# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,2,Mod(99,722)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("722.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.76519902594$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −3.06418 + 2.57115i −0.173648 + 0.984808i −1.50000 + 2.59808i 0.500000 + 0.866025i 1.87939 0.684040i −3.75877 + 1.36808i
245.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 3.75877 1.36808i −0.766044 + 0.642788i −1.50000 2.59808i 0.500000 0.866025i −0.347296 1.96962i 0.694593 + 3.93923i
389.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 3.75877 + 1.36808i −0.766044 0.642788i −1.50000 + 2.59808i 0.500000 + 0.866025i −0.347296 + 1.96962i 0.694593 3.93923i
415.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.694593 + 3.93923i 0.939693 + 0.342020i −1.50000 2.59808i 0.500000 0.866025i −1.53209 + 1.28558i 3.06418 2.57115i
423.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i −3.06418 2.57115i −0.173648 0.984808i −1.50000 2.59808i 0.500000 0.866025i 1.87939 + 0.684040i −3.75877 1.36808i
595.1 −0.766044 + 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.694593 3.93923i 0.939693 0.342020i −1.50000 + 2.59808i 0.500000 + 0.866025i −1.53209 1.28558i 3.06418 + 2.57115i
 $$n$$: 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.d 6
19.b odd 2 1 722.2.e.c 6
19.c even 3 2 inner 722.2.e.d 6
19.d odd 6 2 722.2.e.c 6
19.e even 9 1 722.2.a.b 1
19.e even 9 2 722.2.c.f 2
19.e even 9 3 inner 722.2.e.d 6
19.f odd 18 1 38.2.a.b 1
19.f odd 18 2 722.2.c.d 2
19.f odd 18 3 722.2.e.c 6
57.j even 18 1 342.2.a.d 1
57.l odd 18 1 6498.2.a.y 1
76.k even 18 1 304.2.a.d 1
76.l odd 18 1 5776.2.a.d 1
95.o odd 18 1 950.2.a.b 1
95.r even 36 2 950.2.b.c 2
133.ba even 18 1 1862.2.a.f 1
152.s odd 18 1 1216.2.a.n 1
152.v even 18 1 1216.2.a.g 1
209.p even 18 1 4598.2.a.a 1
228.u odd 18 1 2736.2.a.w 1
247.bl odd 18 1 6422.2.a.b 1
285.bf even 18 1 8550.2.a.u 1
380.bb even 18 1 7600.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 19.f odd 18 1
304.2.a.d 1 76.k even 18 1
342.2.a.d 1 57.j even 18 1
722.2.a.b 1 19.e even 9 1
722.2.c.d 2 19.f odd 18 2
722.2.c.f 2 19.e even 9 2
722.2.e.c 6 19.b odd 2 1
722.2.e.c 6 19.d odd 6 2
722.2.e.c 6 19.f odd 18 3
722.2.e.d 6 1.a even 1 1 trivial
722.2.e.d 6 19.c even 3 2 inner
722.2.e.d 6 19.e even 9 3 inner
950.2.a.b 1 95.o odd 18 1
950.2.b.c 2 95.r even 36 2
1216.2.a.g 1 152.v even 18 1
1216.2.a.n 1 152.s odd 18 1
1862.2.a.f 1 133.ba even 18 1
2736.2.a.w 1 228.u odd 18 1
4598.2.a.a 1 209.p even 18 1
5776.2.a.d 1 76.l odd 18 1
6422.2.a.b 1 247.bl odd 18 1
6498.2.a.y 1 57.l odd 18 1
7600.2.a.h 1 380.bb even 18 1
8550.2.a.u 1 285.bf even 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 $$T_{5}^{6} - 64T_{5}^{3} + 4096$$ T5^6 - 64*T5^3 + 4096 $$T_{7}^{2} + 3T_{7} + 9$$ T7^2 + 3*T7 + 9 $$T_{13}^{6} + T_{13}^{3} + 1$$ T13^6 + T13^3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6} - 64T^{3} + 4096$$
$7$ $$(T^{2} + 3 T + 9)^{3}$$
$11$ $$(T^{2} + 2 T + 4)^{3}$$
$13$ $$T^{6} + T^{3} + 1$$
$17$ $$T^{6} + 27T^{3} + 729$$
$19$ $$T^{6}$$
$23$ $$T^{6} - T^{3} + 1$$
$29$ $$T^{6} + 125 T^{3} + 15625$$
$31$ $$(T^{2} + 8 T + 64)^{3}$$
$37$ $$(T - 2)^{6}$$
$41$ $$T^{6} + 512 T^{3} + 262144$$
$43$ $$T^{6} + 64T^{3} + 4096$$
$47$ $$T^{6} + 512 T^{3} + 262144$$
$53$ $$T^{6} + T^{3} + 1$$
$59$ $$T^{6} - 3375 T^{3} + \cdots + 11390625$$
$61$ $$T^{6} + 8T^{3} + 64$$
$67$ $$T^{6} - 27T^{3} + 729$$
$71$ $$T^{6} - 8T^{3} + 64$$
$73$ $$T^{6} + 729 T^{3} + 531441$$
$79$ $$T^{6} + 1000 T^{3} + \cdots + 1000000$$
$83$ $$(T^{2} - 6 T + 36)^{3}$$
$89$ $$T^{6}$$
$97$ $$T^{6} + 8T^{3} + 64$$