Properties

 Label 1575.2.a.z Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ Inner twists $2$

Related objects

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: $$4$$ Coefficient field: 4.4.174928.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 13$$ x^4 - 9*x^2 + 13 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 3) * q^4 + q^7 + (b3 + 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + \beta_{3} q^{11} + ( - 2 \beta_{2} - 2) q^{13} + \beta_1 q^{14} + (3 \beta_{2} + 6) q^{16} + 2 \beta_1 q^{17} + ( - 2 \beta_{2} + 2) q^{19} + (3 \beta_{2} + 2) q^{22} - \beta_{3} q^{23} + ( - 2 \beta_{3} - 4 \beta_1) q^{26} + (\beta_{2} + 3) q^{28} + ( - \beta_{3} + 2 \beta_1) q^{29} + 6 q^{31} + (\beta_{3} + 5 \beta_1) q^{32} + (2 \beta_{2} + 10) q^{34} - 3 q^{37} - 2 \beta_{3} q^{38} - 4 \beta_1 q^{41} + (2 \beta_{2} + 5) q^{43} + (\beta_{3} + 5 \beta_1) q^{44} + ( - 3 \beta_{2} - 2) q^{46} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + q^{49} + ( - 6 \beta_{2} - 20) q^{52} - 4 \beta_1 q^{53} + (\beta_{3} + 2 \beta_1) q^{56} + ( - \beta_{2} + 8) q^{58} + (2 \beta_{3} + 2 \beta_1) q^{59} + 6 \beta_1 q^{62} + (2 \beta_{2} + 15) q^{64} + (2 \beta_{2} + 5) q^{67} + (2 \beta_{3} + 8 \beta_1) q^{68} + ( - \beta_{3} - 4 \beta_1) q^{71} - 2 \beta_{2} q^{73} - 3 \beta_1 q^{74} + ( - 2 \beta_{2} - 8) q^{76} + \beta_{3} q^{77} + ( - 2 \beta_{2} - 1) q^{79} + ( - 4 \beta_{2} - 20) q^{82} + (2 \beta_{3} - 4 \beta_1) q^{83} + (2 \beta_{3} + 7 \beta_1) q^{86} + (2 \beta_{2} + 23) q^{88} + ( - 2 \beta_{3} + 2 \beta_1) q^{89} + ( - 2 \beta_{2} - 2) q^{91} + ( - \beta_{3} - 5 \beta_1) q^{92} + ( - 8 \beta_{2} - 14) q^{94} + (4 \beta_{2} + 2) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 3) * q^4 + q^7 + (b3 + 2*b1) * q^8 + b3 * q^11 + (-2*b2 - 2) * q^13 + b1 * q^14 + (3*b2 + 6) * q^16 + 2*b1 * q^17 + (-2*b2 + 2) * q^19 + (3*b2 + 2) * q^22 - b3 * q^23 + (-2*b3 - 4*b1) * q^26 + (b2 + 3) * q^28 + (-b3 + 2*b1) * q^29 + 6 * q^31 + (b3 + 5*b1) * q^32 + (2*b2 + 10) * q^34 - 3 * q^37 - 2*b3 * q^38 - 4*b1 * q^41 + (2*b2 + 5) * q^43 + (b3 + 5*b1) * q^44 + (-3*b2 - 2) * q^46 + (-2*b3 - 2*b1) * q^47 + q^49 + (-6*b2 - 20) * q^52 - 4*b1 * q^53 + (b3 + 2*b1) * q^56 + (-b2 + 8) * q^58 + (2*b3 + 2*b1) * q^59 + 6*b1 * q^62 + (2*b2 + 15) * q^64 + (2*b2 + 5) * q^67 + (2*b3 + 8*b1) * q^68 + (-b3 - 4*b1) * q^71 - 2*b2 * q^73 - 3*b1 * q^74 + (-2*b2 - 8) * q^76 + b3 * q^77 + (-2*b2 - 1) * q^79 + (-4*b2 - 20) * q^82 + (2*b3 - 4*b1) * q^83 + (2*b3 + 7*b1) * q^86 + (2*b2 + 23) * q^88 + (-2*b3 + 2*b1) * q^89 + (-2*b2 - 2) * q^91 + (-b3 - 5*b1) * q^92 + (-8*b2 - 14) * q^94 + (4*b2 + 2) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{4} + 4 q^{7}+O(q^{10})$$ 4 * q + 10 * q^4 + 4 * q^7 $$4 q + 10 q^{4} + 4 q^{7} - 4 q^{13} + 18 q^{16} + 12 q^{19} + 2 q^{22} + 10 q^{28} + 24 q^{31} + 36 q^{34} - 12 q^{37} + 16 q^{43} - 2 q^{46} + 4 q^{49} - 68 q^{52} + 34 q^{58} + 56 q^{64} + 16 q^{67} + 4 q^{73} - 28 q^{76} - 72 q^{82} + 88 q^{88} - 4 q^{91} - 40 q^{94}+O(q^{100})$$ 4 * q + 10 * q^4 + 4 * q^7 - 4 * q^13 + 18 * q^16 + 12 * q^19 + 2 * q^22 + 10 * q^28 + 24 * q^31 + 36 * q^34 - 12 * q^37 + 16 * q^43 - 2 * q^46 + 4 * q^49 - 68 * q^52 + 34 * q^58 + 56 * q^64 + 16 * q^67 + 4 * q^73 - 28 * q^76 - 72 * q^82 + 88 * q^88 - 4 * q^91 - 40 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 13$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6\nu$$ v^3 - 6*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6\beta_1$$ b3 + 6*b1

