# Properties

 Label 2738.2.a.p Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{2} + 2) q^{5} - \beta_1 q^{6} + ( - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q + q^2 - b1 * q^3 + q^4 + (b2 + 2) * q^5 - b1 * q^6 + (-b1 + 1) * q^7 + q^8 + (b2 - 1) * q^9 $$q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{2} + 2) q^{5} - \beta_1 q^{6} + ( - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - 1) q^{9} + (\beta_{2} + 2) q^{10} + ( - \beta_{2} + \beta_1 + 3) q^{11} - \beta_1 q^{12} + (\beta_{2} + \beta_1 + 3) q^{13} + ( - \beta_1 + 1) q^{14} + ( - 3 \beta_1 - 1) q^{15} + q^{16} + 3 q^{17} + (\beta_{2} - 1) q^{18} + ( - 3 \beta_1 + 1) q^{19} + (\beta_{2} + 2) q^{20} + (\beta_{2} - \beta_1 + 2) q^{21} + ( - \beta_{2} + \beta_1 + 3) q^{22} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{23} - \beta_1 q^{24} + (3 \beta_{2} + \beta_1 + 1) q^{25} + (\beta_{2} + \beta_1 + 3) q^{26} + (3 \beta_1 - 1) q^{27} + ( - \beta_1 + 1) q^{28} + ( - 4 \beta_{2} + 5 \beta_1) q^{29} + ( - 3 \beta_1 - 1) q^{30} + ( - \beta_1 + 3) q^{31} + q^{32} + ( - \beta_{2} - 2 \beta_1 - 1) q^{33} + 3 q^{34} + (\beta_{2} - 3 \beta_1 + 1) q^{35} + (\beta_{2} - 1) q^{36} + ( - 3 \beta_1 + 1) q^{38} + ( - \beta_{2} - 4 \beta_1 - 3) q^{39} + (\beta_{2} + 2) q^{40} + (2 \beta_{2} + 4 \beta_1 - 2) q^{41} + (\beta_{2} - \beta_1 + 2) q^{42} + (5 \beta_{2} - 6 \beta_1 - 2) q^{43} + ( - \beta_{2} + \beta_1 + 3) q^{44} + \beta_1 q^{45} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{46} + ( - 4 \beta_{2} + 1) q^{47} - \beta_1 q^{48} + (\beta_{2} - 2 \beta_1 - 4) q^{49} + (3 \beta_{2} + \beta_1 + 1) q^{50} - 3 \beta_1 q^{51} + (\beta_{2} + \beta_1 + 3) q^{52} + ( - 4 \beta_{2} + 5 \beta_1 - 3) q^{53} + (3 \beta_1 - 1) q^{54} + (2 \beta_{2} + 2 \beta_1 + 5) q^{55} + ( - \beta_1 + 1) q^{56} + (3 \beta_{2} - \beta_1 + 6) q^{57} + ( - 4 \beta_{2} + 5 \beta_1) q^{58} + ( - 8 \beta_{2} + 3 \beta_1 - 1) q^{59} + ( - 3 \beta_1 - 1) q^{60} + ( - 4 \beta_{2} + 4) q^{61} + ( - \beta_1 + 3) q^{62} + (\beta_{2} - 2) q^{63} + q^{64} + (4 \beta_{2} + 4 \beta_1 + 9) q^{65} + ( - \beta_{2} - 2 \beta_1 - 1) q^{66} + (\beta_{2} + 7) q^{67} + 3 q^{68} + ( - 2 \beta_{2} + 8 \beta_1 - 1) q^{69} + (\beta_{2} - 3 \beta_1 + 1) q^{70} + ( - 