# Properties

 Label 95.1.d.b Level $95$ Weight $1$ Character orbit 95.d Self dual yes Analytic conductor $0.047$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -95 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,1,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, 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("95.94");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 95.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: $$0.0474111762001$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $D_8$ Artin field: Galois closure of 8.2.4286875.1

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + \beta q^{3} + q^{4} - q^{5} - 2 q^{6} + q^{9} +O(q^{10})$$ q - b * q^2 + b * q^3 + q^4 - q^5 - 2 * q^6 + q^9 $$q - \beta q^{2} + \beta q^{3} + q^{4} - q^{5} - 2 q^{6} + q^{9} + \beta q^{10} + \beta q^{12} - \beta q^{13} - \beta q^{15} - q^{16} - \beta q^{18} - q^{19} - q^{20} + q^{25} + 2 q^{26} + 2 q^{30} + \beta q^{32} + q^{36} + \beta q^{37} + \beta q^{38} - 2 q^{39} - q^{45} - \beta q^{48} + q^{49} - \beta q^{50} - \beta q^{52} + \beta q^{53} - \beta q^{57} - \beta q^{60} - q^{64} + \beta q^{65} - \beta q^{67} - 2 q^{74} + \beta q^{75} - q^{76} + 2 \beta q^{78} + q^{80} - q^{81} + \beta q^{90} + q^{95} + 2 q^{96} + \beta q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + b * q^3 + q^4 - q^5 - 2 * q^6 + q^9 + b * q^10 + b * q^12 - b * q^13 - b * q^15 - q^16 - b * q^18 - q^19 - q^20 + q^25 + 2 * q^26 + 2 * q^30 + b * q^32 + q^36 + b * q^37 + b * q^38 - 2 * q^39 - q^45 - b * q^48 + q^49 - b * q^50 - b * q^52 + b * q^53 - b * q^57 - b * q^60 - q^64 + b * q^65 - b * q^67 - 2 * q^74 + b * q^75 - q^76 + 2*b * q^78 + q^80 - q^81 + b * q^90 + q^95 + 2 * q^96 + b * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^5 - 4 * q^6 + 2 * q^9 $$2 q + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{9} - 2 q^{16} - 2 q^{19} - 2 q^{20} + 2 q^{25} + 4 q^{26} + 4 q^{30} + 2 q^{36} - 4 q^{39} - 2 q^{45} + 2 q^{49} - 2 q^{64} - 4 q^{74} - 2 q^{76} + 2 q^{80} - 2 q^{81} + 2 q^{95} + 4 q^{96}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^5 - 4 * q^6 + 2 * q^9 - 2 * q^16 - 2 * q^19 - 2 * q^20 + 2 * q^25 + 4 * q^26 + 4 * q^30 + 2 * q^36 - 4 * q^39 - 2 * q^45 + 2 * q^49 - 2 * q^64 - 4 * q^74 - 2 * q^76 + 2 * q^80 - 2 * q^81 + 2 * q^95 + 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 1.41421 −1.41421
−1.41421 1.41421 1.00000 −1.00000 −2.00000 0 0 1.00000 1.41421
94.2 1.41421 −1.41421 1.00000 −1.00000 −2.00000 0 0 1.00000 −1.41421
 $$n$$: 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.1.d.b 2
3.b odd 2 1 855.1.g.c 2
4.b odd 2 1 1520.1.m.b 2
5.b even 2 1 inner 95.1.d.b 2
5.c odd 4 2 475.1.c.b 2
15.d odd 2 1 855.1.g.c 2
19.b odd 2 1 inner 95.1.d.b 2
19.c even 3 2 1805.1.h.b 4
19.d odd 6 2 1805.1.h.b 4
19.e even 9 6 1805.1.o.b 12
19.f odd 18 6 1805.1.o.b 12
20.d odd 2 1 1520.1.m.b 2
57.d even 2 1 855.1.g.c 2
76.d even 2 1 1520.1.m.b 2
95.d odd 2 1 CM 95.1.d.b 2
95.g even 4 2 475.1.c.b 2
95.h odd 6 2 1805.1.h.b 4
95.i even 6 2 1805.1.h.b 4
95.o odd 18 6 1805.1.o.b 12
95.p even 18 6 1805.1.o.b 12
285.b even 2 1 855.1.g.c 2
380.d even 2 1 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 1.a even 1 1 trivial
95.1.d.b 2 5.b even 2 1 inner
95.1.d.b 2 19.b odd 2 1 inner
95.1.d.b 2 95.d odd 2 1 CM
475.1.c.b 2 5.c odd 4 2
475.1.c.b 2 95.g even 4 2
855.1.g.c 2 3.b odd 2 1
855.1.g.c 2 15.d odd 2 1
855.1.g.c 2 57.d even 2 1
855.1.g.c 2 285.b even 2 1
1520.1.m.b 2 4.b odd 2 1
1520.1.m.b 2 20.d odd 2 1
1520.1.m.b 2 76.d even 2 1
1520.1.m.b 2 380.d even 2 1
1805.1.h.b 4 19.c even 3 2
1805.1.h.b 4 19.d odd 6 2
1805.1.h.b 4 95.h odd 6 2
1805.1.h.b 4 95.i even 6 2
1805.1.o.b 12 19.e even 9 6
1805.1.o.b 12 19.f odd 18 6
1805.1.o.b 12 95.o odd 18 6
1805.1.o.b 12 95.p even 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

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