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Properties

Label 195.2.s.b
Level $195$
Weight $2$
Character orbit 195.s
Analytic conductor $1.557$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,2,Mod(38,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.38");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.s (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} - 24 q^{10} - 24 q^{12} + 24 q^{16} + 24 q^{22} - 8 q^{25} - 16 q^{27} + 36 q^{30} - 8 q^{36} + 16 q^{40} + 12 q^{42} - 64 q^{43} - 20 q^{48} + 16 q^{51} - 72 q^{52} - 80 q^{55} + 8 q^{61} - 72 q^{66} + 44 q^{75} + 84 q^{78} + 112 q^{81} + 48 q^{82} + 20 q^{87} - 8 q^{88} + 44 q^{90} - 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −1.57280 + 1.57280i −1.73164 + 0.0375613i − 2.94741i 2.12590 + 0.693215i 2.66446 − 2.78261i 3.19872 + 3.19872i 1.49009 + 1.49009i 2.99718 − 0.130086i −4.43391 + 2.25333i
38.2 −1.57280 + 1.57280i 0.0375613 − 1.73164i − 2.94741i 2.12590 + 0.693215i 2.66446 + 2.78261i −3.19872 − 3.19872i 1.49009 + 1.49009i −2.99718 − 0.130086i −4.43391 + 2.25333i
38.3 −1.25684 + 1.25684i −1.06336 + 1.36721i − 1.15927i −0.261006 − 2.22078i −0.381882 − 3.05483i −1.60133 − 1.60133i −1.05665 − 1.05665i −0.738515 − 2.90768i 3.11920 + 2.46312i
38.4 −1.25684 + 1.25684i 1.36721 − 1.06336i − 1.15927i −0.261006 − 2.22078i −0.381882 + 3.05483i 1.60133 + 1.60133i −1.05665 − 1.05665i 0.738515 − 2.90768i 3.11920 + 2.46312i
38.5 −0.597551 + 0.597551i −1.67038 + 0.458083i 1.28586i −0.289977 + 2.21719i 0.724408 − 1.27186i −2.39819 − 2.39819i −1.96347 − 1.96347i 2.58032 − 1.53034i −1.15161 − 1.49816i
38.6 −0.597551 + 0.597551i 0.458083 − 1.67038i 1.28586i −0.289977 + 2.21719i 0.724408 + 1.27186i 2.39819 + 2.39819i −1.96347 − 1.96347i −2.58032 − 1.53034i −1.15161 − 1.49816i
38.7 −0.299314 + 0.299314i −0.125002 + 1.72753i 1.82082i 2.19735 − 0.414323i −0.479660 − 0.554490i 0.976029 + 0.976029i −1.14363 − 1.14363i −2.96875 − 0.431892i −0.533684 + 0.781709i
38.8 −0.299314 + 0.299314i 1.72753 − 0.125002i 1.82082i 2.19735 − 0.414323i −0.479660 + 0.554490i −0.976029 − 0.976029i −1.14363 − 1.14363i 2.96875 − 0.431892i −0.533684 + 0.781709i
38.9 0.299314 − 0.299314i −0.125002 + 1.72753i 1.82082i −2.19735 + 0.414323i 0.479660 + 0.554490i −0.976029 − 0.976029i 1.14363 + 1.14363i −2.96875 − 0.431892i −0.533684 + 0.781709i
38.10 0.299314 − 0.299314i 1.72753 − 0.125002i 1.82082i −2.19735 + 0.414323i 0.479660 − 0.554490i 0.976029 + 0.976029i 1.14363 + 1.14363i 2.96875 − 0.431892i −0.533684 + 0.781709i
38.11 0.597551 − 0.597551i −1.67038 + 0.458083i 1.28586i 0.289977 − 2.21719i −0.724408 + 1.27186i 2.39819 + 2.39819i 1.96347 + 1.96347i 2.58032 − 1.53034i −1.15161 − 1.49816i
38.12 0.597551 − 0.597551i 0.458083 − 1.67038i 1.28586i 0.289977 − 2.21719i −0.724408 − 1.27186i −2.39819 − 2.39819i 1.96347 + 1.96347i −2.58032 − 1.53034i −1.15161 − 1.49816i
