# Properties

 Label 1470.2.b.a Level $1470$ Weight $2$ Character orbit 1470.b Analytic conductor $11.738$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} - 864 x^{3} + 891 x^{2} - 972 x + 729$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} -\beta_{1} q^{3} - q^{4} + q^{5} -\beta_{9} q^{6} -\beta_{4} q^{8} + ( -1 + \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} -\beta_{1} q^{3} - q^{4} + q^{5} -\beta_{9} q^{6} -\beta_{4} q^{8} + ( -1 + \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{9} + \beta_{4} q^{10} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{11} + \beta_{1} q^{12} + ( \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{13} -\beta_{1} q^{15} + q^{16} + ( 3 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{17} + ( 1 - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{18} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{19} - q^{20} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{22} + ( -\beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{23} + \beta_{9} q^{24} + q^{25} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{26} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{27} + ( -1 + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{29} -\beta_{9} q^{30} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{31} + \beta_{4} q^{32} + ( -1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{33} + ( -\beta_{1} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{34} + ( 1 - \beta_{1} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{36} + ( 1 + \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{37} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{38} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{39} -\beta_{4} q^{40} + ( \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{41} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{43} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{44} + ( -1 + \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{45} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{46} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{47} -\beta_{1} q^{48} + \beta_{4} q^{50} + ( 1 - 2 \beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{51} + ( -\beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{52} + ( 1 - \beta_{2} - \beta_{4} + 3 \beta_{6} - 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} ) q^{53} + ( 3 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{54} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{55} + ( -6 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{57} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{59} + \beta_{1} q^{60} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} + 4 \beta_{11} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{62} - q^{64} + ( \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{65} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{66} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{67} + ( -3 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{68} + ( -4 - \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{69} + ( 1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{71} + ( -1 + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{11} ) q^{72} + ( 1 + \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} ) q^{73} + ( \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{10} - \beta_{11} ) q^{74} -\beta_{1} q^{75} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{76} + ( 4 - \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{78} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{79} + q^{80} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{81} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{5} - \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{82} + ( 