# Properties

 Label 287.2.c.b Level 287 Weight 2 Character orbit 287.c Analytic conductor 2.292 Analytic rank 0 Dimension 12 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 287.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{8} ) q^{3} + ( 1 - \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( -3 \beta_{8} - \beta_{9} ) q^{6} + \beta_{8} q^{7} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{8} + ( -1 - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{8} ) q^{3} + ( 1 - \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( -3 \beta_{8} - \beta_{9} ) q^{6} + \beta_{8} q^{7} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{8} + ( -1 - \beta_{2} + \beta_{3} ) q^{9} + ( -\beta_{3} + \beta_{4} ) q^{10} + ( 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{11} + ( -2 \beta_{1} - \beta_{6} + \beta_{8} ) q^{12} + ( \beta_{1} - \beta_{8} - \beta_{10} ) q^{13} + \beta_{1} q^{14} + ( \beta_{1} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{15} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{16} + ( \beta_{6} - \beta_{9} - \beta_{10} ) q^{17} + ( -4 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{18} + ( -\beta_{1} - 2 \beta_{8} + \beta_{11} ) q^{19} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{20} + ( 1 + \beta_{2} ) q^{21} + ( \beta_{1} + 3 \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{22} + ( 1 - \beta_{2} + 2 \beta_{5} ) q^{23} + ( 3 \beta_{1} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{24} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{25} + ( -\beta_{1} + 2 \beta_{8} + \beta_{9} ) q^{26} + ( -\beta_{1} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{27} + ( -\beta_{1} + \beta_{8} + \beta_{9} ) q^{28} + ( -\beta_{1} - \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{29} + ( \beta_{1} - 3 \beta_{6} - \beta_{10} ) q^{30} + ( \beta_{2} + \beta_{5} + \beta_{7} ) q^{31} + ( -4 + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{32} + ( 2 + \beta_{2} + 3 \beta_{5} + \beta_{7} ) q^{33} + ( -\beta_{6} + 2 \beta_{9} + \beta_{11} ) q^{34} + ( -\beta_{1} + \beta_{6} + \beta_{10} ) q^{35} + ( -4 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{36} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{37} + ( -3 \beta_{1} + 2 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{38} + ( 1 - \beta_{3} - \beta_{7} ) q^{39} + ( 4 + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - \beta_{7} ) q^{40} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{41} + ( 3 - \beta_{3} ) q^{42} + ( -4 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{43} + ( -\beta_{1} - 3 \beta_{6} + 6 \beta_{8} + 6 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{44} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{45} + ( 1 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{46} + ( \beta_{1} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} + ( -\beta_{1} - 2 \beta_{6} + 6 \beta_{8} + 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{48} - q^{49} + ( 2 + 2 \beta_{2} - 3 \beta_{5} - \beta_{7} ) q^{50} + ( -\beta_{3} + \beta_{4} - \beta_{7} ) q^{51} + ( 2 \beta_{1} + \beta_{6} - \beta_{9} + 2 \beta_{10} ) q^{52} + ( -\beta_{1} + \beta_{6} + 2 \beta_{8} - \beta_{10} ) q^{53} + ( 5 \beta_{1} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{54} + ( 5 \beta_{1} - \beta_{6} + 2 \beta_{8} + \beta_{10} ) q^{55} + ( \beta_{1} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{56} + ( -6 - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{57} + ( 3 \beta_{1} - 3 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{58} + ( -6 - \beta_{2} + \beta_{5} + \beta_{7} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{6} - 