Properties

 Label 912.2.cc.d Level $912$ Weight $2$ Character orbit 912.cc Analytic conductor $7.282$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.cc (of order $$18$$, degree $$6$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + 162 x^{7} + 270 x^{6} + 2916 x^{5} - 4374 x^{4} - 729 x^{3} + 19683$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{3} + ( -\beta_{9} + \beta_{17} ) q^{5} + ( -\beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{16} ) q^{7} + ( -\beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{15} + \beta_{16} ) q^{9} +O(q^{10})$$ $$q + \beta_{14} q^{3} + ( -\beta_{9} + \beta_{17} ) q^{5} + ( -\beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{16} ) q^{7} + ( -\beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{15} + \beta_{16} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{12} - \beta_{14} - \beta_{17} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{10} + \beta_{15} ) q^{13} + ( -\beta_{1} - \beta_{4} - 2 \beta_{8} - \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{17} ) q^{15} + ( -\beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{8} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{17} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} + \beta_{16} ) q^{19} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{8} + \beta_{9} + \beta_{16} - \beta_{17} ) q^{21} + ( 2 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{23} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} ) q^{25} + ( \beta_{1} + 4 \beta_{4} + 2 \beta_{8} - \beta_{9} - \beta_{12} + \beta_{16} + \beta_{17} ) q^{27} + ( \beta_{2} - \beta_{3} + \beta_{5} + \beta_{9} + \beta_{10} - \beta_{13} + 2 \beta_{16} + \beta_{17} ) q^{29} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - 2 \beta_{17} ) q^{31} + ( -2 - \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} ) q^{33} + ( -2 - \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{35} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{37} + ( -1 + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{39} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} ) q^{41} + ( 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{15} - \beta_{17} ) q^{43} + ( 2 - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{45} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} + 2 \beta_{15} - \beta_{17} ) q^{47} + ( \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{17} ) q^{49} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{51} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} - \beta_{17} ) q^{55} + ( \beta_{1} + \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} + 3 \beta_{11} - \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} ) q^{57} + ( -1 - 3 \beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{11} + 2 \beta_{16} + \beta_{17} ) q^{59} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{61} + ( -6 + 2 \beta_{1} - \beta_{4} + \beta_{8} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{17} ) q^{63} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{65} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + 