# Properties

 Label 2850.2.a.q Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.7573645761$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

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 −1.00000 1.00000 0 −1.00000 −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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 2850.2.a.q 1
3.b odd 2 1 8550.2.a.e 1
5.b even 2 1 570.2.a.f 1
5.c odd 4 2 2850.2.d.d 2
15.d odd 2 1 1710.2.a.o 1
20.d odd 2 1 4560.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.f 1 5.b even 2 1
1710.2.a.o 1 15.d odd 2 1
2850.2.a.q 1 1.a even 1 1 trivial
2850.2.d.d 2 5.c odd 4 2
4560.2.a.m 1 20.d odd 2 1
8550.2.a.e 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11}$$ T11 $$T_{13} + 2$$ T13 + 2 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T$$
$19$ $$T - 1$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 2$$
$37$ $$T + 2$$
$41$ $$T$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 6$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T - 2$$
$83$ $$T + 6$$
$89$ $$T + 12$$
$97$ $$T - 10$$