# Properties

 Label 1850.2.a.e Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ 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] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2 q^{7} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 + 2 * q^7 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} + 2 q^{7} - q^{8} - 3 q^{9} - 2 q^{13} - 2 q^{14} + q^{16} + 6 q^{17} + 3 q^{18} - 6 q^{19} - 4 q^{23} + 2 q^{26} + 2 q^{28} - 4 q^{31} - q^{32} - 6 q^{34} - 3 q^{36} + q^{37} + 6 q^{38} - 10 q^{41} + 4 q^{43} + 4 q^{46} + 2 q^{47} - 3 q^{49} - 2 q^{52} - 2 q^{53} - 2 q^{56} - 6 q^{59} + 4 q^{62} - 6 q^{63} + q^{64} - 8 q^{67} + 6 q^{68} + 3 q^{72} - 8 q^{73} - q^{74} - 6 q^{76} + 4 q^{79} + 9 q^{81} + 10 q^{82} + 12 q^{83} - 4 q^{86} + 6 q^{89} - 4 q^{91} - 4 q^{92} - 2 q^{94} + 10 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^4 + 2 * q^7 - q^8 - 3 * q^9 - 2 * q^13 - 2 * q^14 + q^16 + 6 * q^17 + 3 * q^18 - 6 * q^19 - 4 * q^23 + 2 * q^26 + 2 * q^28 - 4 * q^31 - q^32 - 6 * q^34 - 3 * q^36 + q^37 + 6 * q^38 - 10 * q^41 + 4 * q^43 + 4 * q^46 + 2 * q^47 - 3 * q^49 - 2 * q^52 - 2 * q^53 - 2 * q^56 - 6 * q^59 + 4 * q^62 - 6 * q^63 + q^64 - 8 * q^67 + 6 * q^68 + 3 * q^72 - 8 * q^73 - q^74 - 6 * q^76 + 4 * q^79 + 9 * q^81 + 10 * q^82 + 12 * q^83 - 4 * q^86 + 6 * q^89 - 4 * q^91 - 4 * q^92 - 2 * q^94 + 10 * q^97 + 3 * q^98

## 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 0 1.00000 0 0 2.00000 −1.00000 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## 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 1850.2.a.e 1
5.b even 2 1 1850.2.a.j 1
5.c odd 4 2 370.2.b.a 2
15.e even 4 2 3330.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.a 2 5.c odd 4 2
1850.2.a.e 1 1.a even 1 1 trivial
1850.2.a.j 1 5.b even 2 1
3330.2.d.f 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3}$$ T3 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 6$$
$19$ $$T + 6$$
$23$ $$T + 4$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T - 1$$
$41$ $$T + 10$$
$43$ $$T - 4$$
$47$ $$T - 2$$
$53$ $$T + 2$$
$59$ $$T + 6$$
$61$ $$T$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T + 8$$
$79$ $$T - 4$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T - 10$$