# Properties

 Label 1575.2.b.d Level $1575$ Weight $2$ Character orbit 1575.b Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{2} q^{7} - 2 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - b2 * q^7 - 2*b1 * q^8 $$q - \beta_1 q^{2} - \beta_{2} q^{7} - 2 \beta_1 q^{8} - \beta_1 q^{11} + ( - 2 \beta_{2} + 1) q^{13} + ( - \beta_{3} + \beta_1) q^{14} - 4 q^{16} + ( - 2 \beta_{3} + \beta_1) q^{17} + (2 \beta_{2} - 1) q^{19} - 2 q^{22} + 5 \beta_1 q^{23} + ( - 2 \beta_{3} + \beta_1) q^{26} - \beta_1 q^{29} + ( - 2 \beta_{2} + 1) q^{31} + (4 \beta_{2} - 2) q^{34} + 8 q^{37} + (2 \beta_{3} - \beta_1) q^{38} + 5 q^{43} + 10 q^{46} + ( - 2 \beta_{3} + \beta_1) q^{47} + (\beta_{2} - 7) q^{49} - 4 \beta_1 q^{53} + ( - 2 \beta_{3} + 2 \beta_1) q^{56} - 2 q^{58} + ( - 2 \beta_{3} + \beta_1) q^{59} + (2 \beta_{2} - 1) q^{61} + ( - 2 \beta_{3} + \beta_1) q^{62} - 8 q^{64} - 7 q^{67} + 2 \beta_1 q^{71} + ( - 4 \beta_{2} + 2) q^{73} - 8 \beta_1 q^{74} + ( - \beta_{3} + \beta_1) q^{77} + 14 q^{79} + (2 \beta_{3} - \beta_1) q^{83} - 5 \beta_1 q^{86} - 4 q^{88} + ( - 4 \beta_{3} + 2 \beta_1) q^{89} + (\beta_{2} - 14) q^{91} + (4 \beta_{2} - 2) q^{94} + (2 \beta_{2} - 1) q^{97} + (\beta_{3} + 6 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - b2 * q^7 - 2*b1 * q^8 - b1 * q^11 + (-2*b2 + 1) * q^13 + (-b3 + b1) * q^14 - 4 * q^16 + (-2*b3 + b1) * q^17 + (2*b2 - 1) * q^19 - 2 * q^22 + 5*b1 * q^23 + (-2*b3 + b1) * q^26 - b1 * q^29 + (-2*b2 + 1) * q^31 + (4*b2 - 2) * q^34 + 8 * q^37 + (2*b3 - b1) * q^38 + 5 * q^43 + 10 * q^46 + (-2*b3 + b1) * q^47 + (b2 - 7) * q^49 - 4*b1 * q^53 + (-2*b3 + 2*b1) * q^56 - 2 * q^58 + (-2*b3 + b1) * q^59 + (2*b2 - 1) * q^61 + (-2*b3 + b1) * q^62 - 8 * q^64 - 7 * q^67 + 2*b1 * q^71 + (-4*b2 + 2) * q^73 - 8*b1 * q^74 + (-b3 + b1) * q^77 + 14 * q^79 + (2*b3 - b1) * q^83 - 5*b1 * q^86 - 4 * q^88 + (-4*b3 + 2*b1) * q^89 + (b2 - 14) * q^91 + (4*b2 - 2) * q^94 + (2*b2 - 1) * q^97 + (b3 + 6*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^7 $$4 q - 2 q^{7} - 16 q^{16} - 8 q^{22} + 32 q^{37} + 20 q^{43} + 40 q^{46} - 26 q^{49} - 8 q^{58} - 32 q^{64} - 28 q^{67} + 56 q^{79} - 16 q^{88} - 54 q^{91}+O(q^{100})$$ 4 * q - 2 * q^7 - 16 * q^16 - 8 * q^22 + 32 * q^37 + 20 * q^43 + 40 * q^46 - 26 * q^49 - 8 * q^58 - 32 * q^64 - 28 * q^67 + 56 * q^79 - 16 * q^88 - 54 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{2}$$ $$=$$ $$( 3\nu^{2} - 2 ) / 2$$ (3*v^2 - 2) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 6\nu ) / 2$$ (-v^3 + 6*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 3$$ (b3 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + 2 ) / 3$$ (2*b2 + 2) / 3 $$\nu^{3}$$ $$=$$ $$2\beta_1$$ 2*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
 1.22474 + 0.707107i −1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i
1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.2 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 0 0
251.3 1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.4 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 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
3.b odd 2 1 inner
7.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.d 4
3.b odd 2 1 inner 1575.2.b.d 4
5.b even 2 1 1575.2.b.e yes 4
5.c odd 4 2 1575.2.g.b 8
7.b odd 2 1 inner 1575.2.b.d 4
15.d odd 2 1 1575.2.b.e yes 4
15.e even 4 2 1575.2.g.b 8
21.c even 2 1 inner 1575.2.b.d 4
35.c odd 2 1 1575.2.b.e yes 4
35.f even 4 2 1575.2.g.b 8
105.g even 2 1 1575.2.b.e yes 4
105.k odd 4 2 1575.2.g.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.b.d 4 1.a even 1 1 trivial
1575.2.b.d 4 3.b odd 2 1 inner
1575.2.b.d 4 7.b odd 2 1 inner
1575.2.b.d 4 21.c even 2 1 inner
1575.2.b.e yes 4 5.b even 2 1
1575.2.b.e yes 4 15.d odd 2 1
1575.2.b.e yes 4 35.c odd 2 1
1575.2.b.e yes 4 105.g even 2 1
1575.2.g.b 8 5.c odd 4 2
1575.2.g.b 8 15.e even 4 2
1575.2.g.b 8 35.f even 4 2
1575.2.g.b 8 105.k odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} + 2$$ T2^2 + 2 $$T_{37} - 8$$ T37 - 8 $$T_{47}^{2} - 54$$ T47^2 - 54 $$T_{67} + 7$$ T67 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + T + 7)^{2}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$(T^{2} + 27)^{2}$$
$17$ $$(T^{2} - 54)^{2}$$
$19$ $$(T^{2} + 27)^{2}$$
$23$ $$(T^{2} + 50)^{2}$$
$29$ $$(T^{2} + 2)^{2}$$
$31$ $$(T^{2} + 27)^{2}$$
$37$ $$(T - 8)^{4}$$
$41$ $$T^{4}$$
$43$ $$(T - 5)^{4}$$
$47$ $$(T^{2} - 54)^{2}$$
$53$ $$(T^{2} + 32)^{2}$$
$59$ $$(T^{2} - 54)^{2}$$
$61$ $$(T^{2} + 27)^{2}$$
$67$ $$(T + 7)^{4}$$
$71$ $$(T^{2} + 8)^{2}$$
$73$ $$(T^{2} + 108)^{2}$$
$79$ $$(T - 14)^{4}$$
$83$ $$(T^{2} - 54)^{2}$$
$89$ $$(T^{2} - 216)^{2}$$
$97$ $$(T^{2} + 27)^{2}$$