# Properties

 Label 300.3.f.c 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: $$x^{16} - 4 x^{15} + 8 x^{14} - 14 x^{13} + 23 x^{12} - 26 x^{11} + 18 x^{10} - 10 x^{9} + 9 x^{8} - 20 x^{7} + 72 x^{6} - 208 x^{5} + 368 x^{4} - 448 x^{3} + 512 x^{2} - 512 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{20}\cdot 5^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{4} q^{3} + ( 1 + \beta_{5} ) q^{4} + ( -1 - \beta_{9} ) q^{6} + ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{13} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} - \beta_{13} + \beta_{14} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{4} q^{3} + ( 1 + \beta_{5} ) q^{4} + ( -1 - \beta_{9} ) q^{6} + ( \beta_{3} - \beta_{4} - \beta_{8} + \beta_{13} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} - \beta_{13} + \beta_{14} ) q^{8} + 3 q^{9} + ( \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{11} + ( \beta_{3} - \beta_{4} - \beta_{12} ) q^{12} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{8} + \beta_{12} ) q^{13} + ( -3 - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{15} ) q^{14} + ( 4 + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{9} + \beta_{10} - \beta_{11} ) q^{16} + ( 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{12} + 2 \beta_{13} ) q^{17} -3 \beta_{3} q^{18} + ( \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{15} ) q^{19} + ( 3 + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{15} ) q^{21} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{8} - \beta_{12} - \beta_{14} ) q^{22} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{23} + ( 4 - 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{10} ) q^{24} + ( -9 + 4 \beta_{5} + 4 \beta_{6} + \beta_{7} - 3 \beta_{9} + \beta_{15} ) q^{26} -3 \beta_{4} q^{27} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - 2 \beta_{8} + \beta_{12} + 4 \beta_{13} + 2 \beta_{14} ) q^{28} + ( 6 - 5 \beta_{5} + 6 \beta_{9} + 2 \beta_{11} + \beta_{15} ) q^{29} + ( -4 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - 3 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{15} ) q^{31} + ( 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} ) q^{32} + ( -\beta_{2} - 3 \beta_{3} + 2 \beta_{8} + 2 \beta_{13} - 2 \beta_{14} ) q^{33} + ( -16 + 8 \beta_{5} - \beta_{7} - 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{15} ) q^{34} + ( 3 + 3 \beta_{5} ) q^{36} + ( 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 2 \beta_{8} + 2 \beta_{12} + 8 \beta_{14} ) q^{37} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{8} - \beta_{12} - 2 \beta_{14} ) q^{38} + ( -5 \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{15} ) q^{39} + ( -5 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 6 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - \beta_{15} ) q^{41} + ( -\beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{8} - \beta_{12} - 4 \beta_{13} - \beta_{14} ) q^{42} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 