# Properties

 Label 637.2.e.c Level $637$ Weight $2$ Character orbit 637.e Analytic conductor $5.086$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(79,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("637.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.e (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^3 + (-2*z + 2) * q^4 + 3*z * q^5 - z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} - 4 \zeta_{6} q^{12} + q^{13} + 6 q^{15} - 4 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + 7 \zeta_{6} q^{19} + 6 q^{20} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 4 q^{27} - 9 q^{29} + (5 \zeta_{6} - 5) q^{31} - 2 q^{36} - 2 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{39} - 6 q^{41} - q^{43} + ( - 3 \zeta_{6} + 3) q^{45} - 3 \zeta_{6} q^{47} - 8 q^{48} - 12 \zeta_{6} q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + 14 q^{57} + ( - 12 \zeta_{6} + 12) q^{60} + 10 \zeta_{6} q^{61} - 8 q^{64} + 3 \zeta_{6} q^{65} + (14 \zeta_{6} - 14) q^{67} - 12 \zeta_{6} q^{68} - 6 q^{69} - 6 q^{71} + (11 \zeta_{6} - 11) q^{73} + 8 \zeta_{6} q^{75} + 14 q^{76} + \zeta_{6} q^{79} + ( - 12 \zeta_{6} + 12) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 3 q^{83} + 18 q^{85} + (18 \zeta_{6} - 18) q^{87} - 15 \zeta_{6} q^{89} - 6 q^{92} + 10 \zeta_{6} q^{93} + (21 \zeta_{6} - 21) q^{95} - q^{97} +O(q^{100})$$ q + (-2*z + 2) * q^3 + (-2*z + 2) * q^4 + 3*z * q^5 - z * q^9 - 4*z * q^12 + q^13 + 6 * q^15 - 4*z * q^16 + (-6*z + 6) * q^17 + 7*z * q^19 + 6 * q^20 - 3*z * q^23 + (4*z - 4) * q^25 + 4 * q^27 - 9 * q^29 + (5*z - 5) * q^31 - 2 * q^36 - 2*z * q^37 + (-2*z + 2) * q^39 - 6 * q^41 - q^43 + (-3*z + 3) * q^45 - 3*z * q^47 - 8 * q^48 - 12*z * q^51 + (-2*z + 2) * q^52 + (-9*z + 9) * q^53 + 14 * q^57 + (-12*z + 12) * q^60 + 10*z * q^61 - 8 * q^64 + 3*z * q^65 + (14*z - 14) * q^67 - 12*z * q^68 - 6 * q^69 - 6 * q^71 + (11*z - 11) * q^73 + 8*z * q^75 + 14 * q^76 + z * q^79 + (-12*z + 12) * q^80 + (-11*z + 11) * q^81 + 3 * q^83 + 18 * q^85 + (18*z - 18) * q^87 - 15*z * q^89 - 6 * q^92 + 10*z * q^93 + (21*z - 21) * q^95 - q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} + 3 q^{5} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 + 3 * q^5 - q^9 $$2 q + 2 q^{3} + 2 q^{4} + 3 q^{5} - q^{9} - 4 q^{12} + 2 q^{13} + 12 q^{15} - 4 q^{16} + 6 q^{17} + 7 q^{19} + 12 q^{20} - 3 q^{23} - 4 q^{25} + 8 q^{27} - 18 q^{29} - 5 q^{31} - 4 q^{36} - 2 q^{37} + 2 q^{39} - 12 q^{41} - 2 q^{43} + 3 q^{45} - 3 q^{47} - 16 q^{48} - 12 q^{51} + 2 q^{52} + 9 q^{53} + 28 q^{57} + 12 q^{60} + 10 q^{61} - 16 q^{64} + 3 q^{65} - 14 q^{67} - 12 q^{68} - 12 q^{69} - 12 q^{71} - 11 q^{73} + 8 q^{75} + 28 q^{76} + q^{79} + 12 q^{80} + 11 q^{81} + 6 q^{83} + 36 q^{85} - 18 q^{87} - 15 q^{89} - 12 q^{92} + 10 q^{93} - 21 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 + 3 * q^5 - q^9 - 4 * q^12 + 2 * q^13 + 12 * q^15 - 4 * q^16 + 6 * q^17 + 7 * q^19 + 12 * q^20 - 3 * q^23 - 4 * q^25 + 8 * q^27 - 18 * q^29 - 5 * q^31 - 4 * q^36 - 2 * q^37 + 2 * q^39 - 12 * q^41 - 2 * q^43 + 3 * q^45 - 3 * q^47 - 16 * q^48 - 12 * q^51 + 2 * q^52 + 9 * q^53 + 28 * q^57 + 12 * q^60 + 10 * q^61 - 16 * q^64 + 3 * q^65 - 14 * q^67 - 12 * q^68 - 12 * q^69 - 12 * q^71 - 11 * q^73 + 8 * q^75 + 28 * q^76 + q^79 + 12 * q^80 + 11 * q^81 + 6 * q^83 + 36 * q^85 - 18 * q^87 - 15 * q^89 - 12 * q^92 + 10 * q^93 - 21 * q^95 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 + 1.73205i 1.00000 + 1.73205i 1.50000 2.59808i 0 0 0 −0.500000 + 0.866025i 0
508.1 0 1.00000 1.73205i 1.00000 1.73205i 1.50000 + 2.59808i 0 0 0 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.c 2
7.b odd 2 1 637.2.e.b 2
7.c even 3 1 91.2.a.b 1
7.c even 3 1 inner 637.2.e.c 2
7.d odd 6 1 637.2.a.b 1
7.d odd 6 1 637.2.e.b 2
21.g even 6 1 5733.2.a.f 1
21.h odd 6 1 819.2.a.c 1
28.g odd 6 1 1456.2.a.k 1
35.j even 6 1 2275.2.a.d 1
56.k odd 6 1 5824.2.a.f 1
56.p even 6 1 5824.2.a.bd 1
91.r even 6 1 1183.2.a.a 1
91.s odd 6 1 8281.2.a.h 1
91.z odd 12 2 1183.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.b 1 7.c even 3 1
637.2.a.b 1 7.d odd 6 1
637.2.e.b 2 7.b odd 2 1
637.2.e.b 2 7.d odd 6 1
637.2.e.c 2 1.a even 1 1 trivial
637.2.e.c 2 7.c even 3 1 inner
819.2.a.c 1 21.h odd 6 1
1183.2.a.a 1 91.r even 6 1
1183.2.c.a 2 91.z odd 12 2
1456.2.a.k 1 28.g odd 6 1
2275.2.a.d 1 35.j even 6 1
5733.2.a.f 1 21.g even 6 1
5824.2.a.f 1 56.k odd 6 1
5824.2.a.bd 1 56.p even 6 1
8281.2.a.h 1 91.s odd 6 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{3}^{2} - 2T_{3} + 4$$ T3^2 - 2*T3 + 4 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T + 9)^{2}$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$(T + 1)^{2}$$