# Properties

 Label 637.2.a.a Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ 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] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.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: $$5.08647060876$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 3 q^{9}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + 3 * q^5 - 3 * q^9 $$q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 3 q^{9} - 6 q^{10} - 6 q^{11} + q^{13} - 4 q^{16} - 4 q^{17} + 6 q^{18} - 5 q^{19} + 6 q^{20} + 12 q^{22} + 3 q^{23} + 4 q^{25} - 2 q^{26} - 5 q^{29} + 3 q^{31} + 8 q^{32} + 8 q^{34} - 6 q^{36} - 4 q^{37} + 10 q^{38} + 6 q^{41} - q^{43} - 12 q^{44} - 9 q^{45} - 6 q^{46} - 7 q^{47} - 8 q^{50} + 2 q^{52} - 9 q^{53} - 18 q^{55} + 10 q^{58} - 8 q^{59} + 10 q^{61} - 6 q^{62} - 8 q^{64} + 3 q^{65} - 6 q^{67} - 8 q^{68} - 8 q^{71} + 13 q^{73} + 8 q^{74} - 10 q^{76} + 3 q^{79} - 12 q^{80} + 9 q^{81} - 12 q^{82} - 15 q^{83} - 12 q^{85} + 2 q^{86} - 3 q^{89} + 18 q^{90} + 6 q^{92} + 14 q^{94} - 15 q^{95} - 7 q^{97} + 18 q^{99}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + 3 * q^5 - 3 * q^9 - 6 * q^10 - 6 * q^11 + q^13 - 4 * q^16 - 4 * q^17 + 6 * q^18 - 5 * q^19 + 6 * q^20 + 12 * q^22 + 3 * q^23 + 4 * q^25 - 2 * q^26 - 5 * q^29 + 3 * q^31 + 8 * q^32 + 8 * q^34 - 6 * q^36 - 4 * q^37 + 10 * q^38 + 6 * q^41 - q^43 - 12 * q^44 - 9 * q^45 - 6 * q^46 - 7 * q^47 - 8 * q^50 + 2 * q^52 - 9 * q^53 - 18 * q^55 + 10 * q^58 - 8 * q^59 + 10 * q^61 - 6 * q^62 - 8 * q^64 + 3 * q^65 - 6 * q^67 - 8 * q^68 - 8 * q^71 + 13 * q^73 + 8 * q^74 - 10 * q^76 + 3 * q^79 - 12 * q^80 + 9 * q^81 - 12 * q^82 - 15 * q^83 - 12 * q^85 + 2 * q^86 - 3 * q^89 + 18 * q^90 + 6 * q^92 + 14 * q^94 - 15 * q^95 - 7 * q^97 + 18 * 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 0 2.00000 3.00000 0 0 0 −3.00000 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 637.2.a.a 1
3.b odd 2 1 5733.2.a.l 1
7.b odd 2 1 91.2.a.a 1
7.c even 3 2 637.2.e.d 2
7.d odd 6 2 637.2.e.e 2
13.b even 2 1 8281.2.a.l 1
21.c even 2 1 819.2.a.f 1
28.d even 2 1 1456.2.a.g 1
35.c odd 2 1 2275.2.a.h 1
56.e even 2 1 5824.2.a.t 1
56.h odd 2 1 5824.2.a.s 1
91.b odd 2 1 1183.2.a.b 1
91.i even 4 2 1183.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 7.b odd 2 1
637.2.a.a 1 1.a even 1 1 trivial
637.2.e.d 2 7.c even 3 2
637.2.e.e 2 7.d odd 6 2
819.2.a.f 1 21.c even 2 1
1183.2.a.b 1 91.b odd 2 1
1183.2.c.b 2 91.i even 4 2
1456.2.a.g 1 28.d even 2 1
2275.2.a.h 1 35.c odd 2 1
5733.2.a.l 1 3.b odd 2 1
5824.2.a.s 1 56.h odd 2 1
5824.2.a.t 1 56.e even 2 1
8281.2.a.l 1 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{3}$$ T3 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

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