# Properties

 Label 2268.2.t.b Level $2268$ Weight $2$ Character orbit 2268.t Analytic conductor $18.110$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{6}$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{11} q^{5} -\beta_{4} q^{7} +O(q^{10})$$ $$q + \beta_{11} q^{5} -\beta_{4} q^{7} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{10} + \beta_{14} - \beta_{15} ) q^{11} -\beta_{1} q^{13} + ( 2 \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{17} + ( -1 + \beta_{2} - 2 \beta_{7} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{19} + ( -1 - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{23} + ( -\beta_{2} + \beta_{6} + \beta_{15} ) q^{25} + ( 1 + \beta_{1} - \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{29} + ( -3 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{31} + ( \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{35} + ( 1 + \beta_{7} - 2 \beta_{11} + \beta_{12} ) q^{37} + ( 2 - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{15} ) q^{41} + ( -1 - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{6} - 3 \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{14} - 2 \beta_{15} ) q^{43} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{47} + ( -1 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{49} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{53} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} ) q^{55} + ( \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{59} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{15} ) q^{61} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{14} - \beta_{15} ) q^{65} + ( -1 - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{67} + ( -1 - \beta_{1} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} ) q^{71} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{73} + ( -8 - \beta_{1} + \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 4 \beta_{14} - 4 \beta_{15} ) q^{77} + ( -1 - \beta_{7} - \beta_{11} - \beta_{12} ) q^{79} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{83} + ( 4 + \beta_{1} + \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{85} + ( 5 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{91} + ( 7 + \beta_{1} - 7 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{95} + ( \beta_{1} + \beta_{2} + 3 \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 2q^{7} + O(q^{10})$$ $$16q + 2q^{7} + 6q^{11} + 9q^{17} - 21q^{23} - 8q^{25} - 6q^{31} - 15q^{35} + q^{37} + 12q^{41} + 4q^{43} + 18q^{47} - 8q^{49} + 15q^{59} - 3q^{61} - 39q^{65} - 7q^{67} - 48q^{77} - q^{79} - 12q^{85} + 21q^{89} + 9q^{91} + 6q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$19 \nu^{15} + 139 \nu^{14} + 1928 \nu^{13} + 8221 \nu^{12} + 10009 \nu^{11} + 14762 \nu^{10} - 23272 \nu^{9} - 19426 \nu^{8} - 