# Properties

 Label 9.5.d.a Level 9 Weight 5 Character orbit 9.d Analytic conductor 0.930 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 9.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.930329667755$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.39400128.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -3 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{3} + ( 6 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} + ( -3 - \beta_{1} - 3 \beta_{5} ) q^{5} + ( -14 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( 4 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{7} + ( -30 + \beta_{1} + \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{8} + ( 45 - 6 \beta_{1} + 9 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -3 + \beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{3} + ( 6 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{4} + ( -3 - \beta_{1} - 3 \beta_{5} ) q^{5} + ( -14 - 7 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( 4 + 14 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{7} + ( -30 + \beta_{1} + \beta_{2} - 54 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{8} + ( 45 - 6 \beta_{1} + 9 \beta_{2} + 45 \beta_{3} + 3 \beta_{4} ) q^{9} + ( 2 - 8 \beta_{1} + 8 \beta_{2} - \beta_{4} - \beta_{5} ) q^{10} + ( 108 - 2 \beta_{2} + 54 \beta_{3} - 3 \beta_{4} ) q^{11} + ( -8 + 21 \beta_{1} + \beta_{2} - 114 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{12} + ( -25 - 13 \beta_{1} - 26 \beta_{2} - 20 \beta_{3} + 10 \beta_{4} - 5 \beta_{5} ) q^{13} + ( -114 + 2 \beta_{1} + 135 \beta_{3} + 21 \beta_{5} ) q^{14} + ( -175 + 13 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{15} + ( -30 - 42 \beta_{1} - 21 \beta_{2} - 41 \beta_{3} + 15 \beta_{4} - 30 \beta_{5} ) q^{16} + ( -60 - 3 \beta_{1} - 3 \beta_{2} - 162 \beta_{3} - 21 \beta_{4} + 21 \beta_{5} ) q^{17} + ( 336 + 33 \beta_{1} + 3 \beta_{2} + 207 \beta_{3} - 24 \beta_{4} + 3 \beta_{5} ) q^{18} + ( -52 + 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} ) q^{19} + ( 324 - 28 \beta_{2} + 162 \beta_{3} + 24 \beta_{4} ) q^{20} + ( -56 - 74 \beta_{1} + 19 \beta_{2} - 342 \beta_{3} + 15 \beta_{4} - 8 \beta_{5} ) q^{21} + ( -62 + 65 \beta_{1} + 130 \beta_{2} - 60 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{22} + ( -69 + 29 \beta_{1} + 27 \beta_{3} - 42 \beta_{5} ) q^{23} + ( -82 + 16 \beta_{1} + 65 \beta_{2} + 219 \beta_{3} - 13 \beta_{4} + 