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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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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.

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 \( T_{2}^{3} - 6 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 6 T + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 2 - 15 T + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( 2 - 15 T + T^{3} \)
$17$ \( 8 - 24 T + T^{3} \)
$19$ \( 36 + 27 T - 12 T^{2} + T^{3} \)
$23$ \( -32 - 12 T + 6 T^{2} + T^{3} \)
$29$ \( -6 + 33 T - 12 T^{2} + T^{3} \)
$31$ \( 32 - 36 T + 6 T^{2} + T^{3} \)
$37$ \( -246 - 75 T + T^{3} \)
$41$ \( 32 - 72 T - 6 T^{2} + T^{3} \)
$43$ \( 48 - 12 T - 6 T^{2} + T^{3} \)
$47$ \( 328 + 171 T + 24 T^{2} + T^{3} \)
$53$ \( -120 - 48 T + T^{3} \)
$59$ \( -716 + 87 T + 24 T^{2} + T^{3} \)
$61$ \( ( -6 + T )^{3} \)
$67$ \( 4 + 27 T - 12 T^{2} + T^{3} \)
$71$ \( 384 - 48 T - 12 T^{2} + T^{3} \)
$73$ \( -394 + 177 T - 24 T^{2} + T^{3} \)
$79$ \( 256 - 48 T - 12 T^{2} + T^{3} \)
$83$ \( 48 + 60 T + 18 T^{2} + T^{3} \)
$89$ \( 1896 - 84 T - 18 T^{2} + T^{3} \)
$97$ \( -8 + 144 T - 24 T^{2} + T^{3} \)
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