Properties

Label 2214.2.i.b
Level $2214$
Weight $2$
Character orbit 2214.i
Analytic conductor $17.679$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2214,2,Mod(901,2214)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2214, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) 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("2214.901"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2214 = 2 \cdot 3^{3} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2214.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.6788790075\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 738)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{2} - 18 q^{4} - 4 q^{5} - 36 q^{8} - 8 q^{10} - 18 q^{16} - 4 q^{20} + 4 q^{23} - 26 q^{25} + 8 q^{31} + 18 q^{32} - 60 q^{37} + 4 q^{40} + 6 q^{41} + 6 q^{43} + 8 q^{46} + 38 q^{49} + 26 q^{50}+ \cdots + 76 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.84171 + 3.18994i 0 −3.70331 + 2.13810i −1.00000 0 −3.68343
901.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.84171 + 3.18994i 0 3.70331 2.13810i −1.00000 0 −3.68343
901.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.55925 + 2.70071i 0 0.0882889 0.0509736i −1.00000 0 −3.11851
901.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.55925 + 2.70071i 0 −0.0882889 + 0.0509736i −1.00000 0 −3.11851
901.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.13494 + 1.96578i 0 −4.05054 + 2.33858i −1.00000 0 −2.26989
901.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.13494 + 1.96578i 0 4.05054 2.33858i −1.00000 0 −2.26989
901.7 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.714167 + 1.23697i 0 −2.06322 + 1.19120i −1.00000 0 −1.42833
901.8 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.714167 + 1.23697i 0 2.06322 1.19120i −1.00000 0 −1.42833
901.9 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.290704 + 0.503513i 0 −1.28705 + 0.743077i −1.00000 0 −0.581407
901.10 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.290704 + 0.503513i 0 1.28705 0.743077i −1.00000 0 −0.581407
901.11 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.0761831 0.131953i 0 1.94692 1.12405i −1.00000 0 0.152366
901.12 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.0761831 0.131953i 0 −1.94692 + 1.12405i −1.00000 0 0.152366
901.13 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.18467 2.05191i 0 −4.22182 + 2.43747i −1.00000 0 2.36934
901.14 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.18467 2.05191i 0 4.22182 2.43747i −1.00000 0 2.36934
901.15 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.58544 2.74607i 0 1.86289 1.07554i −1.00000 0 3.17089
901.16 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.58544 2.74607i 0 −1.86289 + 1.07554i −1.00000 0 3.17089
901.17 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.69449 2.93494i 0 0.610625 0.352544i −1.00000 0 3.38898
901.18 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.69449 2.93494i 0 −0.610625 + 0.352544i −1.00000 0 3.38898
1639.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.84171 3.18994i 0 −3.70331 2.13810i −1.00000 0 −3.68343
1639.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.84171 3.18994i 0 3.70331 + 2.13810i −1.00000 0 −3.68343
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
41.b even 2 1 inner
369.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2214.2.i.b 36
3.b odd 2 1 738.2.i.b 36
9.c even 3 1 inner 2214.2.i.b 36
9.d odd 6 1 738.2.i.b 36
41.b even 2 1 inner 2214.2.i.b 36
123.b odd 2 1 738.2.i.b 36
369.i even 6 1 inner 2214.2.i.b 36
369.k odd 6 1 738.2.i.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.i.b 36 3.b odd 2 1
738.2.i.b 36 9.d odd 6 1
738.2.i.b 36 123.b odd 2 1
738.2.i.b 36 369.k odd 6 1
2214.2.i.b 36 1.a even 1 1 trivial
2214.2.i.b 36 9.c even 3 1 inner
2214.2.i.b 36 41.b even 2 1 inner
2214.2.i.b 36 369.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 2 T_{5}^{17} + 31 T_{5}^{16} + 52 T_{5}^{15} + 608 T_{5}^{14} + 979 T_{5}^{13} + \cdots + 7056 \) acting on \(S_{2}^{\mathrm{new}}(2214, [\chi])\). Copy content Toggle raw display