# Properties

 Label 637.2.a.h Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.404.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_1 - 2) q^{5} + (\beta_{2} - 3 \beta_1) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{8} + (\beta_{2} + \beta_1 + 3) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (-b2 - 1) * q^3 + (b2 - b1 + 2) * q^4 + (b1 - 2) * q^5 + (b2 - 3*b1) * q^6 + (-2*b2 + 2*b1 - 2) * q^8 + (b2 + b1 + 3) * q^9 $$q + (\beta_1 - 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_1 - 2) q^{5} + (\beta_{2} - 3 \beta_1) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{8} + (\beta_{2} + \beta_1 + 3) q^{9} + (\beta_{2} - 2 \beta_1 + 5) q^{10} + ( - \beta_1 - 1) q^{11} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{12} + q^{13} + (2 \beta_{2} - 3 \beta_1 + 1) q^{15} + (2 \beta_{2} - 4 \beta_1 + 2) q^{16} + ( - \beta_{2} - 1) q^{17} + (5 \beta_1 + 1) q^{18} + (\beta_{2} - 2 \beta_1 - 2) q^{19} + ( - 3 \beta_{2} + 5 \beta_1 - 6) q^{20} + ( - \beta_{2} - \beta_1 - 2) q^{22} + (\beta_{2} - 3 \beta_1 + 1) q^{23} + (2 \beta_{2} - 4 \beta_1 + 10) q^{24} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + (\beta_1 - 1) q^{26} + ( - 4 \beta_1 - 6) q^{27} + (2 \beta_{2} - 3) q^{29} + ( - 5 \beta_{2} + 5 \beta_1 - 8) q^{30} + (2 \beta_{2} + \beta_1) q^{31} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{32} + (\beta_{2} + 3 \beta_1 + 2) q^{33} + (\beta_{2} - 3 \beta_1) q^{34} + (3 \beta_{2} - \beta_1 + 8) q^{36} + ( - 2 \beta_{2} + \beta_1 - 3) q^{37} + ( - 3 \beta_{2} - 3) q^{38} + ( - \beta_{2} - 1) q^{39} + (6 \beta_{2} - 8 \beta_1 + 8) q^{40} + (4 \beta_{2} - 4 \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 2 \beta_1 + 3) q^{43} - 2 \beta_1 q^{44} + ( - \beta_{2} + 4 \beta_1 - 2) q^{45} + ( - 4 \beta_{2} + 3 \beta_1 - 9) q^{46} + (\beta_{2} - 6) q^{47} + ( - 2 \beta_{2} + 10 \beta_1 - 8) q^{48} + ( - 4 \beta_{2} + 4 \beta_1 - 10) q^{50} + (\beta_{2} + \beta_1 + 6) q^{51} + (\beta_{2} - \beta_1 + 2) q^{52} + ( - \beta_{2} - \beta_1 + 5) q^{53} + ( - 4 \beta_{2} - 6 \beta_1 - 6) q^{54} + ( - \beta_{2} - 1) q^{55} + (2 \beta_{2} + 5 \beta_1 - 1) q^{57} + ( - 2 \beta_{2} + \beta_1 + 5) q^{58} + (\beta_{2} + \beta_1 - 8) q^{59} + (6 \beta_{2} - 12 \beta_1 + 16) q^{60} + ( - 2 \beta_{2} + 2 \beta_1 + 8) q^{61} + ( - \beta_{2} + 4 \beta_1 + 5) q^{62} + ( - 4 \beta_1 + 8) q^{64} + (\beta_1 - 2) q^{65} + (2 \beta_{2} + 4 \beta_1 + 