# Properties

 Label 684.3.g.b Level $684$ Weight $3$ Character orbit 684.g Analytic conductor $18.638$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{10} q^{5} + ( \beta_{3} + \beta_{5} - \beta_{11} ) q^{7} + ( 3 - \beta_{3} - \beta_{6} + \beta_{10} - \beta_{12} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{10} q^{5} + ( \beta_{3} + \beta_{5} - \beta_{11} ) q^{7} + ( 3 - \beta_{3} - \beta_{6} + \beta_{10} - \beta_{12} ) q^{8} + ( -1 + \beta_{6} - \beta_{9} ) q^{10} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{11} + ( 5 - 2 \beta_{2} - \beta_{5} - \beta_{10} - \beta_{11} ) q^{13} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{14} + ( 6 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{16} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{17} + \beta_{3} q^{19} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{20} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{22} + ( 1 + \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} ) q^{23} + ( -5 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{25} + ( -4 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + \beta_{6} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{26} + ( 4 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 6 \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{28} + ( -1 + \beta_{1} - \beta_{2} - 4 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} + \beta_{12} - \beta_{13} ) q^{29} + ( 1 - 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{31} + ( -5 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{13} ) q^{32} + ( -6 + 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{34} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{35} + ( 6 - 3 \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{37} + ( \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} ) q^{38} + ( -14 + 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{12} ) q^{40} + ( -18 + 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 7 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} ) q^{41} + ( -1 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{43} + ( 6 - 3 \beta_{1} - \beta_{3} - 3 \beta_{5} + 5 \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{44} + ( 7 + \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - 6 \beta_{11} - 6 \beta_{13} ) q^{46} + ( 1 - 3 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 7 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{47} + ( -18 + 6 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} + 4 \beta_{6} + 6 \beta_{7} - 4 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} ) q^{49} + ( 3 + 7 \beta_{1} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} + 2 \beta_{12} ) q^{50} + ( -23 - 4 \beta_{1} + 9 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - 6 \beta_{12} + 2 \beta_{13} ) q^{52} + ( -5 - 5 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} ) q^{53} + ( -1 + 7 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 2 \beta_{13} ) q^{55} + ( 3 + 6 \beta_{1} - 8 \beta_{2} + 11 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + 4 \beta_{9} + 5 \beta_{10} - 10 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} ) q^{56} + ( -12 + 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} + 10 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} ) q^{58} + ( -5 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} + 5 \beta_{12} - 3 \beta_{13} ) q^{59} + ( -6 - 2 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} - 11 \beta_{10} + 2 \beta_{11} ) q^{61} + ( -32 - 2 \beta_{1} + 10 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{62} + ( -4 + 7 \beta_{1} + \beta_{2} - 10 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 5 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + 5 \beta_{10} + \beta_{11} - 7 \beta_{12} + 3 \beta_{13} ) q^{64} + ( 32 - 8 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 8 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} ) q^{65} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - \beta_{6} - 6 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} + 8 \beta_{12} + 4 \beta_{13} ) q^{67} + ( -8 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{68} + ( 16 + 2 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - \beta_{5} - 8 \beta_{6} - 5 \beta_{7} + 6 \beta_{8} + \beta_{9} + 9 \beta_{10} + 2 \beta_{11} - 5 \beta_{12} + 4 \beta_{13} ) q^{70} + ( -2 + 18 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 4 \beta_{13} ) q^{71} + ( 1 - 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} ) q^{73} + ( 20 - 8 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{74} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{10} ) q^{76} + ( -22 - 18 \beta_{1} + 20 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 10 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} - 15 \beta_{10} - 4 \beta_{11} ) q^{77} + ( -1 - 5 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 6 \beta_{11} + \beta_{12} + 5 \beta_{13} ) q^{79} + ( 24 + 8 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} + 4 \beta_{7} - \beta_{9} - 4 \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{80} + ( -24 + 16 \beta_{1} - 14 \beta_{2} - 7 \beta_{3} + \beta_{4} - 9 \beta_{5} + 15 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + \beta_{9} + 4 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{82} + ( 2 - 10 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} + 8 \beta_{13} ) q^{83} + ( 8 - 22 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} ) q^{85} + ( 8 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} + \beta_{5} + 6 \beta_{6} + 11 \beta_{7} + 2 \beta_{8} - \beta_{9} + 13 \beta_{10} - 8 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} ) q^{86} + ( 30 - 8 \beta_{1} + 5 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} - \beta_{10} - 6 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} ) q^{88} + ( 2 + 13 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - \beta_{6} + 2 \beta_{7} + 8 \beta_{8} + 4 \beta_{9} + 5 \beta_{10} - \beta_{12} + \beta_{13} ) q^{89} + ( 7 - 19 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} + 7 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 7 \beta_{8} - \beta_{10} - 6 \beta_{11} + \beta_{12} + \beta_{13} ) q^{91} + ( -6 + 3 \beta_{1} - 3 \beta_{2} + 10 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + 14 \beta_{6} - 9 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{92} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} - \beta_{4} - \beta_{5} + 6 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + \beta_{9} + 13 \beta_{10} - 8 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{94} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{95} + ( 26 + 22 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 10 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{97} + ( -4 + 22 \beta_{1} - 4 \beta_{2} - 20 \beta_{3} + 4 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} + 8 \beta_{10} - 4 \beta_{12} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 2 q^{2} + 2 q^{4} + 40 q^{8} + O(q^{10})$$ $$14 q - 2 q^{2} + 2 q^{4} + 40 q^{8} - 12 q^{10} + 54 q^{13} - 30 q^{14} + 58 q^{16} - 34 q^{17} - 32 q^{20} + 36 q^{22} - 86 q^{25} + 16 q^{26} + 18 q^{28} - 54 q^{29} - 72 q^{32} - 82 q^{34} + 100 q^{37} - 148 q^{40} - 224 q^{41} + 96 q^{44} + 46 q^{46} - 220 q^{49} + 58 q^{50} - 288 q^{52} - 14 q^{53} - 12 q^{56} - 72 q^{58} + 28 q^{61} - 396 q^{62} - 118 q^{64} + 472 q^{65} - 30 q^{68} + 156 q^{70} + 70 q^{73} + 224 q^{74} - 228 q^{77} + 348 q^{80} - 400 q^{82} + 48 q^{85} + 124 q^{86} + 472 q^{88} - 126 q^{92} - 88 q^{94} + 308 q^{97} - 68 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688$$$$)/20480$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{13} - 17 \nu^{12} + 17 \nu^{11} + 63 \nu^{10} - 352 \nu^{9} + 578 \nu^{8} - 280 \nu^{7} - 2260 \nu^{6} + 6264 \nu^{5} - 5744 \nu^{4} - 5696 \nu^{3} + 32384 \nu^{2} - 47616 \nu + 24576$$$$)/5120$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{13} - 166 \nu^{12} - 129 \nu^{11} + 1194 \nu^{10} - 2006 \nu^{9} - 556 \nu^{8} + 12860 \nu^{7} - 29480 \nu^{6} + 6512 \nu^{5} + 83008 \nu^{4} - 199808 \nu^{3} + 159232 \nu^{2} + 320512 \nu - 991232$$$$)/20480$$ $$\beta_{6}$$ $$=$$ $$($$$$21 \nu^{13} + 6 \nu^{12} - 91 \nu^{11} + 246 \nu^{10} - 354 \nu^{9} - 244 \nu^{8} + 2100 \nu^{7} - 1560 \nu^{6} - 112 \nu^{5} + 3392 \nu^{4} - 10112 \nu^{3} + 4608 \nu^{2} - 13312 \nu + 172032$$$$)/20480$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 768 \nu^{2} - 6144 \nu + 6144$$$$)/1024$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{13} + 2 \nu^{12} - \nu^{11} - 14 \nu^{10} + 42 \nu^{9} - 28 \nu^{8} - 132 \nu^{7} + 440 \nu^{6} - 528 \nu^{5} - 448 \nu^{4} + 2688 \nu^{3} - 2560 \nu^{2} - 1024 \nu + 7168$$$$)/1024$$ $$\beta_{9}$$ $$=$$ $$($$$$-39 \nu^{13} - 34 \nu^{12} - 151 \nu^{11} + 526 \nu^{10} - 394 \nu^{9} - 1604 \nu^{8} + 8260 \nu^{7} - 14200 \nu^{6} - 3632 \nu^{5} + 71232 \nu^{4} - 130432 \nu^{3} + 96768 \nu^{2} + 232448 \nu - 831488$$$$)/20480$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288$$$$)/1024$$ $$\beta_{11}$$ $$=$$ $$($$$$19 \nu^{13} + 9 \nu^{12} - 139 \nu^{11} + 249 \nu^{10} + 84 \nu^{9} - 1506 \nu^{8} + 3600 \nu^{7} - 1260 \nu^{6} - 10328 \nu^{5} + 24368 \nu^{4} - 20928 \nu^{3} - 39808 \nu^{2} + 133632 \nu - 13312$$$$)/5120$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{13} + 2 \nu^{12} + \nu^{11} - 14 \nu^{10} + 36 \nu^{9} - 28 \nu^{8} - 96 \nu^{7} + 296 \nu^{6} - 344 \nu^{5} - 224 \nu^{4} + 1600 \nu^{3} - 1984 \nu^{2} - 256 \nu + 256$$$$)/256$$ $$\beta_{13}$$ $$=$$ $$($$$$-19 \nu^{13} + 11 \nu^{12} + 139 \nu^{11} - 389 \nu^{10} + 396 \nu^{9} + 1146 \nu^{8} - 4560 \nu^{7} + 5820 \nu^{6} + 5848 \nu^{5} - 29168 \nu^{4} + 47808 \nu^{3} - 2432 \nu^{2} - 123392 \nu + 95232$$$$)/5120$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{12} - \beta_{10} + \beta_{6} + \beta_{3} - 3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 3 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{13} + 2 \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta_{1} + 5$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{13} - 7 \beta_{12} + \beta_{11} + 5 \beta_{10} + 4 \beta_{9} + 5 \beta_{8} - 5 \beta_{7} - 2 \beta_{5} + 3 \beta_{4} - 10 \beta_{3} + \beta_{2} + 7 \beta_{1} - 4$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 26 \beta_{6} + \beta_{5} - 7 \beta_{4} - \beta_{3} + 7 \beta_{2} - 8 \beta_{1} - 7$$ $$\nu^{8}$$ $$=$$ $$-7 \beta_{13} - 5 \beta_{12} - 13 \beta_{11} - 15 \beta_{10} + 2 \beta_{9} + 13 \beta_{8} + 11 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} - 