_{6} + 15 \beta_{7} ) q^{25} \) \( + ( -3164 - 2 \beta_{2} - 168 \beta_{3} + 83 \beta_{4} - 40 \beta_{5} + 87 \beta_{6} - 85 \beta_{7} ) q^{26} \) \( + ( -1647 + 918 \beta_{1} - 90 \beta_{2} + 36 \beta_{3} + 171 \beta_{4} + 342 \beta_{5} - 135 \beta_{6} - 45 \beta_{7} ) q^{27} \) \( + ( -250 - 72 \beta_{2} - 24 \beta_{3} - 24 \beta_{4} - 30 \beta_{5} + 120 \beta_{6} - 48 \beta_{7} ) q^{28} \) \( + ( 2998 - 2998 \beta_{1} + 21 \beta_{2} - 72 \beta_{3} - 56 \beta_{4} - 86 \beta_{5} - 9 \beta_{6} + 51 \beta_{7} ) q^{29} \) \( + ( 1314 + 2628 \beta_{1} + 72 \beta_{2} + 42 \beta_{3} - 387 \beta_{4} - 834 \beta_{5} + 3 \beta_{6} + 135 \beta_{7} ) q^{30} \) \( + ( 478 \beta_{1} + 29 \beta_{2} + 98 \beta_{3} - 711 \beta_{4} - 69 \beta_{5} - 109 \beta_{6} - 40 \beta_{7} ) q^{31} \) \( + ( 2223 \beta_{1} + 145 \beta_{2} + 285 \beta_{3} - 49 \beta_{4} - 140 \beta_{5} - 135 \beta_{6} + 5 \beta_{7} ) q^{32} \) \( + ( -246 - 1893 \beta_{1} - 45 \beta_{2} - 25 \beta_{3} + 141 \beta_{4} - 103 \beta_{5} + 161 \beta_{6} - 6 \beta_{7} ) q^{33} \) \( + ( 2583 - 2583 \beta_{1} - 219 \beta_{2} + 321 \beta_{3} + 90 \beta_{4} - 27 \beta_{5} - 336 \beta_{6} - 102 \beta_{7} ) q^{34} \) \( + ( -1966 + 111 \beta_{2} + 243 \beta_{3} - 66 \beta_{4} - 217 \beta_{5} - 288 \beta_{6} + 177 \beta_{7} ) q^{35} \) \( + ( -3213 + 2961 \beta_{1} + 105 \beta_{2} - 165 \beta_{3} - 1032 \beta_{4} - 117 \beta_{5} + 192 \beta_{6} + 210 \beta_{7} ) q^{36} \) \( + ( -2140 - 6 \beta_{2} - 306 \beta_{3} + 150 \beta_{4} - 654 \beta_{5} + 162 \beta_{6} - 156 \beta_{7} ) q^{37} \) \( + ( -2743 + 2743 \beta_{1} + 189 \beta_{2} - 99 \beta_{3} + 452 \beta_{4} + 731 \beta_{5} + 468 \beta_{6} - 90 \beta_{7} ) q^{38} \) \( + ( 2634 - 2862 \beta_{1} - 189 \beta_{2} + 50 \beta_{3} - 279 \beta_{4} + 549 \beta_{5} - 117 \beta_{6} - 18 \beta_{7} ) q^{39} \) \( + ( -2448 \beta_{1} - 280 \beta_{2} - 352 \beta_{3} + 984 \beta_{4} + 72 \beta_{5} - 136 \beta_{6} - 208 \beta_{7} ) q^{40} \) \( + ( 119 \beta_{1} - 72 \beta_{2} - 126 \beta_{3} + 718 \beta_{4} + 54 \beta_{5} + 36 \beta_{6} - 18 \beta_{7} ) q^{41} \) \( + ( 1578 - 438 \beta_{1} + 15 \beta_{2} - 371 \beta_{3} + 240 \beta_{4} + 436 \beta_{5} - 41 \beta_{6} + 45 \beta_{7} ) q^{42} \) \( + ( -1517 + 1517 \beta_{1} + 181 \beta_{2} - 181 \beta_{3} + 205 \beta_{4} + 386 \beta_{5} + 362 \beta_{6} ) q^{43} \) \( + ( 6353 - 16 \beta_{2} - 30 \beta_{3} + 7 \beta_{4} + 709 \beta_{5} + 39 \beta_{6} - 23 \beta_{7} ) q^{44} \) \( + ( 5760 - 1260 \beta_{1} + 90 \beta_{2} + 291 \beta_{3} + 1359 \beta_{4} - 33 \beta_{5} + 264 \beta_{6} - 135 \beta_{7} ) q^{45} \) \( + ( 6372 + 42 \beta_{2} + 628 \beta_{3} - 293 \beta_{4} + 1796 \beta_{5} - 377 \beta_{6} + 335 \beta_{7} ) q^{46} \) \( + ( -188 + 188 \beta_{1} - 190 \beta_{2} + 81 \beta_{3} - 991 \beta_{4} - 1290 \beta_{5} - 489 \beta_{6} + 109 \beta_{7} ) q^{47} \) \( + ( -7614 + 735 \beta_{1} + 81 \beta_{2} - 135 \beta_{3} + 1809 \beta_{4} + 758 \beta_{5} + 83 \beta_{6} - 189 \beta_{7} ) q^{48} \) \( + ( 2355 \beta_{1} + 416 \beta_{2} + 407 \beta_{3} - 339 \beta_{4} + 9 \beta_{5} + 434 \beta_{6} + 425 \beta_{7} ) q^{49} \) \( + ( -14099 \beta_{1} - 335 \beta_{2} - 525 \beta_{3} - 851 \beta_{4} + 190 \beta_{5} + 45 \beta_{6} - 145 \beta_{7} ) q^{50} \) \( + ( -8289 + 14067 \beta_{1} - 9 \beta_{2} + 792 \beta_{3} - 2232 \beta_{4} - 2052 \beta_{5} - 693 \beta_{6} - 153 \beta_{7} ) q^{51} \) \( + ( -8870 + 8870 \beta_{1} + 336 \beta_{2} - 576 \beta_{3} - 2874 \beta_{4} - 2778 \beta_{5} + 432 \beta_{6} + 240 \beta_{7} ) q^{52} \) \( + ( 2052 - 390 \beta_{2} - 450 \beta_{3} + 30 \beta_{4} - 1038 \beta_{5} + 810 \beta_{6} - 420 \beta_{7} ) q^{53} \) \( + ( 9369 - 17199 \beta_{1} - 270 \beta_{2} + 234 \beta_{3} + 297 \beta_{4} - 1791 \beta_{5} - 1224 \beta_{6} - 324 \beta_{7} ) q^{54} \) \( + ( -4482 + 195 \beta_{2} + 43 \beta_{3} + 76 \beta_{4} - 301 \beta_{5} - 314 \beta_{6} + 119 \beta_{7} ) q^{55} \) \( + ( -15218 + 15218 \beta_{1} - 434 \beta_{2} + 450 \beta_{3} - 120 \beta_{4} - 538 \beta_{5} - 852 \beta_{6} - 16 \beta_{7} ) q^{56} \) \( + ( -6351 - 12279 \beta_{1} + 351 \beta_{2} + 16 \beta_{3} + 216 \beta_{4} + 2360 \beta_{5} + 227 \beta_{6} - 27 \beta_{7} ) q^{57} \) \( + ( 1440 \beta_{1} + 341 \beta_{2} + 359 \beta_{3} + 2586 \beta_{4} - 18 \beta_{5} + 305 \beta_{6} + 323 \beta_{7} ) q^{58} \) \( + ( 1019 \beta_{1} + 377 \beta_{2} + 363 \beta_{3} - 2014 \beta_{4} + 14 \beta_{5} + 405 \beta_{6} + 391 \beta_{7} ) q^{59} \) \( + ( 3180 + 16572 \beta_{1} + 198 \beta_{2} + 58 \beta_{3} - 762 \beta_{4} + 2728 \beta_{5} + 1456 \beta_{6} - 84 \beta_{7} ) q^{60} \) \( + ( 12532 - 12532 \beta_{1} - 545 \beta_{2} + 284 \beta_{3} + 1180 \beta_{4} + 374 \beta_{5} - 1351 \beta_{6} + 261 \beta_{7} ) q^{61} \) \( + ( 30550 + 22 \beta_{2} + 732 \beta_{3} - 355 \beta_{4} + 542 \beta_{5} - 399 \beta_{6} + 377 \beta_{7} ) q^{62} \) \( + ( 20016 - 7110 \beta_{1} - 42 \beta_{2} - 1749 \beta_{3} + 1311 \beta_{4} + 876 \beta_{5} + 1107 \beta_{6} - 21 \beta_{7} ) q^{63} \) \( + ( -2735 + 84 \beta_{2} - 1110 \beta_{3} + 597 \beta_{4} + 465 \beta_{5} + 429 \beta_{6} - 513 \beta_{7} ) q^{64} \) \( + ( -2384 + 2384 \beta_{1} + 63 \beta_{2} + 306 \beta_{3} + 4042 \beta_{4} + 4474 \beta_{5} + 495 \beta_{6} - 369 \beta_{7} ) q^{65} \) \( + ( -17073 + 7992 \beta_{1} - 189 \beta_{2} - 93 \beta_{3} - 549 \beta_{4} - 1935 \beta_{5} + 18 \beta_{6} - 72 \beta_{7} ) q^{66} \) \( + ( -1655 \beta_{1} - 955 \beta_{2} - 931 \beta_{3} - 504 \beta_{4} - 24 \beta_{5} - 1003 \beta_{6} - 979 \beta_{7} ) q^{67} \) \( + ( -32085 \beta_{1} + 29 \beta_{2} + 93 \beta_{3} + 5275 \beta_{4} - 64 \beta_{5} - 99 \beta_{6} - 35 \beta_{7} ) q^{68} \) \( + ( -27042 + 21648 \beta_{1} - 66 \beta_{2} - 2027 \beta_{3} + 1509 \beta_{4} - 1463 \beta_{5} - 212 \beta_{6} + 315 \beta_{7} ) q^{69} \) \( + ( -234 + 234 \beta_{1} + 175 \beta_{2} + 311 \beta_{3} + 379 \beta_{4} + 1040 \beta_{5} + 836 \beta_{6} - 486 \beta_{7} ) q^{70} \) \( + ( -7956 + 716 \beta_{2} - 516 \beta_{3} + 616 \beta_{4} + 3188 \beta_{5} - 816 \beta_{6} + 100 \beta_{7} ) q^{71} \) \( + ( 7056 - 3501 \beta_{1} + 45 \beta_{2} + 2001 \beta_{3} - 4077 \beta_{4} + 2100 \beta_{5} + 957 \beta_{6} + 585 \beta_{7} ) q^{72} \) \( + ( 14699 - 873 \beta_{2} + 1161 \beta_{3} - 1017 \beta_{4} - 1845 \beta_{5} + 729 \beta_{6} + 144 \beta_{7} ) q^{73} \) \( + ( -41776 + 41776 \beta_{1} + 1242 \beta_{2} - 1854 \beta_{3} - 6154 \beta_{4} - 5524 \beta_{5} + 1872 \beta_{6} + 612 \beta_{7} ) q^{74} \) \( + ( 7830 - 20883 \beta_{1} - 360 \beta_{2} + 585 \beta_{3} - 3645 \beta_{4} - 3991 \beta_{5} - 706 \beta_{6} + 1125 \beta_{7} ) q^{75} \) \( + ( -10943 \beta_{1} + 163 \beta_{2} + 475 \beta_{3} - 5487 \beta_{4} - 312 \beta_{5} - 461 \beta_{6} - 149 \beta_{7} ) q^{76} \) \( + ( 10658 \beta_{1} + 174 \beta_{2} + 1143 \beta_{3} - 2129 \beta_{4} - 969 \beta_{5} - 1764 \beta_{6} - 795 \beta_{7} ) q^{77} \) \( + ( 38352 - 32046 \beta_{1} - 654 \beta_{2} + 2236 \beta_{3} + 5625 \beta_{4} - 1232 \beta_{5} - 2921 \beta_{6} + 111 \beta_{7} ) q^{78} \) \( + ( -14054 + 14054 \beta_{1} - 392 \beta_{2} + 1685 \beta_{3} + 4201 \beta_{4} + 5102 \beta_{5} + 509 \beta_{6} - 1293 \beta_{7} ) q^{79} \) \( + ( 13648 + 464 \beta_{2} + 672 \beta_{3} - 104 \beta_{4} - 9264 \beta_{5} - 1032 \beta_{6} + 568 \beta_{7} ) q^{80} \) \( + ( 14823 + 20817 \beta_{1} + 1350 \beta_{2} + 1593 \beta_{3} - 2889 \beta_{4} - 1701 \beta_{5} - 2484 \beta_{6} + 1485 \beta_{7} ) q^{81} \) \( + ( -29637 + 180 \beta_{2} - 896 \beta_{3} + 538 \beta_{4} - 1213 \beta_{5} + 178 \beta_{6} - 358 \beta_{7} ) q^{82} \) \( + ( 28918 - 28918 \beta_{1} + 846 \beta_{2} - 765 \beta_{3} - 1097 \beta_{4} - 170 \beta_{5} + 1773 \beta_{6} - 81 \beta_{7} ) q^{83} \) \( + ( -6 + 25926 \beta_{1} - 1242 \beta_{2} - 218 \beta_{3} + 54 \beta_{4} - 1834 \beta_{5} - 538 \beta_{6} - 810 \beta_{7} ) q^{84} \) \( + ( 7920 \beta_{1} - 648 \beta_{2} - 1590 \beta_{3} - 3786 \beta_{4} + 942 \beta_{5} + 1236 \beta_{6} + 294 \beta_{7} ) q^{85} \) \( + ( -2797 \beta_{1} - 386 \beta_{2} - 1110 \beta_{3} - 5451 \beta_{4} + 724 \beta_{5} + 1062 \beta_{6} + 338 \beta_{7} ) q^{86} \) \( + ( -21846 + 8016 \beta_{1} + 270 \beta_{2} + 191 \beta_{3} + 2877 \beta_{4} + 4262 \beta_{5} + 3023 \beta_{6} + 381 \beta_{7} ) q^{87} \) \( + ( 27747 - 27747 \beta_{1} + 311 \beta_{2} - 1139 \beta_{3} + 5948 \beta_{4} + 5431 \beta_{5} - 206 \beta_{6} + 828 \beta_{7} ) q^{88} \) \( + ( -56034 - 716 \beta_{2} - 1788 \beta_{3} + 536 \beta_{4} + 4948 \beta_{5} + 1968 \beta_{6} - 1252 \beta_{7} ) q^{89} \) \( + ( -66798 - 1296 \beta_{1} - 1278 \beta_{2} - 4752 \beta_{3} + 5841 \beta_{4} + 6894 \beta_{5} + 1989 \beta_{6} - 2367 \beta_{7} ) q^{90} \) \( + ( 13702 + 195 \beta_{2} - 97 \beta_{3} + 146 \beta_{4} - 1181 \beta_{5} - 244 \beta_{6} + 49 \beta_{7} ) q^{91} \) \( + ( 37514 - 37514 \beta_{1} - 2436 \beta_{2} + 2124 \beta_{3} + 13274 \beta_{4} + 10526 \beta_{5} - 5184 \beta_{6} + 312 \beta_{7} ) q^{92} \) \( + ( 17484 - 8532 \beta_{1} + 2700 \beta_{2} - 805 \beta_{3} + 1395 \beta_{4} - 2745 \beta_{5} + 1854 \beta_{6} - 261 \beta_{7} ) q^{93} \) \( + ( 41904 \beta_{1} + 1835 \beta_{2} + 2813 \beta_{3} + 2826 \beta_{4} - 978 \beta_{5} - 121 \beta_{6} + 857 \beta_{7} ) q^{94} \) \( + ( 70528 \beta_{1} + 112 \beta_{2} - 300 \beta_{3} + 12364 \beta_{4} + 412 \beta_{5} + 936 \beta_{6} + 524 \beta_{7} ) q^{95} \) \( + ( 36639 - 61614 \beta_{1} + 522 \beta_{2} - 2628 \beta_{3} - 6003 \beta_{4} + 567 \beta_{5} + 1233 \beta_{6} - 1305 \beta_{7} ) q^{96} \) \( + ( 6691 - 6691 \beta_{1} + 154 \beta_{2} - 2296 \beta_{3} - 9200 \beta_{4} - 11188 \beta_{5} - 1834 \beta_{6} + 2142 \beta_{7} ) q^{97} \) \( + ( 9873 - 1682 \beta_{2} + 768 \beta_{3} - 1225 \beta_{4} + 13261 \beta_{5} + 2139 \beta_{6} - 457 \beta_{7} ) q^{98} \) \( + ( -6381 + 20430 \beta_{1} - 2241 \beta_{2} + 1698 \beta_{3} - 666 \beta_{4} - 1011 \beta_{5} - 858 \beta_{6} - 1143 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 49q^{4} \) \(\mathstrut +\mathstrut 78q^{5} \) \(\mathstrut +\mathstrut 171q^{6} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 750q^{8} \) \(\mathstrut -\mathstrut 414q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 49q^{4} \) \(\mathstrut +\mathstrut 78q^{5} \) \(\mathstrut +\mathstrut 