# Properties

 Label 576.2.y.a Level $576$ Weight $2$ Character orbit 576.y Analytic conductor $4.599$ Analytic rank $0$ Dimension $88$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.y (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 144) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q + 4q^{3} - 6q^{5} + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q + 4q^{3} - 6q^{5} + 4q^{7} + 6q^{11} - 2q^{13} + 8q^{19} + 2q^{21} + 12q^{23} + 16q^{27} - 6q^{29} - 8q^{33} - 8q^{37} + 32q^{39} + 2q^{43} + 6q^{45} - 24q^{49} + 12q^{51} + 16q^{55} + 42q^{59} - 2q^{61} - 12q^{65} + 2q^{67} - 10q^{69} + 56q^{75} - 6q^{77} - 8q^{81} - 54q^{83} + 8q^{85} - 48q^{87} - 20q^{91} - 34q^{93} - 4q^{97} - 46q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −1.72804 + 0.117756i 0 −0.961558 + 3.58858i 0 1.29216 2.23809i 0 2.97227 0.406975i 0
47.2 0 −1.68223 0.412428i 0 0.0776974 0.289971i 0 0.374023 0.647827i 0 2.65981 + 1.38760i 0
47.3 0 −1.53620 0.800065i 0 1.00059 3.73424i 0 1.68236 2.91393i 0 1.71979 + 2.45811i 0
47.4 0 −1.38573 + 1.03912i 0 0.310357 1.15827i 0 −0.356047 + 0.616691i 0 0.840471 2.87986i 0
47.5 0 −1.33407 + 1.10465i 0 0.178044 0.664471i 0 −0.645693 + 1.11837i 0 0.559489 2.94737i 0
47.6 0 −1.06447 1.36635i 0 0.619079 2.31044i 0 −2.51270 + 4.35213i 0 −0.733803 + 2.90887i 0
47.7 0 −1.05993 1.36987i 0 −0.746784 + 2.78704i 0 −1.16672 + 2.02082i 0 −0.753089 + 2.90394i 0
47.8 0 −0.819834 + 1.52574i 0 −0.206232 + 0.769670i 0 2.17574 3.76849i 0 −1.65574 2.50170i 0
47.9 0 −0.632098 + 1.61259i 0 −1.02195 + 3.81396i 0 −1.46715 + 2.54117i 0 −2.20090 2.03863i 0
47.10 0 −0.517369 1.65298i 0 −0.521033 + 1.94452i 0 0.322227 0.558114i 0 −2.46466 + 1.71040i 0
47.11 0 0.256373 1.71297i 0 0.546024 2.03779i 0 0.0638076 0.110518i 0 −2.86855 0.878321i 0
47.12 0 0.313101 + 1.70352i 0 0.752772 2.80938i 0 1.02581 1.77675i 0 −2.80394 + 1.06674i 0
47.13 0 0.377192 1.69048i 0 0.282421 1.05401i 0 1.93586 3.35301i 0 −2.71545 1.27527i 0
47.14 0 0.427367 + 1.67850i 0 −0.0458174 + 0.170993i 0 −1.17432 + 2.03397i 0 −2.63471 + 1.43467i 0
47.15 0 0.692394 1.58764i 0 −0.759273 + 2.83365i 0 −1.41719 + 2.45465i 0 −2.04118 2.19854i 0
47.16 0 1.21512 + 1.23430i 0 −0.726024 + 2.70956i 0 0.00424642 0.00735502i 0 −0.0469689 + 2.99963i 0
47.17 0 1.32549 + 1.11494i 0 0.323102 1.20583i 0 −0.140266 + 0.242948i 0 0.513832 + 2.95567i 0
47.18 0 1.49488 0.874828i 0 0.473330 1.76649i 0 −1.40613 + 2.43549i 0 1.46935 2.61553i 0
47.19 0 1.59321 0.679468i 0 −0.642590 + 2.39818i 0 1.93190 3.34616i 0 2.07665 2.16507i 0
47.20 0 1.65787 0.501476i 0 0.997280 3.72190i 0 0.481387 0.833787i 0 2.49704 1.66276i 0
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 527.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
16.f odd 4 1 inner
144.u even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.y.a 88
3.b odd 2 1 1728.2.z.a 88
4.b odd 2 1 144.2.u.a 88
9.c even 3 1 1728.2.z.a 88
9.d odd 6 1 inner 576.2.y.a 88
12.b even 2 1 432.2.v.a 88
16.e even 4 1 144.2.u.a 88
16.f odd 4 1 inner 576.2.y.a 88
36.f odd 6 1 432.2.v.a 88
36.h even 6 1 144.2.u.a 88
48.i odd 4 1 432.2.v.a 88
48.k even 4 1 1728.2.z.a 88
144.u even 12 1 inner 576.2.y.a 88
144.v odd 12 1 1728.2.z.a 88
144.w odd 12 1 144.2.u.a 88
144.x even 12 1 432.2.v.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.u.a 88 4.b odd 2 1
144.2.u.a 88 16.e even 4 1
144.2.u.a 88 36.h even 6 1
144.2.u.a 88 144.w odd 12 1
432.2.v.a 88 12.b even 2 1
432.2.v.a 88 36.f odd 6 1
432.2.v.a 88 48.i odd 4 1
432.2.v.a 88 144.x even 12 1
576.2.y.a 88 1.a even 1 1 trivial
576.2.y.a 88 9.d odd 6 1 inner
576.2.y.a 88 16.f odd 4 1 inner
576.2.y.a 88 144.u even 12 1 inner
1728.2.z.a 88 3.b odd 2 1
1728.2.z.a 88 9.c even 3 1
1728.2.z.a 88 48.k even 4 1
1728.2.z.a 88 144.v odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(576, [\chi])$$.