# Properties

 Label 507.2.f.g Level $507$ Weight $2$ Character orbit 507.f Analytic conductor $4.048$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 24q^{9} - 8q^{16} + 112q^{22} - 84q^{27} + 128q^{40} - 56q^{42} - 188q^{48} + 8q^{55} + 56q^{61} - 92q^{66} - 72q^{81} - 112q^{87} + 296q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1 −1.82526 1.82526i 1.43862 0.964559i 4.66316i 0.624703 + 0.624703i −4.38643 0.865285i −1.18894 1.18894i 4.86096 4.86096i 1.13925 2.77527i 2.28049i
239.2 −1.82526 1.82526i 1.43862 + 0.964559i 4.66316i 0.624703 + 0.624703i −0.865285 4.38643i 1.18894 + 1.18894i 4.86096 4.86096i 1.13925 + 2.77527i 2.28049i
239.3 −1.42721 1.42721i −1.57050 0.730430i 2.07385i 1.72251 + 1.72251i 1.19895 + 3.28391i −2.20041 2.20041i 0.105393 0.105393i 1.93294 + 2.29428i 4.91676i
239.4 −1.42721 1.42721i −1.57050 + 0.730430i 2.07385i 1.72251 + 1.72251i 3.28391 + 1.19895i 2.20041 + 2.20041i 0.105393 0.105393i 1.93294 2.29428i 4.91676i
239.5 −1.38407 1.38407i 0.526444 1.65011i 1.83129i 1.04664 + 1.04664i −3.01250 + 1.55523i 3.17096 + 3.17096i −0.233508 + 0.233508i −2.44571 1.73738i 2.89724i
239.6 −1.38407 1.38407i 0.526444 + 1.65011i 1.83129i 1.04664 + 1.04664i 1.55523 3.01250i −3.17096 3.17096i −0.233508 + 0.233508i −2.44571 + 1.73738i 2.89724i
239.7 −0.928351 0.928351i −1.37245 1.05658i 0.276330i −2.12536 2.12536i 0.293240 + 2.25500i 2.06528 + 2.06528i −2.11323 + 2.11323i 0.767265 + 2.90022i 3.94616i
239.8 −0.928351 0.928351i −1.37245 + 1.05658i 0.276330i −2.12536 2.12536i 2.25500 + 0.293240i −2.06528 2.06528i −2.11323 + 2.11323i 0.767265 2.90022i 3.94616i
239.9 −0.540287 0.540287i 0.0858391 1.72992i 1.41618i 0.996141 + 0.996141i −0.981032 + 0.888277i −1.80254 1.80254i −1.84572 + 1.84572i −2.98526 0.296990i 1.07640i
239.10 −0.540287 0.540287i 0.0858391 + 1.72992i 1.41618i 0.996141 + 0.996141i 0.888277 0.981032i 1.80254 + 1.80254i −1.84572 + 1.84572i −2.98526 + 0.296990i 1.07640i
239.11 −0.249216 0.249216i 0.892053 1.48467i 1.87578i −2.45719 2.45719i −0.592316 + 0.147689i 0.821655 + 0.821655i −0.965906 + 0.965906i −1.40848 2.64881i 1.22474i
239.12 −0.249216 0.249216i 0.892053 + 1.48467i 1.87578i −2.45719 2.45719i 0.147689 0.592316i −0.821655 0.821655i −0.965906 + 0.965906i −1.40848 + 2.64881i 1.22474i
239.13 0.249216 + 0.249216i 0.892053 1.48467i 1.87578i 2.45719 + 2.45719i 0.592316 0.147689i −0.821655 0.821655i 0.965906 0.965906i −1.40848 2.64881i 1.22474i
239.14 0.249216 + 0.249216i 0.892053 + 1.48467i 1.87578i 2.45719 + 2.45719i −0.147689 + 0.592316i 0.821655 + 0.821655i 0.965906 0.965906i −1.40848 + 2.64881i 1.22474i
239.15 0.540287 + 0.540287i 0.0858391 1.72992i 1.41618i −0.996141 0.996141i 0.981032 0.888277i 1.80254 + 1.80254i 1.84572 1.84572i −2.98526 0.296990i 1.07640i
239.16 0.540287 + 0.540287i 0.0858391 + 1.72992i 1.41618i −0.996141 0.996141i −0.888277 + 0.981032i −1.80254 1.80254i 1.84572 1.84572i −2.98526 + 0.296990i 1.07640i
239.17 0.928351 + 0.928351i −1.37245 1.05658i 0.276330i 2.12536 + 2.12536i −0.293240 2.25500i −2.06528 2.06528i 2.11323 2.11323i 0.767265 + 2.90022i 3.94616i
239.18 0.928351 + 0.928351i −1.37245 + 1.05658i 0.276330i 2.12536 + 2.12536i −2.25500 0.293240i 2.06528 + 2.06528i 2.11323 2.11323i 0.767265 2.90022i 3.94616i
239.19 1.38407 + 1.38407i 0.526444 1.65011i 1.83129i −1.04664 1.04664i 3.01250 1.55523i −3.17096 3.17096i 0.233508 0.233508i −2.44571 1.73738i 2.89724i
239.20 1.38407 + 1.38407i 0.526444 + 1.65011i 1.83129i −1.04664 1.04664i −1.55523 + 3.01250i 3.17096 + 3.17096i 0.233508 0.233508i −2.44571 + 1.73738i 2.89724i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 437.24 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 inner
13.b even 2 1 inner
13.d odd 4 2 inner
39.d odd 2 1 inner
39.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.g 48
3.b odd 2 1 inner 507.2.f.g 48
13.b even 2 1 inner 507.2.f.g 48
13.c even 3 2 507.2.k.k 96
13.d odd 4 2 inner 507.2.f.g 48
13.e even 6 2 507.2.k.k 96
13.f odd 12 4 507.2.k.k 96
39.d odd 2 1 inner 507.2.f.g 48
39.f even 4 2 inner 507.2.f.g 48
39.h odd 6 2 507.2.k.k 96
39.i odd 6 2 507.2.k.k 96
39.k even 12 4 507.2.k.k 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.f.g 48 1.a even 1 1 trivial
507.2.f.g 48 3.b odd 2 1 inner
507.2.f.g 48 13.b even 2 1 inner
507.2.f.g 48 13.d odd 4 2 inner
507.2.f.g 48 39.d odd 2 1 inner
507.2.f.g 48 39.f even 4 2 inner
507.2.k.k 96 13.c even 3 2
507.2.k.k 96 13.e even 6 2
507.2.k.k 96 13.f odd 12 4
507.2.k.k 96 39.h odd 6 2
507.2.k.k 96 39.i odd 6 2
507.2.k.k 96 39.k even 12 4

## Hecke kernels

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

 $$T_{2}^{24} + 79 T_{2}^{20} + 1885 T_{2}^{16} + 16327 T_{2}^{12} + 37725 T_{2}^{8} + 11531 T_{2}^{4} + 169$$ $$T_{5}^{24} + 272 T_{5}^{20} + 22390 T_{5}^{16} + 611594 T_{5}^{12} + 4403113 T_{5}^{8} + 10383698 T_{5}^{4} + 4826809$$ $$T_{7}^{24} + 623 T_{7}^{20} + 104321 T_{7}^{16} + 6865831 T_{7}^{12} + 175807737 T_{7}^{8} + 1229789799 T_{7}^{4} + 1698181681$$