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Properties

Label 195.2.o.a
Level $195$
Weight $2$
Character orbit 195.o
Analytic conductor $1.557$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [195,2,Mod(86,195)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) 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("195.86"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.o (of order \(4\), degree \(2\), 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: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{6} - 16 q^{7} + 4 q^{15} - 64 q^{16} + 4 q^{18} - 16 q^{19} - 12 q^{21} + 8 q^{24} - 24 q^{27} + 32 q^{28} + 32 q^{31} - 4 q^{33} - 16 q^{34} + 32 q^{37} - 8 q^{39} + 8 q^{42} - 8 q^{45} - 40 q^{46}+ \cdots - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −1.90946 + 1.90946i 0.460384 − 1.66974i − 5.29209i −0.707107 + 0.707107i 2.30923 + 4.06740i −1.58484 + 1.58484i 6.28612 + 6.28612i −2.57609 − 1.53745i − 2.70039i
86.2 −1.89167 + 1.89167i −1.62902 + 0.588468i − 5.15683i 0.707107 − 0.707107i 1.96838 − 4.19475i 0.157581 − 0.157581i 5.97167 + 5.97167i 2.30741 − 1.91725i 2.67522i
86.3 −1.57927 + 1.57927i 1.73010 + 0.0822389i − 2.98819i 0.707107 − 0.707107i −2.86217 + 2.60241i −2.29263 + 2.29263i 1.56062 + 1.56062i 2.98647 + 0.284563i 2.23343i
86.4 −1.43871 + 1.43871i −0.158548 + 1.72478i − 2.13978i −0.707107 + 0.707107i −2.25335 − 2.70956i −1.21328 + 1.21328i 0.201099 + 0.201099i −2.94973 − 0.546921i − 2.03464i
86.5 −1.35176 + 1.35176i −1.49444 − 0.875587i − 1.65452i −0.707107 + 0.707107i 3.20371 − 0.836541i 3.04341 − 3.04341i −0.467005 − 0.467005i 1.46670 + 2.61702i − 1.91168i
86.6 −1.11881 + 1.11881i 0.455401 − 1.67111i − 0.503463i 0.707107 − 0.707107i 1.36015 + 2.37916i 1.46633 − 1.46633i −1.67434 − 1.67434i −2.58522 − 1.52205i 1.58223i
86.7 −0.789301 + 0.789301i −1.64291 − 0.548503i 0.754007i 0.707107 − 0.707107i 1.72968 − 0.863815i −1.97746 + 1.97746i −2.17374 − 2.17374i 2.39829 + 1.80228i 1.11624i
86.8 −0.483812 + 0.483812i 1.53965 + 0.793397i 1.53185i −0.707107 + 0.707107i −1.12876 + 0.361045i −1.24531 + 1.24531i −1.70875 − 1.70875i 1.74104 + 2.44311i − 0.684213i
86.9 −0.455718 + 0.455718i −0.446465 − 1.67352i 1.58464i −0.707107 + 0.707107i 0.966114 + 0.559191i −2.89711 + 2.89711i −1.63358 − 1.63358i −2.60134 + 1.49434i − 0.644482i
86.10 −0.260415 + 0.260415i 1.18585 + 1.26244i 1.86437i 0.707107 − 0.707107i −0.637571 − 0.0199472i 2.54331 − 2.54331i −1.00634 − 1.00634i −0.187534 + 2.99413i 0.368282i
86.11 0.260415 − 0.260415i 1.18585 − 1.26244i 1.86437i −0.707107 + 0.707107i −0.0199472 − 0.637571i 2.54331 − 2.54331i 1.00634 + 1.00634i −0.187534 − 2.99413i 0.368282i
