# Properties

 Label 637.2.u.e Level $637$ Weight $2$ Character orbit 637.u Analytic conductor $5.086$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(30,637)]

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, 1]))

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

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

chi := DirichletCharacter("637.30");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.u (of order $$6$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2x + 4$$ x^4 - x^3 - x^2 - 2*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2} + 3) q^{6} + (2 \beta_{2} - 1) q^{8} + (\beta_{3} + \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b3 - 1) * q^2 + (b3 + b1) * q^3 + (-2*b3 + b2 + b1) * q^4 + (-b1 + 1) * q^5 + (-b3 - b2 + 3) * q^6 + (2*b2 - 1) * q^8 + (b3 + b1 + 2) * q^9 $$q + (\beta_{3} - 1) q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2} + 3) q^{6} + (2 \beta_{2} - 1) q^{8} + (\beta_{3} + \beta_1 + 2) q^{9} + (\beta_{3} + \beta_1 - 3) q^{10} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 - 2) q^{11} + (5 \beta_{2} - 5) q^{12} + ( - \beta_{2} + 4) q^{13} + ( - \beta_{2} + \beta_1 - 2) q^{15} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{16} + (3 \beta_{2} - 3) q^{17} + (\beta_{3} - \beta_{2} + 1) q^{18} + ( - 3 \beta_{3} + 3 \beta_1) q^{19} + ( - 2 \beta_{3} - \beta_{2} + 4) q^{20} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{22} + ( - 2 \beta_{3} + 4 \beta_1 - 2) q^{23} + ( - 3 \beta_{3} + 4 \beta_{2} + \cdots - 2) q^{24}+ \cdots + ( - 7 \beta_{3} + 18 \beta_{2} + \cdots - 9) q^{99}+O(q^{100})$$ q + (b3 - 1) * q^2 + (b3 + b1) * q^3 + (-2*b3 + b2 + b1) * q^4 + (-b1 + 1) * q^5 + (-b3 - b2 + 3) * q^6 + (2*b2 - 1) * q^8 + (b3 + b1 + 2) * q^9 + (b3 + b1 - 3) * q^10 + (-b3 + 4*b2 + b1 - 2) * q^11 + (5*b2 - 5) * q^12 + (-b2 + 4) * q^13 + (-b2 + b1 - 2) * q^15 + (b3 - 2*b2 - 2*b1 + 1) * q^16 + (3*b2 - 3) * q^17 + (b3 - b2 + 1) * q^18 + (-3*b3 + 3*b1) * q^19 + (-2*b3 - b2 + 4) * q^20 + (-b3 + b2 + 2*b1 - 1) * q^22 + (-2*b3 + 4*b1 - 2) * q^23 + (-3*b3 + 4*b2 + 3*b1 - 2) * q^24 + (b3 - 4*b2 - 2*b1 + 1) * q^25 + (4*b3 - b1 - 3) * q^26 + 5 * q^27 + (2*b3 - 6*b2 - b1 + 5) * q^29 + (-2*b3 - 2*b1 + 5) * q^30 + (-5*b2 + 10) * q^31 + (-3*b2 + b1 - 4) * q^32 + (-5*b3 + 10*b2 + 5*b1 - 5) * q^33 + (-3*b3 + 3*b1) * q^34 + (-4*b3 + 7*b2 + 2*b1 - 5) * q^36 + (4*b2 - 8) * q^37 + (3*b3 + 3*b2 - 6*b1 + 3) * q^38 + (5*b3 - 2*b2 + 2*b1 + 1) * q^39 + (2*b3 - b1 - 1) * q^40 + (4*b2 + 2*b1 + 2) * q^41 + (3*b3 - 7*b2 - 6*b1 + 3) * q^43 + (-3*b2 - 4*b1 + 1) * q^44 + (-b2 - b1) * q^45 + (2*b2 - 6*b1 + 8) * q^46 + (5*b2 + 2*b1 + 3) * q^47 + (b3 - 7*b2 - 2*b1 + 1) * q^48 + (-b2 - b1) * q^50 + (-6*b3 + 6*b2 + 3*b1 - 3) * q^51 + (-7*b3 + 4*b2 + 5*b1 - 1) * q^52 + (4*b3 - 3*b2 - 8*b1 + 4) * q^53 + (5*b3 - 5) * q^54 + (2*b3 - b2 - b1) * q^55 + (3*b3 + 6*b2 - 3*b1 - 3) * q^57 + (3*b3 - 2*b2 - 3*b1 + 1) * q^58 + (-5*b2 - 8*b1 + 3) * q^59 + (5*b3 - 5) * q^60 + (-6*b3 - 6*b1 + 2) * q^61 + (10*b3 - 5*b1 - 5) * q^62 + (-2*b3 - 