# Properties

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

# 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: $$\Q(\sqrt{-3}, \sqrt{-13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 13x^{2} + 169$$ x^4 - 13*x^2 + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 13x^{2} + 169$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 13$$ (v^2) / 13 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 13$$ (v^3) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$13\beta_{2}$$ 13*b2 $$\nu^{3}$$ $$=$$ $$13\beta_{3}$$ 13*b3

## Character values

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

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

## 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
 −3.12250 − 1.80278i 3.12250 + 1.80278i −3.12250 + 1.80278i 3.12250 − 1.80278i
1.50000 0.866025i 0 0.500000 0.866025i −3.12250 1.80278i 0 0 1.73205i −3.00000 −6.24500
30.2 1.50000 0.866025i 0 0.500000 0.866025i 3.12250 + 1.80278i 0 0 1.73205i −3.00000 6.24500
361.1 1.50000 + 0.866025i 0 0.500000 + 0.866025i −3.12250 + 1.80278i 0 0 1.73205i −3.00000 −6.24500
361.2 1.50000 + 0.866025i 0 0.500000 + 0.866025i 3.12250 1.80278i 0 0 1.73205i −3.00000 6.24500
 $$n$$: 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.b odd 2 1 inner
91.p odd 6 1 inner
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.f 4
7.b odd 2 1 inner 637.2.u.f 4
7.c even 3 1 637.2.k.e 4
7.c even 3 1 637.2.q.d 4
7.d odd 6 1 637.2.k.e 4
7.d odd 6 1 637.2.q.d 4
13.e even 6 1 637.2.k.e 4
91.k even 6 1 637.2.q.d 4
91.l odd 6 1 637.2.q.d 4
91.p odd 6 1 inner 637.2.u.f 4
91.t odd 6 1 637.2.k.e 4
91.u even 6 1 inner 637.2.u.f 4
91.w even 12 2 8281.2.a.br 4
91.bd odd 12 2 8281.2.a.br 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.e 4 7.c even 3 1
637.2.k.e 4 7.d odd 6 1
637.2.k.e 4 13.e even 6 1
637.2.k.e 4 91.t odd 6 1
637.2.q.d 4 7.c even 3 1
637.2.q.d 4 7.d odd 6 1
637.2.q.d 4 91.k even 6 1
637.2.q.d 4 91.l odd 6 1
637.2.u.f 4 1.a even 1 1 trivial
637.2.u.f 4 7.b odd 2 1 inner
637.2.u.f 4 91.p odd 6 1 inner
637.2.u.f 4 91.u even 6 1 inner
8281.2.a.br 4 91.w even 12 2
8281.2.a.br 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 3 T + 3)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 13T^{2} + 169$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$T^{4} - 13T^{2} + 169$$
$17$ $$T^{4} + 39T^{2} + 1521$$
$19$ $$(T^{2} + 52)^{2}$$
$23$ $$(T^{2} + 4 T + 16)^{2}$$
$29$ $$(T^{2} + T + 1)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 3 T + 3)^{2}$$
$41$ $$T^{4} - 13T^{2} + 169$$
$43$ $$(T^{2} + 6 T + 36)^{2}$$
$47$ $$T^{4} - 52T^{2} + 2704$$
$53$ $$(T^{2} + 5 T + 25)^{2}$$
$59$ $$T^{4} - 52T^{2} + 2704$$
$61$ $$(T^{2} - 39)^{2}$$
$67$ $$(T^{2} + 192)^{2}$$
$71$ $$(T^{2} + 18 T + 108)^{2}$$
$73$ $$T^{4} - 117 T^{2} + 13689$$
$79$ $$(T^{2} - 6 T + 36)^{2}$$
$83$ $$(T^{2} + 52)^{2}$$
$89$ $$T^{4} - 52T^{2} + 2704$$
$97$ $$T^{4} - 52T^{2} + 2704$$