# Properties

 Label 864.2.p.b Level 864 Weight 2 Character orbit 864.p Analytic conductor 6.899 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{5} + ( -\beta_{4} - \beta_{11} ) q^{7} +O(q^{10})$$ $$q -\beta_{6} q^{5} + ( -\beta_{4} - \beta_{11} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{11} + ( \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{13} + ( -\beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{10} ) q^{19} + ( -\beta_{4} - \beta_{6} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{23} + ( -1 - \beta_{2} - \beta_{3} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{25} + ( -\beta_{4} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{29} + ( \beta_{4} + \beta_{6} - 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{31} + ( 3 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{35} + ( -\beta_{4} - \beta_{6} - \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{37} + ( 4 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{8} + \beta_{10} ) q^{41} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{43} + ( 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{47} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{7} + 2 \beta_{10} ) q^{49} + ( -2 \beta_{4} - \beta_{6} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{53} + ( -\beta_{4} - \beta_{6} - \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{55} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 3 \beta_{8} + 2 \beta_{10} ) q^{59} + ( -\beta_{4} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{61} + ( -1 - 3 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{5} - \beta_{8} ) q^{65} + ( -4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 6 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} ) q^{67} + ( -\beta_{4} - \beta_{6} + \beta_{9} - \beta_{13} - \beta_{14} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{8} ) q^{73} + ( \beta_{6} - \beta_{14} + \beta_{15} ) q^{77} + ( -2 \beta_{6} + \beta_{9} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{79} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - 5 \beta_{7} - 2 \beta_{8} - 5 \beta_{10} ) q^{83} + ( 2 \beta_{4} + \beta_{6} - 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{85} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} + \beta_{7} + \beta_{8} ) q^{89} + ( 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{7} + \beta_{8} - 8 \beta_{10} ) q^{91} + ( \beta_{14} - \beta_{15} ) q^{95} + ( 4 - \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + \beta_{5} - 3 \beta_{7} - 4 \beta_{8} + 3 \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 12q^{11} + 4q^{19} - 14q^{25} + 36q^{41} - 8q^{43} + 10q^{49} + 12q^{59} + 6q^{65} + 16q^{67} - 4q^{73} + 54q^{83} + 36q^{91} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + 68 \nu^{8} + 148 \nu^{7} - 344 \nu^{6} + 440 \nu^{5} - 240 \nu^{4} - 32 \nu^{3} + 608 \nu^{2} - 1152 \nu + 384$$$$)/896$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 25 \nu^{13} + 38 \nu^{12} - 46 \nu^{11} + 24 \nu^{10} - 8 \nu^{9} - 68 \nu^{8} + 244 \nu^{7} - 272 \nu^{6} + 8 \nu^{5} + 128 \nu^{4} - 416 \nu^{3} + 1184 \nu^{2} - 