# Properties

 Label 576.3.o.g Level 576 Weight 3 Character orbit 576.o Analytic conductor 15.695 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 576.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6948632272$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{16}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{6} ) q^{3} + ( -\beta_{1} + \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{9} - \beta_{11} ) q^{7} + ( -1 + 3 \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{6} ) q^{3} + ( -\beta_{1} + \beta_{7} ) q^{5} + ( \beta_{3} + \beta_{9} - \beta_{11} ) q^{7} + ( -1 + 3 \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{9} + ( \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} + ( 6 \beta_{1} - \beta_{15} ) q^{13} + ( -\beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} ) q^{15} + ( 2 - \beta_{4} + 3 \beta_{8} - \beta_{12} ) q^{17} + ( \beta_{2} - \beta_{3} - \beta_{5} - 4 \beta_{6} + \beta_{10} - 4 \beta_{11} + \beta_{13} ) q^{19} + ( 5 - \beta_{7} + 2 \beta_{8} - 2 \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{21} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{11} - 3 \beta_{13} ) q^{23} + ( -1 + \beta_{1} - 3 \beta_{4} + 2 \beta_{8} - 3 \beta_{14} + 2 \beta_{15} ) q^{25} + ( -5 \beta_{2} + 9 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{9} + 5 \beta_{10} + 3 \beta_{11} + 4 \beta_{13} ) q^{27} + ( -4 + 4 \beta_{1} + 3 \beta_{8} - 2 \beta_{14} + 3 \beta_{15} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} - \beta_{11} + 5 \beta_{13} ) q^{31} + ( -9 + \beta_{1} - 3 \beta_{4} + 2 \beta_{7} + 3 \beta_{8} - 5 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{33} + ( -2 \beta_{2} - 7 \beta_{3} - \beta_{5} + \beta_{6} - 3 \beta_{10} + \beta_{11} - 3 \beta_{13} ) q^{35} + ( -1 - 6 \beta_{4} + 3 \beta_{7} + \beta_{8} - 6 \beta_{12} + 3 \beta_{14} ) q^{37} + ( -4 \beta_{2} + 7 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{13} ) q^{39} + ( 10 \beta_{1} + 3 \beta_{7} - 2 \beta_{12} + 3 \beta_{15} ) q^{41} + ( 6 \beta_{2} - \beta_{3} + 2 \beta_{9} - 6 \beta_{10} - 5 \beta_{11} ) q^{43} + ( -16 + 16 \beta_{1} - 6 \beta_{4} + 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{45} + ( -4 \beta_{2} - 5 \beta_{3} - 6 \beta_{6} + 3 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{47} + ( 12 \beta_{1} - 9 \beta_{12} - \beta_{15} ) q^{49} + ( 8 \beta_{2} - 9 \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{9} - 4 \beta_{10} + 7 \beta_{11} - 7 \beta_{13} ) q^{51} + ( 7 - 4 \beta_{4} + \beta_{7} + 3 \beta_{8} - 4 \beta_{12} + \beta_{14} ) q^{53} + ( 16 \beta_{2} - 18 \beta_{3} - 2 \beta_{5} - 16 \beta_{6} - 9 \beta_{10} - 16 \beta_{11} - 9 \beta_{13} ) q^{55} + ( 19 + 13 \beta_{1} - 5 \beta_{4} - 4 \beta_{7} + 4 \beta_{8} + 2 \beta_{12} - 5 \beta_{14} + 2 \beta_{15} ) q^{57} + ( -3 \beta_{2} + 3 \beta_{3} + 6 \beta_{5} - 3 \beta_{6} - 6 \beta_{9} + 6 \beta_{11} - 2 \beta_{13} ) q^{59} + ( 6 - 6 \beta_{1} + 6 \beta_{4} + 7 \beta_{8} - 12 \beta_{14} + 7 \beta_{15} ) q^{61} + ( 16 \beta_{2} - 6 \beta_{3} + 8 \beta_{5} - \beta_{6} - 4 \beta_{9} - 7 \beta_{10} + 3 \beta_{11} - 2 \beta_{13} ) q^{63} + ( -1 + \beta_{1} - \beta_{4} + 3 \beta_{8} - 6 \beta_{14} + 3 \beta_{15} ) q^{65} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 11 \beta_{6} - \beta_{9} - 9 \beta_{11} - 3 \beta_{13} ) q^{67} + ( 1 - 4 \beta_{1} + 4 \beta_{4} + 7 \beta_{7} + \beta_{8} + 6 \beta_{12} + 9 \beta_{14} + \beta_{15} ) q^{69} + ( -21 \beta_{2} - 6 \beta_{3} - 5 \beta_{5} + 17 \beta_{6} - \beta_{10} + 17 \beta_{11} - \beta_{13} ) q^{71} + ( 4 + 3 \beta_{4} + 12 \beta_{7} + 5 \beta_{8} + 3 \beta_{12} + 12 \beta_{14} ) q^{73} + ( 10 \beta_{2} - 2 \beta_{3} - 7 \beta_{5} - 2 \beta_{6} + 5 \beta_{9} - 4 \beta_{10} + 7 \beta_{13} ) q^{75} + ( 45 \beta_{1} + \beta_{7} - 8 \beta_{12} ) q^{77} + ( -6 \beta_{2} + 22 \beta_{3} + 5 \beta_{6} - 4 \beta_{9} + 11 \beta_{10} - 11 \beta_{11} ) q^{79} + ( -5 + \beta_{1} + 4 \beta_{4} + 5 \beta_{7} + 10 \beta_{8} - \beta_{12} - 8 \beta_{14} + 2 \beta_{15} ) q^{81} + ( -21 \beta_{2} - 6 \beta_{3} - 26 \beta_{6} + \beta_{9} - 5 \beta_{10} + \beta_{11} ) q^{83} + ( 13 \beta_{1} - 3 \beta_{7} + 6 \beta_{12} + 9 \beta_{15} ) q^{85} + ( 9 \beta_{2} - 6 \beta_{3} + \beta_{5} - 6 \beta_{6} - 4 \beta_{9} - 4 \beta_{10} + 8 \beta_{11} + 3 \beta_{13} ) q^{87} + ( -25 - 4 \beta_{4} + 3 \beta_{7} - 9 \beta_{8} - 4 \beta_{12} + 3 \beta_{14} ) q^{89} + ( -3 \beta_{2} - 7 \beta_{3} + 4 \beta_{5} - 12 \beta_{6} + 4 \beta_{10} - 12 \beta_{11} + 4 \beta_{13} ) q^{91} + ( 27 + 14 \beta_{1} + 2 \beta_{4} - 7 \beta_{7} - 3 \beta_{8} - 6 \beta_{12} - 9 \beta_{14} - 11 \beta_{15} ) q^{93} + ( -26 \beta_{2} + 26 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{9} + 22 \beta_{11} + 14 \beta_{13} ) q^{95} + ( -16 + 16 \beta_{1} + \beta_{8} - 9 \beta_{14} + \beta_{15} ) q^{97} + ( 34 \beta_{2} - 27 \beta_{3} + 5 \beta_{5} - 16 \beta_{6} - \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - 11 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 6q^{5} + 18q^{9} + O(q^{10})$$ $$16q - 6q^{5} + 18q^{9} + 46q^{13} + 12q^{17} + 66q^{21} - 30q^{25} - 42q^{29} - 168q^{33} - 56q^{37} + 84q^{41} - 174q^{45} + 58q^{49} + 72q^{53} + 366q^{57} + 34q^{61} - 30q^{65} + 54q^{69} + 116q^{73} + 330q^{77} - 102q^{81} + 140q^{85} - 384q^{89} + 486q^{93} - 148q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + 1552 x^{8} - 3648 x^{7} + 6784 x^{6} - 9216 x^{5} + 19456 x^{4} - 30720 x^{3} + 28672 x^{2} - 49152 x + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} - 17 \nu^{14} - 83 \nu^{13} + 394 \nu^{12} - 204 \nu^{11} + 2224 \nu^{10} - 6280 \nu^{9} + 7664 \nu^{8} - 26384 \nu^{7} + 63104 \nu^{6} - 58368 \nu^{5} + 187904 \nu^{4} - 372736 \nu^{3} + 258048 \nu^{2} - 634880 \nu + 1769472$$$$)/540672$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{14} + 5 \nu^{13} - 22 \nu^{12} - 8 \nu^{11} - 72 \nu^{10} + 328 \nu^{9} - 176 \nu^{8} + 1616 \nu^{7} - 3584 \nu^{6} + 3392 \nu^{5} - 13824 \nu^{4} + 26112 \nu^{3} - 7168 \nu^{2} + 57344 \nu - 122880$$$$)/24576$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} + 5 \nu^{14} + 15 \nu^{13} - 8 \nu^{12} - 36 \nu^{11} - 128 \nu^{10} + 136 \nu^{9} + 224 \nu^{8} + 1456 \nu^{7} - 864 \nu^{6} + 192 \nu^{5} - 8704 \nu^{4} + 7680 \nu^{3} + 11264 \nu^{2} + 57344 \nu - 40960$$$$)/24576$$ $$\beta_{4}$$ $$=$$ $$($$$$-27 \nu^{15} + 333 \nu^{14} - 1273 \nu^{13} + 2630 \nu^{12} - 9380 \nu^{11} + 23440 \nu^{10} - 47768 \nu^{9} + 117520 \nu^{8} - 247728 \nu^{7} + 407040 \nu^{6} - 863488 \nu^{5} + 1502720 \nu^{4} - 2111488 \nu^{3} + 3497984 \nu^{2} - 5967872 \nu + 4341760$$$$)/540672$$ $$\beta_{5}$$ $$=$$ $$($$$$-17 \nu^{15} - 179 \nu^{14} + 371 \nu^{13} - 628 \nu^{12} + 3440 \nu^{11} - 6984 \nu^{10} + 9048 \nu^{9} - 37088 \nu^{8} + 62224 \nu^{7} - 105376 \nu^{6} + 234112 \nu^{5} - 334080 \nu^{4} + 241664 \nu^{3} - 929792 \nu^{2} + 786432 \nu - 286720$$$$)/270336$$ $$\beta_{6}$$ $$=$$ $$($$$$-41 \nu^{15} - 37 \nu^{14} - 279 \nu^{13} + 534 \nu^{12} - 180 \nu^{11} + 1600 \nu^{10} - 5448 \nu^{9} - 112 \nu^{8} - 20816 \nu^{7} + 9920 \nu^{6} + 23040 \nu^{5} + 38912 \nu^{4} + 14336 \nu^{3} - 323584 \nu^{2} - 348160 \nu - 622592$$$$)/540672$$ $$\beta_{7}$$ $$=$$ $$($$$$-49 \nu^{15} + 135 \nu^{14} - 723 \nu^{13} + 2762 \nu^{12} - 4100 \nu^{11} + 12352 \nu^{10} - 39144 \nu^{9} + 63664 \nu^{8} - 120656 \nu^{7} + 352128 \nu^{6} - 428416 \nu^{5} + 894464 \nu^{4} - 1829888 \nu^{3} + 1673216 \nu^{2} - 2363392 \nu + 7045120$$$$)/540672$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} + 27 \nu^{14} - 51 \nu^{13} + 176 \nu^{12} - 504 \nu^{11} + 984 \nu^{10} - 2424 \nu^{9} + 5376 \nu^{8} - 8592 \nu^{7} + 18272 \nu^{6} - 33408 \nu^{5} + 45312 \nu^{4} - 76800 \nu^{3} + 135168 \nu^{2} - 98304 \nu + 131072$$$$)/24576$$ $$\beta_{9}$$ $$=$$ $$($$$$71 \nu^{15} + 283 \nu^{14} - 223 \nu^{13} - 1090 \nu^{12} - 652 \nu^{11} + 2256 \nu^{10} + 8344 \nu^{9} + 17296 \nu^{8} - 19088 \nu^{7} - 81088 \nu^{6} - 146048 \nu^{5} + 113664 \nu^{4} + 500736 \nu^{3} + 894976 \nu^{2} - 1241088 \nu - 2179072$$$$)/540672$$ $$\beta_{10}$$ $$=$$ $$($$$$-53 \nu^{15} + \nu^{14} + 23 \nu^{13} + 708 \nu^{12} - 120 \nu^{11} - 488 \nu^{10} - 5128 \nu^{9} + 864 \nu^{8} + 10000 \nu^{7} + 38880 \nu^{6} - 9984 \nu^{5} - 105472 \nu^{4} - 211968 \nu^{3} + 204800 \nu^{2} + 774144 \nu + 786432$$$$)/270336$$ $$\beta_{11}$$ $$=$$ $$($$$$113 \nu^{15} - 147 \nu^{14} + 535 \nu^{13} - 2590 \nu^{12} + 3692 \nu^{11} - 7728 \nu^{10} + 29928 \nu^{9} - 39184 \nu^{8} + 83728 \nu^{7} - 237888 \nu^{6} + 235648 \nu^{5} - 428544 \nu^{4} + 1309696 \nu^{3} - 751616 \nu^{2} + 1183744 \nu - 