# Properties

 Label 2738.2.a.l Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2738 = 2 \cdot 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2738.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.8630400734$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + q^{8} + ( 1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} + ( 2 - 2 \beta ) q^{7} + q^{8} + ( 1 + 3 \beta ) q^{9} + \beta q^{10} -\beta q^{11} + ( 1 + \beta ) q^{12} + ( 1 - \beta ) q^{13} + ( 2 - 2 \beta ) q^{14} + ( 3 + 2 \beta ) q^{15} + q^{16} + 6 q^{17} + ( 1 + 3 \beta ) q^{18} -2 q^{19} + \beta q^{20} + ( -4 - 2 \beta ) q^{21} -\beta q^{22} + ( 3 - 3 \beta ) q^{23} + ( 1 + \beta ) q^{24} + ( -2 + \beta ) q^{25} + ( 1 - \beta ) q^{26} + ( 7 + 4 \beta ) q^{27} + ( 2 - 2 \beta ) q^{28} + ( -3 + 3 \beta ) q^{29} + ( 3 + 2 \beta ) q^{30} + ( -2 + \beta ) q^{31} + q^{32} + ( -3 - 2 \beta ) q^{33} + 6 q^{34} -6 q^{35} + ( 1 + 3 \beta ) q^{36} -2 q^{38} + ( -2 - \beta ) q^{39} + \beta q^{40} + ( 3 + 3 \beta ) q^{41} + ( -4 - 2 \beta ) q^{42} + ( 4 - 2 \beta ) q^{43} -\beta q^{44} + ( 9 + 4 \beta ) q^{45} + ( 3 - 3 \beta ) q^{46} + 2 \beta q^{47} + ( 1 + \beta ) q^{48} + ( 9 - 4 \beta ) q^{49} + ( -2 + \beta ) q^{50} + ( 6 + 6 \beta ) q^{51} + ( 1 - \beta ) q^{52} -6 q^{53} + ( 7 + 4 \beta ) q^{54} + ( -3 - \beta ) q^{55} + ( 2 - 2 \beta ) q^{56} + ( -2 - 2 \beta ) q^{57} + ( -3 + 3 \beta ) q^{58} + ( -6 - 2 \beta ) q^{59} + ( 3 + 2 \beta ) q^{60} + ( 4 - 5 \beta ) q^{61} + ( -2 + \beta ) q^{62} + ( -16 - 2 \beta ) q^{63} + q^{64} -3 q^{65} + ( -3 - 2 \beta ) q^{66} + ( 8 - 5 \beta ) q^{67} + 6 q^{68} + ( -6 - 3 \beta ) q^{69} -6 q^{70} + 6 q^{71} + ( 1 + 3 \beta ) q^{72} + ( -10 - \beta ) q^{73} + q^{75} -2 q^{76} + 6 q^{77} + ( -2 - \beta ) q^{78} + ( 7 - 7 \beta ) q^{79} + \beta q^{80} + ( 16 + 6 \beta ) q^{81} + ( 3 + 3 \beta ) q^{82} + ( 12 - 4 \beta ) q^{83} + ( -4 - 2 \beta ) q^{84} + 6 \beta q^{85} + ( 4 - 2 \beta ) q^{86} + ( 6 + 3 \beta ) q^{87} -\beta q^{88} + 4 \beta q^{89} + ( 9 + 4 \beta ) q^{90} + ( 8 - 2 \beta ) q^{91} + ( 3 - 3 \beta ) q^{92} + q^{93} + 2 \beta q^{94} -2 \beta q^{95} + ( 1 + \beta ) q^{96} + ( -2 + 8 \beta ) q^{97} + ( 9 - 4 \beta ) q^{98} + ( -9 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} + q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} + 12 q^{17} + 5 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} - q^{22} + 3 q^{23} + 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} + 2 q^{28} - 3 q^{29} + 8 q^{30} - 3 q^{31} + 2 q^{32} - 8 q^{33} + 12 q^{34} - 12 q^{35} + 5 q^{36} - 4 q^{38} - 5 q^{39} + q^{40} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} - 3 q^{50} + 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 2 q^{56} - 6 q^{57} - 3 q^{58} - 14 q^{59} + 8 q^{60} + 3 q^{61} - 3 q^{62} - 34 q^{63} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} - 12 q^{70} + 12 q^{71} + 5 q^{72} - 21 q^{73} + 2 q^{75} - 4 q^{76} + 12 q^{77} - 5 q^{78} + 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} + 20 q^{83} - 10 q^{84} + 6 q^{85} + 6 q^{86} + 15 q^{87} - q^{88} + 4 q^{89} + 22 q^{90} + 14 q^{91} + 3 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99} + O(q^{100})$$

## 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
 −1.30278 2.30278
1.00000 −0.302776 1.00000 −1.30278 −0.302776 4.60555 1.00000 −2.90833 −1.30278
1.2 1.00000 3.30278 1.00000 2.30278 3.30278 −2.60555 1.00000 7.90833 2.30278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$37$$ $$1$$

## 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 2738.2.a.l 2
37.b even 2 1 74.2.a.a 2
111.d odd 2 1 666.2.a.j 2
148.b odd 2 1 592.2.a.f 2
185.d even 2 1 1850.2.a.u 2
185.h odd 4 2 1850.2.b.i 4
259.b odd 2 1 3626.2.a.a 2
296.e even 2 1 2368.2.a.s 2
296.h odd 2 1 2368.2.a.ba 2
407.d odd 2 1 8954.2.a.p 2
444.g even 2 1 5328.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 37.b even 2 1
592.2.a.f 2 148.b odd 2 1
666.2.a.j 2 111.d odd 2 1
1850.2.a.u 2 185.d even 2 1
1850.2.b.i 4 185.h odd 4 2
2368.2.a.s 2 296.e even 2 1
2368.2.a.ba 2 296.h odd 2 1
2738.2.a.l 2 1.a even 1 1 trivial
3626.2.a.a 2 259.b odd 2 1
5328.2.a.bf 2 444.g even 2 1
8954.2.a.p 2 407.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{2} - 3 T_{3} - 1$$ $$T_{5}^{2} - T_{5} - 3$$ $$T_{7}^{2} - 2 T_{7} - 12$$ $$T_{13}^{2} - T_{13} - 3$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-1 - 3 T + T^{2}$$
$5$ $$-3 - T + T^{2}$$
$7$ $$-12 - 2 T + T^{2}$$
$11$ $$-3 + T + T^{2}$$
$13$ $$-3 - T + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-27 - 3 T + T^{2}$$
$29$ $$-27 + 3 T + T^{2}$$
$31$ $$-1 + 3 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$-9 - 9 T + T^{2}$$
$43$ $$-4 - 6 T + T^{2}$$
$47$ $$-12 - 2 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$36 + 14 T + T^{2}$$
$61$ $$-79 - 3 T + T^{2}$$
$67$ $$-51 - 11 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$107 + 21 T + T^{2}$$
$79$ $$-147 - 7 T + T^{2}$$
$83$ $$48 - 20 T + T^{2}$$
$89$ $$-48 - 4 T + T^{2}$$
$97$ $$-204 - 4 T + T^{2}$$