# Properties

 Label 1260.2.c.d Level $1260$ Weight $2$ Character orbit 1260.c Analytic conductor $10.061$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + 24 x^{7} - 28 x^{6} + 16 x^{5} + 48 x^{4} - 128 x^{3} + 192 x^{2} - 256 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 420) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + \beta_{8} q^{4} + \beta_{3} q^{5} + ( -\beta_{9} - \beta_{10} ) q^{7} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + \beta_{8} q^{4} + \beta_{3} q^{5} + ( -\beta_{9} - \beta_{10} ) q^{7} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{8} + \beta_{9} q^{10} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{11} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{13} + ( -\beta_{4} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{14} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{12} - \beta_{14} ) q^{16} + ( \beta_{2} - \beta_{4} - \beta_{9} - \beta_{11} ) q^{17} + ( -1 - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{19} + \beta_{14} q^{20} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{22} + ( \beta_{2} + \beta_{4} + 2 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{23} - q^{25} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{26} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{28} + ( -2 + \beta_{2} - \beta_{4} + 2 \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{14} - \beta_{15} ) q^{29} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{31} + ( 1 - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{32} + ( 2 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{34} + ( \beta_{4} + \beta_{7} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{37} + ( -2 - \beta_{2} - 3 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{38} + ( 1 - \beta_{9} - \beta_{10} - \beta_{11} ) q^{40} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{41} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{15} ) q^{43} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{44} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{46} + ( -\beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{47} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{49} -\beta_{4} q^{50} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{52} + ( 3 - \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{53} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{55} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{56} + ( -3 - \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{58} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{59} + ( -3 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{61} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{62} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{15} ) q^{64} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{65} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{67} + ( \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{68} + ( \beta_{2} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{70} + ( 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{71} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{73} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{74} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{76} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - 5 \beta_{14} ) q^{77} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{79} + ( 1 + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{12} + \beta_{13} ) q^{80} + ( 1 + \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{82} + ( -\beta_{1} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{83} + ( -\beta_{2} + \beta_{4} - \beta_{9} - \beta_{11} ) q^{85} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{86} + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - 3 \beta_{13} + \beta_{15} ) q^{88} + ( -\beta_{1} - 4 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{13} - 2 \beta_{14} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{91} + ( -3 - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{92} + ( \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{94} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{95} + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{97} + ( 5 + \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2q^{2} - 2q^{4} - 4q^{7} - 2q^{8} + O(q^{10})$$ $$16q - 2q^{2} - 2q^{4} - 4q^{7} - 2q^{8} + 2q^{14} + 6q^{16} - 24q^{19} - 12q^{22} - 16q^{25} + 12q^{26} + 14q^{28} - 16q^{29} + 8q^{31} + 18q^{32} + 24q^{34} + 24q^{37} - 28q^{38} + 12q^{40} + 8q^{44} - 20q^{46} - 16q^{47} - 16q^{49} + 2q^{50} - 20q^{52} + 32q^{53} - 2q^{56} - 32q^{58} - 8q^{59} - 16q^{62} - 2q^{64} + 8q^{65} - 4q^{68} + 4q^{74} + 16q^{76} + 8q^{77} + 16q^{80} - 4q^{82} - 8q^{83} - 64q^{86} - 52q^{88} + 16q^{91} - 64q^{92} + 16q^{94} + 86q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + 24 x^{7} - 28 x^{6} + 16 x^{5} + 48 x^{4} - 128 x^{3} + 192 x^{2} - 256 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{15} + \nu^{13} - 26 \nu^{12} - 3 \nu^{11} + 19 \nu^{9} + 26 \nu^{8} - 104 \nu^{7} + 56 \nu^{6} - 196 \nu^{5} + 296 \nu^{4} + 256 \nu^{3} + 256 \nu^{2} + 320 \nu - 1152$$$$)/512$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{14} - 5 \nu^{12} + 2 \nu^{11} + 7 \nu^{10} + 17 \nu^{8} - 18 \nu^{7} + 20 \nu^{6} + 8 \nu^{5} + 68 \nu^{4} + 72 \nu^{3} - 112 \nu^{2} + 32 \nu - 384$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} + 3 \nu^{11} + 2 \nu^{10} - 7 \nu^{9} + 12 \nu^{8} - 28 \nu^{7} + 24 \nu^{6} - 28 \nu^{5} + 16 \nu^{4} + 48 \nu^{3} - 128 \nu^{2} + 192 \nu - 256$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{15} + 4 \nu^{14} + 3 \nu^{13} + 14 \nu^{12} + 31 \nu^{11} + 12 \nu^{10} + 25 \nu^{9} + 34 \nu^{8} + 112 \nu^{7} - 8 \nu^{6} - 12 \nu^{5} - 120 \nu^{4} - 480 \nu^{3} - 64 \nu^{2} - 960 \nu - 384$$$$)/512$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{15} - 16 \nu^{14} + 15 \nu^{13} - 38 \nu^{12} + 3 \nu^{11} + 32 \nu^{10} - 3 \nu^{9} + 102 \nu^{8} - 128 \nu^{7} + 120 \nu^{6} - 28 \nu^{5} + 280 \nu^{4} + 800 \nu^{3} - 448 \nu^{2} + 576 \nu - 2176$$$$)/512$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{15} + 12 \nu^{14} - 7 \nu^{13} + 10 \nu^{12} - 19 \nu^{11} - 28 \nu^{10} - 5 \nu^{9} - 58 \nu^{8} + 112 \nu^{7} + 40 \nu^{6} + 60 \nu^{5} - 40 \nu^{4} - 224 \nu^{3} + 704 \nu^{2} - 64 \nu + 1408$$$$)/512$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} - \nu^{13} + \nu^{12} + \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 5 \nu^{8} + 16 \nu^{7} + 4 \nu^{6} + 4 \nu^{5} + 12 \nu^{4} - 64 \nu^{3} + 80 \nu^{2} - 64 \nu + 64$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{15} + 6 \nu^{14} - 3 \nu^{13} + 12 \nu^{12} + \nu^{11} - 10 \nu^{10} + 7 \nu^{9} - 36 \nu^{8} + 50 \nu^{7} - 36 \nu^{6} + 44 \nu^{5} - 64 \nu^{4} - 280 \nu^{3} + 240 \nu^{2} - 352 \nu + 704$$$$)/128$$ $$\beta_{10}$$ $$=$$ $$($$$$17 \nu^{15} + 4 \nu^{14} + 11 \nu^{13} - 34 \nu^{12} - 57 \nu^{11} - 20 \nu^{10} - 79 \nu^{9} + 82 \nu^{8} - 144 \nu^{7} - 168 \nu^{6} - 