# Properties

 Label 637.2.a.g Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + \beta q^{3} + ( - \beta - 3) q^{5} + 2 q^{6} - 2 \beta q^{8} - q^{9} +O(q^{10})$$ q + b * q^2 + b * q^3 + (-b - 3) * q^5 + 2 * q^6 - 2*b * q^8 - q^9 $$q + \beta q^{2} + \beta q^{3} + ( - \beta - 3) q^{5} + 2 q^{6} - 2 \beta q^{8} - q^{9} + ( - 3 \beta - 2) q^{10} - 3 \beta q^{11} + q^{13} + ( - 3 \beta - 2) q^{15} - 4 q^{16} + \beta q^{17} - \beta q^{18} + ( - 3 \beta + 3) q^{19} - 6 q^{22} + (2 \beta - 3) q^{23} - 4 q^{24} + (6 \beta + 6) q^{25} + \beta q^{26} - 4 \beta q^{27} + (2 \beta + 3) q^{29} + ( - 2 \beta - 6) q^{30} + (3 \beta + 1) q^{31} - 6 q^{33} + 2 q^{34} + ( - 3 \beta - 2) q^{37} + (3 \beta - 6) q^{38} + \beta q^{39} + (6 \beta + 4) q^{40} + (2 \beta - 6) q^{41} - 5 q^{43} + (\beta + 3) q^{45} + ( - 3 \beta + 4) q^{46} + ( - \beta - 3) q^{47} - 4 \beta q^{48} + (6 \beta + 12) q^{50} + 2 q^{51} + ( - 2 \beta - 3) q^{53} - 8 q^{54} + (9 \beta + 6) q^{55} + (3 \beta - 6) q^{57} + (3 \beta + 4) q^{58} + ( - 4 \beta - 6) q^{59} - 6 q^{61} + (\beta + 6) q^{62} + 8 q^{64} + ( - \beta - 3) q^{65} - 6 \beta q^{66} + (6 \beta - 6) q^{67} + ( - 3 \beta + 4) q^{69} + (5 \beta - 6) q^{71} + 2 \beta q^{72} + ( - 3 \beta + 5) q^{73} + ( - 2 \beta - 6) q^{74} + (6 \beta + 12) q^{75} + 2 q^{78} + ( - 6 \beta + 7) q^{79} + (4 \beta + 12) q^{80} - 5 q^{81} + ( - 6 \beta + 4) q^{82} + (3 \beta - 9) q^{83} + ( - 3 \beta - 2) q^{85} - 5 \beta q^{86} + (3 \beta + 4) q^{87} + 12 q^{88} + ( - \beta - 3) q^{89} + (3 \beta + 2) q^{90} + (\beta + 6) q^{93} + ( - 3 \beta - 2) q^{94} + (6 \beta - 3) q^{95} + (9 \beta + 1) q^{97} + 3 \beta q^{99} +O(q^{100})$$ q + b * q^2 + b * q^3 + (-b - 3) * q^5 + 2 * q^6 - 2*b * q^8 - q^9 + (-3*b - 2) * q^10 - 3*b * q^11 + q^13 + (-3*b - 2) * q^15 - 4 * q^16 + b * q^17 - b * q^18 + (-3*b + 3) * q^19 - 6 * q^22 + (2*b - 3) * q^23 - 4 * q^24 + (6*b + 6) * q^25 + b * q^26 - 4*b * q^27 + (2*b + 3) * q^29 + (-2*b - 6) * q^30 + (3*b + 1) * q^31 - 6 * q^33 + 2 * q^34 + (-3*b - 2) * q^37 + (3*b - 6) * q^38 + b * q^39 + (6*b + 4) * q^40 + (2*b - 6) * q^41 - 5 * q^43 + (b + 3) * q^45 + (-3*b + 4) * q^46 + (-b - 3) * q^47 - 4*b * q^48 + (6*b + 12) * q^50 + 2 * q^51 + (-2*b - 3) * q^53 - 8 * q^54 + (9*b + 6) * q^55 + (3*b - 6) * q^57 + (3*b + 4) * q^58 + (-4*b - 6) * q^59 - 6 * q^61 + (b + 6) * q^62 + 8 * q^64 + (-b - 3) * q^65 - 6*b * q^66 + (6*b - 6) * q^67 + (-3*b + 4) * q^69 + (5*b - 6) * q^71 + 2*b * q^72 + (-3*b + 5) * q^73 + (-2*b - 6) * q^74 + (6*b + 12) * q^75 + 2 * q^78 + (-6*b + 7) * q^79 + (4*b + 12) * q^80 - 5 * q^81 + (-6*b + 4) * q^82 + (3*b - 9) * q^83 + (-3*b - 2) * q^85 - 5*b * q^86 + (3*b + 4) * q^87 + 12 * q^88 + (-b - 3) * q^89 + (3*b + 2) * q^90 + (b + 6) * q^93 + (-3*b - 2) * q^94 + (6*b - 3) * q^95 + (9*b + 1) * q^97 + 3*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9 $$2 q - 6 q^{5} + 4 q^{6} - 2 q^{9} - 4 q^{10} + 2 q^{13} - 4 q^{15} - 8 q^{16} + 6 q^{19} - 12 q^{22} - 6 q^{23} - 8 q^{24} + 12 q^{25} + 6 q^{29} - 12 q^{30} + 2 q^{31} - 12 q^{33} + 4 q^{34} - 4 q^{37} - 12 q^{38} + 8 q^{40} - 12 q^{41} - 10 q^{43} + 6 q^{45} + 8 q^{46} - 6 q^{47} + 24 q^{50} + 4 q^{51} - 6 q^{53} - 16 q^{54} + 12 q^{55} - 12 q^{57} + 8 q^{58} - 12 q^{59} - 12 q^{61} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 12 q^{67} + 8 q^{69} - 12 q^{71} + 10 q^{73} - 12 q^{74} + 24 q^{75} + 4 q^{78} + 14 q^{79} + 24 q^{80} - 10 q^{81} + 8 q^{82} - 18 q^{83} - 4 q^{85} + 8 q^{87} + 24 q^{88} - 6 q^{89} + 4 q^{90} + 12 q^{93} - 4 q^{94} - 6 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q - 6 * q^5 + 4 * q^6 - 2 * q^9 - 4 * q^10 + 2 * q^13 - 4 * q^15 - 8 * q^16 + 6 * q^19 - 12 * q^22 - 6 * q^23 - 8 * q^24 + 12 * q^25 + 6 * q^29 - 12 * q^30 + 2 * q^31 - 12 * q^33 + 4 * q^34 - 4 * q^37 - 12 * q^38 + 8 * q^40 - 12 * q^41 - 10 * q^43 + 6 * q^45 + 8 * q^46 - 6 * q^47 + 24 * q^50 + 4 * q^51 - 6 * q^53 - 16 * q^54 + 12 * q^55 - 12 * q^57 + 8 * q^58 - 12 * q^59 - 12 * q^61 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 12 * q^67 + 8 * q^69 - 12 * q^71 + 10 * q^73 - 12 * q^74 + 24 * q^75 + 4 * q^78 + 14 * q^79 + 24 * q^80 - 10 * q^81 + 8 * q^82 - 18 * q^83 - 4 * q^85 + 8 * q^87 + 24 * q^88 - 6 * q^89 + 4 * q^90 + 12 * q^93 - 4 * q^94 - 6 * q^95 + 2 * q^97

