# Properties

 Label 287.3.d.a Level 287 Weight 3 Character orbit 287.d Self dual yes Analytic conductor 7.820 Analytic rank 0 Dimension 7 CM discriminant -287 Inner twists 2

# 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: yes Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: 7.7.19468476636329.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( 4 + \beta_{3} + \beta_{4} ) q^{4} + ( 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{6} -7 q^{7} + ( 4 \beta_{1} + 4 \beta_{2} + \beta_{6} ) q^{8} + ( 9 - 5 \beta_{1} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( 4 + \beta_{3} + \beta_{4} ) q^{4} + ( 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{6} -7 q^{7} + ( 4 \beta_{1} + 4 \beta_{2} + \beta_{6} ) q^{8} + ( 9 - 5 \beta_{1} - \beta_{6} ) q^{9} + ( 17 - \beta_{1} + 4 \beta_{4} - 2 \beta_{6} ) q^{12} + ( -\beta_{2} + 3 \beta_{5} ) q^{13} -7 \beta_{2} q^{14} + ( 16 + 7 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} ) q^{16} + ( 11 \beta_{1} - \beta_{6} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{18} + ( -13 \beta_{1} - \beta_{6} ) q^{19} -7 \beta_{4} q^{21} + ( -8 \beta_{3} + \beta_{4} ) q^{23} + ( 17 \beta_{2} - \beta_{3} - 8 \beta_{4} + 4 \beta_{6} ) q^{24} + 25 q^{25} + ( -11 + 5 \beta_{3} - 7 \beta_{4} ) q^{26} + ( -9 \beta_{2} + 9 \beta_{4} - 5 \beta_{5} ) q^{27} + ( -28 - 7 \beta_{3} - 7 \beta_{4} ) q^{28} + ( 16 \beta_{1} + 16 \beta_{2} + 7 \beta_{3} - 8 \beta_{4} + 4 \beta_{6} ) q^{32} + ( 23 \beta_{1} + 11 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{34} + ( 36 - 20 \beta_{1} - 18 \beta_{2} + 17 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{36} + ( -17 \beta_{2} + 3 \beta_{5} ) q^{37} + ( -25 \beta_{1} - 13 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} ) q^{38} + ( 27 \beta_{1} - 17 \beta_{2} - 5 \beta_{5} - \beta_{6} ) q^{39} + 41 q^{41} + ( -14 \beta_{2} + 7 \beta_{5} - 7 \beta_{6} ) q^{42} + ( -5 \beta_{1} + 7 \beta_{6} ) q^{43} + ( -32 \beta_{1} - 14 \beta_{2} - 9 \beta_{5} + \beta_{6} ) q^{46} + ( 23 \beta_{2} + 3 \beta_{5} ) q^{47} + ( 68 - 4 \beta_{1} - 18 \beta_{2} + 17 \beta_{3} + 17 \beta_{4} + 7 \beta_{5} - 8 \beta_{6} ) q^{48} + 49 q^{49} + 25 \beta_{2} q^{50} + ( -9 \beta_{2} + 16 \beta_{3} - 7 \beta_{4} + 11 \beta_{5} ) q^{51} + ( 20 \beta_{1} - 11 \beta_{2} - 7 \beta_{6} ) q^{52} + ( -67 + 18 \beta_{2} - 19 \beta_{3} + \beta_{4} - 9 \beta_{5} + 9 \beta_{6} ) q^{54} + ( -28 \beta_{1} - 28 \beta_{2} - 7 \beta_{6} ) q^{56} + ( -9 \beta_{2} - 8 \beta_{3} + 17 \beta_{4} - 13 \beta_{5} ) q^{57} + ( -63 + 35 \beta_{1} + 7 \beta_{6} ) q^{63} + ( 64 + 28 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} + 15 \beta_{5} - 8 \beta_{6} ) q^{64} + ( 44 \beta_{1} + 14 \beta_{2} + 23 \beta_{3} + 8 \beta_{4} + 15 \beta_{5} - 4 \beta_{6} ) q^{68} + ( 26 - 37 \beta_{1} + 7 \beta_{6} ) q^{69} + ( -143 + 34 \beta_{2} - 20 \beta_{3} - 16 \beta_{4} - 17 \beta_{5} + 17 \beta_{6} ) q^{72} + ( -139 - 11 \beta_{3} - 23 \beta_{4} ) q^{74} + 25 \beta_{4} q^{75} + ( -52 \beta_{1} - 34 \beta_{2} - 25 \beta_{3} + 8 \beta_{4} - 9 \beta_{5} - 4 \beta_{6} ) q^{76} + ( -131 + 55 \beta_{1} - 11 \beta_{4} + 2 \beta_{6} ) q^{78} + ( 81 - 45 \beta_{1} + 7 \beta_{2} + 19 \beta_{5} - 9 \beta_{6} ) q^{81} + 41 \beta_{2} q^{82} + ( -119 + 7 \beta_{1} - 28 \beta_{4} + 14 \beta_{6} ) q^{84} + ( -17 \beta_{1} - 5 \beta_{3} + 28 \beta_{4} - 14 \beta_{6} ) q^{86} + ( -\beta_{2} - 21 \beta_{5} ) q^{89} + ( 7 \beta_{2} - 21 \beta_{5} ) q^{91} + ( -103 - 65 \beta_{1} - 32 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} ) q^{92} + ( 181 + 29 \beta_{3} + 17 \beta_{4} ) q^{94} + ( -151 + 68 \beta_{1} + 68 \beta_{2} - 4 \beta_{3} - 32 \beta_{4} + 17 \beta_{6} ) q^{96} + ( 8 \beta_{3} - 31 \beta_{4} ) q^{97} + 49 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 28q^{4} - 49q^{7} + 63q^{9} + O(q^{10})$$ $$7q + 28q^{4} - 49q^{7} + 63q^{9} + 119q^{12} + 112q^{16} + 175q^{25} - 77q^{26} - 196q^{28} + 252q^{36} + 287q^{41} + 476q^{48} + 343q^{49} - 469q^{54} - 441q^{63} + 448q^{64} + 182q^{69} - 1001q^{72} - 973q^{74} - 917q^{78} + 567q^{81} - 833q^{84} - 721q^{92} + 1267q^{94} - 1057q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 14 x^{5} + 56 x^{3} - 56 x - 15$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 8 \nu^{2} + 6 \nu + 8$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 10 \nu^{3} - 2 \nu^{2} + 20 \nu + 8$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 12 \nu^{4} + 36 \nu^{2} - 4 \nu - 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 8 \beta_{2} + 24$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 10 \beta_{3} + 2 \beta_{2} + 40 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + 12 \beta_{4} + 12 \beta_{3} + 60 \beta_{2} + 4 \beta_{1} + 160$$

