Properties

Label 441.2.o.e
Level $441$
Weight $2$
Character orbit 441.o
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.o (of order \(6\), degree \(2\), 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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{4} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 24q^{4} + 16q^{9} - 24q^{11} - 40q^{15} - 24q^{16} - 16q^{18} - 48q^{23} - 24q^{25} - 24q^{30} + 120q^{32} - 8q^{36} + 88q^{39} + 48q^{50} + 24q^{51} + 80q^{57} - 96q^{60} - 48q^{64} + 120q^{65} + 56q^{72} - 168q^{74} - 88q^{78} - 24q^{79} - 96q^{81} - 24q^{85} + 24q^{86} - 144q^{92} - 32q^{93} + 96q^{95} - 72q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1 −2.34591 1.35441i −1.70185 + 0.322036i 2.66888 + 4.62263i 0.601464 + 1.04177i 4.42857 + 1.54954i 0 9.04141i 2.79259 1.09611i 3.25853i
146.2 −2.34591 1.35441i 1.70185 0.322036i 2.66888 + 4.62263i −0.601464 1.04177i −4.42857 1.54954i 0 9.04141i 2.79259 1.09611i 3.25853i
146.3 −2.05485 1.18637i −1.27902 1.16795i 1.81495 + 3.14358i −1.71774 2.97522i 1.24259 + 3.91735i 0 3.86732i 0.271802 + 2.98766i 8.15151i
146.4 −2.05485 1.18637i 1.27902 + 1.16795i 1.81495 + 3.14358i 1.71774 + 2.97522i −1.24259 3.91735i 0 3.86732i 0.271802 + 2.98766i 8.15151i
146.5 −1.28562 0.742253i −0.729965 + 1.57072i 0.101880 + 0.176462i −0.154215 0.267109i 2.10433 1.47753i 0 2.66653i −1.93430 2.29314i 0.457868i
146.6 −1.28562 0.742253i 0.729965 1.57072i 0.101880 + 0.176462i 0.154215 + 0.267109i −2.10433 + 1.47753i 0 2.66653i −1.93430 2.29314i 0.457868i
146.7 −1.02035 0.589100i −1.61957 + 0.613991i −0.305921 0.529871i 2.16601 + 3.75164i 2.01424 + 0.327604i 0 3.07728i 2.24603 1.98880i 5.10399i
146.8 −1.02035 0.589100i 1.61957 0.613991i −0.305921 0.529871i −2.16601 3.75164i −2.01424 0.327604i 0 3.07728i 2.24603 1.98880i 5.10399i
146.9 −0.850109 0.490811i −0.900030 1.47985i −0.518210 0.897565i 0.940599 + 1.62916i 0.0387990 + 1.69978i 0 2.98061i −1.37989 + 2.66381i 1.84662i
146.10 −0.850109 0.490811i 0.900030 + 1.47985i −0.518210 0.897565i −0.940599 1.62916i −0.0387990 1.69978i 0 2.98061i −1.37989 + 2.66381i 1.84662i
146.11 −0.367369 0.212101i −1.08640 + 1.34897i −0.910027 1.57621i −1.80381 3.12430i 0.685229 0.265143i 0 1.62047i −0.639450 2.93106i 1.53036i
146.12 −0.367369 0.212101i 1.08640 1.34897i −0.910027 1.57621i 1.80381 + 3.12430i −0.685229 + 0.265143i 0 1.62047i −0.639450 2.93106i 1.53036i
146.13 0.105953 + 0.0611722i −0.792362 + 1.54018i −0.992516 1.71909i 0.264715 + 0.458500i −0.178170 + 0.114717i 0 0.487547i −1.74433 2.44076i 0.0647728i
146.14 0.105953 + 0.0611722i 0.792362 1.54018i −0.992516 1.71909i −0.264715 0.458500i 0.178170 0.114717i 0 0.487547i −1.74433 2.44076i 0.0647728i
146.15 0.575298 + 0.332148i −1.69462 0.358137i −0.779355 1.34988i 0.0141520 + 0.0245119i −0.855956 0.768901i 0 2.36404i 2.74348 + 1.21381i 0.0188022i
