# Properties

 Label 507.2.x.a Level $507$ Weight $2$ Character orbit 507.x Analytic conductor $4.048$ Analytic rank $0$ Dimension $48$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.x (of order $$156$$, degree $$48$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{156}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 10q^{7} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 10q^{7} + 6q^{9} - 8q^{16} - 14q^{19} - 18q^{21} + 20q^{28} + 14q^{31} + 2q^{37} + 24q^{39} + 6q^{43} - 18q^{49} - 28q^{52} - 12q^{57} - 24q^{63} - 32q^{67} + 34q^{73} + 30q^{75} + 28q^{76} + 18q^{81} + 12q^{84} - 2q^{91} - 6q^{93} + 38q^{97} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 0 1.38468 1.04052i 1.99838 + 0.0805319i 0 0 −1.33289 + 2.01672i 0 0.834652 2.88155i 0
11.1 0 1.59345 0.678906i −1.06893 1.69038i 0 0 1.78351 3.57116i 0 2.07817 2.16361i 0
20.1 0 1.19983 1.24916i 1.80690 + 0.857385i 0 0 −0.697469 + 4.91487i 0 −0.120798 2.99757i 0
32.1 0 1.72643 0.139372i 1.95958 + 0.400051i 0 0 −0.421909 1.87333i 0 2.96115 0.481234i 0
41.1 0 −0.346455 + 1.69705i −1.92104 0.556435i 0 0 −0.766841 1.06446i 0 −2.75994 1.17590i 0
50.1 0 −1.64291 0.548485i 1.44240 + 1.38545i 0 0 0.680973 + 0.0688013i 0 2.39833 + 1.80223i 0
59.1 0 −1.46391 + 0.925722i 0.320823 + 1.97410i 0 0 5.25731 + 0.105888i 0 1.28608 2.71035i 0
71.1 0 −1.64291 + 0.548485i 1.44240 1.38545i 0 0 0.680973 0.0688013i 0 2.39833 1.80223i 0
98.1 0 1.64291 0.548485i −1.44240 + 1.38545i 0 0 −0.527444 5.22047i 0 2.39833 1.80223i 0
110.1 0 1.46391 0.925722i −0.320823 1.97410i 0 0 0.0119040 0.591030i 0 1.28608 2.71035i 0
119.1 0 1.64291 + 0.548485i −1.44240 1.38545i 0 0 −0.527444 + 5.22047i 0 2.39833 + 1.80223i 0
128.1 0 0.346455 1.69705i 1.92104 + 0.556435i 0 0 4.15936 2.99643i 0 −2.75994 1.17590i 0
137.1 0 −1.72643 + 0.139372i −1.95958 0.400051i 0 0 −4.81030 + 1.08337i 0 2.96115 0.481234i 0
149.1 0 −1.19983 + 1.24916i −1.80690 0.857385i 0 0 1.81421 + 0.257455i 0 −0.120798 2.99757i 0
158.1 0 −1.59345 + 0.678906i 1.06893 + 1.69038i 0 0 3.10761 + 1.55200i 0 2.07817 2.16361i 0
167.1 0 −1.38468 + 1.04052i −1.99838 0.0805319i 0 0 3.92688 + 2.59536i 0 0.834652 2.88155i 0
176.1 0 1.70962 + 0.277840i −0.783933 1.83996i 0 0 1.45291 + 4.07679i 0 2.84561 + 0.950004i 0
197.1 0 0.742517 1.56482i −0.633336 1.89707i 0 0 −0.0345359 + 0.0318602i 0 −1.89734 2.32381i 0
206.1 0 1.72643 + 0.139372i 1.95958 0.400051i 0 0 −0.421909 + 1.87333i 0 2.96115 + 0.481234i 0
215.1 0 −1.59345 0.678906i 1.06893 1.69038i 0 0 3.10761 1.55200i 0 2.07817 + 2.16361i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 500.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
169.l odd 156 1 inner
507.x even 156 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.x.a 48
3.b odd 2 1 CM 507.2.x.a 48
169.l odd 156 1 inner 507.2.x.a 48
507.x even 156 1 inner 507.2.x.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.x.a 48 1.a even 1 1 trivial
507.2.x.a 48 3.b odd 2 1 CM
507.2.x.a 48 169.l odd 156 1 inner
507.2.x.a 48 507.x even 156 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$.