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Properties

Label 231.2.l.a
Level $231$
Weight $2$
Character orbit 231.l
Analytic conductor $1.845$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{3} + 2 \beta_{4} q^{4} + ( - \beta_{7} - \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{3}) q^{7} + (2 \beta_{7} - \beta_{4} + 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{3} + 2 \beta_{4} q^{4} + ( - \beta_{7} - \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{3}) q^{7} + (2 \beta_{7} - \beta_{4} + 2 \beta_{2} + 1) q^{9} + ( - \beta_{7} + 2 \beta_{5} + \beta_1) q^{11} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2} - 2) q^{12} + (2 \beta_{6} + \beta_{3}) q^{13} + ( - \beta_{7} - 2) q^{15} + (4 \beta_{4} - 4) q^{16} + (2 \beta_{7} - 4 \beta_{5} + \cdots - 2 \beta_1) q^{17}+ \cdots + ( - \beta_{7} - 4 \beta_{6} + \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 8 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 8 q^{4} + 4 q^{9} - 8 q^{12} - 16 q^{15} - 16 q^{16} - 12 q^{25} + 40 q^{27} + 4 q^{31} + 4 q^{33} + 16 q^{36} + 28 q^{37} - 16 q^{45} - 32 q^{48} - 28 q^{49} - 8 q^{55} - 16 q^{60} - 64 q^{64} + 4 q^{67} - 64 q^{69} + 12 q^{75} + 28 q^{81} + 84 q^{91} - 4 q^{93} + 64 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 203\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 440\nu^{3} + 81\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 55\nu^{5} + 341\nu^{3} - 81\nu ) / 297 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 4\beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 5\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 23\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{7} - 31\beta_{5} - 24\beta_{2} - 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{3} - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{2} - 203\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−2.23256 − 1.28897i
1.00781 + 0.581861i
2.23256 + 1.28897i
−1.00781 − 0.581861i
−2.23256 + 1.28897i
1.00781 − 0.581861i
2.23256 − 1.28897i
−1.00781 + 0.581861i
0 −0.724745 − 1.57313i 1.00000 − 1.73205i 1.22474 − 0.707107i 0 −1.32288 − 2.29129i 0 −1.94949 + 2.28024i 0
32.2 0 −0.724745 − 1.57313i 1.00000 − 1.73205i 1.22474 − 0.707107i 0 1.32288 + 2.29129i 0 −1.94949 + 2.28024i 0
32.3 0 1.72474 − 0.158919i 1.00000 − 1.73205i −1.22474 + 0.707107i 0 −1.32288 − 2.29129i 0 2.94949 − 0.548188i 0
32.4 0 1.72474 − 0.158919i 1.00000 − 1.73205i −1.22474 + 0.707107i 0 1.32288 + 2.29129i 0 2.94949 − 0.548188i 0
65.1 0 −0.724745 + 1.57313i 1.00000 + 1.73205i 1.22474 + 0.707107i 0 −1.32288 + 2.29129i 0 −1.94949 − 2.28024i 0
65.2 0 −0.724745 + 1.57313i 1.00000 + 1.73205i 1.22474 + 0.707107i 0 1.32288 − 2.29129i 0 −1.94949 − 2.28024i 0
65.3 0 1.72474 + 0.158919i 1.00000 + 1.73205i −1.22474 − 0.707107i 0 −1.32288 + 2.29129i 0 2.94949 + 0.548188i 0
65.4 0 1.72474 + 0.158919i 1.00000 + 1.73205i −1.22474 − 0.707107i 0 1.32288 − 2.29129i 0 2.94949 + 0.548188i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.4
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.a ✓ 8
3.b odd 2 1 inner 231.2.l.a ✓ 8
7.c even 3 1 inner 231.2.l.a ✓ 8
11.b odd 2 1 inner 231.2.l.a ✓ 8
21.h odd 6 1 inner 231.2.l.a ✓ 8
33.d even 2 1 inner 231.2.l.a ✓ 8
77.h odd 6 1 inner 231.2.l.a ✓ 8
231.l even 6 1 inner 231.2.l.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.l.a ✓ 8 1.a even 1 1 trivial
231.2.l.a ✓ 8 3.b odd 2 1 inner
231.2.l.a ✓ 8 7.c even 3 1 inner
231.2.l.a ✓ 8 11.b odd 2 1 inner
231.2.l.a ✓ 8 21.h odd 6 1 inner
231.2.l.a ✓ 8 33.d even 2 1 inner
231.2.l.a ✓ 8 77.h odd 6 1 inner
231.2.l.a ✓ 8 231.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 20 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 42 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 42)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 42)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 189 T^{2} + 35721)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 42)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 242 T^{2} + 58564)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{8} \) Copy content Toggle raw display
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