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Properties

Label 195.3.g.b
Level $195$
Weight $3$
Character orbit 195.g
Analytic conductor $5.313$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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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] = [195,3,Mod(116,195)] 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("195.116"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 195.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] 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: \(5.31336515503\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{3} + 60 q^{4} - 40 q^{9} - 20 q^{10} + 72 q^{12} - 20 q^{13} + 172 q^{16} - 96 q^{22} + 160 q^{25} + 24 q^{27} - 20 q^{30} - 332 q^{36} + 100 q^{39} + 60 q^{40} - 164 q^{42} + 192 q^{43} - 4 q^{48}+ \cdots + 1128 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −3.63232 −0.915357 − 2.85694i 9.19378 −2.23607 3.32487 + 10.3773i − 1.01658i −18.8655 −7.32424 + 5.23025i 8.12212
116.2 −3.63232 −0.915357 + 2.85694i 9.19378 −2.23607 3.32487 − 10.3773i 1.01658i −18.8655 −7.32424 − 5.23025i 8.12212
116.3 −3.47844 2.42404 − 1.76750i 8.09957 2.23607 −8.43187 + 6.14814i − 11.0436i −14.2601 2.75190 − 8.56896i −7.77804
116.4 −3.47844 2.42404 + 1.76750i 8.09957 2.23607 −8.43187 − 6.14814i 11.0436i −14.2601 2.75190 + 8.56896i −7.77804
116.5 −3.05978 0.201191 − 2.99325i 5.36224 2.23607 −0.615600 + 9.15867i 7.98562i −4.16816 −8.91904 − 1.20443i −6.84187
116.6 −3.05978 0.201191 + 2.99325i 5.36224 2.23607 −0.615600 − 9.15867i − 7.98562i −4.16816 −8.91904 + 1.20443i −6.84187
116.7 −2.70715 2.48198 − 1.68517i 3.32869 −2.23607 −6.71910 + 4.56201i 2.05642i 1.81735 3.32042 − 8.36509i 6.05338
116.8 −2.70715 2.48198 + 1.68517i 3.32869 −2.23607 −6.71910 − 4.56201i − 2.05642i 1.81735 3.32042 + 8.36509i 6.05338
116.9 −2.02265 −2.16015 − 2.08177i 0.0911301 2.23607 4.36923 + 4.21070i 3.72274i 7.90629 0.332473 + 8.99386i −4.52279
116.10 −2.02265 −2.16015 + 2.08177i 0.0911301 2.23607 4.36923 − 4.21070i − 3.72274i 7.90629 0.332473 − 8.99386i −4.52279
116.11 −0.701634 0.839006 − 2.88029i −3.50771 −2.23607 −0.588675 + 2.02091i 6.32373i 5.26766 −7.59214 − 4.83316i 1.56890
116.12 −0.701634 0.839006 + 2.88029i −3.50771 −2.23607 −0.588675 − 2.02091i − 6.32373i 5.26766 −7.59214 + 4.83316i 1.56890
116.13 −0.654592 2.58526 − 1.52199i −3.57151 2.23607 −1.69229 + 0.996279i 8.93572i 4.95625 4.36712 − 7.86945i −1.46371
116.14 −0.654592 2.58526 + 1.52199i −3.57151 2.23607 −1.69229 − 0.996279i − 8.93572i 4.95625 4.36712 + 7.86945i −1.46371
116.15 −0.0617129 −2.45596 − 1.72286i −3.99619 2.23607 0.151565 + 0.106323i − 4.03865i 0.493468 3.06351 + 8.46256i −0.137994
116.16 −0.0617129 −2.45596 + 1.72286i −3.99619 2.23607 0.151565 − 0.106323i 4.03865i 0.493468 3.06351 − 8.46256i −0.137994
116.17 0.0617129 −2.45596 − 1.72286i −3.99619 −2.23607 −0.151565 − 0.106323i 4.03865i −0.493468 3.06351 + 8.46256i −0.137994
116.18 0.0617129 −2.45596 + 1.72286i −3.99619 −2.23607 −0.151565 + 0.106323i − 4.03865i −0.493468 3.06351 − 8.46256i −0.137994
116.19 0.654592 2.58526 − 1.52199i −3.57151 −2.23607 1.69229 − 0.996279i − 8.93572i −4.95625 4.36712 − 7.86945i −1.46371
116.20 0.654592 2.58526 + 1.52199i −3.57151 −2.23607 1.69229 + 0.996279i 8.93572i −4.95625 4.36712 + 7.86945i −1.46371
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.3.g.b ✓ 32
3.b odd 2 1 inner 195.3.g.b ✓ 32
13.b even 2 1 inner 195.3.g.b ✓ 32
39.d odd 2 1 inner 195.3.g.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.3.g.b ✓ 32 1.a even 1 1 trivial
195.3.g.b ✓ 32 3.b odd 2 1 inner
195.3.g.b ✓ 32 13.b even 2 1 inner
195.3.g.b ✓ 32 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 47T_{2}^{14} + 865T_{2}^{12} - 7831T_{2}^{10} + 35659T_{2}^{8} - 73097T_{2}^{6} + 47647T_{2}^{4} - 9633T_{2}^{2} + 36 \) acting on \(S_{3}^{\mathrm{new}}(195, [\chi])\). Copy content Toggle raw display