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{60} + ( 6 + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{61} + ( 6 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{62} + ( -\beta_{1} - \beta_{8} - \beta_{9} ) q^{63} + ( 1 - \beta_{2} - 3 \beta_{3} + \beta_{4} - 5 \beta_{5} - 2 \beta_{7} ) q^{64} + ( \beta_{1} - \beta_{6} - 4 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} ) q^{65} + ( 10 - 7 \beta_{3} + 3 \beta_{4} ) q^{66} + ( -5 \beta_{1} + \beta_{6} + 4 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{67} + ( 2 \beta_{6} - \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{68} + ( -\beta_{1} - 2 \beta_{6} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} ) q^{69} + ( \beta_{9} + \beta_{11} ) q^{70} + ( 4 \beta_{1} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{71} + ( \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{72} + ( -6 - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{73} + ( 2 - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{74} + ( -3 \beta_{1} + 4 \beta_{6} - 5 \beta_{8} + \beta_{9} + \beta_{11} ) q^{75} + ( -4 \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{76} + ( -2 + \beta_{3} - \beta_{4} ) q^{77} + ( 3 \beta_{2} + \beta_{5} ) q^{78} + ( -2 \beta_{1} + 2 \beta_{9} + 2 \beta_{10} ) q^{79} + ( -2 + 4 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{80} + ( \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{81} + ( -5 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{9} + 2 \beta_{11} ) q^{82} + ( 6 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{83} + ( -1 + 2 \beta_{2} + \beta_{5} ) q^{84} + ( -\beta_{1} - \beta_{6} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{85} + ( 2 - 6 \beta_{2} - \beta_{4} - \beta_{5} ) q^{86} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 4 \beta_{5} ) q^{87} + ( 8 \beta_{1} + 4 \beta_{6} - 6 \beta_{8} - 7 \beta_{9} - 3 \beta_{11} ) q^{88} + ( -3 \beta_{6} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{89} + ( -4 + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - \beta_{7} ) q^{90} + ( 1 - \beta_{2} + \beta_{7} ) q^{91} + ( 5 - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{92} + ( \beta_{1} - \beta_{6} - 6 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{93} + ( -6 \beta_{1} + \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{94} + ( 7 \beta_{1} - 5 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{95} + ( 4 \beta_{1} + 6 \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{96} + ( 3 \beta_{1} - 4 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} ) q^{97} -\beta_{2} q^{98} + ( -\beta_{1} - 3 \beta_{6} - 6 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{2} + 12q^{4} + 2q^{5} - 24q^{8} - 4q^{9} + O(q^{10})$$ $$12q - 4q^{2} + 12q^{4} + 2q^{5} - 24q^{8} - 4q^{9} - 4q^{10} + 28q^{16} - 40q^{18} + 2q^{20} + 8q^{21} + 16q^{23} + 34q^{25} - 6q^{31} - 42q^{32} + 18q^{33} - 36q^{36} - 10q^{37} + 10q^{39} + 38q^{40} - 2q^{41} + 32q^{42} - 50q^{43} + 6q^{45} - 8q^{46} - 12q^{49} + 18q^{50} - 2q^{51} - 50q^{57} - 70q^{59} + 52q^{61} + 68q^{62} + 8q^{64} + 92q^{66} + 2q^{72} - 64q^{73} + 18q^{74} - 20q^{77} - 12q^{78} - 32q^{80} - 4q^{81} - 56q^{82} + 60q^{83} - 20q^{84} + 48q^{86} - 20q^{87} - 42q^{90} + 14q^{91} + 56q^{92} + 4q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 18 x^{10} + 113 x^{8} + 290 x^{6} + 258 x^{4} + 49 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{10} - 20 \nu^{8} - 13 \nu^{6} + 213 \nu^{4} + 218 \nu^{2} + 13$$$$)/53$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{10} + 20 \nu^{8} + 13 \nu^{6} - 