Properties

Label 231.2.i.a
Level 231
Weight 2
Character orbit 231.i
Analytic conductor 1.845
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 0 −1.00000 −2.00000 1.73205i −3.00000 −0.500000 0.866025i 0
100.1 −0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.00000 −2.00000 + 1.73205i −3.00000 −0.500000 + 0.866025i 0
\(n\): 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.a 2
3.b odd 2 1 693.2.i.e 2
7.c even 3 1 inner 231.2.i.a 2
7.c even 3 1 1617.2.a.g 1
7.d odd 6 1 1617.2.a.h 1
21.g even 6 1 4851.2.a.d 1
21.h odd 6 1 693.2.i.e 2
21.h odd 6 1 4851.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.a 2 1.a even 1 1 trivial
231.2.i.a 2 7.c even 3 1 inner
693.2.i.e 2 3.b odd 2 1
693.2.i.e 2 21.h odd 6 1
1617.2.a.g 1 7.c even 3 1
1617.2.a.h 1 7.d odd 6 1
4851.2.a.d 1 21.g even 6 1
4851.2.a.e 1 21.h odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 5 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 3 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 8 T + 11 T^{2} - 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 7 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 8 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + T + 97 T^{2} )^{2} \)
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