3866624$$$$)/540672$$ $$\beta_{12}$$ $$=$$ $$($$$$-149 \nu^{15} + 459 \nu^{14} - 1103 \nu^{13} + 3442 \nu^{12} - 8044 \nu^{11} + 17744 \nu^{10} - 39528 \nu^{9} + 78896 \nu^{8} - 145104 \nu^{7} + 273024 \nu^{6} - 454400 \nu^{5} + 727552 \nu^{4} - 1042432 \nu^{3} + 1368064 \nu^{2} - 520192 \nu + 524288$$$$)/540672$$ $$\beta_{13}$$ $$=$$ $$($$$$-75 \nu^{15} + 221 \nu^{14} - 593 \nu^{13} + 1918 \nu^{12} - 4124 \nu^{11} + 9984 \nu^{10} - 22904 \nu^{9} + 43280 \nu^{8} - 85744 \nu^{7} + 168064 \nu^{6} - 256384 \nu^{5} + 457728 \nu^{4} - 707584 \nu^{3} + 813056 \nu^{2} - 1658880 \nu + 1146880$$$$)/270336$$ $$\beta_{14}$$ $$=$$ $$($$$$-193 \nu^{15} + 459 \nu^{14} - 1455 \nu^{13} + 4102 \nu^{12} - 10156 \nu^{11} + 24080 \nu^{10} - 59240 \nu^{9} + 101072 \nu^{8} - 237328 \nu^{7} + 435648 \nu^{6} - 712064 \nu^{5} + 1183744 \nu^{4} - 1999872 \nu^{3} + 2381824 \nu^{2} - 4124672 \nu + 2146304$$$$)/540672$$ $$\beta_{15}$$ $$=$$ $$($$$$59 \nu^{15} - 163 \nu^{14} + 277 \nu^{13} - 1510 \nu^{12} + 3126 \nu^{11} - 5420 \nu^{10} + 18080 \nu^{9} - 33120 \nu^{8} + 46048 \nu^{7} - 133440 \nu^{6} + 196512 \nu^{5} - 205312 \nu^{4} + 575744 \nu^{3} - 621056 \nu^{2} + 83968 \nu - 1040384$$$$)/135168$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{13} + \beta_{12} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + \beta_{10} + \beta_{8} + \beta_{6} + \beta_{3} + \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} - \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + 4 \beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 7$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{13} + 3 \beta_{12} + 4 \beta_{11} - \beta_{9} + 3 \beta_{7} + 2 \beta_{6} + \beta_{5} + 6 \beta_{3} - 6 \beta_{2} - 7 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{15} - 3 \beta_{14} - 10 \beta_{11} - 3 \beta_{10} - \beta_{9} + 6 \beta_{8} + 8 \beta_{6} - \beta_{4} + 7 \beta_{3} + 11 \beta_{2} + 19 \beta_{1} - 19$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$9 \beta_{14} + \beta_{13} - 9 \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} - 13 \beta_{5} - 9 \beta_{4} + 5 \beta_{3} + \beta_{2} - 31$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$6 \beta_{15} + 17 \beta_{13} - \beta_{12} + 20 \beta_{11} - 11 \beta_{9} + 33 \beta_{7} - 34 \beta_{6} + 11 \beta_{5} - 14 \beta_{3} + 14 \beta_{2} - 103 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-26 \beta_{15} - 57 \beta_{14} - 46 \beta_{11} - 33 \beta_{10} + 13 \beta_{9} - 26 \beta_{8} + 60 \beta_{6} + 33 \beta_{4} + 13 \beta_{3} + 93 \beta_{2} + 121 \beta_{1} - 121$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-65 \beta_{14} + 39 \beta_{13} + 41 \beta_{12} - 114 \beta_{11} + 39 \beta_{10} - 90 \beta_{8} - 65 \beta_{7} - 114 \beta_{6} - 75 \beta_{5} + 41 \beta_{4} - 45 \beta_{3} - 9 \beta_{2} - 209$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-22 \beta_{15} + 