Properties

Label 195.2.bg.a
Level $195$
Weight $2$
Character orbit 195.bg
Analytic conductor $1.557$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,2,Mod(11,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.bg (of order \(12\), degree \(4\), 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: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{6} + 12 q^{7} - 4 q^{15} + 24 q^{16} - 40 q^{18} - 12 q^{19} - 36 q^{21} - 56 q^{24} + 24 q^{27} - 16 q^{28} - 48 q^{30} - 28 q^{31} - 8 q^{33} + 40 q^{34} + 12 q^{36} - 4 q^{37} + 32 q^{39} + 64 q^{42} - 84 q^{43} + 8 q^{45} + 64 q^{46} + 64 q^{48} - 84 q^{49} - 200 q^{52} + 32 q^{54} - 8 q^{55} + 36 q^{57} - 120 q^{58} - 16 q^{60} - 8 q^{61} - 8 q^{63} + 120 q^{66} + 64 q^{67} + 72 q^{69} + 24 q^{72} + 60 q^{73} - 8 q^{76} + 12 q^{78} - 16 q^{79} + 12 q^{81} + 120 q^{82} + 44 q^{84} - 48 q^{85} + 8 q^{87} - 144 q^{88} - 12 q^{91} - 36 q^{93} + 8 q^{94} + 4 q^{96} + 4 q^{97} - 40 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.45997 0.659148i −0.394058 1.68663i 3.88494 + 2.24297i −0.707107 + 0.707107i −0.142367 + 4.40880i −0.856074 3.19491i −4.47674 4.47674i −2.68944 + 1.32926i 2.20555 1.27338i
11.2 −2.29212 0.614171i 1.55515 + 0.762566i 3.14455 + 1.81551i 0.707107 0.707107i −3.09624 2.70302i −1.17252 4.37590i −2.73676 2.73676i 1.83698 + 2.37181i −2.05506 + 1.18649i
11.3 −2.13499 0.572069i −1.66401 0.480688i 2.49887 + 1.44273i 0.707107 0.707107i 3.27767 + 1.97820i 0.438491 + 1.63647i −1.38389 1.38389i 2.53788 + 1.59974i −1.91418 + 1.10515i
11.4 −2.06524 0.553380i 1.71528 0.240414i 2.22694 + 1.28572i −0.707107 + 0.707107i −3.67552 0.452691i 0.858972 + 3.20573i −0.863953 0.863953i 2.88440 0.824757i 1.85164 1.06905i
11.5 −1.47996 0.396555i −0.525689 + 1.65035i 0.300984 + 0.173773i 0.707107 0.707107i 1.43245 2.23399i 0.218736 + 0.816332i 1.79028 + 1.79028i −2.44730 1.73514i −1.32690 + 0.766085i
11.6 −1.10861 0.297051i −1.55196 + 0.769034i −0.591272 0.341371i −0.707107 + 0.707107i 1.94897 0.391547i −0.0589315 0.219935i 2.17721 + 2.17721i 1.81717 2.38702i 0.993954 0.573859i
11.7 −0.584891 0.156721i 1.60715 + 0.645804i −1.41451 0.816670i 0.707107 0.707107i −0.838798 0.629600i 1.15475 + 4.30957i 1.55569 + 1.55569i 2.16587 + 2.07581i −0.524399 + 0.302762i
11.8 −0.433679 0.116204i 1.50707 0.853659i −1.55748 0.899210i −0.707107 + 0.707107i −0.752784 + 0.195086i −0.961038 3.58664i 1.20590 + 1.20590i 1.54253 2.57305i 0.388826 0.224489i
11.9 −0.424461 0.113734i −1.34964 1.08557i −1.56482 0.903449i −0.707107 + 0.707107i 0.449405 + 0.614282i 0.145569 + 0.543270i 1.18291 + 1.18291i 0.643078 + 2.93026i 0.380561 0.219717i
11.10 0.424461 + 0.113734i −0.265309 1.71161i −1.56482 0.903449i 0.707107 0.707107i 0.0820551 0.756687i 0.145569 + 0.543270i −1.18291 1.18291i −2.85922 + 0.908210i 0.380561 0.219717i
