# Properties

 Label 546.2.q.j 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.

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

## 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 −1.68382 0.405895i −0.500000 0.866025i 0.465645i −1.19343 + 1.25528i 2.62938 + 0.293915i −1.00000 2.67050 + 1.36691i −0.403261 0.232823i
251.2 0.500000 0.866025i −1.63915 + 0.559644i −0.500000 0.866025i 2.94869i −0.334907 + 1.69936i −1.08009 2.41524i −1.00000 2.37360 1.83468i −2.55364 1.47434i
251.3 0.500000 0.866025i −1.48647 + 0.889052i −0.500000 0.866025i 0.655349i 0.0267080 + 1.73184i 0.402891 + 2.61490i −1.00000 1.41917 2.64309i −0.567549 0.327675i
251.4 0.500000 0.866025i −1.17531 + 1.27226i −0.500000 0.866025i 3.58412i 0.514158 + 1.65398i −2.54363 + 0.727985i −1.00000 −0.237306 2.99060i 3.10394 + 1.79206i
251.5 0.500000 0.866025i −0.702948 1.58299i −0.500000 0.866025i 3.28289i −1.72239 0.182725i −0.203837 + 2.63789i −1.00000 −2.01173 + 2.22552i 2.84307 + 1.64144i
251.6 0.500000 0.866025i −0.377999 1.69030i −0.500000 0.866025i 0.188901i −1.65284 0.517794i −1.93422 + 1.80522i −1.00000 −2.71423 + 1.27786i 0.163593 + 0.0944507i
251.7 0.500000 0.866025i 0.377999 + 1.69030i −0.500000 0.866025i 0.188901i 1.65284 + 0.517794i −2.53047 + 0.772472i −1.00000 −2.71423 + 1.27786i −0.163593 0.0944507i
251.8 0.500000 0.866025i 0.702948 + 1.58299i −0.500000 0.866025i 3.28289i 1.72239 + 0.182725i −2.38640 1.14242i −1.00000 −2.01173 + 2.22552i −2.84307 1.64144i
251.9 0.500000 0.866025i 1.17531 1.27226i −0.500000 0.866025i 3.58412i −0.514158 1.65398i −1.90227 + 1.83885i −1.00000 −0.237306 2.99060i −3.10394 1.79206i
251.10 0.500000 0.866025i 1.48647 0.889052i −0.500000 0.866025i 0.655349i −0.0267080 1.73184i −2.06312 1.65636i −1.00000 1.41917 2.64309i 0.567549 + 0.327675i
251.11 0.500000 0.866025i 1.63915 0.559644i −0.500000 0.866025i 2.94869i 0.334907 1.69936i 1.55161 + 2.14301i −1.00000 2.37360 1.83468i 2.55364 + 1.47434i
251.12 0.500000 0.866025i 1.68382 + 0.405895i −0.500000 0.866025i 0.465645i 1.19343 1.25528i 1.06015 2.42406i −1.00000 2.67050 + 1.36691i 0.403261 + 0.232823i
335.1 0.500000 + 0.866025i −1.68382 + 0.405895i −0.500000 + 0.866025i 0.465645i −1.19343 1.25528i 2.62938 0.293915i −1.00000 2.67050 1.36691i −0.403261 + 0.232823i
335.2 0.500000 + 0.866025i −1.63915 0.559644i −0.500000 + 0.866025i 2.94869i −0.334907 1.69936i −1.08009 + 2.41524i −1.00000 2.37360 + 1.83468i −2.55364 + 1.47434i
335.3 0.500000 + 0.866025i −1.48647 0.889052i −0.500000 + 0.866025i 0.655349i 0.0267080 1.73184i 0.402891 2.61490i −1.00000 1.41917 + 2.64309i −0.567549 + 0.327675i
335.4 0.500000 + 0.866025i −1.17531 1.27226i −0.500000 + 0.866025i 3.58412i 0.514158 1.65398i −2.54363 0.727985i −1.00000 −0.237306 + 2.99060i 3.10394 1.79206i
335.5 0.500000 + 0.866025i −0.702948 + 1.58299i −0.500000 + 0.866025i 3.28289i −1.72239 + 0.182725i −0.203837 2.63789i −1.00000 −2.01173 2.22552i 2.84307 1.64144i
335.6 0.500000 + 0.866025i −0.377999 + 1.69030i −0.500000 + 0.866025i 0.188901i −1.65284 + 0.517794i −1.93422 1.80522i −1.00000 −2.71423 1.27786i 0.163593 0.0944507i
335.7 0.500000 + 0.866025i 0.377999 1.69030i −0.500000 + 0.866025i 0.188901i 1.65284 0.517794i −2.53047 0.772472i −1.00000 −2.71423 1.27786i −0.163593 + 0.0944507i
335.8 0.500000 + 0.866025i 0.702948 1.58299i −0.500000 + 0.866025i 3.28289i 1.72239 0.182725i −2.38640 + 1.14242i −1.00000 −2.01173 2.22552i −2.84307 + 1.64144i
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.j yes 24
3.b odd 2 1 546.2.q.i 24
7.b odd 2 1 inner 546.2.q.j yes 24
13.e even 6 1 546.2.q.i 24
21.c even 2 1 546.2.q.i 24
39.h odd 6 1 inner 546.2.q.j yes 24
91.t odd 6 1 546.2.q.i 24
273.u even 6 1 inner 546.2.q.j yes 24

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

## Hecke kernels

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

 $$T_{5}^{12} + 33T_{5}^{10} + 366T_{5}^{8} + 1442T_{5}^{6} + 861T_{5}^{4} + 141T_{5}^{2} + 4$$ T5^12 + 33*T5^10 + 366*T5^8 + 1442*T5^6 + 861*T5^4 + 141*T5^2 + 4 $$T_{11}^{12} + 36 T_{11}^{10} - 96 T_{11}^{9} + 1080 T_{11}^{8} - 2160 T_{11}^{7} + 10464 T_{11}^{6} - 20736 T_{11}^{5} + 74304 T_{11}^{4} - 111744 T_{11}^{3} + 145152 T_{11}^{2} - 82944 T_{11} + 36864$$ T11^12 + 36*T11^10 - 96*T11^9 + 1080*T11^8 - 2160*T11^7 + 10464*T11^6 - 20736*T11^5 + 74304*T11^4 - 111744*T11^3 + 145152*T11^2 - 82944*T11 + 36864 $$T_{17}^{24} + 111 T_{17}^{22} + 8355 T_{17}^{20} + 322054 T_{17}^{18} + 8816949 T_{17}^{16} + 156438945 T_{17}^{14} + 2020732761 T_{17}^{12} + 16041014910 T_{17}^{10} + 87372493479 T_{17}^{8} + \cdots + 102627966736$$ T17^24 + 111*T17^22 + 8355*T17^20 + 322054*T17^18 + 8816949*T17^16 + 156438945*T17^14 + 2020732761*T17^12 + 16041014910*T17^10 + 87372493479*T17^8 + 180214153639*T17^6 + 266911702005*T17^4 + 197535033516*T17^2 + 102627966736