# Properties

 Label 9075.2.a.bo Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.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: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + q^{4} + \beta q^{6} - 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 + q^3 + q^4 + b * q^6 - 2*b * q^7 - b * q^8 + q^9 $$q + \beta q^{2} + q^{3} + q^{4} + \beta q^{6} - 2 \beta q^{7} - \beta q^{8} + q^{9} + q^{12} - \beta q^{13} - 6 q^{14} - 5 q^{16} + \beta q^{17} + \beta q^{18} - 4 \beta q^{19} - 2 \beta q^{21} + 6 q^{23} - \beta q^{24} - 3 q^{26} + q^{27} - 2 \beta q^{28} + \beta q^{29} + 4 q^{31} - 3 \beta q^{32} + 3 q^{34} + q^{36} + 11 q^{37} - 12 q^{38} - \beta q^{39} + \beta q^{41} - 6 q^{42} + 2 \beta q^{43} + 6 \beta q^{46} - 5 q^{48} + 5 q^{49} + \beta q^{51} - \beta q^{52} + 9 q^{53} + \beta q^{54} + 6 q^{56} - 4 \beta q^{57} + 3 q^{58} - 6 q^{59} + 4 \beta q^{62} - 2 \beta q^{63} + q^{64} + 2 q^{67} + \beta q^{68} + 6 q^{69} - 6 q^{71} - \beta q^{72} + 4 \beta q^{73} + 11 \beta q^{74} - 4 \beta q^{76} - 3 q^{78} + q^{81} + 3 q^{82} - 2 \beta q^{84} + 6 q^{86} + \beta q^{87} + 9 q^{89} + 6 q^{91} + 6 q^{92} + 4 q^{93} - 3 \beta q^{96} + 7 q^{97} + 5 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^3 + q^4 + b * q^6 - 2*b * q^7 - b * q^8 + q^9 + q^12 - b * q^13 - 6 * q^14 - 5 * q^16 + b * q^17 + b * q^18 - 4*b * q^19 - 2*b * q^21 + 6 * q^23 - b * q^24 - 3 * q^26 + q^27 - 2*b * q^28 + b * q^29 + 4 * q^31 - 3*b * q^32 + 3 * q^34 + q^36 + 11 * q^37 - 12 * q^38 - b * q^39 + b * q^41 - 6 * q^42 + 2*b * q^43 + 6*b * q^46 - 5 * q^48 + 5 * q^49 + b * q^51 - b * q^52 + 9 * q^53 + b * q^54 + 6 * q^56 - 4*b * q^57 + 3 * q^58 - 6 * q^59 + 4*b * q^62 - 2*b * q^63 + q^64 + 2 * q^67 + b * q^68 + 6 * q^69 - 6 * q^71 - b * q^72 + 4*b * q^73 + 11*b * q^74 - 4*b * q^76 - 3 * q^78 + q^81 + 3 * q^82 - 2*b * q^84 + 6 * q^86 + b * q^87 + 9 * q^89 + 6 * q^91 + 6 * q^92 + 4 * q^93 - 3*b * q^96 + 7 * q^97 + 5*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} + 2 q^{12} - 12 q^{14} - 10 q^{16} + 12 q^{23} - 6 q^{26} + 2 q^{27} + 8 q^{31} + 6 q^{34} + 2 q^{36} + 22 q^{37} - 24 q^{38} - 12 q^{42} - 10 q^{48} + 10 q^{49} + 18 q^{53} + 12 q^{56} + 6 q^{58} - 12 q^{59} + 2 q^{64} + 4 q^{67} + 12 q^{69} - 12 q^{71} - 6 q^{78} + 2 q^{81} + 6 q^{82} + 12 q^{86} + 18 q^{89} + 12 q^{91} + 12 q^{92} + 8 q^{93} + 14 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^9 + 2 * q^12 - 12 * q^14 - 10 * q^16 + 12 * q^23 - 6 * q^26 + 2 * q^27 + 8 * q^31 + 6 * q^34 + 2 * q^36 + 22 * q^37 - 24 * q^38 - 12 * q^42 - 10 * q^48 + 10 * q^49 + 18 * q^53 + 12 * q^56 + 6 * q^58 - 12 * q^59 + 2 * q^64 + 4 * q^67 + 12 * q^69 - 12 * q^71 - 6 * q^78 + 2 * q^81 + 6 * q^82 + 12 * q^86 + 18 * q^89 + 12 * q^91 + 12 * q^92 + 8 * q^93 + 14 * q^97

## 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
 −1.73205 1.73205
−1.73205 1.00000 1.00000 0 −1.73205 3.46410 1.73205 1.00000 0
1.2 1.73205 1.00000 1.00000 0 1.73205 −3.46410 −1.73205 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.bo 2
5.b even 2 1 363.2.a.f 2
11.b odd 2 1 inner 9075.2.a.bo 2
15.d odd 2 1 1089.2.a.o 2
20.d odd 2 1 5808.2.a.ca 2
55.d odd 2 1 363.2.a.f 2
55.h odd 10 4 363.2.e.m 8
55.j even 10 4 363.2.e.m 8
165.d even 2 1 1089.2.a.o 2
220.g even 2 1 5808.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 5.b even 2 1
363.2.a.f 2 55.d odd 2 1
363.2.e.m 8 55.h odd 10 4
363.2.e.m 8 55.j even 10 4
1089.2.a.o 2 15.d odd 2 1
1089.2.a.o 2 165.d even 2 1
5808.2.a.ca 2 20.d odd 2 1
5808.2.a.ca 2 220.g even 2 1
9075.2.a.bo 2 1.a even 1 1 trivial
9075.2.a.bo 2 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{13}^{2} - 3$$ T13^2 - 3 $$T_{17}^{2} - 3$$ T17^2 - 3 $$T_{19}^{2} - 48$$ T19^2 - 48 $$T_{23} - 6$$ T23 - 6 $$T_{37} - 11$$ T37 - 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3$$
$17$ $$T^{2} - 3$$
$19$ $$T^{2} - 48$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 3$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 11)^{2}$$
$41$ $$T^{2} - 3$$
$43$ $$T^{2} - 12$$
$47$ $$T^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$(T + 6)^{2}$$
$61$ $$T^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 48$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 9)^{2}$$
$97$ $$(T - 7)^{2}$$