# Properties

 Label 322.2.a.g Level $322$ Weight $2$ Character orbit 322.a Self dual yes Analytic conductor $2.571$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + ( 1 - \beta_{2} ) q^{5} + ( 1 + \beta_{2} ) q^{6} - q^{7} + q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 + \beta_{2} ) q^{3} + q^{4} + ( 1 - \beta_{2} ) q^{5} + ( 1 + \beta_{2} ) q^{6} - q^{7} + q^{8} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} + ( 1 - \beta_{2} ) q^{10} + ( -2 + \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( \beta_{1} - \beta_{2} ) q^{13} - q^{14} + ( -4 + \beta_{1} + \beta_{2} ) q^{15} + q^{16} + ( 3 - \beta_{1} ) q^{17} + ( 3 - \beta_{1} + \beta_{2} ) q^{18} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( 1 - \beta_{2} ) q^{20} + ( -1 - \beta_{2} ) q^{21} + ( -2 + \beta_{1} - \beta_{2} ) q^{22} - q^{23} + ( 1 + \beta_{2} ) q^{24} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{25} + ( \beta_{1} - \beta_{2} ) q^{26} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} - q^{28} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{29} + ( -4 + \beta_{1} + \beta_{2} ) q^{30} + ( -1 + \beta_{1} ) q^{31} + q^{32} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{33} + ( 3 - \beta_{1} ) q^{34} + ( -1 + \beta_{2} ) q^{35} + ( 3 - \beta_{1} + \beta_{2} ) q^{36} + ( -4 + 2 \beta_{1} ) q^{37} + ( -\beta_{1} - \beta_{2} ) q^{38} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( 1 - \beta_{2} ) q^{40} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -1 - \beta_{2} ) q^{42} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} - \beta_{2} ) q^{44} + ( -1 - 3 \beta_{2} ) q^{45} - q^{46} + ( 1 - \beta_{1} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + q^{49} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{50} + ( 2 - \beta_{1} + 5 \beta_{2} ) q^{51} + ( \beta_{1} - \beta_{2} ) q^{52} -2 \beta_{1} q^{53} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{54} + ( 2 + 2 \beta_{2} ) q^{55} - q^{56} + ( -6 + 2 \beta_{2} ) q^{57} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{58} + ( 3 + 3 \beta_{2} ) q^{59} + ( -4 + \beta_{1} + \beta_{2} ) q^{60} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{61} + ( -1 + \beta_{1} ) q^{62} + ( -3 + \beta_{1} - \beta_{2} ) q^{63} + q^{64} + 4 q^{65} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{66} + ( 2 - \beta_{1} + \beta_{2} ) q^{67} + ( 3 - \beta_{1} ) q^{68} + ( -1 - \beta_{2} ) q^{69} + ( -1 + \beta_{2} ) q^{70} + ( -10 + 2 \beta_{1} ) q^{71} + ( 3 - \beta_{1} + \beta_{2} ) q^{72} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -4 + 2 \beta_{1} ) q^{74} + ( -15 + 2 \beta_{1} + 3 \beta_{2} ) q^{75} + ( -\beta_{1} - \beta_{2} ) q^{76} + ( 2 - \beta_{1} + \beta_{2} ) q^{77} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -8 - \beta_{1} - 5 \beta_{2} ) q^{79} + ( 1 - \beta_{2} ) q^{80} + ( 3 - \beta_{1} + 5 \beta_{2} ) q^{81} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 10 - \beta_{1} + \beta_{2} ) q^{83} + ( -1 - \beta_{2} ) q^{84} + ( 4 - \beta_{1} - 5 \beta_{2} ) q^{85} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{86} + ( 12 - 4 \beta_{1} ) q^{87} + ( -2 + \beta_{1} - \beta_{2} ) q^{88} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{89} + ( -1 - 3 \beta_{2} ) q^{90} + ( -\beta_{1} + \beta_{2} ) q^{91} - q^{92} + ( \beta_{1} - 3 \beta_{2} ) q^{93} + ( 1 - \beta_{1} ) q^{94} + ( 6 - 2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( 1 + \beta_{2} ) q^{96} + ( 7 - \beta_{1} + 4 \beta_{2} ) q^{97} + q^{98} + ( -18 + 3 \beta_{1} - 7 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 2q^{3} + 3q^{4} + 4q^{5} + 2q^{6} - 3q^{7} + 3q^{8} + 7q^{9} + O(q^{10})$$ $$3q + 3q^{2} + 2q^{3} + 3q^{4} + 4q^{5} + 2q^{6} - 3q^{7} + 3q^{8} + 7q^{9} + 4q^{10} - 4q^{11} + 2q^{12} + 2q^{13} - 3q^{14} - 12q^{15} + 3q^{16} + 8q^{17} + 7q^{18} + 4q^{20} - 2q^{21} - 4q^{22} - 3q^{23} + 2q^{24} + 5q^{25} + 2q^{26} + 8q^{27} - 3q^{28} - 10q^{29} - 12q^{30} - 2q^{31} + 3q^{32} - 12q^{33} + 8q^{34} - 4q^{35} + 7q^{36} - 10q^{37} - 8q^{39} + 4q^{40} + 6q^{41} - 2q^{42} - 4q^{43} - 4q^{44} - 3q^{46} + 2q^{47} + 2q^{48} + 3q^{49} + 5q^{50} + 2q^{52} - 2q^{53} + 8q^{54} + 4q^{55} - 3q^{56} - 20q^{57} - 10q^{58} + 6q^{59} - 12q^{60} - 8q^{61} - 2q^{62} - 7q^{63} + 3q^{64} + 12q^{65} - 12q^{66} + 4q^{67} + 8q^{68} - 2q^{69} - 4q^{70} - 28q^{71} + 7q^{72} - 6q^{73} - 10q^{74} - 46q^{75} + 4q^{77} - 8q^{78} - 20q^{79} + 4q^{80} + 3q^{81} + 6q^{82} + 28q^{83} - 2q^{84} + 16q^{85} - 4q^{86} + 32q^{87} - 4q^{88} - 2q^{91} - 3q^{92} + 4q^{93} + 2q^{94} + 20q^{95} + 2q^{96} + 16q^{97} + 3q^{98} - 44q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 6$$$$)/2$$

