Properties

 Label 637.2.u.a Level $637$ Weight $2$ Character orbit 637.u Analytic conductor $5.086$ Analytic rank $0$ 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(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: $$2$$ 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

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

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
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 + 0.866025i 1.00000 0.500000 0.866025i 1.50000 + 0.866025i −1.50000 + 0.866025i 0 1.73205i −2.00000 −3.00000
361.1 −1.50000 0.866025i 1.00000 0.500000 + 0.866025i 1.50000 0.866025i −1.50000 0.866025i 0 1.73205i −2.00000 −3.00000
 $$n$$: 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.a 2
7.b odd 2 1 91.2.u.a yes 2
7.c even 3 1 637.2.k.b 2
7.c even 3 1 637.2.q.b 2
7.d odd 6 1 91.2.k.a 2
7.d odd 6 1 637.2.q.c 2
13.e even 6 1 637.2.k.b 2
21.c even 2 1 819.2.do.c 2
21.g even 6 1 819.2.bm.a 2
91.k even 6 1 637.2.q.b 2
91.l odd 6 1 637.2.q.c 2
91.p odd 6 1 91.2.u.a yes 2
91.t odd 6 1 91.2.k.a 2
91.u even 6 1 inner 637.2.u.a 2
91.w even 12 2 8281.2.a.s 2
91.ba even 12 2 1183.2.e.e 4
91.bc even 12 2 1183.2.e.e 4
91.bd odd 12 2 8281.2.a.w 2
273.u even 6 1 819.2.bm.a 2
273.y even 6 1 819.2.do.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 7.d odd 6 1
91.2.k.a 2 91.t odd 6 1
91.2.u.a yes 2 7.b odd 2 1
91.2.u.a yes 2 91.p odd 6 1
637.2.k.b 2 7.c even 3 1
637.2.k.b 2 13.e even 6 1
637.2.q.b 2 7.c even 3 1
637.2.q.b 2 91.k even 6 1
637.2.q.c 2 7.d odd 6 1
637.2.q.c 2 91.l odd 6 1
637.2.u.a 2 1.a even 1 1 trivial
637.2.u.a 2 91.u even 6 1 inner
819.2.bm.a 2 21.g even 6 1
819.2.bm.a 2 273.u even 6 1
819.2.do.c 2 21.c even 2 1
819.2.do.c 2 273.y even 6 1
1183.2.e.e 4 91.ba even 12 2
1183.2.e.e 4 91.bc even 12 2
8281.2.a.s 2 91.w even 12 2
8281.2.a.w 2 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 $$T_{3} - 1$$ T3 - 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 3T + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 27$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 3$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 3T + 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 9T + 27$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$T^{2} - 15T + 75$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2} - 6T + 12$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} + 75$$
$71$ $$T^{2} - 3T + 3$$
$73$ $$T^{2} - 15T + 75$$
$79$ $$T^{2} - 5T + 25$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} - 12T + 48$$
$97$ $$T^{2} - 9T + 27$$