Properties

Label 546.2.i.i
Level $546$
Weight $2$
Character orbit 546.i
Analytic conductor $4.360$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} - q^{6} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{2} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} - q^{6} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + q^{8} + \beta_{2} q^{9} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} -\beta_{2} q^{12} + q^{13} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{14} + ( 2 - \beta_{3} ) q^{15} + \beta_{2} q^{16} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( -1 - \beta_{2} ) q^{18} + ( -2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -2 + \beta_{3} ) q^{20} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{21} + ( 1 - \beta_{3} ) q^{22} + ( -\beta_{1} - \beta_{3} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{25} + \beta_{2} q^{26} - q^{27} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{28} + ( 1 - 2 \beta_{3} ) q^{29} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{30} -6 \beta_{1} q^{31} + ( -1 - \beta_{2} ) q^{32} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{33} + ( -1 - \beta_{3} ) q^{34} + ( 3 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{35} + q^{36} + ( -\beta_{1} - \beta_{3} ) q^{37} + ( 5 + 2 \beta_{1} + 5 \beta_{2} ) q^{38} + ( 1 + \beta_{2} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{40} -7 \beta_{3} q^{41} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{42} + ( -2 + 6 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{44} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{45} + \beta_{1} q^{46} + \beta_{2} q^{47} - q^{48} + ( 5 - 2 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{49} + ( 1 - 4 \beta_{3} ) q^{50} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -1 - \beta_{2} ) q^{52} + ( -1 + 6 \beta_{1} - \beta_{2} ) q^{53} -\beta_{2} q^{54} + ( -4 + 3 \beta_{3} ) q^{55} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{56} + ( 5 - 2 \beta_{3} ) q^{57} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{58} + ( 5 + 5 \beta_{1} + 5 \beta_{2} ) q^{59} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{60} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{61} -6 \beta_{3} q^{62} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{63} + q^{64} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} ) q^{66} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{67} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{68} -\beta_{3} q^{69} + ( -4 - 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{70} -5 q^{71} + \beta_{2} q^{72} + \beta_{1} q^{73} + \beta_{1} q^{74} + ( -4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{75} + ( -5 + 2 \beta_{3} ) q^{76} + ( 3 + 2 \beta_{1} + 4 \beta_{2} ) q^{77} - q^{78} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{80} + ( -1 - \beta_{2} ) q^{81} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{82} + ( 2 + 4 \beta_{3} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{84} + \beta_{3} q^{85} + ( -6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{86} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{87} + ( -1 - \beta_{1} - \beta_{2} ) q^{88} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{89} + ( -2 + \beta_{3} ) q^{90} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{91} + \beta_{3} q^{92} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{93} + ( -1 - \beta_{2} ) q^{94} + ( -14 - 9 \beta_{1} - 14 \beta_{2} ) q^{95} -\beta_{2} q^{96} + ( -8 + 5 \beta_{3} ) q^{97} + ( -5 + 4 \beta_{1} + 2 \beta_{3} ) q^{98} + ( 1 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} + 4q^{5} - 4q^{6} - 2q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} + 4q^{5} - 4q^{6} - 2q^{7} + 4q^{8} - 2q^{9} + 4q^{10} - 2q^{11} + 2q^{12} + 4q^{13} - 2q^{14} + 8q^{15} - 2q^{16} + 2q^{17} - 2q^{18} + 10q^{19} - 8q^{20} - 4q^{21} + 4q^{22} + 2q^{24} - 2q^{25} - 2q^{26} - 4q^{27} + 4q^{28} + 4q^{29} - 4q^{30} - 2q^{32} + 2q^{33} - 4q^{34} - 8q^{35} + 4q^{36} + 10q^{38} + 2q^{39} + 4q^{40} + 2q^{42} - 8q^{43} - 2q^{44} + 4q^{45} - 2q^{47} - 4q^{48} + 10q^{49} + 4q^{50} - 2q^{51} - 2q^{52} - 2q^{53} + 2q^{54} - 16q^{55} - 2q^{56} + 20q^{57} - 2q^{58} + 10q^{59} - 4q^{60} + 6q^{61} - 2q^{63} + 4q^{64} + 4q^{65} + 2q^{66} + 2q^{67} + 2q^{68} - 8q^{70} - 20q^{71} - 2q^{72} + 2q^{75} - 20q^{76} + 4q^{77} - 4q^{78} + 12q^{79} + 4q^{80} - 2q^{81} + 8q^{83} + 2q^{84} + 4q^{86} + 2q^{87} - 2q^{88} - 8q^{89} - 8q^{90} - 2q^{91} - 2q^{94} - 28q^{95} + 2q^{96} - 32q^{97} - 20q^{98} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.292893 0.507306i −1.00000 1.62132 + 2.09077i 1.00000 −0.500000 + 0.866025i 0.292893 + 0.507306i
79.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.70711 2.95680i −1.00000 −2.62132 0.358719i 1.00000 −0.500000 + 0.866025i 1.70711 + 2.95680i
235.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.292893 + 0.507306i −1.00000 1.62132 2.09077i 1.00000 −0.500000 0.866025i 0.292893 0.507306i
235.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.70711 + 2.95680i −1.00000 −2.62132 + 0.358719i 1.00000 −0.500000 0.866025i 1.70711 2.95680i
\(n\): 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.i 4
3.b odd 2 1 1638.2.j.m 4
7.c even 3 1 inner 546.2.i.i 4
7.c even 3 1 3822.2.a.bn 2
7.d odd 6 1 3822.2.a.bu 2
21.h odd 6 1 1638.2.j.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.i 4 1.a even 1 1 trivial
546.2.i.i 4 7.c even 3 1 inner
1638.2.j.m 4 3.b odd 2 1
1638.2.j.m 4 21.h odd 6 1
3822.2.a.bn 2 7.c even 3 1
3822.2.a.bu 2 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{4} - 4 T_{5}^{3} + 14 T_{5}^{2} - 8 T_{5} + 4 \)
\( T_{17}^{4} - 2 T_{17}^{3} + 5 T_{17}^{2} + 2 T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 289 - 170 T + 83 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( 4 + 2 T^{2} + T^{4} \)
$29$ \( ( -7 - 2 T + T^{2} )^{2} \)
$31$ \( 5184 + 72 T^{2} + T^{4} \)
$37$ \( 4 + 2 T^{2} + T^{4} \)
$41$ \( ( -98 + T^{2} )^{2} \)
$43$ \( ( -68 + 4 T + T^{2} )^{2} \)
$47$ \( ( 1 + T + T^{2} )^{2} \)
$53$ \( 5041 - 142 T + 75 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 625 + 250 T + 125 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( 5 + T )^{4} \)
$73$ \( 4 + 2 T^{2} + T^{4} \)
$79$ \( 16 - 48 T + 140 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( ( -28 - 4 T + T^{2} )^{2} \)
$89$ \( 196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( 14 + 16 T + T^{2} )^{2} \)
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