2*b1 + 9) * q^64 + (-b3 - 3*b1 + 4) * q^65 + 5*b2 * q^66 + (-6*b3 + 2*b2 + 6*b1 - 1) * q^67 + (3*b3 - 3*b2 - 6*b1 + 3) * q^68 + (2*b3 + 6*b2 - 4*b1 + 2) * q^69 + (2*b3 - 2*b2 + 2) * q^71 + (-3*b3 + 8*b2 + 3*b1 - 4) * q^72 + (2*b2 - 4) * q^73 + (-8*b3 + 4*b1 + 4) * q^74 + (3*b3 - 11*b2 - 6*b1 + 3) * q^75 + (-3*b2 + 6*b1 - 9) * q^76 + (-4*b3 - 5*b2 + b1 + 10) * q^78 + (-6*b2 + 6) * q^79 + (b3 + 2*b2 - b1 - 1) * q^80 + (2*b3 + 2*b1 - 6) * q^81 + (2*b3 + 2*b1 - 2) * q^82 + (4*b3 - 6*b2 - 4*b1 + 3) * q^83 + (3*b3 - 3) * q^85 + (-3*b2 + 2*b1 - 5) * q^86 + (10*b3 - 15*b2 - 5*b1 + 10) * q^87 + (-b3 - b1 - 4) * q^88 + (5*b3 - 8*b2 + 11) * q^89 - q^90 + (4*b3 + 4*b1 - 14) * q^92 + (15*b3 - 10*b2 + 5) * q^93 + (3*b3 + 3*b1 - 4) * q^94 + (-6*b3 - 3*b2 + 3*b1 + 6) * q^95 + (-5*b2 - 10*b1 + 5) * q^96 + (3*b3 + 5*b2 - 13) * q^97 + (-7*b3 + 18*b2 + 7*b1 - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + 2 q^{3} + q^{4} + 3 q^{5} + 9 q^{6} + 10 q^{9}+O(q^{10})$$ 4 * q - 3 * q^2 + 2 * q^3 + q^4 + 3 * q^5 + 9 * q^6 + 10 * q^9 $$4 q - 3 q^{2} + 2 q^{3} + q^{4} + 3 q^{5} + 9 q^{6} + 10 q^{9} - 10 q^{10} - 10 q^{12} + 14 q^{13} - 9 q^{15} - q^{16} - 6 q^{17} + 3 q^{18} + 12 q^{20} - q^{22} - 6 q^{23} - 5 q^{25} - 9 q^{26} + 20 q^{27} + 9 q^{29} + 16 q^{30} + 30 q^{31} - 21 q^{32} - 8 q^{36} - 24 q^{37} + 15 q^{38} + 7 q^{39} - 3 q^{40} + 18 q^{41} - 5 q^{43} - 6 q^{44} - 3 q^{45} + 30 q^{46} + 24 q^{47} - 11 q^{48} - 3 q^{50} - 3 q^{51} + 2 q^{52} + 6 q^{53} - 15 q^{54} - q^{55} - 6 q^{59} - 15 q^{60} - 4 q^{61} - 15 q^{62} + 32 q^{64} + 12 q^{65} + 10 q^{66} + 3 q^{68} + 18 q^{69} + 6 q^{71} - 12 q^{73} + 12 q^{74} - 13 q^{75} - 36 q^{76} + 27 q^{78} + 12 q^{79} - 20 q^{81} - 4 q^{82} - 9 q^{85} - 24 q^{86} + 15 q^{87} - 18 q^{88} + 33 q^{89} - 4 q^{90} - 48 q^{92} + 15 q^{93} - 10 q^{94} + 15 q^{95} - 39 q^{97}+O(q^{100})$$ 4 * q - 3 * q^2 + 2 * q^3 + q^4 + 3 * q^5 + 9 * q^6 + 10 * q^9 - 10 * q^10 - 10 * q^12 + 14 * q^13 - 9 * q^15 - q^16 - 6 * q^17 + 3 * q^18 + 12 * q^20 - q^22 - 6 * q^23 - 5 * q^25 - 9 * q^26 + 20 * q^27 + 9 * q^29 + 16 * q^30 + 30 * q^31 - 21 * q^32 - 8 * q^36 - 24 * q^37 + 15 * q^38 + 7 * q^39 - 3 * q^40 + 18 * q^41 - 5 * q^43 - 6 * q^44 - 3 * q^45 + 30 * q^46 + 24 * q^47 - 11 * q^48 - 3 * q^50 - 3 * q^51 + 2 * q^52 + 6 * q^53 - 15 * q^54 - q^55 - 6 * q^59 - 15 * q^60 - 4 * q^61 - 15 * q^62 + 32 * q^64 + 12 * q^65 + 10 * q^66 + 3 * q^68 + 18 * q^69 + 6 * q^71 - 12 * q^73 + 12 * q^74 - 13 * q^75 - 36 * q^76 + 27 * q^78 + 12 * q^79 - 20 * q^81 - 4 * q^82 - 9 * q^85 - 24 * q^86 + 15 * q^87 - 18 * q^88 + 33 * q^89 - 4 * q^90 - 48 * q^92 + 15 * q^93 - 10 * q^94 + 15 * q^95 - 39 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - x^{2} - 2x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu^{2} - \nu - 2 ) / 2$$ (v^3 + v^2 - v - 2) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2$$ (-v^3 + v^2 + v + 2) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2}$$ b3 + b2 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta _1 + 2$$ -b3 + b2 + b1 + 2

