# Properties

 Label 722.2.c.f 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: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) 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.

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

## 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 −0.500000 + 0.866025i −0.500000 0.866025i 2.00000 3.46410i 0.500000 + 0.866025i 3.00000 −1.00000 1.00000 + 1.73205i −2.00000 3.46410i
653.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 2.00000 + 3.46410i 0.500000 0.866025i 3.00000 −1.00000 1.00000 1.73205i −2.00000 + 3.46410i
 $$n$$: 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.f 2
19.b odd 2 1 722.2.c.d 2
19.c even 3 1 722.2.a.b 1
19.c even 3 1 inner 722.2.c.f 2
19.d odd 6 1 38.2.a.b 1
19.d odd 6 1 722.2.c.d 2
19.e even 9 6 722.2.e.d 6
19.f odd 18 6 722.2.e.c 6
57.f even 6 1 342.2.a.d 1
57.h odd 6 1 6498.2.a.y 1
76.f even 6 1 304.2.a.d 1
76.g odd 6 1 5776.2.a.d 1
95.h odd 6 1 950.2.a.b 1
95.l even 12 2 950.2.b.c 2
133.p even 6 1 1862.2.a.f 1
152.l odd 6 1 1216.2.a.n 1
152.o even 6 1 1216.2.a.g 1
209.g even 6 1 4598.2.a.a 1
228.n odd 6 1 2736.2.a.w 1
247.n odd 6 1 6422.2.a.b 1
285.q even 6 1 8550.2.a.u 1
380.s even 6 1 7600.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 19.d odd 6 1
304.2.a.d 1 76.f even 6 1
342.2.a.d 1 57.f even 6 1
722.2.a.b 1 19.c even 3 1
722.2.c.d 2 19.b odd 2 1
722.2.c.d 2 19.d odd 6 1
722.2.c.f 2 1.a even 1 1 trivial
722.2.c.f 2 19.c even 3 1 inner
722.2.e.c 6 19.f odd 18 6
722.2.e.d 6 19.e even 9 6
950.2.a.b 1 95.h odd 6 1
950.2.b.c 2 95.l even 12 2
1216.2.a.g 1 152.o even 6 1
1216.2.a.n 1 152.l odd 6 1
1862.2.a.f 1 133.p even 6 1
2736.2.a.w 1 228.n odd 6 1
4598.2.a.a 1 209.g even 6 1
5776.2.a.d 1 76.g odd 6 1
6422.2.a.b 1 247.n odd 6 1
6498.2.a.y 1 57.h odd 6 1
7600.2.a.h 1 380.s even 6 1
8550.2.a.u 1 285.q even 6 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{7} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$( -3 + T )^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$25 + 5 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$64 + 8 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$64 + 8 T + T^{2}$$
$53$ $$1 + T + T^{2}$$
$59$ $$225 - 15 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$4 - 2 T + T^{2}$$
$73$ $$81 + 9 T + T^{2}$$
$79$ $$100 + 10 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$4 + 2 T + T^{2}$$