Properties

 Label 572.2.bq.a Level $572$ Weight $2$ Character orbit 572.bq Analytic conductor $4.567$ Analytic rank $0$ Dimension $112$ 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.bq (of order $$30$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112q + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q + 20q^{9} - 6q^{11} + 11q^{13} + 30q^{15} + 16q^{17} - 12q^{19} + 6q^{23} + 40q^{25} - 12q^{27} - 5q^{29} + 9q^{33} - 33q^{35} - 45q^{39} - 18q^{41} + 30q^{45} - 16q^{49} + 48q^{51} - 2q^{53} - 20q^{55} - 39q^{59} + 4q^{61} - 102q^{63} - 6q^{65} + 48q^{67} + 34q^{69} + 84q^{71} - 56q^{75} - 22q^{77} - 24q^{79} + 16q^{81} + 60q^{85} - 34q^{87} - 66q^{89} - 41q^{91} + 123q^{93} + 12q^{95} - 15q^{97} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −0.319495 + 3.03979i 0 −0.103194 0.0335297i 0 4.65033 0.488770i 0 −6.20380 1.31866i 0
49.2 0 −0.292873 + 2.78650i 0 −1.71678 0.557815i 0 −3.86952 + 0.406703i 0 −4.74434 1.00844i 0
49.3 0 −0.228377 + 2.17287i 0 −1.96627 0.638880i 0 0.214213 0.0225147i 0 −1.73475 0.368732i 0
49.4 0 −0.140590 + 1.33763i 0 2.78220 + 0.903990i 0 −3.77335 + 0.396595i 0 1.16496 + 0.247620i 0
49.5 0 −0.130188 + 1.23865i 0 3.26836 + 1.06196i 0 −0.124860 + 0.0131234i 0 1.41713 + 0.301220i 0
49.6 0 −0.118461 + 1.12708i 0 1.16420 + 0.378273i 0 3.19565 0.335877i 0 1.67816 + 0.356704i 0
49.7 0 −0.0476950 + 0.453788i 0 −3.72881 1.21156i 0 0.128901 0.0135480i 0 2.73079 + 0.580448i 0
49.8 0 0.0149789 0.142515i 0 −1.45545 0.472905i 0 −3.21049 + 0.337436i 0 2.91436 + 0.619466i 0
49.9 0 0.107949 1.02707i 0 −2.11328 0.686646i 0 0.696123 0.0731655i 0 1.89122 + 0.401992i 0
49.10 0 0.136254 1.29637i 0 2.36621 + 0.768828i 0 2.00645 0.210887i 0 1.27243 + 0.270463i 0
49.11 0 0.188862 1.79690i 0 0.283508 + 0.0921172i 0 0.331103 0.0348003i 0 −0.258739 0.0549966i 0
49.12 0 0.201901 1.92096i 0 1.20386 + 0.391159i 0 −4.99946 + 0.525464i 0 −0.714895 0.151956i 0
49.13 0 0.287109 2.73166i 0 −3.50557 1.13903i 0 2.82501 0.296921i 0 −4.44507 0.944828i 0
49.14 0 0.340624 3.24082i 0 3.52100 + 1.14404i 0 1.92989 0.202840i 0 −7.45247 1.58407i 0
69.1 0 −3.10684 0.660380i 0 −2.01179 + 2.76900i 0 0.661166 + 3.11054i 0 6.47574 + 2.88318i 0
69.2 0 −2.87954 0.612064i 0 0.464082 0.638753i 0 −0.0706912 0.332576i 0 5.17647 + 2.30471i 0
69.3 0 −2.15084 0.457176i 0 2.38726 3.28578i 0 −1.00599 4.73282i 0 1.67648 + 0.746415i 0
69.4 0 −1.52573 0.324303i 0 −1.60954 + 2.21535i 0 −0.454255 2.13710i 0 −0.517971 0.230616i 0
69.5 0 −1.41919 0.301659i 0 0.650916 0.895909i 0 0.629373 + 2.96097i 0 −0.817523 0.363985i 0
69.6 0 −1.18856 0.252636i 0 0.729880 1.00459i 0 0.643353 + 3.02674i 0 −1.39179 0.619666i 0
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 537.14 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.c even 5 1 inner
13.e even 6 1 inner
143.u even 30 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bq.a 112
11.c even 5 1 inner 572.2.bq.a 112
13.e even 6 1 inner 572.2.bq.a 112
143.u even 30 1 inner 572.2.bq.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bq.a 112 1.a even 1 1 trivial
572.2.bq.a 112 11.c even 5 1 inner
572.2.bq.a 112 13.e even 6 1 inner
572.2.bq.a 112 143.u even 30 1 inner

Hecke kernels

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