# Properties

 Label 2738.2.a.i 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: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 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

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

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

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
 −1.73205 1.73205
1.00000 −2.73205 1.00000 1.73205 −2.73205 −2.00000 1.00000 4.46410 1.73205
1.2 1.00000 0.732051 1.00000 −1.73205 0.732051 −2.00000 1.00000 −2.46410 −1.73205
 $$n$$: 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.i 2
37.b even 2 1 2738.2.a.e 2
37.g odd 12 2 74.2.e.b 4
111.m even 12 2 666.2.s.a 4
148.l even 12 2 592.2.w.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.b 4 37.g odd 12 2
592.2.w.e 4 148.l even 12 2
666.2.s.a 4 111.m even 12 2
2738.2.a.e 2 37.b even 2 1
2738.2.a.i 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{2} + 2T_{3} - 2$$ T3^2 + 2*T3 - 2 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{7} + 2$$ T7 + 2 $$T_{13}^{2} - 12$$ T13^2 - 12 $$T_{17}^{2} + 12T_{17} + 33$$ T17^2 + 12*T17 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 2T - 2$$
$5$ $$T^{2} - 3$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 6T + 6$$
$13$ $$T^{2} - 12$$
$17$ $$T^{2} + 12T + 33$$
$19$ $$T^{2} - 6T + 6$$
$23$ $$T^{2} + 6T + 6$$
$29$ $$T^{2} - 75$$
$31$ $$T^{2} + 6T + 6$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 6T - 39$$
$43$ $$T^{2} + 12T - 12$$
$47$ $$T^{2} - 6T + 6$$
$53$ $$T^{2} - 12T + 24$$
$59$ $$T^{2} + 12T + 24$$
$61$ $$T^{2} - 3$$
$67$ $$T^{2} - 10T - 2$$
$71$ $$T^{2} - 12$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} + 30T + 222$$
$83$ $$T^{2} - 6T - 66$$
$89$ $$T^{2} + 24T + 117$$
$97$ $$T^{2} + 12T + 33$$