26486 \nu^{7} - 17106 \nu^{6} - 123732 \nu^{5} - 231723 \nu^{4} - 63747 \nu^{3} + 2064528 \nu^{2} + 2597427 \nu + 4788801$$$$)/103518$$ $$\beta_{2}$$ $$=$$ $$($$$$1307 \nu^{15} - 5068 \nu^{14} - 824 \nu^{13} - 49267 \nu^{12} - 2716 \nu^{11} - 77018 \nu^{10} + 113602 \nu^{9} + 7210 \nu^{8} + 181946 \nu^{7} - 84090 \nu^{6} + 1174032 \nu^{5} - 900801 \nu^{4} + 2054484 \nu^{3} - 11094408 \nu^{2} - 4573017 \nu - 19166868$$$$)/621108$$ $$\beta_{3}$$ $$=$$ $$($$$$-3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + 360098 \nu^{10} - 187876 \nu^{9} + 298928 \nu^{8} - 711356 \nu^{7} + 844320 \nu^{6} - 2586978 \nu^{5} + 9488205 \nu^{4} - 18766647 \nu^{3} + 20620980 \nu^{2} - 33241671 \nu + 46300977$$$$)/1242216$$ $$\beta_{4}$$ $$=$$ $$($$$$-6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + 11630 \nu^{10} - 101824 \nu^{9} + 277628 \nu^{8} - 214640 \nu^{7} + 202764 \nu^{6} - 3624714 \nu^{5} + 7713063 \nu^{4} - 3514995 \nu^{3} + 15997176 \nu^{2} - 16161201 \nu - 5872095$$$$)/1242216$$ $$\beta_{5}$$ $$=$$ $$($$$$1669 \nu^{15} + 1930 \nu^{14} + 23477 \nu^{13} + 5248 \nu^{12} + 44857 \nu^{11} - 82192 \nu^{10} - 18340 \nu^{9} - 86836 \nu^{8} + 54754 \nu^{7} - 419040 \nu^{6} + 118692 \nu^{5} - 910737 \nu^{4} + 6259194 \nu^{3} + 2423439 \nu^{2} + 11432178 \nu - 4533651$$$$)/310554$$ $$\beta_{6}$$ $$=$$ $$($$$$2739 \nu^{15} - 9229 \nu^{14} + 1724 \nu^{13} - 88319 \nu^{12} - 8827 \nu^{11} - 147394 \nu^{10} + 195292 \nu^{9} + 28864 \nu^{8} + 332068 \nu^{7} - 278440 \nu^{6} + 2194674 \nu^{5} - 1523241 \nu^{4} + 4418847 \nu^{3} - 19275408 \nu^{2} - 9423297 \nu - 35170605$$$$)/414072$$ $$\beta_{7}$$ $$=$$ $$($$$$-8405 \nu^{15} - 16535 \nu^{14} - 52096 \nu^{13} - 27863 \nu^{12} - 20561 \nu^{11} + 66614 \nu^{10} + 87524 \nu^{9} + 178940 \nu^{8} + 318940 \nu^{7} + 1092156 \nu^{6} - 981702 \nu^{5} - 2209221 \nu^{4} - 11460447 \nu^{3} - 13165740 \nu^{2} - 15432201 \nu - 11234619$$$$)/621108$$ $$\beta_{8}$$ $$=$$ $$($$$$-4934 \nu^{15} + 6616 \nu^{14} - 21253 \nu^{13} + 86401 \nu^{12} - 24581 \nu^{11} + 193856 \nu^{10} - 186610 \nu^{9} + 38834 \nu^{8} - 305642 \nu^{7} + 605064 \nu^{6} - 2581722 \nu^{5} + 1955178 \nu^{4} - 9141984 \nu^{3} + 17226513 \nu^{2} + 614547 \nu + 34596153$$$$)/310554$$ $$\beta_{9}$$ $$=$$ $$($$$$13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} - 343040 \nu^{10} + 361210 \nu^{9} - 100070 \nu^{8} + 284726 \nu^{7} - 1370658 \nu^{6} + 5698206 \nu^{5} - 63018 \nu^{4} + 18020475 \nu^{3} - 29851092 \nu^{2} - 7676370 \nu - 61443765$$$$)/621108$$ $$\beta_{10}$$ $$=$$ $$($$$$27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} - 442826 \nu^{10} + 157996 \nu^{9} - 326936 \nu^{8} - 426532 \nu^{7} - 3257256 \nu^{6} + 5542434 \nu^{5} + 10347075 \nu^{4} + 36196875 \nu^{3} + 5383908 \nu^{2} + 7598367 \nu - 47062053$$$$)/1242216$$ $$\beta_{11}$$ $$=$$ $$($$$$-16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + 247964 \nu^{10} - 166462 \nu^{9} + 398066 \nu^{8} - 282422 \nu^{7} + 1677798 \nu^{6} - 5939262 \nu^{5} + 5124384 \nu^{4} - 20396205 \nu^{3} + 16524972 \nu^{2} - 21030192 \nu + 23648031$$$$)/621108$$ $$\beta_{12}$$ $$=$$ $$($$$$-19793 \nu^{15} - 1430 \nu^{14} - 108688 \nu^{13} + 147133 \nu^{12} - 121322 \nu^{11} + 529214 \nu^{10} - 286834 \nu^{9} + 407870 \nu^{8} - 442766 \nu^{7} + 2453682 \nu^{6} - 6290388 \nu^{5} + 3336147 \nu^{4} - 34358094 \nu^{3} + 21512304 \nu^{2} - 26451765 \nu + 66410442$$$$)/621108$$ $$\beta_{13}$$ $$=$$ $$($$$$3656 \nu^{15} - 865 \nu^{14} + 15076 \nu^{13} - 31234 \nu^{12} + 13505 \nu^{11} - 75428 \nu^{10} + 55828 \nu^{9} - 61412 \nu^{8} + 79862 \nu^{7} - 379710 \nu^{6} + 1284126 \nu^{5} - 523818 \nu^{4} + 4880493 \nu^{3} - 4856112 \nu^{2} + 1887138 \nu - 10890531$$$$)/103518$$ $$\beta_{14}$$ $$=$$ $$($$$$-27554 \nu^{15} + 18943 \nu^{14} - 80794 \nu^{13} + 318436 \nu^{12} - 60809 \nu^{11} + 523952 \nu^{10} - 614170 \nu^{9} + 383294 \nu^{8} - 708026 \nu^{7} + 2355330 \nu^{6} - 11690478 \nu^{5} + 5227686 \nu^{4} - 29717685 \nu^{3} + 56735154 \nu^{2} + 4219452 \nu + 92446677$$$$)/621108$$ $$\beta_{15}$$ $$=$$ $$($$$$-23045 \nu^{15} + 19169 \nu^{14} - 116460 \nu^{13} + 283569 \nu^{12} - 185713 \nu^{11} + 740670 \nu^{10} - 571592 \nu^{9} + 458764 \nu^{8} - 995208 \nu^{7} + 2760316 \nu^{6} - 9787698 \nu^{5} + 8671095 \nu^{4} - 41432283 \nu^{3} + 51803064 \nu^{2} - 24427089 \nu + 111152817$$$$)/414072$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - \beta_{13} - 2 \beta_{11} + \beta_{10} - \beta_{8} - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{15} + 4 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} - 4 \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 6 \beta_{2} - \beta_{1} - 4$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - \beta_{11} + 5 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{2} + 3 \beta_{1} + 10$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{15} - 14 \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{11} - 10 \beta_{10} - 5 \beta_{8} - 7 \beta_{7} - 11 \beta_{6} - 2 \beta_{5} + 24 \beta_{4} - 17 \beta_{3} - 4 \beta_{2} + \beta_{1} - 3$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{15} + 9 \beta_{14} - 10 \beta_{13} - 13 \beta_{12} - 13 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + 9 \beta_{7} - 12 \beta_{5} - 10 \beta_{4} + 14 \beta_{3} - 24 \beta_{2} - 21 \beta_{1} + 5$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$12 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 15 \beta_{11} + 2 \beta_{10} - 15 \beta_{9} - 26 \beta_{8} - 6 \beta_{7} - 34 \beta_{6} + 7 \beta_{5} + 26 \beta_{4} - 16 \beta_{3} + 87 \beta_{2} - 5 \beta_{1} + 