# Properties

 Label 1575.2.a.x Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $0$ Dimension $3$ 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] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.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: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + q^{7} + (\beta_{2} + 2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1 + 1) * q^4 + q^7 + (b2 + 2*b1 + 2) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + q^{7} + (\beta_{2} + 2 \beta_1 + 2) q^{8} - 2 q^{11} + (\beta_{2} - \beta_1 + 2) q^{13} + \beta_1 q^{14} + (4 \beta_1 + 3) q^{16} + (\beta_{2} - \beta_1) q^{17} + ( - \beta_{2} + \beta_1 + 2) q^{19} - 2 \beta_1 q^{22} + ( - \beta_{2} - \beta_1 + 2) q^{23} + ( - \beta_{2} + 3 \beta_1 - 4) q^{26} + (\beta_{2} + \beta_1 + 1) q^{28} + (2 \beta_{2} + 2 \beta_1 - 2) q^{29} + ( - \beta_{2} - 3 \beta_1 + 2) q^{31} + (2 \beta_{2} + 3 \beta_1 + 8) q^{32} + ( - \beta_{2} + \beta_1 - 4) q^{34} + 4 \beta_1 q^{37} + (\beta_{2} + \beta_1 + 4) q^{38} + (\beta_{2} - 3 \beta_1) q^{41} - 4 \beta_{2} q^{43} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{44} + ( - \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{47} + q^{49} + (\beta_{2} - \beta_1 + 6) q^{52} + ( - \beta_{2} - 3 \beta_1 + 6) q^{53} + (\beta_{2} + 2 \beta_1 + 2) q^{56} + (2 \beta_{2} + 4 \beta_1 + 4) q^{58} + ( - 4 \beta_{2} - 4) q^{59} + (2 \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - 3 \beta_{2} - 3 \beta_1 - 8) q^{62} + (3 \beta_{2} + 7 \beta_1 + 1) q^{64} + (2 \beta_{2} + 2 \beta_1 - 4) q^{67} + ( - \beta_{2} - 3 \beta_1 + 4) q^{68} - 2 q^{71} + ( - \beta_{2} + \beta_1 + 6) q^{73} + (4 \beta_{2} + 4 \beta_1 + 12) q^{74} + (3 \beta_{2} + 5 \beta_1 - 2) q^{76} - 2 q^{77} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{79} + ( - 3 \beta_{2} - \beta_1 - 10) q^{82} + ( - 4 \beta_1 + 4) q^{83} + ( - 8 \beta_1 + 4) q^{86} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{88} + ( - \beta_{2} - \beta_1 - 4) q^{89} + (\beta_{2} - \beta_1 + 2) q^{91} + (\beta_{2} - 3 \beta_1 - 6) q^{92} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{94} + ( - 3 \beta_{2} - 5 \beta_1 + 10) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1 + 1) * q^4 + q^7 + (b2 + 2*b1 + 2) * q^8 - 2 * q^11 + (b2 - b1 + 2) * q^13 + b1 * q^14 + (4*b1 + 3) * q^16 + (b2 - b1) * q^17 + (-b2 + b1 + 2) * q^19 - 2*b1 * q^22 + (-b2 - b1 + 2) * q^23 + (-b2 + 3*b1 - 4) * q^26 + (b2 + b1 + 1) * q^28 + (2*b2 + 2*b1 - 2) * q^29 + (-b2 - 3*b1 + 2) * q^31 + (2*b2 + 3*b1 + 8) * q^32 + (-b2 + b1 - 4) * q^34 + 4*b1 * q^37 + (b2 + b1 + 4) * q^38 + (b2 - 3*b1) * q^41 - 4*b2 * q^43 + (-2*b2 - 2*b1 - 2) * q^44 + (-b2 - b1 - 2) * q^46 + (-2*b2 - 2*b1 + 4) * q^47 + q^49 + (b2 - b1 + 6) * q^52 + (-b2 - 3*b1 + 6) * q^53 + (b2 + 2*b1 + 2) * q^56 + (2*b2 + 4*b1 + 4) * q^58 + (-4*b2 - 4) * q^59 + (2*b2 - 2*b1 - 2) * q^61 + (-3*b2 - 3*b1 - 8) * q^62 + (3*b2 + 7*b1 + 1) * q^64 + (2*b2 + 2*b1 - 4) * q^67 + (-b2 - 3*b1 + 4) * q^68 - 2 * q^71 + (-b2 + b1 + 6) * q^73 + (4*b2 + 4*b1 + 12) * q^74 + (3*b2 + 5*b1 - 2) * q^76 - 2 * q^77 + (-2*b2 + 2*b1 + 4) * q^79 + (-3*b2 - b1 - 10) * q^82 + (-4*b1 + 4) * q^83 + (-8*b1 + 4) * q^86 + (-2*b2 - 4*b1 - 4) * q^88 + (-b2 - b1 - 4) * q^89 + (b2 - b1 + 2) * q^91 + (b2 - 3*b1 - 6) * q^92 + (-2*b2 - 2*b1 - 4) * q^94 + (-3*b2 - 5*b1 + 10) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 5 q^{4} + 3 q^{7} + 9 q^{8}+O(q^{10})$$ 3 * q + q^2 + 5 * q^4 + 3 * q^7 + 9 * q^8 $$3 q + q^{2} + 5 q^{4} + 3 q^{7} + 9 q^{8} - 6 q^{11} + 6 q^{13} + q^{14} + 13 q^{16} + 6 q^{19} - 2 q^{22} + 4 q^{23} - 10 q^{26} + 5 q^{28} - 2 q^{29} + 2 q^{31} + 29 q^{32} - 12 q^{34} + 4 q^{37} + 14 q^{38} - 2 q^{41} - 4 q^{43} - 10 q^{44} - 8 q^{46} + 8 q^{47} + 3 q^{49} + 18 q^{52} + 14 q^{53} + 9 q^{56} + 18 q^{58} - 16 q^{59} - 6 q^{61} - 30 q^{62} + 13 q^{64} - 8 q^{67} + 8 q^{68} - 6 q^{71} + 18 q^{73} + 44 q^{74} + 2 q^{76} - 6 q^{77} + 12 q^{79} - 34 q^{82} + 8 q^{83} + 4 q^{86} - 18 q^{88} - 14 q^{89} + 6 q^{91} - 20 q^{92} - 16 q^{94} + 22 q^{97} + q^{98}+O(q^{100})$$ 3 * q + q^2 + 5 * q^4 + 3 * q^7 + 9 * q^8 - 6 * q^11 + 6 * q^13 + q^14 + 13 * q^16 + 6 * q^19 - 2 * q^22 + 4 * q^23 - 10 * q^26 + 5 * q^28 - 2 * q^29 + 2 * q^31 + 29 * q^32 - 12 * q^34 + 4 * q^37 + 14 * q^38 - 2 * q^41 - 4 * q^43 - 10 * q^44 - 8 * q^46 + 8 * q^47 + 3 * q^49 + 18 * q^52 + 14 * q^53 + 9 * q^56 + 18 * q^58 - 16 * q^59 - 6 * q^61 - 30 * q^62 + 13 * q^64 - 8 * q^67 + 8 * q^68 - 6 * q^71 + 18 * q^73 + 44 * q^74 + 2 * q^76 - 6 * q^77 + 12 * q^79 - 34 * q^82 + 8 * q^83 + 4 * q^86 - 18 * q^88 - 14 * q^89 + 6 * q^91 - 20 * q^92 - 16 * q^94 + 22 * q^97 + q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

