# Properties

 Label 950.2.a.k 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: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{7} - q^{8} + ( 4 + \beta_{1} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{7} - q^{8} + ( 4 + \beta_{1} ) q^{9} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{12} + ( 2 \beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{14} + q^{16} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{17} + ( -4 - \beta_{1} ) q^{18} - q^{19} + ( -2 - 5 \beta_{1} + 3 \beta_{2} ) q^{21} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{22} + ( -5 + \beta_{1} - \beta_{2} ) q^{23} + ( 1 - \beta_{1} - \beta_{2} ) q^{24} + ( -2 \beta_{1} - \beta_{2} ) q^{26} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{27} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{28} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} - q^{32} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{34} + ( 4 + \beta_{1} ) q^{36} + ( 5 + 2 \beta_{1} - 2 \beta_{2} ) q^{37} + q^{38} + ( 9 + 2 \beta_{1} + 3 \beta_{2} ) q^{39} + ( -4 + 4 \beta_{1} ) q^{41} + ( 2 + 5 \beta_{1} - 3 \beta_{2} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 5 - \beta_{1} + \beta_{2} ) q^{46} + ( -3 - 4 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} ) q^{48} + ( 9 - 2 \beta_{1} - 3 \beta_{2} ) q^{49} + ( -5 + 4 \beta_{1} - 3 \beta_{2} ) q^{51} + ( 2 \beta_{1} + \beta_{2} ) q^{52} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{53} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{54} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{56} + ( 1 - \beta_{1} - \beta_{2} ) q^{57} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{59} + ( 8 - 2 \beta_{1} + 4 \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{62} + ( -1 + \beta_{1} - 9 \beta_{2} ) q^{63} + q^{64} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{66} + ( 1 - 3 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{68} + ( 5 - 7 \beta_{1} - 2 \beta_{2} ) q^{69} -2 \beta_{1} q^{71} + ( -4 - \beta_{1} ) q^{72} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -5 - 2 \beta_{1} + 2 \beta_{2} ) q^{74} - q^{76} + ( 18 - 4 \beta_{1} - 6 \beta_{2} ) q^{77} + ( -9 - 2 \beta_{1} - 3 \beta_{2} ) q^{78} + ( 8 + 4 \beta_{2} ) q^{79} + ( -2 + 5 \beta_{1} + \beta_{2} ) q^{81} + ( 4 - 4 \beta_{1} ) q^{82} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -2 - 5 \beta_{1} + 3 \beta_{2} ) q^{84} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{86} + ( 5 + 6 \beta_{1} + 7 \beta_{2} ) q^{87} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{88} -10 q^{89} -7 \beta_{1} q^{91} + ( -5 + \beta_{1} - \beta_{2} ) q^{92} + ( 14 + 2 \beta_{1} ) q^{93} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} - \beta_{2} ) q^{96} -6 \beta_{2} q^{97} + ( -9 + 2 \beta_{1} + 3 \beta_{2} ) q^{98} + ( 6 + 6 \beta_{1} - 8 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 13q^{9} + O(q^{10})$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 13q^{9} + 2q^{11} - 2q^{12} + 2q^{13} + 2q^{14} + 3q^{16} + 4q^{17} - 13q^{18} - 3q^{19} - 11q^{21} - 2q^{22} - 14q^{23} + 2q^{24} - 2q^{26} + 7q^{27} - 2q^{28} + 14q^{29} - 4q^{31} - 3q^{32} - 4q^{33} - 4q^{34} + 13q^{36} + 17q^{37} + 3q^{38} + 29q^{39} - 8q^{41} + 11q^{42} - 2q^{43} + 2q^{44} + 14q^{46} - 13q^{47} - 2q^{48} + 25q^{49} - 11q^{51} + 2q^{52} + 10q^{53} - 7q^{54} + 2q^{56} + 2q^{57} - 14q^{58} - 6q^{59} + 22q^{61} + 4q^{62} - 2q^{63} + 3q^{64} + 4q^{66} + 4q^{68} + 8q^{69} - 2q^{71} - 13q^{72} + 12q^{73} - 17q^{74} - 3q^{76} + 50q^{77} - 29q^{78} + 24q^{79} - q^{81} + 8q^{82} - 12q^{83} - 11q^{84} + 2q^{86} + 21q^{87} - 2q^{88} - 30q^{89} - 7q^{91} - 14q^{92} + 44q^{93} + 13q^{94} + 2q^{96} - 25q^{98} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 0.713538 −1.91223 2.19869
−1.00000 −2.77733 1.00000 0 2.77733 4.69527 −1.00000 4.71354 0
1.2 −1.00000 −2.25561 1.00000 0 2.25561 −4.22547 −1.00000 2.08777 0
1.3 −1.00000 3.03293 1.00000 0 −3.03293 −2.46980 −1.00000 6.19869 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.k 3
3.b odd 2 1 8550.2.a.co 3
4.b odd 2 1 7600.2.a.cb 3
5.b even 2 1 950.2.a.m yes 3
5.c odd 4 2 950.2.b.g 6
15.d odd 2 1 8550.2.a.cj 3
20.d odd 2 1 7600.2.a.bm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 1.a even 1 1 trivial
950.2.a.m yes 3 5.b even 2 1
950.2.b.g 6 5.c odd 4 2
7600.2.a.bm 3 20.d odd 2 1
7600.2.a.cb 3 4.b odd 2 1
8550.2.a.cj 3 15.d odd 2 1
8550.2.a.co 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 9 T_{3} - 19$$ $$T_{7}^{3} + 2 T_{7}^{2} - 21 T_{7} - 49$$ $$T_{11}^{3} - 2 T_{11}^{2} - 32 T_{11} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-19 - 9 T + 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-49 - 21 T + 2 T^{2} + T^{3}$$
$11$ $$24 - 32 T - 2 T^{2} + T^{3}$$
$13$ $$-21 - 23 T - 2 T^{2} + T^{3}$$
$17$ $$-7 - 15 T - 4 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$63 + 57 T + 14 T^{2} + T^{3}$$
$29$ $$-25 + 41 T - 14 T^{2} + T^{3}$$
$31$ $$-152 - 36 T + 4 T^{2} + T^{3}$$
$37$ $$9 + 63 T - 17 T^{2} + T^{3}$$
$41$ $$-64 - 48 T + 8 T^{2} + T^{3}$$
$43$ $$72 - 40 T + 2 T^{2} + T^{3}$$
$47$ $$-525 - 25 T + 13 T^{2} + T^{3}$$
$53$ $$3 + 11 T - 10 T^{2} + T^{3}$$
$59$ $$-175 - 29 T + 6 T^{2} + T^{3}$$
$61$ $$536 + 72 T - 22 T^{2} + T^{3}$$
$67$ $$469 - 131 T + T^{3}$$
$71$ $$-24 - 16 T + 2 T^{2} + T^{3}$$
$73$ $$243 - 27 T - 12 T^{2} + T^{3}$$
$79$ $$320 + 112 T - 24 T^{2} + T^{3}$$
$83$ $$-1544 - 164 T + 12 T^{2} + T^{3}$$
$89$ $$( 10 + T )^{3}$$
$97$ $$-648 - 180 T + T^{3}$$