111 \beta_{13} + 33 \beta_{12} + 108 \beta_{11} - 77 \beta_{9} - 57 \beta_{7} - 494 \beta_{6} + 77 \beta_{5} - 386 \beta_{3} + 386 \beta_{2} + 407 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-294 \beta_{15} - 111 \beta_{14} + 190 \beta_{11} + 73 \beta_{10} + 379 \beta_{9} - 294 \beta_{8} + 244 \beta_{6} - 25 \beta_{4} - 117 \beta_{3} + 171 \beta_{2} + 1567 \beta_{1} - 1567$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-855 \beta_{14} + 929 \beta_{13} + 207 \beta_{12} + 418 \beta_{11} + 929 \beta_{10} + 138 \beta_{8} - 855 \beta_{7} + 418 \beta_{6} - 125 \beta_{5} + 207 \beta_{4} + 181 \beta_{3} - 2095 \beta_{2} - 1543$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-762 \beta_{15} + 1401 \beta_{13} - 265 \beta_{12} + 2548 \beta_{11} - 523 \beta_{9} - 1983 \beta_{7} - 674 \beta_{6} + 523 \beta_{5} + 1874 \beta_{3} - 1874 \beta_{2} + 7313 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$22 \beta_{15} + 1767 \beta_{14} - 2222 \beta_{11} - 1297 \beta_{10} + 3245 \beta_{9} + 22 \beta_{8} - 3028 \beta_{6} - 3039 \beta_{4} + 925 \beta_{3} - 1731 \beta_{2} + 3433 \beta_{1} - 3433$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-2609 \beta_{14} - 1129 \beta_{13} - 3079 \beta_{12} - 1330 \beta_{11} - 1129 \beta_{10} + 9126 \beta_{8} - 2609 \beta_{7} - 1330 \beta_{6} - 347 \beta_{5} - 3079 \beta_{4} - 8733 \beta_{3} - 5145 \beta_{2} - 20225$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-\beta_{1}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 −0.523926 − 1.93016i 1.63139 − 1.15696i −0.710719 + 1.86946i 1.84233 + 0.778342i −1.59523 − 1.20633i −1.26364 + 1.55023i 0.186266 − 1.99131i 1.93353 − 0.511345i −0.523926 + 1.93016i 1.63139 + 1.15696i −0.710719 − 1.86946i 1.84233 − 0.778342i −1.59523 + 1.20633i −1.26364 − 1.55023i 0.186266 + 1.99131i 1.93353 + 0.511345i
0 −2.76570 + 1.16229i 0 4.03104 6.98197i 0 3.90254 2.25313i 0 6.29815 6.42910i 0
319.2 0 −2.67178 1.36441i 0 −3.07403 + 5.32438i 0 −0.511543 + 0.295340i 0 5.27677 + 7.29079i 0
319.3 0 −2.32245 + 1.89900i 0 −1.35609 + 2.34881i 0 −10.0431 + 5.79837i 0 1.78756 8.82069i 0
319.4 0 −0.262217 + 2.98852i 0 −1.10093 + 1.90686i 0 7.23844 4.17912i 0 −8.86248 1.56728i 0
319.5 0 0.262217 2.98852i 0 −1.10093 + 1.90686i 0 −7.23844 + 4.17912i 0 −8.86248 1.56728i 0
319.6 0 2.32245 1.89900i 0 −1.35609 + 2.34881i 0 10.0431 5.79837i 0 1.78756 8.82069i 0
319.7 0 2.67178 + 1.36441i 0 −3.07403 + 5.32438i 0 0.511543 0.295340i 0 5.27677 + 7.29079i 0
319.8 0 2.76570 1.16229i 0 4.03104 6.98197i 0 −3.90254 + 2.25313i 0 6.29815 6.42910i 0
511.1 0 −2.76570 1.16229i 0 4.03104 + 6.98197i 0 3.90254 + 2.25313i 0 6.29815 + 6.42910i 0
511.2 0 −2.67178 + 1.36441i 0 −3.07403 5.32438i 0 −0.511543 0.295340i 0 5.27677 7.29079i 0
511.3 0 −2.32245 1.89900i 0 −1.35609 2.34881i 0 −10.0431 5.79837i 0 1.78756 + 8.82069i 0
