# Properties

 Label 322.2.a.e Level $322$ Weight $2$ Character orbit 322.a Self dual yes Analytic conductor $2.571$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−1.00000 −3.23607 1.00000 1.23607 3.23607 −1.00000 −1.00000 7.47214 −1.23607
1.2 −1.00000 1.23607 1.00000 −3.23607 −1.23607 −1.00000 −1.00000 −1.47214 3.23607
 $$n$$: 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.e 2
3.b odd 2 1 2898.2.a.bd 2
4.b odd 2 1 2576.2.a.t 2
5.b even 2 1 8050.2.a.bf 2
7.b odd 2 1 2254.2.a.k 2
23.b odd 2 1 7406.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.e 2 1.a even 1 1 trivial
2254.2.a.k 2 7.b odd 2 1
2576.2.a.t 2 4.b odd 2 1
2898.2.a.bd 2 3.b odd 2 1
7406.2.a.j 2 23.b odd 2 1
8050.2.a.bf 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(322))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-4 + 2 T + T^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-16 + 4 T + T^{2}$$
$17$ $$20 + 10 T + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-4 - 2 T + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$-16 + 4 T + T^{2}$$
$47$ $$76 + 18 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$76 - 18 T + T^{2}$$
$61$ $$-20 - 10 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-16 - 4 T + T^{2}$$
$73$ $$-180 + T^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-76 - 4 T + T^{2}$$
$89$ $$-100 + 10 T + T^{2}$$
$97$ $$-116 + 6 T + T^{2}$$