38 \beta_{5} ) q^{24} + ( 70 + 2 \beta_{1} + \beta_{2} + 127 \beta_{3} - 35 \beta_{4} + 70 \beta_{5} ) q^{25} + ( -282 - 26 \beta_{1} - 26 \beta_{2} - 486 \beta_{3} + 39 \beta_{4} - 39 \beta_{5} ) q^{26} + ( 27 - 72 \beta_{1} - 162 \beta_{2} + 216 \beta_{3} + 63 \beta_{4} - 27 \beta_{5} ) q^{27} + ( 134 + 84 \beta_{1} - 84 \beta_{2} - 30 \beta_{4} - 30 \beta_{5} ) q^{28} + ( -108 + 127 \beta_{2} - 54 \beta_{3} - 63 \beta_{4} ) q^{29} + ( -168 - 36 \beta_{1} - 114 \beta_{2} + 144 \beta_{3} - 3 \beta_{4} + 21 \beta_{5} ) q^{30} + ( 467 - 55 \beta_{1} - 110 \beta_{2} + 421 \beta_{3} - 92 \beta_{4} + 46 \beta_{5} ) q^{31} + ( -96 - 129 \beta_{1} + 81 \beta_{3} - 15 \beta_{5} ) q^{32} + ( -338 + 83 \beta_{1} - 35 \beta_{2} - 507 \beta_{3} + 91 \beta_{4} - 53 \beta_{5} ) q^{33} + ( -6 - 30 \beta_{1} - 15 \beta_{2} - 189 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{34} + ( 315 + 155 \beta_{1} + 155 \beta_{2} + 702 \beta_{3} + 36 \beta_{4} - 36 \beta_{5} ) q^{35} + ( -672 + 108 \beta_{1} + 255 \beta_{2} - 801 \beta_{3} - 21 \beta_{4} + 84 \beta_{5} ) q^{36} + ( 128 - 126 \beta_{1} + 126 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} ) q^{37} + ( -270 - 115 \beta_{2} - 135 \beta_{3} + 27 \beta_{4} ) q^{38} + ( 683 + 165 \beta_{1} + 152 \beta_{2} + 717 \beta_{3} - 14 \beta_{4} - 10 \beta_{5} ) q^{39} + ( -536 - 10 \beta_{1} - 20 \beta_{2} - 492 \beta_{3} + 88 \beta_{4} - 44 \beta_{5} ) q^{40} + ( 891 + 86 \beta_{1} - 783 \beta_{3} + 108 \beta_{5} ) q^{41} + ( 1940 - 356 \beta_{1} - 226 \beta_{2} + 489 \beta_{3} - 112 \beta_{4} - 55 \beta_{5} ) q^{42} + ( -170 + 332 \beta_{1} + 166 \beta_{2} + 176 \beta_{3} + 85 \beta_{4} - 170 \beta_{5} ) q^{43} + ( 660 - 241 \beta_{1} - 241 \beta_{2} + 1026 \beta_{3} - 147 \beta_{4} + 147 \beta_{5} ) q^{44} + ( -525 - 177 \beta_{1} + 78 \beta_{2} - 18 \beta_{3} - 174 \beta_{4} - 57 \beta_{5} ) q^{45} + ( -832 - 128 \beta_{1} + 128 \beta_{2} + 29 \beta_{4} + 29 \beta_{5} ) q^{46} + ( -2430 - 311 \beta_{2} - 1215 \beta_{3} + 120 \beta_{4} ) q^{47} + ( -532 + 275 \beta_{1} + 50 \beta_{2} + 1431 \beta_{3} - 18 \beta_{4} - 31 \beta_{5} ) q^{48} + ( -1010 - 151 \beta_{1} - 302 \beta_{2} - 891 \beta_{3} + 238 \beta_{4} - 119 \beta_{5} ) q^{49} + ( 192 + 439 \beta_{1} - 189 \beta_{3} + 3 \beta_{5} ) q^{50} + ( 216 + 126 \beta_{1} + 81 \beta_{2} - 783 \beta_{3} - 171 \beta_{4} + 270 \beta_{5} ) q^{51} + ( 108 - 252 \beta_{1} - 126 \beta_{2} + 8 \beta_{3} - 