Properties

 Label 175.1.c.a Level $175$ Weight $1$ Character orbit 175.c Analytic conductor $0.087$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -7 Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,1,Mod(174,175)]

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([1, 1]))

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

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

chi := DirichletCharacter("175.174");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 175.c (of order $$2$$, degree $$1$$, not 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: $$0.0873363772108$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

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

Character values

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

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
174.1
 1.00000i − 1.00000i
1.00000i 0 0 0 0 1.00000i 1.00000i −1.00000 0
174.2 1.00000i 0 0 0 0 1.00000i 1.00000i −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 $$\Q(\sqrt{-7})$$
5.b even 2 1 inner
35.c odd 2 1 inner

Twists

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

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

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(175, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 1$$
$71$ $$(T + 1)^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 1)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$
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