# Properties

 Label 722.2.a.e Level $722$ Weight $2$ Character orbit 722.a Self dual yes Analytic conductor $5.765$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 - q^7 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} - 2 q^{9} - 6 q^{11} - q^{12} - 5 q^{13} - q^{14} + q^{16} + 3 q^{17} - 2 q^{18} + q^{21} - 6 q^{22} + 3 q^{23} - q^{24} - 5 q^{25} - 5 q^{26} + 5 q^{27} - q^{28} - 9 q^{29} + 4 q^{31} + q^{32} + 6 q^{33} + 3 q^{34} - 2 q^{36} - 2 q^{37} + 5 q^{39} + q^{42} + 8 q^{43} - 6 q^{44} + 3 q^{46} - q^{48} - 6 q^{49} - 5 q^{50} - 3 q^{51} - 5 q^{52} + 3 q^{53} + 5 q^{54} - q^{56} - 9 q^{58} - 9 q^{59} - 10 q^{61} + 4 q^{62} + 2 q^{63} + q^{64} + 6 q^{66} - 5 q^{67} + 3 q^{68} - 3 q^{69} + 6 q^{71} - 2 q^{72} - 7 q^{73} - 2 q^{74} + 5 q^{75} + 6 q^{77} + 5 q^{78} + 10 q^{79} + q^{81} - 6 q^{83} + q^{84} + 8 q^{86} + 9 q^{87} - 6 q^{88} + 12 q^{89} + 5 q^{91} + 3 q^{92} - 4 q^{93} - q^{96} + 10 q^{97} - 6 q^{98} + 12 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 - q^7 + q^8 - 2 * q^9 - 6 * q^11 - q^12 - 5 * q^13 - q^14 + q^16 + 3 * q^17 - 2 * q^18 + q^21 - 6 * q^22 + 3 * q^23 - q^24 - 5 * q^25 - 5 * q^26 + 5 * q^27 - q^28 - 9 * q^29 + 4 * q^31 + q^32 + 6 * q^33 + 3 * q^34 - 2 * q^36 - 2 * q^37 + 5 * q^39 + q^42 + 8 * q^43 - 6 * q^44 + 3 * q^46 - q^48 - 6 * q^49 - 5 * q^50 - 3 * q^51 - 5 * q^52 + 3 * q^53 + 5 * q^54 - q^56 - 9 * q^58 - 9 * q^59 - 10 * q^61 + 4 * q^62 + 2 * q^63 + q^64 + 6 * q^66 - 5 * q^67 + 3 * q^68 - 3 * q^69 + 6 * q^71 - 2 * q^72 - 7 * q^73 - 2 * q^74 + 5 * q^75 + 6 * q^77 + 5 * q^78 + 10 * q^79 + q^81 - 6 * q^83 + q^84 + 8 * q^86 + 9 * q^87 - 6 * q^88 + 12 * q^89 + 5 * q^91 + 3 * q^92 - 4 * q^93 - q^96 + 10 * q^97 - 6 * q^98 + 12 * 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 0 −1.00000 −1.00000 1.00000 −2.00000 0
 $$n$$: 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.e 1
3.b odd 2 1 6498.2.a.f 1
4.b odd 2 1 5776.2.a.m 1
19.b odd 2 1 38.2.a.a 1
19.c even 3 2 722.2.c.c 2
19.d odd 6 2 722.2.c.e 2
19.e even 9 6 722.2.e.e 6
19.f odd 18 6 722.2.e.f 6
57.d even 2 1 342.2.a.e 1
76.d even 2 1 304.2.a.c 1
95.d odd 2 1 950.2.a.d 1
95.g even 4 2 950.2.b.b 2
133.c even 2 1 1862.2.a.b 1
152.b even 2 1 1216.2.a.m 1
152.g odd 2 1 1216.2.a.e 1
209.d even 2 1 4598.2.a.p 1
228.b odd 2 1 2736.2.a.n 1
247.d odd 2 1 6422.2.a.h 1
285.b even 2 1 8550.2.a.m 1
380.d even 2 1 7600.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 19.b odd 2 1
304.2.a.c 1 76.d even 2 1
342.2.a.e 1 57.d even 2 1
722.2.a.e 1 1.a even 1 1 trivial
722.2.c.c 2 19.c even 3 2
722.2.c.e 2 19.d odd 6 2
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.d odd 2 1
950.2.b.b 2 95.g even 4 2
1216.2.a.e 1 152.g odd 2 1
1216.2.a.m 1 152.b even 2 1
1862.2.a.b 1 133.c even 2 1
2736.2.a.n 1 228.b odd 2 1
4598.2.a.p 1 209.d even 2 1
5776.2.a.m 1 4.b odd 2 1
6422.2.a.h 1 247.d odd 2 1
6498.2.a.f 1 3.b odd 2 1
7600.2.a.n 1 380.d even 2 1
8550.2.a.m 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 $$T_{5}$$ T5 $$T_{7} + 1$$ T7 + 1 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 6$$
$13$ $$T + 5$$
$17$ $$T - 3$$
$19$ $$T$$
$23$ $$T - 3$$
$29$ $$T + 9$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T$$
$43$ $$T - 8$$
$47$ $$T$$
$53$ $$T - 3$$
$59$ $$T + 9$$
$61$ $$T + 10$$
$67$ $$T + 5$$
$71$ $$T - 6$$
$73$ $$T + 7$$
$79$ $$T - 10$$
$83$ $$T + 6$$
$89$ $$T - 12$$
$97$ $$T - 10$$