# Properties

 Label 252.2.w.a Level $252$ Weight $2$ Character orbit 252.w Analytic conductor $2.012$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 252.w (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{10} q^{3} + \beta_{7} q^{5} + ( \beta_{3} - \beta_{8} ) q^{7} -\beta_{4} q^{9} +O(q^{10})$$ $$q -\beta_{10} q^{3} + \beta_{7} q^{5} + ( \beta_{3} - \beta_{8} ) q^{7} -\beta_{4} q^{9} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{11} + ( -\beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{13} + ( -1 + \beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{15} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{17} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{19} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{21} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{23} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{15} ) q^{25} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{29} + ( \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{31} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{33} + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{35} + ( -\beta_{6} - \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{37} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{39} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{15} ) q^{43} + ( -1 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{12} - \beta_{14} ) q^{45} + ( -2 - 3 \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{47} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{49} + ( -3 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{51} + ( 2 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{53} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{55} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{57} + ( -2 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{59} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{11} + \beta_{15} ) q^{61} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{63} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{15} ) q^{65} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{13} + 2 \beta_{15} ) q^{67} + ( -\beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{14} - 5 \beta_{15} ) q^{69} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{71} + ( \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{73} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{75} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} ) q^{77} + ( 1 - \beta_{1} + \beta_{6} + 3 \beta_{7} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} ) q^{79} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{81} + ( 1 - 3 \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{83} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{85} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 4 \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{87} + ( 