Properties

Label 546.2.a.j
Level $546$
Weight $2$
Character orbit 546.a
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
1.00000 1.00000 1.00000 −0.561553 1.00000 −1.00000 1.00000 1.00000 −0.561553
1.2 1.00000 1.00000 1.00000 3.56155 1.00000 −1.00000 1.00000 1.00000 3.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.a.j 2
3.b odd 2 1 1638.2.a.u 2
4.b odd 2 1 4368.2.a.be 2
7.b odd 2 1 3822.2.a.bo 2
13.b even 2 1 7098.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.j 2 1.a even 1 1 trivial
1638.2.a.u 2 3.b odd 2 1
3822.2.a.bo 2 7.b odd 2 1
4368.2.a.be 2 4.b odd 2 1
7098.2.a.bl 2 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{11}^{2} - T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -4 - T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -38 + T + T^{2} \)
$19$ \( -36 - 3 T + T^{2} \)
$23$ \( 8 + 7 T + T^{2} \)
$29$ \( -38 - T + T^{2} \)
$31$ \( -64 + 4 T + T^{2} \)
$37$ \( 26 + 11 T + T^{2} \)
$41$ \( -8 - 6 T + T^{2} \)
$43$ \( 68 + 17 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( -32 + 12 T + T^{2} \)
$61$ \( -38 - T + T^{2} \)
$67$ \( -8 + 6 T + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( -2 + 3 T + T^{2} \)
$79$ \( 32 + 14 T + T^{2} \)
$83$ \( 152 - 26 T + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( -268 + 4 T + T^{2} \)
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