Properties

Label 546.2.e.c
Level $546$
Weight $2$
Character orbit 546.e
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.e (of order \(2\), degree \(1\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 −1.50000 0.866025i 1.00000 1.73205i −1.50000 0.866025i 2.50000 0.866025i 1.00000 1.50000 + 2.59808i 1.73205i
545.2 1.00000 −1.50000 + 0.866025i 1.00000 1.73205i −1.50000 + 0.866025i 2.50000 + 0.866025i 1.00000 1.50000 2.59808i 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
273.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.e.c yes 2
3.b odd 2 1 546.2.e.b yes 2
7.b odd 2 1 546.2.e.d yes 2
13.b even 2 1 546.2.e.a 2
21.c even 2 1 546.2.e.a 2
39.d odd 2 1 546.2.e.d yes 2
91.b odd 2 1 546.2.e.b yes 2
273.g even 2 1 inner 546.2.e.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.a 2 13.b even 2 1
546.2.e.a 2 21.c even 2 1
546.2.e.b yes 2 3.b odd 2 1
546.2.e.b yes 2 91.b odd 2 1
546.2.e.c yes 2 1.a even 1 1 trivial
546.2.e.c yes 2 273.g even 2 1 inner
546.2.e.d yes 2 7.b odd 2 1
546.2.e.d yes 2 39.d odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 3 \)
\( T_{17} + 3 \)
\( T_{19} + 2 \)
\( T_{71} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( 3 + T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 12 + T^{2} \)
$29$ \( 48 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 3 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 11 + T )^{2} \)
$47$ \( 147 + T^{2} \)
$53$ \( 12 + T^{2} \)
$59$ \( 108 + T^{2} \)
$61$ \( 48 + T^{2} \)
$67$ \( 192 + T^{2} \)
$71$ \( ( -9 + T )^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 12 + T^{2} \)
$97$ \( ( 8 + T )^{2} \)
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