# Properties

 Label 722.2.c.g Level $722$ Weight $2$ Character orbit 722.c Analytic conductor $5.765$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.76519902594$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/722\mathbb{Z}\right)^\times$$.

 $$n$$ $$363$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
429.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.50000 2.59808i −0.500000 0.866025i −1.00000 + 1.73205i −1.50000 2.59808i −3.00000 −1.00000 −3.00000 5.19615i 1.00000 + 1.73205i
653.1 0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 + 0.866025i −1.00000 1.73205i −1.50000 + 2.59808i −3.00000 −1.00000 −3.00000 + 5.19615i 1.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.c.g 2
19.b odd 2 1 722.2.c.a 2
19.c even 3 1 722.2.a.a 1
19.c even 3 1 inner 722.2.c.g 2
19.d odd 6 1 722.2.a.f yes 1
19.d odd 6 1 722.2.c.a 2
19.e even 9 6 722.2.e.h 6
19.f odd 18 6 722.2.e.g 6
57.f even 6 1 6498.2.a.a 1
57.h odd 6 1 6498.2.a.m 1
76.f even 6 1 5776.2.a.a 1
76.g odd 6 1 5776.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.a 1 19.c even 3 1
722.2.a.f yes 1 19.d odd 6 1
722.2.c.a 2 19.b odd 2 1
722.2.c.a 2 19.d odd 6 1
722.2.c.g 2 1.a even 1 1 trivial
722.2.c.g 2 19.c even 3 1 inner
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.f even 6 1
5776.2.a.q 1 76.g odd 6 1
6498.2.a.a 1 57.f even 6 1
6498.2.a.m 1 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(722, [\chi])$$:

 $$T_{3}^{2} - 3T_{3} + 9$$ T3^2 - 3*T3 + 9 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 3T + 9$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 5T + 25$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T - 6)^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 12T + 144$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 15T + 225$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + 12T + 144$$
$83$ $$(T - 2)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} - 12T + 144$$