2 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 5 \beta_{15} ) q^{89} + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{91} + ( -1 - 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} + 3 \beta_{9} - 3 \beta_{11} - \beta_{12} - 3 \beta_{14} + 5 \beta_{15} ) q^{93} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + 3 \beta_{14} - 6 \beta_{15} ) q^{95} + ( -\beta_{1} + 5 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 6 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{97} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - q^{7} + 6q^{9} + O(q^{10})$$ $$16q - q^{7} + 6q^{9} - 6q^{11} - 3q^{13} - 3q^{15} + 9q^{17} + 6q^{21} + 21q^{23} - 8q^{25} + 9q^{27} + 6q^{29} - 15q^{35} + q^{37} - 3q^{39} - 6q^{41} - 2q^{43} - 30q^{45} - 36q^{47} - 5q^{49} - 33q^{51} + 15q^{57} - 30q^{59} - 15q^{63} + 14q^{67} + 21q^{69} - 57q^{75} + 3q^{77} + 2q^{79} + 18q^{81} + 6q^{85} + 48q^{87} + 21q^{89} + 9q^{91} + 21q^{93} - 3q^{97} - 9q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} - 113602 \nu^{9} - 7210 \nu^{8} - 181946 \nu^{7} + 84090 \nu^{6} - 1174032 \nu^{5} + 900801 \nu^{4} - 2054484 \nu^{3} + 11094408 \nu^{2} + 4573017 \nu + 19166868$$$$)/621108$$ $$\beta_{2}$$ $$=$$ $$($$$$3695 \nu^{15} - 20725 \nu^{14} + 51544 \nu^{13} - 99223 \nu^{12} + 215537 \nu^{11} - 360098 \nu^{10} + 187876 \nu^{9} - 298928 \nu^{8} + 711356 \nu^{7} - 844320 \nu^{6} + 2586978 \nu^{5} - 9488205 \nu^{4} + 18766647 \nu^{3} - 20620980 \nu^{2} + 33241671 \nu - 46300977$$$$)/1242216$$ $$\beta_{3}$$ $$=$$ $$($$$$-6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + 11630 \nu^{10} - 101824 \nu^{9} + 277628 \nu^{8} - 214640 \nu^{7} + 202764 \nu^{6} - 3624714 \nu^{5} + 7713063 \nu^{4} - 3514995 \nu^{3} + 15997176 \nu^{2} - 16161201 \nu - 5872095$$$$)/1242216$$ $$\beta_{4}$$ $$=$$ $$($$$$-818 \nu^{15} - 2453 \nu^{14} - 9016 \nu^{13} - 10490 \nu^{12} - 12167 \nu^{11} + 10856 \nu^{10} + 25058 \nu^{9} + 37994 \nu^{8} + 39934 \nu^{7} + 148242 \nu^{6} + 64386 \nu^{5} - 91530 \nu^{4} - 1898397 \nu^{3} - 3112344 \nu^{2} - 4586868 \nu - 2858409$$$$)/103518$$ $$\beta_{5}$$ $$=$$ $$($$$$4659 \nu^{15} - 5459 \nu^{14} + 29980 \nu^{13} - 59671 \nu^{12} + 62083 \nu^{11} - 182738 \nu^{10} + 119048 \nu^{9} - 112444 \nu^{8} + 265832 \nu^{7} - 648764 \nu^{6} + 2006262 \nu^{5} - 2722977 \nu^{4} + 10424025 \nu^{3} - 11026368 \nu^{2} + 9412119 \nu - 25313067$$$$)/414072$$ $$\beta_{6}$$ $$=$$ $$($$$$-7279 \nu^{15} + 11960 \nu^{14} - 9158 \nu^{13} + 168785 \nu^{12} + 36326 \nu^{11} + 302098 \nu^{10} - 390434 \nu^{9} - 30734 \nu^{8} - 575494 \nu^{7} + 625362 \nu^{6} - 4453344 \nu^{5} + 1286793 \nu^{4} - 9909540 \nu^{3} + 36051966 \nu^{2} + 18445887 \nu + 70277058$$$$)/621108$$ $$\beta_{7}$$ $$=$$ $$($$$$-13862 \nu^{15} + 1333 \nu^{14} - 45760 \nu^{13} + 164782 \nu^{12} + 15775 \nu^{11} + 343040 \nu^{10} - 361210 \nu^{9} + 100070 \nu^{8} - 284726 \nu^{7} + 1370658 \nu^{6} - 5698206 \nu^{5} + 63018 \nu^{4} - 18020475 \nu^{3} + 29851092 \nu^{2} + 7676370 \nu + 61443765$$$$)/621108$$ $$\beta_{8}$$ $$=$$ $$($$$$27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} - 442826 \nu^{10} + 157996 \nu^{9} - 326936 \nu^{8} - 426532 \nu^{7} - 3257256 \nu^{6} + 5542434 \nu^{5} + 10347075 \nu^{4} + 36196875 \nu^{3} + 5383908 \nu^{2} + 7598367 \nu - 47062053$$$$)/1242216$$ $$\beta_{9}$$ $$=$$ $$($$$$-7241 \nu^{15} + 4357 \nu^{14} - 32353 \nu^{13} + 38470 \nu^{12} - 78698 \nu^{11} + 72614 \nu^{10} - 29722 \nu^{9} + 229040 \nu^{8} - 92132 \nu^{7} + 637584 \nu^{6} - 2307114 \nu^{5} + 2838753 \nu^{4} - 8018433 \nu^{3} + 3449385 \nu^{2} - 14505642 \nu + 1180980$$$$)/310554$$ $$\beta_{10}$$ $$=$$ $$($$$$5228 \nu^{15} + 1525 \nu^{14} + 20418 \nu^{13} - 46974 \nu^{12} - 5255 \nu^{11} - 122904 \nu^{10} + 100970 \nu^{9} - 49402 \nu^{8} + 85518 \nu^{7} - 572506 \nu^{6} + 1811682 \nu^{5} + 307476 \nu^{4} + 7123329 \nu^{3} - 7628742 \nu^{2} - 1004562 \nu - 19887849$$$$)/207036$$ $$\beta_{11}$$ $$=$$ $$($$$$16952 \nu^{15} - 9175 \nu^{14} + 73804 \nu^{13} - 123904 \nu^{12} + 128807 \nu^{11} - 247964 \nu^{10} + 166462 \nu^{9} - 398066 \nu^{8} + 282422 \nu^{7} - 1677798 \nu^{6} + 5939262 \nu^{5} - 5124384 \nu^{4} + 20396205 \nu^{3} - 16524972 \nu^{2} + 21030192 \nu - 23648031$$$$)/621108$$ $$\beta_{12}$$ $$=$$ $$($$$$11445 \nu^{15} - 10925 \nu^{14} + 48460 \nu^{13} - 170497 \nu^{12} + 54037 \nu^{11} - 403262 \nu^{10} + 367448 \nu^{9} - 147196 \nu^{8} + 581144 \nu^{7} - 1306628 \nu^{6} + 5344746 \nu^{5} - 3588687 \nu^{4} + 19817487 \nu^{3} - 33430320 \nu^{2} + 2317977 \nu - 69748533$$$$)/414072$$ $$\beta_{13}$$ $$=$$ $$($$$$19793 \nu^{15} + 1430 \nu^{14} + 108688 \nu^{13} - 147133 \nu^{12} + 121322 \nu^{11} - 529214 \nu^{10} + 286834 \nu^{9} - 407870 \nu^{8} + 442766 \nu^{7} - 2453682 \nu^{6} + 6290388 \nu^{5} - 3336147 \nu^{4} + 34358094 \nu^{3} - 21512304 \nu^{2} + 26451765 \nu - 66410442$$$$)/621108$$ $$\beta_{14}$$ $$=$$ $$($$$$59447 \nu^{15} + 8969 \nu^{14} + 284176 \nu^{13} - 415519 \nu^{12} + 220139 \nu^{11} - 1262618 \nu^{10} + 758536 \nu^{9} - 1010876 \nu^{8} + 904208 \nu^{7} - 6809388 \nu^{6} + 18944982 \nu^{5} - 4839237 \nu^{4} + 86642541 \nu^{3} - 56752164 \nu^{2} + 46993527 \nu - 158830875$$$$)/1242216$$ $$\beta_{15}$$ $$=$$ $$($$$$24909 \nu^{15} - 6205 \nu^{14} + 135476 \nu^{13} - 180977 \nu^{12} + 218117 \nu^{11} - 566494 \nu^{10} + 299296 \nu^{9} - 605132 \nu^{8} + 530200 \nu^{7} - 2915308 \nu^{6} + 8256786 \nu^{5} - 7163775 \nu^{4} + 40104855 \nu^{3} - 24898752 \nu^{2} + 41926977 \nu - 60703101$$$$)/414072$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{12} - 3 \beta_{10} - \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{1} + 4$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$6 \beta_{15} - \beta_{13} + 2 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 