# Properties

 Label 637.2.a.m Level $637$ Weight $2$ Character orbit 637.a Self dual yes Analytic conductor $5.086$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.4507648.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{5} + \beta_{3} + \beta_1) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b1 - 1) * q^3 + (b4 - b3 - b1 + 1) * q^4 + (-b4 - 1) * q^5 + (b3 + b2 + b1 - 1) * q^6 + (b5 + b3 + b1) * q^8 + (b2 + 3*b1) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{5} + \beta_{3} + \beta_1) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + (2 \beta_{2} + \beta_1 - 1) q^{10} + ( - 2 \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{12}+ \cdots + ( - \beta_{5} - 7 \beta_{4} + \cdots + 5 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b1 - 1) * q^3 + (b4 - b3 - b1 + 1) * q^4 + (-b4 - 1) * q^5 + (b3 + b2 + b1 - 1) * q^6 + (b5 + b3 + b1) * q^8 + (b2 + 3*b1) * q^9 + (2*b2 + b1 - 1) * q^10 + (-2*b5 - b4) * q^11 + (-b5 - b4 + b2 + b1) * q^12 - q^13 + (b5 + 2*b4 + b3 + b2 + b1 + 2) * q^15 + (-b4 - b3 + b2) * q^16 + (b5 + b4 + b3 + b2 + 2*b1 - 3) * q^17 + (-b4 - 2*b3 - 2*b1) * q^18 + (b5 + b4 - b3 + b2) * q^19 + (b3 + b2 + b1 - 3) * q^20 + (-2*b5 - 2*b4 + 4*b3 - b2 + b1 - 3) * q^22 + (2*b5 + b4 - b3 - b1) * q^23 + (-b5 - 2*b4 - 2*b2 - b1 - 2) * q^24 + (b4 + b3 + b1 - 1) * q^25 + b2 * q^26 + (-b3 - 4*b2 - 4*b1 - 2) * q^27 + (-2*b4 + 2*b3 - 1) * q^29 + (-2*b3 - 3*b2 - 3*b1 + 1) * q^30 + (-b5 + 3*b4 + 2*b2 - b1 - 1) * q^31 + (-b5 - b4 - b3 + b2 + b1 - 4) * q^32 + (3*b5 + 4*b4 - b3 + b2 - 2*b1 + 1) * q^33 + (-3*b3 + 3*b2 - 3*b1 + 1) * q^34 + (2*b5 + 2*b3 - b2 - b1 - 3) * q^36 + (2*b5 + b4 - 2*b3 + 2*b2 + 2*b1) * q^37 + (2*b5 - b3 + b1 - 1) * q^38 + (b1 + 1) * q^39 + (-b5 - b4 - b2 - 3*b1) * q^40 + (-b5 - 2*b3 - 2*b2 + 3*b1) * q^41 + (-2*b5 + 2*b4 - 4*b1 + 1) * q^43 + (-2*b5 + b4 + 2*b3 + 3*b2 - 4*b1) * q^44 + (-3*b5 - 3*b4 - 3*b3 - 5*b2 - 4*b1 - 2) * q^45 + (3*b5 + 2*b4 - 3*b3 + b2 + b1 + 2) * q^46 + (-3*b5 - b4 + 3*b3 - 3*b2 - 6) * q^47 + (b5 + 3*b4 + b3 + b2 - b1 + 2) * q^48 + (-b5 - b3 - 3*b1 + 2) * q^50 + (-2*b5 - 4*b4 - 2*b3 - 5*b2 - b1 - 1) * q^51 + (-b4 + b3 + b1 - 1) * q^52 + (-b4 + b3 - 4*b2 - b1 - 2) * q^53 + (b5 + 4*b4 + 2*b2 + b1 + 8) * q^54 + (2*b5 - 3*b3 + 3*b1 + 1) * q^55 + (-2*b5 - 2*b4 - b2) * q^57 + (-2*b5 + 3*b2 - 2) * q^58 + (-2*b5 - 2*b4 + 3*b3 - b2 + b1 - 5) * q^59 + (-b4 - 2*b3 - 3*b2 + 2) * q^60 + (-3*b5 + b4 + 2*b3 - b2 + 3*b1 - 2) * q^61 + (-b5 - 3*b4 + 5*b3 - 3*b2 - 5) * q^62 + (4*b3 + 2*b2 + 2*b1 - 4) * q^64 + (b4 + 1) * q^65 + (4*b5 + 2*b4 - 3*b3 - 2*b2 + 2) * q^66 + (3*b5 + b4 + b3 + b2 - b1 + 4) * q^67 + (b5 - 5*b4 + 4*b3 - 3*b2 + 5*b1 - 6) * q^68 + (-3*b5 - 3*b4 + 2*b3 + b2 + 4*b1 + 1) * q^69 + (3*b5 + 5*b4 - 5*b3 + 4*b2 + b1 + 2) * q^71 + (5*b4 + 5*b2 + 2*b1 + 4) * q^72 + (5*b5 + b4 - b3 + b2 + 2*b1 + 2) * q^73 + (4*b5 - 4*b3 + b2 + b1 - 1) * q^74 + (-b5 - 3*b4 - 2*b3 - 3*b2 - b1 - 2) * q^75 + (b5 - 3*b3 + b2 + 3) * q^76 + (-b3 - b2 - b1 + 1) * q^78 + (3*b4 + 3*b3 - 3*b2 - 2*b1 - 3) * q^79 + (-b5 + 2*b3 - 2*b2 + b1 + 4) * q^80 + (b4 + 5*b3 + 6*b2 + 5*b1 + 6) * q^81 + (b5 + b4 - 3*b3 - b2 - 3*b1 + 8) * q^82 + (2*b5 - b4 - 2*b1 - 7) * q^83 + (-2*b5 + b4 - 2*b3 - 4*b2 - 6*b1) * q^85 + (-2*b5 - 2*b4 + 8*b3 - 5*b2 + 2*b1 - 4) * q^86 + (2*b5 + 2*b4 + b1 + 3) * q^87 + (-b4 + 3*b3 - b2 + 2*b1 - 8) * q^88 + (2*b5 - 3*b4 + b3 - b2 + b1 - 4) * q^89 + (2*b4 + 5*b3 + 2*b2 + 5*b1 + 5) * q^90 + (2*b5 - 4*b3 - b2 + 3*b1 + 3) * q^92 + (-2*b5 - 5*b4 - 6*b3 - 4*b2 + 2) * q^93 + (-6*b5 + 3*b3 + 4*b2 - 5*b1 + 5) * q^94 + (-2*b5 + 2*b3 - 2*b2 - 2*b1 - 1) * q^95 + (2*b5 + 4*b4 + 4) * q^96 + (2*b5 + b4 + 5*b3 - 3*b2 - 3*b1 + 4) * q^97 + (-b5 - 7*b4 - b3 - 2*b2 + 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9}+O(q^{10})$$ 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 $$6 q - 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 12 q^{15} - 16 q^{17} - 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} - 20 q^{27} - 6 q^{29} - 6 q^{31} - 20 q^{32} - 4 q^{33} - 24 q^{36} - 8 q^{38} + 8 q^{39} - 4 q^{40} + 8 q^{41} + 2 q^{43} - 4 q^{44} - 14 q^{45} + 8 q^{46} - 30 q^{47} + 8 q^{48} + 8 q^{50} - 4 q^{51} - 4 q^{52} - 14 q^{53} + 48 q^{54} + 8 q^{55} + 4 q^{57} - 8 q^{58} - 24 q^{59} + 12 q^{60} - 28 q^{62} - 20 q^{64} + 6 q^{65} + 4 q^{66} + 16 q^{67} - 28 q^{68} + 20 q^{69} + 8 q^{71} + 28 q^{72} + 6 q^{73} - 12 q^{74} - 12 q^{75} + 16 q^{76} + 4 q^{78} - 22 q^{79} + 28 q^{80} + 46 q^{81} + 40 q^{82} - 50 q^{83} - 8 q^{85} - 16 q^{86} + 16 q^{87} - 44 q^{88} - 26 q^{89} + 40 q^{90} + 20 q^{92} + 16 q^{93} + 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ 6 * q - 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 - 4 * q^10 + 4 * q^11 + 4 * q^12 - 6 * q^13 + 12 * q^15 - 16 * q^17 - 4 * q^18 - 2 * q^19 - 16 * q^20 - 12 * q^22 - 6 * q^23 - 12 * q^24 - 4 * q^25 - 20 * q^27 - 6 * q^29 - 6 * q^31 - 20 * q^32 - 4 * q^33 - 24 * q^36 - 8 * q^38 + 8 * q^39 - 4 * q^40 + 8 * q^41 + 2 * q^43 - 4 * q^44 - 14 * q^45 + 8 * q^46 - 30 * q^47 + 8 * q^48 + 8 * q^50 - 4 * q^51 - 4 * q^52 - 14 * q^53 + 48 * q^54 + 8 * q^55 + 4 * q^57 - 8 * q^58 - 24 * q^59 + 12 * q^60 - 28 * q^62 - 20 * q^64 + 6 * q^65 + 4 * q^66 + 16 * q^67 - 28 * q^68 + 20 * q^69 + 8 * q^71 + 28 * q^72 + 6 * q^73 - 12 * q^74 - 12 * q^75 + 16 * q^76 + 4 * q^78 - 22 * q^79 + 28 * q^80 + 46 * q^81 + 40 * q^82 - 50 * q^83 - 8 * q^85 - 16 * q^86 + 16 * q^87 - 44 * q^88 - 26 * q^89 + 40 * q^90 + 20 * q^92 + 16 * q^93 + 32 * q^94 - 6 * q^95 + 20 * q^96 + 14 * q^97 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2$$ v^4 - v^3 - 4*v^2 + 2*v + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2$$ v^5 - 2*v^4 - 4*v^3 + 6*v^2 + 4*v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7$$ b4 + b3 + 5*b2 + 6*b1 + 7 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8$$ b5 + 2*b4 + 6*b3 + 8*b2 + 18*b1 + 8

