# Properties

 Label 2738.2.a.g 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2738,2,Mod(1,2738)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$21.8630400734$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
−1.00000 −1.61803 1.00000 −3.85410 1.61803 −3.23607 −1.00000 −0.381966 3.85410
1.2 −1.00000 0.618034 1.00000 2.85410 −0.618034 1.23607 −1.00000 −2.61803 −2.85410
 $$n$$: 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.g 2
37.b even 2 1 74.2.a.b 2
111.d odd 2 1 666.2.a.i 2
148.b odd 2 1 592.2.a.g 2
185.d even 2 1 1850.2.a.t 2
185.h odd 4 2 1850.2.b.j 4
259.b odd 2 1 3626.2.a.s 2
296.e even 2 1 2368.2.a.y 2
296.h odd 2 1 2368.2.a.u 2
407.d odd 2 1 8954.2.a.j 2
444.g even 2 1 5328.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 37.b even 2 1
592.2.a.g 2 148.b odd 2 1
666.2.a.i 2 111.d odd 2 1
1850.2.a.t 2 185.d even 2 1
1850.2.b.j 4 185.h odd 4 2
2368.2.a.u 2 296.h odd 2 1
2368.2.a.y 2 296.e even 2 1
2738.2.a.g 2 1.a even 1 1 trivial
3626.2.a.s 2 259.b odd 2 1
5328.2.a.bc 2 444.g even 2 1
8954.2.a.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{5}^{2} + T_{5} - 11$$ T5^2 + T5 - 11 $$T_{7}^{2} + 2T_{7} - 4$$ T7^2 + 2*T7 - 4 $$T_{13}^{2} + T_{13} - 11$$ T13^2 + T13 - 11 $$T_{17}^{2} - 20$$ T17^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2} + T - 11$$
$7$ $$T^{2} + 2T - 4$$
$11$ $$T^{2} + 5T + 5$$
$13$ $$T^{2} + T - 11$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} - T - 11$$
$29$ $$T^{2} - 3T - 59$$
$31$ $$T^{2} + 17T + 71$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 17T + 71$$
$43$ $$T^{2} - 6T + 4$$
$47$ $$T^{2} - 2T - 4$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} - 14T + 44$$
$61$ $$T^{2} + 19T + 89$$
$67$ $$T^{2} + 9T - 11$$
$71$ $$T^{2} + 12T - 44$$
$73$ $$T^{2} - 3T - 29$$
$79$ $$T^{2} + 3T - 99$$
$83$ $$T^{2} + 20T + 80$$
$89$ $$T^{2} - 12T + 16$$
$97$ $$T^{2} + 8T - 4$$