511.4 0 −0.262217 2.98852i 0 −1.10093 1.90686i 0 7.23844 + 4.17912i 0 −8.86248 + 1.56728i 0
511.5 0 0.262217 + 2.98852i 0 −1.10093 1.90686i 0 −7.23844 4.17912i 0 −8.86248 + 1.56728i 0
511.6 0 2.32245 + 1.89900i 0 −1.35609 2.34881i 0 10.0431 + 5.79837i 0 1.78756 + 8.82069i 0
511.7 0 2.67178 1.36441i 0 −3.07403 5.32438i 0 0.511543 + 0.295340i 0 5.27677 7.29079i 0
511.8 0 2.76570 + 1.16229i 0 4.03104 + 6.98197i 0 −3.90254 2.25313i 0 6.29815 + 6.42910i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.o.g 16
3.b odd 2 1 1728.3.o.g 16
4.b odd 2 1 inner 576.3.o.g 16
8.b even 2 1 36.3.f.c 16
8.d odd 2 1 36.3.f.c 16
9.c even 3 1 inner 576.3.o.g 16
9.d odd 6 1 1728.3.o.g 16
12.b even 2 1 1728.3.o.g 16
24.f even 2 1 108.3.f.c 16
24.h odd 2 1 108.3.f.c 16
36.f odd 6 1 inner 576.3.o.g 16
36.h even 6 1 1728.3.o.g 16
72.j odd 6 1 108.3.f.c 16
72.j odd 6 1 324.3.d.g 8
72.l even 6 1 108.3.f.c 16
72.l even 6 1 324.3.d.g 8
72.n even 6 1 36.3.f.c 16
72.n even 6 1 324.3.d.i 8
72.p odd 6 1 36.3.f.c 16
72.p odd 6 1 324.3.d.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c 16 8.b even 2 1
36.3.f.c 16 8.d odd 2 1
36.3.f.c 16 72.n even 6 1
36.3.f.c 16 72.p odd 6 1
108.3.f.c 16 24.f even 2 1
108.3.f.c 16 24.h odd 2 1
108.3.f.c 16 72.j odd 6 1
108.3.f.c 16 72.l even 6 1
324.3.d.g 8 72.j odd 6 1
324.3.d.g 8 72.l even 6 1
324.3.d.i 8 72.n even 6 1
324.3.d.i 8 72.p odd 6 1
576.3.o.g 16 1.a even 1 1 trivial
576.3.o.g 16 4.b odd 2 1 inner
576.3.o.g 16 9.c even 3 1 inner
576.3.o.g 16 36.f odd 6 1 inner
1728.3.o.g 16 3.b odd 2 1
1728.3.o.g 16 9.d odd 6 1
1728.3.o.g 16 12.b even 2 1
1728.3.o.g 16 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(576, [\chi])$$:

 $$T_{5}^{8} + \cdots$$ $$T_{7}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 9 T^{2} + 66 T^{4} + 1161 T^{6} - 10854 T^{8} + 94041 T^{10} + 433026 T^{12} - 4782969 T^{14} + 43046721 T^{16}$$
$5$ $$( 1 + 3 T - 38 T^{2} + 201 T^{3} + 1495 T^{4} - 6984 T^{5} + 24112 T^{6} + 181380 T^{7} - 866084 T^{8} + 4534500 T^{9} + 15070000 T^{10} - 109125000 T^{11} + 583984375 T^{12} + 1962890625 T^{13} - 9277343750 T^{14} + 18310546875 T^{15} + 152587890625 T^{16} )^{2}$$
$7$ $$1 + 167 T^{2} + 13188 T^{4} + 535927 T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 4977965952336286 T^{18} - 278348169589725074 T^{20} - 9902233810610977584 T^{22} +$$$$15\!\cdots\!25$$$$T^{24} +$$$$42\!\cdots\!27$$$$T^{26} +$$$$25\!\cdots\!88$$$$T^{28} +$$$$76\!\cdots\!67$$$$T^{30} +$$$$11\!\cdots\!01$$$$T^{32}$$
$11$ $$1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 48469876840225802764 T^{18} -$$$$51\!\cdots\!42$$$$T^{20} +$$$$67\!\cdots\!68$$$$T^{22} +$$$$13\!\cdots\!45$$$$T^{24} +$$$$16\!\cdots\!04$$$$T^{26} +$$$$13\!\cdots\!54$$$$T^{28} +$$$$75\!\cdots\!44$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$( 1 - 23 T - 276 T^{2} + 5009 T^{3} + 162641 T^{4} - 1572720 T^{5} - 34351730 T^{6} + 33339262 T^{7} + 8566506504 T^{8} + 5634335278 T^{9} - 981119760530 T^{10} - 7591219050480 T^{11} + 132671260194161 T^{12} + 690533185671641 T^{13} - 6430271493804756 T^{14} - 90559656871083647 T^{15} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 - 3 T + 578 T^{2} - 6549 T^{3} + 169242 T^{4} - 1892661 T^{5} + 48275138 T^{6} - 72412707 T^{7} + 6975757441 T^{8} )^{4}$$