3 \beta_{16} - 3 \beta_{17} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} + 2 \beta_{17} ) q^{69} + ( 3 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{9} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{16} + \beta_{17} ) q^{71} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{73} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{14} + 2 \beta_{15} ) q^{75} + ( -2 - \beta_{2} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 4 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{77} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{79} + ( -2 + \beta_{2} - 3 \beta_{4} + \beta_{5} - 3 \beta_{8} + \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 3 \beta_{17} ) q^{81} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} + 2 \beta_{17} ) q^{83} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{13} + 3 \beta_{14} - 3 \beta_{15} + \beta_{17} ) q^{85} + ( 2 + 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{87} + ( 5 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} + 2 \beta_{15} - \beta_{16} ) q^{89} + ( 3 - 2 \beta_{1} + 4 \beta_{4} - \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{16} + \beta_{17} ) q^{91} + ( -2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} - \beta_{9} - 5 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{93} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + 4 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - 3 \beta_{15} + \beta_{16} + \beta_{17} ) q^{95} + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} - 2 \beta_{17} ) q^{97} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 7 \beta_{8} - \beta_{10} - 3 \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + O(q^{10})$$ $$18q - 12q^{13} + 24q^{15} - 6q^{17} + 6q^{19} - 18q^{25} + 3q^{27} + 6q^{29} - 27q^{33} - 24q^{35} - 6q^{39} - 3q^{41} + 6q^{43} + 54q^{45} + 30q^{47} + 21q^{49} + 33q^{51} + 60q^{53} - 30q^{55} + 12q^{57} + 3q^{59} + 54q^{61} - 84q^{63} - 24q^{65} + 15q^{67} + 24q^{69} + 36q^{71} - 42q^{73} + 6q^{79} + 36q^{83} + 54q^{87} + 60q^{89} + 18q^{91} - 84q^{93} + 6q^{95} + 9q^{97} + 30q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + 162 x^{7} + 270 x^{6} + 2916 x^{5} - 4374 x^{4} - 729 x^{3} + 19683$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{17} + \nu^{14} + 18 \nu^{13} - 36 \nu^{12} - 10 \nu^{11} - 18 \nu^{10} - 90 \nu^{9} + 567 \nu^{8} - 270 \nu^{7} - 162 \nu^{6} - 270 \nu^{5} - 2916 \nu^{4} + 4374 \nu^{3} + 729 \nu^{2}$$$$)/6561$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{17} + 21 \nu^{16} - 81 \nu^{15} + 97 \nu^{14} - 57 \nu^{13} - 198 \nu^{12} + 1568 \nu^{11} - 3075 \nu^{10} + 2313 \nu^{9} - 117 \nu^{8} - 9261 \nu^{7} + 29268 \nu^{6} - 31374 \nu^{5} + 11421 \nu^{4} + 11421 \nu^{3} - 82377 \nu^{2} + 150903 \nu - 65610$$$$)/4374$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{17} + 99 \nu^{16} - 153 \nu^{15} - 22 \nu^{14} + 207 \nu^{13} - 1485 \nu^{12} + 4675 \nu^{11} - 