# Properties

 Label 1950.2.a.x Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4q^{7} + q^{8} + q^{9} + 4q^{11} + q^{12} + q^{13} - 4q^{14} + q^{16} - 4q^{17} + q^{18} + 7q^{19} - 4q^{21} + 4q^{22} + 4q^{23} + q^{24} + q^{26} + q^{27} - 4q^{28} + 5q^{29} + 4q^{31} + q^{32} + 4q^{33} - 4q^{34} + q^{36} + 9q^{37} + 7q^{38} + q^{39} - 5q^{41} - 4q^{42} - 10q^{43} + 4q^{44} + 4q^{46} + 3q^{47} + q^{48} + 9q^{49} - 4q^{51} + q^{52} + 9q^{53} + q^{54} - 4q^{56} + 7q^{57} + 5q^{58} - 6q^{59} + 4q^{61} + 4q^{62} - 4q^{63} + q^{64} + 4q^{66} - 7q^{67} - 4q^{68} + 4q^{69} - 15q^{71} + q^{72} + 12q^{73} + 9q^{74} + 7q^{76} - 16q^{77} + q^{78} + 7q^{79} + q^{81} - 5q^{82} + 6q^{83} - 4q^{84} - 10q^{86} + 5q^{87} + 4q^{88} + 14q^{89} - 4q^{91} + 4q^{92} + 4q^{93} + 3q^{94} + q^{96} - 16q^{97} + 9q^{98} + 4q^{99} + O(q^{100})$$

## 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 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## 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 1950.2.a.x yes 1
3.b odd 2 1 5850.2.a.a 1
5.b even 2 1 1950.2.a.e 1
5.c odd 4 2 1950.2.e.n 2
15.d odd 2 1 5850.2.a.by 1
15.e even 4 2 5850.2.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.e 1 5.b even 2 1
1950.2.a.x yes 1 1.a even 1 1 trivial
1950.2.e.n 2 5.c odd 4 2
5850.2.a.a 1 3.b odd 2 1
5850.2.a.by 1 15.d odd 2 1
5850.2.e.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7} + 4$$ $$T_{11} - 4$$ $$T_{17} + 4$$ $$T_{23} - 4$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-4 + T$$
$13$ $$-1 + T$$
$17$ $$4 + T$$
$19$ $$-7 + T$$
$23$ $$-4 + T$$
$29$ $$-5 + T$$
$31$ $$-4 + T$$
$37$ $$-9 + T$$
$41$ $$5 + T$$
$43$ $$10 + T$$
$47$ $$-3 + T$$
$53$ $$-9 + T$$
$59$ $$6 + T$$
$61$ $$-4 + T$$
$67$ $$7 + T$$
$71$ $$15 + T$$
$73$ $$-12 + T$$
$79$ $$-7 + T$$
$83$ $$-6 + T$$
$89$ $$-14 + T$$
$97$ $$16 + T$$