3 + \beta_{2} + \beta_{6} - \beta_{9} ) q^{83} + ( 3 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{85} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + 3 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{86} + ( -6 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{10} ) q^{87} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{88} + ( 4 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{89} + ( 1 - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{90} + ( \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{92} + ( 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 4 \beta_{11} ) q^{93} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + 4 \beta_{11} ) q^{94} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{95} -\beta_{9} q^{96} + ( 3 - \beta_{1} - 3 \beta_{2} + 6 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{97} + ( 4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{3} - 12q^{4} + 12q^{5} + 2q^{6} - 6q^{9} + O(q^{10})$$ $$12q - 4q^{3} - 12q^{4} + 12q^{5} + 2q^{6} - 6q^{9} + 4q^{12} - 4q^{15} + 12q^{16} + 24q^{17} + 8q^{18} - 12q^{20} - 2q^{24} + 12q^{25} - 8q^{26} + 8q^{27} + 2q^{30} - 20q^{33} + 6q^{36} + 16q^{37} + 16q^{38} + 12q^{39} + 4q^{41} - 6q^{45} - 4q^{46} + 32q^{47} - 4q^{48} + 4q^{51} + 28q^{54} - 36q^{57} - 16q^{58} + 24q^{59} + 4q^{60} - 8q^{62} - 12q^{64} - 20q^{66} + 8q^{67} - 24q^{68} - 50q^{69} - 8q^{72} - 4q^{75} + 32q^{78} + 8q^{79} + 12q^{80} - 10q^{81} + 40q^{83} + 24q^{85} - 56q^{87} + 52q^{89} + 8q^{90} + 28q^{93} + 2q^{96} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} - 864 x^{3} + 891 x^{2} - 972 x + 729$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{11} + \nu^{10} - 11 \nu^{9} + 35 \nu^{8} - 10 \nu^{7} + 123 \nu^{6} - 183 \nu^{5} + 63 \nu^{4} - 630 \nu^{3} + 351 \nu^{2} - 243 \nu + 1701$$$$)/324$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{11} + 17 \nu^{10} - 39 \nu^{9} + 118 \nu^{8} - 153 \nu^{7} + 337 \nu^{6} - 651 \nu^{5} + 678 \nu^{4} - 1413 \nu^{3} + 1521 \nu^{2} - 837 \nu + 1620$$$$)/324$$ $$\beta_{3}$$ $$=$$ $$($$$$8 \nu^{11} - 11 \nu^{10} + 43 \nu^{9} - 145 \nu^{8} + 170 \nu^{7} - 513 \nu^{6} + 891 \nu^{5} - 1017 \nu^{4} + 2502 \nu^{3} - 2457 \nu^{2} + 2403 \nu - 4455$$$$)/324$$ $$\beta_{4}$$ $$=$$ $$($$$$-35 \nu^{11} + 83 \nu^{10} - 265 \nu^{9} + 736 \nu^{8} - 1145 \nu^{7} + 2631 \nu^{6} - 4533 \nu^{5} + 5976 \nu^{4} - 11925 \nu^{3} + 12879 \nu^{2} - 12879 \nu + 17010$$$$)/972$$ $$\beta_{5}$$ $$=$$ $$($$$$-15 \nu^{11} + 32 \nu^{10} - 98 \nu^{9} + 289 \nu^{8} - 415 \nu^{7} + 998 \nu^{6} - 1734 \nu^{5} + 2157 \nu^{4} - 4635 \nu^{3} + 4878 \nu^{2} - 4644 \nu + 6885$$$$)/324$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{11} - 16 \nu^{10} + 57 \nu^{9} - 158 \nu^{8} + 228 \nu^{7} - 560 \nu^{6} + 927 \nu^{5} - 1212 \nu^{4} + 2538 \nu^{3} - 2610 \nu^{2} + 2673 \nu - 3564$$$$)/162$$ $$\beta_{7}$$ $$=$$ $$($$$$16 \nu^{11} - 39 \nu^{10} + 127 \nu^{9} - 351 \nu^{8} + 542 \nu^{7} - 1241 \nu^{6} + 2139 \nu^{5} - 2859 \nu^{4} + 5670 \nu^{3} - 6309 \nu^{2} + 5967 \nu - 7857$$$$)/324$$ $$\beta_{8}$$ $$=$$ $$($$$$16 \nu^{11} - 31 \nu^{10} + 98 \nu^{9} - 284 \nu^{8} + 400 \nu^{7} - 969 \nu^{6} + 1668 \nu^{5} - 2070 \nu^{4} + 4302 \nu^{3} - 4455 \nu^{2} + 4212 \nu - 6075$$$$)/243$$ $$\beta_{9}$$ $$=$$ $$($$$$11 \nu^{11} - 26 \nu^{10} + 76 \nu^{9} - 208 \nu^{8} + 317 \nu^{7} - 708 \nu^{6} + 1230 \nu^{5} - 1638 \nu^{4} + 3123 \nu^{3} - 3348 \nu^{2} + 3078 \nu - 3888$$$$)/162$$ $$\beta_{10}$$ $$=$$ $$($$$$-38 \nu^{11} + 91 \nu^{10} - 279 \nu^{9} + 767 \nu^{8} - 1176 \nu^{7} + 2657 \nu^{6} - 4599 \nu^{5} + 6135 \nu^{4} - 11916 \nu^{3} + 13005 \nu^{2} - 12771 \nu + 15633$$$$)/324$$ $$\beta_{11}$$ $$=$$ $$($$$$-119 \nu^{11} + 287 \nu^{10} - 877 \nu^{9} + 2458 \nu^{8} - 3809 \nu^{7} + 8475 \nu^{6} - 14865 \nu^{5} + 19782 \nu^{4} - 38385 \nu^{3} + 42147 \nu^{2} - 40743 \nu + 51516$$$$)/972$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} - \beta_{9} - \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{9} + \beta_{8} - 2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{11} - \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 5 \beta_{7} - 4 \beta_{6} + 14 \beta_{5} - 16 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} - 8 \beta_{1} + 16$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-11 \beta_{11} + \beta_{10} - 2 \beta_{9} - 9 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} + 8 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 10 \beta_{1} + 4$$ $$\nu^{7}$$ $$=$$ $$($$$$-18 \beta_{11} + 23 \beta_{10} - 5 \beta_{9} + 11 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 14 \beta_{5} - 5 \beta_{4} - 28 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} + 11$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$12 \beta_{11} - 4 \beta_{10} + 16 \beta_{9} - \beta_{8} + 8 \beta_{7} - 48 \beta_{6} + 14 \beta_{5} - 74 \beta_{4} - 8 \beta_{3} + 20 \beta_{2} - 46 \beta_{1} + 83$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$-17 \beta_{11} - 15 \beta_{10} - 12 \beta_{9} - 35 \beta_{8} - \beta_{7} + 8 \beta_{6} + 60 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} - 22 \beta_{2} - 60 \beta_{1} + 11$$ $$\nu^{10}$$ $$=$$ $$($$$$-141 \beta_{11} + 97 \beta_{10} - 65 \beta_{9} + 68 \beta_{8} + 39 \beta_{7} + 38 \beta_{6} + 177 \beta_{5} + 55 \beta_{4} - 21 \beta_{3} + 9 \beta_{2} - 92 \beta_{1} + 140$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-58 \beta_{11} - 45 \beta_{10} - 13 \beta_{9} + 273 \beta_{8} + 65 \beta_{7} - 298 \beta_{6} - 90 \beta_{5} + 337 \beta_{4} - 154 \beta_{3} + 313 \beta_{2} - 208 \beta_{1} + 203$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 1.66557 + 0.475255i −0.384890 + 1.68874i −0.111613 + 1.72845i −1.21252 − 1.23685i 0.312065 − 1.70371i 1.73138 + 0.0481063i 1.66557 − 0.475255i −0.384890 − 1.68874i −0.111613 − 1.72845i −1.21252 + 1.23685i 0.312065 + 1.70371i 1.73138 − 0.0481063i
1.00000i −1.68006 + 0.421203i −1.00000 1.00000 0.421203 + 1.68006i 0 1.00000i 2.64518 1.41529i 1.00000i
881.2 1.00000i −1.17770 + 1.27005i −1.00000 1.00000 1.27005 + 1.17770i 0 1.00000i −0.226058 2.99147i 1.00000i
881.3 1.00000i −0.767566 1.55269i −1.00000 1.00000 −1.55269 + 0.767566i 0 1.00000i −1.82168 + 2.38358i 1.00000i
881.4 1.00000i −0.431645 1.67740i −1.00000 1.00000 −1.67740 + 0.431645i 0 1.00000i −2.62736 + 1.44809i 1.00000i
881.5 1.00000i 0.581597 + 1.63149i −1.00000 1.00000 1.63149 0.581597i 0 1.00000i −2.32349 + 1.89773i 1.00000i
881.6 1.00000i 1.47537 + 0.907353i −1.00000 1.00000 0.907353 1.47537i 0 1.00000i 1.35342 + 2.67736i 1.00000i
881.7 1.00000i −1.68006 0.421203i −1.00000 1.00000 0.421203 1.68006i 0 1.00000i 2.64518 + 1.41529i 1.00000i
881.8 1.00000i −1.17770 1.27005i −1.00000 1.00000 1.27005 1.17770i 0 1.00000i −0.226058 + 2.99147i 1.00000i
881.9 1.00000i −0.767566 + 1.55269i −1.00000 1.00000 −1.55269 0.767566i 0 1.00000i −1.82168 2.38358i 1.00000i
881.10 1.00000i −0.431645 + 1.67740i −1.00000 1.00000 −1.67740 0.431645i 0 1.00000i −2.62736 1.44809i 1.00000i
881.11 1.00000i 0.581597 1.63149i −1.00000 1.00000 1.63149 + 0.581597i 0 1.00000i −2.32349 1.89773i 1.00000i
881.12 1.00000i 1.47537 0.907353i −1.00000 1.00000 0.907353 + 1.47537i 0 1.00000i 1.35342 2.67736i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.b.a 12
3.b odd 2 1 1470.2.b.b 12
7.b odd 2 1 1470.2.b.b 12
7.c even 3 1 210.2.r.b yes 12
7.d odd 6 1 210.2.r.a 12
