# Properties

 Label 2366.2.a.s Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 1 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
1.00000 −2.73205 1.00000 −1.00000 −2.73205 −1.00000 1.00000 4.46410 −1.00000
1.2 1.00000 0.732051 1.00000 −1.00000 0.732051 −1.00000 1.00000 −2.46410 −1.00000
 $$n$$: 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.s 2
13.b even 2 1 2366.2.a.q 2
13.d odd 4 2 2366.2.d.k 4
13.f odd 12 2 182.2.m.a 4
39.k even 12 2 1638.2.bj.c 4
52.l even 12 2 1456.2.cc.b 4
91.w even 12 2 1274.2.v.b 4
91.x odd 12 2 1274.2.o.b 4
91.ba even 12 2 1274.2.o.a 4
91.bc even 12 2 1274.2.m.a 4
91.bd odd 12 2 1274.2.v.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.a 4 13.f odd 12 2
1274.2.m.a 4 91.bc even 12 2
1274.2.o.a 4 91.ba even 12 2
1274.2.o.b 4 91.x odd 12 2
1274.2.v.a 4 91.bd odd 12 2
1274.2.v.b 4 91.w even 12 2
1456.2.cc.b 4 52.l even 12 2
1638.2.bj.c 4 39.k even 12 2
2366.2.a.q 2 13.b even 2 1
2366.2.a.s 2 1.a even 1 1 trivial
2366.2.d.k 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T + 5 T^{2} )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$1 - 2 T + 20 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ 1
$17$ $$1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T + 30 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 52 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 3 T + 29 T^{2} )^{2}$$
$31$ $$1 - 14 T + 108 T^{2} - 434 T^{3} + 961 T^{4}$$
$37$ $$1 + 47 T^{2} + 1369 T^{4}$$
$41$ $$1 - 2 T + 71 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$1 + 14 T + 108 T^{2} + 602 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 62 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 10 T + 119 T^{2} - 530 T^{3} + 2809 T^{4}$$
$59$ $$1 + 18 T + 196 T^{2} + 1062 T^{3} + 3481 T^{4}$$
$61$ $$1 + 20 T + 219 T^{2} + 1220 T^{3} + 3721 T^{4}$$
$67$ $$1 - 6 T + 68 T^{2} - 402 T^{3} + 4489 T^{4}$$
$71$ $$1 - 50 T^{2} + 5041 T^{4}$$
$73$ $$1 + 2 T + 39 T^{2} + 146 T^{3} + 5329 T^{4}$$
$79$ $$1 + 18 T + 212 T^{2} + 1422 T^{3} + 6241 T^{4}$$
$83$ $$1 + 18 T + 220 T^{2} + 1494 T^{3} + 6889 T^{4}$$
$89$ $$1 + 12 T + 202 T^{2} + 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 - 4 T + 186 T^{2} - 388 T^{3} + 9409 T^{4}$$