# Properties

 Label 324.2.i.a Level $324$ Weight $2$ Character orbit 324.i Analytic conductor $2.587$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 324.i (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.58715302549$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + 2700 x^{10} - 4941 x^{9} + 8100 x^{8} - 12150 x^{7} + 17577 x^{6} - 25515 x^{5} + 37179 x^{4} - 48114 x^{3} + 52488 x^{2} - 39366 x + 19683$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{5}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} ) q^{7} + ( 1 - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{13} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{16} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{19} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{17} ) q^{25} + ( 1 + \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} ) q^{29} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + \beta_{6} + 5 \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{35} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - 3 \beta_{16} + 3 \beta_{17} ) q^{37} + ( -3 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{16} ) q^{41} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{43} + ( -3 - 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{17} ) q^{47} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{8} + \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{49} + ( -6 + 4 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - 3 \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} ) q^{53} + ( 3 \beta_{1} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} + 2 \beta_{17} ) q^{55} + ( -4 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{8} - \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - \beta_{14} - 2 \beta_{17} ) q^{59} + ( 1 - \beta_{1} + 2 \beta_{3} + \beta_{5} - 3 \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{61} + ( -2 + 6 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} + 3 \beta_{12} + 6 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{65} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{67} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{71} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{16} + 3 \beta_{17} ) q^{73} + ( 4 + 5 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{77} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{12} + \beta_{14} - 2 \beta_{15} - 2 \beta_{16} ) q^{79} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - 5 \beta_{13} - \beta_{14} - 2 \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{83} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 4 \beta_{14} + 3 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{85} + ( 1 + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + \beta_{12} - 5 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{89} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{7} + 2 \beta_{13} + 3 \beta_{15} ) q^{91} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - 5 \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{95} + ( -3 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{13} + 2 \beta_{14} - \beta_{16} + \beta_{17} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q - 3q^{5} + O(q^{10})$$ $$18q - 3q^{5} - 3q^{11} + 12q^{17} + 30q^{23} + 9q^{25} + 24q^{29} + 9q^{31} + 21q^{35} - 21q^{41} - 9q^{43} - 45q^{47} - 18q^{49} - 66q^{53} - 60q^{59} - 18q^{61} - 33q^{65} - 27q^{67} + 12q^{71} + 9q^{73} + 75q^{77} - 36q^{79} + 45q^{83} - 36q^{85} + 48q^{89} + 9q^{91} - 6q^{95} - 27q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + 2700 x^{10} - 4941 x^{9} + 8100 x^{8} - 12150 x^{7} + 17577 x^{6} - 25515 x^{5} + 37179 x^{4} - 48114 x^{3} + 52488 x^{2} - 39366 x + 19683$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$256 \nu^{17} - 252 \nu^{16} + 819 \nu^{15} - 2946 \nu^{14} + 8433 \nu^{13} - 18684 \nu^{12} + 33978 \nu^{11} - 68400 \nu^{10} + 137493 \nu^{9} - 269973 \nu^{8} + 501201 \nu^{7} - 754920 \nu^{6} + 1160973 \nu^{5} - 1495179 \nu^{4} + 2248965 \nu^{3} - 3514509 \nu^{2} + 4599261 \nu - 3726648$$$$)/1174419$$ $$\beta_{2}$$ $$=$$ $$($$$$463 \nu^{17} + 1365 \nu^{16} - 5913 \nu^{15} + 26250 \nu^{14} - 52794 \nu^{13} + 112482 \nu^{12} - 214746 \nu^{11} + 471456 \nu^{10} - 1038627 \nu^{9} + 1951695 \nu^{8} - 3283119 \nu^{7} + 4760937 \nu^{6} - 6449301 \nu^{5} + 10175868 \nu^{4} - 14715594 \nu^{3} + 23180013 \nu^{2} - 20312856 \nu + 17445699$$$$)/1174419$$ $$\beta_{3}$$ $$=$$ $$($$$$1259 \nu^{17} - 3681 \nu^{16} + 13977 \nu^{15} - 36627 \nu^{14} + 86877 \nu^{13} - 179712 \nu^{12} + 348378 \nu^{11} - 722727 \nu^{10} + 1371402 \nu^{9} - 2518317 \nu^{8} + 4111830 \nu^{7} - 5979501 \nu^{6} + 8620020 \nu^{5} - 11390139 \nu^{4} + 17084844 \nu^{3} - 21347307 \nu^{2} + 24944922 \nu - 16015401$$$$)/1174419$$ $$\beta_{4}$$ $$=$$ $$($$$$1330 \nu^{17} - 5085 \nu^{16} + 18540 \nu^{15} - 38388 \nu^{14} + 80928 \nu^{13} - 154008 \nu^{12} + 336846 \nu^{11} - 701523 \nu^{10} + 1287747 \nu^{9} - 2121012 \nu^{8} + 2980530 \nu^{7} - 4118688 \nu^{6} + 6206625 \nu^{5} - 8889669 \nu^{4} + 13340700 \nu^{3} - 10952496 \nu^{2} + 8122518 \nu + 4494285$$$$)/1174419$$ $$\beta_{5}$$ $$=$$ $$($$$$-155 \nu^{17} - 404 \nu^{16} + 1671 \nu^{15} - 7425 \nu^{14} + 14568 \nu^{13} - 30951 \nu^{12} + 60141 \nu^{11} - 133413 \nu^{10} + 294966 \nu^{9} - 545814 \nu^{8} + 909225 \nu^{7} - 1299564 \nu^{6} + 1770336 \nu^{5} - 2861001 \nu^{4} + 4050810 \nu^{3} - 6268185 \nu^{2} + 5043222 \nu - 4166235$$$$)/130491$$ $$\beta_{6}$$ $$=$$ $$($$$$1495 \nu^{17} - 1329 \nu^{16} + 7407 \nu^{15} + 3399 \nu^{14} - 3645 \nu^{13} + 36558 \nu^{12} - 43035 \nu^{11} + 98253 \nu^{10} - 366705 \nu^{9} + 808542 \nu^{8} - 1912842 \nu^{7} + 2989953 \nu^{6} - 3820284 \nu^{5} + 6479595 \nu^{4} - 7080777 \nu^{3} + 18762273 \nu^{2} - 15241203 \nu + 26204634$$$$)/1174419$$ $$\beta_{7}$$ $$=$$ $$($$$$1537 \nu^{17} - 10986 \nu^{16} + 39195 \nu^{15} - 107463 \nu^{14} + 225909 \nu^{13} - 467478 \nu^{12} + 948396 \nu^{11} - 2011095 \nu^{10} + 3987693 \nu^{9} - 7047351 \nu^{8} + 11172384 \nu^{7} - 16001631 \nu^{6} + 23041665 \nu^{5} - 34234569 \nu^{4} + 48964014 \nu^{3} - 62552574 \nu^{2} + 54849960 \nu - 36577575$$$$)/1174419$$ $$\beta_{8}$$ $$=$$ $$($$$$-2093 \nu^{17} + 3042 \nu^{16} - 4248 \nu^{15} - 21513 \nu^{14} + 46989 \nu^{13} - 112077 \nu^{12} + 171408 \nu^{11} - 409491 \nu^{10} + 1112679 \nu^{9} - 2467800 \nu^{8} + 4675941 \nu^{7} - 6885648 \nu^{6} + 8749863 \nu^{5} - 13899600 \nu^{4} + 20508957 \nu^{3} - 40203621 \nu^{2} + 38014434 \nu - 38565558$$$$)/1174419$$ $$\beta_{9}$$ $$=$$ $$($$$$4 \nu^{17} - 25 \nu^{16} + 84 \nu^{15} - 225 \nu^{14} + 471 \nu^{13} - 981 \nu^{12} + 2001 \nu^{11} - 4215 \nu^{10} + 8208 \nu^{9} - 14337 \nu^{8} + 22626 \nu^{7} - 32481 \nu^{6} + 47061 \nu^{5} - 69579 \nu^{4} + 97686 \nu^{3} - 121014 \nu^{2} + 104976 \nu - 69984$$$$)/2187$$ $$\beta_{10}$$ $$=$$ $$($$$$2600 \nu^{17} - 14172 \nu^{16} + 50355 \nu^{15} - 126681 \nu^{14} + 272574 \nu^{13} - 555255 \nu^{12} + 1151898 \nu^{11} - 2424096 \nu^{10} + 4715325 \nu^{9} - 8230842 \nu^{8} + 12800619 \nu^{7} - 18414540 \nu^{6} + 26763615 \nu^{5} - 39501594 \nu^{4} + 56658609 \nu^{3} - 67587048 \nu^{2} + 60400566 \nu - 32857488$$$$)/1174419$$ $$\beta_{11}$$ $$=$$ $$($$$$-2 \nu^{17} + 12 \nu^{16} - 42 \nu^{15} + 111 \nu^{14} - 237 \nu^{13} + 486 \nu^{12} - 996 \nu^{11} + 2088 \nu^{10} - 4086 \nu^{9} + 7173 \nu^{8} - 11295 \nu^{7} + 16227 \nu^{6} - 23355 \nu^{5} + 34344 \nu^{4} - 48843 \nu^{3} + 60750 \nu^{2} - 53946 \nu + 33534$$$$)/729$$ $$\beta_{12}$$ $$=$$ $$($$$$-3458 \nu^{17} + 25572 \nu^{16} - 91701 \nu^{15} + 251565 \nu^{14} - 529713 \nu^{13} + 1088640 \nu^{12} - 2218407 \nu^{11} + 4685418 \nu^{10} - 9276300 \nu^{9} + 16343451 \nu^{8} - 25771635 \nu^{7} + 36934380 \nu^{6} - 52935687 \nu^{5} + 78954588 \nu^{4} - 111936492 \nu^{3} + 142316838 \nu^{2} - 123058116 \nu + 78745122$$$$)/1174419$$ $$\beta_{13}$$ $$=$$ $$($$$$1334 \nu^{17} - 5391 \nu^{16} + 17655 \nu^{15} - 38283 \nu^{14} + 79776 \nu^{13} - 161046 \nu^{12} + 342663 \nu^{11} - 713466 \nu^{10} + 1311669 \nu^{9} - 2164725 \nu^{8} + 3182814 \nu^{7} - 4494771 \nu^{6} + 6815826 \nu^{5} - 9813555 \nu^{4} + 13725855 \nu^{3} - 13178862 \nu^{2} + 10267965 \nu - 2659392$$$$)/391473$$ $$\beta_{14}$$ $$=$$ $$($$$$4597 \nu^{17} - 31350 \nu^{16} + 111285 \nu^{15} - 296175 \nu^{14} + 622152 \nu^{13} - 1269864 \nu^{12} + 2609760 \nu^{11} - 5504157 \nu^{10} + 10838646 \nu^{9} - 18913284 \nu^{8} + 29571048 \nu^{7} - 42244659 \nu^{6} + 60765471 \nu^{5} - 90553221 \nu^{4} + 128558421 \nu^{3} - 158334426 \nu^{2} + 134139645 \nu - 78627024$$$$)/1174419$$ $$\beta_{15}$$ $$=$$ $$($$$$2059 \nu^{17} - 8138 \nu^{16} + 26538 \nu^{15} - 55365 \nu^{14} + 114774 \nu^{13} - 227628 \nu^{12} + 485571 \nu^{11} - 1003065 \nu^{10} + 1825434 \nu^{9} - 2957823 \nu^{8} + 4251744 \nu^{7} - 5902794 \nu^{6} + 