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Properties

Label 444.2.y.d
Level $444$
Weight $2$
Character orbit 444.y
Analytic conductor $3.545$
Analytic rank $0$
Dimension $68$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(103,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [68,6,34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 6 q^{2} + 34 q^{3} - 6 q^{4} - 4 q^{5} + 6 q^{6} + 12 q^{7} + 12 q^{8} - 34 q^{9} - 12 q^{10} - 4 q^{11} - 6 q^{12} + 18 q^{13} + 14 q^{14} - 8 q^{15} - 14 q^{16} + 10 q^{17} + 6 q^{19} + 32 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −1.40676 + 0.144993i 0.500000 − 0.866025i 1.95795 − 0.407941i 0.493796 + 1.84287i −0.577813 + 1.29079i 2.98228 + 1.72182i −2.69523 + 0.857766i −0.500000 − 0.866025i −0.961856 − 2.52088i
103.2 −1.40462 − 0.164464i 0.500000 − 0.866025i 1.94590 + 0.462019i 0.518417 + 1.93476i −0.844739 + 1.13420i −1.75843 − 1.01523i −2.65726 − 0.968993i −0.500000 − 0.866025i −0.409978 − 2.80286i
103.3 −1.32338 − 0.498669i 0.500000 − 0.866025i 1.50266 + 1.31986i −0.844241 − 3.15075i −1.09355 + 0.896744i −1.11475 − 0.643602i −1.33041 − 2.49600i −0.500000 − 0.866025i −0.453931 + 4.59063i
103.4 −1.04215 + 0.955989i 0.500000 − 0.866025i 0.172171 − 1.99258i 0.727448 + 2.71487i 0.306833 + 1.38053i −4.05196 − 2.33940i 1.72545 + 2.24116i −0.500000 − 0.866025i −3.35350 − 2.13388i
103.5 −0.981854 + 1.01782i 0.500000 − 0.866025i −0.0719271 − 1.99871i −0.225275 − 0.840739i 0.390534 + 1.35922i 2.90003 + 1.67433i 2.10495 + 1.88923i −0.500000 − 0.866025i 1.07691 + 0.596192i
103.6 −0.796228 − 1.16877i 0.500000 − 0.866025i −0.732043 + 1.86121i −0.628369 − 2.34510i −1.41030 + 0.105169i 3.06210 + 1.76790i 2.75820 − 0.626359i −0.500000 − 0.866025i −2.24056 + 2.60165i
103.7 −0.607025 − 1.27731i 0.500000 − 0.866025i −1.26304 + 1.55072i 0.673394 + 2.51314i −1.40970 − 0.112956i −0.445135 − 0.256999i 2.74745 + 0.671969i −0.500000 − 0.866025i 2.80129 − 2.38567i
103.8 −0.254810 − 1.39107i 0.500000 − 0.866025i −1.87014 + 0.708916i −0.393517 − 1.46863i −1.33211 − 0.474863i 0.737379 + 0.425726i 1.46268 + 2.42086i −0.500000 − 0.866025i −1.94269 + 0.921629i
103.9 −0.155927 + 1.40559i 0.500000 − 0.866025i −1.95137 − 0.438339i 0.147502 + 0.550487i 1.13931 + 0.837832i −1.75466 − 1.01305i 0.920397 − 2.67448i −0.500000 − 0.866025i −0.796759 + 0.121492i
103.10 −0.0722074 + 1.41237i 0.500000 − 0.866025i −1.98957 − 0.203967i −0.230426 − 0.859962i 1.18704 + 0.768718i −1.02031 − 0.589076i 0.431739 − 2.79528i −0.500000 − 0.866025i 1.23122 − 0.263351i
103.11 0.801450 + 1.16519i 0.500000 − 0.866025i −0.715357 + 1.86769i −0.0889495 − 0.331964i 1.40981 − 0.111478i 3.65333 + 2.10925i −2.74954 + 0.663329i −0.500000 − 0.866025i 0.315514 − 0.369696i
103.12 0.847149 − 1.13240i 0.500000 − 0.866025i −0.564678 − 1.91863i −0.979264 − 3.65466i −0.557116 − 1.29985i 1.77925 + 1.02725i −2.65103 − 0.985920i −0.500000 − 0.866025i −4.96814 − 1.98712i
103.13 0.905664 + 1.08617i 0.500000 − 0.866025i −0.359544 + 1.96742i −1.03115 − 3.84830i 1.39349 − 0.241242i −2.82667 − 1.63198i −2.46258 + 1.39129i −0.500000 − 0.866025i 3.24604 − 4.60527i
103.14 1.15908 − 0.810266i 0.500000 − 0.866025i 0.686937 − 1.87833i 0.0909304 + 0.339357i −0.122171 − 1.40893i 1.92723 + 1.11269i −0.725732 − 2.73374i −0.500000 − 0.866025i 0.380365 + 0.319664i
103.15 1.30369 + 0.548082i 0.500000 − 0.866025i 1.39921 + 1.42906i 0.0972231 + 0.362842i 1.12650 − 0.854987i 1.55899 + 0.900081i 1.04090 + 2.62993i −0.500000 − 0.866025i −0.0721181 + 0.526319i
103.16 1.39619 − 0.225054i 0.500000 − 0.866025i 1.89870 − 0.628437i −0.397714 − 1.48429i 0.503193 − 1.32166i −2.30569 − 1.33119i 2.50952 − 1.30473i −0.500000 − 0.866025i −0.889330 − 1.98284i
103.17 1.39969 − 0.202183i 0.500000 − 0.866025i 1.91824 − 0.565985i 1.07019 + 3.99402i 0.524748 − 1.31326i 0.543054 + 0.313532i 2.57051 − 1.18004i −0.500000 − 0.866025i 2.30546 + 5.37400i
199.1 −1.36880 − 0.355523i 0.500000 + 0.866025i 1.74721 + 0.973278i 1.66400 + 0.445868i −0.376506 − 1.36317i −1.73463 + 1.00149i −2.04555 − 1.95339i −0.500000 + 0.866025i −2.11916 − 1.20189i
199.2 −1.02814 − 0.971048i 0.500000 + 0.866025i 0.114132 + 1.99674i −3.55837 − 0.953464i 0.326884 − 1.37592i 1.19406 − 0.689390i 1.82159 − 2.16375i −0.500000 + 0.866025i 2.73264 + 4.43564i
199.3 −0.970316 + 1.02883i 0.500000 + 0.866025i −0.116975 − 1.99658i −0.419989 − 0.112536i −1.37615 − 0.325904i −0.315702 + 0.182271i 2.16764 + 1.81696i −0.500000 + 0.866025i 0.523301 − 0.322901i
See all 68 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 103.17
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
148.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.y.d yes 68
4.b odd 2 1 444.2.y.c ✓ 68
37.g odd 12 1 444.2.y.c ✓ 68
148.l even 12 1 inner 444.2.y.d yes 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.y.c ✓ 68 4.b odd 2 1
444.2.y.c ✓ 68 37.g odd 12 1
444.2.y.d yes 68 1.a even 1 1 trivial
444.2.y.d yes 68 148.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):

\( T_{5}^{68} + 4 T_{5}^{67} - 4 T_{5}^{66} - 56 T_{5}^{65} - 660 T_{5}^{64} - 1796 T_{5}^{63} + \cdots + 166181387370496 \) Copy content Toggle raw display
\( T_{7}^{68} - 12 T_{7}^{67} - 53 T_{7}^{66} + 1212 T_{7}^{65} + 483 T_{7}^{64} - 70248 T_{7}^{63} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display