↑

Properties

Label 231.2.j.f
Level $231$
Weight $2$
Character orbit 231.j
Analytic conductor $1.845$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(64,231)]
 
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([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.j (of order \(5\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} - \beta_{3}) q^{2} + (\beta_{7} + \beta_{4} - \beta_{3} + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{4} + 1) q^{4} + ( - 2 \beta_{6} + 2 \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{6}+ \cdots - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} - \beta_{3}) q^{2} + (\beta_{7} + \beta_{4} - \beta_{3} + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{4} + 1) q^{4} + ( - 2 \beta_{6} + 2 \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{6}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{9} + 20 q^{10} - 22 q^{11} - 6 q^{12} - 8 q^{13} + 3 q^{14} - 2 q^{15} + 4 q^{16} - 4 q^{17} - 3 q^{18} + 20 q^{20} + 8 q^{21} - 8 q^{22} - 20 q^{23} - 7 q^{24} - 26 q^{25} - 10 q^{26} + 2 q^{27} + 9 q^{28} + 24 q^{31} - 4 q^{32} - 18 q^{33} + 36 q^{34} - 2 q^{35} + 6 q^{36} + 6 q^{37} + 14 q^{38} + 8 q^{39} + 12 q^{40} + 20 q^{41} + 2 q^{42} - 8 q^{43} - 39 q^{44} - 8 q^{45} - 43 q^{46} - 22 q^{47} + q^{48} - 2 q^{49} + 22 q^{50} + 4 q^{51} + 20 q^{52} - 20 q^{53} - 2 q^{54} + 2 q^{55} + 18 q^{56} - 10 q^{57} - 17 q^{58} + 18 q^{59} - 20 q^{60} - 2 q^{61} + 20 q^{62} + 2 q^{63} + 18 q^{64} - 56 q^{65} + 3 q^{66} - 56 q^{67} - 2 q^{68} - 10 q^{69} + 14 q^{71} + 7 q^{72} + 2 q^{73} - 12 q^{74} - 14 q^{75} - 8 q^{76} + 2 q^{77} + 40 q^{78} + 20 q^{79} + 38 q^{80} - 2 q^{81} + 2 q^{82} - 8 q^{83} + 6 q^{84} + 60 q^{85} + 55 q^{86} - 38 q^{88} - 32 q^{89} - 10 q^{90} - 2 q^{91} - 9 q^{92} - 24 q^{93} + 48 q^{94} - 28 q^{95} + 4 q^{96} + 4 q^{97} + 2 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.418926 − 1.28932i
−0.227943 + 0.701538i
0.418926 + 1.28932i
−0.227943 − 0.701538i
−0.386111 − 0.280526i
1.69513 + 1.23158i
−0.386111 + 0.280526i
1.69513 − 1.23158i
0.0501062 − 0.154211i 0.809017 − 0.587785i 1.59676 + 1.16012i −1.35567 − 4.17234i −0.0501062 − 0.154211i 0.809017 + 0.587785i 0.521270 − 0.378725i 0.309017 − 0.951057i −0.711349
64.2 0.449894 − 1.38463i 0.809017 − 0.587785i −0.0967635 − 0.0703028i 0.737640 + 2.27022i −0.449894 − 1.38463i 0.809017 + 0.587785i 2.21480 − 1.60914i 0.309017 − 0.951057i 3.47528
148.1 0.0501062 + 0.154211i 0.809017 + 0.587785i 1.59676 − 1.16012i −1.35567 + 4.17234i −0.0501062 + 0.154211i 0.809017 − 0.587785i 0.521270 + 0.378725i 0.309017 + 0.951057i −0.711349
148.2 0.449894 + 1.38463i 0.809017 + 0.587785i −0.0967635 + 0.0703028i 0.737640 − 2.27022i −0.449894 + 1.38463i 0.809017 − 0.587785i 2.21480 + 1.60914i 0.309017 + 0.951057i 3.47528
169.1 −1.43376 − 1.04169i −0.309017 + 0.951057i 0.352519 + 1.08494i −0.477260 + 0.346750i 1.43376 − 1.04169i −0.309017 − 0.951057i −0.470553 + 1.44821i −0.809017 − 0.587785i 1.04548
169.2 1.93376 + 1.40496i −0.309017 + 0.951057i 1.14748 + 3.53158i 2.09529 − 1.52232i −1.93376 + 1.40496i −0.309017 − 0.951057i −1.26552 + 3.89486i −0.809017 − 0.587785i 6.19059
190.1 −1.43376 + 1.04169i −0.309017 − 0.951057i 0.352519 − 1.08494i −0.477260 − 0.346750i 1.43376 + 1.04169i −0.309017 + 0.951057i −0.470553 − 1.44821i −0.809017 + 0.587785i 1.04548
190.2 1.93376 − 1.40496i −0.309017 − 0.951057i 1.14748 − 3.53158i 2.09529 + 1.52232i −1.93376 − 1.40496i −0.309017 + 0.951057i −1.26552 − 3.89486i −0.809017 + 0.587785i 6.19059
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.j.f ✓ 8
3.b odd 2 1 693.2.m.f 8
11.c even 5 1 inner 231.2.j.f ✓ 8
11.c even 5 1 2541.2.a.bn 4
11.d odd 10 1 2541.2.a.bm 4
33.f even 10 1 7623.2.a.cl 4
33.h odd 10 1 693.2.m.f 8
33.h odd 10 1 7623.2.a.ci 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f ✓ 8 1.a even 1 1 trivial
231.2.j.f ✓ 8 11.c even 5 1 inner
693.2.m.f 8 3.b odd 2 1
693.2.m.f 8 33.h odd 10 1
2541.2.a.bm 4 11.d odd 10 1
2541.2.a.bn 4 11.c even 5 1
7623.2.a.ci 4 33.h odd 10 1
7623.2.a.cl 4 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + T_{2}^{6} + 4T_{2}^{5} + 9T_{2}^{4} - 8T_{2}^{3} + 39T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 11 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 92416 \) Copy content Toggle raw display
$19$ \( T^{8} + 28 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{4} + 10 T^{3} + \cdots + 109)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 48 T^{6} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{8} - 24 T^{7} + \cdots + 3041536 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 78961 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots + 20214016 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots + 1861)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 22 T^{7} + \cdots + 952576 \) Copy content Toggle raw display
$53$ \( T^{8} + 20 T^{7} + \cdots + 43681 \) Copy content Toggle raw display
$59$ \( T^{8} - 18 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( (T^{4} + 28 T^{3} + \cdots + 1301)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + \cdots + 32041 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( T^{8} - 20 T^{7} + \cdots + 22201 \) Copy content Toggle raw display
$83$ \( T^{8} + 8 T^{7} + \cdots + 8202496 \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + \cdots - 304)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 71639296 \) Copy content Toggle raw display
show more
show less