8987598 \nu^{5} - 12841983 \nu^{4} + 18039591 \nu^{3} - 15774102 \nu^{2} + 11669832 \nu - 258066$$$$)/391473$$ $$\beta_{16}$$ $$=$$ $$($$$$-2492 \nu^{17} + 10743 \nu^{16} - 37734 \nu^{15} + 90852 \nu^{14} - 197352 \nu^{13} + 400671 \nu^{12} - 825684 \nu^{11} + 1724661 \nu^{10} - 3278430 \nu^{9} + 5678019 \nu^{8} - 8764686 \nu^{7} + 12537504 \nu^{6} - 18434628 \nu^{5} + 26300862 \nu^{4} - 37864503 \nu^{3} + 43099209 \nu^{2} - 39663432 \nu + 20177262$$$$)/391473$$ $$\beta_{17}$$ $$=$$ $$($$$$-1499 \nu^{17} + 7184 \nu^{16} - 25138 \nu^{15} + 60936 \nu^{14} - 129453 \nu^{13} + 261534 \nu^{12} - 542742 \nu^{11} + 1138587 \nu^{10} - 2179506 \nu^{9} + 3754026 \nu^{8} - 5749983 \nu^{7} + 8183052 \nu^{6} - 11979036 \nu^{5} + 17468703 \nu^{4} - 25042689 \nu^{3} + 28439991 \nu^{2} - 24616143 \nu + 11744919$$$$)/130491$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{17} + \beta_{15} + \beta_{13} - \beta_{12} + 2 \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{16} + \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 3$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{17} - 2 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{4} - 3 \beta_{2} - 2 \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{16} - 3 \beta_{15} - 3 \beta_{12} + \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + 7 \beta_{7} - \beta_{5} - 2 \beta_{3} + \beta_{2} + 5$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 6 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} - \beta_{11} + 2 \beta_{10} + 4 \beta_{9} + 5 \beta_{7} + \beta_{6} + 7 \beta_{5} + 10 \beta_{4} - 2 \beta_{3} + 17 \beta_{2} + 8 \beta_{1} - 9$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{17} - 3 \beta_{16} + 11 \beta_{15} + 9 \beta_{14} + 2 \beta_{13} + 19 \beta_{12} - 15 \beta_{11} + 10 \beta_{10} + 17 \beta_{9} + 15 \beta_{8} + 3 \beta_{7} + 14 \beta_{6} + 11 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 32 \beta_{2} + 8 \beta_{1} + 3$$ $$\nu^{8}$$ $$=$$ $$33 \beta_{17} - 26 \beta_{16} + 21 \beta_{15} + 41 \beta_{14} + 13 \beta_{13} + 52 \beta_{12} - 11 \beta_{11} + 26 \beta_{10} + 41 \beta_{9} + 34 \beta_{8} - 29 \beta_{7} + 30 \beta_{6} + 7 \beta_{5} - 19 \beta_{4} - 21 \beta_{3} - 8 \beta_{2} + 21 \beta_{1} + 15$$ $$\nu^{9}$$ $$=$$ $$18 \beta_{17} - 3 \beta_{16} + 9 \beta_{15} + 30 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} + 18 \beta_{11} + 21 \beta_{10} - 27 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - 6 \beta_{6} + 6 \beta_{4} + 27 \beta_{3} - 30 \beta_{2} + 21 \beta_{1} - 30$$ $$\nu^{10}$$ $$=$$ $$-78 \beta_{17} + 87 \beta_{16} - 27 \beta_{15} - 21 \beta_{14} - 84 \beta_{13} - 78 \beta_{12} + 51 \beta_{11} - 57 \beta_{10} - 99 \beta_{9} - 66 \beta_{8} + 141 \beta_{7} - 81 \beta_{6} - 99 \beta_{5} + 30 \beta_{4} + 30 \beta_{3} - 9 \beta_{2} + 30 \beta_{1} + 57$$ $$\nu^{11}$$ $$=$$ $$36 \beta_{17} + 45 \beta_{16} + 27 \beta_{15} - 57 \beta_{14} - 33 \beta_{13} - 