# Properties

 Label 507.2.a.b Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + q^5 + q^6 - 2 * q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 3 q^{8} + q^{9} - q^{10} + 2 q^{11} + q^{12} + 2 q^{14} - q^{15} - q^{16} - 7 q^{17} - q^{18} + 6 q^{19} - q^{20} + 2 q^{21} - 2 q^{22} - 6 q^{23} - 3 q^{24} - 4 q^{25} - q^{27} + 2 q^{28} - q^{29} + q^{30} - 4 q^{31} - 5 q^{32} - 2 q^{33} + 7 q^{34} - 2 q^{35} - q^{36} - q^{37} - 6 q^{38} + 3 q^{40} - 9 q^{41} - 2 q^{42} + 6 q^{43} - 2 q^{44} + q^{45} + 6 q^{46} - 6 q^{47} + q^{48} - 3 q^{49} + 4 q^{50} + 7 q^{51} - 9 q^{53} + q^{54} + 2 q^{55} - 6 q^{56} - 6 q^{57} + q^{58} + q^{60} + q^{61} + 4 q^{62} - 2 q^{63} + 7 q^{64} + 2 q^{66} + 2 q^{67} + 7 q^{68} + 6 q^{69} + 2 q^{70} - 6 q^{71} + 3 q^{72} - 11 q^{73} + q^{74} + 4 q^{75} - 6 q^{76} - 4 q^{77} - 4 q^{79} - q^{80} + q^{81} + 9 q^{82} + 14 q^{83} - 2 q^{84} - 7 q^{85} - 6 q^{86} + q^{87} + 6 q^{88} + 14 q^{89} - q^{90} + 6 q^{92} + 4 q^{93} + 6 q^{94} + 6 q^{95} + 5 q^{96} + 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100})$$ q - q^2 - q^3 - q^4 + q^5 + q^6 - 2 * q^7 + 3 * q^8 + q^9 - q^10 + 2 * q^11 + q^12 + 2 * q^14 - q^15 - q^16 - 7 * q^17 - q^18 + 6 * q^19 - q^20 + 2 * q^21 - 2 * q^22 - 6 * q^23 - 3 * q^24 - 4 * q^25 - q^27 + 2 * q^28 - q^29 + q^30 - 4 * q^31 - 5 * q^32 - 2 * q^33 + 7 * q^34 - 2 * q^35 - q^36 - q^37 - 6 * q^38 + 3 * q^40 - 9 * q^41 - 2 * q^42 + 6 * q^43 - 2 * q^44 + q^45 + 6 * q^46 - 6 * q^47 + q^48 - 3 * q^49 + 4 * q^50 + 7 * q^51 - 9 * q^53 + q^54 + 2 * q^55 - 6 * q^56 - 6 * q^57 + q^58 + q^60 + q^61 + 4 * q^62 - 2 * q^63 + 7 * q^64 + 2 * q^66 + 2 * q^67 + 7 * q^68 + 6 * q^69 + 2 * q^70 - 6 * q^71 + 3 * q^72 - 11 * q^73 + q^74 + 4 * q^75 - 6 * q^76 - 4 * q^77 - 4 * q^79 - q^80 + q^81 + 9 * q^82 + 14 * q^83 - 2 * q^84 - 7 * q^85 - 6 * q^86 + q^87 + 6 * q^88 + 14 * q^89 - q^90 + 6 * q^92 + 4 * q^93 + 6 * q^94 + 6 * q^95 + 5 * q^96 + 2 * q^97 + 3 * q^98 + 2 * 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
 0
−1.00000 −1.00000 −1.00000 1.00000 1.00000 −2.00000 3.00000 1.00000 −1.00000
 $$n$$: 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.b 1
3.b odd 2 1 1521.2.a.d 1
4.b odd 2 1 8112.2.a.bc 1
13.b even 2 1 507.2.a.c 1
13.c even 3 2 507.2.e.c 2
13.d odd 4 2 507.2.b.b 2
13.e even 6 2 39.2.e.a 2
13.f odd 12 4 507.2.j.d 4
39.d odd 2 1 1521.2.a.a 1
39.f even 4 2 1521.2.b.c 2
39.h odd 6 2 117.2.g.b 2
52.b odd 2 1 8112.2.a.w 1
52.i odd 6 2 624.2.q.c 2
65.l even 6 2 975.2.i.f 2
65.r odd 12 4 975.2.bb.d 4
156.r even 6 2 1872.2.t.j 2

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

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T + 7$$
$19$ $$T - 6$$
$23$ $$T + 6$$
$29$ $$T + 1$$
$31$ $$T + 4$$
$37$ $$T + 1$$
$41$ $$T + 9$$
$43$ $$T - 6$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T - 1$$
$67$ $$T - 2$$
$71$ $$T + 6$$
$73$ $$T + 11$$
$79$ $$T + 4$$
$83$ $$T - 14$$
$89$ $$T - 14$$
$97$ $$T - 2$$