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Properties

Label 195.2.bf.a
Level $195$
Weight $2$
Character orbit 195.bf
Analytic conductor $1.557$
Analytic rank $0$
Dimension $96$
Inner twists $8$

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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] = [195,2,Mod(17,195)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 2])) 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("195.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} - 12 q^{6} - 12 q^{7} + 4 q^{10} - 12 q^{12} - 24 q^{13} - 6 q^{15} + 16 q^{16} + 20 q^{22} - 16 q^{25} - 32 q^{27} - 36 q^{28} + 30 q^{33} - 4 q^{36} - 84 q^{37} - 152 q^{40} - 48 q^{42}+ \cdots - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −2.61871 + 0.701681i 1.25584 − 1.19284i 4.63323 − 2.67499i 2.05584 + 0.879509i −2.45169 + 4.00490i 1.53607 + 0.411589i −6.42202 + 6.42202i 0.154272 − 2.99603i −6.00077 − 0.860636i
17.2 −2.27913 + 0.610690i 0.904090 + 1.47737i 3.08942 − 1.78368i −1.44565 + 1.70590i −2.96275 − 2.81499i −3.71897 − 0.996495i −2.61503 + 2.61503i −1.36524 + 2.67135i 2.25304 − 4.77081i
17.3 −2.12039 + 0.568157i −1.41092 + 1.00464i 2.44121 − 1.40943i 2.19791 − 0.411355i 2.42092 − 2.93185i −3.37093 − 0.903238i −1.27107 + 1.27107i 0.981406 − 2.83493i −4.42671 + 2.12099i
17.4 −2.09662 + 0.561788i −1.11133 − 1.32851i 2.34816 − 1.35571i −0.261010 − 2.22078i 3.07638 + 2.16105i −0.403690 − 0.108168i −1.09192 + 1.09192i −0.529880 + 2.95283i 1.79485 + 4.50951i
17.5 −1.85180 + 0.496187i 1.71910 − 0.211449i 1.45090 − 0.837676i −1.97286 − 1.05254i −3.07850 + 1.24455i 1.97020 + 0.527913i 0.440097 − 0.440097i 2.91058 − 0.727001i 4.17559 + 0.970178i
17.6 −1.65038 + 0.442218i −0.137150 − 1.72661i 0.796141 − 0.459652i −0.908487 + 2.04320i 0.989888 + 2.78891i 0.109428 + 0.0293212i 1.30565 − 1.30565i −2.96238 + 0.473611i 0.595810 − 3.77380i
17.7 −1.42342 + 0.381405i −0.323695 + 1.70154i 0.148609 − 0.0857994i 1.16842 + 1.90652i −0.188219 − 2.54546i 3.97287 + 1.06453i 1.90523 − 1.90523i −2.79044 − 1.10156i −2.39030 − 2.26814i
17.8 −1.05989 + 0.283997i −1.73203 − 0.00769600i −0.689337 + 0.397989i −1.65169 + 1.50729i 1.83795 − 0.483735i −0.160495 − 0.0430045i 2.16938 − 2.16938i 2.99988 + 0.0266595i 1.32254 − 2.06664i
17.9 −1.01047 + 0.270756i 1.56363 + 0.745014i −0.784302 + 0.452817i 1.92933 − 1.13035i −1.78173 − 0.329454i 0.396013 + 0.106111i 2.14935 − 2.14935i 1.88991 + 2.32986i −1.64348 + 1.66457i
17.10 −0.615188 + 0.164839i −0.106617 + 1.72877i −1.38077 + 0.797186i −1.22475 − 1.87083i −0.219379 − 1.08109i −2.17570 − 0.582977i 1.61872 − 1.61872i −2.97727 − 0.368632i 1.06184 + 0.949023i
17.11 −0.254998 + 0.0683265i −1.72709 + 0.131022i −1.67170 + 0.965154i 0.374569 − 2.20447i 0.431452 − 0.151416i 3.34011 + 0.894980i 0.733676 − 0.733676i 2.96567 − 0.452574i 0.0551097 + 0.587729i
17.12 −0.107570 + 0.0288234i 0.836177 − 1.51684i −1.72131 + 0.993799i −1.90035 − 1.17843i −0.0462274 + 0.187269i −3.86093 − 1.03453i 0.314011 − 0.314011i −1.60161 − 2.53670i 0.238387 + 0.0719893i