1088 \nu + 512$$$$)/896$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{15} - 25 \nu^{14} + 33 \nu^{13} - 44 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + 316 \nu^{8} - 172 \nu^{7} - 24 \nu^{6} + 480 \nu^{5} - 720 \nu^{4} + 1024 \nu^{3} - 1088 \nu^{2} + 128 \nu - 640$$$$)/896$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - 29 \nu^{14} + 59 \nu^{13} - 102 \nu^{12} + 66 \nu^{11} + 24 \nu^{10} - 176 \nu^{9} + 492 \nu^{8} - 540 \nu^{7} + 288 \nu^{6} + 456 \nu^{5} - 1216 \nu^{4} + 2496 \nu^{3} - 2400 \nu^{2} + 1600 \nu - 384$$$$)/896$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{15} + 11 \nu^{14} - 19 \nu^{13} + 16 \nu^{12} + 2 \nu^{11} - 54 \nu^{10} + 116 \nu^{9} - 92 \nu^{8} + 4 \nu^{7} + 192 \nu^{6} - 424 \nu^{5} + 440 \nu^{4} - 352 \nu^{3} - 256 \nu^{2} + 544 \nu - 704$$$$)/448$$ $$\beta_{6}$$ $$=$$ $$($$$$6 \nu^{15} - 8 \nu^{14} + 3 \nu^{13} + 17 \nu^{12} - 39 \nu^{11} + 80 \nu^{10} - 92 \nu^{9} + 16 \nu^{8} + 272 \nu^{7} - 300 \nu^{6} + 260 \nu^{5} - 152 \nu^{4} - 304 \nu^{3} + 1072 \nu^{2} - 1312 \nu + 64$$$$)/448$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{15} + 15 \nu^{14} - 31 \nu^{13} + 4 \nu^{12} + 32 \nu^{11} - 136 \nu^{10} + 176 \nu^{9} - 100 \nu^{8} - 76 \nu^{7} + 552 \nu^{6} - 736 \nu^{5} + 880 \nu^{4} - 256 \nu^{3} - 512 \nu^{2} + 1536 \nu - 1408$$$$)/896$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} + 18 \nu^{14} - 47 \nu^{13} + 100 \nu^{12} - 117 \nu^{11} + 100 \nu^{10} + 60 \nu^{9} - 316 \nu^{8} + 592 \nu^{7} - 648 \nu^{6} + 52 \nu^{5} + 888 \nu^{4} - 2144 \nu^{3} + 2768 \nu^{2} - 3040 \nu + 1536$$$$)/448$$ $$\beta_{9}$$ $$=$$ $$($$$$6 \nu^{15} - 15 \nu^{14} + 24 \nu^{13} - 32 \nu^{12} + 45 \nu^{11} + 24 \nu^{10} - 64 \nu^{9} + 128 \nu^{8} - 92 \nu^{7} + 176 \nu^{6} - 20 \nu^{5} - 600 \nu^{4} + 592 \nu^{3} - 944 \nu^{2} + 928 \nu - 1728$$$$)/448$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + 284 \nu^{8} - 408 \nu^{7} + 212 \nu^{6} + 184 \nu^{5} - 920 \nu^{4} + 1520 \nu^{3} - 1888 \nu^{2} + 1184 \nu - 768$$$$)/448$$ $$\beta_{11}$$ $$=$$ $$($$$$-19 \nu^{15} + 37 \nu^{14} - 13 \nu^{13} - 48 \nu^{12} + 232 \nu^{11} - 384 \nu^{10} + 464 \nu^{9} - 60 \nu^{8} - 740 \nu^{7} + 1608 \nu^{6} - 1696 \nu^{5} + 752 \nu^{4} + 1280 \nu^{3} - 4608 \nu^{2} + 5760 \nu - 4608$$$$)/896$$ $$\beta_{12}$$ $$=$$ $$($$$$-13 \nu^{15} + 29 \nu^{14} - 59 \nu^{13} + 116 \nu^{12} - 150 \nu^{11} + 88 \nu^{10} + 120 \nu^{9} - 324 \nu^{8} + 652 \nu^{7} - 680 \nu^{6} + 104 \nu^{5} + 1216 \nu^{4} - 2720 \nu^{3} + 3968 \nu^{2} - 4288 \nu + 3072$$$$)/448$$ $$\beta_{13}$$ $$=$$ $$($$$$-16 \nu^{15} + 40 \nu^{14} - 99 \nu^{13} + 153 \nu^{12} - 113 \nu^{11} - 36 \nu^{10} + 376 \nu^{9} - 640 \nu^{8} + 768 \nu^{7} - 124 \nu^{6} - 964 \nu^{5} + 2440 \nu^{4} - 3072 \nu^{3} + 3152 \nu^{2} - 2400 \nu - 320$$$$)/448$$ $$\beta_{14}$$ $$=$$ $$($$$$-37 \nu^{15} + 103 \nu^{14} - 183 \nu^{13} + 244 \nu^{12} - 232 \nu^{11} - 64 \nu^{10} + 712 \nu^{9} - 1172 \nu^{8} + 1300 \nu^{7} - 600 \nu^{6} - 1216 \nu^{5} + 4176 \nu^{4} - 6432 \nu^{3} + 6848 \nu^{2} - 3968 \nu + 2816$$$$)/896$$ $$\beta_{15}$$ $$=$$ $$($$$$-57 \nu^{15} + 167 \nu^{14} - 375 \nu^{13} + 528 \nu^{12} - 536 \nu^{11} + 24 \nu^{10} + 1112 \nu^{9} - 2308 \nu^{8} + 2932 \nu^{7} - 1448 \nu^{6} - 1952 \nu^{5} + 7856 \nu^{4} - 12512 \nu^{3} + 14400 \nu^{2} - 12288 \nu + 5888$$$$)/896$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} - 2 \beta_{10}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{8} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 4 \beta_{1} - 4$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 2$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} - 6 \beta_{1}$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 3 \beta_{11} - 10 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} - \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 4 \beta_{1} - 10$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{15} - 6 \beta_{14} - \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 12 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 2$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{15} + \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + 14 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 18 \beta_{1} + 28$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-5 \beta_{15} + 5 \beta_{14} - \beta_{13} + 5 \beta_{12} - \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 12 \beta_{6} + 16 \beta_{5} - \beta_{4} + 18 \beta_{3} + 2 \beta_{2} - 16 \beta_{1} + 2$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$12 \beta_{14} + 9 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{9} - 28 \beta_{8} - 50 \beta_{7} - 6 \beta_{6} + 18 \beta_{5} + 3 \beta_{4} + 18 \beta_{3} + 10 \beta_{2} + 2 \beta_{1} + 10$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-\beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} + 12 \beta_{9} + 18 \beta_{8} - 4 \beta_{7} - 6 \beta_{6} - 18 \beta_{5} + 28 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} - 58 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-3 \beta_{15} + 11 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} + 26 \beta_{10} + 2 \beta_{9} + 26 \beta_{8} + 26 \beta_{7} - 76 \beta_{5} - 21 \beta_{4} + 50 \beta_{3} + 26 \beta_{2} - 84 \beta_{1} + 34$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-38 \beta_{15} - 30 \beta_{14} + 19 \beta_{13} + 49 \beta_{12} + 39 \beta_{11} - 108 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 54 \beta_{7} - 14 \beta_{6} + 2 \beta_{5} - 19 \beta_{4} - 2 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 74$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-41 \beta_{15} + 5 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 16 \beta_{11} - 2 \beta_{10} - 6 \beta_{8} + 22 \beta_{6} - 6 \beta_{5} - 10 \beta_{4} - 108 \beta_{3} - 108 \beta_{2} + 130 \beta_{1} + 44$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$7 \beta_{15} - 7 \beta_{14} - 77 \beta_{13} - 7 \beta_{12} + 91 \beta_{11} - 34 \beta_{10} - 86 \beta_{9} + 102 \beta_{8} + 34 \beta_{7} + 172 \beta_{6} + 8 \beta_{5} + 91 \beta_{4} - 94 \beta_{3} - 102 \beta_{2} - 96 \beta_{1} - 190$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$12 \beta_{14} + 29 \beta_{13} - 17 \beta_{12} - 79 \beta_{11} - 130 \beta_{9} - 