51 \beta_{12} - 90 \beta_{11} + 15 \beta_{10} + 24 \beta_{9} + 15 \beta_{8} + 90 \beta_{7} - 27 \beta_{6} - 39 \beta_{5} + 240 \beta_{4} - 12 \beta_{3} + 81 \beta_{2} + 117 \beta_{1} - 96$$ $$\nu^{12}$$ $$=$$ $$204 \beta_{17} - 81 \beta_{16} - 30 \beta_{15} + 126 \beta_{14} + 240 \beta_{13} + 123 \beta_{12} - 294 \beta_{11} - 42 \beta_{10} + 105 \beta_{9} + 27 \beta_{8} - 33 \beta_{7} + 258 \beta_{6} + 168 \beta_{5} - 12 \beta_{4} - 12 \beta_{3} + 228 \beta_{2} + 138 \beta_{1} - 279$$ $$\nu^{13}$$ $$=$$ $$-102 \beta_{17} - 333 \beta_{16} - 51 \beta_{15} + 126 \beta_{14} - 303 \beta_{13} + 177 \beta_{12} + 18 \beta_{11} - 183 \beta_{10} + 174 \beta_{9} - 45 \beta_{8} - 549 \beta_{7} + 156 \beta_{6} + 147 \beta_{5} - 624 \beta_{4} - 471 \beta_{3} - 33 \beta_{2} + 57 \beta_{1} - 288$$ $$\nu^{14}$$ $$=$$ $$-261 \beta_{17} - 219 \beta_{16} + 108 \beta_{15} - 240 \beta_{14} - 345 \beta_{13} + 177 \beta_{12} - 12 \beta_{11} + 255 \beta_{10} - 366 \beta_{9} + 78 \beta_{8} - 363 \beta_{7} - 270 \beta_{6} + 276 \beta_{5} - 546 \beta_{4} - 297 \beta_{3} + 213 \beta_{2} - 522 \beta_{1} - 144$$ $$\nu^{15}$$ $$=$$ $$387 \beta_{17} + 126 \beta_{16} + 297 \beta_{15} + 711 \beta_{14} + 540 \beta_{13} + 720 \beta_{12} - 441 \beta_{11} + 333 \beta_{10} - 225 \beta_{9} + 801 \beta_{8} + 1035 \beta_{7} + 693 \beta_{6} - 531 \beta_{5} - 1395 \beta_{4} - 261 \beta_{3} + 522 \beta_{2} - 639 \beta_{1} + 2385$$ $$\nu^{16}$$ $$=$$ $$1980 \beta_{17} - 459 \beta_{16} + 1062 \beta_{15} + 1611 \beta_{14} + 1818 \beta_{13} + 891 \beta_{12} - 27 \beta_{11} + 1710 \beta_{10} + 1341 \beta_{9} + 2430 \beta_{8} - 837 \beta_{7} + 1386 \beta_{6} - 486 \beta_{5} + 747 \beta_{4} - 261 \beta_{3} + 405 \beta_{2} + 1863 \beta_{1} - 630$$ $$\nu^{17}$$ $$=$$ $$-1377 \beta_{17} + 2556 \beta_{16} - 621 \beta_{15} - 63 \beta_{14} + 2142 \beta_{13} - 180 \beta_{12} - 90 \beta_{11} - 117 \beta_{10} - 1899 \beta_{9} + 378 \beta_{8} - 1143 \beta_{7} - 972 \beta_{6} - 216 \beta_{5} + 396 \beta_{4} + 2385 \beta_{3} - 414 \beta_{2} + 4266 \beta_{1} - 3411$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$-\beta_{1} - \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.68668 + 0.393823i −0.219955 − 1.71803i 0.472963 + 1.66622i 0.381933 + 1.68942i 1.20201 − 1.24706i −1.34999 − 1.08514i 1.16555 − 1.28120i 0.960398 + 1.44140i −1.29960 − 1.14501i 1.16555 + 1.28120i 0.960398 − 1.44140i −1.29960 + 1.14501i 0.381933 − 1.68942i 1.20201 + 1.24706i −1.34999 + 1.08514i 1.68668 − 0.393823i −0.219955 + 1.71803i 0.472963 − 1.66622i
0 0 0 −2.29878 + 0.836687i 0 −0.775345 4.39720i 0 0 0
37.2 0 0 0 −0.0952805 + 0.0346793i 0 0.165033 + 0.935950i 0 0 0
37.3 0 0 0 3.94709 1.43662i 0 0.610312 + 3.46125i 0 0 0
73.1 0 0 0 −2.26400 + 1.89972i 0 2.50885 0.913148i 0 0 0
73.2 0 0 0 −1.46957 + 1.23312i 0 −3.86125 + 1.40538i 0 0 0
73.3 0 0 0 0.761786 0.639214i 0 1.35240 0.492232i 0 0 0
145.1 0 0 0 −0.583982 + 3.31193i 0 −1.47262 + 1.23568i 0 0 0
