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 $4$

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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.

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

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 −1.68382 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:
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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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

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