# Properties

 Label 507.2.a.j Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(1,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## 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: yes Analytic conductor: $$4.04841538248$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1 - 4) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} + q^{9}+O(q^{10})$$ q - b1 * q^2 + q^3 + b2 * q^4 + (-b2 + b1 - 2) * q^5 - b1 * q^6 + (-b2 + b1 - 4) * q^7 + (-b2 + 2*b1 - 1) * q^8 + q^9 $$q - \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1 - 4) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} + q^{9} + (2 \beta_1 - 1) q^{10} + (4 \beta_{2} - \beta_1 + 2) q^{11} + \beta_{2} q^{12} + (4 \beta_1 - 1) q^{14} + ( - \beta_{2} + \beta_1 - 2) q^{15} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + (\beta_{2} - 2) q^{17} - \beta_1 q^{18} + (\beta_{2} - 4 \beta_1 - 2) q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1 - 4) q^{21} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{22} + ( - 5 \beta_{2} + 3 \beta_1 - 2) q^{23} + ( - \beta_{2} + 2 \beta_1 - 1) q^{24} + (2 \beta_{2} - 3 \beta_1) q^{25} + q^{27} + ( - 2 \beta_{2} - \beta_1) q^{28} + (2 \beta_{2} - 3 \beta_1 - 1) q^{29} + (2 \beta_1 - 1) q^{30} + (2 \beta_{2} + 3 \beta_1 - 3) q^{31} + (4 \beta_{2} - \beta_1 + 3) q^{32} + (4 \beta_{2} - \beta_1 + 2) q^{33} + ( - \beta_{2} + 2 \beta_1 - 1) q^{34} + (4 \beta_{2} - 5 \beta_1 + 9) q^{35} + \beta_{2} q^{36} + ( - \beta_{2} - 5) q^{37} + (3 \beta_{2} + 2 \beta_1 + 7) q^{38} + (\beta_{2} - 4 \beta_1 + 4) q^{40} - \beta_1 q^{41} + (4 \beta_1 - 1) q^{42} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{43} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{44} + ( - \beta_{2} + \beta_1 - 2) q^{45} + (2 \beta_{2} + 2 \beta_1 - 1) q^{46} + ( - 3 \beta_{2} - 6 \beta_1 + 4) q^{47} + ( - 3 \beta_{2} + \beta_1 - 3) q^{48} + (6 \beta_{2} - 7 \beta_1 + 10) q^{49} + (\beta_{2} + 4) q^{50} + (\beta_{2} - 2) q^{51} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{53} - \beta_1 q^{54} + ( - 2 \beta_{2} - 5) q^{55} + (3 \beta_{2} - 8 \beta_1 + 6) q^{56} + (\beta_{2} - 4 \beta_1 - 2) q^{57} + (\beta_{2} + \beta_1 + 4) q^{58} + ( - 6 \beta_{2} + 2 \beta_1 + 2) q^{59} - \beta_1 q^{60} + (3 \beta_{2} + 2 \beta_1 - 4) q^{61} + ( - 5 \beta_{2} + 3 \beta_1 - 8) q^{62} + ( - \beta_{2} + \beta_1 - 4) q^{63} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{66} + (3 \beta_{2} + \beta_1 - 