# Properties

 Label 1950.2.a.i 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: no (minimal twist has level 390) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 0 −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.i 1
3.b odd 2 1 5850.2.a.bs 1
5.b even 2 1 1950.2.a.r 1
5.c odd 4 2 390.2.e.a 2
15.d odd 2 1 5850.2.a.o 1
15.e even 4 2 1170.2.e.d 2
20.e even 4 2 3120.2.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.a 2 5.c odd 4 2
1170.2.e.d 2 15.e even 4 2
1950.2.a.i 1 1.a even 1 1 trivial
1950.2.a.r 1 5.b even 2 1
3120.2.l.a 2 20.e even 4 2
5850.2.a.o 1 15.d odd 2 1
5850.2.a.bs 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7}$$ $$T_{11} + 6$$ $$T_{17}$$ $$T_{23} + 6$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$1 + T$$
$17$ $$T$$
$19$ $$-6 + T$$
$23$ $$6 + T$$
$29$ $$-2 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$6 + T$$
$43$ $$8 + T$$
$47$ $$8 + T$$
$53$ $$-6 + T$$
$59$ $$-10 + T$$
$61$ $$6 + T$$
$67$ $$4 + T$$
$71$ $$8 + T$$
$73$ $$6 + T$$
$79$ $$-16 + T$$
$83$ $$4 + T$$
$89$ $$10 + T$$
$97$ $$2 + T$$