# Properties

 Label 722.2.e.j Level $722$ Weight $2$ Character orbit 722.e Analytic conductor $5.765$ Analytic rank $0$ Dimension $6$ CM no Inner twists $6$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.76519902594$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} - \zeta_{18} q^{3} - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{6} + ( - 4 \zeta_{18}^{3} + 4) q^{7} + \zeta_{18}^{3} q^{8} - 2 \zeta_{18}^{2} q^{9} +O(q^{10})$$ q + (z^4 - z) * q^2 - z * q^3 - z^5 * q^4 + (-z^5 + z^2) * q^6 + (-4*z^3 + 4) * q^7 + z^3 * q^8 - 2*z^2 * q^9 $$q + (\zeta_{18}^{4} - \zeta_{18}) q^{2} - \zeta_{18} q^{3} - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{6} + ( - 4 \zeta_{18}^{3} + 4) q^{7} + \zeta_{18}^{3} q^{8} - 2 \zeta_{18}^{2} q^{9} - 3 \zeta_{18}^{3} q^{11} + (\zeta_{18}^{3} - 1) q^{12} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{13} + 4 \zeta_{18}^{4} q^{14} - \zeta_{18} q^{16} + (6 \zeta_{18}^{4} - 6 \zeta_{18}) q^{17} + 2 q^{18} + (4 \zeta_{18}^{4} - 4 \zeta_{18}) q^{21} + 3 \zeta_{18} q^{22} + 6 \zeta_{18}^{5} q^{23} - \zeta_{18}^{4} q^{24} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{2}) q^{25} + ( - 2 \zeta_{18}^{3} + 2) q^{26} + 5 \zeta_{18}^{3} q^{27} - 4 \zeta_{18}^{2} q^{28} + (2 \zeta_{18}^{3} - 2) q^{31} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{32} + 3 \zeta_{18}^{4} q^{33} - 6 \zeta_{18}^{5} q^{34} + (2 \zeta_{18}^{4} - 2 \zeta_{18}) q^{36} - 10 q^{37} + 2 q^{39} - 9 \zeta_{18} q^{41} - 4 \zeta_{18}^{5} q^{42} - 4 \zeta_{18}^{4} q^{43} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{2}) q^{44} - 6 \zeta_{18}^{3} q^{46} + \zeta_{18}^{2} q^{48} - 9 \zeta_{18}^{3} q^{49} + (5 \zeta_{18}^{3} - 5) q^{50} + ( - 6 \zeta_{18}^{5} + 6 \zeta_{18}^{2}) q^{51} + 2 \zeta_{18}^{4} q^{52} - 6 \zeta_{18}^{5} q^{53} - 5 \zeta_{18} q^{54} + 4 q^{56} + (9 \zeta_{18}^{4} - 9 \zeta_{18}) q^{59} + 4 \zeta_{18}^{5} q^{61} - 2 \zeta_{18}^{4} q^{62} + (8 \zeta_{18}^{5} - 8 \zeta_{18}^{2}) q^{63} + (\zeta_{18}^{3} - 1) q^{64} - 3 \zeta_{18}^{2} q^{66} - 7 \zeta_{18}^{2} q^{67} + 6 \zeta_{18}^{3} q^{68} + ( - 6 \zeta_{18}^{3} + 6) q^{69} - 6 \zeta_{18}^{4} q^{71} - 2 \zeta_{18}^{5} q^{72} + \zeta_{18} q^{73} + ( - 10 \zeta_{18}^{4} + 10 \zeta_{18}) q^{74} - 5 q^{75} - 12 q^{77} + (2 \zeta_{18}^{4} - 2 \zeta_{18}) q^{78} + 4 \zeta_{18} q^{79} + \zeta_{18}^{4} q^{81} + ( - 