39 \beta_{4} + 32 \beta_{3} - 15 \beta_{2} + 3 \beta_{1} - 64$$ $$\nu^{9}$$ $$=$$ $$26 \beta_{13} - 2 \beta_{12} + 30 \beta_{11} + 2 \beta_{10} - 14 \beta_{9} - 27 \beta_{8} + 25 \beta_{7} - 30 \beta_{6} + 21 \beta_{5} - 19 \beta_{4} + 51 \beta_{3} + 35 \beta_{2} - 88 \beta_{1} + 117$$ $$\nu^{10}$$ $$=$$ $$69 \beta_{13} - 33 \beta_{12} + 55 \beta_{11} + 69 \beta_{10} - 46 \beta_{9} + 41 \beta_{8} - 81 \beta_{7} - 22 \beta_{6} + 76 \beta_{5} + 45 \beta_{4} + 32 \beta_{3} - 59 \beta_{2} + 191 \beta_{1} - 136$$ $$\nu^{11}$$ $$=$$ $$74 \beta_{13} + 54 \beta_{12} + 14 \beta_{11} + 42 \beta_{10} - 86 \beta_{9} - 127 \beta_{8} - 171 \beta_{7} + 98 \beta_{6} + 105 \beta_{5} + 9 \beta_{4} - 25 \beta_{3} + 151 \beta_{2} - 32 \beta_{1} + 881$$ $$\nu^{12}$$ $$=$$ $$-151 \beta_{13} + 267 \beta_{12} - 189 \beta_{11} - 159 \beta_{10} - 78 \beta_{9} - 291 \beta_{8} - 53 \beta_{7} + 410 \beta_{6} - 4 \beta_{5} - 279 \beta_{4} + 480 \beta_{3} - 215 \beta_{2} + 1187 \beta_{1} - 1840$$ $$\nu^{13}$$ $$=$$ $$-54 \beta_{13} - 186 \beta_{12} + 110 \beta_{11} - 70 \beta_{10} - 78 \beta_{9} - 347 \beta_{8} + 57 \beta_{7} - 806 \beta_{6} - 219 \beta_{5} - 243 \beta_{4} + 283 \beta_{3} + 1123 \beta_{2} - 1704 \beta_{1} - 3011$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
343.1
 1.94929 + 0.447510i 1.94929 − 0.447510i 1.57398 + 1.23393i 1.57398 − 1.23393i 0.711746 + 1.86907i 0.711746 − 1.86907i 0.645572 + 1.89294i 0.645572 − 1.89294i −0.0607713 + 1.99908i −0.0607713 − 1.99908i −1.89728 + 0.632718i −1.89728 − 0.632718i −1.92254 + 0.551226i −1.92254 − 0.551226i
−1.94929 0.447510i 0 3.59947 + 1.74465i 3.90374 0 2.64664i −6.23566 5.01164i 0 −7.60953 1.74696i
343.2 −1.94929 + 0.447510i 0 3.59947 1.74465i 3.90374 0 2.64664i −6.23566 + 5.01164i 0 −7.60953 + 1.74696i
343.3 −1.57398 1.23393i 0 0.954817 + 3.88437i −3.66290 0 1.93414i 3.29019 7.29209i 0 5.76533 + 4.51977i
343.4 −1.57398 + 1.23393i 0 0.954817 3.88437i −3.66290 0 1.93414i 3.29019 + 7.29209i 0 5.76533 4.51977i
343.5 −0.711746 1.86907i 0 −2.98683 + 2.66060i −4.97973 0 12.2628i 7.09872 + 3.68892i 0 3.54430 + 9.30745i
343.6 −0.711746 + 1.86907i 0 −2.98683 2.66060i −4.97973 0 12.2628i 7.09872 3.68892i 0 3.54430 9.30745i
343.7 −0.645572 1.89294i 0 −3.16647 + 2.44406i 2.38184 0 12.3764i 6.67066 + 4.41614i 0 −1.53765 4.50868i
343.8 −0.645572 + 1.89294i 0 −3.16647 2.44406i 2.38184 0 12.3764i 6.67066 4.41614i 0 −1.53765 + 4.50868i
343.9 0.0607713 1.99908i 0 −3.99261 0.242973i 5.82257 0 5.45132i −0.728358 + 7.96677i 0 0.353845 11.6398i
343.10 0.0607713 + 1.99908i 0 −3.99261 + 0.242973i 5.82257 0 5.45132i −0.728358 7.96677i 0 0.353845 + 11.6398i
343.11 1.89728 0.632718i 0 3.19934 2.40089i −5.79268 0 5.87536i 4.55095 6.57943i 0 −10.9903 + 3.66514i
343.12 1.89728 + 0.632718i 0 3.19934 + 2.40089i −5.79268 0 5.87536i 4.55095 + 6.57943i 0 −10.9903 3.66514i
343.13 1.92254 0.551226i 0 3.39230 2.11950i 2.32715 0 8.62924i 5.35350 5.94475i 0 4.47404 1.28279i
343.14 1.92254 + 0.551226i 0 3.39230 + 2.11950i 2.32715 0 8.62924i 5.35350 + 5.94475i 0 4.47404 + 1.28279i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.14 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.g.b 14
3.b odd 2 1 76.3.b.b 14
4.b odd 2 1 inner 684.3.g.b 14
12.b even 2 1 76.3.b.b 14
24.f even 2 1 1216.3.d.d 14
24.h odd 2 1 1216.3.d.d 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b 14 3.b odd 2 1
76.3.b.b 14 12.b even 2 1
684.3.g.b 14 1.a even 1 1 trivial
684.3.g.b 14 4.b odd 2 1 inner
1216.3.d.d 14 24.f even 2 1
