# Properties

 Label 605.2.j.j Level $605$ Weight $2$ Character orbit 605.j Analytic conductor $4.831$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.j (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.343361479062744140625.1 Defining polynomial: $$x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + 168 x^{7} + 54 x^{6} + 189 x^{5} + 648 x^{4} - 1944 x^{3} + 2187 x^{2} - 2187 x + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} - \beta_{13} ) q^{2} + ( -\beta_{12} - \beta_{14} ) q^{3} + ( -\beta_{3} - \beta_{7} - \beta_{9} ) q^{4} + ( \beta_{1} - \beta_{6} + \beta_{15} ) q^{5} + 2 \beta_{15} q^{6} -2 \beta_{8} q^{7} + ( 3 \beta_{11} + \beta_{12} + \beta_{14} ) q^{8} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{13} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} - \beta_{13} ) q^{2} + ( -\beta_{12} - \beta_{14} ) q^{3} + ( -\beta_{3} - \beta_{7} - \beta_{9} ) q^{4} + ( \beta_{1} - \beta_{6} + \beta_{15} ) q^{5} + 2 \beta_{15} q^{6} -2 \beta_{8} q^{7} + ( 3 \beta_{11} + \beta_{12} + \beta_{14} ) q^{8} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{13} - \beta_{15} ) q^{9} + ( 2 - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{10} + 2 \beta_{5} q^{12} + ( -2 \beta_{2} + 2 \beta_{12} - 2 \beta_{14} ) q^{14} + ( 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{15} + ( -\beta_{6} + \beta_{10} - 3 \beta_{15} ) q^{16} + ( -2 \beta_{6} - 2 \beta_{10} ) q^{17} + ( -\beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{18} -4 \beta_{2} q^{19} + ( 3 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{8} + 4 \beta_{11} - 3 \beta_{13} - 3 \beta_{15} ) q^{20} + ( -4 - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{21} + ( \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{23} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{13} - 2 \beta_{15} ) q^{24} + ( -\beta_{2} + \beta_{11} + 3 \beta_{14} ) q^{25} + ( -2 \beta_{1} + \beta_{6} + \beta_{10} ) q^{27} + ( -6 \beta_{1} + 6 \beta_{6} + 6 \beta_{10} ) q^{28} + ( 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{9} ) q^{29} + ( -2 \beta_{2} - 2 \beta_{11} - 2 \beta_{12} ) q^{30} + ( \beta_{4} + \beta_{13} ) q^{31} + ( -3 \beta_{4} - \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{32} + 4 q^{34} + ( 4 - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{13} - 4 \beta_{15} ) q^{35} + ( 7 \beta_{2} - \beta_{12} + \beta_{14} ) q^{36} + ( -3 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{37} + ( -4 \beta_{1} + 4 \beta_{6} + 4 \beta_{10} ) q^{38} + ( -6 \beta_{3} + \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{40} + ( 2 \beta_{2} - 2 \beta_{12} + 2 \beta_{14} ) q^{41} + ( 4 \beta_{1} + 4 \beta_{5} - 4 \beta_{8} + 4 \beta_{11} ) q^{42} -2 \beta_{5} q^{43} + ( 5 + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{9} + 3 \beta_{10} - \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{45} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{15} ) q^{46} + ( 2 \beta_{11} + 4 \beta_{12} + 4 \beta_{14} ) q^{47} + ( -2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{48} + 5 \beta_{15} q^{49} + ( 5 \beta_{1} - 3 \beta_{6} - \beta_{10} - 4 \beta_{15} ) q^{50} + ( 8 \beta_{3} - 2 \beta_{7} - 2 \beta_{9} ) q^{51} + ( -4 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{8} - 4 \beta_{11} + 4 \beta_{13} ) q^{53} + ( -4 + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{54} + ( -14 + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{56} + ( 4 \beta_{4} - 4 \beta_{13} ) q^{57} + ( -10 \beta_{11} - 6 \beta_{12} - 6 \beta_{14} ) q^{58} + ( 4 \beta_{3} + \beta_{7} + \beta_{9} ) q^{59} + ( -2 \beta_{6} - 4 \beta_{10} + 4 \beta_{15} ) q^{60} + ( -2 \beta_{6} + 2 \beta_{10} + 6 \beta_{15} ) q^{61} + ( -2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{62} + ( 2 \beta_{11} + 6 \beta_{12} + 6 \beta_{14} ) q^{63} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{13} - \beta_{15} ) q^{64} + ( -3 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{67} + ( 4 \beta_{1} + 4 \beta_{5} - 4 \beta_{8} + 4 \beta_{11} ) q^{68} + ( -4 \beta_{2} - \beta_{12} + \beta_{14} ) q^{69} + ( 6 \beta_{3} + 6 \beta_{7} - 8 \beta_{8} - 6 \beta_{9} ) q^{70} + ( 3 \beta_{6} - 3 \beta_{10} ) q^{71} + ( 5 \beta_{1} - 7 \beta_{6} - 7 \beta_{10} ) q^{72} -4 \beta_{8} q^{73} + ( 4 \beta_{2} + 2 \beta_{12} - 2 \beta_{14} ) q^{74} + ( -4 - 3 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{8} - 3 \beta_{11} - 3 \beta_{13} + 4 \beta_{15} ) q^{75} + ( -4 + 4 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} + 4 \beta_{10} + 4 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} ) q^{76} + ( -8 + 8 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{13} + 8 \beta_{15} ) q^{79} + ( -\beta_{2} + 6 \beta_{11} + 5 \beta_{12} + 3 \beta_{14} ) q^{80} + ( -3 \beta_{3} + 2 \beta_{7} + 2 \beta_{9} ) q^{81} + ( 10 \beta_{1} - 6 \beta_{6} - 6 \beta_{10} ) q^{82} + ( -2 \beta_{1} + 4 \beta_{6} + 4 \beta_{10} ) q^{83} -12 \beta_{3} q^{84} + ( 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{14} ) q^{85} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{13} - 2 \beta_{15} ) q^{86} -4 \beta_{5} q^{87} + ( 2 + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{89} + ( -6 + \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} - \beta_{13} + 6 \beta_{15} ) q^{90} + 2 \beta_{8} q^{92} + ( -2 \beta_{1} - \beta_{6} - \beta_{10} ) q^{93} + ( -2 \beta_{6} + 2 \beta_{10} - 10 \beta_{15} ) q^{94} + ( -4 \beta_{3} + 4 \beta_{7} - 4 \beta_{8} ) q^{95} + ( 10 \beta_{2} + 