# Properties

 Label 637.2.e.a Level $637$ Weight $2$ Character orbit 637.e Analytic conductor $5.086$ Analytic rank $0$ Dimension $2$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} - 3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^4 - 3 * q^8 + 3*z * q^9 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} - 3 q^{8} + 3 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + q^{13} + \zeta_{6} q^{16} + ( - 7 \zeta_{6} + 7) q^{17} + ( - 3 \zeta_{6} + 3) q^{18} - 7 \zeta_{6} q^{19} - 3 q^{22} + 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - \zeta_{6} q^{26} - 5 q^{29} + (5 \zeta_{6} - 5) q^{32} - 7 q^{34} + 3 q^{36} - 8 \zeta_{6} q^{37} + (7 \zeta_{6} - 7) q^{38} + 2 q^{43} - 3 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{46} + 7 \zeta_{6} q^{47} - 5 q^{50} + ( - \zeta_{6} + 1) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} + 5 \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} - 7 \zeta_{6} q^{61} + 7 q^{64} + ( - 3 \zeta_{6} + 3) q^{67} - 7 \zeta_{6} q^{68} - 5 q^{71} - 9 \zeta_{6} q^{72} + ( - 14 \zeta_{6} + 14) q^{73} + (8 \zeta_{6} - 8) q^{74} - 7 q^{76} + 6 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - 2 \zeta_{6} q^{86} + (9 \zeta_{6} - 9) q^{88} + 6 q^{92} + ( - 7 \zeta_{6} + 7) q^{94} + 14 q^{97} + 9 q^{99} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^4 - 3 * q^8 + 3*z * q^9 + (-3*z + 3) * q^11 + q^13 + z * q^16 + (-7*z + 7) * q^17 + (-3*z + 3) * q^18 - 7*z * q^19 - 3 * q^22 + 6*z * q^23 + (-5*z + 5) * q^25 - z * q^26 - 5 * q^29 + (5*z - 5) * q^32 - 7 * q^34 + 3 * q^36 - 8*z * q^37 + (7*z - 7) * q^38 + 2 * q^43 - 3*z * q^44 + (-6*z + 6) * q^46 + 7*z * q^47 - 5 * q^50 + (-z + 1) * q^52 + (-3*z + 3) * q^53 + 5*z * q^58 + (7*z - 7) * q^59 - 7*z * q^61 + 7 * q^64 + (-3*z + 3) * q^67 - 7*z * q^68 - 5 * q^71 - 9*z * q^72 + (-14*z + 14) * q^73 + (8*z - 8) * q^74 - 7 * q^76 + 6*z * q^79 + (9*z - 9) * q^81 - 2*z * q^86 + (9*z - 9) * q^88 + 6 * q^92 + (-7*z + 7) * q^94 + 14 * q^97 + 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 + q^4 - 6 * q^8 + 3 * q^9 $$2 q - q^{2} + q^{4} - 6 q^{8} + 3 q^{9} + 3 q^{11} + 2 q^{13} + q^{16} + 7 q^{17} + 3 q^{18} - 7 q^{19} - 6 q^{22} + 6 q^{23} + 5 q^{25} - q^{26} - 10 q^{29} - 5 q^{32} - 14 q^{34} + 6 q^{36} - 8 q^{37} - 7 q^{38} + 4 q^{43} - 3 q^{44} + 6 q^{46} + 7 q^{47} - 10 q^{50} + q^{52} + 3 q^{53} + 5 q^{58} - 7 q^{59} - 7 q^{61} + 14 q^{64} + 3 q^{67} - 7 q^{68} - 10 q^{71} - 9 q^{72} + 14 q^{73} - 8 q^{74} - 14 q^{76} + 6 q^{79} - 9 q^{81} - 2 q^{86} - 9 q^{88} + 12 q^{92} + 7 q^{94} + 28 q^{97} + 18 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^4 - 6 * q^8 + 3 * q^9 + 3 * q^11 + 2 * q^13 + q^16 + 7 * q^17 + 3 * q^18 - 7 * q^19 - 6 * q^22 + 6 * q^23 + 5 * q^25 - q^26 - 10 * q^29 - 5 * q^32 - 14 * q^34 + 6 * q^36 - 8 * q^37 - 7 * q^38 + 4 * q^43 - 3 * q^44 + 6 * q^46 + 7 * q^47 - 10 * q^50 + q^52 + 3 * q^53 + 5 * q^58 - 7 * q^59 - 7 * q^61 + 14 * q^64 + 3 * q^67 - 7 * q^68 - 10 * q^71 - 9 * q^72 + 14 * q^73 - 8 * q^74 - 14 * q^76 + 6 * q^79 - 9 * q^81 - 2 * q^86 - 9 * q^88 + 12 * q^92 + 7 * q^94 + 28 * q^97 + 18 * q^99

## 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
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −3.00000 1.50000 2.59808i 0
508.1 −0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 −3.00000 1.50000 + 2.59808i 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.a 2
7.b odd 2 1 91.2.e.a 2
7.c even 3 1 637.2.a.d 1
7.c even 3 1 inner 637.2.e.a 2
7.d odd 6 1 91.2.e.a 2
7.d odd 6 1 637.2.a.c 1
21.c even 2 1 819.2.j.b 2
21.g even 6 1 819.2.j.b 2
21.g even 6 1 5733.2.a.c 1
21.h odd 6 1 5733.2.a.d 1
28.d even 2 1 1456.2.r.g 2
28.f even 6 1 1456.2.r.g 2
91.b odd 2 1 1183.2.e.b 2
91.r even 6 1 8281.2.a.e 1
91.s odd 6 1 1183.2.e.b 2
91.s odd 6 1 8281.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 7.b odd 2 1
91.2.e.a 2 7.d odd 6 1
637.2.a.c 1 7.d odd 6 1
637.2.a.d 1 7.c even 3 1
637.2.e.a 2 1.a even 1 1 trivial
637.2.e.a 2 7.c even 3 1 inner
819.2.j.b 2 21.c even 2 1
819.2.j.b 2 21.g even 6 1
1183.2.e.b 2 91.b odd 2 1
1183.2.e.b 2 91.s odd 6 1
1456.2.r.g 2 28.d even 2 1
1456.2.r.g 2 28.f even 6 1
5733.2.a.c 1 21.g even 6 1
5733.2.a.d 1 21.h odd 6 1
8281.2.a.e 1 91.r even 6 1
8281.2.a.f 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{3}$$ T3 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 7T + 49$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$T^{2}$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 7T + 49$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2} + 7T + 49$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} - 14T + 196$$
$79$ $$T^{2} - 6T + 36$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 14)^{2}$$