$19$ $$( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 122204089833383 T^{10} + 26492490152025490 T^{12} - 3702875859597687353 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$1 + 2687 T^{2} + 3719652 T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} +$$$$39\!\cdots\!18$$$$T^{14} +$$$$20\!\cdots\!24$$$$T^{16} +$$$$11\!\cdots\!38$$$$T^{18} +$$$$59\!\cdots\!94$$$$T^{20} +$$$$31\!\cdots\!76$$$$T^{22} +$$$$14\!\cdots\!85$$$$T^{24} +$$$$59\!\cdots\!55$$$$T^{26} +$$$$17\!\cdots\!32$$$$T^{28} +$$$$36\!\cdots\!47$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 + 21 T - 2432 T^{2} - 19167 T^{3} + 4062037 T^{4} + 6623136 T^{5} - 4703569190 T^{6} - 3469324686 T^{7} + 4129311885376 T^{8} - 2917702060926 T^{9} - 3326745120272390 T^{10} + 3939595750954656 T^{11} + 2032019438564861557 T^{12} - 8063695540664952567 T^{13} -$$$$86\!\cdots\!12$$$$T^{14} +$$$$62\!\cdots\!01$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} +$$$$40\!\cdots\!98$$$$T^{14} +$$$$41\!\cdots\!28$$$$T^{16} +$$$$37\!\cdots\!58$$$$T^{18} +$$$$31\!\cdots\!94$$$$T^{20} +$$$$25\!\cdots\!88$$$$T^{22} +$$$$18\!\cdots\!37$$$$T^{24} +$$$$11\!\cdots\!27$$$$T^{26} +$$$$66\!\cdots\!64$$$$T^{28} +$$$$27\!\cdots\!15$$$$T^{30} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$( 1 + 14 T + 3040 T^{2} + 66050 T^{3} + 4581118 T^{4} + 90422450 T^{5} + 5697449440 T^{6} + 35920169726 T^{7} + 3512479453921 T^{8} )^{4}$$
$41$ $$( 1 - 42 T - 4424 T^{2} + 125040 T^{3} + 14943835 T^{4} - 232367040 T^{5} - 36025798940 T^{6} + 132403432026 T^{7} + 71285616374608 T^{8} + 222570169235706 T^{9} - 101800297638493340 T^{10} - 1103767662172616640 T^{11} +$$$$11\!\cdots\!35$$$$T^{12} +$$$$16\!\cdots\!40$$$$T^{13} -$$$$99\!\cdots\!44$$$$T^{14} -$$$$15\!\cdots\!62$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} +$$$$45\!\cdots\!90$$$$T^{12} +$$$$10\!\cdots\!64$$$$T^{14} +$$$$19\!\cdots\!28$$$$T^{16} +$$$$34\!\cdots\!64$$$$T^{18} +$$$$53\!\cdots\!90$$$$T^{20} +$$$$75\!\cdots\!08$$$$T^{22} +$$$$90\!\cdots\!65$$$$T^{24} +$$$$94\!\cdots\!36$$$$T^{26} +$$$$84\!\cdots\!02$$$$T^{28} +$$$$56\!\cdots\!92$$$$T^{30} +$$$$18\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 1551988218895697 T^{8} + 4556649670813575888 T^{10} +$$$$11\!\cdots\!54$$$$T^{12} +$$$$26\!\cdots\!06$$$$T^{14} +$$$$58\!\cdots\!04$$$$T^{16} +$$$$12\!\cdots\!86$$$$T^{18} +$$$$27\!\cdots\!94$$$$T^{20} +$$$$52\!\cdots\!08$$$$T^{22} +$$$$87\!\cdots\!37$$$$T^{24} +$$$$11\!\cdots\!79$$$$T^{26} +$$$$12\!\cdots\!40$$$$T^{28} +$$$$85\!\cdots\!63$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$( 1 - 18 T + 10016 T^{2} - 126558 T^{3} + 40472766 T^{4} - 355501422 T^{5} + 79031057696 T^{6} - 398958500322 T^{7} + 62259690411361 T^{8} )^{4}$$