7 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + \beta_{1} + 7$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{15} + 5 \beta_{14} - 7 \beta_{13} + 9 \beta_{12} - 3 \beta_{11} + 10 \beta_{10} - 16 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 10 \beta_{3} + 8 \beta_{2} + 12 \beta_{1} - 4$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{15} + 7 \beta_{14} + 24 \beta_{13} - 25 \beta_{12} - 12 \beta_{11} - 25 \beta_{10} - 3 \beta_{9} - 18 \beta_{8} - 23 \beta_{7} - 30 \beta_{6} + \beta_{5} - 6 \beta_{4} - \beta_{3} - 13 \beta_{2} + 31 \beta_{1} - 13$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$35 \beta_{15} + 4 \beta_{14} - 29 \beta_{13} - 8 \beta_{12} - 27 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 20 \beta_{7} + \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 17 \beta_{3} - 33 \beta_{2} - 82 \beta_{1} + 20$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$73 \beta_{15} - 51 \beta_{14} - 30 \beta_{13} + 4 \beta_{12} - 63 \beta_{11} + 52 \beta_{10} - 24 \beta_{9} - 6 \beta_{8} + 23 \beta_{7} + 21 \beta_{6} + 65 \beta_{5} + 61 \beta_{4} + 39 \beta_{3} - 94 \beta_{2} + 59 \beta_{1} - 26$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-132 \beta_{15} + 4 \beta_{14} - 100 \beta_{13} + 15 \beta_{12} + 48 \beta_{11} + 61 \beta_{10} - 49 \beta_{9} + 90 \beta_{8} + 3 \beta_{7} - 37 \beta_{6} + 235 \beta_{5} - 110 \beta_{4} - 97 \beta_{3} + 85 \beta_{2} + 72 \beta_{1} - 16$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$79 \beta_{15} + 41 \beta_{14} - 17 \beta_{13} - 71 \beta_{12} + 15 \beta_{11} - 174 \beta_{10} + 82 \beta_{9} + 63 \beta_{8} + 17 \beta_{7} + 19 \beta_{6} + 49 \beta_{5} + 36 \beta_{4} + 10 \beta_{3} - 113 \beta_{2} - 472 \beta_{1} - 463$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-5 \beta_{15} + 275 \beta_{14} - 241 \beta_{13} + 185 \beta_{12} - 225 \beta_{11} - 63 \beta_{10} - 28 \beta_{9} + 120 \beta_{8} + 118 \beta_{7} + 197 \beta_{6} - 142 \beta_{5} + 121 \beta_{4} - 152 \beta_{3} + 177 \beta_{2} - 113 \beta_{1} - 80$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$($$$$-995 \beta_{15} + 438 \beta_{14} - 26 \beta_{13} + 764 \beta_{12} + 51 \beta_{11} + 139 \beta_{10} - 452 \beta_{9} - 39 \beta_{8} + 4 \beta_{7} + 331 \beta_{6} + 174 \beta_{5} - 547 \beta_{4} - 387 \beta_{3} + 397 \beta_{2} + 727 \beta_{1} - 198$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$376 \beta_{15} - 4 \beta_{14} + 276 \beta_{13} - 699 \beta_{12} + 408 \beta_{11} - 924 \beta_{10} + 825 \beta_{9} - 129 \beta_{8} - 189 \beta_{7} - 1467 \beta_{6} - 393 \beta_{5} - 625 \beta_{4} - 476 \beta_{3} - 295 \beta_{2} + 558 \beta_{1} - 1935$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$453 \beta_{15} + 679 \beta_{14} - 1199 \beta_{13} - 40 \beta_{12} - 1635 \beta_{11} + 2518 \beta_{10} - 287 \beta_{9} - 732 \beta_{8} + 1228 \beta_{7} + 1309 \beta_{6} - 150 \beta_{5} - 336 \beta_{4} - 781 \beta_{3} - 253 \beta_{2} - 4148 \beta_{1} - 1881$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$1349 \beta_{15} - 1166 \beta_{14} + 1447 \beta_{13} + 472 \beta_{12} - 1887 \beta_{11} + 861 \beta_{10} + 454 \beta_{9} + 1194 \beta_{8} + 2750 \beta_{7} - 320 \beta_{6} + 949 \beta_{5} + 2555 \beta_{4} - 2791 \beta_{3} - 2679 \beta_{2} + 3770 \beta_{1} + 611$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1 + \beta_{1}$$ $$1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 1.08696 + 1.34852i −1.61108 + 0.635951i 1.68042 + 0.419752i −0.268067 − 1.71118i −0.213160 − 1.71888i 1.68124 − 0.416458i −0.811340 + 1.53027i −0.544978 + 1.64408i 1.08696 − 1.34852i −1.61108 − 0.635951i 1.68042 − 0.419752i −0.268067 + 1.71118i −0.213160 + 1.71888i 1.68124 + 0.416458i −0.811340 − 1.53027i −0.544978 − 1.64408i
0 −1.63336 + 0.576322i 0 0.0382122 + 0.0661855i 0 0.232935 2.63548i 0 2.33571 1.88268i 0
5.2 0 −1.43204 0.974295i 0 1.09150 + 1.89054i 0 −1.25859 + 2.32722i 0 1.10150 + 2.79047i 0
5.3 0 −1.36511 + 1.06606i 0 −1.48494 2.57199i 0 −0.200279 + 2.63816i 0 0.727031 2.91057i 0
5.4 0 −0.134439 1.72683i 0 −0.842869 1.45989i 0 −2.27938 1.34329i 0 −2.96385 + 0.464306i 0
5.5 0 0.106783 + 1.72876i 0 1.43402 + 2.48379i 0 2.56899 0.632668i 0 −2.97719 + 0.369204i 0
5.6 0 1.06740 1.36406i 0 0.349828 + 0.605920i 0 2.48683 + 0.903137i 0 −0.721326 2.91199i 0
5.7 0 1.68085 + 0.418028i 0 1.37166 + 2.37578i 0 −2.60476 + 0.463945i 0 2.65051 + 1.40528i 0
5.8 0 1.70992 + 0.276016i 0 −1.95741 3.39033i 0 0.554241 2.58705i 0 2.84763 + 0.943929i 0
101.1 0 −1.63336 0.576322i 0 0.0382122 0.0661855i 0 0.232935 + 2.63548i 0 2.33571 + 1.88268i 0
101.2 0 −1.43204 + 0.974295i 0 1.09150 1.89054i 0 −1.25859 2.32722i 0 1.10150 2.79047i 0
101.3 0 −1.36511 1.06606i 0 −1.48494 + 2.57199i 0 −0.200279 2.63816i 0 0.727031 + 2.91057i 0
101.4 0 −0.134439 + 1.72683i 0 −0.842869 + 1.45989i 0 −2.27938 + 1.34329i 0 −2.96385 0.464306i 0
101.5 0 0.106783 1.72876i 0 1.43402 2.48379i 0 2.56899 + 0.632668i 0 −2.97719 0.369204i 0
101.6 0 1.06740 + 1.36406i 0 0.349828 0.605920i 0 2.48683 0.903137i 0 −0.721326 + 2.91199i 0
101.7 0 1.68085 0.418028i 0 1.37166 2.37578i 0 −2.60476 0.463945i 0 2.65051 1.40528i 0
101.8 0 1.70992 0.276016i 0 −1.95741 + 3.39033i 0 0.554241 + 2.58705i 0 2.84763 0.943929i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a 16 1.a even 1 1 trivial
252.2.w.a 16 63.i even 6 1 inner
252.2.bm.a yes 16 7.d odd 6 1
252.2.bm.a yes 16 9.d odd 6 1
756.2.w.a 16 3.b odd 2 1
756.2.w.a 16 63.t odd 6 1
756.2.bm.a 16 9.c even 3 1
756.2.bm.a 16 21.g even 6 1
1008.2.ca.d 16 4.b odd 2 1
1008.2.ca.d 16 252.r odd 6 1
1008.2.df.d 16 28.f even 6 1
1008.2.df.d 16 36.h even 6 1
1764.2.w.b 16 7.b odd 2 1
1764.2.w.b 16 63.j odd 6 1
1764.2.x.a 16 7.c even 3 1
1764.2.x.a 16 63.s even 6 1
1764.2.x.b 16 7.d odd 6 1
1764.2.x.b 16 63.n odd 6 1