2 \beta_{12} - 2 \beta_{14} ) q^{96} + ( 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{11} - 3 \beta_{13} ) q^{97} + ( 5 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{9} + 5 \beta_{10} - 5 \beta_{12} - 5 \beta_{13} - 5 \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 6q^{4} + 3q^{5} + 8q^{6} + 2q^{9} + O(q^{10})$$ $$16q + 6q^{4} + 3q^{5} + 8q^{6} + 2q^{9} + 40q^{10} - 12q^{14} - q^{15} - 14q^{16} - 16q^{19} + 12q^{20} - 48q^{21} + 4q^{24} - q^{25} - 12q^{29} - 6q^{30} - 2q^{31} + 64q^{34} + 18q^{35} + 30q^{36} + 28q^{40} + 12q^{41} + 72q^{45} - 8q^{46} + 20q^{49} - 18q^{50} - 28q^{51} - 80q^{54} - 240q^{56} - 18q^{59} + 18q^{60} + 20q^{61} + 2q^{64} - 14q^{69} - 24q^{70} + 6q^{71} + 12q^{74} - 15q^{75} - 96q^{76} - 28q^{79} - 6q^{80} + 8q^{81} + 48q^{84} + 2q^{85} + 12q^{86} + 24q^{89} - 28q^{90} - 44q^{94} + 12q^{95} + 36q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + 168 x^{7} + 54 x^{6} + 189 x^{5} + 648 x^{4} - 1944 x^{3} + 2187 x^{2} - 2187 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{15} + 15 \nu^{14} - 40 \nu^{13} + 40 \nu^{12} - 5709 \nu^{11} + 30 \nu^{10} + 280 \nu^{9} - 685 \nu^{8} + 840 \nu^{7} - 90557 \nu^{6} + 945 \nu^{5} + 3240 \nu^{4} - 9720 \nu^{3} + 10935 \nu^{2} - 1444392 \nu + 32805$$$$)/261711$$ $$\beta_{2}$$ $$=$$ $$($$$$-160 \nu^{15} + 400 \nu^{14} + 3467 \nu^{12} - 1040 \nu^{11} - 4960 \nu^{10} + 6480 \nu^{9} - 2480 \nu^{8} + 39680 \nu^{7} - 19440 \nu^{6} - 66960 \nu^{5} + 103680 \nu^{4} - 19440 \nu^{3} + 602640 \nu^{2} - 349920 \nu - 524880$$$$)/785133$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 3 \nu^{12} - 48 \nu^{11} + 93 \nu^{10} + 81 \nu^{9} - 496 \nu^{8} + 93 \nu^{7} - 729 \nu^{6} + 1674 \nu^{5} + 1053 \nu^{4} - 8920 \nu^{3} - 10935 \nu + 13122$$$$)/9693$$ $$\beta_{4}$$ $$=$$ $$($$$$9 \nu^{15} - 40 \nu^{14} + 27 \nu^{13} - 72 \nu^{12} + 72 \nu^{11} + 63 \nu^{10} - 664 \nu^{9} + 504 \nu^{8} - 1233 \nu^{7} + 1512 \nu^{6} + 486 \nu^{5} - 20557 \nu^{4} + 5832 \nu^{3} - 17496 \nu^{2} + 19683 \nu - 19683$$$$)/58158$$ $$\beta_{5}$$ $$=$$ $$($$$$31 \nu^{15} + 718 \nu^{10} + 22258 \nu^{5} + 146286$$$$)/87237$$ $$\beta_{6}$$ $$=$$ $$($$$$-31 \nu^{15} + 93 \nu^{14} - 248 \nu^{13} + 248 \nu^{12} - 501 \nu^{11} + 186 \nu^{10} + 1736 \nu^{9} - 4247 \nu^{8} + 5208 \nu^{7} - 20584 \nu^{6} + 5859 \nu^{5} + 20088 \nu^{4} - 60264 \nu^{3} + 67797 \nu^{2} - 301320 \nu + 203391$$$$)/174474$$ $$\beta_{7}$$ $$=$$ $$($$$$-403 \nu^{15} - 806 \nu^{14} + 1053 \nu^{13} - 403 \nu^{12} + 6448 \nu^{11} - 12493 \nu^{10} - 10881 \nu^{9} + 28336 \nu^{8} - 12493 \nu^{7} + 97929 \nu^{6} - 224874 \nu^{5} - 141453 \nu^{4} + 266409 \nu^{3} + 1468935 \nu - 1762722$$$$)/523422$$ $$\beta_{8}$$ $$=$$ $$($$$$842 \nu^{15} + 1684 \nu^{14} - 324 \nu^{13} + 842 \nu^{12} - 13472 \nu^{11} + 26102 \nu^{10} + 22734 \nu^{9} + 4150 \nu^{8} + 26102 \nu^{7} - 204606 \nu^{6} + 469836 \nu^{5} + 295542 \nu^{4} + 5265 \nu^{3} - 3069090 \nu + 3682908$$$$)/785133$$ $$\beta_{9}$$ $$=$$ $$($$$$-1079 \nu^{15} - 2158 \nu^{14} - 6561 \nu^{13} - 1079 \nu^{12} + 17264 \nu^{11} - 33449 \nu^{10} - 29133 \nu^{9} - 79846 \nu^{8} - 33449 \nu^{7} + 262197 \nu^{6} - 602082 \nu^{5} - 378729 \nu^{4} - 1310985 \nu^{3} + 3932955 \nu - 4719546$$$$)/1570266$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{15} - 3 \nu^{14} + 8 \nu^{13} - 8 \nu^{12} - 7 \nu^{11} - 6 \nu^{10} - 56 \nu^{9} + 137 \nu^{8} - 168 \nu^{7} - 54 \nu^{6} - 189 \nu^{5} - 648 \nu^{4} + 1944 \nu^{3} - 2187 \nu^{2} + 386 \nu - 6561$$$$)/2154$$ $$\beta_{11}$$ $$=$$ $$($$$$646 \nu^{15} - 1615 \nu^{14} + 178 \nu^{12} + 4199 \nu^{11} + 20026 \nu^{10} - 26163 \nu^{9} + 10013 \nu^{8} + 14266 \nu^{7} + 78489 \nu^{6} + 270351 \nu^{5} - 418608 \nu^{4} + 78489 \nu^{3} - 164997 \nu^{2} + 1412802 \nu + 2119203$$$$)/785133$$ $$\beta_{12}$$ $$=$$ $$($$$$-2 \nu^{15} + 5 \nu^{14} - 15 \nu^{12} - 13 \nu^{11} - 62 \nu^{10} + 81 \nu^{9} - 31 \nu^{8} - 222 \nu^{7} - 243 \nu^{6} - 837 \nu^{5} + 1296 \nu^{4} - 243 \nu^{3} - 3955 \nu^{2} - 4374 \nu - 6561$$$$)/2154$$ $$\beta_{13}$$ $$=$$ $$($$$$-599 \nu^{15} - 130 \nu^{14} - 1797 \nu^{13} + 4792 \nu^{12} - 4792 \nu^{11} - 4193 \nu^{10} - 3594 \nu^{9} - 33544 \nu^{8} + 82063 \nu^{7} - 100632 \nu^{6} - 32346 \nu^{5} - 113211 \nu^{4} - 388152 \nu^{3} + 1164456 \nu^{2} - 1310013 \nu + 1310013$$$$)/523422$$ $$\beta_{14}$$ $$=$$ $$($$$$2170 \nu^{15} - 5425 \nu^{14} - 8855 \nu^{12} + 14105 \nu^{11} + 67270 \nu^{10} - 87885 \nu^{9} + 33635 \nu^{8} - 189212 \nu^{7} + 263655 \nu^{6} + 908145 \nu^{5} - 1406160 \nu^{4} + 263655 \nu^{3} - 2066715 \nu^{2} + 4745790 \nu + 7118685$$$$)/1570266$$ $$\beta_{15}$$ $$=$$ $$($$$$-248 \nu^{15} + 744 \nu^{14} - 1984 \nu^{13} + 1984 \nu^{12} - 4008 \nu^{11} + 1488 \nu^{10} + 13888 \nu^{9} - 33976 \nu^{8} + 41664 \nu^{7} - 77435 \nu^{6} + 46872 \nu^{5} + 160704 \nu^{4} - 482112 \nu^{3} + 542376 \nu^{2} - 1014768 \nu + 1627128$$$$)/261711$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{10} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{12} - 3 \beta_{11} - 3 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{9} + \beta_{8} - 2 \beta_{7} - 4 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{11} - \beta_{8} + \beta_{5} - 10 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{13} + 2 \beta_{12} + 2 \beta_{7} - 2 \beta_{6} + 15 \beta_{5} - 15$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{15} - 16 \beta_{10} - 16 \beta_{6} + 8 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-26 \beta_{14} + 35 \beta_{11} - 35 