54 \beta_{4} + 108 \beta_{5} ) q^{52} + ( 1602 - 198 \beta_{1} - 198 \beta_{2} + 3132 \beta_{3} - 36 \beta_{4} + 36 \beta_{5} ) q^{53} + ( -1908 + 180 \beta_{1} + 9 \beta_{2} - 3024 \beta_{3} + 252 \beta_{4} - 234 \beta_{5} ) q^{54} + ( 365 - 29 \beta_{1} + 29 \beta_{2} - 124 \beta_{4} - 124 \beta_{5} ) q^{55} + ( 432 + 718 \beta_{2} + 216 \beta_{3} - 84 \beta_{4} ) q^{56} + ( 1236 - 88 \beta_{1} + 81 \beta_{2} + 507 \beta_{3} - 7 \beta_{4} ) q^{57} + ( 2416 + 8 \beta_{1} + 16 \beta_{2} + 2289 \beta_{3} - 254 \beta_{4} + 127 \beta_{5} ) q^{58} + ( 933 - 802 \beta_{1} - 972 \beta_{3} - 39 \beta_{5} ) q^{59} + ( 944 + 238 \beta_{1} + 38 \beta_{2} + 228 \beta_{3} + 416 \beta_{4} - 214 \beta_{5} ) q^{60} + ( 58 - 166 \beta_{1} - 83 \beta_{2} - 1264 \beta_{3} - 29 \beta_{4} + 58 \beta_{5} ) q^{61} + ( -1596 + 1000 \beta_{1} + 1000 \beta_{2} - 2862 \beta_{3} + 165 \beta_{4} - 165 \beta_{5} ) q^{62} + ( 48 + 192 \beta_{1} - 303 \beta_{2} + 2619 \beta_{3} - 36 \beta_{4} + 462 \beta_{5} ) q^{63} + ( 2074 + 501 \beta_{1} - 501 \beta_{2} + 111 \beta_{4} + 111 \beta_{5} ) q^{64} + ( -1728 - 287 \beta_{2} - 864 \beta_{3} - 81 \beta_{4} ) q^{65} + ( -2202 - 987 \beta_{1} - 519 \beta_{2} - 189 \beta_{3} + 153 \beta_{4} + 48 \beta_{5} ) q^{66} + ( -745 + 554 \beta_{1} + 1108 \beta_{2} - 938 \beta_{3} - 386 \beta_{4} + 193 \beta_{5} ) q^{67} + ( -1380 - 123 \beta_{1} + 999 \beta_{3} - 381 \beta_{5} ) q^{68} + ( -2659 + 85 \beta_{1} + 386 \beta_{2} - 780 \beta_{3} - 40 \beta_{4} - 253 \beta_{5} ) q^{69} + ( 310 + 176 \beta_{1} + 88 \beta_{2} + 3471 \beta_{3} - 155 \beta_{4} + 310 \beta_{5} ) q^{70} + ( -1050 - 876 \beta_{1} - 876 \beta_{2} - 1188 \beta_{3} + 456 \beta_{4} - 456 \beta_{5} ) q^{71} + ( 1428 - 897 \beta_{1} - 804 \beta_{2} - 99 \beta_{3} - 66 \beta_{4} - 21 \beta_{5} ) q^{72} + ( -2734 + 297 \beta_{1} - 297 \beta_{2} - 27 \beta_{4} - 27 \beta_{5} ) q^{73} + ( 5724 - 610 \beta_{2} + 2862 \beta_{3} - 378 \beta_{4} ) q^{74} + ( 2744 - 241 \beta_{1} - 436 \beta_{2} - 1611 \beta_{3} - 102 \beta_{4} + 185 \beta_{5} ) q^{75} + ( -1248 + 43 \beta_{1} + 86 \beta_{2} - 1277 \beta_{3} - 58 \beta_{4} + 29 \beta_{5} ) q^{76} + ( 9 + 1283 \beta_{1} + 216 \beta_{3} + 225 \beta_{5} ) q^{77} + ( -100 + 412 \beta_{1} + 668 \beta_{2} + 3108 \beta_{3} - 139 \beta_{4} + 317 \beta_{5} ) q^{78} + ( -344 - 646 \beta_{1} - 323 \beta_{2} - 2083 \beta_{3} + 172 \beta_{4} - 344 \beta_{5} ) q^{79} + ( -2184 - 410 \beta_{1} - 410 \beta_{2} - 5076 \beta_{3} - 354 \beta_{4} + 354 \beta_{5} ) q^{80} + ( 4617 + 837 \beta_{1} + 648 \beta_{2} + 3807 \beta_{3} + 108 \beta_{4} - 405 \beta_{5} ) q^{81} + ( -1072 - 545 \beta_{1} + 545 \beta_{2} + 86 \beta_{4} + 86 \beta_{5} ) q^{82} + ( -54 + 1297 \beta_{2} - 27 \beta_{3} + 756 \beta_{4} ) q^{83} + ( -3318 + 938 \beta_{1} + 1440 \beta_{2} - 4650 \beta_{3} - 16 \beta_{4} - 342 \beta_{5} ) q^{84} + ( 2958 - 168 \beta_{1} - 336 \beta_{2} + 3456 \beta_{3} + 996 \beta_{4} - 498 \beta_{5} ) q^{85} + ( -3498 - 1087 \beta_{1} + 3996 \beta_{3} + 498 \beta_{5} ) q^{86} + ( -4712 - 628 \beta_{1} - 899 \beta_{2} - 2505 \beta_{3} - 296 \beta_{4} + 244 \beta_{5} ) q^{87} + ( -418 + 310 \beta_{1} + 155 \beta_{2} - 4983 \beta_{3} + 209 \beta_{4} - 418 \beta_{5} ) q^{88} + ( 3102 + 1470 \beta_{1} + 1470 \beta_{2} + 3996 \beta_{3} - 1104 \beta_{4} + 1104 \beta_{5} ) q^{89} + ( 3870 + 432 \beta_{1} + 414 \beta_{2} + 594 \beta_{3} - 333 \beta_{4} - 99 \beta_{5} ) q^{90} + ( 6227 - 421 \beta_{1} + 421 \beta_{2} - 218 \beta_{4} - 218 \beta_{5} ) q^{91} + ( 6588 - 1042 \beta_{2} + 3294 \beta_{3} + 288 \beta_{4} ) q^{92} + ( -235 + 1101 \beta_{1} - 64 \beta_{2} - 5736 \beta_{3} - 158 \beta_{4} - 145 \beta_{5} ) q^{93} + ( -6122 - 1264 \beta_{1} - 2528 \beta_{2} - 5811 \beta_{3} + 622 \beta_{4} - 311 \beta_{5} ) q^{94} + ( -1626 + 196 \beta_{1} + 1512 \beta_{3} - 114 \beta_{5} ) q^{95} + ( 522 + 27 \beta_{1} - 909 \beta_{2} + 2862 \beta_{3} - 225 \beta_{4} + 117 \beta_{5} ) q^{96} + ( 244 - 784 \beta_{1} - 392 \beta_{2} + 9383 \beta_{3} - 122 \beta_{4} + 244 \beta_{5} ) q^{97} + ( -2910 - 1509 \beta_{1} - 1509 \beta_{2} - 4914 \beta_{3} + 453 \beta_{4} - 453 \beta_{5} ) q^{98} + ( 984 + 141 \beta_{1} + 1542 \beta_{2} + 3771 \beta_{3} + 165 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{2} - 3q^{3} + 15q^{4} - 12q^{5} - 99q^{6} + 12q^{7} + 99q^{9} + O(q^{10})$$ $$6q - 3q^{2} - 3q^{3} + 15q^{4} - 12q^{5} - 99q^{6} + 12q^{7} + 99q^{9} - 36q^{10} + 483q^{11} + 330q^{12} - 6q^{13} - 1146q^{14} - 1026q^{15} + 15q^{16} + 1404q^{18} - 258q^{19} + 1614q^{20} + 480q^{21} - 369q^{22} - 282q^{23} - 1449q^{24} - 273q^{25} + 54q^{27} + 1308q^{28} - 1056q^{29} - 1278q^{30} + 1290q^{31} - 1161q^{32} + 279q^{33} + 513q^{34} - 2385q^{36} + 12q^{37} - 789q^{38} + 1974q^{39} - 1314q^{40} + 7629q^{41} + 9612q^{42} - 285q^{43} - 4212q^{45} - 5760q^{46} - 9642q^{47} - 6771q^{48} - 1863q^{49} + 