8) q^{66} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{67} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{68} + ( - \beta_{2} + 8 \beta_1 - 3) q^{69} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{71} + ( - 4 \beta_{2} + 4 \beta_1 - 10) q^{72} + (\beta_{2} + 6 \beta_1 - 4) q^{73} + (3 \beta_{2} - 7 \beta_1 + 4) q^{74} + ( - 2 \beta_{2} + 8 \beta_1 - 4) q^{75} + (\beta_{2} - 5 \beta_1 + 4) q^{76} + (\beta_{2} - 3 \beta_1) q^{78} + (2 \beta_{2} + 2 \beta_1 - 1) q^{79} + ( - 8 \beta_{2} + 10 \beta_1 - 14) q^{80} + (3 \beta_{2} + 9 \beta_1 + 1) q^{81} + ( - 8 \beta_{2} + 6 \beta_1 - 6) q^{82} + ( - 2 \beta_{2} + 3 \beta_1 - 8) q^{83} + (2 \beta_{2} - 3 \beta_1 + 1) q^{85} + (4 \beta_{2} - \beta_1 + 1) q^{86} + (3 \beta_{2} - 2 \beta_1 - 7) q^{87} + (2 \beta_1 - 2) q^{88} + (3 \beta_{2} - 4 \beta_1 + 4) q^{89} + (5 \beta_{2} - 4 \beta_1 + 13) q^{90} + (5 \beta_{2} - 11 \beta_1 + 12) q^{92} + ( - 5 \beta_1 - 11) q^{93} + ( - \beta_{2} - 4 \beta_1 + 7) q^{94} + ( - 4 \beta_{2} + 2 \beta_1 - 1) q^{95} + (8 \beta_{2} - 4 \beta_1 + 16) q^{96} + (\beta_{2} - 4 \beta_1 + 2) q^{97} + ( - 2 \beta_{2} - 7 \beta_1 - 7) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (-b2 - 1) * q^3 + (b2 - b1 + 2) * q^4 + (b1 - 2) * q^5 + (b2 - 3*b1) * q^6 + (-2*b2 + 2*b1 - 2) * q^8 + (b2 + b1 + 3) * q^9 + (b2 - 2*b1 + 5) * q^10 + (-b1 - 1) * q^11 + (-2*b2 + 2*b1 - 6) * q^12 + q^13 + (2*b2 - 3*b1 + 1) * q^15 + (2*b2 - 4*b1 + 2) * q^16 + (-b2 - 1) * q^17 + (5*b1 + 1) * q^18 + (b2 - 2*b1 - 2) * q^19 + (-3*b2 + 5*b1 - 6) * q^20 + (-b2 - b1 - 2) * q^22 + (b2 - 3*b1 + 1) * q^23 + (2*b2 - 4*b1 + 10) * q^24 + (b2 - 3*b1 + 2) * q^25 + (b1 - 1) * q^26 + (-4*b1 - 6) * q^27 + (2*b2 - 3) * q^29 + (-5*b2 + 5*b1 - 8) * q^30 + (2*b2 + b1) * q^31 + (-2*b2 + 2*b1 - 8) * q^32 + (b2 + 3*b1 + 2) * q^33 + (b2 - 3*b1) * q^34 + (3*b2 - b1 + 8) * q^36 + (-2*b2 + b1 - 3) * q^37 + (-3*b2 - 3) * q^38 + (-b2 - 1) * q^39 + (6*b2 - 8*b1 + 8) * q^40 + (4*b2 - 4*b1 - 2) * q^41 + (-2*b2 + 2*b1 + 3) * q^43 - 2*b1 * q^44 + (-b2 + 4*b1 - 2) * q^45 + (-4*b2 + 3*b1 - 9) * q^46 + (b2 - 6) * q^47 + (-2*b2 + 10*b1 - 8) * q^48 + (-4*b2 + 4*b1 - 10) * q^50 + (b2 + b1 + 6) * q^51 + (b2 - b1 + 2) * q^52 + (-b2 - b1 + 5) * q^53 + (-4*b2 - 6*b1 - 6) * q^54 + (-b2 - 1) * q^55 + (2*b2 + 5*b1 - 1) * q^57 + (-2*b2 + b1 + 5) * q^58 + (b2 + b1 - 8) * q^59 + (6*b2 - 12*b1 + 16) * q^60 + (-2*b2 + 2*b1 + 8) * q^61 + (-b2 + 4*b1 + 5) * q^62 + (-4*b1 + 8) * q^64 + (b1 - 2) * q^65 + (2*b2 + 4*b1 + 