# Properties

 Label 1849.2.a.l 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.

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

## 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
−0.801938 3.24698 −1.35690 2.00000 −2.60388 1.19806 2.69202 7.54288 −1.60388
1.2 0.554958 0.198062 −1.69202 2.00000 0.109916 2.55496 −2.04892 −2.96077 1.10992
1.3 2.24698 1.55496 3.04892 2.00000 3.49396 4.24698 2.35690 −0.582105 4.49396
 $$n$$: 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.l 3
43.b odd 2 1 1849.2.a.i 3
43.f odd 14 2 43.2.e.b 6
129.j even 14 2 387.2.u.a 6
172.j even 14 2 688.2.u.c 6

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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1849))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 8 T^{5} + 8 T^{6}$$
$3$ $$1 - 5 T + 15 T^{2} - 31 T^{3} + 45 T^{4} - 45 T^{5} + 27 T^{6}$$
$5$ $$( 1 - 2 T + 5 T^{2} )^{3}$$
$7$ $$1 - 8 T + 40 T^{2} - 125 T^{3} + 280 T^{4} - 392 T^{5} + 343 T^{6}$$
$11$ $$1 + 12 T^{2} - 7 T^{3} + 132 T^{4} + 1331 T^{6}$$
$13$ $$1 - 6 T + 44 T^{2} - 157 T^{3} + 572 T^{4} - 1014 T^{5} + 2197 T^{6}$$
$17$ $$1 - 4 T + 47 T^{2} - 128 T^{3} + 799 T^{4} - 1156 T^{5} + 4913 T^{6}$$
$19$ $$1 + 12 T + 98 T^{2} + 485 T^{3} + 1862 T^{4} + 4332 T^{5} + 6859 T^{6}$$
$23$ $$1 - 11 T + 107 T^{2} - 547 T^{3} + 2461 T^{4} - 5819 T^{5} + 12167 T^{6}$$
$29$ $$1 + T + 71 T^{2} + 71 T^{3} + 2059 T^{4} + 841 T^{5} + 24389 T^{6}$$
$31$ $$1 - 5 T + 71 T^{2} - 213 T^{3} + 2201 T^{4} - 4805 T^{5} + 29791 T^{6}$$
$37$ $$1 - 2 T + 26 T^{2} - 399 T^{3} + 962 T^{4} - 2738 T^{5} + 50653 T^{6}$$
$41$ $$1 + 6 T + 86 T^{2} + 311 T^{3} + 3526 T^{4} + 10086 T^{5} + 68921 T^{6}$$
$43$ 1
$47$ $$( 1 + T + 47 T^{2} )^{3}$$
$53$ $$1 - 9 T + 165 T^{2} - 925 T^{3} + 8745 T^{4} - 25281 T^{5} + 148877 T^{6}$$
$59$ $$1 + 24 T + 348 T^{2} + 3169 T^{3} + 20532 T^{4} + 83544 T^{5} + 205379 T^{6}$$
$61$ $$1 - T + 139 T^{2} - 205 T^{3} + 8479 T^{4} - 3721 T^{5} + 226981 T^{6}$$
$67$ $$1 + 2 T + 172 T^{2} + 281 T^{3} + 11524 T^{4} + 8978 T^{5} + 300763 T^{6}$$
$71$ $$1 + 129 T^{2} + 56 T^{3} + 9159 T^{4} + 357911 T^{6}$$
$73$ $$1 - 14 T + 212 T^{2} - 1743 T^{3} + 15476 T^{4} - 74606 T^{5} + 389017 T^{6}$$
$79$ $$1 - 3 T + 212 T^{2} - 391 T^{3} + 16748 T^{4} - 18723 T^{5} + 493039 T^{6}$$
$83$ $$1 - 19 T + 248 T^{2} - 2231 T^{3} + 20584 T^{4} - 130891 T^{5} + 571787 T^{6}$$
$89$ $$1 - 26 T + 350 T^{2} - 3439 T^{3} + 31150 T^{4} - 205946 T^{5} + 704969 T^{6}$$
$97$ $$1 - 50 T + 1122 T^{2} - 14291 T^{3} + 108834 T^{4} - 470450 T^{5} + 912673 T^{6}$$