# Properties

 Label 570.2.n.a Level $570$ Weight $2$ Character orbit 570.n Analytic conductor $4.551$ Analytic rank $0$ Dimension $80$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 40 q^{4}+O(q^{10})$$ 80 * q + 40 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 40 q^{4} + 30 q^{15} - 40 q^{16} + 8 q^{19} + 8 q^{25} - 4 q^{30} + 48 q^{39} + 12 q^{45} - 128 q^{49} - 36 q^{54} + 12 q^{55} + 30 q^{60} - 24 q^{61} - 80 q^{64} + 4 q^{66} + 36 q^{70} + 16 q^{76} + 24 q^{79} + 32 q^{81} - 8 q^{85} - 54 q^{90} + 24 q^{91} - 60 q^{99}+O(q^{100})$$ 80 * q + 40 * q^4 + 30 * q^15 - 40 * q^16 + 8 * q^19 + 8 * q^25 - 4 * q^30 + 48 * q^39 + 12 * q^45 - 128 * q^49 - 36 * q^54 + 12 * q^55 + 30 * q^60 - 24 * q^61 - 80 * q^64 + 4 * q^66 + 36 * q^70 + 16 * q^76 + 24 * q^79 + 32 * q^81 - 8 * q^85 - 54 * q^90 + 24 * q^91 - 60 * 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1 −0.866025 0.500000i −1.69438 0.359253i 0.500000 + 0.866025i −0.161263 2.23025i 1.28775 + 1.15831i 1.18214i 1.00000i 2.74187 + 1.21743i −0.975465 + 2.01208i
179.2 −0.866025 0.500000i −1.68343 + 0.407506i 0.500000 + 0.866025i 2.16632 + 0.554116i 1.66165 + 0.488805i 1.83012i 1.00000i 2.66788 1.37202i −1.59903 1.56304i
179.3 −0.866025 0.500000i −1.66274 0.485072i 0.500000 + 0.866025i −1.14569 + 1.92026i 1.19744 + 1.25145i 4.59876i 1.00000i 2.52941 + 1.61310i 1.95233 1.09015i
179.4 −0.866025 0.500000i −1.53736 + 0.797834i 0.500000 + 0.866025i −2.22475 + 0.224660i 1.73031 + 0.0777329i 0.0202833i 1.00000i 1.72692 2.45311i 2.03902 + 0.917815i
179.5 −0.866025 0.500000i −1.25145 1.19744i 0.500000 + 0.866025i −1.09015 + 1.95233i 0.485072 + 1.66274i 4.59876i 1.00000i 0.132279 + 2.99708i 1.92026 1.14569i
179.6 −0.866025 0.500000i −1.15831 1.28775i 0.500000 + 0.866025i 2.01208 0.975465i 0.359253 + 1.69438i 1.18214i 1.00000i −0.316615 + 2.98325i −2.23025 0.161263i
179.7 −0.866025 0.500000i −0.802721 + 1.53481i 0.500000 + 0.866025i 0.971414 + 2.01404i 1.46258 0.927823i 3.47395i 1.00000i −1.71128 2.46405i 0.165751 2.22992i
179.8 −0.866025 0.500000i −0.540671 + 1.64550i 0.500000 + 0.866025i −0.636160 2.14367i 1.29099 1.15471i 4.14066i 1.00000i −2.41535 1.77935i −0.520902 + 2.17455i
179.9 −0.866025 0.500000i −0.488805 1.66165i 0.500000 + 0.866025i −1.56304 1.59903i −0.407506 + 1.68343i 1.83012i 1.00000i −2.52214 + 1.62444i 0.554116 + 2.16632i
179.10 −0.866025 0.500000i −0.105626 + 1.72883i 0.500000 + 0.866025i −0.264142 + 2.22041i 0.955888 1.44440i 2.53373i 1.00000i −2.97769 0.365218i 1.33896 1.79086i
179.11 −0.866025 0.500000i −0.0777329 1.73031i 0.500000 + 0.866025i 0.917815 + 2.03902i −0.797834 + 1.53736i 0.0202833i 1.00000i −2.98792 + 0.269003i 0.224660 2.22475i
179.12 −0.866025 0.500000i 0.457417 + 1.67056i 0.500000 + 0.866025i −1.92017 1.14583i 0.439145 1.67546i 4.60944i 1.00000i −2.58154 + 1.52829i 1.09000 + 1.95241i
179.13 −0.866025 0.500000i 0.813860 + 1.52893i 0.500000 + 0.866025i 0.951780 2.02339i 0.0596429 1.73102i 1.16784i 1.00000i −1.67527 + 2.48867i −1.83596 + 1.27642i
179.14 −0.866025 0.500000i 0.927823 1.46258i 0.500000 + 0.866025i −2.22992 + 0.165751i −1.53481 + 0.802721i 3.47395i 1.00000i −1.27829 2.71403i 2.01404 + 0.971414i
179.15 −0.866025 0.500000i 1.08728 + 1.34827i 0.500000 + 0.866025i 2.16634 + 0.554035i −0.267475 1.71127i 1.36390i 1.00000i −0.635658 + 2.93188i −1.59909 1.56298i
179.16 −0.866025 0.500000i 1.15471 1.29099i 0.500000 + 0.866025i 2.17455 0.520902i −1.64550 + 0.540671i 4.14066i 1.00000i −0.333286 2.98143i −2.14367 0.636160i
179.17 −0.866025 0.500000i 1.44440 0.955888i 0.500000 + 0.866025i −1.79086 + 1.33896i −1.72883 + 0.105626i 2.53373i 1.00000i 1.17255 2.76136i 2.22041 0.264142i
179.18 −0.866025 0.500000i 1.67546 0.439145i 0.500000 + 0.866025i 1.95241 + 1.09000i −1.67056 0.457417i 4.60944i 1.00000i 2.61430 1.47153i −1.14583 1.92017i
179.19 −0.866025 0.500000i 1.71127 + 0.267475i 0.500000 + 0.866025i −1.56298 1.59909i −1.34827 1.08728i 1.36390i 1.00000i 2.85691 + 0.915446i 0.554035 + 2.16634i
179.20 −0.866025 0.500000i 1.73102 0.0596429i 0.500000 + 0.866025i 1.27642 1.83596i −1.52893 0.813860i 1.16784i 1.00000i 2.99289 0.206487i −2.02339 + 0.951780i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.40 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
5.b even 2 1 inner
15.d odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner
95.h odd 6 1 inner
285.q even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.n.a 80
3.b odd 2 1 inner 570.2.n.a 80
5.b even 2 1 inner 570.2.n.a 80
15.d odd 2 1 inner 570.2.n.a 80
19.d odd 6 1 inner 570.2.n.a 80
57.f even 6 1 inner 570.2.n.a 80
95.h odd 6 1 inner 570.2.n.a 80
285.q even 6 1 inner 570.2.n.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.n.a 80 1.a even 1 1 trivial
570.2.n.a 80 3.b odd 2 1 inner
570.2.n.a 80 5.b even 2 1 inner
570.2.n.a 80 15.d odd 2 1 inner
570.2.n.a 80 19.d odd 6 1 inner
570.2.n.a 80 57.f even 6 1 inner
570.2.n.a 80 95.h odd 6 1 inner
570.2.n.a 80 285.q even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(570, [\chi])$$.