# Properties

 Label 507.2.b.f Level $507$ Weight $2$ Character orbit 507.b Analytic conductor $4.048$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(337,507)]

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, 1]))

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, 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: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} - q^{3} + ( - \beta_{4} - \beta_{2} - 3) q^{4} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{6} + ( - \beta_{5} + \beta_1) q^{7} + ( - 5 \beta_{5} - 3 \beta_{3}) q^{8} + q^{9}+O(q^{10})$$ q + (b5 + b3 + b1) * q^2 - q^3 + (-b4 - b2 - 3) * q^4 + (b5 - 2*b3 + b1) * q^5 + (-b5 - b3 - b1) * q^6 + (-b5 + b1) * q^7 + (-5*b5 - 3*b3) * q^8 + q^9 $$q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} - q^{3} + ( - \beta_{4} - \beta_{2} - 3) q^{4} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{6} + ( - \beta_{5} + \beta_1) q^{7} + ( - 5 \beta_{5} - 3 \beta_{3}) q^{8} + q^{9} + ( - 4 \beta_{4} + 2 \beta_{2} + 1) q^{10} + ( - 3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{11} + (\beta_{4} + \beta_{2} + 3) q^{12} + (2 \beta_{4} + 2 \beta_{2} - 3) q^{14} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{15} + (5 \beta_{4} + 6 \beta_{2}) q^{16} + (\beta_{4} + 3 \beta_{2} - 1) q^{17} + (\beta_{5} + \beta_{3} + \beta_1) q^{18} + ( - 2 \beta_{5} + \beta_{3}) q^{19} + ( - \beta_{5} + 7 \beta_{3} - 5 \beta_1) q^{20} + (\beta_{5} - \beta_1) q^{21} + (5 \beta_{4} + 7 \beta_{2} - 7) q^{22} + (\beta_{4} - 4 \beta_{2} + 1) q^{23} + (5 \beta_{5} + 3 \beta_{3}) q^{24} + (2 \beta_{4} + 5 \beta_{2} - 6) q^{25} - q^{27} + (3 \beta_{5} + \beta_{3} - 3 \beta_1) q^{28} + (2 \beta_{4} - \beta_{2} - 1) q^{29} + (4 \beta_{4} - 2 \beta_{2} - 1) q^{30} + ( - 8 \beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{31} + (12 \beta_{5} + 7 \beta_{3} - 7 \beta_1) q^{32} + (3 \beta_{5} + \beta_{3} - 3 \beta_1) q^{33} + (7 \beta_{5} + 7 \beta_{3} - 6 \beta_1) q^{34} + ( - 4 \beta_{4} - \beta_{2} + 3) q^{35} + ( - \beta_{4} - \beta_{2} - 3) q^{36} + ( - 7 \beta_{5} + \beta_{3}) q^{37} + (5 \beta_{4} + \beta_{2} - 2) q^{38} + (\beta_{4} - 7 \beta_{2} + 3) q^{40} + (5 \beta_{5} + \beta_{3} - 3 \beta_1) q^{41} + ( - 2 \beta_{4} - 2 \beta_{2} + 3) q^{42} + (2 \beta_{4} + 4 \beta_{2} + 3) q^{43} + (11 \beta_{5} + 7 \beta_{3} - 10 \beta_1) q^{44} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{45} + ( - 5 \beta_{5} - 12 \beta_{3} + 10 \beta_1) q^{46} + ( - 2 \beta_{5} + \beta_{3}) q^{47} + ( - 5 \beta_{4} - 6 \beta_{2}) q^{48} + (2 \beta_{4} + \beta_{2} + 4) q^{49} + (8 \beta_{5} + 7 \beta_{3} - 14 \beta_1) q^{50} + ( - \beta_{4} - 3 \beta_{2} + 1) q^{51} + ( - 4 \beta_{4} - \beta_{2} - 4) q^{53} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{54} + ( - 10 \beta_{4} - 2 \beta_{2} + 5) q^{55} + ( - \beta_{4} - 3 \beta_{2} + 