2637 \nu^{10} - 4167 \nu^{9} + 10098 \nu^{8} - 38475 \nu^{7} + 51273 \nu^{6} + 24003 \nu^{5} - 60507 \nu^{4} + 91125 \nu^{3} - 237654 \nu^{2} + 63423 \nu + 334611$$$$)/13122$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{17} + 2 \nu^{16} - 27 \nu^{15} + 50 \nu^{14} - 47 \nu^{13} + 9 \nu^{12} + 409 \nu^{11} - 1366 \nu^{10} + 1629 \nu^{9} - 822 \nu^{8} - 1809 \nu^{7} + 11025 \nu^{6} - 19359 \nu^{5} + 12420 \nu^{4} - 81 \nu^{3} - 24786 \nu^{2} + 75087 \nu - 72171$$$$)/1458$$ $$\beta_{6}$$ $$=$$ $$($$$$-16 \nu^{17} + 81 \nu^{16} + 36 \nu^{15} - 335 \nu^{14} + 477 \nu^{13} - 1422 \nu^{12} + 1406 \nu^{11} + 6165 \nu^{10} - 12420 \nu^{9} + 12609 \nu^{8} - 20493 \nu^{7} - 25272 \nu^{6} + 127062 \nu^{5} - 103761 \nu^{4} + 67554 \nu^{3} - 41553 \nu^{2} - 373977 \nu + 603612$$$$)/13122$$ $$\beta_{7}$$ $$=$$ $$($$$$-11 \nu^{17} - 3 \nu^{16} - 6 \nu^{15} + 92 \nu^{14} + 48 \nu^{13} - 255 \nu^{12} - 137 \nu^{11} - 1425 \nu^{10} + 3108 \nu^{9} + 1350 \nu^{8} - 504 \nu^{7} + 3645 \nu^{6} - 36045 \nu^{5} + 26001 \nu^{4} + 10854 \nu^{3} + 8262 \nu^{2} + 74358 \nu - 225261$$$$)/4374$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{17} - 105 \nu^{16} + 90 \nu^{15} + 275 \nu^{14} - 318 \nu^{13} + 1296 \nu^{12} - 4181 \nu^{11} - 2247 \nu^{10} + 12492 \nu^{9} - 11367 \nu^{8} + 36126 \nu^{7} - 25110 \nu^{6} - 114507 \nu^{5} + 125469 \nu^{4} - 81648 \nu^{3} + 210681 \nu^{2} + 183708 \nu - 787320$$$$)/13122$$ $$\beta_{9}$$ $$=$$ $$($$$$-11 \nu^{17} + 147 \nu^{16} - 216 \nu^{15} - 43 \nu^{14} + 375 \nu^{13} - 2259 \nu^{12} + 6451 \nu^{11} - 3183 \nu^{10} - 6660 \nu^{9} + 15525 \nu^{8} - 52947 \nu^{7} + 64233 \nu^{6} + 40041 \nu^{5} - 88857 \nu^{4} + 125388 \nu^{3} - 292329 \nu^{2} + 50301 \nu + 452709$$$$)/13122$$ $$\beta_{10}$$ $$=$$ $$($$$$-17 \nu^{17} - 5 \nu^{16} + 99 \nu^{15} - 136 \nu^{14} + 284 \nu^{13} - 405 \nu^{12} - 1655 \nu^{11} + 4369 \nu^{10} - 4167 \nu^{9} + 5472 \nu^{8} + 5508 \nu^{7} - 41229 \nu^{6} + 46683 \nu^{5} - 25569 \nu^{4} + 13851 \nu^{3} + 103518 \nu^{2} - 237654 \nu + 63423$$$$)/4374$$ $$\beta_{11}$$ $$=$$ $$($$$$-4 \nu^{17} + 17 \nu^{16} - 10 \nu^{15} - 23 \nu^{14} + 82 \nu^{13} - 332 \nu^{12} + 536 \nu^{11} + 197 \nu^{10} - 1234 \nu^{9} + 2619 \nu^{8} - 5760 \nu^{7} + 2430 \nu^{6} + 9666 \nu^{5} - 12825 \nu^{4} + 18198 \nu^{3} - 24057 \nu^{2} - 20412 \nu + 56862$$$$)/1458$$ $$\beta_{12}$$ $$=$$ $$($$$$-23 \nu^{17} + 10 \nu^{16} + 6 \nu^{15} + 86 \nu^{14} + 161 \nu^{13} - 717 \nu^{12} - 41 \nu^{11} - 908 \nu^{10} + 2814 \nu^{9} + 4716 \nu^{8} - 4941 \nu^{7} - 1377 \nu^{6} - 32373 \nu^{5} + 23436 \nu^{4} + 35640 \nu^{3} + 7290 \nu^{2} + 26973 \nu - 247131$$$$)/4374$$ $$\beta_{13}$$ $$=$$ $$($$$$-9 \nu^{17} - 4 \nu^{16} + 39 \nu^{15} - 21 \nu^{14} + 94 \nu^{13} - 174 \nu^{12} - 717 \nu^{11} + 1220 \nu^{10} - 393 \nu^{9} + 1797 \nu^{8} + 2520 \nu^{7} - 14598 \nu^{6} + 6345 \nu^{5} + 270 \nu^{4} + 5913 \nu^{3} + 41553 \nu^{2} - 67068 \nu - 34992$$$$)/1458$$ $$\beta_{14}$$ $$=$$ $$($$$$-76 \nu^{17} - 108 \nu^{16} + 459 \nu^{15} - 194 \nu^{14} + 