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Properties

Label 552.2.j.d
Level $552$
Weight $2$
Character orbit 552.j
Analytic conductor $4.408$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [552,2,Mod(323,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("552.323"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,0,2,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 2 q^{3} + 4 q^{4} + 8 q^{5} + 4 q^{6} - 9 q^{8} + 2 q^{9} - 4 q^{10} - 21 q^{12} - 4 q^{14} - 8 q^{15} - 12 q^{16} - 11 q^{18} + 4 q^{19} - 2 q^{20} + 8 q^{21} + 18 q^{22} + 42 q^{23} + 28 q^{24}+ \cdots + 64 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.41381 − 0.0336448i 0.124259 − 1.72759i 1.99774 + 0.0951350i 1.31688 −0.233803 + 2.43831i 2.00880i −2.82122 − 0.201717i −2.96912 − 0.429336i −1.86182 − 0.0443062i
323.2 −1.41381 + 0.0336448i 0.124259 + 1.72759i 1.99774 − 0.0951350i 1.31688 −0.233803 − 2.43831i − 2.00880i −2.82122 + 0.201717i −2.96912 + 0.429336i −1.86182 + 0.0443062i
323.3 −1.40766 − 0.136034i −1.50629 − 0.855045i 1.96299 + 0.382978i −2.98020 2.00402 + 1.40851i − 1.36209i −2.71112 − 0.806135i 1.53780 + 2.57588i 4.19509 + 0.405409i
323.4 −1.40766 + 0.136034i −1.50629 + 0.855045i 1.96299 − 0.382978i −2.98020 2.00402 − 1.40851i 1.36209i −2.71112 + 0.806135i 1.53780 − 2.57588i 4.19509 − 0.405409i
323.5 −1.33226 − 0.474424i 0.873799 − 1.49549i 1.54984 + 1.26411i −1.38336 −1.87362 + 1.57783i − 5.18286i −1.46507 − 2.41942i −1.47295 − 2.61351i 1.84300 + 0.656301i
323.6 −1.33226 + 0.474424i 0.873799 + 1.49549i 1.54984 − 1.26411i −1.38336 −1.87362 − 1.57783i 5.18286i −1.46507 + 2.41942i −1.47295 + 2.61351i 1.84300 − 0.656301i
323.7 −1.30785 − 0.538067i −1.71722 + 0.226158i 1.42097 + 1.40743i 3.49837 2.36757 + 0.628199i − 1.83575i −1.10113 − 2.60529i 2.89771 − 0.776727i −4.57536 − 1.88236i
323.8 −1.30785 + 0.538067i −1.71722 − 0.226158i 1.42097 − 1.40743i 3.49837 2.36757 − 0.628199i 1.83575i −1.10113 + 2.60529i 2.89771 + 0.776727i −4.57536 + 1.88236i
323.9 −1.07807 − 0.915298i 1.49079 + 0.881788i 0.324458 + 1.97351i 1.83050 −0.800072 − 2.31514i 4.30356i 1.45656 − 2.42455i 1.44490 + 2.62912i −1.97340 − 1.67545i
323.10 −1.07807 + 0.915298i 1.49079 − 0.881788i 0.324458 − 1.97351i 1.83050 −0.800072 + 2.31514i − 4.30356i 1.45656 + 2.42455i 1.44490 − 2.62912i −1.97340 + 1.67545i
323.11 −0.975618 − 1.02380i 1.69183 + 0.371101i −0.0963383 + 1.99768i −1.56730 −1.27065 − 2.09415i − 0.668312i 2.13922 − 1.85034i 2.72457 + 1.25568i 1.52908 + 1.60460i
323.12 −0.975618 + 1.02380i 1.69183 − 0.371101i −0.0963383 − 1.99768i −1.56730 −1.27065 + 2.09415i 0.668312i 2.13922 + 1.85034i 2.72457 − 1.25568i 1.52908 − 1.60460i
323.13 −0.964429 − 1.03435i −0.513686 + 1.65412i −0.139754 + 1.99511i 2.51134 2.20635 − 1.06396i − 2.77384i 2.19842 − 1.77959i −2.47225 − 1.69940i −2.42201 − 2.59760i
323.14 −0.964429 + 1.03435i −0.513686 − 1.65412i −0.139754 − 1.99511i 2.51134 2.20635 + 1.06396i 2.77384i 2.19842 + 1.77959i −2.47225 + 1.69940i −2.42201 + 2.59760i
323.15 −0.734855 − 1.20830i −1.73179 − 0.0302019i −0.919975 + 1.77585i −0.871841 1.23612 + 2.11471i 3.09231i 2.82181 − 0.193388i 2.99818 + 0.104607i 0.640677 + 1.05344i
323.16 −0.734855 + 1.20830i −1.73179 + 0.0302019i −0.919975 − 1.77585i −0.871841 1.23612 − 2.11471i − 3.09231i 2.82181 + 0.193388i 2.99818 − 0.104607i 0.640677 − 1.05344i
323.17 −0.242573 − 1.39325i 1.04780 + 1.37917i −1.88232 + 0.675933i −2.15601 1.66736 − 1.79441i − 2.22228i 1.39835 + 2.45858i −0.804216 + 2.89020i 0.522991 + 3.00387i
323.18 −0.242573 + 1.39325i 1.04780 − 1.37917i −1.88232 − 0.675933i −2.15601 1.66736 + 1.79441i 2.22228i 1.39835 − 2.45858i −0.804216 − 2.89020i 0.522991 − 3.00387i
323.19 −0.218758 − 1.39719i 0.262136 − 1.71210i −1.90429 + 0.611292i −0.130526 −2.44948 + 0.00827965i 3.77863i 1.27067 + 2.52693i −2.86257 − 0.897607i 0.0285535 + 0.182369i
323.20 −0.218758 + 1.39719i 0.262136 + 1.71210i −1.90429 − 0.611292i −0.130526 −2.44948 − 0.00827965i − 3.77863i 1.27067 − 2.52693i −2.86257 + 0.897607i 0.0285535 − 0.182369i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.42
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.j.d yes 42
3.b odd 2 1 552.2.j.c ✓ 42
4.b odd 2 1 2208.2.j.d 42
8.b even 2 1 2208.2.j.c 42
8.d odd 2 1 552.2.j.c ✓ 42
12.b even 2 1 2208.2.j.c 42
24.f even 2 1 inner 552.2.j.d yes 42
24.h odd 2 1 2208.2.j.d 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.j.c ✓ 42 3.b odd 2 1
552.2.j.c ✓ 42 8.d odd 2 1
552.2.j.d yes 42 1.a even 1 1 trivial
552.2.j.d yes 42 24.f even 2 1 inner
2208.2.j.c 42 8.b even 2 1
2208.2.j.c 42 12.b even 2 1
2208.2.j.d 42 4.b odd 2 1
2208.2.j.d 42 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{21} - 4 T_{5}^{20} - 50 T_{5}^{19} + 192 T_{5}^{18} + 1064 T_{5}^{17} - 3716 T_{5}^{16} + \cdots + 24576 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display