156 \beta_{8} - 218 \beta_{7} + 130 \beta_{6} + 202 \beta_{5} - 129 \beta_{4} + 202 \beta_{3} - 46 \beta_{2} - 326 \beta_{1} - 62$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$-1$$ $$-\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 1.40985 + 0.111062i −0.186766 − 1.40183i 0.867527 + 1.11687i −1.37702 − 0.322193i −0.409484 + 1.35363i −0.533474 − 1.30973i 1.12063 + 0.862658i 0.608741 − 1.27649i 1.40985 − 0.111062i −0.186766 + 1.40183i 0.867527 − 1.11687i −1.37702 + 0.322193i −0.409484 − 1.35363i −0.533474 + 1.30973i 1.12063 − 0.862658i 0.608741 + 1.27649i
0 0 0 −1.74322 3.01934i 0 −1.80802 1.04386i 0 0 0
143.2 0 0 0 −1.60936 2.78750i 0 1.82223 + 1.05206i 0 0 0
143.3 0 0 0 −0.895377 1.55084i 0 2.08793 + 1.20546i 0 0 0
143.4 0 0 0 −0.565188 0.978934i 0 3.71499 + 2.14485i 0 0 0
143.5 0 0 0 0.565188 + 0.978934i 0 −3.71499 2.14485i 0 0 0
143.6 0 0 0 0.895377 + 1.55084i 0 −2.08793 1.20546i 0 0 0
143.7 0 0 0 1.60936 + 2.78750i 0 −1.82223 1.05206i 0 0 0
143.8 0 0 0 1.74322 + 3.01934i 0 1.80802 + 1.04386i 0 0 0
719.1 0 0 0 −1.74322 + 3.01934i 0 −1.80802 + 1.04386i 0 0 0
719.2 0 0 0 −1.60936 + 2.78750i 0 1.82223 1.05206i 0 0 0
719.3 0 0 0 −0.895377 + 1.55084i 0 2.08793 1.20546i 0 0 0
719.4 0 0 0 −0.565188 + 0.978934i 0 3.71499 2.14485i 0 0 0
719.5 0 0 0 0.565188 0.978934i 0 −3.71499 + 2.14485i 0 0 0
719.6 0 0 0 0.895377 1.55084i 0 −2.08793 + 1.20546i 0 0 0
719.7 0 0 0 1.60936 2.78750i 0 −1.82223 + 1.05206i 0 0 0
719.8 0 0 0 1.74322 3.01934i 0 1.80802 1.04386i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 719.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.p.b 16
3.b odd 2 1 288.2.p.b 16
4.b odd 2 1 216.2.l.b 16
8.b even 2 1 216.2.l.b 16
8.d odd 2 1 inner 864.2.p.b 16
9.c even 3 1 288.2.p.b 16
9.c even 3 1 2592.2.f.b 16
9.d odd 6 1 inner 864.2.p.b 16
9.d odd 6 1 2592.2.f.b 16
12.b even 2 1 72.2.l.b 16
24.f even 2 1 288.2.p.b 16
24.h odd 2 1 72.2.l.b 16
36.f odd 6 1 72.2.l.b 16
36.f odd 6 1 648.2.f.b 16
36.h even 6 1 216.2.l.b 16
36.h even 6 1 648.2.f.b 16
72.j odd 6 1 216.2.l.b 16
72.j odd 6 1 648.2.f.b 16
72.l even 6 1 inner 864.2.p.b 16
72.l even 6 1 2592.2.f.b 16
72.n even 6 1 72.2.l.b 16
72.n even 6 1 648.2.f.b 16
72.p odd 6 1 288.2.p.b 16
72.p odd 6 1 2592.2.f.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.b 16 12.b even 2 1
72.2.l.b 16 24.h odd 2 1
72.2.l.b 16 36.f odd 6 1
72.2.l.b 16 72.n even 6 1
216.2.l.b 16 4.b odd 2 1
216.2.l.b 16 8.b even 2 1
216.2.l.b 16 36.h even 6 1
216.2.l.b 16 72.j odd 6 1
288.2.p.b 16 3.b odd 2 1
288.2.p.b 16 9.c even 3 1
288.2.p.b 16 24.f even 2 1
288.2.p.b 16 72.p odd 6 1
648.2.f.b 16 36.f odd 6 1
648.2.f.b 16 36.h even 6 1
648.2.f.b 16 72.j odd 6 1
648.2.f.b 16 72.n even 6 1
864.2.p.b 16 1.a even 1 1 trivial
864.2.p.b 16 8.d odd 2 1 inner
864.2.p.b 16 9.d odd 6 1 inner
864.2.p.b 16 72.l even 6 1 inner
2592.2.f.b 16 9.c even 3 1
2592.2.f.b 16 9.d odd 6 1
2592.2.f.b 16 72.l even 6 1
2592.2.f.b 16 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 13 T^{2} + 48 T^{4} + 103 T^{6} - 1099 T^{8} + 3648 T^{10} - 5222 T^{12} - 177298 T^{14} + 1667616 T^{16} - 4432450 T^{18} - 3263750 T^{20} + 57000000 T^{22} - 429296875 T^{24} + 1005859375 T^{26} + 11718750000 T^{28} - 79345703125 T^{30} + 152587890625 T^{32}$$
