Properties

Label 231.2.i.d
Level $231$
Weight $2$
Character orbit 231.i
Analytic conductor $1.845$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.207107 0.358719i −0.500000 + 0.866025i 0.914214 1.58346i 1.00000 + 1.73205i 0.414214 1.00000 + 2.44949i −1.58579 −0.500000 0.866025i 0.414214 0.717439i
67.2 1.20711 + 2.09077i −0.500000 + 0.866025i −1.91421 + 3.31552i 1.00000 + 1.73205i −2.41421 1.00000 2.44949i −4.41421 −0.500000 0.866025i −2.41421 + 4.18154i
100.1 −0.207107 + 0.358719i −0.500000 0.866025i 0.914214 + 1.58346i 1.00000 1.73205i 0.414214 1.00000 2.44949i −1.58579 −0.500000 + 0.866025i 0.414214 + 0.717439i
100.2 1.20711 2.09077i −0.500000 0.866025i −1.91421 3.31552i 1.00000 1.73205i −2.41421 1.00000 + 2.44949i −4.41421 −0.500000 + 0.866025i −2.41421 4.18154i
\(n\): 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.d 4
3.b odd 2 1 693.2.i.f 4
7.c even 3 1 inner 231.2.i.d 4
7.c even 3 1 1617.2.a.n 2
7.d odd 6 1 1617.2.a.m 2
21.g even 6 1 4851.2.a.bd 2
21.h odd 6 1 693.2.i.f 4
21.h odd 6 1 4851.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.d 4 1.a even 1 1 trivial
231.2.i.d 4 7.c even 3 1 inner
693.2.i.f 4 3.b odd 2 1
693.2.i.f 4 21.h odd 6 1
1617.2.a.m 2 7.d odd 6 1
1617.2.a.n 2 7.c even 3 1
4851.2.a.bd 2 21.g even 6 1
4851.2.a.be 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 4 - 2 T + T^{2} )^{2} \)
$7$ \( ( 7 - 2 T + T^{2} )^{2} \)
$11$ \( ( 1 + T + T^{2} )^{2} \)
$13$ \( ( -4 - 4 T + T^{2} )^{2} \)
$17$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( ( 49 - 7 T + T^{2} )^{2} \)
$29$ \( ( -17 + 2 T + T^{2} )^{2} \)
$31$ \( 1024 + 32 T^{2} + T^{4} \)
$37$ \( 5041 + 142 T + 75 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( ( 8 + 8 T + T^{2} )^{2} \)
$43$ \( ( 31 + 14 T + T^{2} )^{2} \)
$47$ \( 1681 + 574 T + 155 T^{2} + 14 T^{3} + T^{4} \)
$53$ \( 8464 - 1840 T + 308 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( 529 + 138 T + 59 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( ( 16 - 4 T + T^{2} )^{2} \)
$67$ \( 784 - 336 T + 116 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( ( 41 + 14 T + T^{2} )^{2} \)
$73$ \( 16 + 48 T + 140 T^{2} + 12 T^{3} + T^{4} \)
$79$ \( 15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( ( -8 + T^{2} )^{2} \)
$89$ \( 40000 + 200 T^{2} + T^{4} \)
$97$ \( ( -63 + 6 T + T^{2} )^{2} \)
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