## 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
 0.470683 2.34292 −1.81361
1.00000 −2.24914 1.00000 4.24914 −2.24914 −1.00000 1.00000 2.05863 4.24914
1.2 1.00000 1.14637 1.00000 0.853635 1.14637 −1.00000 1.00000 −1.68585 0.853635
1.3 1.00000 3.10278 1.00000 −1.10278 3.10278 −1.00000 1.00000 6.62721 −1.10278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

## 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 322.2.a.g 3
3.b odd 2 1 2898.2.a.be 3
4.b odd 2 1 2576.2.a.w 3
5.b even 2 1 8050.2.a.bh 3
7.b odd 2 1 2254.2.a.p 3
23.b odd 2 1 7406.2.a.x 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.g 3 1.a even 1 1 trivial
2254.2.a.p 3 7.b odd 2 1
2576.2.a.w 3 4.b odd 2 1
2898.2.a.be 3 3.b odd 2 1
7406.2.a.x 3 23.b odd 2 1
8050.2.a.bh 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(322))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8$$ $$T_{5}^{3} - 4 T_{5}^{2} - 2 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$8 - 6 T - 2 T^{2} + T^{3}$$
$5$ $$4 - 2 T - 4 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-16 - 12 T + 4 T^{2} + T^{3}$$
$13$ $$16 - 16 T - 2 T^{2} + T^{3}$$
$17$ $$44 + 6 T - 8 T^{2} + T^{3}$$
$19$ $$-16 - 28 T + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$-352 - 32 T + 10 T^{2} + T^{3}$$
$31$ $$-32 - 14 T + 2 T^{2} + T^{3}$$
$37$ $$-344 - 28 T + 10 T^{2} + T^{3}$$
$41$ $$344 - 100 T - 6 T^{2} + T^{3}$$
$43$ $$-128 - 64 T + 4 T^{2} + T^{3}$$
$47$ $$32 - 14 T - 2 T^{2} + T^{3}$$
$53$ $$136 - 60 T + 2 T^{2} + T^{3}$$
$59$ $$216 - 54 T - 6 T^{2} + T^{3}$$
$61$ $$-524 - 74 T + 8 T^{2} + T^{3}$$
$67$ $$16 - 12 T - 4 T^{2} + T^{3}$$
$71$ $$64 + 200 T + 28 T^{2} + T^{3}$$
$73$ $$-1448 - 220 T + 6 T^{2} + T^{3}$$
$79$ $$-2432 - 92 T + 20 T^{2} + T^{3}$$
$83$ $$-656 + 244 T - 28 T^{2} + T^{3}$$
$89$ $$76 - 34 T + T^{3}$$
$97$ $$172 - 26 T - 16 T^{2} + T^{3}$$