747 \nu^{13} - 522 \nu^{12} - 9724 \nu^{11} + 13104 \nu^{10} - 1521 \nu^{9} + 9774 \nu^{8} + 50193 \nu^{7} - 167832 \nu^{6} + 45090 \nu^{5} + 39366 \nu^{4} - 13851 \nu^{3} + 546750 \nu^{2} - 649539 \nu - 551124$$$$)/13122$$ $$\beta_{15}$$ $$=$$ $$($$$$11 \nu^{17} - 17 \nu^{16} - 3 \nu^{15} + 13 \nu^{14} - 145 \nu^{13} + 525 \nu^{12} - 283 \nu^{11} - 413 \nu^{10} + 807 \nu^{9} - 4125 \nu^{8} + 5787 \nu^{7} + 2817 \nu^{6} - 5103 \nu^{5} + 7695 \nu^{4} - 26811 \nu^{3} + 7047 \nu^{2} + 37179 \nu + 10935$$$$)/1458$$ $$\beta_{16}$$ $$=$$ $$($$$$103 \nu^{17} - 99 \nu^{16} - 27 \nu^{15} - 157 \nu^{14} - 1026 \nu^{13} + 4140 \nu^{12} - 1265 \nu^{11} + 621 \nu^{10} - 3555 \nu^{9} - 30429 \nu^{8} + 39960 \nu^{7} + 12150 \nu^{6} + 60615 \nu^{5} - 24057 \nu^{4} - 216513 \nu^{3} + 22599 \nu^{2} + 74358 \nu + 669222$$$$)/13122$$ $$\beta_{17}$$ $$=$$ $$($$$$-17 \nu^{17} + 10 \nu^{16} + 27 \nu^{15} - 10 \nu^{14} + 188 \nu^{13} - 576 \nu^{12} - 269 \nu^{11} + 874 \nu^{10} - 351 \nu^{9} + 4680 \nu^{8} - 3078 \nu^{7} - 10746 \nu^{6} + 1161 \nu^{5} - 702 \nu^{4} + 26973 \nu^{3} + 20412 \nu^{2} - 55404 \nu - 78732$$$$)/1458$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{16} + \beta_{15} + \beta_{9} - \beta_{8} - 2 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{14} - 4 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{17} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 3 \beta_{2} + \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{17} - 2 \beta_{16} + \beta_{15} + 3 \beta_{13} + 3 \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 6 \beta_{1} - 12$$ $$\nu^{6}$$ $$=$$ $$-9 \beta_{16} + 10 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 2 \beta_{7} + 2 \beta_{6} + 5 \beta_{3} + \beta_{2} - 9 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} + 15 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 9 \beta_{9} + 18 \beta_{8} - 12 \beta_{7} + 7 \beta_{5} + 36 \beta_{4} + 15 \beta_{2} - 3 \beta_{1} - 2$$ $$\nu^{8}$$ $$=$$ $$-18 \beta_{17} - \beta_{16} + 15 \beta_{15} - 9 \beta_{14} + 21 \beta_{12} + 24 \beta_{11} + 21 \beta_{10} - 11 \beta_{9} + 5 \beta_{8} + 36 \beta_{7} + 18 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} - 14 \beta_{3} - 18 \beta_{2} + 9 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$-27 \beta_{17} - 27 \beta_{16} + 11 \beta_{15} + \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - 30 \beta_{11} + 6 \beta_{10} - 27 \beta_{9} - 36 \beta_{8} + 74 \beta_{7} + 76 \beta_{6} - 7 \beta_{5} + 13 \beta_{3} + 44 \beta_{2} + 27 \beta_{1} - 30$$ $$\nu^{10}$$ $$=$$ $$33 \beta_{17} - 54 \beta_{16} + 63 \beta_{15} + 24 \beta_{14} - 89 \beta_{13} + \beta_{12} + 2 \beta_{11} - 10 \beta_{10} - 27 \beta_{9} + 18 \beta_{8} + 9 \beta_{7} + 66 \beta_{6} - 76 \beta_{5} - 18 \beta_{4} + 63 \beta_{3} + 9 \beta_{2} - 67 \beta_{1} + 140$$ $$\nu^{11}$$ $$=$$ $$-15 \beta_{17} + 100 \beta_{16} - 103 \beta_{15} + 126 \beta_{14} - 129 \beta_{13} + 102 \beta_{12} - 30 \beta_{11} - 63 \beta_{10} - \beta_{9} - 2 \beta_{8} - 18 \beta_{7} + 