# Properties

 Label 175.1.d.b Level $175$ Weight $1$ Character orbit 175.d Self dual yes Analytic conductor $0.087$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -7 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,1,Mod(76,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("175.76");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 175.d (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: yes Analytic conductor: $$0.0873363772108$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.175.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.153125.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^7 - q^8 + q^9 $$q + q^{2} - q^{7} - q^{8} + q^{9} - q^{11} - q^{14} - q^{16} + q^{18} - q^{22} + q^{23} - q^{29} + q^{37} + q^{43} + q^{46} + q^{49} - 2 q^{53} + q^{56} - q^{58} - q^{63} + q^{64} + q^{67} - q^{71} - q^{72} + q^{74} + q^{77} - q^{79} + q^{81} + q^{86} + q^{88} + q^{98} - q^{99}+O(q^{100})$$ q + q^2 - q^7 - q^8 + q^9 - q^11 - q^14 - q^16 + q^18 - q^22 + q^23 - q^29 + q^37 + q^43 + q^46 + q^49 - 2 * q^53 + q^56 - q^58 - q^63 + q^64 + q^67 - q^71 - q^72 + q^74 + q^77 - q^79 + q^81 + q^86 + q^88 + q^98 - q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0
1.00000 0 0 0 0 −1.00000 −1.00000 1.00000 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 $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.1.d.b yes 1
3.b odd 2 1 1575.1.h.a 1
4.b odd 2 1 2800.1.f.b 1
5.b even 2 1 175.1.d.a 1
5.c odd 4 2 175.1.c.a 2
7.b odd 2 1 CM 175.1.d.b yes 1
7.c even 3 2 1225.1.i.a 2
7.d odd 6 2 1225.1.i.a 2
15.d odd 2 1 1575.1.h.c 1
15.e even 4 2 1575.1.e.a 2
20.d odd 2 1 2800.1.f.a 1
20.e even 4 2 2800.1.p.a 2
21.c even 2 1 1575.1.h.a 1
28.d even 2 1 2800.1.f.b 1
35.c odd 2 1 175.1.d.a 1
35.f even 4 2 175.1.c.a 2
35.i odd 6 2 1225.1.i.b 2
35.j even 6 2 1225.1.i.b 2
35.k even 12 4 1225.1.j.a 4
35.l odd 12 4 1225.1.j.a 4
105.g even 2 1 1575.1.h.c 1
105.k odd 4 2 1575.1.e.a 2
140.c even 2 1 2800.1.f.a 1
140.j odd 4 2 2800.1.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 5.c odd 4 2
175.1.c.a 2 35.f even 4 2
175.1.d.a 1 5.b even 2 1
175.1.d.a 1 35.c odd 2 1
175.1.d.b yes 1 1.a even 1 1 trivial
175.1.d.b yes 1 7.b odd 2 1 CM
1225.1.i.a 2 7.c even 3 2
1225.1.i.a 2 7.d odd 6 2
1225.1.i.b 2 35.i odd 6 2
1225.1.i.b 2 35.j even 6 2
1225.1.j.a 4 35.k even 12 4
1225.1.j.a 4 35.l odd 12 4
1575.1.e.a 2 15.e even 4 2
1575.1.e.a 2 105.k odd 4 2
1575.1.h.a 1 3.b odd 2 1
1575.1.h.a 1 21.c even 2 1
1575.1.h.c 1 15.d odd 2 1
1575.1.h.c 1 105.g even 2 1
2800.1.f.a 1 20.d odd 2 1
2800.1.f.a 1 140.c even 2 1
2800.1.f.b 1 4.b odd 2 1
2800.1.f.b 1 28.d even 2 1
2800.1.p.a 2 20.e even 4 2
2800.1.p.a 2 140.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{1}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 1$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 1$$
$29$ $$T + 1$$
$31$ $$T$$
$37$ $$T - 1$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T$$
$53$ $$T + 2$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T - 1$$
$71$ $$T + 1$$
$73$ $$T$$
$79$ $$T + 1$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$