# Properties

 Label 190.2.b.a Level $190$ Weight $2$ Character orbit 190.b Analytic conductor $1.517$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [190,2,Mod(39,190)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(190, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("190.39");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$190 = 2 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 190.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: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{2} q^{2} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{3} - q^{4} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} + \zeta_{8} - 1) q^{6} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{8} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{9} +O(q^{10})$$ q - z^2 * q^2 + (z^3 - z^2 + z) * q^3 - q^4 + (z^3 + 2*z) * q^5 + (-z^3 + z - 1) * q^6 + (z^3 + 3*z^2 + z) * q^7 + z^2 * q^8 + (-2*z^3 + 2*z) * q^9 $$q - \zeta_{8}^{2} q^{2} + (\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{3} - q^{4} + (\zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{3} + \zeta_{8} - 1) q^{6} + (\zeta_{8}^{3} + 3 \zeta_{8}^{2} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{8} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{9} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{10} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}) q^{12} + ( - 2 \zeta_{8}^{3} + \cdots - 2 \zeta_{8}) q^{13} + \cdots - 4 q^{99} +O(q^{100})$$ q - z^2 * q^2 + (z^3 - z^2 + z) * q^3 - q^4 + (z^3 + 2*z) * q^5 + (-z^3 + z - 1) * q^6 + (z^3 + 3*z^2 + z) * q^7 + z^2 * q^8 + (-2*z^3 + 2*z) * q^9 + (-2*z^3 + z) * q^10 + (z^3 - z) * q^11 + (-z^3 + z^2 - z) * q^12 + (-2*z^3 - 3*z^2 - 2*z) * q^13 + (-z^3 + z + 3) * q^14 + (-2*z^3 + z^2 + z - 3) * q^15 + q^16 + z^2 * q^17 + (-2*z^3 - 2*z) * q^18 + q^19 + (-z^3 - 2*z) * q^20 + (2*z^3 - 2*z + 1) * q^21 + (z^3 + z) * q^22 + (3*z^3 - 5*z^2 + 3*z) * q^23 + (z^3 - z + 1) * q^24 + (3*z^2 - 4) * q^25 + (2*z^3 - 2*z - 3) * q^26 + (z^3 + z^2 + z) * q^27 + (-z^3 - 3*z^2 - z) * q^28 + (2*z^3 - 2*z + 3) * q^29 + (-z^3 + 3*z^2 - 2*z + 1) * q^30 + (-3*z^3 + 3*z + 2) * q^31 - z^2 * q^32 + (z^3 - 2*z^2 + z) * q^33 + q^34 + (6*z^3 + z^2 - 3*z - 3) * q^35 + (2*z^3 - 2*z) * q^36 + (-6*z^3 - 6*z) * q^37 - z^2 * q^38 + (-z^3 + z + 1) * q^39 + (2*z^3 - z) * q^40 + (3*z^3 - 3*z) * q^41 + (2*z^3 - z^2 + 2*z) * q^42 + (3*z^3 - 6*z^2 + 3*z) * q^43 + (-z^3 + z) * q^44 + (6*z^2 + 2) * q^45 + (-3*z^3 + 3*z - 5) * q^46 + (z^3 - z^2 + z) * q^48 + (6*z^3 - 6*z - 4) * q^49 + (4*z^2 + 3) * q^50 + (z^3 - z + 1) * q^51 + (2*z^3 + 3*z^2 + 2*z) * q^52 + (-6*z^3 + 3*z^2 - 6*z) * q^53 + (-z^3 + z + 1) * q^54 + (-3*z^2 - 1) * q^55 + (z^3 - z - 3) * q^56 + (z^3 - z^2 + z) * q^57 + (2*z^3 - 3*z^2 + 2*z) * q^58 + (7*z^3 - 7*z + 3) * q^59 + (2*z^3 - z^2 - z + 3) * q^60 + (-3*z^3 + 3*z + 10) * q^61 + (-3*z^3 - 2*z^2 - 3*z) * q^62 + (6*z^3 + 4*z^2 + 6*z) * q^63 - q^64 + (-6*z^3 - 2*z^2 + 3*z + 6) * q^65 + (-z^3 + z - 2) * q^66 + (-3*z^3 + 9*z^2 - 3*z) * q^67 - z^2 * q^68 + (-8*z^3 + 8*z - 11) * q^69 + (3*z^3 + 3*z^2 + 6*z + 1) * q^70 + (z^3 - z - 12) * q^71 + (2*z^3 + 2*z) * q^72 + (-6*z^3 - 3*z^2 - 6*z) * q^73 + (6*z^3 - 6*z) * q^74 + (-z^3 + 4*z^2 - 7*z + 3) * q^75 - q^76 + (-3*z^3 - 2*z^2 - 3*z) * q^77 + (-z^3 - z^2 - z) * q^78 + (-6*z^3 + 6*z - 2) * q^79 + (z^3 + 2*z) * q^80 + (-6*z^3 + 6*z - 1) * q^81 + (3*z^3 + 3*z) * q^82 + (-6*z^3 - 6*z^2 - 6*z) * q^83 + (-2*z^3 + 2*z - 1) * q^84 + (2*z^3 - z) * q^85 + (-3*z^3 + 3*z - 6) * q^86 + (5*z^3 - 7*z^2 + 5*z) * q^87 + (-z^3 - z) * q^88 + (5*z^3 - 5*z) * q^89 + (-2*z^2 + 6) * q^90 + (-9*z^3 + 9*z + 13) * q^91 + (-3*z^3 + 5*z^2 - 3*z) * q^92 + (-z^3 + 4*z^2 - z) * q^93 + (z^3 + 2*z) * q^95 + (-z^3 + z - 1) * q^96 + (4*z^3 - 6*z^2 + 4*z) * q^97 + (6*z^3 + 4*z^2 + 6*z) * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 $$4 q - 4 q^{4} - 4 q^{6} + 12 q^{14} - 12 q^{15} + 4 q^{16} + 4 q^{19} + 4 q^{21} + 4 q^{24} - 16 q^{25} - 12 q^{26} + 12 q^{29} + 4 q^{30} + 8 q^{31} + 4 q^{34} - 12 q^{35} + 4 q^{39} + 8 q^{45} - 20 q^{46} - 16 q^{49} + 12 q^{50} + 4 q^{51} + 4 q^{54} - 4 q^{55} - 12 q^{56} + 12 q^{59} + 12 q^{60} + 40 q^{61} - 4 q^{64} + 24 q^{65} - 8 q^{66} - 44 q^{69} + 4 q^{70} - 48 q^{71} + 12 q^{75} - 4 q^{76} - 8 q^{79} - 4 q^{81} - 4 q^{84} - 24 q^{86} + 24 q^{90} + 52 q^{91} - 4 q^{96} - 16 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 + 12 * q^14 - 12 * q^15 + 4 * q^16 + 4 * q^19 + 4 * q^21 + 4 * q^24 - 16 * q^25 - 12 * q^26 + 12 * q^29 + 4 * q^30 + 8 * q^31 + 4 * q^34 - 12 * q^35 + 4 * q^39 + 8 * q^45 - 20 * q^46 - 16 * q^49 + 12 * q^50 + 4 * q^51 + 4 * q^54 - 4 * q^55 - 12 * q^56 + 12 * q^59 + 12 * q^60 + 40 * q^61 - 4 * q^64 + 24 * q^65 - 8 * q^66 - 44 * q^69 + 4 * q^70 - 48 * q^71 + 12 * q^75 - 4 * q^76 - 8 * q^79 - 4 * q^81 - 4 * q^84 - 24 * q^86 + 24 * q^90 + 52 * q^91 - 4 * q^96 - 16 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/190\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
1.00000i 2.41421i −1.00000 −0.707107 2.12132i −2.41421 1.58579i 1.00000i −2.82843 −2.12132 + 0.707107i
39.2 1.00000i 0.414214i −1.00000 0.707107 + 2.12132i 0.414214 4.41421i 1.00000i 2.82843 2.12132 0.707107i
39.3 1.00000i 0.414214i −1.00000 0.707107 2.12132i 0.414214 4.41421i 1.00000i 2.82843 2.12132 + 0.707107i
39.4 1.00000i 2.41421i −1.00000 −0.707107 + 2.12132i −2.41421 1.58579i 1.00000i −2.82843 −2.12132 0.707107i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.b.a 4
3.b odd 2 1 1710.2.d.c 4
4.b odd 2 1 1520.2.d.e 4
5.b even 2 1 inner 190.2.b.a 4
5.c odd 4 1 950.2.a.f 2
5.c odd 4 1 950.2.a.g 2
15.d odd 2 1 1710.2.d.c 4
15.e even 4 1 8550.2.a.bn 2
15.e even 4 1 8550.2.a.cb 2
20.d odd 2 1 1520.2.d.e 4
20.e even 4 1 7600.2.a.v 2
20.e even 4 1 7600.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 1.a even 1 1 trivial
190.2.b.a 4 5.b even 2 1 inner
950.2.a.f 2 5.c odd 4 1
950.2.a.g 2 5.c odd 4 1
1520.2.d.e 4 4.b odd 2 1
1520.2.d.e 4 20.d odd 2 1
1710.2.d.c 4 3.b odd 2 1
1710.2.d.c 4 15.d odd 2 1
7600.2.a.v 2 20.e even 4 1
7600.2.a.bg 2 20.e even 4 1
8550.2.a.bn 2 15.e even 4 1
8550.2.a.cb 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 6T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(190, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4} + 8T^{2} + 25$$
$7$ $$T^{4} + 22T^{2} + 49$$
$11$ $$(T^{2} - 2)^{2}$$
$13$ $$T^{4} + 34T^{2} + 1$$
$17$ $$(T^{2} + 1)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 86T^{2} + 49$$
$29$ $$(T^{2} - 6 T + 1)^{2}$$
$31$ $$(T^{2} - 4 T - 14)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T^{2} - 18)^{2}$$
$43$ $$T^{4} + 108T^{2} + 324$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 162T^{2} + 3969$$
$59$ $$(T^{2} - 6 T - 89)^{2}$$
$61$ $$(T^{2} - 20 T + 82)^{2}$$
$67$ $$T^{4} + 198T^{2} + 3969$$
$71$ $$(T^{2} + 24 T + 142)^{2}$$
$73$ $$T^{4} + 162T^{2} + 3969$$
$79$ $$(T^{2} + 4 T - 68)^{2}$$
$83$ $$T^{4} + 216T^{2} + 1296$$
$89$ $$(T^{2} - 50)^{2}$$
$97$ $$T^{4} + 136T^{2} + 16$$