# Properties

 Label 637.2.a.k Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $5$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.746052.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 7x^{3} + 8x + 2$$ x^5 - x^4 - 7*x^3 + 8*x + 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + \beta_{4} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{4} + \beta_{2} q^{5} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 2) q^{8}+ \cdots + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + b4 * q^3 + (-b4 + b3 - b1 + 2) * q^4 + b2 * q^5 + (b4 + b3 + b1 + 1) * q^6 + (-b4 + b3 - b2 - 2*b1 + 2) * q^8 + (b2 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{2} + \beta_{4} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{4} + \beta_{2} q^{5} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 2) q^{8}+ \cdots + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \cdots + 4) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + b4 * q^3 + (-b4 + b3 - b1 + 2) * q^4 + b2 * q^5 + (b4 + b3 + b1 + 1) * q^6 + (-b4 + b3 - b2 - 2*b1 + 2) * q^8 + (b2 + 1) * q^9 + (-b3 + b2 + b1 + 1) * q^10 + (-b3 + 2) * q^11 + (b4 + b3 - b2 - b1 - 1) * q^12 - q^13 + (2*b4 + b2 + 2*b1) * q^15 + (b3 - 2*b2 - 3*b1 + 2) * q^16 + (-b4 - b2 - 2*b1 + 1) * q^17 + (-b3 + b2 + 2) * q^18 + (-b4 - b3 - b2 - 2*b1 - 2) * q^19 + (-3*b3 + b1 - 1) * q^20 + (-b4 - b3 + b2 - b1 + 2) * q^22 + (-b4 + 2) * q^23 + (-b4 + 2*b3 - 2*b2 - 2*b1) * q^24 + (2*b4 - 2*b3 - b2 + 1) * q^25 + (b1 - 1) * q^26 + (b2 + 2*b1) * q^27 + (b4 + 2*b1 - 1) * q^29 + (4*b4 - b3 + b2 + 3*b1 - 3) * q^30 + (b4 + 2*b2 + 2) * q^31 + (4*b3 - b2 - b1 + 5) * q^32 + (3*b4 + 2*b1 - 2) * q^33 + (-3*b4 + 2*b3 - b2 - 3*b1 + 5) * q^34 + (-b4 - 2*b3 + 1) * q^36 + (-b4 - 2*b3 + 2*b1) * q^37 + (-4*b4 + b3 + b1 + 2) * q^38 - b4 * q^39 + (-2*b4 - 2*b3 + b2 + 2*b1 - 6) * q^40 + (3*b4 - 2*b3 - b2 + 2*b1 - 4) * q^41 + (b4 - 2*b3 - b2 - 2*b1) * q^43 + (-3*b4 + 2*b2 - b1 + 1) * q^44 + (2*b4 - 2*b3 + 6) * q^45 + (-b4 - b3 - 3*b1 + 1) * q^46 + (-4*b4 + b3) * q^47 + (-3*b4 + 3*b3 - 2*b2 - 3*b1 + 5) * q^48 + (b3 + b2 + 2*b1 + 2) * q^50 + (-b4 + 2*b3 - 2*b2 - 2) * q^51 + (b4 - b3 + b1 - 2) * q^52 + (2*b4 + 2*b3 + 4*b1 + 3) * q^53 + (2*b4 - 3*b3 + b2 + b1 - 5) * q^54 + (-2*b4 + 2*b3 + 3*b2 - 2*b1 + 2) * q^55 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^57 + (3*b4 - b3 + 2*b1 - 6) * q^58 + (b3 - 2) * q^59 + (2*b4 - b3 + 5*b1 - 7) * q^60 + (-2*b4 + 4*b3 + 3) * q^61 + (b4 - b3 + 2*b2 + b1 + 5) * q^62 + (3*b4 + 4*b3 - b2 - 4*b1 + 3) * q^64 - b2 * q^65 + (5*b4 + b3 + 5*b1 - 5) * q^66 + (b4 + 3*b3 + 3*b2 + 2*b1 + 4) * q^67 + (-2*b4 + 3*b3 - b2 - 7*b1 + 8) * q^68 + (2*b4 - b2 - 4) * q^69 + (b4 - b3 + 4*b1 + 2) * q^71 + (-3*b4 - b3 - 4) * q^72 + (2*b3 - b2) * q^73 + (-b4 - 5*b3 + 2*b2 + b1 - 7) * q^74 + (b4 + b2 + 2*b1 + 4) * q^75 + (-2*b3 + b2 - 3*b1 - 1) * q^76 + (-b4 - b3 - b1 - 1) * q^78 + (-2*b4 + 2*b3 - b2 + 2*b1) * q^79 + (-2*b4 - b3 + 3*b2 + 5*b1 - 11) * q^80 + (2*b4 - 2*b3 - 2*b2 - 5) * q^81 + (3*b4 + b2 + 8*b1 - 8) * q^82 + (-b4 - 3*b2 - 2*b1 - 2) * q^83 + (-4*b4 + b2 - 4) * q^85 + (-3*b4 + 2*b3 + b2 + 2*b1 + 6) * q^86 + (-b4 - 2*b3 + b2 - 2*b1 + 2) * q^87 + (-2*b4 - 2*b3 - 1) * q^88 + (3*b4 - 2*b3 - 2*b2 - 2*b1) * q^89 + (2*b2 - 2*b1 + 8) * q^90 + (-3*b4 + b3 + b2 - b1 + 5) * q^92 + (6*b4 + 3*b2 + 4*b1 + 4) * q^93 + (-3*b4 - 3*b3 - b2 - 5*b1 - 4) * q^94 + (-6*b4 + 2*b3 - b2 - 2*b1 - 2) * q^95 + (-b4 + b3 - b2 - 9*b1 + 9) * q^96 + (b4 - 2*b3 - b2 - 2*b1 + 2) * q^97 + (-2*b4 + b3 + 3*b2 - 2*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q + 4 * q^2 + 8 * q^4 - 2 * q^5 + 5 * q^6 + 9 * q^8 + 3 * q^9 $$5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} - 5 q^{13} + 10 q^{16} + 5 q^{17} + 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 4 q^{26} - 3 q^{29} - 13 q^{30} + 6 q^{31} + 22 q^{32} - 8 q^{33} + 22 q^{34} + 7 q^{36} + 4 q^{37} + 10 q^{38} - 28 q^{40} - 14 q^{41} + 2 q^{43} + 32 q^{45} + 3 q^{46} - q^{47} + 23 q^{48} + 9 q^{50} - 8 q^{51} - 8 q^{52} + 17 q^{53} - 23 q^{54} - 16 q^{57} - 27 q^{58} - 11 q^{59} - 29 q^{60} + 11 q^{61} + 23 q^{62} + 9 q^{64} + 2 q^{65} - 21 q^{66} + 13 q^{67} + 32 q^{68} - 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} + 20 q^{75} - 8 q^{76} - 5 q^{78} + 2 q^{79} - 55 q^{80} - 19 q^{81} - 34 q^{82} - 6 q^{83} - 22 q^{85} + 28 q^{86} + 8 q^{87} - 3 q^{88} + 4 q^{89} + 34 q^{90} + 21 q^{92} + 18 q^{93} - 20 q^{94} - 12 q^{95} + 37 q^{96} + 12 q^{97} + 11 q^{99}+O(q^{100})$$ 5 * q + 4 * q^2 + 8 * q^4 - 2 * q^5 + 5 * q^6 + 9 * q^8 + 3 * q^9 + 5 * q^10 + 11 * q^11 - 5 * q^12 - 5 * q^13 + 10 * q^16 + 5 * q^17 + 9 * q^18 - 9 * q^19 - q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 - 4 * q^26 - 3 * q^29 - 13 * q^30 + 6 * q^31 + 22 * q^32 - 8 * q^33 + 22 * q^34 + 7 * q^36 + 4 * q^37 + 10 * q^38 - 28 * q^40 - 14 * q^41 + 2 * q^43 + 32 * q^45 + 3 * q^46 - q^47 + 23 * q^48 + 9 * q^50 - 8 * q^51 - 8 * q^52 + 17 * q^53 - 23 * q^54 - 16 * q^57 - 27 * q^58 - 11 * q^59 - 29 * q^60 + 11 * q^61 + 23 * q^62 + 9 * q^64 + 2 * q^65 - 21 * q^66 + 13 * q^67 + 32 * q^68 - 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 + 20 * q^75 - 8 * q^76 - 5 * q^78 + 2 * q^79 - 55 * q^80 - 19 * q^81 - 34 * q^82 - 6 * q^83 - 22 * q^85 + 28 * q^86 + 8 * q^87 - 3 * q^88 + 4 * q^89 + 34 * q^90 + 21 * q^92 + 18 * q^93 - 20 * q^94 - 12 * q^95 + 37 * q^96 + 12 * q^97 + 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 7x^{3} + 8x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 4\nu + 2$$ v^3 - 2*v^2 - 4*v + 2 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 4\nu^{2} + 3\nu + 1$$ v^4 - 2*v^3 - 4*v^2 + 3*v + 1 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 5\nu^{2} + 4\nu + 4$$ v^4 - 2*v^3 - 5*v^2 + 4*v + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta _1 + 3$$ -b4 + b3 + b1 + 3 $$\nu^{3}$$ $$=$$ $$-2\beta_{4} + 2\beta_{3} + \beta_{2} + 6\beta _1 + 4$$ -2*b4 + 2*b3 + b2 + 6*b1 + 4 $$\nu^{4}$$ $$=$$ $$-8\beta_{4} + 9\beta_{3} + 2\beta_{2} + 13\beta _1 + 19$$ -8*b4 + 9*b3 + 2*b2 + 13*b1 + 19