$7$ $$1 + 23 T^{2} + 204 T^{4} + 1399 T^{6} + 12029 T^{8} + 45000 T^{10} - 343382 T^{12} - 4039462 T^{14} - 24407256 T^{16} - 197933638 T^{18} - 824460182 T^{20} + 5294205000 T^{22} + 69344791229 T^{24} + 395182873351 T^{26} + 2823622589004 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32}$$
$11$ $$( 1 - 9 T + 41 T^{2} - 108 T^{3} + 276 T^{4} - 1188 T^{5} + 4961 T^{6} - 11979 T^{7} + 14641 T^{8} )^{2}( 1 + 3 T + 38 T^{2} + 87 T^{3} + 606 T^{4} + 957 T^{5} + 4598 T^{6} + 3993 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$1 + 47 T^{2} + 1116 T^{4} + 17239 T^{6} + 168353 T^{8} + 463056 T^{10} - 26916578 T^{12} - 784598686 T^{14} - 12569281176 T^{16} - 132597177934 T^{18} - 768764384258 T^{20} + 2235082868304 T^{22} + 137330714072513 T^{24} + 2376542540984911 T^{26} + 26000662996688796 T^{28} + 185056690127866583 T^{30} + 665416609183179841 T^{32}$$
$17$ $$( 1 - 101 T^{2} + 4882 T^{4} - 146891 T^{6} + 2998666 T^{8} - 42451499 T^{10} + 407749522 T^{12} - 2437894469 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - T + 64 T^{2} - 49 T^{3} + 1726 T^{4} - 931 T^{5} + 23104 T^{6} - 6859 T^{7} + 130321 T^{8} )^{4}$$
$23$ $$1 - 85 T^{2} + 3432 T^{4} - 80441 T^{6} + 1017209 T^{8} + 1764696 T^{10} - 386280290 T^{12} + 9323832998 T^{14} - 183562178736 T^{16} + 4932307655942 T^{18} - 108097062633890 T^{20} + 261238341174744 T^{22} + 79658639026700729 T^{24} - 3332389988537139209 T^{26} + 75210991050693741672 T^{28} -$$$$98\!\cdots\!65$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 97 T^{2} + 3780 T^{4} - 80561 T^{6} + 1632089 T^{8} - 54786000 T^{10} + 1038848902 T^{12} + 18974156858 T^{14} - 1329857348520 T^{16} + 15957265917578 T^{18} + 734758090255462 T^{20} - 32587990464306000 T^{22} + 816446667883105529 T^{24} - 33892595421897492761 T^{26} +$$$$13\!\cdots\!80$$$$T^{28} -$$$$28\!\cdots\!57$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 + 131 T^{2} + 7152 T^{4} + 312151 T^{6} + 16524593 T^{8} + 706250232 T^{10} + 22761450214 T^{12} + 818587141454 T^{14} + 29243993208000 T^{16} + 786662242937294 T^{18} + 21020677263083494 T^{20} + 626799680607103992 T^{22} + 14093677267060286513 T^{24} +$$$$25\!\cdots\!51$$$$T^{26} +$$$$56\!\cdots\!72$$$$T^{28} +$$$$99\!\cdots\!51$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$( 1 - 140 T^{2} + 8596 T^{4} - 317540 T^{6} + 10470934 T^{8} - 434712260 T^{10} + 16110287956 T^{12} - 359201697260 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 18 T + 292 T^{2} - 3312 T^{3} + 34963 T^{4} - 303732 T^{5} + 2503888 T^{6} - 17908938 T^{7} + 122450608 T^{8} - 734266458 T^{9} + 4209035728 T^{10} - 20933513172 T^{11} + 98797081843 T^{12} - 383715737712 T^{13} + 1387030438372 T^{14} - 3505576929858 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 4 T - 96 T^{2} - 868 T^{3} + 4061 T^{4} + 55182 T^{5} + 78652 T^{6} - 1341518 T^{7} - 8451684 T^{8} - 57685274 T^{9} + 145427548 T^{10} + 4387355274 T^{11} + 13883750861 T^{12} - 127603328524 T^{13} - 606850852704 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 - 265 T^{2} + 38172 T^{4} - 3663305 T^{6} + 256619789 T^{8} - 13483362840 T^{10} + 543517455226 T^{12} - 17957421673750 T^{14} + 672442507889160 T^{16} - 39667944477313750 T^{18} + 2652191799434662906 T^{20} -$$$$14\!