# Properties

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

# 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta + 1) q^{7} + (\beta - 2) q^{9}+O(q^{10})$$ q + b * q^3 + (-b + 1) * q^7 + (b - 2) * q^9 $$q + \beta q^{3} + ( - \beta + 1) q^{7} + (\beta - 2) q^{9} + (3 \beta - 2) q^{11} + (5 \beta - 1) q^{13} + ( - 5 \beta + 2) q^{17} + ( - 3 \beta + 3) q^{19} - q^{21} + q^{23} + ( - 4 \beta + 1) q^{27} + ( - 2 \beta - 6) q^{29} + ( - 5 \beta - 1) q^{31} + (\beta + 3) q^{33} - 4 \beta q^{37} + (4 \beta + 5) q^{39} + (7 \beta - 8) q^{41} + ( - 6 \beta + 6) q^{47} + ( - \beta - 5) q^{49} + ( - 3 \beta - 5) q^{51} + ( - 4 \beta + 6) q^{53} - 3 q^{57} + ( - 6 \beta + 8) q^{59} + ( - 7 \beta + 2) q^{61} + (2 \beta - 3) q^{63} + (4 \beta + 8) q^{67} + \beta q^{69} + (5 \beta - 4) q^{71} - 2 \beta q^{73} + (2 \beta - 5) q^{77} + (4 \beta - 8) q^{79} + ( - 6 \beta + 2) q^{81} + ( - 8 \beta + 6) q^{83} + ( - 8 \beta - 2) q^{87} + ( - 4 \beta - 4) q^{89} + (\beta - 6) q^{91} + ( - 6 \beta - 5) q^{93} + (\beta - 14) q^{97} + ( - 5 \beta + 7) q^{99} +O(q^{100})$$ q + b * q^3 + (-b + 1) * q^7 + (b - 2) * q^9 + (3*b - 2) * q^11 + (5*b - 1) * q^13 + (-5*b + 2) * q^17 + (-3*b + 3) * q^19 - q^21 + q^23 + (-4*b + 1) * q^27 + (-2*b - 6) * q^29 + (-5*b - 1) * q^31 + (b + 3) * q^33 - 4*b * q^37 + (4*b + 5) * q^39 + (7*b - 8) * q^41 + (-6*b + 6) * q^47 + (-b - 5) * q^49 + (-3*b - 5) * q^51 + (-4*b + 6) * q^53 - 3 * q^57 + (-6*b + 8) * q^59 + (-7*b + 2) * q^61 + (2*b - 3) * q^63 + (4*b + 8) * q^67 + b * q^69 + (5*b - 4) * q^71 - 2*b * q^73 + (2*b - 5) * q^77 + (4*b - 8) * q^79 + (-6*b + 2) * q^81 + (-8*b + 6) * q^83 + (-8*b - 2) * q^87 + (-4*b - 4) * q^89 + (b - 6) * q^91 + (-6*b - 5) * q^93 + (b - 14) * q^97 + (-5*b + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^7 - 3 * q^9 $$2 q + q^{3} + q^{7} - 3 q^{9} - q^{11} + 3 q^{13} - q^{17} + 3 q^{19} - 2 q^{21} + 2 q^{23} - 2 q^{27} - 14 q^{29} - 7 q^{31} + 7 q^{33} - 4 q^{37} + 14 q^{39} - 9 q^{41} + 6 q^{47} - 11 q^{49} - 13 q^{51} + 8 q^{53} - 6 q^{57} + 10 q^{59} - 3 q^{61} - 4 q^{63} + 20 q^{67} + q^{69} - 3 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{79} - 2 q^{81} + 4 q^{83} - 12 q^{87} - 12 q^{89} - 11 q^{91} - 16 q^{93} - 27 q^{97} + 9 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^7 - 3 * q^9 - q^11 + 3 * q^13 - q^17 + 3 * q^19 - 2 * q^21 + 2 * q^23 - 2 * q^27 - 14 * q^29 - 7 * q^31 + 7 * q^33 - 4 * q^37 + 14 * q^39 - 9 * q^41 + 6 * q^47 - 11 * q^49 - 13 * q^51 + 8 * q^53 - 6 * q^57 + 10 * q^59 - 3 * q^61 - 4 * q^63 + 20 * q^67 + q^69 - 3 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^79 - 2 * q^81 + 4 * q^83 - 12 * q^87 - 12 * q^89 - 11 * q^91 - 16 * q^93 - 27 * q^97 + 9 * 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 −0.618034 0 0 0 1.61803 0 −2.61803 0
1.2 0 1.61803 0 0 0 −0.618034 0 −0.381966 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.bu 2
4.b odd 2 1 1150.2.a.j 2
5.b even 2 1 1840.2.a.l 2
20.d odd 2 1 230.2.a.c 2
20.e even 4 2 1150.2.b.i 4
40.e odd 2 1 7360.2.a.bh 2
40.f even 2 1 7360.2.a.bn 2
60.h even 2 1 2070.2.a.u 2
460.g even 2 1 5290.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 20.d odd 2 1
1150.2.a.j 2 4.b odd 2 1
1150.2.b.i 4 20.e even 4 2
1840.2.a.l 2 5.b even 2 1
2070.2.a.u 2 60.h even 2 1
5290.2.a.o 2 460.g even 2 1
7360.2.a.bh 2 40.e odd 2 1
7360.2.a.bn 2 40.f even 2 1
9200.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9200))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 1$$
$11$ $$T^{2} + T - 11$$
$13$ $$T^{2} - 3T - 29$$
$17$ $$T^{2} + T - 31$$
$19$ $$T^{2} - 3T - 9$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 14T + 44$$
$31$ $$T^{2} + 7T - 19$$
$37$ $$T^{2} + 4T - 16$$
$41$ $$T^{2} + 9T - 41$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 6T - 36$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 10T - 20$$
$61$ $$T^{2} + 3T - 59$$
$67$ $$T^{2} - 20T + 80$$
$71$ $$T^{2} + 3T - 29$$
$73$ $$T^{2} + 2T - 4$$
$79$ $$T^{2} + 12T + 16$$
$83$ $$T^{2} - 4T - 76$$
$89$ $$T^{2} + 12T + 16$$
$97$ $$T^{2} + 27T + 181$$