9 \zeta_{18}^{5} + 9 \zeta_{18}^{2}) q^{82} + (3 \zeta_{18}^{3} - 3) q^{83} + 4 \zeta_{18}^{3} q^{84} + 4 \zeta_{18}^{2} q^{86} + ( - 3 \zeta_{18}^{3} + 3) q^{88} + (6 \zeta_{18}^{5} - 6 \zeta_{18}^{2}) q^{89} + 8 \zeta_{18}^{5} q^{91} + 6 \zeta_{18} q^{92} + ( - 2 \zeta_{18}^{4} + 2 \zeta_{18}) q^{93} - q^{96} + ( - 17 \zeta_{18}^{4} + 17 \zeta_{18}) q^{97} + 9 \zeta_{18} q^{98} + 6 \zeta_{18}^{5} q^{99} +O(q^{100})$$ q + (z^4 - z) * q^2 - z * q^3 - z^5 * q^4 + (-z^5 + z^2) * q^6 + (-4*z^3 + 4) * q^7 + z^3 * q^8 - 2*z^2 * q^9 - 3*z^3 * q^11 + (z^3 - 1) * q^12 + (2*z^5 - 2*z^2) * q^13 + 4*z^4 * q^14 - z * q^16 + (6*z^4 - 6*z) * q^17 + 2 * q^18 + (4*z^4 - 4*z) * q^21 + 3*z * q^22 + 6*z^5 * q^23 - z^4 * q^24 + (-5*z^5 + 5*z^2) * q^25 + (-2*z^3 + 2) * q^26 + 5*z^3 * q^27 - 4*z^2 * q^28 + (2*z^3 - 2) * q^31 + (-z^5 + z^2) * q^32 + 3*z^4 * q^33 - 6*z^5 * q^34 + (2*z^4 - 2*z) * q^36 - 10 * q^37 + 2 * q^39 - 9*z * q^41 - 4*z^5 * q^42 - 4*z^4 * q^43 + (3*z^5 - 3*z^2) * q^44 - 6*z^3 * q^46 + z^2 * q^48 - 9*z^3 * q^49 + (5*z^3 - 5) * q^50 + (-6*z^5 + 6*z^2) * q^51 + 2*z^4 * q^52 - 6*z^5 * q^53 - 5*z * q^54 + 4 * q^56 + (9*z^4 - 9*z) * q^59 + 4*z^5 * q^61 - 2*z^4 * q^62 + (8*z^5 - 8*z^2) * q^63 + (z^3 - 1) * q^64 - 3*z^2 * q^66 - 7*z^2 * q^67 + 6*z^3 * q^68 + (-6*z^3 + 6) * q^69 - 6*z^4 * q^71 - 2*z^5 * q^72 + z * q^73 + (-10*z^4 + 10*z) * q^74 - 5 * q^75 - 12 * q^77 + (2*z^4 - 2*z) * q^78 + 4*z * q^79 + z^4 * q^81 + (-9*z^5 + 9*z^2) * q^82 + (3*z^3 - 3) * q^83 + 4*z^3 * q^84 + 4*z^2 * q^86 + (-3*z^3 + 3) * q^88 + (6*z^5 - 6*z^2) * q^89 + 8*z^5 * q^91 + 6*z * q^92 + (-2*z^4 + 2*z) * q^93 - q^96 + (-17*z^4 + 17*z) * q^97 + 9*z * q^98 + 6*z^5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 12 q^{7} + 3 q^{8}+O(q^{10})$$ 6 * q + 12 * q^7 + 3 * q^8 $$6 q + 12 q^{7} + 3 q^{8} - 9 q^{11} - 3 q^{12} + 12 q^{18} + 6 q^{26} + 15 q^{27} - 6 q^{31} - 60 q^{37} + 12 q^{39} - 18 q^{46} - 27 q^{49} - 15 q^{50} + 24 q^{56} - 3 q^{64} + 18 q^{68} + 18 q^{69} - 30 q^{75} - 72 q^{77} - 9 q^{83} + 12 q^{84} + 9 q^{88} - 6 q^{96}+O(q^{100})$$ 6 * q + 12 * q^7 + 3 * q^8 - 9 * q^11 - 3 * q^12 + 12 * q^18 + 6 * q^26 + 15 * q^27 - 6 * q^31 - 60 * q^37 + 12 * q^39 - 18 * q^46 - 27 * q^49 - 15 * q^50 + 24 * q^56 - 3 * q^64 + 18 * q^68 + 18 * q^69 - 30 * q^75 - 72 * q^77 - 9 * q^83 + 12 * q^84 + 9 * q^88 - 6 * q^96

