Properties

Label 462.2.i.c
Level $462$
Weight $2$
Character orbit 462.i
Analytic conductor $3.689$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

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

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 1.50000 + 2.59808i −1.00000 2.50000 + 0.866025i −1.00000 −0.500000 0.866025i −1.50000 + 2.59808i
331.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.50000 2.59808i −1.00000 2.50000 0.866025i −1.00000 −0.500000 + 0.866025i −1.50000 2.59808i
\(n\): 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 462.2.i.c 2
3.b odd 2 1 1386.2.k.c 2
7.c even 3 1 inner 462.2.i.c 2
7.c even 3 1 3234.2.a.h 1
7.d odd 6 1 3234.2.a.g 1
21.g even 6 1 9702.2.a.bd 1
21.h odd 6 1 1386.2.k.c 2
21.h odd 6 1 9702.2.a.cf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.c 2 1.a even 1 1 trivial
462.2.i.c 2 7.c even 3 1 inner
1386.2.k.c 2 3.b odd 2 1
1386.2.k.c 2 21.h odd 6 1
3234.2.a.g 1 7.d odd 6 1
3234.2.a.h 1 7.c even 3 1
9702.2.a.bd 1 21.g even 6 1
9702.2.a.cf 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 9 + 3 T + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( 3 + T )^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 81 - 9 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 25 + 5 T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 256 - 16 T + T^{2} \)
$79$ \( 289 + 17 T + T^{2} \)
$83$ \( ( -9 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -17 + T )^{2} \)
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