146.16 0.575298 + 0.332148i 1.69462 + 0.358137i −0.779355 1.34988i −0.0141520 0.0245119i 0.855956 + 0.768901i 0 2.36404i 2.74348 + 1.21381i 0.0188022i
146.17 1.58658 + 0.916012i −1.44264 0.958541i 0.678156 + 1.17460i 0.322784 + 0.559079i −1.41082 2.84227i 0 1.17925i 1.16240 + 2.76565i 1.18270i
146.18 1.58658 + 0.916012i 1.44264 + 0.958541i 0.678156 + 1.17460i −0.322784 0.559079i 1.41082 + 2.84227i 0 1.17925i 1.16240 + 2.76565i 1.18270i
146.19 1.61855 + 0.934468i −1.01272 1.40514i 0.746462 + 1.29291i −1.25287 2.17003i −0.326074 3.22063i 0 0.947692i −0.948811 + 2.84601i 4.68306i
146.20 1.61855 + 0.934468i 1.01272 + 1.40514i 0.746462 + 1.29291i 1.25287 + 2.17003i 0.326074 + 3.22063i 0 0.947692i −0.948811 + 2.84601i 4.68306i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o 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.o.e 48
3.b odd 2 1 1323.2.o.e 48
7.b odd 2 1 inner 441.2.o.e 48
7.c even 3 1 441.2.i.d 48
7.c even 3 1 441.2.s.d 48
7.d odd 6 1 441.2.i.d 48
7.d odd 6 1 441.2.s.d 48
9.c even 3 1 1323.2.o.e 48
9.d odd 6 1 inner 441.2.o.e 48
21.c even 2 1 1323.2.o.e 48
21.g even 6 1 1323.2.i.d 48
21.g even 6 1 1323.2.s.d 48
21.h odd 6 1 1323.2.i.d 48
21.h odd 6 1 1323.2.s.d 48
63.g even 3 1 1323.2.i.d 48
63.h even 3 1 1323.2.s.d 48
63.i even 6 1 441.2.s.d 48
63.j odd 6 1 441.2.s.d 48
63.k odd 6 1 1323.2.i.d 48
63.l odd 6 1 1323.2.o.e 48
63.n odd 6 1 441.2.i.d 48
63.o even 6 1 inner 441.2.o.e 48
63.s even 6 1 441.2.i.d 48
63.t odd 6 1 1323.2.s.d 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 63.n odd 6 1
441.2.i.d 48 63.s even 6 1
441.2.o.e 48 1.a even 1 1 trivial
441.2.o.e 48 7.b odd 2 1 inner
441.2.o.e 48 9.d odd 6 1 inner
441.2.o.e 48 63.o even 6 1 inner
441.2.s.d 48 7.c even 3 1
441.2.s.d 48 7.d odd 6 1
441.2.s.d 48 63.i even 6 1
441.2.s.d 48 63.j odd 6 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.g even 3 1
1323.2.i.d 48 63.k odd 6 1
1323.2.o.e 48 3.b odd 2 1
1323.2.o.e 48 9.c even 3 1
1323.2.o.e 48 21.c even 2 1
1323.2.o.e 48 63.l odd 6 1
1323.2.s.d 48 21.g even 6 1
1323.2.s.d 48 21.h odd 6 1
1323.2.s.d 48 63.h even 3 1
1323.2.s.d 48 63.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 \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\(T_{2}^{24} - \cdots\)
\(11\!\cdots\!90\)\( T_{5}^{28} + \)\(49\!\cdots\!44\)\( T_{5}^{26} + \)\(16\!\cdots\!48\)\( T_{5}^{24} + \)\(42\!\cdots\!76\)\( T_{5}^{22} + \)\(79\!\cdots\!44\)\( T_{5}^{20} + \)\(10\!\cdots\!12\)\( T_{5}^{18} + \)\(95\!\cdots\!91\)\( T_{5}^{16} + \)\(52\!\cdots\!64\)\( T_{5}^{14} + \)\(20\!\cdots\!80\)\( T_{5}^{12} + \)\(50\!\cdots\!24\)\( T_{5}^{10} + 859511233110 T_{5}^{8} + 68412222392 T_{5}^{6} + 3795958686 T_{5}^{4} + 3040548 T_{5}^{2} + 2401 \)">\(T_{5}^{48} + \cdots\)