## Character values

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

 $$n$$ $$363$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.173648 − 0.984808i −0.766044 − 0.642788i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i 0.939693 + 0.342020i
0.939693 + 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i 0 −0.173648 + 0.984808i 2.00000 3.46410i 0.500000 + 0.866025i 1.87939 0.684040i 0
245.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0 −0.766044 + 0.642788i 2.00000 + 3.46410i 0.500000 0.866025i −0.347296 1.96962i 0
389.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0 −0.766044 0.642788i 2.00000 3.46410i 0.500000 + 0.866025i −0.347296 + 1.96962i 0
415.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i 0 0.939693 + 0.342020i 2.00000 + 3.46410i 0.500000 0.866025i −1.53209 + 1.28558i 0
423.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i 0 −0.173648 0.984808i 2.00000 + 3.46410i 0.500000 0.866025i 1.87939 + 0.684040i 0
595.1 −0.766044 + 0.642788i −0.939693 0.342020i 0.173648 0.984808i 0 0.939693 0.342020i 2.00000 3.46410i 0.500000 + 0.866025i −1.53209 1.28558i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 595.1 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 2 inner
19.e even 9 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.e.j 6
19.b odd 2 1 722.2.e.i 6
19.c even 3 2 inner 722.2.e.j 6
19.d odd 6 2 722.2.e.i 6
19.e even 9 2 38.2.c.a 2
19.e even 9 1 722.2.a.c 1
19.e even 9 3 inner 722.2.e.j 6
19.f odd 18 1 722.2.a.d 1
19.f odd 18 2 722.2.c.b 2
19.f odd 18 3 722.2.e.i 6
57.j even 18 1 6498.2.a.e 1
57.l odd 18 2 342.2.g.b 2
57.l odd 18 1 6498.2.a.s 1
76.k even 18 1 5776.2.a.n 1
76.l odd 18 2 304.2.i.c 2
76.l odd 18 1 5776.2.a.g 1
95.p even 18 2 950.2.e.d 2
95.q odd 36 4 950.2.j.e 4
152.t even 18 2 1216.2.i.h 2
152.u odd 18 2 1216.2.i.d 2
228.v even 18 2 2736.2.s.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 19.e even 9 2
304.2.i.c 2 76.l odd 18 2
342.2.g.b 2 57.l odd 18 2
722.2.a.c 1 19.e even 9 1
722.2.a.d 1 19.f odd 18 1
722.2.c.b 2 19.f odd 18 2
722.2.e.i 6 19.b odd 2 1
722.2.e.i 6 19.d odd 6 2
722.2.e.i 6 19.f odd 18 3
722.2.e.j 6 1.a even 1 1 trivial
722.2.e.j 6 19.c even 3 2 inner
722.2.e.j 6 19.e even 9 3 inner
950.2.e.d 2 95.p even 18 2
950.2.j.e 4 95.q odd 36 4
1216.2.i.d 2 152.u odd 18 2
1216.2.i.h 2 152.t even 18 2
2736.2.s.m 2 228.v even 18 2
5776.2.a.g 1 76.l odd 18 1
5776.2.a.n 1 76.k even 18 1
6498.2.a.e 1 57.j even 18 1
6498.2.a.s 1 57.l odd 18 1

## Hecke kernels

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

 $$T_{3}^{6} + T_{3}^{3} + 1$$ T3^6 + T3^3 + 1 $$T_{5}$$ T5 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16 $$T_{13}^{6} + 8T_{13}^{3} + 64$$ T13^6 + 8*T13^3 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6}$$
$7$ $$(T^{2} - 4 T + 16)^{3}$$
$11$ $$(T^{2} + 3 T + 9)^{3}$$
$13$ $$T^{6} + 8T^{3} + 64$$
$17$ $$T^{6} - 216 T^{3} + 46656$$
$19$ $$T^{6}$$
$23$ $$T^{6} - 216 T^{3} + 46656$$
$29$ $$T^{6}$$
$31$ $$(T^{2} + 2 T + 4)^{3}$$
$37$ $$(T + 10)^{6}$$
$41$ $$T^{6} + 729 T^{3} + 531441$$
$43$ $$T^{6} - 64T^{3} + 4096$$
$47$ $$T^{6}$$
$53$ $$T^{6} + 216 T^{3} + 46656$$
$59$ $$T^{6} - 729 T^{3} + 531441$$
$61$ $$T^{6} - 64T^{3} + 4096$$
$67$ $$T^{6} - 343 T^{3} + 117649$$
$71$ $$T^{6} - 216 T^{3} + 46656$$
$73$ $$T^{6} - T^{3} + 1$$
$79$ $$T^{6} - 64T^{3} + 4096$$
$83$ $$(T^{2} + 3 T + 9)^{3}$$
$89$ $$T^{6} + 216 T^{3} + 46656$$
$97$ $$T^{6} + 4913 T^{3} + \cdots + 24137569$$