4 \beta_{2} + 6 \beta_1 + 8) q^{71} + (\beta_{2} - 1) q^{72} + (2 \beta_{2} - 3 \beta_1 - 6) q^{73} + ( - \beta_{2} - 4 \beta_1 - 5) q^{75} + ( - 3 \beta_1 + 1) q^{76} + ( - 2 \beta_{2} - \beta_1 + 2) q^{77} + ( - \beta_{2} - 4 \beta_1 - 3) q^{78} + (2 \beta_{2} + 4 \beta_1 + 1) q^{79} + (\beta_{2} + 2) q^{80} + ( - 6 \beta_{2} + \beta_1 - 3) q^{81} + (2 \beta_{2} + 4 \beta_1 - 2) q^{82} + (\beta_{2} + 3 \beta_1 - 3) q^{83} + (\beta_{2} - \beta_1 + 2) q^{84} + (3 \beta_{2} + 6) q^{85} + (5 \beta_{2} - 6 \beta_1 - 2) q^{86} + ( - 5 \beta_{2} + 4 \beta_1 - 6) q^{87} + ( - \beta_{2} + \beta_1 + 3) q^{88} + (5 \beta_{2} - 8 \beta_1 - 1) q^{89} + \beta_1 q^{90} - 3 \beta_1 q^{91} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{92} + (\beta_{2} - 3 \beta_1 + 2) q^{93} + ( - 4 \beta_{2} + 1) q^{94} + (\beta_{2} - 9 \beta_1 - 1) q^{95} - \beta_1 q^{96} + (2 \beta_{2} + 14) q^{97} + (\beta_{2} - 2 \beta_1 - 4) q^{98} + (5 \beta_{2} - \beta_1 - 4) q^{99}+O(q^{100})$$ q + q^2 - b1 * q^3 + q^4 + (b2 + 2) * q^5 - b1 * q^6 + (-b1 + 1) * q^7 + q^8 + (b2 - 1) * q^9 + (b2 + 2) * q^10 + (-b2 + b1 + 3) * q^11 - b1 * q^12 + (b2 + b1 + 3) * q^13 + (-b1 + 1) * q^14 + (-3*b1 - 1) * q^15 + q^16 + 3 * q^17 + (b2 - 1) * q^18 + (-3*b1 + 1) * q^19 + (b2 + 2) * q^20 + (b2 - b1 + 2) * q^21 + (-b2 + b1 + 3) * q^22 + (-3*b2 + 2*b1 - 5) * q^23 - b1 * q^24 + (3*b2 + b1 + 1) * q^25 + (b2 + b1 + 3) * q^26 + (3*b1 - 1) * q^27 + (-b1 + 1) * q^28 + (-4*b2 + 5*b1) * q^29 + (-3*b1 - 1) * q^30 + (-b1 + 3) * q^31 + q^32 + (-b2 - 2*b1 - 1) * q^33 + 3 * q^34 + (b2 - 3*b1 + 1) * q^35 + (b2 - 1) * q^36 + (-3*b1 + 1) * q^38 + (-b2 - 4*b1 - 3) * q^39 + (b2 + 2) * q^40 + (2*b2 + 4*b1 - 2) * q^41 + (b2 - b1 + 2) * q^42 + (5*b2 - 6*b1 - 2) * q^43 + (-b2 + b1 + 3) * q^44 + b1 * q^45 + (-3*b2 + 2*b1 - 5) * q^46 + (-4*b2 + 1) * q^47 - b1 * q^48 + (b2 - 2*b1 - 4) * q^49 + (3*b2 + b1 + 1) * q^50 - 3*b1 * q^51 + (b2 + b1 + 3) * q^52 + (-4*b2 + 5*b1 - 3) * q^53 + (3*b1 - 1) * q^54 + (2*b2 + 2*b1 + 5) * q^55 + (-b1 + 1) * q^56 + (3*b2 - b1 + 6) * q^57 + (-4*b2 + 5*b1) * q^58 + (-8*b2 + 3*b1 - 1) * q^59 + (-3*b1 - 1) * q^60 + (-4*b2 + 4) * q^61 + (-b1 + 3) * q^62 + (b2 - 2) * q^63 + q^64 + (4*b2 + 4*b1 + 9) * q^65 + (-b2 - 2*b1 - 1) * q^66 + (b2 + 7) * q^67 + 3 * q^68 + (-2*b2 + 8*b1 - 1) * q^69 + (b2 - 3*b1 + 1) * q^70 + (-4*b2 + 6*b1 + 8) * q^71 + (b2 - 1) * q^72 + (2*b2 - 3*b1 - 6) * q^73 + (-b2 - 4*b1 - 5) * q^75 + (-3*b1 + 1) * q^76 + (-2*b2 - b1 + 2) * q^77 + (-b2 - 4*b1 - 3) * q^78 + (2*b2 + 4*b1 + 1) * q^79 + (b2 + 2) * q^80 + (-6*b2 + b1 - 3) * q^81 + (2*b2 + 4*b1 - 2) * q^82 + (b2 + 3*b1 - 3) * q^83 + (b2 - b1 + 2) * q^84 + (3*b2 + 6) * q^85 + (5*b2 - 6*b1 - 2) * q^86 + (-5*b2 + 4*b1 - 6) * q^87 + (-b2 + b1 + 3) * q^88 + (5*b2 - 8*b1 - 1) * q^89 + b1 * q^90 - 3*b1 * q^91 + (-3*b2 + 2*b1 - 5) * q^92 + (b2 - 3*b1 + 2) * q^93 + (-4*b2 + 1) * q^94 + (b2 - 9*b1 - 1) * q^95 - b1 * q^96 + (2*b2 + 14) * q^97 + (b2 - 2*b1 - 4) * q^98 + (5*b2 - b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 - 3 * q^9 $$3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{9} + 6 q^{10} + 9 q^{11} + 9 q^{13} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 9 q^{17} - 3 q^{18} + 3 q^{19} + 6 q^{20} + 6 q^{21} + 9 q^{22} - 15 q^{23} + 3 q^{25} + 9 q^{26} - 3 q^{27} + 3 q^{28} - 3 q^{30} + 9 q^{31} + 3 q^{32} - 3 q^{33} + 9 q^{34} + 3 q^{35} - 3 q^{36} + 3 q^{38} - 9 q^{39} + 6 q^{40} - 6 q^{41} + 6 q^{42} - 6 q^{43} + 9 q^{44} - 15 q^{46} + 3 q^{47} - 12 q^{49} + 3 q^{50} + 9 q^{52} - 9 q^{53} - 3 q^{54} + 15 q^{55} + 3 q^{56} + 18 q^{57} - 3 q^{59} - 3 q^{60} + 12 q^{61} + 9 q^{62} - 6 q^{63} + 3 q^{64} + 27 q^{65} - 3 q^{66} + 21 q^{67} + 9 q^{68} - 3 q^{69} + 3 q^{70} + 24 q^{71} - 3 q^{72} - 18 q^{73} - 15 q^{75} + 3 q^{76} + 6 q^{77} - 9 q^{78} + 3 q^{79} + 6 q^{80} - 9 q^{81} - 6 q^{82} - 9 q^{83} + 6 q^{84} + 18 q^{85} - 6 q^{86} - 18 q^{87} + 9 q^{88} - 3 q^{89} - 15 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{95} + 42 q^{97} - 12 q^{98} - 12 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 - 3 * q^9 + 6 * q^10 + 9 * q^11 + 9 * q^13 + 3 * q^14 - 3 * q^15 + 3 * q^16 + 9 * q^17 - 3 * q^18 + 3 * q^19 + 6 * q^20 + 6 * q^21 + 9 * q^22 - 15 * q^23 + 3 * q^25 + 9 * q^26 - 3 * q^27 + 3 * q^28 - 3 * q^30 + 9 * q^31 + 3 * q^32 - 3 * q^33 + 9 * q^34 + 3 * q^35 - 3 * q^36 + 3 * q^38 - 9 * q^39 + 6 * q^40 - 6 * q^41 + 6 * q^42 - 6 * q^43 + 9 * q^44 - 15 * q^46 + 3 * q^47 - 12 * q^49 + 3 * q^50 + 9 * q^52 - 9 * q^53 - 3 * q^54 + 15 * q^55 + 3 * q^56 + 18 * q^57 - 3 * q^59 - 3 * q^60 + 12 * q^61 + 9 * q^62 - 6 * q^63 + 3 * q^64 + 27 * q^65 - 3 * q^66 + 21 * q^67 + 9 * q^68 - 3 * q^69 + 3 * q^70 + 24 * q^71 - 3 * q^72 - 18 * q^73 - 15 * q^75 + 3 * q^76 + 6 * q^77 - 9 * q^78 + 3 * q^79 + 6 * q^80 - 9 * q^81 - 6 * q^82 - 9 * q^83 + 6 * q^84 + 18 * q^85 - 6 * q^86 - 18 * q^87 + 9 * q^88 - 3 * q^89 - 15 * q^92 + 6 * q^93 + 3 * q^94 - 3 * q^95 + 42 * q^97 - 12 * q^98 - 12 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.87939 −0.347296 −1.53209
1.00000 −1.87939 1.00000 3.53209 −1.87939 −0.879385 1.00000 0.532089 3.53209
1.2 1.00000 0.347296 1.00000 0.120615 0.347296 1.34730 1.00000 −2.87939 0.120615
1.3 1.00000 1.53209 1.00000 2.34730 1.53209 2.53209 1.00000 −0.652704 2.34730
 $$n$$: 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.p 3
37.b even 2 1 2738.2.a.m 3
37.h even 18 2 74.2.f.a 6
111.n odd 18 2 666.2.x.c 6
148.o odd 18 2 592.2.bc.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.a 6 37.h even 18 2
592.2.bc.b 6 148.o odd 18 2
666.2.x.c 6 111.n odd 18 2
2738.2.a.m 3 37.b even 2 1
2738.2.a.p 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{3} - 3T_{3} + 1$$ T3^3 - 3*T3 + 1 $$T_{5}^{3} - 6T_{5}^{2} + 9T_{5} - 1$$ T5^3 - 6*T5^2 + 9*T5 - 1 $$T_{7}^{3} - 3T_{7}^{2} + 3$$ T7^3 - 3*T7^2 + 3 $$T_{13}^{3} - 9T_{13}^{2} + 18T_{13} - 9$$ T13^3 - 9*T13^2 + 18*T13 - 9 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - 3T + 1$$
$5$ $$T^{3} - 6 T^{2} + 9 T - 1$$
$7$ $$T^{3} - 3T^{2} + 3$$
$11$ $$T^{3} - 9 T^{2} + 24 T - 17$$
$13$ $$T^{3} - 9 T^{2} + 18 T - 9$$
$17$ $$(T - 3)^{3}$$
$19$ $$T^{3} - 3 T^{2} - 24 T + 53$$
$23$ $$T^{3} + 15 T^{2} + 54 T + 3$$
$29$ $$T^{3} - 63T + 171$$
$31$ $$T^{3} - 9 T^{2} + 24 T - 17$$
$37$ $$T^{3}$$
$41$ $$T^{3} + 6 T^{2} - 72 T - 456$$
$43$ $$T^{3} + 6 T^{2} - 81 T - 467$$
$47$ $$T^{3} - 3 T^{2} - 45 T - 17$$
$53$ $$T^{3} + 9 T^{2} - 36 T + 9$$
$59$ $$T^{3} + 3 T^{2} - 144 T - 829$$
$61$ $$T^{3} - 12T^{2} + 64$$
$67$ $$T^{3} - 21 T^{2} + 144 T - 321$$
$71$ $$T^{3} - 24 T^{2} + 108 T + 456$$
$73$ $$T^{3} + 18 T^{2} + 87 T + 53$$
$79$ $$T^{3} - 3 T^{2} - 81 T - 213$$
$83$ $$T^{3} + 9 T^{2} - 12 T - 179$$
$89$ $$T^{3} + 3 T^{2} - 144 T - 829$$
$97$ $$T^{3} - 42 T^{2} + 576 T - 2568$$