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Properties

Label 416.3.y.b
Level $416$
Weight $3$
Character orbit 416.y
Analytic conductor $11.335$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [416,3,Mod(95,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.95"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 416.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,0,0,0,0,4] 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: \(11.3351789974\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{7} + 42 q^{9} - 12 q^{11} - 14 q^{13} + 12 q^{15} - 6 q^{17} + 16 q^{19} - 104 q^{25} - 10 q^{29} + 40 q^{31} - 48 q^{35} - 30 q^{37} - 136 q^{39} + 90 q^{41} + 324 q^{43} - 90 q^{45} - 232 q^{47}+ \cdots - 368 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −4.30255 − 2.48408i 0 − 2.22967i 0 2.07167 + 3.58824i 0 7.84127 + 13.5815i 0
95.2 0 −3.54956 − 2.04934i 0 − 4.62317i 0 4.55443 + 7.88850i 0 3.89957 + 6.75426i 0
95.3 0 −3.46258 − 1.99912i 0 8.68214i 0 0.235119 + 0.407237i 0 3.49296 + 6.04998i 0
95.4 0 −2.83173 − 1.63490i 0 − 5.42742i 0 −6.16303 − 10.6747i 0 0.845805 + 1.46498i 0
95.5 0 −2.33560 − 1.34846i 0 8.92114i 0 −5.61609 − 9.72736i 0 −0.863316 − 1.49531i 0
95.6 0 −1.41864 − 0.819050i 0 5.60939i 0 6.19551 + 10.7309i 0 −3.15831 − 5.47036i 0
95.7 0 −0.823793 − 0.475617i 0 − 7.23083i 0 −2.45649 − 4.25477i 0 −4.04758 − 7.01061i 0
95.8 0 0.382104 + 0.220608i 0 2.51554i 0 0.389552 + 0.674724i 0 −4.40266 − 7.62564i 0
95.9 0 1.46007 + 0.842973i 0 − 0.159636i 0 −2.56203 − 4.43756i 0 −3.07879 − 5.33262i 0
95.10 0 1.73167 + 0.999781i 0 2.97421i 0 −1.05608 − 1.82919i 0 −2.50088 − 4.33165i 0
95.11 0 2.73217 + 1.57742i 0 − 8.34541i 0 −1.62187 − 2.80915i 0 0.476511 + 0.825341i 0
95.12 0 2.84943 + 1.64512i 0 − 2.62827i 0 5.08252 + 8.80318i 0 0.912841 + 1.58109i 0
95.13 0 4.57107 + 2.63911i 0 3.44994i 0 −1.71508 − 2.97061i 0 9.42979 + 16.3329i 0
95.14 0 4.99792 + 2.88555i 0 1.95616i 0 4.66188 + 8.07461i 0 12.1528 + 21.0493i 0
127.1 0 −4.30255 + 2.48408i 0 2.22967i 0 2.07167 − 3.58824i 0 7.84127 − 13.5815i 0
127.2 0 −3.54956 + 2.04934i 0 4.62317i 0 4.55443 − 7.88850i 0 3.89957 − 6.75426i 0
127.3 0 −3.46258 + 1.99912i 0 − 8.68214i 0 0.235119 − 0.407237i 0 3.49296 − 6.04998i 0
127.4 0 −2.83173 + 1.63490i 0 5.42742i 0 −6.16303 + 10.6747i 0 0.845805 − 1.46498i 0
127.5 0 −2.33560 + 1.34846i 0 − 8.92114i 0 −5.61609 + 9.72736i 0 −0.863316 + 1.49531i 0
127.6 0 −1.41864 + 0.819050i 0 − 5.60939i 0 6.19551 − 10.7309i 0 −3.15831 + 5.47036i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.3.y.b yes 28
4.b odd 2 1 416.3.y.a ✓ 28
13.e even 6 1 416.3.y.a ✓ 28
52.i odd 6 1 inner 416.3.y.b yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.3.y.a ✓ 28 4.b odd 2 1
416.3.y.a ✓ 28 13.e even 6 1
416.3.y.b yes 28 1.a even 1 1 trivial
416.3.y.b yes 28 52.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 84 T_{3}^{26} + 4390 T_{3}^{24} + 1908 T_{3}^{23} - 142392 T_{3}^{22} - 106020 T_{3}^{21} + \cdots + 276827508736 \) acting on \(S_{3}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display