# Properties

 Label 6027.2.a.bh Level 6027 Weight 2 Character orbit 6027.a Self dual Yes Analytic conductor 48.126 Analytic rank 0 Dimension 13 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6027 = 3 \cdot 7^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6027.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$48.1258372982$$ Analytic rank: $$0$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{12}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{8} ) q^{5} + \beta_{1} q^{6} + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{8} ) q^{5} + \beta_{1} q^{6} + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{8} + q^{9} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + ( 1 + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{13} + ( -1 + \beta_{8} ) q^{15} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{16} + ( 2 + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} ) q^{17} -\beta_{1} q^{18} + ( -1 + \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{19} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{22} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{23} + ( \beta_{1} + \beta_{8} + \beta_{9} ) q^{24} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{11} ) q^{25} + ( 3 - 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{26} - q^{27} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{30} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{11} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} ) q^{32} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{10} + \beta_{12} ) q^{33} + ( -\beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( 2 - \beta_{1} - \beta_{3} - \beta_{6} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{37} + ( 1 - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{38} + ( -1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{39} + ( 4 - \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} ) q^{40} - q^{41} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{43} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{44} + ( 1 - \beta_{8} ) q^{45} + ( -1 + 4 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{46} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{47} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{48} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{50} + ( -2 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{51} + ( 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{8} - \beta_{9} + 3 \beta_{11} + 3 \beta_{12} ) q^{52} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{53} + \beta_{1} q^{54} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{55} + ( 1 - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{57} + ( 7 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 6 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{58} + ( 2 \beta_{1} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{59} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{60} + ( 5 - 3 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{61} + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{62} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} ) q^{64} + ( -1 - \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{65} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{11} + \beta_{12} ) q^{66} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{10} + \beta_{12} ) q^{67} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{12} ) q^{69} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{8} - 3 \beta_{9} + 2 \beta_{12} ) q^{71} + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{72} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{73} + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{74} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} + \beta_{11} ) q^{75} + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{76} + ( -3 