145.2 0 0 0 0.103132 0.584890i 0 2.18780 1.83578i 0 0 0
145.3 0 0 0 0.399598 2.26623i 0 −0.715176 + 0.600104i 0 0 0
181.1 0 0 0 −0.583982 3.31193i 0 −1.47262 1.23568i 0 0 0
181.2 0 0 0 0.103132 + 0.584890i 0 2.18780 + 1.83578i 0 0 0
181.3 0 0 0 0.399598 + 2.26623i 0 −0.715176 0.600104i 0 0 0
253.1 0 0 0 −2.26400 1.89972i 0 2.50885 + 0.913148i 0 0 0
253.2 0 0 0 −1.46957 1.23312i 0 −3.86125 1.40538i 0 0 0
253.3 0 0 0 0.761786 + 0.639214i 0 1.35240 + 0.492232i 0 0 0
289.1 0 0 0 −2.29878 0.836687i 0 −0.775345 + 4.39720i 0 0 0
289.2 0 0 0 −0.0952805 0.0346793i 0 0.165033 0.935950i 0 0 0
289.3 0 0 0 3.94709 + 1.43662i 0 0.610312 3.46125i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.i.a 18
3.b odd 2 1 108.2.i.a 18
9.c even 3 1 972.2.i.b 18
9.c even 3 1 972.2.i.d 18
9.d odd 6 1 972.2.i.a 18
9.d odd 6 1 972.2.i.c 18
12.b even 2 1 432.2.u.d 18
27.e even 9 1 inner 324.2.i.a 18
27.e even 9 1 972.2.i.b 18
27.e even 9 1 972.2.i.d 18
27.e even 9 1 2916.2.a.c 9
27.e even 9 2 2916.2.e.d 18
27.f odd 18 1 108.2.i.a 18
27.f odd 18 1 972.2.i.a 18
27.f odd 18 1 972.2.i.c 18
27.f odd 18 1 2916.2.a.d 9
27.f odd 18 2 2916.2.e.c 18
108.l even 18 1 432.2.u.d 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.i.a 18 3.b odd 2 1
108.2.i.a 18 27.f odd 18 1
324.2.i.a 18 1.a even 1 1 trivial
324.2.i.a 18 27.e even 9 1 inner
432.2.u.d 18 12.b even 2 1
432.2.u.d 18 108.l even 18 1
972.2.i.a 18 9.d odd 6 1
972.2.i.a 18 27.f odd 18 1
972.2.i.b 18 9.c even 3 1
972.2.i.b 18 27.e even 9 1
972.2.i.c 18 9.d odd 6 1
972.2.i.c 18 27.f odd 18 1
972.2.i.d 18 9.c even 3 1
972.2.i.d 18 27.e even 9 1
2916.2.a.c 9 27.e even 9 1
2916.2.a.d 9 27.f odd 18 1
2916.2.e.c 18 27.f odd 18 2
2916.2.e.d 18 27.e even 9 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$T^{18}$$
$5$ $$729 + 13122 T + 65610 T^{2} - 1701 T^{3} + 195372 T^{4} + 37665 T^{5} + 7452 T^{6} + 84807 T^{7} + 116964 T^{8} + 72711 T^{9} + 34182 T^{10} + 12636 T^{11} + 2772 T^{12} - 27 T^{13} - 171 T^{14} - 48 T^{15} + 3 T^{17} + T^{18}$$
$7$ $$1456849 - 54315 T + 544968 T^{2} - 274575 T^{3} + 351396 T^{4} - 1457190 T^{5} + 908187 T^{6} + 105147 T^{7} - 11907 T^{8} - 98359 T^{9} + 49284 T^{10} - 9558 T^{11} + 915 T^{12} + 423 T^{13} - 162 T^{14} + 6 T^{15} + 9 T^{16} + T^{18}$$
$11$ $$23357889 + 36798462 T + 157549293 T^{2} + 104237280 T^{3} + 17891118 T^{4} + 494019 T^{5} - 928584 T^{6} - 138267 T^{7} + 1480194 T^{8} + 155088 T^{9} - 149850 T^{10} + 2187 T^{11} + 10872 T^{12} - 81 T^{13} + 45 T^{14} - 102 T^{15} + 3 T^{17} + T^{18}$$
$13$ $$4068289 + 44456697 T + 163132866 T^{2} + 197048472 T^{3} + 198727038 T^{4} + 128316231 T^{5} + 65384862 T^{6} + 23024529 T^{7} + 6132105 T^{8} + 825770 T^{9} + 109116 T^{10} + 405 T^{11} + 1725 T^{12} - 468 T^{13} + 1053 T^{14} + 33 T^{15} - 45 T^{16} + T^{18}$$