213 \nu^{4} - 165 \nu^{2} + 146$$$$)/53$$ $$\beta_{4}$$ $$=$$ $$($$$$-10 \nu^{10} - 153 \nu^{8} - 754 \nu^{6} - 1320 \nu^{4} - 447 \nu^{2} + 171$$$$)/53$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{10} - 133 \nu^{8} - 741 \nu^{6} - 1586 \nu^{4} - 1036 \nu^{2} - 107$$$$)/53$$ $$\beta_{6}$$ $$=$$ $$($$$$-15 \nu^{11} - 256 \nu^{9} - 1502 \nu^{7} - 3570 \nu^{5} - 2976 \nu^{3} - 618 \nu$$$$)/53$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{10} + 236 \nu^{8} + 1489 \nu^{6} + 3783 \nu^{4} + 3141 \nu^{2} + 366$$$$)/53$$ $$\beta_{8}$$ $$=$$ $$($$$$13 \nu^{11} + 236 \nu^{9} + 1489 \nu^{7} + 3783 \nu^{5} + 3141 \nu^{3} + 419 \nu$$$$)/53$$ $$\beta_{9}$$ $$=$$ $$($$$$-41 \nu^{11} - 728 \nu^{9} - 4480 \nu^{7} - 11136 \nu^{5} - 9205 \nu^{3} - 1191 \nu$$$$)/53$$ $$\beta_{10}$$ $$=$$ $$($$$$40 \nu^{11} + 718 \nu^{9} + 4500 \nu^{7} + 11587 \nu^{5} + 10533 \nu^{3} + 2178 \nu$$$$)/53$$ $$\beta_{11}$$ $$=$$ $$($$$$48 \nu^{11} + 851 \nu^{9} + 5241 \nu^{7} + 13120 \nu^{5} + 11092 \nu^{3} + 1437 \nu$$$$)/53$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} + 2 \beta_{8} - \beta_{6} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 7 \beta_{3} - 8 \beta_{2} + 16$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{11} + \beta_{10} - 11 \beta_{9} - 20 \beta_{8} + 9 \beta_{6} + 29 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{7} + 15 \beta_{5} - 11 \beta_{4} + 49 \beta_{3} + 57 \beta_{2} - 97$$ $$\nu^{7}$$ $$=$$ $$26 \beta_{11} - 11 \beta_{10} + 98 \beta_{9} + 165 \beta_{8} - 71 \beta_{6} - 183 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-26 \beta_{7} - 150 \beta_{5} + 97 \beta_{4} - 351 \beta_{3} - 405 \beta_{2} + 630$$ $$\nu^{9}$$ $$=$$ $$-247 \beta_{11} + 97 \beta_{10} - 802 \beta_{9} - 1287 \beta_{8} + 545 \beta_{6} + 1218 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$247 \beta_{7} + 1296 \beta_{5} - 792 \beta_{4} + 2555 \beta_{3} + 2910 \beta_{2} - 4286$$ $$\nu^{11}$$ $$=$$ $$2088 \beta_{11} - 792 \beta_{10} + 6294 \beta_{9} + 9806 \beta_{8} - 4139 \beta_{6} - 8414 \beta_{1}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/287\mathbb{Z}\right)^\times$$.

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
204.1
 2.72816i − 2.72816i 2.16363i − 2.16363i − 0.473876i 0.473876i 0.152182i − 0.152182i 1.12463i − 1.12463i 2.08886i − 2.08886i
−2.72816 1.72816i 5.44285 −1.58817 4.71469i 1.00000i −9.39263 0.0134697 4.33279
204.2 −2.72816 1.72816i 5.44285 −1.58817 4.71469i 1.00000i −9.39263 0.0134697 4.33279
204.3 −2.16363 1.16363i 2.68131 2.90865 2.51768i 1.00000i −1.47412 1.64595 −6.29325
204.4 −2.16363 1.16363i 2.68131 2.90865 2.51768i 1.00000i −1.47412 1.64595 −6.29325
204.5 −0.473876 0.526124i −1.77544 −1.62273 0.249318i 1.00000i 1.78909 2.72319 0.768972
204.6 −0.473876 0.526124i −1.77544 −1.62273 0.249318i 1.00000i 1.78909 2.72319 0.768972
204.7 0.152182 1.15218i −1.97684 3.83685 0.175342i 1.00000i −0.605204 1.67248 0.583900
204.8 0.152182 1.15218i −1.97684 3.83685 0.175342i 1.00000i −0.605204 1.67248 0.583900
204.9 1.12463 2.12463i −0.735213 −4.04675 2.38941i 1.00000i −3.07610 −1.51404 −4.55109
204.10 1.12463 2.12463i −0.735213 −4.04675 2.38941i 1.00000i −3.07610 −1.51404 −4.55109
204.11 2.08886 3.08886i 2.36333 1.51216 6.45219i 1.00000i 0.758953 −6.54105 3.15868
204.12 2.08886 3.08886i 2.36333 1.51216 6.45219i 1.00000i 0.758953 −6.54105 3.15868
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 204.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
41.b Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2 T_{2}^{5} - 7 T_{2}^{4} - 10 T_{2}^{3} + 12 T_{2}^{2} + 5 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$.