## 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
 −1.41421 1.41421
−1.41421 −1.41421 0 −1.58579 2.00000 0 2.82843 −1.00000 2.24264
1.2 1.41421 1.41421 0 −4.41421 2.00000 0 −2.82843 −1.00000 −6.24264
 $$n$$: 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.g 2
3.b odd 2 1 5733.2.a.s 2
7.b odd 2 1 91.2.a.c 2
7.c even 3 2 637.2.e.g 4
7.d odd 6 2 637.2.e.f 4
13.b even 2 1 8281.2.a.v 2
21.c even 2 1 819.2.a.h 2
28.d even 2 1 1456.2.a.q 2
35.c odd 2 1 2275.2.a.j 2
56.e even 2 1 5824.2.a.bk 2
56.h odd 2 1 5824.2.a.bl 2
91.b odd 2 1 1183.2.a.d 2
91.i even 4 2 1183.2.c.d 4

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

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{17}^{2} - 2$$ T17^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} + 6T + 7$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 18$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} - 6T - 9$$
$23$ $$T^{2} + 6T + 1$$
$29$ $$T^{2} - 6T + 1$$
$31$ $$T^{2} - 2T - 17$$
$37$ $$T^{2} + 4T - 14$$
$41$ $$T^{2} + 12T + 28$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} + 6T + 7$$
$53$ $$T^{2} + 6T + 1$$
$59$ $$T^{2} + 12T + 4$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 12T - 36$$
$71$ $$T^{2} + 12T - 14$$
$73$ $$T^{2} - 10T + 7$$
$79$ $$T^{2} - 14T - 23$$
$83$ $$T^{2} + 18T + 63$$
$89$ $$T^{2} + 6T + 7$$
$97$ $$T^{2} - 2T - 161$$