# Properties

 Label 2738.2.a.n Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 3$$ v^2 - 2*v - 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 2$$ (-b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 4$$ b1 + 4

## 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
 −1.65544 2.86620 −0.210756
−1.00000 −3.05137 1.00000 −1.25951 3.05137 3.05137 −1.00000 6.31088 1.25951
1.2 −1.00000 0.517304 1.00000 4.21509 −0.517304 −0.517304 −1.00000 −2.73240 −4.21509
1.3 −1.00000 2.53407 1.00000 −3.95558 −2.53407 −2.53407 −1.00000 3.42151 3.95558
 $$n$$: 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.n 3
37.b even 2 1 2738.2.a.o 3
37.e even 6 2 74.2.c.c 6
111.h odd 6 2 666.2.f.j 6
148.j odd 6 2 592.2.i.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.c 6 37.e even 6 2
592.2.i.e 6 148.j odd 6 2
666.2.f.j 6 111.h odd 6 2
2738.2.a.n 3 1.a even 1 1 trivial
2738.2.a.o 3 37.b even 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 8T_{3} + 4$$ T3^3 - 8*T3 + 4 $$T_{5}^{3} + T_{5}^{2} - 17T_{5} - 21$$ T5^3 + T5^2 - 17*T5 - 21 $$T_{7}^{3} - 8T_{7} - 4$$ T7^3 - 8*T7 - 4 $$T_{13} + 2$$ T13 + 2 $$T_{17}^{3} - 3T_{17}^{2} - 29T_{17} + 63$$ T17^3 - 3*T17^2 - 29*T17 + 63

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} - 8T + 4$$
$5$ $$T^{3} + T^{2} - 17 T - 21$$
$7$ $$T^{3} - 8T - 4$$
$11$ $$T^{3} - 4 T^{2} - 16 T + 48$$
$13$ $$(T + 2)^{3}$$
$17$ $$T^{3} - 3 T^{2} - 29 T + 63$$
$19$ $$T^{3} + 8 T^{2} - 8 T - 112$$
$23$ $$T^{3} + 8 T^{2} + 4 T - 12$$
$29$ $$T^{3} - 3 T^{2} - 5 T + 3$$
$31$ $$T^{3} + 4 T^{2} - 24 T + 16$$
$37$ $$T^{3}$$
$41$ $$T^{3} + 7 T^{2} - 13 T - 3$$
$43$ $$T^{3} + 8 T^{2} - 20 T - 148$$
$47$ $$T^{3} + 16 T^{2} + 56 T + 48$$
$53$ $$T^{3} - 22 T^{2} + 140 T - 216$$
$59$ $$T^{3} + 4 T^{2} - 160 T - 672$$
$61$ $$T^{3} + 9 T^{2} + 19 T + 7$$
$67$ $$T^{3} + 16 T^{2} + 64 T + 16$$
$71$ $$T^{3} + 12 T^{2} + 40 T + 36$$
$73$ $$T^{3} - 2 T^{2} - 20 T - 8$$
$79$ $$T^{3} - 4 T^{2} - 72 T - 144$$
$83$ $$T^{3} - 4 T^{2} - 64 T - 132$$
$89$ $$T^{3} + 9 T^{2} - 149 T - 549$$
$97$ $$T^{3} + 13 T^{2} + 27 T + 7$$