# Properties

 Label 1950.2.a.c 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

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

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

## 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.c 1
3.b odd 2 1 5850.2.a.bj 1
5.b even 2 1 1950.2.a.z 1
5.c odd 4 2 390.2.e.c 2
15.d odd 2 1 5850.2.a.t 1
15.e even 4 2 1170.2.e.a 2
20.e even 4 2 3120.2.l.g 2

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

 $$T_{7} + 2$$ $$T_{11} - 2$$ $$T_{17} + 2$$ $$T_{23}$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$2 + T$$
$11$ $$-2 + T$$
$13$ $$-1 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$T$$
$29$ $$-4 + T$$
$31$ $$-8 + T$$
$37$ $$6 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$2 + T$$
$59$ $$-10 + T$$
$61$ $$14 + T$$
$67$ $$-16 + T$$
$71$ $$4 + T$$
$73$ $$8 + T$$
$79$ $$8 + T$$
$83$ $$12 + T$$
$89$ $$-6 + T$$
$97$ $$12 + T$$