Properties

Label 546.2.i.e
Level $546$
Weight $2$
Character orbit 546.i
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 1.73205i −1.00000 −2.50000 + 0.866025i −1.00000 −0.500000 + 0.866025i −1.00000 1.73205i
235.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 + 1.73205i −1.00000 −2.50000 0.866025i −1.00000 −0.500000 0.866025i −1.00000 + 1.73205i
\(n\): 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 546.2.i.e 2
3.b odd 2 1 1638.2.j.a 2
7.c even 3 1 inner 546.2.i.e 2
7.c even 3 1 3822.2.a.k 1
7.d odd 6 1 3822.2.a.i 1
21.h odd 6 1 1638.2.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.e 2 1.a even 1 1 trivial
546.2.i.e 2 7.c even 3 1 inner
1638.2.j.a 2 3.b odd 2 1
1638.2.j.a 2 21.h odd 6 1
3822.2.a.i 1 7.d odd 6 1
3822.2.a.k 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{17}^{2} + 5 T_{17} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 25 + 5 T + T^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( 1 - T + T^{2} \)
$53$ \( 9 - 3 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 25 - 5 T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( ( 1 + T )^{2} \)
$73$ \( 144 + 12 T + T^{2} \)
$79$ \( 36 - 6 T + T^{2} \)
$83$ \( ( -16 + T )^{2} \)
$89$ \( 196 - 14 T + T^{2} \)
$97$ \( ( -4 + T )^{2} \)
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