9 \beta_{4} - 4 \beta_{8} - 3 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{43} + ( -6 - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} - 6 \beta_{10} - 2 \beta_{11} ) q^{44} + ( -12 - 4 \beta_{5} - \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} - 4 \beta_{15} ) q^{46} + ( -6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 18 \beta_{4} - 6 \beta_{8} + 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} ) q^{47} + ( -3 \beta_{1} - \beta_{2} - 5 \beta_{3} - 8 \beta_{4} - \beta_{8} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{48} + ( 2 + 2 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + 12 \beta_{9} + 4 \beta_{10} - 6 \beta_{15} ) q^{49} + ( -3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} + 3 \beta_{10} - \beta_{11} - 4 \beta_{15} ) q^{51} + ( 5 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} - \beta_{8} + 2 \beta_{12} - 5 \beta_{14} ) q^{52} + ( \beta_{2} + 5 \beta_{3} - 4 \beta_{8} - 4 \beta_{13} - 4 \beta_{14} ) q^{53} + ( -3 - 3 \beta_{9} ) q^{54} + ( -24 + 2 \beta_{5} - \beta_{6} - \beta_{7} - 8 \beta_{9} - 2 \beta_{10} + 8 \beta_{15} ) q^{56} + ( 2 \beta_{1} + 5 \beta_{2} - \beta_{3} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{57} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} + 22 \beta_{4} - 3 \beta_{8} + 7 \beta_{12} + 8 \beta_{13} - 3 \beta_{14} ) q^{58} + ( -2 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{15} ) q^{59} + ( 23 - 8 \beta_{6} - \beta_{7} + 18 \beta_{9} - \beta_{10} + 7 \beta_{11} - 4 \beta_{15} ) q^{61} + ( -\beta_{1} - 8 \beta_{2} + 2 \beta_{3} + 20 \beta_{4} + 2 \beta_{8} - 7 \beta_{12} + 2 \beta_{14} ) q^{62} + ( 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{8} + 3 \beta_{13} ) q^{63} + ( -4 - 2 \beta_{7} - 6 \beta_{10} - 2 \beta_{11} - 8 \beta_{15} ) q^{64} + ( -12 - 8 \beta_{6} + \beta_{7} + 6 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{15} ) q^{66} + ( -11 \beta_{1} - 3 \beta_{2} - 18 \beta_{3} + 3 \beta_{4} - 4 \beta_{8} + \beta_{12} - \beta_{13} + 5 \beta_{14} ) q^{67} + ( 2 \beta_{1} + 10 \beta_{2} + 12 \beta_{3} - 14 \beta_{4} + 6 \beta_{8} - 8 \beta_{13} + 6 \beta_{14} ) q^{68} + ( 14 + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 6 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} - \beta_{15} ) q^{69} + ( -\beta_{5} + 3 \beta_{6} - 5 \beta_{7} + 4 \beta_{9} + \beta_{10} - 5 \beta_{11} - 4 \beta_{15} ) q^{71} + ( -6 \beta_{3} - 6 \beta_{4} - 3 \beta_{13} + 3 \beta_{14} ) q^{72} + ( -10 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} + 6 \beta_{8} - 10 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} ) q^{73} + ( 10 + 8 \beta_{5} + 8 \beta_{6} - 14 \beta_{9} + 10 \beta_{10} - 10 \beta_{11} + 10 \beta_{15} ) q^{74} + ( 15 + \beta_{5} + 4 \beta_{6} - 9 \beta_{7} + 4 \beta_{9} + \beta_{10} - 5 \beta_{11} ) q^{76} + ( -2 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + 6 \beta_{8} - 2 \beta_{12} + 4 \beta_{13} + 8 \beta_{14} ) q^{77} + ( -6 \beta_{1} - 3 \beta_{2} + \beta_{3} + 6 \beta_{4} + 3 \beta_{8} - 4 \beta_{12} + 3 \beta_{14} ) q^{78} + ( -4 \beta_{5} + 2 \beta_{7} - 8 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 4 \beta_{15} ) q^{79} + 9 q^{81} + ( 3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} + 38 \beta_{4} + 9 \beta_{8} + 7 \beta_{12} - 15 \beta_{14} ) q^{82} + ( 4 \beta_{1} + 4 \beta_{3} + 22 \beta_{4} + 16 \beta_{8} + 4 \beta_{12} - 12 \beta_{13} - 4 \beta_{14} ) q^{83} + ( 19 - \beta_{5} + 2 \beta_{7} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 8 \beta_{15} ) q^{84} + ( 27 + 4 \beta_{5} - 7 \beta_{7} + 9 \beta_{9} - 9 \beta_{10} + \beta_{11} + 8 \beta_{15} ) q^{86} + ( \beta_{1} - \beta_{2} + 12 \beta_{3} + 2 \beta_{4} + 4 \beta_{8} + 5 \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{87} + ( -4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} - 32 \beta_{4} - 4 \beta_{8} + 8 \beta_{12} + 6 \beta_{13} - 10 \beta_{14} ) q^{88} + ( -16 - 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{15} ) q^{89} + ( -15 \beta_{6} + 3 \beta_{7} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + 15 \beta_{15} ) q^{91} + ( -12 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} - 26 \beta_{4} + 4 \beta_{8} - 6 \beta_{12} - 4 \beta_{13} - 8 \beta_{14} ) q^{92} + ( 3 \beta_{1} - 8 \beta_{2} - 18 \beta_{3} + \beta_{8} + 3 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} ) q^{93} + ( -2 + 8 \beta_{5} - 18 \beta_{9} + 4 \beta_{10} + 12 \beta_{11} + 12 \beta_{15} ) q^{94} + ( 12 + 4 \beta_{5} - 6 \beta_{6} - 4 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} + 4 \beta_{15} ) q^{96} + ( -8 \beta_{1} + 4 \beta_{2} + 28 \beta_{3} - 8 \beta_{12} - 8 \beta_{13} + \beta_{14} ) q^{97} + ( -6 \beta_{1} - 2 \beta_{2} + 36 \beta_{4} + 6 \beta_{8} + 2 \beta_{12} - 16 \beta_{13} + 6 \beta_{14} ) q^{98} + ( 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 6 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{4} - 12q^{6} + 48q^{9} + O(q^{10})$$ $$16q + 16q^{4} - 12q^{6} + 48q^{9} - 44q^{14} + 80q^{16} + 48q^{21} + 72q^{24} - 132q^{26} + 64q^{29} - 248q^{34} + 48q^{36} - 32q^{41} - 80q^{44} - 152q^{46} - 32q^{49} - 36q^{54} - 344q^{56} + 272q^{61} - 32q^{64} - 216q^{66} + 192q^{69} + 216q^{74} + 240q^{76} + 144q^{81} + 288q^{84} + 428q^{86} - 256q^{89} - 24q^{94} + 192q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 8 x^{14} - 14 x^{13} + 23 x^{12} - 26 x^{11} + 18 x^{10} - 10 x^{9} + 9 x^{8} - 20 x^{7} + 72 x^{6} - 208 x^{5} + 368 x^{4} - 448 x^{3} + 512 x^{2} - 512 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} + 14 \nu^{14} - 2 \nu^{13} - 22 \nu^{12} - 11 \nu^{11} + 12 \nu^{10} + 20 \nu^{9} + 30 \nu^{8} - 113 \nu^{7} + 42 \nu^{6} + 182 \nu^{5} - 100 \nu^{4} - 88 \nu^{3} + 144 \nu^{2} + 480 \nu + 1408$$$$)/320$$ $$\beta_{2}$$ $$=$$ $$($$$$11 \nu^{15} - 21 \nu^{14} + 6 \nu^{13} + 2 \nu^{12} - 29 \nu^{11} + 71 \nu^{10} - 82 \nu^{9} + 16 \nu^{8} + 113 \nu^{7} - 137 \nu^{6} + 190 \nu^{5} - 164 \nu^{4} - 632 \nu^{3} + 1296 \nu^{2} - 1312 \nu + 1152$$$$)/320$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{15} - 15 \nu^{14} + 26 \nu^{13} - 40 \nu^{12} + 52 \nu^{11} - 17 \nu^{10} - 4 \nu^{9} + 14 \nu^{8} + 24 \nu^{7} - 139 \nu^{6} + 306 \nu^{5} - 708 \nu^{4} + 888 \nu^{3} - 752 \nu^{2} + 736 \nu - 512$$$$)/320$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{15} + 12 \nu^{14} - 17 \nu^{13} + 32 \nu^{12} - 47 \nu^{11} + 32 \nu^{10} - 13 \nu^{9} + 14 \nu^{8} - 31 \nu^{7} + 98 \nu^{6} - 221 \nu^{5} + 562 \nu^{4} - 684 \nu^{3} + 648 \nu^{2} - 880 \nu + 640$$$$)/160$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{15} - 22 \nu^{14} + 52 \nu^{13} - 86 \nu^{12} + 147 \nu^{11} - 144 \nu^{10} + 82 \nu^{9} - 14 \nu^{8} - 43 \nu^{7} - 78 \nu^{6} + 452 \nu^{5} - 1496 \nu^{4} + 2256 \nu^{3} - 2496 \nu^{2} + 2432 \nu - 2368$$$$)/320$$ $$\beta_{6}$$ $$=$$ $$($$$$14 \nu^{15} - 17 \nu^{14} + 14 \nu^{13} - 52 \nu^{12} + 24 \nu^{11} + 9 \nu^{10} - 4 \nu^{9} + 30 \nu^{8} + 100 \nu^{7} - 141 \nu^{6} + 382 \nu^{5} - 960 \nu^{4} + 360 \nu^{3} - 432 \nu^{2} + 1120 \nu + 928$$$$)/160$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{15} - 62 \nu^{14} + 68 \nu^{13} - 46 \nu^{12} + 199 \nu^{11} - 144 \nu^{10} - 54 \nu^{9} + 26 \nu^{8} + 225 \nu^{7} - 438 \nu^{6} + 628 \nu^{5} - 1832 \nu^{4} + 3728 \nu^{3} - 1216 \nu^{2} + 1920 \nu - 5184$$$$)/320$$ $$\beta_{8}$$ $$=$$ $$($$$$33 \nu^{15} - 33 \nu^{14} + 48 \nu^{13} - 150 \nu^{12} + 45 \nu^{11} + 67 \nu^{10} - 48 \nu^{9} + 52 \nu^{8} - 13 \nu^{7} - 401 \nu^{6} + 1504 \nu^{5} - 2168 \nu^{4} + 320 \nu^{3} - 2016 \nu^{2} + 1600 \nu + 4480$$$$)/320$$ $$\beta_{9}$$ $$=$$ $$($$$$-20 \nu^{15} + 57 \nu^{14} - 102 \nu^{13} + 172 \nu^{12} - 226 \nu^{11} + 179 \nu^{10} - 88 \nu^{9} + 10 \nu^{8} - 98 \nu^{7} + 429 \nu^{6} - 1310 \nu^{5} + 2692 \nu^{4} - 3752 \nu^{3} + 3952 \nu^{2} - 4064 \nu + 2624$$$$)/320$$ $$\beta_{10}$$ $$=$$ $$($$$$21 \nu^{15} - 55 \nu^{14} + 106 \nu^{13} - 210 \nu^{12} + 357 \nu^{11} - 315 \nu^{10} + 234 \nu^{9} - 120 \nu^{8} - 57 \nu^{7} - 35 \nu^{6} + 1106 \nu^{5} - 3116 \nu^{4} + 4792 \nu^{3} - 6320 \nu^{2} + 8288 \nu - 6016$$$$)/320$$ $$\beta_{11}$$ $$=$$ $$($$$$52 \nu^{15} - 57 \nu^{14} + 102 \nu^{13} - 236 \nu^{12} + 194 \nu^{11} - 115 \nu^{10} + 152 \nu^{9} - 74 \nu^{8} + 130 \nu^{7} - 429 \nu^{6} + 1822 \nu^{5} - 3844 \nu^{4} + 2728 \nu^{3} - 4976 \nu^{2} + 6112 \nu + 64$$$$)/320$$ $$\beta_{12}$$ $$=$$ $$($$$$52 \nu^{15} - 139 \nu^{14} + 244 \nu^{13} - 432 \nu^{12} + 582 \nu^{11} - 469 \nu^{10} + 126 \nu^{9} + 34 \nu^{8} + 114 \nu^{7} - 867 \nu^{6} + 3012 \nu^{5} - 7224 \nu^{4} + 9584 \nu^{3} - 10272 \nu^{2} + 10560 \nu - 6400$$$$)/320$$ $$\beta_{13}$$ $$=$$ $$($$$$32 \nu^{15} - 99 \nu^{14} + 143 \nu^{13} - 302 \nu^{12} + 446 \nu^{11} - 343 \nu^{10} + 223 \nu^{9} - 130 \nu^{8} + 84 \nu^{7} - 461 \nu^{6} + 1875 \nu^{5} - 4902 \nu^{4} + 6740 \nu^{3} - 7272 \nu^{2} + 9680 \nu - 6144$$$$)/160$$ $$\beta_{14}$$ $$=$$ $$($$$$17 \nu^{15} - 42 \nu^{14} + 66 \nu^{13} - 138 \nu^{12} + 195 \nu^{11} - 144 \nu^{10} + 84 \nu^{9} - 54 \nu^{8} + 57 \nu^{7} - 198 \nu^{6} + 906 \nu^{5} - 2268 \nu^{4} + 2904 \nu^{3} - 3216 \nu^{2} + 4128 \nu - 2432$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$-66 \nu^{15} + 177 \nu^{14} - 270 \nu^{13} + 524 \nu^{12} - 740 \nu^{11} + 559 \nu^{10} - 292 \nu^{9} + 202 \nu^{8} - 344 \nu^{7} + 941 \nu^{6} - 3454 \nu^{5} + 8832 \nu^{4} - 11240 \nu^{3} + 11824 \nu^{2} - 15712 \nu + 10144$$$$)/160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{15} - 3 \beta_{14} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 6$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{8} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_{1}$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{15} + 5 \beta_{14} + 5 \beta_{13} + 5 \beta_{12} + 5 \beta_{11} - 11 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} - \beta_{7} - 5 \beta_{6} - 16 \beta_{5} + 20 \beta_{4} + 15 \beta_{3} + 5 \beta_{1} + 24$$$$)/40$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{11} + 3 \beta_{10} - 10 \beta_{9} - 3 \beta_{7} - 12 \beta_{5} + 2$$$$)/20$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{15} + 13 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} + 5 \beta_{11} + 5 \beta_{10} - 22 \beta_{9} + 8 \beta_{8} + 17 \beta_{7} - 5 \beta_{6} - 10 \beta_{5} + 16 \beta_{4} - 57 \beta_{3} + 4 \beta_{2} + 11 \beta_{1} - 68$$$$)/40$$ $$\nu^{6}$$ $$=$$ $$($$$$12 \beta_{14} - 5 \beta_{13} - 5 \beta_{8} + 70 \beta_{4} + 15 \beta_{2}$$$$)/20$$ $$\nu^{7}$$ $$=$$ $$($$$$-6 \beta_{15} + 14 \beta_{14} - 20 \beta_{13} - 15 \beta_{12} - 15 \beta_{11} + \beta_{10} - 34 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 20 \beta_{6} + \beta_{5} + 24 \beta_{4} - 8 \beta_{3} - 19 \beta_{2} - 21 \beta_{1} + 28$$$$)/20$$ $$\nu^{8}$$ $$=$$ $$($$$$9 \beta_{15} - 5 \beta_{11} + 18 \beta_{10} - 19 \beta_{9} + 6 \beta_{7} + 25 \beta_{6} + 8 \beta_{5} - 64$$$$)/10$$ $$\nu^{9}$$ $$=$$ $$($$$$61 \beta_{15} + 25 \beta_{14} + 45 \beta_{13} - 55 \beta_{12} + 55 \beta_{11} + 3 \beta_{10} - 126 \beta_{9} - 50 \beta_{8} + 13 \beta_{7} + 45 \beta_{6} + 108 \beta_{5} + 20 \beta_{4} + 55 \beta_{3} + 60 \beta_{2} + 5 \beta_{1} + 8$$$$)/40$$ $$\nu^{10}$$ $$=$$ $$($$$$16 \beta_{14} + 5 \beta_{13} - 40 \beta_{12} + 7 \beta_{8} + 134 \beta_{4} + 282 \beta_{3} - 19 \beta_{2} - 26 \beta_{1}$$$$)/20$$ $$\nu^{11}$$ $$=$$ $$($$$$91 \beta_{15} + 141 \beta_{14} - 95 \beta_{13} - 105 \beta_{12} - 105 \beta_{11} + 175 \beta_{10} - 106 \beta_{9} + 66 \beta_{8} + 31 \beta_{7} + 95 \beta_{6} + 150 \beta_{5} - 508 \beta_{4} + 131 \beta_{3} + 38 \beta_{2} - 53 \beta_{1} + 496$$$$)/40$$ $$\nu^{12}$$ $$=$$ $$($$$$8 \beta_{15} + 27 \beta_{11} + 13 \beta_{10} + 16 \beta_{9} + 27 \beta_{7} - 8 \beta_{6} + 4 \beta_{5} + 114$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-117 \beta_{15} - 118 \beta_{14} - 