# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
1.00000 −2.73205 1.00000 −0.732051 −2.73205 1.00000 1.00000 4.46410 −0.732051
1.2 1.00000 0.732051 1.00000 2.73205 0.732051 1.00000 1.00000 −2.46410 2.73205
 $$n$$: 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.f 2
3.b odd 2 1 2898.2.a.w 2
4.b odd 2 1 2576.2.a.u 2
5.b even 2 1 8050.2.a.x 2
7.b odd 2 1 2254.2.a.o 2
23.b odd 2 1 7406.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.f 2 1.a even 1 1 trivial
2254.2.a.o 2 7.b odd 2 1
2576.2.a.u 2 4.b odd 2 1
2898.2.a.w 2 3.b odd 2 1
7406.2.a.s 2 23.b odd 2 1
8050.2.a.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(322))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-2 + 2 T + T^{2}$$
$5$ $$-2 - 2 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$-2 - 2 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$-2 + 10 T + T^{2}$$
$37$ $$4 - 8 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$-18 + 6 T + T^{2}$$
$53$ $$-12 + T^{2}$$
$59$ $$-66 + 6 T + T^{2}$$
$61$ $$-18 - 6 T + T^{2}$$
$67$ $$52 + 16 T + T^{2}$$
$71$ $$-104 + 4 T + T^{2}$$
$73$ $$4 + 8 T + T^{2}$$
$79$ $$-12 + 12 T + T^{2}$$
$83$ $$-92 - 8 T + T^{2}$$
$89$ $$54 - 18 T + T^{2}$$
$97$ $$46 + 14 T + T^{2}$$
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