# Properties

 Label 546.2.q.i Level $546$ Weight $2$ Character orbit 546.q Analytic conductor $4.360$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{2} - 12q^{4} - 18q^{7} + 24q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 12q^{2} - 12q^{4} - 18q^{7} + 24q^{8} - 12q^{9} + 6q^{15} - 12q^{16} + 6q^{18} - 12q^{21} + 36q^{23} - 12q^{25} + 18q^{28} - 12q^{30} - 12q^{32} + 36q^{35} + 6q^{36} + 36q^{37} + 12q^{39} + 24q^{42} - 42q^{43} - 36q^{46} + 6q^{50} + 12q^{51} - 18q^{56} + 36q^{57} + 6q^{60} + 24q^{63} + 24q^{64} - 66q^{65} - 54q^{67} - 48q^{71} - 12q^{72} - 36q^{74} - 42q^{78} - 48q^{79} + 84q^{81} - 12q^{84} + 84q^{86} + 36q^{91} - 18q^{93} + 36q^{95} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1 −0.500000 + 0.866025i −1.72239 + 0.182725i −0.500000 0.866025i 3.28289i 0.702948 1.58299i −0.203837 + 2.63789i 1.00000 2.93322 0.629446i 2.84307 + 1.64144i
251.2 −0.500000 + 0.866025i −1.65284 + 0.517794i −0.500000 0.866025i 0.188901i 0.377999 1.69030i −1.93422 + 1.80522i 1.00000 2.46378 1.71166i 0.163593 + 0.0944507i
251.3 −0.500000 + 0.866025i −1.19343 1.25528i −0.500000 0.866025i 0.465645i 1.68382 0.405895i 2.62938 + 0.293915i 1.00000 −0.151472 + 2.99617i −0.403261 0.232823i
251.4 −0.500000 + 0.866025i −0.514158 + 1.65398i −0.500000 0.866025i 3.58412i −1.17531 1.27226i −1.90227 + 1.83885i 1.00000 −2.47128 1.70081i −3.10394 1.79206i
251.5 −0.500000 + 0.866025i −0.334907 1.69936i −0.500000 0.866025i 2.94869i 1.63915 + 0.559644i −1.08009 2.41524i 1.00000 −2.77567 + 1.13826i −2.55364 1.47434i
251.6 −0.500000 + 0.866025i −0.0267080 + 1.73184i −0.500000 0.866025i 0.655349i −1.48647 0.889052i −2.06312 1.65636i 1.00000 −2.99857 0.0925084i 0.567549 + 0.327675i
251.7 −0.500000 + 0.866025i 0.0267080 1.73184i −0.500000 0.866025i 0.655349i 1.48647 + 0.889052i 0.402891 + 2.61490i 1.00000 −2.99857 0.0925084i −0.567549 0.327675i
251.8 −0.500000 + 0.866025i 0.334907 + 1.69936i −0.500000 0.866025i 2.94869i −1.63915 0.559644i 1.55161 + 2.14301i 1.00000 −2.77567 + 1.13826i 2.55364 + 1.47434i
251.9 −0.500000 + 0.866025i 0.514158 1.65398i −0.500000 0.866025i 3.58412i 1.17531 + 1.27226i −2.54363 + 0.727985i 1.00000 −2.47128 1.70081i 3.10394 + 1.79206i
251.10 −0.500000 + 0.866025i 1.19343 + 1.25528i −0.500000 0.866025i 0.465645i −1.68382 + 0.405895i 1.06015 2.42406i 1.00000 −0.151472 + 2.99617i 0.403261 + 0.232823i
251.11 −0.500000 + 0.866025i 1.65284 0.517794i −0.500000 0.866025i 0.188901i −0.377999 + 1.69030i −2.53047 + 0.772472i 1.00000 2.46378 1.71166i −0.163593 0.0944507i
251.12 −0.500000 + 0.866025i 1.72239 0.182725i −0.500000 0.866025i 3.28289i −0.702948 + 1.58299i −2.38640 1.14242i 1.00000 2.93322 0.629446i −2.84307 1.64144i
335.1 −0.500000 0.866025i −1.72239 0.182725i −0.500000 + 0.866025i 3.28289i 0.702948 + 1.58299i −0.203837 2.63789i 1.00000 2.93322 + 0.629446i 2.84307 1.64144i
335.2 −0.500000 0.866025i −1.65284 0.517794i −0.500000 + 0.866025i 0.188901i 0.377999 + 1.69030i −1.93422 1.80522i 1.00000 2.46378 + 1.71166i 0.163593 0.0944507i
335.3 −0.500000 0.866025i −1.19343 + 1.25528i −0.500000 + 0.866025i 0.465645i 1.68382 + 0.405895i 2.62938 0.293915i 1.00000 −0.151472 2.99617i −0.403261 + 0.232823i
335.4 −0.500000 0.866025i −0.514158 1.65398i −0.500000 + 0.866025i 3.58412i −1.17531 + 1.27226i −1.90227 1.83885i 1.00000 −2.47128 + 1.70081i −3.10394 + 1.79206i
335.5 −0.500000 0.866025i −0.334907 + 1.69936i −0.500000 + 0.866025i 2.94869i 1.63915 0.559644i −1.08009 + 2.41524i 1.00000 −2.77567 1.13826i −2.55364 + 1.47434i
335.6 −0.500000 0.866025i −0.0267080 1.73184i −0.500000 + 0.866025i 0.655349i −1.48647 + 0.889052i −2.06312 + 1.65636i 1.00000 −2.99857 + 0.0925084i 0.567549 0.327675i
335.7 −0.500000 0.866025i 0.0267080 + 1.73184i −0.500000 + 0.866025i 0.655349i 1.48647 0.889052i 0.402891 2.61490i 1.00000 −2.99857 + 0.0925084i −0.567549 + 0.327675i
335.8 −0.500000 0.866025i 0.334907 1.69936i −0.500000 + 0.866025i 2.94869i −1.63915 + 0.559644i 1.55161 2.14301i 1.00000 −2.77567 1.13826i 2.55364 1.47434i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 335.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
39.h odd 6 1 inner
273.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.q.i 24
3.b odd 2 1 546.2.q.j yes 24
7.b odd 2 1 inner 546.2.q.i 24
13.e even 6 1 546.2.q.j yes 24
21.c even 2 1 546.2.q.j yes 24
39.h odd 6 1 inner 546.2.q.i 24
91.t odd 6 1 546.2.q.j yes 24
273.u even 6 1 inner 546.2.q.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.i 24 1.a even 1 1 trivial
546.2.q.i 24 7.b odd 2 1 inner
546.2.q.i 24 39.h odd 6 1 inner
546.2.q.i 24 273.u even 6 1 inner
546.2.q.j yes 24 3.b odd 2 1
546.2.q.j yes 24 13.e even 6 1
546.2.q.j yes 24 21.c even 2 1
546.2.q.j yes 24 91.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{12} + 33 T_{5}^{10} + 366 T_{5}^{8} + 1442 T_{5}^{6} + 861 T_{5}^{4} + 141 T_{5}^{2} + 4$$ $$T_{11}^{12} + \cdots$$ $$T_{17}^{24} + \cdots$$