Properties

 Label 637.2.f.d 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: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

Character values

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

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

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
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i 1.36603 2.36603i −0.500000 0.866025i −1.73205 2.36603 + 4.09808i 0 −1.73205 −2.23205 3.86603i 1.50000 2.59808i
295.2 0.866025 1.50000i −0.366025 + 0.633975i −0.500000 0.866025i 1.73205 0.633975 + 1.09808i 0 1.73205 1.23205 + 2.13397i 1.50000 2.59808i
393.1 −0.866025 1.50000i 1.36603 + 2.36603i −0.500000 + 0.866025i −1.73205 2.36603 4.09808i 0 −1.73205 −2.23205 + 3.86603i 1.50000 + 2.59808i
393.2 0.866025 + 1.50000i −0.366025 0.633975i −0.500000 + 0.866025i 1.73205 0.633975 1.09808i 0 1.73205 1.23205 2.13397i 1.50000 + 2.59808i
 $$n$$: 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.d 4
7.b odd 2 1 91.2.f.b 4
7.c even 3 1 637.2.g.d 4
7.c even 3 1 637.2.h.e 4
7.d odd 6 1 637.2.g.e 4
7.d odd 6 1 637.2.h.d 4
13.c even 3 1 inner 637.2.f.d 4
13.c even 3 1 8281.2.a.r 2
13.e even 6 1 8281.2.a.t 2
21.c even 2 1 819.2.o.b 4
28.d even 2 1 1456.2.s.o 4
91.g even 3 1 637.2.h.e 4
91.h even 3 1 637.2.g.d 4
91.m odd 6 1 637.2.h.d 4
91.n odd 6 1 91.2.f.b 4
91.n odd 6 1 1183.2.a.f 2
91.t odd 6 1 1183.2.a.e 2
91.v odd 6 1 637.2.g.e 4
91.bc even 12 2 1183.2.c.e 4
273.bn even 6 1 819.2.o.b 4
364.v even 6 1 1456.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 7.b odd 2 1
91.2.f.b 4 91.n odd 6 1
637.2.f.d 4 1.a even 1 1 trivial
637.2.f.d 4 13.c even 3 1 inner
637.2.g.d 4 7.c even 3 1
637.2.g.d 4 91.h even 3 1
637.2.g.e 4 7.d odd 6 1
637.2.g.e 4 91.v odd 6 1
637.2.h.d 4 7.d odd 6 1
637.2.h.d 4 91.m odd 6 1
637.2.h.e 4 7.c even 3 1
637.2.h.e 4 91.g even 3 1
819.2.o.b 4 21.c even 2 1
819.2.o.b 4 273.bn even 6 1
1183.2.a.e 2 91.t odd 6 1
1183.2.a.f 2 91.n odd 6 1
1183.2.c.e 4 91.bc even 12 2
1456.2.s.o 4 28.d even 2 1
1456.2.s.o 4 364.v even 6 1
8281.2.a.r 2 13.c even 3 1
8281.2.a.t 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{4} + 3T_{2}^{2} + 9$$ T2^4 + 3*T2^2 + 9 $$T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4$$ T3^4 - 2*T3^3 + 6*T3^2 + 4*T3 + 4 $$T_{5}^{2} - 3$$ T5^2 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$13$ $$T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169$$
$17$ $$T^{4} + 12 T^{3} + 111 T^{2} + \cdots + 1089$$
$19$ $$(T^{2} - 2 T + 4)^{2}$$
$23$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$(T^{2} - 2 T - 26)^{2}$$
$37$ $$(T^{2} - 7 T + 49)^{2}$$
$41$ $$T^{4} + 27T^{2} + 729$$
$43$ $$T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4$$
$47$ $$(T^{2} + 12 T - 12)^{2}$$
$53$ $$(T^{2} + 6 T - 39)^{2}$$
$59$ $$T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084$$
$61$ $$T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676$$
$71$ $$(T^{2} + 6 T + 36)^{2}$$
$73$ $$(T^{2} + 4 T - 23)^{2}$$
$79$ $$(T^{2} - 22 T + 94)^{2}$$
$83$ $$(T^{2} + 6 T - 18)^{2}$$
$89$ $$T^{4} - 12 T^{3} + 156 T^{2} + \cdots + 144$$
$97$ $$T^{4} + 8 T^{3} + 156 T^{2} + \cdots + 8464$$