Properties

Label 546.2.i.k
Level $546$
Weight $2$
Character orbit 546.i
Analytic conductor $4.360$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
Defining polynomial: \(x^{6} - 3 x^{5} + 24 x^{4} - 43 x^{3} + 138 x^{2} - 117 x + 73\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 24 x^{4} - 43 x^{3} + 138 x^{2} - 117 x + 73\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 30 \nu^{3} + 40 \nu^{2} - 70 \nu + 13 \)\()/31\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 18 \nu^{4} - 15 \nu^{3} + 144 \nu^{2} + 151 \nu - 164 \)\()/62\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} + 47 \nu^{3} - 197 \nu^{2} + 461 \nu - 133 \)\()/62\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} + 16 \nu^{3} - 166 \nu^{2} + 151 \nu - 102 \)\()/31\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 8\)
\(\nu^{3}\)\(=\)\(-3 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 8 \beta_{1} - 7\)
\(\nu^{4}\)\(=\)\(16 \beta_{5} - 16 \beta_{4} + 20 \beta_{3} - 9 \beta_{2} - 28 \beta_{1} + 75\)
\(\nu^{5}\)\(=\)\(45 \beta_{5} - 60 \beta_{4} + 40 \beta_{3} - 33 \beta_{2} + 55 \beta_{1} + 139\)

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
0.500000 + 0.679547i
0.500000 3.05087i
0.500000 + 3.23735i
0.500000 0.679547i
0.500000 + 3.05087i
0.500000 3.23735i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.90280 + 3.29575i 1.00000 −2.64411 0.0932392i −1.00000 −0.500000 + 0.866025i 1.90280 + 3.29575i
79.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.30661 + 2.26312i 1.00000 1.77890 1.95845i −1.00000 −0.500000 + 0.866025i 1.30661 + 2.26312i
79.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.70942 2.96080i 1.00000 2.36521 + 1.18566i −1.00000 −0.500000 + 0.866025i −1.70942 2.96080i
235.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.90280 3.29575i 1.00000 −2.64411 + 0.0932392i −1.00000 −0.500000 0.866025i 1.90280 3.29575i
235.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.30661 2.26312i 1.00000 1.77890 + 1.95845i −1.00000 −0.500000 0.866025i 1.30661 2.26312i
235.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.70942 + 2.96080i 1.00000 2.36521 1.18566i −1.00000 −0.500000 0.866025i −1.70942 + 2.96080i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.3
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.k 6
3.b odd 2 1 1638.2.j.q 6
7.c even 3 1 inner 546.2.i.k 6
7.c even 3 1 3822.2.a.bv 3
7.d odd 6 1 3822.2.a.bw 3
21.h odd 6 1 1638.2.j.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.k 6 1.a even 1 1 trivial
546.2.i.k 6 7.c even 3 1 inner
1638.2.j.q 6 3.b odd 2 1
1638.2.j.q 6 21.h odd 6 1
3822.2.a.bv 3 7.c even 3 1
3822.2.a.bw 3 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{6} + 3 T_{5}^{5} + 21 T_{5}^{4} + 32 T_{5}^{3} + 246 T_{5}^{2} + 408 T_{5} + 1156 \)
\( T_{17}^{6} + 6 T_{17}^{5} + 99 T_{17}^{4} + 386 T_{17}^{3} + 6261 T_{17}^{2} + 24066 T_{17} + 145924 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{3} \)
$3$ \( ( 1 - T + T^{2} )^{3} \)
$5$ \( 1156 + 408 T + 246 T^{2} + 32 T^{3} + 21 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( 343 - 147 T - 42 T^{2} + 47 T^{3} - 6 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( 7921 - 2403 T + 996 T^{2} - 97 T^{3} + 36 T^{4} - 3 T^{5} + T^{6} \)
$13$ \( ( -1 + T )^{6} \)
$17$ \( 145924 + 24066 T + 6261 T^{2} + 386 T^{3} + 99 T^{4} + 6 T^{5} + T^{6} \)
$19$ \( 1024 + 96 T + 201 T^{2} + 46 T^{3} + 39 T^{4} + 6 T^{5} + T^{6} \)
$23$ \( 3600 - 1800 T + 900 T^{2} - 120 T^{3} + 30 T^{4} + T^{6} \)
$29$ \( ( 3 + T )^{6} \)
$31$ \( 1600 - 2400 T + 3000 T^{2} - 820 T^{3} + 165 T^{4} - 15 T^{5} + T^{6} \)
$37$ \( 1024 + 576 T + 516 T^{2} - 44 T^{3} + 54 T^{4} + 6 T^{5} + T^{6} \)
$41$ \( ( -56 + 78 T - 18 T^{2} + T^{3} )^{2} \)
$43$ \( ( -16 - 12 T + 12 T^{2} + T^{3} )^{2} \)
$47$ \( 44944 - 13356 T + 5241 T^{2} - 46 T^{3} + 99 T^{4} - 6 T^{5} + T^{6} \)
$53$ \( 1681 - 4797 T + 13566 T^{2} - 433 T^{3} + 126 T^{4} + 3 T^{5} + T^{6} \)
$59$ \( 2401 - 1323 T + 876 T^{2} - 17 T^{3} + 36 T^{4} - 3 T^{5} + T^{6} \)
$61$ \( 40000 + 15000 T + 5625 T^{2} + 400 T^{3} + 75 T^{4} + T^{6} \)
$67$ \( 64 - 984 T + 15081 T^{2} - 754 T^{3} + 159 T^{4} + 6 T^{5} + T^{6} \)
$71$ \( ( 1538 + 417 T + 36 T^{2} + T^{3} )^{2} \)
$73$ \( 44944 + 34344 T + 21156 T^{2} + 3464 T^{3} + 414 T^{4} + 24 T^{5} + T^{6} \)
$79$ \( 10000 + 1500 T^{2} + 200 T^{3} + 225 T^{4} + 15 T^{5} + T^{6} \)
$83$ \( ( 108 - 108 T - 9 T^{2} + T^{3} )^{2} \)
$89$ \( 462400 + 183600 T + 52500 T^{2} + 6740 T^{3} + 630 T^{4} + 30 T^{5} + T^{6} \)
$97$ \( ( -166 - 72 T - 3 T^{2} + T^{3} )^{2} \)
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