# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,2,Mod(429,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("722.429");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.c (of order $$3$$, degree $$2$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ 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

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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 $$S_{2}^{\mathrm{new}}(722, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{5}^{2} - 4T_{5} + 16$$ T5^2 - 4*T5 + 16 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

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