# Properties

 Label 2738.2.a.k Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.79129 2.79129
1.00000 −1.79129 1.00000 0.791288 −1.79129 −2.00000 1.00000 0.208712 0.791288
1.2 1.00000 2.79129 1.00000 −3.79129 2.79129 −2.00000 1.00000 4.79129 −3.79129
 $$n$$: 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.k 2
37.b even 2 1 2738.2.a.h 2
37.d odd 4 2 74.2.b.a 4
111.g even 4 2 666.2.c.b 4
148.g even 4 2 592.2.g.c 4
185.f even 4 2 1850.2.c.g 4
185.j odd 4 2 1850.2.d.e 4
185.k even 4 2 1850.2.c.h 4
296.j even 4 2 2368.2.g.h 4
296.m odd 4 2 2368.2.g.j 4
444.j odd 4 2 5328.2.h.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 37.d odd 4 2
592.2.g.c 4 148.g even 4 2
666.2.c.b 4 111.g even 4 2
1850.2.c.g 4 185.f even 4 2
1850.2.c.h 4 185.k even 4 2
1850.2.d.e 4 185.j odd 4 2
2368.2.g.h 4 296.j even 4 2
2368.2.g.j 4 296.m odd 4 2
2738.2.a.h 2 37.b even 2 1
2738.2.a.k 2 1.a even 1 1 trivial
5328.2.h.m 4 444.j odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} - 5$$ T3^2 - T3 - 5 $$T_{5}^{2} + 3T_{5} - 3$$ T5^2 + 3*T5 - 3 $$T_{7} + 2$$ T7 + 2 $$T_{13}^{2} + 3T_{13} - 3$$ T13^2 + 3*T13 - 3 $$T_{17}^{2} - 6T_{17} - 12$$ T17^2 - 6*T17 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - T - 5$$
$5$ $$T^{2} + 3T - 3$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 3T - 3$$
$13$ $$T^{2} + 3T - 3$$
$17$ $$T^{2} - 6T - 12$$
$19$ $$T^{2} + 6T - 12$$
$23$ $$T^{2} + 3T - 3$$
$29$ $$T^{2} + 3T - 3$$
$31$ $$T^{2} - 3T - 45$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 15T + 51$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 6T - 12$$
$53$ $$T^{2} - 6T - 12$$
$59$ $$T^{2} + 6T - 12$$
$61$ $$T^{2} + 21T + 105$$
$67$ $$T^{2} - T - 47$$
$71$ $$T^{2} - 84$$
$73$ $$T^{2} + 5T - 41$$
$79$ $$T^{2} - 21T + 105$$
$83$ $$T^{2} - 12T - 48$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 18T + 60$$