Properties

 Label 9075.2.a.cd Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 3) q^{8} + q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + q^3 + (b2 + 3) * q^4 + (b1 - 1) * q^6 + (b2 + b1 + 1) * q^7 + (-2*b2 + b1 - 3) * q^8 + q^9 $$q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{2} + 3) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - 2 \beta_{2} + \beta_1 - 3) q^{8} + q^{9} + (\beta_{2} + 3) q^{12} + (\beta_{2} + 2) q^{13} + ( - \beta_{2} + 2 \beta_1 + 1) q^{14} + (3 \beta_{2} - 2 \beta_1 + 5) q^{16} + (2 \beta_{2} + \beta_1 - 1) q^{17} + (\beta_1 - 1) q^{18} + (\beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} + \beta_1 + 1) q^{21} + ( - \beta_{2} + 2 \beta_1 - 1) q^{23} + ( - 2 \beta_{2} + \beta_1 - 3) q^{24} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{26} + q^{27} + (2 \beta_{2} + \beta_1 + 7) q^{28} - 2 \beta_1 q^{29} + (\beta_{2} + 6) q^{31} + ( - 4 \beta_{2} + \beta_1 - 13) q^{32} + ( - 3 \beta_{2} + 1) q^{34} + (\beta_{2} + 3) q^{36} + (3 \beta_{2} + 1) q^{37} + ( - 3 \beta_{2} - 7) q^{38} + (\beta_{2} + 2) q^{39} + ( - \beta_1 - 1) q^{41} + ( - \beta_{2} + 2 \beta_1 + 1) q^{42} + ( - 3 \beta_{2} + 2 \beta_1 + 4) q^{43} + (4 \beta_{2} + \beta_1 + 11) q^{46} + ( - \beta_{2} + 3) q^{47} + (3 \beta_{2} - 2 \beta_1 + 5) q^{48} + 2 \beta_1 q^{49} + (2 \beta_{2} + \beta_1 - 1) q^{51} + (4 \beta_{2} - 2 \beta_1 + 12) q^{52} + (4 \beta_{2} + 4 \beta_1 + 2) q^{53} + (\beta_1 - 1) q^{54} + ( - \beta_{2} + 4 \beta_1 - 9) q^{56} + (\beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{58} + (\beta_{2} - 2 \beta_1 - 1) q^{59} + (\beta_{2} + 4 \beta_1) q^{61} + ( - 2 \beta_{2} + 6 \beta_1 - 8) q^{62} + (\beta_{2} + \beta_1 + 1) q^{63} + (3 \beta_{2} - 8 \beta_1 + 15) q^{64} + (\beta_{2} + 6 \beta_1 - 4) q^{67} + (2 \beta_{2} - \beta_1 + 7) q^{68} + ( - \beta_{2} + 2 \beta_1 - 1) q^{69} + (\beta_{2} + 9) q^{71} + ( - 2 \beta_{2} + \beta_1 - 3) q^{72} + (2 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - 6 \beta_{2} + \beta_1 - 7) q^{74} + (4 \beta_{2} - 5 \beta_1 + 11) q^{76} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{78} + (2 \beta_{2} - \beta_1 - 1) q^{79} + q^{81} + ( - \beta_{2} - 2 \beta_1 - 3) q^{82} + ( - 6 \beta_1 + 4) q^{83} + (2 \beta_{2} + \beta_1 + 7) q^{84} + (8 \beta_{2} + 6 \beta_1 + 10) q^{86} - 2 \beta_1 q^{87} + ( - 4 \beta_1 + 2) q^{89} + (\beta_{2} + 6) q^{91} + ( - 5 \beta_{2} + 8 \beta_1 - 13) q^{92} + (\beta_{2} + 6) q^{93} + (2 \beta_{2} + 3 \beta_1 - 1) q^{94} + ( - 4 \beta_{2} + \beta_1 - 13) q^{96} + (2 \beta_{2} + 6 \beta_1 - 11) q^{97} + (2 \beta_{2} + 2 \beta_1 + 8) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + q^3 + (b2 + 3) * q^4 + (b1 - 1) * q^6 + (b2 + b1 + 1) * q^7 + (-2*b2 + b1 - 3) * q^8 + q^9 + (b2 + 3) * q^12 + (b2 + 2) * q^13 + (-b2 + 2*b1 + 1) * q^14 + (3*b2 - 2*b1 + 5) * q^16 + (2*b2 + b1 - 1) * q^17 + (b1 - 1) * q^18 + (b2 - b1 + 1) * q^19 + (b2 + b1 + 1) * q^21 + (-b2 + 2*b1 - 1) * q^23 + (-2*b2 + b1 - 3) * q^24 + (-2*b2 + 2*b1 - 4) * q^26 + q^27 + (2*b2 + b1 + 7) * q^28 - 2*b1 * q^29 + (b2 + 6) * q^31 + (-4*b2 + b1 - 13) * q^32 + (-3*b2 + 1) * q^34 + (b2 + 3) * q^36 + (3*b2 + 1) * q^37 + (-3*b2 - 7) * q^38 + (b2 + 2) * q^39 + (-b1 - 1) * q^41 + (-b2 + 2*b1 + 1) * q^42 + (-3*b2 + 2*b1 + 4) * q^43 + (4*b2 + b1 + 11) * q^46 + (-b2 + 3) * q^47 + (3*b2 - 2*b1 + 5) * q^48 + 2*b1 * q^49 + (2*b2 + b1 - 1) * q^51 + (4*b2 - 2*b1 + 12) * q^52 + (4*b2 + 4*b1 + 2) * q^53 + (b1 - 1) * q^54 + (-b2 + 4*b1 - 9) * q^56 + (b2 - b1 + 1) * q^57 + (-2*b2 - 2*b1 - 8) * q^58 + (b2 - 2*b1 - 1) * q^59 + (b2 + 4*b1) * q^61 + (-2*b2 + 6*b1 - 8) * q^62 + (b2 + b1 + 1) * q^63 + (3*b2 - 8*b1 + 15) * q^64 + (b2 + 6*b1 - 4) * q^67 + (2*b2 - b1 + 7) * q^68 + (-b2 + 2*b1 - 1) * q^69 + (b2 + 9) * q^71 + (-2*b2 + b1 - 3) * q^72 + (2*b2 - 2*b1 - 2) * q^73 + (-6*b2 + b1 - 7) * q^74 + (4*b2 - 5*b1 + 11) * q^76 + (-2*b2 + 2*b1 - 4) * q^78 + (2*b2 - b1 - 1) * q^79 + q^81 + (-b2 - 2*b1 - 3) * q^82 + (-6*b1 + 4) * q^83 + (2*b2 + b1 + 7) * q^84 + (8*b2 + 6*b1 + 10) * q^86 - 2*b1 * q^87 + (-4*b1 + 2) * q^89 + (b2 + 6) * q^91 + (-5*b2 + 8*b1 - 13) * q^92 + (b2 + 6) * q^93 + (2*b2 + 3*b1 - 1) * q^94 + (-4*b2 + b1 - 13) * q^96 + (2*b2 + 6*b1 - 11) * q^97 + (2*b2 + 2*b1 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 $$3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} + 3 q^{27} + 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 8 q^{36} - 18 q^{38} + 5 q^{39} - 4 q^{41} + 6 q^{42} + 17 q^{43} + 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} - 4 q^{51} + 30 q^{52} + 6 q^{53} - 2 q^{54} - 22 q^{56} + q^{57} - 24 q^{58} - 6 q^{59} + 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} - 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} - 14 q^{74} + 24 q^{76} - 8 q^{78} - 6 q^{79} + 3 q^{81} - 10 q^{82} + 6 q^{83} + 20 q^{84} + 28 q^{86} - 2 q^{87} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} - 2 q^{94} - 34 q^{96} - 29 q^{97} + 24 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 8 * q^12 + 5 * q^13 + 6 * q^14 + 10 * q^16 - 4 * q^17 - 2 * q^18 + q^19 + 3 * q^21 - 6 * q^24 - 8 * q^26 + 3 * q^27 + 20 * q^28 - 2 * q^29 + 17 * q^31 - 34 * q^32 + 6 * q^34 + 8 * q^36 - 18 * q^38 + 5 * q^39 - 4 * q^41 + 6 * q^42 + 17 * q^43 + 30 * q^46 + 10 * q^47 + 10 * q^48 + 2 * q^49 - 4 * q^51 + 30 * q^52 + 6 * q^53 - 2 * q^54 - 22 * q^56 + q^57 - 24 * q^58 - 6 * q^59 + 3 * q^61 - 16 * q^62 + 3 * q^63 + 34 * q^64 - 7 * q^67 + 18 * q^68 + 26 * q^71 - 6 * q^72 - 10 * q^73 - 14 * q^74 + 24 * q^76 - 8 * q^78 - 6 * q^79 + 3 * q^81 - 10 * q^82 + 6 * q^83 + 20 * q^84 + 28 * q^86 - 2 * q^87 + 2 * q^89 + 17 * q^91 - 26 * q^92 + 17 * q^93 - 2 * q^94 - 34 * q^96 - 29 * q^97 + 24 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

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
 −1.76156 −0.363328 3.12489
−2.76156 1.00000 5.62620 0 −2.76156 1.86464 −10.0140 1.00000 0
1.2 −1.36333 1.00000 −0.141336 0 −1.36333 −2.50466 2.91934 1.00000 0
1.3 2.12489 1.00000 2.51514 0 2.12489 3.64002 1.09461 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

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 9075.2.a.cd 3
5.b even 2 1 9075.2.a.cj 3
11.b odd 2 1 825.2.a.m yes 3
33.d even 2 1 2475.2.a.z 3
55.d odd 2 1 825.2.a.i 3
55.e even 4 2 825.2.c.f 6
165.d even 2 1 2475.2.a.bd 3
165.l odd 4 2 2475.2.c.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.i 3 55.d odd 2 1
825.2.a.m yes 3 11.b odd 2 1
825.2.c.f 6 55.e even 4 2
2475.2.a.z 3 33.d even 2 1
2475.2.a.bd 3 165.d even 2 1
2475.2.c.q 6 165.l odd 4 2
9075.2.a.cd 3 1.a even 1 1 trivial
9075.2.a.cj 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 5T_{2} - 8$$ T2^3 + 2*T2^2 - 5*T2 - 8 $$T_{7}^{3} - 3T_{7}^{2} - 7T_{7} + 17$$ T7^3 - 3*T7^2 - 7*T7 + 17 $$T_{13}^{3} - 5T_{13}^{2} + 8$$ T13^3 - 5*T13^2 + 8 $$T_{17}^{3} + 4T_{17}^{2} - 25T_{17} + 22$$ T17^3 + 4*T17^2 - 25*T17 + 22 $$T_{19}^{3} - T_{19}^{2} - 19T_{19} - 25$$ T19^3 - T19^2 - 19*T19 - 25 $$T_{23}^{3} - 43T_{23} + 58$$ T23^3 - 43*T23 + 58 $$T_{37}^{3} - 75T_{37} - 34$$ T37^3 - 75*T37 - 34

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 3 T^{2} + \cdots + 17$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 5T^{2} + 8$$
$17$ $$T^{3} + 4 T^{2} + \cdots + 22$$
$19$ $$T^{3} - T^{2} + \cdots - 25$$
$23$ $$T^{3} - 43T + 58$$
$29$ $$T^{3} + 2 T^{2} + \cdots + 16$$
$31$ $$T^{3} - 17 T^{2} + \cdots - 136$$
$37$ $$T^{3} - 75T - 34$$
$41$ $$T^{3} + 4T^{2} - T - 2$$
$43$ $$T^{3} - 17 T^{2} + \cdots + 1100$$
$47$ $$T^{3} - 10 T^{2} + \cdots - 8$$
$53$ $$T^{3} - 6 T^{2} + \cdots + 824$$
$59$ $$T^{3} + 6 T^{2} + \cdots - 136$$
$61$ $$T^{3} - 3 T^{2} + \cdots - 244$$
$67$ $$T^{3} + 7 T^{2} + \cdots - 1588$$
$71$ $$T^{3} - 26 T^{2} + \cdots - 580$$
$73$ $$T^{3} + 10 T^{2} + \cdots - 472$$
$79$ $$T^{3} + 6 T^{2} + \cdots - 212$$
$83$ $$T^{3} - 6 T^{2} + \cdots + 1328$$
$89$ $$T^{3} - 2 T^{2} + \cdots + 328$$
$97$ $$T^{3} + 29 T^{2} + \cdots - 2153$$