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$39 \beta_{8} + 70 \beta_{7} + 39 \beta_{3}$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$68 \beta_{15} - 74 \beta_{13} + 37 \beta_{11} - 37 \beta_{8} + 37 \beta_{5} + 74 \beta_{4} - 68 \beta_{3} + 68 \beta_{2} + 37 \beta_{1} - 68$$ $$\nu^{10}$$ $$=$$ $$($$$$62 \beta_{14} - 62 \beta_{10} - 62 \beta_{9} - 253 \beta_{5} - 62 \beta_{4} - 253$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-93 \beta_{15} + 506 \beta_{6} - 93 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$160 \beta_{14} + 160 \beta_{12} + 80 \beta_{11} + 679 \beta_{2}$$ $$\nu^{13}$$ $$=$$ $$($$$$-1198 \beta_{9} - 1079 \beta_{8} + 1079 \beta_{3}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-1797 \beta_{15} + 2158 \beta_{13} - 1797 \beta_{11} + 1797 \beta_{8} - 1797 \beta_{5} + 1797 \beta_{3} - 1797 \beta_{2} - 1797 \beta_{1} + 1797$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$-718 \beta_{14} - 718 \beta_{13} - 718 \beta_{12} + 718 \beta_{10} + 718 \beta_{9} - 718 \beta_{7} + 718 \beta_{6} + 359 \beta_{5} + 718 \beta_{4} + 3596$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 1.56693 + 0.738055i −0.144291 − 1.72603i −0.897801 − 1.48120i −0.833856 + 1.51812i −0.217724 − 1.71831i 1.59696 + 0.670602i 1.13127 + 1.31158i −1.70149 + 0.323920i 1.56693 − 0.738055i −0.144291 + 1.72603i −0.897801 + 1.48120i −0.833856 − 1.51812i −0.217724 + 1.71831i 1.59696 − 0.670602i 1.13127 − 1.31158i −1.70149 − 0.323920i
−2.40079 0.780063i 0.465695 + 0.640974i 3.53725 + 2.56996i −1.81200 + 1.31021i −0.618034 1.90211i 2.03615 2.80252i −3.51992 4.84475i 0.733075 2.25617i 5.37228 1.73205i
9.2 −0.753510 0.244830i 1.48377 + 2.04223i −1.11020 0.806607i −0.228670 2.22434i −0.618034 1.90211i −2.03615 + 2.80252i 1.57045 + 2.16154i −1.04209 + 3.20723i −0.372281 + 1.73205i
9.3 0.753510 + 0.244830i −1.48377 2.04223i −1.11020 0.806607i −1.12244 1.93394i −0.618034 1.90211i 2.03615 2.80252i −1.57045 2.16154i −1.04209 + 3.20723i −0.372281 1.73205i
9.4 2.40079 + 0.780063i −0.465695 0.640974i 3.53725 + 2.56996i 2.23606 0.00508966i −0.618034 1.90211i −2.03615 + 2.80252i 3.51992 + 4.84475i 0.733075 2.25617i 5.37228 + 1.73205i
124.1 −1.48377 2.04223i −0.753510 + 0.244830i −1.35111 + 4.15829i −1.80602 + 1.31844i 1.61803 + 1.17557i −3.29456 1.07047i 5.69534 1.85053i −1.91922 + 1.39439i 5.37228 + 1.73205i
124.2 −0.465695 0.640974i −2.40079 + 0.780063i 0.424058 1.30512i 2.04481 + 0.904839i 1.61803 + 1.17557i 3.29456 + 1.07047i −2.54105 + 0.825636i 2.72823 1.98218i −0.372281 1.73205i
124.3 0.465695 + 0.640974i 2.40079 0.780063i 0.424058 1.30512i 1.49244 + 1.66512i 1.61803 + 1.17557i −3.29456 1.07047i 2.54105 0.825636i 2.72823 1.98218i −0.372281 + 1.73205i
