Properties

 Label 9200.2.a.bo Level $9200$ Weight $2$ Character orbit 9200.a Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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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: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + (3 \beta - 3) q^{7} + (\beta - 2) q^{9} +O(q^{10})$$ q - b * q^3 + (3*b - 3) * q^7 + (b - 2) * q^9 $$q - \beta q^{3} + (3 \beta - 3) q^{7} + (\beta - 2) q^{9} + (\beta + 4) q^{11} + (\beta + 1) q^{13} + (3 \beta - 4) q^{17} + ( - 3 \beta + 5) q^{19} - 3 q^{21} - q^{23} + (4 \beta - 1) q^{27} - 6 \beta q^{29} + ( - 3 \beta + 7) q^{31} + ( - 5 \beta - 1) q^{33} - 6 \beta q^{37} + ( - 2 \beta - 1) q^{39} + ( - \beta - 4) q^{41} + ( - 2 \beta - 8) q^{43} + ( - 6 \beta + 8) q^{47} + ( - 9 \beta + 11) q^{49} + (\beta - 3) q^{51} - 2 q^{53} + ( - 2 \beta + 3) q^{57} + 6 q^{59} + (3 \beta - 2) q^{61} + ( - 6 \beta + 9) q^{63} + ( - 2 \beta - 2) q^{67} + \beta q^{69} + (\beta - 2) q^{71} + ( - 4 \beta - 10) q^{73} + (12 \beta - 9) q^{77} + (6 \beta - 2) q^{79} + ( - 6 \beta + 2) q^{81} + ( - 6 \beta + 2) q^{83} + (6 \beta + 6) q^{87} + (6 \beta - 6) q^{89} + 3 \beta q^{91} + ( - 4 \beta + 3) q^{93} + (13 \beta - 8) q^{97} + (3 \beta - 7) q^{99} +O(q^{100})$$ q - b * q^3 + (3*b - 3) * q^7 + (b - 2) * q^9 + (b + 4) * q^11 + (b + 1) * q^13 + (3*b - 4) * q^17 + (-3*b + 5) * q^19 - 3 * q^21 - q^23 + (4*b - 1) * q^27 - 6*b * q^29 + (-3*b + 7) * q^31 + (-5*b - 1) * q^33 - 6*b * q^37 + (-2*b - 1) * q^39 + (-b - 4) * q^41 + (-2*b - 8) * q^43 + (-6*b + 8) * q^47 + (-9*b + 11) * q^49 + (b - 3) * q^51 - 2 * q^53 + (-2*b + 3) * q^57 + 6 * q^59 + (3*b - 2) * q^61 + (-6*b + 9) * q^63 + (-2*b - 2) * q^67 + b * q^69 + (b - 2) * q^71 + (-4*b - 10) * q^73 + (12*b - 9) * q^77 + (6*b - 2) * q^79 + (-6*b + 2) * q^81 + (-6*b + 2) * q^83 + (6*b + 6) * q^87 + (6*b - 6) * q^89 + 3*b * q^91 + (-4*b + 3) * q^93 + (13*b - 8) * q^97 + (3*b - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 3 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 3 * q^7 - 3 * q^9 $$2 q - q^{3} - 3 q^{7} - 3 q^{9} + 9 q^{11} + 3 q^{13} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 2 q^{23} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 7 q^{33} - 6 q^{37} - 4 q^{39} - 9 q^{41} - 18 q^{43} + 10 q^{47} + 13 q^{49} - 5 q^{51} - 4 q^{53} + 4 q^{57} + 12 q^{59} - q^{61} + 12 q^{63} - 6 q^{67} + q^{69} - 3 q^{71} - 24 q^{73} - 6 q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{83} + 18 q^{87} - 6 q^{89} + 3 q^{91} + 2 q^{93} - 3 q^{97} - 11 q^{99}+O(q^{100})$$ 2 * q - q^3 - 3 * q^7 - 3 * q^9 + 9 * q^11 + 3 * q^13 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 2 * q^23 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 7 * q^33 - 6 * q^37 - 4 * q^39 - 9 * q^41 - 18 * q^43 + 10 * q^47 + 13 * q^49 - 5 * q^51 - 4 * q^53 + 4 * q^57 + 12 * q^59 - q^61 + 12 * q^63 - 6 * q^67 + q^69 - 3 * q^71 - 24 * q^73 - 6 * q^77 + 2 * q^79 - 2 * q^81 - 2 * q^83 + 18 * q^87 - 6 * q^89 + 3 * q^91 + 2 * q^93 - 3 * q^97 - 11 * 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
 1.61803 −0.618034
0 −1.61803 0 0 0 1.85410 0 −0.381966 0
1.2 0 0.618034 0 0 0 −4.85410 0 −2.61803 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.bo 2
4.b odd 2 1 1150.2.a.l 2
5.b even 2 1 9200.2.a.by 2
5.c odd 4 2 1840.2.e.c 4
20.d odd 2 1 1150.2.a.n 2
20.e even 4 2 230.2.b.a 4
60.l odd 4 2 2070.2.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 20.e even 4 2
1150.2.a.l 2 4.b odd 2 1
1150.2.a.n 2 20.d odd 2 1
1840.2.e.c 4 5.c odd 4 2
2070.2.d.c 4 60.l odd 4 2
9200.2.a.bo 2 1.a even 1 1 trivial
9200.2.a.by 2 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{7}^{2} + 3T_{7} - 9$$ T7^2 + 3*T7 - 9 $$T_{11}^{2} - 9T_{11} + 19$$ T11^2 - 9*T11 + 19

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3T - 9$$
$11$ $$T^{2} - 9T + 19$$
$13$ $$T^{2} - 3T + 1$$
$17$ $$T^{2} + 5T - 5$$
$19$ $$T^{2} - 7T + 1$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 6T - 36$$
$31$ $$T^{2} - 11T + 19$$
$37$ $$T^{2} + 6T - 36$$
$41$ $$T^{2} + 9T + 19$$
$43$ $$T^{2} + 18T + 76$$
$47$ $$T^{2} - 10T - 20$$
$53$ $$(T + 2)^{2}$$
$59$ $$(T - 6)^{2}$$
$61$ $$T^{2} + T - 11$$
$67$ $$T^{2} + 6T + 4$$
$71$ $$T^{2} + 3T + 1$$
$73$ $$T^{2} + 24T + 124$$
$79$ $$T^{2} - 2T - 44$$
$83$ $$T^{2} + 2T - 44$$
$89$ $$T^{2} + 6T - 36$$
$97$ $$T^{2} + 3T - 209$$
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