# Properties

 Label 300.3.f.b Level $300$ Weight $3$ Character orbit 300.f Analytic conductor $8.174$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 5 x^{14} + 12 x^{12} + 25 x^{10} + 53 x^{8} + 100 x^{6} + 192 x^{4} + 320 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{24}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{12} q^{2} -\beta_{9} q^{3} + ( -1 + \beta_{4} ) q^{4} + ( -1 - \beta_{2} ) q^{6} + ( \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{7} + ( -\beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{12} q^{2} -\beta_{9} q^{3} + ( -1 + \beta_{4} ) q^{4} + ( -1 - \beta_{2} ) q^{6} + ( \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{7} + ( -\beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{8} + 3 q^{9} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + 2 \beta_{10} ) q^{11} + ( \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{12} + ( \beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{13} + ( 3 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 4 \beta_{5} + 2 \beta_{7} ) q^{14} + ( 5 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{10} ) q^{16} + ( -\beta_{6} - \beta_{8} + 2 \beta_{9} + 3 \beta_{11} - 3 \beta_{12} + 3 \beta_{14} - \beta_{15} ) q^{17} + 3 \beta_{12} q^{18} + ( 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 5 \beta_{5} + 3 \beta_{7} + 2 \beta_{10} ) q^{19} + ( -6 - \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{7} - \beta_{10} ) q^{21} + ( -\beta_{6} - 3 \beta_{8} - 3 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{22} + ( -3 \beta_{6} + 3 \beta_{8} - 6 \beta_{9} + 2 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{23} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{10} ) q^{24} + ( -2 + \beta_{1} + 3 \beta_{3} - 4 \beta_{5} - 5 \beta_{7} + \beta_{10} ) q^{26} -3 \beta_{9} q^{27} + ( -4 \beta_{6} + 5 \beta_{8} + 4 \beta_{9} - 3 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} + 7 \beta_{14} - 5 \beta_{15} ) q^{28} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} - 8 \beta_{7} - 3 \beta_{10} ) q^{29} + ( 2 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - \beta_{5} + 3 \beta_{7} - 2 \beta_{10} ) q^{31} + ( -\beta_{6} + 5 \beta_{8} + 6 \beta_{9} + 2 \beta_{11} + 6 \beta_{12} - 5 \beta_{13} + \beta_{14} ) q^{32} + ( -\beta_{8} + 2 \beta_{9} + \beta_{11} - 5 \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{33} + ( 12 + 3 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} - \beta_{7} + 5 \beta_{10} ) q^{34} + ( -3 + 3 \beta_{4} ) q^{36} + ( 3 \beta_{6} + 9 \beta_{8} - 4 \beta_{9} - 7 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} - 7 \beta_{14} + \beta_{15} ) q^{37} + ( -2 \beta_{6} - 2 \beta_{8} + 6 \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - 6 \beta_{13} + 10 \beta_{14} - 6 \beta_{15} ) q^{38} + ( -\beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 6 \beta_{10} ) q^{39} + ( -6 - 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( \beta_{6} - 9 \beta_{8} - \beta_{9} - 3 \beta_{11} - 9 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{42} + ( 