Properties

 Label 1638.2.a.x Level $1638$ Weight $2$ Character orbit 1638.a Self dual yes Analytic conductor $13.079$ Analytic rank $0$ 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] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + b * q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + \beta q^{5} - q^{7} + q^{8} + \beta q^{10} - \beta q^{11} - q^{13} - q^{14} + q^{16} + ( - \beta + 2) q^{17} + (\beta + 4) q^{19} + \beta q^{20} - \beta q^{22} + (\beta + 6) q^{23} + (\beta + 3) q^{25} - q^{26} - q^{28} + ( - \beta + 4) q^{29} + 2 \beta q^{31} + q^{32} + ( - \beta + 2) q^{34} - \beta q^{35} + (\beta - 2) q^{37} + (\beta + 4) q^{38} + \beta q^{40} + ( - 2 \beta + 4) q^{41} + ( - 3 \beta + 4) q^{43} - \beta q^{44} + (\beta + 6) q^{46} + (2 \beta + 6) q^{47} + q^{49} + (\beta + 3) q^{50} - q^{52} + ( - 2 \beta + 4) q^{53} + ( - \beta - 8) q^{55} - q^{56} + ( - \beta + 4) q^{58} + 2 q^{59} + ( - 3 \beta - 2) q^{61} + 2 \beta q^{62} + q^{64} - \beta q^{65} - 4 \beta q^{67} + ( - \beta + 2) q^{68} - \beta q^{70} + 2 \beta q^{71} + (\beta - 6) q^{73} + (\beta - 2) q^{74} + (\beta + 4) q^{76} + \beta q^{77} - 2 \beta q^{79} + \beta q^{80} + ( - 2 \beta + 4) q^{82} - 6 q^{83} + (\beta - 8) q^{85} + ( - 3 \beta + 4) q^{86} - \beta q^{88} + ( - 2 \beta - 8) q^{89} + q^{91} + (\beta + 6) q^{92} + (2 \beta + 6) q^{94} + (5 \beta + 8) q^{95} + ( - 4 \beta - 2) q^{97} + q^{98} +O(q^{100})$$ q + q^2 + q^4 + b * q^5 - q^7 + q^8 + b * q^10 - b * q^11 - q^13 - q^14 + q^16 + (-b + 2) * q^17 + (b + 4) * q^19 + b * q^20 - b * q^22 + (b + 6) * q^23 + (b + 3) * q^25 - q^26 - q^28 + (-b + 4) * q^29 + 2*b * q^31 + q^32 + (-b + 2) * q^34 - b * q^35 + (b - 2) * q^37 + (b + 4) * q^38 + b * q^40 + (-2*b + 4) * q^41 + (-3*b + 4) * q^43 - b * q^44 + (b + 6) * q^46 + (2*b + 6) * q^47 + q^49 + (b + 3) * q^50 - q^52 + (-2*b + 4) * q^53 + (-b - 8) * q^55 - q^56 + (-b + 4) * q^58 + 2 * q^59 + (-3*b - 2) * q^61 + 2*b * q^62 + q^64 - b * q^65 - 4*b * q^67 + (-b + 2) * q^68 - b * q^70 + 2*b * q^71 + (b - 6) * q^73 + (b - 2) * q^74 + (b + 4) * q^76 + b * q^77 - 2*b * q^79 + b * q^80 + (-2*b + 4) * q^82 - 6 * q^83 + (b - 8) * q^85 + (-3*b + 4) * q^86 - b * q^88 + (-2*b - 8) * q^89 + q^91 + (b + 6) * q^92 + (2*b + 6) * q^94 + (5*b + 8) * q^95 + (-4*b - 2) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} - q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} + 9 q^{19} + q^{20} - q^{22} + 13 q^{23} + 7 q^{25} - 2 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 3 q^{34} - q^{35} - 3 q^{37} + 9 q^{38} + q^{40} + 6 q^{41} + 5 q^{43} - q^{44} + 13 q^{46} + 14 q^{47} + 2 q^{49} + 7 q^{50} - 2 q^{52} + 6 q^{53} - 17 q^{55} - 2 q^{56} + 7 q^{58} + 4 q^{59} - 7 q^{61} + 2 q^{62} + 2 q^{64} - q^{65} - 4 q^{67} + 3 q^{68} - q^{70} + 2 q^{71} - 11 q^{73} - 3 q^{74} + 9 q^{76} + q^{77} - 2 q^{79} + q^{80} + 6 q^{82} - 12 q^{83} - 15 q^{85} + 5 q^{86} - q^{88} - 18 q^{89} + 2 q^{91} + 13 q^{92} + 14 q^{94} + 21 q^{95} - 8 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8 + q^10 - q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 3 * q^17 + 9 * q^19 + q^20 - q^22 + 13 * q^23 + 7 * q^25 - 2 * q^26 - 2 * q^28 + 7 * q^29 + 2 * q^31 + 2 * q^32 + 3 * q^34 - q^35 - 3 * q^37 + 9 * q^38 + q^40 + 6 * q^41 + 5 * q^43 - q^44 + 13 * q^46 + 14 * q^47 + 2 * q^49 + 7 * q^50 - 2 * q^52 + 6 * q^53 - 17 * q^55 - 2 * q^56 + 7 * q^58 + 4 * q^59 - 7 * q^61 + 2 * q^62 + 2 * q^64 - q^65 - 4 * q^67 + 3 * q^68 - q^70 + 2 * q^71 - 11 * q^73 - 3 * q^74 + 9 * q^76 + q^77 - 2 * q^79 + q^80 + 6 * q^82 - 12 * q^83 - 15 * q^85 + 5 * q^86 - q^88 - 18 * q^89 + 2 * q^91 + 13 * q^92 + 14 * q^94 + 21 * q^95 - 8 * q^97 + 2 * q^98

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
 −2.37228 3.37228
1.00000 0 1.00000 −2.37228 0 −1.00000 1.00000 0 −2.37228
1.2 1.00000 0 1.00000 3.37228 0 −1.00000 1.00000 0 3.37228
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$7$$ $$+1$$
$$13$$ $$+1$$

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 1638.2.a.x yes 2
3.b odd 2 1 1638.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.a.v 2 3.b odd 2 1
1638.2.a.x yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1638))$$:

 $$T_{5}^{2} - T_{5} - 8$$ T5^2 - T5 - 8 $$T_{11}^{2} + T_{11} - 8$$ T11^2 + T11 - 8 $$T_{17}^{2} - 3T_{17} - 6$$ T17^2 - 3*T17 - 6 $$T_{19}^{2} - 9T_{19} + 12$$ T19^2 - 9*T19 + 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T - 8$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + T - 8$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 3T - 6$$
$19$ $$T^{2} - 9T + 12$$
$23$ $$T^{2} - 13T + 34$$
$29$ $$T^{2} - 7T + 4$$
$31$ $$T^{2} - 2T - 32$$
$37$ $$T^{2} + 3T - 6$$
$41$ $$T^{2} - 6T - 24$$
$43$ $$T^{2} - 5T - 68$$
$47$ $$T^{2} - 14T + 16$$
$53$ $$T^{2} - 6T - 24$$
$59$ $$(T - 2)^{2}$$
$61$ $$T^{2} + 7T - 62$$
$67$ $$T^{2} + 4T - 128$$
$71$ $$T^{2} - 2T - 32$$
$73$ $$T^{2} + 11T + 22$$
$79$ $$T^{2} + 2T - 32$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 18T + 48$$
$97$ $$T^{2} + 8T - 116$$