Properties

Label 441.2.s.d
Level $441$
Weight $2$
Character orbit 441.s
Analytic conductor $3.521$
Analytic rank $0$
Dimension $48$
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.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{4} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{4} - 8q^{9} - 40q^{15} - 24q^{16} + 32q^{18} + 48q^{25} + 48q^{30} - 120q^{32} - 8q^{36} - 32q^{39} + 96q^{44} + 48q^{50} + 48q^{53} + 80q^{57} - 72q^{60} - 48q^{64} - 120q^{65} + 32q^{72} - 88q^{78} - 24q^{79} + 120q^{81} - 24q^{85} - 144q^{92} + 16q^{93} - 96q^{95} - 72q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1 −2.23278 + 1.28910i −0.625134 1.61530i 2.32354 4.02449i 2.33189 3.47807 + 2.80076i 0 6.82470i −2.21841 + 2.01956i −5.20660 + 3.00603i
362.2 −2.23278 + 1.28910i 0.625134 + 1.61530i 2.32354 4.02449i −2.33189 −3.47807 2.80076i 0 6.82470i −2.21841 + 2.01956i 5.20660 3.00603i
362.3 −1.80506 + 1.04215i −1.73189 0.0239080i 1.17216 2.03024i 3.30465 3.15107 1.76173i 0 0.717672i 2.99886 + 0.0828118i −5.96509 + 3.44395i
362.4 −1.80506 + 1.04215i 1.73189 + 0.0239080i 1.17216 2.03024i −3.30465 −3.15107 + 1.76173i 0 0.717672i 2.99886 + 0.0828118i 5.96509 3.44395i
362.5 −1.61855 + 0.934468i −0.710525 + 1.57961i 0.746462 1.29291i 2.50573 −0.326074 3.22063i 0 0.947692i −1.99031 2.24470i −4.05565 + 2.34153i
362.6 −1.61855 + 0.934468i 0.710525 1.57961i 0.746462 1.29291i −2.50573 0.326074 + 3.22063i 0 0.947692i −1.99031 2.24470i 4.05565 2.34153i
362.7 −1.58658 + 0.916012i −0.108803 + 1.72863i 0.678156 1.17460i −0.645568 −1.41082 2.84227i 0 1.17925i −2.97632 0.376160i 1.02425 0.591348i
362.8 −1.58658 + 0.916012i 0.108803 1.72863i 0.678156 1.17460i 0.645568 1.41082 + 2.84227i 0 1.17925i −2.97632 0.376160i −1.02425 + 0.591348i
362.9 −0.575298 + 0.332148i −0.537154 1.64665i −0.779355 + 1.34988i 0.0283039 0.855956 + 0.768901i 0 2.36404i −2.42293 + 1.76901i −0.0162832 + 0.00940110i
362.10 −0.575298 + 0.332148i 0.537154 + 1.64665i −0.779355 + 1.34988i −0.0283039 −0.855956 0.768901i 0 2.36404i −2.42293 + 1.76901i 0.0162832 0.00940110i
362.11 −0.105953 + 0.0611722i −1.73002 + 0.0838860i −0.992516 + 1.71909i 0.529430 0.178170 0.114717i 0 0.487547i 2.98593 0.290249i −0.0560949 + 0.0323864i
362.12 −0.105953 + 0.0611722i 1.73002 0.0838860i −0.992516 + 1.71909i −0.529430 −0.178170 + 0.114717i 0 0.487547i 2.98593 0.290249i 0.0560949 0.0323864i
362.13 0.367369 0.212101i −1.71145 0.266368i −0.910027 + 1.57621i −3.60763 −0.685229 + 0.265143i 0 1.62047i 2.85810 + 0.911750i −1.32533 + 0.765180i
362.14 0.367369 0.212101i 1.71145 + 0.266368i −0.910027 + 1.57621i 3.60763 0.685229 0.265143i 0 1.62047i 2.85810 + 0.911750i 1.32533 0.765180i
362.15 0.850109 0.490811i −0.831570 + 1.51937i −0.518210 + 0.897565i −1.88120 0.0387990 + 1.69978i 0 2.98061i −1.61698 2.52693i −1.59922 + 0.923312i
362.16 0.850109 0.490811i 0.831570 1.51937i −0.518210 + 0.897565i 1.88120 −0.0387990 1.69978i 0 2.98061i −1.61698 2.52693i 1.59922 0.923312i
362.17 1.02035 0.589100i −1.34152 1.09560i −0.305921 + 0.529871i 4.33202 −2.01424 0.327604i 0 3.07728i 0.599340 + 2.93952i 4.42019 2.55200i
362.18 1.02035 0.589100i 1.34152 + 1.09560i −0.305921 + 0.529871i −4.33202 2.01424 + 0.327604i 0 3.07728i 0.599340 + 2.93952i −4.42019 + 2.55200i
362.19 1.28562 0.742253i −1.72526 + 0.153190i 0.101880 0.176462i −0.308431 −2.10433 + 1.47753i 0 2.66653i 2.95307 0.528586i −0.396525 + 0.228934i
362.20 1.28562 0.742253i 1.72526 0.153190i 0.101880 0.176462i 0.308431 2.10433 1.47753i 0 2.66653i 2.95307 0.528586i 0.396525 0.228934i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.s.d 48
3.b odd 2 1 1323.2.s.d 48
7.b odd 2 1 inner 441.2.s.d 48
7.c even 3 1 441.2.i.d 48
7.c even 3 1 441.2.o.e 48
7.d odd 6 1 441.2.i.d 48
7.d odd 6 1 441.2.o.e 48
9.c even 3 1 1323.2.i.d 48
9.d odd 6 1 441.2.i.d 48
21.c even 2 1 1323.2.s.d 48
21.g even 6 1 1323.2.i.d 48
21.g even 6 1 1323.2.o.e 48
21.h odd 6 1 1323.2.i.d 48
21.h odd 6 1 1323.2.o.e 48
63.g even 3 1 1323.2.s.d 48
63.h even 3 1 1323.2.o.e 48
63.i even 6 1 441.2.o.e 48
63.j odd 6 1 441.2.o.e 48
63.k odd 6 1 1323.2.s.d 48
63.l odd 6 1 1323.2.i.d 48
63.n odd 6 1 inner 441.2.s.d 48
63.o even 6 1 441.2.i.d 48
63.s even 6 1 inner 441.2.s.d 48
63.t odd 6 1 1323.2.o.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.i.d 48 7.c even 3 1
441.2.i.d 48 7.d odd 6 1
441.2.i.d 48 9.d odd 6 1
441.2.i.d 48 63.o even 6 1
441.2.o.e 48 7.c even 3 1
441.2.o.e 48 7.d odd 6 1
441.2.o.e 48 63.i even 6 1
441.2.o.e 48 63.j odd 6 1
441.2.s.d 48 1.a even 1 1 trivial
441.2.s.d 48 7.b odd 2 1 inner
441.2.s.d 48 63.n odd 6 1 inner
441.2.s.d 48 63.s even 6 1 inner
1323.2.i.d 48 9.c even 3 1
1323.2.i.d 48 21.g even 6 1
1323.2.i.d 48 21.h odd 6 1
1323.2.i.d 48 63.l odd 6 1
1323.2.o.e 48 21.g even 6 1
1323.2.o.e 48 21.h odd 6 1
1323.2.o.e 48 63.h even 3 1
1323.2.o.e 48 63.t odd 6 1
1323.2.s.d 48 3.b odd 2 1
1323.2.s.d 48 21.c even 2 1
1323.2.s.d 48 63.g even 3 1
1323.2.s.d 48 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{24} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).