86.12 0.455718 − 0.455718i −0.446465 + 1.67352i 1.58464i 0.707107 − 0.707107i 0.559191 + 0.966114i −2.89711 + 2.89711i 1.63358 + 1.63358i −2.60134 − 1.49434i − 0.644482i
86.13 0.483812 − 0.483812i 1.53965 − 0.793397i 1.53185i 0.707107 − 0.707107i 0.361045 − 1.12876i −1.24531 + 1.24531i 1.70875 + 1.70875i 1.74104 − 2.44311i − 0.684213i
86.14 0.789301 − 0.789301i −1.64291 + 0.548503i 0.754007i −0.707107 + 0.707107i −0.863815 + 1.72968i −1.97746 + 1.97746i 2.17374 + 2.17374i 2.39829 − 1.80228i 1.11624i
86.15 1.11881 − 1.11881i 0.455401 + 1.67111i − 0.503463i −0.707107 + 0.707107i 2.37916 + 1.36015i 1.46633 − 1.46633i 1.67434 + 1.67434i −2.58522 + 1.52205i 1.58223i
86.16 1.35176 − 1.35176i −1.49444 + 0.875587i − 1.65452i 0.707107 − 0.707107i −0.836541 + 3.20371i 3.04341 − 3.04341i 0.467005 + 0.467005i 1.46670 − 2.61702i − 1.91168i
86.17 1.43871 − 1.43871i −0.158548 − 1.72478i − 2.13978i 0.707107 − 0.707107i −2.70956 − 2.25335i −1.21328 + 1.21328i −0.201099 − 0.201099i −2.94973 + 0.546921i − 2.03464i
86.18 1.57927 − 1.57927i 1.73010 − 0.0822389i − 2.98819i −0.707107 + 0.707107i 2.60241 − 2.86217i −2.29263 + 2.29263i −1.56062 − 1.56062i 2.98647 − 0.284563i 2.23343i
86.19 1.89167 − 1.89167i −1.62902 − 0.588468i − 5.15683i −0.707107 + 0.707107i −4.19475 + 1.96838i 0.157581 − 0.157581i −5.97167 − 5.97167i 2.30741 + 1.91725i 2.67522i
86.20 1.90946 − 1.90946i 0.460384 + 1.66974i − 5.29209i 0.707107 − 0.707107i 4.06740 + 2.30923i −1.58484 + 1.58484i −6.28612 − 6.28612i −2.57609 + 1.53745i − 2.70039i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.20
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.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.o.a ✓ 40
3.b odd 2 1 inner 195.2.o.a ✓ 40
5.b even 2 1 975.2.o.p 40
5.c odd 4 1 975.2.n.q 40
5.c odd 4 1 975.2.n.r 40
13.d odd 4 1 inner 195.2.o.a ✓ 40
15.d odd 2 1 975.2.o.p 40
15.e even 4 1 975.2.n.q 40
15.e even 4 1 975.2.n.r 40
39.f even 4 1 inner 195.2.o.a ✓ 40
65.f even 4 1 975.2.n.q 40
65.g odd 4 1 975.2.o.p 40
65.k even 4 1 975.2.n.r 40
195.j odd 4 1 975.2.n.r 40
195.n even 4 1 975.2.o.p 40
195.u odd 4 1 975.2.n.q 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.o.a ✓ 40 1.a even 1 1 trivial
195.2.o.a ✓ 40 3.b odd 2 1 inner
195.2.o.a ✓ 40 13.d odd 4 1 inner
195.2.o.a ✓ 40 39.f even 4 1 inner
975.2.n.q 40 5.c odd 4 1
975.2.n.q 40 15.e even 4 1
975.2.n.q 40 65.f even 4 1
975.2.n.q 40 195.u odd 4 1
975.2.n.r 40 5.c odd 4 1
975.2.n.r 40 15.e even 4 1
975.2.n.r 40 65.k even 4 1
975.2.n.r 40 195.j odd 4 1
975.2.o.p 40 5.b even 2 1
975.2.o.p 40 15.d odd 2 1
975.2.o.p 40 65.g odd 4 1
975.2.o.p 40 195.n even 4 1

Hecke kernels

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