# Properties

 Label 211.2.a.a Level 211 Weight 2 Character orbit 211.a Self dual yes Analytic conductor 1.685 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$211$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 211.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.68484348265$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.618034 1.61803
−0.618034 0.381966 −1.61803 3.23607 −0.236068 1.61803 2.23607 −2.85410 −2.00000
1.2 1.61803 2.61803 0.618034 −1.23607 4.23607 −0.618034 −2.23607 3.85410 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$211$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.2.a.a 2
3.b odd 2 1 1899.2.a.d 2
4.b odd 2 1 3376.2.a.f 2
5.b even 2 1 5275.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.a.a 2 1.a even 1 1 trivial
1899.2.a.d 2 3.b odd 2 1
3376.2.a.f 2 4.b odd 2 1
5275.2.a.d 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(211))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$3$ $$1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4}$$
$5$ $$1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ $$1 - T + 13 T^{2} - 7 T^{3} + 49 T^{4}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 8 T + 37 T^{2} - 104 T^{3} + 169 T^{4}$$
$17$ $$1 - 11 T + 63 T^{2} - 187 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4}$$
$29$ $$1 + 53 T^{2} + 841 T^{4}$$
$31$ $$1 + 11 T + 61 T^{2} + 341 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 9 T + 43 T^{2} )^{2}$$
$47$ $$1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4}$$
$53$ $$1 - 13 T + 147 T^{2} - 689 T^{3} + 2809 T^{4}$$
$59$ $$1 + 73 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + 3 T + 61 T^{2} )^{2}$$
$67$ $$( 1 + 12 T + 67 T^{2} )^{2}$$
$71$ $$1 + 6 T + 26 T^{2} + 426 T^{3} + 5041 T^{4}$$
$73$ $$1 + 7 T + 127 T^{2} + 511 T^{3} + 5329 T^{4}$$
$79$ $$1 + 10 T + 138 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 8 T + 162 T^{2} - 664 T^{3} + 6889 T^{4}$$
$89$ $$1 - 15 T + 223 T^{2} - 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 - T + 183 T^{2} - 97 T^{3} + 9409 T^{4}$$