Properties

Label 546.2.e.g
Level $546$
Weight $2$
Character orbit 546.e
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{4} + 16 \nu^{2} + 45 \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu \)\()/54\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153 \)\()/72\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu \)\()/216\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{6} - 16 \nu^{4} - 8 \nu^{2} - 81 \)\()/72\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 45 \nu \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{4} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 4 \beta_{5} - \beta_{3} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{4} + 5 \beta_{2} - 3\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + 8 \beta_{5} + 8 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} + 8 \beta_{4} - 16 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(24 \beta_{5} - 24 \beta_{3} - 13 \beta_{1}\)

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
−1.26217 1.18614i
−1.26217 + 1.18614i
−0.396143 1.68614i
−0.396143 + 1.68614i
0.396143 1.68614i
0.396143 + 1.68614i
1.26217 1.18614i
1.26217 + 1.18614i
1.00000 −1.26217 1.18614i 1.00000 3.37228i −1.26217 1.18614i −2.52434 0.792287i 1.00000 0.186141 + 2.99422i 3.37228i
545.2 1.00000 −1.26217 + 1.18614i 1.00000 3.37228i −1.26217 + 1.18614i −2.52434 + 0.792287i 1.00000 0.186141 2.99422i 3.37228i
545.3 1.00000 −0.396143 1.68614i 1.00000 2.37228i −0.396143 1.68614i −0.792287 + 2.52434i 1.00000 −2.68614 + 1.33591i 2.37228i
545.4 1.00000 −0.396143 + 1.68614i 1.00000 2.37228i −0.396143 + 1.68614i −0.792287 2.52434i 1.00000 −2.68614 1.33591i 2.37228i
545.5 1.00000 0.396143 1.68614i 1.00000 2.37228i 0.396143 1.68614i 0.792287 2.52434i 1.00000 −2.68614 1.33591i 2.37228i
545.6 1.00000 0.396143 + 1.68614i 1.00000 2.37228i 0.396143 + 1.68614i 0.792287 + 2.52434i 1.00000 −2.68614 + 1.33591i 2.37228i
545.7 1.00000 1.26217 1.18614i 1.00000 3.37228i 1.26217 1.18614i 2.52434 + 0.792287i 1.00000 0.186141 2.99422i 3.37228i
545.8 1.00000 1.26217 + 1.18614i 1.00000 3.37228i 1.26217 + 1.18614i 2.52434 0.792287i 1.00000 0.186141 + 2.99422i 3.37228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
39.d odd 2 1 inner
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.g yes 8
3.b odd 2 1 546.2.e.e 8
7.b odd 2 1 inner 546.2.e.g yes 8
13.b even 2 1 546.2.e.e 8
21.c even 2 1 546.2.e.e 8
39.d odd 2 1 inner 546.2.e.g yes 8
91.b odd 2 1 546.2.e.e 8
273.g even 2 1 inner 546.2.e.g yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.e 8 3.b odd 2 1
546.2.e.e 8 13.b even 2 1
546.2.e.e 8 21.c even 2 1
546.2.e.e 8 91.b odd 2 1
546.2.e.g yes 8 1.a even 1 1 trivial
546.2.e.g yes 8 7.b odd 2 1 inner
546.2.e.g yes 8 39.d odd 2 1 inner
546.2.e.g yes 8 273.g even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 17 T_{5}^{2} + 64 \)
\( T_{17}^{4} - 7 T_{17}^{2} + 4 \)
\( T_{19}^{4} - 63 T_{19}^{2} + 324 \)
\( T_{71} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( 81 + 45 T^{2} + 16 T^{4} + 5 T^{6} + T^{8} \)
$5$ \( ( 64 + 17 T^{2} + T^{4} )^{2} \)
$7$ \( 2401 - 34 T^{4} + T^{8} \)
$11$ \( ( -33 + T^{2} )^{4} \)
$13$ \( ( 169 - 22 T^{2} + T^{4} )^{2} \)
$17$ \( ( 4 - 7 T^{2} + T^{4} )^{2} \)
$19$ \( ( 324 - 63 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1 + 46 T^{2} + T^{4} )^{2} \)
$29$ \( ( 324 + 63 T^{2} + T^{4} )^{2} \)
$31$ \( ( 576 - 51 T^{2} + T^{4} )^{2} \)
$37$ \( ( 1681 + 94 T^{2} + T^{4} )^{2} \)
$41$ \( ( 484 + 77 T^{2} + T^{4} )^{2} \)
$43$ \( ( 22 + 11 T + T^{2} )^{4} \)
$47$ \( ( 2304 + 129 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1024 + 112 T^{2} + T^{4} )^{2} \)
$59$ \( ( 576 + 84 T^{2} + T^{4} )^{2} \)
$61$ \( ( 841 + 74 T^{2} + T^{4} )^{2} \)
$67$ \( ( 256 + 43 T^{2} + T^{4} )^{2} \)
$71$ \( ( 2 + T )^{8} \)
$73$ \( ( 1369 - 118 T^{2} + T^{4} )^{2} \)
$79$ \( ( 34 + 13 T + T^{2} )^{4} \)
$83$ \( ( 1024 + 68 T^{2} + T^{4} )^{2} \)
$89$ \( ( 100 + T^{2} )^{4} \)
$97$ \( ( 4624 - 139 T^{2} + T^{4} )^{2} \)
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