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

## 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
 0.311108 −1.48119 2.17009
−1.90321 0 1.62222 0 0 1.00000 0.719004 0 0
1.2 0.193937 0 −1.96239 0 0 1.00000 −0.768452 0 0
1.3 2.70928 0 5.34017 0 0 1.00000 9.04945 0 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 1575.2.a.x 3
3.b odd 2 1 525.2.a.j 3
5.b even 2 1 1575.2.a.w 3
5.c odd 4 2 315.2.d.e 6
12.b even 2 1 8400.2.a.dg 3
15.d odd 2 1 525.2.a.k 3
15.e even 4 2 105.2.d.b 6
20.e even 4 2 5040.2.t.v 6
21.c even 2 1 3675.2.a.bi 3
35.f even 4 2 2205.2.d.l 6
60.h even 2 1 8400.2.a.dj 3
60.l odd 4 2 1680.2.t.k 6
105.g even 2 1 3675.2.a.bj 3
105.k odd 4 2 735.2.d.b 6
105.w odd 12 4 735.2.q.f 12
105.x even 12 4 735.2.q.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 15.e even 4 2
315.2.d.e 6 5.c odd 4 2
525.2.a.j 3 3.b odd 2 1
525.2.a.k 3 15.d odd 2 1
735.2.d.b 6 105.k odd 4 2
735.2.q.e 12 105.x even 12 4
735.2.q.f 12 105.w odd 12 4
1575.2.a.w 3 5.b even 2 1
1575.2.a.x 3 1.a even 1 1 trivial
1680.2.t.k 6 60.l odd 4 2
2205.2.d.l 6 35.f even 4 2
3675.2.a.bi 3 21.c even 2 1
3675.2.a.bj 3 105.g even 2 1
5040.2.t.v 6 20.e even 4 2
8400.2.a.dg 3 12.b even 2 1
8400.2.a.dj 3 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1$$ T2^3 - T2^2 - 5*T2 + 1 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{3} - 6T_{13}^{2} - 4T_{13} + 8$$ T13^3 - 6*T13^2 - 4*T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 5T + 1$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$(T - 1)^{3}$$
$11$ $$(T + 2)^{3}$$
$13$ $$T^{3} - 6 T^{2} + \cdots + 8$$
$17$ $$T^{3} - 16T - 16$$
$19$ $$T^{3} - 6 T^{2} + \cdots + 40$$
$23$ $$T^{3} - 4 T^{2} + \cdots + 16$$
$29$ $$T^{3} + 2 T^{2} + \cdots - 40$$
$31$ $$T^{3} - 2 T^{2} + \cdots + 184$$
$37$ $$T^{3} - 4 T^{2} + \cdots + 64$$
$41$ $$T^{3} + 2 T^{2} + \cdots - 200$$
$43$ $$T^{3} + 4 T^{2} + \cdots - 832$$
$47$ $$T^{3} - 8 T^{2} + \cdots + 128$$
$53$ $$T^{3} - 14 T^{2} + \cdots + 296$$
$59$ $$T^{3} + 16 T^{2} + \cdots - 1280$$
$61$ $$T^{3} + 6 T^{2} + \cdots - 248$$
$67$ $$T^{3} + 8 T^{2} + \cdots - 128$$
$71$ $$(T + 2)^{3}$$
$73$ $$T^{3} - 18 T^{2} + \cdots - 104$$
$79$ $$T^{3} - 12 T^{2} + \cdots + 320$$
$83$ $$T^{3} - 8 T^{2} + \cdots + 256$$
$89$ $$T^{3} + 14 T^{2} + \cdots + 40$$
$97$ $$T^{3} - 22 T^{2} + \cdots + 1864$$
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