18 \beta_{6} - 9 \beta_{5} + 68 \beta_{4} - 51 \beta_{3} + 63 \beta_{2} + 66 \beta_{1} + 120$$ $$\nu^{12}$$ $$=$$ $$-90 \beta_{17} - 81 \beta_{16} + 136 \beta_{15} - 154 \beta_{14} + 33 \beta_{13} + 249 \beta_{12} + 129 \beta_{11} + 111 \beta_{10} + 36 \beta_{9} - 342 \beta_{8} + 148 \beta_{7} + 2 \beta_{6} - 34 \beta_{5} - 612 \beta_{4} + 17 \beta_{3} - 155 \beta_{2} + 81 \beta_{1} - 51$$ $$\nu^{13}$$ $$=$$ $$-62 \beta_{17} - 171 \beta_{16} - 189 \beta_{15} + 153 \beta_{14} - 19 \beta_{13} - 100 \beta_{12} - 362 \beta_{11} - 101 \beta_{10} - 99 \beta_{9} - 648 \beta_{8} - 96 \beta_{7} - 6 \beta_{6} - 29 \beta_{5} - 486 \beta_{4} + 432 \beta_{3} + 255 \beta_{2} + 44 \beta_{1} - 2$$ $$\nu^{14}$$ $$=$$ $$372 \beta_{17} - 106 \beta_{16} + 206 \beta_{15} + 243 \beta_{14} - 426 \beta_{13} + 9 \beta_{12} + 450 \beta_{11} - 39 \beta_{10} - 80 \beta_{9} + 380 \beta_{8} - 1026 \beta_{7} - 756 \beta_{6} - 525 \beta_{5} - 29 \beta_{4} + 710 \beta_{3} - 810 \beta_{2} - 393 \beta_{1} + 354$$ $$\nu^{15}$$ $$=$$ $$-351 \beta_{17} + 765 \beta_{16} - 802 \beta_{15} + 916 \beta_{14} + 46 \beta_{13} + 327 \beta_{12} + 402 \beta_{11} - 624 \beta_{10} + 351 \beta_{9} - 378 \beta_{8} - 376 \beta_{7} - 974 \beta_{6} + 522 \beta_{5} + 270 \beta_{4} - 752 \beta_{3} - 298 \beta_{2} + 279 \beta_{1} - 2208$$ $$\nu^{16}$$ $$=$$ $$-878 \beta_{17} - 630 \beta_{16} + 927 \beta_{15} - 1554 \beta_{14} + 1726 \beta_{13} + 430 \beta_{12} + 2318 \beta_{11} + 998 \beta_{10} - 423 \beta_{9} - 558 \beta_{8} + 390 \beta_{7} - 351 \beta_{6} + 326 \beta_{5} - 3060 \beta_{4} + 729 \beta_{3} - 852 \beta_{2} - 1284 \beta_{1} - 4210$$ $$\nu^{17}$$ $$=$$ $$-918 \beta_{17} + 406 \beta_{16} - 3786 \beta_{15} + 1656 \beta_{14} - 378 \beta_{13} - 3126 \beta_{12} - 1536 \beta_{11} - 1074 \beta_{10} - 3040 \beta_{9} + 1750 \beta_{8} + 504 \beta_{7} + 738 \beta_{6} - 192 \beta_{5} + 5150 \beta_{4} + 1436 \beta_{3} + 1125 \beta_{2} - 1566 \beta_{1} + 54$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\beta_{7}$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.0786547 − 1.73026i 1.47158 + 0.913487i −1.72388 − 0.168030i 0.0786547 + 1.73026i 1.47158 − 0.913487i −1.72388 + 0.168030i 1.40849 + 1.00804i −1.73189 + 0.0237018i −0.442647 − 1.67453i 1.40849 − 1.00804i −1.73189 − 0.0237018i −0.442647 + 1.67453i −0.396613 + 1.68603i 1.69944 − 0.334495i −0.363139 − 1.69356i −0.396613 − 1.68603i 1.69944 + 0.334495i −0.363139 + 1.69356i
0 −1.53778 0.797015i 0 0.258510 0.710252i 0 −0.777943 + 1.34744i 0 1.72953 + 2.45127i 0
257.2 0 0.0553136 + 1.73117i 0 0.882820 2.42553i 0 1.58376 2.74316i 0 −2.99388 + 0.191514i 0
257.3 0 0.716422 1.57694i 0 −1.14133 + 3.13578i 0 1.07356 1.85947i 0 −1.97348 2.25951i 0
401.1 0 −1.53778 + 0.797015i 0 0.258510 + 0.710252i 0 −0.777943 1.34744i 0 1.72953 2.45127i 0
401.2 0 0.0553136 1.73117i 0 0.882820 + 2.42553i 0 1.58376 + 2.74316i 0 −2.99388 0.191514i 0
401.3 0 0.716422 + 1.57694i 0 −1.14133 3.13578i 0 1.07356 + 1.85947i 0 −1.97348 + 2.25951i 0
