# Properties

 Label 950.2.a.i Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.58578819202$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 2 + \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} ) q^{12} + ( -3 - \beta_{2} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + q^{16} + ( 1 + \beta_{2} ) q^{17} + ( -2 - \beta_{2} ) q^{18} + q^{19} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{21} + ( \beta_{1} + \beta_{2} ) q^{22} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 - \beta_{1} ) q^{24} + ( 3 + \beta_{2} ) q^{26} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{27} + ( -1 - \beta_{1} ) q^{28} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{29} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{31} - q^{32} + ( -2 - \beta_{1} + \beta_{2} ) q^{33} + ( -1 - \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( -4 - 2 \beta_{1} ) q^{37} - q^{38} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 3 \beta_{1} + \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{42} + ( -6 - \beta_{1} - \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{44} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{46} + ( 4 - 2 \beta_{2} ) q^{47} + ( -1 + \beta_{1} ) q^{48} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{49} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( -3 - \beta_{2} ) q^{52} + ( -5 + \beta_{2} ) q^{53} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{54} + ( 1 + \beta_{1} ) q^{56} + ( -1 + \beta_{1} ) q^{57} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{58} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{59} + ( -10 - \beta_{1} - \beta_{2} ) q^{61} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{62} -2 \beta_{1} q^{63} + q^{64} + ( 2 + \beta_{1} - \beta_{2} ) q^{66} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 1 + \beta_{2} ) q^{68} + ( -9 - 5 \beta_{2} ) q^{69} + ( -4 + 5 \beta_{1} + \beta_{2} ) q^{71} + ( -2 - \beta_{2} ) q^{72} + ( -5 + 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 4 + 2 \beta_{1} ) q^{74} + q^{76} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{77} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -2 + 6 \beta_{1} ) q^{79} + ( -5 - 2 \beta_{1} ) q^{81} + ( -3 \beta_{1} - \beta_{2} ) q^{82} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{84} + ( 6 + \beta_{1} + \beta_{2} ) q^{86} + ( 1 - 5 \beta_{1} + 4 \beta_{2} ) q^{87} + ( \beta_{1} + \beta_{2} ) q^{88} + ( -4 - \beta_{1} + \beta_{2} ) q^{89} + ( 1 + 3 \beta_{1} ) q^{91} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{92} + ( -4 + 3 \beta_{1} - 5 \beta_{2} ) q^{93} + ( -4 + 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} ) q^{96} + ( -2 + 4 \beta_{2} ) q^{97} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{98} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} - 3q^{8} + 5q^{9} + O(q^{10})$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 4q^{7} - 3q^{8} + 5q^{9} - 2q^{12} - 8q^{13} + 4q^{14} + 3q^{16} + 2q^{17} - 5q^{18} + 3q^{19} - 10q^{21} + 2q^{24} + 8q^{26} - 2q^{27} - 4q^{28} - 8q^{29} + 4q^{31} - 3q^{32} - 8q^{33} - 2q^{34} + 5q^{36} - 14q^{37} - 3q^{38} + 10q^{39} + 2q^{41} + 10q^{42} - 18q^{43} + 14q^{47} - 2q^{48} - 3q^{49} - 6q^{51} - 8q^{52} - 16q^{53} + 2q^{54} + 4q^{56} - 2q^{57} + 8q^{58} + 2q^{59} - 30q^{61} - 4q^{62} - 2q^{63} + 3q^{64} + 8q^{66} - 2q^{67} + 2q^{68} - 22q^{69} - 8q^{71} - 5q^{72} - 10q^{73} + 14q^{74} + 3q^{76} + 8q^{77} - 10q^{78} - 17q^{81} - 2q^{82} + 6q^{83} - 10q^{84} + 18q^{86} - 6q^{87} - 14q^{89} + 6q^{91} - 4q^{93} - 14q^{94} + 2q^{96} - 10q^{97} + 3q^{98} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.76156 −0.363328 3.12489
−1.00000 −2.76156 1.00000 0 2.76156 0.761557 −1.00000 4.62620 0
1.2 −1.00000 −1.36333 1.00000 0 1.36333 −0.636672 −1.00000 −1.14134 0
1.3 −1.00000 2.12489 1.00000 0 −2.12489 −4.12489 −1.00000 1.51514 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 950.2.a.i 3
3.b odd 2 1 8550.2.a.cl 3
4.b odd 2 1 7600.2.a.cd 3
5.b even 2 1 950.2.a.n 3
5.c odd 4 2 190.2.b.b 6
15.d odd 2 1 8550.2.a.ck 3
15.e even 4 2 1710.2.d.d 6
20.d odd 2 1 7600.2.a.bi 3
20.e even 4 2 1520.2.d.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 5.c odd 4 2
950.2.a.i 3 1.a even 1 1 trivial
950.2.a.n 3 5.b even 2 1
1520.2.d.j 6 20.e even 4 2
1710.2.d.d 6 15.e even 4 2
7600.2.a.bi 3 20.d odd 2 1
7600.2.a.cd 3 4.b odd 2 1
8550.2.a.ck 3 15.d odd 2 1
8550.2.a.cl 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 8$$ $$T_{7}^{3} + 4 T_{7}^{2} - T_{7} - 2$$ $$T_{11}^{3} - 10 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-8 - 5 T + 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-2 - T + 4 T^{2} + T^{3}$$
$11$ $$-8 - 10 T + T^{3}$$
$13$ $$-2 + 13 T + 8 T^{2} + T^{3}$$
$17$ $$4 - 7 T - 2 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-122 - 49 T + T^{3}$$
$29$ $$-410 - 51 T + 8 T^{2} + T^{3}$$
$31$ $$232 - 62 T - 4 T^{2} + T^{3}$$
$37$ $$16 + 40 T + 14 T^{2} + T^{3}$$
$41$ $$-100 - 50 T - 2 T^{2} + T^{3}$$
$43$ $$148 + 98 T + 18 T^{2} + T^{3}$$
$47$ $$64 + 32 T - 14 T^{2} + T^{3}$$
$53$ $$106 + 77 T + 16 T^{2} + T^{3}$$
$59$ $$80 - 29 T - 2 T^{2} + T^{3}$$
$61$ $$892 + 290 T + 30 T^{2} + T^{3}$$
$67$ $$-64 - 61 T + 2 T^{2} + T^{3}$$
$71$ $$-1016 - 122 T + 8 T^{2} + T^{3}$$
$73$ $$164 - 95 T + 10 T^{2} + T^{3}$$
$79$ $$-880 - 228 T + T^{3}$$
$83$ $$8 - 28 T - 6 T^{2} + T^{3}$$
$89$ $$-20 + 46 T + 14 T^{2} + T^{3}$$
$97$ $$-488 - 100 T + 10 T^{2} + T^{3}$$