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Properties

Label 231.2.l.b
Level $231$
Weight $2$
Character orbit 231.l
Analytic conductor $1.845$
Analytic rank $0$
Dimension $48$
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] = [231,2,Mod(32,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.l (of order \(6\), 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: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9} + 14 q^{12} + 4 q^{15} - 28 q^{16} - 24 q^{22} - 36 q^{27} - 4 q^{31} - 26 q^{33} + 56 q^{34} + 12 q^{36} - 28 q^{37} + 96 q^{42} + 30 q^{45} - 72 q^{48} + 24 q^{49} - 80 q^{55} + 16 q^{58} - 32 q^{60} + 256 q^{64} - 42 q^{66} + 16 q^{67} + 36 q^{69} - 76 q^{70} + 34 q^{75} + 104 q^{78} - 26 q^{81} - 4 q^{82} - 16 q^{88} - 36 q^{91} + 52 q^{93} - 144 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −1.36539 − 2.36492i −0.516280 − 1.65332i −2.72858 + 4.72604i −1.96785 + 1.13614i −3.20504 + 3.47838i −1.90462 − 1.83641i 9.44073 −2.46691 + 1.70715i 5.37375 + 3.10254i
32.2 −1.36539 − 2.36492i 1.68995 − 0.379546i −2.72858 + 4.72604i 1.96785 − 1.13614i −3.20504 − 3.47838i 1.90462 + 1.83641i 9.44073 2.71189 − 1.28283i −5.37375 − 3.10254i
32.3 −1.19873 − 2.07626i −1.49003 + 0.883063i −1.87390 + 3.24569i −0.707611 + 0.408540i 3.61961 + 2.03514i 2.64562 + 0.0267152i 4.19028 1.44040 − 2.63159i 1.69647 + 0.979456i
32.4 −1.19873 − 2.07626i −0.0197381 + 1.73194i −1.87390 + 3.24569i 0.707611 − 0.408540i 3.61961 − 2.03514i −2.64562 − 0.0267152i 4.19028 −2.99922 − 0.0683703i −1.69647 − 0.979456i
32.5 −0.939478 − 1.62722i −1.69195 − 0.370556i −0.765237 + 1.32543i 2.63713 − 1.52255i 0.986571 + 3.10131i −0.738610 − 2.54056i −0.882219 2.72538 + 1.25392i −4.95505 − 2.86080i
32.6 −0.939478 − 1.62722i 1.16688 + 1.27999i −0.765237 + 1.32543i −2.63713 + 1.52255i 0.986571 − 3.10131i 0.738610 + 2.54056i −0.882219 −0.276760 + 2.98721i 4.95505 + 2.86080i
32.7 −0.794537 − 1.37618i 0.668961 − 1.59765i −0.262580 + 0.454801i 1.14794 − 0.662766i −2.73017 + 0.348784i 2.48641 − 0.904309i −2.34363 −2.10498 − 2.13753i −1.82417 − 1.05318i
32.8 −0.794537 − 1.37618i 1.04913 − 1.37816i −0.262580 + 0.454801i −1.14794 + 0.662766i −2.73017 − 0.348784i −2.48641 + 0.904309i −2.34363 −0.798666 − 2.89174i 1.82417 + 1.05318i
32.9 −0.468581 − 0.811606i −1.03189 + 1.39112i 0.560864 − 0.971445i 2.73832 − 1.58097i 1.61256 + 0.185636i −0.0402023 + 2.64545i −2.92556 −0.870410 − 2.87096i −2.56625 − 1.48162i
32.10 −0.468581 − 0.811606i −0.688798 + 1.58920i 0.560864 − 0.971445i −2.73832 + 1.58097i 1.61256 − 0.185636i 0.0402023 − 2.64545i −2.92556 −2.05112 − 2.18928i 2.56625 + 1.48162i
