Properties

Label 231.2.i.b
Level $231$
Weight $2$
Character orbit 231.i
Analytic conductor $1.845$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(67,231)]
 
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([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.i (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - 4 \zeta_{6} q^{5} + q^{6} + (2 \zeta_{6} - 3) q^{7} + 3 q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - 4 \zeta_{6} q^{5} + q^{6} + (2 \zeta_{6} - 3) q^{7} + 3 q^{8} - \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{10} + ( - \zeta_{6} + 1) q^{11} - \zeta_{6} q^{12} + ( - \zeta_{6} - 2) q^{14} - 4 q^{15} + \zeta_{6} q^{16} + ( - 7 \zeta_{6} + 7) q^{17} + ( - \zeta_{6} + 1) q^{18} + 5 \zeta_{6} q^{19} - 4 q^{20} + (3 \zeta_{6} - 1) q^{21} + q^{22} + 9 \zeta_{6} q^{23} + ( - 3 \zeta_{6} + 3) q^{24} + (11 \zeta_{6} - 11) q^{25} - q^{27} + (3 \zeta_{6} - 1) q^{28} + q^{29} - 4 \zeta_{6} q^{30} + (2 \zeta_{6} - 2) q^{31} + ( - 5 \zeta_{6} + 5) q^{32} - \zeta_{6} q^{33} + 7 q^{34} + (4 \zeta_{6} + 8) q^{35} - q^{36} + 3 \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{38} - 12 \zeta_{6} q^{40} - 2 q^{41} + (2 \zeta_{6} - 3) q^{42} - q^{43} - \zeta_{6} q^{44} + (4 \zeta_{6} - 4) q^{45} + (9 \zeta_{6} - 9) q^{46} - 7 \zeta_{6} q^{47} + q^{48} + ( - 8 \zeta_{6} + 5) q^{49} - 11 q^{50} - 7 \zeta_{6} q^{51} - \zeta_{6} q^{54} - 4 q^{55} + (6 \zeta_{6} - 9) q^{56} + 5 q^{57} + \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} + (4 \zeta_{6} - 4) q^{60} + 10 \zeta_{6} q^{61} - 2 q^{62} + (\zeta_{6} + 2) q^{63} + 7 q^{64} + ( - \zeta_{6} + 1) q^{66} + ( - 12 \zeta_{6} + 12) q^{67} - 7 \zeta_{6} q^{68} + 9 q^{69} + (12 \zeta_{6} - 4) q^{70} - 15 q^{71} - 3 \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + (3 \zeta_{6} - 3) q^{74} + 11 \zeta_{6} q^{75} + 5 q^{76} + (3 \zeta_{6} - 1) q^{77} + 8 \zeta_{6} q^{79} + ( - 4 \zeta_{6} + 4) q^{80} + (\zeta_{6} - 1) q^{81} - 2 \zeta_{6} q^{82} + 4 q^{83} + (\zeta_{6} + 2) q^{84} - 28 q^{85} - \zeta_{6} q^{86} + ( - \zeta_{6} + 1) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} - 12 \zeta_{6} q^{89} - 4 q^{90} + 9 q^{92} + 2 \zeta_{6} q^{93} + ( - 7 \zeta_{6} + 7) q^{94} + ( - 20 \zeta_{6} + 20) q^{95} - 5 \zeta_{6} q^{96} - q^{97} + ( - 3 \zeta_{6} + 8) q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + q^{4} - 4 q^{5} + 2 q^{6} - 4 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + q^{4} - 4 q^{5} + 2 q^{6} - 4 q^{7} + 6 q^{8} - q^{9} + 4 q^{10} + q^{11} - q^{12} - 5 q^{14} - 8 q^{15} + q^{16} + 7 q^{17} + q^{18} + 5 q^{19} - 8 q^{20} + q^{21} + 2 q^{22} + 9 q^{23} + 3 q^{24} - 11 q^{25} - 2 q^{27} + q^{28} + 2 q^{29} - 4 q^{30} - 2 q^{31} + 5 q^{32} - q^{33} + 14 q^{34} + 20 q^{35} - 2 q^{36} + 3 q^{37} - 5 q^{38} - 12 q^{40} - 4 q^{41} - 4 q^{42} - 2 q^{43} - q^{44} - 4 q^{45} - 9 q^{46} - 7 q^{47} + 2 q^{48} + 2 q^{49} - 22 q^{50} - 7 q^{51} - q^{54} - 8 q^{55} - 12 q^{56} + 10 q^{57} + q^{58} - 7 q^{59} - 4 q^{60} + 10 q^{61} - 4 q^{62} + 5 q^{63} + 14 q^{64} + q^{66} + 12 q^{67} - 7 q^{68} + 18 q^{69} + 4 q^{70} - 30 q^{71} - 3 q^{72} + 4 q^{73} - 3 q^{74} + 11 q^{75} + 10 q^{76} + q^{77} + 8 q^{79} + 4 q^{80} - q^{81} - 2 q^{82} + 8 q^{83} + 5 q^{84} - 56 q^{85} - q^{86} + q^{87} + 3 q^{88} - 12 q^{89} - 8 q^{90} + 18 q^{92} + 2 q^{93} + 7 q^{94} + 20 q^{95} - 5 q^{96} - 2 q^{97} + 13 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0.500000 0.866025i 0.500000 0.866025i −2.00000 3.46410i 1.00000 −2.00000 + 1.73205i 3.00000 −0.500000 0.866025i 2.00000 3.46410i
100.1 0.500000 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i −2.00000 + 3.46410i 1.00000 −2.00000 1.73205i 3.00000 −0.500000 + 0.866025i 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.i.b 2
3.b odd 2 1 693.2.i.b 2
7.c even 3 1 inner 231.2.i.b 2
7.c even 3 1 1617.2.a.c 1
7.d odd 6 1 1617.2.a.d 1
21.g even 6 1 4851.2.a.q 1
21.h odd 6 1 693.2.i.b 2
21.h odd 6 1 4851.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.b 2 1.a even 1 1 trivial
231.2.i.b 2 7.c even 3 1 inner
693.2.i.b 2 3.b odd 2 1
693.2.i.b 2 21.h odd 6 1
1617.2.a.c 1 7.c even 3 1
1617.2.a.d 1 7.d odd 6 1
4851.2.a.n 1 21.h odd 6 1
4851.2.a.q 1 21.g even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 15)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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