# Properties

 Label 190.2.a.b Level $190$ Weight $2$ Character orbit 190.a Self dual yes Analytic conductor $1.517$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [190,2,Mod(1,190)]

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([0, 0]))

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

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

chi := DirichletCharacter("190.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$190 = 2 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 190.a (trivial)

## 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: $$1.51715763840$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 −3.00000 1.00000 −1.00000 −3.00000 −5.00000 1.00000 6.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.a.b 1
3.b odd 2 1 1710.2.a.g 1
4.b odd 2 1 1520.2.a.j 1
5.b even 2 1 950.2.a.c 1
5.c odd 4 2 950.2.b.a 2
7.b odd 2 1 9310.2.a.u 1
8.b even 2 1 6080.2.a.x 1
8.d odd 2 1 6080.2.a.b 1
15.d odd 2 1 8550.2.a.bm 1
19.b odd 2 1 3610.2.a.e 1
20.d odd 2 1 7600.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.b 1 1.a even 1 1 trivial
950.2.a.c 1 5.b even 2 1
950.2.b.a 2 5.c odd 4 2
1520.2.a.j 1 4.b odd 2 1
1710.2.a.g 1 3.b odd 2 1
3610.2.a.e 1 19.b odd 2 1
6080.2.a.b 1 8.d odd 2 1
6080.2.a.x 1 8.b even 2 1
7600.2.a.a 1 20.d odd 2 1
8550.2.a.bm 1 15.d odd 2 1
9310.2.a.u 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(190))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 3$$
$5$ $$T + 1$$
$7$ $$T + 5$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T - 7$$
$29$ $$T + 3$$
$31$ $$T + 2$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T - 6$$
$47$ $$T$$
$53$ $$T + 13$$
$59$ $$T + 9$$
$61$ $$T + 12$$
$67$ $$T + 3$$
$71$ $$T$$
$73$ $$T - 11$$
$79$ $$T + 2$$
$83$ $$T + 10$$
$89$ $$T - 2$$
$97$ $$T + 2$$