# Properties

 Label 637.2.h.a Level $637$ Weight $2$ Character orbit 637.h 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(165,637)]

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([4, 4]))

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

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

chi := DirichletCharacter("637.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.h (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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 \zeta_{6} q^{3} - q^{4} + 3 \zeta_{6} q^{5} - 3 \zeta_{6} q^{6} - 3 q^{8} + (6 \zeta_{6} - 6) q^{9} +O(q^{10})$$ q + q^2 - 3*z * q^3 - q^4 + 3*z * q^5 - 3*z * q^6 - 3 * q^8 + (6*z - 6) * q^9 $$q + q^{2} - 3 \zeta_{6} q^{3} - q^{4} + 3 \zeta_{6} q^{5} - 3 \zeta_{6} q^{6} - 3 q^{8} + (6 \zeta_{6} - 6) q^{9} + 3 \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} + 3 \zeta_{6} q^{12} + (4 \zeta_{6} - 1) q^{13} + ( - 9 \zeta_{6} + 9) q^{15} - q^{16} + 2 q^{17} + (6 \zeta_{6} - 6) q^{18} + (\zeta_{6} - 1) q^{19} - 3 \zeta_{6} q^{20} + 3 \zeta_{6} q^{22} + 9 \zeta_{6} q^{24} + (4 \zeta_{6} - 4) q^{25} + (4 \zeta_{6} - 1) q^{26} + 9 q^{27} + (7 \zeta_{6} - 7) q^{29} + ( - 9 \zeta_{6} + 9) q^{30} + ( - 3 \zeta_{6} + 3) q^{31} + 5 q^{32} + ( - 9 \zeta_{6} + 9) q^{33} + 2 q^{34} + ( - 6 \zeta_{6} + 6) q^{36} + 2 q^{37} + (\zeta_{6} - 1) q^{38} + ( - 9 \zeta_{6} + 12) q^{39} - 9 \zeta_{6} q^{40} + ( - 3 \zeta_{6} + 3) q^{41} + 7 \zeta_{6} q^{43} - 3 \zeta_{6} q^{44} - 18 q^{45} + \zeta_{6} q^{47} + 3 \zeta_{6} q^{48} + (4 \zeta_{6} - 4) q^{50} - 6 \zeta_{6} q^{51} + ( - 4 \zeta_{6} + 1) q^{52} + (3 \zeta_{6} - 3) q^{53} + 9 q^{54} + (9 \zeta_{6} - 9) q^{55} + 3 q^{57} + (7 \zeta_{6} - 7) q^{58} + 4 q^{59} + (9 \zeta_{6} - 9) q^{60} + (13 \zeta_{6} - 13) q^{61} + ( - 3 \zeta_{6} + 3) q^{62} + 7 q^{64} + (9 \zeta_{6} - 12) q^{65} + ( - 9 \zeta_{6} + 9) q^{66} + 3 \zeta_{6} q^{67} - 2 q^{68} - 13 \zeta_{6} q^{71} + ( - 18 \zeta_{6} + 18) q^{72} + (13 \zeta_{6} - 13) q^{73} + 2 q^{74} + 12 q^{75} + ( - \zeta_{6} + 1) q^{76} + ( - 9 \zeta_{6} + 12) q^{78} + 3 \zeta_{6} q^{79} - 3 \zeta_{6} q^{80} - 9 \zeta_{6} q^{81} + ( - 3 \zeta_{6} + 3) q^{82} + 6 \zeta_{6} q^{85} + 7 \zeta_{6} q^{86} + 21 q^{87} - 9 \zeta_{6} q^{88} - 6 q^{89} - 18 q^{90} - 9 q^{93} + \zeta_{6} q^{94} - 3 q^{95} - 15 \zeta_{6} q^{96} - 5 \zeta_{6} q^{97} - 18 q^{99} +O(q^{100})$$ q + q^2 - 3*z * q^3 - q^4 + 3*z * q^5 - 3*z * q^6 - 3 * q^8 + (6*z - 6) * q^9 + 3*z * q^10 + 3*z * q^11 + 3*z * q^12 + (4*z - 1) * q^13 + (-9*z + 9) * q^15 - q^16 + 2 * q^17 + (6*z - 6) * q^18 + (z - 1) * q^19 - 3*z * q^20 + 3*z * q^22 + 9*z * q^24 + (4*z - 4) * q^25 + (4*z - 1) * q^26 + 9 * q^27 + (7*z - 7) * q^29 + (-9*z + 9) * q^30 + (-3*z + 3) * q^31 + 5 * q^32 + (-9*z + 9) * q^33 + 2 * q^34 + (-6*z + 6) * q^36 + 2 * q^37 + (z - 1) * q^38 + (-9*z + 12) * q^39 - 9*z * q^40 + (-3*z + 3) * q^41 + 7*z * q^43 - 3*z * q^44 - 18 * q^45 + z * q^47 + 3*z * q^48 + (4*z - 4) * q^50 - 6*z * q^51 + (-4*z + 1) * q^52 + (3*z - 3) * q^53 + 9 * q^54 + (9*z - 9) * q^55 + 3 * q^57 + (7*z - 7) * q^58 + 4 * q^59 + (9*z - 9) * q^60 + (13*z - 13) * q^61 + (-3*z + 3) * q^62 + 7 * q^64 + (9*z - 12) * q^65 + (-9*z + 9) * q^66 + 3*z * q^67 - 2 * q^68 - 13*z * q^71 + (-18*z + 18) * q^72 + (13*z - 13) * q^73 + 2 * q^74 + 12 * q^75 + (-z + 1) * q^76 + (-9*z + 12) * q^78 + 3*z * q^79 - 3*z * q^80 - 9*z * q^81 + (-3*z + 3) * q^82 + 6*z * q^85 + 7*z * q^86 + 21 * q^87 - 9*z * q^88 - 6 * q^89 - 18 * q^90 - 9 * q^93 + z * q^94 - 3 * q^95 - 15*z * q^96 - 5*z * q^97 - 18 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 2 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^8 - 6 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 6 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} + 2 q^{13} + 9 q^{15} - 2 q^{16} + 4 q^{17} - 6 q^{18} - q^{19} - 3 q^{20} + 3 q^{22} + 9 q^{24} - 4 q^{25} + 2 q^{26} + 18 q^{27} - 7 q^{29} + 9 q^{30} + 3 q^{31} + 10 q^{32} + 9 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{37} - q^{38} + 15 q^{39} - 9 q^{40} + 3 q^{41} + 7 q^{43} - 3 q^{44} - 36 q^{45} + q^{47} + 3 q^{48} - 4 q^{50} - 6 q^{51} - 2 q^{52} - 3 q^{53} + 18 q^{54} - 9 q^{55} + 6 q^{57} - 7 q^{58} + 8 q^{59} - 9 q^{60} - 13 q^{61} + 3 q^{62} + 14 q^{64} - 15 q^{65} + 9 q^{66} + 3 q^{67} - 4 q^{68} - 13 q^{71} + 18 q^{72} - 13 q^{73} + 4 q^{74} + 24 q^{75} + q^{76} + 15 q^{78} + 3 q^{79} - 3 q^{80} - 9 q^{81} + 3 q^{82} + 6 q^{85} + 7 q^{86} + 42 q^{87} - 9 q^{88} - 12 q^{89} - 36 q^{90} - 18 q^{93} + q^{94} - 6 q^{95} - 15 q^{96} - 5 q^{97} - 36 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 2 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^8 - 6 * q^9 + 3 * q^10 + 3 * q^11 + 3 * q^12 + 2 * q^13 + 9 * q^15 - 2 * q^16 + 4 * q^17 - 6 * q^18 - q^19 - 3 * q^20 + 3 * q^22 + 9 * q^24 - 4 * q^25 + 2 * q^26 + 18 * q^27 - 7 * q^29 + 9 * q^30 + 3 * q^31 + 10 * q^32 + 9 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^37 - q^38 + 15 * q^39 - 9 * q^40 + 3 * q^41 + 7 * q^43 - 3 * q^44 - 36 * q^45 + q^47 + 3 * q^48 - 4 * q^50 - 6 * q^51 - 2 * q^52 - 3 * q^53 + 18 * q^54 - 9 * q^55 + 6 * q^57 - 7 * q^58 + 8 * q^59 - 9 * q^60 - 13 * q^61 + 3 * q^62 + 14 * q^64 - 15 * q^65 + 9 * q^66 + 3 * q^67 - 4 * q^68 - 13 * q^71 + 18 * q^72 - 13 * q^73 + 4 * q^74 + 24 * q^75 + q^76 + 15 * q^78 + 3 * q^79 - 3 * q^80 - 9 * q^81 + 3 * q^82 + 6 * q^85 + 7 * q^86 + 42 * q^87 - 9 * q^88 - 12 * q^89 - 36 * q^90 - 18 * q^93 + q^94 - 6 * q^95 - 15 * q^96 - 5 * q^97 - 36 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 −1.50000 2.59808i −1.00000 1.50000 + 2.59808i −1.50000 2.59808i 0 −3.00000 −3.00000 + 5.19615i 1.50000 + 2.59808i
471.1 1.00000 −1.50000 + 2.59808i −1.00000 1.50000 2.59808i −1.50000 + 2.59808i 0 −3.00000 −3.00000 5.19615i 1.50000 2.59808i
 $$n$$: 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
91.h 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.h.a 2
7.b odd 2 1 91.2.h.a yes 2
7.c even 3 1 637.2.f.a 2
7.c even 3 1 637.2.g.a 2
7.d odd 6 1 91.2.g.a 2
7.d odd 6 1 637.2.f.b 2
13.c even 3 1 637.2.g.a 2
21.c even 2 1 819.2.s.a 2
21.g even 6 1 819.2.n.c 2
91.g even 3 1 637.2.f.a 2
91.h even 3 1 inner 637.2.h.a 2
91.h even 3 1 8281.2.a.j 1
91.k even 6 1 8281.2.a.g 1
91.l odd 6 1 8281.2.a.c 1
91.m odd 6 1 637.2.f.b 2
91.m odd 6 1 1183.2.e.a 2
91.n odd 6 1 91.2.g.a 2
91.n odd 6 1 1183.2.e.a 2
91.p odd 6 1 1183.2.e.c 2
91.t odd 6 1 1183.2.e.c 2
91.v odd 6 1 91.2.h.a yes 2
91.v odd 6 1 8281.2.a.i 1
273.r even 6 1 819.2.s.a 2
273.bn even 6 1 819.2.n.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 7.d odd 6 1
91.2.g.a 2 91.n odd 6 1
91.2.h.a yes 2 7.b odd 2 1
91.2.h.a yes 2 91.v odd 6 1
637.2.f.a 2 7.c even 3 1
637.2.f.a 2 91.g even 3 1
637.2.f.b 2 7.d odd 6 1
637.2.f.b 2 91.m odd 6 1
637.2.g.a 2 7.c even 3 1
637.2.g.a 2 13.c even 3 1
637.2.h.a 2 1.a even 1 1 trivial
637.2.h.a 2 91.h even 3 1 inner
819.2.n.c 2 21.g even 6 1
819.2.n.c 2 273.bn even 6 1
819.2.s.a 2 21.c even 2 1
819.2.s.a 2 273.r even 6 1
1183.2.e.a 2 91.m odd 6 1
1183.2.e.a 2 91.n odd 6 1
1183.2.e.c 2 91.p odd 6 1
1183.2.e.c 2 91.t odd 6 1
8281.2.a.c 1 91.l odd 6 1
8281.2.a.g 1 91.k even 6 1
8281.2.a.i 1 91.v odd 6 1
8281.2.a.j 1 91.h even 3 1

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 7T + 49$$
$31$ $$T^{2} - 3T + 9$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 13T + 169$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$T^{2} + 13T + 169$$
$73$ $$T^{2} + 13T + 169$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 5T + 25$$