# Properties

 Label 637.2.e.g Level $637$ Weight $2$ Character orbit 637.e 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(79,637)]

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, 0]))

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

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

chi := DirichletCharacter("637.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.e (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: $$4$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

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

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 $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 + 1.22474i −0.707107 1.22474i 0 2.20711 3.82282i 2.00000 0 −2.82843 0.500000 0.866025i 3.12132 + 5.40629i
79.2 0.707107 1.22474i 0.707107 + 1.22474i 0 0.792893 1.37333i 2.00000 0 2.82843 0.500000 0.866025i −1.12132 1.94218i
508.1 −0.707107 1.22474i −0.707107 + 1.22474i 0 2.20711 + 3.82282i 2.00000 0 −2.82843 0.500000 + 0.866025i 3.12132 5.40629i
508.2 0.707107 + 1.22474i 0.707107 1.22474i 0 0.792893 + 1.37333i 2.00000 0 2.82843 0.500000 + 0.866025i −1.12132 + 1.94218i
 $$n$$: 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.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.e.g 4
7.b odd 2 1 637.2.e.f 4
7.c even 3 1 637.2.a.g 2
7.c even 3 1 inner 637.2.e.g 4
7.d odd 6 1 91.2.a.c 2
7.d odd 6 1 637.2.e.f 4
21.g even 6 1 819.2.a.h 2
21.h odd 6 1 5733.2.a.s 2
28.f even 6 1 1456.2.a.q 2
35.i odd 6 1 2275.2.a.j 2
56.j odd 6 1 5824.2.a.bl 2
56.m even 6 1 5824.2.a.bk 2
91.r even 6 1 8281.2.a.v 2
91.s odd 6 1 1183.2.a.d 2
91.bb even 12 2 1183.2.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 7.d odd 6 1
637.2.a.g 2 7.c even 3 1
637.2.e.f 4 7.b odd 2 1
637.2.e.f 4 7.d odd 6 1
637.2.e.g 4 1.a even 1 1 trivial
637.2.e.g 4 7.c even 3 1 inner
819.2.a.h 2 21.g even 6 1
1183.2.a.d 2 91.s odd 6 1
1183.2.c.d 4 91.bb even 12 2
1456.2.a.q 2 28.f even 6 1
2275.2.a.j 2 35.i odd 6 1
5733.2.a.s 2 21.h odd 6 1
5824.2.a.bk 2 56.m even 6 1
5824.2.a.bl 2 56.j odd 6 1
8281.2.a.v 2 91.r even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 2T_{2}^{2} + 4$$ T2^4 + 2*T2^2 + 4 $$T_{3}^{4} + 2T_{3}^{2} + 4$$ T3^4 + 2*T3^2 + 4 $$T_{5}^{4} - 6T_{5}^{3} + 29T_{5}^{2} - 42T_{5} + 49$$ T5^4 - 6*T5^3 + 29*T5^2 - 42*T5 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$T^{4} + 2T^{2} + 4$$
$5$ $$T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 18T^{2} + 324$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} + 2T^{2} + 4$$
$19$ $$T^{4} + 6 T^{3} + 45 T^{2} - 54 T + 81$$
$23$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$29$ $$(T^{2} - 6 T + 1)^{2}$$
$31$ $$T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289$$
$37$ $$T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196$$
$41$ $$(T^{2} + 12 T + 28)^{2}$$
$43$ $$(T + 5)^{4}$$
$47$ $$T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49$$
$53$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$59$ $$T^{4} - 12 T^{3} + 140 T^{2} + \cdots + 16$$
$61$ $$(T^{2} - 6 T + 36)^{2}$$
$67$ $$T^{4} - 12 T^{3} + 180 T^{2} + \cdots + 1296$$
$71$ $$(T^{2} + 12 T - 14)^{2}$$
$73$ $$T^{4} + 10 T^{3} + 93 T^{2} + 70 T + 49$$
$79$ $$T^{4} + 14 T^{3} + 219 T^{2} + \cdots + 529$$
$83$ $$(T^{2} + 18 T + 63)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49$$
$97$ $$(T^{2} - 2 T - 161)^{2}$$