Properties

Label 462.2.i.d
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.

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

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.d 2
3.b odd 2 1 1386.2.k.f 2
7.c even 3 1 inner 462.2.i.d 2
7.c even 3 1 3234.2.a.c 1
7.d odd 6 1 3234.2.a.j 1
21.g even 6 1 9702.2.a.bq 1
21.h odd 6 1 1386.2.k.f 2
21.h odd 6 1 9702.2.a.bv 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.d 2 1.a even 1 1 trivial
462.2.i.d 2 7.c even 3 1 inner
1386.2.k.f 2 3.b odd 2 1
1386.2.k.f 2 21.h odd 6 1
3234.2.a.c 1 7.c even 3 1
3234.2.a.j 1 7.d odd 6 1
9702.2.a.bq 1 21.g even 6 1
9702.2.a.bv 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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( -4 + T )^{2} \)
$17$ \( 1 - T + T^{2} \)
$19$ \( 9 - 3 T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( 36 + 6 T + T^{2} \)
$37$ \( 9 - 3 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 49 + 7 T + T^{2} \)
$61$ \( 36 + 6 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( ( 15 + T )^{2} \)
$73$ \( 144 + 12 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( 64 - 8 T + T^{2} \)
$97$ \( ( -7 + T )^{2} \)
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