# Properties

 Label 722.2.a.a Level $722$ Weight $2$ Character orbit 722.a Self dual yes Analytic conductor $5.765$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.76519902594$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
−1.00000 −3.00000 1.00000 2.00000 3.00000 −3.00000 −1.00000 6.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$19$$ $$1$$

## 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 722.2.a.a 1
3.b odd 2 1 6498.2.a.m 1
4.b odd 2 1 5776.2.a.q 1
19.b odd 2 1 722.2.a.f yes 1
19.c even 3 2 722.2.c.g 2
19.d odd 6 2 722.2.c.a 2
19.e even 9 6 722.2.e.h 6
19.f odd 18 6 722.2.e.g 6
57.d even 2 1 6498.2.a.a 1
76.d even 2 1 5776.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.a 1 1.a even 1 1 trivial
722.2.a.f yes 1 19.b odd 2 1
722.2.c.a 2 19.d odd 6 2
722.2.c.g 2 19.c even 3 2
722.2.e.g 6 19.f odd 18 6
722.2.e.h 6 19.e even 9 6
5776.2.a.a 1 76.d even 2 1
5776.2.a.q 1 4.b odd 2 1
6498.2.a.a 1 57.d even 2 1
6498.2.a.m 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(722))$$:

 $$T_{3} + 3$$ T3 + 3 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 3$$ T7 + 3 $$T_{13} - 3$$ T13 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 3$$
$5$ $$T - 2$$
$7$ $$T + 3$$
$11$ $$T + 2$$
$13$ $$T - 3$$
$17$ $$T + 1$$
$19$ $$T$$
$23$ $$T - 5$$
$29$ $$T - 3$$
$31$ $$T - 6$$
$37$ $$T + 6$$
$41$ $$T + 12$$
$43$ $$T + 10$$
$47$ $$T + 8$$
$53$ $$T - 3$$
$59$ $$T + 3$$
$61$ $$T$$
$67$ $$T + 15$$
$71$ $$T$$
$73$ $$T + 11$$
$79$ $$T - 12$$
$83$ $$T - 2$$
$89$ $$T + 6$$
$97$ $$T + 12$$