3027q^{50} + 2457q^{51} - 240q^{52} - 405q^{54} + 2016q^{55} - 462q^{56} + 5367q^{57} + 6462q^{58} + 6225q^{59} + 7470q^{60} + 3630q^{61} - 7578q^{63} + 15450q^{64} - 7158q^{65} - 13734q^{66} - 5055q^{67} - 10503q^{68} - 13878q^{69} - 9684q^{70} + 8451q^{72} - 14622q^{73} + 26454q^{74} + 21021q^{75} - 4047q^{76} + 2580q^{77} - 12060q^{78} + 4764q^{79} + 18387q^{81} - 9702q^{82} - 1866q^{83} - 6486q^{84} + 12366q^{85} - 37731q^{86} - 21564q^{87} + 14787q^{88} + 20790q^{90} + 34836q^{91} + 33636q^{92} + 19254q^{93} - 12708q^{94} - 13362q^{95} - 3672q^{96} - 28959q^{97} - 9126q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 11 x^{4} + 14 x^{3} + 98 x^{2} + 20 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 11 \nu^{4} - 121 \nu^{3} + 98 \nu^{2} + 1118 \nu - 220$$$$)/1098$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 11 \nu^{4} + 121 \nu^{3} - 98 \nu^{2} + 529 \nu + 220$$$$)/549$$ $$\beta_{3}$$ $$=$$ $$($$$$55 \nu^{5} - 56 \nu^{4} + 616 \nu^{3} + 649 \nu^{2} + 5488 \nu + 22$$$$)/1098$$ $$\beta_{4}$$ $$=$$ $$($$$$373 \nu^{5} - 260 \nu^{4} + 3958 \nu^{3} + 6817 \nu^{2} + 37558 \nu + 15082$$$$)/1098$$ $$\beta_{5}$$ $$=$$ $$($$$$-406 \nu^{5} + 623 \nu^{4} - 4657 \nu^{3} - 3583 \nu^{2} - 36898 \nu + 6206$$$$)/1098$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} - 21 \beta_{3} + 2 \beta_{2} + \beta_{1} - 22$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 11 \beta_{2} - 11 \beta_{1} - 26$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$22 \beta_{5} - 11 \beta_{4} + 237 \beta_{3} - 23 \beta_{2} - 46 \beta_{1} + 22$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$23 \beta_{5} - 46 \beta_{4} + 549 \beta_{3} - 270 \beta_{2} - 135 \beta_{1} + 572$$$$)/3$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 1.89154 − 3.27625i −0.102534 + 0.177594i −1.28901 + 2.23263i 1.89154 + 3.27625i −0.102534 − 0.177594i −1.28901 − 2.23263i
−5.67463 3.27625i −1.11837 8.93024i 13.4676 + 23.3266i 10.2044 5.89150i −22.9114 + 54.3399i 26.6364 46.1356i 71.6534i −78.4985 + 19.9746i −77.2081
2.2 0.307601 + 0.177594i 8.32172 + 3.42768i −7.93692 13.7472i −30.0804 + 17.3669i 1.95104 + 2.53225i 15.6054 27.0294i 11.3212i 57.5020 + 57.0484i −12.3370
2.3 3.86703 + 2.23263i −8.70335 2.29167i 1.96929 + 3.41090i 13.8760 8.01130i −28.5397 28.2933i −36.2418 + 62.7727i 53.8574i 70.4965 + 39.8904i 71.5451
5.1 −5.67463 + 3.27625i −1.11837 + 8.93024i 13.4676 23.3266i 10.2044 + 5.89150i −22.9114 54.3399i 26.6364 + 46.1356i 71.6534i −78.4985 19.9746i −77.2081
