# Properties

 Label 2366.2.a.bb Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 1 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a root $$\beta$$ of the polynomial $$x^{3} - x^{2} - 2 x + 1$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
1.00000 −3.04892 1.00000 −4.24698 −3.04892 −1.00000 1.00000 6.29590 −4.24698
1.2 1.00000 1.35690 1.00000 −1.19806 1.35690 −1.00000 1.00000 −1.15883 −1.19806
1.3 1.00000 1.69202 1.00000 −2.55496 1.69202 −1.00000 1.00000 −0.137063 −2.55496
 $$n$$: 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.bb yes 3
13.b even 2 1 2366.2.a.w 3
13.d odd 4 2 2366.2.d.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.w 3 13.b even 2 1
2366.2.a.bb yes 3 1.a even 1 1 trivial
2366.2.d.o 6 13.d odd 4 2

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3}^{3} - 7 T_{3} + 7$$ $$T_{5}^{3} + 8 T_{5}^{2} + 19 T_{5} + 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{3}$$
$3$ $$1 + 2 T^{2} + 7 T^{3} + 6 T^{4} + 27 T^{6}$$
$5$ $$1 + 8 T + 34 T^{2} + 93 T^{3} + 170 T^{4} + 200 T^{5} + 125 T^{6}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$1 + 9 T + 53 T^{2} + 211 T^{3} + 583 T^{4} + 1089 T^{5} + 1331 T^{6}$$
$13$ 1
$17$ $$1 - 10 T + 40 T^{2} - 117 T^{3} + 680 T^{4} - 2890 T^{5} + 4913 T^{6}$$
$19$ $$1 + 7 T + 57 T^{2} + 259 T^{3} + 1083 T^{4} + 2527 T^{5} + 6859 T^{6}$$
$23$ $$1 - 3 T + 65 T^{2} - 125 T^{3} + 1495 T^{4} - 1587 T^{5} + 12167 T^{6}$$
$29$ $$1 - 5 T - 5 T^{2} + 87 T^{3} - 145 T^{4} - 4205 T^{5} + 24389 T^{6}$$
$31$ $$1 + 24 T + 278 T^{2} + 1951 T^{3} + 8618 T^{4} + 23064 T^{5} + 29791 T^{6}$$
$37$ $$1 - 4 T + 72 T^{2} - 337 T^{3} + 2664 T^{4} - 5476 T^{5} + 50653 T^{6}$$
$41$ $$1 - 3 T + 77 T^{2} - 289 T^{3} + 3157 T^{4} - 5043 T^{5} + 68921 T^{6}$$
$43$ $$1 + 9 T + 149 T^{2} + 787 T^{3} + 6407 T^{4} + 16641 T^{5} + 79507 T^{6}$$
$47$ $$1 + 17 T + 235 T^{2} + 1767 T^{3} + 11045 T^{4} + 37553 T^{5} + 103823 T^{6}$$
$53$ $$1 + 10 T + 176 T^{2} + 1019 T^{3} + 9328 T^{4} + 28090 T^{5} + 148877 T^{6}$$
$59$ $$1 - 3 T + 131 T^{2} - 355 T^{3} + 7729 T^{4} - 10443 T^{5} + 205379 T^{6}$$
$61$ $$1 + 5 T + 105 T^{2} + 779 T^{3} + 6405 T^{4} + 18605 T^{5} + 226981 T^{6}$$
$67$ $$1 - 5 T + 95 T^{2} - 867 T^{3} + 6365 T^{4} - 22445 T^{5} + 300763 T^{6}$$
$71$ $$1 + 24 T + 356 T^{2} + 3577 T^{3} + 25276 T^{4} + 120984 T^{5} + 357911 T^{6}$$
$73$ $$1 + T + 189 T^{2} + 103 T^{3} + 13797 T^{4} + 5329 T^{5} + 389017 T^{6}$$
$79$ $$1 - 10 T + 128 T^{2} - 489 T^{3} + 10112 T^{4} - 62410 T^{5} + 493039 T^{6}$$
$83$ $$1 - 7 T + 95 T^{2} - 371 T^{3} + 7885 T^{4} - 48223 T^{5} + 571787 T^{6}$$
$89$ $$1 + 2 T + 252 T^{2} + 327 T^{3} + 22428 T^{4} + 15842 T^{5} + 704969 T^{6}$$
$97$ $$1 - 27 T + 527 T^{2} - 5911 T^{3} + 51119 T^{4} - 254043 T^{5} + 912673 T^{6}$$