# Properties

 Label 722.2.c.c 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: $$\Q(\sqrt{-3})$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{7} + q^{8} + 2 \zeta_{6} q^{9} -6 q^{11} - q^{12} + 5 \zeta_{6} q^{13} + ( 1 - \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} -2 q^{18} + ( -1 + \zeta_{6} ) q^{21} + ( 6 - 6 \zeta_{6} ) q^{22} -3 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + 5 \zeta_{6} q^{25} -5 q^{26} + 5 q^{27} + \zeta_{6} q^{28} + 9 \zeta_{6} q^{29} + 4 q^{31} -\zeta_{6} q^{32} + ( -6 + 6 \zeta_{6} ) q^{33} -3 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{36} -2 q^{37} + 5 q^{39} -\zeta_{6} q^{42} + ( -8 + 8 \zeta_{6} ) q^{43} + 6 \zeta_{6} q^{44} + 3 q^{46} + \zeta_{6} q^{48} -6 q^{49} -5 q^{50} + 3 \zeta_{6} q^{51} + ( 5 - 5 \zeta_{6} ) q^{52} -3 \zeta_{6} q^{53} + ( -5 + 5 \zeta_{6} ) q^{54} - q^{56} -9 q^{58} + ( 9 - 9 \zeta_{6} ) q^{59} + 10 \zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} -2 \zeta_{6} q^{63} + q^{64} -6 \zeta_{6} q^{66} + 5 \zeta_{6} q^{67} + 3 q^{68} -3 q^{69} + ( -6 + 6 \zeta_{6} ) q^{71} + 2 \zeta_{6} q^{72} + ( 7 - 7 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + 5 q^{75} + 6 q^{77} + ( -5 + 5 \zeta_{6} ) q^{78} + ( -10 + 10 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -6 q^{83} + q^{84} -8 \zeta_{6} q^{86} + 9 q^{87} -6 q^{88} -12 \zeta_{6} q^{89} -5 \zeta_{6} q^{91} + ( -3 + 3 \zeta_{6} ) q^{92} + ( 4 - 4 \zeta_{6} ) q^{93} - q^{96} + ( -10 + 10 \zeta_{6} ) q^{97} + ( 6 - 6 \zeta_{6} ) q^{98} -12 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 12q^{11} - 2q^{12} + 5q^{13} + q^{14} - q^{16} - 3q^{17} - 4q^{18} - q^{21} + 6q^{22} - 3q^{23} + q^{24} + 5q^{25} - 10q^{26} + 10q^{27} + q^{28} + 9q^{29} + 8q^{31} - q^{32} - 6q^{33} - 3q^{34} + 2q^{36} - 4q^{37} + 10q^{39} - q^{42} - 8q^{43} + 6q^{44} + 6q^{46} + q^{48} - 12q^{49} - 10q^{50} + 3q^{51} + 5q^{52} - 3q^{53} - 5q^{54} - 2q^{56} - 18q^{58} + 9q^{59} + 10q^{61} - 4q^{62} - 2q^{63} + 2q^{64} - 6q^{66} + 5q^{67} + 6q^{68} - 6q^{69} - 6q^{71} + 2q^{72} + 7q^{73} + 2q^{74} + 10q^{75} + 12q^{77} - 5q^{78} - 10q^{79} - q^{81} - 12q^{83} + 2q^{84} - 8q^{86} + 18q^{87} - 12q^{88} - 12q^{89} - 5q^{91} - 3q^{92} + 4q^{93} - 2q^{96} - 10q^{97} + 6q^{98} - 12q^{99} + O(q^{100})$$

## 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 0 0.500000 + 0.866025i −1.00000 1.00000 1.00000 + 1.73205i 0
653.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 1.00000 1.00000 1.73205i 0
 $$n$$: 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.c 2
19.b odd 2 1 722.2.c.e 2
19.c even 3 1 722.2.a.e 1
19.c even 3 1 inner 722.2.c.c 2
19.d odd 6 1 38.2.a.a 1
19.d odd 6 1 722.2.c.e 2
19.e even 9 6 722.2.e.e 6
19.f odd 18 6 722.2.e.f 6
57.f even 6 1 342.2.a.e 1
57.h odd 6 1 6498.2.a.f 1
76.f even 6 1 304.2.a.c 1
76.g odd 6 1 5776.2.a.m 1
95.h odd 6 1 950.2.a.d 1
95.l even 12 2 950.2.b.b 2
133.p even 6 1 1862.2.a.b 1
152.l odd 6 1 1216.2.a.e 1
152.o even 6 1 1216.2.a.m 1
209.g even 6 1 4598.2.a.p 1
228.n odd 6 1 2736.2.a.n 1
247.n odd 6 1 6422.2.a.h 1
285.q even 6 1 8550.2.a.m 1
380.s even 6 1 7600.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 19.d odd 6 1
304.2.a.c 1 76.f even 6 1
342.2.a.e 1 57.f even 6 1
722.2.a.e 1 19.c even 3 1
722.2.c.c 2 1.a even 1 1 trivial
722.2.c.c 2 19.c even 3 1 inner
722.2.c.e 2 19.b odd 2 1
722.2.c.e 2 19.d odd 6 1
722.2.e.e 6 19.e even 9 6
722.2.e.f 6 19.f odd 18 6
950.2.a.d 1 95.h odd 6 1
950.2.b.b 2 95.l even 12 2
1216.2.a.e 1 152.l odd 6 1
1216.2.a.m 1 152.o even 6 1
1862.2.a.b 1 133.p even 6 1
2736.2.a.n 1 228.n odd 6 1
4598.2.a.p 1 209.g even 6 1
5776.2.a.m 1 76.g odd 6 1
6422.2.a.h 1 247.n odd 6 1
6498.2.a.f 1 57.h odd 6 1
7600.2.a.n 1 380.s even 6 1
8550.2.a.m 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 $$S_{2}^{\mathrm{new}}(722, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$( 1 + T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ 1
$23$ $$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$
$29$ $$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4}$$
$61$ $$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$
$71$ $$1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 6 T + 83 T^{2} )^{2}$$
$89$ $$1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 + 10 T + 3 T^{2} + 970 T^{3} + 9409 T^{4}$$