Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−1.00000 −1.00000 −1.00000 −2.00000 1.00000 1.00000 3.00000 1.00000 2.00000
\(n\): 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.a 1
3.b odd 2 1 693.2.a.d 1
4.b odd 2 1 3696.2.a.t 1
5.b even 2 1 5775.2.a.t 1
7.b odd 2 1 1617.2.a.e 1
11.b odd 2 1 2541.2.a.h 1
21.c even 2 1 4851.2.a.p 1
33.d even 2 1 7623.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.a 1 1.a even 1 1 trivial
693.2.a.d 1 3.b odd 2 1
1617.2.a.e 1 7.b odd 2 1
2541.2.a.h 1 11.b odd 2 1
3696.2.a.t 1 4.b odd 2 1
4851.2.a.p 1 21.c even 2 1
5775.2.a.t 1 5.b even 2 1
7623.2.a.f 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( 1 + T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 + T \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 10 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 16 T + 79 T^{2} \)
$83$ \( 1 - 4 T + 83 T^{2} \)
$89$ \( 1 - 18 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
show more
show less