171q^{6} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 750q^{8} \) \(\mathstrut -\mathstrut 414q^{9} \) \(\mathstrut +\mathstrut 60q^{10} \) \(\mathstrut +\mathstrut 444q^{11} \) \(\mathstrut +\mathstrut 2724q^{12} \) \(\mathstrut -\mathstrut 182q^{13} \) \(\mathstrut +\mathstrut 1392q^{14} \) \(\mathstrut -\mathstrut 2052q^{15} \) \(\mathstrut -\mathstrut 289q^{16} \) \(\mathstrut -\mathstrut 4356q^{17} \) \(\mathstrut -\mathstrut 8100q^{18} \) \(\mathstrut +\mathstrut 952q^{19} \) \(\mathstrut +\mathstrut 6684q^{20} \) \(\mathstrut +\mathstrut 8670q^{21} \) \(\mathstrut +\mathstrut 1011q^{22} \) \(\mathstrut +\mathstrut 8844q^{23} \) \(\mathstrut +\mathstrut 549q^{24} \) \(\mathstrut -\mathstrut 1654q^{25} \) \(\mathstrut -\mathstrut 24888q^{26} \) \(\mathstrut -\mathstrut 10152q^{27} \) \(\mathstrut -\mathstrut 1604q^{28} \) \(\mathstrut +\mathstrut 12018q^{29} \) \(\mathstrut +\mathstrut 22104q^{30} \) \(\mathstrut +\mathstrut 1132q^{31} \) \(\mathstrut +\mathstrut 8703q^{32} \) \(\mathstrut -\mathstrut 8820q^{33} \) \(\mathstrut +\mathstrut 10125q^{34} \) \(\mathstrut -\mathstrut 16224q^{35} \) \(\mathstrut -\mathstrut 14589q^{36} \) \(\mathstrut -\mathstrut 15176q^{37} \) \(\mathstrut -\mathstrut 11145q^{38} \) \(\mathstrut +\mathstrut 8220q^{39} \) \(\mathstrut -\mathstrut 8736q^{40} \) \(\mathstrut +\mathstrut 1248q^{41} \) \(\mathstrut +\mathstrut 10098q^{42} \) \(\mathstrut -\mathstrut 6092q^{43} \) \(\mathstrut +\mathstrut 49530q^{44} \) \(\mathstrut +\mathstrut 43038q^{45} \) \(\mathstrut +\mathstrut 45960q^{46} \) \(\mathstrut -\mathstrut 60q^{47} \) \(\mathstrut -\mathstrut 57405q^{48} \) \(\mathstrut +\mathstrut 9090q^{49} \) \(\mathstrut -\mathstrut 57057q^{50} \) \(\mathstrut -\mathstrut 9396q^{51} \) \(\mathstrut -\mathstrut 32510q^{52} \) \(\mathstrut +\mathstrut 20952q^{53} \) \(\mathstrut +\mathstrut 8181q^{54} \) \(\mathstrut -\mathstrut 36120q^{55} \) \(\mathstrut -\mathstrut 61170q^{56} \) \(\mathstrut -\mathstrut 104298q^{57} \) \(\mathstrut +\mathstrut 8328q^{58} \) \(\mathstrut +\mathstrut 2076q^{59} \) \(\mathstrut +\mathstrut 88308q^{60} \) \(\mathstrut +\mathstrut 48142q^{61} \) \(\mathstrut +\mathstrut 241764q^{62} \) \(\mathstrut +\mathstrut 133524q^{63} \) \(\mathstrut -\mathstrut 20926q^{64} \) \(\mathstrut -\mathstrut 13146q^{65} \) \(\mathstrut -\mathstrut 100998q^{66} \) \(\mathstrut -\mathstrut 7148q^{67} \) \(\mathstrut -\mathstrut 123129q^{68} \) \(\mathstrut -\mathstrut 125982q^{69} \) \(\mathstrut -\mathstrut 654q^{70} \) \(\mathstrut -\mathstrut 71856q^{71} \) \(\mathstrut +\mathstrut 35451q^{72} \) \(\mathstrut +\mathstrut 122452q^{73} \) \(\mathstrut -\mathstrut 160320q^{74} \) \(\mathstrut -\mathstrut 18732q^{75} \) \(\mathstrut -\mathstrut 49571q^{76} \) \(\mathstrut +\mathstrut 39534q^{77} \) \(\mathstrut +\mathstrut 181422q^{78} \) \(\mathstrut -\mathstrut 59516q^{79} \) \(\mathstrut +\mathstrut 124512q^{80} \) \(\mathstrut +\mathstrut 194562q^{81} \) \(\mathstrut -\mathstrut 233598q^{82} \) \(\mathstrut +\mathstrut 117696q^{83} \) \(\mathstrut +\mathstrut 108354q^{84} \) \(\mathstrut +\mathstrut 28836q^{85} \) \(\mathstrut -\mathstrut 15915q^{86} \) \(\mathstrut -\mathstrut 142956q^{87} \) \(\mathstrut +\mathstrut 104523q^{88} \) \(\mathstrut -\mathstrut 451728q^{89} \) \(\mathstrut -\mathstrut 539892q^{90} \) \(\mathstrut +\mathstrut 111392q^{91} \) \(\mathstrut +\mathstrut 134034q^{92} \) \(\mathstrut +\mathstrut 113898q^{93} \) \(\mathstrut +\mathstrut 169464q^{94} \) \(\mathstrut +\mathstrut 294888q^{95} \) \(\mathstrut +\mathstrut 42768q^{96} \) \(\mathstrut +\mathstrut 33976q^{97} \) \(\mathstrut +\mathstrut 57654q^{98} \) \(\mathstrut +\mathstrut 33696q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(40\) \(x^{6}\mathstrut +\mathstrut \) \(568\) \(x^{4}\mathstrut +\mathstrut \) \(3363\) \(x^{2}\mathstrut +\mathstrut \) \(7056\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 40 \nu^{5} + 484 \nu^{3} + 1683 \nu + 84 \)\()/168\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} - 4 \nu^{6} - 96 \nu^{5} - 88 \nu^{4} - 984 \nu^{3} - 592 \nu^{2} - 3273 \nu - 1644 \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} - 158 \nu^{5} - 42 \nu^{4} - 1580 \nu^{3} - 798 \nu^{2} - 4929 \nu - 3108 \)\()/28\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{7} + 28 \nu^{6} - 276 \nu^{5} + 868 \nu^{4} - 2592 \nu^{3} + 8428 \nu^{2} - 7167 \nu + 25116 \)\()/168\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 31 \nu^{4} - 301 \nu^{2} - 897 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 31 \nu^{4} + 310 \nu^{2} - 9 \nu + 987 \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 51 \nu^{7} - 28 \nu^{6} + 1620 \nu^{5} - 1372 \nu^{4} + 16368 \nu^{3} - 18508 \nu^{2} + 51477 \nu - 67452 \)\()/168\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(180\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut +\mathstrut \) \(20\) \(\beta_{5}\mathstrut +\mathstrut \) \(22\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(57\) \(\beta_{6}\mathstrut -\mathstrut \) \(74\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(40\) \(\beta_{2}\mathstrut +\mathstrut \) \(2088\)\()/18\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(199\) \(\beta_{7}\mathstrut -\mathstrut \) \(471\) \(\beta_{6}\mathstrut -\mathstrut \) \(632\) \(\beta_{5}\mathstrut -\mathstrut \) \(793\) \(\beta_{4}\mathstrut +\mathstrut \) \(126\) \(\beta_{3}\mathstrut -\mathstrut \) \(272\) \(\beta_{2}\mathstrut +\mathstrut \) \(1080\) \(\beta_{1}\mathstrut -\mathstrut \) \(540\)\()/18\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(113\) \(\beta_{7}\mathstrut +\mathstrut \) \(432\) \(\beta_{6}\mathstrut +\mathstrut \) \(518\) \(\beta_{5}\mathstrut -\mathstrut \) \(206\) \(\beta_{4}\mathstrut +\mathstrut \) \(93\) \(\beta_{3}\mathstrut -\mathstrut \) \(319\) \(\beta_{2}\mathstrut -\mathstrut \) \(13347\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(2867\) \(\beta_{7}\mathstrut +\mathstrut \) \(6465\) \(\beta_{6}\mathstrut +\mathstrut \) \(9286\) \(\beta_{5}\mathstrut +\mathstrut \) \(12107\) \(\beta_{4}\mathstrut -\mathstrut \) \(2136\) \(\beta_{3}\mathstrut +\mathstrut \) \(3598\) \(\beta_{2}\mathstrut -\mathstrut \) \(22752\) \(\beta_{1}\mathstrut +\mathstrut \) \(11376\)\()/18\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
3.62198i
2.34949i
3.84183i
2.56934i
3.62198i
2.34949i
3.84183i
2.56934i
−3.77467 6.53793i −15.5883 + 0.0716805i −12.4963 + 21.6443i 43.7155 75.7175i 59.3094 + 101.645i 20.6033 + 35.6859i −52.9007 242.990 2.23475i −660.048
4.2 −1.47673 2.55778i 11.3240 10.7129i 11.6385 20.1585i −32.4255 + 56.1625i −44.1239 13.1441i 80.0952 + 138.729i −163.259 13.4657 242.627i 191.535
4.3 1.78978 + 3.09998i 1.34189 + 15.5306i 9.59340 16.6162i 4.05388 7.02152i −45.7429 + 31.9561i −87.7139 151.925i 183.226 −239.399 + 41.6808i 29.0221
4.4 4.96163 + 8.59380i −3.07760 15.2816i −33.2356 + 57.5657i 23.6560 40.9735i 116.057 102.270i 1.01546 + 1.75882i −342.066 −224.057 + 94.0614i 469.490
7.1 −3.77467 + 6.53793i −15.5883 0.0716805i −12.4963 21.6443i 43.7155 + 75.7175i 59.3094 101.645i 20.6033 35.6859i −52.9007 242.990 + 2.23475i −660.048
7.2 −1.47673 + 2.55778i 11.3240 + 10.7129i 11.6385 + 20.1585i −32.4255 56.1625i −44.1239 + 13.1441i 80.0952 138.729i −163.259 13.4657 + 242.627i 191.535
7.3 1.78978 3.09998i 1.34189 15.5306i 9.59340 + 16.6162i 4.05388 + 7.02152i −45.7429 31.9561i −87.7139 + 151.925i 183.226 −239.399 41.6808i 29.0221
7.4 4.96163 8.59380i −3.07760 + 15.2816i −33.2356 57.5657i 23.6560 + 40.9735i 116.057 + 102.270i 1.01546 1.75882i −342.066 −224.057 94.0614i 469.490
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{6}^{\mathrm{new}}(9, [\chi])\).