# Properties

 Label 211.2.a.b Level 211 Weight 2 Character orbit 211.a Self dual yes Analytic conductor 1.685 Analytic rank 1 Dimension 3 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.80194 −1.24698 0.445042
−2.24698 −1.80194 3.04892 −1.19806 4.04892 3.93900 −2.35690 0.246980 2.69202
1.2 −0.554958 1.24698 −1.69202 −4.24698 −0.692021 1.91185 2.04892 −1.44504 2.35690
1.3 0.801938 −0.445042 −1.35690 −2.55496 −0.356896 −3.85086 −2.69202 −2.80194 −2.04892
 $$n$$: 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.b 3
3.b odd 2 1 1899.2.a.f 3
4.b odd 2 1 3376.2.a.j 3
5.b even 2 1 5275.2.a.i 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2 T_{2}^{2} - T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(211))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 8 T^{5} + 8 T^{6}$$
$3$ $$1 + T + 7 T^{2} + 5 T^{3} + 21 T^{4} + 9 T^{5} + 27 T^{6}$$
$5$ $$1 + 8 T + 34 T^{2} + 93 T^{3} + 170 T^{4} + 200 T^{5} + 125 T^{6}$$
$7$ $$1 - 2 T + 6 T^{2} + T^{3} + 42 T^{4} - 98 T^{5} + 343 T^{6}$$
$11$ $$1 + 2 T + 4 T^{2} - 27 T^{3} + 44 T^{4} + 242 T^{5} + 1331 T^{6}$$
$13$ $$1 + 3 T + 35 T^{2} + 79 T^{3} + 455 T^{4} + 507 T^{5} + 2197 T^{6}$$
$17$ $$1 - 6 T + 56 T^{2} - 205 T^{3} + 952 T^{4} - 1734 T^{5} + 4913 T^{6}$$
$19$ $$1 + 7 T + 57 T^{2} + 259 T^{3} + 1083 T^{4} + 2527 T^{5} + 6859 T^{6}$$
$23$ $$1 + 19 T + 187 T^{2} + 1113 T^{3} + 4301 T^{4} + 10051 T^{5} + 12167 T^{6}$$
$29$ $$1 + 6 T + 8 T^{2} - 29 T^{3} + 232 T^{4} + 5046 T^{5} + 24389 T^{6}$$
$31$ $$1 + 5 T + 71 T^{2} + 297 T^{3} + 2201 T^{4} + 4805 T^{5} + 29791 T^{6}$$
$37$ $$1 + 2 T + 68 T^{2} + 231 T^{3} + 2516 T^{4} + 2738 T^{5} + 50653 T^{6}$$
$41$ $$1 + 18 T + 203 T^{2} + 1468 T^{3} + 8323 T^{4} + 30258 T^{5} + 68921 T^{6}$$
$43$ $$1 - 4 T + 118 T^{2} - 343 T^{3} + 5074 T^{4} - 7396 T^{5} + 79507 T^{6}$$
$47$ $$1 - 11 T + 165 T^{2} - 1047 T^{3} + 7755 T^{4} - 24299 T^{5} + 103823 T^{6}$$
$53$ $$1 + 10 T + 134 T^{2} + 935 T^{3} + 7102 T^{4} + 28090 T^{5} + 148877 T^{6}$$
$59$ $$1 - 5 T + 99 T^{2} - 759 T^{3} + 5841 T^{4} - 17405 T^{5} + 205379 T^{6}$$
$61$ $$1 + 23 T + 322 T^{2} + 2987 T^{3} + 19642 T^{4} + 85583 T^{5} + 226981 T^{6}$$
$67$ $$1 - 7 T + 201 T^{2} - 889 T^{3} + 13467 T^{4} - 31423 T^{5} + 300763 T^{6}$$
$71$ $$1 - 18 T + 272 T^{2} - 2429 T^{3} + 19312 T^{4} - 90738 T^{5} + 357911 T^{6}$$
$73$ $$1 + 2 T + 43 T^{2} + 956 T^{3} + 3139 T^{4} + 10658 T^{5} + 389017 T^{6}$$
$79$ $$1 - 8 T + 137 T^{2} - 696 T^{3} + 10823 T^{4} - 49928 T^{5} + 493039 T^{6}$$
$83$ $$1 + 21 T + 347 T^{2} + 3535 T^{3} + 28801 T^{4} + 144669 T^{5} + 571787 T^{6}$$
$89$ $$1 + 31 T + 571 T^{2} + 6471 T^{3} + 50819 T^{4} + 245551 T^{5} + 704969 T^{6}$$
$97$ $$1 + 7 T + 193 T^{2} + 721 T^{3} + 18721 T^{4} + 65863 T^{5} + 912673 T^{6}$$