$59$ $$1 + 18092 T^{2} + 175903662 T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} +$$$$35\!\cdots\!18$$$$T^{12} +$$$$21\!\cdots\!16$$$$T^{14} -$$$$83\!\cdots\!68$$$$T^{16} +$$$$26\!\cdots\!76$$$$T^{18} +$$$$52\!\cdots\!78$$$$T^{20} +$$$$30\!\cdots\!68$$$$T^{22} +$$$$11\!\cdots\!61$$$$T^{24} +$$$$29\!\cdots\!84$$$$T^{26} +$$$$55\!\cdots\!82$$$$T^{28} +$$$$69\!\cdots\!32$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 - 17 T - 3582 T^{2} + 125549 T^{3} - 18305593 T^{4} + 350178696 T^{5} - 12273736208 T^{6} - 2060229775052 T^{7} + 599123836260396 T^{8} - 7666114992968492 T^{9} - 169940200011910928 T^{10} + 18041337511166813256 T^{11} -$$$$35\!\cdots\!33$$$$T^{12} +$$$$89\!\cdots\!49$$$$T^{13} -$$$$95\!\cdots\!22$$$$T^{14} -$$$$16\!\cdots\!97$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} +$$$$12\!\cdots\!60$$$$T^{10} +$$$$66\!\cdots\!50$$$$T^{12} +$$$$31\!\cdots\!76$$$$T^{14} +$$$$14\!\cdots\!68$$$$T^{16} +$$$$63\!\cdots\!96$$$$T^{18} +$$$$27\!\cdots\!50$$$$T^{20} +$$$$10\!\cdots\!60$$$$T^{22} +$$$$35\!\cdots\!25$$$$T^{24} +$$$$10\!\cdots\!00$$$$T^{26} +$$$$22\!\cdots\!66$$$$T^{28} +$$$$33\!\cdots\!68$$$$T^{30} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 23176426971828216728 T^{10} +$$$$88\!\cdots\!80$$$$T^{12} -$$$$21\!\cdots\!88$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 3307960763 T^{5} + 220767925534 T^{6} - 4388692562381 T^{7} + 806460091894081 T^{8} )^{4}$$
$79$ $$1 + 23147 T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} +$$$$59\!\cdots\!06$$$$T^{12} -$$$$27\!\cdots\!10$$$$T^{14} -$$$$25\!\cdots\!44$$$$T^{16} -$$$$10\!\cdots\!10$$$$T^{18} +$$$$90\!\cdots\!66$$$$T^{20} +$$$$53\!\cdots\!92$$$$T^{22} +$$$$49\!\cdots\!89$$$$T^{24} +$$$$28\!\cdots\!59$$$$T^{26} +$$$$11\!\cdots\!16$$$$T^{28} +$$$$31\!\cdots\!67$$$$T^{30} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} -$$$$32\!\cdots\!46$$$$T^{12} +$$$$11\!\cdots\!22$$$$T^{14} +$$$$52\!\cdots\!80$$$$T^{16} +$$$$55\!\cdots\!62$$$$T^{18} -$$$$73\!\cdots\!86$$$$T^{20} +$$$$96\!\cdots\!28$$$$T^{22} +$$$$63\!\cdots\!17$$$$T^{24} -$$$$57\!\cdots\!09$$$$T^{26} +$$$$32\!\cdots\!12$$$$T^{28} +$$$$96\!\cdots\!95$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 + 96 T + 27716 T^{2} + 1940016 T^{3} + 308634966 T^{4} + 15366866736 T^{5} + 1738963951556 T^{6} + 47710203932256 T^{7} + 3936588805702081 T^{8} )^{4}$$
$97$ $$( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 701815562883455698 T^{9} -$$$$71\!\cdots\!00$$$$T^{10} +$$$$17\!\cdots\!44$$$$T^{11} +$$$$47\!\cdots\!03$$$$T^{12} -$$$$11\!\cdots\!72$$$$T^{13} -$$$$20\!\cdots\!88$$$$T^{14} +$$$$48\!\cdots\!06$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$