32.11 −0.463987 − 0.803650i −1.71325 − 0.254482i 0.569432 − 0.986284i −1.53501 + 0.886239i 0.590414 + 1.49493i −2.26798 + 1.36244i −2.91279 2.87048 + 0.871983i 1.42445 + 0.822407i
32.12 −0.463987 − 0.803650i 1.07701 + 1.35648i 0.569432 − 0.986284i 1.53501 − 0.886239i 0.590414 − 1.49493i 2.26798 − 1.36244i −2.91279 −0.680079 + 2.92190i −1.42445 − 0.822407i
32.13 0.463987 + 0.803650i −1.71325 − 0.254482i 0.569432 − 0.986284i −1.53501 + 0.886239i −0.590414 − 1.49493i 2.26798 − 1.36244i 2.91279 2.87048 + 0.871983i −1.42445 − 0.822407i
32.14 0.463987 + 0.803650i 1.07701 + 1.35648i 0.569432 − 0.986284i 1.53501 − 0.886239i −0.590414 + 1.49493i −2.26798 + 1.36244i 2.91279 −0.680079 + 2.92190i 1.42445 + 0.822407i
32.15 0.468581 + 0.811606i −1.03189 + 1.39112i 0.560864 − 0.971445i 2.73832 − 1.58097i −1.61256 − 0.185636i 0.0402023 − 2.64545i 2.92556 −0.870410 − 2.87096i 2.56625 + 1.48162i
32.16 0.468581 + 0.811606i −0.688798 + 1.58920i 0.560864 − 0.971445i −2.73832 + 1.58097i −1.61256 + 0.185636i −0.0402023 + 2.64545i 2.92556 −2.05112 − 2.18928i −2.56625 − 1.48162i
32.17 0.794537 + 1.37618i 0.668961 − 1.59765i −0.262580 + 0.454801i 1.14794 − 0.662766i 2.73017 − 0.348784i −2.48641 + 0.904309i 2.34363 −2.10498 − 2.13753i 1.82417 + 1.05318i
32.18 0.794537 + 1.37618i 1.04913 − 1.37816i −0.262580 + 0.454801i −1.14794 + 0.662766i 2.73017 + 0.348784i 2.48641 − 0.904309i 2.34363 −0.798666 − 2.89174i −1.82417 − 1.05318i
32.19 0.939478 + 1.62722i −1.69195 − 0.370556i −0.765237 + 1.32543i 2.63713 − 1.52255i −0.986571 − 3.10131i 0.738610 + 2.54056i 0.882219 2.72538 + 1.25392i 4.95505 + 2.86080i
32.20 0.939478 + 1.62722i 1.16688 + 1.27999i −0.765237 + 1.32543i −2.63713 + 1.52255i −0.986571 + 3.10131i −0.738610 − 2.54056i 0.882219 −0.276760 + 2.98721i −4.95505 − 2.86080i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
11.b odd 2 1 inner
21.h odd 6 1 inner
33.d even 2 1 inner
77.h odd 6 1 inner
231.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.l.b ✓ 48
3.b odd 2 1 inner 231.2.l.b ✓ 48
7.c even 3 1 inner 231.2.l.b ✓ 48
11.b odd 2 1 inner 231.2.l.b ✓ 48
21.h odd 6 1 inner 231.2.l.b ✓ 48
33.d even 2 1 inner 231.2.l.b ✓ 48
77.h odd 6 1 inner 231.2.l.b ✓ 48
231.l even 6 1 inner 231.2.l.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.l.b ✓ 48 1.a even 1 1 trivial
231.2.l.b ✓ 48 3.b odd 2 1 inner
231.2.l.b ✓ 48 7.c even 3 1 inner
231.2.l.b ✓ 48 11.b odd 2 1 inner
231.2.l.b ✓ 48 21.h odd 6 1 inner
231.2.l.b ✓ 48 33.d even 2 1 inner
231.2.l.b ✓ 48 77.h odd 6 1 inner
231.2.l.b ✓ 48 231.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 21 T_{2}^{22} + 275 T_{2}^{20} + 2244 T_{2}^{18} + 13377 T_{2}^{16} + 56240 T_{2}^{14} + \cdots + 83521 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display