Properties

 Label 9200.2.a.bm Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ 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] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.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: $$73.4623698596$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 115) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta - 1) q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + (b - 1) * q^7 - 2 * q^9 $$q - q^{3} + (\beta - 1) q^{7} - 2 q^{9} + (\beta + 1) q^{11} + (\beta + 4) q^{13} + ( - 2 \beta + 2) q^{17} + (3 \beta - 1) q^{19} + ( - \beta + 1) q^{21} - q^{23} + 5 q^{27} + (2 \beta - 5) q^{29} + ( - \beta - 2) q^{31} + ( - \beta - 1) q^{33} + (3 \beta + 3) q^{37} + ( - \beta - 4) q^{39} + ( - 2 \beta - 3) q^{41} + (3 \beta - 3) q^{43} + (2 \beta + 5) q^{47} + ( - 2 \beta - 1) q^{49} + (2 \beta - 2) q^{51} + 6 q^{53} + ( - 3 \beta + 1) q^{57} - 4 \beta q^{59} + (5 \beta + 1) q^{61} + ( - 2 \beta + 2) q^{63} + (3 \beta + 3) q^{67} + q^{69} + (\beta + 4) q^{71} + 3 \beta q^{73} + 4 q^{77} + (\beta - 11) q^{79} + q^{81} + ( - 2 \beta - 2) q^{83} + ( - 2 \beta + 5) q^{87} + ( - \beta + 5) q^{89} + (3 \beta + 1) q^{91} + (\beta + 2) q^{93} + (5 \beta - 5) q^{97} + ( - 2 \beta - 2) q^{99} +O(q^{100})$$ q - q^3 + (b - 1) * q^7 - 2 * q^9 + (b + 1) * q^11 + (b + 4) * q^13 + (-2*b + 2) * q^17 + (3*b - 1) * q^19 + (-b + 1) * q^21 - q^23 + 5 * q^27 + (2*b - 5) * q^29 + (-b - 2) * q^31 + (-b - 1) * q^33 + (3*b + 3) * q^37 + (-b - 4) * q^39 + (-2*b - 3) * q^41 + (3*b - 3) * q^43 + (2*b + 5) * q^47 + (-2*b - 1) * q^49 + (2*b - 2) * q^51 + 6 * q^53 + (-3*b + 1) * q^57 - 4*b * q^59 + (5*b + 1) * q^61 + (-2*b + 2) * q^63 + (3*b + 3) * q^67 + q^69 + (b + 4) * q^71 + 3*b * q^73 + 4 * q^77 + (b - 11) * q^79 + q^81 + (-2*b - 2) * q^83 + (-2*b + 5) * q^87 + (-b + 5) * q^89 + (3*b + 1) * q^91 + (b + 2) * q^93 + (5*b - 5) * q^97 + (-2*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^7 - 4 * q^9 $$2 q - 2 q^{3} - 2 q^{7} - 4 q^{9} + 2 q^{11} + 8 q^{13} + 4 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 10 q^{27} - 10 q^{29} - 4 q^{31} - 2 q^{33} + 6 q^{37} - 8 q^{39} - 6 q^{41} - 6 q^{43} + 10 q^{47} - 2 q^{49} - 4 q^{51} + 12 q^{53} + 2 q^{57} + 2 q^{61} + 4 q^{63} + 6 q^{67} + 2 q^{69} + 8 q^{71} + 8 q^{77} - 22 q^{79} + 2 q^{81} - 4 q^{83} + 10 q^{87} + 10 q^{89} + 2 q^{91} + 4 q^{93} - 10 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^7 - 4 * q^9 + 2 * q^11 + 8 * q^13 + 4 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 10 * q^27 - 10 * q^29 - 4 * q^31 - 2 * q^33 + 6 * q^37 - 8 * q^39 - 6 * q^41 - 6 * q^43 + 10 * q^47 - 2 * q^49 - 4 * q^51 + 12 * q^53 + 2 * q^57 + 2 * q^61 + 4 * q^63 + 6 * q^67 + 2 * q^69 + 8 * q^71 + 8 * q^77 - 22 * q^79 + 2 * q^81 - 4 * q^83 + 10 * q^87 + 10 * q^89 + 2 * q^91 + 4 * q^93 - 10 * q^97 - 4 * 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
 −0.618034 1.61803
0 −1.00000 0 0 0 −3.23607 0 −2.00000 0
1.2 0 −1.00000 0 0 0 1.23607 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$1$$

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 9200.2.a.bm 2
4.b odd 2 1 575.2.a.g 2
5.b even 2 1 1840.2.a.p 2
12.b even 2 1 5175.2.a.bb 2
20.d odd 2 1 115.2.a.b 2
20.e even 4 2 575.2.b.c 4
40.e odd 2 1 7360.2.a.bt 2
40.f even 2 1 7360.2.a.bf 2
60.h even 2 1 1035.2.a.m 2
140.c even 2 1 5635.2.a.n 2
460.g even 2 1 2645.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.b 2 20.d odd 2 1
575.2.a.g 2 4.b odd 2 1
575.2.b.c 4 20.e even 4 2
1035.2.a.m 2 60.h even 2 1
1840.2.a.p 2 5.b even 2 1
2645.2.a.d 2 460.g even 2 1
5175.2.a.bb 2 12.b even 2 1
5635.2.a.n 2 140.c even 2 1
7360.2.a.bf 2 40.f even 2 1
7360.2.a.bt 2 40.e odd 2 1
9200.2.a.bm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7}^{2} + 2T_{7} - 4$$ T7^2 + 2*T7 - 4 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 4$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} - 8T + 11$$
$17$ $$T^{2} - 4T - 16$$
$19$ $$T^{2} + 2T - 44$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 10T + 5$$
$31$ $$T^{2} + 4T - 1$$
$37$ $$T^{2} - 6T - 36$$
$41$ $$T^{2} + 6T - 11$$
$43$ $$T^{2} + 6T - 36$$
$47$ $$T^{2} - 10T + 5$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 80$$
$61$ $$T^{2} - 2T - 124$$
$67$ $$T^{2} - 6T - 36$$
$71$ $$T^{2} - 8T + 11$$
$73$ $$T^{2} - 45$$
$79$ $$T^{2} + 22T + 116$$
$83$ $$T^{2} + 4T - 16$$
$89$ $$T^{2} - 10T + 20$$
$97$ $$T^{2} + 10T - 100$$