300 \nu^{5} - 312 \nu^{4} + 1056 \nu^{3} + 320 \nu^{2} + 2112 \nu + 128$$$$)/512$$ $$\beta_{11}$$ $$=$$ $$($$$$17 \nu^{15} - 28 \nu^{14} + 27 \nu^{13} - 66 \nu^{12} - 9 \nu^{11} + 44 \nu^{10} - 31 \nu^{9} + 178 \nu^{8} - 256 \nu^{7} + 184 \nu^{6} - 236 \nu^{5} + 264 \nu^{4} + 1120 \nu^{3} - 960 \nu^{2} + 2368 \nu - 3456$$$$)/512$$ $$\beta_{12}$$ $$=$$ $$($$$$15 \nu^{15} - 36 \nu^{14} + 5 \nu^{13} - 62 \nu^{12} + 9 \nu^{11} + 84 \nu^{10} - 33 \nu^{9} + 206 \nu^{8} - 224 \nu^{7} + 264 \nu^{6} + 12 \nu^{5} + 504 \nu^{4} + 1376 \nu^{3} - 1088 \nu^{2} + 1472 \nu - 4224$$$$)/512$$ $$\beta_{13}$$ $$=$$ $$($$$$3 \nu^{15} + 5 \nu^{13} - 2 \nu^{12} - 7 \nu^{11} - 17 \nu^{9} + 18 \nu^{8} - 20 \nu^{7} - 8 \nu^{6} - 68 \nu^{5} - 72 \nu^{4} + 112 \nu^{3} - 32 \nu^{2} + 384 \nu$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$11 \nu^{15} - 10 \nu^{14} + 9 \nu^{13} - 32 \nu^{12} - 15 \nu^{11} + 18 \nu^{10} - 37 \nu^{9} + 104 \nu^{8} - 164 \nu^{7} + 64 \nu^{6} - 164 \nu^{5} + 784 \nu^{3} - 288 \nu^{2} + 1152 \nu - 1408$$$$)/256$$ $$\beta_{15}$$ $$=$$ $$($$$$-21 \nu^{15} + 40 \nu^{14} - 23 \nu^{13} + 78 \nu^{12} - 11 \nu^{11} - 104 \nu^{10} + 11 \nu^{9} - 302 \nu^{8} + 440 \nu^{7} - 264 \nu^{6} + 156 \nu^{5} - 312 \nu^{4} - 1792 \nu^{3} + 2176 \nu^{2} - 1984 \nu + 5504$$$$)/512$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} - 2 \beta_{9} + 2 \beta_{6} - 2 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_{1} + 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - \beta_{1} - 2$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{14} - 3 \beta_{10} - \beta_{9} + 5 \beta_{7} - \beta_{6} + \beta_{4} - 4 \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-8 \beta_{14} + 4 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 6 \beta_{3} - 3 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-5 \beta_{15} + 6 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} + \beta_{10} + 9 \beta_{9} + 14 \beta_{8} - 5 \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 14 \beta_{3} - 2 \beta_{2} - \beta_{1} + 14$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-6 \beta_{15} + 2 \beta_{14} - 6 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 18 \beta_{4} - 12 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 10$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$7 \beta_{15} + 6 \beta_{14} - 4 \beta_{13} + 12 \beta_{11} - 7 \beta_{10} + 3 \beta_{9} - 36 \beta_{8} + 13 \beta_{7} - 5 \beta_{6} + 16 \beta_{5} - 15 \beta_{4} + 12 \beta_{2} + 11 \beta_{1} + 12$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$4 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 20 \beta_{12} - 18 \beta_{11} + 12 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 16 \beta_{7} + 14 \beta_{6} + 12 \beta_{5} + 36 \beta_{4} - 42 \beta_{3} + 4 \beta_{2} + 13 \beta_{1} + 28$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-5 \beta_{15} + 12 \beta_{14} + 34 \beta_{13} + 2 \beta_{12} - 50 \beta_{11} - 35 \beta_{10} - 3 \beta_{9} + 34 \beta_{8} - 37 \beta_{7} + 29 \beta_{6} - 34 \beta_{5} - 27 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 27 \beta_{1} - 30$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-2 \beta_{15} - 18 \beta_{14} + 16 \beta_{13} - 6 \beta_{12} + 8 \beta_{11} - 6 \beta_{10} + 30 \beta_{9} + 2 \beta_{8} - 8 \beta_{7} + 58 \beta_{6} - 10 \beta_{5} + 2 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 9 \beta_{1} - 2$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-73 \beta_{15} - 118 \beta_{14} + 8 \beta_{13} + 48 \beta_{12} - 40 \beta_{11} + 77 \beta_{10} + 47 \beta_{9} - 8 \beta_{8} + 77 \beta_{7} - \beta_{6} + 16 \beta_{5} + 89 \beta_{4} - 44 \beta_{3} + 56 \beta_{2} - 33 \beta_{1} - 40$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-56 \beta_{15} - 128 \beta_{14} + 32 \beta_{13} + 132 \beta_{12} + 10 \beta_{11} + 52 \beta_{10} + 90 \beta_{9} + 72 \beta_{8} - 36 \beta_{7} - 50 \beta_{6} - 44 \beta_{5} + 40 \beta_{4} - 158 \beta_{3} + 40 \beta_{2} - 115 \beta_{1} + 24$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
811.1
 1.40936 − 0.117062i 1.40936 + 0.117062i 1.10145 − 0.887017i 1.10145 + 0.887017i 1.07312 − 0.921096i 1.07312 + 0.921096i 0.309204 − 1.38000i 0.309204 + 1.38000i −0.102186 − 1.41052i −0.102186 + 1.41052i −0.449546 − 1.34086i −0.449546 + 1.34086i −0.947441 − 1.04993i −0.947441 + 1.04993i −1.39396 − 0.238466i −1.39396 + 0.238466i
−1.40936 0.117062i 0 1.97259 + 0.329965i 1.00000i 0 0.776136 2.52935i −2.74147 0.695955i 0 0.117062 1.40936i
811.2 −1.40936 + 0.117062i 0 1.97259 0.329965i 1.00000i 0 0.776136 + 2.52935i −2.74147 + 0.695955i 0 0.117062 + 1.40936i
811.3 −1.10145 0.887017i 0 0.426402 + 1.95402i 1.00000i 0 0.391948 + 2.61656i 1.26358 2.53049i 0 0.887017 1.10145i
811.4 −1.10145 + 0.887017i 0 0.426402 1.95402i 1.00000i 0 0.391948 2.61656i 1.26358 + 2.53049i 0 0.887017 + 1.10145i
811.5 −1.07312 0.921096i 0 0.303166 + 1.97689i 1.00000i 0 −1.82575 1.91485i 1.49557 2.40068i 0 −0.921096 + 1.07312i
811.6 −1.07312 + 0.921096i 0 0.303166 1.97689i 1.00000i 0 −1.82575 + 1.91485i 1.49557 + 2.40068i 0 −0.921096 1.07312i
811.7 −0.309204 1.38000i 0 −1.80879 + 0.853401i 1.00000i 0 −2.64459 + 0.0785232i 1.73698 + 2.23224i 0 1.38000 0.309204i
811.8 −0.309204 + 1.38000i 0 −1.80879 0.853401i 1.00000i 0 −2.64459 0.0785232i 1.73698 2.23224i 0 1.38000 + 0.309204i
811.9 0.102186 1.41052i 0 −1.97912 0.288270i 1.00000i 0 −0.178143 2.63975i −0.608847 + 2.76212i 0 −1.41052 0.102186i
811.10 0.102186 + 1.41052i 0 −1.97912 + 0.288270i 1.00000i 0 −0.178143 + 2.63975i −0.608847 2.76212i 0 −1.41052 + 0.102186i
811.11 0.449546 1.34086i 0 −1.59582 1.20556i 1.00000i 0 1.40015 + 2.24490i −2.33388 + 1.59781i 0 −1.34086 0.449546i
811.12 0.449546 + 1.34086i 0 −1.59582 + 1.20556i 1.00000i 0 1.40015 2.24490i −2.33388 1.59781i 0 −1.34086 + 0.449546i
811.13 0.947441 1.04993i 0 −0.204711 1.98950i 1.00000i 0 −2.29670 1.31346i −2.28279 1.67000i 0 1.04993 + 0.947441i
811.14 0.947441 + 1.04993i 0 −0.204711 + 1.98950i 1.00000i 0 −2.29670 + 1.31346i −2.28279 + 1.67000i 0 1.04993 0.947441i
811.15 1.39396 0.238466i 0 1.88627 0.664826i 1.00000i 0 2.37694 1.16196i 2.47085 1.37655i 0 0.238466 + 1.39396i
811.16 1.39396 + 0.238466i 0 1.88627 + 0.664826i 1.00000i 0 2.37694 + 1.16196i 2.47085 + 1.37655i 0 0.238466 1.39396i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.c.d 16
3.b odd 2 1 420.2.c.a 16
4.b odd 2 1 1260.2.c.e 16
7.b odd 2 1 1260.2.c.e 16
12.b even 2 1 420.2.c.b yes 16
21.c even 2 1 420.2.c.b yes 16
28.d even 2 1 inner 1260.2.c.d 16
84.h odd 2 1 420.2.c.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.c.a 16 3.b odd 2 1
420.2.c.a 16 84.h odd 2 1
420.2.c.b yes 16 12.b even 2 1
420.2.c.b yes 16 21.c even 2 1
1260.2.c.d 16 1.a even 1 1 trivial
1260.2.c.d 16 28.d even 2 1 inner
1260.2.c.e 16 4.b odd 2 1
1260.2.c.e 16 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{16} + \cdots$$ $$T_{19}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 256 T + 192 T^{2} + 128 T^{3} + 48 T^{4} - 16 T^{5} - 28 T^{6} - 24 T^{7} - 28 T^{8} - 12 T^{9} - 7 T^{10} - 2 T^{11} + 3 T^{12} + 4 T^{13} + 3 T^{14} + 2 T^{15} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1 + T^{2} )^{8}$$
