↑

Properties

Label 231.2.s.a
Level $231$
Weight $2$
Character orbit 231.s
Analytic conductor $1.845$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(8,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.s (of order \(10\), 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: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{3} - 24 q^{4} - 10 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{3} - 24 q^{4} - 10 q^{6} - 20 q^{9} - 20 q^{12} - 32 q^{16} - 10 q^{18} - 60 q^{19} + 30 q^{24} + 12 q^{25} - 40 q^{28} + 60 q^{30} + 40 q^{31} - 44 q^{33} + 8 q^{34} + 2 q^{36} + 16 q^{37} - 10 q^{39} - 96 q^{45} - 40 q^{46} + 48 q^{48} + 24 q^{49} - 30 q^{51} - 40 q^{52} + 28 q^{55} - 30 q^{57} + 36 q^{58} - 32 q^{60} + 40 q^{61} - 28 q^{64} + 118 q^{66} - 56 q^{67} + 26 q^{69} + 20 q^{70} + 150 q^{72} + 40 q^{73} + 2 q^{75} - 20 q^{78} - 40 q^{79} + 8 q^{81} - 16 q^{82} + 40 q^{84} - 60 q^{85} - 156 q^{88} + 100 q^{90} + 36 q^{91} - 36 q^{93} + 160 q^{96} - 88 q^{97} - 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.24267 + 1.62939i 1.07372 − 1.35909i 1.75661 − 5.40628i 0.476086 − 0.655276i −0.193509 + 4.79750i −0.951057 − 0.309017i 3.15622 + 9.71386i −0.694246 − 2.91857i 2.24530i
8.2 −2.11971 + 1.54006i 0.569669 + 1.63569i 1.50336 − 4.62686i 0.853755 − 1.17509i −3.72660 − 2.58986i 0.951057 + 0.309017i 2.31965 + 7.13914i −2.35095 + 1.86360i 3.80570i
8.3 −1.67788 + 1.21905i −1.27625 + 1.17097i 0.711169 − 2.18875i 0.764082 − 1.05167i 0.713926 − 3.52058i −0.951057 − 0.309017i 0.193159 + 0.594483i 0.257645 − 2.98892i 2.69603i
8.4 −1.65451 + 1.20207i −1.40677 − 1.01044i 0.674386 − 2.07555i 0.537508 − 0.739816i 3.54213 − 0.0192577i −0.951057 − 0.309017i 0.115245 + 0.354686i 0.958019 + 2.84292i 1.87015i
8.5 −1.65450 + 1.20206i −0.288346 − 1.70788i 0.674376 − 2.07552i −1.12543 + 1.54901i 2.53005 + 2.47908i 0.951057 + 0.309017i 0.115224 + 0.354623i −2.83371 + 0.984921i − 3.91568i
8.6 −1.61087 + 1.17037i 1.32659 + 1.11363i 0.607116 − 1.86851i −2.20177 + 3.03048i −3.44031 − 0.241324i −0.951057 − 0.309017i −0.0217402 − 0.0669093i 0.519657 + 2.95465i − 7.45860i
8.7 −1.32709 + 0.964186i 1.73091 + 0.0627395i 0.213475 − 0.657007i 1.68710 − 2.32209i −2.35757 + 1.58566i 0.951057 + 0.309017i −0.663628 − 2.04244i 2.99213 + 0.217193i 4.70830i
8.8 −0.723708 + 0.525805i −1.64598 − 0.539221i −0.370751 + 1.14105i −0.915619 + 1.26024i 1.47473 − 0.475224i 0.951057 + 0.309017i −0.884520 − 2.72227i 2.41848 + 1.77509i − 1.39348i
8.9 −0.699320 + 0.508086i 0.0766728 + 1.73035i −0.387136 + 1.19148i −1.26605 + 1.74257i −0.932787 − 1.17111i 0.951057 + 0.309017i −0.868877 − 2.67413i −2.98824 + 0.265342i − 1.86187i
8.10 −0.487641 + 0.354292i −1.64748 + 0.534620i −0.505763 + 1.55658i 2.33078 − 3.20804i 0.613966 − 0.844391i 0.951057 + 0.309017i −0.677377 − 2.08475i 2.42836 − 1.76155i 2.39015i
8.11 −0.372543 + 0.270668i −0.457102 − 1.67065i −0.552507 + 1.70044i 1.99416 − 2.74473i 0.622481 + 0.498664i −0.951057 − 0.309017i −0.539021 − 1.65893i −2.58212 + 1.52731i 1.56229i
8.12 −0.0636021 + 0.0462097i 1.52227 + 0.826261i −0.616124 + 1.89623i 0.403747 − 0.555710i −0.135001 + 0.0177914i −0.951057 − 0.309017i −0.0970253 − 0.298613i 1.63459 + 2.51558i 0.0540014i
8.13 0.0636021 − 0.0462097i −0.745875 + 1.56322i −0.616124 + 1.89623i −0.403747 + 0.555710i 0.0247968 + 0.133891i −0.951057 − 0.309017i 0.0970253 + 0.298613i −1.88734 − 2.33194i 0.0540014i
8.14 0.372543 − 0.270668i −0.612178 − 1.62026i −0.552507 + 1.70044i −1.99416 + 2.74473i −0.666615 − 0.437919i −0.951057 − 0.309017i 0.539021 + 1.65893i −2.25048 + 1.98377i 1.56229i
8.15 0.487641 − 0.354292i 1.64708 − 0.535846i −0.505763 + 1.55658i −2.33078 + 3.20804i 0.613338 − 0.844848i 0.951057 + 0.309017i 0.677377 + 2.08475i 2.42574 − 1.76516i 2.39015i
8.16 0.699320 − 0.508086i 0.955046 + 1.44495i −0.387136 + 1.19148i 1.26605 − 1.74257i 1.40204 + 0.525239i 0.951057 + 0.309017i 0.868877 + 2.67413i −1.17577 + 2.75999i − 1.86187i
8.17 0.723708 − 0.525805i 1.01468 − 1.40372i −0.370751 + 1.14105i 0.915619 − 1.26024i −0.00375218 − 1.54941i 0.951057 + 0.309017i 0.884520 + 2.72227i −0.940859 − 2.84865i − 1.39348i
8.18 1.32709 − 0.964186i −1.36346 + 1.06816i 0.213475 − 0.657007i −1.68710 + 2.32209i −0.779526 + 2.73218i 0.951057 + 0.309017i 0.663628 + 2.04244i 0.718055 − 2.91280i 4.70830i
8.19 1.61087 − 1.17037i −0.418655 + 1.68069i 0.607116 − 1.86851i 2.20177 − 3.03048i 1.29263 + 3.19736i −0.951057 − 0.309017i 0.0217402 + 0.0669093i −2.64946 − 1.40726i − 7.45860i
8.20 1.65450 − 1.20206i −0.770590 − 1.55119i 0.674376 − 2.07552i 1.12543 − 1.54901i −3.13957 − 1.64014i 0.951057 + 0.309017i −0.115224 − 0.354623i −1.81238 + 2.39066i − 3.91568i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.s.a ✓ 96
3.b odd 2 1 inner 231.2.s.a ✓ 96
11.d odd 10 1 inner 231.2.s.a ✓ 96
33.f even 10 1 inner 231.2.s.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.s.a ✓ 96 1.a even 1 1 trivial
231.2.s.a ✓ 96 3.b odd 2 1 inner
231.2.s.a ✓ 96 11.d odd 10 1 inner
231.2.s.a ✓ 96 33.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).