Properties

Label 546.2.i.j
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

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

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
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
1.32288 + 2.29129i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.822876 + 1.42526i −1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i 0.822876 + 1.42526i
79.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.82288 3.15731i −1.00000 1.32288 2.29129i −1.00000 −0.500000 + 0.866025i −1.82288 3.15731i
235.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.822876 1.42526i −1.00000 −1.32288 2.29129i −1.00000 −0.500000 0.866025i 0.822876 1.42526i
235.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.82288 + 3.15731i −1.00000 1.32288 + 2.29129i −1.00000 −0.500000 0.866025i −1.82288 + 3.15731i
\(n\): 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.j 4
3.b odd 2 1 1638.2.j.k 4
7.c even 3 1 inner 546.2.i.j 4
7.c even 3 1 3822.2.a.bk 2
7.d odd 6 1 3822.2.a.bi 2
21.h odd 6 1 1638.2.j.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.j 4 1.a even 1 1 trivial
546.2.i.j 4 7.c even 3 1 inner
1638.2.j.k 4 3.b odd 2 1
1638.2.j.k 4 21.h odd 6 1
3822.2.a.bi 2 7.d odd 6 1
3822.2.a.bk 2 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36 \)
\( T_{17}^{4} - 8 T_{17}^{3} + 55 T_{17}^{2} - 72 T_{17} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 49 + 7 T^{2} + T^{4} \)
$11$ \( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 81 - 72 T + 55 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( ( 25 + 5 T + T^{2} )^{2} \)
$23$ \( 324 - 180 T + 82 T^{2} - 10 T^{3} + T^{4} \)
$29$ \( ( -27 + 2 T + T^{2} )^{2} \)
$31$ \( 576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( 4 + 12 T + 34 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( ( 18 + 10 T + T^{2} )^{2} \)
$43$ \( ( -28 + T^{2} )^{2} \)
$47$ \( ( 9 + 3 T + T^{2} )^{2} \)
$53$ \( ( 9 - 3 T + T^{2} )^{2} \)
$59$ \( 3969 + 63 T^{2} + T^{4} \)
$61$ \( 2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 10609 + 618 T + 139 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 93 - 22 T + T^{2} )^{2} \)
$73$ \( 12996 - 2508 T + 370 T^{2} - 22 T^{3} + T^{4} \)
$79$ \( ( 100 - 10 T + T^{2} )^{2} \)
$83$ \( ( 36 + 16 T + T^{2} )^{2} \)
$89$ \( 324 + 324 T + 306 T^{2} + 18 T^{3} + T^{4} \)
$97$ \( ( -14 + 14 T + T^{2} )^{2} \)
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