Properties

Label 441.2.f.h
Level $441$
Weight $2$
Character orbit 441.f
Analytic conductor $3.521$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 4q^{2} - 12q^{4} - 24q^{8} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 4q^{2} - 12q^{4} - 24q^{8} + 8q^{9} + 20q^{11} + 4q^{15} - 12q^{16} + 4q^{18} + 32q^{23} - 12q^{25} + 16q^{29} + 48q^{32} - 4q^{36} + 24q^{37} + 32q^{39} - 112q^{44} - 48q^{46} - 4q^{50} - 56q^{51} - 64q^{53} - 12q^{57} - 88q^{60} + 96q^{64} + 60q^{65} - 12q^{67} - 112q^{71} + 168q^{72} + 68q^{74} - 60q^{78} + 12q^{79} + 80q^{81} + 12q^{85} + 76q^{86} + 16q^{92} - 80q^{93} + 64q^{95} + 20q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1 −1.08816 1.88474i −1.68791 0.388551i −1.36816 + 2.36973i 0.634145 1.09837i 1.10439 + 3.60407i 0 1.60248 2.69806 + 1.31167i −2.76019
148.2 −1.08816 1.88474i 1.68791 + 0.388551i −1.36816 + 2.36973i −0.634145 + 1.09837i −1.10439 3.60407i 0 1.60248 2.69806 + 1.31167i 2.76019
148.3 −0.649936 1.12572i −0.0514049 1.73129i 0.155166 0.268756i 1.76292 3.05347i −1.91554 + 1.18309i 0 −3.00314 −2.99472 + 0.177994i −4.58314
148.4 −0.649936 1.12572i 0.0514049 + 1.73129i 0.155166 0.268756i −1.76292 + 3.05347i 1.91554 1.18309i 0 −3.00314 −2.99472 + 0.177994i 4.58314
148.5 −0.0341870 0.0592136i −1.69514 0.355671i 0.997662 1.72800i −1.33190 + 2.30691i 0.0368912 + 0.112535i 0 −0.273176 2.74700 + 1.20582i 0.182134
148.6 −0.0341870 0.0592136i 1.69514 + 0.355671i 0.997662 1.72800i 1.33190 2.30691i −0.0368912 0.112535i 0 −0.273176 2.74700 + 1.20582i −0.182134
148.7 0.551407 + 0.955065i −1.67475 + 0.441824i 0.391901 0.678793i 0.0527330 0.0913363i −1.34544 1.35587i 0 3.07001 2.60958 1.47989i 0.116309
148.8 0.551407 + 0.955065i 1.67475 0.441824i 0.391901 0.678793i −0.0527330 + 0.0913363i 1.34544 + 1.35587i 0 3.07001 2.60958 1.47989i −0.116309
148.9 0.863305 + 1.49529i −1.09452 + 1.34239i −0.490592 + 0.849731i 1.75616 3.04175i −2.95217 0.477737i 0 1.75910 −0.604030 2.93856i 6.06439
148.10 0.863305 + 1.49529i 1.09452 1.34239i −0.490592 + 0.849731i −1.75616 + 3.04175i 2.95217 + 0.477737i 0 1.75910 −0.604030 2.93856i −6.06439
148.11 1.35757 + 2.35137i −0.521588 1.65165i −2.68597 + 4.65224i −0.793197 + 1.37386i 3.17555 3.46867i 0 −9.15528 −2.45589 + 1.72296i −4.30727
148.12 1.35757 + 2.35137i 0.521588 + 1.65165i −2.68597 + 4.65224i 0.793197 1.37386i −3.17555 + 3.46867i 0 −9.15528 −2.45589 + 1.72296i 4.30727
295.1 −1.08816 + 1.88474i −1.68791 + 0.388551i −1.36816 2.36973i 0.634145 + 1.09837i 1.10439 3.60407i 0 1.60248 2.69806 1.31167i −2.76019
295.2 −1.08816 + 1.88474i 1.68791 0.388551i −1.36816 2.36973i −0.634145 1.09837i −1.10439 + 3.60407i 0 1.60248 2.69806 1.31167i 2.76019