1) q^{56} + (2 \beta_{5} - \beta_{3}) q^{57} + (\beta_{5} - 6 \beta_{3} + 3 \beta_1) q^{58} + ( - 4 \beta_{3} + 2 \beta_1) q^{59} + (\beta_{5} - 7 \beta_{3} + 5 \beta_1) q^{60} + ( - 5 \beta_{4} + \beta_{2} - 3) q^{61} + (13 \beta_{4} + 16 \beta_{2} - 9) q^{62} + ( - \beta_{5} + \beta_1) q^{63} + ( - 7 \beta_{4} - 14 \beta_{2} + 7) q^{64} + ( - 5 \beta_{4} - 7 \beta_{2} + 7) q^{66} + ( - 4 \beta_{3} + 7 \beta_1) q^{67} + ( - 5 \beta_{4} - 14 \beta_{2} + 2) q^{68} + ( - \beta_{4} + 4 \beta_{2} - 1) q^{69} + ( - 7 \beta_{5} + 4 \beta_{3} + \beta_1) q^{70} + ( - 4 \beta_{5} - 7 \beta_{3} + 5 \beta_1) q^{71} + ( - 5 \beta_{5} - 3 \beta_{3}) q^{72} + ( - 7 \beta_{5} - 6 \beta_{3} + 9 \beta_1) q^{73} + (15 \beta_{4} + 6 \beta_{2} - 2) q^{74} + ( - 2 \beta_{4} - 5 \beta_{2} + 6) q^{75} + (6 \beta_{5} - 2 \beta_{3} + \beta_1) q^{76} + (4 \beta_{4} + 2 \beta_{2} - 7) q^{77} + 3 \beta_{4} q^{79} + ( - 11 \beta_{5} - 5 \beta_{3} + 8 \beta_1) q^{80} + q^{81} + ( - 9 \beta_{4} - 9 \beta_{2} + 7) q^{82} + (3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{83} + ( - 3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{84} + ( - 5 \beta_{5} + \beta_{3} - 3 \beta_1) q^{85} + (15 \beta_{5} + 13 \beta_{3} - 3 \beta_1) q^{86} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{87} + ( - 5 \beta_{4} - 14 \beta_{2} + 2) q^{88} + ( - 2 \beta_{5} - 6 \beta_{3} - \beta_1) q^{89} + ( - 4 \beta_{4} + 2 \beta_{2} + 1) q^{90} + (19 \beta_{2} - 4) q^{92} + (8 \beta_{5} + 3 \beta_{3} - 5 \beta_1) q^{93} + (5 \beta_{4} + \beta_{2} - 2) q^{94} + ( - 4 \beta_{4} - 5 \beta_{2} + 10) q^{95} + ( - 12 \beta_{5} - 7 \beta_{3} + 7 \beta_1) q^{96} + (3 \beta_{5} + 12 \beta_{3} - 2 \beta_1) q^{97} + (10 \beta_{5} + 5 \beta_{3} + 4 \beta_1) q^{98} + ( - 3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{99}+O(q^{100})$$ q + (b5 + b3 + b1) * q^2 - q^3 + (-b4 - b2 - 3) * q^4 + (b5 - 2*b3 + b1) * q^5 + (-b5 - b3 - b1) * q^6 + (-b5 + b1) * q^7 + (-5*b5 - 3*b3) * q^8 + q^9 + (-4*b4 + 2*b2 + 1) * q^10 + (-3*b5 - b3 + 3*b1) * q^11 + (b4 + b2 + 3) * q^12 + (2*b4 + 2*b2 - 3) * q^14 + (-b5 + 2*b3 - b1) * q^15 + (5*b4 + 6*b2) * q^16 + (b4 + 3*b2 - 1) * q^17 + (b5 + b3 + b1) * q^18 + (-2*b5 + b3) * q^19 + (-b5 + 7*b3 - 5*b1) * q^20 + (b5 - b1) * q^21 + (5*b4 + 7*b2 - 7) * q^22 + (b4 - 4*b2 + 1) * q^23 + (5*b5 + 3*b3) * q^24 + (2*b4 + 5*b2 - 6) * q^25 - q^27 + (3*b5 + b3 - 3*b1) * q^28 + (2*b4 - b2 - 1) * q^29 + (4*b4 - 2*b2 - 1) * q^30 + (-8*b5 - 3*b3 + 5*b1) * q^31 + (12*b5 + 7*b3 - 7*b1) * q^32 + (3*b5 + b3 - 3*b1) * q^33 + (7*b5 + 7*b3 - 6*b1) * q^34 + (-4*b4 - b2 + 3) * q^35 + (-b4 - b2 - 3) * q^36 + (-7*b5 + b3) * q^37 + (5*b4 + b2 - 2) * q^38 + (b4 - 7*b2 + 3) * q^40 + (5*b5 + b3 - 3*b1) * q^41 + (-2*b4 - 2*b2 + 3) * q^42 + (2*b4 + 4*b2 + 3) * q^43 + (11*b5 + 7*b3 - 10*b1) * q^44 + (b5 - 2*b3 + b1) * q^45 + (-5*b5 - 12*b3 + 