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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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