Properties

Label 462.2.i.b
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.

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

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 0 −1.00000 −2.00000 + 1.73205i −1.00000 −0.500000 0.866025i 0
331.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 −1.00000 −2.00000 1.73205i −1.00000 −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 462.2.i.b 2
3.b odd 2 1 1386.2.k.d 2
7.c even 3 1 inner 462.2.i.b 2
7.c even 3 1 3234.2.a.l 1
7.d odd 6 1 3234.2.a.f 1
21.g even 6 1 9702.2.a.bn 1
21.h odd 6 1 1386.2.k.d 2
21.h odd 6 1 9702.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.b 2 1.a even 1 1 trivial
462.2.i.b 2 7.c even 3 1 inner
1386.2.k.d 2 3.b odd 2 1
1386.2.k.d 2 21.h odd 6 1
3234.2.a.f 1 7.d odd 6 1
3234.2.a.l 1 7.c even 3 1
9702.2.a.bl 1 21.h odd 6 1
9702.2.a.bn 1 21.g even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( 4 + 2 T + T^{2} \)
$37$ \( 49 - 7 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -11 + T )^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( ( -3 + T )^{2} \)
$73$ \( 16 - 4 T + T^{2} \)
$79$ \( 256 - 16 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 1 + T )^{2} \)
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