# Properties

 Label 1850.2.a.d 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 - q^7 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} - q^{7} - q^{8} - 3 q^{9} - 3 q^{11} + 4 q^{13} + q^{14} + q^{16} + 3 q^{17} + 3 q^{18} + 3 q^{22} + 8 q^{23} - 4 q^{26} - q^{28} - 3 q^{29} - 7 q^{31} - q^{32} - 3 q^{34} - 3 q^{36} + q^{37} + 11 q^{41} - 11 q^{43} - 3 q^{44} - 8 q^{46} - 4 q^{47} - 6 q^{49} + 4 q^{52} - 11 q^{53} + q^{56} + 3 q^{58} - 12 q^{59} - 15 q^{61} + 7 q^{62} + 3 q^{63} + q^{64} + 4 q^{67} + 3 q^{68} + 6 q^{71} + 3 q^{72} - 2 q^{73} - q^{74} + 3 q^{77} - 8 q^{79} + 9 q^{81} - 11 q^{82} - 12 q^{83} + 11 q^{86} + 3 q^{88} - 4 q^{91} + 8 q^{92} + 4 q^{94} + q^{97} + 6 q^{98} + 9 q^{99}+O(q^{100})$$ q - q^2 + q^4 - q^7 - q^8 - 3 * q^9 - 3 * q^11 + 4 * q^13 + q^14 + q^16 + 3 * q^17 + 3 * q^18 + 3 * q^22 + 8 * q^23 - 4 * q^26 - q^28 - 3 * q^29 - 7 * q^31 - q^32 - 3 * q^34 - 3 * q^36 + q^37 + 11 * q^41 - 11 * q^43 - 3 * q^44 - 8 * q^46 - 4 * q^47 - 6 * q^49 + 4 * q^52 - 11 * q^53 + q^56 + 3 * q^58 - 12 * q^59 - 15 * q^61 + 7 * q^62 + 3 * q^63 + q^64 + 4 * q^67 + 3 * q^68 + 6 * q^71 + 3 * q^72 - 2 * q^73 - q^74 + 3 * q^77 - 8 * q^79 + 9 * q^81 - 11 * q^82 - 12 * q^83 + 11 * q^86 + 3 * q^88 - 4 * q^91 + 8 * q^92 + 4 * q^94 + q^97 + 6 * q^98 + 9 * 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.

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 −1.00000 −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.d 1
5.b even 2 1 1850.2.a.l 1
5.c odd 4 2 370.2.b.b 2
15.e even 4 2 3330.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.b 2 5.c odd 4 2
1850.2.a.d 1 1.a even 1 1 trivial
1850.2.a.l 1 5.b even 2 1
3330.2.d.c 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 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 3$$
$13$ $$T - 4$$
$17$ $$T - 3$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T + 3$$
$31$ $$T + 7$$
$37$ $$T - 1$$
$41$ $$T - 11$$
$43$ $$T + 11$$
$47$ $$T + 4$$
$53$ $$T + 11$$
$59$ $$T + 12$$
$61$ $$T + 15$$
$67$ $$T - 4$$
$71$ $$T - 6$$
$73$ $$T + 2$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T$$
$97$ $$T - 1$$