# Properties

 Label 2366.2.a.x Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2366.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.8926051182$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1384.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + ( -1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} - q^{8} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} + ( -1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} - q^{8} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} + ( 1 - \beta_{2} ) q^{10} + ( -3 + \beta_{1} + \beta_{2} ) q^{11} + \beta_{1} q^{12} + q^{14} + 2 \beta_{2} q^{15} + q^{16} + 2 \beta_{2} q^{17} + ( -4 + \beta_{1} - \beta_{2} ) q^{18} + ( -1 - \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{20} -\beta_{1} q^{21} + ( 3 - \beta_{1} - \beta_{2} ) q^{22} + ( 1 - \beta_{1} + \beta_{2} ) q^{23} -\beta_{1} q^{24} + ( 3 + 2 \beta_{1} - 2 \beta_{2} ) q^{25} + ( -7 + 3 \beta_{1} + \beta_{2} ) q^{27} - q^{28} + ( 2 + 2 \beta_{1} ) q^{29} -2 \beta_{2} q^{30} + ( 3 - \beta_{1} - \beta_{2} ) q^{31} - q^{32} + ( 7 - 3 \beta_{1} + 3 \beta_{2} ) q^{33} -2 \beta_{2} q^{34} + ( 1 - \beta_{2} ) q^{35} + ( 4 - \beta_{1} + \beta_{2} ) q^{36} + ( -3 + \beta_{1} - \beta_{2} ) q^{37} + ( 1 + \beta_{2} ) q^{38} + ( 1 - \beta_{2} ) q^{40} + ( 5 + \beta_{1} - \beta_{2} ) q^{41} + \beta_{1} q^{42} + ( -2 - 2 \beta_{2} ) q^{43} + ( -3 + \beta_{1} + \beta_{2} ) q^{44} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{45} + ( -1 + \beta_{1} - \beta_{2} ) q^{46} + ( -5 - \beta_{1} - \beta_{2} ) q^{47} + \beta_{1} q^{48} + q^{49} + ( -3 - 2 \beta_{1} + 2 \beta_{2} ) q^{50} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{51} -6 q^{53} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{54} + ( 10 + 2 \beta_{1} - 2 \beta_{2} ) q^{55} + q^{56} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{57} + ( -2 - 2 \beta_{1} ) q^{58} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{59} + 2 \beta_{2} q^{60} + ( 2 - \beta_{1} ) q^{61} + ( -3 + \beta_{1} + \beta_{2} ) q^{62} + ( -4 + \beta_{1} - \beta_{2} ) q^{63} + q^{64} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{66} + ( -5 + \beta_{1} - \beta_{2} ) q^{67} + 2 \beta_{2} q^{68} + ( -7 + 3 \beta_{1} + \beta_{2} ) q^{69} + ( -1 + \beta_{2} ) q^{70} + 4 \beta_{1} q^{71} + ( -4 + \beta_{1} - \beta_{2} ) q^{72} + ( 3 + \beta_{1} + \beta_{2} ) q^{73} + ( 3 - \beta_{1} + \beta_{2} ) q^{74} + ( 14 - \beta_{1} - 2 \beta_{2} ) q^{75} + ( -1 - \beta_{2} ) q^{76} + ( 3 - \beta_{1} - \beta_{2} ) q^{77} + ( -3 - \beta_{1} + 3 \beta_{2} ) q^{79} + ( -1 + \beta_{2} ) q^{80} + ( 9 - 6 \beta_{1} + 2 \beta_{2} ) q^{81} + ( -5 - \beta_{1} + \beta_{2} ) q^{82} + ( -5 + 2 \beta_{1} + 3 \beta_{2} ) q^{83} -\beta_{1} q^{84} + ( 14 + 4 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 2 + 2 \beta_{2} ) q^{86} + ( 14 + 2 \beta_{2} ) q^{87} + ( 3 - \beta_{1} - \beta_{2} ) q^{88} + ( 8 + 2 \beta_{2} ) q^{89} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} ) q^{92} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 5 + \beta_{1} + \beta_{2} ) q^{94} + ( -6 - 2 \beta_{1} ) q^{95} -\beta_{1} q^{96} + ( 7 - \beta_{1} + \beta_{2} ) q^{97} - q^{98} + ( -12 + 10 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + q^{3} + 3q^{4} - 2q^{5} - q^{6} - 3q^{7} - 3q^{8} + 12q^{9} + O(q^{10})$$ $$3q - 3q^{2} + q^{3} + 3q^{4} - 2q^{5} - q^{6} - 3q^{7} - 3q^{8} + 12q^{9} + 2q^{10} - 7q^{11} + q^{12} + 3q^{14} + 2q^{15} + 3q^{16} + 2q^{17} - 12q^{18} - 4q^{19} - 2q^{20} - q^{21} + 7q^{22} + 3q^{23} - q^{24} + 9q^{25} - 17q^{27} - 3q^{28} + 8q^{29} - 2q^{30} + 7q^{31} - 