11.11 0.433679 + 0.116204i −1.49283 + 0.878333i −1.55748 0.899210i 0.707107 0.707107i −0.749473 + 0.207442i −0.961038 3.58664i −1.20590 1.20590i 1.45706 2.62240i 0.388826 0.224489i
11.12 0.584891 + 0.156721i −0.244293 + 1.71474i −1.41451 0.816670i −0.707107 + 0.707107i −0.411620 + 0.964648i 1.15475 + 4.30957i −1.55569 1.55569i −2.88064 0.837797i −0.524399 + 0.302762i
11.13 1.10861 + 0.297051i 1.44198 0.959522i −0.591272 0.341371i 0.707107 0.707107i 1.88363 0.635393i −0.0589315 0.219935i −2.17721 2.17721i 1.15864 2.76723i 0.993954 0.573859i
11.14 1.47996 + 0.396555i 1.69209 + 0.369915i 0.300984 + 0.173773i −0.707107 + 0.707107i 2.35754 + 1.21847i 0.218736 + 0.816332i −1.79028 1.79028i 2.72633 + 1.25186i −1.32690 + 0.766085i
11.15 2.06524 + 0.553380i −1.06585 + 1.36527i 2.22694 + 1.28572i 0.707107 0.707107i −2.95675 + 2.22980i 0.858972 + 3.20573i 0.863953 + 0.863953i −0.727941 2.91034i 1.85164 1.06905i
11.16 2.13499 + 0.572069i 0.415718 1.68142i 2.49887 + 1.44273i −0.707107 + 0.707107i 1.84944 3.35200i 0.438491 + 1.63647i 1.38389 + 1.38389i −2.65436 1.39800i −1.91418 + 1.10515i
11.17 2.29212 + 0.614171i −0.117173 + 1.72808i 3.14455 + 1.81551i −0.707107 + 0.707107i −1.32991 + 3.88901i −1.17252 4.37590i 2.73676 + 2.73676i −2.97254 0.404970i −2.05506 + 1.18649i
11.18 2.45997 + 0.659148i −1.26364 1.18458i 3.88494 + 2.24297i 0.707107 0.707107i −2.32769 3.74695i −0.856074 3.19491i 4.47674 + 4.47674i 0.193547 + 2.99375i 2.20555 1.27338i
41.1 −0.698381 + 2.60640i −1.27348 1.17399i −4.57351 2.64052i −0.707107 0.707107i 3.94925 2.49931i 2.70283 0.724221i 6.26025 6.26025i 0.243511 + 2.99010i 2.33683 1.34917i
41.2 −0.649589 + 2.42430i −1.00663 + 1.40950i −3.72321 2.14960i 0.707107 + 0.707107i −2.76316 3.35597i −2.77349 + 0.743154i 4.08041 4.08041i −0.973391 2.83769i −2.17357 + 1.25491i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k 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.bg.a 72
3.b odd 2 1 inner 195.2.bg.a 72
5.b even 2 1 975.2.bo.e 72
5.c odd 4 1 975.2.bp.h 72
5.c odd 4 1 975.2.bp.i 72
13.f odd 12 1 inner 195.2.bg.a 72
15.d odd 2 1 975.2.bo.e 72
15.e even 4 1 975.2.bp.h 72
15.e even 4 1 975.2.bp.i 72
39.k even 12 1 inner 195.2.bg.a 72
65.o even 12 1 975.2.bp.h 72
65.s odd 12 1 975.2.bo.e 72
65.t even 12 1 975.2.bp.i 72
195.bc odd 12 1 975.2.bp.i 72
195.bh even 12 1 975.2.bo.e 72
195.bn odd 12 1 975.2.bp.h 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bg.a 72 1.a even 1 1 trivial
195.2.bg.a 72 3.b odd 2 1 inner
195.2.bg.a 72 13.f odd 12 1 inner
195.2.bg.a 72 39.k even 12 1 inner
975.2.bo.e 72 5.b even 2 1
975.2.bo.e 72 15.d odd 2 1
975.2.bo.e 72 65.s odd 12 1
975.2.bo.e 72 195.bh even 12 1
975.2.bp.h 72 5.c odd 4 1
975.2.bp.h 72 15.e even 4 1
975.2.bp.h 72 65.o even 12 1
975.2.bp.h 72 195.bn odd 12 1
975.2.bp.i 72 5.c odd 4 1
975.2.bp.i 72 15.e even 4 1
975.2.bp.i 72 65.t even 12 1
975.2.bp.i 72 195.bc odd 12 1

Hecke kernels

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