# Properties

 Label 287.3.d.c Level 287 Weight 3 Character orbit 287.d Analytic conductor 7.820 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta_{4} q^{3} -3 q^{4} + \beta_{6} q^{5} + \beta_{4} q^{6} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} + 7 q^{8} + ( 1 - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q - q^{2} -\beta_{4} q^{3} -3 q^{4} + \beta_{6} q^{5} + \beta_{4} q^{6} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} + 7 q^{8} + ( 1 - 2 \beta_{5} ) q^{9} -\beta_{6} q^{10} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{11} + 3 \beta_{4} q^{12} + \beta_{4} q^{13} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{14} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{15} + 5 q^{16} + ( -\beta_{1} - 6 \beta_{4} ) q^{17} + ( -1 + 2 \beta_{5} ) q^{18} + ( -8 \beta_{1} - 7 \beta_{4} ) q^{19} -3 \beta_{6} q^{20} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{21} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{22} + ( -13 - 3 \beta_{5} ) q^{23} -7 \beta_{4} q^{24} + ( -8 - 5 \beta_{5} ) q^{25} -\beta_{4} q^{26} + ( -4 \beta_{1} - 2 \beta_{4} ) q^{27} + ( -6 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} ) q^{28} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{7} ) q^{29} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{30} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{31} -33 q^{32} + ( 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{4} - 4 \beta_{6} ) q^{33} + ( \beta_{1} + 6 \beta_{4} ) q^{34} + ( 4 \beta_{1} - 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{7} ) q^{35} + ( -3 + 6 \beta_{5} ) q^{36} + ( -28 + 2 \beta_{5} ) q^{37} + ( 8 \beta_{1} + 7 \beta_{4} ) q^{38} + ( -10 + 2 \beta_{5} ) q^{39} + 7 \beta_{6} q^{40} + ( 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{41} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{42} + ( 8 - 6 \beta_{5} ) q^{43} + ( -3 \beta_{1} - 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} ) q^{44} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 5 \beta_{6} ) q^{45} + ( 13 + 3 \beta_{5} ) q^{46} + ( -9 \beta_{1} + 10 \beta_{4} ) q^{47} -5 \beta_{4} q^{48} + ( -11 - \beta_{1} + 2 \beta_{2} - \beta_{4} + 8 \beta_{5} - 4 \beta_{6} ) q^{49} + ( 8 + 5 \beta_{5} ) q^{50} + ( 56 - 12 \beta_{5} ) q^{51} -3 \beta_{4} q^{52} + ( 3 \beta_{1} + 6 \beta_{3} + 3 \beta_{4} + 5 \beta_{7} ) q^{53} + ( 4 \beta_{1} + 2 \beta_{4} ) q^{54} + ( -14 \beta_{1} - 17 \beta_{4} ) q^{55} + ( 14 \beta_{1} + 7 \beta_{3} + 7 \beta_{4} ) q^{56} + ( 38 - 14 \beta_{5} ) q^{57} + ( 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} ) q^{58} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{59} + ( 3 \beta_{1} + 6 \beta_{3} + 3 \beta_{4} ) q^{60} + ( 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{61} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{62} + ( -8 \beta_{1} + 7 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} ) q^{63} + 13 q^{64} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{65} + ( -3 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} + 4 \beta_{6} ) q^{66} + ( -4 \beta_{1} - 8 \beta_{3} - 4 \beta_{4} - \beta_{7} ) q^{67} + ( 3 \beta_{1} + 18 \beta_{4} ) q^{68} + ( -6 \beta_{1} - 2 \beta_{4} ) q^{69} + ( -4 \beta_{1} + 2 \beta_{3} + 5 \beta_{4} - 3 \beta_{7} ) q^{70} + ( -3 \beta_{1} - 6 \beta_{3} - 3 \beta_{4} - 12 \beta_{7} ) q^{71} + ( 7 - 14 \beta_{5} ) q^{72} -6 \beta_{6} q^{73} + ( 28 - 2 \beta_{5} ) q^{74} + ( -10 \beta_{1} - 17 \beta_{4} ) q^{75} + ( 24 \beta_{1} + 21 \beta_{4} ) q^{76} + ( -57 + 17 \beta_{5} + 5 \beta_{6} ) q^{77} + ( 10 - 2 \beta_{5} ) q^{78} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} + 6 \beta_{7} ) q^{79} + 5 \beta_{6} q^{80} + ( -5 + 14 \beta_{5} ) q^{81} + ( -6 \beta_{1} + 3 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} ) q^{82} + ( -7 \beta_{1} + 14 \beta_{2} - 7 \beta_{4} - \beta_{6} ) q^{83} + ( -3 - 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} ) q^{84} + ( -5 \beta_{1} - 10 \beta_{3} - 5 \beta_{4} - 2 \beta_{7} ) q^{85} + ( -8 + 6 \beta_{5} ) q^{86} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + 8 \beta_{6} ) q^{87} + ( 7 \beta_{1} + 14 \beta_{3} + 7 \beta_{4} + 7 \beta_{7} ) q^{88} + ( 23 \beta_{1} + 21 \beta_{4} ) q^{89} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} + 5 \beta_{6} ) q^{90} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{91} + ( 39 + 9 \beta_{5} ) q^{92} + ( -4 \beta_{1} - 8 \beta_{3} - 4 \beta_{4} - 4 \beta_{7} ) q^{93} + ( 9 \beta_{1} - 10 \beta_{4} ) q^{94} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} - 16 \beta_{7} ) q^{95} + 33 \beta_{4} q^{96} + ( 19 \beta_{1} - 19 \beta_{4} ) q^{97} + ( 11 + \beta_{1} - 2 \beta_{2} + \beta_{4} - 8 \beta_{5} + 4 \beta_{6} ) q^{98} + ( 13 \beta_{1} + 26 \beta_{3} + 13 \beta_{4} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{2} - 24q^{4} + 56q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{2} - 24q^{4} + 56q^{8} + 8q^{9} + 40q^{16} - 8q^{18} + 8q^{21} - 104q^{23} - 64q^{25} - 264q^{32} - 24q^{36} - 224q^{37} - 80q^{39} - 8q^{42} + 64q^{43} + 104q^{46} - 88q^{49} + 64q^{50} + 448q^{51} + 304q^{57} + 104q^{64} + 56q^{72} + 224q^{74} - 456q^{77} + 80q^{78} - 40q^{81} - 24q^{84} - 64q^{86} - 8q^{91} + 312q^{92} + 88q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 120 x^{6} + 2874 x^{4} + 16920 x^{2} + 19881$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-83 \nu^{6} - 10524 \nu^{4} - 286059 \nu^{2} - 1258848$$$$)/191196$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} + 5 \nu^{6} - 506 \nu^{5} + 506 \nu^{4} - 3795 \nu^{3} + 3795 \nu^{2} + 146076 \nu - 50478$$$$)/31866$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} - 5 \nu^{6} + 506 \nu^{5} - 506 \nu^{4} + 3795 \nu^{3} - 3795 \nu^{2} - 82344 \nu + 50478$$$$)/31866$$ $$\beta_{4}$$ $$=$$ $$($$$$143 \nu^{6} + 16596 \nu^{4} + 331599 \nu^{2} + 653112$$$$)/191196$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{6} + 225 \nu^{4} + 4230 \nu^{2} + 12285$$$$)/1356$$ $$\beta_{6}$$ $$=$$ $$($$$$-365 \nu^{7} - 42249 \nu^{5} - 882489 \nu^{3} - 3564621 \nu$$$$)/382392$$ $$\beta_{7}$$ $$=$$ $$($$$$59 \nu^{7} + 7033 \nu^{5} + 161623 \nu^{3} + 675813 \nu$$$$)/42488$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$8 \beta_{5} - 21 \beta_{4} - 9 \beta_{1} - 60$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-20 \beta_{7} - 36 \beta_{6} - 21 \beta_{4} - 105 \beta_{3} - 63 \beta_{2} - 21 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-420 \beta_{5} + 978 \beta_{4} + 258 \beta_{1} + 2163$$ $$\nu^{5}$$ $$=$$ $$($$$$2280 \beta_{7} + 3960 \beta_{6} + 1956 \beta_{4} + 9291 \beta_{3} + 5379 \beta_{2} + 1956 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$78936 \beta_{5} - 175635 \beta_{4} - 39015 \beta_{1} - 372060$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-215556 \beta_{7} - 373428 \beta_{6} - 175635 \beta_{4} - 831339 \beta_{3} - 480069 \beta_{2} - 175635 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
286.1
 − 9.49661i 9.49661i 1.25038i − 1.25038i 4.66647i − 4.66647i − 2.54461i 2.54461i
−1.00000 −4.37780 −3.00000 3.17602i 4.37780 0.818350 6.95200i 7.00000 10.1652 3.17602i
286.2 −1.00000 −4.37780 −3.00000 3.17602i 4.37780 0.818350 + 6.95200i 7.00000 10.1652 3.17602i
286.3 −1.00000 −0.913701 −3.00000 7.47749i 0.913701 −6.10985 3.41609i 7.00000 −8.16515 7.47749i
286.4 −1.00000 −0.913701 −3.00000 7.47749i 0.913701 −6.10985 + 3.41609i 7.00000 −8.16515 7.47749i
286.5 −1.00000 0.913701 −3.00000 7.47749i −0.913701 6.10985 + 3.41609i 7.00000 −8.16515 7.47749i
286.6 −1.00000 0.913701 −3.00000 7.47749i −0.913701 6.10985 3.41609i 7.00000 −8.16515 7.47749i
286.7 −1.00000 4.37780 −3.00000 3.17602i −4.37780 −0.818350 + 6.95200i 7.00000 10.1652 3.17602i
286.8 −1.00000 4.37780 −3.00000 3.17602i −4.37780 −0.818350 6.95200i 7.00000 10.1652 3.17602i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 286.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.b even 2 1 inner
287.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.d.c 8
7.b odd 2 1 inner 287.3.d.c 8
41.b even 2 1 inner 287.3.d.c 8
287.d odd 2 1 inner 287.3.d.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.d.c 8 1.a even 1 1 trivial
287.3.d.c 8 7.b odd 2 1 inner
287.3.d.c 8 41.b even 2 1 inner
287.3.d.c 8 287.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(287, [\chi])$$:

 $$T_{2} + 1$$ $$T_{3}^{4} - 20 T_{3}^{2} + 16$$