60$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$61 \beta_{15} - 25 \beta_{14} + 25 \beta_{13} - 41 \beta_{12} + 6 \beta_{11} - 46 \beta_{10} + 54 \beta_{9} - 56 \beta_{8} + 19 \beta_{7} - 61 \beta_{6} + \beta_{5} + 64 \beta_{4} - 64 \beta_{3} + \beta_{2} + 40 \beta_{1} + 15$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-56 \beta_{15} + 13 \beta_{14} - 67 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 118 \beta_{10} - 136 \beta_{9} - 47 \beta_{8} + 9 \beta_{7} + 40 \beta_{6} - 62 \beta_{5} - 110 \beta_{4} + 54 \beta_{3} - 66 \beta_{2} - 28 \beta_{1} - 81$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-185 \beta_{15} + 207 \beta_{14} + 14 \beta_{13} + 99 \beta_{12} - 61 \beta_{11} + 112 \beta_{10} - 169 \beta_{9} + 19 \beta_{8} - 68 \beta_{7} + 25 \beta_{6} - 125 \beta_{5} - 197 \beta_{4} + 301 \beta_{3} + 412 \beta_{2} - 49 \beta_{1} - 632$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-7 \beta_{15} - 106 \beta_{14} + 234 \beta_{13} + 286 \beta_{12} + 274 \beta_{11} + 48 \beta_{10} - 81 \beta_{9} - 117 \beta_{8} - 125 \beta_{7} - 136 \beta_{6} + 78 \beta_{5} - 46 \beta_{4} - 193 \beta_{3} + 28 \beta_{2} + 72 \beta_{1} - 63$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$($$$$-144 \beta_{15} - 439 \beta_{14} + 36 \beta_{13} - 147 \beta_{12} + 673 \beta_{11} - 220 \beta_{10} - 419 \beta_{9} - 28 \beta_{8} - 150 \beta_{7} - 40 \beta_{6} - 13 \beta_{5} + 52 \beta_{4} + 290 \beta_{3} - 858 \beta_{2} + 181 \beta_{1} - 502$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$-373 \beta_{15} + 399 \beta_{14} - 651 \beta_{13} - 44 \beta_{12} - 632 \beta_{11} + 544 \beta_{10} - 32 \beta_{9} + 210 \beta_{8} + 794 \beta_{7} + 475 \beta_{6} + 426 \beta_{5} - 875 \beta_{4} + 1117 \beta_{3} - 1357 \beta_{2} - 673 \beta_{1} - 517$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$1073 \beta_{15} - 1398 \beta_{14} - 11 \beta_{13} + 363 \beta_{12} + 530 \beta_{11} - 367 \beta_{10} - 807 \beta_{9} - 1190 \beta_{8} - 1674 \beta_{7} - 1065 \beta_{6} + 1111 \beta_{5} + 617 \beta_{4} - 1560 \beta_{3} + 4110 \beta_{2} - 365 \beta_{1} - 1014$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$1627 \beta_{15} - 544 \beta_{14} + 1472 \beta_{13} - 2368 \beta_{12} + 2547 \beta_{11} - 361 \beta_{10} + 886 \beta_{9} - 838 \beta_{8} + 1875 \beta_{7} + 188 \beta_{6} + 998 \beta_{5} - 2247 \beta_{4} + 157 \beta_{3} - 3141 \beta_{2} + 1555 \beta_{1} + 1741$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1781.1
 −0.544978 − 1.64408i 1.68042 − 0.419752i −0.268067 + 1.71118i 1.08696 − 1.34852i 1.68124 + 0.416458i −1.61108 − 0.635951i −0.811340 − 1.53027i −0.213160 + 1.71888i −0.544978 + 1.64408i 1.68042 + 0.419752i −0.268067 − 1.71118i 1.08696 + 1.34852i 1.68124 − 0.416458i −1.61108 + 0.635951i −0.811340 + 1.53027i −0.213160 − 1.71888i
