# Properties

 Label 950.2.a.j Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.993.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( -1 + \beta_{2} ) q^{7} - q^{8} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( -1 + \beta_{1} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{6} + ( -1 + \beta_{2} ) q^{7} - q^{8} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} + 2 \beta_{1} q^{11} + ( -1 + \beta_{1} ) q^{12} + ( 2 - \beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + q^{16} + ( -4 + \beta_{1} - \beta_{2} ) q^{17} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{18} + q^{19} + ( 2 + \beta_{1} ) q^{21} -2 \beta_{1} q^{22} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{23} + ( 1 - \beta_{1} ) q^{24} + ( -2 + \beta_{1} - \beta_{2} ) q^{26} + ( -6 + 3 \beta_{1} - 2 \beta_{2} ) q^{27} + ( -1 + \beta_{2} ) q^{28} + ( 2 + \beta_{1} - \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} ) q^{31} - q^{32} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( 4 - \beta_{1} + \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{36} + ( 1 - 2 \beta_{1} ) q^{37} - q^{38} + ( -5 + 5 \beta_{1} - \beta_{2} ) q^{39} + ( -2 - \beta_{1} ) q^{42} + ( 4 + 2 \beta_{1} ) q^{43} + 2 \beta_{1} q^{44} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{46} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -1 + \beta_{1} ) q^{48} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{49} + ( 7 - 7 \beta_{1} + \beta_{2} ) q^{51} + ( 2 - \beta_{1} + \beta_{2} ) q^{52} + ( 5 - 4 \beta_{1} - \beta_{2} ) q^{53} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{54} + ( 1 - \beta_{2} ) q^{56} + ( -1 + \beta_{1} ) q^{57} + ( -2 - \beta_{1} + \beta_{2} ) q^{58} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{59} -2 \beta_{2} q^{61} + ( -2 - 2 \beta_{1} ) q^{62} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{63} + q^{64} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{66} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{67} + ( -4 + \beta_{1} - \beta_{2} ) q^{68} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{69} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{72} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{73} + ( -1 + 2 \beta_{1} ) q^{74} + q^{76} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 5 - 5 \beta_{1} + \beta_{2} ) q^{78} + ( 8 - 4 \beta_{1} ) q^{79} + ( 10 - 7 \beta_{1} ) q^{81} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 2 + \beta_{1} ) q^{84} + ( -4 - 2 \beta_{1} ) q^{86} + ( 1 - \beta_{1} + \beta_{2} ) q^{87} -2 \beta_{1} q^{88} + ( 6 - 4 \beta_{1} ) q^{89} + ( 6 + 2 \beta_{1} - \beta_{2} ) q^{91} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{92} + ( 6 + 2 \beta_{2} ) q^{93} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} ) q^{96} + ( 8 + 6 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{98} + ( -14 + 8 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 5q^{9} + O(q^{10})$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 5q^{9} + 2q^{11} - 2q^{12} + 6q^{13} + 2q^{14} + 3q^{16} - 12q^{17} - 5q^{18} + 3q^{19} + 7q^{21} - 2q^{22} + 2q^{23} + 2q^{24} - 6q^{26} - 17q^{27} - 2q^{28} + 6q^{29} + 8q^{31} - 3q^{32} + 24q^{33} + 12q^{34} + 5q^{36} + q^{37} - 3q^{38} - 11q^{39} - 7q^{42} + 14q^{43} + 2q^{44} - 2q^{46} - 3q^{47} - 2q^{48} + 9q^{49} + 15q^{51} + 6q^{52} + 10q^{53} + 17q^{54} + 2q^{56} - 2q^{57} - 6q^{58} + 6q^{59} - 2q^{61} - 8q^{62} + 14q^{63} + 3q^{64} - 24q^{66} - 4q^{67} - 12q^{68} - 6q^{71} - 5q^{72} + 12q^{73} - q^{74} + 3q^{76} + 10q^{77} + 11q^{78} + 20q^{79} + 23q^{81} + 4q^{83} + 7q^{84} - 14q^{86} + 3q^{87} - 2q^{88} + 14q^{89} + 19q^{91} + 2q^{92} + 20q^{93} + 3q^{94} + 2q^{96} + 28q^{97} - 9q^{98} - 36q^{99} + O(q^{100})$$

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

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

## 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
 −2.25342 0.480031 2.77339
−1.00000 −3.25342 1.00000 0 3.25342 0.0778929 −1.00000 7.58473 0
1.2 −1.00000 −0.519969 1.00000 0 0.519969 −4.76957 −1.00000 −2.72963 0
1.3 −1.00000 1.77339 1.00000 0 −1.77339 2.69168 −1.00000 0.144903 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.j 3
3.b odd 2 1 8550.2.a.cp 3
4.b odd 2 1 7600.2.a.bz 3
5.b even 2 1 950.2.a.l yes 3
5.c odd 4 2 950.2.b.h 6
15.d odd 2 1 8550.2.a.ci 3
20.d odd 2 1 7600.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 1.a even 1 1 trivial
950.2.a.l yes 3 5.b even 2 1
950.2.b.h 6 5.c odd 4 2
7600.2.a.bk 3 20.d odd 2 1
7600.2.a.bz 3 4.b odd 2 1
8550.2.a.ci 3 15.d odd 2 1
8550.2.a.cp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 3$$ $$T_{7}^{3} + 2 T_{7}^{2} - 13 T_{7} + 1$$ $$T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-3 - 5 T + 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$1 - 13 T + 2 T^{2} + T^{3}$$
$11$ $$24 - 24 T - 2 T^{2} + T^{3}$$
$13$ $$35 - 3 T - 6 T^{2} + T^{3}$$
$17$ $$-9 + 33 T + 12 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-111 - 51 T - 2 T^{2} + T^{3}$$
$29$ $$9 - 3 T - 6 T^{2} + T^{3}$$
$31$ $$56 - 4 T - 8 T^{2} + T^{3}$$
$37$ $$1 - 25 T - T^{2} + T^{3}$$
$41$ $$T^{3}$$
$43$ $$24 + 40 T - 14 T^{2} + T^{3}$$
$47$ $$45 - 57 T + 3 T^{2} + T^{3}$$
$53$ $$867 - 105 T - 10 T^{2} + T^{3}$$
$59$ $$1431 - 201 T - 6 T^{2} + T^{3}$$
$61$ $$-120 - 56 T + 2 T^{2} + T^{3}$$
$67$ $$-75 - 23 T + 4 T^{2} + T^{3}$$
$71$ $$-1512 - 192 T + 6 T^{2} + T^{3}$$
$73$ $$317 - 27 T - 12 T^{2} + T^{3}$$
$79$ $$320 + 32 T - 20 T^{2} + T^{3}$$
$83$ $$-168 - 108 T - 4 T^{2} + T^{3}$$
$89$ $$312 - 36 T - 14 T^{2} + T^{3}$$
$97$ $$2440 + 44 T - 28 T^{2} + T^{3}$$