124.4 1.48377 + 2.04223i 0.753510 0.244830i −1.35111 + 4.15829i 0.695822 2.12505i 1.61803 + 1.17557i 3.29456 + 1.07047i −5.69534 + 1.85053i −1.91922 + 1.39439i 5.37228 1.73205i
269.1 −2.40079 + 0.780063i 0.465695 0.640974i 3.53725 2.56996i −1.81200 1.31021i −0.618034 + 1.90211i 2.03615 + 2.80252i −3.51992 + 4.84475i 0.733075 + 2.25617i 5.37228 + 1.73205i
269.2 −0.753510 + 0.244830i 1.48377 2.04223i −1.11020 + 0.806607i −0.228670 + 2.22434i −0.618034 + 1.90211i −2.03615 2.80252i 1.57045 2.16154i −1.04209 3.20723i −0.372281 1.73205i
269.3 0.753510 0.244830i −1.48377 + 2.04223i −1.11020 + 0.806607i −1.12244 + 1.93394i −0.618034 + 1.90211i 2.03615 + 2.80252i −1.57045 + 2.16154i −1.04209 3.20723i −0.372281 + 1.73205i
269.4 2.40079 0.780063i −0.465695 + 0.640974i 3.53725 2.56996i 2.23606 + 0.00508966i −0.618034 + 1.90211i −2.03615 2.80252i 3.51992 4.84475i 0.733075 + 2.25617i 5.37228 1.73205i
444.1 −1.48377 + 2.04223i −0.753510 0.244830i −1.35111 4.15829i −1.80602 1.31844i 1.61803 1.17557i −3.29456 + 1.07047i 5.69534 + 1.85053i −1.91922 1.39439i 5.37228 1.73205i
444.2 −0.465695 + 0.640974i −2.40079 0.780063i 0.424058 + 1.30512i 2.04481 0.904839i 1.61803 1.17557i 3.29456 1.07047i −2.54105 0.825636i 2.72823 + 1.98218i −0.372281 + 1.73205i
444.3 0.465695 0.640974i 2.40079 + 0.780063i 0.424058 + 1.30512i 1.49244 1.66512i 1.61803 1.17557i −3.29456 + 1.07047i 2.54105 + 0.825636i 2.72823 + 1.98218i −0.372281 1.73205i
444.4 1.48377 2.04223i 0.753510 + 0.244830i −1.35111 4.15829i 0.695822 + 2.12505i 1.61803 1.17557i 3.29456 1.07047i −5.69534 1.85053i −1.91922 1.39439i 5.37228 + 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 444.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 3 inner
55.j even 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.j 16
5.b even 2 1 inner 605.2.j.j 16
11.b odd 2 1 605.2.j.i 16
11.c even 5 1 605.2.b.c 4
11.c even 5 3 inner 605.2.j.j 16
11.d odd 10 1 55.2.b.a 4
11.d odd 10 3 605.2.j.i 16
33.f even 10 1 495.2.c.a 4
44.g even 10 1 880.2.b.h 4
55.d odd 2 1 605.2.j.i 16
55.h odd 10 1 55.2.b.a 4
55.h odd 10 3 605.2.j.i 16
55.j even 10 1 605.2.b.c 4
55.j even 10 3 inner 605.2.j.j 16
55.k odd 20 2 3025.2.a.ba 4
55.l even 20 2 275.2.a.h 4
165.r even 10 1 495.2.c.a 4
165.u odd 20 2 2475.2.a.bi 4
220.o even 10 1 880.2.b.h 4
220.w odd 20 2 4400.2.a.cc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 11.d odd 10 1
55.2.b.a 4 55.h odd 10 1
275.2.a.h 4 55.l even 20 2
495.2.c.a 4 33.f even 10 1
495.2.c.a 4 165.r even 10 1
605.2.b.c 4 11.c even 5 1
605.2.b.c 4 55.j even 10 1
605.2.j.i 16 11.b odd 2 1
605.2.j.i 16 11.d odd 10 3
605.2.j.i 16 55.d odd 2 1
605.2.j.i 16 55.h odd 10 3
605.2.j.j 16 1.a even 1 1 trivial
605.2.j.j 16 5.b even 2 1 inner
605.2.j.j 16 11.c even 5 3 inner
605.2.j.j 16 55.j even 10 3 inner