0 0 0 −1.95741 3.39033i 0 1.96333 1.77351i 0 0 0
1781.2 0 0 0 −1.48494 2.57199i 0 −2.18457 + 1.49253i 0 0 0
1781.3 0 0 0 −0.842869 1.45989i 0 2.30301 + 1.30235i 0 0 0
1781.4 0 0 0 0.0382122 + 0.0661855i 0 2.16592 1.51947i 0 0 0
1781.5 0 0 0 0.349828 + 0.605920i 0 −2.02556 1.70209i 0 0 0
1781.6 0 0 0 1.09150 + 1.89054i 0 −1.38614 + 2.25358i 0 0 0
1781.7 0 0 0 1.37166 + 2.37578i 0 0.900590 + 2.48776i 0 0 0
1781.8 0 0 0 1.43402 + 2.48379i 0 −0.736590 2.54115i 0 0 0
2105.1 0 0 0 −1.95741 + 3.39033i 0 1.96333 + 1.77351i 0 0 0
2105.2 0 0 0 −1.48494 + 2.57199i 0 −2.18457 1.49253i 0 0 0
2105.3 0 0 0 −0.842869 + 1.45989i 0 2.30301 1.30235i 0 0 0
2105.4 0 0 0 0.0382122 0.0661855i 0 2.16592 + 1.51947i 0 0 0
2105.5 0 0 0 0.349828 0.605920i 0 −2.02556 + 1.70209i 0 0 0
2105.6 0 0 0 1.09150 1.89054i 0 −1.38614 2.25358i 0 0 0
2105.7 0 0 0 1.37166 2.37578i 0 0.900590 2.48776i 0 0 0
2105.8 0 0 0 1.43402 2.48379i 0 −0.736590 + 2.54115i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2105.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.t.b 16
3.b odd 2 1 2268.2.t.a 16
7.d odd 6 1 2268.2.t.a 16
9.c even 3 1 252.2.w.a 16
9.c even 3 1 756.2.bm.a 16
9.d odd 6 1 252.2.bm.a yes 16
9.d odd 6 1 756.2.w.a 16
21.g even 6 1 inner 2268.2.t.b 16
36.f odd 6 1 1008.2.ca.d 16
36.f odd 6 1 3024.2.df.d 16
36.h even 6 1 1008.2.df.d 16
36.h even 6 1 3024.2.ca.d 16
63.g even 3 1 1764.2.x.a 16
63.g even 3 1 5292.2.w.b 16
63.h even 3 1 1764.2.bm.a 16
63.h even 3 1 5292.2.x.b 16
63.i even 6 1 756.2.bm.a 16
63.i even 6 1 1764.2.x.a 16
63.j odd 6 1 1764.2.x.b 16
63.j odd 6 1 5292.2.bm.a 16
63.k odd 6 1 756.2.w.a 16
63.k odd 6 1 1764.2.x.b 16
63.l odd 6 1 1764.2.w.b 16
63.l odd 6 1 5292.2.bm.a 16
63.n odd 6 1 1764.2.w.b 16
63.n odd 6 1 5292.2.x.a 16
63.o even 6 1 1764.2.bm.a 16
63.o even 6 1 5292.2.w.b 16
63.s even 6 1 252.2.w.a 16
63.s even 6 1 5292.2.x.b 16
63.t odd 6 1 252.2.bm.a yes 16
63.t odd 6 1 5292.2.x.a 16
252.n even 6 1 3024.2.ca.d 16
252.r odd 6 1 3024.2.df.d 16
252.bj even 6 1 1008.2.df.d 16
252.bn odd 6 1 1008.2.ca.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a 16 9.c even 3 1
252.2.w.a 16 63.s even 6 1
252.2.bm.a yes 16 9.d odd 6 1
252.2.bm.a yes 16 63.t odd 6 1
756.2.w.a 16 9.d odd 6 1
756.2.w.a 16 63.k odd 6 1
756.2.bm.a 16 9.c even 3 1
756.2.bm.a 16 63.i even 6 1
1008.2.ca.d 16 36.f odd 6 1
1008.2.ca.d 16 252.bn odd 6 1
1008.2.df.d 16 36.h even 6 1
1008.2.df.d 16 252.bj even 6 1
1764.2.w.b 16 63.l odd 6 1
1764.2.w.b 16 63.n odd 6 1
1764.2.x.a 16 63.g even 3 1
1764.2.x.a 16 63.i even 6 1
1764.2.x.b 16 63.j odd 6 1
1764.2.x.b 16 63.k odd 6 1
1764.2.bm.a 16 63.h even 3 1
1764.2.bm.a 16 63.o even 6 1
2268.2.t.a 16 3.b odd 2 1
2268.2.t.a 16 7.d odd 6 1
2268.2.t.b 16 1.a even 1 1 trivial
2268.2.t.b 16 21.g even 6 1 inner