## 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
 3.00852 1.19566 −0.265608 −1.21332 −1.72525
−2.00852 −1.75906 2.03417 −0.905722 3.53311 0 −0.0686323 0.0942784 1.81916
1.2 −0.195656 0.259788 −1.96172 −3.93251 −0.0508292 0 0.775135 −2.93251 0.769420
1.3 1.26561 2.62728 −0.398235 2.90260 3.32511 0 −3.03523 3.90260 3.67356
1.4 2.21332 −2.47443 2.89879 2.12280 −5.47671 0 1.98932 3.12280 4.69843
1.5 2.72525 1.34642 5.42699 −2.18716 3.66932 0 9.33940 −1.18716 −5.96057
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.k 5
3.b odd 2 1 5733.2.a.bm 5
7.b odd 2 1 637.2.a.l 5
7.c even 3 2 637.2.e.m 10
7.d odd 6 2 91.2.e.c 10
13.b even 2 1 8281.2.a.bx 5
21.c even 2 1 5733.2.a.bl 5
21.g even 6 2 819.2.j.h 10
28.f even 6 2 1456.2.r.p 10
91.b odd 2 1 8281.2.a.bw 5
91.s odd 6 2 1183.2.e.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 7.d odd 6 2
637.2.a.k 5 1.a even 1 1 trivial
637.2.a.l 5 7.b odd 2 1
637.2.e.m 10 7.c even 3 2
819.2.j.h 10 21.g even 6 2
1183.2.e.f 10 91.s odd 6 2
1456.2.r.p 10 28.f even 6 2
5733.2.a.bl 5 21.c even 2 1
5733.2.a.bm 5 3.b odd 2 1
8281.2.a.bw 5 91.b odd 2 1
8281.2.a.bx 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{5} - 4T_{2}^{4} - T_{2}^{3} + 17T_{2}^{2} - 12T_{2} - 3$$ T2^5 - 4*T2^4 - T2^3 + 17*T2^2 - 12*T2 - 3 $$T_{3}^{5} - 9T_{3}^{3} + 16T_{3} - 4$$ T3^5 - 9*T3^3 + 16*T3 - 4 $$T_{17}^{5} - 5T_{17}^{4} - 22T_{17}^{3} + 106T_{17}^{2} + 93T_{17} - 429$$ T17^5 - 5*T17^4 - 22*T17^3 + 106*T17^2 + 93*T17 - 429

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 4 T^{4} + \cdots - 3$$
$3$ $$T^{5} - 9 T^{3} + \cdots - 4$$
$5$ $$T^{5} + 2 T^{4} + \cdots + 48$$
$7$ $$T^{5}$$
$11$ $$T^{5} - 11 T^{4} + \cdots + 33$$
$13$ $$(T + 1)^{5}$$
$17$ $$T^{5} - 5 T^{4} + \cdots - 429$$
$19$ $$T^{5} + 9 T^{4} + \cdots + 223$$
$23$ $$T^{5} - 10 T^{4} + \cdots + 12$$
$29$ $$T^{5} + 3 T^{4} + \cdots - 108$$
$31$ $$T^{5} - 6 T^{4} + \cdots + 356$$
$37$ $$T^{5} - 4 T^{4} + \cdots - 7036$$
$41$ $$T^{5} + 14 T^{4} + \cdots + 1584$$
$43$ $$T^{5} - 2 T^{4} + \cdots + 64$$
$47$ $$T^{5} + T^{4} + \cdots + 5169$$
$53$ $$T^{5} - 17 T^{4} + \cdots + 19959$$
$59$ $$T^{5} + 11 T^{4} + \cdots - 33$$
$61$ $$T^{5} - 11 T^{4} + \cdots + 8461$$
$67$ $$T^{5} - 13 T^{4} + \cdots - 22699$$
$71$ $$T^{5} - 15 T^{4} + \cdots - 6336$$
$73$ $$T^{5} - 75 T^{3} + \cdots + 712$$
$79$ $$T^{5} - 2 T^{4} + \cdots - 1000$$
$83$ $$T^{5} + 6 T^{4} + \cdots + 7488$$
$89$ $$T^{5} - 4 T^{4} + \cdots - 7692$$
$97$ $$T^{5} - 12 T^{4} + \cdots + 2384$$