# Properties

 Label 1950.2.a.q Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $1$ 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: $$1$$ 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

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

## 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 −1.00000 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.q yes 1
3.b odd 2 1 5850.2.a.j 1
5.b even 2 1 1950.2.a.m 1
5.c odd 4 2 1950.2.e.b 2
15.d odd 2 1 5850.2.a.bt 1
15.e even 4 2 5850.2.e.x 2

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

 $$T_{7} + 1$$ $$T_{11} + 3$$ $$T_{17} + 1$$ $$T_{23} - 4$$ $$T_{31} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$3 + T$$
$13$ $$-1 + T$$
$17$ $$1 + T$$
$19$ $$8 + T$$
$23$ $$-4 + T$$
$29$ $$7 + T$$
$31$ $$-1 + T$$
$37$ $$-4 + T$$
$41$ $$6 + T$$
$43$ $$12 + T$$
$47$ $$-3 + T$$
$53$ $$-5 + T$$
$59$ $$9 + T$$
$61$ $$-5 + T$$
$67$ $$11 + T$$
$71$ $$-8 + T$$
$73$ $$T$$
$79$ $$8 + T$$
$83$ $$-7 + T$$
$89$ $$8 + T$$
$97$ $$-6 + T$$