Properties

Label 546.2.e.h
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: 8.0.10070523904.11
Defining polynomial: \(x^{8} - 10 x^{4} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 7 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 19 \nu^{2} \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 3 \nu^{5} + 9 \nu^{4} + \nu^{3} - \nu^{2} - 3 \nu - 45 \)\()/36\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 3 \nu^{5} + 9 \nu^{4} - \nu^{3} + \nu^{2} + 3 \nu - 45 \)\()/36\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 10 \nu^{3} \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} + 5\)
\(\nu^{5}\)\(=\)\(6 \beta_{3} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{4} + 19 \beta_{2}\)
\(\nu^{7}\)\(=\)\(-3 \beta_{7} + 20 \beta_{6} - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_{2} + 20 \beta_{1}\)

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
−1.68014 0.420861i
−1.68014 + 0.420861i
−0.420861 1.68014i
−0.420861 + 1.68014i
0.420861 1.68014i
0.420861 + 1.68014i
1.68014 0.420861i
1.68014 + 0.420861i
1.00000 −1.68014 0.420861i 1.00000 0.841723i −1.68014 0.420861i −0.595188 + 2.57794i 1.00000 2.64575 + 1.41421i 0.841723i
545.2 1.00000 −1.68014 + 0.420861i 1.00000 0.841723i −1.68014 + 0.420861i −0.595188 2.57794i 1.00000 2.64575 1.41421i 0.841723i
545.3 1.00000 −0.420861 1.68014i 1.00000 3.36028i −0.420861 1.68014i −2.37608 1.16372i 1.00000 −2.64575 + 1.41421i 3.36028i
545.4 1.00000 −0.420861 + 1.68014i 1.00000 3.36028i −0.420861 + 1.68014i −2.37608 + 1.16372i 1.00000 −2.64575 1.41421i 3.36028i
545.5 1.00000 0.420861 1.68014i 1.00000 3.36028i 0.420861 1.68014i 2.37608 + 1.16372i 1.00000 −2.64575 1.41421i 3.36028i
545.6 1.00000 0.420861 + 1.68014i 1.00000 3.36028i 0.420861 + 1.68014i 2.37608 1.16372i 1.00000 −2.64575 + 1.41421i 3.36028i
545.7 1.00000 1.68014 0.420861i 1.00000 0.841723i 1.68014 0.420861i 0.595188 2.57794i 1.00000 2.64575 1.41421i 0.841723i
545.8 1.00000 1.68014 + 0.420861i 1.00000 0.841723i 1.68014 + 0.420861i 0.595188 + 2.57794i 1.00000 2.64575 + 1.41421i 0.841723i
\(n\): 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.h yes 8
3.b odd 2 1 546.2.e.f 8
7.b odd 2 1 inner 546.2.e.h yes 8
13.b even 2 1 546.2.e.f 8
21.c even 2 1 546.2.e.f 8
39.d odd 2 1 inner 546.2.e.h yes 8
91.b odd 2 1 546.2.e.f 8
273.g even 2 1 inner 546.2.e.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.f 8 3.b odd 2 1
546.2.e.f 8 13.b even 2 1
546.2.e.f 8 21.c even 2 1
546.2.e.f 8 91.b odd 2 1
546.2.e.h yes 8 1.a even 1 1 trivial
546.2.e.h yes 8 7.b odd 2 1 inner
546.2.e.h yes 8 39.d odd 2 1 inner
546.2.e.h 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{4} + 12 T_{5}^{2} + 8 \)
\( T_{17}^{4} - 80 T_{17}^{2} + 1152 \)
\( T_{19}^{4} - 52 T_{19}^{2} + 648 \)
\( T_{71}^{2} + 8 T_{71} - 96 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( 81 - 10 T^{4} + T^{8} \)
$5$ \( ( 8 + 12 T^{2} + T^{4} )^{2} \)
$7$ \( 2401 + 196 T^{2} - 10 T^{4} + 4 T^{6} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 28561 + 310 T^{4} + T^{8} \)
$17$ \( ( 1152 - 80 T^{2} + T^{4} )^{2} \)
$19$ \( ( 648 - 52 T^{2} + T^{4} )^{2} \)
$23$ \( ( 16 + 64 T^{2} + T^{4} )^{2} \)
$29$ \( ( 144 + 32 T^{2} + T^{4} )^{2} \)
$31$ \( ( 288 - 40 T^{2} + T^{4} )^{2} \)
$37$ \( ( 144 + 32 T^{2} + T^{4} )^{2} \)
$41$ \( ( 11552 + 216 T^{2} + T^{4} )^{2} \)
$43$ \( ( -8 + T )^{8} \)
$47$ \( ( 1152 + 80 T^{2} + T^{4} )^{2} \)
$53$ \( ( 16 + 64 T^{2} + T^{4} )^{2} \)
$59$ \( ( 72 + 20 T^{2} + T^{4} )^{2} \)
$61$ \( ( 648 + 52 T^{2} + T^{4} )^{2} \)
$67$ \( ( 576 + 176 T^{2} + T^{4} )^{2} \)
$71$ \( ( -96 + 8 T + T^{2} )^{4} \)
$73$ \( ( 1568 - 168 T^{2} + T^{4} )^{2} \)
$79$ \( ( 8 + 12 T + T^{2} )^{4} \)
$83$ \( ( 6728 + 180 T^{2} + T^{4} )^{2} \)
$89$ \( ( 1568 + 168 T^{2} + T^{4} )^{2} \)
$97$ \( ( 11552 - 216 T^{2} + T^{4} )^{2} \)
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