Properties

Label 231.2.i.c
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: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.00000 + 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i 0 2.00000 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
100.1 1.00000 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i 0 2.00000 2.50000 0.866025i 0 −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.c 2
3.b odd 2 1 693.2.i.a 2
7.c even 3 1 inner 231.2.i.c 2
7.c even 3 1 1617.2.a.a 1
7.d odd 6 1 1617.2.a.b 1
21.g even 6 1 4851.2.a.s 1
21.h odd 6 1 693.2.i.a 2
21.h odd 6 1 4851.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.c 2 1.a even 1 1 trivial
231.2.i.c 2 7.c even 3 1 inner
693.2.i.a 2 3.b odd 2 1
693.2.i.a 2 21.h odd 6 1
1617.2.a.a 1 7.c even 3 1
1617.2.a.b 1 7.d odd 6 1
4851.2.a.r 1 21.h odd 6 1
4851.2.a.s 1 21.g even 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 4 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 9 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 2 T + 97 T^{2} )^{2} \)
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