4 \beta_{6} - 4 \beta_{8} - 24 \beta_{9} + 8 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} - 8 \beta_{14} + 8 \beta_{15} ) q^{43} + ( -21 + 6 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + \beta_{10} ) q^{44} + ( 20 + 6 \beta_{1} - 2 \beta_{3} + 8 \beta_{4} - 12 \beta_{5} + 6 \beta_{7} - 2 \beta_{10} ) q^{46} + ( 3 \beta_{6} - 3 \beta_{8} - 12 \beta_{9} + 4 \beta_{11} + 4 \beta_{12} - 6 \beta_{13} - 4 \beta_{14} - 3 \beta_{15} ) q^{47} + ( \beta_{6} + 4 \beta_{8} - 4 \beta_{9} - 7 \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{48} + ( 7 + 4 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} + 12 \beta_{7} + 12 \beta_{10} ) q^{49} + ( -5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{51} + ( 10 \beta_{8} + 24 \beta_{9} - 14 \beta_{11} - 8 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{52} + ( 2 \beta_{6} + 22 \beta_{8} + \beta_{11} + \beta_{12} + 4 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{53} + ( -3 - 3 \beta_{2} ) q^{54} + ( 17 + 4 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 14 \beta_{5} - 9 \beta_{7} - 3 \beta_{10} ) q^{56} + ( \beta_{6} - 9 \beta_{8} - 6 \beta_{11} - 6 \beta_{12} + 2 \beta_{13} - 6 \beta_{14} - \beta_{15} ) q^{57} + ( -7 \beta_{6} - 11 \beta_{8} + 23 \beta_{9} - 13 \beta_{11} - 19 \beta_{12} - \beta_{13} - 5 \beta_{14} - 3 \beta_{15} ) q^{58} + ( 5 \beta_{1} + 4 \beta_{2} + \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - \beta_{7} - 6 \beta_{10} ) q^{59} + ( -26 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 14 \beta_{7} + 6 \beta_{10} ) q^{61} + ( -6 \beta_{6} + 6 \beta_{8} - 22 \beta_{9} - 2 \beta_{11} + 6 \beta_{12} - 10 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} ) q^{62} + ( 3 \beta_{9} + 3 \beta_{11} + 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{63} + ( 13 - 3 \beta_{1} + \beta_{2} - 8 \beta_{3} + 6 \beta_{4} + 9 \beta_{5} - 4 \beta_{7} - 3 \beta_{10} ) q^{64} + ( 15 + \beta_{1} + 3 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} + 4 \beta_{10} ) q^{66} + ( -2 \beta_{6} + 2 \beta_{8} - 6 \beta_{9} - 2 \beta_{11} - 8 \beta_{12} + 10 \beta_{13} + 2 \beta_{14} + 8 \beta_{15} ) q^{67} + ( 4 \beta_{6} - 12 \beta_{8} + 4 \beta_{9} - 4 \beta_{11} + 12 \beta_{12} - 10 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{68} + ( 8 - 4 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 7 \beta_{7} ) q^{69} + ( -6 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{71} + ( -3 \beta_{6} - 3 \beta_{8} + 6 \beta_{9} - 6 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{72} + ( 6 \beta_{6} - 11 \beta_{8} - 8 \beta_{9} - 6 \beta_{11} + 18 \beta_{12} + 4 \beta_{13} - 6 \beta_{14} + 2 \beta_{15} ) q^{73} + ( -16 - 7 \beta_{1} + 11 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} - 7 \beta_{7} - 17 \beta_{10} ) q^{74} + ( -2 - 12 \beta_{2} + 24 \beta_{5} - 10 \beta_{7} + 6 \beta_{10} ) q^{76} + ( 20 \beta_{8} - 4 \beta_{9} + 12 \beta_{11} + 24 \beta_{12} - 4 \beta_{13} + 12 \beta_{14} + 4 \beta_{15} ) q^{77} + ( \beta_{6} - 16 \beta_{8} + 5 \beta_{9} + 4 \beta_{11} - \beta_{12} + 3 \beta_{13} + 6 \beta_{14} + 6 \beta_{15} ) q^{78} + ( -6 \beta_{1} - 9 \beta_{2} + 17 \beta_{3} - \beta_{4} + 15 \beta_{5} - 17 \beta_{7} - 18 \beta_{10} ) q^{79} + 9 q^{81} + ( 8 \beta_{6} - 8 \beta_{8} - 24 \beta_{9} + 8 \beta_{11} + 2 \beta_{12} ) q^{82} + ( 10 \beta_{6} - 10 \beta_{8} - 18 \beta_{9} + 14 \beta_{11} + 4 \beta_{12} - 10 \beta_{13} - 14 \beta_{14} ) q^{83} + ( 1 - 6 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 9 \beta_{7} + 7 \beta_{10} ) q^{84} + ( -32 - 4 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 24 \beta_{5} + 12 \beta_{7} + 4 \beta_{10} ) q^{86} + ( 7 \beta_{6} - 7 \beta_{8} + \beta_{9} + 7 \beta_{11} - 12 \beta_{12} + 5 \beta_{13} - 7 \beta_{14} + 12 \beta_{15} ) q^{87} + ( 2 \beta_{6} - 3 \beta_{8} + 14 \beta_{9} + 13 \beta_{11} - 22 \beta_{12} - 8 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{88} + ( -14 - 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + 20 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 22 \beta_{10} ) q^{89} + ( -10 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{5} - 8 \beta_{7} - 8 \beta_{10} ) q^{91} + ( -8 \beta_{6} - 4 \beta_{8} + 52 \beta_{9} + 8 \beta_{11} + 12 \beta_{12} + 28 \beta_{14} - 16 \beta_{15} ) q^{92} + ( -3 \beta_{6} + 7 \beta_{8} + 4 \beta_{9} - 10 \beta_{11} - 22 \beta_{12} - 2 \beta_{13} - 10 \beta_{14} - \beta_{15} ) q^{93} + ( 6 - 16 \beta_{1} - 24 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + 10 \beta_{10} ) q^{94} + ( -24 - 3 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 7 \beta_{5} - 3 \beta_{7} - 12 \beta_{10} ) q^{96} + ( -4 \beta_{6} - 33 \beta_{8} + 12 \beta_{9} + 12 \beta_{11} - 24 \beta_{12} + 4 \beta_{13} + 12 \beta_{14} - 8 \beta_{15} ) q^{97} + ( 4 \beta_{6} + 44 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} + 27 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} + 12 \beta_{15} ) q^{98} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} + 6 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 20q^{4} - 12q^{6} + 48q^{9} + O(q^{10})$$ $$16q - 20q^{4} - 12q^{6} + 48q^{9} + 40q^{14} + 68q^{16} - 96q^{21} - 36q^{24} - 72q^{26} - 128q^{29} + 184q^{34} - 60q^{36} - 32q^{41} - 344q^{44} + 304q^{46} + 112q^{49} - 36q^{54} + 232q^{56} - 352q^{61} + 220q^{64} + 216q^{66} + 192q^{69} - 264q^{74} - 48q^{76} + 144q^{81} + 72q^{84} - 400q^{86} - 160q^{89} + 192q^{94} - 348q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5 x^{14} + 12 x^{12} + 25 x^{10} + 53 x^{8} + 100 x^{6} + 192 x^{4} + 320 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{12} - 8 \nu^{10} - 25 \nu^{8} - 17 \nu^{6} - 96 \nu^{4} - 160 \nu^{2} - 128$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{14} - 3 \nu^{12} + 16 \nu^{10} + 5 \nu^{8} - 67 \nu^{6} + 24 \nu^{4} + 248 \nu^{2} - 336$$$$)/304$$ $$\beta_{3}$$ $$=$$ $$($$$$-17 \nu^{14} - 81 \nu^{12} - 176 \nu^{10} - 321 \nu^{8} - 593 \nu^{6} - 1176 \nu^{4} - 2120 \nu^{2} - 3296$$$$)/304$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{14} - 16 \nu^{12} - 35 \nu^{10} - 81 \nu^{8} - 180 \nu^{6} - 271 \nu^{4} - 552 \nu^{2} - 804$$$$)/76$$ $$\beta_{5}$$ $$=$$ $$($$$$165 \nu^{14} + 357 \nu^{12} + 680 \nu^{10} + 1533 \nu^{8} + 2805 \nu^{6} + 4896 \nu^{4} + 8640 \nu^{2} + 9280$$$$)/2432$$ $$\beta_{6}$$ $$=$$ $$($$$$41 \nu^{15} + 97 \nu^{13} + 192 \nu^{11} + 193 \nu^{9} + 241 \nu^{7} - 1384 \nu^{5} + 1760 \nu^{3} + 832 \nu$$$$)/2432$$ $$\beta_{7}$$ $$=$$ $$($$$$267 \nu^{14} + 843 \nu^{12} + 1432 \nu^{10} + 3763 \nu^{8} + 6971 \nu^{6} + 11040 \nu^{4} + 25920 \nu^{2} + 32704$$$$)/2432$$ $$\beta_{8}$$ $$=$$ $$($$$$71 \nu^{15} + 231 \nu^{13} + 440 \nu^{11} + 1135 \nu^{9} + 1815 \nu^{7} + 3168 \nu^{5} + 7200 \nu^{3} + 9152 \nu$$$$)/2432$$ $$\beta_{9}$$ $$=$$ $$($$$$15 \nu^{15} + 47 \nu^{13} + 88 \nu^{11} + 215 \nu^{9} + 479 \nu^{7} + 704 \nu^{5} + 1536 \nu^{3} + 1984 \nu$$$$)/512$$ $$\beta_{10}$$ $$=$$ $$($$$$-407 \nu^{14} - 1367 \nu^{12} - 2488 \nu^{10} - 5727 \nu^{8} - 11783 \nu^{6} - 20832 \nu^{4} - 43200 \nu^{2} - 56128$$$$)/2432$$ $$\beta_{11}$$ $$=$$ $$($$$$-353 \nu^{15} - 1217 \nu^{13} - 2984 \nu^{11} - 4761 \nu^{9} - 10257 \nu^{7} - 19904 \nu^{5} - 38272 \nu^{3} - 53312 \nu$$$$)/9728$$ $$\beta_{12}$$ $$=$$ $$($$$$-103 \nu^{15} - 303 \nu^{13} - 512 \nu^{11} - 1167 \nu^{9} - 2207 \nu^{7} - 4264 \nu^{5} - 8544 \nu^{3} - 9920 \nu$$$$)/2432$$ $$\beta_{13}$$ $$=$$ $$($$$$-449 \nu^{15} - 1889 \nu^{13} - 3048 \nu^{11} - 7289 \nu^{9} - 15537 \nu^{7} - 27904 \nu^{5} - 62976 \nu^{3} - 99392 \nu$$$$)/9728$$ $$\beta_{14}$$ $$=$$ $$($$$$527 \nu^{15} + 1295 \nu^{13} + 2872 \nu^{11} + 5847 \nu^{9} + 10175 \nu^{7} + 21408 \nu^{5} + 54016 \nu^{3} + 48064 \nu$$$$)/9728$$ $$\beta_{15}$$ $$=$$ $$($$$$427 \nu^{15} + 1051 \nu^{13} + 2248 \nu^{11} + 5139 \nu^{9} + 9995 \nu^{7} + 19408 \nu^{5} + 39936 \nu^{3} + 56000 \nu$$$$)/4864$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{9} + \beta_{6}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{7} - 4 \beta_{5} + \beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + 3 \beta_{14} - \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{9} - \beta_{8}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{10} - 7 \beta_{7} + 4 \beta_{5} + 4 \beta_{4} + \beta_{3} - 6 \beta_{2} - 5 \beta_{1}$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{13} - 7 \beta_{12} - 2 \beta_{11} - 5 \beta_{9} - 2 \beta_{8} - 7 \beta_{6}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{7} - 6 \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_{1} - 10$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + 5 \beta_{12} + 41 \beta_{9} - 20 \beta_{8} + \beta_{6}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$21 \beta_{10} + 27 \beta_{7} + 4 \beta_{5} - 5 \beta_{3} - 20 \beta_{2} - 21 \beta_{1} - 12$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$\beta_{15} - 3 \beta_{14} + 5 \beta_{13} + 10 \beta_{12} + 7 \beta_{11} - 5 \beta_{9} + 41 \beta_{8} - 4 \beta_{6}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$21 \beta_{10} - 9 \beta_{7} - 4 \beta_{5} - 36 \beta_{4} - 33 \beta_{3} + 54 \beta_{2} + 21 \beta_{1} - 16$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$23 \beta_{13} + 47 \beta_{12} - 78 \beta_{11} - 19 \beta_{9} + 18 \beta_{8} + 15 \beta_{6}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$-12 \beta_{10} + 14 \beta_{7} + 54 \beta_{4} - 14 \beta_{3} - 31 \beta_{2} + 17 \beta_{1} - 46$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-94 \beta_{15} - 30 \beta_{14} - 67 \beta_{13} - 213 \beta_{12} - 73 \beta_{9} - 12 \beta_{8} + 15 \beta_{6}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$-29 \beta_{10} - 51 \beta_{7} + 188 \beta_{5} + 93 \beta_{3} + 100 \beta_{2} + 29 \beta_{1} + 476$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$135 \beta_{15} - 53 \beta_{14} + 63 \beta_{13} - 134 \beta_{12} + 73 \beta_{11} - 63 \beta_{9} - 153 \beta_{8} + 72 \beta_{6}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 1.28061 − 0.600040i 1.28061 + 0.600040i −0.120653 + 1.40906i −0.120653 − 1.40906i 0.422403 − 1.34966i 0.422403 + 1.34966i −0.957636 − 1.04064i −0.957636 + 1.04064i 0.957636 − 1.04064i 0.957636 + 1.04064i −0.422403 − 1.34966i −0.422403 + 1.34966i 0.120653 + 1.40906i 0.120653 − 1.40906i −1.28061 − 0.600040i −1.28061 + 0.600040i
−1.95141 0.438172i 1.73205 3.61601 + 1.71011i 0 −3.37994 0.758935i −6.33166 −6.30701 4.92155i 3.00000 0
199.2 −1.95141 + 0.438172i 1.73205 3.61601 1.71011i 0 −3.37994 + 0.758935i −6.33166 −6.30701 + 4.92155i 3.00000 0
199.3 −1.08539 1.67986i −1.73205 −1.64388 + 3.64660i 0 1.87994 + 2.90961i 0.596540 7.91002 1.19648i 3.00000 0
199.4 −1.08539 + 1.67986i −1.73205 −1.64388 3.64660i 0 1.87994 2.90961i 0.596540 7.91002 + 1.19648i 3.00000 0
199.5 −0.696577 1.87477i 1.73205 −3.02956 + 2.61185i 0 −1.20651 3.24721i 5.46770 7.00695 + 3.86039i 3.00000 0
199.6 −0.696577 + 1.87477i 1.73205 −3.02956 2.61185i 0 −1.20651 + 3.24721i 5.46770 7.00695 3.86039i 3.00000 0
199.7 −0.169449 1.99281i 1.73205 −3.94257 + 0.675358i 0 −0.293494 3.45165i −12.3959 2.01392 + 7.74236i 3.00000 0
199.8 −0.169449 + 1.99281i 1.73205 −3.94257 0.675358i 0 −0.293494 + 3.45165i −12.3959 2.01392 7.74236i 3.00000 0
199.9 0.169449 1.99281i −1.73205 −3.94257 0.675358i 0 −0.293494 + 3.45165i 12.3959 −2.01392 + 7.74236i 3.00000 0
199.10 0.169449 + 1.99281i −1.73205 −3.94257 + 0.675358i 0 −0.293494 3.45165i 12.3959 −2.01392 7.74236i 3.00000 0
199.11 0.696577 1.87477i −1.73205 −3.02956 2.61185i 0 −1.20651 + 3.24721i −5.46770 −7.00695 + 3.86039i 3.00000 0
199.12 0.696577 + 1.87477i −1.73205 −3.02956 + 2.61185i 0 −1.20651 3.24721i −5.46770 −7.00695 3.86039i 3.00000 0
199.13 1.08539 1.67986i 1.73205 −1.64388 3.64660i 0 1.87994 2.90961i −0.596540 −7.91002 1.19648i 3.00000 0
199.14 1.08539 + 1.67986i 1.73205 −1.64388 + 3.64660i 0 1.87994 + 2.90961i −0.596540 −7.91002 + 1.19648i 3.00000 0
199.15 1.95141 0.438172i −1.73205 3.61601 1.71011i 0 −3.37994 + 0.758935i 6.33166 6.30701 4.92155i 3.00000 0
199.16 1.95141 + 0.438172i −1.73205 3.61601 + 1.71011i 0 −3.37994 0.758935i 6.33166 6.30701 + 4.92155i 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.f.b 16
3.b odd 2 1 900.3.f.f 16
4.b odd 2 1 inner 300.3.f.b 16
5.b even 2 1 inner 300.3.f.b 16
5.c odd 4 1 60.3.c.a 8