21.c even 2 1 inner 1470.2.b.a 12
21.g even 6 1 210.2.r.b yes 12
21.h odd 6 1 210.2.r.a 12
35.i odd 6 1 1050.2.s.g 12
35.j even 6 1 1050.2.s.f 12
35.k even 12 1 1050.2.u.f 12
35.k even 12 1 1050.2.u.g 12
35.l odd 12 1 1050.2.u.e 12
35.l odd 12 1 1050.2.u.h 12
105.o odd 6 1 1050.2.s.g 12
105.p even 6 1 1050.2.s.f 12
105.w odd 12 1 1050.2.u.e 12
105.w odd 12 1 1050.2.u.h 12
105.x even 12 1 1050.2.u.f 12
105.x even 12 1 1050.2.u.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.r.a 12 7.d odd 6 1
210.2.r.a 12 21.h odd 6 1
210.2.r.b yes 12 7.c even 3 1
210.2.r.b yes 12 21.g even 6 1
1050.2.s.f 12 35.j even 6 1
1050.2.s.f 12 105.p even 6 1
1050.2.s.g 12 35.i odd 6 1
1050.2.s.g 12 105.o odd 6 1
1050.2.u.e 12 35.l odd 12 1
1050.2.u.e 12 105.w odd 12 1
1050.2.u.f 12 35.k even 12 1
1050.2.u.f 12 105.x even 12 1
1050.2.u.g 12 35.k even 12 1
1050.2.u.g 12 105.x even 12 1
1050.2.u.h 12 35.l odd 12 1
1050.2.u.h 12 105.w odd 12 1
1470.2.b.a 12 1.a even 1 1 trivial
1470.2.b.a 12 21.c even 2 1 inner
1470.2.b.b 12 3.b odd 2 1
1470.2.b.b 12 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 $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{12} + 68 T_{11}^{10} + 1590 T_{11}^{8} + 14828 T_{11}^{6} + 50113 T_{11}^{4} + 36360 T_{11}^{2} + 1296$$ $$T_{17}^{6} - 12 T_{17}^{5} + 32 T_{17}^{4} + 96 T_{17}^{3} - 572 T_{17}^{2} + 816 T_{17} - 288$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{6}$$
$3$ $$729 + 972 T + 891 T^{2} + 540 T^{3} + 279 T^{4} + 144 T^{5} + 78 T^{6} + 48 T^{7} + 31 T^{8} + 20 T^{9} + 11 T^{10} + 4 T^{11} + T^{12}$$
$5$ $$( -1 + T )^{12}$$
$7$ $$T^{12}$$
$11$ $$1296 + 36360 T^{2} + 50113 T^{4} + 14828 T^{6} + 1590 T^{8} + 68 T^{10} + T^{12}$$
$13$ $$8433216 + 6772128 T^{2} + 1697017 T^{4} + 156660 T^{6} + 6638 T^{8} + 132 T^{10} + T^{12}$$
$17$ $$( -288 + 816 T - 572 T^{2} + 96 T^{3} + 32 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$19$ $$242985744 + 74536920 T^{2} + 8321497 T^{4} + 450588 T^{6} + 12734 T^{8} + 180 T^{10} + T^{12}$$
$23$ $$5103081 + 3446838 T^{2} + 849007 T^{4} + 95572 T^{6} + 5031 T^{8} + 118 T^{10} + T^{12}$$
$29$ $$12194064 + 21665952 T^{2} + 4309528 T^{4} + 335392 T^{6} + 12009 T^{8} + 190 T^{10} + T^{12}$$
$31$ $$301925376 + 81341952 T^{2} + 8694160 T^{4} + 471168 T^{6} + 13544 T^{8} + 192 T^{10} + T^{12}$$
$37$ $$( 64 - 1232 T + 189 T^{2} + 392 T^{3} - 54 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$41$ $$( 549 - 5802 T + 4627 T^{2} + 260 T^{3} - 145 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$43$ $$( -2972 + 392 T + 1588 T^{2} - 196 T^{3} - 127 T^{4} + T^{6} )^{2}$$
$47$ $$( 3924 + 2472 T - 5507 T^{2} + 1696 T^{3} - 58 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$53$ $$11664 + 2774952 T^{2} + 2991841 T^{4} + 662676 T^{6} + 42326 T^{8} + 396 T^{10} + T^{12}$$
$59$ $$( 21312 - 10176 T - 4496 T^{2} + 2208 T^{3} - 148 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$61$ $$2158903296 + 525004800 T^{2} + 48097024 T^{4} + 2068800 T^{6} + 42209 T^{8} + 366 T^{10} + T^{12}$$
$67$ $$( 576 + 1152 T + 624 T^{2} + 24 T^{3} - 47 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$71$ $$8680276224 + 2846784384 T^{2} + 285221776 T^{4} + 9033600 T^{6} + 108824 T^{8} + 552 T^{10} + T^{12}$$
$73$ $$89884836864 + 12257063424 T^{2} + 548134672 T^{4} + 11285312 T^{6} + 115656 T^{8} + 560 T^{10} + T^{12}$$
$79$ $$( 2896 + 4304 T - 5964 T^{2} + 2032 T^{3} - 204 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$83$ $$( -1152 + 1344 T - 128 T^{2} - 328 T^{3} + 137 T^{4} - 20 T^{5} + T^{6} )^{2}$$
$89$ $$( 125712 + 14400 T - 22088 T^{2} + 3176 T^{3} + 41 T^{4} - 26 T^{5} + T^{6} )^{2}$$
$97$ $$2449062144 + 6309688704 T^{2} + 526939024 T^{4} + 14087168 T^{6} + 154776 T^{8} + 680 T^{10} + T^{12}$$