# Properties

 Label 1849.2.a.j Level $1849$ Weight $2$ Character orbit 1849.a Self dual yes Analytic conductor $14.764$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7643393337$$ Analytic rank: $$0$$ 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: no (minimal twist has level 43) 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.

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

## 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 0.445042 −1.24698
−1.80194 1.24698 1.24698 −2.24698 −2.24698 −1.69202 1.35690 −1.44504 4.04892
1.2 −0.445042 −1.80194 −1.80194 0.801938 0.801938 3.04892 1.69202 0.246980 −0.356896
1.3 1.24698 −0.445042 −0.445042 −0.554958 −0.554958 −1.35690 −3.04892 −2.80194 −0.692021
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$-1$$

## 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 1849.2.a.j 3
43.b odd 2 1 1849.2.a.k 3
43.f odd 14 2 43.2.e.a 6
129.j even 14 2 387.2.u.c 6
172.j even 14 2 688.2.u.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.e.a 6 43.f odd 14 2
387.2.u.c 6 129.j even 14 2
688.2.u.b 6 172.j even 14 2
1849.2.a.j 3 1.a even 1 1 trivial
1849.2.a.k 3 43.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 2 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1849))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 4 T^{2} + 3 T^{3} + 8 T^{4} + 4 T^{5} + 8 T^{6}$$
$3$ $$1 + T + 7 T^{2} + 5 T^{3} + 21 T^{4} + 9 T^{5} + 27 T^{6}$$
$5$ $$1 + 2 T + 14 T^{2} + 19 T^{3} + 70 T^{4} + 50 T^{5} + 125 T^{6}$$
$7$ $$1 + 14 T^{2} - 7 T^{3} + 98 T^{4} + 343 T^{6}$$
$11$ $$1 - 2 T + 18 T^{2} - 57 T^{3} + 198 T^{4} - 242 T^{5} + 1331 T^{6}$$
$13$ $$1 + T + 9 T^{2} + 67 T^{3} + 117 T^{4} + 169 T^{5} + 2197 T^{6}$$
$17$ $$1 + 2 T + 43 T^{2} + 60 T^{3} + 731 T^{4} + 578 T^{5} + 4913 T^{6}$$
$19$ $$1 - 13 T + 111 T^{2} - 565 T^{3} + 2109 T^{4} - 4693 T^{5} + 6859 T^{6}$$
$23$ $$1 + 3 T + 44 T^{2} + 55 T^{3} + 1012 T^{4} + 1587 T^{5} + 12167 T^{6}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{3}$$
$31$ $$1 - 11 T + 47 T^{2} - 135 T^{3} + 1457 T^{4} - 10571 T^{5} + 29791 T^{6}$$
$37$ $$1 - 7 T + 97 T^{2} - 427 T^{3} + 3589 T^{4} - 9583 T^{5} + 50653 T^{6}$$
$41$ $$1 - 22 T + 275 T^{2} - 2132 T^{3} + 11275 T^{4} - 36982 T^{5} + 68921 T^{6}$$
$43$ 1
$47$ $$1 + 23 T + 301 T^{2} + 2469 T^{3} + 14147 T^{4} + 50807 T^{5} + 103823 T^{6}$$
$53$ $$1 - 6 T + 108 T^{2} - 707 T^{3} + 5724 T^{4} - 16854 T^{5} + 148877 T^{6}$$
$59$ $$1 + 8 T + 182 T^{2} + 943 T^{3} + 10738 T^{4} + 27848 T^{5} + 205379 T^{6}$$
$61$ $$1 + 6 T + 167 T^{2} + 740 T^{3} + 10187 T^{4} + 22326 T^{5} + 226981 T^{6}$$
$67$ $$1 - 12 T + 221 T^{2} - 1504 T^{3} + 14807 T^{4} - 53868 T^{5} + 300763 T^{6}$$
$71$ $$1 - 22 T + 309 T^{2} - 3228 T^{3} + 21939 T^{4} - 110902 T^{5} + 357911 T^{6}$$
$73$ $$1 - 15 T + 245 T^{2} - 2119 T^{3} + 17885 T^{4} - 79935 T^{5} + 389017 T^{6}$$
$79$ $$1 + 10 T + 254 T^{2} + 1581 T^{3} + 20066 T^{4} + 62410 T^{5} + 493039 T^{6}$$
$83$ $$1 + 4 T + 154 T^{2} + 747 T^{3} + 12782 T^{4} + 27556 T^{5} + 571787 T^{6}$$
$89$ $$1 + 2 T + 112 T^{2} + 579 T^{3} + 9968 T^{4} + 15842 T^{5} + 704969 T^{6}$$
$97$ $$1 + 21 T + 347 T^{2} + 3577 T^{3} + 33659 T^{4} + 197589 T^{5} + 912673 T^{6}$$