\cdots\!60$$$$T^{22} +$$$$61\!\cdots\!29$$$$T^{24} -$$$$19\!\cdots\!45$$$$T^{26} +$$$$44\!\cdots\!52$$$$T^{28} -$$$$68\!\cdots\!85$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$( 1 + 196 T^{2} + 21556 T^{4} + 1665484 T^{6} + 98315734 T^{8} + 4678344556 T^{10} + 170087208436 T^{12} + 4344214781284 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 - 6 T + 178 T^{2} - 996 T^{3} + 15871 T^{4} - 89238 T^{5} + 1194346 T^{6} - 6437052 T^{7} + 79012984 T^{8} - 379786068 T^{9} + 4157518426 T^{10} - 18327611202 T^{11} + 192314636431 T^{12} - 712064601804 T^{13} + 7508134988098 T^{14} - 14931908908914 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$1 + 299 T^{2} + 43416 T^{4} + 4465735 T^{6} + 393470093 T^{8} + 31461221184 T^{10} + 2332722797890 T^{12} + 164022774523334 T^{14} + 10603540284680400 T^{16} + 610328744001325814 T^{18} + 32298508956660075490 T^{20} +$$$$16\!\cdots\!24$$$$T^{22} +$$$$75\!\cdots\!33$$$$T^{24} +$$$$31\!\cdots\!35$$$$T^{26} +$$$$11\!\cdots\!36$$$$T^{28} +$$$$29\!\cdots\!59$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$( 1 - 8 T - 138 T^{2} + 1052 T^{3} + 11279 T^{4} - 57198 T^{5} - 930218 T^{6} + 1298482 T^{7} + 71382744 T^{8} + 86998294 T^{9} - 4175748602 T^{10} - 17203042074 T^{11} + 227284493759 T^{12} + 1420331612564 T^{13} - 12483256739322 T^{14} - 48485692842584 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 + 400 T^{2} + 73324 T^{4} + 8374048 T^{6} + 685441990 T^{8} + 42213575968 T^{10} + 1863286097644 T^{12} + 51240113568400 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + T + 214 T^{2} - 5 T^{3} + 20758 T^{4} - 365 T^{5} + 1140406 T^{6} + 389017 T^{7} + 28398241 T^{8} )^{4}$$
$79$ $$1 + 383 T^{2} + 74724 T^{4} + 9593503 T^{6} + 916548293 T^{8} + 73505234952 T^{10} + 5745220732498 T^{12} + 475438917658202 T^{14} + 38877973133064792 T^{16} + 2967214285104838682 T^{18} +$$$$22\!\cdots\!38$$$$T^{20} +$$$$17\!\cdots\!92$$$$T^{22} +$$$$13\!\cdots\!73$$$$T^{24} +$$$$90\!\cdots\!03$$$$T^{26} +$$$$44\!\cdots\!84$$$$T^{28} +$$$$14\!\cdots\!23$$$$T^{30} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$( 1 - 27 T + 544 T^{2} - 8127 T^{3} + 107593 T^{4} - 1304076 T^{5} + 14403022 T^{6} - 148987620 T^{7} + 1398163588 T^{8} - 12365972460 T^{9} + 99222418558 T^{10} - 745653703812 T^{11} + 5106183131353 T^{12} - 32012583305661 T^{13} + 177855563112736 T^{14} - 732673376719929 T^{15} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 440 T^{2} + 100348 T^{4} - 14946776 T^{6} + 1566592486 T^{8} - 118393412696 T^{10} + 6296058399868 T^{12} - 218671768022840 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 4 T - 198 T^{2} - 320 T^{3} + 21017 T^{4} + 106308 T^{5} - 1078406 T^{6} - 6931984 T^{7} + 64836612 T^{8} - 672402448 T^{9} - 10146722054 T^{10} + 97024441284 T^{11} + 1860619898777 T^{12} - 2747948882240 T^{13} - 164928456975942 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} )^{2}$$