1) q^{67} + ( - 3 \beta_{2} + \beta_1 + 1) q^{68} + ( - 5 \beta_{2} + 3 \beta_1 - 2) q^{69} + (\beta_{2} - 9 \beta_1 + 6) q^{70} + ( - 2 \beta_{2} + 7 \beta_1 - 1) q^{71} + ( - \beta_{2} + 2 \beta_1 - 1) q^{72} + (\beta_{2} + \beta_1 - 6) q^{73} + (\beta_{2} + 5 \beta_1 + 1) q^{74} + (2 \beta_{2} - 3 \beta_1) q^{75} + ( - 7 \beta_{2} + \beta_1 - 3) q^{76} + ( - 10 \beta_{2} + 2 \beta_1 - 9) q^{77} + ( - 5 \beta_{2} + \beta_1 - 5) q^{79} + (3 \beta_{2} - 2 \beta_1 + 7) q^{80} + q^{81} + (\beta_{2} + 2) q^{82} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{83} + ( - 2 \beta_{2} - \beta_1) q^{84} + (2 \beta_{2} - 3 \beta_1 + 4) q^{85} + (4 \beta_{2} + \beta_1 + 6) q^{86} + (2 \beta_{2} - 3 \beta_1 - 1) q^{87} + (5 \beta_{2} + \beta_1 - 1) q^{88} + (\beta_{2} - 3 \beta_1 + 3) q^{89} + (2 \beta_1 - 1) q^{90} + (6 \beta_{2} - 5 \beta_1 - 2) q^{92} + (2 \beta_{2} + 3 \beta_1 - 3) q^{93} + (9 \beta_{2} - 4 \beta_1 + 15) q^{94} + (2 \beta_{2} + 5 \beta_1) q^{95} + (4 \beta_{2} - \beta_1 + 3) q^{96} + ( - 10 \beta_{2} + 6 \beta_1 - 7) q^{97} + (\beta_{2} - 10 \beta_1 + 8) q^{98} + (4 \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100})$$ q - b1 * q^2 + q^3 + b2 * q^4 + (-b2 + b1 - 2) * q^5 - b1 * q^6 + (-b2 + b1 - 4) * q^7 + (-b2 + 2*b1 - 1) * q^8 + q^9 + (2*b1 - 1) * q^10 + (4*b2 - b1 + 2) * q^11 + b2 * q^12 + (4*b1 - 1) * q^14 + (-b2 + b1 - 2) * q^15 + (-3*b2 + b1 - 3) * q^16 + (b2 - 2) * q^17 - b1 * q^18 + (b2 - 4*b1 - 2) * q^19 - b1 * q^20 + (-b2 + b1 - 4) * q^21 + (-3*b2 - 2*b1 - 2) * q^22 + (-5*b2 + 3*b1 - 2) * q^23 + (-b2 + 2*b1 - 1) * q^24 + (2*b2 - 3*b1) * q^25 + q^27 + (-2*b2 - b1) * q^28 + (2*b2 - 3*b1 - 1) * q^29 + (2*b1 - 1) * q^30 + (2*b2 + 3*b1 - 3) * q^31 + (4*b2 - b1 + 3) * q^32 + (4*b2 - b1 + 2) * q^33 + (-b2 + 2*b1 - 1) * q^34 + (4*b2 - 5*b1 + 9) * q^35 + b2 * q^36 + (-b2 - 5) * q^37 + (3*b2 + 2*b1 + 7) * q^38 + (b2 - 4*b1 + 4) * q^40 - b1 * q^41 + (4*b1 - 1) * q^42 + (-2*b2 - 2*b1 - 1) * q^43 + (-3*b2 + 4*b1 + 3) * q^44 + (-b2 + b1 - 2) * q^45 + (2*b2 + 2*b1 - 1) * q^46 + (-3*b2 - 6*b1 + 4) * q^47 + (-3*b2 + b1 - 3) * q^48 + (6*b2 - 7*b1 + 10) * q^49 + (b2 + 4) * q^50 + (b2 - 2) * q^51 + (-2*b2 + 3*b1 - 6) * q^53 - b1 * q^54 + (-2*b2 - 5) * q^55 + (3*b2 - 8*b1 + 6) * q^56 + (b2 - 4*b1 - 2) * q^57 + (b2 + b1 + 4) * q^58 + (-6*b2 + 2*b1 + 2) * q^59 - b1 * q^60 + (3*b2 + 2*b1 - 4) * q^61 + (-5*b2 + 3*b1 - 8) * q^62 + (-b2 + b1 - 4) * q^63 + (3*b2 - 5*b1 + 4) * q^64 + (-3*b2 - 2*b1 - 