Properties

Label 546.2.i.d
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.

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

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 0 −1.00000 −2.50000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0
235.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 −2.50000 0.866025i 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 546.2.i.d 2
3.b odd 2 1 1638.2.j.i 2
7.c even 3 1 inner 546.2.i.d 2
7.c even 3 1 3822.2.a.u 1
7.d odd 6 1 3822.2.a.bf 1
21.h odd 6 1 1638.2.j.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.d 2 1.a even 1 1 trivial
546.2.i.d 2 7.c even 3 1 inner
1638.2.j.i 2 3.b odd 2 1
1638.2.j.i 2 21.h odd 6 1
3822.2.a.u 1 7.c even 3 1
3822.2.a.bf 1 7.d odd 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{17}^{2} + 7 T_{17} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 49 + 7 T + T^{2} \)
$19$ \( 49 + 7 T + T^{2} \)
$23$ \( 4 + 2 T + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 16 + 4 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( 1 + T + T^{2} \)
$59$ \( 49 + 7 T + T^{2} \)
$61$ \( 169 + 13 T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( ( -9 + T )^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 196 + 14 T + T^{2} \)
$83$ \( ( -16 + T )^{2} \)
$89$ \( 144 - 12 T + T^{2} \)
$97$ \( ( -6 + T )^{2} \)
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