3q^{32} + 21q^{33} - 2q^{34} + 2q^{35} + 12q^{36} - 9q^{37} + 4q^{38} + 2q^{40} + 15q^{41} + q^{42} - 8q^{43} - 7q^{44} + 12q^{45} - 3q^{46} - 17q^{47} + q^{48} + 3q^{49} - 9q^{50} + 6q^{51} - 18q^{53} + 17q^{54} + 30q^{55} + 3q^{56} - 4q^{57} - 8q^{58} - 10q^{59} + 2q^{60} + 5q^{61} - 7q^{62} - 12q^{63} + 3q^{64} - 21q^{66} - 15q^{67} + 2q^{68} - 17q^{69} - 2q^{70} + 4q^{71} - 12q^{72} + 11q^{73} + 9q^{74} + 39q^{75} - 4q^{76} + 7q^{77} - 7q^{79} - 2q^{80} + 23q^{81} - 15q^{82} - 10q^{83} - q^{84} + 44q^{85} + 8q^{86} + 44q^{87} + 7q^{88} + 26q^{89} - 12q^{90} + 3q^{92} - 21q^{93} + 17q^{94} - 20q^{95} - q^{96} + 21q^{97} - 3q^{98} - 26q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 10 x + 14$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 7$$

## 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
 −3.30590 1.52023 2.78567
−1.00000 −3.30590 1.00000 −0.376939 3.30590 −1.00000 −1.00000 7.92896 0.376939
1.2 −1.00000 1.52023 1.00000 −4.16867 −1.52023 −1.00000 −1.00000 −0.688899 4.16867
1.3 −1.00000 2.78567 1.00000 2.54561 −2.78567 −1.00000 −1.00000 4.75994 −2.54561
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 2366.2.a.x 3
13.b even 2 1 2366.2.a.bc 3
13.d odd 4 2 182.2.d.b 6
39.f even 4 2 1638.2.c.i 6
52.f even 4 2 1456.2.k.b 6
91.i even 4 2 1274.2.d.l 6
91.z odd 12 4 1274.2.n.m 12
91.bb even 12 4 1274.2.n.n 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.d.b 6 13.d odd 4 2
1274.2.d.l 6 91.i even 4 2
1274.2.n.m 12 91.z odd 12 4
1274.2.n.n 12 91.bb even 12 4
1456.2.k.b 6 52.f even 4 2
1638.2.c.i 6 39.f even 4 2
2366.2.a.x 3 1.a even 1 1 trivial
2366.2.a.bc 3 13.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3}^{3} - T_{3}^{2} - 10 T_{3} + 14$$ $$T_{5}^{3} + 2 T_{5}^{2} - 10 T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$1 - T - T^{2} + 8 T^{3} - 3 T^{4} - 9 T^{5} + 27 T^{6}$$
$5$ $$1 + 2 T + 5 T^{2} + 16 T^{3} + 25 T^{4} + 50 T^{5} + 125 T^{6}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$1 + 7 T + 25 T^{2} + 66 T^{3} + 275 T^{4} + 847 T^{5} + 1331 T^{6}$$
$13$ 1
$17$ $$1 - 2 T + 7 T^{2} - 12 T^{3} + 119 T^{4} - 578 T^{5} + 4913 T^{6}$$
$19$ $$1 + 4 T + 51 T^{2} + 136 T^{3} + 969 T^{4} + 1444 T^{5} + 6859 T^{6}$$
$23$ $$1 - 3 T + 53 T^{2} - 106 T^{3} + 1219 T^{4} - 1587 T^{5} + 12167 T^{6}$$
$29$ $$1 - 8 T + 67 T^{2} - 288 T^{3} + 1943 T^{4} - 6728 T^{5} + 24389 T^{6}$$
$31$ $$1 - 7 T + 85 T^{2} - 346 T^{3} + 2635 T^{4} - 6727 T^{5} + 29791 T^{6}$$
$37$ $$1 + 9 T + 119 T^{2} + 622 T^{3} + 4403 T^{4} + 12321 T^{5} + 50653 T^{6}$$
$41$ $$1 - 15 T + 179 T^{2} - 1274 T^{3} + 7339 T^{4} - 25215 T^{5} + 68921 T^{6}$$
$43$ $$1 + 8 T + 105 T^{2} + 560 T^{3} + 4515 T^{4} + 14792 T^{5} + 79507 T^{6}$$
$47$ $$1 + 17 T + 213 T^{2} + 1686 T^{3} + 10011 T^{4} + 37553 T^{5} + 103823 T^{6}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{3}$$
$59$ $$1 + 10 T + 83 T^{2} + 296 T^{3} + 4897 T^{4} + 34810 T^{5} + 205379 T^{6}$$
$61$ $$1 - 5 T + 181 T^{2} - 608 T^{3} + 11041 T^{4} - 18605 T^{5} + 226981 T^{6}$$
$67$ $$1 + 15 T + 257 T^{2} + 2026 T^{3} + 17219 T^{4} + 67335 T^{5} + 300763 T^{6}$$
$71$ $$1 - 4 T + 53 T^{2} + 328 T^{3} + 3763 T^{4} - 20164 T^{5} + 357911 T^{6}$$
$73$ $$1 - 11 T + 235 T^{2} - 1610 T^{3} + 17155 T^{4} - 58619 T^{5} + 389017 T^{6}$$
$79$ $$1 + 7 T + 149 T^{2} + 1254 T^{3} + 11771 T^{4} + 43687 T^{5} + 493039 T^{6}$$
$83$ $$1 + 10 T + 123 T^{2} + 408 T^{3} + 10209 T^{4} + 68890 T^{5} + 571787 T^{6}$$
$89$ $$1 - 26 T + 447 T^{2} - 4860 T^{3} + 39783 T^{4} - 205946 T^{5} + 704969 T^{6}$$
$97$ $$1 - 21 T + 419 T^{2} - 4270 T^{3} + 40643 T^{4} - 197589 T^{5} + 912673 T^{6}$$