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Properties

Label 444.2.y.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 6 q^{2} - 34 q^{3} - 6 q^{4} - 4 q^{5} - 6 q^{6} - 12 q^{7} + 12 q^{8} - 34 q^{9} - 12 q^{10} + 4 q^{11} + 6 q^{12} + 18 q^{13} - 18 q^{14} + 8 q^{15} - 14 q^{16} + 10 q^{17} - 6 q^{19} - 32 q^{20}+ \cdots - 2 q^{99}+O(q^{100}) \) 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 −1.40307 − 0.177192i −0.500000 + 0.866025i 1.93721 + 0.497226i 0.0972231 + 0.362842i 0.854987 − 1.12650i −1.55899 − 0.900081i −2.62993 − 1.04090i −0.500000 − 0.866025i −0.0721181 − 0.526319i
103.2 −1.32741 + 0.487821i −0.500000 + 0.866025i 1.52406 − 1.29508i −1.03115 − 3.84830i 0.241242 − 1.39349i 2.82667 + 1.63198i −1.39129 + 2.46258i −0.500000 − 0.866025i 3.24604 + 4.60527i
103.3 −1.27667 + 0.608363i −0.500000 + 0.866025i 1.25979 − 1.55336i −0.0889495 − 0.331964i 0.111478 − 1.40981i −3.65333 − 2.10925i −0.663329 + 2.74954i −0.500000 − 0.866025i 0.315514 + 0.369696i
103.4 −1.11107 − 0.874939i −0.500000 + 0.866025i 0.468965 + 1.94424i 1.07019 + 3.99402i 1.31326 − 0.524748i −0.543054 − 0.313532i 1.18004 − 2.57051i −0.500000 − 0.866025i 2.30546 − 5.37400i
103.5 −1.09661 − 0.892998i −0.500000 + 0.866025i 0.405108 + 1.95854i −0.397714 − 1.48429i 1.32166 − 0.503193i 2.30569 + 1.33119i 1.30473 − 2.50952i −0.500000 − 0.866025i −0.889330 + 1.98284i
103.6 −0.643651 + 1.25925i −0.500000 + 0.866025i −1.17143 − 1.62104i −0.230426 − 0.859962i −0.768718 − 1.18704i 1.02031 + 0.589076i 2.79528 − 0.431739i −0.500000 − 0.866025i 1.23122 + 0.263351i
103.7 −0.598660 − 1.28125i −0.500000 + 0.866025i −1.28321 + 1.53407i 0.0909304 + 0.339357i 1.40893 + 0.122171i −1.92723 − 1.11269i 2.73374 + 0.725732i −0.500000 − 0.866025i 0.380365 − 0.319664i
103.8 −0.567759 + 1.29524i −0.500000 + 0.866025i −1.35530 − 1.47077i 0.147502 + 0.550487i −0.837832 − 1.13931i 1.75466 + 1.01305i 2.67448 − 0.920397i −0.500000 − 0.866025i −0.796759 − 0.121492i
103.9 −0.167450 − 1.40427i −0.500000 + 0.866025i −1.94392 + 0.470289i −0.979264 − 3.65466i 1.29985 + 0.557116i −1.77925 − 1.02725i 0.985920 + 2.65103i −0.500000 − 0.866025i −4.96814 + 1.98712i
103.10 0.341399 + 1.37239i −0.500000 + 0.866025i −1.76689 + 0.937062i −0.225275 − 0.840739i −1.35922 − 0.390534i −2.90003 − 1.67433i −1.88923 − 2.10495i −0.500000 − 0.866025i 1.07691 − 0.596192i
103.11 0.424538 + 1.34899i −0.500000 + 0.866025i −1.63954 + 1.14539i 0.727448 + 2.71487i −1.38053 − 0.306833i 4.05196 + 2.33940i −2.24116 − 1.72545i −0.500000 − 0.866025i −3.35350 + 2.13388i
103.12 0.916206 − 1.07730i −0.500000 + 0.866025i −0.321133 − 1.97405i −0.393517 − 1.46863i 0.474863 + 1.33211i −0.737379 − 0.425726i −2.42086 − 1.46268i −0.500000 − 0.866025i −1.94269 − 0.921629i
103.13 1.14579 + 0.828948i −0.500000 + 0.866025i 0.625690 + 1.89961i 0.493796 + 1.84287i −1.29079 + 0.577813i −2.98228 − 1.72182i −0.857766 + 2.69523i −0.500000 − 0.866025i −0.961856 + 2.52088i
103.14 1.16435 − 0.802670i −0.500000 + 0.866025i 0.711441 − 1.86918i 0.673394 + 2.51314i 0.112956 + 1.40970i 0.445135 + 0.256999i −0.671969 − 2.74745i −0.500000 − 0.866025i 2.80129 + 2.38567i
103.15 1.27394 − 0.614070i −0.500000 + 0.866025i 1.24584 − 1.56457i −0.628369 − 2.34510i −0.105169 + 1.41030i −3.06210 − 1.76790i 0.626359 − 2.75820i −0.500000 − 0.866025i −2.24056 − 2.60165i
103.16 1.29867 + 0.559879i −0.500000 + 0.866025i 1.37307 + 1.45419i 0.518417 + 1.93476i −1.13420 + 0.844739i 1.75843 + 1.01523i 0.968993 + 2.65726i −0.500000 − 0.866025i −0.409978 + 2.80286i
103.17 1.39541 + 0.229829i −0.500000 + 0.866025i 1.89436 + 0.641413i −0.844241 − 3.15075i −0.896744 + 1.09355i 1.11475 + 0.643602i 2.49600 + 1.33041i −0.500000 − 0.866025i −0.453931 − 4.59063i
199.1 −1.35473 − 0.405834i −0.500000 − 0.866025i 1.67060 + 1.09959i −0.419989 − 0.112536i 0.325904 + 1.37615i 0.315702 − 0.182271i −1.81696 − 2.16764i −0.500000 + 0.866025i 0.523301 + 0.322901i
199.2 −1.06302 − 0.932734i −0.500000 − 0.866025i 0.260015 + 1.98303i −1.55697 − 0.417189i −0.276262 + 1.38697i 1.97652 − 1.14115i 1.57323 − 2.35052i −0.500000 + 0.866025i 1.26596 + 1.89572i
199.3 −1.00765 + 0.992290i −0.500000 − 0.866025i 0.0307195 − 1.99976i 1.66400 + 0.445868i 1.36317 + 0.376506i 1.73463 − 1.00149i 1.95339 + 2.04555i −0.500000 + 0.866025i −2.11916 + 1.20189i
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.c ✓ 68
4.b odd 2 1 444.2.y.d yes 68
37.g odd 12 1 444.2.y.d yes 68
148.l even 12 1 inner 444.2.y.c ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.y.c ✓ 68 1.a even 1 1 trivial
444.2.y.c ✓ 68 148.l even 12 1 inner
444.2.y.d yes 68 4.b odd 2 1
444.2.y.d yes 68 37.g odd 12 1

Hecke kernels

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

\( T_{5}^{68} + 4 T_{5}^{67} - 4 T_{5}^{66} - 56 T_{5}^{65} - 660 T_{5}^{64} - 1796 T_{5}^{63} + \cdots + 166181387370496 \) 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