1216.3.d.d 14 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{7} - 66 T_{5}^{5} + 28 T_{5}^{4} + 1337 T_{5}^{3} - 1348 T_{5}^{2} - 8400 T_{5} + 13312$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16384 + 8192 T + 1024 T^{2} - 3584 T^{3} - 2688 T^{4} - 448 T^{5} + 528 T^{6} + 440 T^{7} + 132 T^{8} - 28 T^{9} - 42 T^{10} - 14 T^{11} + T^{12} + 2 T^{13} + T^{14}$$
$3$ $$T^{14}$$
$5$ $$( 13312 - 8400 T - 1348 T^{2} + 1337 T^{3} + 28 T^{4} - 66 T^{5} + T^{7} )^{2}$$
$7$ $$46106276299 + 23021007551 T^{2} + 3578469343 T^{4} + 212956211 T^{6} + 5808593 T^{8} + 75725 T^{10} + 453 T^{12} + T^{14}$$
$11$ $$94488337600 + 207279189936 T^{2} + 72889781124 T^{4} + 2781706433 T^{6} + 41107428 T^{8} + 280822 T^{10} + 880 T^{12} + T^{14}$$
$13$ $$( -1265920 + 2184704 T - 907488 T^{2} + 7824 T^{3} + 9959 T^{4} - 293 T^{5} - 27 T^{6} + T^{7} )^{2}$$
$17$ $$( 69101055 - 18411921 T - 417581 T^{2} + 355811 T^{3} - 9955 T^{4} - 1107 T^{5} + 17 T^{6} + T^{7} )^{2}$$
$19$ $$( 19 + T^{2} )^{7}$$
$23$ $$683970816311296 + 864035000025088 T^{2} + 39218066795008 T^{4} + 551803020800 T^{6} + 2798046987 T^{8} + 5204683 T^{10} + 3881 T^{12} + T^{14}$$
$29$ $$( 112127472 - 91750704 T + 10060400 T^{2} + 839992 T^{3} - 75791 T^{4} - 2901 T^{5} + 27 T^{6} + T^{7} )^{2}$$
$31$ $$112006799137177600 + 7005784227446784 T^{2} + 136835964338176 T^{4} + 1042205249536 T^{6} + 3570928640 T^{8} + 5704128 T^{10} + 4048 T^{12} + T^{14}$$
$37$ $$( 914894720 + 490075456 T - 48142944 T^{2} - 85712 T^{3} + 100072 T^{4} - 1508 T^{5} - 50 T^{6} + T^{7} )^{2}$$
$41$ $$( -18882293760 + 1089553152 T + 442902400 T^{2} - 1190928 T^{3} - 433472 T^{4} - 1776 T^{5} + 112 T^{6} + T^{7} )^{2}$$
$43$ $$58\!\cdots\!24$$$$+ 9894601275600773376 T^{2} + 51208545244013840 T^{4} + 109716719259377 T^{6} + 112918336940 T^{8} + 57363094 T^{10} + 13244 T^{12} + T^{14}$$
$47$ $$6486650127800008704 + 2593987636485123072 T^{2} + 16602143603532784 T^{4} + 38825334689281 T^{6} + 44446706748 T^{8} + 26838726 T^{10} + 8204 T^{12} + T^{14}$$
$53$ $$( -18228603200 + 4696986240 T - 390622104 T^{2} + 10269704 T^{3} + 111981 T^{4} - 6645 T^{5} + 7 T^{6} + T^{7} )^{2}$$
$59$ $$50\!\cdots\!00$$$$+$$$$25\!\cdots\!04$$$$T^{2} + 4608193003005303072 T^{4} + 3925759870481904 T^{6} + 1663620366787 T^{8} + 336602299 T^{10} + 30513 T^{12} + T^{14}$$
$61$ $$( 522558358600 - 16277769076 T - 1000269562 T^{2} + 29414873 T^{3} + 485080 T^{4} - 14158 T^{5} - 14 T^{6} + T^{7} )^{2}$$
$67$ $$29\!\cdots\!76$$$$+$$$$25\!\cdots\!00$$$$T^{2} + 5120555266235997504 T^{4} + 3973974955017136 T^{6} + 1503100062371 T^{8} + 293178683 T^{10} + 27889 T^{12} + T^{14}$$
$71$ $$49\!\cdots\!00$$$$+$$$$57\!\cdots\!76$$$$T^{2} + 15805923995875204096 T^{4} + 11221518796559104 T^{6} + 3467793570368 T^{8} + 526066128 T^{10} + 37812 T^{12} + T^{14}$$
$73$ $$( -77971926925 - 6681670753 T + 6909415 T^{2} + 11130931 T^{3} + 95497 T^{4} - 5907 T^{5} - 35 T^{6} + T^{7} )^{2}$$
$79$ $$18\!\cdots\!00$$$$+$$$$22\!\cdots\!64$$$$T^{2} + 7079503618001244160 T^{4} + 6799300948930560 T^{6} + 2745532366528 T^{8} + 498656432 T^{10} + 38740 T^{12} + T^{14}$$
$83$ $$34\!\cdots\!76$$$$+$$$$85\!\cdots\!08$$$$T^{2} +$$$$17\!\cdots\!88$$$$T^{4} + 81697848347602944 T^{6} + 15885346272448 T^{8} + 1466300592 T^{10} + 62788 T^{12} + T^{14}$$
$89$ $$( 88310345728 + 32804608000 T + 2083382784 T^{2} + 23700672 T^{3} - 1100800 T^{4} - 23640 T^{5} + T^{7} )^{2}$$
$97$ $$( 4410892984320 - 293552295936 T - 20455196416 T^{2} + 115812736 T^{3} + 4374632 T^{4} - 27396 T^{5} - 154 T^{6} + T^{7} )^{2}$$
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