# Properties

 Label 570.2.bf.a Level $570$ Weight $2$ Character orbit 570.bf Analytic conductor $4.551$ Analytic rank $0$ Dimension $240$ 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.bf (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$40$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 30q^{15} + 48q^{19} + 12q^{25} - 48q^{39} + 36q^{45} - 72q^{46} + 216q^{49} - 180q^{51} + 36q^{54} + 60q^{55} + 6q^{60} - 72q^{61} + 120q^{64} - 156q^{66} + 48q^{79} - 120q^{81} - 108q^{84} - 48q^{85} - 36q^{90} - 24q^{91} - 216q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.984808 + 0.173648i −1.70539 0.302753i 0.939693 0.342020i 0.247201 2.22236i 1.73205 + 0.00201678i 4.20953 + 2.43037i −0.866025 + 0.500000i 2.81668 + 1.03262i 0.142463 + 2.23153i
29.2 −0.984808 + 0.173648i −1.64517 + 0.541678i 0.939693 0.342020i 2.21549 0.302666i 1.52611 0.819129i −2.76521 1.59649i −0.866025 + 0.500000i 2.41317 1.78230i −2.12927 + 0.682783i
29.3 −0.984808 + 0.173648i −1.62527 0.598755i 0.939693 0.342020i −2.17153 + 0.533326i 1.70455 + 0.307434i 0.260757 + 0.150548i −0.866025 + 0.500000i 2.28299 + 1.94627i 2.04593 0.902307i
29.4 −0.984808 + 0.173648i −1.37297 + 1.05591i 0.939693 0.342020i 1.98695 + 1.02568i 1.16876 1.27828i 3.69762 + 2.13482i −0.866025 + 0.500000i 0.770112 2.89947i −2.13487 0.665072i
29.5 −0.984808 + 0.173648i −1.23853 1.21080i 0.939693 0.342020i 1.86795 1.22913i 1.42997 + 0.977341i −1.22822 0.709111i −0.866025 + 0.500000i 0.0679076 + 2.99923i −1.62614 + 1.53482i
29.6 −0.984808 + 0.173648i −1.20820 + 1.24107i 0.939693 0.342020i −1.35867 1.77595i 0.974336 1.43202i −1.28021 0.739129i −0.866025 + 0.500000i −0.0805041 2.99892i 1.64642 + 1.51304i
29.7 −0.984808 + 0.173648i −1.03267 + 1.39054i 0.939693 0.342020i −1.21793 + 1.87527i 0.775516 1.54873i −1.48423 0.856923i −0.866025 + 0.500000i −0.867188 2.87193i 0.873794 2.05827i
29.8 −0.984808 + 0.173648i −0.977341 1.42997i 0.939693 0.342020i 0.640862 + 2.14226i 1.21080 + 1.23853i 1.22822 + 0.709111i −0.866025 + 0.500000i −1.08961 + 2.79513i −1.00313 1.99843i
29.9 −0.984808 + 0.173648i −0.307434 1.70455i 0.939693 0.342020i −1.32068 1.80439i 0.598755 + 1.62527i −0.260757 0.150548i −0.866025 + 0.500000i −2.81097 + 1.04807i 1.61394 + 1.54764i
29.10 −0.984808 + 0.173648i −0.00201678 1.73205i 0.939693 0.342020i −1.23914 + 1.86133i 0.302753 + 1.70539i −4.20953 2.43037i −0.866025 + 0.500000i −2.99999 + 0.00698632i 0.897098 2.04822i
29.11 −0.984808 + 0.173648i 0.144472 + 1.72602i 0.939693 0.342020i −2.21146 0.330832i −0.441997 1.67471i 2.42894 + 1.40235i −0.866025 + 0.500000i −2.95826 + 0.498722i 2.23531 0.0582100i
29.12 −0.984808 + 0.173648i 0.438663 + 1.67558i 0.939693 0.342020i 2.09802 0.773495i −0.722960 1.57395i 1.27073 + 0.733656i −0.866025 + 0.500000i −2.61515 + 1.47003i −1.93183 + 1.12606i
29.13 −0.984808 + 0.173648i 0.819129 1.52611i 0.939693 0.342020i 1.50261 + 1.65594i −0.541678 + 1.64517i 2.76521 + 1.59649i −0.866025 + 0.500000i −1.65805 2.50017i −1.76734 1.36986i
29.14 −0.984808 + 0.173648i 0.917943 + 1.46880i 0.939693 0.342020i 0.501847 2.17902i −1.15905 1.28709i −3.34801 1.93297i −0.866025 + 0.500000i −1.31476 + 2.69655i −0.115839 + 2.23307i
29.15 −0.984808 + 0.173648i 1.27828 1.16876i 0.939693 0.342020i 2.18139 + 0.491467i −1.05591 + 1.37297i −3.69762 2.13482i −0.866025 + 0.500000i 0.268009 2.98800i −2.23359 0.105206i
29.16 −0.984808 + 0.173648i 1.28709 + 1.15905i 0.939693 0.342020i −1.01621 + 1.99181i −1.46880 0.917943i 3.34801 + 1.93297i −0.866025 + 0.500000i 0.313195 + 2.98361i 0.654901 2.13801i
29.17 −0.984808 + 0.173648i 1.43202 0.974336i 0.939693 0.342020i −2.18236 + 0.487123i −1.24107 + 1.20820i 1.28021 + 0.739129i −0.866025 + 0.500000i 1.10134 2.79053i 2.06462 0.858686i
29.18 −0.984808 + 0.173648i 1.54873 0.775516i 0.939693 0.342020i 0.272408 2.21941i −1.39054 + 1.03267i 1.48423 + 0.856923i −0.866025 + 0.500000i 1.79715 2.40214i 0.117127 + 2.23300i
29.19 −0.984808 + 0.173648i 1.57395 + 0.722960i 0.939693 0.342020i 1.10999 + 1.94112i −1.67558 0.438663i −1.27073 0.733656i −0.866025 + 0.500000i 1.95466 + 2.27581i −1.43019 1.71888i
29.20 −0.984808 + 0.173648i 1.67471 + 0.441997i 0.939693 0.342020i −1.90673 1.16807i −1.72602 0.144472i −2.42894 1.40235i −0.866025 + 0.500000i 2.60928 + 1.48043i 2.08060 + 0.819221i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 509.40 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.f odd 18 1 inner
57.j even 18 1 inner
95.o odd 18 1 inner
285.bf even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bf.a 240
3.b odd 2 1 inner 570.2.bf.a 240
5.b even 2 1 inner 570.2.bf.a 240
15.d odd 2 1 inner 570.2.bf.a 240
19.f odd 18 1 inner 570.2.bf.a 240
57.j even 18 1 inner 570.2.bf.a 240
95.o odd 18 1 inner 570.2.bf.a 240
285.bf even 18 1 inner 570.2.bf.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bf.a 240 1.a even 1 1 trivial
570.2.bf.a 240 3.b odd 2 1 inner
570.2.bf.a 240 5.b even 2 1 inner
570.2.bf.a 240 15.d odd 2 1 inner
570.2.bf.a 240 19.f odd 18 1 inner
570.2.bf.a 240 57.j even 18 1 inner
570.2.bf.a 240 95.o odd 18 1 inner
570.2.bf.a 240 285.bf even 18 1 inner

## Hecke kernels

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