8) * q^66 + (-4*b2 + 2*b1 - 4) * q^67 + (-2*b2 + 2*b1 - 6) * q^68 + (-b2 + 8*b1 - 3) * q^69 + (-2*b2 + 3*b1 + 1) * q^71 + (-4*b2 + 4*b1 - 10) * q^72 + (b2 + 6*b1 - 4) * q^73 + (3*b2 - 7*b1 + 4) * q^74 + (-2*b2 + 8*b1 - 4) * q^75 + (b2 - 5*b1 + 4) * q^76 + (b2 - 3*b1) * q^78 + (2*b2 + 2*b1 - 1) * q^79 + (-8*b2 + 10*b1 - 14) * q^80 + (3*b2 + 9*b1 + 1) * q^81 + (-8*b2 + 6*b1 - 6) * q^82 + (-2*b2 + 3*b1 - 8) * q^83 + (2*b2 - 3*b1 + 1) * q^85 + (4*b2 - b1 + 1) * q^86 + (3*b2 - 2*b1 - 7) * q^87 + (2*b1 - 2) * q^88 + (3*b2 - 4*b1 + 4) * q^89 + (5*b2 - 4*b1 + 13) * q^90 + (5*b2 - 11*b1 + 12) * q^92 + (-5*b1 - 11) * q^93 + (-b2 - 4*b1 + 7) * q^94 + (-4*b2 + 2*b1 - 1) * q^95 + (8*b2 - 4*b1 + 16) * q^96 + (b2 - 4*b1 + 2) * q^97 + (-2*b2 - 7*b1 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{8} + 11 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 4 * q^3 + 6 * q^4 - 5 * q^5 - 2 * q^6 - 6 * q^8 + 11 * q^9 $$3 q - 2 q^{2} - 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{8} + 11 q^{9} + 14 q^{10} - 4 q^{11} - 18 q^{12} + 3 q^{13} + 2 q^{15} + 4 q^{16} - 4 q^{17} + 8 q^{18} - 7 q^{19} - 16 q^{20} - 8 q^{22} + q^{23} + 28 q^{24} + 4 q^{25} - 2 q^{26} - 22 q^{27} - 7 q^{29} - 24 q^{30} + 3 q^{31} - 24 q^{32} + 10 q^{33} - 2 q^{34} + 26 q^{36} - 10 q^{37} - 12 q^{38} - 4 q^{39} + 22 q^{40} - 6 q^{41} + 9 q^{43} - 2 q^{44} - 3 q^{45} - 28 q^{46} - 17 q^{47} - 16 q^{48} - 30 q^{50} + 20 q^{51} + 6 q^{52} + 13 q^{53} - 28 q^{54} - 4 q^{55} + 4 q^{57} + 14 q^{58} - 22 q^{59} + 42 q^{60} + 24 q^{61} + 18 q^{62} + 20 q^{64} - 5 q^{65} + 30 q^{66} - 14 q^{67} - 18 q^{68} - 2 q^{69} + 4 q^{71} - 30 q^{72} - 5 q^{73} + 8 q^{74} - 6 q^{75} + 8 q^{76} - 2 q^{78} + q^{79} - 40 q^{80} + 15 q^{81} - 20 q^{82} - 23 q^{83} + 2 q^{85} + 6 q^{86} - 20 q^{87} - 4 q^{88} + 11 q^{89} + 40 q^{90} + 30 q^{92} - 38 q^{93} + 16 q^{94} - 5 q^{95} + 52 q^{96} + 3 q^{97} - 30 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 4 * q^3 + 6 * q^4 - 5 * q^5 - 2 * q^6 - 6 * q^8 + 11 * q^9 + 14 * q^10 - 4 * q^11 - 18 * q^12 + 3 * q^13 + 2 * q^15 + 4 * q^16 - 4 * q^17 + 8 * q^18 - 7 * q^19 - 16 * q^20 - 8 * q^22 + q^23 + 28 * q^24 + 4 * q^25 - 2 * q^26 - 22 * q^27 - 7 * q^29 - 24 * q^30 + 3 * q^31 - 24 * q^32 + 10 * q^33 - 2 * q^34 + 26 * q^36 - 10 * q^37 - 12 * q^38 - 4 * q^39 + 22 * q^40 - 6 * q^41 + 9 * q^43 - 2 * q^44 - 3 * q^45 - 28 * q^46 - 17 * q^47 - 16 * q^48 - 30 * q^50 + 20 * q^51 + 6 * q^52 + 13 * q^53 - 28 * q^54 - 4 * q^55 + 4 * q^57 + 14 * q^58 - 22 * q^59 + 42 * q^60 + 24 * q^61 + 18 * q^62 + 20 * q^64 - 5 * q^65 + 30 * q^66 - 14 * q^67 - 18 * q^68 - 2 * q^69 + 4 * q^71 - 30 * q^72 - 5 * q^73 + 8 * q^74 - 6 * q^75 + 8 * q^76 - 2 * q^78 + q^79 - 40 * q^80 + 15 * q^81 - 20 * q^82 - 23 * q^83 + 2 * q^85 + 6 * q^86 - 20 * q^87 - 4 * q^88 + 11 * q^89 + 40 * q^90 + 30 * q^92 - 38 * q^93 + 16 * q^94 - 5 * q^95 + 52 * q^96 + 3 * q^97 - 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 −1.65544 −0.210756 2.86620
−2.65544 −2.39593 5.05137 −3.65544 6.36226 0 −8.10275 2.74049 9.70682
1.2 −1.21076 1.74483 −0.534070 −2.21076 −2.11256 0 3.06814 0.0444180 2.67669
1.3 1.86620 −3.34889 1.48270 0.866198 −6.24970 0 −0.965392 8.21509 1.61650
 $$n$$: 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.h 3
3.b odd 2 1 5733.2.a.be 3
7.b odd 2 1 637.2.a.i yes 3
7.c even 3 2 637.2.e.l 6
7.d odd 6 2 637.2.e.k 6
13.b even 2 1 8281.2.a.bh 3
21.c even 2 1 5733.2.a.bd 3
91.b odd 2 1 8281.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 1.a even 1 1 trivial
637.2.a.i yes 3 7.b odd 2 1
637.2.e.k 6 7.d odd 6 2
637.2.e.l 6 7.c even 3 2
5733.2.a.bd 3 21.c even 2 1
5733.2.a.be 3 3.b odd 2 1
8281.2.a.bh 3 13.b even 2 1
8281.2.a.bk 3 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 4T_{2} - 6$$ T2^3 + 2*T2^2 - 4*T2 - 6 $$T_{3}^{3} + 4T_{3}^{2} - 2T_{3} - 14$$ T3^3 + 4*T3^2 - 2*T3 - 14 $$T_{17}^{3} + 4T_{17}^{2} - 2T_{17} - 14$$ T17^3 + 4*T17^2 - 2*T17 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} + \cdots - 6$$
$3$ $$T^{3} + 4 T^{2} + \cdots - 14$$
$5$ $$T^{3} + 5 T^{2} + \cdots - 7$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 4T^{2} - 2$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} + 4 T^{2} + \cdots - 14$$
$19$ $$T^{3} + 7 T^{2} + \cdots - 63$$
$23$ $$T^{3} - T^{2} + \cdots - 43$$
$29$ $$T^{3} + 7 T^{2} + \cdots - 3$$
$31$ $$T^{3} - 3 T^{2} + \cdots + 49$$
$37$ $$T^{3} + 10 T^{2} + \cdots - 82$$
$41$ $$T^{3} + 6 T^{2} + \cdots - 504$$
$43$ $$T^{3} - 9 T^{2} + \cdots + 101$$
$47$ $$T^{3} + 17 T^{2} + \cdots + 147$$
$53$ $$T^{3} - 13 T^{2} + \cdots + 9$$
$59$ $$T^{3} + 22 T^{2} + \cdots + 252$$
$61$ $$T^{3} - 24 T^{2} + \cdots - 224$$
$67$ $$T^{3} + 14 T^{2} + \cdots - 648$$
$71$ $$T^{3} - 4 T^{2} + \cdots + 194$$
$73$ $$T^{3} + 5 T^{2} + \cdots - 1561$$
$79$ $$T^{3} - T^{2} + \cdots - 99$$
$83$ $$T^{3} + 23 T^{2} + \cdots + 203$$
$89$ $$T^{3} - 11 T^{2} + \cdots - 21$$
$97$ $$T^{3} - 3 T^{2} + \cdots + 7$$