5.2 0.307601 0.177594i 8.32172 3.42768i −7.93692 + 13.7472i −30.0804 17.3669i 1.95104 2.53225i 15.6054 + 27.0294i 11.3212i 57.5020 57.0484i −12.3370
5.3 3.86703 2.23263i −8.70335 + 2.29167i 1.96929 3.41090i 13.8760 + 8.01130i −28.5397 + 28.2933i −36.2418 62.7727i 53.8574i 70.4965 39.8904i 71.5451
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.5.d.a 6
3.b odd 2 1 27.5.d.a 6
4.b odd 2 1 144.5.q.a 6
9.c even 3 1 27.5.d.a 6
9.c even 3 1 81.5.b.a 6
9.d odd 6 1 inner 9.5.d.a 6
9.d odd 6 1 81.5.b.a 6
12.b even 2 1 432.5.q.a 6
36.f odd 6 1 432.5.q.a 6
36.f odd 6 1 1296.5.e.c 6
36.h even 6 1 144.5.q.a 6
36.h even 6 1 1296.5.e.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.d.a 6 1.a even 1 1 trivial
9.5.d.a 6 9.d odd 6 1 inner
27.5.d.a 6 3.b odd 2 1
27.5.d.a 6 9.c even 3 1
81.5.b.a 6 9.c even 3 1
81.5.b.a 6 9.d odd 6 1
144.5.q.a 6 4.b odd 2 1
144.5.q.a 6 36.h even 6 1
432.5.q.a 6 12.b even 2 1
432.5.q.a 6 36.f odd 6 1
1296.5.e.c 6 36.f odd 6 1
1296.5.e.c 6 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(9, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 21 T^{2} + 54 T^{3} + 150 T^{4} + 1284 T^{5} + 556 T^{6} + 20544 T^{7} + 38400 T^{8} + 221184 T^{9} + 1376256 T^{10} + 3145728 T^{11} + 16777216 T^{12}$$
$3$ $$1 + 3 T - 45 T^{2} - 162 T^{3} - 3645 T^{4} + 19683 T^{5} + 531441 T^{6}$$
$5$ $$1 + 12 T + 1146 T^{2} + 13176 T^{3} + 624786 T^{4} + 24253584 T^{5} + 478052638 T^{6} + 15158490000 T^{7} + 244057031250 T^{8} + 3216796875000 T^{9} + 174865722656250 T^{10} + 1144409179687500 T^{11} + 59604644775390625 T^{12}$$
$7$ $$1 - 12 T - 2598 T^{2} - 158692 T^{3} + 1689750 T^{4} + 261282744 T^{5} + 12756026130 T^{6} + 627339868344 T^{7} + 9741072489750 T^{8} - 2196501548501092 T^{9} - 86339153619823398 T^{10} - 957507195571344012 T^{11} +$$$$19\!\cdots\!01$$$$T^{12}$$
$11$ $$1 - 483 T + 146469 T^{2} - 33184998 T^{3} + 6185106393 T^{4} - 950746069515 T^{5} + 124654869741274 T^{6} - 13919873203769115 T^{7} + 1325832485269426233 T^{8} -$$$$10\!\cdots\!58$$$$T^{9} +$$$$67\!\cdots\!09$$$$T^{10} -$$$$32\!\cdots\!83$$$$T^{11} +$$$$98\!\cdots\!41$$$$T^{12}$$
$13$ $$1 + 6 T - 68352 T^{2} - 1958440 T^{3} + 2716582572 T^{4} + 62453281302 T^{5} - 85607901136722 T^{6} + 1783728167266422 T^{7} + 2215999860113594412 T^{8} - 45627901827271689640 T^{9} -$$$$45\!\cdots\!32$$$$T^{10} +$$$$11\!\cdots\!06$$$$T^{11} +$$$$54\!\cdots\!61$$$$T^{12}$$