10*b1) * q^46 + (-2*b5 + b3) * q^47 + (-5*b4 - 6*b2) * q^48 + (2*b4 + b2 + 4) * q^49 + (8*b5 + 7*b3 - 14*b1) * q^50 + (-b4 - 3*b2 + 1) * q^51 + (-4*b4 - b2 - 4) * q^53 + (-b5 - b3 - b1) * q^54 + (-10*b4 - 2*b2 + 5) * q^55 + (-b4 - 3*b2 + 1) * q^56 + (2*b5 - b3) * q^57 + (b5 - 6*b3 + 3*b1) * q^58 + (-4*b3 + 2*b1) * q^59 + (b5 - 7*b3 + 5*b1) * q^60 + (-5*b4 + b2 - 3) * q^61 + (13*b4 + 16*b2 - 9) * q^62 + (-b5 + b1) * q^63 + (-7*b4 - 14*b2 + 7) * q^64 + (-5*b4 - 7*b2 + 7) * q^66 + (-4*b3 + 7*b1) * q^67 + (-5*b4 - 14*b2 + 2) * q^68 + (-b4 + 4*b2 - 1) * q^69 + (-7*b5 + 4*b3 + b1) * q^70 + (-4*b5 - 7*b3 + 5*b1) * q^71 + (-5*b5 - 3*b3) * q^72 + (-7*b5 - 6*b3 + 9*b1) * q^73 + (15*b4 + 6*b2 - 2) * q^74 + (-2*b4 - 5*b2 + 6) * q^75 + (6*b5 - 2*b3 + b1) * q^76 + (4*b4 + 2*b2 - 7) * q^77 + 3*b4 * q^79 + (-11*b5 - 5*b3 + 8*b1) * q^80 + q^81 + (-9*b4 - 9*b2 + 7) * q^82 + (3*b5 - b3 + 2*b1) * q^83 + (-3*b5 - b3 + 3*b1) * q^84 + (-5*b5 + b3 - 3*b1) * q^85 + (15*b5 + 13*b3 - 3*b1) * q^86 + (-2*b4 + b2 + 1) * q^87 + (-5*b4 - 14*b2 + 2) * q^88 + (-2*b5 - 6*b3 - b1) * q^89 + (-4*b4 + 2*b2 + 1) * q^90 + (19*b2 - 4) * q^92 + (8*b5 + 3*b3 - 5*b1) * q^93 + (5*b4 + b2 - 2) * q^94 + (-4*b4 - 5*b2 + 10) * q^95 + (-12*b5 - 7*b3 + 7*b1) * q^96 + (3*b5 + 12*b3 - 2*b1) * q^97 + (10*b5 + 5*b3 + 4*b1) * q^98 + (-3*b5 - b3 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} - 22 q^{4} + 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^3 - 22 * q^4 + 6 * q^9 $$6 q - 6 q^{3} - 22 q^{4} + 6 q^{9} + 2 q^{10} + 22 q^{12} - 10 q^{14} + 22 q^{16} + 2 q^{17} - 18 q^{22} - 22 q^{25} - 6 q^{27} - 4 q^{29} - 2 q^{30} + 8 q^{35} - 22 q^{36} + 6 q^{40} + 10 q^{42} + 30 q^{43} - 22 q^{48} + 30 q^{49} - 2 q^{51} - 34 q^{53} + 6 q^{55} - 2 q^{56} - 26 q^{61} + 4 q^{62} + 18 q^{66} - 26 q^{68} + 30 q^{74} + 22 q^{75} - 30 q^{77} + 6 q^{79} + 6 q^{81} + 6 q^{82} + 4 q^{87} - 26 q^{88} + 2 q^{90} + 14 q^{92} + 42 q^{95}+O(q^{100})$$ 6 * q - 6 * q^3 - 22 * q^4 + 6 * q^9 + 2 * q^10 + 22 * q^12 - 10 * q^14 + 22 * q^16 + 2 * q^17 - 18 * q^22 - 22 * q^25 - 6 * q^27 - 4 * q^29 - 2 * q^30 + 8 * q^35 - 22 * q^36 + 6 * q^40 + 10 * q^42 + 30 * q^43 - 22 * q^48 + 30 * q^49 - 2 * q^51 - 34 * q^53 + 6 * q^55 - 2 * q^56 - 26 * q^61 + 4 * q^62 + 18 * q^66 - 26 * q^68 + 30 * q^74 + 22 * q^75 - 30 * q^77 + 6 * q^79 + 6 * q^81 + 6 * q^82 + 4 * q^87 - 26 * q^88 + 2 * q^90 + 14 * q^92 + 42 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5x^{4} + 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3\nu^{2} + 1$$ v^4 + 3*v^2 + 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4\nu^{3} + 3\nu$$ v^5 + 4*v^3 + 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_1$$ b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3\beta_{2} + 5$$ b4 - 3*b2 + 5 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + 9\beta_1$$ b5 - 4*b3 + 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 0.445042i − 1.80194i − 1.24698i 1.24698i 1.80194i 0.445042i
