# Properties

 Label 722.2.a.b Level $722$ Weight $2$ Character orbit 722.a Self dual yes Analytic conductor $5.765$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,2,Mod(1,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.a (trivial)

## 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: yes Analytic conductor: $$5.76519902594$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 1.00000 −4.00000 −1.00000 3.00000 −1.00000 −2.00000 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.a.b 1
3.b odd 2 1 6498.2.a.y 1
4.b odd 2 1 5776.2.a.d 1
19.b odd 2 1 38.2.a.b 1
19.c even 3 2 722.2.c.f 2
19.d odd 6 2 722.2.c.d 2
19.e even 9 6 722.2.e.d 6
19.f odd 18 6 722.2.e.c 6
57.d even 2 1 342.2.a.d 1
76.d even 2 1 304.2.a.d 1
95.d odd 2 1 950.2.a.b 1
95.g even 4 2 950.2.b.c 2
133.c even 2 1 1862.2.a.f 1
152.b even 2 1 1216.2.a.g 1
152.g odd 2 1 1216.2.a.n 1
209.d even 2 1 4598.2.a.a 1
228.b odd 2 1 2736.2.a.w 1
247.d odd 2 1 6422.2.a.b 1
285.b even 2 1 8550.2.a.u 1
380.d even 2 1 7600.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 19.b odd 2 1
304.2.a.d 1 76.d even 2 1
342.2.a.d 1 57.d even 2 1
722.2.a.b 1 1.a even 1 1 trivial
722.2.c.d 2 19.d odd 6 2
722.2.c.f 2 19.c even 3 2
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.d odd 2 1
950.2.b.c 2 95.g even 4 2
1216.2.a.g 1 152.b even 2 1
1216.2.a.n 1 152.g odd 2 1
1862.2.a.f 1 133.c even 2 1
2736.2.a.w 1 228.b odd 2 1
4598.2.a.a 1 209.d even 2 1
5776.2.a.d 1 4.b odd 2 1
6422.2.a.b 1 247.d odd 2 1
6498.2.a.y 1 3.b odd 2 1
7600.2.a.h 1 380.d even 2 1
8550.2.a.u 1 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(722))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} + 4$$ T5 + 4 $$T_{7} - 3$$ T7 - 3 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

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