# Properties

 Label 2366.2.a.f Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 0 Dimension 1 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
−1.00000 1.00000 1.00000 3.00000 −1.00000 1.00000 −1.00000 −2.00000 −3.00000
 $$n$$: 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.f 1
13.b even 2 1 2366.2.a.n 1
13.c even 3 2 182.2.g.b 2
13.d odd 4 2 2366.2.d.f 2
39.i odd 6 2 1638.2.r.c 2
52.j odd 6 2 1456.2.s.e 2
91.g even 3 2 1274.2.h.j 2
91.h even 3 2 1274.2.e.d 2
91.m odd 6 2 1274.2.h.i 2
91.n odd 6 2 1274.2.g.g 2
91.v odd 6 2 1274.2.e.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.b 2 13.c even 3 2
1274.2.e.d 2 91.h even 3 2
1274.2.e.i 2 91.v odd 6 2
1274.2.g.g 2 91.n odd 6 2
1274.2.h.i 2 91.m odd 6 2
1274.2.h.j 2 91.g even 3 2
1456.2.s.e 2 52.j odd 6 2
1638.2.r.c 2 39.i odd 6 2
2366.2.a.f 1 1.a even 1 1 trivial
2366.2.a.n 1 13.b even 2 1
2366.2.d.f 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3} - 1$$ $$T_{5} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 - T + 3 T^{2}$$
$5$ $$1 - 3 T + 5 T^{2}$$
$7$ $$1 - T$$
$11$ $$1 + 11 T^{2}$$
$13$ 1
$17$ $$1 - 6 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 - 3 T + 23 T^{2}$$
$29$ $$1 - 6 T + 29 T^{2}$$
$31$ $$1 + 10 T + 31 T^{2}$$
$37$ $$1 - 8 T + 37 T^{2}$$
$41$ $$1 + 41 T^{2}$$
$43$ $$1 - 8 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 - 12 T + 53 T^{2}$$
$59$ $$1 - 3 T + 59 T^{2}$$
$61$ $$1 - 11 T + 61 T^{2}$$
$67$ $$1 - 2 T + 67 T^{2}$$
$71$ $$1 + 3 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 + 4 T + 79 T^{2}$$
$83$ $$1 + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 - 2 T + 97 T^{2}$$