# Properties

 Label 1575.2.b.a Level $1575$ Weight $2$ Character orbit 1575.b Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(251,1575)]

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([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 8x^{2} + 9$$ x^4 + 8*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{2} q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - 2) * q^4 + b2 * q^7 + (b3 - 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{2} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + (\beta_{3} + \beta_1) q^{11} + (\beta_{3} - 2 \beta_1) q^{14} + ( - 2 \beta_{2} + 3) q^{16} + ( - \beta_{2} - 5) q^{22} + (\beta_{3} - 3 \beta_1) q^{23} + ( - 2 \beta_{2} + 7) q^{28} + ( - \beta_{3} - 3 \beta_1) q^{29} + 3 \beta_1 q^{32} - 4 \beta_{2} q^{37} + 2 \beta_{2} q^{43} + (\beta_{3} - \beta_1) q^{44} + ( - 5 \beta_{2} + 11) q^{46} + 7 q^{49} + ( - \beta_{3} + 5 \beta_1) q^{53} + 7 \beta_1 q^{56} + ( - \beta_{2} + 13) q^{58} + ( - \beta_{2} - 6) q^{64} + 4 q^{67} + (3 \beta_{3} - \beta_1) q^{71} + ( - 4 \beta_{3} + 8 \beta_1) q^{74} + (3 \beta_{3} + \beta_1) q^{77} + 8 q^{79} + (2 \beta_{3} - 4 \beta_1) q^{86} + ( - 5 \beta_{2} - 7) q^{88} + ( - 3 \beta_{3} + 15 \beta_1) q^{92} + 7 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - 2) * q^4 + b2 * q^7 + (b3 - 2*b1) * q^8 + (b3 + b1) * q^11 + (b3 - 2*b1) * q^14 + (-2*b2 + 3) * q^16 + (-b2 - 5) * q^22 + (b3 - 3*b1) * q^23 + (-2*b2 + 7) * q^28 + (-b3 - 3*b1) * q^29 + 3*b1 * q^32 - 4*b2 * q^37 + 2*b2 * q^43 + (b3 - b1) * q^44 + (-5*b2 + 11) * q^46 + 7 * q^49 + (-b3 + 5*b1) * q^53 + 7*b1 * q^56 + (-b2 + 13) * q^58 + (-b2 - 6) * q^64 + 4 * q^67 + (3*b3 - b1) * q^71 + (-4*b3 + 8*b1) * q^74 + (3*b3 + b1) * q^77 + 8 * q^79 + (2*b3 - 4*b1) * q^86 + (-5*b2 - 7) * q^88 + (-3*b3 + 15*b1) * q^92 + 7*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4}+O(q^{10})$$ 4 * q - 8 * q^4 $$4 q - 8 q^{4} + 12 q^{16} - 20 q^{22} + 28 q^{28} + 44 q^{46} + 28 q^{49} + 52 q^{58} - 24 q^{64} + 16 q^{67} + 32 q^{79} - 28 q^{88}+O(q^{100})$$ 4 * q - 8 * q^4 + 12 * q^16 - 20 * q^22 + 28 * q^28 + 44 * q^46 + 28 * q^49 + 52 * q^58 - 24 * q^64 + 16 * q^67 + 32 * q^79 - 28 * q^88

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6\beta_1$$ b3 - 6*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 − 2.57794i − 1.16372i 1.16372i 2.57794i
2.57794i 0 −4.64575 0 0 −2.64575 6.82058i 0 0
251.2 1.16372i 0 0.645751 0 0 2.64575 3.07892i 0 0
251.3 1.16372i 0 0.645751 0 0 2.64575 3.07892i 0 0
251.4 2.57794i 0 −4.64575 0 0 −2.64575 6.82058i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.b.a 4
3.b odd 2 1 inner 1575.2.b.a 4
5.b even 2 1 63.2.c.a 4
5.c odd 4 2 1575.2.g.d 8
7.b odd 2 1 CM 1575.2.b.a 4
15.d odd 2 1 63.2.c.a 4
15.e even 4 2 1575.2.g.d 8
20.d odd 2 1 1008.2.k.a 4
21.c even 2 1 inner 1575.2.b.a 4
35.c odd 2 1 63.2.c.a 4
35.f even 4 2 1575.2.g.d 8
35.i odd 6 2 441.2.p.b 8
35.j even 6 2 441.2.p.b 8
40.e odd 2 1 4032.2.k.b 4
40.f even 2 1 4032.2.k.c 4
45.h odd 6 2 567.2.o.f 8
45.j even 6 2 567.2.o.f 8
60.h even 2 1 1008.2.k.a 4
105.g even 2 1 63.2.c.a 4
105.k odd 4 2 1575.2.g.d 8
105.o odd 6 2 441.2.p.b 8
105.p even 6 2 441.2.p.b 8
120.i odd 2 1 4032.2.k.c 4
120.m even 2 1 4032.2.k.b 4
140.c even 2 1 1008.2.k.a 4
280.c odd 2 1 4032.2.k.c 4
280.n even 2 1 4032.2.k.b 4
315.z even 6 2 567.2.o.f 8
315.bg odd 6 2 567.2.o.f 8
420.o odd 2 1 1008.2.k.a 4
840.b odd 2 1 4032.2.k.b 4
840.u even 2 1 4032.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 5.b even 2 1
63.2.c.a 4 15.d odd 2 1
63.2.c.a 4 35.c odd 2 1
63.2.c.a 4 105.g even 2 1
441.2.p.b 8 35.i odd 6 2
441.2.p.b 8 35.j even 6 2
441.2.p.b 8 105.o odd 6 2
441.2.p.b 8 105.p even 6 2
567.2.o.f 8 45.h odd 6 2
567.2.o.f 8 45.j even 6 2
567.2.o.f 8 315.z even 6 2
567.2.o.f 8 315.bg odd 6 2
1008.2.k.a 4 20.d odd 2 1
1008.2.k.a 4 60.h even 2 1
1008.2.k.a 4 140.c even 2 1
1008.2.k.a 4 420.o odd 2 1
1575.2.b.a 4 1.a even 1 1 trivial
1575.2.b.a 4 3.b odd 2 1 inner
1575.2.b.a 4 7.b odd 2 1 CM
1575.2.b.a 4 21.c even 2 1 inner
1575.2.g.d 8 5.c odd 4 2
1575.2.g.d 8 15.e even 4 2
1575.2.g.d 8 35.f even 4 2
1575.2.g.d 8 105.k odd 4 2
4032.2.k.b 4 40.e odd 2 1
4032.2.k.b 4 120.m even 2 1
4032.2.k.b 4 280.n even 2 1
4032.2.k.b 4 840.b odd 2 1
4032.2.k.c 4 40.f even 2 1
4032.2.k.c 4 120.i odd 2 1
4032.2.k.c 4 280.c odd 2 1
4032.2.k.c 4 840.u even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 8T_{2}^{2} + 9$$ T2^4 + 8*T2^2 + 9 $$T_{37}^{2} - 112$$ T37^2 - 112 $$T_{47}$$ T47 $$T_{67} - 4$$ T67 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 8T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} + 44T^{2} + 36$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 92T^{2} + 324$$
$29$ $$T^{4} + 116T^{2} + 2916$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 112)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 28)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 212T^{2} + 36$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$(T - 4)^{4}$$
$71$ $$T^{4} + 284 T^{2} + 12996$$
$73$ $$T^{4}$$
$79$ $$(T - 8)^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$