# Properties

 Label 1862.2.a.f Level $1862$ Weight $2$ Character orbit 1862.a Self dual yes Analytic conductor $14.868$ Analytic rank $0$ 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] = [1862,2,Mod(1,1862)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1862, 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("1862.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1862 = 2 \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1862.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: $$14.8681448564$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + 4 q^{5} + q^{6} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + 4 * q^5 + q^6 + q^8 - 2 * q^9 $$q + q^{2} + q^{3} + q^{4} + 4 q^{5} + q^{6} + q^{8} - 2 q^{9} + 4 q^{10} + 2 q^{11} + q^{12} + q^{13} + 4 q^{15} + q^{16} - 3 q^{17} - 2 q^{18} + q^{19} + 4 q^{20} + 2 q^{22} - q^{23} + q^{24} + 11 q^{25} + q^{26} - 5 q^{27} - 5 q^{29} + 4 q^{30} + 8 q^{31} + q^{32} + 2 q^{33} - 3 q^{34} - 2 q^{36} - 2 q^{37} + q^{38} + q^{39} + 4 q^{40} + 8 q^{41} + 4 q^{43} + 2 q^{44} - 8 q^{45} - q^{46} - 8 q^{47} + q^{48} + 11 q^{50} - 3 q^{51} + q^{52} - q^{53} - 5 q^{54} + 8 q^{55} + q^{57} - 5 q^{58} - 15 q^{59} + 4 q^{60} - 2 q^{61} + 8 q^{62} + q^{64} + 4 q^{65} + 2 q^{66} + 3 q^{67} - 3 q^{68} - q^{69} + 2 q^{71} - 2 q^{72} - 9 q^{73} - 2 q^{74} + 11 q^{75} + q^{76} + q^{78} - 10 q^{79} + 4 q^{80} + q^{81} + 8 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} - 5 q^{87} + 2 q^{88} - 8 q^{90} - q^{92} + 8 q^{93} - 8 q^{94} + 4 q^{95} + q^{96} + 2 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + 4 * q^5 + q^6 + q^8 - 2 * q^9 + 4 * q^10 + 2 * q^11 + q^12 + q^13 + 4 * q^15 + q^16 - 3 * q^17 - 2 * q^18 + q^19 + 4 * q^20 + 2 * q^22 - q^23 + q^24 + 11 * q^25 + q^26 - 5 * q^27 - 5 * q^29 + 4 * q^30 + 8 * q^31 + q^32 + 2 * q^33 - 3 * q^34 - 2 * q^36 - 2 * q^37 + q^38 + q^39 + 4 * q^40 + 8 * q^41 + 4 * q^43 + 2 * q^44 - 8 * q^45 - q^46 - 8 * q^47 + q^48 + 11 * q^50 - 3 * q^51 + q^52 - q^53 - 5 * q^54 + 8 * q^55 + q^57 - 5 * q^58 - 15 * q^59 + 4 * q^60 - 2 * q^61 + 8 * q^62 + q^64 + 4 * q^65 + 2 * q^66 + 3 * q^67 - 3 * q^68 - q^69 + 2 * q^71 - 2 * q^72 - 9 * q^73 - 2 * q^74 + 11 * q^75 + q^76 + q^78 - 10 * q^79 + 4 * q^80 + q^81 + 8 * q^82 + 6 * q^83 - 12 * q^85 + 4 * q^86 - 5 * q^87 + 2 * q^88 - 8 * q^90 - q^92 + 8 * q^93 - 8 * q^94 + 4 * q^95 + q^96 + 2 * q^97 - 4 * 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 1.00000 1.00000 4.00000 1.00000 0 1.00000 −2.00000 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 1862.2.a.f 1
7.b odd 2 1 38.2.a.b 1
21.c even 2 1 342.2.a.d 1
28.d even 2 1 304.2.a.d 1
35.c odd 2 1 950.2.a.b 1
35.f even 4 2 950.2.b.c 2
56.e even 2 1 1216.2.a.g 1
56.h odd 2 1 1216.2.a.n 1
77.b even 2 1 4598.2.a.a 1
84.h odd 2 1 2736.2.a.w 1
91.b odd 2 1 6422.2.a.b 1
105.g even 2 1 8550.2.a.u 1
133.c even 2 1 722.2.a.b 1
133.m odd 6 2 722.2.c.d 2
133.p even 6 2 722.2.c.f 2
133.y odd 18 6 722.2.e.c 6
133.ba even 18 6 722.2.e.d 6
140.c even 2 1 7600.2.a.h 1
399.h odd 2 1 6498.2.a.y 1
532.b odd 2 1 5776.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 7.b odd 2 1
304.2.a.d 1 28.d even 2 1
342.2.a.d 1 21.c even 2 1
722.2.a.b 1 133.c even 2 1
722.2.c.d 2 133.m odd 6 2
722.2.c.f 2 133.p even 6 2
722.2.e.c 6 133.y odd 18 6
722.2.e.d 6 133.ba even 18 6
950.2.a.b 1 35.c odd 2 1
950.2.b.c 2 35.f even 4 2
1216.2.a.g 1 56.e even 2 1
1216.2.a.n 1 56.h odd 2 1
1862.2.a.f 1 1.a even 1 1 trivial
2736.2.a.w 1 84.h odd 2 1
4598.2.a.a 1 77.b even 2 1
5776.2.a.d 1 532.b odd 2 1
6422.2.a.b 1 91.b odd 2 1
6498.2.a.y 1 399.h odd 2 1
7600.2.a.h 1 140.c even 2 1
8550.2.a.u 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1862))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} - 4$$ T5 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 4$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T + 1$$
$29$ $$T + 5$$
$31$ $$T - 8$$
$37$ $$T + 2$$
$41$ $$T - 8$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T + 1$$
$59$ $$T + 15$$
$61$ $$T + 2$$
$67$ $$T - 3$$
$71$ $$T - 2$$
$73$ $$T + 9$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T$$
$97$ $$T - 2$$