# Properties

 Label 572.2.bc.a Level $572$ Weight $2$ Character orbit 572.bc Analytic conductor $4.567$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.bc (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 28q^{9} + 4q^{11} + 8q^{15} - 12q^{23} - 24q^{27} - 4q^{31} - 10q^{33} - 12q^{37} - 64q^{45} - 8q^{47} + 40q^{53} + 22q^{55} + 48q^{59} - 36q^{67} - 48q^{71} + 120q^{75} + 28q^{81} + 28q^{89} + 36q^{91} + 20q^{93} - 68q^{97} - 44q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1 0 −1.56324 2.70762i 0 0.477789 0.477789i 0 −1.02740 3.83429i 0 −3.38745 + 5.86724i 0
197.2 0 −1.56324 2.70762i 0 0.477789 0.477789i 0 1.02740 + 3.83429i 0 −3.38745 + 5.86724i 0
197.3 0 −1.02435 1.77422i 0 −1.58636 + 1.58636i 0 −0.799284 2.98297i 0 −0.598575 + 1.03676i 0
197.4 0 −1.02435 1.77422i 0 −1.58636 + 1.58636i 0 0.799284 + 2.98297i 0 −0.598575 + 1.03676i 0
197.5 0 −0.425370 0.736762i 0 2.35532 2.35532i 0 −0.337440 1.25934i 0 1.13812 1.97128i 0
197.6 0 −0.425370 0.736762i 0 2.35532 2.35532i 0 0.337440 + 1.25934i 0 1.13812 1.97128i 0
197.7 0 −0.308749 0.534769i 0 −1.59856 + 1.59856i 0 −0.412876 1.54087i 0 1.30935 2.26786i 0
197.8 0 −0.308749 0.534769i 0 −1.59856 + 1.59856i 0 0.412876 + 1.54087i 0 1.30935 2.26786i 0
197.9 0 0.591694 + 1.02484i 0 0.335170 0.335170i 0 −0.719487 2.68516i 0 0.799795 1.38529i 0
197.10 0 0.591694 + 1.02484i 0 0.335170 0.335170i 0 0.719487 + 2.68516i 0 0.799795 1.38529i 0
197.11 0 1.26462 + 2.19039i 0 −2.02978 + 2.02978i 0 −0.531478 1.98350i 0 −1.69853 + 2.94193i 0
197.12 0 1.26462 + 2.19039i 0 −2.02978 + 2.02978i 0 0.531478 + 1.98350i 0 −1.69853 + 2.94193i 0
197.13 0 1.46539 + 2.53814i 0 2.04642 2.04642i 0 −1.22814 4.58348i 0 −2.79476 + 4.84067i 0
197.14 0 1.46539 + 2.53814i 0 2.04642 2.04642i 0 1.22814 + 4.58348i 0 −2.79476 + 4.84067i 0
241.1 0 −1.56324 + 2.70762i 0 0.477789 + 0.477789i 0 −1.02740 + 3.83429i 0 −3.38745 5.86724i 0
241.2 0 −1.56324 + 2.70762i 0 0.477789 + 0.477789i 0 1.02740 3.83429i 0 −3.38745 5.86724i 0
241.3 0 −1.02435 + 1.77422i 0 −1.58636 1.58636i 0 −0.799284 + 2.98297i 0 −0.598575 1.03676i 0
241.4 0 −1.02435 + 1.77422i 0 −1.58636 1.58636i 0 0.799284 2.98297i 0 −0.598575 1.03676i 0
241.5 0 −0.425370 + 0.736762i 0 2.35532 + 2.35532i 0 −0.337440 + 1.25934i 0 1.13812 + 1.97128i 0
241.6 0 −0.425370 + 0.736762i 0 2.35532 + 2.35532i 0 0.337440 1.25934i 0 1.13812 + 1.97128i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 505.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
13.f odd 12 1 inner
143.o even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bc.a 56
11.b odd 2 1 inner 572.2.bc.a 56
13.f odd 12 1 inner 572.2.bc.a 56
143.o even 12 1 inner 572.2.bc.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bc.a 56 1.a even 1 1 trivial
572.2.bc.a 56 11.b odd 2 1 inner
572.2.bc.a 56 13.f odd 12 1 inner
572.2.bc.a 56 143.o even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(572, [\chi])$$.