# Properties

 Label 6422.2.a.b Level $6422$ Weight $2$ Character orbit 6422.a Self dual yes Analytic conductor $51.280$ 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] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6422 = 2 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6422.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: $$51.2799281781$$ 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 $$q - q^{2} - q^{3} + q^{4} + 4 q^{5} + q^{6} - 3 q^{7} - q^{8} - 2 q^{9} - 4 q^{10} - 2 q^{11} - q^{12} + 3 q^{14} - 4 q^{15} + q^{16} + 3 q^{17} + 2 q^{18} + q^{19} + 4 q^{20} + 3 q^{21} + 2 q^{22} - q^{23} + q^{24} + 11 q^{25} + 5 q^{27} - 3 q^{28} - 5 q^{29} + 4 q^{30} + 8 q^{31} - q^{32} + 2 q^{33} - 3 q^{34} - 12 q^{35} - 2 q^{36} + 2 q^{37} - q^{38} - 4 q^{40} + 8 q^{41} - 3 q^{42} + 4 q^{43} - 2 q^{44} - 8 q^{45} + q^{46} - 8 q^{47} - q^{48} + 2 q^{49} - 11 q^{50} - 3 q^{51} - q^{53} - 5 q^{54} - 8 q^{55} + 3 q^{56} - q^{57} + 5 q^{58} - 15 q^{59} - 4 q^{60} + 2 q^{61} - 8 q^{62} + 6 q^{63} + q^{64} - 2 q^{66} - 3 q^{67} + 3 q^{68} + q^{69} + 12 q^{70} - 2 q^{71} + 2 q^{72} - 9 q^{73} - 2 q^{74} - 11 q^{75} + q^{76} + 6 q^{77} - 10 q^{79} + 4 q^{80} + q^{81} - 8 q^{82} + 6 q^{83} + 3 q^{84} + 12 q^{85} - 4 q^{86} + 5 q^{87} + 2 q^{88} + 8 q^{90} - q^{92} - 8 q^{93} + 8 q^{94} + 4 q^{95} + q^{96} + 2 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100})$$ 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 + 3 * q^14 - 4 * q^15 + q^16 + 3 * q^17 + 2 * q^18 + q^19 + 4 * q^20 + 3 * q^21 + 2 * q^22 - q^23 + q^24 + 11 * q^25 + 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^38 - 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^53 - 5 * q^54 - 8 * q^55 + 3 * q^56 - q^57 + 5 * q^58 - 15 * q^59 - 4 * q^60 + 2 * q^61 - 8 * q^62 + 6 * q^63 + q^64 - 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 + q^76 + 6 * q^77 - 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 - q^92 - 8 * q^93 + 8 * q^94 + 4 * q^95 + 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
$$2$$ $$1$$
$$13$$ $$1$$
$$19$$ $$-1$$

## 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 6422.2.a.b 1
13.b even 2 1 38.2.a.b 1
39.d odd 2 1 342.2.a.d 1
52.b odd 2 1 304.2.a.d 1
65.d even 2 1 950.2.a.b 1
65.h odd 4 2 950.2.b.c 2
91.b odd 2 1 1862.2.a.f 1
104.e even 2 1 1216.2.a.n 1
104.h odd 2 1 1216.2.a.g 1
143.d odd 2 1 4598.2.a.a 1
156.h even 2 1 2736.2.a.w 1
195.e odd 2 1 8550.2.a.u 1
247.d odd 2 1 722.2.a.b 1
247.n odd 6 2 722.2.c.f 2
247.q even 6 2 722.2.c.d 2
247.bl odd 18 6 722.2.e.d 6
247.bn even 18 6 722.2.e.c 6
260.g odd 2 1 7600.2.a.h 1
741.d even 2 1 6498.2.a.y 1
988.g even 2 1 5776.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 13.b even 2 1
304.2.a.d 1 52.b odd 2 1
342.2.a.d 1 39.d odd 2 1
722.2.a.b 1 247.d odd 2 1
722.2.c.d 2 247.q even 6 2
722.2.c.f 2 247.n odd 6 2
722.2.e.c 6 247.bn even 18 6
722.2.e.d 6 247.bl odd 18 6
950.2.a.b 1 65.d even 2 1
950.2.b.c 2 65.h odd 4 2
1216.2.a.g 1 104.h odd 2 1
1216.2.a.n 1 104.e even 2 1
1862.2.a.f 1 91.b odd 2 1
2736.2.a.w 1 156.h even 2 1
4598.2.a.a 1 143.d odd 2 1
5776.2.a.d 1 988.g even 2 1
6422.2.a.b 1 1.a even 1 1 trivial
6498.2.a.y 1 741.d even 2 1
7600.2.a.h 1 260.g odd 2 1
8550.2.a.u 1 195.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

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

## Hecke characteristic polynomials

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