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
 −2.68190 −1.34440 1.34440 2.68190
−2.68190 0 5.19258 0 0 1.00000 −8.56218 0 0
1.2 −1.34440 0 −0.192582 0 0 1.00000 2.94771 0 0
1.3 1.34440 0 −0.192582 0 0 1.00000 −2.94771 0 0
1.4 2.68190 0 5.19258 0 0 1.00000 8.56218 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

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.a.z yes 4
3.b odd 2 1 inner 1575.2.a.z yes 4
5.b even 2 1 1575.2.a.y 4
5.c odd 4 2 1575.2.d.l 8
15.d odd 2 1 1575.2.a.y 4
15.e even 4 2 1575.2.d.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.a.y 4 5.b even 2 1
1575.2.a.y 4 15.d odd 2 1
1575.2.a.z yes 4 1.a even 1 1 trivial
1575.2.a.z yes 4 3.b odd 2 1 inner
1575.2.d.l 8 5.c odd 4 2
1575.2.d.l 8 15.e even 4 2

Hecke kernels

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

 $$T_{2}^{4} - 9T_{2}^{2} + 13$$ T2^4 - 9*T2^2 + 13 $$T_{11}^{4} - 42T_{11}^{2} + 325$$ T11^4 - 42*T11^2 + 325 $$T_{13}^{2} + 2T_{13} - 28$$ T13^2 + 2*T13 - 28

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 9T^{2} + 13$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4} - 42T^{2} + 325$$
$13$ $$(T^{2} + 2 T - 28)^{2}$$
$17$ $$T^{4} - 36T^{2} + 208$$
$19$ $$(T^{2} - 6 T - 20)^{2}$$
$23$ $$T^{4} - 42T^{2} + 325$$
$29$ $$T^{4} - 74T^{2} + 325$$
$31$ $$(T - 6)^{4}$$
$37$ $$(T + 3)^{4}$$
$41$ $$T^{4} - 144T^{2} + 3328$$
$43$ $$(T^{2} - 8 T - 13)^{2}$$
$47$ $$T^{4} - 212 T^{2} + 10192$$
$53$ $$T^{4} - 144T^{2} + 3328$$
$59$ $$T^{4} - 212 T^{2} + 10192$$
$61$ $$T^{4}$$
$67$ $$(T^{2} - 8 T - 13)^{2}$$
$71$ $$T^{4} - 194T^{2} + 13$$
$73$ $$(T^{2} - 2 T - 28)^{2}$$
$79$ $$(T^{2} - 29)^{2}$$
$83$ $$T^{4} - 296T^{2} + 5200$$
$89$ $$T^{4} - 196T^{2} + 208$$
$97$ $$(T^{2} - 116)^{2}$$