Properties

 Label 3648.2.a.bn Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ 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] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.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: $$29.1294266574$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^5 + (-b + 2) * q^7 + q^9 $$q - q^{3} + \beta q^{5} + ( - \beta + 2) q^{7} + q^{9} + ( - \beta + 2) q^{11} - 6 q^{13} - \beta q^{15} + \beta q^{17} + q^{19} + (\beta - 2) q^{21} - 4 q^{23} + (\beta + 5) q^{25} - q^{27} - 2 q^{29} + (2 \beta - 4) q^{31} + (\beta - 2) q^{33} + (\beta - 10) q^{35} + ( - 2 \beta - 2) q^{37} + 6 q^{39} + (2 \beta + 2) q^{41} + ( - 3 \beta + 2) q^{43} + \beta q^{45} + ( - \beta - 2) q^{47} + ( - 3 \beta + 7) q^{49} - \beta q^{51} + 6 q^{53} + (\beta - 10) q^{55} - q^{57} - 4 q^{59} + ( - \beta - 4) q^{61} + ( - \beta + 2) q^{63} - 6 \beta q^{65} - 12 q^{67} + 4 q^{69} + ( - \beta + 4) q^{73} + ( - \beta - 5) q^{75} + ( - 3 \beta + 14) q^{77} - 4 \beta q^{79} + q^{81} + 4 \beta q^{83} + (\beta + 10) q^{85} + 2 q^{87} + (4 \beta - 2) q^{89} + (6 \beta - 12) q^{91} + ( - 2 \beta + 4) q^{93} + \beta q^{95} - 6 q^{97} + ( - \beta + 2) q^{99} +O(q^{100})$$ q - q^3 + b * q^5 + (-b + 2) * q^7 + q^9 + (-b + 2) * q^11 - 6 * q^13 - b * q^15 + b * q^17 + q^19 + (b - 2) * q^21 - 4 * q^23 + (b + 5) * q^25 - q^27 - 2 * q^29 + (2*b - 4) * q^31 + (b - 2) * q^33 + (b - 10) * q^35 + (-2*b - 2) * q^37 + 6 * q^39 + (2*b + 2) * q^41 + (-3*b + 2) * q^43 + b * q^45 + (-b - 2) * q^47 + (-3*b + 7) * q^49 - b * q^51 + 6 * q^53 + (b - 10) * q^55 - q^57 - 4 * q^59 + (-b - 4) * q^61 + (-b + 2) * q^63 - 6*b * q^65 - 12 * q^67 + 4 * q^69 + (-b + 4) * q^73 + (-b - 5) * q^75 + (-3*b + 14) * q^77 - 4*b * q^79 + q^81 + 4*b * q^83 + (b + 10) * q^85 + 2 * q^87 + (4*b - 2) * q^89 + (6*b - 12) * q^91 + (-2*b + 4) * q^93 + b * q^95 - 6 * q^97 + (-b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + q^5 + 3 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + 3 q^{11} - 12 q^{13} - q^{15} + q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 11 q^{25} - 2 q^{27} - 4 q^{29} - 6 q^{31} - 3 q^{33} - 19 q^{35} - 6 q^{37} + 12 q^{39} + 6 q^{41} + q^{43} + q^{45} - 5 q^{47} + 11 q^{49} - q^{51} + 12 q^{53} - 19 q^{55} - 2 q^{57} - 8 q^{59} - 9 q^{61} + 3 q^{63} - 6 q^{65} - 24 q^{67} + 8 q^{69} + 7 q^{73} - 11 q^{75} + 25 q^{77} - 4 q^{79} + 2 q^{81} + 4 q^{83} + 21 q^{85} + 4 q^{87} - 18 q^{91} + 6 q^{93} + q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + q^5 + 3 * q^7 + 2 * q^9 + 3 * q^11 - 12 * q^13 - q^15 + q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 11 * q^25 - 2 * q^27 - 4 * q^29 - 6 * q^31 - 3 * q^33 - 19 * q^35 - 6 * q^37 + 12 * q^39 + 6 * q^41 + q^43 + q^45 - 5 * q^47 + 11 * q^49 - q^51 + 12 * q^53 - 19 * q^55 - 2 * q^57 - 8 * q^59 - 9 * q^61 + 3 * q^63 - 6 * q^65 - 24 * q^67 + 8 * q^69 + 7 * q^73 - 11 * q^75 + 25 * q^77 - 4 * q^79 + 2 * q^81 + 4 * q^83 + 21 * q^85 + 4 * q^87 - 18 * q^91 + 6 * q^93 + q^95 - 12 * q^97 + 3 * 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
 −2.70156 3.70156
0 −1.00000 0 −2.70156 0 4.70156 0 1.00000 0
1.2 0 −1.00000 0 3.70156 0 −1.70156 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$19$$ $$-1$$

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 3648.2.a.bn 2
4.b odd 2 1 3648.2.a.bs 2
8.b even 2 1 912.2.a.o 2
8.d odd 2 1 456.2.a.e 2
24.f even 2 1 1368.2.a.l 2
24.h odd 2 1 2736.2.a.bb 2
152.b even 2 1 8664.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.e 2 8.d odd 2 1
912.2.a.o 2 8.b even 2 1
1368.2.a.l 2 24.f even 2 1
2736.2.a.bb 2 24.h odd 2 1
3648.2.a.bn 2 1.a even 1 1 trivial
3648.2.a.bs 2 4.b odd 2 1
8664.2.a.v 2 152.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3648))$$:

 $$T_{5}^{2} - T_{5} - 10$$ T5^2 - T5 - 10 $$T_{7}^{2} - 3T_{7} - 8$$ T7^2 - 3*T7 - 8 $$T_{11}^{2} - 3T_{11} - 8$$ T11^2 - 3*T11 - 8 $$T_{23} + 4$$ T23 + 4 $$T_{31}^{2} + 6T_{31} - 32$$ T31^2 + 6*T31 - 32

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - T - 10$$
$7$ $$T^{2} - 3T - 8$$
$11$ $$T^{2} - 3T - 8$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2} - T - 10$$
$19$ $$(T - 1)^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} + 6T - 32$$
$37$ $$T^{2} + 6T - 32$$
$41$ $$T^{2} - 6T - 32$$
$43$ $$T^{2} - T - 92$$
$47$ $$T^{2} + 5T - 4$$
$53$ $$(T - 6)^{2}$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} + 9T + 10$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 7T + 2$$
$79$ $$T^{2} + 4T - 160$$
$83$ $$T^{2} - 4T - 160$$
$89$ $$T^{2} - 164$$
$97$ $$(T + 6)^{2}$$