3024.2.ca.d 16 36.h even 6 1
3024.2.ca.d 16 252.n even 6 1
3024.2.df.d 16 36.f odd 6 1
3024.2.df.d 16 252.r odd 6 1
5292.2.w.b 16 63.g even 3 1
5292.2.w.b 16 63.o even 6 1
5292.2.x.a 16 63.n odd 6 1
5292.2.x.a 16 63.t odd 6 1
5292.2.x.b 16 63.h even 3 1
5292.2.x.b 16 63.s even 6 1
5292.2.bm.a 16 63.j odd 6 1
5292.2.bm.a 16 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$324 - 4698 T + 62289 T^{2} - 89424 T^{3} + 143289 T^{4} - 45738 T^{5} + 71361 T^{6} - 21573 T^{7} + 23103 T^{8} - 4167 T^{9} + 3600 T^{10} - 423 T^{11} + 405 T^{12} - 24 T^{13} + 24 T^{14} + T^{16}$$
$7$ $$5764801 - 1647086 T + 705894 T^{2} - 319333 T^{3} + 314531 T^{4} - 72030 T^{5} + 41209 T^{6} - 13517 T^{7} + 8451 T^{8} - 1931 T^{9} + 841 T^{10} - 210 T^{11} + 131 T^{12} - 19 T^{13} + 6 T^{14} - 2 T^{15} + T^{16}$$
$11$ $$26244 + 13122 T - 269001 T^{2} - 135594 T^{3} + 2920374 T^{4} - 4745790 T^{5} + 3147093 T^{6} - 725355 T^{7} - 137538 T^{8} + 80433 T^{9} + 7047 T^{10} - 8019 T^{11} + 711 T^{12} + 234 T^{13} - 27 T^{14} - 6 T^{15} + T^{16}$$
$13$ $$3337929 + 15159798 T^{2} + 15881103 T^{4} + 6461154 T^{6} + 1147599 T^{8} + 102429 T^{10} + 4788 T^{12} + 111 T^{14} + T^{16}$$
$17$ $$13549761 + 35746191 T + 94137876 T^{2} + 25350003 T^{3} + 32433372 T^{4} - 1254096 T^{5} + 8181405 T^{6} - 690741 T^{7} + 912402 T^{8} - 115920 T^{9} + 73566 T^{10} - 8910 T^{11} + 3321 T^{12} - 420 T^{13} + 105 T^{14} - 9 T^{15} + T^{16}$$
$19$ $$2099601 + 29655234 T + 166640004 T^{2} + 381649968 T^{3} + 320108679 T^{4} - 85622103 T^{5} - 40684761 T^{6} + 9517095 T^{7} + 4276089 T^{8} - 316089 T^{9} - 188478 T^{10} + 7974 T^{11} + 6156 T^{12} - 93 T^{14} + T^{16}$$
$23$ $$15198451524 - 14629258530 T + 2044217331 T^{2} + 2550348180 T^{3} - 604378908 T^{4} - 340227216 T^{5} + 112333068 T^{6} + 20560716 T^{7} - 7612947 T^{8} - 1006263 T^{9} + 350730 T^{10} + 37422 T^{11} - 7767 T^{12} - 1008 T^{13} + 99 T^{14} + 21 T^{15} + T^{16}$$
$29$ $$15752961 + 369712350 T^{2} + 2176008867 T^{4} + 525653469 T^{6} + 44217981 T^{8} + 1677753 T^{10} + 31626 T^{12} + 288 T^{14} + T^{16}$$
$31$ $$3910251024 + 4355979120 T - 2625353532 T^{2} - 4726500660 T^{3} + 4449942117 T^{4} - 478336266 T^{5} - 309231837 T^{6} + 48203910 T^{7} + 16995015 T^{8} - 2259819 T^{9} - 490239 T^{10} + 57834 T^{11} + 11187 T^{12} - 792 T^{13} - 120 T^{14} + 6 T^{15} + T^{16}$$
$37$ $$52765696 + 3719168 T + 115382016 T^{2} + 5193472 T^{3} + 232814336 T^{4} + 12859776 T^{5} + 40260736 T^{6} + 1907864 T^{7} + 5129568 T^{8} + 175166 T^{9} + 263287 T^{10} - 10395 T^{11} + 8996 T^{12} - 167 T^{13} + 108 T^{14} - T^{15} + T^{16}$$
