# Properties

 Label 507.2.a.g Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + (\beta + 2) q^{4} + (\beta - 2) q^{5} + \beta q^{6} + ( - \beta - 1) q^{7} + (\beta + 4) q^{8} + q^{9}+O(q^{10})$$ q + b * q^2 + q^3 + (b + 2) * q^4 + (b - 2) * q^5 + b * q^6 + (-b - 1) * q^7 + (b + 4) * q^8 + q^9 $$q + \beta q^{2} + q^{3} + (\beta + 2) q^{4} + (\beta - 2) q^{5} + \beta q^{6} + ( - \beta - 1) q^{7} + (\beta + 4) q^{8} + q^{9} + ( - \beta + 4) q^{10} - 2 q^{11} + (\beta + 2) q^{12} + ( - 2 \beta - 4) q^{14} + (\beta - 2) q^{15} + 3 \beta q^{16} + \beta q^{17} + \beta q^{18} + ( - 2 \beta + 4) q^{19} + \beta q^{20} + ( - \beta - 1) q^{21} - 2 \beta q^{22} + 2 q^{23} + (\beta + 4) q^{24} + ( - 3 \beta + 3) q^{25} + q^{27} + ( - 4 \beta - 6) q^{28} + ( - 3 \beta + 2) q^{29} + ( - \beta + 4) q^{30} + ( - \beta + 1) q^{31} + (\beta + 4) q^{32} - 2 q^{33} + (\beta + 4) q^{34} - 2 q^{35} + (\beta + 2) q^{36} + ( - \beta + 6) q^{37} + (2 \beta - 8) q^{38} + (3 \beta - 4) q^{40} + \beta q^{41} + ( - 2 \beta - 4) q^{42} + ( - \beta + 3) q^{43} + ( - 2 \beta - 4) q^{44} + (\beta - 2) q^{45} + 2 \beta q^{46} + ( - 4 \beta + 2) q^{47} + 3 \beta q^{48} + (3 \beta - 2) q^{49} - 12 q^{50} + \beta q^{51} + (3 \beta + 4) q^{53} + \beta q^{54} + ( - 2 \beta + 4) q^{55} + ( - 6 \beta - 8) q^{56} + ( - 2 \beta + 4) q^{57} + ( - \beta - 12) q^{58} + ( - 2 \beta - 6) q^{59} + \beta q^{60} + (2 \beta + 7) q^{61} - 4 q^{62} + ( - \beta - 1) q^{63} + ( - \beta + 4) q^{64} - 2 \beta q^{66} + ( - \beta + 3) q^{67} + (3 \beta + 4) q^{68} + 2 q^{69} - 2 \beta q^{70} + 14 q^{71} + (\beta + 4) q^{72} + (2 \beta - 7) q^{73} + (5 \beta - 4) q^{74} + ( - 3 \beta + 3) q^{75} - 2 \beta q^{76} + (2 \beta + 2) q^{77} + (\beta + 7) q^{79} + ( - 3 \beta + 12) q^{80} + q^{81} + (\beta + 4) q^{82} + ( - 2 \beta - 4) q^{83} + ( - 4 \beta - 6) q^{84} + ( - \beta + 4) q^{85} + (2 \beta - 4) q^{86} + ( - 3 \beta + 2) q^{87} + ( - 2 \beta - 8) q^{88} + (2 \beta + 8) q^{89} + ( - \beta + 4) q^{90} + (2 \beta + 4) q^{92} + ( - \beta + 1) q^{93} + ( - 2 \beta - 16) q^{94} + (6 \beta - 16) q^{95} + (\beta + 4) q^{96} + (\beta - 7) q^{97} + (\beta + 12) q^{98} - 2 q^{99} +O(q^{100})$$ q + b * q^2 + q^3 + (b + 2) * q^4 + (b - 2) * q^5 + b * q^6 + (-b - 1) * q^7 + (b + 4) * q^8 + q^9 + (-b + 4) * q^10 - 2 * q^11 + (b + 2) * q^12 + (-2*b - 4) * q^14 + (b - 2) * q^15 + 3*b * q^16 + b * q^17 + b * q^18 + (-2*b + 4) * q^19 + b * q^20 + (-b - 