# Properties

 Label 2366.2.a.o 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

# Learn more about

## 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

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

## 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
 0
1.00000 3.00000 1.00000 0 3.00000 −1.00000 1.00000 6.00000 0
 $$n$$: 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.o 1
13.b even 2 1 182.2.a.b 1
13.d odd 4 2 2366.2.d.i 2
39.d odd 2 1 1638.2.a.q 1
52.b odd 2 1 1456.2.a.b 1
65.d even 2 1 4550.2.a.o 1
91.b odd 2 1 1274.2.a.a 1
91.r even 6 2 1274.2.f.m 2
91.s odd 6 2 1274.2.f.u 2
104.e even 2 1 5824.2.a.a 1
104.h odd 2 1 5824.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.a.b 1 13.b even 2 1
1274.2.a.a 1 91.b odd 2 1
1274.2.f.m 2 91.r even 6 2
1274.2.f.u 2 91.s odd 6 2
1456.2.a.b 1 52.b odd 2 1
1638.2.a.q 1 39.d odd 2 1
2366.2.a.o 1 1.a even 1 1 trivial
2366.2.d.i 2 13.d odd 4 2
4550.2.a.o 1 65.d even 2 1
5824.2.a.a 1 104.e even 2 1
5824.2.a.be 1 104.h odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ $$1 + 5 T^{2}$$
$7$ $$1 + T$$
$11$ $$1 - 5 T + 11 T^{2}$$
$13$ 1
$17$ $$1 + 4 T + 17 T^{2}$$
$19$ $$1 + 2 T + 19 T^{2}$$
$23$ $$1 - 5 T + 23 T^{2}$$
$29$ $$1 - 4 T + 29 T^{2}$$
$31$ $$1 + T + 31 T^{2}$$
$37$ $$1 + 7 T + 37 T^{2}$$
$41$ $$1 - 9 T + 41 T^{2}$$
$43$ $$1 + 12 T + 43 T^{2}$$
$47$ $$1 - 7 T + 47 T^{2}$$
$53$ $$1 + 4 T + 53 T^{2}$$
$59$ $$1 - 6 T + 59 T^{2}$$
$61$ $$1 - 13 T + 61 T^{2}$$
$67$ $$1 + 11 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 + 7 T + 73 T^{2}$$
$79$ $$1 + 17 T + 79 T^{2}$$
$83$ $$1 + 4 T + 83 T^{2}$$
$89$ $$1 + 14 T + 89 T^{2}$$
$97$ $$1 + 5 T + 97 T^{2}$$
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