# Properties

 Label 1849.2.a.e Level $1849$ Weight $2$ Character orbit 1849.a Self dual yes Analytic conductor $14.764$ Analytic rank $1$ Dimension $2$ 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: $$1$$ 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: 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.61803 −0.381966 4.85410 −3.23607 1.00000 0.236068 −7.47214 −2.85410 8.47214
1.2 −0.381966 −2.61803 −1.85410 1.23607 1.00000 −4.23607 1.47214 3.85410 −0.472136
 $$n$$: 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.e 2
43.b odd 2 1 1849.2.a.h 2
43.c even 3 2 43.2.c.b 4
129.f odd 6 2 387.2.h.d 4
172.g odd 6 2 688.2.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.b 4 43.c even 3 2
387.2.h.d 4 129.f odd 6 2
688.2.i.e 4 172.g odd 6 2
1849.2.a.e 2 1.a even 1 1 trivial
1849.2.a.h 2 43.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$3$ $$1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}$$
$5$ $$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$11$ $$1 + 5 T + 27 T^{2} + 55 T^{3} + 121 T^{4}$$
$13$ $$1 - 5 T + 31 T^{2} - 65 T^{3} + 169 T^{4}$$
$17$ $$1 - T + 3 T^{2} - 17 T^{3} + 289 T^{4}$$
$19$ $$1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4}$$
$23$ $$1 - 11 T + 75 T^{2} - 253 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 3 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$1 + 3 T + 65 T^{2} + 111 T^{3} + 1369 T^{4}$$
$41$ $$1 - 10 T + 87 T^{2} - 410 T^{3} + 1681 T^{4}$$
$43$ 1
$47$ $$1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4}$$
$53$ $$1 + 5 T + 111 T^{2} + 265 T^{3} + 2809 T^{4}$$
$59$ $$1 + T + 87 T^{2} + 59 T^{3} + 3481 T^{4}$$
$61$ $$1 + T + 111 T^{2} + 61 T^{3} + 3721 T^{4}$$
$67$ $$1 + T + 123 T^{2} + 67 T^{3} + 4489 T^{4}$$
$71$ $$1 - 3 T - 7 T^{2} - 213 T^{3} + 5041 T^{4}$$
$73$ $$1 + 3 T + 137 T^{2} + 219 T^{3} + 5329 T^{4}$$
$79$ $$1 + 5 T + 163 T^{2} + 395 T^{3} + 6241 T^{4}$$
$83$ $$1 - 3 T - 43 T^{2} - 249 T^{3} + 6889 T^{4}$$
$89$ $$1 - 3 T + 169 T^{2} - 267 T^{3} + 7921 T^{4}$$
$97$ $$1 + 14 T + 238 T^{2} + 1358 T^{3} + 9409 T^{4}$$