# Properties

 Label 2366.4.a.h Level $2366$ Weight $4$ Character orbit 2366.a Self dual yes Analytic conductor $139.599$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2366.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$139.598519074$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 8q^{3} + 4q^{4} + 14q^{5} + 16q^{6} + 7q^{7} + 8q^{8} + 37q^{9} + O(q^{10})$$ $$q + 2q^{2} + 8q^{3} + 4q^{4} + 14q^{5} + 16q^{6} + 7q^{7} + 8q^{8} + 37q^{9} + 28q^{10} + 28q^{11} + 32q^{12} + 14q^{14} + 112q^{15} + 16q^{16} + 74q^{17} + 74q^{18} - 80q^{19} + 56q^{20} + 56q^{21} + 56q^{22} - 112q^{23} + 64q^{24} + 71q^{25} + 80q^{27} + 28q^{28} + 190q^{29} + 224q^{30} - 72q^{31} + 32q^{32} + 224q^{33} + 148q^{34} + 98q^{35} + 148q^{36} + 346q^{37} - 160q^{38} + 112q^{40} - 162q^{41} + 112q^{42} - 412q^{43} + 112q^{44} + 518q^{45} - 224q^{46} - 24q^{47} + 128q^{48} + 49q^{49} + 142q^{50} + 592q^{51} + 318q^{53} + 160q^{54} + 392q^{55} + 56q^{56} - 640q^{57} + 380q^{58} + 200q^{59} + 448q^{60} - 198q^{61} - 144q^{62} + 259q^{63} + 64q^{64} + 448q^{66} + 716q^{67} + 296q^{68} - 896q^{69} + 196q^{70} - 392q^{71} + 296q^{72} - 538q^{73} + 692q^{74} + 568q^{75} - 320q^{76} + 196q^{77} + 240q^{79} + 224q^{80} - 359q^{81} - 324q^{82} + 1072q^{83} + 224q^{84} + 1036q^{85} - 824q^{86} + 1520q^{87} + 224q^{88} - 810q^{89} + 1036q^{90} - 448q^{92} - 576q^{93} - 48q^{94} - 1120q^{95} + 256q^{96} - 1354q^{97} + 98q^{98} + 1036q^{99} + O(q^{100})$$

## 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
2.00000 8.00000 4.00000 14.0000 16.0000 7.00000 8.00000 37.0000 28.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 2366.4.a.h 1
13.b even 2 1 14.4.a.a 1
39.d odd 2 1 126.4.a.h 1
52.b odd 2 1 112.4.a.a 1
65.d even 2 1 350.4.a.l 1
65.h odd 4 2 350.4.c.b 2
91.b odd 2 1 98.4.a.a 1
91.r even 6 2 98.4.c.d 2
91.s odd 6 2 98.4.c.f 2
104.e even 2 1 448.4.a.b 1
104.h odd 2 1 448.4.a.o 1
143.d odd 2 1 1694.4.a.g 1
156.h even 2 1 1008.4.a.s 1
273.g even 2 1 882.4.a.i 1
273.w odd 6 2 882.4.g.b 2
273.ba even 6 2 882.4.g.k 2
364.h even 2 1 784.4.a.s 1
455.h odd 2 1 2450.4.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 13.b even 2 1
98.4.a.a 1 91.b odd 2 1
98.4.c.d 2 91.r even 6 2
98.4.c.f 2 91.s odd 6 2
112.4.a.a 1 52.b odd 2 1
126.4.a.h 1 39.d odd 2 1
350.4.a.l 1 65.d even 2 1
350.4.c.b 2 65.h odd 4 2
448.4.a.b 1 104.e even 2 1
448.4.a.o 1 104.h odd 2 1
784.4.a.s 1 364.h even 2 1
882.4.a.i 1 273.g even 2 1
882.4.g.b 2 273.w odd 6 2
882.4.g.k 2 273.ba even 6 2
1008.4.a.s 1 156.h even 2 1
1694.4.a.g 1 143.d odd 2 1
2366.4.a.h 1 1.a even 1 1 trivial
2450.4.a.bo 1 455.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3} - 8$$ $$T_{5} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$-8 + T$$
$5$ $$-14 + T$$
$7$ $$-7 + T$$
$11$ $$-28 + T$$
$13$ $$T$$
$17$ $$-74 + T$$
$19$ $$80 + T$$
$23$ $$112 + T$$
$29$ $$-190 + T$$
$31$ $$72 + T$$
$37$ $$-346 + T$$
$41$ $$162 + T$$
$43$ $$412 + T$$
$47$ $$24 + T$$
$53$ $$-318 + T$$
$59$ $$-200 + T$$
$61$ $$198 + T$$
$67$ $$-716 + T$$
$71$ $$392 + T$$
$73$ $$538 + T$$
$79$ $$-240 + T$$
$83$ $$-1072 + T$$
$89$ $$810 + T$$
$97$ $$1354 + T$$