Properties

 Label 1859.2.a.g 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: 3.3.361.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 7$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} - \beta_{2} ) q^{2} + \beta_{1} q^{3} + ( 2 + \beta_{1} ) q^{4} + \beta_{2} q^{5} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{6} + ( -2 + \beta_{2} ) q^{7} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} - \beta_{2} ) q^{2} + \beta_{1} q^{3} + ( 2 + \beta_{1} ) q^{4} + \beta_{2} q^{5} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{6} + ( -2 + \beta_{2} ) q^{7} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{8} + ( 1 + \beta_{2} ) q^{9} + ( -2 - \beta_{1} + \beta_{2} ) q^{10} - q^{11} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{12} + ( -4 + \beta_{1} + 3 \beta_{2} ) q^{14} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{15} + ( 2 \beta_{1} + \beta_{2} ) q^{16} + ( -2 - \beta_{1} ) q^{17} + ( -1 - 2 \beta_{1} ) q^{18} + ( 1 - 3 \beta_{1} - 2 \beta_{2} ) q^{19} + ( -3 + 2 \beta_{1} + 3 \beta_{2} ) q^{20} + ( -3 + \beta_{2} ) q^{21} + ( -1 + \beta_{1} + \beta_{2} ) q^{22} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( 2 - 5 \beta_{1} - 3 \beta_{2} ) q^{24} + ( -\beta_{1} - \beta_{2} ) q^{25} + ( -3 + \beta_{2} ) q^{27} + ( -7 + 3 \beta_{2} ) q^{28} + ( -3 + 4 \beta_{1} + \beta_{2} ) q^{29} -7 q^{30} + ( -1 + 2 \beta_{2} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} ) q^{32} -\beta_{1} q^{33} + ( -1 + 3 \beta_{1} + 4 \beta_{2} ) q^{34} + ( 5 - \beta_{1} - 3 \beta_{2} ) q^{35} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{36} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{37} + ( 8 + 4 \beta_{1} + 3 \beta_{2} ) q^{38} -7 q^{40} + ( 2 - 3 \beta_{1} - 4 \beta_{2} ) q^{41} + ( -5 + 2 \beta_{1} + 4 \beta_{2} ) q^{42} + ( 1 - 2 \beta_{1} - 5 \beta_{2} ) q^{43} + ( -2 - \beta_{1} ) q^{44} + ( 5 - \beta_{1} ) q^{45} + ( 7 + \beta_{1} - 2 \beta_{2} ) q^{46} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{47} + ( 5 + 2 \beta_{1} + 3 \beta_{2} ) q^{48} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{49} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{51} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{53} + ( -5 + 2 \beta_{1} + 4 \beta_{2} ) q^{54} -\beta_{2} q^{55} + ( -5 + 2 \beta_{1} + 4 \beta_{2} ) q^{56} + ( -6 - 3 \beta_{1} - 5 \beta_{2} ) q^{57} + ( -9 - 2 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -11 + 2 \beta_{1} + \beta_{2} ) q^{59} + ( -1 + 3 \beta_{1} + 5 \beta_{2} ) q^{60} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -5 - \beta_{1} + 3 \beta_{2} ) q^{62} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{63} + ( -3 - 2 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{66} + ( -5 - \beta_{1} + \beta_{2} ) q^{67} + ( -8 - 4 \beta_{1} - \beta_{2} ) q^{68} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{69} + ( 12 - \beta_{1} - 6 \beta_{2} ) q^{70} + ( 3 + \beta_{1} + \beta_{2} ) q^{71} + ( -8 - \beta_{1} - 2 \beta_{2} ) q^{72} + 2 q^{73} + ( -9 + 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( -4 - 9 \beta_{1} - 9 \beta_{2} ) q^{76} + ( 2 - \beta_{2} ) q^{77} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{80} + ( -6 - \beta_{1} - 2 \beta_{2} ) q^{81} + ( 13 + 5 \beta_{1} ) q^{82} + ( 3 + 4 \beta_{1} + \beta_{2} ) q^{83} + ( -9 - \beta_{1} + 3 \beta_{2} ) q^{84} + ( 3 - 2 \beta_{1} - 3 \beta_{2} ) q^{85} + ( 13 + 6 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 13 - \beta_{1} + 5 \beta_{2} ) q^{87} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{88} + ( 6 - \beta_{2} ) q^{89} + ( 6 - 4 \beta_{1} - 3 \beta_{2} ) q^{90} + ( 6 - 4 \beta_{1} - 7 \beta_{2} ) q^{92} + ( -6 + 3 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -3 - 4 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -1 - 4 \beta_{1} ) q^{95} -7 q^{96} + ( -5 + 4 \beta_{1} ) q^{97} + ( 13 + 4 \beta_{1} - 5 \beta_{2} ) q^{98} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + q^{3} + 7 q^{4} + q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 4 q^{9} + O(q^{10})$$ $$3 q + q^{2} + q^{3} + 7 q^{4} + q^{5} - 6 q^{6} - 5 q^{7} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 3 q^{11} + 15 q^{12} - 8 q^{14} - 6 q^{15} + 3 q^{16} - 7 q^{17} - 5 q^{18} - 2 q^{19} - 4 q^{20} - 8 q^{21} - q^{22} + 3 q^{23} - 2 q^{24} - 2 q^{25} - 8 q^{27} - 18 q^{28} - 4 q^{29} - 21 q^{30} - q^{31} - 6 q^{32} - q^{33} + 4 q^{34} + 11 q^{35} + 3 q^{36} - 12 q^{37} + 31 q^{38} - 21 q^{40} - q^{41} - 9 q^{42} - 4 q^{43} - 7 q^{44} + 14 q^{45} + 20 q^{46} + 8 q^{47} + 20 q^{48} + 12 q^{50} - 15 q^{51} - 9 q^{53} - 9 q^{54} - q^{55} - 9 q^{56} - 26 q^{57} - 33 q^{58} - 30 q^{59} + 5 q^{60} - 18 q^{61} - 13 q^{62} + 6 q^{63} - 8 q^{64} + 6 q^{66} - 15 q^{67} - 29 q^{68} + q^{69} + 29 q^{70} + 11 q^{71} - 27 q^{72} + 6 q^{73} - 23 q^{74} - 7 q^{75} - 30 q^{76} + 5 q^{77} + 12 q^{79} + q^{80} - 21 q^{81} + 44 q^{82} + 14 q^{83} - 25 q^{84} + 4 q^{85} + 43 q^{86} + 43 q^{87} + 6 q^{88} + 17 q^{89} + 11 q^{90} + 7 q^{92} - 13 q^{93} - 10 q^{94} - 7 q^{95} - 21 q^{96} - 11 q^{97} + 38 q^{98} - 4 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x + 7$$:

