# Properties

 Label 2366.2.a.t Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 0 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: $$0$$ 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.

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

## 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
 −1.73205 1.73205
1.00000 −1.73205 1.00000 0.267949 −1.73205 −1.00000 1.00000 0 0.267949
1.2 1.00000 1.73205 1.00000 3.73205 1.73205 −1.00000 1.00000 0 3.73205
 $$n$$: 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.t 2
13.b even 2 1 2366.2.a.r 2
13.c even 3 2 182.2.g.e 4
13.d odd 4 2 2366.2.d.l 4
39.i odd 6 2 1638.2.r.v 4
52.j odd 6 2 1456.2.s.l 4
91.g even 3 2 1274.2.h.o 4
91.h even 3 2 1274.2.e.o 4
91.m odd 6 2 1274.2.h.p 4
91.n odd 6 2 1274.2.g.l 4
91.v odd 6 2 1274.2.e.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.e 4 13.c even 3 2
1274.2.e.o 4 91.h even 3 2
1274.2.e.p 4 91.v odd 6 2
1274.2.g.l 4 91.n odd 6 2
1274.2.h.o 4 91.g even 3 2
1274.2.h.p 4 91.m odd 6 2
1456.2.s.l 4 52.j odd 6 2
1638.2.r.v 4 39.i odd 6 2
2366.2.a.r 2 13.b even 2 1
2366.2.a.t 2 1.a even 1 1 trivial
2366.2.d.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3}^{2} - 3$$ $$T_{5}^{2} - 4 T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ 1
$17$ $$1 + 22 T^{2} + 289 T^{4}$$
$19$ $$1 + 26 T^{2} + 361 T^{4}$$
$23$ $$1 - 10 T + 59 T^{2} - 230 T^{3} + 529 T^{4}$$
$29$ $$1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 12 T + 98 T^{2} + 444 T^{3} + 1369 T^{4}$$
$41$ $$1 - 12 T + 106 T^{2} - 492 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 4 T + 50 T^{2} + 188 T^{3} + 2209 T^{4}$$
$53$ $$1 + 58 T^{2} + 2809 T^{4}$$
$59$ $$1 - 16 T + 155 T^{2} - 944 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T + 99 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T + 90 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$1 - 2 T + 131 T^{2} - 142 T^{3} + 5041 T^{4}$$
$73$ $$1 + 8 T + 150 T^{2} + 584 T^{3} + 5329 T^{4}$$
$79$ $$1 - 16 T + 174 T^{2} - 1264 T^{3} + 6241 T^{4}$$
$83$ $$1 + 58 T^{2} + 6889 T^{4}$$
$89$ $$1 + 24 T + 310 T^{2} + 2136 T^{3} + 7921 T^{4}$$
$97$ $$1 + 20 T + 246 T^{2} + 1940 T^{3} + 9409 T^{4}$$