Properties

Label 231.2.a.d
Level $231$
Weight $2$
Character orbit 231.a
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(1,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

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: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} - \beta_1 q^{6} - q^{7} + (2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} - \beta_1 q^{6} - q^{7} + (2 \beta_1 + 1) q^{8} + q^{9} + (\beta_{2} - 2 \beta_1 + 3) q^{10} + q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{2} + \beta_1) q^{13} - \beta_1 q^{14} + (\beta_{2} - \beta_1) q^{15} + (\beta_1 + 4) q^{16} - 2 \beta_1 q^{17} + \beta_1 q^{18} + ( - \beta_{2} - \beta_1 + 4) q^{19} + (3 \beta_1 - 7) q^{20} + q^{21} + \beta_1 q^{22} + ( - 2 \beta_1 - 2) q^{23} + ( - 2 \beta_1 - 1) q^{24} + ( - \beta_{2} - 3 \beta_1 + 5) q^{25} + (\beta_{2} - 2 \beta_1 + 3) q^{26} - q^{27} + ( - \beta_{2} - 2) q^{28} + (\beta_{2} - \beta_1 + 4) q^{29} + ( - \beta_{2} + 2 \beta_1 - 3) q^{30} + (2 \beta_{2} - 2) q^{31} + (\beta_{2} + 2) q^{32} - q^{33} + ( - 2 \beta_{2} - 8) q^{34} + (\beta_{2} - \beta_1) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{2} + 2 \beta_1 - 5) q^{38} + (\beta_{2} - \beta_1) q^{39} + (\beta_{2} - 3 \beta_1 + 6) q^{40} + (2 \beta_{2} + 2 \beta_1 + 2) q^{41} + \beta_1 q^{42} + ( - 2 \beta_1 + 2) q^{43} + (\beta_{2} + 2) q^{44} + ( - \beta_{2} + \beta_1) q^{45} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{46} + (\beta_{2} + \beta_1 - 8) q^{47} + ( - \beta_1 - 4) q^{48} + q^{49} + ( - 3 \beta_{2} + 3 \beta_1 - 13) q^{50} + 2 \beta_1 q^{51} + (3 \beta_1 - 7) q^{52} - 2 \beta_{2} q^{53} - \beta_1 q^{54} + ( - \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_1 - 1) q^{56} + (\beta_{2} + \beta_1 - 4) q^{57} + ( - \beta_{2} + 6 \beta_1 - 3) q^{58} + ( - 3 \beta_{2} + \beta_1 - 8) q^{59} + ( - 3 \beta_1 + 7) q^{60} + 6 q^{61} + (2 \beta_1 + 2) q^{62} - q^{63} + (2 \beta_1 - 7) q^{64} + ( - \beta_{2} - 3 \beta_1 + 10) q^{65} - \beta_1 q^{66} + (\beta_{2} + \beta_1 + 4) q^{67} + ( - 8 \beta_1 - 2) q^{68} + (2 \beta_1 + 2) q^{69} + ( - \beta_{2} + 2 \beta_1 - 3) q^{70} + ( - 4 \beta_1 + 4) q^{71} + (2 \beta_1 + 1) q^{72} + (\beta_{2} - \beta_1 + 8) q^{73} + (3 \beta_{2} + 2 \beta_1 + 13) q^{74} + (\beta_{2} + 3 \beta_1 - 5) q^{75} + (4 \beta_{2} - 5 \beta_1 - 1) q^{76} - q^{77} + ( - \beta_{2} + 2 \beta_1 - 3) q^{78} + (4 \beta_1 + 4) q^{79} + ( - 3 \beta_{2} + 2 \beta_1 + 3) q^{80} + q^{81} + (2 \beta_{2} + 6 \beta_1 + 10) q^{82} + (2 \beta_{2} - 6) q^{83} + (\beta_{2} + 2) q^{84} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{85} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{86} + ( - \beta_{2} + \beta_1 - 4) q^{87} + (2 \beta_1 + 1) q^{88} + (4 \beta_{2} + 6) q^{89} + (\beta_{2} - 2 \beta_1 + 3) q^{90} + (\beta_{2} - \beta_1) q^{91} + ( - 2 \beta_{2} - 8 \beta_1 - 6) q^{92} + ( - 2 \beta_{2} + 2) q^{93} + (\beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 7 \beta_{2} + 5 \beta_1 + 4) q^{95} + ( - \beta_{2} - 2) q^{96} + (2 \beta_{2} + 8) q^{97} + \beta_1 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.36147
−0.167449
2.52892
−2.36147 −1.00000 3.57653 −3.93800 2.36147 −1.00000 −3.72294 1.00000 9.29947
1.2 −0.167449 −1.00000 −1.97196 3.80451 0.167449 −1.00000 0.665102 1.00000 −0.637062
1.3 2.52892 −1.00000 4.39543 0.133492 −2.52892 −1.00000 6.05784 1.00000 0.337590
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)
\(11\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.a.d 3
3.b odd 2 1 693.2.a.m 3
4.b odd 2 1 3696.2.a.bp 3
5.b even 2 1 5775.2.a.bw 3
7.b odd 2 1 1617.2.a.s 3
11.b odd 2 1 2541.2.a.bi 3
21.c even 2 1 4851.2.a.bp 3
33.d even 2 1 7623.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 1.a even 1 1 trivial
693.2.a.m 3 3.b odd 2 1
1617.2.a.s 3 7.b odd 2 1
2541.2.a.bi 3 11.b odd 2 1
3696.2.a.bp 3 4.b odd 2 1
4851.2.a.bp 3 21.c even 2 1
5775.2.a.bw 3 5.b even 2 1
7623.2.a.cb 3 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 6T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 6T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 15T + 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 15T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 24T + 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 12 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} + \cdots - 6 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 75T - 246 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} + \cdots + 48 \) Copy content Toggle raw display
$47$ \( T^{3} + 24 T^{2} + \cdots + 328 \) Copy content Toggle raw display
$53$ \( T^{3} - 48T - 120 \) Copy content Toggle raw display
$59$ \( T^{3} + 24 T^{2} + \cdots - 716 \) Copy content Toggle raw display
$61$ \( (T - 6)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + \cdots + 384 \) Copy content Toggle raw display
$73$ \( T^{3} - 24 T^{2} + \cdots - 394 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + \cdots + 48 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + \cdots + 1896 \) Copy content Toggle raw display
$97$ \( T^{3} - 24 T^{2} + \cdots - 8 \) Copy content Toggle raw display
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