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

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
 2.28514 −2.50702 1.22188
−2.50702 2.28514 4.28514 1.22188 −5.72889 −0.778124 −5.72889 2.22188 −3.06327
1.2 1.22188 −2.50702 −0.507019 2.28514 −3.06327 0.285142 −3.06327 3.28514 2.79216
1.3 2.28514 1.22188 3.22188 −2.50702 2.79216 −4.50702 2.79216 −1.50702 −5.72889
 $$n$$: 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.g 3
13.b even 2 1 1859.2.a.f 3
13.e even 6 2 143.2.e.b 6

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

Hecke kernels

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

 $$T_{2}^{3} - T_{2}^{2} - 6 T_{2} + 7$$ $$T_{7}^{3} + 5 T_{7}^{2} + 2 T_{7} - 1$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$7 - 6 T - T^{2} + T^{3}$$
$3$ $$7 - 6 T - T^{2} + T^{3}$$
$5$ $$7 - 6 T - T^{2} + T^{3}$$
$7$ $$-1 + 2 T + 5 T^{2} + T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$T^{3}$$
$17$ $$-7 + 10 T + 7 T^{2} + T^{3}$$
$19$ $$77 - 43 T + 2 T^{2} + T^{3}$$
$23$ $$-1 - 16 T - 3 T^{2} + T^{3}$$
$29$ $$-49 - 77 T + 4 T^{2} + T^{3}$$
$31$ $$31 - 25 T + T^{2} + T^{3}$$
$37$ $$-31 + 29 T + 12 T^{2} + T^{3}$$
$41$ $$31 - 82 T + T^{2} + T^{3}$$
$43$ $$-581 - 115 T + 4 T^{2} + T^{3}$$
$47$ $$259 - 61 T - 8 T^{2} + T^{3}$$
$53$ $$-49 + 8 T + 9 T^{2} + T^{3}$$
$59$ $$791 + 281 T + 30 T^{2} + T^{3}$$
$61$ $$7 + 51 T + 18 T^{2} + T^{3}$$
$67$ $$11 + 56 T + 15 T^{2} + T^{3}$$
$71$ $$-31 + 34 T - 11 T^{2} + T^{3}$$
$73$ $$( -2 + T )^{3}$$
$79$ $$88 - 28 T - 12 T^{2} + T^{3}$$
$83$ $$341 - 17 T - 14 T^{2} + T^{3}$$
$89$ $$-151 + 90 T - 17 T^{2} + T^{3}$$
$97$ $$-7 - 61 T + 11 T^{2} + T^{3}$$