2) * q^66 + (3*b2 + b1 - 1) * q^67 + (-3*b2 + b1 + 1) * q^68 + (-5*b2 + 3*b1 - 2) * q^69 + (b2 - 9*b1 + 6) * q^70 + (-2*b2 + 7*b1 - 1) * q^71 + (-b2 + 2*b1 - 1) * q^72 + (b2 + b1 - 6) * q^73 + (b2 + 5*b1 + 1) * q^74 + (2*b2 - 3*b1) * q^75 + (-7*b2 + b1 - 3) * q^76 + (-10*b2 + 2*b1 - 9) * q^77 + (-5*b2 + b1 - 5) * q^79 + (3*b2 - 2*b1 + 7) * q^80 + q^81 + (b2 + 2) * q^82 + (-3*b2 + 4*b1 + 3) * q^83 + (-2*b2 - b1) * q^84 + (2*b2 - 3*b1 + 4) * q^85 + (4*b2 + b1 + 6) * q^86 + (2*b2 - 3*b1 - 1) * q^87 + (5*b2 + b1 - 1) * q^88 + (b2 - 3*b1 + 3) * q^89 + (2*b1 - 1) * q^90 + (6*b2 - 5*b1 - 2) * q^92 + (2*b2 + 3*b1 - 3) * q^93 + (9*b2 - 4*b1 + 15) * q^94 + (2*b2 + 5*b1) * q^95 + (4*b2 - b1 + 3) * q^96 + (-10*b2 + 6*b1 - 7) * q^97 + (b2 - 10*b1 + 8) * q^98 + (4*b2 - b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{3} - q^{4} - 4 q^{5} - q^{6} - 10 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 + 3 * q^3 - q^4 - 4 * q^5 - q^6 - 10 * q^7 + 3 * q^9 $$3 q - q^{2} + 3 q^{3} - q^{4} - 4 q^{5} - q^{6} - 10 q^{7} + 3 q^{9} - q^{10} + q^{11} - q^{12} + q^{14} - 4 q^{15} - 5 q^{16} - 7 q^{17} - q^{18} - 11 q^{19} - q^{20} - 10 q^{21} - 5 q^{22} + 2 q^{23} - 5 q^{25} + 3 q^{27} + q^{28} - 8 q^{29} - q^{30} - 8 q^{31} + 4 q^{32} + q^{33} + 18 q^{35} - q^{36} - 14 q^{37} + 20 q^{38} + 7 q^{40} - q^{41} + q^{42} - 3 q^{43} + 16 q^{44} - 4 q^{45} - 3 q^{46} + 9 q^{47} - 5 q^{48} + 17 q^{49} + 11 q^{50} - 7 q^{51} - 13 q^{53} - q^{54} - 13 q^{55} + 7 q^{56} - 11 q^{57} + 12 q^{58} + 14 q^{59} - q^{60} - 13 q^{61} - 16 q^{62} - 10 q^{63} + 4 q^{64} - 5 q^{66} - 5 q^{67} + 7 q^{68} + 2 q^{69} + 8 q^{70} + 6 q^{71} - 18 q^{73} + 7 q^{74} - 5 q^{75} - q^{76} - 15 q^{77} - 9 q^{79} + 16 q^{80} + 3 q^{81} + 5 q^{82} + 16 q^{83} + q^{84} + 7 q^{85} + 15 q^{86} - 8 q^{87} - 7 q^{88} + 5 q^{89} - q^{90} - 17 q^{92} - 8 q^{93} + 32 q^{94} + 3 q^{95} + 4 q^{96} - 5 q^{97} + 13 q^{98} + q^{99}+O(q^{100})$$ 3 * q - q^2 + 3 * q^3 - q^4 - 4 * q^5 - q^6 - 10 * q^7 + 3 * q^9 - q^10 + q^11 - q^12 + q^14 - 4 * q^15 - 5 * q^16 - 7 * q^17 - q^18 - 11 * q^19 - q^20 - 10 * q^21 - 5 * q^22 + 2 * q^23 - 5 * q^25 + 3 * q^27 + q^28 - 8 * q^29 - q^30 - 8 * q^31 + 4 * q^32 + q^33 + 18 * q^35 - q^36 - 14 * q^37 + 20 * q^38 + 7 * q^40 - q^41 + q^42 - 3 * q^43 + 16 * q^44 - 4 * q^45 - 3 * q^46 + 9 * q^47 - 5 * q^48 + 17 * q^49 + 11 * q^50 - 7 * q^51 - 13 * q^53 - q^54 - 13 * q^55 + 7 * q^56 - 11 * q^57 + 12 * q^58 + 14 * q^59 - q^60 - 13 * q^61 - 16 * q^62 - 10 * q^63 + 4 * q^64 - 5 * q^66 - 5 * q^67 + 7 * q^68 + 2 * q^69 + 8 * q^70 + 6 * q^71 - 18 * q^73 + 7 * q^74 - 5 * q^75 - q^76 - 15 * q^77 - 9 * q^79 + 16 * q^80 + 3 * q^81 + 5 * q^82 + 16 * q^83 + q^84 + 7 * q^85 + 15 * q^86 - 8 * q^87 - 7 * q^88 + 5 * q^89 - q^90 - 17 * q^92 - 8 * q^93 + 32 * q^94 + 3 * q^95 + 4 * q^96 - 5 * q^97 + 13 * q^98 + q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.80194 0.445042 −1.24698
−1.80194 1.00000 1.24698 −1.44504 −1.80194 −3.44504 1.35690 1.00000 2.60388
1.2 −0.445042 1.00000 −1.80194 0.246980 −0.445042 −1.75302 1.69202 1.00000 −0.109916
1.3 1.24698 1.00000 −0.445042 −2.80194 1.24698 −4.80194 −3.04892 1.00000 −3.49396
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.a.j 3
3.b odd 2 1 1521.2.a.q 3
4.b odd 2 1 8112.2.a.by 3
13.b even 2 1 507.2.a.k yes 3
13.c even 3 2 507.2.e.k 6
13.d odd 4 2 507.2.b.g 6
13.e even 6 2 507.2.e.j 6
13.f odd 12 4 507.2.j.h 12
39.d odd 2 1 1521.2.a.p 3
39.f even 4 2 1521.2.b.m 6
52.b odd 2 1 8112.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 1.a even 1 1 trivial
507.2.a.k yes 3 13.b even 2 1
507.2.b.g 6 13.d odd 4 2
507.2.e.j 6 13.e even 6 2
507.2.e.k 6 13.c even 3 2
507.2.j.h 12 13.f odd 12 4
1521.2.a.p 3 39.d odd 2 1
1521.2.a.q 3 3.b odd 2 1
1521.2.b.m 6 39.f even 4 2
8112.2.a.by 3 4.b odd 2 1
8112.2.a.cf 3 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1$$ T2^3 + T2^2 - 2*T2 - 1 $$T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1$$ T5^3 + 4*T5^2 + 3*T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 2T - 1$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} + 4 T^{2} + 3 T - 1$$
$7$ $$T^{3} + 10 T^{2} + 31 T + 29$$
$11$ $$T^{3} - T^{2} - 30 T + 43$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 7 T^{2} + 14 T + 7$$
$19$ $$T^{3} + 11 T^{2} + 10 T - 113$$
$23$ $$T^{3} - 2 T^{2} - 43 T - 83$$
$29$ $$T^{3} + 8 T^{2} + 5 T - 43$$
$31$ $$T^{3} + 8 T^{2} - 23 T - 197$$
$37$ $$T^{3} + 14 T^{2} + 63 T + 91$$
$41$ $$T^{3} + T^{2} - 2T - 1$$
$43$ $$T^{3} + 3 T^{2} - 25 T + 29$$
$47$ $$T^{3} - 9 T^{2} - 120 T + 911$$
$53$ $$T^{3} + 13 T^{2} + 40 T + 29$$
$59$ $$T^{3} - 14T^{2} + 56$$
$61$ $$T^{3} + 13 T^{2} + 12 T - 223$$
$67$ $$T^{3} + 5 T^{2} - 22 T - 97$$
$71$ $$T^{3} - 6 T^{2} - 79 T + 461$$
$73$ $$T^{3} + 18 T^{2} + 101 T + 167$$
$79$ $$T^{3} + 9 T^{2} - 22 T - 169$$
$83$ $$T^{3} - 16 T^{2} + 55 T + 43$$
$89$ $$T^{3} - 5 T^{2} - 8 T - 1$$
$97$ $$T^{3} + 5 T^{2} - 169 T - 1189$$