# Properties

 Label 1950.2.a.n 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

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

## 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.n 1
3.b odd 2 1 5850.2.a.c 1
5.b even 2 1 390.2.a.c 1
5.c odd 4 2 1950.2.e.e 2
15.d odd 2 1 1170.2.a.n 1
15.e even 4 2 5850.2.e.m 2
20.d odd 2 1 3120.2.a.a 1
60.h even 2 1 9360.2.a.bc 1
65.d even 2 1 5070.2.a.u 1
65.g odd 4 2 5070.2.b.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.c 1 5.b even 2 1
1170.2.a.n 1 15.d odd 2 1
1950.2.a.n 1 1.a even 1 1 trivial
1950.2.e.e 2 5.c odd 4 2
3120.2.a.a 1 20.d odd 2 1
5070.2.a.u 1 65.d even 2 1
5070.2.b.i 2 65.g odd 4 2
5850.2.a.c 1 3.b odd 2 1
5850.2.e.m 2 15.e even 4 2
9360.2.a.bc 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7} + 4$$ $$T_{11}$$ $$T_{17} - 2$$ $$T_{23} + 8$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

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