75 \beta_{13} - 35 \beta_{12} + 35 \beta_{11} + 77 \beta_{10} + 12 \beta_{9} - 9 \beta_{8} - 14 \beta_{7} - 75 \beta_{6} + 57 \beta_{5} + 242 \beta_{4} + 396 \beta_{3} + 38 \beta_{2} + 67 \beta_{1} + 686$$$$)/20$$ $$\nu^{14}$$ $$=$$ $$($$$$65 \beta_{14} - 90 \beta_{13} - 15 \beta_{12} - 16 \beta_{8} - 212 \beta_{4} + 84 \beta_{3} - 93 \beta_{2} + 58 \beta_{1}$$$$)/10$$ $$\nu^{15}$$ $$=$$ $$($$$$453 \beta_{15} + 335 \beta_{14} + 75 \beta_{13} + 435 \beta_{12} + 435 \beta_{11} + 99 \beta_{10} + 222 \beta_{9} - 570 \beta_{8} - 51 \beta_{7} - 75 \beta_{6} - 756 \beta_{5} - 1260 \beta_{4} + 45 \beta_{3} + 480 \beta_{2} + 195 \beta_{1} + 64$$$$)/40$$

## 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
 −0.485936 − 1.32811i −0.485936 + 1.32811i 1.21868 − 0.717516i 1.21868 + 0.717516i 1.26238 − 0.637499i 1.26238 + 0.637499i −0.152947 − 1.40592i −0.152947 + 1.40592i 1.40592 + 0.152947i 1.40592 − 0.152947i −0.637499 + 1.26238i −0.637499 − 1.26238i 0.717516 − 1.21868i 0.717516 + 1.21868i −1.32811 − 0.485936i −1.32811 + 0.485936i
−1.99209 0.177680i −1.73205 3.93686 + 0.707911i 0 3.45040 + 0.307751i −1.19501 −7.71680 2.10973i 3.00000 0
199.2 −1.99209 + 0.177680i −1.73205 3.93686 0.707911i 0 3.45040 0.307751i −1.19501 −7.71680 + 2.10973i 3.00000 0
199.3 −1.92737 0.534079i 1.73205 3.42952 + 2.05874i 0 −3.33830 0.925051i 11.9716 −5.51043 5.79958i 3.00000 0
199.4 −1.92737 + 0.534079i 1.73205 3.42952 2.05874i 0 −3.33830 + 0.925051i 11.9716 −5.51043 + 5.79958i 3.00000 0
199.5 −1.49110 1.33290i 1.73205 0.446749 + 3.97497i 0 −2.58266 2.30865i −6.56834 4.63210 6.52255i 3.00000 0
199.6 −1.49110 + 1.33290i 1.73205 0.446749 3.97497i 0 −2.58266 + 2.30865i −6.56834 4.63210 + 6.52255i 3.00000 0
199.7 −0.305673 1.97650i 1.73205 −3.81313 + 1.20833i 0 −0.529441 3.42340i 0.329898 3.55383 + 7.16731i 3.00000 0
199.8 −0.305673 + 1.97650i 1.73205 −3.81313 1.20833i 0 −0.529441 + 3.42340i 0.329898 3.55383 7.16731i 3.00000 0
199.9 0.305673 1.97650i −1.73205 −3.81313 1.20833i 0 −0.529441 + 3.42340i −0.329898 −3.55383 + 7.16731i 3.00000 0
199.10 0.305673 + 1.97650i −1.73205 −3.81313 + 1.20833i 0 −0.529441 3.42340i −0.329898 −3.55383 7.16731i 3.00000 0
199.11 1.49110 1.33290i −1.73205 0.446749 3.97497i 0 −2.58266 + 2.30865i 6.56834 −4.63210 6.52255i 3.00000 0
199.12 1.49110 + 1.33290i −1.73205 0.446749 + 3.97497i 0 −2.58266 2.30865i 6.56834 −4.63210 + 6.52255i 3.00000 0
199.13 1.92737 0.534079i −1.73205 3.42952 2.05874i 0 −3.33830 + 0.925051i −11.9716 5.51043 5.79958i 3.00000 0
199.14 1.92737 + 0.534079i −1.73205 3.42952 + 2.05874i 0 −3.33830 0.925051i −11.9716 5.51043 + 5.79958i 3.00000 0
199.15 1.99209 0.177680i 1.73205 3.93686 0.707911i 0 3.45040 0.307751i 1.19501 7.71680 2.10973i 3.00000 0
199.16 1.99209 + 0.177680i 1.73205 3.93686 + 0.707911i 0 3.45040 + 0.307751i 1.19501 7.71680 + 2.10973i 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.c 16
3.b odd 2 1 900.3.f.h 16
4.b odd 2 1 inner 300.3.f.c 16
5.b even 2 1 inner 300.3.f.c 16