$17$ $$1 - 346011 T^{2} + 53617620939 T^{4} - 5294214329064626 T^{6} +$$$$37\!\cdots\!99$$$$T^{8} -$$$$16\!\cdots\!91$$$$T^{10} +$$$$33\!\cdots\!21$$$$T^{12}$$
$19$ $$( 1 + 129 T + 372939 T^{2} + 32427790 T^{3} + 48601783419 T^{4} + 2190879632289 T^{5} + 2213314919066161 T^{6} )^{2}$$
$23$ $$1 + 282 T + 648876 T^{2} + 175507776 T^{3} + 227497773480 T^{4} + 72727641623526 T^{5} + 70224828144985186 T^{6} + 20352175959569139366 T^{7} +$$$$17\!\cdots\!80$$$$T^{8} +$$$$38\!\cdots\!96$$$$T^{9} +$$$$39\!\cdots\!36$$$$T^{10} +$$$$48\!\cdots\!82$$$$T^{11} +$$$$48\!\cdots\!41$$$$T^{12}$$
$29$ $$1 + 1056 T + 1968402 T^{2} + 1686104640 T^{3} + 1760514009810 T^{4} + 1015827714042684 T^{5} + 1221310573219134286 T^{6} +$$$$71\!\cdots\!04$$$$T^{7} +$$$$88\!\cdots\!10$$$$T^{8} +$$$$59\!\cdots\!40$$$$T^{9} +$$$$49\!\cdots\!42$$$$T^{10} +$$$$18\!\cdots\!56$$$$T^{11} +$$$$12\!\cdots\!81$$$$T^{12}$$
$31$ $$1 - 1290 T - 570606 T^{2} + 1377538196 T^{3} + 182213722890 T^{4} - 535899775714218 T^{5} + 81606055489213290 T^{6} -$$$$49\!\cdots\!78$$$$T^{7} +$$$$15\!\cdots\!90$$$$T^{8} +$$$$10\!\cdots\!56$$$$T^{9} -$$$$41\!\cdots\!86$$$$T^{10} -$$$$86\!\cdots\!90$$$$T^{11} +$$$$62\!\cdots\!21$$$$T^{12}$$
$37$ $$( 1 - 6 T + 3399531 T^{2} + 1254253444 T^{3} + 6371268418491 T^{4} - 21074876723526 T^{5} + 6582952005840035281 T^{6} )^{2}$$
$41$ $$1 - 7629 T + 33279051 T^{2} - 105879107016 T^{3} + 271025936843037 T^{4} - 581054677368191211 T^{5} +$$$$10\!\cdots\!02$$$$T^{6} -$$$$16\!\cdots\!71$$$$T^{7} +$$$$21\!\cdots\!77$$$$T^{8} -$$$$23\!\cdots\!96$$$$T^{9} +$$$$21\!\cdots\!91$$$$T^{10} -$$$$13\!\cdots\!29$$$$T^{11} +$$$$50\!\cdots\!61$$$$T^{12}$$
$43$ $$1 + 285 T - 4416855 T^{2} + 6696709682 T^{3} + 5626286140005 T^{4} - 17612350767962595 T^{5} + 17128129577980595658 T^{6} -$$$$60\!\cdots\!95$$$$T^{7} +$$$$65\!\cdots\!05$$$$T^{8} +$$$$26\!\cdots\!82$$$$T^{9} -$$$$60\!\cdots\!55$$$$T^{10} +$$$$13\!\cdots\!85$$$$T^{11} +$$$$15\!\cdots\!01$$$$T^{12}$$
$47$ $$1 + 9642 T + 52651236 T^{2} + 208863538416 T^{3} + 680262116291880 T^{4} + 1905117042381356886 T^{5} +$$$$45\!\cdots\!66$$$$T^{6} +$$$$92\!\cdots\!66$$$$T^{7} +$$$$16\!\cdots\!80$$$$T^{8} +$$$$24\!\cdots\!56$$$$T^{9} +$$$$29\!\cdots\!56$$$$T^{10} +$$$$26\!\cdots\!42$$$$T^{11} +$$$$13\!\cdots\!81$$$$T^{12}$$
$53$ $$1 - 20649822 T^{2} + 267970470920223 T^{4} -$$$$22\!\cdots\!72$$$$T^{6} +$$$$16\!\cdots\!03$$$$T^{8} -$$$$80\!\cdots\!62$$$$T^{10} +$$$$24\!\cdots\!81$$$$T^{12}$$
