# Properties

 Label 637.2.f.e Level $637$ Weight $2$ Character orbit 637.f Analytic conductor $5.086$ Analytic rank $0$ Dimension $4$ 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(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: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.207107 + 0.358719i 0.707107 1.22474i 0.914214 + 1.58346i 1.82843 0.292893 + 0.507306i 0 −1.58579 0.500000 + 0.866025i −0.378680 + 0.655892i
295.2 1.20711 2.09077i −0.707107 + 1.22474i −1.91421 3.31552i −3.82843 1.70711 + 2.95680i 0 −4.41421 0.500000 + 0.866025i −4.62132 + 8.00436i
393.1 −0.207107 0.358719i 0.707107 + 1.22474i 0.914214 1.58346i 1.82843 0.292893 0.507306i 0 −1.58579 0.500000 0.866025i −0.378680 0.655892i
393.2 1.20711 + 2.09077i −0.707107 1.22474i −1.91421 + 3.31552i −3.82843 1.70711 2.95680i 0 −4.41421 0.500000 0.866025i −4.62132 8.00436i
 $$n$$: 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.e 4
7.b odd 2 1 637.2.f.f yes 4
7.c even 3 1 637.2.g.g 4
7.c even 3 1 637.2.h.c 4
7.d odd 6 1 637.2.g.f 4
7.d odd 6 1 637.2.h.b 4
13.c even 3 1 inner 637.2.f.e 4
13.c even 3 1 8281.2.a.o 2
13.e even 6 1 8281.2.a.y 2
91.g even 3 1 637.2.h.c 4
91.h even 3 1 637.2.g.g 4
91.m odd 6 1 637.2.h.b 4
91.n odd 6 1 637.2.f.f yes 4
91.n odd 6 1 8281.2.a.p 2
91.t odd 6 1 8281.2.a.x 2
91.v odd 6 1 637.2.g.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.e 4 1.a even 1 1 trivial
637.2.f.e 4 13.c even 3 1 inner
637.2.f.f yes 4 7.b odd 2 1
637.2.f.f yes 4 91.n odd 6 1
637.2.g.f 4 7.d odd 6 1
637.2.g.f 4 91.v odd 6 1
637.2.g.g 4 7.c even 3 1
637.2.g.g 4 91.h even 3 1
637.2.h.b 4 7.d odd 6 1
637.2.h.b 4 91.m odd 6 1
637.2.h.c 4 7.c even 3 1
637.2.h.c 4 91.g even 3 1
8281.2.a.o 2 13.c even 3 1
8281.2.a.p 2 91.n odd 6 1
8281.2.a.x 2 91.t odd 6 1
8281.2.a.y 2 13.e even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ T2^4 - 2*T2^3 + 5*T2^2 + 2*T2 + 1 $$T_{3}^{4} + 2T_{3}^{2} + 4$$ T3^4 + 2*T3^2 + 4 $$T_{5}^{2} + 2T_{5} - 7$$ T5^2 + 2*T5 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$T^{4} + 2T^{2} + 4$$
$5$ $$(T^{2} + 2 T - 7)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$T^{4} + 6 T^{3} + 35 T^{2} + 6 T + 1$$
$19$ $$(T^{2} - 6 T + 36)^{2}$$
$23$ $$T^{4} + 2T^{2} + 4$$
$29$ $$T^{4} + 14 T^{3} + 155 T^{2} + \cdots + 1681$$
$31$ $$(T^{2} + 8 T + 14)^{2}$$
$37$ $$T^{4} - 2 T^{3} + 75 T^{2} + \cdots + 5041$$
$41$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$43$ $$T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4$$
$47$ $$(T^{2} + 4 T - 28)^{2}$$
$53$ $$(T + 3)^{4}$$
$59$ $$T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324$$
$61$ $$T^{4} - 14 T^{3} + 155 T^{2} + \cdots + 1681$$
$67$ $$T^{4} + 18T^{2} + 324$$
$71$ $$T^{4} - 12 T^{3} + 140 T^{2} + \cdots + 16$$
$73$ $$(T^{2} - 10 T - 7)^{2}$$
$79$ $$(T^{2} - 12 T + 18)^{2}$$
$83$ $$(T^{2} - 12 T - 14)^{2}$$
$89$ $$T^{4} - 8 T^{3} + 176 T^{2} + \cdots + 12544$$
$97$ $$T^{4} + 16 T^{3} + 200 T^{2} + \cdots + 3136$$