# Properties

 Label 2850.2.a.g Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 + 4 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} - q^{8} + q^{9} - q^{12} + 4 q^{13} - 4 q^{14} + q^{16} - 6 q^{17} - q^{18} + q^{19} - 4 q^{21} + 6 q^{23} + q^{24} - 4 q^{26} - q^{27} + 4 q^{28} + 6 q^{29} + 2 q^{31} - q^{32} + 6 q^{34} + q^{36} + 4 q^{37} - q^{38} - 4 q^{39} + 6 q^{41} + 4 q^{42} + 4 q^{43} - 6 q^{46} - 6 q^{47} - q^{48} + 9 q^{49} + 6 q^{51} + 4 q^{52} - 6 q^{53} + q^{54} - 4 q^{56} - q^{57} - 6 q^{58} - 12 q^{59} + 14 q^{61} - 2 q^{62} + 4 q^{63} + q^{64} - 8 q^{67} - 6 q^{68} - 6 q^{69} - q^{72} - 14 q^{73} - 4 q^{74} + q^{76} + 4 q^{78} - 10 q^{79} + q^{81} - 6 q^{82} + 12 q^{83} - 4 q^{84} - 4 q^{86} - 6 q^{87} - 6 q^{89} + 16 q^{91} + 6 q^{92} - 2 q^{93} + 6 q^{94} + q^{96} + 10 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 + 4 * q^7 - q^8 + q^9 - q^12 + 4 * q^13 - 4 * q^14 + q^16 - 6 * q^17 - q^18 + q^19 - 4 * q^21 + 6 * q^23 + q^24 - 4 * q^26 - q^27 + 4 * q^28 + 6 * q^29 + 2 * q^31 - q^32 + 6 * q^34 + q^36 + 4 * q^37 - q^38 - 4 * q^39 + 6 * q^41 + 4 * q^42 + 4 * q^43 - 6 * q^46 - 6 * q^47 - q^48 + 9 * q^49 + 6 * q^51 + 4 * q^52 - 6 * q^53 + q^54 - 4 * q^56 - q^57 - 6 * q^58 - 12 * q^59 + 14 * q^61 - 2 * q^62 + 4 * q^63 + q^64 - 8 * q^67 - 6 * q^68 - 6 * q^69 - q^72 - 14 * q^73 - 4 * q^74 + q^76 + 4 * q^78 - 10 * q^79 + q^81 - 6 * q^82 + 12 * q^83 - 4 * q^84 - 4 * q^86 - 6 * q^87 - 6 * q^89 + 16 * q^91 + 6 * q^92 - 2 * q^93 + 6 * q^94 + q^96 + 10 * q^97 - 9 * 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 4.00000 −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.g 1
3.b odd 2 1 8550.2.a.bj 1
5.b even 2 1 114.2.a.c 1
5.c odd 4 2 2850.2.d.p 2
15.d odd 2 1 342.2.a.c 1
20.d odd 2 1 912.2.a.c 1
35.c odd 2 1 5586.2.a.u 1
40.e odd 2 1 3648.2.a.bc 1
40.f even 2 1 3648.2.a.i 1
60.h even 2 1 2736.2.a.o 1
95.d odd 2 1 2166.2.a.a 1
285.b even 2 1 6498.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.c 1 5.b even 2 1
342.2.a.c 1 15.d odd 2 1
912.2.a.c 1 20.d odd 2 1
2166.2.a.a 1 95.d odd 2 1
2736.2.a.o 1 60.h even 2 1
2850.2.a.g 1 1.a even 1 1 trivial
2850.2.d.p 2 5.c odd 4 2
3648.2.a.i 1 40.f even 2 1
3648.2.a.bc 1 40.e odd 2 1
5586.2.a.u 1 35.c odd 2 1
6498.2.a.t 1 285.b even 2 1
8550.2.a.bj 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4 $$T_{23} - 6$$ T23 - 6

## Hecke characteristic polynomials

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