880.2.b.h 4 44.g even 10 1
880.2.b.h 4 220.o even 10 1
2475.2.a.bi 4 165.u odd 20 2
3025.2.a.ba 4 55.k odd 20 2
4400.2.a.cc 4 220.w odd 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{16} - \cdots$$ $$T_{19}^{4} + 4 T_{19}^{3} + 16 T_{19}^{2} + 64 T_{19} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 - 448 T^{2} + 720 T^{4} - 1148 T^{6} + 1829 T^{8} - 287 T^{10} + 45 T^{12} - 7 T^{14} + T^{16}$$
$3$ $$256 - 448 T^{2} + 720 T^{4} - 1148 T^{6} + 1829 T^{8} - 287 T^{10} + 45 T^{12} - 7 T^{14} + T^{16}$$
$5$ $$390625 - 234375 T + 78125 T^{2} - 56250 T^{3} + 33750 T^{4} - 18375 T^{5} + 9250 T^{6} - 4410 T^{7} + 2021 T^{8} - 882 T^{9} + 370 T^{10} - 147 T^{11} + 54 T^{12} - 18 T^{13} + 5 T^{14} - 3 T^{15} + T^{16}$$
$7$ $$( 20736 - 1728 T^{2} + 144 T^{4} - 12 T^{6} + T^{8} )^{2}$$
$11$ $$T^{16}$$
$13$ $$T^{16}$$
$17$ $$16777216 - 7340032 T^{2} + 2949120 T^{4} - 1175552 T^{6} + 468224 T^{8} - 18368 T^{10} + 720 T^{12} - 28 T^{14} + T^{16}$$
$19$ $$( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$23$ $$( 4 + 7 T^{2} + T^{4} )^{4}$$
$29$ $$( 331776 - 82944 T + 34560 T^{2} - 12096 T^{3} + 4464 T^{4} + 504 T^{5} + 60 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$31$ $$( 4096 - 512 T + 576 T^{2} - 136 T^{3} + 89 T^{4} + 17 T^{5} + 9 T^{6} + T^{7} + T^{8} )^{2}$$
$37$ $$429981696 - 367276032 T^{2} + 310728960 T^{4} - 262863792 T^{6} + 222371649 T^{8} - 1825443 T^{10} + 14985 T^{12} - 123 T^{14} + T^{16}$$
$41$ $$( 331776 + 82944 T + 34560 T^{2} + 12096 T^{3} + 4464 T^{4} - 504 T^{5} + 60 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$43$ $$( 12 + T^{2} )^{8}$$
$47$ $$( 3748096 - 85184 T^{2} + 1936 T^{4} - 44 T^{6} + T^{8} )^{2}$$
$53$ $$1099511627776 - 120259084288 T^{2} + 12079595520 T^{4} - 1203765248 T^{6} + 119865344 T^{8} - 1175552 T^{10} + 11520 T^{12} - 112 T^{14} + T^{16}$$
$59$ $$( 20736 + 15552 T + 9936 T^{2} + 6156 T^{3} + 3789 T^{4} + 513 T^{5} + 69 T^{6} + 9 T^{7} + T^{8} )^{2}$$
$61$ $$( 4096 + 5120 T + 6912 T^{2} + 9280 T^{3} + 12464 T^{4} - 1160 T^{5} + 108 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$67$ $$( 36 + 87 T^{2} + T^{4} )^{4}$$
$71$ $$( 26873856 + 1119744 T + 419904 T^{2} + 33048 T^{3} + 7209 T^{4} - 459 T^{5} + 81 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$73$ $$( 5308416 - 110592 T^{2} + 2304 T^{4} - 48 T^{6} + T^{8} )^{2}$$
$79$ $$( 65536 + 57344 T + 46080 T^{2} + 36736 T^{3} + 29264 T^{4} + 2296 T^{5} + 180 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$83$ $$( 3748096 - 85184 T^{2} + 1936 T^{4} - 44 T^{6} + T^{8} )^{2}$$
$89$ $$( -6 - 3 T + T^{2} )^{8}$$
$97$ $$110075314176 - 9746251776 T^{2} + 671846400 T^{4} - 42565824 T^{6} + 2602449 T^{8} - 73899 T^{10} + 2025 T^{12} - 51 T^{14} + T^{16}$$