1) * q^21 - 2*b * q^22 + 2 * q^23 + (b + 4) * q^24 + (-3*b + 3) * q^25 + q^27 + (-4*b - 6) * q^28 + (-3*b + 2) * q^29 + (-b + 4) * q^30 + (-b + 1) * q^31 + (b + 4) * q^32 - 2 * q^33 + (b + 4) * q^34 - 2 * q^35 + (b + 2) * q^36 + (-b + 6) * q^37 + (2*b - 8) * q^38 + (3*b - 4) * q^40 + b * q^41 + (-2*b - 4) * q^42 + (-b + 3) * q^43 + (-2*b - 4) * q^44 + (b - 2) * q^45 + 2*b * q^46 + (-4*b + 2) * q^47 + 3*b * q^48 + (3*b - 2) * q^49 - 12 * q^50 + b * q^51 + (3*b + 4) * q^53 + b * q^54 + (-2*b + 4) * q^55 + (-6*b - 8) * q^56 + (-2*b + 4) * q^57 + (-b - 12) * q^58 + (-2*b - 6) * q^59 + b * q^60 + (2*b + 7) * q^61 - 4 * q^62 + (-b - 1) * q^63 + (-b + 4) * q^64 - 2*b * q^66 + (-b + 3) * q^67 + (3*b + 4) * q^68 + 2 * q^69 - 2*b * q^70 + 14 * q^71 + (b + 4) * q^72 + (2*b - 7) * q^73 + (5*b - 4) * q^74 + (-3*b + 3) * q^75 - 2*b * q^76 + (2*b + 2) * q^77 + (b + 7) * q^79 + (-3*b + 12) * q^80 + q^81 + (b + 4) * q^82 + (-2*b - 4) * q^83 + (-4*b - 6) * q^84 + (-b + 4) * q^85 + (2*b - 4) * q^86 + (-3*b + 2) * q^87 + (-2*b - 8) * q^88 + (2*b + 8) * q^89 + (-b + 4) * q^90 + (2*b + 4) * q^92 + (-b + 1) * q^93 + (-2*b - 16) * q^94 + (6*b - 16) * q^95 + (b + 4) * q^96 + (b - 7) * q^97 + (b + 12) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 3 * q^7 + 9 * q^8 + 2 * q^9 $$2 q + q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} + 9 q^{8} + 2 q^{9} + 7 q^{10} - 4 q^{11} + 5 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + q^{18} + 6 q^{19} + q^{20} - 3 q^{21} - 2 q^{22} + 4 q^{23} + 9 q^{24} + 3 q^{25} + 2 q^{27} - 16 q^{28} + q^{29} + 7 q^{30} + q^{31} + 9 q^{32} - 4 q^{33} + 9 q^{34} - 4 q^{35} + 5 q^{36} + 11 q^{37} - 14 q^{38} - 5 q^{40} + q^{41} - 10 q^{42} + 5 q^{43} - 10 q^{44} - 3 q^{45} + 2 q^{46} + 3 q^{48} - q^{49} - 24 q^{50} + q^{51} + 11 q^{53} + q^{54} + 6 q^{55} - 22 q^{56} + 6 q^{57} - 25 q^{58} - 14 q^{59} + q^{60} + 16 q^{61} - 8 q^{62} - 3 q^{63} + 7 q^{64} - 2 q^{66} + 5 q^{67} + 11 q^{68} + 4 q^{69} - 2 q^{70} + 28 q^{71} + 9 q^{72} - 12 q^{73} - 3 q^{74} + 3 q^{75} - 2 q^{76} + 6 q^{77} + 15 q^{79} + 21 q^{80} + 2 q^{81} + 9 q^{82} - 10 q^{83} - 16 q^{84} + 7 q^{85} - 6 q^{86} + q^{87} - 18 q^{88} + 18 q^{89} + 7 q^{90} + 10 q^{92} + q^{93} - 34 q^{94} - 26 q^{95} + 9 q^{96} - 13 q^{97} + 25 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 3 * q^7 + 9 * q^8 + 2 * q^9 + 7 * q^10 - 4 * q^11 + 5 * q^12 - 10 * q^14 - 3 * q^15 + 3 * q^16 + q^17 + q^18 + 6 * q^19 + q^20 - 3 * q^21 - 2 * q^22 + 4 * q^23 + 9 * q^24 + 3 * q^25 + 2 * q^27 - 16 * q^28 + q^29 + 7 * q^30 + q^31 + 9 * q^32 - 4 * q^33 + 9 * q^34 - 4 * q^35 + 5 * q^36 + 11 * q^37 - 14 * q^38 - 5 * q^40 + q^41 - 10 * q^42 + 5 * q^43 - 10 * q^44 - 3 * q^45 + 2 * q^46 + 3 * q^48 - q^49 - 24 * q^50 + q^51 + 11 * q^53 + q^54 + 6 * q^55 - 22 * q^56 + 6 * q^57 - 25 * q^58 - 14 * q^59 + q^60 + 16 * q^61 - 8 * q^62 - 3 * q^63 + 7 * q^64 - 2 * q^66 + 5 * q^67 + 11 * q^68 + 4 * q^69 - 2 * q^70 + 28 * q^71 + 9 * q^72 - 12 * q^73 - 3 * q^74 + 3 * q^75 - 2 * q^76 + 6 * q^77 + 15 * q^79 + 21 * q^80 + 2 * q^81 + 9 * q^82 - 10 * q^83 - 16 * q^84 + 7 * q^85 - 6 * q^86 + q^87 - 18 * q^88 + 18 * q^89 + 7 * q^90 + 10 * q^92 + q^93 - 34 * q^94 - 26 * q^95 + 9 * q^96 - 13 * q^97 + 25 * q^98 - 4 * q^99

## 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
 −1.56155 2.56155
−1.56155 1.00000 0.438447 −3.56155 −1.56155 0.561553 2.43845 1.00000 5.56155
1.2 2.56155 1.00000 4.56155 0.561553 2.56155 −3.56155 6.56155 1.00000 1.43845
 $$n$$: 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.g 2
3.b odd 2 1 1521.2.a.g 2
4.b odd 2 1 8112.2.a.bk 2
13.b even 2 1 507.2.a.d 2
13.c even 3 2 39.2.e.b 4
13.d odd 4 2 507.2.b.d 4
13.e even 6 2 507.2.e.g 4
13.f odd 12 4 507.2.j.g 8
39.d odd 2 1 1521.2.a.m 2
39.f even 4 2 1521.2.b.h 4
39.i odd 6 2 117.2.g.c 4
52.b odd 2 1 8112.2.a.bo 2
52.j odd 6 2 624.2.q.h 4
65.n even 6 2 975.2.i.k 4
65.q odd 12 4 975.2.bb.i 8
156.p even 6 2 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.c even 3 2
117.2.g.c 4 39.i odd 6 2
507.2.a.d 2 13.b even 2 1
507.2.a.g 2 1.a even 1 1 trivial
507.2.b.d 4 13.d odd 4 2
507.2.e.g 4 13.e even 6 2
507.2.j.g 8 13.f odd 12 4
624.2.q.h 4 52.j odd 6 2
975.2.i.k 4 65.n even 6 2
975.2.bb.i 8 65.q odd 12 4
1521.2.a.g 2 3.b odd 2 1
1521.2.a.m 2 39.d odd 2 1
1521.2.b.h 4 39.f even 4 2
1872.2.t.r 4 156.p even 6 2
8112.2.a.bk 2 4.b odd 2 1
8112.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$T^{2} + 3T - 2$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} - 6T - 8$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - T - 4$$
$37$ $$T^{2} - 11T + 26$$
$41$ $$T^{2} - T - 4$$
$43$ $$T^{2} - 5T + 2$$
$47$ $$T^{2} - 68$$
$53$ $$T^{2} - 11T - 8$$
$59$ $$T^{2} + 14T + 32$$
$61$ $$T^{2} - 16T + 47$$
$67$ $$T^{2} - 5T + 2$$
$71$ $$(T - 14)^{2}$$
$73$ $$T^{2} + 12T + 19$$
$79$ $$T^{2} - 15T + 52$$
$83$ $$T^{2} + 10T + 8$$
$89$ $$T^{2} - 18T + 64$$
$97$ $$T^{2} + 13T + 38$$