# Properties

 Label 6253.2.a.c Level $6253$ Weight $2$ Character orbit 6253.a Self dual yes Analytic conductor $49.930$ 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] = [6253,2,Mod(1,6253)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6253, 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("6253.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6253 = 13^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6253.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: $$49.9304563839$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} + q^{7} + 6 q^{9}+O(q^{10})$$ q + 2 * q^2 - 3 * q^3 + 2 * q^4 + 2 * q^5 - 6 * q^6 + q^7 + 6 * q^9 $$q + 2 q^{2} - 3 q^{3} + 2 q^{4} + 2 q^{5} - 6 q^{6} + q^{7} + 6 q^{9} + 4 q^{10} + 5 q^{11} - 6 q^{12} + 2 q^{14} - 6 q^{15} - 4 q^{16} + 12 q^{18} + 4 q^{20} - 3 q^{21} + 10 q^{22} + 2 q^{23} - q^{25} - 9 q^{27} + 2 q^{28} + 6 q^{29} - 12 q^{30} + 4 q^{31} - 8 q^{32} - 15 q^{33} + 2 q^{35} + 12 q^{36} + q^{37} + 9 q^{41} - 6 q^{42} + 2 q^{43} + 10 q^{44} + 12 q^{45} + 4 q^{46} + 9 q^{47} + 12 q^{48} - 6 q^{49} - 2 q^{50} + q^{53} - 18 q^{54} + 10 q^{55} + 12 q^{58} - 8 q^{59} - 12 q^{60} - 8 q^{61} + 8 q^{62} + 6 q^{63} - 8 q^{64} - 30 q^{66} - 8 q^{67} - 6 q^{69} + 4 q^{70} - 9 q^{71} + q^{73} + 2 q^{74} + 3 q^{75} + 5 q^{77} + 4 q^{79} - 8 q^{80} + 9 q^{81} + 18 q^{82} + 15 q^{83} - 6 q^{84} + 4 q^{86} - 18 q^{87} - 4 q^{89} + 24 q^{90} + 4 q^{92} - 12 q^{93} + 18 q^{94} + 24 q^{96} - 4 q^{97} - 12 q^{98} + 30 q^{99}+O(q^{100})$$ q + 2 * q^2 - 3 * q^3 + 2 * q^4 + 2 * q^5 - 6 * q^6 + q^7 + 6 * q^9 + 4 * q^10 + 5 * q^11 - 6 * q^12 + 2 * q^14 - 6 * q^15 - 4 * q^16 + 12 * q^18 + 4 * q^20 - 3 * q^21 + 10 * q^22 + 2 * q^23 - q^25 - 9 * q^27 + 2 * q^28 + 6 * q^29 - 12 * q^30 + 4 * q^31 - 8 * q^32 - 15 * q^33 + 2 * q^35 + 12 * q^36 + q^37 + 9 * q^41 - 6 * q^42 + 2 * q^43 + 10 * q^44 + 12 * q^45 + 4 * q^46 + 9 * q^47 + 12 * q^48 - 6 * q^49 - 2 * q^50 + q^53 - 18 * q^54 + 10 * q^55 + 12 * q^58 - 8 * q^59 - 12 * q^60 - 8 * q^61 + 8 * q^62 + 6 * q^63 - 8 * q^64 - 30 * q^66 - 8 * q^67 - 6 * q^69 + 4 * q^70 - 9 * q^71 + q^73 + 2 * q^74 + 3 * q^75 + 5 * q^77 + 4 * q^79 - 8 * q^80 + 9 * q^81 + 18 * q^82 + 15 * q^83 - 6 * q^84 + 4 * q^86 - 18 * q^87 - 4 * q^89 + 24 * q^90 + 4 * q^92 - 12 * q^93 + 18 * q^94 + 24 * q^96 - 4 * q^97 - 12 * q^98 + 30 * 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
2.00000 −3.00000 2.00000 2.00000 −6.00000 1.00000 0 6.00000 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
$$13$$ $$+1$$
$$37$$ $$-1$$

## 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 6253.2.a.c 1
13.b even 2 1 37.2.a.a 1
39.d odd 2 1 333.2.a.d 1
52.b odd 2 1 592.2.a.e 1
65.d even 2 1 925.2.a.e 1
65.h odd 4 2 925.2.b.b 2
91.b odd 2 1 1813.2.a.a 1
104.e even 2 1 2368.2.a.q 1
104.h odd 2 1 2368.2.a.b 1
143.d odd 2 1 4477.2.a.b 1
156.h even 2 1 5328.2.a.r 1
195.e odd 2 1 8325.2.a.e 1
481.d even 2 1 1369.2.a.e 1
481.j odd 4 2 1369.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 13.b even 2 1
333.2.a.d 1 39.d odd 2 1
592.2.a.e 1 52.b odd 2 1
925.2.a.e 1 65.d even 2 1
925.2.b.b 2 65.h odd 4 2
1369.2.a.e 1 481.d even 2 1
1369.2.b.c 2 481.j odd 4 2
1813.2.a.a 1 91.b odd 2 1
2368.2.a.b 1 104.h odd 2 1
2368.2.a.q 1 104.e even 2 1
4477.2.a.b 1 143.d odd 2 1
5328.2.a.r 1 156.h even 2 1
6253.2.a.c 1 1.a even 1 1 trivial
8325.2.a.e 1 195.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6253))$$.

## Hecke characteristic polynomials

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