# Properties

 Label 1850.2.a.k 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} + q^{8} - 3 q^{9} - 4 q^{11} - 2 q^{13} + q^{16} + 2 q^{17} - 3 q^{18} - 4 q^{19} - 4 q^{22} - 2 q^{26} - 6 q^{29} - 4 q^{31} + q^{32} + 2 q^{34} - 3 q^{36} + q^{37} - 4 q^{38} - 6 q^{41} - 4 q^{43} - 4 q^{44} + 8 q^{47} - 7 q^{49} - 2 q^{52} - 10 q^{53} - 6 q^{58} + 4 q^{59} + 10 q^{61} - 4 q^{62} + q^{64} + 8 q^{67} + 2 q^{68} - 3 q^{72} - 10 q^{73} + q^{74} - 4 q^{76} - 4 q^{79} + 9 q^{81} - 6 q^{82} - 4 q^{86} - 4 q^{88} + 2 q^{89} + 8 q^{94} - 6 q^{97} - 7 q^{98} + 12 q^{99}+O(q^{100})$$ q + q^2 + q^4 + q^8 - 3 * q^9 - 4 * q^11 - 2 * q^13 + q^16 + 2 * q^17 - 3 * q^18 - 4 * q^19 - 4 * q^22 - 2 * q^26 - 6 * q^29 - 4 * q^31 + q^32 + 2 * q^34 - 3 * q^36 + q^37 - 4 * q^38 - 6 * q^41 - 4 * q^43 - 4 * q^44 + 8 * q^47 - 7 * q^49 - 2 * q^52 - 10 * q^53 - 6 * q^58 + 4 * q^59 + 10 * q^61 - 4 * q^62 + q^64 + 8 * q^67 + 2 * q^68 - 3 * q^72 - 10 * q^73 + q^74 - 4 * q^76 - 4 * q^79 + 9 * q^81 - 6 * q^82 - 4 * q^86 - 4 * q^88 + 2 * q^89 + 8 * q^94 - 6 * q^97 - 7 * q^98 + 12 * 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 0 1.00000 0 0 0 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.k 1
5.b even 2 1 370.2.a.b 1
5.c odd 4 2 1850.2.b.d 2
15.d odd 2 1 3330.2.a.w 1
20.d odd 2 1 2960.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.b 1 5.b even 2 1
1850.2.a.k 1 1.a even 1 1 trivial
1850.2.b.d 2 5.c odd 4 2
2960.2.a.g 1 20.d odd 2 1
3330.2.a.w 1 15.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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