38.13 1.25684 − 1.25684i −1.06336 + 1.36721i − 1.15927i 0.261006 + 2.22078i 0.381882 + 3.05483i 1.60133 + 1.60133i 1.05665 + 1.05665i −0.738515 − 2.90768i 3.11920 + 2.46312i
38.14 1.25684 − 1.25684i 1.36721 − 1.06336i − 1.15927i 0.261006 + 2.22078i 0.381882 − 3.05483i −1.60133 − 1.60133i 1.05665 + 1.05665i 0.738515 − 2.90768i 3.11920 + 2.46312i
38.15 1.57280 − 1.57280i −1.73164 + 0.0375613i − 2.94741i −2.12590 − 0.693215i −2.66446 + 2.78261i −3.19872 − 3.19872i −1.49009 − 1.49009i 2.99718 − 0.130086i −4.43391 + 2.25333i
38.16 1.57280 − 1.57280i 0.0375613 − 1.73164i − 2.94741i −2.12590 − 0.693215i −2.66446 − 2.78261i 3.19872 + 3.19872i −1.49009 − 1.49009i −2.99718 − 0.130086i −4.43391 + 2.25333i
77.1 −1.57280 − 1.57280i −1.73164 − 0.0375613i 2.94741i 2.12590 − 0.693215i 2.66446 + 2.78261i 3.19872 − 3.19872i 1.49009 − 1.49009i 2.99718 + 0.130086i −4.43391 − 2.25333i
77.2 −1.57280 − 1.57280i 0.0375613 + 1.73164i 2.94741i 2.12590 − 0.693215i 2.66446 − 2.78261i −3.19872 + 3.19872i 1.49009 − 1.49009i −2.99718 + 0.130086i −4.43391 − 2.25333i
77.3 −1.25684 − 1.25684i −1.06336 − 1.36721i 1.15927i −0.261006 + 2.22078i −0.381882 + 3.05483i −1.60133 + 1.60133i −1.05665 + 1.05665i −0.738515 + 2.90768i 3.11920 − 2.46312i
77.4 −1.25684 − 1.25684i 1.36721 + 1.06336i 1.15927i −0.261006 + 2.22078i −0.381882 − 3.05483i 1.60133 − 1.60133i −1.05665 + 1.05665i 0.738515 + 2.90768i 3.11920 − 2.46312i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.16
Significant digits:
Format:

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.b even 2 1 inner
15.e even 4 1 inner
39.d odd 2 1 inner
65.h odd 4 1 inner
195.s 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.s.b ✓ 32
3.b odd 2 1 inner 195.2.s.b ✓ 32
5.b even 2 1 975.2.s.e 32
5.c odd 4 1 inner 195.2.s.b ✓ 32
5.c odd 4 1 975.2.s.e 32
13.b even 2 1 inner 195.2.s.b ✓ 32
15.d odd 2 1 975.2.s.e 32
15.e even 4 1 inner 195.2.s.b ✓ 32
15.e even 4 1 975.2.s.e 32
39.d odd 2 1 inner 195.2.s.b ✓ 32
65.d even 2 1 975.2.s.e 32
65.h odd 4 1 inner 195.2.s.b ✓ 32
65.h odd 4 1 975.2.s.e 32
195.e odd 2 1 975.2.s.e 32
195.s even 4 1 inner 195.2.s.b ✓ 32
195.s even 4 1 975.2.s.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.s.b ✓ 32 1.a even 1 1 trivial
195.2.s.b ✓ 32 3.b odd 2 1 inner
195.2.s.b ✓ 32 5.c odd 4 1 inner
195.2.s.b ✓ 32 13.b even 2 1 inner
195.2.s.b ✓ 32 15.e even 4 1 inner
195.2.s.b ✓ 32 39.d odd 2 1 inner
195.2.s.b ✓ 32 65.h odd 4 1 inner
195.2.s.b ✓ 32 195.s even 4 1 inner
975.2.s.e 32 5.b even 2 1
975.2.s.e 32 5.c odd 4 1
975.2.s.e 32 15.d odd 2 1
975.2.s.e 32 15.e even 4 1
975.2.s.e 32 65.d even 2 1
975.2.s.e 32 65.h odd 4 1
975.2.s.e 32 195.e odd 2 1
975.2.s.e 32 195.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 35T_{2}^{12} + 263T_{2}^{8} + 133T_{2}^{4} + 4 \) acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\). Copy content Toggle raw display