$41$ $$( 302733 + 9954 T - 120663 T^{2} - 15975 T^{3} + 7299 T^{4} + 615 T^{5} - 150 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$43$ $$( 167359 - 7598 T - 100592 T^{2} + 5131 T^{3} + 10921 T^{4} + 151 T^{5} - 203 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$47$ $$1971620372736 - 2703471458688 T + 2461461067440 T^{2} - 1330775271864 T^{3} + 540676969353 T^{4} - 148161520953 T^{5} + 32201307486 T^{6} - 4775169699 T^{7} + 712315611 T^{8} - 79062246 T^{9} + 10379754 T^{10} - 846819 T^{11} + 84231 T^{12} - 4884 T^{13} + 438 T^{14} - 18 T^{15} + T^{16}$$
$53$ $$531441 + 18068994 T + 191318760 T^{2} - 457747848 T^{3} + 292233501 T^{4} + 102922407 T^{5} - 60026589 T^{6} - 16579647 T^{7} + 9705177 T^{8} + 1451439 T^{9} - 466074 T^{10} - 56376 T^{11} + 19764 T^{12} - 153 T^{14} + T^{16}$$
$59$ $$165574120464 - 266986987488 T + 484571771388 T^{2} + 11513862288 T^{3} + 77431736565 T^{4} - 9743030109 T^{5} + 7073934444 T^{6} - 677695707 T^{7} + 309484062 T^{8} - 42107274 T^{9} + 8617212 T^{10} - 720099 T^{11} + 79218 T^{12} - 3882 T^{13} + 393 T^{14} - 15 T^{15} + T^{16}$$
$61$ $$1475481744 - 5807894400 T + 9308572164 T^{2} - 6644786400 T^{3} + 1891535733 T^{4} + 166075839 T^{5} - 176063652 T^{6} + 1463967 T^{7} + 11523348 T^{8} - 569754 T^{9} - 366912 T^{10} + 19287 T^{11} + 8784 T^{12} - 342 T^{13} - 111 T^{14} + 3 T^{15} + T^{16}$$
$67$ $$2114953586944 + 2140118586496 T + 2010327590448 T^{2} + 427777738280 T^{3} + 163978476161 T^{4} + 24180967287 T^{5} + 7944661591 T^{6} + 951833224 T^{7} + 231304086 T^{8} + 20606821 T^{9} + 4411018 T^{10} + 318924 T^{11} + 50012 T^{12} + 1859 T^{13} + 270 T^{14} + 7 T^{15} + T^{16}$$
$71$ $$780959242139904 + 134744717006208 T^{2} + 7956570857364 T^{4} + 234856231407 T^{6} + 3949834995 T^{8} + 39561129 T^{10} + 233316 T^{12} + 747 T^{14} + T^{16}$$
$73$ $$7523023152969 + 2752955922474 T - 1191205799235 T^{2} - 558789791238 T^{3} + 238887968904 T^{4} + 213060248478 T^{5} + 62170270956 T^{6} + 8564672457 T^{7} + 317278665 T^{8} - 59680611 T^{9} - 5521941 T^{10} + 432909 T^{11} + 71235 T^{12} - 306 T^{14} + T^{16}$$
$79$ $$10549504 + 14992768 T + 30606480 T^{2} + 29157800 T^{3} + 44789945 T^{4} + 38599749 T^{5} + 35635933 T^{6} + 16316434 T^{7} + 7394544 T^{8} + 943669 T^{9} + 497152 T^{10} + 68454 T^{11} + 18914 T^{12} + 779 T^{13} + 144 T^{14} + T^{15} + T^{16}$$
$83$ $$( -818037 - 2519676 T - 703134 T^{2} + 80568 T^{3} + 28188 T^{4} - 636 T^{5} - 330 T^{6} + T^{8} )^{2}$$
$89$ $$7161826993281 - 1051545832029 T + 1294382305422 T^{2} - 236676836007 T^{3} + 180123305796 T^{4} - 30425509008 T^{5} + 9647366571 T^{6} - 1422741915 T^{7} + 335911536 T^{8} - 43169922 T^{9} + 6642648 T^{10} - 611550 T^{11} + 69741 T^{12} - 5130 T^{13} + 459 T^{14} - 21 T^{15} + T^{16}$$
$97$ $$22864161681 + 494765680599 T^{2} + 1271094188526 T^{4} + 87910050978 T^{6} + 2409676065 T^{8} + 32778594 T^{10} + 229563 T^{12} + 777 T^{14} + T^{16}$$
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