## 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
 −1.66745 2.35100 −1.20475 1.90903 −0.146243 0.758419
−2.44785 0.667452 3.99195 −0.910286 −1.63382 0 −4.87599 −2.55451 2.22824
1.2 −1.17619 −3.35100 −0.616586 −3.14862 3.94140 0 3.07759 8.22917 3.70337
1.3 −0.656184 0.204753 −1.56942 1.35996 −0.134356 0 2.34220 −2.95808 −0.892385
1.4 0.264627 −2.90903 −1.92997 1.43515 −0.769807 0 −1.03998 5.46247 0.379780
1.5 1.83237 −0.853757 1.35758 −2.62555 −1.56440 0 −1.17715 −2.27110 −4.81098
1.6 2.18322 −1.75842 2.76645 −2.11065 −3.83901 0 1.67333 0.0920365 −4.60802
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.m 6
3.b odd 2 1 5733.2.a.bu 6
7.b odd 2 1 637.2.a.n yes 6
7.c even 3 2 637.2.e.o 12
7.d odd 6 2 637.2.e.n 12
13.b even 2 1 8281.2.a.cc 6
21.c even 2 1 5733.2.a.br 6
91.b odd 2 1 8281.2.a.cd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 1.a even 1 1 trivial
637.2.a.n yes 6 7.b odd 2 1
637.2.e.n 12 7.d odd 6 2
637.2.e.o 12 7.c even 3 2
5733.2.a.br 6 21.c even 2 1
5733.2.a.bu 6 3.b odd 2 1
8281.2.a.cc 6 13.b even 2 1
8281.2.a.cd 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} + 4T_{2} - 2$$ T2^6 - 8*T2^4 + 14*T2^2 + 4*T2 - 2 $$T_{3}^{6} + 8T_{3}^{5} + 20T_{3}^{4} + 12T_{3}^{3} - 12T_{3}^{2} - 8T_{3} + 2$$ T3^6 + 8*T3^5 + 20*T3^4 + 12*T3^3 - 12*T3^2 - 8*T3 + 2 $$T_{17}^{6} + 16T_{17}^{5} + 56T_{17}^{4} - 324T_{17}^{3} - 2792T_{17}^{2} - 6792T_{17} - 5294$$ T17^6 + 16*T17^5 + 56*T17^4 - 324*T17^3 - 2792*T17^2 - 6792*T17 - 5294

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 8 T^{4} + \cdots - 2$$
$3$ $$T^{6} + 8 T^{5} + \cdots + 2$$
$5$ $$T^{6} + 6 T^{5} + \cdots + 31$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 4 T^{5} + \cdots - 562$$
$13$ $$(T + 1)^{6}$$
$17$ $$T^{6} + 16 T^{5} + \cdots - 5294$$
$19$ $$T^{6} + 2 T^{5} + \cdots - 73$$
$23$ $$T^{6} + 6 T^{5} + \cdots + 529$$
$29$ $$T^{6} + 6 T^{5} + \cdots + 529$$
$31$ $$T^{6} + 6 T^{5} + \cdots - 44249$$
$37$ $$T^{6} - 82 T^{4} + \cdots + 254$$
$41$ $$T^{6} - 8 T^{5} + \cdots + 28784$$
$43$ $$T^{6} - 2 T^{5} + \cdots + 35153$$
$47$ $$T^{6} + 30 T^{5} + \cdots - 135617$$
$53$ $$T^{6} + 14 T^{5} + \cdots - 1319$$
$59$ $$T^{6} + 24 T^{5} + \cdots + 1532$$
$61$ $$T^{6} - 246 T^{4} + \cdots - 216584$$
$67$ $$T^{6} - 16 T^{5} + \cdots + 6112$$
$71$ $$T^{6} - 8 T^{5} + \cdots - 1206162$$
$73$ $$T^{6} - 6 T^{5} + \cdots - 142657$$
$79$ $$T^{6} + 22 T^{5} + \cdots + 7913$$
$83$ $$T^{6} + 50 T^{5} + \cdots - 167041$$
$89$ $$T^{6} + 26 T^{5} + \cdots + 9959$$
$97$ $$T^{6} - 14 T^{5} + \cdots + 217287$$