# Properties

 Label 95.3.d.a Level $95$ Weight $3$ Character orbit 95.d Analytic conductor $2.589$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,3,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, 3, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ 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: no Analytic conductor: $$2.58856251142$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \frac{1}{2}(1 + \sqrt{-19})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 q^{4} + ( - \beta - 4) q^{5} + (6 \beta - 3) q^{7} - 9 q^{9}+O(q^{10})$$ q - 4 * q^4 + (-b - 4) * q^5 + (6*b - 3) * q^7 - 9 * q^9 $$q - 4 q^{4} + ( - \beta - 4) q^{5} + (6 \beta - 3) q^{7} - 9 q^{9} - 3 q^{11} + 16 q^{16} + ( - 14 \beta + 7) q^{17} - 19 q^{19} + (4 \beta + 16) q^{20} + (16 \beta - 8) q^{23} + (9 \beta + 11) q^{25} + ( - 24 \beta + 12) q^{28} + ( - 27 \beta + 42) q^{35} + 36 q^{36} + (6 \beta - 3) q^{43} + 12 q^{44} + (9 \beta + 36) q^{45} + (26 \beta - 13) q^{47} - 122 q^{49} + (3 \beta + 12) q^{55} - 103 q^{61} + ( - 54 \beta + 27) q^{63} - 64 q^{64} + (56 \beta - 28) q^{68} + (66 \beta - 33) q^{73} + 76 q^{76} + ( - 18 \beta + 9) q^{77} + ( - 16 \beta - 64) q^{80} + 81 q^{81} + ( - 64 \beta + 32) q^{83} + (63 \beta - 98) q^{85} + ( - 64 \beta + 32) q^{92} + (19 \beta + 76) q^{95} + 27 q^{99}+O(q^{100})$$ q - 4 * q^4 + (-b - 4) * q^5 + (6*b - 3) * q^7 - 9 * q^9 - 3 * q^11 + 16 * q^16 + (-14*b + 7) * q^17 - 19 * q^19 + (4*b + 16) * q^20 + (16*b - 8) * q^23 + (9*b + 11) * q^25 + (-24*b + 12) * q^28 + (-27*b + 42) * q^35 + 36 * q^36 + (6*b - 3) * q^43 + 12 * q^44 + (9*b + 36) * q^45 + (26*b - 13) * q^47 - 122 * q^49 + (3*b + 12) * q^55 - 103 * q^61 + (-54*b + 27) * q^63 - 64 * q^64 + (56*b - 28) * q^68 + (66*b - 33) * q^73 + 76 * q^76 + (-18*b + 9) * q^77 + (-16*b - 64) * q^80 + 81 * q^81 + (-64*b + 32) * q^83 + (63*b - 98) * q^85 + (-64*b + 32) * q^92 + (19*b + 76) * q^95 + 27 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} - 9 q^{5} - 18 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 - 9 * q^5 - 18 * q^9 $$2 q - 8 q^{4} - 9 q^{5} - 18 q^{9} - 6 q^{11} + 32 q^{16} - 38 q^{19} + 36 q^{20} + 31 q^{25} + 57 q^{35} + 72 q^{36} + 24 q^{44} + 81 q^{45} - 244 q^{49} + 27 q^{55} - 206 q^{61} - 128 q^{64} + 152 q^{76} - 144 q^{80} + 162 q^{81} - 133 q^{85} + 171 q^{95} + 54 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 - 9 * q^5 - 18 * q^9 - 6 * q^11 + 32 * q^16 - 38 * q^19 + 36 * q^20 + 31 * q^25 + 57 * q^35 + 72 * q^36 + 24 * q^44 + 81 * q^45 - 244 * q^49 + 27 * q^55 - 206 * q^61 - 128 * q^64 + 152 * q^76 - 144 * q^80 + 162 * q^81 - 133 * q^85 + 171 * q^95 + 54 * q^99

## 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
 0.5 + 2.17945i 0.5 − 2.17945i
0 0 −4.00000 −4.50000 2.17945i 0 13.0767i 0 −9.00000 0
94.2 0 0 −4.00000 −4.50000 + 2.17945i 0 13.0767i 0 −9.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.3.d.a 2
3.b odd 2 1 855.3.g.a 2
5.b even 2 1 inner 95.3.d.a 2
5.c odd 4 2 475.3.c.c 2
15.d odd 2 1 855.3.g.a 2
19.b odd 2 1 CM 95.3.d.a 2
57.d even 2 1 855.3.g.a 2
95.d odd 2 1 inner 95.3.d.a 2
95.g even 4 2 475.3.c.c 2
285.b even 2 1 855.3.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.d.a 2 1.a even 1 1 trivial
95.3.d.a 2 5.b even 2 1 inner
95.3.d.a 2 19.b odd 2 1 CM
95.3.d.a 2 95.d odd 2 1 inner
475.3.c.c 2 5.c odd 4 2
475.3.c.c 2 95.g even 4 2
855.3.g.a 2 3.b odd 2 1
855.3.g.a 2 15.d odd 2 1
855.3.g.a 2 57.d even 2 1
855.3.g.a 2 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 9T + 25$$
$7$ $$T^{2} + 171$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 931$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} + 1216$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 171$$
$47$ $$T^{2} + 3211$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 103)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 20691$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 19456$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$