# Properties

 Label 1850.2.a.f 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 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

 $$f(q)$$ $$=$$ $$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 2q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 2q^{7} - q^{8} + q^{9} + 2q^{12} - 2q^{13} + 2q^{14} + q^{16} - 6q^{17} - q^{18} + 2q^{19} - 4q^{21} - 2q^{24} + 2q^{26} - 4q^{27} - 2q^{28} + 6q^{29} - 10q^{31} - q^{32} + 6q^{34} + q^{36} - q^{37} - 2q^{38} - 4q^{39} - 6q^{41} + 4q^{42} + 4q^{43} + 6q^{47} + 2q^{48} - 3q^{49} - 12q^{51} - 2q^{52} - 6q^{53} + 4q^{54} + 2q^{56} + 4q^{57} - 6q^{58} - 6q^{59} - 10q^{61} + 10q^{62} - 2q^{63} + q^{64} - 2q^{67} - 6q^{68} - q^{72} - 2q^{73} + q^{74} + 2q^{76} + 4q^{78} - 10q^{79} - 11q^{81} + 6q^{82} + 6q^{83} - 4q^{84} - 4q^{86} + 12q^{87} - 6q^{89} + 4q^{91} - 20q^{93} - 6q^{94} - 2q^{96} - 2q^{97} + 3q^{98} + O(q^{100})$$

## 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.

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 −2.00000 −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.f 1
5.b even 2 1 370.2.a.d 1
5.c odd 4 2 1850.2.b.b 2
15.d odd 2 1 3330.2.a.d 1
20.d odd 2 1 2960.2.a.m 1

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

 $$T_{3} - 2$$ $$T_{7} + 2$$

## Hecke characteristic polynomials

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