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{78} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{9} ) q^{79} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{80} + q^{81} + \beta_{1} q^{82} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{11} ) q^{83} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{85} + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} + 3 \beta_{12} ) q^{86} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( 6 - \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{90} + ( -7 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{10} - \beta_{11} + 3 \beta_{12} ) q^{92} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{11} ) q^{93} + ( 2 - 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} + \beta_{10} - 4 \beta_{11} - 4 \beta_{12} ) q^{94} + ( -2 + 5 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{95} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{96} + ( 3 - \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{97} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$13q - 4q^{2} - 13q^{3} + 12q^{4} + 8q^{5} + 4q^{6} - 12q^{8} + 13q^{9} + O(q^{10})$$ $$13q - 4q^{2} - 13q^{3} + 12q^{4} + 8q^{5} + 4q^{6} - 12q^{8} + 13q^{9} + q^{10} - 10q^{11} - 12q^{12} + 16q^{13} - 8q^{15} + 26q^{16} + 12q^{17} - 4q^{18} + 11q^{19} + 6q^{20} + q^{22} - 15q^{23} + 12q^{24} + 15q^{25} + 18q^{26} - 13q^{27} - 8q^{29} - q^{30} + 9q^{31} - 23q^{32} + 10q^{33} - 7q^{34} + 12q^{36} - 2q^{37} + 20q^{38} - 16q^{39} + 49q^{40} - 13q^{41} - 7q^{43} - 22q^{44} + 8q^{45} - 4q^{46} + 26q^{47} - 26q^{48} - 15q^{50} - 12q^{51} + 24q^{52} + 4q^{53} + 4q^{54} + q^{55} - 11q^{57} + 39q^{58} - 3q^{59} - 6q^{60} + 28q^{61} + 7q^{62} + 2q^{64} - 20q^{65} - q^{66} + 7q^{67} + 55q^{68} + 15q^{69} - 40q^{71} - 12q^{72} - 2q^{73} + q^{74} - 15q^{75} - 26q^{76} - 18q^{78} + 13q^{79} + 22q^{80} + 13q^{81} + 4q^{82} + 14q^{83} + 48q^{85} - 49q^{86} + 8q^{87} + 20q^{88} + 35q^{89} + q^{90} - 105q^{92} - 9q^{93} - 2q^{94} + 7q^{95} + 23q^{96} + 64q^{97} - 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{13} - 4 x^{12} - 11 x^{11} + 56 x^{10} + 26 x^{9} - 263 x^{8} + 50 x^{7} + 478 x^{6} - 174 x^{5} - 311 x^{4} + 84 x^{3} + 69 x^{2} - 12 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$-105 \nu^{12} + 402 \nu^{11} + 1471 \nu^{10} - 6122 \nu^{9} - 7424 \nu^{8} + 32643 \nu^{7} + 19248 \nu^{6} - 71846 \nu^{5} - 35248 \nu^{4} + 56201 \nu^{3} + 35530 \nu^{2} - 1463 \nu - 4612$$$$)/2162$$ $$\beta_{4}$$ $$=$$ $$($$$$944 \nu^{12} - 3120 \nu^{11} - 12772 \nu^{10} + 44868 \nu^{9} + 58087 \nu^{8} - 219999 \nu^{7} - 110536 \nu^{6} + 428762 \nu^{5} + 119516 \nu^{4} - 298916 \nu^{3} - 91598 \nu^{2} + 44430 \nu + 11577$$$$)/1081$$ $$\beta_{5}$$ $$=$$ $$($$$$2617 \nu^{12} - 8228 \nu^{11} - 37363 \nu^{10} + 120936 \nu^{9} + 187114 \nu^{8} - 614931 \nu^{7} - 418518 \nu^{6} + 1279188 \nu^{5} + 499382 \nu^{4} - 1003841 \nu^{3} - 321158 \nu^{2} + 182903 \nu + 44488$$$$)/2162$$ $$\beta_{6}$$ $$=$$ $$($$$$1537 \nu^{12} - 5483 \nu^{11} - 19453 \nu^{10} + 78197 \nu^{9} + 75852 \nu^{8} - 378472 \nu^{7} - 93135 \nu^{6} + 723044 \nu^{5} + 43093 \nu^{4} - 495112 \nu^{3} - 63961 \nu^{2} + 80294 \nu + 9559$$$$)/1081$$ $$\beta_{7}$$ $$=$$ $$($$$$1680 \nu^{12} - 6432 \nu^{11} - 19212 \nu^{10} + 89304 \nu^{9} + 55005 \nu^{8} - 412026 \nu^{7} + 21737 \nu^{6} + 714974 \nu^{5} - 146249 \nu^{4} - 398713 \nu^{3} + 5531 \nu^{2} + 50433 \nu + 7851$$$$)/1081$$ $$\beta_{8}$$ $$=$$ $$($$$$-4041 \nu^{12} + 14730 \nu^{11} + 49447 \nu^{10} - 207380 \nu^{9} - 177062 \nu^{8} + 981221 \nu^{7} + 151906 \nu^{6} - 1793936 \nu^{5} + 9716 \nu^{4} + 1126967 \nu^{3} + 153280 \nu^{2} - 171323 \nu - 31592$$$$)/2162$$ $$\beta_{9}$$ $$=$$ $$($$$$4041 \nu^{12} - 14730 \nu^{11} - 49447 \nu^{10} + 207380 \nu^{9} + 177062 \nu^{8} - 981221 \nu^{7} - 151906 \nu^{6} + 1793936 \nu^{5} - 9716 \nu^{4} - 1124805 \nu^{3} - 153280 \nu^{2} + 160513 \nu + 31592$$$$)/2162$$ $$\beta_{10}$$ $$=$$ $$($$$$-2764 \nu^{12} + 10088 \nu^{11} + 33585 \nu^{10} - 141614 \nu^{9} - 117946 \nu^{8} + 666901 \nu^{7} + 90681 \nu^{6} - 1209623 \nu^{5} + 23336 \nu^{4} + 751304 \nu^{3} + 104974 \nu^{2} - 113389 \nu - 23055$$$$)/1081$$ $$\beta_{11}$$ $$=$$ $$($$$$-5977 \nu^{12} + 21092 \nu^{11} + 75787 \nu^{10} - 299544 \nu^{9} - 297124 \nu^{8} + 1438983 \nu^{7} + 375044 \nu^{6} - 2709136 \nu^{5} - 204722 \nu^{4} + 1799105 \nu^{3} + 294962 \nu^{2} - 275121 \nu - 47218$$$$)/2162$$ $$\beta_{12}$$ $$=$$ $$($$$$6745 \nu^{12} - 23044 \nu^{11} - 88523 \nu^{10} + 329744 \nu^{9} + 376628 \nu^{8} - 1604297 \nu^{7} - 611328 \nu^{6} + 3091452 \nu^{5} + 548460 \nu^{4} - 2142659 \nu^{3} - 502024 \nu^{2} + 347655 \nu + 78110$$$$)/2162$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + \beta_{8} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + 7 \beta_{2} + \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{12} + 2 \beta_{11} + 9 \beta_{9} + 9 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 30 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$\beta_{12} + 11 \beta_{11} + 12 \beta_{9} + 12 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} + 8 \beta_{5} + \beta_{4} + 2 \beta_{3} + 44 \beta_{2} + 14 \beta_{1} + 82$$ $$\nu^{7}$$ $$=$$ $$25 \beta_{12} + 26 \beta_{11} + 66 \beta_{9} + 70 \beta_{8} + 3 \beta_{7} + 14 \beta_{6} - 13 \beta_{5} - 8 \beta_{4} + 10 \beta_{3} + 14 \beta_{2} + 191 \beta_{1} - 6$$ $$\nu^{8}$$ $$=$$ $$17 \beta_{12} + 98 \beta_{11} - 4 \beta_{10} + 104 \beta_{9} + 112 \beta_{8} + 67 \beta_{7} + 32 \beta_{6} + 49 \beta_{5} + 17 \beta_{4} + 24 \beta_{3} + 274 \beta_{2} + 141 \beta_{1} + 471$$ $$\nu^{9}$$ $$=$$ $$229 \beta_{12} + 251 \beta_{11} - 8 \beta_{10} + 454 \beta_{9} + 529 \beta_{8} + 49 \beta_{7} + 147 \beta_{6} - 121 \beta_{5} - 41 \beta_{4} + 76 \beta_{3} + 140 \beta_{2} + 1259 \beta_{1} - 11$$ $$\nu^{10}$$ $$=$$ $$196 \beta_{12} + 811 \beta_{11} - 75 \beta_{10} + 797 \beta_{9} + 963 \beta_{8} + 488 \beta_{7} + 357 \beta_{6} + 266 \beta_{5} + 196 \beta_{4} + 207 \beta_{3} + 1727 \beta_{2} + 1254 \beta_{1} + 2806$$ $$\nu^{11}$$ $$=$$ $$1867 \beta_{12} + 2161 \beta_{11} - 166 \beta_{10} + 3042 \beta_{9} + 3979 \beta_{8} + 550 \beta_{7} + 1364 \beta_{6} - 995 \beta_{5} - 100 \beta_{4} + 534 \beta_{3} + 1226 \beta_{2} + 8497 \beta_{1} + 202$$ $$\nu^{12}$$ $$=$$ $$1927 \beta_{12} + 6467 \beta_{11} - 937 \beta_{10} + 5748 \beta_{9} + 7966 \beta_{8} + 3595 \beta_{7} + 3425 \beta_{6} + 1281 \beta_{5} + 1932 \beta_{4} + 1587 \beta_{3} + 11075 \beta_{2} + 10469 \beta_{1} + 17210$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.73393 2.58904 2.20098 1.84587 1.21702 0.467051 0.441481 −0.196565 −0.558375 −0.646272 −1.46649 −2.15509 −2.47258
−2.73393 −1.00000 5.47440 −2.29019 2.73393 0 −9.49877 1.00000 6.26122
1.2 −2.58904 −1.00000 4.70310 3.45213 2.58904 0 −6.99843 1.00000 −8.93769
1.3 −2.20098 −1.00000 2.84432 −2.77621 2.20098 0 −1.85834 1.00000 6.11039
1.4 −1.84587 −1.00000 1.40722 1.41311 1.84587 0 1.09419 1.00000 −2.60841
1.5 −1.21702 −1.00000 −0.518850 3.45145 1.21702 0 3.06550 1.00000 −4.20050
1.6 −0.467051 −1.00000 −1.78186 −0.377698 0.467051 0 1.76632 1.00000 0.176404
1.7 −0.441481 −1.00000 −1.80509 3.83895 0.441481 0 1.67988 1.00000 −1.69482
1.8 0.196565 −1.00000 −1.96136 1.00666 −0.196565 0 −0.778664 1.00000 0.197874
1.9 0.558375 −1.00000 −1.68822 0.297196 −0.558375 0 −2.05941 1.00000 0.165947
1.10 0.646272 −1.00000 −1.58233 −2.20733 −0.646272 0 −2.31516 1.00000 −1.42653
1.11 1.46649 −1.00000 0.150578 −1.27945 −1.46649 0 −2.71215 1.00000 −1.87630
1.12 2.15509 −1.00000 2.64442 −0.784696 −2.15509 0 1.38879 1.00000 −1.69109
1.13 2.47258 −1.00000 4.11367 4.25608 −2.47258 0 5.22624 1.00000 10.5235
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$
$$41$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6027))$$:

 $$T_{2}^{13} + \cdots$$ $$T_{5}^{13} - \cdots$$ $$T_{13}^{13} - \cdots$$