$59$ $$1 - 6225 T + 34368069 T^{2} - 133533682650 T^{3} + 415161196812177 T^{4} - 1308007205289880689 T^{5} +$$$$36\!\cdots\!46$$$$T^{6} -$$$$15\!\cdots\!29$$$$T^{7} +$$$$60\!\cdots\!17$$$$T^{8} -$$$$23\!\cdots\!50$$$$T^{9} +$$$$74\!\cdots\!29$$$$T^{10} -$$$$16\!\cdots\!25$$$$T^{11} +$$$$31\!\cdots\!61$$$$T^{12}$$
$61$ $$1 - 3630 T - 31672896 T^{2} + 38334701264 T^{3} + 1025452935692820 T^{4} - 698091809532688782 T^{5} -$$$$14\!\cdots\!30$$$$T^{6} -$$$$96\!\cdots\!62$$$$T^{7} +$$$$19\!\cdots\!20$$$$T^{8} +$$$$10\!\cdots\!44$$$$T^{9} -$$$$11\!\cdots\!56$$$$T^{10} -$$$$18\!\cdots\!30$$$$T^{11} +$$$$70\!\cdots\!41$$$$T^{12}$$
$67$ $$1 + 5055 T - 13352055 T^{2} + 5710009118 T^{3} + 246985082323965 T^{4} - 2158812454985978265 T^{5} -$$$$15\!\cdots\!22$$$$T^{6} -$$$$43\!\cdots\!65$$$$T^{7} +$$$$10\!\cdots\!65$$$$T^{8} +$$$$46\!\cdots\!98$$$$T^{9} -$$$$22\!\cdots\!55$$$$T^{10} +$$$$16\!\cdots\!55$$$$T^{11} +$$$$66\!\cdots\!21$$$$T^{12}$$
$71$ $$1 - 87967842 T^{2} + 4070274316914543 T^{4} -$$$$12\!\cdots\!12$$$$T^{6} +$$$$26\!\cdots\!23$$$$T^{8} -$$$$36\!\cdots\!82$$$$T^{10} +$$$$26\!\cdots\!81$$$$T^{12}$$
$73$ $$( 1 + 7311 T + 93929019 T^{2} + 399497247430 T^{3} + 2667418918455579 T^{4} + 5896029731837626191 T^{5} +$$$$22\!\cdots\!21$$$$T^{6} )^{2}$$
$79$ $$1 - 4764 T - 88341198 T^{2} + 209973394004 T^{3} + 6244109393857182 T^{4} - 7045428820224395952 T^{5} -$$$$25\!\cdots\!90$$$$T^{6} -$$$$27\!\cdots\!12$$$$T^{7} +$$$$94\!\cdots\!02$$$$T^{8} +$$$$12\!\cdots\!64$$$$T^{9} -$$$$20\!\cdots\!58$$$$T^{10} -$$$$42\!\cdots\!64$$$$T^{11} +$$$$34\!\cdots\!81$$$$T^{12}$$
$83$ $$1 + 1866 T + 10594740 T^{2} + 17604008208 T^{3} + 123292807619064 T^{4} + 3108136851733394166 T^{5} -$$$$11\!\cdots\!18$$$$T^{6} +$$$$14\!\cdots\!86$$$$T^{7} +$$$$27\!\cdots\!24$$$$T^{8} +$$$$18\!\cdots\!88$$$$T^{9} +$$$$53\!\cdots\!40$$$$T^{10} +$$$$44\!\cdots\!66$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12}$$
$89$ $$1 - 92525118 T^{2} + 4141804686688959 T^{4} -$$$$27\!\cdots\!72$$$$T^{6} +$$$$16\!\cdots\!79$$$$T^{8} -$$$$14\!\cdots\!98$$$$T^{10} +$$$$61\!\cdots\!41$$$$T^{12}$$
$97$ $$1 + 28959 T + 315999867 T^{2} + 3434149167476 T^{3} + 57450550006007937 T^{4} +$$$$60\!\cdots\!37$$$$T^{5} +$$$$48\!\cdots\!90$$$$T^{6} +$$$$53\!\cdots\!97$$$$T^{7} +$$$$45\!\cdots\!57$$$$T^{8} +$$$$23\!\cdots\!16$$$$T^{9} +$$$$19\!\cdots\!07$$$$T^{10} +$$$$15\!\cdots\!59$$$$T^{11} +$$$$48\!\cdots\!81$$$$T^{12}$$