1764.2.bm.a 16 7.c even 3 1
1764.2.bm.a 16 63.o even 6 1
2268.2.t.a 16 9.d odd 6 1
2268.2.t.a 16 63.k odd 6 1
2268.2.t.b 16 9.c even 3 1
2268.2.t.b 16 63.s even 6 1
3024.2.ca.d 16 12.b even 2 1
3024.2.ca.d 16 252.bj even 6 1
3024.2.df.d 16 36.f odd 6 1
3024.2.df.d 16 84.j odd 6 1
5292.2.w.b 16 21.c even 2 1
5292.2.w.b 16 63.h even 3 1
5292.2.x.a 16 21.h odd 6 1
5292.2.x.a 16 63.k odd 6 1
5292.2.x.b 16 21.g even 6 1
5292.2.x.b 16 63.g even 3 1
5292.2.bm.a 16 21.h odd 6 1
5292.2.bm.a 16 63.l odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$6561 - 2187 T^{2} - 729 T^{3} - 243 T^{5} + 324 T^{6} + 54 T^{7} - 72 T^{8} + 18 T^{9} + 36 T^{10} - 9 T^{11} - 3 T^{13} - 3 T^{14} + T^{16}$$
$5$ $$324 - 4698 T + 62289 T^{2} - 89424 T^{3} + 143289 T^{4} - 45738 T^{5} + 71361 T^{6} - 21573 T^{7} + 23103 T^{8} - 4167 T^{9} + 3600 T^{10} - 423 T^{11} + 405 T^{12} - 24 T^{13} + 24 T^{14} + T^{16}$$
$7$ $$5764801 + 823543 T + 352947 T^{2} - 117649 T^{3} - 196882 T^{4} - 48363 T^{5} - 980 T^{6} + 7777 T^{7} + 5328 T^{8} + 1111 T^{9} - 20 T^{10} - 141 T^{11} - 82 T^{12} - 7 T^{13} + 3 T^{14} + T^{15} + T^{16}$$
$11$ $$26244 - 13122 T - 269001 T^{2} + 135594 T^{3} + 2920374 T^{4} + 4745790 T^{5} + 3147093 T^{6} + 725355 T^{7} - 137538 T^{8} - 80433 T^{9} + 7047 T^{10} + 8019 T^{11} + 711 T^{12} - 234 T^{13} - 27 T^{14} + 6 T^{15} + T^{16}$$
$13$ $$3337929 - 17462466 T + 38097783 T^{2} - 40000230 T^{3} + 18330786 T^{4} + 831303 T^{5} - 3499173 T^{6} + 321543 T^{7} + 494064 T^{8} - 66258 T^{9} - 35379 T^{10} + 4608 T^{11} + 1980 T^{12} - 162 T^{13} - 51 T^{14} + 3 T^{15} + T^{16}$$
$17$ $$13549761 + 35746191 T + 94137876 T^{2} + 25350003 T^{3} + 32433372 T^{4} - 1254096 T^{5} + 8181405 T^{6} - 690741 T^{7} + 912402 T^{8} - 115920 T^{9} + 73566 T^{10} - 8910 T^{11} + 3321 T^{12} - 420 T^{13} + 105 T^{14} - 9 T^{15} + T^{16}$$
$19$ $$2099601 + 29655234 T + 166640004 T^{2} + 381649968 T^{3} + 320108679 T^{4} - 85622103 T^{5} - 40684761 T^{6} + 9517095 T^{7} + 4276089 T^{8} - 316089 T^{9} - 188478 T^{10} + 7974 T^{11} + 6156 T^{12} - 93 T^{14} + T^{16}$$
$23$ $$15198451524 + 14629258530 T + 2044217331 T^{2} - 2550348180 T^{3} - 604378908 T^{4} + 340227216 T^{5} + 112333068 T^{6} - 20560716 T^{7} - 7612947 T^{8} + 1006263 T^{9} + 350730 T^{10} - 37422 T^{11} - 7767 T^{12} + 1008 T^{13} + 99 T^{14} - 21 T^{15} + T^{16}$$
$29$ $$15752961 - 9001692 T - 182284263 T^{2} + 105142212 T^{3} + 2135730888 T^{4} + 464916834 T^{5} - 210471048 T^{6} - 53675541 T^{7} + 20028303 T^{8} + 2348109 T^{9} - 577287 T^{10} - 58239 T^{11} + 12753 T^{12} + 828 T^{13} - 126 T^{14} - 6 T^{15} + T^{16}$$
$31$ $$3910251024 + 10103222664 T^{2} + 4911390369 T^{4} + 724329054 T^{6} + 47767077 T^{8} + 1632609 T^{10} + 29898 T^{12} + 276 T^{14} + T^{16}$$
