# Properties

 Label 2366.2.a.bd 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: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ 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 basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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 −0.801938 1.00000 −3.04892 −0.801938 −1.00000 1.00000 −2.35690 −3.04892
1.2 1.00000 0.554958 1.00000 1.35690 0.554958 −1.00000 1.00000 −2.69202 1.35690
1.3 1.00000 2.24698 1.00000 1.69202 2.24698 −1.00000 1.00000 2.04892 1.69202
 $$n$$: 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.bd yes 3
13.b even 2 1 2366.2.a.y 3
13.d odd 4 2 2366.2.d.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.y 3 13.b even 2 1
2366.2.a.bd yes 3 1.a even 1 1 trivial
2366.2.d.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{3}$$
$3$ $$1 - 2 T + 8 T^{2} - 11 T^{3} + 24 T^{4} - 18 T^{5} + 27 T^{6}$$
$5$ $$1 + 8 T^{2} + 7 T^{3} + 40 T^{4} + 125 T^{6}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$1 - T + 31 T^{2} - 21 T^{3} + 341 T^{4} - 121 T^{5} + 1331 T^{6}$$
$13$ 1
$17$ $$1 - 2 T + 36 T^{2} - 81 T^{3} + 612 T^{4} - 578 T^{5} + 4913 T^{6}$$
$19$ $$1 - 15 T + 125 T^{2} - 653 T^{3} + 2375 T^{4} - 5415 T^{5} + 6859 T^{6}$$
$23$ $$1 - T + 53 T^{2} - 59 T^{3} + 1219 T^{4} - 529 T^{5} + 12167 T^{6}$$
$29$ $$1 - 5 T + 65 T^{2} - 193 T^{3} + 1885 T^{4} - 4205 T^{5} + 24389 T^{6}$$
$31$ $$1 - 4 T + 54 T^{2} - 289 T^{3} + 1674 T^{4} - 3844 T^{5} + 29791 T^{6}$$
$37$ $$1 - 28 T + 370 T^{2} - 2863 T^{3} + 13690 T^{4} - 38332 T^{5} + 50653 T^{6}$$
$41$ $$1 - 13 T + 149 T^{2} - 1067 T^{3} + 6109 T^{4} - 21853 T^{5} + 68921 T^{6}$$
$43$ $$1 + T + 85 T^{2} + 169 T^{3} + 3655 T^{4} + 1849 T^{5} + 79507 T^{6}$$
$47$ $$1 - 7 T + 43 T^{2} - 21 T^{3} + 2021 T^{4} - 15463 T^{5} + 103823 T^{6}$$
$53$ $$1 + 12 T + 186 T^{2} + 1259 T^{3} + 9858 T^{4} + 33708 T^{5} + 148877 T^{6}$$
$59$ $$1 + 3 T + 117 T^{2} + 481 T^{3} + 6903 T^{4} + 10443 T^{5} + 205379 T^{6}$$
$61$ $$1 + 3 T + 95 T^{2} + 479 T^{3} + 5795 T^{4} + 11163 T^{5} + 226981 T^{6}$$
$67$ $$1 + 5 T + 151 T^{2} + 545 T^{3} + 10117 T^{4} + 22445 T^{5} + 300763 T^{6}$$
$71$ $$1 + 164 T^{2} - 91 T^{3} + 11644 T^{4} + 357911 T^{6}$$
$73$ $$1 - 21 T + 149 T^{2} - 707 T^{3} + 10877 T^{4} - 111909 T^{5} + 389017 T^{6}$$
$79$ $$1 + 2 T + 12 T^{2} - 931 T^{3} + 948 T^{4} + 12482 T^{5} + 493039 T^{6}$$
$83$ $$1 + 5 T + 241 T^{2} + 789 T^{3} + 20003 T^{4} + 34445 T^{5} + 571787 T^{6}$$
$89$ $$1 - 14 T + 176 T^{2} - 2191 T^{3} + 15664 T^{4} - 110894 T^{5} + 704969 T^{6}$$
$97$ $$1 + 15 T + 359 T^{2} + 3007 T^{3} + 34823 T^{4} + 141135 T^{5} + 912673 T^{6}$$