497.1 0 −1.57724 0.715766i 0 0.262261 + 0.0462437i 0 −0.604656 1.04730i 0 1.97536 + 2.25787i 0
497.2 0 0.845418 + 1.51171i 0 −2.22841 0.392929i 0 1.16829 + 2.02354i 0 −1.57054 + 2.55605i 0
497.3 0 1.67151 0.453924i 0 1.96615 + 0.346685i 0 −0.910931 1.57778i 0 2.58791 1.51748i 0
545.1 0 −1.57724 + 0.715766i 0 0.262261 0.0462437i 0 −0.604656 + 1.04730i 0 1.97536 2.25787i 0
545.2 0 0.845418 1.51171i 0 −2.22841 + 0.392929i 0 1.16829 2.02354i 0 −1.57054 2.55605i 0
545.3 0 1.67151 + 0.453924i 0 1.96615 0.346685i 0 −0.910931 + 1.57778i 0 2.58791 + 1.51748i 0
641.1 0 −1.26184 + 1.18649i 0 −1.86241 + 2.21954i 0 −0.562083 0.973556i 0 0.184473 2.99432i 0
641.2 0 −0.560041 1.63901i 0 −0.343148 + 0.408948i 0 0.716507 + 1.24103i 0 −2.37271 + 1.83583i 0
641.3 0 1.64823 0.532290i 0 2.20556 2.62849i 0 −1.68651 2.92113i 0 2.43333 1.75467i 0
737.1 0 −1.26184 1.18649i 0 −1.86241 2.21954i 0 −0.562083 + 0.973556i 0 0.184473 + 2.99432i 0
737.2 0 −0.560041 + 1.63901i 0 −0.343148 0.408948i 0 0.716507 1.24103i 0 −2.37271 1.83583i 0
737.3 0 1.64823 + 0.532290i 0 2.20556 + 2.62849i 0 −1.68651 + 2.92113i 0 2.43333 + 1.75467i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 737.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.cc.d 18
3.b odd 2 1 912.2.cc.c 18
4.b odd 2 1 114.2.l.a 18
12.b even 2 1 114.2.l.b yes 18
19.f odd 18 1 912.2.cc.c 18
57.j even 18 1 inner 912.2.cc.d 18
76.k even 18 1 114.2.l.b yes 18
228.u odd 18 1 114.2.l.a 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.l.a 18 4.b odd 2 1
114.2.l.a 18 228.u odd 18 1
114.2.l.b yes 18 12.b even 2 1
114.2.l.b yes 18 76.k even 18 1
912.2.cc.c 18 3.b odd 2 1
912.2.cc.c 18 19.f odd 18 1
912.2.cc.d 18 1.a even 1 1 trivial
912.2.cc.d 18 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{18} + \cdots$$ $$T_{7}^{18} + \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$19683 - 729 T^{3} - 1458 T^{5} + 270 T^{6} - 162 T^{7} + 270 T^{8} + 81 T^{9} + 90 T^{10} - 18 T^{11} + 10 T^{12} - 18 T^{13} - T^{15} + T^{18}$$
$5$ $$1728 - 10368 T + 11232 T^{2} - 72 T^{3} + 70344 T^{4} - 60552 T^{5} + 141193 T^{6} - 5202 T^{7} - 30258 T^{8} + 1458 T^{9} - 1458 T^{10} - 324 T^{11} + 46 T^{12} + 162 T^{13} + 108 T^{14} + 9 T^{16} + T^{18}$$
$7$ $$87616 + 113664 T + 216720 T^{2} + 148720 T^{3} + 195300 T^{4} + 100062 T^{5} + 117161 T^{6} + 41064 T^{7} + 43740 T^{8} + 9622 T^{9} + 11985 T^{10} + 1494 T^{11} + 2222 T^{12} + 90 T^{13} + 297 T^{14} + 4 T^{15} + 21 T^{16} + T^{18}$$
$11$ $$35769627 - 149822217 T + 272171448 T^{2} - 263848509 T^{3} + 131458797 T^{4} - 15125949 T^{5} - 15944255 T^{6} + 4590198 T^{7} + 3068313 T^{8} - 2063799 T^{9} + 285702 T^{10} + 95760 T^{11} - 27296 T^{12} - 3933 T^{13} + 1884 T^{14} - 51 T^{16} + T^{18}$$
$13$ $$2365632 - 7384608 T + 22638528 T^{2} - 26484192 T^{3} + 27392256 T^{4} - 4914324 T^{5} + 14891337 T^{6} - 1225044 T^{7} - 1056429 T^{8} + 175320 T^{9} - 36207 T^{10} - 42480 T^{11} - 360 T^{12} + 3708 T^{13} + 324 T^{14} - 84 T^{15} + 27 T^{16} + 12 T^{17} + T^{18}$$