$7$ $$5764801 + 3294172 T + 1882384 T^{2} + 1142876 T^{3} + 451388 T^{4} + 160524 T^{5} + 65072 T^{6} + 22988 T^{7} + 6854 T^{8} + 3284 T^{9} + 1328 T^{10} + 468 T^{11} + 188 T^{12} + 68 T^{13} + 16 T^{14} + 4 T^{15} + T^{16}$$
$11$ $$16384 + 172032 T^{2} + 627712 T^{4} + 947712 T^{6} + 549248 T^{8} + 75904 T^{10} + 4292 T^{12} + 108 T^{14} + T^{16}$$
$13$ $$1982464 + 60882944 T^{2} + 38963200 T^{4} + 10193408 T^{6} + 1398080 T^{8} + 108192 T^{10} + 4724 T^{12} + 108 T^{14} + T^{16}$$
$17$ $$589824 + 3407872 T^{2} + 4907008 T^{4} + 2729984 T^{6} + 641792 T^{8} + 71296 T^{10} + 3952 T^{12} + 104 T^{14} + T^{16}$$
$19$ $$( 1408 - 6784 T + 1696 T^{2} + 2400 T^{3} - 352 T^{4} - 288 T^{5} + 6 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$23$ $$110166016 + 264994816 T^{2} + 196538368 T^{4} + 59332608 T^{6} + 7611136 T^{8} + 463744 T^{10} + 13792 T^{12} + 192 T^{14} + T^{16}$$
$29$ $$( -475392 - 357376 T + 36736 T^{2} + 45056 T^{3} + 2800 T^{4} - 1264 T^{5} - 132 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$31$ $$( 435072 + 44288 T - 101408 T^{2} - 9088 T^{3} + 6616 T^{4} + 352 T^{5} - 150 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$37$ $$( -843008 + 540928 T + 21504 T^{2} - 52800 T^{3} + 2352 T^{4} + 1504 T^{5} - 108 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$41$ $$186292371456 + 317850714112 T^{2} + 145118531584 T^{4} + 12630601728 T^{6} + 463714816 T^{8} + 8764288 T^{10} + 89680 T^{12} + 472 T^{14} + T^{16}$$
$43$ $$32480690176 + 89542098944 T^{2} + 28063666176 T^{4} + 2828883968 T^{6} + 132512000 T^{8} + 3322752 T^{10} + 46176 T^{12} + 336 T^{14} + T^{16}$$
$47$ $$( 1243904 - 411392 T - 177344 T^{2} + 41984 T^{3} + 9264 T^{4} - 1152 T^{5} - 176 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$53$ $$( 311424 + 756608 T + 129344 T^{2} - 72352 T^{3} - 5288 T^{4} + 2704 T^{5} - 106 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$59$ $$( -323072 - 815104 T - 52032 T^{2} + 97280 T^{3} + 14640 T^{4} - 1440 T^{5} - 256 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$61$ $$334139490304 + 337266016256 T^{2} + 80774565888 T^{4} + 8235839488 T^{6} + 408406528 T^{8} + 9941632 T^{10} + 110800 T^{12} + 552 T^{14} + T^{16}$$
$67$ $$1665379926016 + 1333371076608 T^{2} + 235939749888 T^{4} + 16515217408 T^{6} + 559284992 T^{8} + 10131840 T^{10} + 100080 T^{12} + 504 T^{14} + T^{16}$$
$71$ $$57232008953856 + 36387631611904 T^{2} + 3186748889088 T^{4} + 120727758336 T^{6} + 2458985024 T^{8} + 28845536 T^{10} + 194196 T^{12} + 692 T^{14} + T^{16}$$
$73$ $$34021064704 + 39953014784 T^{2} + 16064015360 T^{4} + 2638006272 T^{6} + 177219264 T^{8} + 5348384 T^{10} + 74260 T^{12} + 460 T^{14} + T^{16}$$
$79$ $$57415827456 + 81825366016 T^{2} + 29610151936 T^{4} + 4426000384 T^{6} + 291090176 T^{8} + 7517952 T^{10} + 89072 T^{12} + 488 T^{14} + T^{16}$$
$83$ $$( -30015488 - 23134208 T - 2523136 T^{2} + 457728 T^{3} + 65920 T^{4} - 2592 T^{5} - 468 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$89$ $$278378643456 + 1212059189248 T^{2} + 286828687360 T^{4} + 23481739264 T^{6} + 804129536 T^{8} + 13741952 T^{10} + 123744 T^{12} + 560 T^{14} + T^{16}$$
$97$ $$970650566656 + 5353125814272 T^{2} + 2353403653120 T^{4} + 174997760000 T^{6} + 4227116736 T^{8} + 48005920 T^{10} + 283540 T^{12} + 844 T^{14} + T^{16}$$