$17$ $$5247698481 + 58677210 T + 2875839390 T^{2} + 345993120 T^{3} + 1104080706 T^{4} + 111260952 T^{5} + 200310894 T^{6} + 3235059 T^{7} + 24516594 T^{8} - 920700 T^{9} + 2095146 T^{10} - 223398 T^{11} + 129753 T^{12} - 16659 T^{13} + 5769 T^{14} - 798 T^{15} + 153 T^{16} - 12 T^{17} + T^{18}$$
$19$ $$49014001 + 326575647 T + 2074057056 T^{2} + 1056361713 T^{3} + 1474984368 T^{4} - 300138966 T^{5} + 616337016 T^{6} - 55225710 T^{7} + 62754345 T^{8} - 3074488 T^{9} + 4390776 T^{10} - 112293 T^{11} + 181824 T^{12} - 1332 T^{13} + 5481 T^{14} - 12 T^{15} + 90 T^{16} + T^{18}$$
$23$ $$5597583489 + 3375518589 T - 2765907648 T^{2} + 1025621595 T^{3} + 1693288395 T^{4} - 2078424684 T^{5} + 1458449226 T^{6} - 574251039 T^{7} + 173163096 T^{8} - 47773125 T^{9} + 11619855 T^{10} - 2502738 T^{11} + 635382 T^{12} - 172287 T^{13} + 35118 T^{14} - 4785 T^{15} + 459 T^{16} - 30 T^{17} + T^{18}$$
$29$ $$451152679041 + 834533618661 T + 650948532435 T^{2} + 175653471942 T^{3} + 5064830289 T^{4} + 5658914745 T^{5} + 2614462434 T^{6} - 362237670 T^{7} + 477233856 T^{8} - 84579741 T^{9} + 20666583 T^{10} - 3980178 T^{11} + 476136 T^{12} - 69606 T^{13} + 9792 T^{14} - 1209 T^{15} + 216 T^{16} - 24 T^{17} + T^{18}$$
$31$ $$7983601201 + 5914589445 T + 498465891 T^{2} + 2367008934 T^{3} + 2471476689 T^{4} + 861159366 T^{5} + 1536450135 T^{6} + 414363879 T^{7} + 287013852 T^{8} + 59591918 T^{9} + 8642277 T^{10} + 161586 T^{11} - 244893 T^{12} - 34920 T^{13} + 4788 T^{14} + 816 T^{15} - 45 T^{16} - 9 T^{17} + T^{18}$$
$37$ $$9420061249 + 46633365018 T + 290089928232 T^{2} - 320069115984 T^{3} + 303489272520 T^{4} - 110713915692 T^{5} + 39640117686 T^{6} - 7645807071 T^{7} + 1983326472 T^{8} - 278564290 T^{9} + 67713390 T^{10} - 6165504 T^{11} + 1355325 T^{12} - 69075 T^{13} + 20061 T^{14} - 498 T^{15} + 171 T^{16} + T^{18}$$
$41$ $$29274867801 + 48894619032 T + 366142339314 T^{2} + 993983544180 T^{3} + 1271305510803 T^{4} + 943409691144 T^{5} + 453232129104 T^{6} + 147541511259 T^{7} + 32979939912 T^{8} + 4943026836 T^{9} + 455442912 T^{10} + 16220898 T^{11} - 1681785 T^{12} - 300537 T^{13} - 13617 T^{14} + 1491 T^{15} + 270 T^{16} + 21 T^{17} + T^{18}$$
$43$ $$5055621837841 + 8978252629608 T + 6998595237927 T^{2} + 2542300090917 T^{3} + 346038412563 T^{4} - 25728733683 T^{5} - 13077347277 T^{6} - 2212516440 T^{7} + 194828580 T^{8} + 140815721 T^{9} + 32885667 T^{10} + 4975812 T^{11} + 644568 T^{12} + 63378 T^{13} + 6606 T^{14} + 816 T^{15} + 99 T^{16} + 9 T^{17} + T^{18}$$
$47$ $$941480149401 + 15320449703889 T + 107289832893309 T^{2} + 50280258634155 T^{3} + 11531522434905 T^{4} + 1857114492453 T^{5} + 192676368681 T^{6} - 3120621552 T^{7} - 4087640781 T^{8} - 424718316 T^{9} + 70205616 T^{10} + 38210535 T^{11} + 8783154 T^{12} + 1369467 T^{13} + 163377 T^{14} + 15084 T^{15} + 1026 T^{16} + 45 T^{17} + T^{18}$$
$53$ $$( -9249336 - 5813532 T + 905094 T^{2} + 861291 T^{3} + 6210 T^{4} - 31653 T^{5} - 2463 T^{6} + 234 T^{7} + 33 T^{8} + T^{9} )^{2}$$