5.c odd 4 1 300.3.c.d 8
12.b even 2 1 900.3.f.f 16
15.d odd 2 1 900.3.f.f 16
15.e even 4 1 180.3.c.b 8
15.e even 4 1 900.3.c.u 8
20.d odd 2 1 inner 300.3.f.b 16
20.e even 4 1 60.3.c.a 8
20.e even 4 1 300.3.c.d 8
40.i odd 4 1 960.3.e.c 8
40.k even 4 1 960.3.e.c 8
60.h even 2 1 900.3.f.f 16
60.l odd 4 1 180.3.c.b 8
60.l odd 4 1 900.3.c.u 8
120.q odd 4 1 2880.3.e.j 8
120.w even 4 1 2880.3.e.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 5.c odd 4 1
60.3.c.a 8 20.e even 4 1
180.3.c.b 8 15.e even 4 1
180.3.c.b 8 60.l odd 4 1
300.3.c.d 8 5.c odd 4 1
300.3.c.d 8 20.e even 4 1
300.3.f.b 16 1.a even 1 1 trivial
300.3.f.b 16 4.b odd 2 1 inner
300.3.f.b 16 5.b even 2 1 inner
300.3.f.b 16 20.d odd 2 1 inner
900.3.c.u 8 15.e even 4 1
900.3.c.u 8 60.l odd 4 1
900.3.f.f 16 3.b odd 2 1
900.3.f.f 16 12.b even 2 1
900.3.f.f 16 15.d odd 2 1
900.3.f.f 16 60.h even 2 1
960.3.e.c 8 40.i odd 4 1
960.3.e.c 8 40.k even 4 1
2880.3.e.j 8 120.q odd 4 1
2880.3.e.j 8 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 224 T_{7}^{6} + 12032 T_{7}^{4} - 188416 T_{7}^{2} + 65536$$ acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$65536 + 40960 T^{2} + 8448 T^{4} - 640 T^{6} - 592 T^{8} - 40 T^{10} + 33 T^{12} + 10 T^{14} + T^{16}$$
$3$ $$( -3 + T^{2} )^{8}$$
$5$ $$T^{16}$$
$7$ $$( 65536 - 188416 T^{2} + 12032 T^{4} - 224 T^{6} + T^{8} )^{2}$$
$11$ $$( 10496 + 208 T^{2} + T^{4} )^{4}$$
$13$ $$( 155351296 + 21781248 T^{2} + 290528 T^{4} + 1008 T^{6} + T^{8} )^{2}$$
$17$ $$( 77721856 + 7269632 T^{2} + 162144 T^{4} + 848 T^{6} + T^{8} )^{2}$$
$19$ $$( 6544162816 + 173686784 T^{2} + 925952 T^{4} + 1696 T^{6} + T^{8} )^{2}$$
$23$ $$( 101419319296 - 1884176384 T^{2} + 4397312 T^{4} - 3616 T^{6} + T^{8} )^{2}$$
$29$ $$( 1334416 - 34688 T - 2152 T^{2} + 32 T^{3} + T^{4} )^{4}$$
$31$ $$( 59895709696 + 2731491328 T^{2} + 7432448 T^{4} + 5408 T^{6} + T^{8} )^{2}$$
$37$ $$( 59919206656 + 2291918592 T^{2} + 8020448 T^{4} + 6192 T^{6} + T^{8} )^{2}$$
$41$ $$( 87184 - 7264 T - 1800 T^{2} + 8 T^{3} + T^{4} )^{4}$$
$43$ $$( 33624411406336 - 62108155904 T^{2} + 40259072 T^{4} - 10816 T^{6} + T^{8} )^{2}$$
$47$ $$( 1056981385216 - 9701752832 T^{2} + 15726848 T^{4} - 8032 T^{6} + T^{8} )^{2}$$
$53$ $$( 228545188096 + 35356046592 T^{2} + 37352288 T^{4} + 11472 T^{6} + T^{8} )^{2}$$
$59$ $$( 173909016576 + 2174459904 T^{2} + 6273792 T^{4} + 4896 T^{6} + T^{8} )^{2}$$
$61$ $$( -2142704 - 273568 T - 2536 T^{2} + 88 T^{3} + T^{4} )^{4}$$
$67$ $$( 281086590976 - 15044755456 T^{2} + 57554432 T^{4} - 16064 T^{6} + T^{8} )^{2}$$
$71$ $$( 16079971680256 + 101402017792 T^{2} + 64237568 T^{4} + 13952 T^{6} + T^{8} )^{2}$$
$73$ $$( 24622079140096 + 121864272128 T^{2} + 90628704 T^{4} + 17552 T^{6} + T^{8} )^{2}$$
$79$ $$( 3198642669223936 + 2420601929728 T^{2} + 550899968 T^{4} + 41888 T^{6} + T^{8} )^{2}$$
$83$ $$( 4284940379815936 - 2381453017088 T^{2} + 464465408 T^{4} - 36928 T^{6} + T^{8} )^{2}$$
$89$ $$( 70652944 - 757600 T - 20584 T^{2} + 40 T^{3} + T^{4} )^{4}$$
$97$ $$( 3514328965714176 + 2030236372224 T^{2} + 414794592 T^{4} + 34896 T^{6} + T^{8} )^{2}$$