Properties

 Label 1859.2.a.e Level $1859$ Weight $2$ Character orbit 1859.a Self dual yes Analytic conductor $14.844$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1859.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$14.8441897358$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.756.1 Defining polynomial: $$x^{3} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

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
 −2.26180 −0.339877 2.60168
−1.00000 −2.26180 −1.00000 −2.11575 2.26180 −3.37755 3.00000 2.11575 2.11575
1.2 −1.00000 −0.339877 −1.00000 2.88448 0.339877 3.54461 3.00000 −2.88448 −2.88448
1.3 −1.00000 2.60168 −1.00000 −3.76873 −2.60168 −0.167055 3.00000 3.76873 3.76873
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$13$$ $$1$$

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 1859.2.a.e 3
13.b even 2 1 1859.2.a.h 3
13.e even 6 2 143.2.e.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.a 6 13.e even 6 2
1859.2.a.e 3 1.a even 1 1 trivial
1859.2.a.h 3 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1859))$$:

 $$T_{2} + 1$$ $$T_{7}^{3} - 12 T_{7} - 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-2 - 6 T + T^{3}$$
$5$ $$-23 - 9 T + 3 T^{2} + T^{3}$$
$7$ $$-2 - 12 T + T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$T^{3}$$
$17$ $$-49 - 21 T + 3 T^{2} + T^{3}$$
$19$ $$-66 - 12 T + 6 T^{2} + T^{3}$$
$23$ $$16 - 24 T + T^{3}$$
$29$ $$57 - 39 T + 3 T^{2} + T^{3}$$
$31$ $$-536 - 84 T + 6 T^{2} + T^{3}$$
$37$ $$-279 - 81 T + 3 T^{2} + T^{3}$$
$41$ $$73 - 63 T + 3 T^{2} + T^{3}$$
$43$ $$-24 - 12 T + 6 T^{2} + T^{3}$$
$47$ $$122 - 54 T - 6 T^{2} + T^{3}$$
$53$ $$63 - 45 T - 3 T^{2} + T^{3}$$
$59$ $$182 + 42 T - 18 T^{2} + T^{3}$$
$61$ $$189 - 27 T - 9 T^{2} + T^{3}$$
$67$ $$-202 - 18 T + 12 T^{2} + T^{3}$$
$71$ $$( 6 + T )^{3}$$
$73$ $$-2483 - 249 T + 9 T^{2} + T^{3}$$
$79$ $$22 - 36 T + 12 T^{2} + T^{3}$$
$83$ $$-294 - 84 T + T^{3}$$
$89$ $$-756 + 18 T^{2} + T^{3}$$
$97$ $$116 + 84 T + 18 T^{2} + T^{3}$$