# Properties

 Label 507.2.k.k Level $507$ Weight $2$ Character orbit 507.k Analytic conductor $4.048$ Analytic rank $0$ Dimension $96$ CM no Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.k (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$24$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$96q + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q + 24q^{9} + 8q^{16} - 112q^{22} - 168q^{27} + 256q^{40} + 56q^{42} + 188q^{48} - 8q^{55} - 56q^{61} - 184q^{66} + 72q^{81} + 112q^{87} - 296q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1 −0.668092 + 2.49335i −1.55464 0.763602i −4.03841 2.33158i 0.624703 + 0.624703i 2.94257 3.36612i 1.62412 0.435181i 4.86096 4.86096i 1.83382 + 2.37425i −1.97496 + 1.14025i
80.2 −0.668092 + 2.49335i 0.116023 1.72816i −4.03841 2.33158i 0.624703 + 0.624703i 4.23140 + 1.44386i −1.62412 + 0.435181i 4.86096 4.86096i −2.97308 0.401012i −1.97496 + 1.14025i
80.3 −0.522394 + 1.94960i 0.152679 + 1.72531i −1.79600 1.03692i 1.72251 + 1.72251i −3.44342 0.603628i 3.00582 0.805406i 0.105393 0.105393i −2.95338 + 0.526837i −4.25804 + 2.45838i
80.4 −0.522394 + 1.94960i 1.41782 + 0.994878i −1.79600 1.03692i 1.72251 + 1.72251i −2.68028 + 2.24447i −3.00582 + 0.805406i 0.105393 0.105393i 1.02043 + 2.82112i −4.25804 + 2.45838i
80.5 −0.506604 + 1.89067i −1.69226 + 0.369140i −1.58594 0.915644i 1.04664 + 1.04664i 0.159382 3.38651i −4.33161 + 1.16065i −0.233508 + 0.233508i 2.72747 1.24936i −2.50908 + 1.44862i
80.6 −0.506604 + 1.89067i 1.16581 1.28097i −1.58594 0.915644i 1.04664 + 1.04664i 1.83128 + 2.85311i 4.33161 1.16065i −0.233508 + 0.233508i −0.281758 2.98674i −2.50908 + 1.44862i
80.7 −0.339800 + 1.26815i −0.228800 + 1.71687i 0.239309 + 0.138165i −2.12536 2.12536i −2.09951 0.873546i −2.82123 + 0.755945i −2.11323 + 2.11323i −2.89530 0.785641i 3.41747 1.97308i
80.8 −0.339800 + 1.26815i 1.60126 + 0.660289i 0.239309 + 0.138165i −2.12536 2.12536i −1.38145 + 1.80627i 2.82123 0.755945i −2.11323 + 2.11323i 2.12804 + 2.11458i 3.41747 1.97308i
80.9 −0.197759 + 0.738045i −1.54108 + 0.790622i 1.22645 + 0.708090i 0.996141 + 0.996141i −0.278754 1.29374i 2.46232 0.659775i −1.84572 + 1.84572i 1.74983 2.43682i −0.932193 + 0.538202i
80.10 −0.197759 + 0.738045i 1.45524 0.939300i 1.22645 + 0.708090i 0.996141 + 0.996141i 0.405460 + 1.25979i −2.46232 + 0.659775i −1.84572 + 1.84572i 1.23543 2.73381i −0.932193 + 0.538202i
80.11 −0.0912193 + 0.340435i −1.73179 0.0302056i 1.62448 + 0.937892i −2.45719 2.45719i 0.168255 0.586806i −1.12240 + 0.300747i −0.965906 + 0.965906i 2.99818 + 0.104619i 1.06066 0.612371i
80.12 −0.0912193 + 0.340435i 0.839735 1.51487i 1.62448 + 0.937892i −2.45719 2.45719i 0.439116 + 0.424061i 1.12240 0.300747i −0.965906 + 0.965906i −1.58969 2.54419i 1.06066 0.612371i
80.13 0.0912193 0.340435i −1.73179 0.0302056i 1.62448 + 0.937892i 2.45719 + 2.45719i −0.168255 + 0.586806i 1.12240 0.300747i 0.965906 0.965906i 2.99818 + 0.104619i 1.06066 0.612371i
80.14 0.0912193 0.340435i 0.839735 1.51487i 1.62448 + 0.937892i 2.45719 + 2.45719i −0.439116 0.424061i −1.12240 + 0.300747i 0.965906 0.965906i −1.58969 2.54419i 1.06066 0.612371i
80.15 0.197759 0.738045i −1.54108 + 0.790622i 1.22645 + 0.708090i −0.996141 0.996141i 0.278754 + 1.29374i −2.46232 + 0.659775i 1.84572 1.84572i 1.74983 2.43682i −0.932193 + 0.538202i
80.16 0.197759 0.738045i 1.45524 0.939300i 1.22645 + 0.708090i −0.996141 0.996141i −0.405460 1.25979i 2.46232 0.659775i 1.84572 1.84572i 1.23543 2.73381i −0.932193 + 0.538202i
80.17 0.339800 1.26815i −0.228800 + 1.71687i 0.239309 + 0.138165i 2.12536 + 2.12536i 2.09951 + 0.873546i 2.82123 0.755945i 2.11323 2.11323i −2.89530 0.785641i 3.41747 1.97308i
80.18 0.339800 1.26815i 1.60126 + 0.660289i 0.239309 + 0.138165i 2.12536 + 2.12536i 1.38145 1.80627i −2.82123 + 0.755945i 2.11323 2.11323i 2.12804 + 2.11458i 3.41747 1.97308i
80.19 0.506604 1.89067i −1.69226 + 0.369140i −1.58594 0.915644i −1.04664 1.04664i −0.159382 + 3.38651i 4.33161 1.16065i 0.233508 0.233508i 2.72747 1.24936i −2.50908 + 1.44862i
80.20 0.506604 1.89067i 1.16581 1.28097i −1.58594 0.915644i −1.04664 1.04664i −1.83128 2.85311i −4.33161 + 1.16065i 0.233508 0.233508i −0.281758 2.98674i −2.50908 + 1.44862i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 488.24 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.c even 3 1 inner
13.d odd 4 2 inner
13.e even 6 1 inner
13.f odd 12 2 inner
39.d odd 2 1 inner
39.f even 4 2 inner
39.h odd 6 1 inner
39.i odd 6 1 inner
39.k even 12 2 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{48} - \cdots$$ $$T_{5}^{24} + 272 T_{5}^{20} + 22390 T_{5}^{16} + 611594 T_{5}^{12} + 4403113 T_{5}^{8} + 10383698 T_{5}^{4} + 4826809$$ $$28\!\cdots\!69$$$$T_{7}^{24} -$$$$95\!\cdots\!52$$$$T_{7}^{20} +$$$$22\!\cdots\!99$$$$T_{7}^{16} -$$$$19\!\cdots\!41$$$$T_{7}^{12} +$$$$12\!\cdots\!04$$$$T_{7}^{8} -$$$$20\!\cdots\!19$$$$T_{7}^{4} +$$$$28\!\cdots\!61$$">$$T_{7}^{48} - \cdots$$