# Properties

 Label 637.2.f.a Level $637$ Weight $2$ Character orbit 637.f Analytic conductor $5.086$ Analytic rank $1$ 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(295,637)]

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([0, 4]))

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

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

chi := DirichletCharacter("637.295");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.f (of order $$3$$, degree $$2$$, 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: $$1$$ 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.

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

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i −3.00000 −1.50000 2.59808i 0 −3.00000 −3.00000 5.19615i 1.50000 2.59808i
393.1 −0.500000 0.866025i −1.50000 2.59808i 0.500000 0.866025i −3.00000 −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
13.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.f.a 2
7.b odd 2 1 637.2.f.b 2
7.c even 3 1 637.2.g.a 2
7.c even 3 1 637.2.h.a 2
7.d odd 6 1 91.2.g.a 2
7.d odd 6 1 91.2.h.a yes 2
13.c even 3 1 inner 637.2.f.a 2
13.c even 3 1 8281.2.a.j 1
13.e even 6 1 8281.2.a.g 1
21.g even 6 1 819.2.n.c 2
21.g even 6 1 819.2.s.a 2
91.g even 3 1 637.2.h.a 2
91.h even 3 1 637.2.g.a 2
91.l odd 6 1 1183.2.e.c 2
91.m odd 6 1 91.2.h.a yes 2
91.m odd 6 1 1183.2.e.a 2
91.n odd 6 1 637.2.f.b 2
91.n odd 6 1 8281.2.a.i 1
91.p odd 6 1 1183.2.e.c 2
91.t odd 6 1 8281.2.a.c 1
91.v odd 6 1 91.2.g.a 2
91.v odd 6 1 1183.2.e.a 2
273.r even 6 1 819.2.n.c 2
273.bf even 6 1 819.2.s.a 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.v odd 6 1
91.2.h.a yes 2 7.d odd 6 1
91.2.h.a yes 2 91.m odd 6 1
637.2.f.a 2 1.a even 1 1 trivial
637.2.f.a 2 13.c even 3 1 inner
637.2.f.b 2 7.b odd 2 1
637.2.f.b 2 91.n odd 6 1
637.2.g.a 2 7.c even 3 1
637.2.g.a 2 91.h even 3 1
637.2.h.a 2 7.c even 3 1
637.2.h.a 2 91.g even 3 1
819.2.n.c 2 21.g even 6 1
819.2.n.c 2 273.r even 6 1
819.2.s.a 2 21.g even 6 1
819.2.s.a 2 273.bf even 6 1
1183.2.e.a 2 91.m odd 6 1
1183.2.e.a 2 91.v odd 6 1
1183.2.e.c 2 91.l odd 6 1
1183.2.e.c 2 91.p odd 6 1
8281.2.a.c 1 91.t odd 6 1
8281.2.a.g 1 13.e even 6 1
8281.2.a.i 1 91.n odd 6 1
8281.2.a.j 1 13.c even 3 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 7T + 49$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$(T + 1)^{2}$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} + 13T + 169$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$T^{2} + 13T + 169$$
$73$ $$(T - 13)^{2}$$
$79$ $$(T + 3)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} + 5T + 25$$