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Properties

Label 416.4.i.h
Level $416$
Weight $4$
Character orbit 416.i
Analytic conductor $24.545$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [416,4,Mod(289,416)] 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("416.289"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 416.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,0,10,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.5447945624\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 10 q^{5} + 26 q^{7} - 117 q^{9} + 22 q^{11} + 37 q^{13} + 86 q^{15} - 109 q^{17} + 140 q^{19} + 20 q^{21} + 292 q^{23} + 812 q^{25} - 60 q^{27} + 101 q^{29} - 988 q^{31} - 226 q^{33} + 216 q^{35}+ \cdots - 8040 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −5.10450 − 8.84125i 0 2.27672 0 12.0705 − 20.9067i 0 −38.6118 + 66.8776i 0
289.2 0 −3.41763 − 5.91950i 0 −12.6950 0 −16.0117 + 27.7331i 0 −9.86034 + 17.0786i 0
289.3 0 −2.49442 − 4.32046i 0 1.37064 0 −3.31979 + 5.75004i 0 1.05573 − 1.82858i 0
289.4 0 −2.42316 − 4.19703i 0 19.6217 0 2.38297 − 4.12742i 0 1.75662 − 3.04255i 0
289.5 0 −1.29452 − 2.24218i 0 −21.8029 0 14.5235 − 25.1555i 0 10.1484 − 17.5776i 0
289.6 0 0.184475 + 0.319519i 0 2.59639 0 11.4964 − 19.9124i 0 13.4319 − 23.2648i 0
289.7 0 1.10524 + 1.91434i 0 15.5670 0 −13.4497 + 23.2956i 0 11.0569 − 19.1511i 0
289.8 0 1.87807 + 3.25292i 0 −9.07236 0 −3.65304 + 6.32726i 0 6.44569 − 11.1643i 0
289.9 0 2.54575 + 4.40937i 0 1.57332 0 −1.02975 + 1.78358i 0 0.538320 − 0.932398i 0
289.10 0 4.36146 + 7.55428i 0 −11.4139 0 −4.58130 + 7.93504i 0 −24.5448 + 42.5128i 0
289.11 0 4.65922 + 8.07001i 0 16.9784 0 14.5719 − 25.2393i 0 −29.9167 + 51.8173i 0
321.1 0 −5.10450 + 8.84125i 0 2.27672 0 12.0705 + 20.9067i 0 −38.6118 − 66.8776i 0
321.2 0 −3.41763 + 5.91950i 0 −12.6950 0 −16.0117 − 27.7331i 0 −9.86034 − 17.0786i 0
321.3 0 −2.49442 + 4.32046i 0 1.37064 0 −3.31979 − 5.75004i 0 1.05573 + 1.82858i 0
321.4 0 −2.42316 + 4.19703i 0 19.6217 0 2.38297 + 4.12742i 0 1.75662 + 3.04255i 0
321.5 0 −1.29452 + 2.24218i 0 −21.8029 0 14.5235 + 25.1555i 0 10.1484 + 17.5776i 0
321.6 0 0.184475 − 0.319519i 0 2.59639 0 11.4964 + 19.9124i 0 13.4319 + 23.2648i 0
321.7 0 1.10524 − 1.91434i 0 15.5670 0 −13.4497 − 23.2956i 0 11.0569 + 19.1511i 0
321.8 0 1.87807 − 3.25292i 0 −9.07236 0 −3.65304 − 6.32726i 0 6.44569 + 11.1643i 0
321.9 0 2.54575 − 4.40937i 0 1.57332 0 −1.02975 − 1.78358i 0 0.538320 + 0.932398i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.4.i.h yes 22
4.b odd 2 1 416.4.i.g ✓ 22
13.c even 3 1 inner 416.4.i.h yes 22
52.j odd 6 1 416.4.i.g ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.4.i.g ✓ 22 4.b odd 2 1
416.4.i.g ✓ 22 52.j odd 6 1
416.4.i.h yes 22 1.a even 1 1 trivial
416.4.i.h yes 22 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 207 T_{3}^{20} + 20 T_{3}^{19} + 28546 T_{3}^{18} + 5482 T_{3}^{17} + 2159843 T_{3}^{16} + \cdots + 30667140235264 \) acting on \(S_{4}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display