# Properties

 Label 1850.2.a.o Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + q^7 + q^8 + q^9 $$q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9} + 3 q^{11} + 2 q^{12} + 4 q^{13} + q^{14} + q^{16} - 3 q^{17} + q^{18} + 2 q^{19} + 2 q^{21} + 3 q^{22} - 6 q^{23} + 2 q^{24} + 4 q^{26} - 4 q^{27} + q^{28} + 3 q^{29} + 5 q^{31} + q^{32} + 6 q^{33} - 3 q^{34} + q^{36} - q^{37} + 2 q^{38} + 8 q^{39} + 3 q^{41} + 2 q^{42} + q^{43} + 3 q^{44} - 6 q^{46} - 12 q^{47} + 2 q^{48} - 6 q^{49} - 6 q^{51} + 4 q^{52} - 3 q^{53} - 4 q^{54} + q^{56} + 4 q^{57} + 3 q^{58} - q^{61} + 5 q^{62} + q^{63} + q^{64} + 6 q^{66} + 4 q^{67} - 3 q^{68} - 12 q^{69} + 6 q^{71} + q^{72} + 16 q^{73} - q^{74} + 2 q^{76} + 3 q^{77} + 8 q^{78} + 8 q^{79} - 11 q^{81} + 3 q^{82} + 12 q^{83} + 2 q^{84} + q^{86} + 6 q^{87} + 3 q^{88} - 6 q^{89} + 4 q^{91} - 6 q^{92} + 10 q^{93} - 12 q^{94} + 2 q^{96} - 17 q^{97} - 6 q^{98} + 3 q^{99}+O(q^{100})$$ q + q^2 + 2 * q^3 + q^4 + 2 * q^6 + q^7 + q^8 + q^9 + 3 * q^11 + 2 * q^12 + 4 * q^13 + q^14 + q^16 - 3 * q^17 + q^18 + 2 * q^19 + 2 * q^21 + 3 * q^22 - 6 * q^23 + 2 * q^24 + 4 * q^26 - 4 * q^27 + q^28 + 3 * q^29 + 5 * q^31 + q^32 + 6 * q^33 - 3 * q^34 + q^36 - q^37 + 2 * q^38 + 8 * q^39 + 3 * q^41 + 2 * q^42 + q^43 + 3 * q^44 - 6 * q^46 - 12 * q^47 + 2 * q^48 - 6 * q^49 - 6 * q^51 + 4 * q^52 - 3 * q^53 - 4 * q^54 + q^56 + 4 * q^57 + 3 * q^58 - q^61 + 5 * q^62 + q^63 + q^64 + 6 * q^66 + 4 * q^67 - 3 * q^68 - 12 * q^69 + 6 * q^71 + q^72 + 16 * q^73 - q^74 + 2 * q^76 + 3 * q^77 + 8 * q^78 + 8 * q^79 - 11 * q^81 + 3 * q^82 + 12 * q^83 + 2 * q^84 + q^86 + 6 * q^87 + 3 * q^88 - 6 * q^89 + 4 * q^91 - 6 * q^92 + 10 * q^93 - 12 * q^94 + 2 * q^96 - 17 * q^97 - 6 * q^98 + 3 * 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 2.00000 1.00000 0 2.00000 1.00000 1.00000 1.00000 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.o 1
5.b even 2 1 370.2.a.a 1
5.c odd 4 2 1850.2.b.g 2
15.d odd 2 1 3330.2.a.v 1
20.d odd 2 1 2960.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.a 1 5.b even 2 1
1850.2.a.o 1 1.a even 1 1 trivial
1850.2.b.g 2 5.c odd 4 2
2960.2.a.j 1 20.d odd 2 1
3330.2.a.v 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 3$$
$13$ $$T - 4$$
$17$ $$T + 3$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T - 3$$
$31$ $$T - 5$$
$37$ $$T + 1$$
$41$ $$T - 3$$
$43$ $$T - 1$$
$47$ $$T + 12$$
$53$ $$T + 3$$
$59$ $$T$$
$61$ $$T + 1$$
$67$ $$T - 4$$
$71$ $$T - 6$$
$73$ $$T - 16$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 17$$