295.3 −0.649936 + 1.12572i −0.0514049 + 1.73129i 0.155166 + 0.268756i 1.76292 + 3.05347i −1.91554 1.18309i 0 −3.00314 −2.99472 0.177994i −4.58314
295.4 −0.649936 + 1.12572i 0.0514049 1.73129i 0.155166 + 0.268756i −1.76292 3.05347i 1.91554 + 1.18309i 0 −3.00314 −2.99472 0.177994i 4.58314
295.5 −0.0341870 + 0.0592136i −1.69514 + 0.355671i 0.997662 + 1.72800i −1.33190 2.30691i 0.0368912 0.112535i 0 −0.273176 2.74700 1.20582i 0.182134
295.6 −0.0341870 + 0.0592136i 1.69514 0.355671i 0.997662 + 1.72800i 1.33190 + 2.30691i −0.0368912 + 0.112535i 0 −0.273176 2.74700 1.20582i −0.182134
295.7 0.551407 0.955065i −1.67475 0.441824i 0.391901 + 0.678793i 0.0527330 + 0.0913363i −1.34544 + 1.35587i 0 3.07001 2.60958 + 1.47989i 0.116309
295.8 0.551407 0.955065i 1.67475 + 0.441824i 0.391901 + 0.678793i −0.0527330 0.0913363i 1.34544 1.35587i 0 3.07001 2.60958 + 1.47989i −0.116309
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
63.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.f.h 24
3.b odd 2 1 1323.2.f.h 24
7.b odd 2 1 inner 441.2.f.h 24
7.c even 3 1 441.2.g.h 24
7.c even 3 1 441.2.h.h 24
7.d odd 6 1 441.2.g.h 24
7.d odd 6 1 441.2.h.h 24
9.c even 3 1 inner 441.2.f.h 24
9.c even 3 1 3969.2.a.bh 12
9.d odd 6 1 1323.2.f.h 24
9.d odd 6 1 3969.2.a.bi 12
21.c even 2 1 1323.2.f.h 24
21.g even 6 1 1323.2.g.h 24
21.g even 6 1 1323.2.h.h 24
21.h odd 6 1 1323.2.g.h 24
21.h odd 6 1 1323.2.h.h 24
63.g even 3 1 441.2.h.h 24
63.h even 3 1 441.2.g.h 24
63.i even 6 1 1323.2.g.h 24
63.j odd 6 1 1323.2.g.h 24
63.k odd 6 1 441.2.h.h 24
63.l odd 6 1 inner 441.2.f.h 24
63.l odd 6 1 3969.2.a.bh 12
63.n odd 6 1 1323.2.h.h 24
63.o even 6 1 1323.2.f.h 24
63.o even 6 1 3969.2.a.bi 12
63.s even 6 1 1323.2.h.h 24
63.t odd 6 1 441.2.g.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.h 24 1.a even 1 1 trivial
441.2.f.h 24 7.b odd 2 1 inner
441.2.f.h 24 9.c even 3 1 inner
441.2.f.h 24 63.l odd 6 1 inner
441.2.g.h 24 7.c even 3 1
441.2.g.h 24 7.d odd 6 1
441.2.g.h 24 63.h even 3 1
441.2.g.h 24 63.t odd 6 1
441.2.h.h 24 7.c even 3 1
441.2.h.h 24 7.d odd 6 1
441.2.h.h 24 63.g even 3 1
441.2.h.h 24 63.k odd 6 1
1323.2.f.h 24 3.b odd 2 1
1323.2.f.h 24 9.d odd 6 1
1323.2.f.h 24 21.c even 2 1
1323.2.f.h 24 63.o even 6 1
1323.2.g.h 24 21.g even 6 1
1323.2.g.h 24 21.h odd 6 1
1323.2.g.h 24 63.i even 6 1
1323.2.g.h 24 63.j odd 6 1
1323.2.h.h 24 21.g even 6 1
1323.2.h.h 24 21.h odd 6 1
1323.2.h.h 24 63.n odd 6 1
1323.2.h.h 24 63.s even 6 1
3969.2.a.bh 12 9.c even 3 1
3969.2.a.bh 12 63.l odd 6 1
3969.2.a.bi 12 9.d odd 6 1
3969.2.a.bi 12 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\(T_{2}^{12} - \cdots\)
\(T_{5}^{24} + \cdots\)