$59$ $$14164767759321 - 14872228773435 T + 3613868674650 T^{2} - 775181890233 T^{3} + 893265966576 T^{4} + 425340219582 T^{5} + 231921757251 T^{6} + 58557015153 T^{7} + 13084921683 T^{8} + 1067931243 T^{9} - 39207726 T^{10} - 38822166 T^{11} - 5076657 T^{12} + 303669 T^{13} + 171198 T^{14} + 22890 T^{15} + 1593 T^{16} + 60 T^{17} + T^{18}$$
$61$ $$173474749009 + 348268146516 T + 100468039086 T^{2} - 110032310145 T^{3} + 47340702441 T^{4} - 26571475053 T^{5} + 9885571782 T^{6} - 1788939288 T^{7} + 354085362 T^{8} - 43847170 T^{9} + 2540979 T^{10} + 1547226 T^{11} + 106521 T^{12} + 7740 T^{13} + 1791 T^{14} + 321 T^{15} + 144 T^{16} + 18 T^{17} + T^{18}$$
$67$ $$3568321 + 46820754 T + 7189055910 T^{2} - 19059869622 T^{3} + 16026308433 T^{4} - 3392981136 T^{5} + 729941982 T^{6} - 174501 T^{7} + 52876908 T^{8} - 39440014 T^{9} + 14087844 T^{10} + 931878 T^{11} - 478569 T^{12} - 90909 T^{13} + 13365 T^{14} + 3507 T^{15} + 378 T^{16} + 27 T^{17} + T^{18}$$
$71$ $$135419769 + 104813016012 T + 81076683544182 T^{2} + 36503093872872 T^{3} + 25611294223818 T^{4} - 896949481590 T^{5} + 1520508888594 T^{6} - 19636008903 T^{7} + 45283321026 T^{8} - 843673860 T^{9} + 878149674 T^{10} - 9177948 T^{11} + 10748367 T^{12} - 249723 T^{13} + 81747 T^{14} - 1986 T^{15} + 423 T^{16} - 12 T^{17} + T^{18}$$
$73$ $$13254226129 - 38888519076 T + 95286448350 T^{2} - 65131003524 T^{3} + 45095175747 T^{4} - 15219004698 T^{5} + 8088201096 T^{6} - 2387321109 T^{7} + 916431966 T^{8} - 190233394 T^{9} + 52530543 T^{10} - 8391960 T^{11} + 1967802 T^{12} - 221958 T^{13} + 34596 T^{14} - 1425 T^{15} + 234 T^{16} - 9 T^{17} + T^{18}$$
$79$ $$1826490081529 + 1137973366494 T - 29869592958 T^{2} - 408685469178 T^{3} + 188530809822 T^{4} + 61626679218 T^{5} - 3587682063 T^{6} - 4410148464 T^{7} - 913219236 T^{8} + 86052512 T^{9} + 107594838 T^{10} + 32790996 T^{11} + 6588915 T^{12} + 977832 T^{13} + 110664 T^{14} + 9861 T^{15} + 702 T^{16} + 36 T^{17} + T^{18}$$
$83$ $$289679140521369 - 361047385532673 T + 300312480958515 T^{2} - 144637176949308 T^{3} + 53704781887653 T^{4} - 5966326194642 T^{5} + 1408627076049 T^{6} - 363467911689 T^{7} + 43995638430 T^{8} - 9754014168 T^{9} + 2338056819 T^{10} - 250252578 T^{11} + 11609487 T^{12} - 704538 T^{13} + 122148 T^{14} - 13104 T^{15} + 945 T^{16} - 45 T^{17} + T^{18}$$
$89$ $$76986883963089 - 11969093668257 T + 30929640313518 T^{2} + 8815415247 T^{3} + 9346422037590 T^{4} - 478535879808 T^{5} + 717107163174 T^{6} - 113884567596 T^{7} + 45231614757 T^{8} - 7033398534 T^{9} + 1642261716 T^{10} - 272219859 T^{11} + 44001864 T^{12} - 5025402 T^{13} + 465741 T^{14} - 30444 T^{15} + 1530 T^{16} - 48 T^{17} + T^{18}$$
$97$ $$736742449 - 98913760596 T + 5642550696429 T^{2} + 5834447083563 T^{3} + 3440730089133 T^{4} + 1532081912013 T^{5} + 554459359965 T^{6} + 152587943784 T^{7} + 30685482990 T^{8} + 4591887221 T^{9} + 523915029 T^{10} + 38962782 T^{11} + 1121874 T^{12} - 170298 T^{13} - 25002 T^{14} - 1506 T^{15} + 171 T^{16} + 27 T^{17} + T^{18}$$