# Properties

 Label 1849.2.a.i Level $1849$ Weight $2$ Character orbit 1849.a Self dual yes Analytic conductor $14.764$ Analytic rank $1$ 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: $$1$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta^{2} ) q^{2} + ( -3 - \beta + \beta^{2} ) q^{3} + ( -2 + \beta + \beta^{2} ) q^{4} -2 q^{5} + ( -3 + 2 \beta^{2} ) q^{6} + ( -1 - \beta^{2} ) q^{7} + ( -2 - 2 \beta + \beta^{2} ) q^{8} + ( 7 + 3 \beta - 4 \beta^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta^{2} ) q^{2} + ( -3 - \beta + \beta^{2} ) q^{3} + ( -2 + \beta + \beta^{2} ) q^{4} -2 q^{5} + ( -3 + 2 \beta^{2} ) q^{6} + ( -1 - \beta^{2} ) q^{7} + ( -2 - 2 \beta + \beta^{2} ) q^{8} + ( 7 + 3 \beta - 4 \beta^{2} ) q^{9} + ( -2 + 2 \beta^{2} ) q^{10} + ( 1 - 3 \beta ) q^{11} + ( 5 - 3 \beta^{2} ) q^{12} + ( 5 + \beta - 2 \beta^{2} ) q^{13} + ( -2 + \beta + 3 \beta^{2} ) q^{14} + ( 6 + 2 \beta - 2 \beta^{2} ) q^{15} + ( 1 - \beta ) q^{16} + ( -2 + 2 \beta^{2} ) q^{17} + ( 6 + \beta - 2 \beta^{2} ) q^{18} + ( 1 - \beta + 2 \beta^{2} ) q^{19} + ( 4 - 2 \beta - 2 \beta^{2} ) q^{20} + ( 3 + 2 \beta ) q^{21} + ( -2 + 3 \beta + 2 \beta^{2} ) q^{22} + ( 4 - \beta ) q^{23} + ( 8 + 3 \beta - 3 \beta^{2} ) q^{24} - q^{25} + ( 4 + \beta - 2 \beta^{2} ) q^{26} + ( -15 - 3 \beta + 8 \beta^{2} ) q^{27} + ( 4 - 4 \beta - 3 \beta^{2} ) q^{28} + ( 5 + \beta - 3 \beta^{2} ) q^{29} + ( 6 - 4 \beta^{2} ) q^{30} + ( -4 - 3 \beta + 4 \beta^{2} ) q^{31} + ( 4 + 5 \beta - 2 \beta^{2} ) q^{32} + ( 2 \beta + \beta^{2} ) q^{33} + ( -2 \beta - 2 \beta^{2} ) q^{34} + ( 2 + 2 \beta^{2} ) q^{35} + ( -9 - 5 \beta + 5 \beta^{2} ) q^{36} + ( 2 + 7 \beta - 3 \beta^{2} ) q^{37} + ( 2 - \beta - 4 \beta^{2} ) q^{38} + ( -16 - 4 \beta + 7 \beta^{2} ) q^{39} + ( 4 + 4 \beta - 2 \beta^{2} ) q^{40} + ( 5 + 4 \beta - 5 \beta^{2} ) q^{41} + ( 5 - 2 \beta - 5 \beta^{2} ) q^{42} + ( 1 + \beta - 5 \beta^{2} ) q^{44} + ( -14 - 6 \beta + 8 \beta^{2} ) q^{45} + ( 3 + \beta - 3 \beta^{2} ) q^{46} - q^{47} + ( -2 + \beta^{2} ) q^{48} + ( -7 + \beta + 5 \beta^{2} ) q^{49} + ( -1 + \beta^{2} ) q^{50} + ( 6 - 4 \beta^{2} ) q^{51} + ( -7 - \beta + 3 \beta^{2} ) q^{52} + ( 4 - 3 \beta ) q^{53} + ( -10 - 5 \beta + 2 \beta^{2} ) q^{54} + ( -2 + 6 \beta ) q^{55} + ( 1 + 5 \beta ) q^{56} + ( -2 - 2 \beta - \beta^{2} ) q^{57} + ( 3 + 2 \beta ) q^{58} + ( -13 + 3 \beta^{2} ) q^{59} + ( -10 + 6 \beta^{2} ) q^{60} + ( 3 + 5 \beta - 3 \beta^{2} ) q^{61} + ( -3 - \beta - \beta^{2} ) q^{62} + ( -8 - 5 \beta + 6 \beta^{2} ) q^{63} + ( 5 - \beta - 5 \beta^{2} ) q^{64} + ( -10 - 2 \beta + 4 \beta^{2} ) q^{65} + ( 3 - 3 \beta - 4 \beta^{2} ) q^{66} + ( -7 - \beta + 4 \beta^{2} ) q^{67} + ( 4 \beta + 2 \beta^{2} ) q^{68} + ( -11 - 3 \beta + 4 \beta^{2} ) q^{69} + ( 4 - 2 \beta - 6 \beta^{2} ) q^{70} + ( 2 - 6 \beta ) q^{71} + ( -21 - 2 \beta + 8 \beta^{2} ) q^{72} + ( -8 + 5 \beta + \beta^{2} ) q^{73} + ( 6 - 4 \beta - 3 \beta^{2} ) q^{74} + ( 3 + \beta - \beta^{2} ) q^{75} + ( -5 + 7 \beta + 3 \beta^{2} ) q^{76} + ( -4 + 9 \beta + 2 \beta^{2} ) q^{77} + ( -13 - 3 \beta + 6 \beta^{2} ) q^{78} + ( 3 + 4 \beta - 2 \beta^{2} ) q^{79} + ( -2 + 2 \beta ) q^{80} + ( 27 + \beta - 11 \beta^{2} ) q^{81} + ( 4 + \beta + \beta^{2} ) q^{82} + ( -5 - 6 \beta + 8 \beta^{2} ) q^{83} + ( -8 + 3 \beta + 7 \beta^{2} ) q^{84} + ( 4 - 4 \beta^{2} ) q^{85} + ( -16 - 3 \beta + 8 \beta^{2} ) q^{87} + ( 1 - 2 \beta + 4 \beta^{2} ) q^{88} + ( -17 + 5 \beta + 4 \beta^{2} ) q^{89} + ( -12 - 2 \beta + 4 \beta^{2} ) q^{90} + ( -6 - \beta + 2 \beta^{2} ) q^{91} + ( -7 + 4 \beta + 2 \beta^{2} ) q^{92} + ( 15 + 3 \beta - 8 \beta^{2} ) q^{93} + ( -1 + \beta^{2} ) q^{94} + ( -2 + 2 \beta - 4 \beta^{2} ) q^{95} + ( -17 - 7 \beta + 6 \beta^{2} ) q^{96} + ( 15 + \beta^{2} ) q^{97} + ( -1 - 6 \beta - 4 \beta^{2} ) q^{98} + ( -5 + 6 \beta - \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 −1.24698 0.445042
−2.24698 −1.55496 3.04892 −2.00000 3.49396 −4.24698 −2.35690 −0.582105 4.49396
1.2 −0.554958 −0.198062 −1.69202 −2.00000 0.109916 −2.55496 2.04892 −2.96077 1.10992
1.3 0.801938 −3.24698 −1.35690 −2.00000 −2.60388 −1.19806 −2.69202 7.54288 −1.60388
 $$n$$: 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.i 3
43.b odd 2 1 1849.2.a.l 3
43.e even 7 2 43.2.e.b 6
129.l odd 14 2 387.2.u.a 6
172.k odd 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.e even 7 2
387.2.u.a 6 129.l odd 14 2
688.2.u.c 6 172.k odd 14 2
1849.2.a.i 3 1.a even 1 1 trivial
1849.2.a.l 3 43.b odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1849))$$.

## 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
$47$ $$( 1 + T + 47 T^{2} )^{3}$$
$53$ $$1 - 9 T + 165 T^{2} - 925 T^{3} + 8745 T^{4} - 25281 T^{5} + 148877 T^{6}$$
$59$ $$1 + 24 T + 348 T^{2} + 3169 T^{3} + 20532 T^{4} + 83544 T^{5} + 205379 T^{6}$$
$61$ $$1 + T + 139 T^{2} + 205 T^{3} + 8479 T^{4} + 3721 T^{5} + 226981 T^{6}$$
$67$ $$1 + 2 T + 172 T^{2} + 281 T^{3} + 11524 T^{4} + 8978 T^{5} + 300763 T^{6}$$
$71$ $$1 + 129 T^{2} - 56 T^{3} + 9159 T^{4} + 357911 T^{6}$$
$73$ $$1 + 14 T + 212 T^{2} + 1743 T^{3} + 15476 T^{4} + 74606 T^{5} + 389017 T^{6}$$
$79$ $$1 - 3 T + 212 T^{2} - 391 T^{3} + 16748 T^{4} - 18723 T^{5} + 493039 T^{6}$$
$83$ $$1 - 19 T + 248 T^{2} - 2231 T^{3} + 20584 T^{4} - 130891 T^{5} + 571787 T^{6}$$
$89$ $$1 + 26 T + 350 T^{2} + 3439 T^{3} + 31150 T^{4} + 205946 T^{5} + 704969 T^{6}$$
$97$ $$1 - 50 T + 1122 T^{2} - 14291 T^{3} + 108834 T^{4} - 470450 T^{5} + 912673 T^{6}$$