2.69202i −1.00000 −5.24698 1.04892i 2.69202i 0.554958i 8.74094i 1.00000 2.82371
337.2 2.35690i −1.00000 −3.55496 3.69202i 2.35690i 0.801938i 3.66487i 1.00000 −8.70171
337.3 2.04892i −1.00000 −2.19806 3.35690i 2.04892i 2.24698i 0.405813i 1.00000 6.87800
337.4 2.04892i −1.00000 −2.19806 3.35690i 2.04892i 2.24698i 0.405813i 1.00000 6.87800
337.5 2.35690i −1.00000 −3.55496 3.69202i 2.35690i 0.801938i 3.66487i 1.00000 −8.70171
337.6 2.69202i −1.00000 −5.24698 1.04892i 2.69202i 0.554958i 8.74094i 1.00000 2.82371
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.b.f 6
3.b odd 2 1 1521.2.b.k 6
13.b even 2 1 inner 507.2.b.f 6
13.c even 3 2 507.2.j.i 12
13.d odd 4 1 507.2.a.i 3
13.d odd 4 1 507.2.a.l yes 3
13.e even 6 2 507.2.j.i 12
13.f odd 12 2 507.2.e.i 6
13.f odd 12 2 507.2.e.l 6
39.d odd 2 1 1521.2.b.k 6
39.f even 4 1 1521.2.a.n 3
39.f even 4 1 1521.2.a.s 3
52.f even 4 1 8112.2.a.cg 3
52.f even 4 1 8112.2.a.cp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.d odd 4 1
507.2.a.l yes 3 13.d odd 4 1
507.2.b.f 6 1.a even 1 1 trivial
507.2.b.f 6 13.b even 2 1 inner
507.2.e.i 6 13.f odd 12 2
507.2.e.l 6 13.f odd 12 2
507.2.j.i 12 13.c even 3 2
507.2.j.i 12 13.e even 6 2
1521.2.a.n 3 39.f even 4 1
1521.2.a.s 3 39.f even 4 1
1521.2.b.k 6 3.b odd 2 1
1521.2.b.k 6 39.d odd 2 1
8112.2.a.cg 3 52.f even 4 1
8112.2.a.cp 3 52.f even 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 17T_{2}^{4} + 94T_{2}^{2} + 169$$ T2^6 + 17*T2^4 + 94*T2^2 + 169 $$T_{5}^{6} + 26T_{5}^{4} + 181T_{5}^{2} + 169$$ T5^6 + 26*T5^4 + 181*T5^2 + 169

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 17 T^{4} + 94 T^{2} + \cdots + 169$$
$3$ $$(T + 1)^{6}$$
$5$ $$T^{6} + 26 T^{4} + 181 T^{2} + \cdots + 169$$
$7$ $$T^{6} + 6 T^{4} + 5 T^{2} + 1$$
$11$ $$T^{6} + 41 T^{4} + 474 T^{2} + \cdots + 1681$$
$13$ $$T^{6}$$
$17$ $$(T^{3} - T^{2} - 16 T - 13)^{2}$$
$19$ $$T^{6} + 21 T^{4} + 98 T^{2} + 49$$
$23$ $$(T^{3} - 49 T - 91)^{2}$$
$29$ $$(T^{3} + 2 T^{2} - 15 T - 29)^{2}$$
$31$ $$T^{6} + 174 T^{4} + 7985 T^{2} + \cdots + 38809$$
$37$ $$T^{6} + 166 T^{4} + 8693 T^{2} + \cdots + 142129$$
$41$ $$T^{6} + 73 T^{4} + 1214 T^{2} + \cdots + 841$$
$43$ $$(T^{3} - 15 T^{2} + 47 T - 41)^{2}$$
$47$ $$T^{6} + 21 T^{4} + 98 T^{2} + 49$$
$53$ $$(T^{3} + 17 T^{2} + 66 T - 41)^{2}$$
$59$ $$T^{6} + 68 T^{4} + 1504 T^{2} + \cdots + 10816$$
$61$ $$(T^{3} + 13 T^{2} - 16 T - 167)^{2}$$
$67$ $$T^{6} + 213 T^{4} + 1214 T^{2} + \cdots + 1681$$
$71$ $$T^{6} + 182 T^{4} + 8281 T^{2} + \cdots + 41209$$
$73$ $$T^{6} + 306 T^{4} + 29301 T^{2} + \cdots + 851929$$
$79$ $$(T^{3} - 3 T^{2} - 18 T + 27)^{2}$$
$83$ $$T^{6} + 62 T^{4} + 649 T^{2} + \cdots + 1849$$
$89$ $$T^{6} + 201 T^{4} + 10226 T^{2} + \cdots + 12769$$
$97$ $$T^{6} + 587 T^{4} + 95331 T^{2} + \cdots + 2679769$$