5.c odd 4 1 300.3.c.e 8
5.c odd 4 1 300.3.c.g yes 8
12.b even 2 1 900.3.f.h 16
15.d odd 2 1 900.3.f.h 16
15.e even 4 1 900.3.c.n 8
15.e even 4 1 900.3.c.t 8
20.d odd 2 1 inner 300.3.f.c 16
20.e even 4 1 300.3.c.e 8
20.e even 4 1 300.3.c.g yes 8
60.h even 2 1 900.3.f.h 16
60.l odd 4 1 900.3.c.n 8
60.l odd 4 1 900.3.c.t 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.c.e 8 5.c odd 4 1
300.3.c.e 8 20.e even 4 1
300.3.c.g yes 8 5.c odd 4 1
300.3.c.g yes 8 20.e even 4 1
300.3.f.c 16 1.a even 1 1 trivial
300.3.f.c 16 4.b odd 2 1 inner
300.3.f.c 16 5.b even 2 1 inner
300.3.f.c 16 20.d odd 2 1 inner
900.3.c.n 8 15.e even 4 1
900.3.c.n 8 60.l odd 4 1
900.3.c.t 8 15.e even 4 1
900.3.c.t 8 60.l odd 4 1
900.3.f.h 16 3.b odd 2 1
900.3.f.h 16 12.b even 2 1
900.3.f.h 16 15.d odd 2 1
900.3.f.h 16 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 188 T_{7}^{6} + 6470 T_{7}^{4} - 9532 T_{7}^{2} + 961$$ acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$65536 - 32768 T^{2} + 3072 T^{4} + 1280 T^{6} - 496 T^{8} + 80 T^{10} + 12 T^{12} - 8 T^{14} + T^{16}$$
$3$ $$( -3 + T^{2} )^{8}$$
$5$ $$T^{16}$$
$7$ $$( 961 - 9532 T^{2} + 6470 T^{4} - 188 T^{6} + T^{8} )^{2}$$
$11$ $$( 30824704 + 6673408 T^{2} + 135008 T^{4} + 704 T^{6} + T^{8} )^{2}$$
$13$ $$( 2430481 + 14031732 T^{2} + 202934 T^{4} + 852 T^{6} + T^{8} )^{2}$$
$17$ $$( 17529760000 + 224000000 T^{2} + 957024 T^{4} + 1664 T^{6} + T^{8} )^{2}$$
$19$ $$( 1099651921 + 32129228 T^{2} + 312950 T^{4} + 1132 T^{6} + T^{8} )^{2}$$
$23$ $$( 1611540736 - 86660096 T^{2} + 934496 T^{4} - 1984 T^{6} + T^{8} )^{2}$$
$29$ $$( 93616 + 29824 T - 1600 T^{2} - 16 T^{3} + T^{4} )^{4}$$
$31$ $$( 819736484449 + 5243206492 T^{2} + 9582374 T^{4} + 5660 T^{6} + T^{8} )^{2}$$
$37$ $$( 20688524759296 + 42686293248 T^{2} + 30434144 T^{4} + 9168 T^{6} + T^{8} )^{2}$$
$41$ $$( 3504448 + 37664 T - 4968 T^{2} + 8 T^{3} + T^{4} )^{4}$$
$43$ $$( 974581609681 - 6860196428 T^{2} + 13866230 T^{4} - 6892 T^{6} + T^{8} )^{2}$$
$47$ $$( 13610196640000 - 54677446400 T^{2} + 43512416 T^{4} - 12016 T^{6} + T^{8} )^{2}$$
$53$ $$( 24469088997376 + 63523350528 T^{2} + 52132544 T^{4} + 13968 T^{6} + T^{8} )^{2}$$
$59$ $$( 195562066176 + 4566129408 T^{2} + 10470240 T^{4} + 6192 T^{6} + T^{8} )^{2}$$
$61$ $$( 9745129 + 324236 T - 8098 T^{2} - 68 T^{3} + T^{4} )^{4}$$
$67$ $$( 23066205847441 - 258205504012 T^{2} + 134011190 T^{4} - 21548 T^{6} + T^{8} )^{2}$$
$71$ $$( 11235904000000 + 28259891200 T^{2} + 24817856 T^{4} + 8816 T^{6} + T^{8} )^{2}$$
$73$ $$( 1901137194529024 + 1487063354624 T^{2} + 345328992 T^{4} + 31568 T^{6} + T^{8} )^{2}$$
$79$ $$( 2278988775424 + 24693882880 T^{2} + 45364736 T^{4} + 14528 T^{6} + T^{8} )^{2}$$
$83$ $$( 120362665464064 - 3496365751040 T^{2} + 748225376 T^{4} - 48688 T^{6} + T^{8} )^{2}$$
$89$ $$( -1507328 - 237568 T - 3328 T^{2} + 64 T^{3} + T^{4} )^{4}$$
$97$ $$( 243922954789089 + 1287733480740 T^{2} + 361856934 T^{4} + 33636 T^{6} + T^{8} )^{2}$$