$37$ $$52765696 + 3719168 T + 115382016 T^{2} + 5193472 T^{3} + 232814336 T^{4} + 12859776 T^{5} + 40260736 T^{6} + 1907864 T^{7} + 5129568 T^{8} + 175166 T^{9} + 263287 T^{10} - 10395 T^{11} + 8996 T^{12} - 167 T^{13} + 108 T^{14} - T^{15} + T^{16}$$
$41$ $$91647269289 - 3013404282 T + 36627754095 T^{2} - 8471239848 T^{3} + 12508926552 T^{4} - 1968463728 T^{5} + 1038978252 T^{6} - 28504467 T^{7} + 44578395 T^{8} - 430317 T^{9} + 1135899 T^{10} + 20637 T^{11} + 18891 T^{12} + 330 T^{13} + 186 T^{14} + 6 T^{15} + T^{16}$$
$43$ $$28009034881 + 1271593682 T + 16892706132 T^{2} + 953140042 T^{3} + 8330008163 T^{4} + 324910827 T^{5} + 1058091937 T^{6} - 87622111 T^{7} + 97936317 T^{8} - 3925843 T^{9} + 2048842 T^{10} - 18162 T^{11} + 30590 T^{12} - 104 T^{13} + 207 T^{14} + 2 T^{15} + T^{16}$$
$47$ $$( 1404144 + 1925352 T + 887031 T^{2} + 134271 T^{3} - 8811 T^{4} - 3468 T^{5} - 114 T^{6} + 18 T^{7} + T^{8} )^{2}$$
$53$ $$531441 + 18068994 T + 191318760 T^{2} - 457747848 T^{3} + 292233501 T^{4} + 102922407 T^{5} - 60026589 T^{6} - 16579647 T^{7} + 9705177 T^{8} + 1451439 T^{9} - 466074 T^{10} - 56376 T^{11} + 19764 T^{12} - 153 T^{14} + T^{16}$$
$59$ $$( -406908 - 656136 T + 132849 T^{2} + 92961 T^{3} - 2979 T^{4} - 3201 T^{5} - 168 T^{6} + 15 T^{7} + T^{8} )^{2}$$
$61$ $$1475481744 + 4244295672 T^{2} + 2010944961 T^{4} + 341040105 T^{6} + 25911891 T^{8} + 1015731 T^{10} + 21420 T^{12} + 231 T^{14} + T^{16}$$
$67$ $$( -1454288 + 1471592 T - 106757 T^{2} - 93061 T^{3} + 10750 T^{4} + 1703 T^{5} - 221 T^{6} - 7 T^{7} + T^{8} )^{2}$$
$71$ $$780959242139904 + 134744717006208 T^{2} + 7956570857364 T^{4} + 234856231407 T^{6} + 3949834995 T^{8} + 39561129 T^{10} + 233316 T^{12} + 747 T^{14} + T^{16}$$
$73$ $$7523023152969 + 2752955922474 T - 1191205799235 T^{2} - 558789791238 T^{3} + 238887968904 T^{4} + 213060248478 T^{5} + 62170270956 T^{6} + 8564672457 T^{7} + 317278665 T^{8} - 59680611 T^{9} - 5521941 T^{10} + 432909 T^{11} + 71235 T^{12} - 306 T^{14} + T^{16}$$
$79$ $$( -3248 + 4616 T + 2863 T^{2} - 6523 T^{3} + 1996 T^{4} + 461 T^{5} - 143 T^{6} - T^{7} + T^{8} )^{2}$$
$83$ $$669184533369 - 2061188196012 T + 5773577517018 T^{2} - 1903485074616 T^{3} + 720461504880 T^{4} - 85919425596 T^{5} + 25248534300 T^{6} - 2208157416 T^{7} + 614588409 T^{8} - 32727636 T^{9} + 8300268 T^{10} - 290448 T^{11} + 80712 T^{12} - 1272 T^{13} + 330 T^{14} + T^{16}$$
$89$ $$7161826993281 - 1051545832029 T + 1294382305422 T^{2} - 236676836007 T^{3} + 180123305796 T^{4} - 30425509008 T^{5} + 9647366571 T^{6} - 1422741915 T^{7} + 335911536 T^{8} - 43169922 T^{9} + 6642648 T^{10} - 611550 T^{11} + 69741 T^{12} - 5130 T^{13} + 459 T^{14} - 21 T^{15} + T^{16}$$
$97$ $$22864161681 - 104576900445 T - 8224102287 T^{2} + 766864072890 T^{3} + 1208668941423 T^{4} - 71751605004 T^{5} - 41675283648 T^{6} + 2243025486 T^{7} + 1044816804 T^{8} - 41968233 T^{9} - 12357171 T^{10} + 394884 T^{11} + 109872 T^{12} - 1161 T^{13} - 384 T^{14} + 3 T^{15} + T^{16}$$