$17$ $$3878307 - 11297232 T - 5531679 T^{2} + 14004081 T^{3} + 29829231 T^{4} + 9789921 T^{5} - 1848509 T^{6} + 9106179 T^{7} + 3836457 T^{8} - 255264 T^{9} + 1118904 T^{10} + 32520 T^{11} + 45985 T^{12} + 9255 T^{13} + 1224 T^{14} + 309 T^{15} + 21 T^{16} + 6 T^{17} + T^{18}$$
$19$ $$322687697779 - 101901378246 T - 26816152170 T^{2} + 12937617275 T^{3} + 757686294 T^{4} - 835096968 T^{5} - 27477154 T^{6} + 50433144 T^{7} + 899004 T^{8} - 2942011 T^{9} + 47316 T^{10} + 139704 T^{11} - 4006 T^{12} - 6408 T^{13} + 306 T^{14} + 275 T^{15} - 30 T^{16} - 6 T^{17} + T^{18}$$
$23$ $$123187392 - 294127200 T + 617992416 T^{2} - 696487032 T^{3} + 1062671292 T^{4} - 860428764 T^{5} + 689100193 T^{6} - 273220578 T^{7} + 71012562 T^{8} - 10843848 T^{9} + 2793426 T^{10} - 516060 T^{11} - 93302 T^{12} + 15768 T^{13} + 3864 T^{14} - 108 T^{15} - 105 T^{16} + T^{18}$$
$29$ $$52719833664 - 16391255904 T - 31128894864 T^{2} - 3628092384 T^{3} + 12691761048 T^{4} + 6298040142 T^{5} + 1080612909 T^{6} - 72681030 T^{7} - 18102744 T^{8} - 3826152 T^{9} + 188226 T^{10} + 424692 T^{11} + 16692 T^{12} - 9324 T^{13} + 3798 T^{14} - 606 T^{15} + 111 T^{16} - 6 T^{17} + T^{18}$$
$31$ $$6231379854528 - 6774606004608 T + 2135586670416 T^{2} + 347326742880 T^{3} - 278203568436 T^{4} - 4904234262 T^{5} + 22269166245 T^{6} - 850196898 T^{7} - 1103329728 T^{8} + 54695574 T^{9} + 41003091 T^{10} - 1667142 T^{11} - 1030362 T^{12} + 25614 T^{13} + 18621 T^{14} - 165 T^{16} + T^{18}$$
$37$ $$7868768303808 + 3301605756576 T^{2} + 585487601676 T^{4} + 57645174609 T^{6} + 3468414222 T^{8} + 131833359 T^{10} + 3146832 T^{12} + 44991 T^{14} + 342 T^{16} + T^{18}$$
$41$ $$1846709769969 + 3383671593780 T + 3442117775661 T^{2} + 2055695641578 T^{3} + 936326856420 T^{4} + 212090870691 T^{5} + 24457418397 T^{6} + 5050712961 T^{7} + 1414453590 T^{8} + 183830688 T^{9} + 35530407 T^{10} + 6446142 T^{11} + 578637 T^{12} - 6291 T^{13} + 567 T^{14} - 237 T^{15} - 81 T^{16} + 3 T^{17} + T^{18}$$
$43$ $$390621250009 + 141475570914 T + 2502920005845 T^{2} + 3193760135025 T^{3} + 1945100254026 T^{4} + 642244530381 T^{5} + 125197505055 T^{6} + 13788062184 T^{7} + 834745182 T^{8} - 259072400 T^{9} - 85583703 T^{10} - 4253289 T^{11} + 1635219 T^{12} + 201903 T^{13} - 2391 T^{14} - 1233 T^{15} - 21 T^{16} - 6 T^{17} + T^{18}$$
$47$ $$3499077312 - 22163691744 T + 234020746080 T^{2} - 235677357072 T^{3} + 228877829676 T^{4} - 41796892746 T^{5} - 20048430563 T^{6} + 20956649406 T^{7} - 819624165 T^{8} - 1305742308 T^{9} + 224832402 T^{10} + 436206 T^{11} - 4210880 T^{12} + 526350 T^{13} - 9561 T^{14} - 3696 T^{15} + 513 T^{16} - 30 T^{17} + T^{18}$$
$53$ $$3426463296 - 33524972064 T + 116975876544 T^{2} - 162987312888 T^{3} + 145477389636 T^{4} - 95821382268 T^{5} + 51044044113 T^{6} - 22246047792 T^{7} + 8030520198 T^{8} - 2415368574 T^{9} + 599567868 T^{10} - 122186682 T^{11} + 20675985 T^{12} - 2895858 T^{13} + 325836 T^{14} - 27750 T^{15} + 1650 T^{16} - 60 T^{17} + T^{18}$$