## 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
 −0.292301 −0.957283 1.48399 2.01727 −2.38176 −2.67770 2.80779
−3.91456 5.59495 11.3238 0 −21.9018 −7.00000 −28.6694 22.3034 0
286.2 −3.08361 −3.35781 5.50864 0 10.3542 −7.00000 −4.65206 2.27491 0
286.3 −1.79777 0.867827 −0.768030 0 −1.56015 −7.00000 8.57181 −8.24688 0
286.4 0.0693632 −4.10058 −3.99519 0 −0.284429 −7.00000 −0.554572 7.81476 0
286.5 1.67278 −5.98117 −1.20181 0 −10.0052 −7.00000 −8.70148 26.7744 0
286.6 3.17010 5.18274 6.04955 0 16.4298 −7.00000 6.49729 17.8608 0
286.7 3.88369 1.79404 11.0831 0 6.96751 −7.00000 27.5084 −5.78141 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 286.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
287.d odd 2 1 CM by $$\Q(\sqrt{-287})$$

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{7} - 28 T_{2}^{5} + 224 T_{2}^{3} - 448 T_{2} + 31$$ $$T_{3}^{7} - 63 T_{3}^{5} + 1134 T_{3}^{3} - 5103 T_{3} + 3718$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 31 T^{7} + 16384 T^{14}$$
$3$ $$1 + 3718 T^{7} + 4782969 T^{14}$$
$5$ $$( 1 - 5 T )^{7}( 1 + 5 T )^{7}$$
$7$ $$( 1 + 7 T )^{7}$$
$11$ $$( 1 - 11 T )^{7}( 1 + 11 T )^{7}$$
$13$ $$1 + 24950438 T^{7} + 3937376385699289 T^{14}$$
$17$ $$1 + 800009246 T^{7} + 168377826559400929 T^{14}$$
$19$ $$1 + 305833574 T^{7} + 799006685782884121 T^{14}$$
$23$ $$1 - 6483516722 T^{7} + 11592836324538749809 T^{14}$$
$29$ $$( 1 - 29 T )^{7}( 1 + 29 T )^{7}$$
$31$ $$( 1 - 31 T )^{7}( 1 + 31 T )^{7}$$
$37$ $$1 - 101889625898 T^{7} +$$$$90\!\cdots\!89$$$$T^{14}$$
$41$ $$( 1 - 41 T )^{7}$$
$43$ $$1 - 358855177802 T^{7} +$$$$73\!\cdots\!49$$$$T^{14}$$
$47$ $$1 - 987930187618 T^{7} +$$$$25\!\cdots\!69$$$$T^{14}$$
$53$ $$( 1 - 53 T )^{7}( 1 + 53 T )^{7}$$
$59$ $$( 1 - 59 T )^{7}( 1 + 59 T )^{7}$$
$61$ $$( 1 - 61 T )^{7}( 1 + 61 T )^{7}$$
$67$ $$( 1 - 67 T )^{7}( 1 + 67 T )^{7}$$
$71$ $$( 1 - 71 T )^{7}( 1 + 71 T )^{7}$$
$73$ $$( 1 - 73 T )^{7}( 1 + 73 T )^{7}$$
$79$ $$( 1 - 79 T )^{7}( 1 + 79 T )^{7}$$
$83$ $$( 1 - 83 T )^{7}( 1 + 83 T )^{7}$$
$89$ $$1 - 87666289649746 T^{7} +$$$$19\!\cdots\!41$$$$T^{14}$$
$97$ $$1 - 74677982531458 T^{7} +$$$$65\!\cdots\!69$$$$T^{14}$$