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
 −0.895644 − 1.09445i 1.39564 + 0.228425i −0.895644 + 1.09445i 1.39564 − 0.228425i
−1.89564 + 1.09445i −1.79129 1.39564 2.41733i 1.89564 + 1.09445i 3.39564 1.96048i 0 1.73205i 0.208712 −4.79129
30.2 0.395644 0.228425i 2.79129 −0.895644 + 1.55130i −0.395644 0.228425i 1.10436 0.637600i 0 1.73205i 4.79129 −0.208712
361.1 −1.89564 1.09445i −1.79129 1.39564 + 2.41733i 1.89564 1.09445i 3.39564 + 1.96048i 0 1.73205i 0.208712 −4.79129
361.2 0.395644 + 0.228425i 2.79129 −0.895644 1.55130i −0.395644 + 0.228425i 1.10436 + 0.637600i 0 1.73205i 4.79129 −0.208712
 $$n$$: 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.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.u.e 4
7.b odd 2 1 637.2.u.d 4
7.c even 3 1 637.2.k.d 4
7.c even 3 1 637.2.q.e 4
7.d odd 6 1 637.2.k.f 4
7.d odd 6 1 637.2.q.f yes 4
13.e even 6 1 637.2.k.d 4
91.k even 6 1 637.2.q.e 4
91.l odd 6 1 637.2.q.f yes 4
91.p odd 6 1 637.2.u.d 4
91.t odd 6 1 637.2.k.f 4
91.u even 6 1 inner 637.2.u.e 4
91.w even 12 2 8281.2.a.bq 4
91.bd odd 12 2 8281.2.a.bs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.d 4 7.c even 3 1
637.2.k.d 4 13.e even 6 1
637.2.k.f 4 7.d odd 6 1
637.2.k.f 4 91.t odd 6 1
637.2.q.e 4 7.c even 3 1
637.2.q.e 4 91.k even 6 1
637.2.q.f yes 4 7.d odd 6 1
637.2.q.f yes 4 91.l odd 6 1
637.2.u.d 4 7.b odd 2 1
637.2.u.d 4 91.p odd 6 1
637.2.u.e 4 1.a even 1 1 trivial
637.2.u.e 4 91.u even 6 1 inner
8281.2.a.bq 4 91.w even 12 2
8281.2.a.bs 4 91.bd odd 12 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + \cdots + 1$$
$3$ $$(T^{2} - T - 5)^{2}$$
$5$ $$T^{4} - 3 T^{3} + \cdots + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 17T^{2} + 25$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} + 45T^{2} + 81$$
$23$ $$T^{4} + 6 T^{3} + \cdots + 144$$
$29$ $$T^{4} - 9 T^{3} + \cdots + 225$$
$31$ $$(T^{2} - 15 T + 75)^{2}$$
$37$ $$(T^{2} + 12 T + 48)^{2}$$
$41$ $$T^{4} - 18 T^{3} + \cdots + 400$$
$43$ $$T^{4} + 5 T^{3} + \cdots + 1681$$
$47$ $$T^{4} - 24 T^{3} + \cdots + 1681$$
$53$ $$T^{4} - 6 T^{3} + \cdots + 5625$$
$59$ $$T^{4} + 6 T^{3} + \cdots + 11881$$
$61$ $$(T^{2} + 2 T - 188)^{2}$$
$67$ $$T^{4} + 150T^{2} + 2601$$
$71$ $$T^{4} - 6 T^{3} + \cdots + 16$$
$73$ $$(T^{2} + 6 T + 12)^{2}$$
$79$ $$(T^{2} - 6 T + 36)^{2}$$
$83$ $$T^{4} + 62T^{2} + 625$$
$89$ $$T^{4} - 33 T^{3} + \cdots + 2209$$
$97$ $$T^{4} + 39 T^{3} + \cdots + 12321$$