$59$ $$38983402581561 + 19432108592010 T - 5011803296289 T^{2} - 2532697771320 T^{3} + 775696949469 T^{4} + 8631478017 T^{5} - 8041532733 T^{6} - 7047029133 T^{7} + 4427702217 T^{8} - 1300488633 T^{9} + 244996146 T^{10} - 29118996 T^{11} + 3016905 T^{12} - 280800 T^{13} + 15030 T^{14} - 879 T^{15} + 147 T^{16} - 3 T^{17} + T^{18}$$
$61$ $$65033160256 + 837712259040 T + 2998086559584 T^{2} - 151176769792 T^{3} + 3334220844156 T^{4} - 2696150385294 T^{5} + 1082915269121 T^{6} - 274879113276 T^{7} + 50528749842 T^{8} - 7207731646 T^{9} + 892275015 T^{10} - 120249426 T^{11} + 18467003 T^{12} - 2581020 T^{13} + 285744 T^{14} - 23830 T^{15} + 1419 T^{16} - 54 T^{17} + T^{18}$$
$67$ $$5641379998084323 + 1581247049832693 T - 290433671288019 T^{2} - 114565691996118 T^{3} - 7956330419556 T^{4} + 1657979849619 T^{5} + 452104179777 T^{6} + 14656134762 T^{7} + 14609136726 T^{8} - 676511640 T^{9} + 122075937 T^{10} - 45891909 T^{11} + 1625472 T^{12} - 347733 T^{13} + 51813 T^{14} - 2271 T^{15} + 261 T^{16} - 15 T^{17} + T^{18}$$
$71$ $$404099233344 - 22573947081024 T + 367044508201824 T^{2} - 367155605037456 T^{3} + 184787209022256 T^{4} - 60645595428594 T^{5} + 14509501175097 T^{6} - 2655253780470 T^{7} + 379169768331 T^{8} - 43380742482 T^{9} + 4239884412 T^{10} - 360192096 T^{11} + 26973540 T^{12} - 2019816 T^{13} + 163431 T^{14} - 12294 T^{15} + 783 T^{16} - 36 T^{17} + T^{18}$$
$73$ $$192753487369 - 791600833443 T + 379924057347 T^{2} + 1263361473954 T^{3} + 1256598051213 T^{4} + 701270458347 T^{5} + 270776321586 T^{6} + 64815302400 T^{7} + 8018230623 T^{8} - 260359090 T^{9} - 225519060 T^{10} - 27864642 T^{11} + 1456731 T^{12} + 926085 T^{13} + 167232 T^{14} + 16047 T^{15} + 987 T^{16} + 42 T^{17} + T^{18}$$
$79$ $$21259626441408 + 7167426056640 T - 962723212560 T^{2} + 5052547513800 T^{3} + 4243049245944 T^{4} + 1218226937454 T^{5} + 190286398989 T^{6} + 11521887120 T^{7} - 2597238567 T^{8} - 446160798 T^{9} - 37037088 T^{10} - 3097980 T^{11} + 468732 T^{12} - 51786 T^{13} + 15705 T^{14} - 12 T^{15} + 105 T^{16} - 6 T^{17} + T^{18}$$
$83$ $$176145902499843 + 534178903806279 T + 535595856380577 T^{2} - 13302831266856 T^{3} - 93320828111484 T^{4} + 1882524564045 T^{5} + 13153839521878 T^{6} - 3730676567724 T^{7} + 355076915166 T^{8} + 9426572775 T^{9} - 3658557183 T^{10} + 13537359 T^{11} + 36121867 T^{12} - 2042253 T^{13} - 83529 T^{14} + 8856 T^{15} + 186 T^{16} - 36 T^{17} + T^{18}$$
$89$ $$381874169643201 + 44793101479392 T + 237935902096233 T^{2} - 42655301651403 T^{3} - 3165140950848 T^{4} + 2162662551999 T^{5} - 341040944421 T^{6} - 47606747436 T^{7} + 29902263615 T^{8} - 3747657384 T^{9} + 4114944 T^{10} - 1600794 T^{11} + 14602713 T^{12} - 3384864 T^{13} + 410364 T^{14} - 32745 T^{15} + 1776 T^{16} - 60 T^{17} + T^{18}$$
$97$ $$17447631785307 - 76196333175057 T + 112977824953914 T^{2} - 70055543626245 T^{3} + 29093451072057 T^{4} - 8236950703911 T^{5} + 2166057349236 T^{6} - 461848162446 T^{7} + 80427211497 T^{8} - 10544774805 T^{9} + 1116199845 T^{10} - 93383199 T^{11} + 5172174 T^{12} + 3240 T^{13} - 41472 T^{14} + 4230 T^{15} - 144 T^{16} - 9 T^{17} + T^{18}$$