17.13 0.107570 − 0.0288234i −1.48257 − 0.895534i −1.72131 + 0.993799i 1.90035 + 1.17843i −0.185293 − 0.0536002i −3.86093 − 1.03453i −0.314011 + 0.314011i 1.39604 + 2.65539i 0.238387 + 0.0719893i
17.14 0.254998 − 0.0683265i 1.56121 − 0.750075i −1.67170 + 0.965154i −0.374569 + 2.20447i 0.346856 − 0.297940i 3.34011 + 0.894980i −0.733676 + 0.733676i 1.87477 − 2.34206i 0.0551097 + 0.587729i
17.15 0.615188 − 0.164839i 0.956716 + 1.44385i −1.38077 + 0.797186i 1.22475 + 1.87083i 0.826563 + 0.730534i −2.17570 − 0.582977i −1.61872 + 1.61872i −1.16939 + 2.76270i 1.06184 + 0.949023i
17.16 1.01047 − 0.270756i −0.981641 + 1.42702i −0.784302 + 0.452817i −1.92933 + 1.13035i −0.605549 + 1.70775i 0.396013 + 0.106111i −2.14935 + 2.14935i −1.07276 − 2.80164i −1.64348 + 1.66457i
17.17 1.05989 − 0.283997i 1.49614 − 0.872682i −0.689337 + 0.397989i 1.65169 − 1.50729i 1.33790 − 1.34985i −0.160495 − 0.0430045i −2.16938 + 2.16938i 1.47685 − 2.61130i 1.32254 − 2.06664i
17.18 1.42342 − 0.381405i 1.13110 + 1.31172i 0.148609 − 0.0857994i −1.16842 − 1.90652i 2.11032 + 1.43573i 3.97287 + 1.06453i −1.90523 + 1.90523i −0.441245 + 2.96737i −2.39030 − 2.26814i
17.19 1.65038 − 0.442218i −0.744530 − 1.56387i 0.796141 − 0.459652i 0.908487 − 2.04320i −1.92033 − 2.25172i 0.109428 + 0.0293212i −1.30565 + 1.30565i −1.89135 + 2.32869i 0.595810 − 3.77380i
17.20 1.85180 − 0.496187i −1.59450 + 0.676428i 1.45090 − 0.837676i 1.97286 + 1.05254i −2.61706 + 2.04378i 1.97020 + 0.527913i −0.440097 + 0.440097i 2.08489 − 2.15713i 4.17559 + 0.970178i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
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
5.c odd 4 1 inner
13.e even 6 1 inner
15.e even 4 1 inner
39.h odd 6 1 inner
65.r odd 12 1 inner
195.bf even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.bf.a ✓ 96
3.b odd 2 1 inner 195.2.bf.a ✓ 96
5.b even 2 1 975.2.bn.d 96
5.c odd 4 1 inner 195.2.bf.a ✓ 96
5.c odd 4 1 975.2.bn.d 96
13.e even 6 1 inner 195.2.bf.a ✓ 96
15.d odd 2 1 975.2.bn.d 96
15.e even 4 1 inner 195.2.bf.a ✓ 96
15.e even 4 1 975.2.bn.d 96
39.h odd 6 1 inner 195.2.bf.a ✓ 96
65.l even 6 1 975.2.bn.d 96
65.r odd 12 1 inner 195.2.bf.a ✓ 96
65.r odd 12 1 975.2.bn.d 96
195.y odd 6 1 975.2.bn.d 96
195.bf even 12 1 inner 195.2.bf.a ✓ 96
195.bf even 12 1 975.2.bn.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bf.a ✓ 96 1.a even 1 1 trivial
195.2.bf.a ✓ 96 3.b odd 2 1 inner
195.2.bf.a ✓ 96 5.c odd 4 1 inner
195.2.bf.a ✓ 96 13.e even 6 1 inner
195.2.bf.a ✓ 96 15.e even 4 1 inner
195.2.bf.a ✓ 96 39.h odd 6 1 inner
195.2.bf.a ✓ 96 65.r odd 12 1 inner
195.2.bf.a ✓ 96 195.bf even 12 1 inner
975.2.bn.d 96 5.b even 2 1
975.2.bn.d 96 5.c odd 4 1
975.2.bn.d 96 15.d odd 2 1
975.2.bn.d 96 15.e even 4 1
975.2.bn.d 96 65.l even 6 1
975.2.bn.d 96 65.r odd 12 1
975.2.bn.d 96 195.y odd 6 1
975.2.bn.d 96 195.bf even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(195, [\chi])\).